Statistical Mechanics of Dilute Solid SolutionsR. F. Brebrick Citation: Journal of Applied Physics 33, 422 (1962); doi: 10.1063/1.1777134 View online: http://dx.doi.org/10.1063/1.1777134 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Statistical Mechanics of Adsorption from Dilute Liquid Solution J. Chem. Phys. 57, 714 (1972); 10.1063/1.1678304 Statistical Mechanics of Quenched Solid Solutions with Application to Magnetically Dilute Alloys J. Math. Phys. 5, 1401 (1964); 10.1063/1.1704075 Statistical Mechanics of Dilute Polymer Solutions. III. Ternary Mixtures of Two Polymers and a Solvent J. Chem. Phys. 20, 873 (1952); 10.1063/1.1700586 Statistical Mechanics of Dilute Polymer Solutions. II J. Chem. Phys. 18, 1086 (1950); 10.1063/1.1747866 Statistical Mechanics of Dilute Polymer Solutions J. Chem. Phys. 17, 1347 (1949); 10.1063/1.1747184

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JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 33, NO.1 JANUARY, 1962

Statistical Mechanics of Dilute Solid Solutions

R. F. BREBRICK Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington 73, Massachusetts

An apparently quite general model for an essentially-ordered semiconductor compound containing im­purity traces consists: (1) of atomic point defects randomly distributed over appropriate, equivalent sites and contributing to the internal energy of the crystal by terms linear in the concentration of each type of atomic point defect, and (2) an electronic energy band structure in which the concentration and type of "impurity" levels is determined by the concentration and type of atomic point defects. The assignment of donor or acceptor character to the native interstitials and vacancies is predicted in a specific case by analogy with the alkali halides. Otherwise the nature of the binding is irrelevant, provided the un-ionized impurity level associated with each substitutional atomic point defect bears the same charge as that on the sub­stituted atom. From the appropriate quasi-grand partition function, one obtains the usual Fermi-Dirac distribution for electrons as well as distribution functions for the atomic point defects. In addition, one obtains expressions for the chemical potentials of the thermodynamic components. The latter are utilized in a discussion of those aspects of the M-N phase diagram pertinent to the semiconductor compound MN and in a discussion of amphoteric impurities.

INTRODUCTION

T HE physical chemistry of crystalline semicon­ductors is presently interpreted in terms of a

model which, although varying in specific details as applied to different materials, still maintains certain basic features. Admittedly, some of these often rest upon an extrapolated empirical basis as the model is applied to different classes of semiconductor materials. However, having a model, its behavior can be deduced for comparison with experiment by applying statistical mechanics to construct a quasi-grand partition func­tion and thereby to obtain the distribution functions for the various point-defect concentrations and the expressions for the chemical potentials of the thermo­dynamic components. A final step desirable for the purposes of this comparison consists of eliminating the Fermi level from the distribution functions to obtain the concentration of each point defect as a function of the independent, external variables. For the vapor­crystal equilibrium in a binary system these are the temperature and partial pressure of one species of one component.

It is the purpose here to discuss the basic features of the model that must be prescribed in order to construct a partition function. Next, the expressions for the chemical potentials and a few problems, whose solution is relatively straightforward using these, are discussed for compound semiconductors. The distribution func­tions and the conditions under which these lead to mass-action law-type equations,! the final solution ob­tained by expressing the concentration of each point defect in terms of the external variables,2 and the special case of elemental semiconductors3,4 have been

* Operated with support from the U. S. Army, Navy, and Air Force.

I R. F. Brebrick, J. Phys. Chern. Solids 11, 43 (1959). • F. A. Kroger and H. J. Vink, Soli<!, State Physics, edited by

F. Seitz and D. Turnbull (Academic Press Inc., New York, 1956), Vol. III, p. 310; and J. Phys. Chern. Solids 5, 208 (1958).

3 H. Reiss, J. Chern. Phys. 21, 1209 (1953). 4 R. L. Longini and R. F. Greene, Phys. Rev. 102, 992 (1956).

discussed elsewhere, and in other papers at this Con­ference.'

Basic Features of the Model

Consider an essentially ordered semiconductor com­pound MN, containing small concentrations of foreign impurities. Practically all of the M atoms occupy sites of one sublattice, while the N atoms occupy the sites of the other sublattice. However atomic point defects, of which foreign impurities are one class, are present. The first basic feature of the model is that the atomic point defects are distributed at random over appro­priate, equivalent sites and increase the internal energy by terms linear in the concentration of each type of atomic point defect. This implies that the concentra­tions of defects are small compared to that of the lattice sites, and that there is no clustering. As thus stated, this first feature is unnecessarily restrictive since defect clustering, and in particular pairing, is important in many cases and has been taken account of by various authors.6- 8 Since pairing is discussed in other papers at this Conference, it will be omitted here for simplicity. As far as pairing theories indicate sufficiently broad conditions under which pairing is not important, this procedure still yields results of interest.

The second basic feature of the model is that the electrons distribute themselves over a semiconductor band structure, the atomic point defects serving to introduce donor or acceptor levels. The questions then arise as to what are the predominant atomic point de­fects native in the crystal; are these donors or acceptors; are foreign impurities substitutional or interstitial; and are these donors or acceptors? Experiment can generally be relied upon to answer the last two questions con­cerning impurities without great difficulty. In many cases, however, the concentrations of native atomic

5 See papers by C. S. Fuller and J. S. Prenner in this issue. 6 A. B. Lidiard, Phys. Rev. 94, 29 (1954). 7 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J.

35, 535 (1956); H. Reiss, J. Chern. Phys. 25, 400 (1956). 8 J. S. Prenner, J. Chern. Phys. 25, 1294 (1956).

422

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MECHANICS OF DILUTED SOLID SOLUTIONS 423

point defects are so small that they are detected only by indirect means which do not distinguish between the native atomic point defects and foreign impurities. Therefore, in practice one considers all plausible possi­bilities for the native defects, identifies their donor or acceptor character by analogy with the behavior of the ionic alkali halides, and then uses experiment to decide among these possibilities. By analogy with the alkali halides, vacancies in the sublattice of the more metallic component M, of the compound MN as well as inter­stitial N atoms, are acceptors. Vacancies in the sub­lattice of the less metallic component N, as well as interstitial M atoms, are donors. This sort of assign­ment seems to be consistent with the experimental findings for a number of compound semiconductors such as Ag2S, Ag2Se, Cu2Se, CdTe, CdS, ZnO, PbS, PbSe, and PbTe. Whether or not they will prove to be con­sistent with the behavior of the III-V semiconductors in which the metallic Group III element carries a nega­tive formal charge remains to be seen, although there is some evidence that a gallium vacancy in GaAs is an acceptor.9 For the place-exchange type of atomic point defect, supposedly predominant in Bi2Tea, an excess of the metallic component, incorporated in the structure by putting metallic atoms on nonmetal sites where they are acceptors, makes the semiconductor more P type. This is in contrast to the vacancy-interstitial mechanisms and provides a ready means of distinguish­ing the place-exchange mechanism.

The effect of the atomic point defects upon the lattice vibrations has been accounted forlo .a by assum­ing that each vacancy results in the loss of three modes of a given frequency, w, and the replacement of a small number of modes of the same frequency by an equal number of lower frequency, w'. An interstitial atom adds three modes of some frequency. In the high­temperature limit this leads to a composition-inde­pendent term in the expressions for the chemical po­tentials. Finally, two constraints are imposed upon the concentrations of the point defects appearing in the partition function. First, there is a definite concentra­tion of lattice sites in a crystalline material essentially ordered with respect to at least one component. Second, the crystal must be electrically neutral. It was in writ­ing a general expression for this equation that includes electrons, holes, ionized foreign donors, and acceptors, as well as the ionized atomic point defects native to the pure crystal, that Kroger and Vink2 combined Wagner's concept of controlled imperfections applicable to the alkali halides and Verwey's concept of controlled valency into a single model. However, the equation for electrical neutrality is interesting in another respect also. Suppose that in the compound MN, the M atoms bear a charge of +e and the N atoms a charge of - E.

9 J. M. Whelan and C. S. Fuller, J. App!. Phys. 31,1507 (1960). 10 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic

Crystals (Oxford University Press, New York, 1950), p. 30. n R. F. Brebrick, J. Phys. Chern. Solids 18, 116 (1961).

It is desirable that this charge drop out of the equation for electrical neutrality and hence appear nowhere in the equations descriptive of the model, i.e., that in a sense the model be independent of the degree of ionic character of the crystal bonding. For this to be so, it is sufficient that the un-ionized atomic point defects occupying M sites (N sites) bear the same charge as the M atoms (N atoms). A simple example is a chlorine vacancy in N aCI with an electron trapped in the associ­ated donor level, which bears the same charge of -1 as does the chloride ion in N aCI. Thus, in general, all un-ionized, substitutional atomic point defects will be charged on the basis of this assumption.

Chemical Potentials of the Major Components

For the model described, one can obtain the dis­tribution functions for each point defect and the ex­pressions for the chemical potentials. If vacancies are present in the crystal, the chemical potentials of the crystal components are given byl

J,LM=RTlnSjVM+Ef+J,LMO(T)=RTlnPM, (1a)

J,LN=RTlnSjVN-Ef+J,LNO(T)=RTlnpN, (1b)

where S is the concentration of lattice sites in each sub­lattice, V M and V N are respectively the concentrations of singly ionized M vacancies (acceptors) and singly ionized N vacancies (donors), E f is the Fermi level, and PM and PN are respectively the partial pressures of monoatomic M and monoatomic N in equilibrium with the crystal, the vapor phase being assumed to be ideal. The composition-independent terms J,LMO(T) and J,LNO(T) contain the energies to create the un-ionized vacancies, the energy of the associated donor or acceptor level, a contribution from the change in vibrational entropy due to the presence of the atomic point defects, and the chemical potentials of the monoatomic gases at 1 atmosphere relative to the isolated atoms in their lowest quantum states. These terms are discussed else­wherell and are of importance for such questions as the positions of the solidus lines for the compound and the partial pressures over the pure, intrinsic material. The fairly general nature of Eqs. (1a) and (1b) should be stressed. The same equations apply independent of the fraction of M-vacancy acceptor levels or N-vacancy donor levels that are ionized, regardless of whether these donor and acceptor levels are capable of multiple ionization or not, and regardless of the degree of de­generacy of the electron distribution. They also apply if interstitials are present. One then has another equa­tion for the chemical potential of component M if M interstitials are present. The 2 equations must be simul­taneously satisfied and together determine the fraction of interstitial donors and vacancy acceptors. The situation of course is analogous if N interstitials are present.

The simplest version of the model leading to Eqs. (1a) and (ib) is one in which the donors and acceptors

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424 R. F. BREBRICK

are capable of a single stage of ionization only, and in which the electron distribution is nondegenerate. The Fermi level can then be expressed in terms of the carrier concentrations using the Boltzmann distribution law. The concentrations of ionized vacancies can also be obtained in terms of the carrier concentrations, using the equation for electrical neutrality and the mass ac­tion law equation for ionized vacancies,

The resulting expressions areH :

E f = E/+ RT sinh-I[ (n- P )/2ni], (2)

In(V N/k.s!) = -In(V M/ks !) =sinh-I[(n-p-D)/2ks!], (3)

where nand p are the concentrations of conduction band electrons and valence band holes, respectively, ni is the intrinsic-carrier concentration, and D is the concentration of ionized foreign donors minus that of ionized foreign acceptors. Assuming the vapor in equi­librium with the crystal is ideal, the partial pressure PM r of r-atomic M component or that of s-atomic N component, PN, in equilibrium with the crystal can be obtained from Eqs. (1a), (1b), (2), and (3) asH:

1 1 -In(pMJp~d =-In(pN// px.,) r s

(n-p-D) (Il-P)

= sinh-I ---, - +sinh-I -- , 2ks' 2ni

where PMri and PN,i are the partial pressures over the

pure, intrinsic crystal, i.e., for n= p, D=O. Equation (4) also gives the solidus lines of the compound MN and hence its field of stability if the partial pressures at the 3-phase line, where crystal, liquid, and vapor co­exist, are known. Alternatively, one can use partial pressure-composition-temperature data to determine the parameters, ks, ni, and PMr

i • The application of Eq. (4) is of course equivalent to the mass action law analysis used by Kroger and Vink2 and applied by Bloem and Krogerl2 to PbS. It should be emphasized again that such an analysis would be formally the same for a more general model in which M-interstitial donors and N-interstitial acceptors are present in addition to vacancies, so long as the former are also capable of only a single stage of ionization. In this case, the Schottky constant ks, appearing in Eq. (4), is replaced by an effective equilibrium constant K E , given by

(5)

where kl is the equilibrium constant equal to the product of ionized M-vacancy acceptors and ionized M-interstitial donors, and k2 is the analogous quantity

12 ]. Bloem and F. A. Kroger, Z. physik Chern. 7, 1 (1956); and J. Bloem, Philips Research Repts. 11, 273 (1956).

for N interstitials and vacancies. The energy parameters included in the intrinsic partial pressures then also apply to interstit ials as well as to vacancies. Thus, while a fit of the experimental data on PbS indicatesl2

that the native atomic point defects predominant in Pb(S)-rich PbS are isolated donors (acceptors) showing only a single state of ionization, these defects are not necessarily vacancies, as indeed Bloem and Kroger have pointed out.

While Eq. (4) describes those aspects of the M-N phase diagram related to the compound MN, it is of interest to consider the slopes of the solidus lines for the pure compound explicitly, using a somewhat dif­ferent approach. Starting from the criterion for thermo­dynamic equilibrium that the chemical potential of each component must be the same in every phase at constant temperature and pressure, the temperature derivatives of the solidus lines of the compound MN are given by a thermodynamic development in terms of the mole fraction of component N on the solidus line x; the mole fraction of component N in the co­existing, condensed phase x'; the integral molar Gibbs free energy of the compound F; the heat involved in reversibly transferring one mole of component M from the compound to the coexisting phase ~M, and the analogous heat term for component N, ~N. The equa­tion is l3

(dX) ~l-x'2~M+x' ~N (6)

- dT C)('X (x'-x)Ta2F/ax2

The first derivative of the Gibbs free energy is simply the difference in the chemical potentials of components Nand M in the compound. The chemical potentials arc given by Eqs. (1a) and (1b), For the special case of a nondegenerate semiconductor and no multiple ioniza­tion of donors or acceptors, the Fermi level and ionized vacancy concentrations appearing therein satisfy Eqs, (2) and (3), respectively. For the purpose of simplicity, it is also assumed that the donors and acceptors are all ionized so that there is a simple relation between the mole fraction of component N in the compound and the carrier concentrations given by

2x= 1- (n- P )/25, (7)

where, as before, 5 is the concentration of lattice sites in each sublattice. Equation (6) can then be rewritten as

/ (1) (l-x')~M+x'~N -dln(ln-pl) d - = ,

, T 2R(x' -0.5)! (8)

where it has been assumed that the mole fraction in the solid compound can be taken as equal to 0.5 without substantial error (in many cases, this will be so except

13 Carl Wagner, Thermodynamics oj Alloys (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1952); J. L. Meijering, Philips Research Repts. 3, 281 (1948).

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MECHANICS OF DILUTED SOLID SOLUTIONS 425

near the invariant melting point) and where

In-pl/2ni In-pl/2k8! f + . (9)

[( I n-pi /2ni)2+ 1J! [( I n-pi /2k.!)2+ 1J!

Thus the slope of In( \ n-p\) along a solidus line vs ljT is not constant in general. The function f goes from values near zero when the argument \ n-p I is small compared to the 2 parameters, to values near unity when \ n-p \ is intermediate in value between those of the 2 parameters, to values approaching 2 when I n-p \ becomes large compared to the 2 parameters. Therefore, even in the simplest situation when the compound is in equilibrium with one of the essentially pure ele­ments, say ~, so that x' ~N in the numerator of Eq. (8) is negligible, the slope of a In( \ n-pi) vs l/T plot along a straight line portion may be either ~M/ R or ~M/2R, depending upon whether I n-p \ is intermediate in value between 2ni and 2k,! or large compared to both.

Chemical Potentials of Impurities

The chemical potential of a donor impurity substitut­ing for an M atom is obtained from the quasi-grand partition function as

tLD = tLM+ RTln (:V D/ S)+ Ef+tLDO(T), (10)

where :V D is the concentration of singly ionized im­purities, and tLDO(T) is a composition-independent term involving the energy of the donor level, etc. The equa­tion for an M-substituent acceptor impurity differs only in that the sign before the Fermi level Ef is negative rather than positive. Finally, the equations for N­substituent donors or acceptors are analogous to those for the M-substituent donors or acceptors, respectively, except that the term tLM is replaced by tLN. The chemical potentials of the major components enter the expres­sions for the chemical potentials of the substituent im­purities because of the requirement that the concen­tration of lattice sites S is constant. Otherwise, Eq. (8) is equivalent to the expressions given for impurities in elemental semiconductors.3,4 Although the sum of the chemical potentials tLM and tLN in the compound MN is assumed to vary with composition by only a negligible fraction of the molar Gibbs free energy, the individual chemical potentials and their difference can vary significantly.

As an illustration of the above, suppose that a par­ticular impurity can occupy an M site as a donor or an N site as an acceptor. Silicon in GaAs is an example. If for a fixed total impurity content the impurity dis­tributes between the M and N sites, while the tempera­ture and chemical potential of component N are held constant, the distribution can be obtained by equating the chemical potential of the impurity in the donor

state to that in the acceptor state. Rearranging the results into a form convenient for comparison with experiment gives

N 1 + (:V diN a)i exp[2 (Efi- Ef )/ RTJ

D 1- (:Vd/iVa)i exp[2(Ef i_Ef )/RTJ' (11)

where N is the total concentration of the impurity in the singly ionized state, D is the difference in the COn­centrations of ionized donor impurities and ionized acceptor impurities, and the effective, intrinsic donor­acceptor ratio is given by

(N diN a)i= (:Yd/:V a)iO exp[2(tLN-tLN i)/ RTJ

= (Nd/Na)iO(PN,/pNri)2/r. (12)

Here the ratio (Nd/:Va)iO is defined for intrinsic ma­terial, n= p, in the limit of vanishingly small total impurity content and, in contrast to the effective, intrinsic ratio, is a function of temperature only. The behavior of silicon in GaAs equilibrated at 1200°C under a fixed arsenic pressure has been successfully analyzedl4 by means of Eq. (10). However, the deriva­tion was based upon expressions for the chemical po­tential of an amphoteric impurity in an elemental semi­conductor in which the terms tLN and tLM do not appear.4 The behavior as a function of arsenic pressure was not studied experimentally. The considerations here would predict that the effective, intrinsic ratio given by Eq. (12) would be about 4.5 times as large for As-saturated GaAs at 1200°C as for Ga-saturated GaAs, based upon the corresponding arsenic-partial pressures being, re­spectively, 4 and 0.2 atmospheres.15 At 900°C the corre­sponding arsenic pressures are 33 and 0.02 atmospheres and the effective, intrinsic ratio varies by a factor of about 40 according to Eq. (12). The effective, intrinsic ratio for a germanium impurity in GaAs at 900°C has been studied qualitatively and found to increase with increasing arsenic pressure,t6 and an analog of Eq. (12) that appears to be in error has been given.

Finally in the considerations leading to Eqs. (10), (11), and (12), no detailed picture of the native atomic point defects or their activity as donors or acceptors is necessary. One must be more explicit only if it is de­sired to express the chemical potentials tLM and tLN appearing in Eqs. (10) and (12) in terms of, say, the carrier concentrations. Thus, the validity of Eqs. (11) and (12) for GaAs is independent of whether vacancies are the predominant native atomic point defects or not and independent of whether these are donor, acceptors, or neither.

14 J. M. Whelan, J. D. Struthers, and J. A. Ditzenberger, J. Phys. Chern. Solids (to be published).

15 J. van den Boorngaard and K. Schol, Philips Research Repts. 12, 127 (1957).

16 J. O. McCaldin and Roy Harada, J. App!. Phys. 31, 2065 (1960).

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