PHYSICAL REVIEW B VOLUME 50, NUMBER 15 15 OCTOBER 1994-I Electronic properties of cubic and hexagonal SiC polytypes from ab initio calculations P. Kackell, B. Wenzien, and F. Bechstedt Priedrich Sc-hiller Un-ioersitat, Institut fiir Eesthorpertheorie and Theoretische Optih, Mcz-Mien Pla-tz I, 077/8 Jena, Germany (Received 21 April 1994) Ab initio total-energy studies are used to determine the lattice constants and the atomic positions within the unit cells for 3C-, 6H-, 4H-, and 2H-SiC. The electronic structures are calculated for the atomic geometries obtained theoretically within the density-functional theory (DFT) and the local- density approximation (LDA). We state more precisely the ordering of the conduction-band minima and derive effective masses. By adding quasiparticle corrections to the DFT-I DA band structures we 6nd indirect fundamental energy gaps in agreement with the experiment. A physical explanation of the empirical Choyke-Hamilton-Patrick relation is given. Band discontinuities, bandwidths, crystal- field splittings, and ionic gaps are discussed versus hexagonality. I. INTRODUCTION The physics of silicon carbide and the possibilities of its use in devices have been subjects of considerable interest because of its strong chemical bonding, physical stability, and other attractive electrical, optical, and thermal prop- erties. Many applications require a detailed knowledge of the electronic structure of the material, which itself de- pends on the atomic geometry. The latter one is rather complicated since SiC crystallizes in more than hundred difFerent modifications, i. e. , polytypes. The band struc- ture of SiC and, in particular, the indirect fundamental gap vary remarkably with the polytype. In numerous theoretical studies in the last twenty years ab initio pseudopotential methods are extensively applied to the ground-state properties of 3C-SiC, z s 2H-SiC, v ~o and also 4H- or 6H-SiC. s ~~ However, results of ab ini- tio density-functional-theory (DFT) calculations based on the local-density approximation (LDA) for the elec- tronic structure are only available for 3C (Refs. 12 — 14) and 2H (Refs. 12 and 14) polytypes. Only in a very re- cent work~s also 4H and 6H have been studied. However, the DFT-LDA band gaps of semiconductors and insula- tors have been consistently underestimated by 30 — 50% compared with the experiment. ~s ~s Despite the discrep- ancies between the calculated and measured band gaps, the dispersion of single bands as well as the energetical ordering of the conduction-band mmima come out nearly correctly from the calculations. When many-body quasi- particle (QP) effects~s ~s are taken into account addition- ally, a reliable indirect energy gap may be predicted, at least for cubic 3C-SiC ~v ~s o The majority of the ab initio DFT-LDA calcula- tions concerning SiC (Refs. 2 — 5 and 7 — 15) expand the wave functions in terms of plane waves. The only exception concerns the full-potential linear-muon-tin- orbital (LMTO) method. s In the case of electronic- structure LMTO calculations for SiC polytypes usually the semiempirical atomic-sphere approximation (ASA) is additionally introduced. In general, the gap prob- lem. appears also within the LMTO-ASA description. However, it is remarkably reduced by varying the num- ber and the radii of the atomic spheres for the dif- ferent polytypes. s~'2s Electronic band structure studies for zinc-blende and wurtzite SiC before 1987 have ap- plied empirical methods, e. g. , the empirical pseudopo- tential method2s 2s or semiempirical linear combination of atomic orbitals methods. 27 zs Recently, Backes et al. s suggested an interesting interpretation of the trends in the electronic structure versus the polytype by consider- ing the polytypes as structures of mutually twisted Si-C bilayers and by interface matching of the electronic wave functions. More or less direct experimental studies of the electronic structure of the SiC polytypes are rather rare. Besides x-ray emission spectra3 ' only one band structure mapping exists for 3C-SiC. In the present paper, we present results of parameter- free pseudopotential calculations. The lattice constants a, c (for the hexagonal polytypes 2H, 4H, and 6H), or the length ao of the characteristic cube (for zinc-blende 3C-SiC) are determined by means of total-energy min- imizations. The atomic positions within the hexagonal unit cells are also optimized. ~~ Then, the band struc- tures are calculated for the theoretical equilibrium atomic structures. Otherwise, polytypes under pressure would be considered. s4 The infiuence of the atomic relaxations within the unit cells on the electronic structure is dis- cussed. We study the dispersion of the resulting bands in more detail. The energetical ordering of the conduction- band minima is studied versus the "percentage hexag- onality" of the polytypes. We discuss reasons for the indirectness of the energy gaps. With the physical expla- nation of the linear relation between the electronic energy gap and the percentage hexagonality up to 50'%%uo, i.e. , the Choyke-Hamilton-Patrick rule, a long-standing, but un- resolved problem is attacked. I'urthermore, the resulting effective masses are compared with experimental results and their structural trends are derived. Q163-1829/94/50{15)/10761{8)/$06. 00 50 10 761 Oc1994 The American Physical Society