Applications of the CPA to elemental tetrahedrally bonded amorphous semiconductors

Solid State Communications, Vol. 38, pp. 147-150. Pergamon Press Ltd. 198 I. Printed in Great Britain.

0038-1098/81/020147-04502.00/0

APPLICATIONS OF THE CPA TO ELEMENTAL TETRAHEDRALLY BONDED AMORPHOUS SEMICONDUCTORS C.T. White and W.E. Carlos* Naval Research Laboratory, Washington, DC 20375, U.S.A.

(Received 21 March 1980 by R.A. Cowley) It is pointed out that the CPA can easily be coupled with multiband, nearest-neighbor tight-binding Bethe-lattice Hamiltonians to fruitfully study the electronic structure of tetrahedrally bonded elemental amorphous semiconductors. Results are presented which illustrate the effects on the model semiconducting gap of (i) a rectangular distribution of site diagonal terms in the Hamiltohian, and (ii) vacancies distributed randomly over the lattice. The relationship of these results to experiment is briefly explored. IN RECENT YEARS there has been considerable experimental [ 1 - 4 ] and theoretical [ 1 - 3 , 5 - 1 0 ] effort directed toward understanding the origin and effects of intrinsic states in the gap of tetrahedrally bonded amorphous-semiconductors. This work has, of course, been strongly stimulated because of the possible technological importance of these systems in semiconductor device applications. A significant general development in the theory of disordered and/or correlated systems was the coherentpotential-approximation (CPA) of Taylor [11 ], Soven [12] and Hubbard [13]. Here we outline how the CPA can be applied to tight-binding multiband Bethe-lattice [ 14] (BL) models of homopolar tetrahedrally bonded a-semiconductors. The resultant formalism is then used to obtain CPA results which show the effects on the model density of states (DOS) of: (1) a continuous distribution of site energies, and (2) various concentrations of vacancies arranged randomly over the lattice. The basic tight-binding Hamiltonian, ~ , we employ has been described in detail for the crystalline case by Hirabayashi [15]. This model takes into account all nearest neighbor interactions between s, p valence states. In an obvious notation [16], the matrix elements entering this Hamiltonian for the periodic case are completely specified by the one and two center parameters,E,,Ep and E~, E~,, E~,~, E~y, respectively. We use the CPA to determine the DOS per atom of this model for the case that E s and Ep are not the same at each site i as in the crystalline case but are rather random variables which we will denote by 4 and 8 , respectively. This DOS is a statistically sharply distributed quantity and can be expressed in the usual way as p(E) = (-- 1/rr) Im [tr (goo(E +/13+))], where the Green function g obeys the

equation of motiong = [Z _ ~ ] - t and goo denotes the four by four matrix composed of the sixteen matrix elements o f g between sqike and p-like states associated with the atom labelled o. Furthermore, the symbol ( ) here and throughout indicates an average over the random variables {e~} and {~}. To obtain (g) we employ the CPA. Hence, we first introduce an effective Hamiltonian H and the corresponding Green function G = [Z - - H ] q derived from Jf by replacing the random potentials ~ and d'/ at each site i by the as yet unspecifield effective potentials ~;s and Zp, respectively. Then by use of Dyson's equation (g) can be expressed as (g) = G + G(T)G, where the scattering matrix, T, arises because of fluctuations of the actual random system around the effective system. Since for the present problem ~f - - H = 2;,, Vn (where the subscript n indicates that the only non-zero matrix elements of the operator V involves s and p states associated with the site n), we can expand T in the usual way [ 17] in terms of the single site scattering matrices tn = (1 -- VnG)-I V n We now make the CPA which simply involves the replacement of (goo) in the expression for o(E) by Goo subject to the self-consistency conditions: (to) = 0, which for the cases we treat guarantees that corrections to this approximation enter only in fourth order in the tn's. Hence the present multiband CPA approach will reproduce all the various limits usually detailed [18J for the single band CPA. For the present problem, since Vn and Gn,~ are both diagonal matrices and the p matrix elements of each are internally equal, the set of equations, (to) = 0, reduce to simply the two: <(E~ -- ~s)/[1 - (E~ - Zs)Ggo]> = <(Eg -- ~p)/[1

- (Eg -- 2;p)ggo]> = O, * N R C - N R L Resident Research Associate. 147

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APPLICATIONS OF THE CPA TO AMORPHOUS SEMICONDUCTORS

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Fig. 1. Illustrates the asymmetric way states are introduced into the non-random model gap as the width W (in eV) of the rectangular probability distribution is increased. Vertical lines denote positions of the unperturbed band edges. where, e.g. Ggo - (Po~lGooiPox). Equation (1) shows after the fact the reason for introducing only two coherent potentials; Zs and Y.p. In order to solve these equations for Y's and Zp we must obtain G~o(Es, Ep), Ggo(F_,s, Ep) which can be done by employing the transfer matrLx approach recently generalized to treat the present nearest-neighbor multiband BL system [7, 8]. More specifically, these transfer matrix techniques yield a set of five equations [8], widch we solve together with the two CPA equations by numerical iteration to find Zs(Z) and Xp(Z) and hence the CPA DOS. As a first application of the above outlined formalism, we have studied the CPA model DOS in the neighborhood o f the ideal (nonrandom) BL gap assuming that the random variables {esiJ, {~} are statistically independent with each obeying a continuous probability density centered at some so of the form, P(s) = W - i if Is -- sol < WI2 and zero otherwise. The probability densities for ~ and ~ were centered at their respective crystalline values Ep and E 8. One might expect these choices for ~ , d ' / t o crudely but reasonably model the effects of the local environment varying from atom to atom in the amorphous system, as can be seen, e.g. through a bond-orbital approximation [19]. In Figure i we display the DOS in the neighborhood

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Fig. 2. Shows the effects of different concentrations of vacancies on the DOS per atom in the neighborhood of the mobility gap. of the ideal BL gap obtained by assuming that W = 0, 5 and 8 eV. The parameters (in eV) used in obtaining these figures have been used in previous work [20] and are appropriate to germanium namely; E s = -- 6.3, Ep = 2.1,Ess = - 1.7,E~ = 1.33,Exx = 0.66 and Exy = 1.71. These calculations were made principally to investigate further the suggestion made in [7] that because the states at the valence band edge (VBE) are p-bonding like while those at the conduction band edge (CBE) are s-antibonding like one might expect that randomness might have greater affect on the VBE than the CBE. These anticipated differences in effects could then provide a partial explanation for the experimental results [4] which seem to indicate that there is an asymmetry DOS in the mobility gap with the larger DOS closer to the VBE. Indeed, our results shown in Fig. 1 are consistent with this picture and we have obtained essentially similar results using a parameter set appropriate for Si. Note the CPA DOS of Fig. 1 exhibits sharp band edges. This however is expected to be an artifact of the CPA with the actual model DOS tailing into the semiconducting gap from the CPA band edges as in a M o t t CFO picture [21]. The CPA does not reproduce such expected tail states because these states are quite localized and hence scattering effects higher than third order in the tn's (ignored in the CPA) are essential to their description and reproduction. These band edge tail states occurring when W 4 : 0 may close the gap of Fig. (1) in

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APPLICATIONS OF THE CPA TO AMORPHOUS SEMICONDUCTORS

the sense that the DOS is no longer precisely zero anywhere in the nonrandom energy gap region. However because of the localization of such states a mobility gap should remain [21 ]. Furthermore, an analogy to single band results [22] the mobility edges for W even as large as 8 eV are expected not to lie too far from our CPA determined band edges. This leads us to expect [23], e.g. that the actual DOS on the scale of Fig. 1 for W = 5 and 8 eV would appear quite similar to the corresponding CPA results of this figure with the most easily visible modification being small tails extending into the gap. As a second application of our CPA approach, we treat the problem of vacancies randomly distributed over the lattice. Amorphous semiconductors are thought to contain a relatively large number of voids and the like and so this problem is of considerable interest. Mathematically a vacancy at some site n can be treated simply by letting the associated site diagonal elements of our Hamiltonian at that site (,J~nn,~.nn)s I9 recede to positive infinity. Thus, if we assume that the vacancies are uncorrelated this problem is especially suited to our present approach with the probability density for each e,s., ~ adopting the binary delta function form P(eSi, ~ ) = PvS(esii -- e~),5(e~ -- e~ ) + (1 --P,~)8(e~ - - E s ) 6 ( ~ - - E p ) , where P~ denotes the concentration of vacancies and e= ~ + By employing the same parameter set as before, we have obtained the DOS for this model for values of Pv ranging from 10 -4 tO 10-1. In all cases we were able to resolve departures in the CPA averaged DOS from the ideal BL case. These changes are essentially concentrated in the neighborhood of the ideal semiconductor gap and results for that region are shown in Fig. 2 for several values ofPv. From these results we see that a relatively small percentage of vacancies produces noticeable effects. Indeed a concentration of 1% vacancies is sufficient to remove large-portions of the ideal gap while concentrations of 10% fills in this gap rather uniformly. It is mobility gap states such as shown in Fig. 2 that are, e.g. presumably compensated by H in, e.g. hydrogenated a-silicon to produce a dopable system. Ptease note, that since we find the vacancy induced states for low Pv to lie deep in the ideal BL gap and "isolated" [24] multiple scattering effects of higher than third order in the tn's between vacancies in close proximity will be important in the description of the associated states. Of course the probability of such configurations approaches zero asPv -+ 0 and the measure of the associated states becomes negligible as the scale of Fig. 2 forPv ~< 0.01. On the other hand the measure of such states on the scale of typical dopant concentrations may not be negligible and to study these effects it is necessary to generalize the CPA to a cluster

149

CPA approach along, e.g. lines laid down earlier [25] for single band models. For a number of additional reasons [26] application of these CPA vacancy model results beyond semiquantitative comparisons to the actual physical system is at this time unwarranted. Perhaps one of the most important of these is our ne~ect of manybody effects which should become important in the description of the states we find introduced deep in the ideal gap because of the expected greater localization of these states; especially for low values of Pv. These manybody effects arise from Hubbard-like electron-electron repulsive interactions between elect tons in these states as well as local electron-lattice reconstruction interactions. As has been stressed, e.g. by one of us in the past [6, 27], reconstruction effects associated with defects in tetrahedrally bonded systems can be very large. In fact, calculations [27] have shown that these effects may actually overcome the Coulomb repulsive terms leading to an effective attractive interaction between spin up and spin down electrons in the vicinity of dangling Si bonds as well as strained [28] Si-Si bonds. Now within this context suppose that reconstruction effects dominate to the point of producing an attractive (Anderson-like [29] negative U) electron-electron interaction associated with the CPA gap states shown in Fig. 2. In this instance all of these states would be either doubly occupied or doubly empty. In addition, those states which are doubly occupied (empty) will be shifted downward (upward) in energy from the Fermi-level, EF. Since in the uncompensated case we f'md EF to be essentially at the center of the gap band DOS of Fig. 2, these effects would deplete states from the neighborhood of EF and probably generate a "gap" in this DOS, especially at low values of Pv. On the other hand, if these states were appropriately described by a positive U model one would again expect a reduction of the one-electron DOS in the vicinity o f E F through the complete or partial formation of a primarily Hubbard-like [ 13] gap. Thus we see that in either case the one-electron DOS shown in the CPA gap of Fig. 2 could be appreciably altered. Finally, in this vein it is noteworthy that because o f the lattice coordinates involved in both the negative and positive U cases, these one-electron levels would not in general coincide with where an experiment relying on a well-defined EF would determine them to be. REFERENCES i.

See, e.g. Structure and Excitations o f A morphous Solids, AIP Conf. Proceedings 31, (Edited by G. Lucovsky & F i . Galeener) AIP (1976) and refs. therein.

150 -.1

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

APPLICATIONS OF THE CPA TO AMORPHOUS SEMICONDUCTORS See, e.g. Proc. 7th Int. Conf. on Amorpl~ and Liq. Semicond. (Edited by W.E. Spear), U. Edinburgh Press (1976) and refs. therein. See, e.g. Proc. 8th Int. Conf. on Amorph. and Liq. Semicond., J. Non-Cryst. Solids (in press). A. Madan, P.G. Le Comber & W.E. Spear,J. NonCryst. Solids 20,239 (1976). M.F. Thorpe, D. Weaire & R. Alben,Phy~ Rev. B7, 3777 (1973). C.T. White & K.L. Ngai,Proc. 2nd[nt. Conf. on Electronic Prop. of 2 D System. Berchtesgaden (1977);Surf. Sci 73, 116 (1978). V.T. Rajan & F. Yndurain, Solid State CommurL 20,309 (1976). J.D. Joannopoulos,Phys. Rev. B16, 2764 (1977). W.Y. Ching, D.J. Lain & C.C. Lin, Phys. Rev. Lett. 42,805 (1979). J.C. Phillips, Phys. Rev. Lett. 42, 1151 (1979). D.W. Taylor,Phys. Rev. 156, 1017 (1967). P. Soven,Phys. Rev. 156,809 (1967). J. Hubbard,Proc. R. Soc. 23 (London) A281,401 (1964). A Bethe lattice is a lattice with no closed loops which has been used previously in, e.g. [5, 7, and 8] as the starting point in studying a-tetrahedral solids. K. Hirabayashi,J. Phys. Soc. Japan 27, 1475 (1969), J.C. Slater & G.F. Koster.,Phys. Rev. 94, 1498 (1954). B. Velicky, S. Kirkpatrick & H. Ehreneich, Phys. Rev. 175,747 (1968). R.J. Elliott, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46,465 (1974).

19. 20. 21. 22. 23.

24. 25. 26.

27. 28. 29.

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W.A. Harrison, Electronic Structure and the Properties of Solids, p. 59. W.H. Freeman and Co., San Francisco (1980). D.J. Chadi & M.L. Cohen,Phys. StatusSotidi[b) 68,405 (1975). N.F. Mott & E.A. Davis, Electronic Processes in Non-Crystalline Materials, p. 199. Clarendon Press, Oxford (197 I). See, e.g.D.C. Licciardello & E.N. Economou, Phys. Rev. B l l , 3697 (1974). This is plausible because states in the extended region of the spectrum being spread out over the system should see an effective (motionaIly narrowed) potential of the type self-consistently determined by the CPA. Tail states of small measure not reproduced by the CPA are expected in this case as well. See, e.g.C.T. White & E.N. Economou, Phys. Rev. BI5, 3742 (1977). These include: (i) the neglect of any ring structure through the use of the BL, (ii) the inclusion of only nearest neighbor interactions, (iii) the assumption of statistical independence in the vacancy positions, and (iv) supposing W = 0 in studying the vacancy problem. C.T. White & K.L. Ngai,J. Vac. Sci. Technol. 16, 1412 (1979). A strained bond is one with the bonding strength different and usually appreciably less than the ideal Si-Si bulk bond strength. P.W. Anderson, Phys. Rev. Lett. 34,953 (1975).