Energy-gap reduction in heavily doped silicon: Causes and consequences

SoldSrrrrr E/e~rm,,r~s Vol. 2X. Nos. l/2.

pp. 17-24.

0038-1101/85 %3.00+ 00 Pergamon Press Ltd

1985

Printed inGreat Britain

ENERGY-GAP

REDUCTION IN HEAVILY DOPED SILICON: CAUSES AND CONSEQUENCES

SOKRATES T. PANTELIDES,ANNABELL.ASELLONI~ and ROBERTOCARS IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. Abstract-The authors review briefly the existing theoretical treatments of the various effects that contribute to the reduction of the energy gap in heavily doped Si, namely electron-electron and electron-impurity interactions and the effect of disorder in the impurity distribution. They then turn to the longstanding question why energy-gap reductions extracted from three different types of experiments have persistently produced values with substantial discrepancies, making it impossible to compare with theoretical values. First, they demonstrate that a meaningful comparison between theory and experiment can indeed be made if theoretical calculations are carried out for actual quantities that experiments measure, e.g. luminescence spectra, as recently done by Selloni and Pantelides. Then, they demonstrate that, independent of any theoretical calculations, the optical absorption spectra are fully consistent with the luminescence spectra and that the discrepancies in the energy-gap reductions extracted from the two sets of spectra are caused entirely by the curve-fitting procedures used in analyzing optical-absorption data. Finally, they show explicitly that, as already believed by many authors, energy-gap reductions extracted from electrical measurements on transistors do not correspond to true gap reductions. They identify two corrections that must be added to the values extracted from the electrical data in order to arrive at the true gap reductions and show that the resulting values are in good overall agreemen: with luminescence and absorption data. They, therefore, demonstrate that the observed reduction in emitter injection efficiency in bipolar transistors is not strictly due to a gap reduction, as generally believed, but to three very different effects.

1. INTRODUCTION

By a detailed analysis of optical-absorption spectra, it was possible to extract values of the energy-gap reduction as a function of impurity concentration. In the late 196Os, band-gap narrowing in heavilydoped Si was used by Kaufmann and Bergh[6] to explain an observed discrepancy between the calculated and measured injection efficiency of bipolar transistors. Since then, the dependence of the energy gap on impurity concentration has been an indispensable input in the modeling of electronic devices with heavily doped regions. In 1976, Slotboom and de Graaff[7] measured the I-V characteristics of npn bipolar transistors with different doping levels in the base region. They analyzed their data and extracted values for the band-gap reduction as a function of concentration. Their values, however, differed signiticantly from those extracted from optical-absorption data[5,6]. All the available data were summarized by Keyes[S], who pointed out that the serious discrepancies in the experimental values of the gap reduction and the lack of a comprehensive theory to account for the dependence of the gap reduction on the dopant concentration constituted an important problem for solid state physics.

Certain impurities (e.g. P, As, B, etc., in Si) are known to introduce hydrogenic localized states near one of the edges of the fundamental energy gap of a semiconductor. At high concentrations, these states broaden into a band which eventually merges with the nearby band continuum. At concentrations which exceed this critical value, there are no longer any discrete energy levels. Instead, one has a new set of energy bands and the Fermi level at T= 0 K lies inside one of these bands. For example, in heavilydoped n-type Si, the resulting valence bands are completely full and the conduction bands are full up to the Fermi level EF. If the forbidden energy gap were to have the same value ERO as in the intrinsic material, the new optical edge would lie at Ego + EF. This idea enabled Burstein[l] in 1954 to interpret the observed “anomalous” absorption edges in heavilydoped n-type InSb. The shift of the optical gap to higher energies became known as the Burstein shift. Soon afterwards, theoretical calculations of impurity bands in InAs and InSb by Stem and Talley[2], showed that the forbidden energy gap may have a value Eg that is smaller than E,,, leading to smaller Burstein shifts. Optical-absorption experiments in Ge and Si in the 196Os[3-51 indeed revealed that the Burstein shift was significantly smaller than was predicted on the basis of the known effective masses and impurity concentrations and could even be negative.

In subsequent years, there was a proliferation of optical-absorption and electrical measurements and the values reported in the literature varied widely. During the same years, the theory of the energy-gap narrowing as a function of impurity concentration was developed in a rigorous way[9-111 (see Section 2), leading to detailed numerical calculations. Different approximations, however, led to significantly different values. The state of the art in 1981 is

l-Present address: Institute of Physics, University of Rome, Piazzale Alto Moro 5, Rome, Italy. *Present address: ISAS, Strada Costiera 11, Trieste, Italy.

17

S. T. PANTELIDES et ~11

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The values extracted from these measurements were significantly different from those extracted from either optical-absorption or electrical measurements. In Fig. 2, we collected representative values extracted from optical-absorption[13], photoluminescence[l2] and electrical measurements[7] and plotted them

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Energy-gap

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against the impurity concentration. We see that the discrepancies between the values extracted from the three types of data are indeed significant. It is evident from the above discussion that experiments do not measure the band-gap reduction directly, but extract it from data by making a number of assumptions. Clearly a more meaningful comparison between theory and experiment would be possible if one were to carry out theoretical calculations of the very same quantities that are measured by a particular experiment. In 1982, Selloni and Pantelides adopted this point of view. They calculated theoretical photoluminescence spectra and demonstrated that very good agreement with the data could be obtained. Nevertheless, the origins of the discrepancies in the values of the energy-gap reductions extracted from the three different types of experiments (Fig. 2) could not be reconciled. In this paper, we will do the following. In Section 2, we will review briefly the causes of the band-gap reduction in heavily doped semiconductors and the various methods that have been used to carry out calculations. We will also review briefly the results of Selloni and Pantelides[14] which resulted in the first direct comparison of theory and experiment in Si and discuss some theoretical issues that have since been raised by Berggren and Semelius[lS] regarding the inclusion of impurity disorder in the calculations. In Section 3, without making use of any theoretical calculations, we will demonstrate that the opticalabsorption and photoluminescence data themselves are fully consistent with each other[16]. We will further trace the causes of the serious discrepancies in the gap-reduction values extracted from these two types of spectra to erroneous fitting procedures used in analyzing optical-absorption data. Finally, we will show that the gap-reduction values extracted from electrical measurements do not correspond to the reduction of the actual energy gap, but some other “effective” energy gap, as already suggested by many authors. We will go one important step further, however, and identify the corrections that must be added to the gap-reduction values extracted from the electrical data in order to make them correspond to reductions of the true energy gap. We show that these corrections are of the right order of magnitude to account completely for the purported discrepancies with the optical-absorption and photoluminescence data. Accurate calculations of one of these corrections are, however, very difficult. 2.

BRIEF REVIEW OF THEORY

In early theories, the energy bands of heavily doped semiconductors were viewed to arise from a broadened impurity band which merges with the continuum[2]. An alternative point of view, however, turned out to be more useful. Let us illustrate the idea by assuming n-type material. One starts with a perfect crystal and then inserts all the extra donor electrons in the conduction bands, adding a cornpensating positive background. Electron-electron in-

in silicon

19

teractions will result in shifts of the energy bands. Finally, the positive background is replaced by discrete positively charged impurity ions. The electron-impurity interactions then induce additional shifts. The procedure is very similar to that used to study simple metals. It was conceived quite early (see; e.g. Haas[3]) and has since been used for actual calculations by many authors. A brief review of these calculations by Mahan[lO], revealed that most calculations of the 1960s and 1970s were plagued by errors. Inkson[9] was apparently the first to recognize that all bands are shifted by the presence of the extra electrons or holes and introduced diagrammatic techniques for the calculation of the shifts induced by electron-electron interactions. Mahan[lO] was the first to carry out detailed numerical calculations of all the energy shifts that arise from electron-electron and electron-impurity interactions for heavily doped Si and Ge. More extensive calculations were reported later by Berggren and Semelius[ll]. The approach of all these theories is basically a many-body approach coupled with second-order perturbation theory. One calculates the “self-energies” of the electrons and holes resulting from electron-electron interactions and electronimpurity interactions. Mahan’s[lO] calculations assumed an ordered fee array of impurities. Berggren and Semelius[ll] carried out calculations with both an fee array and a model that corresponds to complete randomness as in an ideal gas. The results of the ordered-array calculation were in quite good agreement with those of Mahan[lO], but the results of the random-distribution calculation were significantly different. Because of the enormous scatter in “experimental” gap reductions, however, as we noted in the Introduction, no meaningful comparison of theory and experiment could be made (see Fig. 1). Selloni and Pantelides[14] adopted the self-energy calculations of Mahan[lO] and of Berggren and Semelius[ll] for the electron-electron interactions, which shift the bands essentially rigidly. For the inclusion of electron-impurity interactions, however, they formulated the problem in a single-particle approximation. The advantage of this approach is that, at least for ordered arrays of impurities, perturbation theory can be avoided. Instead, the powerful methods of band theory can be brought to bear. By calculating the complete band structure in the entire Brillouin zone, one can calculate the full density of states and luminescence or absorption spectra. In addition, the single-particle formulation made practical the inclusion of multivalley interactions which arise from the fact that the lowest conduction band of Si has six equivalent minima and which had been neglected by all earlier theories. It was found that, in the single-valley approximation, electron-impurity interactions produce negligible band shifts as found in the ordered-array calculations of Mahan[lO] and of Berggren and Semelius[ll]. Inclusion of multivalley interactions, however, produced large shifts, com-

S. T. PANTELIDES et (11.

itself cannot be extracted from the data without fitting the experimental spectra with calculated spectra. In addition, the precise position of the band edge is ill-defined because of band tails[l7]. Figure 4 actually shows two theoretical values for E, + EF, with and without multivalley interactions. Clearly, the latter are very important in achieving agreement with the experimental values.

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to those induced by electron-electron interactions. Selloni and Pantelides [14] calculated the photoluminescence spectra of heavily doped n-type Si and showed that very good agreement with experimental data was obtained without any adjustable parameter for the position of the spectra on the energy axis (Fig. 3). In order to appreciate the magnitude of the discrepancies between theory and experiment Selloni and Pantelides[l4] also plotted the theoretical and experimental values of E + EF against electron density (Fig. 4). The exper!mental values were simply read off the spectra as the points of inflection in the high-energy steep drop-offs. These are clearly fairly unambiguous experimental values, whereas the gap par&k

Since the work of Selloni and Pantelides[14], Berggren and Semelius[lS] re-examined their theoretical values and constructed a plot similar to Fig. 4 (Fig. 5). The results of their ordered-array calculations are virtually identical with the single-valley results of Selloni and Pantelides, indicating that second-order perturbation theory, the only additional approximation made by Berggren and Semelius [ll, 151 is adequate in that case. The results of their calculations for a random distribution of impurities, on the other hand, are virtually identical with the ordered-array, multivalley results of Selloni and Pantelides[14] and thus agree with experiment without the need to include multivalley terms (Fig. 5). Multivalley contributions calculated by Berggren and Semelius [ 151 using second-order perturbation theory were found to be negligible when a random distribution of impurities is assumed. Thus, it appears that comparable agreement with experimental data is obtained by two theories which

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similar to Fig. 3. from [15]. Solid dots are the points (same as those shown in Fig. 4), open the theoretical values of [14], including multiCurve (iv) is the single-valley calculation of [ll the complete-randomness model, and curve (v) curve after inclusion of multivalley terms by perturbation theory.

21

Energy-gap reduction in silicon attribute the energy shifts to very different effects. The Selloni-Pantelides theory asserts that randomness is not very important and most of the resulting energy shift comes from multivalley interactions, The Berggren-Semelius theory asserts that “complete randomness” is a more appropriate assumption, and most of the resulting energy shift comes from intravalley interactions. One might worry that both approaches calculate the same thing and give it different names, but that is not true. The two theories can actually be tested by experimental data in semiconductors with a single-valley conduction band such as GaAs, InSb, etc. We have not, however, been able to locate appropriate data. In the Appendix we present arguments in favor of the Sellom-Pantelides assumptions. 3. RECONCILIATION OF DISCREPANCIES GAP-REDUCTION VALUES

IN

Our first task in this section is to demonstrate that the photoluminescence spectra of [12] are fully consistent [16] with the optical-absorption spectra of [13]. As we already saw in Section 2, the photoluminescence spectra allow a fairly unambiguous extraction of the Fermi level, i.e. the quantity Eg + EF. Optical absorption ought to begin at precisely that point. In Fig. 6, we show the optical-absorption spectra of Schmid[l3]. The short vertical lines denote the positions of Eg + E,, as extracted from the luminescence data of [12]. We note that they are just to the left of the dip formed by the superposition of the decreasing free-carrier intraband absorption and the increasing interband absorption of interest here. The small insert illustrates why Eg + EF is to be expected just to the left of the dip. We conclude that the optical-absorption and luminescence spectra are fully consistent with each other. We now turn to the question of band-gap reductions. Once the Fermi level is located, one needs to know the density of states in order to locate the band edge and thus extract a gap reduction. The values of the gap reductions from luminescence shown in Fig. I

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2 were extracted by assuming the state density of a simple parabolic band with an appropriate effective mass. It turns out that, if the more precise state density calculated by Selloni and Pantelides[14] is used, the resulting gap reductions are only slightly different. The luminescence values shown in Fig. 2 can, therefore, be used as the standard. What, then, went wrong with the analysis of the optical-absorption spectra that yielded the values plotted in Fig. 2? That analysis used a simple line-shape theory and fitted the data at fairly high energies (see Fig. 3 of [13]) using the band gap and, hence, the Fermi level as an adjustable parameter. For example, this type of fit for Si doped with an As concentration of 6 x 10ix cmm3 produced a zero gap reduction, which implies that the optical edge is to the right of the optical edge in pure Si. Figure 6, however, shows clearly and unambiguously that the optical edge in the doped material is to the left of the optical edge of pure Si by a significant amount. We conclude that the curve-fitting procedure that led to the gap reductions shown in Fig. 2 is erroneous. In order to extract a meaningful gap reduction from optical-absorption spectra one must first carefully locate the Fermi level. In the absence of luminescence spectra, the task is not quite straightforward because of the presence of the free-carrier absorption and other effects that normally occur at optical-absorption thresholds (phonon assisted processes, etc.). For our purposes here, it suffices to say that the Fermi levels extracted from luminescence spectra are consistent with the absorption spectra (Fig. 6). Hence, we can continue to view the luminescence values of Fig. 2 as the standard and discard all published gap reductions extracted previously from optical-absorption spectra as incorrect. Finally we examine the electrical measurements. Slotboom and de Graaff[7] effectively measured the pn product in heavily doped Si[lg]. They then assumed that pn is given by pn = CT3 exp ( -

.$/kT)

,

(1)

where C is independent of temperature and J?~ is viewed as the actual energy gap. They noted that this equation is valid for low impurity concentrations, but then asserted that calculations prove that the pn product for high impurity concentrations can also be described by eqn (l), where .& continues to stand for the actual, concentration-dependent energy gap. They then carried out a rather complex processing of the data and extracted values of the gap, and, hence, gap reductions as functions of impurity concentration. The gap reductions are shown in Fig. 2 where we see that they disagree seriously with the standard we established above, namely the luminescence values. Support for the use of eqn (1) was given in a separate paper by Slotboom[19]. He calculated the pn product numerically by assuming a set of model

S. T.

22

PANTELIDES

bands and performing the appropriate integrals. He then processed these numerical values in the same way the experimental values were processed in [7], by assuming the validity of eqn (1). He thus extracted gap reductions from the theoretical pn products and found them to agree with those extracted from the experimental pn products. He concluded that this agreement provides justification for using eqn (1) in heavily doped semiconductors. In fact, no such conclusion is warranted. The agreement between the two sets of gap reductions means only that the theoretical pn products happen to be in good agreement with the experimental pn products. Identical processing of the theoretical and experimental pn products is then guaranteed to yield gap reductions which agree with each other. Thus, in [19], Slotboom showed that he had a good model for the calculation of the pn product, but he provided no support for the use of eqn (1) in heavily doped semiconductors. It has been noted by many authors that eqn (1) is not valid and that the gap values extracted by using eqn (1) represent some sort of effective gap. It has not been possible, however, to relate this effective gap to the true gap. The first, somewhat obvious shortcoming of eqn (1) is that it is derived by assuming Boltzmann statistics for the carriers. In fact, in degenerate semiconductors, one must use FermiDirac statistics, which yield a very different expression for the prz product. Thus, if one insists on using eqn (1). one must realize that it is just an arbitrary expression and the quantity E, does not correspond to the actual band gap. It is possible, however, to relate the quantity &, appearing in eqn (1) to the true band gap by calculating the correct expression for the pn product and equating it to the right-hand side of eqn (1). The resulting equation can then be solved to obtain an expression for the true band gap in terms of the parameter ks of eqn (1) and other quantities. If we use Fermi--Dirac statistics, we get

L,,=E,-t

S(T).

(2)

where S(T)=E,:-kTlog(P(T)),

(3)

where

P(T)=2/m,&dE--~J;;

0

P<(E_)/P(6)

1 +exp[(

E-

E,)/kT] (4)

Here, we have assumed the zero of energy to be at the conduction-band edge, p,(E) is the actual density of states of the conduction bands at the relevant concentration and po( E) is the density of states of a free-electron parabola with the effective-mass of the conduction-band edge of the pure crystal. The Fermi level E,. in eqn (4) also depends on temperature and

etaI

must be determined self-consistently by requiring charge conservation. If we assume p, ( E) = p,,( E). P(T) reduces to F,,,( EF/kT), where F,, 2 is the Fermi-Dirac integral. The need to include this quantity as a correction to Slotboom and de Graaf’s[7] gap-reduction values was recognized by Keyes[g] and by Abram, Rees and Wilson[20]. Evaluation of this quantity shows that S(T) is always positive ‘anti increases with increasing concentration. Thus, it makes the discrepancies with the luminescence gapreduction values worse, as illustrated by the short vertical arrows in Fig. 2. Clearly, it does not suffice to reconcile electrical measurements with the !uminescence and optical-absorption spectra. We propose here that, in addition to Fermi~ Dirac statistics, another effect needs to be included in a correct calculation of the pn product in a heavily doped semiconductor for comparison with data such as those of [7]: the existence of fluctuations in the value of the band gap. Fluctuations were already noted by Schmid[13] to affect the value of the p/r product, We note that the pn product is extracted from values of the current that passes through the emitter. Let us assume that, over the area of the emitter, the band gap has the value EX t A, where the values of A are given by some distribution which we assume has the Gaussian form x( A, n) = (l/a&) exp (- Az/o’). (Compare, e.g. [17]). If. in the absence of fluctuations, the pn product has some functional dependence on the band gap given by pn =f(

E,)?

the final form of the pri product P”=l~~f(E,+A)R(A.“)dA 0

(5) is given by (6)

Once more, this expression can be equated to ths right-hand side of eqn (1) so that an exprcsxion for the parameter ,k, can be obtained. We can write &=E,+S(T)+~F(T).

(7)

where S(T) is the term arising from statistics alone in the absence of fluctuations, as described above. and F(T) arises from the inclusion of fluctuations. Calculation of F(T) with full Fermi--Dirac statistics must be done numerically. For purposes of illustration, we use Boltzmann statistics, in which ax. F(T) has the extremely simple form F( T) = -- o-/4kT.

(8)

We note now that F(T) has the correct sign so that. when added to the electrical values of Fig. 2. the resulting gap reductions decrease toward the standard luminescence values. For quantitative results. one needs a theory for (J, and, of course, a full calculation of F(T) using Fermi-Dirac statistics. If we use the simple formula eqn (8) WC see that the

Energy-gap

reduction

electrical gap reductions can be made to agree with the standard luminescence values if values of o approx. 35-55 meV are used, which appear quite reasonable. Our conclusion, however, is that the electrical measurements of the pn product cannot be easily used to extract values of the actual band-gap reductions. 4. CONCLUSIONS

We have demonstrated that the photoluminescence spectra, the optical absorption spectra and the electrical measurements of the pn product in heavily doped Si are fully consistent with each other. The luminescence spectra provide the simplest way to extract energy-gap reductions. The optical-absorption spectra can also be used to extract gap reductions, but the task is somewhat more delicate because the need to locate the Fermi level is complicated by free-carrier absorption and phonon-assisted processes. Finally, though in principle one can extract energy-gap reductions from electrical measurements, the task is complicated tremendously by the need to have a theory of fluctuations and to incorporate Fermi-Dirac statistics. Our analysis of the electrical measurements has a significant consequence in our understanding of the behavior of bipolar transistors. It has been generally believed that the observed reduction of the emitter injection efficiency is due to a reduction of the energy gap caused by heavy doping. We have now demonstrated that the reduction of the emitter injection efficiency is actually due to three factors caused by heavy doping: a reduction in the value of the energy gap, a contribution arising from the fact that now carriers obey Fermi-Dirac statistics instead of Boltzmann statistics and a contribution arising from fluctuations in the value of the band gap. Since the objective of transistor design is to increase emitter efficiency, our results suggest that it is desirable to have Fermi-Dirac statistics because they result in a larger effective energy gap. This observation suggests that the injection efficiency ought to increase at lower temperatures because the Fermi level moves deeper into the degenerate band. Furthermore, our results reveal that a homogeneous material with small fluctuations in the value of the band gap will have an apparent smaller gap reduction. Thus, dopants that have a smaller tendency to cluster might be more suitable for heavily doped regions of bipolar transistors. Acknowledgement-This work was supported ONR Contract No. NOOO14-80-C-0679.

in part

by

REFERENCES

1. E. Burstein, Phys. Rev. 93, 632 (1954). 2. F. Stem and R.-M. Talley, Phys. Rev. 100,1638 (1955). 3. C. Haas. Phvs. Rev. 125. 1965 (1962); J. I. Pankove and P. Aigram, Phys. Rev. 126, 936 (1562). 4. A. A. Volfson and V. K. Subashiev, Sov. Phys. Semic. 1, 327 (1967).

in silicon

23

A. Aziza and E. Amzallag, Phvs. Stat. 5. M. Balkanski, Solidi 31, 323 (1969). and A. A. Be@, IEEE Trans. ED-l% 6. W. L. Kaufmann 732 (1968). 7. J. W. Slotboom and H. C. de GraalT, Solid-St. Electron. 19, 857 (1976). 8. R. W. Keyes, Comm. Sol. State Phys. 7, 149 (1977). 9. J. C. Inkson, J. Phys. C 9, 1177 (1976). 10. G. D. Mahan, J. Appl. Phys. 15, 2634 (1980). 11. K.-F. Berggren and B. E. Semelius, Phys. Rev. B 24, 1971 (1981). 12. P. E. Schmid, M. L. W. Thewalt and W. P. Dumke, So/id-St. Commun. 38, 1091 (1981); W. P. Dumke, Appl. Phvs. Lett. 42, 196 (1983). 13. P: E. S&mid, Phys. Rev. B 23, 5531 (1981). 14. A. Selloni and S. T. Pantelides. Phvs. , Rev. L&t. 49, 586 (1982). 15. K.-F. Berggren and B. E. Semelius, Phys. Rev. B 29, 5575 (1984). was reached independently by J. 16. The same conclusion Wagner, Phys. Rev. B 29, 2002 (1984) on the basis of his own data. See also his paper in this volume. 17. E. 0. Kane, Phys. Rev. 131, 79 (1963); see also this volume. 18. They actually measure ppn, where ).t is the minoritycarrier mobility, which is not known accurately. We will not consider this uncertainty in the analysis of [7] further. Solid-St. Electron. 20, 279 (1977). 19. J. W. Slotboom, 20. R. A. Abram, G. J. Rees and B. L. H. Wilson, Adv. Phys. 27, 799 (1978). 21. See, e.g. S. T. Pantelides, Rev. Mod. Ph_vs. SO, 797 (1978). 22 A. Selloni and S. T. Pantelides, unpublished. APPENDIX

In[15], Berggren and Semelius concluded that, for the “complete-randomness” model, intervalley interactions are suppressed. As a result, the bulk of the band-edge shift comes from intravalley interactions, We wish to argue here that the suppression of intervalley interactions found in [15] is an artifact of second-order perturbation and that a full calculation would yield large contributions. When added to the already large intravalley terms, the resulting total shift would be far in excess of experimental values. We will argue that such a result is a consequence of unphysical aspects of the complete-randomness model. We first examine the magnitude of multivalley terms. It is well known that they are large and important in the dilute limit for isolated impurities[21]. They are large because the intervalley wave vectors allow the sampling of strong parts of the impurity potential near the core where screening is not very effective. As the concentration increases, this effect persists even in the metallic-screening regime because the dielectric constants of both pure Si and a free-electron gas have roughly the same small value at wave vectors corresponding to the intervalley separations. The SellomPantelides[l4] ordered-array calculations demonstrate this fact. There is no obvious reason that the intervalley interactions should be suppressed by randomness since the intervalley wave vectors are still there and ought to sample the weakly screened part of the core impurity potential. We suggest that the suppression found in [16] is actually an artifact of perturbation theory which does not allow an adequate localization of the wave functions around the impurity core. In support of this suggestion, we note that, in the Sellom-Pantelides ordered-array calculations it was necessary to include additional localized basis functions in order to get large intervalley contributions[22]. In contrast, use of the basis set that proved adequate for the intravalley terms produces negligible intervalley terms in accord with the findings of [15]. We conclude that intervalley terms ought to be large even in the complete-randomness model

24

S. T. PANTELIDES et 01.

of [ll, 151, so that the final predictions of that model would be substantially larger than the experimental values, as already suggested in [14]. Finally, we wish to suggest reasons why the completerandomness model of [11,15] fails to provide a correct picture by yielding too large gap reductions, whereas the ordered-array calculations is more appropriate. The impuri ties in a real crystal are probably distributed randomly with some average in-terimpurity spacing corresponding to their concentration. Randomness is largely in the angular distribution of impurities around any one impurity. Assuming an ordered-array seems to capture the essence of such a state of affairs. Angular randomness can then be simulated by

assuming different ordered-arrays and averaging the results. Such calculations have, in fact, been done and reveal that angular randomness broadens the ordered-array state densities[22]. In contrast, the complete-randomness model of [ll. 1.51 assumes that the probability of tinding two impurities a certain distance apart is independent of that distance. In other words, the impurities have equal probability of being extremely close, around their average spacing. or very far apart. We suggest that the large energy shifts produced by the complete-randomness model of [ll. 151 are a comequence of the fact that the model “allows” the impurities to get very close with a substantial amount of prohabilit~, i c. equal to the probability of being at the average spacing.