PHYSICAL REVIEW 8 VOLUME 31, NUMBER 6 15 MARCH 1985 Band-structure calculation for GaAs and Si beyond the local-density approximation F. Manghi, G. Riegler, * and C. M. Bertoni Dipartimento di Fisiea dell' Universita degli Studi di Modena, I-41100 Modena, Ita1y G. B. Bachelet~ hfax Plan-ck Insti-tut fiir Festkorperforschung, D 7000 -Stuttgart 80, Federal Republic of Germany (Received 3 August 1984; revised manuscript received 13 November 1984) We have performed pseudopotential self-consistent band-structure calculations for bulk solids in the framework of the Kohn-Sham density-functional theory, applying the nonlocal exchange- correlation functional of Gunnarsson and Jones to valence electrons with no further approximation. We present our method together with representative results for GaAs and Si, the main interest being focused on single-particle energy eigenvalues and gaps. An accurate 'comparison of the results ob- tained with this approach and the local-density-approximation results is given. The dependence of such a comparison on the choice of the basis set (localized Gaussian orbitals or plane waves) is also examined, and found to be unimportant. The reliability of the method, the changes in band struc- ture, their k dependence, and the behavior of the exchange-correlation potential throughout the unit cell are discussed. We find that the inclusion of nonlocality in the description of exchange and correlation does not change valence-band states significantly and increases by a very small amount the energy difference between some conduction-band states and the top of the valence band. The in- crease ( (0. 2 eV) is, however, insufficient to solve the problem of the local-density-approximation minimum band gaps, which remain much smaller than the measured energy gaps. I. INTRODUCTION The local-density approximation (LDA) to exchange and correlation is the most widely used version of the density-functional theory for the determination of the electronic properties of solids, ' and it has been successful- ly applied to the calculation of many ground-state proper- ties of semiconductors, in spite of the inhomogeneity of their electronic charge distribution. The band structure of diamond- and zinc-blende- structure semiconductors has been obtained in the I. DA scheme from first principles, both through pseudopoten- tial ' and all-electron methods. ' ' The agreement be- tween full-core calculations and first-principles pseudopo- tentials " is excellent, ' but both of these methods fail in the description of the band gaps, which are systemati- cally underestimated with respect to the experimental values. Such a deficiency has been avoided in the past by using either empirical pseudopotentials designed to repro- duce optical data, ' or by an artificial enhancement of the exchange and correlation potential. ' ' The failure of first-principles band-structure calculations in predicting the energy separation between occupied valence states and unoccupied conduction states seems to reflect fundamen- tal limitations of the approach; how many of these limita- tions are intrinsic to density-functional theory and how many to its local approximation is a question which must still be answered. Recently, it has been pointed out that even an exact formulation of the density-functional theory would lead to a nonzero difference between the fundamen- tal gap of a semiconductor and the one-particle energy gap obtained from the solution of the Kohn-Sham equa- tions. ' ' The order of magnitude of this difference is, however, unknown, and its relevance can be indirectly es- timated by calculations with improved approximations of the density functional, beyond the LDA. A straightforward approach would be the direct evalua- tion of quasiparticle energies, avoiding the Kohn-Sham equations through a self-energy calculation. ' '' This method leads to qualitatively correct results, but until now it has only been implemented for simplified models, and no practical prescription for a quantitative calculation has been put forward. The other conceivable source of inaccuracies in the density-functional description of the first excitation in semiconductors is the local approximation. This approxi- mation is expected to fail for real systems since they show substantia1 departures from homogeneity. In this respect improvements of the exchange-correlation potential, such as gradient corrections and modifications of the exchange-correlation, hole ' have been proposed. Vhthin the spirit of the latter approach, our paper presents the results of a first-principle pseudopotential calculation for GaAs and Si in which the nonlocal ap- proximation for the exchange and correlation (NLXC) po- tential outlined in Refs. 23 and 24 is adopted for the valence electrons. %e will show that, within our approxi- mations, no evidence is found that a better inclusion of in- homogeneity can give any substantial improvement of the minimum gap. Another ingredient of the calculation which may affect the band structure (and specifically the energy gaps) is the choice of basis functions, which is incomplete in most variational calculations. 31 3680 1985 The American Physical Society