(PDF) Applications of isotope effects in solids

J O U R N A L O F M A T E R I A L S S C I E N C E 3 8 (2 0 0 3 ) 3341 – 3429

Review

Applications of isotope effects in solids

V. G. PLEKHANOVFonoriton Science Lab., Garon Ltd., PO Box 2632, Tallinn, 13813, EstoniaE-mail: [emailprotected]

This article reviews the current status of the employment of the isotope effect in solids.Diffusion, self-diffusion processes with different isotopes in pure materials andheterostructures (quantum wells), neutron transmutation doping of differentsemiconducting crystals, optical fiber as well as use isotope-mixed crystals (C, LiH) as thegenerator of the coherent radiation in the ultraviolet range of the spectrum are the mainmodern applications of isotope science and engineering. There are briefly discussed theother future applications including modern personal computer, isotope-based quantumcomputer as well as information storage. We hope to give sufficient references topublished work so that the interested reader can easily find the primary literature sourcesto this rapidly expanding field of solid state physics. C© 2003 Kluwer Academic Publishers

IntroductionThe availability of isotopically pure crystals with lowcarrier and impurity concentrations has allowed in thelast three decades the investigation of isotope effectson lattice dynamical and electronic properties of solids[1]. The results of experimental and theoretical stud-ies of the fundamental properties of the objects of re-search that earlier were simply in accessible (naturallywith exception of LiHx D1−x crystals) briefly are pre-sented in the reviews [2–4]. The use of such objectsallows the investigation of not only the isotope ef-fects in lattice dynamics (elastic, thermal and vibra-tional properties) but also the influence of such effectson the electronic states via electron-phonon coupling(the renormalization of the band-to-band transition en-ergy Eg, the exciton binding energy EB and the sizeof the longitudinal-transverse splitting �LT). The ther-mal conductivity enhancement in the isotopically en-riched materials amounts (C, Ge, Si) to almost 60%at room temperature and is close to a factor six at thethermal conductivity maximum around 20 K (Si-case)(see also [3, 4]). The change in the lattice constant is�a/a ∼ 10−3 ÷ 10−4, while the change δcik in the elas-tic constants amounts to several percent. In addition,crystals of different isotopic compositions possess dif-ferent Debye temperatures. This difference between aLiH crystal and its deuteride exceeds hundred degrees.Of the same order of magnitude is the difference be-tween Debye temperatures for diamond crystals. Verypronounced and general effects of isotopic substitutionare observed in phonon spectra. The Raman lines in iso-topically mixed crystals are not only shifted (the shift ofLO phonon lines exceeds 100 cm−1) but are also broad-ened. This broadening is related to the isotopic disorderof a crystal lattice. It is shown that the degree of changein the scattering potential is different for different

isotopic mixed crystals [1]. In the case of semiconduct-ing crystals (C, Ge, Si, α-Sn etc.), phonon scattering isweak, which allows one to successfully apply the coher-ent potential approximation (CPA) for describing shiftand broadening of scattering lines in Raman spectra[2, 3]. In the case of LiH, the change in the scatteringpotential is so strong that it results in phonon localiza-tion, which is directly observed in experiments [1, 4].

Substituting a light isotope with a heavy one in-creases the interband transition energy Eg (excludingCu-salts) and the binding energy of the Wannier-Mottexciton EB as well as the magnitude of the longitudinal-transverse splitting �LT [5, 6]. The nonlinear variationof these quantites with the isotope concentration is dueto the isotopic disordering of the crystal lattice and isconsistent with the concentration dependence of linehalfwidth in exciton reflection and luminescence spec-tra. A comparative study of the temperature and iso-topic shift of the edge of fundamental absorption for alarge number of different semiconducting and insulat-ing crystals indicates that the main (but not the only)contribution to this shift comes from zero oscillationswhose magnitude may be quite considerable and com-parable with the energy of LO phonons. The theoreticaldescription of the experimentally observed dependenceof the binding energy of the Wannier-Mott exciton EBon the nuclear mass requires the simultaneous consid-eration of the exchange of LO phonons between theelectron and hole in the exciton, and the separate inter-actions of carriers with LO phonons (see also [1]). Theexperimental dependence EB ∼ f (x) for LiHx D1−x

crystals fits in well enough with the calculation accord-ing to the model of large-radius exciton in a disorderedmedium; hence it follows that the fluctuation smear-ing of the band edges is caused by isotopic disorder-ing of the crystal lattice. Due to zero-point motion, the

atoms in a solid feel the anharmonicity [7] of the inter-atomic potential even at low temperatures. Therefore,the lattice parameters of two chemically identical crys-tals formed by different isotopes do not coincide heav-ier isotopes having smaller zero-point delocalization(as expected in a harmonic approximation) and smallerlattice parameters (an anharmonic effect). Moreover,phonon—related properties such as thermal conductiv-ity, thermal expansion or melting temperature, are ex-pected to depend on the isotope mass (details see [1]).

Our brief discussion, we start with a fact that phononfrequency are directly affected by changes of the aver-age mass of the whole crystal or its sublattice (VCA-model), even if we look upon them as noninteractingparticles, i.e., as harmonic oscillators. The direct influ-ence of the isotope mass on the frequencies of coupledphonon modes may been used to determine their eigen-vectors. Secondly, the mean square amplitude 〈u2〉 ofphonons depend on the isotope masses only at low tem-perature, while they are determined by the temperatureT only, once T becomes larger than Debye temperature.A refinement of these effects must take place when tak-ing interactions among phonons into account. Theseinteractions lead to finite phonon lifetimes and addi-tional frequency renormalization. The underlying pro-cesses can be divided into two classes: (1) anharmonicinteractions in which a zone center phonon decays intotwo phonons or more with wave-vector and energy con-servation, and (2) elastic scattering in which a phononscatters into phonons of similar energies but differentwave-vectors. While the former processes arise fromcubic and quartic terms in the expansion of lattice po-tential [7], the latter are due to the relaxed wave-vectorconservation rule in samples that are isotopically dis-ordered and thus not strictly translationally invariant.Since the vast majority of compounds derive from el-ements having more than one stable isotope, it is clearthat both processes are present most of the time. Unfor-tunately, their absolute sizes and relative importancecannot be predicted easily. However, isotope enrich-ment allows one to suppress the elastic scattering in-duced by isotope disorder. In contrast, the anharmonicphonon-phonon interaction cannot be suppressed, sothat isotope-disorder-induced effects can only be stud-ied against a background contribution from anharmonicprocesses. However, if one assumes that the two pro-cesses are independent of each other one can measurethe disorder-induced renormalization by comparison ofphonon energies and linewidth of isotopically pure sam-ples with those gained from disordered ones.

The isotopic composition affects the band-gapsthrough the electron-phonon coupling and through thechange of volume with isotopic mass. Although theelectronic properties of different isotopes of a givenatom are, to a very good approximation, the same,isotope substitution in a crystal modifies the phononspectrum which, in turn, modifies the electron energybands through electron-phonon interaction. Measuringthe energy gaps in samples with different isotopic com-position then yields the difference in the changes ofthe valence- and conduction-band renormalization. Thereason for the changes lies in the fundamental quantum-

mechanical concept of zero-point motion—the vibra-tional energy that the atoms in the crystal have, evenat low temperatures. If we excite an electron from oneelectronic state to another, we actually excite the wholecrystal. In other words, we move the crystal from aground state made up of low-energy electrons plus zero-point vibrations to an excited state in which there isone excited electron plus the zero-point vibrations ofthe crystal. The values of the zero-point energy in thetwo electronic states are slightly different because thevibrational frequencies depend on the chemical bond-ing, which is changed by exciting an electron. If theaverage mass of the vibrating atoms is increased, thenthe vibrational frequencies will be reduced. As a result,the difference in zero-point motions will be smaller,and the transition energy will therefore increase withincreasing mass [1].

Present review is devoted to description of differ-ent applications of the isotope effect in solids. Inthe Chapter 1 we detail analyze the process of self-diffusion in isotope pure materials and heterostructure.Interest in diffusion in solids is as old as metallurgy orceramics, but the scientific study of the phenomenonmay probably be dated some sixth-seven decades ago.As is well-known, the measured diffusion coefficientdepends on the chemistry and structure of the sam-ple on which it is measured. We have organized thechapter around general principles that are applicableto all materials. We are briefly discussed the SIMS(secondary ion mass spectrometry) technique. In SIMStechnique, the sample is bombarded by reactive ions,and the sputtered-off molecules are ionized in a plasmaand fed into a mass-spectrometer. Self-diffusion is themigration of constituent atoms in materials. This pro-cess is mediated by native defects in solids and thus canbe used to study the dynamics and kinetics of these de-fects. The knowledge obtained in these studies is pivotalfor the understanding of many important mass transportprocesses such as impurity diffusion in solids.

Chapter 2 describes the new reactor technology—neutron transmutation doping (NTD). Capture of ther-mal neutrons by isotope nuclei followed by nuclear de-cay produces new elements, resulting in a very numberof possibilities for isotope selective doping of solids.There are presented different facilities which use in thisreactor technology. The feasibility of constructing re-actors dedicated to the production of NTD silicon, ger-manium (and other compounds) was analyzed in termsof technical and economic viability and the practicalityof such a proposal is examined. The importance of thistechnology for studies of the semiconductor doping aswell as metal-insulator transitions and neutral impu-rity scattering process is underlined. The introductionof particle irradiation into processing of semiconduc-tor materials and devices creates a new need for addi-tional understanding of atomic-displacement-produceddefects in semiconductors. It is shown that measure-ment of decay rates of induced radioactivity and thesystem of clearance and certification such as to allowthe solids to be internationally transported as “ExemtMaterial”, as defined in IAEA Regulations, are dealtwith.

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The use of the isotopes in a theory and technologyof the optical fibers we considered in Chapter 3. Thischapter is addressed to readers who wish to learn aboutfiber communications systems and, particular, about theproperties of optical fibers. Very briefly in this chapterwe describe the Maxwell equations as well as waveelectromagnetic equation. In this chapters we describenot only the properties of optical fibres but also the ma-terials for optical fiber and fiber technology. For the firsttime it was shown also the influence of the isotopes onproperties of the optical fibers.

Chapter 4 is devoted the application of isotope effectin laser physics. There is short description of theoryand practice of semiconductor lasers. The discovery ofthe linear luminescence of free excitons observed overa wide temperature range has placed lithium hydride,as well as crystals of diamond in line as prospectivesources of coherent radiation in the UV spectral range.For LiH isotope tuning of the exciton emission has alsobeen shown [1].

The last chapter of this review is devoted to descrip-tion of the other unexplored applications of isotopic en-gineering. In the first place we considered the materialsfor information storage in modern personal computersas well as in biology. In this chapter is shown that iso-topic substitution has made it possible to produce theobjects of research that earlier were simply inaccessi-ble (with exception of the LiH-LiD system). The useof such objects allows the investigation of not only theisotope effects in lattice dynamics (elastic, thermal andvibrational properties) but also the influence of such ef-fects on the electronic states (the renormalization of theband-to-band transition energy Eg, the exciton bindingenergy EB, and the size of the longitudinal-transversesplitting �LT). Very perspective is isotope-based quan-tum computer. We should add here that the strength ofthe hyperfine interaction is proportional to the probabil-ity density of the electron wavefunction at the nucleus.In semiconductors, the electron wavefunction extendsover large distances through the crystal lattice. Twonuclear spins can consequently interact with the sameelectron, leading to electron-mediated or indirect nu-clear spin coupling. Because the electron is sensitive toexternally applied electric fields, the hyperfine interac-tion and electron-mediated nuclear spin interaction canbe controlled by voltages applied to metallic gates ina semiconductor device, enabling the external manip-ulation of nuclear spin dynamics that is necessary forquantum computation in quantum computers (detailssee [383, 384]).

A brief summary is presented in the conclusion. Thedifficult and unsolvable problems of isotope effects insolids are considered there. The main aim of this reviewis to familiarize readers with present and some futureapplications in isotope science and engineering.

Chapter 1. Process of self-diffusion inisotope pure materials andheterostructures

1.1. General remarksInterest in diffusion is as old as metallurgy or ceramics.The first measurement of diffusion in the solid state was

made by Roberts–Austen in 1896 [8]. Many measure-ments, especially of chemical diffusion in metals, weremade in the 1930s; the field was reviewed by Mehl[9], Jost [10], and Seith [11]. Diffusion research in-creased after World War II; the increase was motivatedby the connection among diffusion, defects, and radi-ation damage and helped by the availability of manyartificial radiotracers. These researchers were the firstto attempt to identify the basic underlying atomisticmechanisms responsible for mass transport throughsolids by a quantitative investigations and theoreticalanalysis of the activation energies required for diffu-sion by exchange, interstitial, and vacancy mechanismsin solids. Prior to this time, there had been little con-cern with treating diffusional phenomena on a micro-scopic basis, and most research was concerned withfairly crude observation of overall bulk transfer pro-cesses at junctions between regions with strong compo-sitional differences. It was at this time that suggestionson how to carry out high-precision, highly reproduciblediffusion experiments were first put forward (Slifkinet al. [12], Tomizuka [13]). The three major factorsthat determine the quality of a diffusion measurementare

1. the method used,2. the care taken in the measurement, and3. the extent to which the material is specified [14].

The most accurate method has, in general, been con-sidered to be radiotracer sectioning (Tomizuka [13]),and most of this article is devoted to this method, es-pecially to points for which special care must be taken;these are the measurement of temperature, the accu-racy of sectioning, and the reproducibility of count-ing the radioactivity. The importance of specifying thematerial cannot be overstated. The measured diffusioncoefficient depends on the chemistry and structure ofthe sample on which it is measured. Impurities, non-stoichiometry of compounds, grain boundaries, anddislocations can give apparent values of the diffusioncoefficient that are different from, and usually largerthan, the true value (see also [15, 16]). The objectiveof this chapter is to describe some experimental re-sults as well as their theoretical analysis that are re-ceived in last decade. We have organized the chap-ter around general principles that are applicable toall materials, and then listed the particulars. The ma-terials we consider are mainly inorganic solids, es-pecially semiconductor and insulator materials. Theeffects of pressure on diffusion is omitted. Previousreviews covering mainly metals and inorganic materi-als have been given by Hoffman [17], Tomizuka [13],Cadek and Janda [18], Adda and Pholibert [19], Lundy[20], Beniere [21] and last two book of Academic Press[22, 23].

Radioactive tracers are essential to many of the exper-iments described in this chapter [24–27]. Radioactivetracers are hazardous materials, and the experimenterwho uses them is under the strongest moral obligationto avoid exposure of his colleagues and contaminationof his environment.

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1.2. The relation of diffusion experimentsto the mathematics of diffusion

For measurable diffusion to take place a gradient ofsome kind is necessary. Diffusion is a consequence ofthe hopping motion of atoms through a solid. The dif-fusion coefficient D is defined in Fick’s first law (Fick[28]),

�J = −D �∇C + C �V (1)

where �J is the flux of atoms, C their concentration, and�V the velocity of the center of mass, which moves dueto the application of a force such as an electric field or athermal gradient (see also [29]). A number of differentdiffusion coefficients exist, e.g., for the diffusion of aradioactive tracer in a chemically hom*ogeneous solidin the absence of external forces,

�J ∗ = −D∗ �∇C∗ (2a)

where the asterisk denotes the radioactive species. Fordiffusion in a chemical gradient,

�J = −D �∇C (2b)

where D is the interdiffusion or chemical diffusion co-efficient. Any of these equations can be combined withthe equation of continuity

∂C/∂t = −�∇ · �J (3)

to yield Fick’s second law

∂C/∂t = �∇ · (D∇C) (4a)

where the mass flow term has been omitted. For a tracerin a hom*ogeneous system,

∂C∗/∂t = −�∇∗ · �∇2C∗. (4b)

Equations 4a and 4b describe the types of diffusionexperiments discussed in this chapter.

The tracer diffusion coefficient is given also in theatomistic form

D∗ = γ a2 f (5)

where γ is a geometric factor, a the jump distance, the atomic jump frequency, and f the correlationfactor (see, e.g. [29]). It is thus possible, in principle, tomeasure D∗ by mesuring in a resonance experimentof some kind [30, 31].

We are concerned here with diffusion measurementswhere the diffusion coefficient is obtained via Fick’ssecond law, i.e., from a solution of the diffusion equa-tion (see, also [1]). Fick’s second law is used rather thanhis first because concentrations are easier to measurethan fluxes and because of D in the solid state are sosmall that the required steady state is seldom reached.In order to obtain a solution of the diffusion equation,the initial and boundary conditions (IC and BC) must

be known. The IC correspond to the distribution of thediffusing substance in the sample before the diffusionanneal, and the BC describe what happens to the dif-fusing substance at the boundaries of the sample dur-ing the diffusion anneal. If the experimental IC and BCcorrespond to the mathematical conditions, the mathe-matical solution to the diffusion equation C(x, y, z, t)will describe the distribution of the diffusing substanceas a function of position in the sample and of annealingtime. The diffusion coefficient is finally obtained by fit-ting the experimentally determined C(x, y, z, t) to theappropriate solution of the diffusion equation with Das a parameter.

Most laboratory experiments are arranged so that dif-fusion takes place in one dimension. The solution ofthe diffusion equation is then C(x, t). One most oftendetermines C(x) at constant t , i.e., the concentrationdistribution along the diffusion direction after a diffu-sion annealing time t . It is also possible to determineC(t) at a constant x (e.g., the concentration at a surface)or

∫∫C(x, t)dxdt (e.g., the weight gain of a sample as

a function of time). The IC, BC, and solutions to thediffusion equation (for D = const) for some commongeometries are described below. These, and solutionsfor other cases, are given by Crank [32] and Carslawand Jaeger [33].

1. Thin Layer or Instantaneous Source Geometry(Fig. 1a). An infititesimally thin layer (�(Dt)1/2) ofdiffusing substance is deposited on one surface of asemi-infinitive (�(Dt)1/2) solid. The initial conditionsis

C(x, 0) = Mδ(x) (6)

where δ is the Dirac delta function and M the strengthof the source in atoms per unit area. The boundary con-dition is

∂C(0, t)/∂t = 0 (7)

i.e., there is no flux through the surface. The solution is

C(x, t) = (M/√

π Dt) exp(−x2/4Dt). (8)

One determines C(x) for constant t .2. Thick Layer Geometry (see Fig. 1b). Similar to

the above, except that the layer thickness h is of orderof the diffusion distance:

IC : C(x, 0) = C0, h ≥ x ≥ 0(7a)

C(x, 0) = 0, x > h.

BC : ∂C(0, t)/∂x = 0 (8a)

solution:

C(x, t) = C0

[erf

(x + h

2√

)− erf

(x−h

2√

)], (9)

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Figure 1 Concentration distributions for different initial conditions.Dotted line is for t = 0, solid line is for a finite t . (a) Thin layer ge-ometry [case (1)]; (b) thick layer geometry [case (2)]; solid curve forDt = h2; (c) infinite couple [case (3)] (after Crank [32]).

where

erf(λ) = 2√π

∫ λ

0exp(−η2)dη. (10a)

Measure C(x) for constant t . Note:

erfc(λ) ≡ 1 − erf(λ). (10b)

3. Infinite Couple (see Fig. 1c). A sample of uniformconcentration C0 is welded to a sample of uniform con-centration C1. The weld plane is situated at x = 0.Couple containing a volatile

IC: C(x, 0) = C1, x < 0(11)

C(x, 0) = C0, x > 0

Solution:

C ′(x, t) ≡ C(x, t)−C0

C1−C0=

[1 − erf

2√

)]. (12)

Measure C(x) for constant t .4. Vapor-Solid Coule. A semi-infinite couple con-

taining a volatile component component is placed intoa dynamic vacuum at t = 0:

IC : C(x, 0) = C0, x > 0 (13a)

BC : C(0, t) = 0, t > 0. (13b)

Solution:

C(x, t) = C0erf(x/2√

Dt). (14a)

Exposing a sample initially devoid of volatile com-ponent to a vapor of the volatile componenet at apressure in equilibrium with C0 gives the analogousmathematics:

IC : C(x, 0) = 0, x > 0 (14b)

BC : C(0, t) = C0, t > 0. (15a)

Solution:

C(x, t) = C0[1 − erf(x/2√

Dt]

= C0erfc(x/2√

Dt). (15b)

The same equations apply to isotopic exchange be-tween solid and vapor. Measure either C(x) at constantt or integral weight gain (loss)

∫ ∞

∫ t

0C(x, t)dtdx .

5. Grain Boundary Diffusion. The mathematics inthis case are more complicated (see, e.g. [34]), owingto the coupled lattice diffusion, but one still measuresC(x) at constant t .

6. Exchange experiment [35]. This technique is usedfor materials for which a massive sample cannot beprepared. It involves diffusion exchange between anassembly of powder and a gas of limited volume, fromwhich very small aliquots are drawn at different times.

In the first three sample configurations two bodiesof widely different composition are brought into con-tact. The assumption implicit in the BC is that diffus-ing material passes across the resulting interface with-out hindrance, i.e., it is not held up by surface oxides,low solubility, chemical reactions, etc. Nonfulfillmentof this condition leads to deviation of the experimentalC(x) from solution of the diffusion equation (detailssee [36]).

In the vapor-solid couple and the exchange exper-iment, the assumption implicit in the BC is that thesurface of the solid equilibrates with the gas phase in-stantaneously. However, optical measurements of thechange of the surface concentration at low temperaturehave indicated that the attainment of solid-gas equilib-rium can be slow process (see also [37]).

In this connection we should add, that the thin ge-ometry has several advantages. The thin layer can be

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deposited without straining the sample, which is es-sential for single crystal samples. A thin layer also al-lows the use of high specific radioisotopes, and thusmeasurements of diffusion without a chemical gradi-ent. Diffusion under large chemical gradient can leadto deformation of the sample and generation of defects(see, also [38–41]). For the above reasons, the thin layergeometry is most often used in experiments in whichdiffusion is measured in order to study the fundamen-tals of diffusion and defect behavior in solids. Suchexperiments usually concern diffusion as a function oftemperature, pressure, or concentration, and small dif-ferences in D are import, in contrast to engineeringexperiments in which the magnitude of the penetrationof one material into another is of interest.

It should be noted that all the solutions to the diffusionequation considered above are expressed in terms ofthe dimensionless variable x/(2

√Dt). The length is

a kind 2√

Dt is a kind of mean penetration distance,and this has to be the same order of magnitude as thecharacteristic distance associated with an experiment.For sectioning experiment, the characteristic distanceis the section thickness. For ion-beam depth profiling,it is the ion range, etc. [27].

In the ordinary thin-layer sectioning experiment, onewishes to measure diffusion over a drop in specific ac-tivity C of ∼103; any effects due to diffusion alongshort-circuiting paths are likely to show up as curva-ture in the penetration plot over such a range, whilethey may not be visible if the measurement is only overfactor of 6 in C [42]. Usually twenty sections suffice todefine a penetration plot; from Equation 8, the sectionthickness required to get a drop of 103 in C over 20sections is

θ ≈√

Dt/3.8. (16)

A preliminary estimate of D is useful in planning anexperiment.

If the isotope decays significantly during the time ofthe experiment, more radioisotope has to be deposited.Under the conditions of θ ≈ √

Dt/3.8, the specific ac-tivity drops by a factor of ∼2 per section at the twentiethsection. These last points on the penetration plot havethe greatest weight in determining D, so the countingstatistics must be maintained and the penetration plotextended as far as possible. This implies use of an in-tense source of radioisotope, on the other hand, toomuch activity poses an unnecessary health hazard aswell as increasing the dead-time correction. The radio-tracer may rapidly reach the side surfaces of the sampleby surface diffusion or evaporation, and then diffuseinward. To keep the diffusion one-dimensional, one re-moves ≈ 6

√Dt from the sides of the sample before

sectioning.

1.3. Self-diffusion processAs is well-known, in all diffusion mechanisms theatoms under consideration have to carry out jumps be-tween different sites (see, e.g. [43]). If the extreme caseof coherent tunneling [44] is left aside, the diffusional

jumps are assisted by the thermal movement of theatoms. In the standard situation the jump rate is en-tirely determined by the temperature T (apart from theeffects of hydrostatic pressure, which may be incorpo-rated by formulating the theory in terms of enthalpyand Gibbs free energy). For the purposes of the presentchapter, we may in the first approximation disregardquantum mechanical contribution to the diffusity (nat-urally excluding the self-diffusion in LiH), so that incubic crystals the diffusion coefficient under standardconditions may be written as an Arrhenius expression

Dα = Dα0 exp( − H M

/kT

)(17)

with the preexponential factor

Dα0 = gαa20να0 exp

(SM

/k)

(18)

Here H Mα denotes the enthalpy and SM

α the entropy ofmigration, a0 the lattice constant, and να0 the attemptfrequency, k has its usual meaning as Boltzmann’s con-stant, and gα is a factor that takes into account the ge-ometry of the crystal structure and the atomistic detailsof the different process. The subscript α refers to thedefect species controlling the diffusion process, i.e.,in the case of the direct interstitial mechanism it indi-cates the chemical nature, geometrical configuration,etc., of the interstitial involved, whereas in the case ofindirect diffusion it characterizes the intrinsic defectsacting as diffusion vehicles. In the latter case, we shouldwrite β instead of α if we wish to indicate that these in-trinsic defects are monovacancies or monointerstitials.

The tracer self-diffusion coefficient, i.e., the diffusityof radioactive self-atoms under thermal-equilibriumconditions, is given by (see also [43])

DT =∑

β=I,V

fβ DSDβ =

∑β=I,V

fβ DβCeqβ (19)

where

Ceqβ = exp

(SF

/k)

exp(−H F

/kT

)(20)

are the concentrations of self-interstitials (β = I ) andmonovacancies (β = V ) in thermal equilibrium. Asit is clear, in Equation 19, contributions by clusters ofI or V are neglected. The fβ denote correlation fac-tors, DSD

β ≡ DβCeqβ contributions to the uncorrelated

self-diffusion coefficient∑

β=I,V DSDβ , and SF

β and H Fβ

entropies and enthalpies of formation, respectively. In-sertion of Equations 17, 18 and 20 into Equation 19yields

DT =∑

β=I,V

DTβ =

∑β=I,V

fβgβa20νβ0 exp

(−GSDβ

/kT

)

=∑

β=I,V

DTβ0 exp

(−H SDβ

/kT

)(21)

with the preexponential factors

DTβ0 = fβgβa2

0νβ0 exp(−H SD

/kT

)(22)

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and the Gibbs free energy of self-diffusion

GSDβ = H SD

β − T SSDβ (23)

the self-diffusion enthalpy

H SDβ = H F

β + H Mβ (24)

and the self-diffusion entropy

SSDβ = SF

β + SMβ (25)

The diffusion coefficient Ds of foreign substitutionalatoms in thermal equilibrium may be derived fromEquations 19 or 21 by inserting factor hβ under thesummation signs. These factors account for the inter-action between the intrinsic thermal equilibrium defectsand the substitutional atoms. They depend on tempera-ture and the atomic fraction of the substitutional atoms,unless this is small compared to unity. For more de-tailed and complete discussion in this field we refer thereader to reviews by Frank et al. [43](see also [27]).

Compared with metals, self-diffusion in semiconduc-tors is very slow process. For the elemental semicon-ductors this is illustrated in Fig. 2, in which the self-diffusivities of the cubic semiconductors Si and Ge andof the trigonal semiconductors Te and Se are comparedwith those of typical metals such as Cu, Ag, and Au ona temperature scale normalized to the melting temper-ature Tm. Fig. 2 reveals the following differences be-tween metals and semiconductors, already emphasizedby Seeger and Chik [48].

Figure 2 Comparison between the self-diffusivities of the cubic semi-conductors Ge and Si (Frank et al. [43]), the trigonal semiconductorsTe (Ghoshtagore [45], Werner et al. [46]) and Se (Bratter and Gobrecht[47]) and the typical metals Cu, Ag, Au (after [43]).

1. Near the melting temperatures the self-diffusion insemiconductors is several orders of magnitude slowerthan in typical metals.

2. At lower normalized temperatures the ratio ofthe self-diffusivities of metals and semiconductors be-comes even larger.

Generally speaking, the origin of these differenceslies in the hom*opolar bonding of the semiconductors(details see [43]).

As is well-know, the conventional and well-established techniques of determining the tracer self-diffusion coefficient DT based on studying the redistri-bution of radioactive or stable tracers initially depositedon the specimen surface of serial sectioning methods.In the case of radioactive isotopes, the redistributionmay be investigated with radiation detection methods;for stable isotopes—secondary ion mass spectroscopy(SIMS) may be used (see also below).

1.4. SIMS-techniqueThe most complete description of the experimentaltechnique for study the diffusion processes in solids thereaders may be found in the excellent review by Roth-man [36]. Here we are briefly discussed the peculiarityof sputtering and SIMS techniques.

We define microsectioning [49] as the cutting of sec-tions a few hundreds nm or less in thickness, so thatthe surface on which the tracer is deposited, the “front”surface, is not necessary flat on the scale of

√Dt , and

so the thickness of the individual sections are not deter-mined separately. The isoconcentration contours thenfollow the contour of the front surface, and one mustremove sections parallel to this nonflat surface, ratherthan parallel to a flat surface. If this condition is met andif the indulations in the front surface are gentle (radiusof curvature ρ � √

Dt), one can treat the sections as ifthey were flat (see, however [50]).

Simple chemical dissolution followed by countingthe solution has been used for metals [51], alloys [52],and silicate glasses [53, 54]. One uses a constant volumeof solvent in the counting vial, rinses the sample so thatthe rinse flows into the same vial, and then adjusts thetotal volume of solution so that it is the same for allsections.

In sputtering, material is removed by ion bombard-ment owing to the transfer of momentum from the bom-barding ions to the atoms of the targets. A depth profilecan be constructed by (1) analyzing the sputtered-offmaterial in a mass spectrometer (SIMS), (2) collectingand analyzing the sputtered-off material, (3) determin-ing the concentration of the diffusing material in the re-maining surface by, e.g., Auger electron spectroscopy(see, e.g. [55–58]), or (4) counting the residual activ-ity of the entire sample [56]. As a rule, noble gas ions,especially Ar+, are accelerated to a few hundred eVor more, with current densities ≤1 mA cm−2. This iscalled physical sputtering, in contrast to bombardmentwith reactive ions, which is called chemical sputtering.Typical removal rates are of the order of 10 nm min−1

for 1 mA cm−2 of 500-eV Ar+ ions. There are two

3347

excellent reviews of the subject of sputtering (Behrish[60], Chapman [61]), and the reader is referred to thesefor an understanding of the process. All equipment forsputtering includes a vacuum chamber, pumping equip-ment, and a controlled gas leak such as a micrometerneedle valve. A hohg-speed pumping system is neededas gas is passed continuously and there are bursts ofdesorbed gases to cope with. Cold-trapped diffusionpumps, cryopumps, or turbopumps have all been used.All sputtering equipment has a gaseous discharge in it.Common glow discharges are not suitable, as too higha gas pressure is required, with resulting low mean-freepaths and back diffusion of the sputtered atoms. There-fore, either an ion gun an rf power source is used. Twotypes of ion sources have been used in sectioning exper-iments, the custom-modified duoplasmatron of Maierand Schule (see [43]), and a commercially availableKauffman-type gun [62, 63]. Almost any ion sourceused in ion milling should be usable. The main require-ment is that the source put out ion currents ≥1 mA cm−2

at ∼1 kV over ∼4 cm2 area in a reasonably uniformbeam (±10% except at the very edge), and that thecurrent stay constant over period of several hours. Thelength of a run is limited by life of the filament. Inaddition to the ion source, chamber, pumps, and valv-ing, one needs a collector and a sample holder. Theseare usually custom made. Designs have been given byGupta and Tsui [64] and Atkinson and Taylor [65] for rfsystems as well as Mundy and Rothman [60] for ion gunsystems. The collector is either a carousel, with six Alplanchets, which allow six sections to be taken beforethe chamber is opened [67], or a device like a camerasback, on which poleester film is rolled; the latter allows32 sections to be taken.

In the SIMS technique, the sample is bombarded byreactive ions, and the sputtered-off molecules are ion-ized in a plasma and fed into a mass spectrometer. Themass spectrum is scanned and the ion current for tracerand host atoms can be recorded simultaneously. Thebeam is swept over the sample and, in effect, digs acrater, the bottom of which is more or less flat; anaperture prevents ions originating from the edges ofthe crater from reaching the mass spectrometer (seeFig. 3). The penetration plot is constructed from theplots of istaneous tracer/host atom ratio versus sputter-ing time and of distance sputtering time. The distance isobtained by using interferometric measurement of thetotal crater depth under the assumption that material is

Figure 3 Schematic diagram of crater caused by sputtering in a SIMSapparatus (after Dorner et al. [66]).

removed uniformly as a function of time. Large changesof chemical composition along the diffusion directioncan invalidate that assumption. The limitations of theSIMS technique have been discussed by Liebl [67] andReuter and Baglin [68], and a detailed description ofits application to diffusion has been given by Seran[69] and Macht and Naundorf [70]. A recent paper [66]shows the quality of results that can be obtained (seealso [71]). In general, the resolution is no worse thanthat obtained by sputtering and counting the sections,and the sensitivity is no worse than that of countingtechniques. The major disadvantage of SIMS is its cost.The SIMS apparatus is commercially made (see e.g.[36] and references therein) but represents a large capi-tal investiment. Not withstanding the cost of the appara-tus careful contros must be applied to the measurementsand artifacts [72] must be avoided.

If the entire of the sample is section the depth ofmaterial removed is best determined by weighing ona microbalance. With care, a sample can be weighedto ±3 µg, which corresponds to ±150 nm for cross-sectional area of 0.1 cm2 which is about the minimumuseful area, and a density of 2 g cm−3. For larger areas ordensities, even better sensitivities are obtained, down toperhaps ±10 nm (details see review [36] and referencestherein).

1.5. Self-diffusion of Li and H in LiH crystalsSelf-diffusion is the migration of constituent atoms inmaterials. This process is mediated by native defects insolids and thus can be used to study the dynamics andkinetics of these defects. The knowledge obtained inthese studies is pivotal for the understanding of manyimportant mass transport processes such as impuritydiffusion in materials. Self-diffusion of D(H ) and Liin LiH crystals is studied in papers [73, 74] and [75,76] respectively. As was above shown the gas-solid iso-tope exchange method has been used for the measure-ment of self-diffusion coefficients in solids. Two papers[74, 75] have reported on a thermogravimetric study ofthe pressure and temperature dependence of diffusioncoefficient of the deuteride ion in LiH crystals. As iswell-known, in this method a crystal of the compoundof interest is equilibrated in a furnace with a gas usu-ally containing an isotopic species of the diffusant. Theweight change of the crystal due to the permutationprocess from the gas to the solid is then monitored asa function of time. By assuming that the mass uptakeis due to the isotopic exchange process with deuteriumgas and subsequent diffusion of the deuteride ion intothe crystal and that the rate of the exchange process isdiffusion controlled, the mass gain of the crystal wascalculated from solution of Fick’s law. The best leastsquares fit of the data obtained in papers [74, 75] to one-dimensional and three-dimensional models was used tofind the diffusion coefficients and the activation energyfor the deuteride ion.

According to [74] solution of Fick’s second law [29]

∂C/∂t = ∂2C/∂x2 (26)

3348

subject to the boundary conditions

C(x, 0) = 0(27)

C(0, t) = Cs,

where Cs is the surface concentration of the diffusingspecies and

lim C(, t) → 0, x → 0

gives, for Q, the total amount of diffusing substancewhich has entered the solid at time t ,

Q = 2AsCs(D/π )1/2t1/2 + B (28)

where As is the total surface area of the crystal, D is thediffusion coefficient, and B a constant that accounts forthe initial condition that Q �= 0 at t = 0. Q is expressedas the ratio of the number of moles of deuteride iondiffusing to total moles of LiH contained in the crystal.

Equation 28 may be written as

Q = C ′t1/2 + B, (29)

where C ′ is defined as

C ′ = 2AsCs(D/π )1/2. (30)

With the assumption that the surface coverage Cs canbe written in terms of an adsorption isotherm, θ , C ′ wasrewritten as

C ′ = 0.8748(Asθ/ρV )D1/2, (31)

where ρ is the density of LiH at a given temperature,V is the volume of the crystal, and the constant in-cludes the necessary factors for consistency of units.Rearrangements gives

C = C ′(As/ρV )−1 = 0.8748θ D1/2. (32)

The values for C ′ were found from the fit of Equa-tion 29 to the data at each temperature (see, Fig. 4). Allfits were exceptionally good (see also Table I). Valuesof C calculated from C ′ by using the relation given inEquation 26 are also given in Table I.

Equation 32 may be written in the form

C = 0.8748θ D1/20 e−Ea/2RT (33)

by substituting for D,

D = D0e−Ea/2RT , (34)

where D0 is a constant and Ea the activation energy.According to Equation 33, the temperature variance

of C is determined by the exponential term involv-ing temperature and the temperature dependence of the

Q = θ

{1 −

[ ∞∑u=0

(2u + 1)2 π2exp

(− (2u + 1)2 π2 Dt

4a2

)]×

[ ∞∑v=0

(2u + 1)2 π2exp

(− (2v + 1)2 π2 Dt

4b2

)]

×{ ∞∑

w=0

(2w + 1)2 π2exp

(− (2w + 1)2 π2 Dt

4c2

)}+ B

}, (36)

TABLE I Summary of one-dimensional semi-infinite solid data (after[75])

Sample C × 104 g cm−2

no. T ◦C C ′ × 103, min−1 min−1/2 Ea, kcal

8 550 4.74 3.96 22.7 ± 2.8a

9 524 4.37 3.49 22 ± 2b

10 500 2.80 2.5211 450 1.50 1.5312 410 1.37 0.9613 399 1.09 0.86

a Ea as found from the best least squares fit of Equation 28 to C withθ = 0.80. The error reported is the 95% confidence level fit to the data.b Ea from Ref. [77].

Figure 4 Q as a function of time for data of samples 9, 10 and 13.The solid curves are the best least squares fit of Equation 29 to theexperimental data (after [75]).

fractional surface coverage θ . If θ is known as a functionof temperature, C may be fit to Equation 33 to give D0and the activation energy. Lacking knowledge of theexact variation of θ with temperature, Equation 27 maybe rewritten in the more convenient form

C/θ = 0.8748D1/20 e−Ea/2RT (35)

and θ assumed to vary with temperature according tothe expected behavior outlined above. The quantity C/θ

was next calculated using θ = T/(b + T ) and the fit ofthe data to Equation 29 found, giving the activation en-ergy as 22.1 ± 3.0 kcal and D0 as 4.01 × 10−3 cm2 s−1.The avctivation energy thus found is in good agreementwith the value obtained by Funkee and Richtering [77]from NMR measurements, 22 ± 2 kcal.

In three-dimensional bulk diffusion model the solu-tion to Fick’s law [28] for a finite solid has the nextrelation

3349

Figure 5 Temperature dependence of self-diffusion coefficients Li (1,3) and H (2, 4–6) in LiH single crystals; 1, 2 [76]; 3, 6—[77]; 4,5—[75](after [76]).

where the symbols are as previouly defined and 2a, 2band 2c are the dimensions of the crystal in the x , y andz directions. The best fit of the experimental data ofpapers [74, 75] gives for the diffusion coefficient as afunction of temperature, D = 2.41 × 10−2e−24.3×103/RT

cm2 s−1. The diffusion coefficient for deuteride ion inlithium hydride at 465◦C reported in paper [74] tohave been found from three dimensional model was(1.9 ± 0.6) × 10−9 cm2 s−1 (see also Fig. 5). The dif-fusion coefficient calculated from Equation 36 was1.6 × 10−9 cm2 s−1. A comparison of tracer diffusioncoefficients as calculated from one- and three dimen-sional models is given in Table II, and a plot of − log Dversus 103/T is given in Fig. 6.

The activation energies for various migrating specieshave been theoretically calculated by Dellin et al. [78].They find activation energies for interstitial H− diffu-sion to lie in the range 11.5 to 23 kcal, while activationenergy for H− vacancy migration is calculated to be2.3 kcal. From these calculations interstitial H− mi-gration would seem possible based on the activationenergy of about 24 kcal found by Spencer et al. [75].However, Dellin et al. [78] in agreement with Pretzelet al. [79] find that interstitial H− is an unstable speciesin LiH and thus could not be the difffusing species.

To the conclusion of this part, we should indicateonce more that the activation energy found by Spenceret al. [75] ∼24 kcal, is in excellent agreement withthe 22 kcal determined for H− sel-diffusion in LiH byNMR [77]. This agreement, plus the consistencies of

T ABL E I I Comparison of tracer diffusion coefficients as found fromone- and three-dimensional models (after [75])

Sampleno. T ◦, C 1-Da × 109 cm2 s−1 3-Db × 109 cm2 s−1 3-Dc

8 550 5.8 8.7 0.2 ± 0.19 524 3.7 5.3

10 500 2.3 3.411 450 0.84 1.112 410 0.34 0.4113 399 0.26 0.31

aCalculated from Equation 28.bCalculated from Equation 30.c3-D is the data from [76].

Figure 6 Arrhenius plot of the diffusity of D− in LiH in the temperatureregion 400–550◦C for θ = 0.8. The activation energy as determinedfrom this plot is 24.3 ± 2.6 kcal. (after [75]).

previous work [74], make D− vacancy migration stillthe most likely species and mode of migration (see also[76]).

1.6. Self-diffusion in intrinsic GeIn intrinsic germanium the temperature dependenceof the tracer self-diffusion coefficient of the radioac-tive isotope 71Ge has been measured by several groups[80–85] by means of different techniques (see Fig. 7).With the exception of the latest experiments, precisiongrinding techniques were used to remove sections withthickness of the order of 1 µm from the diffusion zoneof the annealed specimens. As a consequence, the tem-perature range covered by the earlier experiments israther limited. By means of a sputtering technique forserial sectioning [84, 85] have been able to extend therange of self-diffusion studies in Ge to diffusivities aslow as 10−22 m2 s−1.

The overall agreement between Ge self-diffusiondata of different authors is good. In the region of overlapa small difference between the data of Vogel et al. [84]and those of the earlier workers may be seen. We tendto attribute this to problems in determining small diffu-sion coefficients during the earlier work. Widmer andGunther-Mohr [82] used Gruzin’s [86] or Steigmann’s[87] methods, both of which are known to be lessreliable than the layer-counting method since thesemethods require a precise knowledge of the absorp-tion coefficient of the radiation involved. In the work ofValenta and Ramasastry [81], the condition δ �

√DTt

(δ − thickness of the deposited tracer layer) was not al-ways fulfilled. Since, nevertheless, these authors usedthe thin-film solution of the diffusion equation to de-duce tracer diffusion coefficients, the obtained valuesare likely to be somewhat larger than the true DT values.As may be seen in Fig. 7, the temperature dependenceof the DT data of Ge is well described by an Arrheniuslaw (the preexponential factors DT

0 and the self-diffusion enthalpies H SD obtained from measurements

3350

Figure 7 Tracer self-diffusion coefficient of Ge as a function of temper-ature: �—[80]; �—[81]; ◦—[82]; �—[83]; •—[84]; x—[85] (after[43]).

of different authors are compiled in Table III). Seegerand Chik [88] argued that this result may be accountedfor in terms of an indirect self-diffusion mechanisminvolving one type of intrinsic defect. Guided by fur-ther observations, they suggest that it is the vacancymechanism (details see [43]) that controls self-diffusionin Ge. Table III shows that the preexponential factorDT

0 of Ge is considerably larger than the DT0 values

T ABL E I I I Self-diffusion data for germanium and silicon (after [43])

DT0 HSD Temperature

Element (104 m2 s−1) (eV) range (K) Technique References

Ge 7.7 2.95 1039–1201 SG [80]32 3.1 1023–1143 SG [81]44 3.12 1004–1188 SM + GM [82]10.8 2.9924.8 3.14 822–1164 SS [84]13.6 3.09 808–1177 SS [85]1.2 × 10−3 3.05 543–690 SIMS [97]

Si 1800 4.77 1473–1673 HL [389]1200 4.72 1451–1573 CS [90]9000 5.13 1373–1573 ES [91], [92]1460 5.02 1320–1660 SS [93]8 4.1 1173–1373 R [94]154 4.65 1128–1448 SIMS, 30Si [95]20 4.4 1103–1473 R [96]

SG = sectioning by grinding; SAM = Steigmann’s method; GM =Gruzin’s method; SS = sectioning by sputtering; HL = hand lapping;CS = chemical sectioning, n activation of 30Si; ES = electrochemicalsectioning; R = (p,γ ) resonance of 30Si.

typical for metals (10−6 m2 s−1 � DT0 � 10−4 m2 s−1)

[43]. Arguing that for an ordinary mechanism the prod-uct fvgva2

0νv0 in Equation 16 for DTv0 (≡ DT

0 ) shouldbe of the same order of magnitude for Ge and metals,Seeger and Chik [88] interpreted the large DT

0 value ofGe in terms of a large self-diffusion entropy of the va-cancy in Ge, SSD

v ≈ 10 k. They suggested that this largeSSD

v value arises from a spreading out of the vacancyover several atomic volumes.

As we can see from Table III the published valueof fundamental quantities such as the diffusion coeffi-cient vary by several orders of magnitude for variousauthors (see also [43, 88]). Such a spread in the experi-mental data makes it difficult to determine conclusivelythe underlying physical processes. Reliable diffusiondata are therefore crucial to clarify the diffusion mech-anisms and to accurately determine the correspondingmaterial parameters. The conventional technique (see,e.g., Table III) to determine the self-diffusion coeffi-cient DSD in semiconductors is to deposit thin layerof radioactive tracer on the surface of the crystal (e.g.,71Ge, 31Si). In a subsequent annealing step the trac-ers diffuse into the crystal. The depth profile of thetracer atoms is then determined by serial sectioningand measurements of the corresponding radioactivity.There are several experimental difficulties arising fromthis method (see also [97]).

1. Traditionally, lapping and grinding was used forthe serial sectioning. This requires that the mean pen-etration distance (DSDt)1/2 of the tracer atoms duringthe time t of a diffusion anneal has to be in the µmrange. Especially in silicon, the large distance and theshort half-life (2.6 h for 31Si) limit this method to be ap-plicable only to higher temperatures (larger DSD). Ger-manium is more convenient in this respect (the half-lifeperiod of 71Ge is 11.2 days), but it was not until micro-sectioning technique (e.g., sputtering) were inventedthat the measurements could be extended to lower tem-peratures in recent years (see [285]).

2. Surface effects such as oxidation, contamination,strain, etc. might influence the tracer diffusion substan-tially (e.g., through the formation of intrinsic defects).

Fuchs et al. [97], recently reported results of a veryaccurate method to measure the self-diffusion coeffi-cient of Ge which circumvents many of the experimen-tal problems encountered in the conventional methods.These authors used germanium isotopic heterostruc-tures (stable isotopes), grown by molecular-beam epi-taxy (MBE) (details see [98–100]). As is well-known,isotope heterostructures consist of layers of pure (e.g.,70Ge, 74Ge) or deliberately mixed isotopes of a chem-ical element. Fig. 8 shows the schematic of the partic-ular samples used by Fuchs et al. [97]. At the interfaceonly the atomic mass is changing, while (to first or-der) all the other physical properties stay the same. Inthe as grown samples, this interface is atomically flatwith layer thickness fluctuations of about two atomicML (details see [99]). Upon annealing, the isotopes dif-fuse into each other (self-diffusion) with a rate whichdepends strongly on temperature. The concentration

3351

Figure 8 Schematic of the isotope heterostructure used by Fuchs et al.(after [97]).

profiles in paper [97] were measured with SIMS, af-ter pieces of the same samples have been separatelyannealed at different temperatures. This allows an ac-curate determination of the self-diffusion enthalpy aswell as the corresponding entropy. The isotopic het-erostructures are unique for the self-diffusion studiesin several aspects (see also [1]).

1. The interdiffusion of germanium isotopes takesplace at the isotopic interface inside the crystal, un-affected by possible surface effects (e.g., oxidation,impurities and strain) encountered in the conventionaltechnique.

2. One sample annealed at one temperature providesfive more or less independent measurements (Ge con-sists of five stable isotopes). Their initial respective con-centrations vary for the different layers of the as-grownisotope heterostructure. After annealing, the concen-tration profile of each of the five isotopes can be ana-lyzed separately to obtain five data points for each an-nealing temperature. The samples were cut into severalpieces. One piece was kept in paper [97] for reference(as-grown), the were separately annealed at five differ-ent temperatures (543, 586, 605, 636, and 690◦C). Thetemperature controller permitted a variation of the tem-perature of 1–2◦C. The recording of the concentrationdepth profiles of all five stable Ge isotopes was per-formed with SIMS. The oxygen primary beam had animpact energy of 8 keV per incident ion. The beam wasrastered over a square area of about 200 µm in size andthe detected secondary ions extracted from the central30µm diameter region of the crater. The precision of theSIMS data was estimated to be within ±5%. The depthresolution of the system was determined from profilestaken from the as-grown samples with an atomically flatinterface. What theoretically should be a step functionin the concentration profile appeared as a slope of about4 nm per decade of the measured atomic fraction at theleading edge of a layer, and about 16 nm per decade atthe falling edge (details see [97]).

As is well-known, diffusion in the crystals occursthrough jumping thermally activated between differ-ent sites in the lattice [15, 16]. In principle, there aremany possibilities for such jumps (substitutional or

interstitial sites, vacancies, etc. (details see [43])). InGe crystals, however, it is known that the only processof significance for the migration of germanium atoms isthrough the vacancy mechanism (see also [88]) In thiscase the self-diffusion coefficient DSD can be writtenas as Arrhenius expression [97] (see Equation 22)

DSD = g f a2ν0 exp

[−GSD

]

= D0 exp

[−H SD

], (37)

wher GSD is the Gibbs free energy of self-diffusion,

GSD = H SD = T SSD, (38)

H SD is the self-diffusion enthalpy, and the preexponem-ntial factor

D0 = g f a2ν0 exp

[SSD

](39)

contains the self-diffusion entropy SSD, the correlationfactor f ( f = 1/2 for the vacancy mechanism in thediamond lattice [101], the attempt frequency ν0, thegeometric factor g (g = 1/8 for vacancies in Ge andthe lattice constant a, k is Boltzmann’s constant (seealso part 1.3) The enthalpy H SD and the entropy SSD

depend on the formation (subscript F) as well as themigration (subscript M) of the vacancy:

H SD = H SDF + H SD

M and SSD = SSDF + SSD

M . (40)

The quantity which we can extract from the dataof paper [97] is primarily the self-diffusion coefficientDSD as a function of annealing temperature T . This wasdone in citing paper by fitting of experimental depthprofiles to theory, with DSD being the only fitting pa-rameter. Equation 31 then allows to determine the self-diffusion enthalpy H SD, and the self-diffusion entropySSD is deduced using Equation 40. Solving Fick’s diffu-sion equation for the specific geometry of samples usedin indicated paper (see Fig. 8), these authors obtain theatomic fraction ci of a given germanium isotope i interms of error functions (erf) (see Equation 36):

ci(x) ={

c0,Ii − c0,II

2erf

[h/2 + x

2√

DSDt

]+ c0,I

}

c0,IIi − c0,III

2erf

[h/2 − x

2√

DSDt

]+ c0,III

}, (41)

where h is the layer thickness (110 or 200 in Ge sam-ples, see Fig. 8), and c0,I

i , c0,IIi , and c0,III

i are the initialconcentrations of the isotope i in the enriched 74Gelayer, in the enriched 70Ge layer, and in the substrate,respectively. Fig. 9 shows the profiles of all five isotopesof an annealed sample (586◦C for 55, 55 h), togetherwith a fit of the data to Equation 41. For clarity onlythe fit to the 70Ge profile is shown, but other profilescan be independently fitted as well. The excellent qual-ity of the fit over four orders of magnitude displays

3352

Figure 9 Experimental depth profile of the atomic fraction of 70Ge,72Ge, 73Ge, 74Ge and 76Ge (symbols) of a diffusion annealed sample(annealed at 586◦C for 55.55 h). The solid line is a fit of the 70Ge dataof Equation 41. For clarity, only the fit to the 70Ge data is shown (after[97]).

Figure 10 Experimental depth profiles of the same sample as Fig. 9, butbefore annealing (after [97]).

the remarkable accuracy of the method used by Fuchset al. As a reference, the corresponding concentrationprofiles for as-grown sample are displayed in Fig. 10.The annealing time was purposefully chosen such thatthe plateaus in the annealed samples (around 300 and100 nm) correspond to the original concentrations inthe isotopically enriched layers.

The values for the self-diffusion coefficient DSD ob-tained at 543, 586, 605, 636 and 690◦C are presented

Figure 11 Arrhenius plot of the self-diffusion coefficient as a functionof temperature. Data of Fuchs et al. [99] agrees favorably well with themost recent data ([84, 85]). The older data [80–82] might be less accurate(after [97]).

in an Arrhenius plot in Fig. 11. The lines in Fig. 11represent the results of previous authors [43]. The vari-ation in DSD obtained from different groups is com-parable with the scatter of the data within the workof each of the publications. Fitting the experimentalvalues of DSD to Equation 37 Fuchs et al. obtain theself-diffusion enthalpy H SD equals 3.0(5) eV. As cansee from Table III this in excellent agreement with pre-viously published values of 2.95–3.14 eV. The value ofexperimental preexponential factor D0 is 1.2 × 10−3 m2

s−1. This compares to previously published values of(0.78–4.4) ×10−3 m2 s−1. Converting D0 into the self-diffusion entropy SSD through Equation 39 they obtainSSD ≈ 9 k (using ν0 = 8 × 1012 s−1 and a = 0.565 nm).The self-diffusion entropy for Ge is larger than for met-als (2–4) k. As an explanation, Seeger and Chik [88]invoked the idea of extended (spread-out) defects, andBourgoin and Lanoo [102] have proposed that vacancyin Ge is strongly relaxed.

Finally, we want to mention the effect of the isotopicmass on the self-diffusion coefficient (see also [43]).The many-body treatment of atomic jump processesleads to an expression for the strength of the isotopeeffect in terms of the correlation factor f of Equation 19and the fraction �K of the kinetic energy which is asso-ciated with the motion in the jump direction [103, 104].

[ DI

DII

]−1[mII

]1/2−1= f �K . (42)

In previous Ge self-diffusion experiments, Campbell[83] found f �K values between 0.26 and 0.30, whichtranslates into a ratio of D70Ge

SD /D74GeSD between 1.007

3353

and 1.008 [105]. This small difference, however is be-low the precision of the Fuchs et al. work. When fittingthe experimental depth profilies to Equation 41, theycould indeed not detect any appreciable difference be-tween the different isotopes. In addition, such smalldeviations would be insignificant in the Arrhenius plot(logarithmic scale of DSD in Fig. 11) for the determina-tion of the self-diffusion enthalpy H SDand entropy SSD.

1.7. Self- and interdiffusion of Ga and Alin isotope pure and dopedheterostructures

Self-diffusion is the most fundamental matter trans-port process in solids. Understanding this process ispivotal to understanding all diffusion phenomena insolids, including those for native defects and impuri-ties. As was noted above, compared to metals [106],self-diffusion processes in semiconductors are signifi-cantly more complex (see also [10, 11, 29, 107, 108]).This is due to the much richer spectrum of native de-fects and to the much larger effects of small concentra-tions of defects on the Fermi level position and otherproperties [43]. In III–V compounds, experiments aremore difficult to perform because of the high partialvapor pressure of the group-V elements and the depen-dence of native defect species and concentrations onstoichiometry [109].

Over the past thirty years, there have been only a fewattempts to directly study Ga self-diffusion in GaAsusing isotopes [110–112]. Goldstein [110] and Palfreyet al. [111] diffused radioactive 72Ga into bulk GaAsat elevated temperatures and obtained depth profilesof 72Ga by mechanical sectioning and radioactive as-saying. With a rather limited temperature range inves-tigated, they reported activation enthalpies of 5.6 and2.6 eV, respectively. Tan et al. [112] studied the disor-dering of 69GaAs/71GaAs isotope superlattice structureand found an activation enthalpy of 4 eV. However,arguing that the heavily Si-doped substrates in theirsamples affected the result, these authors discardedthis value in favor of their earlier estimate of a 6 eVactivation enthalpy [109]. In view of this controversyand the fact that our knowledge on self-diffusion inGaAs is primarily derived from studies of interdiffusion(see also below) of Ga and Al in Gax Al1−x As systems[113, 115], it is of great interest to study diffusion ofGa isotopes with some new approaches.

Wang et al. [114] reported results from Ga self-diffusion studies in GaAs. They measured concentra-tion profiles of 69Ga and 71Ga using SIMS and deter-mined the activation enthalpy and entropy by analyzingthe diffusion coefficients obtained between 800 and1225◦C as well as examined effects of substrate doping.71GaAs and 69GaAs layers of 200 nm each were grownusing molecular beam epitaxy (MBE) at 580◦C onGaAs substrates of natural isotope composition (69Ga:71Ga = 60.2:38.8) The nominal isotope purity in theepilayers is 99.6%.

Fig. 12 shows the SIMS depth profiles of 69Ga and71Ga after annealing at T = 974◦C for 3321 s. Excellentfits over 2.5 orders of magnitude in concentration are

Figure 12 SIMS depth profiles of 69Ga and 71Ga in GaAs isotope epi-layers annealed at 974◦C for 3321 s. The circles are theoretical fits (after[114]).

obtained for this and all the other depth profiles takenfrom samples with smooth surfaces.

The characteristic diffusion length R is defined as

R = 2√

Dt, (43)

where D (as usually) is the Ga self-diffusion coefficientand t is the annealing time. The Ga self-diffusion coeffi-cient D can be derived from Equation 37. An Arrheniusplot for D is presented in Fig. 14 (see below). The Dvalues span 6 orders of magnitude in the temperaturerange from 800 to 1225◦C.

Expanding the research on Ga self-diffusion beyondGaAs to other III–V compound semiconductors can bequite instructive in elucidateng the microscopic mech-anism. This was the main reason an investigation [116]on Ga in gallium phosphide (GaP) using the same SIMStechnique. For the experiment, 71GaP and 69GaP epi-taxial layers 200 nm thick were grown by solid sourcemolecular beam epitaxy (MBE) at 700◦C on undopedGaP substrates. The isotope composition was the sameas in GaAs. Fig. 13 shows the SIMS profiles (solid

Figure 13 SIMS depth profiles of 69Ga and 71Ga in GaP isotope epi-layers annealed at 1111◦C for 231 min. The filled circles represent thecalculated 69Ga concentration profile (after [116]).

3354

Figure 14 Arrhenius plots of Ga self-diffusion coefficients in GaAs(filled circles) and GaP (filled squares) (after [116]).

lines) and the calculated C(x) of 69Ga (circles) and71Ga (continuous line) in a sample annealed at 1111◦Cfor 3 h and 51 min. Excellent agreement is obtained be-tween the measured and the calculated profiles over twoand a half orders of magnitude in concentration. Thisagreement strongly supports the assumptions made inEquation 12. The fitting procedure leads to an accuratedetermination of D through Equation 43.

In Fig. 14, D is plotted versus temperature T . TheD values span over two orders of magnitude in thetemperature range from 1000 to 1190◦C. They can rep-resented by Equation 17. From this the cited authorsare determined the activation enthalpy H SD and thepreexponential factor D0 to be 4.5 eV and 2.0 cm2 s−1,respectively. The self-diffusion entropy can be obtainedused the Equation 39. In GaP g ∼ 1, a = 5.45 A, andν0 = 1.2 ×1013 Hz. Using these values Wang et al. [116]obtained SSD = 4 k. The Ga self-diffusion coefficientsin GaAs from [114] are also shown in Fig. 14 for com-parison. The activation enthalpy and entropy for GaAsare 4.24 eV and 7.5 k, respectively.

As in the case of GaAs, the Ga self-diffusion coef-ficients in GaP follow an Arrhenius relation describedby Equation 37, indicating that a single type of nativedefect is most likely responsible for mediating the Gaself-diffusion in GaP over studied temperature range. Inintrinsic GaAs, the defect mediating Ga self-diffusionhas been ascertained to be the triply negatively chargedgallium vacancy acceptor, V 3−

Ga [109]. Such assign-ment may still be premature for GaP. From a recentpositron annihilation study in GaP, Krause-Rehberget al. [117, 118] reported that positron trapping by va-cancies behaves similarly in GaP as in GaAs. Vacanciesare detected by these authors only in n-type GaP at roomtemperature, with a detection limit of 2 × 1015 cm−3.These findings neither support nor exclude the possi-bility that in intrinsic GaP it is also the acceptor-likeVGa that mediates Ga self-diffusion.

Fig. 14 shows that the Ga self-diffusion coefficient inGaP is about two orders of magnitude lower than thatin GaAs. The decreased cation diffusion in III–V com-pounds, with phosphorus replacing arsenic as anion,

has been previously observed [117, 118]. Interdiffu-sion of Ga and Al was determined to be two orders ofmagnitude slower in the AlGaInP/GaInP superlatticesystem than in the AlGaAs/GaAs system (details seebelow). The change has been attributed to the strongerGa P bond compared to the Ga As bond [118]. Al-though Wang et al. [116] measured a higher activationenthalpy in GaP (4.5 eV) than in GaAs (4.24 eV), thedifference is not large enough to be outside the exper-imental uncertainty. More reliable is the difference inthe preexponential factor D0, or the entropy term SSD,between GaP (4 k) and GaAs (7.5 k). As was shownabove, this entropy term is the sum of the formationentropy SF and migration entropy SM for the native de-fect mediating the self-diffusion. SF or SM representsthe number of equivalent formation configurations ormigration jumps. The significant difference in SSD in-dicates profound variations in the way that the medi-ating native defects are formed or migrate in GaP ascompared to GaAs (see also [1]). The small value ofS supports a simple native defect species as the majordiffusion vehicle (details see [114]).

Bracht et al. [115] were used three undoped iso-tope heterostructures of Al71

x Ga1−x As/Al69y Ga1−yAs/

71GaAs with (X , Y ) = (0.41, 0.62) (a); (0.62, 0.85) (b);and (0.68, 0.88) (c) and in addition an AlAs/71GaAs(d) layer structures for Ga self- and Al-Ga interdiffu-sion experiments. The thickness of the layers lie be-tween 100 and 200 nm. The structure were grown byMBE at about 600◦C on a 200 nm thick undoped naturalGaAs buffer layer which was deposited on (100) ori-ented GaAs substrate wafers. A natural GaAs cappinglayer, about 200 nm thick, was grown om top of thestructure to protect the AlGaAs layer against oxidationin air. Concentration profiles of Al, 69Ga, 71Ga, and Asin the annealed samples were measured with SIMS us-ing a Cs+ ion beam with an energy of 5.5 keV. The depthof the craters left from the analysis were determinedwith a Tencor P-10 surface profilometer. The measuredsecondary ion counts were converted into concentra-tions taking into account the count rates obtained on anAl0.56Ga0.44As standard.

Concentration profiles of Al, 69Ga, and 71Ga of theas-grown structure b are shown in Fig. 15a. Fig. 15b il-lustrates the corresponding distribution after annealingof sample b at 1050◦C for 1800 s. Concentration profilesof 69Ga (see also [115]) which lie within the Al71GaAsand 71GaAs layers, are accurately described by thesolution of Fick’s law (see above) for self-diffusionacross an interface taking into account a concentration-independent diffusion coefficient. The measured 69Gaprofiles within the 71GaAs layer of samples a to d allyield the same Ga self-diffusion coefficient within ex-perimental accuracy even though the Al concentrationin the 71GaAs layer varies from 1018 cm−3 (detectionlimit) in e.g., sample d up to 1020 cm−3 in sample bdue to in-diffusion of Al from the adjacent AlGaAslayer. The temperature dependence of Ga self-diffusionin Alx Ga1−x As with X = 0, 0.41, 0.62, and 0.68 isshown in Fig. 16. The activation enthalpy H SD of Gaself-diffusion in Alx Ga1−x As and the correspondingpreexponential factor D0 are summarizes in Table IV.

3355

Figure 15 SIMS depth profiles of Al(◦), 69Ga (�), and 71Ga (+) inthe as-grown Al71GaAs/Al69GaAs/71GaAs heterostructure b [see (a)]and after annealing at 1050◦C for 1800s [see (b)]. Solid lines in (a)connect the data to guide the eye. Solid lines in (b) show best fits to theexperimental profiles. For clarity, only every fourth data point is plottedin (a) (b) (after [115]).

Figure 16 Temperature dependence of the diffusion coefficient D of Gain Alx Ga1−x As with x = 0 (◦); 0.41 (*); 0.62 (�); 0.68 (�) and 1.0 (•)and of Al in GaAs (�) (after [115]).

Recently Wang et al. [114] determined an activationenthalpy of (4.24 ± 0.06) eV for Ga self-diffusion inGaAs which deviates from the present result of (3.71 ±0.07) eV. The authors [114] favor the activation en-thalpy of 3.71 eV for Ga self-diffusion in GaAs, becausesimultaneous annealing of the former 69GaAs/71GaAsheterostructure with sample a has revealed that the Gaprofile near the interface of the 71GaAs layer and theGaAs substrate, which was considered by Wang et al.for the self-diffusion study, deviates from the expectederror function solution. This may be caused by surfacecontamination of the GaAs substrate wafer. The tem-perature dependence of Ga self-diffusion in AlGaAsreveals that Ga diffusion decreases with increasing Alcontent whereas H SD appears to be constant within theexperimental error (see Fig. 16 and Table IV).

The Al and Ga profiles near the Al69GaAs/71GaAsinterface result from Al-Ga interdiffusion. The Al-Ga interdifussion coefficient D can be expressed as

TABLE IV Activation enthalpy HSD and natural logarithm of thepreexponential factor D0 for Al and Ga diffusion in Alx Ga1−x As forintrinsic and As-rich (pAs ∼ 1 atm) conditions (after [115])

Element X of Alx Ga1−x As HSD ln (D0/cm2 s−1)

Al 0.0 3.50 ± 0.08 −1.77 ± 0.80Ga 0.0 3.71 ± 0.07 −0.45 ± 0.67Ga 0.41 3.70 ± 0.21 −1.05 ± 1.98Ga 0.62 3.60 ± 0.13 −2.51 ± 1.24Ga 0.68 3.51 ± 0.19 −3.96 ± 1.78Ga 1.0 3.48 ± 0.23 −4.51 ± 2.16

(see [107])

D = (XAl DGa + XGa DAl)�S, (44)

where XAl and XGa are the mole fractions of Al andGa. DGa and DAl represent the Ga and Al diffusioncoefficients in AlAs and GaAs, respectively. � is thethermodynamic favtor and S the vacancy wind fac-tor which takes vacancy interaction and correlation ef-fects into account. For modeling Al-Ga interdiffusionauthors [115] assume the simplest possible values of� = 1 (ideal solution) and of S = 1. On this basis,Fick’s second law was solved numerically. The mea-sured Al and 71Ga profiles shown in Fig. 15b can bothbe described with the composition dependent interdif-fusion coefficient D according to Equation 44 whichtakes into account the actual mole fraction of Al and Gaas a function of depth. According Bracht et al. [115] ,all interdiffusion profiles of samples a to d, which wereannealed at the same temperature, are accurately de-scribed with data for DGa and DAl which are consistentwithin 40%. The temperature dependence of Al diffu-sion in GaAs and of Ga diffusion in AlAs is shown inFig. 16. Fitting Equation 37 to these results yields datafor H SDand D0 which are listed in Table IV.

The values for H SD shown in Table IV all lie in therange of (3.6 ± 0.1) eV. Recently Wee et al. [119] re-ported an activation enthalpy of (3.6 ± 0.2) eV and apre-exponential factor of 0.2 cm2 s−1 for interdiffusionof Al0.2Ga0.8As/GaAs at temperatures between 750 and1150◦C. Their data are consistent with data of paper[115] on Ga diffusion in AlGaAs and its dependenceon the Al content. Tan et al. (see [112] and referencestherein) have proposed 6 eV for Ga self-diffusion. Thisresult is based on a compilation of Ga self-diffusionand Al-Ga interdiffusion data obtained under variousexperimental conditions which includes AlGaAs/GaAsheterostructures with Al contents up to 100%. The acti-vation enthalpy of 6 eV appears now to be questionablesince Bracht et al. results unambiguously shows that Al-Ga interdiffusion does not represent Ga self-diffusion.

The single activation enthalpy found for Ga self- andAl-Ga interdiffusion suggests that the diffusion is medi-ated by the same native defect. Vacancies on the sublat-tice of the group-III atoms are assumed to mediate theGa self- and Al-Ga interdiffusion under intrinsic con-ditions [120]. In this case the self-diffusion coefficientis given by (see also (37))

DSD = fVCeqV DV = fVga2ν exp

(−GSDV

/kT

), (45)

3356

where fV is a correlation factor and CeqV and DV

the thermal equilibrium concentration of vacanciesand their diffusivity. Ceq

V DV can be expressed by thegeometry factor g, the jump distance a, the jumpattempt frequency ν, and the Gibbs free energy GSD

V ofself-diffusion via vacancies. ν is proportional to 1/

√m,

where m represents the atomic mass of the jumpingatom. Different jump frequencies of 27Al and 71Gacaused by the difference in their masses are proposedto be the cause for the experimentally observed higherAl diffusion in GaAs compared to Ga self-diffusion.The experimentally determined ratio between the Aland the Ga diffusion coefficient in GaAs is consistentwith (mGa/mAl)0.5. The decreasing Ga diffusivity withincreasing Al content in AlGaAs can be interpretedwith a different location of the intrinsic Fermi energylevel with respect to the vacancy charge transitionstate [121]. This causes different thermal equilibriumconcentrations of vacancies in AlGaAs for differentAl compositions.

In the next we follow Bracht et al. papers considerthe experimental ratio between Ga diffusion in GaAsand AlAs. This equals the ratio between Ceq

V DV forGaAs and AlAs if the binding energy between Ga anda vacancy in AlAs is negligible. Assuming only a singlenegatively charged vacancy, the total concentration ofvacancies in thermal equilibrium is given by [122]

CeqV = Ceq

[1 + exp

(E in

F −EV−/0/kT)]

, (46)

where CeqV0 is the equilibrium concentration of a neutral

vacancy, E inF the Fermi level under intrinsic condition,

and EV−/0 the acceptor energy level of a singly chargedvacancy. The Fermi-level position is given by

E inF = 0.5Eg + 0.75kT ln(m∗

V/m∗C), (47)

where Eg is the band gap energy and m∗V and m∗

Care the effective density of state masses for holes andelectrons in GaAs and AlAs, respectively [108]. UsingEquations 46 and 47, the ratio between Ceq

V in GaAsand AlAs is estimated to be 6.5 × exp · [0.077 eV/(kT)] assuming that Ceq

V0 in GaAs and AlAs are similarwith respect to the valence-band position of AlAsas energy reference. Bracht et al. obtain a ratio of12.8 at, e.g., 1050◦C, which is consistent with theirexperimental result of 11.1.

Dopant enhanced as well as reduced layer disorder-ing of semiconductor heterostructures are phenomenathat have been reported frequently in the literature. Su-perlattice structure doped with Si during growth [123,124] or by implantation [125, 126] reveal an enhancedAl-Ga interdiffusion after annealing compared to in-trinsic conditions. Conversely, Be doping if Si-dopedAlAs/GaAs has been found to suppress the superlatticedisordering when the Be doping level exceeds that ofSi [127, 128]. The dopant induced change in the posi-tion of the Fermi level, which strongly alters the con-centrations of charged native point defects and hencethe self- and interdiffusion, is generally considered tobe possible for this diffusion phenomenon [129]. Sofar, the relative contributions of the various charged

native point defects to the self- and interdiffusion,which are required to predict the disordering of GaAsbased superlattice structures for different doping lev-els, are not accurately known. Total energy calculationsprovide one way to determine the nature of native de-fects in group III–V compound semiconductors. Thesecalculations predict that the triply negatively chargedvacancy on the gallium sublattice (V 3−

Ga ) is the domi-nant native defect in GaAs, both for intrinsic and n-typedoping conditions under an As-rich ambient [130]. Onthe other hand, the charge state of native defects can bededuced exp erimentally from the doping dependenceof Ga self-diffusion in GaAs. Generally, Al-Ga interdif-fusion in AlAs/GaAs superlattices is considered to sim-ulate the self-diffusion process of the group-III atom.Tan and Gosele [131] analyzed data of Al-Ga interdif-fusion in Si doped AlAs/GaAs superlattices [132] andfound, in agreement with the theoretical calculationsof paper [130], that the self-diffusion on the Ga sublat-tice is mediated by V 3−

Ga . In contrast to these seeminglyconsistent theoretical and experimental studies showthat singly negatively charged Ga vacancies V −

Ga medi-ate the Ga self-diffusion and the Al-Ga interdiffusionunder n-type doping conditions [132, 134]. Some ofthe data of Mei et al. [132] , which have been used tosupport the triply charged defect hypothesis [131], canalso be described assuming a singly charged vacancy[134]. Furthermore, Ga self-diffusion in GaAs has beenproposed to be governed by triply charged defects athigh Si doping levels, whereas at low doping concen-trations a Fermi level independent mechanism was as-sumed [135, 136]. These discrepancies in the dopingdependence of group-III atom diffusion in the AlGaAsmaterial system forced Bracht et al. [137] to investigatewith SIMS technique the Ga self-diffusion in undoped,Si- and Be-doped GaAs by using 71GaAs/natGaAs iso-tope heterostructures.

Three Al71GaAs/Al69GaAs/71GaAs/natGaAs isotopeheterostructure with different Al content and oneAlAs/71GaAs/nat GaAs structure were used for the dif-fusion experiments [137]. One set of samples was keptundoped and another set was doped either with Si orBe, i.e., altogether 12 different isotope structures wereused. Each of these samples contains a 71GaAs/natGaAsinterface on which was focused on in the present self-diffusion study.

SIMS measurements on the undoped set of isotopeheterostructures have revealed a change in the SIMSsputter rate with increasing Al concentration [138].Since all samples used for the Ga self-diffusion experi-ments contain AlGaAs layers, a decrease in the sputterrate with increasing Al content results an apparentlythicker AlGaAs layers and thinner GaAs layers. There-fore a point by point correction of the penetration depthof the SIMS profile was performed taking into accountthe measured sputter-rate dependence. According tocited authors the thickness of the AlGaAs and GaAslayers corrected in this way are consistent within 10%or better with the thickness determined directly withtransmission electron microscopy.

Concentration profiles of 69Ga near the 71GaAs/natGaAs interface measured with SIMS after annealing

3357

Figure 17 SIMS depth profiles of 69Ga, 28Si and 9Be after annealingof (a) Si-doped and (b) Be doped 71GaAs/natGaAs isotope structuresat various temperatures and times as indicated. The solid lines showthe best fits to experimental data. Ga self-diffusion coefficients deducedfrom each profile are listed in the figure. The numbers 1, 2, and 3 at thetop X-axis indicate the position of the 71GaAs/natGaAs interface of theparticular sample. For clarity only the dopant profile (dashed line) fromthe Ga profile 3 is shown (after [137]).

of Si- and Be doped samples are shown in Figs 17a,band 18. The Si- and Be profiles reveal a nearly constantdopant concentration across the 71GaAs/natGaAs inter-face. A few sufficiently large Si-doped samples werealso analyzed with C-V profiling to determine the freecarrier concentration after annealing. The carrier con-centration profile of one sample is shown in Fig. 18together with the corresponding SIMS profiles of SIand Ga. C-V profiling revealed a Si donor concentra-tion, CSi, of about 2 × 1018 cm−3 along the 69Ga profile.Similar results were obtained from the analysis of othersamples. The electron concentration n after annealingis smaller than CSi of the as-grown structures. This de-crease is also apparent in the observed effect of dopingon Ga self-diffusion. The hole concentration p of Be-doped samples could not be accurately determined byC-V profiling after annealing. In all cases the cratersleft behind by the electrochemical etching were verynon-uniform. Consequently, no reliable data could beobtained for p. SIMS analysis shows that the Be con-centration along the Ga self-diffusion profile equals theconcentration of the as-grown structure. Therefore, it isassumed that the free carrier concentration p due to Be

Figure 18 SIMS depth profiles of 69Ga and 28Si of a Si doped71GaAs/natGaAs structure annealed at 800◦C for 2 h. The solid lineshows the best fit to the experimental data which yields the Ga self-diffusion coefficient listed in the figure. The room temperature electronconcentration, which was measured by C-V profiling, is shown as dashedline (after [137]).

doping is not significantly affected by the thermal an-nealing. This is supportted by the doping dependenceof Ga self-diffusion, which is accurately described tak-ing into account the Be doping level of the as-grownsamples (details see [137]).

Ge self-diffusion profiles shown in Figs 17 and 18are accurately reproduced by the solution of Fick’s lawfor self-diffusion across an interface (see above). Allexperimental 69Ga profiles were described on the basisof Equation 41, in order to take into account the diffu-sion of 69Ga from the adjacent Al69GaAs layer into the71GaAs layer. The thickness d = x1 − x2 determined inthis way for the 71GaAs layer is consistent within 10%with the thickness in the as-grown structure. The un-certainty results mainly from the accuracy of the craterdepth measurements.

The self-diffusion coefficients DGa extracted fromthe analysis of all Ga diffusion profiles in undoped, Si-and Be-doped isotope samples are listed in Table V.Each value of DGa is the average of at least four diffu-sion coefficients where each of them has been deducedfrom a different sample. The accuracy given for DGarepresents the standard deviation of all data which be-long to the same temperature and doping level. Thetemperature dependence of DGa for intrinsic, Si- andBe-doped GaAs is depicted in Fig. 64. Experimentalresults [138] for intrinsic conditions are accurately de-scribed by (see also [115])

DGa = 0.64 exp

(−3.71 eV

)cm2 s−1. (48)

Solid lines in Fig. 19 are best fits to the experimentaldata which accurately reproduce the temperature de-pendence of Ga sel-diffusion under the different dopingconditions.

According to Bracht et al. [137], the Ga self-diffusioncoefficient is given by the sum of the transport coeffi-cients of vacancies in various charge states

DGa = 1

3∑r=0

frCeqV r−

GaDV r−

Ga(49)

3358

T ABL E V Ga self-diffusion coefficients DGa in undoped, Si- and Be-doped 71GaAs/natGaAs isotope heterostructures. The numbers in parenthesisrepresent the number of samples which have been annealed at the particular temperature T and time t . A free electron and hole concentration of both(3.0 ± 0.5) × 1018 cm−3 was determined by Hall effect measurements for the as-grown Si- and Be-doped samples. ni, E in

F and n/ni represent theintrinsic carrier concentration [140], the position of the Fermi level under intrinsic conditions [140], and the ratio of the electron concentration in thedoped and undoped samples, respectively (after [137])

T ◦, C t(s) DGa(cm2 s−1) Type of doping ni(cm−3) E inF (eV) n/na

1160 270 (4) (6.12 ± 0.17) × 10−14 Undoped 1.56 × 1018 0.512 1.01050 1800 (4) (4.76 ± 0.80) × 10−15 Undoped 7.41 × 1017 0.545 1.01050 3600(1); 4800(4) (1.46 ± 0.20) × 10−15 Be-doped 7.41 × 1017 0.545 0.233

955 12600(1); 15120 (3) (3.84 ± 0.61) × 10−16 Undoped 3.54 × 1017 0.572 1.0955 10800 (1); 86400(4) (8.28 ± 2.50) × 10−17 Be-doped 3.54 × 1017 0.572 0.116872 260400(4); 432000(2) (2.47 ± 0.17) × 10−17 Undoped 1.70 × 1017 0.595 1.0872 496800(1); 1814400(4) (2.37 ± 0.59) × 10−18 Be-doped 1.70 × 1017 0.595 0.056872 1680(5) (3.67 ± 0.93) × 10−15 Si-doped 1.70 × 1017 0.595 14.8800 1209600(2); 1370700(2) (2.84 ± 0.36) × 10−18 Undoped 8.32 × 1016 0.615 1.0800 7200(4); 10800(1) (9.86 ± 1.65) × 10−16 Si-doped 8.32 × 1016 0.615 26.6736 – 1.88 × 10−19(b) Undoped 4.09 × 1016 0.632 1.0736 25200(1); 30600(4) (1.45 ± 0.21) × 10−16 Si-doped 4.09 × 1016 0.632 45.9

aData correspond to fit #3 (see Table VI).bCalculated with followed equation DGa = 0.64 exp(− 2.71eV

kB T ) cm2 s−1.

Figure 19 Temperature dependence of the Ga self-diffusion coefficientDGa in undoped (x), Si-doped (�), and Be-doped (•) GaAs for PAs4 = 1atm. The influence of doping on DGa is best reproduced (see solid lines) ifthe effect of the Fermi level together with a compensation of Si donors bynegatively charged vacancies is taken into account. The long-dashed lineis expected if the compensation by vacancies is ignored. Short-dashedlines represent the contribution of the doubly positive charged Ga self-interstitial I2+

Ga to Ga self-diffusion. Lower dashed line: I2+Ga contribution

for intrinsic conditions and PAs4 = 1 atm; upper dashed line: I2+Ga con-

tribution calcula ted for a hole concentration of 3 × 1018 cm−3 (after[137]).

where C0, fr, CeqV r−

Garepresent the Ga atom density in

GaAs (C0 = 2.215 × 1022 cm−3), the diffusion correla-tion factor, and the thermal equilibrium concentrationand the diffusion coefficient of the vacancy DV r−

Gawith

the charge r ∈ [0, 1, 2, 3], respectively. The correla-tion factor contains information about the microscopicjump mechanism [29]. The different vacancy configu-rations can introduce energy levels within the energyband-gap of GaAs. Occupation of these energy statesdepends on the position of the Fermi level. Under ex-trinsic conditions, i.e., when the hole or the electronconcentration introduced by doping exceeds the intrin-sic carrier concentration, the Fermi level deviates fromits intrinsic position. As a consequence, the ratio of the

charged to neutral vacancy concentrations is changed[122]. For vacancies which can introduce single (V −

Ga),double (V 2−

Ga ), and triple (V 3−Ga ) acceptor states with en-

ergy levels at EV −Ga

, EV 2−Ga

, EV 3−Ga

above the valence bandedge EV, the rations are given by

CeqV −

CeqV 0

= gV −Ga

exp

(EF − EV −

)(50)

CeqV 2−

CeqV 0

= gV 2−Ga

exp

( EF − EV 2−Ga

− EV −Ga

)(51)

CeqV 3−

CeqV 0

= gV 3−Ga

exp

( EF − EV 3−Ga

− EV 2−Ga

− EV −Ga

)(52)

The thermal equilibrium concentration of the neutralGa vacancy Ceq

V 0Ga

is independent of the position of theFermi level EF. However, if EF lies above EV r−

Ga, the for-

mation of negatively charged vacancies is energeticallyfavored compared to the formation of neutral defects.In Equations 49–51 the degeneracy factors gV r−

Ga, which

take into account a spin degeneracy of the defect anda degeneracy of the GaAs valence band, have all beenset to one.

The total concentration of VGa in thermal equilib-rium, CV eq

Ga, is given by the sum of the corresponding

concentrations of the various charge states of galliumvacancies

CeqVGa

=3∑

r=0

CeqV r−

Ga= Ceq

V 0Ga

3∑r=0

CeqV r−

CeqV 0

. (53)

Based on Equations 49–51 it becomes clear, that CeqVGa

changes when the position of the Fermi level changes.Naturally, the Ga self-diffusion coefficient given byEquation 49 may be written as

DGa = 1

C0Ceq

V 0Ga

3∑r=0

CeqV r−

CeqV 0

DV r−Ga

. (54)

3359

Recent published molecular dynamic calculations[139] show that the migration enthalpy of a Ga is nearlyindependent on its charge state. For accuracy migra-tion vis second-nearest neighbor hopping, an enthalpyof 1.7 eV was determined for V 0

Ga compared to 1.9 eVfor V 3−

Ga [139]. In addition, in paper [138] was assumedthat the entropy change associated with the V r−

Ga mi-gration and the correlation factor fr are similar for allcharge states, i.e., DV r−

Ga= DVGa and fr ≈ f . With these

assumptions, it follows that [138]

DGa = 1

C0f Ceq

V 0Ga

DVGa

3∑r=0

CeqV r−

CeqV 0

. (55)

Taking into account Equations 50–52, and the well-known expression

ni= exp

(EF − E i

), (56)

the following relationship between the Ga self-diffusion coefficient under intrinsic and extrinsic con-ditions have next form

DGa(n)

DGa(ni)= 1 + ∑3

m=1

(nni

)mexp

m E iF−

∑3m=1 EV m−

1 + ∑3m=1 exp

m E iF−

∑3m=1 EV m−

. (57)

In Equations 56 and 57, ni. E iF, n, and EF define the

free electron concentration and the Fermi level positionunder intrinsic and extrinsic conditions, respectively. Ifonly one charge state r dominates Ga self-diffusion,Equation 57 is reduced to

DGa(n)

DGa(ni)=

(58)

with r ∈ [0, 1, 2, 3]. This simplified relationship hasbeen generally used to analyze the doping dependenceof Ga self-diffusion and Al-Ga interdiffusion [131].Bracht et al. [137] prefer more general Equation 57,since this equation takes into account that the chargestate of the vacancy mediating Ga self-diffusion maychange with doping and temperature.

Fitting of Equation 57 to experimental results re-quired data for E i

F and ni. For these quantities thecited authors used the data reported by Blakemore [140]which are listed in Table V. The electron concentration nof the Si- and Be-doped GaAs samples were calculatedvia the charge balance equation yielding [137]

n = 1

(CSi −

3∑m=0

mCeqV m−

)

+√√√√n2

i + 1

(CSi −

3∑m=0

mCeqV m−

)(59)

and

n2i

n= p = 1

(CBe −

3∑m=0

mCeqV m−

)

+√√√√n2

i + 1

(CBe −

3∑m=0

mCeqV m−

), (60)

respectively. CSi—represents the Si donor concentra-tion and CBe—the acceptor concentration due to Bedoping. Both concentrations are equal the free carrierconcentration of 3 × 1018 cm−3, measured at room tem-perature. Negatively charged vacancies affect the freecarrier concentration are taken into account in bothEquations 59 and 60.

The free carrier concentration given by Equations 58and 60 also depends via Equations 49–51 and 56 onthe vacancy-related energy levels. Therefore Equations57, 59 and 60 were solved by Bracht et al. simultane-ously. Ceq

V m−Ga

(r ∈ [0, 1, 2, 3]) were calculated with Equa-tions 44–46 using a thermal equilibrium concentrationof neutral vacancies which is given by [141, 142].

CeqV 0

Ga= C∗ P1/4

As4T −5/8 exp

(H f

V 0Ga

), (61)

where C∗ is a preexponential factor and H fV 0

Gathe forma-

tion enthalpy of the neutral vacancy. Equa-tion 55 includes the influence of the As4 vapor phaseon Ceq

V 0Ga

[141, 142]. So far equilibrium concentrationsof V 0

Ga have been estimated with an uncertainty factorof at least 10 [141]. In order to accurately describe thedoping dependence of Ga self-diffusion on the basis ofEquation 51, not only EV m−

Ga(m ∈ [0, 1, 2, 3]) but also

C∗ were used as fit parameters. The formation enthalpyof neutral vacancies was set to a value of (1.9 ± 0.2) eV.This value equals the difference between the activationenthalpy of (3.71 ± 0.07) eV for Ga self-diffusionand the vacancy migration enthalpy of (1.8 ± 0.2) eVwhich was deduced in paper [136] from Al-Ga inter-diffusion in nonstoichiometric AlAs/GaAs quantumwells (see also Equation 48). Authors [137] concludedthat H f

V 0Ga

= (1.9 ± 0.2) eV is believed to be fairlyreliable.

The temperature and doping dependences of Ga self-diffusion in intrinsic, n-type and p-type GaAs whichwere calculated in paper [137] on the basis of the pa-rameters given under fit #3 in Table VI, are shown bythe solid lines in Fig. 19. The long-dashed line in Fig. 19displays the temperature dependence of DGa which isexpected if compensation via negatively charged va-cancies is ignored in the n-type GaAs with CSi =3 × 1018 cm−3.

A representation of DGa data [137] versus the ration/ni is given in Fig. 20. Solid lines have been calculatedwith Equation 57 taking into account the n/ni valueslisted in Table V and the results for EV m−

Gaof fit #3 (see

Table VI). The doping dependence of DGa at 872◦C,which was calculated on the basis of the theoreticalresults for EV m−

Ga(m ∈ [0, 1, 2, 3]) reported by Baraff

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T ABL E VI Parameter values obtained from fitting Equation 51 to the experimental results. The Ga self-diffusion is best described by neutral,singly and doubly negatively charged vacancies with relative contributions of the various charge states which change with temperature and doping.Additionally, a compensation of the Si donor concentration by negatively charged Ga vacancies was taken into account; it becomes especially significantat low temperatures (after [137])

Fit parametera Fit #1b Fit #2c Fit #3 Fit #4 Fit #5

EV−Ga

− EV (eV) 0.38 ± 0.11 0.42 ± 0.04 0.42 0.42 0.42

EV2−Ga

− EV (eV) 0.72 ± 0.19 0.60 ± 0.04 0.60 0.60 0.60

EV3−Ga

− EV (eV) 1.33 ± 4.33 – – – –

H f

V0Ga

(eV) 1.9 1.9 1.9 1.7 2.1

C∗(cm−3) 0.293 0.293 162 ± 54 18.6 ± 5.8 1385 ± 487

a EV denotes the valence band edge.bC∗ and H f

V0Ga

according to Tan [141].cV 3−

Ga contribution ignored, C∗ and H f

V0Ga

according to Tan [141].

Figure 20 Ga self-diffusion coefficients DGa versus the ration n/ni be-tween the free carrier concentration under extrinsic and intrinsic dopingconditions PAs4 = 1 atm. and different temperatures as indicated. Solidlines were calcuated via Equation 57 taking into account EVm

Gagiven

by fit #3 of Table XII. The dashed line shows the doping dependenceof DGa for 872◦C which was calculated for EV−

Ga− EV ≈ 0.20 eV,

EV2−Ga

− EV ≈ 0.52 eV, and EV3−Ga

− EV ≈ 0.72 eV according to Baraff

and Schluter [130] (after [137]).

and Schluter [138], is shown in Fig. 20 as a dashedline. The strong deviation from data of paper [137]clearly indicates that in contrast to the theoretical calcu-lations the triply charged Ga vacancy does not mediatethe Ga self-diffusion under intrinsic and n-type dopingconditions.

As was noted above, Muraki and Horikoshi [135]have studied, with the help of photoluminescence spec-troscopy, the Al-Ga interdiffusion of Si- and Be-dopedAl0.34Ga0.66As/GaAs superlattice structures. They haveproposed that Al-Ga interdiffusion is mediated bysingly negatively charged Ga vacancies in both n- andp-type material. The Al-Ga interdiffusion in these struc-tures with an Al content of 34 at.% is expected to simu-late the Ga self-diffusion in GaAs. This is supported byrecent results obtained in paper [115] on the Al compo-sition dependence of Ga self-diffusion in AlGaAs (seealso above) which show that for Al concentrations of41 at.% DGa in AlGaAs is only a factor of two smallerthan DGa in GaAs. The interdiffusion coefficients givenby Muraki and Horikoshi for different diffusion tem-peratures and doping levels are illustrated in Fig. 21 asa function of the ratio n/ni. A compensation of the Si

Figure 21 Al-Ga interdiffusion coefficient DAl−Ga reported by Murakiand Horikoshi [135] versus n/ni. Solid lines were calvulated with Equa-tion 51 taking into account DGa(ni) of Muraki and Horikoshi and resultsof Bracht et al. [137] for EV−

Ga(after [137]).

donors by charged vacancies was also considered bytaking into account the equilibrium concentrations ofvacancies which, consistent with results of paper [137],were obtained for the particular Si doping levels. Solidlines shown in Fig. 21 were computed via Equation51 with DGa (n/ni = 1) of Muraki and Horikoshi andthe results of Bracht et al. [137] for EV m−

Ga(see also

fit #3, Table VI). No adjustable parameter were used.The dependence on As4 pressure cancels out becauseall diffusion experiments were performed under identi-cal conditions. The deviation between the experimentaldata and the corresponding solid line for 800◦C mayindicate that the Si donor concentration is additionallyreduced by the formation of SiGa—acceptor pairs suchas SiGa–SiAs complexes or other compensation centers.This is supported by the fact that after annealing Murakiand Horikoshi have observed broad photoluminescencesignals in the Si-doped samples which they attributedto deep centers.

Self-diffusion coefficients for 900◦C reported byMuraki and Horikoshi are displayed in Fig. 22 as afunction of n/ni. The solid line in Fig. 22 was cal-culated on the basis of the results of Bracht et al.and accurately describes the Ga self-diffusion data forn/ni > 1. Small differences (n/ni > 1) between the as-grown and annealed structures can also be caused by

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Figure 22 Ga self-diffusion coefficient DGa reported by Muraki andHorikoshi [135] versus n/ni. The data in parenthesis are considered tobe not reliable. The solid line was calculated via Equation 51 taking intoaccount results of Bracht et al. for EV−

Gaand a value DGa(ni) which is

a factor of 5 smaller than the corresponding data given by Muraki andHorikoshi (after [137]).

Figure 23 Al-Ga interdiffusion coefficients DAl-Ga (n) of Mei et al.[133] normalized by DGa(ni) [115] as a function of n/ni. The solid linerepresents the best fit which is reproduced by DAl-Ga (n)/DGa(ni) = 3.4(n/ni)2.3 (after [137]).

SIMS broadening effects associated with a degradationof the surface quality during annealing under As-poorconditions.

Fig. 23 illustrates the ratio between the reduced in-terdiffusion coefficients DAl-Ga(n, PAs4 = 1 atm) andDGa(n, PAs4 = 1 atm) as a function of n/ni. In thisdouble logarithmic representation, the slope of the ex-perimental data equals the exponent r . The best fit [137]yields r = 2.3 ± 0.1 showing that V 2−

Ga rather than V 3−Ga

mediate the Al-Ga interdiffusion in agreement with theresults of Bracht et al. Based on this reanalysis, the V3−

Gamediated self- and interdiffusion in n-type GaAs, andthe activation enthalpy of 6 eV for Ga self-diffusionunder intrinsic conditions proposed by Tan and Gosele[131], are found to be incorrect (details see [137, 143]).

In summary, the doping dependence of group-IIIatom diffusion in the AlGaAs material system ([135,136, 141, 142] and references therein) can be consis-tently explained with the result presented by Brachtet al. [137]. Neutral, singly and doubly charged Gavacancies all contribute to the self-diffusion in un-doped, p-type and n-type material with relative con-tributions which depend on temperature and doping.The lower power dependence for the doping effects of

self-diffusion in samples containing group-VI donorsis proposed to be caused by the formation of next near-est neighbor complexes between the dopant and thevacancy.

Chapter 2. Neutron transmutation doping2.1. The NTD process—a new reactor

technologyThe neutron transmutation doping (NTD) process in-volves the cooperation of semiconductor materials spe-cialists, device producers, radiation damage and defectspecialists and reactor personnel. Of all possible in-teractions among these groups, those with the reactorcommunity have traditionally been the weakest. Reac-tor personnel have, therefore, had the greatest learningcurves to overcome. It is to the credit of both the reactorcommunity and the semiconductor industry that thesedifficulties have been overcome so readily in the fewyears since 1975 when NTD silicon first appeared onthe market. The transmutation doping process simplyinvolves irradiation of an undoped semiconductor witha thermal neutron flux. The major advantages of theNTD process are illustrated schematically in Fig. 24.The hom*ogeneity in NTD-Si is a result of a hom*oge-neous distribution of silicon isotopes in the target ma-terial and the long range of neutrons in silicon. Dopingaccuracy is a result of careful neutron flux integration.The material improvements offered by the NTD processform the basis for semiconductor device improvement(details see [145]).

As is well-known, research reactor facilities providethe best source of thermal neutrons for this purpose atthe present time (see e.g. [146–149]). These reactors areideally suited for such projects because they have usu-ally been constructed with sample irradiation as one ofthe prime design requirements. Although these reactor

Figure 24 Advantages of NTD process. Histogram of irradiation t arg etaccuracy obtained for commercial sample lot at NURR. Insert is aschematic representation of spreading resistance traces across a waferdiameter for conventionally doped and NTD Si (after [144]).

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facilities provide a source of thermal (E ∼ 0.025 eV)neutrons, this thermal flux is always accompanied by afast neutron component which is not useful in providingdoping transmutations, but does produce radiation dam-age (displacements of atoms from their normal latticesites) which must be repaired by annealing, the pro-cess of heating the irradiated material to temperaturessufficiently high that the irradiation produced defectsbecome mobile and can removed.

To understand the process further, we must be con-cerned with the interactions of neutrons, both thermaland fast, with the target material to be doped. Becauseneutrons are neutral particles, their range of penetrationin most materials is usually very long. They interactonly very weakly with atomic electrons through theirmagnetic movements. Being neutral, neutrons see noCoulombic barrier at the target nuclei and, thereforeeven very slow neutrons may reach into the nucleuswithout difficulty. In fact, the slower the neutron veloc-ity, the greater is the time of interaction between theneutron and the target nucleus. We, therefore, expectthe probability of neutron capture by the target nucleito be enhanced at low neutron energies.

This interaction is described in terms of a capturecross-section, σc, where the number of captures per unitvolume, N , is given by

N = NTσc�, (62)

where NT is the number of target nuclei per unit volume,σc the capture cross-section and � = φt is the influence(flux times time) given in n/cm2. Fig. 25 shows thecapture cross-section as a function of neutron energy for

Figure 25 Neutron capture cross-section as a function of neutron energyfor natural silicon (after [144]).

Figure 26 Typical neutron activation analysis (NAA) gamma-ray spectrum to search for trace substances deposited on an air filter after 1 min.irradiation at MURR (after [152]).

silicon as averaged over all three stable silicon isotopes[150]. Similar behavior is found individually for eachsilicon isotope. It can be seen in Fig. 25 that for lowenergies:

σc ∼ E−1/2 ∼ 1/V . (63)

For a given nuclear radius, (1/V ) is proportional tothe interaction time. Therefore, the cross-section repre-sents a probability of interaction between the nucleusand neutron.

After neutron capture, the target nucleus differs fromthe initial nucleus by the addition of the nucleon andis a new isotope in an excited state which must relaxby the emission of energy in some form. This emis-sion is usually in the form of electromagnetic radia-tion (photons) of high energy usually called gammas(see e.g. [151] and references therein). The time fordecay of this excess energy by gamma emission canbe very short (prompt gammas) or can take an appre-ciable time in which case a half-life a factor of two,can be measured. The gamma emission spectrum ischaracteristic of the nuclear energy levels of the trans-muted target nuclei and can be used as a powerful tracesubstance technique called neutron activation analysis(NAA), to detect quantitavely impurity levels as low as109 atoms/cm3 [152]. A typical trace substance NAAgamma spectrum is shown in Fig. 26. Each emissionline is characteristic of a particular nuclear transitionof a particular isotope. The absorption of a neutron andthe emission of gammas is represented by the notation:

AX(n,γ ) A + 1X, (64)

where (n, γ ) represents (absorption, emission), A isthe initial number of nucleons in the target elementX before neutron absorption while A + 1 is the num-ber after absorption. It is possible for the product iso-tope A + 1X to be naturally occurring and stable. Inmany cases, however, the product isotope is unstable.Unstable isotopes further decay by various modes in-volving the emission of electrons (β-decay), protons,α-particles, K -shell electron capture or internal conver-sion until a stable isotopic state is reached (details see,e.g. [153]). These decays produce radioactivity and canbe characterized by their half-lives T1/2. In the case of

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silicon, three stable target isotopes are transformed by(n, γ ) reactions [154] as follows:

(92.3%) 28Si (n, γ ) 29Si, σc = 0.08 b;

(4.7%) 29Si (n, γ ) 30Si, σc = 0.28 b;

(3.1%) 30Si (n, γ ) 31Si → 31P + β−,

σc = 0.11 b; (T1/2 = 2.62 h). (65)

The relarive abundance of each stable silicon isotopeis shown in parenthesis (see also [151]). The cross-sections are expressed in barns (1 barn = 10−24 cm2).The first two reactions produce no dopants and onlyredistribute the relative abundances slightly. The thirdreaction produces 31P, the desired donor dopant [155],at a rate of about 3.355 ppb per 1018 nth cm−2 [144].This production is calculated using (63), the 30Si (NT

∼=5 × 1022 Si · cm−3 × 0.031).

In addition to the desired phosphorus production re-action and its relatively short half-life for β− decay, thereaction

31P (n, γ ) 32P → 32S + β−,

σc = 0.19 b(T1/2 = 14.3 d) (66)

occurs as a secondary undesirable effect. The decay of32P is the primary source of radioactivity in NTD floatfine Si. Of course, any undesirable trace impurities inthe silicon starting material can lead to abnormally longhalf-life activities which may require that material beheld out of production until exempt limits are reached.These factors have stimulated on the subject of radiationprotection. Once the dopant phosphorus has been addedsilicon ingot by transmutation of the 30Si isotope, theproblems remains to make this radiation damaged andhighly disordered material useful from an electronicdevice point of view. Several radiation damage mech-anisms contribute to the displacement of the siliconatoms from their normal lattice position (details seebelow). These are:

1. Fast neutron knock-on displacements,2. Fission gamma induced damage,3. Gamma recoil damage,4. Beta recoil damage,5. Charged particle knock-on from (n, p); (n, α) etc

reactions (see details [151]).

Estimates can be made of the rate at which Si atomdisplacements are produced by these various mecha-nisms, once a detailed neutron energy spectrum of theirradiation is known, and these rates compared to therate at which phosphorus is produced.

The number of displaced atom per unit volume persecond, ND, is estimated from the equation

dND/dt = NTσφν, (67)

where NT is the number of target atoms per unit volume,φ is the flux of damaging particles and ν is the numberof displacements per incident damaging particle. The

cross-section for gamma induced displacements in sil-icon is small while the cross-sections for (n, p); (n, α),etc., are of the order of millibarns and have thresh-olds in the MeV range. The fast neutron knock-on dis-placements can be calculated from the elastic neutronscattering cross-section once the reactor neutron en-ergy spectrum is known. Estimates of fission spectraand graphite moderated fission spectra can be found inthe literature [156].

Even if the fast neutron damage could be completelyeliminated, the recoil damage mechanisms, which arecaused by thermal neutron capture, still would pro-duce massive numbers of displacements compared tothe number of phosphorus atoms produced. In the caseof gamma recoil, a gamma of energy hω carries a mo-mentum hω/c which must equal the Si isotope recoilmomentum MV . The recoil energy

ER = 1

2MV 2 = 1

(hω)2

MC2(68)

is, therefore, departed to the silicon atom of mass Mfor each gamma emitted. An average over all possi-ble silicon isotope gamma emission and cross-sectionsyields an average recoil energy of 780 eV [157] whichis significantly higher than the Si displacement energy.A similar effect is encountered for 31Si β− decay. Theβ− carries a momentum

p = 1

√E2

β−(m0c2)2 ≡ MV . (69)

Therefore,

ER = 1

2MV 2 = 1

[E2

β−(m0c2)2]/(Mc2). (70)

For a β− emitted with an energy of 1.5 MeV, ER =33.2 eV or roughly twice the displacement threshold.

From the above considerations, a very crude estimateof the numbers of displacements per phosphorus atomproduced can be made. The results of the estimationare shown in Table VII. While the absolute numbersof displacements should not be taken literally, the rela-tive magnitudes of the amounts of damage produced bythese various mechanisms are probably order of mag-nitude correct. An inspection of Table VII indicatesthat the gamma recoil mechanism is significant relativeto the quantity of phosphorous produced even in highlymoderated reactors. We are led to the inescapable con-clusion that transmutation doping will always producesignificant amounts of radiation damage which must be

TABLE VII Number of displaced silicon atoms per phosphorus pro-duced for various damage mechanisms shown for an in-core fission spec-trum and a graphite moderated spectrum (after [144])

Damage particle/position In core In pool

Fast neutron 4.06 × 106 1.38 × 104

Fission gamma 3.64 × 103 36.4Gamma recoil 1.29 × 103 1.29 × 103

Beta recoil 2.76 2.76Total DISP/(P) 4.06 × 106 1.51 × 104

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repaired in some way. These defects introduce defectlevels into the band gap which cause free carrier re-moval and a reduction in carrier mobility and minoritycarrier lifetime (see, e.g. [157]).

The defects produced by neutron irradiation are re-moved by thermal annealing as discussed previously.It is at this point in the process where disagreementas to the best procedure is likely to be the greatest.The spectrum of possible defect structures and theirenergetics is impressively large and incompletely un-derstood. Therefore, annealing procedures accordingMeese [144] will be based on art rather than exact sci-ence. They will also tend to become proprietary for thisreason.

This is unfortunate since it is precisely in this areathat fundamental knowledge is needed to produce thebest possible product. Although carrier concentrationand mobility recovery are easily obtainable by variousannealing procedures, minority carrier life-time recov-ery is very elusive at present.

So, neutron transmutation offers both advantages anddisadvantages over conventionally doped silicon (de-tails see [151]) .

Advantages

1. Precision target doping (= 1% or better).2. Better axial and radial uniformity.3. No microresistivity structure.

Disadvantages

1. Irradiation costs.2. Reduction in minority carrier life-time.3. Radioactive safeguards considerations.

Figure 27 Typical neutron flux profiles (after [158]).

The steady growth of the NTD-silicon (and oth-ers NTD-semiconductors) market suggests (see alsobelow) that the advantages are outweighing thedisadvantages.

2.2. Reactor facilities for transmutationdoping

Irradiation of silicon for the purpose of phosphorus dop-ing has been carried out in the Harwell (England) re-search reactors since 1975 [158]. At Harwell silicon isirradiated in the twin, heavy-water, materials-testing re-actors DIDO and PLUTO. Both reactors operate contin-uously throughout the year and each actieves an avail-ability of greater than 86% of calender time. As thereactors are D2O moderated and cooled the irradiationconditions are particularly good for silicon doping. Theratio of thermal to fast neutrons is in excess of 1000:1,which minimizes the damage which has to be removedby annealing. In considering the accuracy which canbe achieved in the neutron doping process it is neces-sary to consider the neutron flux profiles and gradients.Fig. 27 shows a typical unperturbed flux profile for anirradiation position over a length of 50 cm spaced aboutthe maximum flux value. It will be noted that the maxi-mum and minimum flux values differ by 8–12% of themaximum, and the gradient at the lower end is particu-larly steep. To reduce this variation and to smooth theprofile, flux flatteners or neutron screens, in the form ofstainless steel tubes are fitted to the facility liners withthe result shown as “modified profile” in Fig. 27. Aswas shown by Smith, the severe gradient arising fromthe 8–12% variation has been reduced to 2–3% and theoverall variation reduced to 5%.

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Figure 28 Resistivity variations (after [158]).

The “modified profile” can now be examined in moredetail, and this is shown in Fig. 28. The profile can beconsidered in terms of “resistivity distribution” whichis of interest to the customer and can form the basis forthe technical specification of the product. The “AverageTarget Resistivity” or A.T.R. 50 [158] is, as the namesuggests, the average of the resistivity over a length of50 cm. Achievement of this value is subject to variationsarising from the irradiation timing, the measurement ofthe mean flux, and the distribution shape; therefore indi-cating authors apply a tolerance of ±5% to the A.T.R.50value. The exact shape of the distribution is also subjectto variations due to disturbances in the reactor such ascontrol-absorber movements and other irradiations andexperiments; according to [158] tolerances are there-fore also applied to the “Resistivity Distribution” of5% maximum greater, and 10% maximum less, thanA.T.R.50.

Researchers of Harwell offer irradiation of volume.Fig. 29 illustrates this and shows the volume which forconvenience Smith describes as a “batch”. It is cylindri-cal volume of 90 mm diameter and 500 mm in length.Although radial gradients are small, crystals are rotatedduring irradiation, and a maximum variation of ±1%on a diameter of 10 cm (4 in.) is guaranteed. In practice,according to Smith, variations are less than can be mea-sured within the accuracy of a conventional four-pointprobe. The average annual dose is typically 6.5 × 1017ncm−2, which corresponds to a resistivity of 35 �-cm.In many of the literature references on the neutron dop-ing of silicon one finds the statement that “three days,or at most a week, after irradiation, silicon is save totransport and to handle”. This is, of course, a relativestatement and it is necessary to define what is meantby “safe”. In the I.A.E.A. publication “Regulations forthe Safe Transport of Radioactive Materials, 1977” itstates that to qualify as “Exempt” or safe material the

Figure 29 Batch dimensions (after [158]).

following conditions must be met:

1. The radiation level at any point on the externalsurface of the package shall not exceed 0. mRem/h;and

2. The non-fixed radioactive contamination of anyexternal surface shall not exceed 10−4µCi cm−2. This

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Figure 30 Section and layout of the irradiation of the irradiation facility (after [160]).

level is permissible when averaged over any area of300 cm2 of any part of the surface.

These are internationally accepted standards for thetransport of packages and Harwell complies them. Tomeet this obligation Smith write that each crystal, orpiece, of silicon complies with the following criteriabefore certification for dispatch:

1. Radiation is less than 0.1 mRem/h, a factor of 5less than that necessary for the package according toI.A.E.A. Regulations. and

2. Contamination is less than 5 × 10−5µCi cm−2, afactor of 2 less than that necessary for the package.

As a result of the application of these low levelswhich we consider to be essential only three days delayprior to shipment os not always possible [404], partic-ularly for material irradiated down to low resistivities(see also [159]).

The tendency of the NTD silicon producers to in-crease their capacity and the extending range of de-vices in which NTD silicon is being used, call for spe-cial reactor irradiation facilities. The JRC heavy watermoderated ESSOR reactor (Ispra, Italy) is describedby Bourdon and Restelli [160]. In this article, atten-tion has been especialy devoted to obtain an automatedoperation of the facility, and to optimize the character-istics of the irradiation volume. The height of 50 cm,with respect to a core vertical dimension of 150 cm,has been selected in order to obtain a minimum axialspread of the neutron flux (±4%). A thermal neutronflux of (2.7 ÷ 3) × 1014 n cm−2 s−1 is available at theirradiation position with a thermal to fast (>100 keV)neutron flux ratio larger than 400.

The silicon crystals are loaded into a transport unitwhich can locate up to 100 ingots 77 mm in diameterby 500 mm length (Fig. 30). The ingots are loadedprotected by a bored plastic capsule which definesthe irradiation volume indicated above. The capsulesare then loaded into the reactor through a lock betweenthe transport unit and the channel for transfer from theair to the heavy water circuit. The heavy water circula-tion assures efficient cooling of the silicon ingot duringirradiation (the maximum crystal temperature shouldnot exceed 70◦C) and induces, by use of a suitableshape of the capsule (see Fig. 31), a slow rotation of theingot for minimizing radial dispersion of the neutronfluency. At the end of the irradiation, determined bythe control system, the capsule is transferred into thetransport unit where it stays for at least 4 days beforebeing disharged thus providing for decay time betweenthe irradiation and the extraction time. The plastic ma-terial for the capsule must be chosen as a function of itsqualities of mechanical behavior, radiation resistance,heat resistance and low radioactivation. On the basis ofthe three first requirements, some commercial plasticshave been selected and samples of each irradiated inthe HFR reactor of Petten (Netherland) or in ESSOR,in order to choose the best material (see also Fig. 32).None of the tested materials is completely satisfactoryso that Bourdon and Restelli have foreseen the use ofNoryl 731 for the preparation of capsules to be usedfor short irradiation times, and polysterene or PPO forlong irradiation times.

The high thermal to fast neutron flux ratio assuresfewer lattice defects as demonstrated by the fact thata thermal annealing of 5 min. at 750–800◦C has beenfound sufficient to achieve complete recovery of thefinal resistivity (see Fig. 33).

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Figure 31 Irradiation capsule (after [160]).

Concluding this paragraph we should remember oncemore the advantages offered by the ESSOR reactor.These advantages are a high thermal neutron flux den-sity, very uniform and easily controllable over a largeirradiation volume, a good thermal to fast neutron fluxratio in conjunction with the possibility to irradiate thecapsule immersed in D2O which assures an efficientcooling of the Si crystal.

The General Electric Test Reactor (GETR)(Schenectady, U.S.A.) was designed and constructedto provide large irradiation volumes outside the reactorpressure vessel in a surrounding water pool. Thethermal neutron flux available for silicon irradiationapans four decades, 1011 to >1014 nv (see Fig. 34).The large irradiation volume permits the inclusionof flux flattening and spectral softening devices ifdesired [161]. As is well-known, NTD silicon offerssignificant technical advantages over chemically dopedsilicon (see also [162]). In particular, NTD siliconhas a more uniform phosphorus concentration acrossthe radius of an ingot or wafer than chemically dopedmaterial. The uniformity could approach 1% for a

Figure 32 Radioactivation of some plastic materials (after [160]).

Figure 33 Number of carriers (and Hall mobility) versus annealing tem-perature (isochronal anneals) (after [160]).

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Figure 34 GETR 50 MW neutron flux pool position Z-6 (after [161]).

three inch diameter wafer activated in the 50 MWGETR. Availability of such uniform NTD siliconwould make it possible to manufacture higher powerdensity thyristors for high voltage applications (detailssee below).

A program of silicon irradiation is being carried outat the National Bureu of Standards (NBS) (Washington,DC, U.S.A.) 10 MW, heavy water moderated reactor.A plan view of the NDBS reactor core showing sev-eral of the irradiation locations is shown in Fig. 35. Aset of five pneumatic rabbit tubes, useful for irradiatingsilicon chips to analyze for impurities or to study irra-diation damage, provide a range of thermal fluxes from2 × 1011 n cm−2 s−1 (copper-cadmium ratio of 3400)to 6 × 1013 (copper-cadmium ratio of 46) [164]. Thoseresearchers interested in long term silicon doping irradi-ations can currently use two vertical facilities designedG2 and G4 [163]. Both facilities are D2O filled andare completely isolated from the reactor coolant. Sincethey are isolated, encapsulation of the silicon is unnec-essary and only an aluminium hardness is needed tohold the sample. The G2 tube will accept samples up to1.6 inches in diameter and has a neutron flux at the core

midplane of 1.1 × 1014 (copper-cadmium ratio of 55)[163]. A vertical flux profile of this facility is depictedin Fig. 36. Irradiation of samples in G2 for periods ofone day to six weeks has been done for Oak RidgeNational Laboratory (ORNL). The predicted phospho-rus doping rate of 7.5 × 1013 atoms cm−3 hr−1 yieldeda concentration in excellent agreement with that mea-sured by ORNL. The G4 tube is located at the cen-ter of the reactor core and will accept samples up to3 inches in diameter. Its neutron flux has roughly thesame shape as that in G2 but is about 28% greater. Aone kilogram silicon sample has been irradiated in G4to a measured phosphorus concentration of 1.4 × 1017

atoms cm−3 (details see [163]).The absolute differential neutron-energy spectrum

for the low temperature fast-neutron irradiation facilityin the CP-5 reactor by means of a 20-foil activation tech-nique was determined by Kirk and Greenwood [165].Fig. 37 shows a simplified schematic of VT53, the cryo-genic fast-neutron irradiation facility at CP-5 (a moredetailed description of this equipment see in paper [165]and references cited therein). The elements silicon,nickel, niobium, and gold were selected to illustrate

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Figure 35 Plan view of NBSW reactor core and irradiation facilities(after [163]).

Figure 36 Vertical flux profile of position G2 (after [163]).

the results of these calculations. Table VIII gives thegroup distributions (partially collapsed from the orig-inal, more detailed group structure) of primary recoilsand the total primary recoil cross-sections+ for thesefour elements. Also displayed in this table (for goldonly) is an error of one standard deviation for recoils ineach group, based on the covariant error matrix gener-ated during the neutron spectrum error analysis by theSANDANL code. To facilitate comparison among thefour elements, Fig. 38 shows the integral distributionof primary recoils (see also [166]). Using the Robinson

Figure 37 Simplified schematic of the cryogenic fast-neutron facility inCP-5 (VT53) (after [165]).

analytical approximation [167] to the Linfhard et al.[168] theory of electronic energy losses, it is also pos-sible to calculate the damage energy distribution andspectrum averaged total damage energy cross-sections[169, 170]. The number of Frenkel defects (interstitialand vacancy pairs) (see also [157]) produced by a pri-mary recoil of energy T is generally proportional tothe damage energy available from this recoil, whichis just the total recoil energy, T , minus the electronicenergy losses at this recoil energy. The distribution ofdamage energy over the recoil energy groups thus givesa good indication of how the Frenkel defects are dis-tributed with primary recoil events. As an example,Table IX gives the distributions of damage energy inrecoil energy groups and the spectrum averaged dam-age energy cross-sections for the same four elementsshown in Table IX. Fig. 39 graphically illustrates thecorresponding integral damage energy distributions.

The integrated neutron flux determined by Kirkand Greenwood (2.2 × 1016 n/m2s ±13%, for En >

0.1 MeV) for the low temperature fast neutron facilityin CP-5 is 70% greater than that determined less accu-rately early [171]. The amount of 235U burn up in thefuel cylinder over this period of time is not known withcertainty. However, based on a comparison of resistivitydamage rate measurements in cooper made over a com-parable period of time [172, 173], the burn-up is about5% for an 8 year time period. Thus, it is reasonableto assume that the flux (corrected for burnup) in thisfacility has remained constant in time within the uncer-tainty of the present measurement. It is indeed foundthat the major cause of the difference between the fluxmeasurements is the improvement in accuracy of thecross-section data on which these flux determinationsare based. Therefore, any use of data from previous ex-periments in this facility will employ the presently de-termined neutron spectrum and integrated flux values.One caution that is perhaps obvious should be noted.In comparing experimental data from different neutronirradiation facilities, one must be careful when using

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T ABL E VII I Group distributions of recoils and spectrum averagedcross-sections in gold, niobium, nickel and silicon irradiated in the VT53 fast neutron energy spectrum (after [165])

Primary recoil/ Si Ni Nb Auenergy group PRD (%) PRD (%) PRD (%) PRD (%) +/−

0.5 eV ↓ 2.4 4.0 2.6 11.8 1.45–10 0.4 1.7 1.2 5.1 0.9

10–20 0.6 2.9 1.9 5.6 1.020–50 1.5 6.2 4.3 6.7 0.650–100 1.9 6.8 3.9 5.4 0.3

100–200 2.4 8.7 4.9 7.2 0.6200–400 2.7 10.1 6.1 9.4 0.9400–600 1.8 8.0 4.7 6.4 0.7600–800 1.5 6.8 4.1 4.7 0.6800–1000 1.3 4.4 3.5 3.7 0.6

1–1.5 keV ↓ 2.8 5.4 7.2 6.6 1.41.5–2 2.4 3.9 5.5 4.2 1.1

2–3 3.9 5.3 8.4 5.4 1.53–5 6.7 5.8 11.5 5.8 1.35–7 5.5 3.6 7.2 3.0 0.57–10 6.9 3.9 6.7 2.6 0.4

10–20 18.8 5.7 9.0 3.9 0.420–40 18.1 3.3 4.7 2.1 0.140–60 6.7 1.3 1.5 0.3 0.0260–80 3.5 0.8 0.6 0.08 0.004

PRD = Primary recoil distributions.Spectrum averaged cross-sections (barns)σelastic 2.86 8.82 6.77 9.35 ±12%σinelastic 0.05 0.12 0.40 0.80 ±9%σn,2n 0.0001 0.001 0.001 0.002 21%σn,p 0.002 0.02 0.0001 – –σn,α 0.001 0.001σtotal 2.91 8.96 7.17 10.15 ±11%

Figure 38 Integral distributions of primary recoils in Au, Nb, Ni and Siirradiated in VT53 (after [165]).

integrated flux values that have been determined at dif-ferent times, since cross-sections have changed withtime (details see [165, 174]).

2.3. Nuclear reaction under influenceof the charged particles

According to the modern concept (see, e.g. [153]), thenuclear reactions leading to the formation doped impu-rities, may be proceed under the influence of chargedparticles (protons, deutrons, α-particles, etc) and neu-trons and γ -quantes. In such a case the energy of bom-bardering on the nuclear particle must be sufficientlyenough for overcoming mutual Coulomb repulsion ofparticle and nuclear. And in case of following par-ticle flying out from excited compound nuclear theother charged particle should also obtain in compound

TABLE IX Group distributions os damage energy and spectrum av-eraged damage energy cross-sections (〈σ Td〉) in gold, niobium, nickel,and silicon irradiated in the VT 53 fast neutron energy spectrum (after[165])

Primary recoil/ Si Ni Nb Auenergy group DED (%) DED (%) DED (%) DED (%)

0–5 eV ↓ 0.00 0.001 0.001 0.015–10 0.0002 0.003 0.002 0.02

10–20 0.0005 0.01 0.006 0.0420–50 0.003 0.05 0.03 0.1150–100 0.008 0.1 0.05 0.2

100–200 0.02 0.3 0.14 0.5200–400 0.04 0.6 0.3 1.2400–600 0.05 0.8 0.4 1.4600–800 0.06 1.0 0.5 1.4800–1000 0.06 0.8 0.6 1.5

1–1.5 keV ↓ 0.2 1.4 1.6 3.61.5–2 0.2 1.4 1.7 3.2

2–3 0.5 2.6 3.6 5.73–5 1.3 4.3 7.6 9.55–7 1.6 4.1 7.1 7.47–10 2.8 6.2 9.2 8.9

10–20 12.7 14.7 20.7 22.620–40 12.6 10.7 10.9 6.260–80 8.9 9.5 6.3 2.0

DED = Damage energy distribution (%)Spectrum Averaged damage energy cross-sections (keV-barns)〈σ Td〉 42.8 36.0 33.9 19.5(±8.7%)

Figure 39 Integral damage energy distributions for Au, Nb, Ni and Siirradiated in VT53 (after [165]).

nuclear the energy sufficient for overcoming thisCoulomb barrier. If rn is a nuclear radii and e is theelectron charge, in such a case the barrier height [151]has the following relation

Bb = Zx Zae2

rn� Zx Za

A1/3MeV (71)

and achieves of 5–10 MeV for light nuclears, 10–20 MeV for middle nuclears and 20–30 MeV for heavynuclears [151, 153].

Historically the first sourced of charged particles (α-particles) were the radioactive elements of Ra, Rn, Po,Pu and others, which in result of radioactive decay emitin 1 s on 1 gr emitter until 1010–1011 α-particles [175]with energies of 4–8 MeV [153, 175]. The main ques-tion is how the beam of α-particles passes on its ownenergy to the atoms of irradiated material. It is appearedthat owing to Coulomb interaction of the particles withmaterial the kinetic energy of charged particles is spenton the ionization and excitation the atoms of irradiated

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substance. The estimations show [153] that the cross-section of the ionization process (∼10−21 cm2) approx-imately ∼1000 times more than the cross-section of thenuclear interaction (∼10−24 cm2). Strictly speaking formonoenergetic beam of particles R0 (the distance ofparticle run) has the meaning of the middle run dis-tance relatively to which it has the Gauss distribution.During one act of ionization the particle energy is di-minished approximately 3.5 eV [153] and probabilityof nuclear interaction depends on the particle energy atthe moment of the particle and nuclear collision. Thenumber of nuclear interactions in thin layer dx at thedepth x from the surface of the target has the form

dν = η(x)Nσ (x)dx � η0 Nσ (x)dx, (72)

where N is concentration of target nucleus, η0 and η

are the particles beam on the surface and in depth x , re-spectively; σ (x) is the cross-section of interaction. Thefull number of interactions in the depth of substance,the thickness of which is more than R0, is equal

ν = η0 N∫ R0

0σ (x)dx = η0 N

∫ E0

σ (E)dE∣∣ dEdx

∣∣ , (73)

where E0 is the start energy of particle. Yield of nuclearreaction which is determined by the part of particlesundergoes the nuclear interaction.

V (E) = ν

η0= N

∫ E0

σ (E)dE∣∣ dEdx

∣∣ . (74)

From the last formula, it is followed that the yield ofnuclear reaction at the energy of charged particle E isdetermined by the cross-section σ (E) and specific ion-ization of particle dE/dx . And, vice versa, it is knownthe functions dE/dx and dV/dE from (74) allow tofind the cross-section of interaction

σ (E) = 1

∣∣∣∣dE

∣∣∣∣. (75)

It is known (see e.g. [153]) that for the charged parti-cle the magnitude dE/dx is proportional to the squareroot of particle charge, concentration of electrons insubstance (ne) and some function of the velocity f (v) ∼1/v and doesn’t depend on particle mass:

dx∼ Z2ne f (v). (76)

The dependence dE/dx is permitted to recount thedata on the motion of one concrete particle in concretesubstance on the motion of another particles in anothersubstances (see also [152]). So far dE/dx = f (E), thenin such a case taking the integral, we can obtain the fullrun of particle.

R =∫ E0

f (E). (77)

For example, α-particles (Rα,x) in the substance XAZ

can be defined on the run in air (Rα,air) with an assis-tance of empirical formula [153]

Rα,x(E) = 0.56Rα,air(E)A1/3, (78)

where Rα,air in cm and relates in the air at the tem-perature 15◦C and ambient pressure and Rα,x will beobtained in mg/cm2.

The run of protons is connected with the run ofα-particle of another formula, which works at E ≥0.5 MeV:

Rp = 1.007Rα(3.972)E − 0.2, (79)

where Rα(3.972) is a run of α particles with the en-ergy of 3.972. The run of other charged particles withmass Mx (exclude electron) is connected with the runof protons in the next relation

Rx(E) = Mx

MpRp

(Mp

MxE

). (80)

Owing to the fast retardation of charged particles,they can dope the layer of small thickness with notuniformly distribution of doped impurities on the depth.The possible reactions obtained with charged particlesare described by Smirnov [151], where it was indicatedthe half-time decay. To conclusion of this part we shouldnote that regular experiences in this field at the presenttime is absent.

2.4. Nuclear reaction under actionof the γ -rays

The reactions of (γ , n)-, (γ , p)- and (γ , α) belongto reactions of splitting nuclears irradiated by γ -rays.These reactions are endoenergetical and have some en-ergetic threshold. In common case the probability of(γ , n) reaction is more than the probability of (γ , p)-and (γ , α) reactions. The energy of γ -rays ∼10 MeVis called the (γ , n) reactions. At the Eγ ∼100 MeV thereactions with the creation of several particles (γ , 2n;γ , pn, etc.) are possible [151]. Inasmuch as the im-purities can be produced in practically any substanceswith the assistance of photonuclear reactions, thereforethey are represent the interest for semiconductor dop-ing. The main reason of this circ*mstance is large pen-etrating capability compared to the charged particles[151]. This circ*mstance can guarantee the uniformdoping of the large volume substance for some impurityatoms. The perspective of this direction at the presenttime is also supported by the possibility of receivingof the γ -quantes with any energy at retarding electronemission. For that the monoenergetical electrons withthe energy 25–60 MeV received by means of the accel-erator, is directed on the target from heavy metals (Pb,Bi, W, U etc.). As a result the retarding electrons createthe continuous spectrum of γ -emission. The maximumenergy of this continuum is equal to the kinetic energyof electrons Ee and intensity of γ -emission approxi-mately inversely proportional to the γ -quantes energy

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Figure 40 The spectrum of γ -emission is created under electrons re-tarding with Ee on the target of the accelerator (after [151]).

(see Fig. 40). The increase of γ -emission use for ir-radiation Si and other semiconductors has been noted[176].

2.5. Nuclear reactions under the influenceof neutron

The main hope in the field of nuclear doping of semi-conductors at the present time is placed on the neutrons(see also [177]) inasmuch as these particles are neutralparticles and they possess a large penetrating capabil-ity. Furthermore neutron can interact practically withall nuclears. At the preset time there are the differentsources of neutrons with different intensity acceptedand approached (see above). The history of techniquesby means of which was received the first neutronsbeams can be found in paper [151]. The main sourceof the neutron beam is still nuclear reactors of differ-ent types (see also above). Typical neutron spectrumof the research water-water reactor WWR is given inreview [178]. In this spectrum there are the neutronswith energy 0.01 ≤ E ≤ 107 eV. At the irradiationof neutron beam the cross-section of compound nu-clear σx essence depends on the degree of resonanceto intrinsic magnitude of energy levels of compoundnuclear.

For neutrons with the energy En ∼ 10−2 ÷ 104 eV[178] we can write σ (n, γ ) for cross-section of nuclearreaction

σ (n, γ ) = σ0(E0/En)1/2 = σ0(v0/v), (81)

where σ0, v0 and E0 are some constants of cross-sectionof the nuclear reaction, velocity and neutron energytaken as a starting point. As is well-known for NTDthermal neutrons are of greater interest. On this inter-val there are almost all number of neutrons in the spec-trum of nuclear reactor which is described by Maxwelldistribution on the velocities:

n(v) = 4√π

(Mn

2kT

)3/2

v2 exp(−Mnv2/2kT ), (82)

where as usually k is the Boltzman constant and T istemperature.

The average cross-section on the spectrum n(v)provides

σ =∫

σ0v0

vn(v)vdv∫

n(v)vdv= σ0v0

v, (83)

where v = √8kT/π Mn is the average magnitude of

velocity. It is conveniently to use the next formula ofmore probable velocity of neutrons

vmp =√

2kT

Mn= v

√π

2= v/1.228. (84)

This correspond to the maximum of distribution ofn(v). Taking into account that v0 = vmp and σ0 = σmpand at ambient temperature (T = 293 K) vmp is equalvmp = 2200 m/s. In such a case we obtain the relationconnected with the middle cross-section (n, γ ) reactionon the thermal neutron with the magnitude of cross-section usually measurable on the neutrons with moreprobable velocity

σ = σmpvmp

v= σmp

1.128= σ2200/1.128. (85)

The energetical dependence on the cross-section forSi can be described by following approximated equa-tion [179]

σ = σa + σs = σa + σfa(1 − e−2w)

= 0.8√En + 2.25(1 − eC EnT )

, (86)

where En is the energy of neutrons (MeV), σa is thecross-section of absorption which is equal 160 × 10−3

b at En = 25 MeV, σs is the cross-section of the scat-tering, σfa is the cross-section of free atom, which isequal 2.25 b, e−2w is the Debye-Waller factor; C =1.439 × 10−5 is the normalized constant of the depen-dence σ (E) to the magnitude σ = 0.55 b at En =50 MeV at 300 K. Calculations on the ground of Equa-tion 80 the energetical dependence on the cross-sectionof Si at different temperatures as well as some eksper-imental data are shown in Fig. 41. As it is seen for

Figure 41 The dependence of the calculated on the formula the cross-section of the thermal neutrons in polycrystal (1) and monocrystal (2) ofSi at different temperatures (after [151]).

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polycrystals (curve 1) Si the cross-section in thermalfield doesn’t depend on the energy and for monocrys-tals (curve 2) this dependence is changed according tothe law 1/v and depends on the temperature. The lastfact indicates that the NTD of different semiconductorsand one semiconductor with different impurities canirradiate at different temperature (details see [151]).

2.6. Influence of the dopantsAccording to common scheme of the nuclear reactionsthe primary products of (n, γ ) reactions present the nu-clear XA+1

Z per one atomic unit heavier outcome ones.These can be heavier isotopes of the same or neigh-bor element of the periodical table. If these isotopeshave in natural mixture isotopes irradiated material andappear stable ones, in such a case nuclear transforma-tion leads to the change of the initial concentration ofstable isotopes in the irradiated substance. In such acase this prices doesn’t create impurities of neighborelements of the periodical table. The possibility of theexchange of electrophysics properties with the assis-tance of impurities which are created by the method ofnuclear doping at irradiation of neutrons proves for nu-merous semiconductors (see e.g., Table 2.6 in [151]).In this way there are different properties of semicon-ductors to connect with the nature and concentrationsof the predominance impurities after respective anneal-ing: for example Sn and Te in InSb [180–182], Ge andSe in GaAs [183–185], In in CdS [186] and phosphorusin Si [154, 181, 182, 187–190] etc.

For the estimation of the character the distribution ofimpurities on the thickness of the doped materials we’lloutcome from fact that the attenuation of the intensityof collimated neutron flux by the layer of substance ofthe thickness x is confirmed by the well-know law:

I = I0 exp(−Nσ x) = I0 exp(−µx)

= I0 exp(−x/ ll), (87)

where N is the amount of the atoms of the irradiatedmaterial per 1 cm3; ll is the middle length of the absorp-tion [191] connected with the macroscopical absorptioncoefficient of the material µ = Nσ next relation

ll = 1/Nσ = 1/µ, (88)

where ll characterizes also the layer thickness of thematerial on which the neutron flux and respectively

T ABL E X The efficiency of the attenuation of the neutrons and γ -rays by the different semiconductor materials (after [151])

SN SN SN GR GR GR

Material Density, ρ( gcm3 ) Nσ (cm−1) ln (cm) L (cm) µ/ρ( cm2

g ) µ (cm−1) lγ (cm)

Si 2.42 0.008 125.0 22.2 0.024 0.058 17.2Ge 5.46 0.25 4.0 4.7 0.39 0.213 4.7GaAs 5.4 0.36 2.8 – 0.039 0.210 4.8InSb 5.78 7.0 0.14 – 0.052 0.391 3.43CdS 4.82 115.0 0.01 – 0.045 0.217 4.6SN = Slow neutrons; GR = for γ -rays with energy E = 20 30 MeV.

the impurities concentration is diminished in e = 2.72times.

The mentioned above relations are correct, if to as-sume that the neutron cross-section of scattering issmall compared to the absorption cross-section. In com-mon case when there are both absorption and scatteringof neutrons it is necessary to use the conclusions of thecommon theory of neutron diffusion [191]:

I = I0 exp(−x/L), (89)

where L is the diffusion length.For the compound semiconductors having some sorts

of atoms, the attenuation of the emission beam can beconsidered as additive property of medium. Taking intoaccount this fact we represent the Equation 81 in thefollowing form

I = I0 exp

(−µ

ρm

). (90)

Here ρ is the density of substance, µ/ρ is the masscoefficient of the attenuation; m is a mass of substancewith the cross-section 1 cm2 and thickness x . Then forcompound substance we can write

µ/ρ =∑

Ci(µ/ρ)i, (91)

where Ci is the weight concentration of the i-elementof the mixture. Equations 81–85 apply also to the at-tenuation of the narrow beam of γ -quantes. In thiscase the linear coefficient of the attenuation is deter-mined (see also [175] and references therein) by thesum of the contributions of photo- and Compton effects,and also from creation process of electron-positronpairs, e.g.

µ= N (σph + σc + σp), (92)

where σph, σc and σp are the cross-sections of the ir-radiated processes of the interaction of γ -quantes withsubstance in estimation per one atom. The efficiency ofthe neutron absorption and γ -rays absorption of differ-ent semiconductor materials at one side irradiation is re-flect in Table X. The magnitude of µ/ρ and respectivelyvalues ln and lγ are calculated using the Equation 85and the knowledges about the cross-section of neu-trons [192] and extrapolation data of mass coefficientsof absorption of γ -radiation for some elements. The

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comparison of the data mentioned above in the Table Xshows that for neutrons at increasing absorption abilityfrom Si to CdS rather sharply decreases in the thicknessof the semiconductor material layer in the boundary ofwhich the distribution of impurities can be regardedhom*ogenous. At that in Si the natural doping is inho-mogeneous caused by the absorption and scattering ofneutrons and has small and for real ingot with diameter50 mm doesn’t exceed 1% [155] and for crystal withdiameter 80 mm the relation of concentrations in center(Cmin) and at the end (Cmax) of the ingot is combined

Cmin/Cmax � 0.956 [193] (details see[151]).

2.7. Atomic displacement effects in NTDRecoil atoms from γ -rays or particle emissions af-ter thermal neutron capture, and recoil atoms fromelastically and inelastically scattered fast neutrons,produce atomic displacements in solids [194]. Therelative importance of the displacement damage pro-duced by thermal and fast neutrons can be estimatedby using isotope concentrations, capture or scatteringcross-sections, and recoil energies. Such an estimatehas been made for silicon and the results are presentedin Table XI. The first three columns list the siliconisotopes, isotope concentrations, and cross-sections forthermal neutron capture [194] and for fast neutron scat-tering [195]. The probability of interaction (product ofisotope concentration and capture or scattering cross-section) is listed in column 4. The energies availablefrom silicon recoils for producing atomic displace-ments are given in column 5. For thermal neutrons,these energies were obtained from a product of theprobability for thermal neutron capture and the averagerecoil energy [194] of 780 eV. Kirk and Greenwood[165] used 474 eV for the average silicon recoil energy,

Figure 42 Shown in the upper left is a track for a 50 keV Si atom recoiling in Si according to calculation by Van Lint et al. [196]. Upper right is adisplacement cascade for a 10 keV Ge atom recoiling in Ge according to calculations by Yoshida [197], where the open and solid circles representinterstitials and vacancies, respectively. The lower part of the figure shows the energy deposited per A for different Si recoil energies Er, accordingto the formulations of Brice [198] (after [195]).

TABLE XI Energy available for producing displacement damage insilicon by neutron transmutation doping (after [195])

Concentration Cross-section Nσ νNσ

Silicon isotope (1022 cm−3) (10−24 cm2) (cm−1) (eV/cm)

a) Thermal Neutron Capture Recoil (Energy into displacement,ν = 780 eV)

28Si (n, γ )29Si 4.61 0.08 0.0037 2.8829Si (n, γ )30Si 0.23 0.27 0.00062 0.4930Si (n, γ )31Si 0.15 0.12 0.00018 3.51β(1.5 MeV)→ Total 3.512.61 hr → 31P

b) Fast Neutron Knock-on Recoil (Average recoil energy of50 keV assumed, ν = 25 keV)

All 5 3(avg) 0.15 3.8 × 103

but the lower energy only emphasizes further the dom-inance of the fast neutrons for producing displacementdamage (see also [195]). The details of the damage pro-duced by recoiling 31Si and 31P atoms may, however,be important in determining the lattice location of 31Pintroduced by NTD. Column 5 of Table XI shows thatthe energy available for displacements from an incidentfast neutron is 103 times that from an incident thermalneutron. Fast neutrons will, therefore, dominate the dis-placement damage until thermal-to-fast ratios exceed1000:1 (see also above). Thermal neutron capture cross-sections for germanium and gallium arsenide [195] aremuch larger than those for silicon (see also Table X).Consequently, displacement damage by thermal neu-trons relative to fast neutrons is expected to be moreimportant in these materials than it is in silicon. Anatom recoiling in a host material creates a high-defectdensity (cluster of defects) along the recoil track [157].The upper part of Fig. 42 illustrates clusters formed byrecoil tracks calculated for a 50 keV silicon atom re-coiling in silicon [196], and for a 10 keV germanium

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atom recoiling in germanium [197]. The track in sili-con as a side view and the authors [196] emphasize thehigh damage density in subclusters expected near theend of range for each recoiling silicon atom. These sub-cluster regions are similar to the capture-recoil damageregions produced by thermal neutron capture. There-fore, the kind of damage regions produced by thermalneutrons is included in fast neutron damage. The trackin germanium is an end view where the open circles,which represent interstitials, are shown to be concen-trated on the periphery of the track. The lower part ofFig. 42 shows results of calculating the depth distribu-tion of displacement damage by averaging over a largenumber of tracks for silicon atoms recoiling with en-ergy Er in silicon [198]. The peak damage density firstincreases with Er but then decreases because the dam-age clusters become more diffuse. Knock-on recoilsin reactor-neutron irradiations have an average Er ≈50 keV. The areas under the curves give the energyspent in collision processes, and is ≈ 25 keV for Er =50 keV. The other half of Er is spent in ionization [198].Assuming 25 eV/displacement, there is sufficient en-ergy to produce 1000 displacements.

One of the most graphic early examples of defectclusters produced by neutron bombardment of semi-conductors was obtained by Bertolotti et al. [199] us-ing etched surface replication transmission electron mi-croscopy (TEM). Fig. 43 is a sketch taken from suchresults obtained on 14 MeV neutron irradiated silicon[200]. Most people would agree that the central regionis probably due to the core of the displacement-damageclusters. There is less agreement on the interpretation ofthe outer zone. Direct TEM mesurement [201, 202] in-dicate a strained region around the damage core so thatstrain-induced differential etching may have caused theouter zone observed in papers [199, 200]. Model byNelson [203] postulates trapping of mobile defects by

Figure 43 Sketch of a region observed on an etched surface of Si fol-lowing 14 MeV neutron irradiation. Central core (500 A) is attributed torecoil damage (after [195]).

damage clusters so that the differential etching mayhave caused by ab excess of trapped vacancies or inter-stitials. However, the interpretation that has been mostextensively used to explain experimental data is due toGossick [204] and Crawford and Cleland [205]. In thismodel, the outer zone represents a space charge regionsurrounding a p-type germanium. The damage clusterin silicon is nearly intrinsic so that a space charge re-gion would be found in both n- and p-type silicon. Thespace charge model has been used to interpret minoritycarrier lifetime data [206], changes in carrier concen-tration and carrier mobility [207], the light sensitivityof neutron-produced electrical changes [207], photo-conductivity [208] and EPR observations on specificdefects in neutron irradiated silicon (see also [182] andreferences therein).

Thus calculations of energy deposition into displace-ment processes show that fast neutrons will dominatedefect production in NTD processing of silicon unlessthermal-to-fast neutron ratios exceed 1000:1. Defectclusters are produced by silicon-atom recoils from fastneutron collisions. Using an experimental value for theenergy needed per unit volume to form amorphous ma-terial, it is argued in paper [195] that amorphous zoneformation in silicon NTD is highly unlikely.

2.8. Experimental results2.8.1. GeAs was noted above, neutron transmutation (NT) is es-pecially intriguing for semiconductors for several rea-sons. First, the NT process can create new elementsremoved by just one atomic number. Considering forthe moment the elemental group-IV semiconductorsGe and Si, this means that the donors As and P willbe created, respectively, following neutron capture andβ-decay of isotopes of these semiconductor elements.The new elements are, of course, the prototypi-cal donors. Neutron capture leads to NTD. Second,the number of new atoms A+1

Z+1N created is simply (seealso 2.1)

A+1Z N = nσn

AZ N, (93)

where n being the total neutron fluency (cm−2), σn thecross-section for thermal neutron capture (cm2), andAZ N the atom concentration of the specific isotope in thegiven isotope mixture (cm−3) (either natural or manmade). Considering that the values of σn lie in the10−23 ÷ 10−24 cm−2 range (see above), it recognizesthat very large neutron fluences are required to trans-mute a significant number of atoms of one element intoanother. Whereas this may pose problems to fulfill themedieval alchemist’s dream, it is just perfectly suitedfor the person who wants to dope semiconductors. Withthe thermal neutron fluences available in modern nu-clear reactor (see also 2.2) one can dope Ge up to themetal-insulator (MI) transition (2 to 3 × 1017 cm−3)while Si can be doped with phosphorus to several times1015 cm−3 [182]. As will be shown below this is dueto the small atom concentration of 30

14Si and the mod-est value of the thermal capture cross-section. Third,there are elements which have light isotopes which

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Figure 44 The dependence of the phosphorus atoms concentration onthe neutron irradiation doze of Si crystals and followed annealing at800◦C during 1 h. The dependence was measured by Hall effect (after[215]).

upon neutron capture transmute to a lower Z elementeither by electron capture or by positron decay. In thiscase acceptors are created. A classical case is the trans-mutation of 70

32Ge into 7131Ga.

The main advantage of the NTD method, as we knowat present, is the precision doping which is connectedwith the linear dependence of concentration of dopingimpurities on the doze of neutron irradiation. Such de-pendence is numerous observed in the different experi-ments (see, e.g. [209–214]). As an example, in Fig. 44there is shown the dependence of the concentrationsdoped phosphorus on the doze of irradiation the Si crys-

Figure 45 The dependence of the concentration of free electrons in 74GeNTD on the irradiation of the thermal neutron doze and followed anneal-ing at 460◦C during 24 (1), 50 (2) and 100 (3) hs. (after [216]).

TABL E XII Characteristics of the transmutation process of germanium (after [209])

Isotope Abundance (%) NCCS (barn) NCDR Dopant type

7032Ge 20.5 3.4; 3.2; 3.25 70

32Ge(n, γ ) → 7132Ge →71

31Ga p7232Ge 27.4 0.98; 1.0; 1.0 72

32Ge(n, γ ) → 7332Ge

7332Ge 7.8 14.0; 14.0; 15.0 73

32Ge(n, γ ) → 7432Ge

7432Ge 36.5 0.62; 0.5; 0.52 74

32Ge(n, γ ) → 7532Ge → 75

33As n7632Ge 7.8 0.36; 0.2; 0.16 76

32Ge(n, γ ) → 7732Ge → 77

33As →7534Se n

NCCS = Neutron capture cros-sections.NCDR = Neutron capture and decay reactions.

tal in nuclear reactor. This dependence was measuredwith the help of Hall effect [198]. However, at the largedoze of neutron irradiation there is observed the non-linear dependence. On Fig. 45 is shown the results ofpaper [216] where was observed the deviation from lin-ear law at the large doze of neutron irradiation of thesample of 74Ge which was annealed after irradiation atT = 460◦C during different time (see also caption ofFig. 45). More amazing effect was observed at the sec-ond irradiation of the samples of 74Ge previously strongdoped with As by NTD method. Instead expectable in-crease of the concentration free charges (electrons) nthere is observed the decrease n. This decrease wasdirect proportional to the neutron irradiation doze of74Ge crystals. Both effects are rather details analyzedin papers [213, 216].

The transmutation of the stable germanium isotopesvia capture of thermal neutrons is well understood.Table XII contains all the information relevant to NTDof germanium. In paper of Haller et al. [209] quotedthe values of the thermal neutron capture cross-sectionσn of three sources [181, 190, 217]. The informationof Table XII permits the computation of the acceptorand donor concentrations for a known neutron expo-sure. Not only are these concentrations important butthe ratio of the sum of all minority dopants (donors)and the sum of all majority dopants (acceptors) i.e.,the compensation K , is crucial for the low temperatureconduction. For the case of germanium, one obtains Kfrom the following equation (see also [209])

K =( ∑

donors · cm−3)/(∑

acceptors · cm3)

= (NAs + NSe)/NGa. (94)

The substitutional selenium impurities are dou-ble donors providing two electrons for compensa-tion. Therefore they are counted twice in the sum ofdonors. Using the different values for σn, one findsK ranging from 0.322 to 0.405 for crystals with neg-ligible initial donor and acceptor concentrations. Itwould be of great help for both the basic under-standing of the hopping conduction [218] as well asfor application of neutron-transmutation-doped germa-nium as, for example, bolometer material [209], ifthese cross-sections could be accurately evaluated inone or more well characterized nuclear reactors (seeabove).

In order to obtain the above K values and thus takefull advantage of NTD, Haller et al. choose the purest

3377

available Ge crystals as a starting material. Germaniumis, in this respect, ideally suited for NTD because itcan be purified at present time down to concentrationsof �1011 cm−3 (see e.g. [219]). Such low concentra-tions are negligible when compared with the dopantconcentrations after NTD in the low 1016 cm−3 range.According to Haller et al. [209] the concentrations odelectrically inactive impurities such as hydrogen, car-bon, oxygen and silicon can be as high as 1014 cm−3.Of all the isotopes of these impurities only 30

14Si trans-mutes to an electrically active impurity, phosphorus,a shallow donor. With only one silicon atom in ev-ery 4.4 × 108 germanium atoms and only 3% os allsilicon atoms being 30

14Si which has a neutron capturecross-section much smaller than the germanium isotopecross-sections, Haller et al. can estimate that less thanone phosphorus donor is produced for every 1011 gal-lium majority acceptors during the NTD process. Theseauthors concluded that ultra-pure germanium crystalsare virtually perfect starting material. For the NTDstudy, they have chosen an ultra-pure germanium sin-gle crystal which they have grown at the crystal growthfacility described early (see also [209, 219] and refer-ences therein).

The measured resistivities (ρ) in paper [209] are pre-sented in Fig. 46. The results of these measurements

Figure 46 Resistivity as a function of 1000/T for NTD and uncompensated germanium samples. Each curve is labeled by the gallium concentrationobtained by either NTD or melt doping (after [209]).

yield the mobility µ:

ρ = (pµe)−1,

RH = (pe)−1

(95)µ = RH/ρ,

where p is free hole concentration, e is charge ofthe electron and RH is the magnitude from Hallmeasurement.

The mobility values are only useful down to thetemperature where hopping conduction sets in. Themobility values agree well with published values formelt-doped material in the temperature range above thehopping regime. This indicates that the concentration ofresidual radiation damage or other free-carrier scatter-ing centers must be very small. Fig. 46 shows the log(resistivity) versus 1000/T dependence for six NTDgermanium samples. The number next to each curvecorresponds to the acceptor (gallium) concentration ineach sample. For comparison Haller et al. have alsomeasured gallium-doped germanium samples whichhave extremely small values of K . These so-called un-compensated samples were cut from crystals whichwere doped in the melt and were grown in the ultra-pure germanium crystal-growing equipment, and notNTD doped. The compensating donor concentration

3378

in these crystals is estimated to be less than 1011 to1012 cm−3. The resulting K is of the order of 10−4 to10−5. The resistivity-temperature dependence of theseNTD samples is characterized by three regimes. At hightemperatures (room temperature down to about 50 K),the resistivity decreases because the carrier mobilityincreases. Below about 50 K carrier freeze out beginsand reduces the free hole concentration rapidly. Theslope of the freeze out in highly-compensated materialos proportional to the acceptor binding energy EA −EV � 11 meV. At still lower temperatures, the appear-ance of hopping conduction causes the resistivity to in-crease only very slowly. All six NTD germanium sam-ples show these three resistivity regimes very clearly.The low-compensated samples show different log(ρ)versus 1/T dependences. A third conduction mecha-nism has been proposed for such material [220]. It isbased on the idea that carriers can “hop” from neutralto a neighboring neutral acceptor thereby forming apositively charged acceptor. The NTD process in high-purity germanium leads to a fixed compensation whichin turn results in a certain slope of the log(ρ) versus1/T dependence for a given neutron exposure.

2.8.1.1. Metal-insulator transition. In next part of thisparagraph let us briefly discussed the metal-insulatortransition (MIT) [210, 218, 221, 222] in transmutedGe. In the literature there is an intensive debate whetherMIT is a phase transition of first or second order andwhat are the experimental conditions to obtain it at finitetemperatures and in real (disordered) system (see, e.g.[211, 212, 214, 223]). If the MIT is as second orderphase transition a further challenge is the solution of theso called puzzle of the critical index, µ for the scalingbehavior of the metallic conductivity near the MIT, i.e.,just above the critical impurity concentration Nc andas small compensation, K . According to the scalingtheory of the MIT for doped semiconductors [222], theconductivity at zero temperature σ (0) = σ (T → 0),when plotted as a function of impurity concentrationN , is equal to zero on the insulating side of the MITand remains finite on the metallic side, obeying a powerlaw in the vicinity of the transition,

σ (0) ∝ [(N/N c) − 1]µ, (96)

where Nc is the critical impurity concentration of thegiven system and µ is the critical conductivity expo-nent. The value of µ, determined experimentally, iscompared with theoretical predictions. Up to presenttime µ ≈ 0.5 has been obtained with nominally un-compensated semiconductors (Si:P [224], Si:As [225,226], Ge:As [227], Si:B [228]) while µ ≈ 1 has beenfound with compensated semiconductors (Ge:Sb [229],Si:P,B [230], Ge:Ga, As [231]) and amorphous al-loys [232–234]. Exceptions are uncompensated Ge:Sbwith µ ≈ 1 [210] and Gax Ar1−x amorphous alloyswith µ ≈ 0.5 [235, 236]. As was shown in [211] thevalueµ≈0.5 obtained with simple systems like uncom-pensated semiconductors turns out to be inconsistentwith theoretical prediction [221, 222, 234]. In his orig-inal theory Mott considered only the electron-electron

(e−–e−) interaction (Mott transition) and predicted adiscontinuous transition of σ (0) at Nc [237]. Althoughthere is much evidence for the importance of e−–e−-interactions, no experimental observation of such anabrupt transition has been reported. Anderson’s ides ofMIT is based solely on the disordered potential arisingfrom randomly distributed dopants (Anderson transi-tion) [238]. This lead to the development of the well-known “scaling theory” which predicted µ ≈ 1 forthree dimensional systems (see also [222] and refer-ences therein). More recently, higher order calculationsof the scaling theory (exclusively with disorder and nointeractions) predict µ ≈ 1.3 [239], and more impor-tantly, this value is shown to be independent of timereversal invariance [240] and of strength of spin-orbitinteractions [241] (see also [212]). It is therefore clearthat the effect of disorder alone cannot explain the ex-perimental results of µ ≈ 0.5 or 1. Chayes et al. com-bined the theories of Mott and Anderson and success-fully set the lowest limit µ > 2/3 [242]. This resultpermits µ ≈ 1 obtained with compensated semicon-ductors and amorphous alloys. However, there still isno theory which convincingly explain µ ≈ 0.5 foundfor uncompensated semiconductors.

Even with today’s advanced semiconductor technol-ogy, melt-doping of bulk semiconductors always leadsto inhom*ogeneous dopant distributions due to impuritysegregation and striation during crystal growth [223].In papers [211, 223] this difficulties have overcome byapplying the NTD technique to a chemically pure, iso-topically enriched 70Ge [211] and 74Ge [223] crystals.The 70Ge crystal of isotopic composition [70Ge] = 96.2at.% and [72Ge] = 3.8 at.% was grown in paper [211]using the Czochralski method developed for ultra-pureGe [219]. The as-grown crystal was free of dislocations,p-type with an electrically active net-impurity concen-tration less than 5 × 1011 cm−3. In paper [223] was usedisotopically engineered germanium which was grownfrom pure 74Ge, enriched up to 94%, or by addition ofa controlled portion of Ge with natural isotopic contentto the 74Ge material. In this way both, the doping aswell as the compensation, are very hom*ogeneous dueto the NTD and the compensation by controlled mix-tures of 74Ge and 70Ge which transmute to 75As donorsand 71Ga acceptors. Four series of n-type NTD-Ge withdifferent K were grown [223]. The values of K are pro-portional to the product of the isotopic abundance andthe thermal neutron cross-section of all isotopes pro-ducing impurities (see above): K = NGa/(NAs + NSe),whereas the impurity concentration is additionally pro-portional to the irradiation doze. A very small fractionof 72Ge becomes 73Ge which is stable, i.e., no otheracceptors or donors are introduced. Use NTD sinceit is known [145, 243] to produce the most hom*oge-neous, perfectly random dopant distribution down tothe atomic level. Fig. 47 shows the temperature depen-dence of the resistivities (ρ) of 14 insulating samples inthe range N = 0.16–0.99Nc for NTD 70Ge:Ga crystals.The analogous picture for NTD 74Ge:Ga is shown inFig. 48. All curves become linear only when lnρ is plot-ted against T −1/2 in good agreement with theory of vari-able range hopping conduction for strongly interacting

3379

Figure 47 The logarithm of the resistivity plotted as a function of T −1/2

for 14 insulating NTD 70Ge:Ga samples. Gallium concentration from topto bottom in units 1016 cm−3 are 3.02; 8.00; 9.36; 14.50; 17.17; 17.52;17.61; 17.68; 17.70; 17.79; 17.96; 18.05; 18.23 and 18.40 (after [211]).

Figure 48 Typical temperature dependences of the resistivity as a func-tion of T −1/2 for four samples NTD 70Ge:Ga crystals (after [223]).

electrons [218]:

ρ = ρ0 exp(T0/T )1/2, (97)

where ρ0 is a prefactor and T0 is given by

T0 ≈ 2.8e2/k(N )ξ (N ), (98)

where k(N ) and ξ (N ) are the dielectric constantand localization length depending on N , respec-tively. Moreover, k(N ) ∝ [Nc/(Nc−N )]s and ξ (N ) ∝[Nc/(Nc−N )]ζ as N approaches Nc from the insulatingside so that T0 becomes [218]

T0 = A[(Nc−N )Nc]α. (99)

Here α = s + ζ is to be determined experimentally[211].

Fig. 49 shows the dependence of T0 as function ofNd = n/(1 − K ) for different K of 74Ge:Ga [223]. Asearly these authors used the intersection point of thesedependencies with the x-axis as a tool for the detrmina-tion of Nc(K ). The left half of Fig. 50 shows the exper-imentally determined T0 versus [Ga] (filled diamonds)

Figure 49 Determination of Nc from the extrapolation T0 → 0 in therange T0 > T (after [223]).

Figure 50 The left side shows (70Ge) as a function of Ga concentration(♦). The solid curve is the best fit obtained with Equation 99 (withα ≈ 1). The right side shows the zero temperature conductivity σ (◦)obtained from the extrapolation in Fig. 51 for the metallic samples as afunction of Ga concentration (•). The solid curve is the best fit obtainedwith Equation 99 (after [211]).

together with the result of a three-parameter-fitting us-ing A, Nc and α as variables in Equation 99 (solid curve)[211]. These authors deduced [Ga] for sample using fol-lowing equation [71Ga](cm−3) = 0.1155 × n (cm−2),since it was known the precise neutron fluency used ineach irradiation. The best fit of T0 with Equation 99 wasobtained with the values α = 1.03 ± 0.038 and Nc =(1.855 ± 0.012) × 1017 cm−3. A much larger value ofα ≈ 2 has been reported for Ge:As using only threesamples with the highest N being far from the transi-tion 0.56Nc) [244]. In paper [211], it has been obtainedα = 1 with 14 hom*ogeneously doped samples of [Ga] =0.16–0.99Nc, all demonstrating the lhρ ∝ T −1/2 depen-dence, i.e., this data set should be considered to be thefirst reliable determination of the hoping conductivityexponent α for a particular semiconductor system.

Fig. 51 shows the conductivity σ according to theresults of paper [211] in ten metallic samples plottedagainst T 1/2. Extrapolation of each curve to T = 0 K,i.e., the determination of the zero temperature conduc-tivity σ (0), yields a very small error since the depen-dence of σ on T for all samples is very weak. The righthalf of Fig. 50 shows σ (0) as a function of [Ga] (filledcircles) together with a fit obtained by the scaling ex-pression Equation 99 (solid curve). The values of theparameters determined in paper [211] from this fit are

3380

Figure 51 Conductivity plotted as a function of T 1/2 for 10 metallicNTD 70Ge:Ga samples. Sold lines indicate extrapolation to T = 0 K.Gallium concentration from top to bottom in units of 1016 cm−3 are18.61; 19.33; 20.04; 20.76; 21.47; 22.19; 22.90; 23.62; 24.50 and 26.25(after [211]).

µ = 0.502 ± 0.025 and Nc = (1.856 ± 0.003) × 1017

cm−3. This value present µ ≈ 0.5 for uncompensatedGe:Ga semiconductors with high confidence, since thetwo values of Nc obtained from the scaling of T0 [Equa-tion 99] and σ (0) [Equation 96] agree perfectly (detailssee also [214, 223]).

2.8.1.2. Neutral-impurities scattering. The low-temperature mobility of free carriers in semiconductorsis mainly determined by ionized- and neutral-impurityscattering. The ionized-impurity scattering mechanismhas been extensively studied (see e.g. [245] andreferences therein), and various aspects of this processare now quite well understood. Scattering by neutralimpurities (see also [177]) is much less than byionized centers, i.e., its contribution is significantonly in crystals with low compensation and at verylow temperatures where most of the free carriers arefrozen on the impurity sites. The availability of highlyenriched isotopes of Ge which can be purified toresidual dopant levels <1012 cm−3 has provided thefirst opportunity to measure neutral impurity scatteringover a wide temperature range. Three Ge isotopestransmute into shallow acceptors (Ga), shallow donors(As) and double donors (Se) (see also above):

7032Ge + n → 71

32GeEC(t1/2=11.2 days) → 7132Ga + νe,

7432Ge + n → 75

32Geβ−(t1/2=82.2 min) → 7532As + β− + νe,

7632Ge + n → 77

32Geβ−(t1/2=11.3 h) → β− + νe

+ 7732Asβ−(t1/2=38.8 h) → 77

32Se + β− + νe. (100)

The isotopes 72Ge and 73Ge are transmuted into thestable 73 Ge and 74Ge respectively. Controlling the ra-tio of 70Ge and 74 Ge in bulk Ge crystals allows finetuning of the majority- as well as the minority carrier

TABLE XII I Carrier concentration of the Ge crystals used in thework of Fuchs et al. [105]

p-type NA − ND ND K = ND/NA

Ge:Ga #1 3.1 × 1014 3 × 1012 9 × 10−3

Ge:Ga #2 7.7 × 1015 9 × 1013 1.2 × 10−2

Ge:Ga #3 1.7 × 1016 2 × 1014 1.2 × 10−2

Ge:Ga #4 1.0 × 1015 1.2 × 1013 1.2 × 10−2

n-type ND − NA NA K = NA/ND

Ge:As #1 3.5 × 1014 8.5 × 1012 2.4 × 10−2

Ge:As #2 1.2 × 1015 1.2 × 1013 1.0 × 10−2

concentration. Currently, this is the best method to varythe free-carrier concentration independently from com-pensation ratio. As opposed to other doping methods,NTD yields a very hom*ogeneous, perfectly random dis-tribution of the dopants down to the atomic levels [246].Thus isotopically controlled crystals offer a unique pos-sibility to study systematically the scattering mech-anism of the charge carriers in semiconductors. Ex-tensive Hall-effect and resistivity measurements fromroom temperature down to 4.2 K yielded very accuratefree-carrier concentrations and mobilities as a functionof temperature and doping level were done in papers[105, 247, 248]. Itoh et al. have performed temperature-dependent Hall measurements on four different p-typeand two-different n-type Ge crystals. The n-type crys-tals were obtained through NTD of isotopically en-riched 74Ge, and the p-type crystals correspondinglyfrom NTD of isotopically enriched 70Ge. The neutroncross-section for the neutron capture of the isotope forthese irradiations were determined to be σc(70Ge) =2.5(5) × 10−24 cm2 and σc(74Ge) = 0.6(1) × 10−24 cm2

by Itoh et al. [249]. To remove structural defects due tothe unintentional irradiation with fast neutrons, all sam-ples had to be thermally annealed at 650◦C for 10 s in arapid thermal annealer. Hall mobility obtained from theconductivity and free-carrier concentration data (listedin the Table XIII) are displayed in Fig. 52. A magneticfield of 3 kG was used, that is, for the temperature rangeof interest for the neutral impurity scattering the high-field limit µB �1 is satisfied and the Hall mobility canbe equated with the drift mobility.

Fuchs et al. [105] analyzed the mobility data ofFig. 52 in terms of scattering of the carriers fromphonons (µ1), ionized impurities (µi) and neutral im-purities (µn) assuming next rule

µ= 1

µ1+ 1

µi+ 1

µn(101)

To extract the neutral impurity scattering contribu-tion, they subtracted 1/µ1 + 1/µi from the measured1/µ. The relative contributions of phonon scattering(1/µ1), ionized impurity scattering (1/µi) and the re-sulting neutral impurity scattering (1/µ−1/µ1−1/µi)are plotted in Fig. 53 (data Ge:Ga #2). For T > 80 K,phonon scattering is the dominant scattering mecha-nism. On comparison of Figs 52 and 53, it becomesclear that the “dip” in the carrier mobility around 50 Kis caused by scattering from ionized impurities, whichdominate the scattering of the carriers between 20 and

3381

Figure 52 Temperature dependence of the carrier mobility of (a) p-type and (b) n-type NTD Ge crystals (after [105]).

Figure 53 Temperature dependence of the relative contributions to themobility. Note that the mobility is dominated by neutral impurity scat-tering below 20 K (70Ge:Ga #2 crystal) (after [105]).

80 K. The flattering and saturation of the mobilitiesbelow 20 K originate from neutral impurity scattering,which can only be observed in crystals with very highcrystalline quality and low compensation like isotopi-cally enriched NTD Ge crystals used in paper [247].

The experimental data, obtained in [247] allow theseauthors quantitative comparison with theory. Accord-ing to Erginsoy [250], the inverse relaxation timeτ−1, the scattering rate, for neutral-impurity scatteringequals:

τ−1 = 20k NN h3

m∗2e2, (102)

where k is dielectric constant, e is the electron charge,NN is the neutral-impurity concentration, and m∗ is theelectron effective mass. Equation 102 can be consideredonly as a first-order approximation because the prefac-tor 20 is an empirically determined constant and onlythe lowest s partial wave is taken into account in thephase-shift calculation (see also [251, 252]). McGilland Baron [253] have used for τ−1

neutral the followingequation:

τ−1neutral = 4π NN he

2km∗EB

∞∑l=0

(l + 1)

4w1/2[3 sin2(δ−

l − δ−l+1)

+ sin2(δ+l − δ+

l+1)], (103)

where EB is the binding energy of the scattering cen-tres, w ≡ E/EB and E is the incident electron energy,and δ+

l and δ−l are the lth partial shift for the singlet

and triplet states respectively. Authors [253] graphi-cally showed the accurate τ−1

neutral as a function of w forneutral-impurity scattering in semiconductors. This re-sult has been considered as an appropriate model forneutral-impurity scattering in semiconductors and hasbeen discussed in detail in many standard textbooks(see, e.g. [254]).

Meyer and Bartoli reevaluated this task and obtainedan analytical expression that is essentially the same asthe graphical solution of authors of paper [253] butcovering a wider incident-electron energy range (seealso [255]):

τ−1neutral = A(w)k N N h3

m∗2H e2

, (104)

with

A(w) = 35.2

w1/2

(1 + e−5w)(1 + 80.6W + 23.7w2)

(1 + 41.3w + 133w2)

×[

wln(1 + w)− (1 + 0.5w − 1.7w2)

(1 + w)3

(105)

Here m∗H is the hydrogenic effective mass given by

m∗H = EBk2m0

EH. (106)

In last equation m0 is the electron rest mass and EH =13.6 eV is the binding energy of hydrogen. In total-mobility calculation Itoh et al. [247] employ a standardrelaxation-time approximation. This approach is validbecause they are limiting to low temperatures (T <

25 K) where the inelastic optical-phonon deformation-potential scattering is negligible. Three scatteringmechanisms are considered: neutral-impurity, ionized-impurity, and acoustic-phonon deformation-potentialscattering. The neutral-impurity scattering contributionwas calculated using both Equations 102 and 104 so it

3382

can compare the models of Erginsoy and Meyer andBartoli with experimental results of paper [247]. Theconcentration of neutral-impurity centres as a functionof temperature NN(T ) in each sample is given by nextrelation

NN(T ) = NMJ − NMN − n(T ). (107)

Here NMJ, NMN and n(T ) are the majority-impurity,minority-impurity, and free-carrier concentrations, re-spectively. For the ionized-impurity scattering, Itohet al. employ the Brooks-Herring expression [255,256]:

τ−1ion = π NIe4(kBT )−3/2x−3/2

(2m∗con)1/2k1/2

[ln

[1+4x

]

− 4x/a

1 + 4x/a

], (108)

n(T ) = 2(NMJ−N MN)

{[1 + (NMN/gN B)exp(EMJ/kBT )]+√[1 + (NMNgN B)exp(EMJ/kBT )]2 + (4/gN B)(NMJ−N MN) exp(EMJkBT )} , (114)

where

a = 2π h2e2n

m∗kk2BT 2

, (109)

and x = E/kBT (E is the incident electron energy),m∗

con is the average conductivity effective mass, and NIis the ionized-impurity concentration. The temperature-dependent NI in each sample is given by

NI(T ) = n(T ) + 2NMN. (110)

For the acoustic-phonon deformation-potential scat-tering [257]:

τ−1ac = Bac(m∗

conT )3/2x1/2, (111)

where the constant Bac has well-established values forn- and p-type Ge as shown in Table XIV. Having foundτ−1 of all three scattering mechanisms, Itoh et al. cal-culated an average 〈τ 〉 using the Maxwell-Boltzmanintegration:

〈τ 〉 = 4

3√

∫ ∞

x3/2exp(−x)

τ−1ac + τ−1

ion + τ−1neutral

dx . (112)

T ABL E XIV Parameters used in the total-mobility calculations (after[247])

Ga:As (n-type) Ge:HGa (p-type)

k 16 16m∗

con 0.12 m0 0.28 m0a Bac 1.08 × 1010 g3/2K−3/2 9.50 × 108 g3/2K−3/2

BB(theoretical) 12.5 meB b 11.2 meVc

aThe values of B are determined experimentally using ultrapure n- andp-type Ge of NMJ ∼ NMN ∼ 3 × 1011 cm−3.bM. Altarelli, W. Y. Hsu and R. A. Sabatini, J. Phys. C 10, L605 (1977).cA. Baldareshi and N. O. Lipari, in Proc. 13th Inern. Conf. Phys. Semi-cond. (F. G. Fumi, ed, North-Holland, Amsterdam, 1976) p. 595.

Finally the total mobility µtot was given by

µtot = e〈τ 〉/m∗con. (113)

All parameters required for the mobility calculationsare well-known in Ge (see Table XIV). The only un-known material parameters at this point are sample-dependent NMJ, NMN and n(T ) in Equations 107 and110. All three parameters as will shown below, can bedetermined precisely for each sample by performingvariable-temperature Hall-effect measurements. Con-sequently all mobility calculations are performed with-out any adjusable or scaling parameters. The exper-imental curves are fitted with the following standardsemiconductor statistics [258], which describes thetemperature dependence of the free-carrier concentra-tion in semiconductors doped by shallow majority im-purities NMJ and compensated by minority impuritiesNMN:

where g = 1/2 (g = 4) is the spin degeneracy fora donor (acceptor), NB is the effective conduction-(valence-) band density of states, and EMJ are the exper-imentally determined ionization energies: 14 and 11.07meV for As and Ga, respectively (Table XIV).

Fig. 53 shows the relative strength of the scatteringfrom the ionized and the neutral impurities. There isonly a relatively small temperature region in which thescattering from the neutral impurities dominates. Thisrange extends to higher temperatures as the free-carrierconcentration is increased. The calculated “transitiontemperatures” above which the ionized impurities arethe main scattering centres (see also [259]) comparevery well with experimental results of Itoh et al. [247](see also Fig. 54).

Figure 54 Data points represent experimentally measured carrier mo-bility in (a) four 74Ge:As and (b) two 70Ge:Ga samples. For a directcomparison theoretically calculated mobility using Erginsoy’s model(broken line) and the model of Meyer and Bartoli (solid line) is shownfor each sample. The contributions of the different scattering mecha-nisms to the total mobility of the 70Ge:Ga—1 sample are shown in theupper half of (b) (after [247]).

3383

We now turn the attention to the low-temperatureregime where mobilities are dominated ny neutral-impurity scattering. Fig. 54 shows a direct comparisonof the experimental results with theoretical total-mobility curves calculated. For each sample two the-oretical total-mobility curves are calculated: one usingErginsoy’s model [Equation 102] and other usingMeyer and Bartoli model [Equation 104]. A strikinglygood agreement was obtained between the experimen-tal and theoretical mobilities calculated with the modelof Meyer and Bartoli for all samples (see Fig. 54).

In order to demonstrate the importance of the ho-mogeneous dopant distribution, Itoh et al. have per-formed the same study on samples cut from Ge:Ga crys-tals grown by the conventional Czochralski method,where Ga impurities were introduced to Ge melt dur-ing the crystal growth. These authors observed devi-ations of the measured mobility from the theoreticalcalculations, which are most likely due to inhom*o-geneous Ga impurity distributions in melt-doped Ge.Only the use of NTD semiconductors with randomlydistributed dopants allows for an accurate test of theneutral impurity-scattering models.

2.8.2. SiliconIt is well-known that doping of silicon single crystalsby incorporation of impurities from the melt during so-lidification in most cases leads to an inhom*ogeneousdistribution of impurities in the solids [155, 236]. Thisis due to the fact that nearly all impurities in silicon havethermal equilibrium distribution coefficients much lessthan unity and that the solidification or crystal growsat each position of the interface is characterized by adifferent state of thermal inequilibrium leading to dis-tribution coefficients that in space and time continu-ously change and result in a nonuniform impurity dis-tribution [162, 260]. In actual crystal production thenonuniformity is further enhanced by lack of controlof exactly constant melt volume and feed of the dop-ing impurity. The most widely used doping elementsin silicon are boron and phosphorus. Boron has a dis-tribution coefficient between 0.9 and 1 which makesa doping uniformity of ±10% easily obtainable (see,e.g. [236]). The thermal equilibrium distribution coef-ficient for phosphorus of approximately 0.3 leads ingeneral to the above mentioned large doping variationsboth on a macroscale (center to periphery) and on a mi-croscale (striations). No other n-type doping elementhas a larger distribution coefficient. Because fast dif-fusing p-type dopants (Ga, Al) are available, becauseelectron mobility is greater than hole mobility, and be-cause contact alloying technology is reasonable, n-typesilicon is generally used for solid state power devices[162, 261]. With avalanche breakdown voltages beingdetermined from areas with lower resistivities, use ofa conventionally doped material results in hot-spot for-mation prior to breakdown and too high forward voltagedrop leading to excessive heat dissipation because of asafe punch through design [155, 236, 262].

Phosphorus doping by means of NTD was suggestedby Lark-Horovitz [180] and Tanenbaum and Mills [154]

for hom*ogeneity purposes and has been applied forhigh-power thyristor manufacturing in [155, 236, 261].Hill et al. [261] were demonstrated how such a hom*o-geneous phosphorus doping may result in a “theoreti-cal design” possibility for high-power components (seealso below).

The process used for fractional transmutation of sil-icon into phosphorus and thereby performing n-typedoping

3014Si(n, γ ) = 31

14Siβ−

2.62 h → 3115P (115)

was first pointed out by Lark-Horovitz in 1951 [180].Apart from special applications [263] and research, theabove process was, however, not utilized to any extentuntil the early seventies, at which time manufacturersof high-power thyristors and rectifiers for high-voltagedirect current transmission lines, in particular, initi-ated usage of the transmutation doping process [193,261, 264]. The reasons for not using the neutron dop-ing method throughout the sixties may be found in thelack of a processing technology which could benefitfrom a more uniform doping, insufficient availabilityof high resistivity starting material, and the lack of nu-clear reactors with irradiation capacities in excess ofthat needed for testing fuel and materials for nuclearpower stations.

Let us, for the following discussion, assume that com-pletely uniform neutron doping may be accomplished.The hom*ogeneity of the doped silicon is in this casedetermined by the background doping, i.e., the distri-bution of impurities in the starting material, where thenet impurity concentration may be of either donor oracceptor type. Let us further, for simplicity, considerstarting material of one conductivity type and assumecomplete n-type conduction after irradiation and an-nealing. With CS being the net impurity concentrationof the starting material and CD the resulting donor con-centration after irradiation we have, for both n- andp-type material,

CmaxD − Cmin

D = CmaxS − Cmin

S . (116)

In such case we may define

1. the hom*ogeneity factors for the starting material(αS) and for the neutron doped material (αD), respec-tively

αS = CminS

CmaxS

(117)

and

αD = CminD

CmaxD

(118)

and2. the doping factor

fD = CmaxD

CmaxS

. (119)

3384

T ABL E XV Values for hom*ogeneity factor αD as function of hom*o-geneity factor αs of starting material and doping factor fD as defined intext (after [193])

αS/ fD .1 .2 .3 .4 .5 .6 .7 .8 .91 .1 .2 .3 .4 .5 .6 .7 .8 .92 .55 .6 .65 .7 .75 .8 .85 .9 .955 .82 .84 .86 .88 .9 .92 .94 .96 .987 .87 .89 .90 .91 .93 .94 .96 .97 .99

10 .91 .92 .93 .94 .95 .96 .97 .98 .9920 .955 .96 .965 .97 .975 .98 .985 .99 .99550 .98 .98 .99 .99 .99 .99 .99 .996 .998

100 .991 .992 .993 .994 .995 .996 .997 .998 .999

From this is easily derived

1 − αD = 1−αS

fD. (120)

Table XV summarizes values of αD as a function ofαS and fD. It is seen that in order to obtain neutron-doped silicon with, for instance, a hom*ogeneity factorgreater than 0.9, it is necessary to use a doping factor ofat least 7 when starting from “undoped” n-type materialin which the hom*ogeneity factor is typically not greaterthan 0.3 when taking the microcavitations (striations)into account. An examples of such neutron-doped sil-icon are shown in Figs 55 and 56. It should be notedthat in terms of resistivity, which is often used for im-purity characterization, a doping factor fD means use

Figure 55 Spreading resistance measurements of a thermal neutron ir-radiation doped silicon slice. Step-length on scan 1 and 2 is 250 µmand on scan 3 step-length is 50 µm. Starting material has been selectedgreater than 1500� cm n-type (after [193]).

of starting material with minimum resistivity a factorfD or 2.8 fD greater than the resistivity after neutrondoping for n- and p-type starting material, respectively.The difference is due to the electron mobility being2.8 times greater than the hole mobility. In conclusionof this section it should be generally noted that in orderto make neutron-doped silicon with significantly moreuniform resistivity than conventionally doped material,a doping factor fD = 5 or more should be applied.

Following Janus and Malmros [193] let us considerfurther the theoretical case where a cylindrical siliconcrystal is surrounded by a material with the same neu-tron absorption and scattering efficiency as the siliconitself (see Fig. 57). Let us furthermore assume a thermalneutron flux gradient along an x axis perpendicular tothe crystal axis with the neutrons coming from an ex-ternal source. In this case the neutron flux will have theform

� = �0 · exp

(− x

), (121)

where b, the decay length, may be obtained from theformula

b = (3 · σSi · σSi,t · C2

)−0.5. (122)

σSi = 0.16 · 10−24 cm2 is the mean of the absorp-tion cross-sections for the 3 silicon isotopes, 28Si,29Si and 30Si weighted with their abundances. σSi,t =2.3 · 10−24 cm2 is the total cross-section (absorption+ scattering) and CSi = 4.96 · 1022 cm−3 is the totalnumber of silicon atoms in 1 cm3. Hence b may becalculated:

bsilicon = 19 cm. (123)

In order to improve the doping hom*ogeneity in thecylindrical crystal this will be slowly rotated around itsaxis. The time average of this flux at the distance r fromthis axis is

� = 1

∫ π

0�0 exp

[− r

bcost

]dt

= �0

[1+1

+ · · · · ·]. (124)

The ratio between the neutron dose at the peripheryand at the axis of the crystal cylinder will then be

�(a)

�(0)� 1 + 1

, (125)

where a is the crystal radius (Fig. 57).For intrinsic starting material the irradiation doped

silicon will thus have a hom*ogeneity factor of

αD � 1 − 1

� 0.956 (126)

for an 80-mm-diameter crystal, i.e., the absorption lim-iting factor for the obtainable radial variations.

3385

Figure 56 Typical lateral microscopic resistivity distributions in conventionally doped silicon and in silicon doped by neutron irradiation (after [236]).

Figure 57 Irradiation configuration. (a) Top view of the facility with thecylindrical crystal situated outside the core rotating around its cylindricalaxis. Arrows indicate overall direction of neutrons. The flux does notvary along the cylindrical axis. (b) The neutron flux is a function of thedistance from the reactor core (after [193]).

In the above analysis we have neglected the effects offast neutron moderation in the silicon. By comparison,however, of irradiations performed in reactors with fastneutron fluxes from 10−4 to 1 times the thermal flux andwith different flux gradients, the authors [193] haveobserved no influence on the resistivity hom*ogeneitydue to fast neutron moderation in the silicon.

In irradiated silicon crystals for semiconductor de-vice applications only two isotopes 31Si and 32P areof importance in connection with radioactivity of neu-tron doped material. For thermal neutron doses less than1019 neutron/cm2, no other elements have been detectedemitting radiation. Futhermore, 31Si, having a halflifeof 2.62 h, decays to an undetectable level in 3–5 days.For this reason, it will be discussed the radioactivetyonly of the 32P isotope. Fig. 58 pictures the 32P activityas a function of final resistivity for a variety of thermalneutron flux levels typical for the nuclear test reactorsin use. As was shown in [193] absolute flux determina-tion to 1% accuracy has proven obtainable for instance

Figure 58 The radioactivity of the 32P isotope in silicon after 4 days ofcool down subsequent to irradiation. It may be observed that the activityas function of the resistivity obtained depends on the neutron flux used(after [193]).

by means of calorimetric boron carbide monitors.

3115P(n, γ )32

15Pβ−

14.3d → 3216S (127)

as a secondary one with 31P concentration at each in-stant in time being dependent on the neutron dose re-ceived and the time allowed for the 31

14Si β−

2.62 h → 3215P

decay.From Fig. 58 it may be observed that neutron doping

below 5 � · cm can be performed only when accept-ing cool down periods corresponding to the 32P halflifeof 14.3 days. The exempt limit for inactivity of 2 ×10−3 µCi/g shown on the figure is representative formost European countries, as well as being the value

3386

recommended by the International Atomic EnergyAgency (IAEA), Vienna, Austria [265]. It should beadded that careful cleaning of the silicon prior to in-sertion in a nuclear reactor is vital to avoid radioactivesurface contamination. For the safety of the personneland the end product users, a double check upon ship-ping from the reactor sites and upon reception in thesilicon plant, respectively is carried out to secure thatonly inactive material (below the exempt limit) is beingfurther processed after the neutron doping. In general,this implies shipment from the reactor not earlier than4 days after irradiation.

The use of NTD is of particular interest to thyris-tor manufacturers where n-type starting material is re-quired for the basic p-n-p structure [259, 260]. Someadvantages for high power device design and perfor-mance include:

1. more precise control of avalanche breakdownvoltage,

2. more uniform avalanche breakdown, i.e., greatercapacity to withstand overvoltages,

3. more uniform current flow in forward direction,i.e., greater surge current capacity, and

4. narrower neutral zone and therefore narrower baseand lower forward voltage drop Vf.

The summary of some points concerning the prepa-ration of NTD silicon for special applications on an Rand D scale describe in papers [260–261]. The pro-duction of large quantities of NTD silicon for powerdevices is described in [190]. More recently (see, e.g.[266]) the NTD technique has been also proposed forthe effectual doping of P in a-Si:H films (see also [4]).The results of [266] are shown that NTD technique isan excellent method for doping of P in a-Si:H.

Despite intensive study over many years and consid-erable progress, no clear understanding has emergedof one of the fundamental issues regarding theMIT in doped semiconductors and amorphous metal-semiconductor mixtures: whether and under what cir-c*mstances the Hall coefficient diverges as the tran-sitions is approached (see above for Ge). As iswell-known in the localized regime the spatial behaviorof the wave functions is usually described by an expo-nential decay length reflecting the spatial extent of thewave function (see e.g. [267–269]). Dai et al. recentlyare shown that the Hall coefficient of Si:P diverges atthe transitions, as it does in Si:B [270] and Ge:Sb [271].The difference in the behavior of MIT according theseauthors may be connected with a different degree ofcompensation. It is also possible that the MIT is differ-ent in a persistent photoconductor, where the disorderis particularly strong and the concentration of shallowdonors is varied and controlled through illumination.(details see also [268, 269]).

2.8.3. Other compoundsThe NTD method have used with success in studyof compound semiconductors: GaAs [184, 272–274]and GaP [275, 276]. NTD of GaAs is based on the

following thermal neutron capture nuclear reactions(see also [184]):

69Ga (n, γ ) 70Gaβ−

21.1 min → 70Ge, (128)

71Ga (n, γ ) 72Gaβ−

14.1 h → 72Ge, (129)

75As (n, γ ) 76Asβ−

26.3 h → 76Se. (130)

The relative abundances of the isotopes involved inthe reactions and the cross-sections for these reactionsare such that the ratio of Se and Ge concentrations pro-duced is

NSe/NGe = 1.46. (131)

Selenium is a typically shallow substitutional donorin GaAs with an electronic energy level a few meV fromthe conduction bans edge [277]. Germanium in GaAs isan amphoteric impurity which acts as a shallow donor(also a few meV from the conduction band) is situatedon a Ga site and as an acceptor level at EV + 0.04 eVif situated on an As site [278]. Since, if electronicallyactive, all of the Se atoms and some portion of the Geatoms are expected to act as donors, NTD of GaAs isexpected to dope GaAs more n-type. The addition ofdonors moves the Fermi level (EF) away from the va-lence band (EV) to the conduction band (EC). If a suf-ficiently high concentration of donors is added, EF willmove to the upper half of the bandgap and the GaAs willbe converted to n-type. Analysis of Hall effect data as afunction of temperature provides a means of measuringthe donor content in irradiated GaAs samples. Younget al. were thus able to compare electrically active addeddonor content to the NTD-produced impurity concen-trations determined from nuclear measurements. TheHall effect analysis also allows them to determine con-centrations and energy levels (E) of impurities or de-fects in the p-type GaAs samples if the Fermi level inthe material moves near E at some temperature over therange of measurements. This technique thus provides ameans of identifying and measuring undercompensatedacceptor content in the samples. The low temperaturephotoluminescence technique used in paper [184] mea-sured donor-to-acceptor or conduction-band-acceptorluminescence. It provides an accurate determination ofthe position of acceptor electronic levels in the GaAs,permitting positive identification of impurities or de-fects with known luminescence lines. Identifications oflines due to specific impurities or defects can be madeusing luminescence techniques regardless of the posi-tion of the Fermi level in material. Little detailed infor-mation concerning an acceptor level can be obtainedfrom Hall effect if that acceptor is overcompensated.However, the presence of specific acceptors can be de-tected by luminescence techniques even in n-type sam-ples. On other hand, luminescence data do not providethe quantitative information obtainable from Hall effectmeasurements.

The results of room temperature measurement ofthe electrical properties of eight annealed NTD GaAssamples are summarized in Table XVI. The total NTDdose (NSe + NGe), the carrier concentration and carrier

3387

T ABL E XVI Room temperature results for Hall effect samples ofGaAs annealed 830◦C/20 min (after [184])

n (−) orSample no NTD dose/cm3 P (+) in cm−3 µ, cm2 V−1 s−1

1, 2 3.8 × 1015 +2.3 × 1016 3603 8.5 × 1015 +2.4 × 1016 3414 1.7 × 1016 +1.9 × 1016 337

10 2.7 × 1016 +8.6 × 1015 24212 7 × 1016 −1.6 × 1016 125120 1.5 × 1017 −7.7 × 1016 396015 2.8 × 1017 −2.3 × 1017 363116, 18 6.3 × 1017 −4.9 × 1017 3110

type (negative values of concentration indicate n-type),along with carrier mobility at room temperature are in-dicated in Table XXII. Note that following an NTD dosesufficient to produce 7 · 1016 atoms/cm3 initially presentin the samples. Therefore, 7 × 1016 donors/cm3 wouldindeed be expected to just overcompensate the p-typematerial. The results presented in Table XVI show thatthe p-type samples become progressively less p-typeand the n-type samples progressively more n-type withincreasing NTD dose. Because the donor levels in GaAsare very shallow, they remain fully ionized in the tem-perature range of Young et al. experiments, so that themeasured electron concentration is practically temper-ature independent (see Fig. 4 in [184]). This measuredn for each sample is approximately equal to total donorminus total acceptor concentration.

Fig. 59 shows the measured in [184] added electri-cally active donor concentration in eight NTD samplesas a function of NSe and of (NSe + NGe) added bytransmutation as determined from nuclear activity mea-surements. The uncertainty in determining added donorcontent in the p-type samples is large because of thecomplexity of analyzing material with multiple inde-pendent acceptor levels in closely compensated cases.The added donors can be much more accurately deter-mined in the more highly doped n-type samples. Theresults shown in Fig. 59 imply that all of the selenium

Figure 59 Measured added donors vs. NTD produced impurity content(after [184]).

Figure 60 Relative photoluminescence spectra for four n-type NTDsamples. The four spectra are not normalized with respect to each other(after [184]).

and a substantial fraction of the Ge atoms introduced bytransmutation act as donors following the 830◦C/20 minanneal. As will be shown below from photolumines-cence measurements a fraction of Ge atoms producedby transmutation are on acceptor rather than donor sitesin GaAs samples.

Fig. 60 show relative luminescence spectra for thefour n-type samples respectively. The spectral positionsindicated by arrows for carbon acceptor, the Ge ac-ceptor, and 0.07 eV acceptor correspond to donor (orband) to acceptor luminescence lines. The most impor-tant conclusion to be drawn from a comparison of thespectra for the control and eight NTD samples is that Geacceptors not present in the “starting material” controlsample are introduce by the NTD process. The increasein intensity of the Ge acceptor line with increasing doserelative to both the carbon and 0.07 eV acceptor linesindicates that Ge acceptor content increases with in-creasing transmutation doping. Therefore, some of theGe atoms produced by NTD in these samples are act-ing as acceptors rather than donors. Photoluminescencemeasurement studies of the control and eight annealedNTD samples at longer wavelengths indicate anothernew line present only in NTD samples at about 9450 A.The intensity of this line increases with increasing NTDdose.

The characteristic lifetimes of radioactive isotopescan be used to label and identify defect levels in semi-conductors which can be detected by photolumines-cence [273] and Raman-scattering spectroscopy [274].Magerle et al. [273] show photoluminescence spectraof GaAs doped with 111In that decays to 111Cd. 111In isisoelectronic to Ga and hence occupies Ga lattice sitesin GaAs. It decays to 111Cd with a lifetime τ111In = 98 hby electron capture [279]. Since the recoil energy of the

3388

Figure 61 Photoluminescence spectra of undoped and 111In doped GaAs successively taken 4 h; 7 h; 12 h; 22 h; 2 d; 4 d and 9 d after doping. Allspectra are normalized to the intensity of the (e, C) peak. In the inset, the height ICd/IC of the (e, Cd) peak in these spectra is shown as a function oftime after doping with 111In. The solid line is a fit to the data using Equation 139 (after [273]).

Cd nucleus due to the emission of the neutrino is muchsmaller than the typical displacement energy in GaAs[280], 111Cd atoms on Ga sites (CdGa) are created bythe decay of 111In on Ga sites (111InGa) and act thereas shallow acceptors. This chemical transmutation wasmonitored by photoluminescence spectroscopy. Fig. 61shows successively taken photolumunescence spectrafrom the 111In doped sample. A spectrum from theundoped part is also shown. The photoluminescencespectrum of the undoped part of the sample shows thefeatures well known for undoped MBE-grown GaAs[281]. The peaks FX and AX around 819 nm are due tothe recombination of free and bound excitons. The peak(e,C) at 830 nm and its LO phonon replica (e,C)-LO at850 nm are due to recombination of electrons from theconduction band into C acceptor states. The recombi-nation of electrons from donor states into C acceptorstates appears as a small shoulder at the right-hand sidesof either of these two peaks. C is a residual impurityin GaAs present in MBE-grown material with a typi-cal concentration between 1014 and 1015 cm−3 [281].Magerle et al. determined the height ICd/IC of the(e, Cd) peak normalized to IC as the function of timeafter doping. These was done by substracting the nor-malized spectrum of the undoped part from the normal-ized spectra of the 111In doped part. The height ICd/ICof the (e,Cd) peak remaining in these difference spectrais displayed in the insert of Fig. 61. Indicated authorsfitted these data by

ICd

IC(t) = ICd

IC(t =∞)

(1 − e− t

)(132)

and obtained a time constant τ = 52(17) h, which isnot the nuclear lifetime τ111In = 98 h of 111In. EvidentlyICd/IC is not proportional to NCd. The photolumines-cence intensity ICd is proportianal to the recombina-tion rate of excess carriers per unit area through Cdacceptors states �nL BCd NCd, where BCd is a recom-bination coefficient. The excess sheet carrier concen-tration in the implanted layer �nL can be expressed interms of the total carrier lifetime in the implanted layer

τL and the generation rate of excess carriers per unitarea in the implanted layer fLG by using the first of thetwo equilibrium conditions

fLG = �nL

τLand fBG = �nB

τB(133)

The second one describes the balabce between thegeneration rate fBG and the recombination rate of ex-cess carriers �nB

τBin the bulk. The total generation rate

G is proportional to the incident photon flux and fL +fB = 1. To get an expression for τL, cited authors as-sumed two additional recombination processes in theimplanted layer: the radiative recombination via Cd ac-ceptors and nonradiative recombination due to residualimplantation damage, and write the recombination ratein the small single approximation (see, e.g. [282]) as

�nL

τL= �nL

τB+ �nL BCd NCd + �nL Bnr fnr NCd (134)

Here �nL Bnr fnr NCd is the nonradiative recombina-tion rate per unit area due to residual implantation dam-age, fnr NCd is the concentration of these nonradiativerecombination centers, and Bnr is the corresponding re-combination coefficient. Hence �nL and �nB can beexpressed as a function of NCd and the recombinationrates through all the different recombination channelsand thereby the relative photoluminescence peak inten-sities can be deduced. IC is proportional to the sum ofthe (e,C) recombination rates per unit area in the im-planted layer and the bulk and within this model it canobtain

IC ∝ �nL+�nB

τC= G

τB

τC

(fL

1 + �Cd/ f Bb+ fB

). (135)

Here �Cd is the dose between 109 and 1013 cm2.Thereby τC = 1/BC NC is an effective lifetime describ-ing the recombination probability through C acceptorstates and b is a constant defined below. With help ofEquations 133 and 135 it can be obtain (assuming that

3389

the detection efficiencies of both peaks are equal) thefollowing relation between ICd/IC and �Cd:

ICd

IC= �nL BCd NCd

(�nL + �nB)/τC= a

1 + b/�Cd, (136)

with

a = fL

BCd

(Bnr fnr + BCd)

τC

τB

and

(137)b = d

fB(Bnr fnr + BCd)τB.

This model describes quantitatively the dependenceof (e, Cd) intensity of NCd and cited authors use it todescribe the increase of ICd/IC with time in the 111In-doped sample. Authors [273] model the change of thecarrier lifetime τL with time t in the 111In doped sampleas

τL= 1

τB+ BCd NIn

(1 − e− t

)Bnr fnr, (138)

where NIn = �In/d is the initial 111In concentration, τ =τ111In = 98.0 h is the nuclear lifetime of 111In, and BCd,Bnr and fnr are the same constants as above. Thereby weassume following Magerle et al. that the same kinds ofnonradiative recombination centers are produced by Indoping as by Cd doping and that the Cd concentrationare identical to the 111In concentration profile. Takinginto account all above saying we can write

ICd

IC= a

1 + b/�In(1 − e− t

) + c/(

e− tτ −1

) , (139)

where a and b are the same constants as above andc = Bnr fnr/(Bnr fnr + BCd). This c term accounts forthe fact that the 111In doped sample the concentration ofnonradiative centers is not changing with Cd concen-tration. Magerle et al. fitted Equation 139 to the datashown in inset of Fig. 61, keeping τ = 98.0 h, a = 1.25and b = 3.0 × 1011 cm−2, and obtained �In = 4.49 ×1011 cm−2 and c = 0.5 (2). This fit is shown as a solidline and agrees perfectly with the experimental data. Inconclusion of this part it should note that this identifica-tion technique is applicable to a large variety of defectlevels since for most elements suitable radioactive iso-tope exist (details see [279]).

Coupling between the LO phonon mode and the lon-gitudinal plasma mode in NTD semi-insulating GaAswas studied in paper [274] using Raman-scatteringspectroscopy and a Fourier-transform infrared spec-trometer. Raman spectra are shown in Fig. 62 for unir-radiated, as-irradiated and annealed samples. The re-markable feature is the low intensity and asymmetriclinewidth of the Lo-phonon spectrum observed in an-nealed samples, which are annealed above 600◦C. Thebehavior is not understood by considering the only LOphonon. We should pay attention to the electrical acti-vation of NTD impurities, which begin to activate elec-trically around 600◦C. In the long-wavelength limit, thevalence electrons, the polar lattice vibrations, and the

Figure 62 Raman spectra at room temperature taken for the variousannealing temperatures of (100)—oriented NTD GaAs irradiated withneutron dozes (athermal neutron of 1.5 × 1018 cm−2 and a fast neutronof 7 × 1017 cm−2). The coupling L+ mode is observed at annealingtemperatures above 600◦C (see Table XVII) (after [274]).

conduction electrons make additive contributions to thetotal dielectric response function [283]:

εT(0, ω) = ε∞ + (ε0−ε∞)/[(

1−ω2/ω2t

) − ω2pε∞/ω2].

(140)

The high-frequency value (L+) of the mixed LO-phonon-plasmon modes is calculated from the roots ofthe dielectric constant of Equation 132. The frequen-cies of the L+ mod e and of the longitudinal plasmamode ωp = (4πne2/ε∞m∗)1/2 for various annealingtemperatures are listed in Table XVII. Here n is the elec-tron concentration, m∗ the effective mass in the conduc-tion band (= 0.07m0), and ε∞ (= 11.3) the optical di-electric constant. The mixed LO-phonon-plasma modeappears around 300 cm−1 for electron concentrationof (0.8–2) × 1017 cm−3. The phonon strength [283]

TABLE XVII Electron concentrations and the coupling modes ofNTD GaAs (after [274])

LO-phononfrequency L+ mode PF

Sample EC (cm−3) (cm−1) (cm−1) (cm−1)

Unirradiated 1 ∼ 2 × 107 296.6As-irradiated a 295.6500◦C annealed a 297.8600◦C annealed 8.2 × 1016 296.0 299 96.4650◦C annealed 2.2 × 1017 296.6 304 158700◦C annealed 2.5 × 1017 296.2 305 168

aSince the conduction is dominated by Mott-type hopping conduction(M. Satoh and K. Kuriyama, Phys. Rev. B40, 3473 (1989), the electronconcentration can not be measured by the van der Pauw method.EC = Electron concentration; PF = Plasma frequency.

3390

Figure 63 Infrared-absorption spectra at room temperature taken for thevarious annealing temperatures of the NTD GaAs used for the Raman-scattering esperiments (after [274]).

for the high-frequency mode (L+) of the interactingplasmon-LO-phonon mode is about 0.95 for an elec-tron concentration of 1 × 1017 cm−3, while that for thelow-frequency mode (L−) is below 0.1. Therefore, theasymmetric linewidth of the Raman spectrum observedin the annealed NTD GaAs arises from both the LO-phonon and L+ modes, but the L− mode is not ob-served because of a very weak phonon strength. As a re-sult, the LO-phonon intensity decreases with increasingcoupling, and L+ mode appears beside the LO-phononpeak.

The absorption spectra in the various annealing tem-peratures for NTD GaAs are shown in Fig. 63. In unir-radiated samples, an absorption around 2350 cm−1 isassigned as the antisymmetric stretching vibration ofCO2 arising from CO2 in an ambient atmosphere.Theabsorption peaks observed around 500 cm−1 are alsoassigned as a two-phonon overtone scattering [284]of transverse optical phonons (TO); these were ob-served at 493 cm−1 [2TO(X)], 508 cm−1 [2TO(L)], and524 cm−1 [2TO()], respectively. In as-irradiated sam-ples, a continuous absorption extending to the higherenergy was observed. Although this origin cannot be at-tributed to interstitial anion clusters as discussed in neu-tron irradiated GaP [285]. In samples annealed above600◦C, the remarkable absorption was observed at wavenumbers below 1450 cm−1. The absorption increaseswith increasing annealing temperature (see Fig. 109).This behavior arises from the fre-electron absorptiondue to the activation of NTD impurities, which occur atannealing temperatures above 600◦C. The free-electronabsorption observed is consistent with a collective mo-tion as a plasmon mode described in Raman-scatteringstudies.

Kuriyama et al. [276] were studied by a photolu-minescence method the transmuted impurities Ge an

S in NTD semi-insulating Gap. In NTD GaP, Ge andS impurities are transmuted from Ga and P atoms by(n, γ ) reactions, respectively. Ge in GaP is an ampho-teric impurity for which both the donor and acceptorstates appear to be deep. The ratio between transmutedimpurities Ge and S is about 16:1. Unfortunately, afterthe transmutation reactions, the transmuted atoms areusually not in their original positions but displaced intointerstitial positions due to the recoil produced by the γ

and β particles in the nuclear reactions. In addition, thedefects induced by the fast neutron irradiation disturbthe electrical activation of transmuted impurities. How-ever, Frenkel type defects [275, 286] in NTD GaP wereannealed out between 200 and 300◦C, while P antisite(PGa) defects of ∼ 1018 cm−3 annihilated at annealingtemperatures between 600 and 650◦C. Therefore, trans-muted impurities, Ge and S, would be substituted on Gaand/or P lattice sites by annealing at around 650◦C.

Fig. 64 shows the photoluminescence (PL) spectra ofunirradiated and NTD GaP. The PL spectrum (peak 1)of unirradiated samples shows signature of the DA pairrecombination involving S donor and carbon acceptor[288]. Two (peaks 2 and 3) of the replicas occur at ener-gies consistent with electronic transitions accompaniedby zone-center optical phonons with energies 50.1 meV(LO) and 100.2 (2LO). Sulfur, silicon and carbonin GaP are the most common as the residual impuri-ties [288]. In NTD-GaP the main transition energy wasobserved 1.65 eV. Since Ge in GaP is the amphotericimpurity with deep acceptor and donor levels, strongphonon co-operation will also occur. But optical transi-tion rates will be significant only for associates. Similarsituation has been proposed for Si in GaP [287], form-ing a nearest-neighbor SiGa-SiP complex. Therefore,the broad emission would be expected to arise from

Figure 64 Photoluminescence (PL) spectra taken at 15 K for unirradi-ated and NTD-GaP. PL peaks 1, 2 and 3 in unirradiated GaP representSp − Cp DA pair recombination, its LO—phonon replica, and 2LO—phonon replica, respectively. 1.65 and 1.87 eV emissions in NTD-GaPare attributed to GeGa − GeP complex and SP − GeP DA pair recombi-nations, respectively (after [276]).

3391

Figure 65 Variation of the half-width W with the square root of thetemperature T for the 1.65 eV band in NTD-GaP. Theoretical curve is aplot of Equation 141 with hν = 0.025 eV (after [276]).

a nearest-neighbor GeGa-GeP coupled strongly to thelattice. To confirm the presence of the GeGa-GeP com-pose, the temperature dependence of the half-width, W ,of the broad emission was measured. If the localizedelectron transitions from the excited state to the groundstate of this complex center produce the characteris-tics luminescence, the dependence would be appear tofollow the configuration-coordinate (CC) [289] modelequation:

W = A[coth(hν/2kT )]1/2, (141)

where A is a constant whose value is equal to W asthe temperature approaches 0 K and hν is the energyof the vibrational mode of the excited state. In Fig. 65,Equation 141 has been fitted to the experimental valuefor NTD-GaP. For the estimation of W , the spectrumof the 1.65 eV band was substracted from that of the1.87 eV band. The value of hν used was 0.025 eV.The good fit to this equation that was found for theGeGa-GeP center in NTD-GaP shows the validity ofapplying the CC model. Results of paper [276] indicatethat NTD method is a useful one for introducing Gedonor, resulting from a fact the Ge atoms are transmutedfrom Ga lattice sites in GaP. The obtained results areconsistent with the presence of the GeGa-GeP complexas described earlier.

Chapter 3. Optical fiber3.1. IntroductionOptical communication using fibers is a major newtechnology which will profoundly impact telephonesystems, computer interconnections and instrumenta-tion (internet). Fiber links provide several major ad-vantages over conventional electronic communicationssystems. These include immunity to electromagneticinterference, thinner and lighter cables, lower transmis-sion losses (especially for very data rates) and potentialkilometer-long link capabilities extending to the giga-hertz region.

An optical waveguide is a dielectric structure thattransports energy at wavelengths in the infrared or vis-ible ranges [290, 291] of the electromagnetic spec-trum. In practice, waveguides used for optical commu-nications are highly flexible fibers composed of nearly

Figure 66 Nomenclature, profiles and ranges of dimensions for typicaloptical fibers, where ρ is the core radius, λ is the free-space wavelengthof light and � = (1 − n2

cl/n2co)/2 (after [292]).

transparent dielectric materials. The cross-section ofthese fibers is small—comparable to the thick of a hu-man hair- and generally is divisible into three layers asshown in Fig. 66. The central region is the core, which issurrounded by the cladding, which in turn is surroundedby a protective jacket. Within the core, the refractive-index profile n can be uniform or graded, while thecladding index is typically uniform [293]. The two sit-uation correspond to the step-index and graded-indexprofiles shown in the insets in Fig. 66. It is necessarythat the core index be greater than the cladding index[294], at least in some region of the cross-section, ifguidance is to take place. For the majority of applica-tions, most of the light energy propagates in the coreand only a small fraction travels in the cladding. Thejacket is almost optically isolated from the core, so forthis reason we usually ignore its effect and assume anunbounded cladding for simplicity in the analysis.

As usually, optical waveguides can be convenientlydivided into two subclasses called multimode waveg-uides (with comparatively large cores) and single-mode waveguides (with comparatively small cores).The demarcation between the two is below. Multi-mode waveguides obey the condition (see e.g. [295])(2πρ/λ)(n2

co−n2cl)

1/2 � 1, where ρ is a linear dimen-sion in the core, e.g., the radius of the fiber core, λ isthe wavelength of light in free space, nco is the maxi-mum refractive index in the core and ncl is the uniformrefractive index in the cladding.

As will be shown below electromagnetic propaga-tion along optical waveguides is described exactly byMaxwell’s equations. However, it is well-known thatclassical geometric optics provides an approximate de-scription of light propagation in regions where the re-fractive index varies only slightly over a distance com-parable to the wavelength of light. This is typical ofmultimode optical waveguides used for communica-tion. Thus, the most direct and conceptually simple wayto describe light propagation in multimode waveguidesis by tracing rays along the core (see also [296, 297]). By

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using classical geometric optics, we should ignored allwave effects. In multimode waveguides, wave effectsare usually negligible [292], but there are exceptionalsituations when such effects accumulate exponentiallywith the distance light travels. Naturally in this cases,wave effects must be retained, since they can have asignificant influence in long waveguides. In each suchsituation, we modify the classical geometric optics de-scription by taking into account the local plane wavenature of light. The phenomenon of greatest practicalinterest in fibers used for long-distance communica-tions is the spread of pulses as they propagate along thefiber. For idealized multimode fibers, pulse spreadingis easily described by classical geometric optics. Butsince propagation in multimode guides is so complexthat simple models and physical understanding are gen-erally of much greater assistance than a precise, exactanalysis [298].

3.2. Maxwell’s equationsThe spatial dependence of the electric field �E(x, y, z)and the magnetic field �H (x, y, z) of an optical waveg-uide is determined by Maxwell’s equations. Further weassume an implicit time dependence exp(−iωt) in thefield vectors, current density �J and charge density σ .The dielectric constant ε(x, y, z) is related to the refrac-tive index n(x, y, z) by ε = n2ε0, where ε0 is the dielec-tric constant of free space. For the nonmagnetic mate-rials which normally constitute an optical waveguide,the magnetic permeability µ is very nearly equal to thefree-space value µ0. Under these conditions, Maxwell’sequations are expressible in the form (see, e.g. [299])

�∇ × �E = −∂ �B∂t

= −µµ0∂ �H∂t

(142)

�∇ × �H = ∂ �D∂t

= εε0∂ �E∂t

(143)

�D = εε0 �E (144)

�B = µµ0 �H (145)

�∇ · �D = 0 (no free charges) (146)

�∇ · �B = 0 (no free poles). (147)

We will assume further that our guiding structurealong the z direction. Thus we look for solutions toproblems in which the z dependence of the field is ofthe form

�E = �E0(x, y) exp i(ωt − zk · z) (148)

�H = �H 0(x, y) exp i(ωt − zk · z) (149)

In this expression we note that ω is related to ν, thefrequency, by the relation ω = 2πν. Furthermore, inwriting the time and space variation of the field in termsof the complex exponential, it is understood, but notwritten, throughout this chapter that when a field is tobe evaluated (e.g., for measurement) the only physicalmeanengful part of this complex expression is the real

part. Thus, if we denote a real measurable field by ε

and we wish to relate this to the theoretically derivedfield �E , which is given by an expression of the form ofEquation 148, then

ε = Re[ �E]. (150)

The advantage of this approach is that expressionsinvolving the exponential of a complex quantity aremore readily manipulated than the equivalent expres-sions involving sine and cosine. In next we will de-rive expressions for the field components in planar andcylindrical geometries before studying the solutions forguided waves in planar and cylindrical waveguides (seealso [298]).

3.2.1. Planar geometryThe expression �∇x in rectangular Cartesian coordinatesis

�∇ × �A =

∣∣∣∣∣∣∣∣�i �j �k

∂

∂x

∂

∂y

∂

∂zxAyAzA

∣∣∣∣∣∣∣∣, (151)

where �i , �j , �k are unit vectors in the x, y, z directions.Thus from Equation 142 we obtain for the x directedcomponent of �∇x �E

(∂z E

∂y−∂y E

∂z

)= −µµ0

∂x Hx

∂t. (152)

Now we can write similar equations for ∂y H/

∂t ; ∂z H/∂t ; ∂x E/∂t ; ∂y E/∂t and ∂z E/∂t . If then sub-stitute the field expressions of Equations 148 and149 into these expressions, we obtain the followingresults:

∂z E

∂y+ i zky E = −iµµ0ω

x H, (153)

i zkx E + ∂z E

∂x= iµµ0ω

y H, (154)

∂y E

∂x− ∂x E

∂y= −iµµ0ω

z H, (155)

∂z H

∂y+ i zky H = −iεε0ω

x E, (156)

−i zkx H − ∂z H

∂x= iεε0ω

y E, (157)

∂y H

∂x− ∂x H

∂y= iεε0ω

z E . (158)

Analysis of planar structures will be restricted to in-finite films that lie in the y–z plane. Thus, in additionto the assumption that the fields have the z dependencealready postulated, it can further assume that the partialderivative with respect to y vanishes (hereafter ∂/∂y =0) for an infinite plane wave traveling in the z direction.With this assumption the above equations simplify and

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demonstrate a fundamental relationship about the fieldsin such a structure. Indicated relations have forms:

i zky E = −iµµ0ωx H (TE group), (159)

i zkx E + ∂z E

∂x= iµµ0ω

y H (TM group), (160)

∂y E

∂x= −iµµ0ω

z H (TE group), (161)

i zky H = iεε0ωx E (TM group), (162)

−i zkx H − ∂z H

∂x= iεε0ω

y E (TE group), (163)

∂y H

∂x= iεε0ω

z E (TM group). (164)

We can see that the fields have now split into twoseparate groups, namely, y E , x H and z H are coupledand y H , x E and z E are also coupled. The guided wavesformed by the first group are described as TE modes(for transverse electric), and the latter are known as TMmodes (for transverse magnetic). We can now use theabove relations to derive simpler expressions for thetransverse field components in terms of the z E and z Hcomponents only e.g., eliminate y H between Equations160, 162 and 164 to obtain a relation for x E . This yeildsthe following

x E =( −i zk

εε0µµ0ω2 − zk2

)∂z E

∂x, (165)

y E =( −iωµµ0

εε0µµ0ω2 − zk2

)∂z H

∂x, (166)

x H =( −i zk

εε0µµ0ω2 − zk2

)∂z H

∂x, (167)

y H =( −iωεε0

εε0µµ0ω2 − zk2

)∂z E

∂x, (168)

Substituting these expressions into Equations 159 to164 yields two wave equations for propagation in the xdirection:

∂2z E

∂x2− (ω2εε0µµ0 − zk2)z E = 0, (169)

∂2z H

∂x2− (ω2εε0µµ0 − zk2)z H = 0. (170)

These indicate that for the transverse dependenceof the fields we should seek solutions of the formexp(ixkx), where

xk2 = (ω2εε0µµ0 −z k2) = −(xγ )2. (171)

The significance of the variable xγ introduced herewill become apparent later. Notice that Equation 171could have been obtained much more straightforwardlyby deriving the wave equation directly from Equations142 to 147, setting ∂/∂y = 0, and substituting thefield Equations 148–149. However, that route wouldnot have yielded detailed interrelationship between thevector components of the field that we will need for

finding the conditions for guided waves. In conclusion,derived relationships between the vector components ofthe fields for a planar structure lying in the y–z plane,with a wave propagating in the z direction are sum-marized in Equations 159 to 164. In addition we haveshown that fields in such a structure take the generalform

�E = �E(x) exp i(ωt − zkz ± xkx), (172)

�H = �H (x) exp i(ωt − zkz ± xkx), (173)

xk2 = ω2εε0µµ0 − zk2, (174)

n2k20 = ω2εε0µµ0. (175)

3.2.2. Cylindrical geometryWe now repeat the analysis of Section 3.2.1 but in cylin-drical polar coordinates, since these are more appropri-ate to the analysis of optical fiber guides. The coordi-nates x , y and z are now replaced by r , φ and z. Theseare related to the coordinates as follows:

x = r cos φ, (176)

y = r sin φ, (177)

z = z. (178)

Since we are still concerned with a structure that isexpected to guide waves in the z direction, we shouldpostulated fields of the form

�E = �E(r, φ) exp i(ωt − zkz), (179)

�H = �H (r, φ) exp i(ωt − zkz). (180)

The relation for �∇ × �A in polar coordinates is asfollows:

�∇ × �A =

∣∣∣∣∣∣∣�rr φ

�kr

∂∂r

∂∂φ

∂∂z

r ArφAzA

∣∣∣∣∣∣∣ , (181)

where �k is the unit vector in the z direction. We will nowderive the expressions for the field components by useof Maxwell Equations 142–147. We obtain the set of re-lations equivalent to equations in Cartesian coordinatesfor the planar case:

[∂z E

∂φ+ i zk(rφ E)

]= −iµµ0ω

r H, (182)

i zkr E + ∂z E

∂r= iµµ0ω

φ H, (183)

[r∂φ E

∂r− ∂ r E

∂φ

]= −iµµ0ω

z H, (184)

[∂z H

∂φ+ i zk(rφ H )

]= iεε0ω

r E, (185)

[r∂φ H

∂r− ∂ r H

∂φ

]= iεε0ω

z E . (186)

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Solving these system of equations, we obtain expres-sions for the r and φ components only in terms of thez components:

r E = −i

Tk2

k∂z E

∂r+ µµ0ω

∂z H

∂φ

], (187)

φ E = i

Tk2

[ zk

∂z E

∂φ− µµ0ω

∂z H

∂r

], (188)

r H = −i

Tk2

k∂z H

∂r− εε0ω

∂z E

∂φ

], (189)

φ H = −i

Tk2

[ zk

∂z H

∂φ+ εε0ω

∂z E

∂r

], (190)

Tk2 = ω2εε0µµ0 − zk2 = n2k20 − zk2. (191)

Here Tk is the total transverse component of the �k inthe waveguide.

3.2.3. The electromagnetic wave equationBelow we derive the standard derivation of the waveequation. Moreover, we give the form of the Lapla-cian operator for rectangular Cartesian and polar coor-dinates. If we take the curl of the first Maxwell equation,then we obtain:

�∇ × ( �∇ × �E) = −µµ0

[�∇ × ∂ �H

∂t

]. (193)

Differentiating Equation 143 with respect to timeyields

�∇ × ∂ �H∂t

= εε0∂2 �E∂t2

. (194)

We then make use of the vector identity

�∇ × ( �∇ × �E) = −�∇( �∇ × �E) − �∇2 �E = −�∇2 �E (195)

since �∇ · �E = 0 (see also Equation 146). Then it followsdirectly by substitution that

�∇2 �E = εε0µµ0∂2 �E∂t2

(196)

and likewise

�∇2 �H = εε0µµ0∂2 �H∂t2

. (197)

These equations are both of the general form

�∇2 �A = 1

V 2

∂2 �A∂t2

, (198)

where V is the velocity of propagation (phase velocity)of the wave in the medium. It follows as usually that

Vp = 1√εε0µµ0

(199)

and that, for free space, we have the velocity of light,c, given by

c = 1√ε0µ0

. (200)

For planar waveguides, described by rectangularCartesian coordinates, or circular fibers, described bycylindrical polar coordinates, the Laplacian operatorhas the forms

∇2 A = ∂2 A

∂x2+ ∂2 A

∂y2+ ∂2 A

∂z2, (201)

∇2 A = 1

∂

∂r

(r∂ A

∂r

)+ 1

∂2 A

∂φ2+ ∂2 A

∂z2. (202)

We should stressed here that the modes considered infiber optics are exact solution of Maxwell’s equations(details see [292–294]).

3.3. Geometrical optics of fibersAs was noted above the mechanism of light propagationalong fibers as small as a few wavelength in diametercan be understood almost entirely using the ray the-ory and well-known principles of geometrical optics.It is interesting to note that, even for smaller-diameterdielectric cylinders, which act as waveguides, the geo-metrical optical theory, with some modification, helpsin the understanding of the complex mechanisms. Webegin ray analysis of multimode optical waveguideswith the planar, or slab waveguide, which is the sim-plest dielectric structure for illustrating the principlesinvolved, and has application in integrated optics [290,291]. Since we can analyze its light transmission char-acteristics in terms of a superposition of ray paths, itis important to fully appreciate the behavior of individ-ual rays. We will study the trajectories of rays withinplanar waveguides, concentrating on those rays—thebound rays—which propagate without loss of energyon a nonabsorbing waveguide, and can, therefore prop-agate arbitrarily large distances. The planar, or slab,waveguide is illustrated in Fig. 67. It consists typicallyof a core layer of thickness 2ρ sandwiched between twolayers which form the cladding. As explained in the in-troduction, we assume, for simplicity, that the claddingis unbounded The planes x = ±ρ are the core-claddinginterfaces. Since the waveguide extends indefinitely inall directions orthogonal to the x-axis, the problem istwo dimensional [300]. The z-axis is located along the

Figure 67 Nomenclature and coordinares for describing planar wave-guides. A representative graded profile varies over the core and is uniformover cladding, assumed unbounded (after [292]).

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axis of the waveguide midway between interfaces. Therefractive-index profile n(x) in Fig. 67 can be uniformor graded across the core, and assumes uniform valuencl in the cladding. It is necessary that the core refrac-tive index make some values greater than ncl for thewaveguide to have guidance properties. Furthermore,we assume that the profile does not vary with z, so thatthe waveguide is translationally invariant, or cylindri-cally symmetric. The parameters defined in Fig. 67 canbe combined with the free-space wavelength λ of thelight propagating along the waveguide to form a singledimensionless parameter V , known as the waveguideparameter, or waveguide frequency. If nco is the max-imum value of n(x), which need not conduct with theon-axis value n(0), then it will define

V = 2πρ

(n2

co−n2cl

)1/2. (203)

The ray theory considered here is restricted to mul-timode waveguides i.e., waveguides satisfying V � 1.The step-index planar waveguide, according to Fig. 68,has the refractive-index profile defined by

n(x) = nco, −ρ < x < ρ;(204)

n(x) = ncl, |x | > ρ,

where nco and ncl are constants and nco > ncl. One ofthe most important problems is to determine the con-ditions necessary for a ray to be bound, i.e., the raypropagates along the nonabsorbing waveguide withoutloss of power.

Propagating within the uniform core of the step-indexwaveguide of Fig. 68 is along straight lines. If a rayoriginates at P on one interface and makes angle θzwith the waveguide axis, it will meet the opposite in-terface at Q as shown in Fig. 68. The situation at Qin more details is pictured in Fig. 69 reflection in thissituation is governed by Snell’s law [293, 300]. While

Figure 68 Propagation along a straight line between interfaces in thecore of a step-profile planar waveguide (after [292]).

Figure 69 Reflection at a planar interface unbounded regions of re-fractive indices nco and ncl, showing (a) total internal reflection and(b) partial reflection and refraction (after [292]).

these laws are usually expressed in terms of angles rel-ative to the normal QN, we following [292] prefer toretain the complementary angle θz. Thus, in terms ofcomplementary angles, the incident ray at Q is totallyinternally reflected if 0 % θz ≺ θc, and is partly refractedif θc ≺ θz % π/2, where θc is the complement of thecritical angle, defined by [292]

θc = cos−1[

ncl

nco

]= sin−1

[1− n2

n2co

]1/2

. (205)

In the first case, Fig. 115a shows the reflected rayleaving the interface at the same angle θz as the incidentray, while in the second case (see Fig. 69b) shows thatthe ray bifurcates, part of it being reflected at angleθz and part of it being transmitted into the cladding atangle θt to the interface, which satisfies Snell’s law

nco cos θz = ncl cos θt. (206)

Only total internal reflection returns all the ray power,i.e., the energy flowing along the ray, back into the coremedium. A ray is reflected from the interface back intothe core at angle θz regardless of whether partial or totalreflection occur. If we repeat this procedure at succes-sive reflections from the interfaces (see Fig. 70), in suchway we construct the zig-zag paths, or trajectories. Pathdepicted in Fig 116a is a ray that is totally reflected atevery reflection. We refer to this as a bound ray, sinceits path is entirely confined within the core. Path (seeFig. 70b) is for a ray that is partly reflected at each reflec-tion. We refer to this as a refracting ray. The rays maybe categorized by the value of θz according to [292]

Bound rays : 0 % θz ≺ θc, (207a)

Refracting rays : θc % θz % π/2. (207b)

Since the power of a bound ray is totally reflectedback into the core at every reflection, the ray can prop-agate indefinitely without any loss of power. A refract-ing ray loses a fraction of its power at each reflectionand therefore attenuates as it propagates.

As was shown in whole raw of textbooks (see, e.g.[292–298]), it is useful to introduce parameters thatcharacterize ray propagation, as it is these parameters,rather than the spatial dependence of the ray path, thatare important. As can be seen from Fig. 71 the raytrajectory is fully characterized once the angle θz is

Figure 70 Zig-zag paths within the core of a step-profile planar waveg-uide for (a) bound rays and (b) refracting rays (after [292]).

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Figure 71 Path length and ray half-period zp for a ray in the core of astep-profile planar waveguide (after [292]).

prescribed. We define the path length Lp between suc-cessive reflections to be the distance PQ. Accordinggeometry picture we have

Lp = 2ρ

sin θz= 2ρnco(

n2co−β2

)1/2 , (208)

where β = nco cos θz = ncl cos θt is a ray invariant. Forthe ray transit time below, we require the optical pathlength L0. In a hom*ogeneous medium this is given bythe product of path length and refractive index:

L0 = ncoLp = 2ρnco

sin θz= 2ρn2

co(n2

co−β2)1/2 , (209)

A quantity which will appear frequently in attenu-ation problems is the ray half-period zp. This is dis-tance between successive reflections , mesured alongthe waveguide axis:

zp = 2ρ

tan θz= Lp cos θz = 2ρβ(

n2co−β2

)1/2 . (210)

Closely related is the number of reflection N per unitlength of the waveguide, which is the reciprocal of theray half-period. Hence

N = 1

zp= tan θz

2ρ.. (211)

It is clear from these definitions that over arbitrarydistance z along the waveguide, the accumulated pathlength, optical path length and number of reflectionsare given proportionally by

zpLp;

zpL0; Nz = z

zp, (212)

respectively. These parameters are indicated inTable I-1 of [292]).

The most important quantity required to describepulse spreading is the ray transit time t . This is the timea ray takes to propagate distance z along the waveguide,following the the zig-zag ray path (see also Fig. 70). Thevelocity of light vg in fiber along the path is given by[300]

vg = c/nco, (213)

where c is the fre-space speed of light. The transit timedescribes the following relation:

t = z

vg= zL0

zpc= zn2

cβ= znco

c cos θz, (214)

so that the greater θz, the longer the transit time.We can account for material dispersion, which occurs

when the refractive index varies with the wavelength oflight λ, i.e., nco = nco (λ). This requires more sophisti-cated reasoning relying on treating a ray as if it were aplane wave in local regions. Ray energy propagates atthe group velocity vg, which is given by Equation 213in a dispersionless medium. but, allowing for materialdispersion, it has the more general form [299]

vg = c

[nco(λ)−λ

dnco(λ)

dλ

]−1

. (215)

It is convenient to introduce the group index ng, de-scribed by

ng = nco(λ) − λdnco (λ)

dλ, (216)

in which case, the transit time is expressible as

t = zng

c cos θz= zngnco

cβ, (217)

and varies with both θz and λ.Early we establish the basic concepts for the ray anal-

ysis of planar waveguides. Now we extend the analysisto optical fibers, which are used for high-capacity com-munication over long distances. As far as ray tracingis concerned, the only difference between fibers andplanar waveguides is the introduction of the third di-mension. Thus, although the ray concepts are the sameas early, the analysis and resulting expression are gen-erally more complicated because of the fiber geometry[297]. Nevertheless, one of the important results of fiberoptics shows that the ray transit times for step and cladpower-law profile fibers of both circular and noncir-cular are identical to those of the corresponding planarwaveguides. If this remarkable simplification is accept-able without proof, then pulse spreading in such fiberscan be studied directly. An optical fiber is illustrated inFig. 72. Unless otherwise stated, the core is assumed

Figure 72 Nomenclature for describing circular fibers. Cartesian coor-dinates x , y, z and cylindrical coordinates r , φ, z are oriented so thatthe z-axis lies along the fiber axis. A representative graded profile variesover the core and is uniform over the cladding, assumed unbounded (after[292]).

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to have a circularly symmetric cross-section of radiusρ, surrounded by the cladding, which, for simplicityis assumed unbounded. The core-cladding interface isthe cylindrical surface r = ρ. Over the core, the ax-isymmetric refractive-index profile n(r ) is either uni-form or graded, and it takes the uniform value ncl inthe cladding. The dimensionless parameter V of Equa-tion 203 also applies to fibers, and will br referred asthe fiber parameter, where ρ is the core radius. Thequantity (n2

co − n2cl)

1/2 is often referred to as the nu-merical aperture of the fiber, while a related expression[n2

co(r )−n2cl]

1/2 is sometimes called the local numericalaperture (details see [297, 298]).

3.4. Waveguide mode propagationIn a light pipe electromagnetic energy is propagateddown the pipe by reflection from the walls of the struc-ture. If the transverse dimensions are comparable tothe wavelength of the light only certain field distribu-tions (modes) will satisfy Maxwells equations and theboundary conditions. In this case the light pipe is moreappropriately considered as a waveguide. Even in verylarge structures there are so many of them, their numberincreasing as the area, that in most cases a geometricaloptics description is more fruitful (see above).

As is well-known, the distinction between metallicand dielectric waveguides is in the reflection mecha-nism responsible for confining the energy. The metallicguide does so by reflection from a good conductor atthe boundary. In the dielectric waveguide, this is ac-complished by total internal reflection, which is gottenby having the central dielectric made of a material ofhigher index of refraction than the surrounding dielec-tric. The two regions will henceforth be referred to asthe core and cladding. In a metallic guide there aretwo sets of solutions, the transverse electric and trans-verse magnetic modes. In the dielectric guide all butthe cylindrically symmetric modes TE0m and TM0m arehybrid, i.e., they have both electric and magnetic z com-ponents [301–303]. In general, one would expect twosets of such hybrid modes, because the boundary condi-tions give a characteristic equation which is quadratic inthe Bessel functions describing the field in the centraldielectric (details see [293, 297, 298] and referencestherein).

The cylindrical dielectric waveguide consists of acore of high refractive index nco and radius a surroundedby a cladding of lower refractive index ncl. Let thecladding material of index ncl extend to infinity. Weshall use both cartesian (x, y) and cylindrical polar co-ordinates (r , φ). The propagation constant β of anymode of this fiber is limited within the interval nco ≥β ≥ nclk, where k = 2π/λ is the wavenumber in freespace. If we define parameters

u = a(k2n2

co − β)1/2

, (218)

w = a(β − k2n2

)1/2, (219)

the mode field can be expressed by Bessel functionJ (ur/a) inside the core and modified Hankel function

K (wr/a) outside the core [293, 302]. The quadraticsummation

v2 = u2 + w2 (220)

leads to a third parameter

v = ak(n2

co − n2cl

)1/2, (221)

which can be considered as a normalized frequency. Bymatching the fields at the core-cladding interface, weobtain characteristic functions u(v) or w(v) for everymode; the propagation constant and all other parametersof interest can be derived from these functions. Forweak guidance, we have

� = (nco − ncl)/ncl � 1. (222)

In this case, we can construct modes whose trans-verse field is essentially polarized in one direction. Be-low we postulate transverse field components (see also[297])

y E = z H = Z0/nco

= El[Jl(ur/a)/J l(u)] cos l φ (223a)y E = z H = Z0/ncl

= El[Kl(wr/a)/K l(w)] cos l φ. (223b)

Here, as in following, the (a) holds for the core andthe (b) for the cladding; Z0 is the plane wave impedancein vacuum, and El the electrical field strength at the in-terface. Fig. 73b–e illustrate the case l = 1. Since we

Figure 73 Sketch of the fiber cross-section and the four possible distri-butions of LPl1 (after [303]).

3398

have the freedom of choosing sin lφ or cos lφ in Equa-tion 223 and two orthogonal states of polarization, wecan construct a set of four modes for every l as long asl > 0. For l = 0, we have only a set of two modes polar-ized orthogonally with respect to each other. The longi-tudinal components can be obtained from the equations[304]

z E = i Z0

n2co

]∂z H

∂y, (224a)

z E = i Z0

n2cl

]∂z H

∂y, (224b)

and

z H = (i/k Z0)(∂y E/∂x). (224c)

By introducing (446), we have

z E = −i E l

2ka

nco

Jl+1(ur/a)

Jl(u)sin(l + 1)φ

+ u

nco

Jl−1(ur/a)

Jl(u)sin(l − 1)φ

], (225a)

z E = −i E l

2ka

ncl

Kl+1(wr/a)

Kl(w)sin(l + 1)φ

+ w

ncl

Kl−1(wr/a)

Kl(w)sin(l − 1)φ

], (225b)

z H = −i E l

2k Z0a

Jl+1(ur/a)

Jl(u)cos(l + 1)φ

− uJl−1(ur/a)

Jl(u)cos(l − 1)φ

], (225c)

z H = −i E l

2k Z0a

Kl+1(wr/a)

Kl(w)cos(l + 1)φ

+ wKl−1(wr/a)

Kl(w)cos(l − 1)φ

]. (225d)

For small �, the longitudinal components [Equa-tions 225a–d] are small compared to the transversecomponents. The factors involved are u/ak and w/akwhich because of Equations 218 and 219 are bothof the order �1/2. Repeated differentiation of Equa-tions 225a–d) leads to transverse components which arenot identical with the postulated field [Equation 222]but small of order � compared to it. We shall neglectthese fields in the following. It is this approximationthat determines the accuracy of assumption of linearlypolarized modes (see also [298]).

To match the fields at the interface let us write Equa-tion 222 in terms of cylindrical components. We thenhave

φ E = El

2[Jl(ur/a)/J l(u)][cos(l + 1)φ

+ cos(l − 1)φ], (226a)

φ E = El

2[Kl(wr/a)/J l(w)][cos(l + 1)φ

+ cos(l − 1)φ], (226b)

Figure 74 The regions of the parameter u for modes of order l = 0; 1(after [298]).

φ H = − El

2Z0[nco Jl(ur/a)/J l(u)][sin(l + 1)φ

− sin(l − 1)φ], (226c)

φ H = − El

2Z0[ncl Kl(wr/a)/J l(w)][sin(l + 1)φ

− sin(l − 1)φ], (226d)

If we set nco = ncl in Equations 226–227 and usethe recurrence relations for Jl and Kl, we can match alltangential field components at the interface by the oneequation

u[Jl−1(u)/J l(u)] = −w[Kl−1(w)/K l(w)]. (227)

This is the characteristic equation for the linearlypolarized (LP) modes. Setting w = 0 yields the cutoffvalues Jl−1(u) = 0. For l = 0, this includes the roots ofthe Bessel function J−1 (u) = −J1(u), which we shallcount so as to include J1(0) = 0 as the first root. In suchway Gloge [303] obtain the cutoff values indicated inFig. 74 for LP0m and LP1m. In the limit of w → ∞ wehave J1(0) = 0. Thus, the solution for u are between thezeros of Jl−1(u) and Jl (u). Every solution is associatedwith one set of modes designed LPlm. For l ≥ 1, eachset composes four modes.

The accuracy of the characteristic equation can beimproved if we retain ncl and nco as different in Equa-tions 216 and 217. In this case, as was shown by Gloge[303], terms with (l + 1)φ and (l − 1)φ satisfy twodifferent characteristic equations:

(u/nco)[Jl±1(u)/J l(u)]

= ±(w/ncl)[Kl±1(w)/K l(w)] (228)

By using the recurrence relations for Jl and Kl, Glogehave shown that these two equations converge intoEquation 227 for nco = ncl. For ncl �= nco, this degen-eracy ceases to exist; each mode LPlm breaks up intomodes with terms (l + 1)φ, which can be identified asHEl+1,m or TEm and TMm (see also [292, 302]). A morerigorous proof of this results is given below.

As mentioned above, the problem of the dielectriccylinder with sharp index step can be solved exactly.

3399

Using above done the nomenclature, one can write theexact characteristic equation in the form

(Q − D − 2�{[(l ± 1)/ω2] ± [Kl(ω)/K l±1(ω)]})× (Q − D) = Q2[1 − 2�(u2/v2)], (229)

where

Q = (l ± 1)(v2/u2w2). (230)

D = [Jl(u)/u J l±1(u)] ∓ [Kl(w)/wK l±1(w)], (231)

and

2� = (n2

co−n2cl

)/n2

co. (232)

The upper sign holds for HEl+1 modes and the lowersign for EHl−1, TM and TE. Equation 232 agrees withEquation 222 in the case of small index differences. If� is set to zero in Equation 229, we find D = 0; andEquation 231 then becomes the simplified characteris-tic Equation 227. For small �, D is also small. Nowwe simplify Equation 220 to the extent that we retainterms linear in � or D. This result in

D = �{Q(u2/v2)−[(l ± 1)/w2]

∓ [Kl(w)/wK l±1(w)]} (231)

and with Equation 230

D = ∓�[Kl(w)/wK l±1(w)]. (232)

By introducing this into Equation 231 and insertingEquation 232, we find

(u/nco)[Jl±1(u)/J l(u)]

= ±(w/ncl)[Kl±1(w)/K l(w)] (233)

This is exactly the characteristic Equation 228.Evidently the guided wave traveling along the cir-

cular guide carries energy. The respective amounts arereadily calculated using the Poynting vector to estimatethe energy flow [298]. The Poynting vector in axial di-rection can be calculated from the cross product of thetransverse fields given in Equation 222. Integration overthe cross-section of core and cladding leads to tabulatedintegrals (see also [298, 301, 305]); the results are

Pco = [1 + (w2/u2)(1/k)](πa2/2)(Z0/nco)E2l (234)

and

Pcl = [(1/k)−1](πa2/2)(Z0/ncl)E2l (235)

for the power flow in core and cladding, respectively.If we ignore the small difference between nco and ncl,the total power in a certain mode becomes

P = Pco + Pcl

= (v2/u2)(1/k)(πa2/2)(Z0/nco)E2l . (236)

Practical fibers have small heat and scattering losseswhich cause significant attenuation over long distances.In general, these losses are attributable to certain partsof the fiber and proportional to the power propagating inthis part. For considerations of this kind, it is convenientto use the power fractions

Pco/P = 1 − (u2/v2)(1 − k) (237)

and

Pcl/P = (u2/v2)(1 − k). (238)

As expected, the mode power is concentration in thecore far away from cutoff. As cutoff is approached, thepower of lower order modes (l = 0, 1) withdraws intothe cladding, whereas modes with l ≥ 2 maintain a fixedratio of (l−1) between the power in core and cladding atcutoff. The power density is related to the mode powerP by

p(r ) = ku2

πa2

[J 2

l (ur/a)/J 2l (u)

]cos2 l φ (239a)

p(r ) = ku2

πa2

[K 2

l (wr/a)/K 2l (w)

]cos2 l φ. (239b)

By averaging over φ at r = a, we obtain the meandensity

p(r ) = k(u2/v2)P

πa2

[J 2

l (ur/a)/J 2l (u)

](240a)

p(r ) = k(u2/v2)P

πa2

[K 2

l (wr/a)/K 2l (w)

](240b)

At the core-cladding interface, we have r = a and

p(a) = k(u2/v2)P

πa2. (241)

The normalized density πa2 p(a)/P is plotted inFig. 75. For modes of order l = 0, 1 this density ap-proaches zero both at cutoff and far away from it, hav-ing a maximum in between. Modes with l ≥ 2 havep(a) = [1−(1/ l)] P/πa2 at cutoff. For r � a/w, wecan replace the K functions in Equation 241 by theirapproximation for large argument and obtain

p(r ) ≈ k(u2/v2)(P/πar ) exp[−2w(r − a)/a],

for r � a, (242)

as long as w is not too small. The power density de-creases exponentially with the distance from the inter-face (see also [296]). It decreases sharply as cutoff isapproached and is zero at cutoff. For sufficiently smallw we may set u = v and replace the K functions inEquation 241 by their approximation for small argu-ment, obtaining

p(r ) ≈ kl(P/πa2)(a/r )l (243)

3400

Figure 75 Normalized power density at the core-cladding interface plotted vs v (after [294]).

for r > a, w = 0. This function describes the cutoffpower distribution in the cladding. It decreases with thedistance from the axis for all but the lowest azimuthalorder, whose cladding field is independent of theradius.

3.5. Pulse spreadingThe transmission of information along optical fibers isnormally achieved by sending out a sequence of pulsesof light energy. However, as an individual pulse prop-agates, it spreads out, due to the dispersive propertiesof the fiber. Clearly if this spread becomes sufficientlylarge the pulse will overlap with adjacent pulses, lead-ing to a decrease in information-carrying capacity be-cause of the loss of resolution at the end of the fiber.

Below a formalism is presented for describing thepropagation characteristics of graded-index, multi-mode fibers. The index profiles of cylindrically sym-metric waveguides can be conveniently specified by theequation

n2(r ) = n21[1 − 2� f (r/a)], (244)

where n(r ) is the refractive index of the waveguide as afunction of distance r from the axis, and n1 is the indexalong the axis. The profile function f (r/a) is definedso that it is zero on axis

f (0) = 0 (245)

and becomes equal to unity at the core-cladding bound-ary located at r = a,

f (r/a) = 1, for r ≥ a. (246)

The cladding index n2 (= ncl) is thus defined to be

n2 = n1(1 − 2�)1/2. (247)

The quantity � provides a useful measure of the core-cladding index difference. From Equation 247 one finds

� = (n2

1−n22

)/2n2

1. (248)

Each mode of the waveguide can be specified bythe pair of integers µ and ν, which, respectively spec-ify the number of radial nodes and azimuthal nodesin the transverse electromagnetic fields of that mode.The propagation constant βµν of each mode dependsexplicitly on all quantities that specify the waveguidestructure and on the wavelength λ of the propagatinglight (see also [303]),

βµν = βµν(n1,�, a,λ). (249)

The propagation constants depend as usually on thewavelength explicitly and implicitly through the wave-length variation of n1 and �. Although the index profilef (r/a) may also vary slightly with wavelength, sucheffects will not considered.

For analyzing pulse transmission, one is concernedwith the group delay time per unit length for the modeµ, ν. This is given

τµν = dβµν

dω. (250)

If the free space propagation constant, k = 2π/λ, isintroduced, Equation 250 can be rewritten as

τµν = 1

dβµν

dk. (251)

3401

If the fiber is excited by an impulse excitation andif there is no mode coupling, the impulse response,P(t, z, λ), for spectral component λ at position z canbe written as

P(t, z, λ) =∑

Pµν(λ, z)δ[t − zτµν(λ)], (252)

where the summation extends over all guided modes.The distribution functions Pµν(λ, z) describe the powerin mode µ, ν as a function of wavelength and position.At z = 0 the distribution function will be determinedby the spatial, angular, and spectral distribution of thesource, as well as by the source-fiber coupling configu-ration. As the impulse propagates along the waveguidethe power in each mode will change according to theattenuation occurring in that mode. Further it will beassumed that no mode coupling occurs.

In general, only the total power integrated over allsource wavelengths will be detected. Hence, the quan-tity of practical interest is the full impulse response

P(t, z) =∫ ∞

0dλP(t, z, λ). (253)

The propagation characteristics of the fiber can bedescribed by specifying the moments Mn(z) of the fullimpulse response. These moments are defined by

Mn(z) =∫ ∞

0dt tn P(t, z). (254)

In some situations, knowledge of only the first fewmoments is sufficient for system design considerations.If this is the case, the required amount of pulse broad-ening information is reduced considerably. Equations252–254 can be combined to yield

Mn(z) = zn∫ ∞

0dλ

∑Pµν(λ, z)τ n

µν(λ). (255)

The predominant wavelength dependence of the dis-tribution function Pµν(λ, z) is determined by the spec-tral distribution S(λ) of the source. Even for the rel-atively broad LED sources, S(λ) is a sharply peakedfunction whose rms width, at most, a few percent of themean source wavelength. One can rhus define a newdistribution function pµν(λ, z) by the expression

Pµν(λ, z) = S(λ)pµν(λ, z), (256)

where pµν(λ, z) is a slowly varying function of λ overthe range where S(λ) is nonzero.

Proceedings with the analysis, it can be assumed thatS(λ) is normalized so that∫ ∞

0dλS(λ) = 1. (257)

Consequently, the mean source wavelength λ0 isgiven by

λ0 =∫ ∞

0dλλS(λ), (258)

and the root mean square (rms) spectral width of thesource σs is given by

σs =[ ∫ ∞

0dλ(λ − λ0)2S(λ)

]1/2

. (259)

The influence of the source spectral distribution onthe fiber’s transmission properties can be studied byexpanding the delay per unit length of the µ, νth mode,τµν(λ), in a Taylor series about λ0. Substituting thisseries into Equation 255 and using Equation 256 give

Mn(z) = zn∫ ∞

0dλS(λ)

∑µν

p(z){τ nµν(λ0)

+ n(λ − λ0)τ n−1µν (λ0)τ ′

µν(λ0)

+ n(λ − λ0)2/2τ n−2µν (λ0)τ ′′

µν(λ0) + }. (260)

Treating pµν as independent of λ, Equations 255–259can be used to integrate Equation 260 to find that

Mn(z) = zn∑

pµν(z)τ nµν(λ0) + σ 2

/(2λ2

)× {

nτ n−1µν (λ0)λ2

0τ′′µν(λ0)

+ n(n − 1)τ n−2µν (λ0)[λ0τ

′µν(λ0)]2})

+ 0(σ 3

s /λ30

). (261)

The small size of σs/λ0 allows the neglect of higherorder terms. The following quantities are most usefulin describing the energy distribution at z. By definition,the total power arriving at z is given by

M0(z) =∑

pµν(z), (262)

the mean delay time of the pulse τ (z) is given by

τ (z) = M1(z)/M0(z), (263)

and the rms pulse width σ (z) by

σ (z) = [M2(z)/M0(z) − τ 2(z)]1/2. (264)

Combinations of higher moments further describethe power distribution, but the first three are the mostimportant. To simplify the notation required in the fol-lowing expressions, the symbol 〈〉 will be used to indi-cate the average value of a quantity with respect to thedistribution pµν so, for example

〈A〉 ≡∑

pµν(z)Aµν/M0. (265)

From Equations 252–255, the full pulse delay timeis found to be

τ (z) = z[〈τ (λ0)〉 + σ 2

/(2λ2

)⟨λ2

0τ′′(λ0)

⟩]. (266)

For the purpose of specifying the fiber bandwidth fordigital systems, Personick [306] has shown that one is

3402

primarily concerned with the rms width σ (z). This isgiven by

σ (z) = (σ 2

intermodal + σ 2intramodal

)1/2 + 0(σ 3

s /λ30

), (267)

where the definitions

σ 2intermodal = z2{〈τ 2(λ0)〉 − 〈τ (λ0)〉2

+ σ 2s

λ20

[λ2

0τ′′(λ0)τ (λ0)

− ⟨λ2

0τ′′(λ0)

⟩〈τ (λ0)〉]} (268)

σ 2intramodal = z2 σ 2

λ20

〈[λ0τ′(λ0)]2〉 (269)

have been introduced. We can see, that the square ofrms width has been separated into an intermodal andan intramodal component. The intermodal term (262)results from delay differences among the modes andvanishes only if all delay differences vanish. This termis found to contain a dominant term and a small cor-rection that is proportional to the square of the relativesource spectral width (σs/λ0). For the refractive indexprofiles considered below, this term is found to be negli-gible. The intramodal term (269) represents an averageof the pulse broadening within each mode. It becomesthe only term present in the dispersion of a single-modewaveguide. The intramodal dispersion arises from twodistinct effects, a pure material effect that correspondsto the pulse broadening in bulk material and waveg-uide effect. This separation can be made by writing themodal delay time in the form

τµν = N1/c + δτµν, (270)

where N1 is material group index,

N1 = n1λdn1/dλ, (271)

and δτµν represents the correction to this introducedby the waveguide structure. The derivative τ ′

µν can bewritten as

τ ′µν = −λn′′

1 + δτ ′µν. (272)

Since the intramodal contribution to the total rmspulse width is obtained by squaring and averaging overEquation 273, one can write the intramodal contribu-tions as

σ 2intramodal = z2 σ 2

λ20

[(λ2

0n′′1

)2 − 2λ20n′′

1〈λ0δτ′〉

+ 〈(λ0δτ′)2〉]. (273)

Hence, the total intramodal contribution has a purematerial component, a waveguide component, and amixed component arising from the cross product. Forgraded-index waveguides the derivative of the inter-modal delay differences δτ ′

µν is also small, therefore

the intramodal contribution is then dominated by thepure material term

σintramodal ≈ σs

λ0

(λ2

0n′′1

). (274)

This quantity represents the ultimate lower limit ofthe pulse broadening. According to [307] the meaningof σintermodal has next expression

σintermodal

= L N 1�

α + 1

(α + 2

3α + 2

)1/2

×[

C21 + 4C1C2�(α + 1)

2α + 1+ 4�2C2

2 (2α + 2)2

(5α + 2)(3α + 2)

]1/2

(275)

where

C1 = α − 2 − ε

α + 2, (276)

and

C2 = 3α − 2 − 2ε

2(α + 2). (277)

As was shown in [308], the minimum of the inter-modal rms width occurs for (see also [298])

αc = 2 + ε − �(4 + ε)(3 + ε)

(5 + 2ε). (278)

Here ε = −2n1λ�′N�

and α = 2 + ε. The last term onthe right of Equation 278 represents a small correctionto the optimal α. This correction results from a partialcancellation that occurs between the two mode depen-dent terms, if α − 2 − ε is of order �. The size of thiscorrection changes as the distribution of power amongthe modes varies.

Fig. 76A shows the fitted values of n1(λ) and n2(λ)in the range from 0.5 µm to 1.1 µm. Figs 76B and76C, respectively, show the first and second derivativesof the refractive indices of these two glasses. Fig. 77shows the values of � and λ�′ determined from thedata. It can be observed in Fig. 77A that, over the rangeof wavelength shown, � decreases by about 15%. Fromthe index data on Figs 76 and 77, ε(λ) can be calculated,and a plot of αc vs. λ is shown in Fig. 78. The value ofαc minimizing the intermodal rms pulse width departssignificantly from the optimal profile α = 2(1 − 6�/5),which is predicted if the effect of material dispersion isignored. The optimal α varies quite strongly with wave-length, decreasing from about 2.5 at 500 nm to 2.2 at1000 nm. A waveguide with a given index profile α isthus predicted to show different pulse widths accord-ing to the source wavelength used. This wavelength-dependent pulse broadening provides a tool for observ-ing these effects and is discussed at length in [307].

According to [307] the delay time can be used to eval-uateλτ ′

n , which is required for predicting the intramodal

3403

Figure 76 Refractive index data for 3.4 wt% TiO2 doped silica (n2) andfused silica (n1) are shown for (A) Sellmeir fit to the refractive index,(B)—λdn/dλ, (C)—λ2d2n/dλ2 (after [307]).

Figure 77 The index difference � determined from the data of Fig. 122Ais shown in (A), and the derivative λd�/dλ in (B) (after [307]).

Figure 78 The α value that minimizes the pulse broadening as a functionof wavelength (after [307]).

broadening has next form

λτ ′n = −λ2n′′

1 + N1�

(α − 2 − ε

α + 2

)

×(

2α

α + 2

)(n

)α/α + 2

. (279)

Since λ2n′′1 and � are the same order of magnitude,

both terms contribute to λn′n for large α (but with op-

posite signs). For α near αc, the pure material termin Equation 273 dominates. This reflect the fact thatwhen the intermodal delay differences are small, thederivatives of these differences are also small. Assum-ing equal excitation of the modes, one can perform thesummation over all modes required by Equation 259and find that

σintramodal

= σλ

[(−λ2n′′

1)2 − 2λ2n′′1(N1�)

×(

α − 2 − ε

α + 2

)(2α

2α + 2

)

+ (N1�)2(

α − 2 − ε

α + 2

)2( 2α

3α + 2

)]1/2

. (280)

The total rms pulse widths for α profiles can be pre-dicted from Equations 275 and 280. In Fig. 79, thisrms pulse width is shown as a function of α for threetypes of GaAs sources, all operating at λ = 0.9 µm,but having differential spectral bandwidths. The threecurves correspond to an LED, an injection laser, and adistributed-feedback laser having typical rms spectralwidths of 150 A, 10 A and 2 A, respectively. The waveg-uide is assumed to have n1(λ) and n2(λ) of Fig. 76 andto propagate equal power in all modes. For compari-son, a dashed curve represents the rms width predictedif material dispersion and intramodal broadening areignored. We shoul add, that for the LED source, pulsebroadening of less than 1.5 ns/km can be achieved if α iswithin 25% of the optimal value. For the injection laser,an α within 5% of the optimal will give pulse widthsless than 0.2 ns/km, and for the distributed-feedbacklaser, widths of 0.05 ns/km are predicted if 1% controlon α can be achieved.

3404

Figure 79 Assuming equal power in all modes, the rms pulse width isshown as a function of α for three different sources, all operating at0.9 µm. The sources are taken to be an LED, a gallium arsenide injec-tion laser, and a distributed feedback laser having rms spectral widths of150 A, 10 A and 2 A, respectively. The dashed curve shows the pulsewidth that would be predicted if al material dispersion effects were ne-glected (after [298]).

In the preceding sections it was found that the α

value that minimizes the intermodal pulse dispersiondeparts significantly from the parabolic profile and thatthe magnitude of this departure depends on the wave-length of the source. In Fig. 79, one sees that the cal-culated rms pulse width depends very strongly on thedifference between the α of the waveguide and the opti-mal α. In Fig. 80, the calculated rms width is shown as afunction of source wavelength for waveguides with in-dex profiles in the range 1.5 ≤ α ≤ 2.9. For the purposeof illustration, it was assumed [307] in these calculationthat the rms spectral width of the source is 2 A. As isknown, for this range of α values both the intermodaland intramodal contributions must be considered. Thewavelength dependence of the intermodal term dependson the specific α value. For α ≺ 2.2, the intermodalpulse width decreases as λ increases, because for theseα′s the difference |α − αc| decreases with λ. The op-posite occurs for α ( 2.5, and the intermodal pulsewidth increases with λ. For waveguides with 2.2 ≺ α ≺2.5, the minimal intermodal broadening occurs at onewavelength in the range 500 nm ≺ λ ≺ 1100 nm. As λ

is varied through this range, the intermodal broadeningdecreases until the optimal wavelength is reached, andthereafter it increases. The intramodal contribution tothe rms width is dominated by material dispersion. Itis largest at the shorter wavelength and decreases quiterapidly with wavelength. For a source spectral width of2 A, the rms width decreases from 0.1 ns/km to 0.006 ns/km between 500 nm and 1100 nm. The pulse width be-havior shown in Fig. 80 reflects the combined effect ofintermodal and intramodal pulse broadening.

Figure 80 For a source with 2 A spectral width, the rms pulse width isshown as a function of wavelength for several different values of α (after[307]).

3.6. Materials for optical fibersThe choice of materials to be used in the fabricationof fibers is influenced by the need to satisfy simultane-ously many requirements. Obviously the material mustbe formable into a fine filament, transparent and avail-able with two different refractive indices for core andcladding, respectively. These requirements alone moreor less limit the field to plastics or glasses, although aliquid has been used to form the core of a fiber drawnfrom a hollow tube of glass. Many plastics are excludedfrom further consideration because the presence of hy-drogen in their structures give rise to very high lossesand because their molecular size leads to large scatter-ing losses. And within the infinite number of possibleglasses, most are rule out by other consideration. To ap-preciate this situation more fully, we need to examine inmore detail the physical mechanisms involved, partic-ularly those controlling optical loss since most opticalcommunications systems require fibers of exception-ally low attenuation at the optical carrier frequency.Usually less than 20 dB/km is sought [309, 310].

Glasses are formed from fused mixtures of metal ox-ides, sulfides or selenides. Because they are fused mix-tures rather than fixed compounds with crystal struc-tures, their compositions are infinitely variable withincertain regions of their respective phase diagrams, andlarge numbers of different glasses are manufactured byindustry. Most of these fall into the category of oxideglasses, since these are the optically transparent ones,the sulfide and selenide glasses being used in the in-frared region approximately 0.6 µm to 14 µm or more.

Of the oxide glasses, by far the most common aresilica (SiO2); sodium calcium silicates, frequently usedfor plate and window glass; sodium borosilicates, often

3405

used for oven ware and chemical apparatus; and leadsilicates, which are the crystal glasses having relativelyhigh refractive indices and thus appearing “shiny”. Typ-ical starting materials for these glasses are sodiumand calcium carbonates, boric oxide (B2O3) or boricacid, silica (sand), and lead oxide. For the typical fibermaterials, new conditions arise, namely, the need forvery low loss, which means very high chemical purity,so that the materials sources used are usually differ-ent from those serving large scale industry. The struc-ture of glasses is noticeably different from that of thesolids usually encountered in the electronic industry,namely, crystals. In the latter the individual atoms arewell defined in space according to very precise andrepeated patterns, lying in exact three-dimensional lat-tices. Glass, on other hand, consists of a loosely con-nected network formed by groups, which can be addedto or modified by other components. For example, theaddition of sodium (a network modifier) tends to breakup the SiO2 network, as shown in Fig. 81, in a sodiumsilicate glass. The result is less strongly bound than pure

Figure 81 Networks involving SiO2 groups, shown schematically in twodimensions. (a) A regular SiO2 lattice (two-dimensional quartz crystal).(b) The effect on the lattice of the addition of a network modifier (after[298]).

Figure 82 Viscosity versus temperature for a number of glasses (after[298]).

silica, and the melting temperature is thereby lowered.Since B2O3 is also a network former like SiO2, a se-ries of glasses such as sodium borates exist, parallelingsodium silicates but having very much lower meltingtemperatures since the B O bond is much weaker thanthe Si O bond.

Fig. 82 shows viscosity-temperature curves for somecommonly manufactured glasses. Particularly note-worth is the fact that viscosity of a pure silica glassis much higher than that of multicomponent glasses atthe same temperature. Just as the viscosity of a glass is afunction of its composition, so too are the refractive in-dex and the thermal expansion coefficient. Since there isan infinity of glass compositions, full data are not avail-able for all glasses; in fact, comprehensive data are re-stricted to a few glass groups that have been extensivelystudied because of large scale commercial applications.The sodium calcium silicate (NCS) group is one such.A second glass system of great current interest is thesodium borosilicate (NBS) group of glasses. Anothergroup of glasses that may be of interest in fiber manufac-ture is lead silicates. These are of potential interest sincethey allow large refractive index differences to be ob-tained between core and cladding glasses. For example,a pair with indices of 1.5 and 1.65 showing a 10% indexdifference can probably be obtained by using the highlypolarizable lead ions to dope the higher index material.Fibers with such large index difference will be charac-terized by a higher level of Rayleigh scattering from thelead (of the order 10 dB/km, compared to 1 dB/km forsilica at 900 nm) and have not yet been produced withvery low loss (i.e., under 15 or 20 dB/km). But for manyshort link systems applications these disadvantages aremore than offset by the large acceptance angle for sucha high index difference fiber. All these three groups ofglasses are of interest because they can be made bynearly conventional glass melting techniques in largequantities at low cost. They are formed by mixing theappropriate powders and fusing them in a crucible to

3406

form a glass. But there is another group of materials,as already noted, all based upon pure silica with smalladditions of one or more other oxides.

3.6.1. Absorptive losses in glassesOptical fiber choose to operate in one of two wavelengthbands, the GaAs device region from 800 to 900 nmor the Nd laser region at 1.06 µm, with remote pos-sibility of an extension to perhaps 1.2 or 1.3 µm us-ing other semiconductor sources (see also [309, 310])This choice is dictated by good physical reasons. Wave-lengths shorter than about 600 to 650 nm are ruled outbecause Rayleigh scattering in the glass becomes se-vere and impurity absorption spilling over the bandedge absorption is a problem. Beyond about 1.3 µm thefirst overtone of the OH stretching vibration appears at1.4 µm, giving heavy absorption. Some spillover frommultiphonon bands from the glass constituents proba-bly adds to this, and detectors become far less efficientbecause the quantum energy hν rapidly approaches kTunder conditions at room temperature operation.

Five absorption mechanisms are of concern in a glassthat is considered for use in optical fibers. The first twoare associated with the basic glass constituents them-selves, typically a combination of oxides of silicon,sodium. boron, calcium, germanium and so on. Theglass has a band edge absorption somewhere in the ul-traviolet (UV) region of the spectrum. Such absorptionis extremely intense; and although the wavelength ofinterest for operation of a system is a considerable dis-tance away, there has been serious debate as to whetherthe tail of the band edge could provide a significant lossmechanism (see also [311–316]).

Fig. 83 shows the losses of some fibers with very lowwater contents made with different constituents in the

Figure 83 Losses versus wavelength of some low water content fibers showing the matrix influence on the infrared transmission (after [318]).

core by the chemical vapor deposition technique [317],and in Fig. 84 the infrared spectra of some of these samematerials, showing the relative size and position of theinfrared absorptions, are presented. The work suggeststhat GeO2 doping for the core is the most favorablebecause of the longer wavelength at which the GeO2stretching vibration occurs, and this leads to an estimate[318] for the lowest loss for this type of fiber, at about1.5 microns, of about 0.3 dB/km, as shown in Fig. 85.Evidently, to obtain figures even remotely approachingthese, very low water contents must be achieved. Theremaining absorption mechanisms are all related to im-purities or defects in the glass [314] and are thereforenot intrinsic. Besides that, a problem with most opticalfibers is to reduce the hydroxyl ion (OH) content to asufficiently low level, usually a few parts per million(details see [298]).

3.6.2. Rayleigh scatteringThe phenomena of Rayleigh scattering is well-knownto all of us as the mechanism responsible for blue sky.It is scattering of light from microirregularities in thedielectric medium through which the electromagneticwave is propagating. The physical scale of the irregu-larities is of the order of one-tenth wavelength or less,so that each irregularity acts as a point source for scat-tered radiation. The resulting relation pattern from theinduced dipole is doughnut shaped, being uniform in theplane perpendicular to the dipole and varying as sin φ

in the plane containing the dipole, where φ is the anglebetween the observation direction and the dipole axis.In the sky, Rayleigh scattering arises from the minutedensity fluctuation in the atmospheric gas caused bythe constant thermal fluctuation of the medium. whilethe sin φ component causes it to be highly polarized at

3407

Figure 84 Infrared transmission spectra of the materials associated with the fibers of Fig. 83, showing the various absorption bands involved (after[318]).

Figure 85 Predicted losses arising both the ultraviolet edge absorptionand the infrared absorption for GeO2 − SiO2 fiber (after [317]).

ground level. In glass, Rayleigh scattering can arisefrom two separate effects, density and compositionfluctuations.

We have already seen that glasses are disorderedstructures, loosely connected in a largely random se-quence. Evidently, in such a structure there are localregions in which the average density is higher than inother regions. The dense and less dense regions can be

traced back to thermally driven fluctuations in densityarising from the Brownian movement of the liquid glassbefore it froze. The magnitude of the fluctuation is thusexpected to be related to the freezing temperature—thehigher the temperature, the greater the density fluctu-ation. One might therefore expect that a high meltingmaterial such as silica would show a higher Rayleighscattering loss than a lead glass of lower melting tem-perature. But this is not so for another reason.

The magnitude of the density fluctuation scatteringis given by the following expression [319]:

αscat.ρ = 8π3n3

3λ4(n8 p2)(kT f)βT. (280a)

Here p is the photoelastic coefficient for the glass; Tfis the fictive temperature, defined as the temperature atwhich it becomes possible for the glass to reach a stateof thermal equilibrium and closely related to the annealtemperature; k is Boltzmann’s constant; and βT is theisothermal compressibility. Certainly, the energy kTf isthe driving energy for the fluctuation giving rise to theloss mechanism described by αscat.ρ .

The expression for the scattering due to compositionis more complex, taking the form [319]:

αscat,c = 32π3n2

3λ4ρNA

m∑j=1

[(∂n

∂xj

)Tf,x i �=xj

∂n

∂ρ

)Tf,x i

(∂ρ

∂xj

)P,T f,x i �=xj

Mjxj. (281)

In this formula, Mj and xj are, respectively, the molec-ular weight and the weight fraction of the j th modifier,and NA is Avagadro’s number. The partial derivatives

3408

of the refractive index and the density, ρ, may be deter-mined from experimental data so that the summationcan be performed. Schroeder [320] has derived an ex-pression for the mean composition fluctuation in theglass, which takes the form:

〈�C2〉Tf = ρ0

(kT f

)(∂µ

∂C

)−1

C0,T f

, (282)

where C is the mole fraction of one of the constituentsin the composition C0, V is the molar volume, and µ

is the chemical potential difference between the majorand minor constituents of th (binary)glass. Evidently,the composition fluctuation is minimized if ∂µ/∂C islarge and the fictive temperature is small.

Since the refractive index of a glass is a sensitivefunction of composition, it is frequently the case thatthe composition fluctuation a swamps the density fluc-tuation component. Thus a low melting glass may wellhave a much higher scatter loss than silica, and ingeneral this is bound to be the case. However, someparticular glass compositions have been found to leadto exceptionally low total Rayleigh scattering losses.The potassium silicates [320] are one such group whichshow a minimum scattering loss at one composition, asshown in Fig. 86. In Table XVIII it can list the Rayleighscattering at 900 nm wavelength for a number of glassesthat are of interest for optical fiber systems.

3.7. Fiber preparationIn this part we briefly consider the methods by whichthe different materials can be formed into fibers withthe desired properties. Several techniques have been de-scribed for melting glasses that are specifically intended

Figure 86 Rayleigh scattering loss at 633 nm versus composition for asodium silicate glass (after [320]).

TABLE XVII I Rayleigh scattering losses for a number of fiberglasses (quoted for 900 nm) (after [298])

Glass Loss (dB/km)Silica ∼1.2Sodium borosilicatea ∼1.5Phosphosilicateb ∼1.6Germania silicatec ∼1.2Selfocd ∼2.0

aRef. [298].bD. B. Keck, R. D. Maurer and P. C. Schultz, Appl. Phys. Lett. 22, 307(1973) (0.7% index difference from silica).cM. Kawachi, A. Kawara and T. Miyash*ta, Electron. Lett. 13, 442 (1977)(0.18% index difference from silica).dG. W. Morey, The Properties of Glass, 2nd ed. (Reinhold, New York,1954).

for use in the preparation of optical fibers. Each is basedupon the use of powders which are premixed, heated ina crucible until they fuse, and then agitated to producea hom*ogeneous mix. The heating may be applied to thecrucible through black-body radiative coupling fromthe walls of an electrically heated furnace; it may begenerated by the coupling of radio-frequency (RF) ra-diation to the crucible when the crucible is made of aconducting material such as platinum; or, alternatively,RF energy may be coupled directly to the melt glass,provided that is preheated to a temperature at whichit begins to conduct. In the last case, the crucible re-mains relatively cold, thus helping to reduce problemsof crucible contamination. In all cases it is necessary touse for the melting an enclosure which excludes con-tamination from the laboratory environment. Typically,silica liners are used to provide such isolation.

In the electric furnace, energy is transferred from thefurnace wall to the crucible by black-body radiation. Agreat deal of early melting work was done using plat-inum crucibles contained within silica enclosures forclean-lines, but with the heat applied through RF energycoupled from an encircling coil, as shown in Fig. 87.This technique has the attraction that there is no refrac-tory insulating anywhere near the melt [322]. However,precise temperature control is generally more difficult,the thermal time constant of the apparatus being muchshorter, and the apparatus is far more expensive. It alsorequires a conducting crucible, ruling out the use ofsilica.

The production of fiber using a double-crucible ap-paratus is conceptually extremely simple [323, 324].The basic apparatus is illustrated in Fig. 88. Two con-centric crucibles are held with their axes vertical. Eachcrucible has in its base a central circular nozzle, theinner being carefully aligned to be concentric with theouter and perhaps 1 cm above it. The inner crucible isfilled with core composition glass, and the outer withcladding composition glass. With the glasses molten,the core glass flows through the inner nozzle into thecladding stream and is subsequently carried out of thecladding outer nozzle surrounded by cladding glass, sothat a composite glass flow is produced. The moltenglass exudes into the space outside the outer crucible,and a filament is pulled from the exudant to form a coredglass fiber. In practice, the production of low loss fiberby this process requires a great deal of careful attention

3409

Figure 87 Radio frequency heating of a platinum crucible for a glassmelting in a clean enclosure (after [321]).

Figure 88 Apparatus for double crucible fiber pulling (after [321]).

to detail at every stage of the process; the glass prepa-ration, cleaning of crucibles, assembly of apparatus,loading of the glass, atmosphere control, stabilizationof temperature and so on.

We should note that increasing interest centers uponthe deposition of more than one dopant for a given fiberand also upon depositing multiple layers, the concentra-tion in each layer being carefully monitored and variedin a predetermined manner. In the former category atypical deposition might be of B2O3-SiO2 to form acladding and GeO2-SiO2 to form the core. Such a paircould be chosen to enable the viscosities of the twolayers to be more equally matched, thus facilitating thecollapse stage. Also, it is found that the presence ofsome of the dopants allows greatly increased gas flowand reaction rates to be used, a factor of considerablecommercial importance in implementing a process.

The presence of layers arising from the depositionprocess is sometimes clearly evident in the preformsand fibers. This appears to be particularly true of CVDfiber and less so of outside deposition soor fiber. In-terferometric analysis of the index profiles of preformsshows the presence of an “oscillating” index profile fol-lowing a mean curve of the form desired, as illustratedschematically in Fig. 89. To conclusion of this part wepay attention to new class of fibers—the holey fibers,which have the other technology (details see [325] andreferences therein).

3.8. Isotopes in fibersAs a first example of possible major applications of iso-topic engineering it will be considered isotopic fiber-optics and isotopic optoelectronics at large (see also[1, 100, 177]). It is known that for typical solids the lat-tice constant variations of isotopically different samplesare usually within the limits

�d

d∼ 10−3 ÷ 10−4. (283)

Let us define an isotopic fiber as a structure in whichcore and cladding have the same chemical content butdifferent isotopic composition. The boundary betweendifferent isotopic regions form an isotopic interface.The difference in the refractive index on both sides ofthe isotopic interface could lead to the possibility of to-tal internal reflection of light and, consequently, could

Figure 89 A schematic picture of the index distribution of a graded corfiber produced by layer deposition (after [298]).

3410

Figure 90 Light guidance in an optical waveguide by total internal reflection.

provide an alternative route to the confinement of light.For a quantitative estimate let us consider a boundarybetween SiO2 (the main component of silica) wherebody sides are identical chemically and structurally buthave a different isotopic content—e.g., 28Si16O2 and30Si18O2 respectively (Fig. 90). In the first approxima-tion the refractive index n is proportional to the numberof light scatterers in the unit volume. From the Clausius-Mosotti relation for the refractive index one can deducethe following proportion ( at �n � n)

�n

n� 3c

�d

d, (284)

where c is a dimensionless adjustment factor of the or-der of unity. Substituting Equation 283 to Equation 284we can obtain

�n

n∼ 3 × 10−3 ÷ 10−4. (285)

Using the Snell law of light refraction we obtain thefollowing expression for the ray bending angle � whenthe light travels through the refractive boundary

θ � arc sin

(n1

n2sin α0

), (286)

where α is the angle between the falling ray and thedirection normal to the interface. For a sliding ray (α �90◦), which is the control case for light confinement infibers, the combining of Equations 285 and 286 leadsto an estimate

θ ∼ 1.5 ÷ 4.5◦.

Thus, the isotopic fibers in which core and claddingare made of different isotopes the half-angle of theacceptance-cone could be up to several degrees. Theresulting lattice mismatch and strains at the isotopicboundaries are correspondingly one part per few thou-sand [1] and, therefore, could be tolerated. Further ad-vancement of this “isotopic option” could open the wayfor an essentially monolitic optical chip with built-inisotopic channels inside the fully integrated and chem-ically uniform structure.

Besides that we should pay attention to the fact thatcomposition (different isotopes) fluctuation are subjectto the restoring force of the total free energy of theglass system which will also seek to minimize itself.Usin g isotope pure materials for core and cladding weshould receive significant less Rayleigh scattering (seee.g. [4]).

Chapter 4. Laser materials4.1. Some general remarksAs is well-known, the word laser is an acronym for“light amplification by the stimulated emission of radi-ation”, a phrase which covers most, though not all, ofthe key physical processes inside a laser. Unfortunately,that concise definition may not be very enlightening tothe nonspecialist who wants to use a laser but has lessconcern about the internal physics than the externalcharacteristics. A general knowledge of laser physicsis as helpful to the laser user as a general understand-ing of semiconductor physics is to the circuit designer.From a practical standpoint, a laser can be consideredas a source of a narrow beam of monochromatic, co-herent light in the visible, infrared or UV parts of spec-trum. The power in a continuous beam can range froma fraction of a milliwatt to around 20 kilowatts (kW) incommercial lasers, and up to more than a megawatt inspecial military lasers. Pulsed lasers can deliver muchhigher peak powers during a pulse, although the averagepower levels (including intervals while the laser is offand on) are comparable to those of continuous lasers.

The range of laser devices is broad. The lasermedium, or material emitting the laser beam, can bea gas, liquid, crystalline solid, or semiconductor crys-tal and can range in size from a grain of salt to fillingthe inside of a moderate-sized building. Not every laserproduces a narrow beam of monochromatic, coherentlight. A typical laser beam has a divergence angle ofaround a milliradian, meaning that it will spread to onemeter in diameter after traveling a kilometer. This fig-ure can vary widely depending on the type of laser andthe optics used with it, but in any case it serves to con-centrate the output power onto a small area. Semicon-ductor diode lasers, for example, produce beams thatspread out over an angle of 20 to 40◦ hardly a pencil-thin beam. Liquid dye lasers emit at a range of wave-lengths broad or narrow depending on the optics usedwith them. Other types emit at a number of spectrallines, producing light is neither truly monochromaticnor coherent. Practically speaking, lasers contain threekey elements. One is the laser medium itself, whichgenerates the laser light. A second is the power supply,which delivers energy to the laser medium in the formneeded to excite it to emit light. The third is the opti-cal cavity or resonator, which concentrates the light tostimulate the emission of laser radiation. All three ele-ments can take various forms, and although they are notalways immediately evident in all types of lasers, theirfunctions are essential. Like most other light sources,lasers are inefficient in converting input energy intolight; efficiencies can range from under 0.01 to around20% [326–329].

3411

4.2. Absorption and induced emissionThe idea of stimulated emission of radiation, as well-known, originated with Albert Einstein [330]. Until thattime, physicists had believed that a photon could inter-act with an atom in only two ways: it could be ab-sorbed and raise the atom to a higher energy level orbe emitted as the atom dropped to a lower energy level.Einstein proposed a third possibility-that a photon withenergy corresponding to that of an energy—level tran-sition could stimulate an atom in the upper level to dropto the lower level, in the process stimulating the emis-sion of another photon with the same energy as first. Inthe normal world, stimulated emission is unlikely be-cause at thermodynamic equilibrium more atoms are inlower energy levels than in higher ones. Thus a photonis much more likely to encounter an atom in a lowerlevel and be absorbed than to encounter one in a higherlevel and stimulate emission.

Below we consider a cavity whose walls are at tem-perature T , containing radiation and an ensemble ofatoms and let each atom be represented by a two-levelquantum mechanical system with an energy level sepa-ration of hω0. In thermal equilibrium the energy densityper unit angular frequency range at ω0 is given by [331]

ρω0 = hω30

π2c3

e-hω0/kT − 1= ω2

π2c3

hω0n3

e-hω0/kT − 1. (287)

Then

ρω0 (e-hω0/kT − 1) = hω3

0n3

π2c3. (288)

In addition, because of detailed balance

AN e2 + B21ρω0 N e

2 = B12ρω0 N e1 , (289)

where N e1 and N e

2 are the equilibrium populations ofatoms in the lower and upper levels, respectively, andB21ρω0 and B12ρω0 are the probabilities per unit time forinduced downward and upward transitions, respectivelyand A is Einstein coefficient.

Now we can write

AN e

N e1

+ B21ρω0

N e2

N e1

= B12ρω0 (290)

and

Ae− hω0kT + B21ρω0 e

− hω0kT = B12ρω0 (291)

ρω0

(B12e− hω0

kT − B21) = A. (292)

Set

B12 = B21 = B. (293)

ρω0

(e− hω0

kT −1)B = A. (294)

B= ρω0

(e− hω0

kT − 1) = hω3

0n3

π2c3(295)

B = π2c3

hω30n3

A = π2c3

n3 hω30

4nω30|µ|2

3hc3= 4π2

3h2n2|µ|2

(296)

A = 4nω30

3hc3|µ|2 (s−1);

B = 4π2

3h2n2|µ|2 (cm3erg−1 s−2). (297)

B= hω3

0n3

π2c3(erg · s · cm−3) (298)

Let us call w(ω)dω the probability per unit time thatan atom undergoes an induced transition by absorbingor emitting a photon with angular frequency in (ω, ω +dω). It is

w(ω)dω = Bg(ω)ρωdω = 4π2

3h2n2|µ|2ρωg(ω)dω

= 4π2

3nch2|µ|2 I (ω)g(ω)dω, (299)

where

I (ω)dω = ρω

ndω

= intensity of radiation with angular

frequency in (ω, ω + dω). (300)

Now we can write∫w(ω)dω = 4π2

3h2n2|µ|2ρω = 4π2

3h2nc|µ|2 I (ω0) (301)

Further let us assume that a plane wave goes througha certain medium in the x-direction. Let the mediumconsist of atoms which have two possible energy levelsand let N1(N2) be the concentration of atoms in thelower (higher) energy level (see also [327])

dI (ω) = −w(ω)(N1 − N2)hωdx

(erg

cm2

)(302)

But from (301) we have

w(ω) = 4π2

3nch2|µ|2 I (ω)g(ω). (303)

Then

dI (ω) = −[

4π2

3nch2|µ|2 I (ω)g(ω)

](N1 − N2)hωdx

= −[

4π2(N1 − N2)

3nch2|µ|2 I (ω)g(ω)

]I (ω)dx

= −α(ω)I (ω)dx, (304)

3412

where

α(ω) = 4π2(N1 − N2)

3nch2|µ|2ωg(ω)

= nhω

cB(ω)(N1 − N2).

(absorption coefficient.) (305)

The solution of (304) is,

I (ω, x) = I (ω, x = 0)e−αx . (306)

We define the sbsorption cross-section of the radia-tive transition as follows (details see also [328])

σ (ω) = α(ω)

N1 − N2= 4π2

3nch2|µ|2ωg(ω) (cm2) (307)

Note the following

∫α(ω)dω = 4π2(N1 − N2)

3nch2|µ|2ω0

= nhω0

cB(N1 − N2)(cm−1 s−1). (308)

4.3. Semiconductor lasers4.3.1. Heterojunction lasersSemiconductor lasers, like other lasers, have popula-tion inversions which lead to stimulated emission ofphotons. Semiconductor laser is different from anotherlasers primarily because the energy levels in semicon-ductors must be treated as continuous distributions oflevels rather than as discrete levels [332]. We shall as-sume that the densities of states in the conduction andvalence bands of the semiconductor are known func-tions of energy, and that the occupations of these levelsare characterized by quasi-Fermi levels [257]. Then theprobability that the state of energy E in the conductionband is occupied by an electron is

fc(E) = 1/{1 + exp[(E − Fc)/kT ]}, (309)

where Fc is the quasi-Fermi level for the conductionband, k is Boltzmann’s constant, and T is absolute tem-perature. A corresponding expression applies in the va-lence band, with quasi-Fermi level Fv. For a system inthermal equilibrium, the quasi-Fermi levels are equalto each and become the Fermi level EF. In an excitedsystem we have Fc ( Fv, and we can use the separationof the quasi-Fermi levels as a measure of the excita-tion. The use of quasi-Fermi levels greatly simplifies thetreatment of systems with many energy levels, or withcontinuous distributions of levels, because one quantityrepresents the occupation probability of many levels.The concept of quasi-Fermi level in an excited systemis valid provided the scattering rate of carriers withina band is rapid compared to the recombination rate be-tween bands, i.e., if the carriers within the conduction

band and within the valence band rapidly establish aquasi-equilibrium among themselves although the con-duction band and the valence band are not in equilib-rium with each other For semiconductors with substan-tial numbers of free carriers, carrier-carrier scatteringwill lead to the establishment of quasi-equilibrium (seee.g. [333, 334]).

The original semiconductor lasers were p-n junc-tions prepared by diffusion of acceptor impurities inton-type GaAs, and this is still one of the most commonstructures. Semiconductors with �k—conserving transi-tions at the energy gap are strongly favored for lasing[335], but some impurity levels can lead to stimulatedemission in indirect-gap semiconductors, e.g. [336].All the p-n junction lasers are excited by passingcurrent through the p-n junction, and the excitation rateis characterized by the current density. When a forwardcurrent flows, electrons are injected into the p-typematerial and holes are injected into the n-type material,te latter to a much smaller extent partly because ofthe lower hole mobility. In heterojunctions, potentialbarriers play a major role in the injection of carriers[337]. The excess of electron and hole concentrationsover their equilibrium values creates a local populationinversion and leads to stimulated emission of photonsat sufficiently high excitation levels. The layer nearthe p-n junction where this occurs is called the activeregion or active layer of the device. Fig. 91 showsthat the effective thickness of the active layer ingraded junction lasers increases as the current densityincreases. This leads to smaller quasi-Fermi levelseparations and to less efficient use of the excitation forlasing. Heterostructure lasers (see also [337]) containbuilt-in potential barriers for the electrons which tendto confine them to regions of fixed width. Thus theexcitation can be used most effectively.

A second class of excitation methods involves excita-tion with photons [339] or with an electron beam [338].For optical excitation, the active layer thickness will beof the order of 1/α, where α is the absorption coefficientof the incident photons. For electron beam excitation,the active layer thickness will be of the order of the pen-etration depth of the electrons, which is a function oftheir energy (see, e.g. [340]). In both cases, diffusion ofcarriers will add a distance of the order of the diffusionlength to the thickness given. If sufficiently thin samplesare used, the excitation state may be relatively uniformthrough the sample, provided surface recombination isunimportant. The excitation rate for the externally ex-cited structure can be converted to an equivalent cur-rent density. For photons, the absorbed photon flux ismultiplied by the electronic charge provided that eachabsorbed photon give rise to an electron-hole pair. Itshould be added that values of about 3Eg are necessaryto create one e-h pair [339]. This means that the mainpart of the incident pump energy is transferred into heat.This is one of the disadvantages of e-beam pumping.

The value of the gain coefficient gth at the lasingthreshold is given by a very simple calculation (see also[338]). If the laser cavity is bounded by parallel surfaceswith reflectivities R1 and R2, separated by a distanceL , then the amplification of the photon intensity on a

3413

Figure 91 Spatial distribution of the recombination rate for graded GaAs p-n junction lasers with the indicated forward current densities at 300 K.The donor concentration is 3×1018 cm−3, and the acceptor gradients at the junction are (a) 1022 cm−4 and (b) 1023 cm−4. The ordinate is the nominalcurrent density defined in Equation 317, in A/cm2, equivalent to 6.2 × 1022 recombination/cm3s. Note the increase in effective active layer thicknessas the current density increases, particularly in (a). (after [338]).

round trip through the device is R1R2exp[2(g − α)L],where α is an effective absorption coefficient, e.g., forfree-carrier absorption. The lasing condition is that thisamplification factor be unity. Thus

gth = α +(

)log

), (310)

where R = (R1 R2)1/2. The current density for whichlasing occurs, i.e., the lowest current density for which greaches gth at any photon energy, is the threshold currentdensity Jth, and is one of the principal characteristicsof a semiconductor laser.

The rate at which photons are emitted per unit volumecan be written as (see, e.g., [341])

r (E)dE = [rspon(E) + Nrstim(E)]dE, (311)

where N is the average number of photons per mode,given for thermal equilibrium by

N (E) = 1

exp(E/kT ) − 1. (312)

The term rspon (311) gives the rate of spontaneousdownward transitions of the electronic system, and rstimis the difference between the stimulated rates of down-ward and upward transitions.

The spontaneous and stimulated functions can bewritten

rspon(E) = a∫

ρc(Eu)ρv(E1)|M |2 fu(1 − f 1)dEu,

(313a)

rstim(E) = a∫

ρc(Eu)ρv(E1)|M |2( fu− f 1)dEu,

(313b)

a = 4Ne2 E/m2 h2c3, (313c)

where fu and f1 are the probabilities that the upper andlower states involved in the transition are occupied, Nis the index of refraction, and E1 = E − Eg − Eu withthe convention for measuring energies from nominalband edges separated by the nominal energy gap Eg.The squared matrix element |M |2 is averaged over allpolarizations of the radiation, and is a function of bothEu and E1. Equation 313 ignores Coulomb interactionbetween electron and hole (see below), which shouldbe relatively weak when large carrier concentrations arepresent. Lasing in materials for which exciton effectsare important has been treated by Haug [342].

When the populations are characterized by quasi-Fermi levels as in Equation 309, the relation betweenrstim and rspon is

rstim(E) = rspon{1 − exp[(E − �F)/kT ], (314)

where �F = Fc − Fv is the difference of the quasi-Fermi levels for the conduction band and the valenceband. The relation between the local gain coefficientglocal and rstim is

glocal(E) = −α(E) = (πch3/2/N E)2rstim(E), (315)

where the minus sign arises because we have definedrstim and glocal to be positive when radiation is emit-ted, while the absorption coefficient α is positive when

3414

radiation is absorbed. Combine (314) and (315) weobtain

rspon(E) = (N E/πch3/2)2

× α(E/{exp[(E − �F)]/kT ) − 1}. (316)

in thermal equilibrium, when �F = 0, this reduces tothe well-known result of Van Roosbroeck and Shokley[343]. For semiconductor lasers the term in �F is ex-tremely important. Equation 314 shows that there canbe gain only for photon energies E ≺ �F .

The local gain coefficient is a function of photon en-ergy, but for most laser cases we are interested primarilyin its peak value gmax. In Fig. 92 it plots gmax vs. exci-tation rate, which is characterized for this purpose bythe nominal current density

Jnom = 10−4e∫

rspon(E)dE, (317)

the current density required to maintain the actual ex-citation rate in a hom*ogeneous layer 1 µm thick. The

Figure 92 Maximum local gain coefficient (the value of glocal(E) at thephoton energy with maximum gain) vs nominal current density, definedin Equation 317, for various temperature and for (a) ND = 3 × 1018,NA = 6 × 1018 cm−3; (b) ND = 3 × 1018, NA = 4 × 1018 cm−3;(c) ND = 1019, NA = 1.3 × 1019 cm−3. The ion screening temperatureis 30,000 K (after [338]).

Figure 93 Temperature dependence of the nominal current density re-quired to reach a gain coefficient gmax = 50 cm−1, taken from resultsof Fig. 92. The curves are labeled by the values of donor and accep-tor concentration, respectively, in units of 1018 cm−3. The temperaturedependence of the threshold current density is also a function of thetemperature dependence of the losses and of the effective active layerthickness (after [338]).

curves of Fig. 92 are given for several sets of valuesof donor and acceptor concentrations, and for severaltemperatures from absolute zero to 300 K. Note thatincreasing the compensation, i.e. increasing ND + NAfor a given |ND + NA|, raises the nominal current den-sity required to reach a specified gain at low tempera-ture, but lowers it at room temperature (cf. Figs 92a–c). Fig. 93 shows the nominal current density requiredto reach a gain of 50 cm−1 for each case in Fig. 92.It is an indication of the temperature dependence ofthe threshold current density for typical semiconductorjunction lasers but makes no allowance for the effect ofthe structure or for losses such as free carrier absorption.A similar analysis has also been made by Hwang [344] ,whose curves of gain versus current density are similarto those given in Fig. 93. Both calculations agree wellwth experiment. The photon energy at which lasing oc-curs is found by Hwang [344] to be in good agreementwith experimental results (details see [338, 339, 342]).

4.3.2. Study of excitons lasingWith increasing excitation intensity, frequently laser ac-tion is observed in the excitonic luminescence. How-ever, the direct recombination of an exciton can nevergive rise to laser action, because the coupled exciton-photon system corresponds in the resonant approxima-tion to two linearly coupled harmonic oscillators. Theequations of motion of this system do not contain thenonlinearity which is necessary to describe laser action.The participation of a third field is required in order tointroduce the possibility of laser action [342], i.e., thelaser action in exciton systems has to be a parametricprocess in which a pump field, a signal field and an idlerfield participate.

Below we describe a scheme for lasing action in-volving excitons in a pure crystal. In Refs. [345, 346]a general theory of various spontaneous and stimulatedexciton recombination processes has been developed.In our analysis we assume a two-band model for thecrystal. Atoms at lattice sites interact with each otherand an electromagnetic field. The analysis is addressed

3415

to tight-binding excitons in the first step. According to[346] the weak-binding case of Wannier-Mott excitonsfollows a similar description but is more involved. Innext section we follow very close the results of Liu andLiboff [346].

The Hamiltonian of the entire system of crystal andelectromagnetic field is [346]

H = Hxtal + Hem + Hint (318)

The Hamiltonian of the crystal can be written in theWannier representation

Hxtal =∑

Eec†ρcρ +∑

Ehd†ρdρ

+ h∑ρ1,ρ2

W (ρ1 − ρ2)c†ρ1cρ1 d

†ρ2

dρ2 . (319)

In this expression cρ and dρ operators refer, respec-tively to electrons and holes at site ρ. Their respectiveenergies are Ee and Eh. Indicated operators obey theanticommutation relations:

{cρ, c†ρ ′ } = {dρ, d†ρ ′ } = δρ,ρ

′ ,

{cρ, dρ ′ } = {cρ, d†ρ ′ } = {c†ρ ′, dρ ′ }

= {c†ρ , d†ρ ′ } = 0. (320)

The function in Equation 319 represents the interac-tion between electrons and holes and has the integralrepresentation

hW (ρ1 − ρ2) =∫ ∫

d�rd�r ′w∗c (�r − �ρ2)w∗

v(�r ′ − �ρ2)

× e2

|�r − �r ′|wc(�r ′ − �ρ2)wv(�r − �ρ1),

(321)

where wc(�r − �ρ) and wv(�r − �ρ) are Wannier func-tions relevant to the conduction and valence bands,respectively.

The Hamiltonian Hxtal may be diagonalized as fol-lows (see also [347]). For this purpose we define theoperators

B†ρ = c†ρd†

ρ , (322a)

Bρ = dρcρ. (322b)

The operator B†ρ creates an electron-hole pair

whereas Bρ annihilates the pair. The operator

µρ = 1

2(c†ρcρ + d†

ρdρ) (323)

indicates if the atom at the site ρ is excited or not.Its eigenvalues are (1, 0). Combining the latter threeequations gives

[Bρ′ , B†

ρ] = δρ,ρ′ (1 − 2µρ). (324)

We further define the operator

b†q = 1√

∑ρ

eiqρ B†ρ, (325)

which represents the creation of an exciton in the prop-agating state �q. The total number of atoms in the crystalis N . The commutation relation for exciton operatorsthe appears as [342]

[bq, b†q′ ] = δq,q ′ − 2

∑ρ

ei(q−q′)µρ. (326)

In this manner wereach the important result that ex-citon operators are boson operators only when the num-ber of excited atoms in the crystal is small comparedto the total number of atoms N . In this even, the sec-ond term of Equation 326 can be neglected, giving[bq, b†

q′ ] = δq,q ′ . The transformation Equation 325 thendiagonalizes Hxtal as

Hxtal =∑

Eqb†qbq, (327a)

where

Eq = Ee + Eh + εq, (327b)

and

εq = h∑

W (ρ)e−iqρ. (327c)

The diagonalization Equation 327a demonstrate thatexcitons are collective excitations of the entire crystal.

In the limit of intense lasing, cited authors [346] re-placed the exciton and photon amplitudes with theirclassical counterparts (see also [348–350]:

b†q = βq exp

[i(ω0

q + �1 + iη)t], (328)

a†j = aj

[i(ω0

j + �2 + iη)t], (329)

where hω0q = Eq . The operator a†

j creates a photonof frequency ωj and γ is the decay rate of this mode.The coefficient η is introduced to describe the decay ofexcitons. The frequency �1 and �2 remain arbitrary.Equations for a†

j and βq have following expressions:

(i�1 − η)βq = − i√N

∑ρ

Gρσρa†j eiqρ, (330)

(i�2 − γ )a†j = i√

∑ρ

∑q

G∗ρβqe−iqρ. (331)

Substituting Equation 330 into 331, and take intoaccount of the resonance condition, we have

ω0q + �1 = ω0

j + �2, (332)

3416

and equating the real and imaginary parts of the result-ing equation gives

γ = η

η2 + �21

∑ρ

|Gρ |2σρ, (333)

�2 = − �1

η2 + �21

∑ρ

|Gρ |2σρ. (334)

It follows that

�1

�2= − η

γ. (335)

While the difference frequency

� ≡ �1 − �2, (336)

Equation 560 yields

�1 = �

1 + γ

, �2 = − �

1 + γ

. (337)

Substituting the exponential forms (322) and (329)into next equation

dtσρ = 1

τ(s0 − σρ) + 2i√

[ ∑q

Gρbqa†j eiqρ

−∑

G∗ρb†

qaje−iqρ

](338)

in steady state we find

0 = 1

τ(s0 − σρ) + 2i√

[ ∑q

Gρb∗qa∗

j eiqρ

−∑

G∗ρbqaje

−iqρ

]. (339)

Inserting Equation 330 into the latter equation gives

σρ = s0 − 4τ

η[1 + �2/(η + γ )2]|Gρ |2〈n〉σρ, (340)

where s0 is an asymptotic inversion value. Here, wehave written

n = 〈a∗j aj〉 (341)

for the average number of photons in the jth mode.Solving Equation 340 for the local inversion, we

obtain

σρ = s0

[1 + 8πτ 〈n〉e2d2

hω0j m2V η{1 + �2/(η + γ )2

sin2�kj�ρ]−1

(342)

For the average inversion

〈σ 〉 = 1

∑ρ

σρ, (343)

we find

〈σ 〉 = s0

×∑

[1 + 8πτ 〈n〉e2d2

hω0j m2V η{1 + �2/(η + γ )2

sin2�kj�ρ]−1

(344)

In the continuum limit Equation 344 becomes

〈σ 〉 = s0

∫dr

1 + (4τnγ /ρS) sin2 �k�r

], (345)

where N = ρV and 〈n〉= nV . Thus, ρ is atomic densityand n is photon density. Furthermore,

S = hω0j m2γ η

2πe2ρδ2f

[1 + �2

(η + γ )2

]. (346)

Substituting the form for γ given by Equation 343into the last equation, together with the continuum formof next equation

Gρ = − e

√2π

hωjVsin(�kj�ρ)ejδj (347)

gives the following relation

S = s0

∫dr

sin2�k�r1 + (4τnγ /ρS)sin2�k�r , (348)

with∫

dr V , Equations 345 and 347 may be manipu-lated to yield

n = ρs0

4πγ

(1−〈σ 〉

). (349)

Finally, integrating Equation 345 we obtain

〈σ 〉s0

= 1√1 + 4τnγ /ρS

. (350)

With Equation 574 this expression becomes

〈σ 〉 = S

(1+

√1 + 4s0

). (351)

For laser action to be realized, one must have the pho-ton density (at frequencyωj), n > 0. From Equation 349this condition is satisfied providing s0 > 〈σ 〉, or fromEquation 287, s0 > S, which with Equation 349 gives

s0 >hω0

j m2γ η

πe2ρδ2f

[1 + �2

(η + γ )2

]. (352)

At resonance, � = 0 and the right-hand side ofthe preceding expression is minimized. In this limitEquation 348 reduces to the same form given by the

3417

Schawlow-Townes criterion [351] for randomly dis-tributed atoms. However, in Equation 348, the matrix el-ement δf ≡〈vρ|p|cρ〉 is appropriate to excitons whereasthe matrix element in the Schawlow-Townes expressionis relevant to a single atom.

For Wannier-Mott excitons the criterion Equa-tion 348 maintains with δf replaced by

δW−M ≡∑ρ ′

F(ρ ′)∫

drw∗v(r − ρ)pwc(r − ρ − ρ ′).

(353)

In this expression F(ρ) is the wavefunction of aWannier-Mott exciton [352]. In preceding investiga-tion Liu and Liboff have considered rigid lattices. Iflattice vibrations are present, exciton-phonon interac-tions will cause exciton diffusion [352] which may di-minish coherence for lasing. However, lasing will stilloccur [343] providing the relaxation time for excitondiffusion is longer than the exciton relaxation time, τ

in Equation 328. This will be the case for sufficientlyweak excito-phonon interaction (details see below).

As was shown above optical transitions in pure III–V compounds which can be used for laser action areband-band transitions. In II–VI compounds (as well asLiH [1] and etc.), the recombination process of elec-trons and holes via exciton states is more favorablethan the band-band transition [352–354]. During lastfour decades laser action has been obtained in II–VIcompounds by electron beam bombardment [355–357],by optical excitation [353, 358–360]. The laser transi-tions involve the A1 - nLO phonons, where n = 1,2. Gain measurements [357, 359] and simultaneousmeasurements of the emission intensities of the A1line (direct A - exciton recombination [353]) and theA1 - LO line also confirm the statement that in CdS theA1 - LO (A1 - 2LO) line starts to lase for sufficientlyhigh pump rates (see also Fig. 94) (details see [356]).In cited papers Haug [342] calculated the temperature

Figure 94 The onset of stimulated emission in CdSe at 77 K (after[356]).

Figure 95 Maximum gain frequency �max vs temperature (after [342]).

dependence of the maximum gain frequency at thresh-old (see Fig. 95). The result is simple in the low-temperature limit

�max → 3

2kT (354a)

and also in the high-temperature limit

�max → (3κkT/B)2/5 exp(−2hν/5kT, (354b)

where 2κ = 1.25 × 1012 s−1, corresponding to lossesof 100 cm−1, B = 1.55 × 1035 erg−3/2 s−1 for CdScrystals. These limiting results have also been givenby Mashkevich et al. [360]. The typical experimentalgain Ithr (T ) dependence, obtained in paper [359] ispresented in Fig. 96. There are shown some spectra ofstimulated emission at different temperatures. Authorsof [359] indicated some contradiction of their experi-mental results with theoretical description.

4.4. Nonlinear properties of excitons inisotope-mixed crystals

Another application of isotope pure and isotope mixedcrystals that will be discussed here is related to the pos-sibility of using an isotopically mixed medium (e.g.,LiHx D1−x or 12Cx

13C1−x ) as an oscillator of coherentradiation in the ultraviolet spectral range [361, 362].To achieve this, the use of indirect electron transitionsinvolving, say, LO phonons was planned [342, 363].As was shown above using indirect electron transi-tions involving phonons to degenerate coherent radi-ation in semiconductors was originally proposed byBasov et al. (see [355] and reference therein). Kulevskyand Prokhorov [358] were the first to observe stimu-lated radiation using emission lines of LO phonon rep-etitions in CdS crystals on two photon excitation (seealso [364]). The detection of LO phonon replicas offree-exciton luminescence in wide-gap insulators at-tracted considerable attention to these crystals (see e.g.Plekhanov [5], Plekhanov [6]). At the same time it iaallowed one to pose a question about the possibilityof obtaining stimulated emission in UV (VUV) region(4–6 eV) of the spectrum, where no solid state sourcesfor coherent radiation exist yet. In the first place this

3418

Figure 96 The dependence Ithr(T) and some examples of (above-threshold) lasing spectra (in the range of A1–1LO; A1–2LO phonons) at differenttemperature (after [359]).

related to the emitters working on the transitions ofthe intrinsic electronic excitation (exciton). The lastone provides the high energetical yield of the coher-ent emission per unit volume of the substance. The re-sults obtained on solidified xenon (Basov et al. [365])and argon (Schwenter et al. [366]) under electron beamexcitation with following excimer molecules emissionform an exception.

In this part we will discuss the investigation results ofthe influence of the excitation light density on the reso-nant secondary emission spectra of the free-exciton inthe wide-gap insulator LiHx D1−x (LiH1−x Fx ) crystals.The cubic LiH crystals are typical wide-gap ionic in-sulator with Eg = 4.992 eV [1] with relatively weakexciton-phonon interaction however: EB/hωLO = 0.29where EB and hωLO are exciton binding energy and

Figure 97 Photoluminescence spectra of free excitons at 4.2 K: 1—LiH;2—LiHx D1−x and 3—LiD crystals (after [5]).

longitudinal optical phonon’s energy, respectively. Be-sides it might be pointed out that the analogous rela-tion for CdS, diamond and NaI is 0.73, 0.45 and 12.7,respectively (Plekhanov [367]). Fig. 97 depicts, as anexample, the exciton luminescence spectrum of pure(LiH and LiD) and mixed (LiHx D1−x ) crystals at a lowtemperature. Analogous results for 12Cx

13C1−x mixeddiamond crystals are shown in Fig. 98. A common fea-ture of all three spectra depicted in Fig. 97 is a phonon-less emission line of free excitons and its 1LO and 2LOphonon replicas. An increase in the density of the ex-citing light causes a burst of the radiation energy in thelong-wave wing of the emission of the 1LO and 2LOrepetitions (see Fig. 99) at a rate is higher for the 1LOrepetion line [361]. A detailed dependence of the lu-minescence intensity and the shape of the 2LO phononreplica line are presented in Fig. 100 and Fig. 101, re-spectively. The further investigations have shown [369]

Figure 98 Cathode—luminescence spectra of isotopically modified di-amond at 36 K. Intrinsic photo-assisted recombination peaks are labelledin the top spectrum, those from boron-bound excitons in that at the bot-tom (after [368]).

3419

Figure 99 Luminescence spectra of free excitons in LiH crystals in theregion of the emission lines of 1LO and 2LO phonon repetitions at 4.2 Kfor low (1) and high (2) density of excitations of 4.99 eV photons. Thescales of the different curves are different (after [369]).

Figure 100 The dependence of the intensity in the maximum (1) andon the long-wavelength side (2) of 2LO replica emission line of LiHcrystals on the excitation light intensity (after [369]).

that with the increase of the excitation light intensity atthe beginning a certain narrowing can be observed, fol-lowed by widening of the line of 2LO phonon replicawith a simultaneous appearance of a characteristics,probably mode structure. From Fig. 100 it can be seenthat the coupling between longwavelength lumines-cence intensity and excitation light intensity is not onlylinear, but, in fact, of a threshold character as in caseof other crystals [332, 364]. A proximity of the excitonparameters of LiH and CdS (ZnO) crystals allowed tocarry out the interpretation of the density effects in LiHon the analogy with these semiconducting compounds.Coming from this in the paper [369] it was shownthat for the observed experimental picture on LiH crys-

Figure 101 The dependence of the shape of 2LO replica line on theexcitation intensity (I0) light: 1—0.05I0; 2—0.09I0; 3—0.35I0; 4—I0

(after [369]).

tals to suppose the exciton-phonon mechanism of lightgeneration [342] is enough the excitons density about1015 cm−3. This is reasonable value, if the high qualityof the resonator mirrow—the crystal cleavage “in situ”and relatively large exciton radius (r = 40 A [5] istaken into account. To this light mechanism generationmust be also promoting a large value of the LO phononenergy (hωLO = 140 meV) . Owing to this the radiativetransition is being realized in the spectral region with asmall value of the absorption coefficient, and thus witha small losses in resonator (details see also [367]).

In the present part we briefly analyse the shift offree exciton luminescence on the crystal lattice depen-dence in the first step it will be considered F doped ofLiH crystals. The reflectance spectra of the investigatedcrystals with clean surface (cleaved in LHeT) had a dis-tinctly expressed excitonic structure. Typical reflectionspectra of LiH1−x Fx with mirror surface is depicted inFig. 102 [370]. The crystal cleavage is carried in super-fluid helium of helium cryostat bath. In the Fig. 102 forthe comparison is shown the reflection spectrum of purecrystal of LiH (curve 1). All spectra possess the identi-cal longwavelength exciton structure: we can see 1 s and2 s exciton states. It’s clearly seen, that the F additionin LiH crystals leads, as it naturally expected, to short-wavelength shift of the spectrum as the whole. With thegrowth x is increased the energetic interval between 1sand 2s exciton states and at the same time their short-wavelength shift is different and has saturated character

3420

Figure 102 Reflection spectra of LiH (1); LiH1−x Fx (x = 0.06%) (2);and (x = 1.6%) (3) at 78 and 4.2 K (2′) (after [370]).

[363]. At the maximum value of x ≤ 2.5% the excitonRydberg (EB), obtained on the hydrogen-like formu-lae is equal 75 ± 3 meV (EB = 40 meV for LiH). ForLiD1−x Fx crystals at x ≤ 1.6% exciton binding energyis equal EB = 57 ± 2 meV. Supposing the linear depen-dence EB on the x concentration in LiH1−x Fx crystals,we obtain EB = 62 ± 25 meV for x = 1.6% and theexperimental meaning of this parameter is equal 67 ±3 meV. Such a large error at the theoretical extrapola-tion is connected with the large errors when determin-ing EB for LiF crystals [371, 372]. However, despite ofthe identical structure of all free-excitons luminescencespectra, it is necessary to note a rather big variation ofthe luminescence intensity of the crystals from the dif-ferent batches observed in the experiment. Thereforethe crystals possessing the maximum value of the freeexciton luminescence quantum yield were chosen formeasurements of the density effects.

The luminescence spectra of virgin and mixed crys-tals are very likely and consist of narrow zero phononline and its more wider LO replicas [6, 369]. As wellas in the reflection spectra (see above) the dopant ofLiH crystals with deuterium or fluorine is drawn toshortwavelength shift of the luminescence spectrumas a whole (Figs 97 and 103). The increasing of thedeuterium concentrations leads to the widening of theluminescence line (see also [5]). The increasing ofthe fluorine concentration is causing (Figs 103, 104),except the spectrum shift, the sharp ignition of zerophonon line intensity in comparison with the lines ofLo-replicas intensity (see also [370]). Except indicatedeffects, the fluorine activation of the LiH (LiD) crys-tals is shifting the temperature quenching of the freeexcitons luminescence in the more high region. As on

Figure 103 Free excitons luminescence spectra of LiH (1);LiH0.984F0.016 and LiD0.992F0.008 crystals cleaved in liquid helium(after [373]).

Figure 104 Reflection (1) and luminescence (2, 2′) spectra at 4.2 K and260 K (3) of LiH1−x Fx mixed crystals cleaved in liquid helium (after[373]).

example on the Fig. 104 is depicted the reflection andluminescence spectra of LiH1−x Fx crystals in the widetemperature range [5]. It can be seen that free exci-tons zero-phonon emission line in these crystals is reli-ably registrated practically up to the room temperature.Taking into account the mixed crystals lattice poten-tial relief it could (compare Fig. 105) not be excludedabsolutely the possibility to obtain the laser generationon the zero-phonon line emission that was already ob-tained in the paper [374].

In conclusion of this section we should underlinedthat if the observable mode structure is really causedby the laser generation it may be smoothly tuned inthe region of energies 4.5 ± 5.1 eV owing to smoothtransition of the line emission energy in the LiHx D1−x

(LiHx F1−x ; LiDx F1−x ) mixed crystals [5, 6] as well asin the range 5.35–5.10 eV in 12Cx

13C1−x mixed crystals(see also Fig. 20 in [1]).

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Figure 105 Emission spectra of CdS0.9Se0.1 at different levels of excita-tion: 102(1); 104(2) and 106 (3) Wt/cm2. 0—zero-phonon line; I—III—LO—phonon replicas and dashed line is the absorption spectra, T = 2 K(after [374]).

Chapter 5. Other unexplored applicationsof isotopic engineering

5.1. Isotopic information storageThe current rapid progress in the technology of high-density optical storage makes the mere announcingof any other thinkable alternatives a rather unthankfultask. An obvious query ‘who needs it and what for?’has, nevertheless, served very little purpose in the pastand should not be used to veto the discussion of non-orthodox technological possibilities. One such possi-bility, namely the technology of isotopic informationstorage (IIS) is discussed in this paragraph.

Isotopic information storage may consist in assigningthe information ‘zero’ or ‘one’ to mono-isotopic mi-croislands (or even to a single atoms) within a bulk crys-talline (or thin film) structure. This technique could leadto a very high density of ROM-type (read-only memoryor permanent storage) information storage, probablyup to 1020 bits per cm3. The details are discussed byBerezin et al. [100, 375, 376]: here it notes only that theuse of tri-isotopic systems (e.g., 28Si; 29Si; 30Si) ratherthan di-isotopic (e.g., 12C; 13C) could naturally leadto direct three dimensional color imaging without theneed for complicated redigitizing (it is known that anyvisible color can be simulated by a properly weightedcombination of three prime colors, but not of two).

Indeed, let us assume that the characteristic size ofone information-bearing isotopic unit (several atoms)is 100 A. Then 1 cm3 of crystalline structure, e.g. dia-mond, is able to store roughly (108)3/100 = 1022 bitsof information [376]. This capacity greatly exceeds thatneed to store the information content of all literatureever published (∼= 1017 bits), including all newspapers.

The main potential advantage of isotope-mixed crys-tals lies in the fact that the information is incorporatedin the chemically hom*ogeneous matrix. There are no

chemically different impurities (like in optical storagewith color centres) or grain boundaries between islandsof different magnetization (like in common magneticstorage). The information in isotope-mixed crystals ex-ists as a part of the regular crystals lattice. Therefore,the stored information in isotope-mixed crystals is pro-tected by the rigidity of the crystal itself. There areno “weak points” in the structure (impurities, domainwells, lattice strain etc) which can lead to the infor-mation loss due to bond strains, enhanced diffusion,remagnetization, etc. Differences in the bond lengthsbetween different isotopes (e.g., 28Si–29Si or 29Si–30Si;H–D and so on) are due to the anharmonicity of zero-point vibrations (see, e.g., [28]). This is not enoughfor the development of any noticeable lattice strainsalthough these differences are sufficiently large to bedistinguishably detected in IIS-reading).

The mechanism potentially available in IIS for thewriting-in of the information may be divided into twogeneral categories. The first category refers to all tech-niques which are able to direct externally a particularatom to a specified position on the surface of the grow-ing crystal structure. Any beam technique with abilityof focusing on 1 A scale could, in principle, appear fea-sible for such purpose. The second category relates toall ’internally operated’ possibilities, i.e., delivering ofthe required isotope as a part of the molecule and de-positing it in a particular position through some chem-ical process (e.g., exchange reaction, chemisorption,etc). This group of possibilities is, in fact, similar to theDNA-RNA mechanism in actual biological informationtransfer in living systems. Some chemically very simplecrystals have, nevertheless, a very complex lattice struc-ture. One known example is elementary boron [377,378], which can crystallize in a beta-rhombohedralstructure with 105 atoms in a unit cell with 15 crystallo-graphically nonequivalent positions. Moreover, variousatoms have 3 different coordination numbers: 91 atomshave the coordination number 6; 12 atoms, 8 and 2“special” atoms have 9 nearest neighbors [378]. Thispeculiarity of the crystalline boron is rather surprisingin view of the fact that it is an elementary (monoatomic)crystal. Similar possibilities exist for lattices with 2 el-ements, e.g., the silicides of manganese are known toform very complex structures [379]. It is therefore, le-gitimate to consider the ability of such structures withcomplex crystal lattice to provide the basis for compactinformation storage within the frameworks of the modelof an alternative (nonorganic) genetic code. In princi-ple, isotopic combinations could provide the basis forthe storage information even in “simple” crystals (e.g.,in carbon or silicon-based structures) and not only justin crystals with complex unit cells. It is possible evento raise the question what (if any) effects could be con-nected with isotopic permutations in “regular” biology(e.g., 12C- and 13C combinations in various fragmentsof DNA) Nevertheless, in crystals with large and so-phisticatedly constructed unit cells the already “preex-isting” significant level of structural complexity makesthem, apparently, more preferable candidates for theevolution game of isotopic information-bearing arrays(details see [378]).

3422

The possible key to a 3D-access could, in principle,be provided by any method which is able to probe thenuclear mass and/or magnetic moment of a single atomat a particular lattice site below surface. Without de-tailed elaborations the following possibilities presentthemselves:

1. Spectroscopy of localized crystal vibrations (seealso, [1] which generally contains the information onthe vibrational frequencies of an individual atom (due tosquare-root-of mass) dependence of the vibrational fre-quencies this could be a spectroscopically pronouncedeffect.

2. Recoil phenomena (e.g., Rutherford backscatter-ing).

3. Nuclear magnetic resonance (NMR).4. Spin-sensitive neutron scattering.

Of course, there has to be a great deal of perfection-ing refinement to these or other techniques before theycan actually be applied for their use in isotope-mixedcrystals.

5.2. Isotopic structuring for fundamentalstudies

Isotopic substitution has made it possible to produce theobjects of research that earlier were simply inaccesible(with the exception of the LiH-LiD system). The use ofsuch objects allows the investigation of not only theisotope effects in lattice dynamics (elastic, thermal, andvibrational properties (see reviews [2–6]) but also theinfluence of such effects on the electronic states (therenormalization of the band-to-band transition energyEg, the exciton binding energy EB, and the size of thelongitudinal-transverse splitting �LT [5, 6]).

Furthermore, it is widely known that the melting andboiling points of ordinary water and heavy water (D2O)differ by a few degrees centigrade. For elements heav-ier than hydrogen the isotopic differences in meltingpoints (�T ) of elemental and complex solids are gen-erally smaller but also detectable. It is quite surprising,however, that there are almost no reports of direct mea-surement of these differences in the literature.

Another noticeable fact is that sometimes the isotopeeffect shows a drastic “self-amplification,” e.g., isotopicreplacements of Ba and Ti in BaTiO3 (both are heavyelements) can shift the phase transitions temperaturesby as much as 200 K [380]. The reason(s) for suchselective anomalies are not yet clearly established. Thispart are widely considered early in the book [1].

5.3. Other possibilitiesHere we shall briefly list a few additional possibilitiesof isotopic structuring (see also [2–4]).

1. Very perspective direction of isotope engineeringcould be based on exploiting the differences in thermalconductivity (see above) between isotopically pure andisotopically mixed solids for purposes such as phononfocusing, precise thermometry based on isotopically-gradiented structures, etc.

2. The use of isotopically structured Ni-films forneutron interference filters has been reported byAntonov et al. [381].

3. Isotopically structured light devices. This couldslightly shift the spectral characteristics and lead tosome changes in the kinetics of energy transfer, modifythe lifetimes, recombination rate, etc.

4. Since the speed of sound is proportional to√

variations in isotopically structured acousto-electronicdevices (transducers, surface acoustic wave devices,etc.) could be significant, especially in achieving phasedifferences over the relatively short isotopically distin-guished paths (see also [177]).

5. The possibility to get at rather low pressure thetransition of metal-insulator with metallic conductiv-ity on the zone genetic related with hydrogen in LiHcrystals (see also [177] and references therein).

6. The use of the isotope boundary for Mossbauerfiltration of synchrotron radiation, since this makes itpossible to get rid of the background noise caused bythe interaction between synchrotron radiation and theelectrons in matter [382].

7. Isotope-based quantum computers (see e.g. [177,383, 384]).

Above we have outlined several, mostly untested pos-sibilities arising from exploiting differences in variousstable isotopes and purposeful isotopic structuring. Theabove examples of the potential capabilities of isotopicengineering by no means an exhaustive list.

6. ConclusionIn this review, we have presented briefly the results ofexperimental and theoretical studies of the objects of re-search that earlier were simply in accessible (naturallywith exception of LiHx D1−x crystals). The use of suchobjects allows the investigation of not only the isotopeeffects in lattice dynamics (elastic, thermal and vibra-tional properties) but also the influence of such effectson the electronic states via electron-phonon coupling(the renormalization of the band-to-band transition en-ergy Eg, the exciton binding energy EB and the size ofthe longitudinal-transverse splitting �LT).

Substituting a light isotope with a heavy one in-creases the interband transition energy Eg (excludingCu-salts) and the binding energy of the Wannier-Mottexciton EB as well as the magnitude of the longitudinal-transverse splitting �LT. The nonlinear variation ofthese quantites with the isotope concentration is dueto the isotopic disordering of the crystal lattice and isconsistent with the concentration dependence of linehalfwidth in exciton reflection and luminescence spec-tra. A comparative study of the temperature and iso-topic shift of the edge of fundamental absorption for alarge number of different semiconducting and insulat-ing crystals indicates that the main (but not the only)contribution to this shift comes from zero oscillationswhose magnitude may be quite considerable and com-parable with the energy of LO phonons. The theoreticaldescription of the experimentally observed dependenceof the binding energy of the Wannier-Mott exciton EB

3423

on the nuclear mass requires the simultaneous consid-eration of the exchange of LO phonons between theelectron and hole in the exciton, and the separate inter-actions of carriers with LO phonons. The experimentaldependence EB ∼ f (x) for LiHx D1−x crystals fits inwell enough with the calculation according to the modelof large-radius excitonin a disordered medium; hence itfollows that the fluctuation smearing of the band edgesis caused by isotopic disordering of the crystal lattice.

Details analysis the process of the self-diffusionin isotope pure and heterostructures was done inChapter 1. This chapter was organized around generalprinciples that are applicable to all materials. There isbriefly discussed very popular in nowadays the SIMStechnique. As is well-known self-diffusion is the migra-tion of constituent atoms (isotopes) in materials. Theknowledge obtained in self-diffusion studies is pivotalfor the understanding of many important mass transportprocess including impurity diffusion in solids to use indifferent semiconductor devices.

The new reactor technology-neutron transmutationdoping (NTD) of semiconductors was described inChapter 2. Capture of thermal neutrons by isotope nu-clei followed by nuclear decay produces new elements,resulting in a very number of possibilities for isotopeselective doping of solids. There are presented differentfacilities which use in this reactor technology. The fea-sibility of constructing reactors dedicated to the produc-tion of NTD silicon, germanium (and other compounds)was analyzed in terms of technical and economic viabil-ity and the practicality of such a proposal is examined.The importance of this technology for studies of thesemiconductor doping (materials for different devices)as well as metal-insulator transitions and neutral impu-rity scattering process is underlined.

The use of the isotopes in a theory and technologyof the optical fibers we considered in Chapter 3. Thischapter is addressed to readers who wish to learn aboutfiber communications systems and, particular, about theproperties of optical fibers. Very briefly in this chapterwe describe the Maxwell equations as well as waveelectromagnetic equation. In this chapters we describenot only the properties of optical fibres but also thematerials for optical fiber and fiber technology. It wasshown also the influence of the isotopes on propertiesof the optical fibers.

Chapter 4 is devoted the application of isotope effectin laser physics. There is short description of theory andpractice of semiconductor lasers. The discovery of thelinear luminescence of free excitons observed over awide temperature range has placed lithium hydride [1],as well as crystals of diamond in line as prospectivesources of coherent radiation in the UV spectral range.For LiH isotope tuning of the exciton emission has alsobeen shown.

The last chapter of this book is devoted to descriptionof the other unexplored applications of isotopic engi-neering. In the first place we considered the materialsfor information storage in modern personal computersas well as in biology. Large perspective has the isotope-base quantum computers. We should add here that thestrength of the hyperfine interaction is proportional to

the probability density of the electron wavefunctionat the nucleus. In semiconductors, the electron wave-function extends over large distances through the crys-tal lattice. Two nuclear spins can consequently interactwith the same electron, leading to electron-mediatedor indirect nuclear spin coupling. Because the electronis sensitive to externally applied electric fields, the hy-perfine interaction and electron-mediated nuclear spininteraction can be controlled by voltages applied tometallic gates in a semiconductor device, enabling theexternal manipulation of nuclear spin dynamics that isnecessary for quantum computation in quantum com-puters (details see [177, 383]).

The wide possibilities of isotopic engineering dis-cussed in this review hold the greatest promise for appli-cation in solid-state and quantum electronics, biology,human memory, optoelectronics, different electronicdevices, electronic and quantum computers, and manyother modern and new technologies that are even nowdifficult to imaginate. And we should repeat that themain aim of this review is to familiarize readers withpresent and some future applications in isotope scienceand engineering.

AcknowledgmentI would like to express my deep thanks to many authorsand publishers whose Figures and Tables I used in myreview. Many thanks are due to my son N. V. Plekhanovfor technical assistance, especially in graphical materi-als. Many thanks are due to A. DePina and K. Costellofor their editorial assistance.

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Received 28 Januaryand accepted 13 May 2003

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(PDF) Applications of isotope effects in solids - DOKUMEN.TIPS (2024)

FAQs

What are the 10 uses of isotopes? ›

Uses of Isotopes

Isotope	Uses
Phosphorus-32	Study of plant metabolism
Uranium-235	Generating electricity through the nuclear power generator
Carbon-14	Estimation of artefacts or fossils' age
Lead-210	In determining the age of sand and earth layers up to 80 years

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What does the number next to isotopes signify? ›

The number next to isotopes signifies the mass number of the isotope. For example, H-1 is the symbol for the isotope of hydrogen containing only one proton. H-2 is the isotope of hydrogen that contains one proton and one neutron. Isotopes of the same element always contain the same number of protons.

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How do you identify an isotope? ›

Isotopes are notated in multiple ways. Most commonly, they are specified by the name or symbol of the particular element, immediately following by a hyphen and the mass number (e.g., carbon-14 or C-14).

What are isotopes class 9? ›

Atoms with the same number of protons but different numbers of neutrons are called isotopes. They share almost the same chemical properties, but differ in mass and therefore in physical properties.

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What is the use of isotopes in everyday life? ›

Uses of Isotopes

The isotopes of Uranium such as U-235 are used as a fossil fuel in nuclear reactors. Radioactive isotopes are generally used for medicinal purposes, for example, for detecting cancerous cells. Iodine is an isotope of carbon which is used in the treatment of goitre.

What is the most common use for isotopes? ›

Isotopes have unique properties, and these properties make them useful in diagnostics and treatment applications. They are important in nuclear medicine, oil and gas exploration, basic research, and national security.

What are the magic numbers for isotopes? ›

The magic numbers for nuclei are 2, 8, 20, 28, 50, 82, and 126. Thus, tin (atomic number 50), with 50 protons in its nucleus, has 10 stable isotopes, whereas indium (atomic number 49) and antimony (atomic number 51) have only 2 stable isotopes apiece.

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What is the difference between C12 and C13? ›

Carbon 12, 13 and 14 are carbon isotopes, meaning that they have additional neutrons: Carbon 12 has exactly 6 protons and 6 neutrons ( hence the 12 ) Carbon 13 has 6 protons and 7 neutrons.

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What does the top and bottom number mean for isotopes? ›

For nuclear notation, the mass number of the isotope goes on top and the atomic number goes on the bottom.

What is the only way to tell isotopes apart? ›

Answer and Explanation:

Isotopes can be told apart by their mass numbers. Isotopes of the same element have the same quantity of protons in their nuclei. They have different quantities of neutrons, though. The mass number of an isotope is the sum of its protons and neutrons.

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What can isotopes tell us? ›

By measuring the ratios of different isotopes in bones or teeth and using scientific knowledge about how they occur in nature to trace them back to the sources that they came from, archaeologists can find out many things about an individual, such as what their diet was like and the environment they grew up in.

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How can you tell if an isotope will be radioactive? ›

An atom is stable if the forces among the particles that makeup the nucleus are balanced. An atom is unstable (radioactive) if these forces are unbalanced; if the nucleus has an excess of internal energy. Instability of an atom's nucleus may result from an excess of either neutrons or protons.

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What is isotopes one word answer? ›

Isotopes can be defined as the variants of chemical elements that possess the same number of protons and electrons, but a different number of neutrons.

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What is an isotope for dummies? ›

Isotopes are atoms of an element that have the same number of protons and electrons, but a different number of neutrons. Because isotopes have different numbers of neutrons, they also have different atomic masses, the amount of mass in an atom based on the number of protons and neutrons.

What makes an isotope unstable? ›

Unstable isotopes

When an isotope holds too many or too few neutrons to maintain its stability, the atom decays and produces radiation. We talk about alpha, beta and gamma rays.

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What are isotopes Class 10 examples? ›

A group of isotopes of any element will always have the same number of protons and electrons. They will differ in the number of neutrons held by their respective nuclei. An example of a group of isotopes is hydrogen-1 (protium), hydrogen-2 (deuterium), and hydrogen-3 (tritium).

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What are the 10 uses of radioactivity? ›

Uses of Radioactivity

Medical procedures including diagnosis and treatment of cancer.
Sterilising food (irradiating food)
Sterilising medical equipment.
Determining the age of ancient artefacts.
Checking the thickness of materials.
Smoke detectors (alarms)

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What element has 10 isotopes? ›

Of the chemical elements, only 1 element (tin) has 10 such stable isotopes, 5 have 7 stable isotopes, 7 have 6 stable isotopes, 11 have 5 stable isotopes, 9 have 4 stable isotopes, 5 have 3 stable isotopes, 16 have 2 stable isotopes, and 26 have 1 stable isotope.

What are the 5 uses of radioactive isotopes? ›

11.5: Uses of Radioisotopes

Diagnostic Medical Applications. PET Scanning. PET Scan. Other Isotopic Tests.
Therapeutic Radiation. Radiation therapy and Chemotherapy: Two different treatment procedures. External Beam Therapy (Photon and Proton Therapy) Brachtherapy.

Aug 10, 2022

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What are the 10 uses of isotopes? ›

What can isotopes tell us? ›

References