Optical and loss spectra of SiC polytypes from ab initio calculations B. Adolph, K. Tenelsen, V. I. Gavrilenko, and F. Bechstedt Friedrich-Schiller-Universita ¨t, Institut fu ¨r Festko ¨rpertheorie und Theoretische Optik, Max-Wien-Platz 1, 07743 Jena, Germany ~Received 9 July 1996; revised manuscript received 24 September 1996! Frequency-dependent dielectric functions are calculated for the hexagonal polytypes 2 H, 4 H, and 6 H as well as the cubic modification 3 C of silicon carbide. The calculations are based on the ab initio pseudopotential-plane-wave method and random-phase approximation. We find a remarkable redistribution of the optical absorption and related spectra as reflectivity and energy loss due to the variation of the crystal structure and polarization direction. The relation of spectra and underlying electronic structures as well as the influence of nonlocality and quasiparticle effects are discussed. The numerical results are compared with experimental data available. @S0163-1829~97!00103-3# I. INTRODUCTION Silicon carbide ~SiC! occurs in about 200 polytypes. 1 The two most extreme polytypes are zinc blende (3 C ) with pure cubic ( C ) stacking of the Si-C double layers in the @111# direction and wurtzite (2 H ) with pure hexagonal stacking in the @0001# direction. The other hexagonal ( H ) and rhombo- hedral ( R ) polytypes, nH and nR , represent combinations of these stacking sequences with a periodicity of n double lay- ers in the stacking direction. 2 It is well known that the poly- typism is of strong influence on the physical and chemical properties. For example, the energy gaps and the location of the conduction-band minima in k space vary with the crystal structure ~cf. Ref. 3 and references therein!. With a change of the indirect energy gap of about 1 eV between the 3 C and 2 H polytypes, 4,5 SiC represents an extraordinary example for the polytype influence on the electronic structure. Optical spectroscopy should make visible the drastic changes in the electronic properties with the polytype. Re- cent advances in crystal growth of SiC have allowed the study of the optical properties of different polytypes. A sys- tematic investigation of the vacuum-ultraviolet reflectivity has been presented for 3 C ,4 H ,6 H , and 15R crystals. 6 Pre- vious reports of reflectivity, 7 electroreflectivity, 8 and spectro- scopic ellipsometry 9 were mainly restricted to the zinc blende polytype, 3 C -SiC. Recent calculations 6 of the reflec- tivity within the linear-muffin-tin-orbital method and the atomic-sphere approximation need a calibration of the abso- lute reflectivity to compare with the experimental data. Ab- sorption spectra have been calculated using ab initio pseudo- potentials but neglect the effect of their nonlocal contribution to the optical transition operator. 10 In the past simplified physical descriptions such as the empirical pseudopotential method and the orthogonalized-plane wave method have been applied to 3 C -SiC ~Refs. 11,12! and 2 H -SiC. 12 In this work we calculate the optical and dielectric prop- erties of the hexagonal 2 H -, 4 H -, and 6 H -SiC polytypes and compare them with those for cubic 3 C -SiC. The ab initio calculations start from the atomic and electronic structures obtained within the density-functional theory ~DFT! and local-density approximation ~LDA!. The influence of nonlo- cality and quasiparticle effects is discussed. We present the frequency dependence of the dielectric tensors as well as of reflectivity and energy-loss functions. By means of the band structures we give explanations for the shifts and splittings of peak structures in the dielectric functions with the polytype. Furthermore, we compare our results with the experimental data. II. METHOD In order to examine the optical and energy-loss properties of SiC polytypes we have first calculated the imaginary parts of the elements of the second-rank dielectric tensor « ab ( v ) in the optical limit, 13 Im« ab ~ v ! 5 8 p 2 e 2 \ 2 V ( c , v ( k ^ c ku v a u v k&^ c ku v b u v k& * @ « c ~ k! 2« v ~ k!# 2 3d @ « c ~ k! 2« v ~ k! 2\ v # . ~1! Herein the Bloch eigenfunction u n k& belonging to a band index n ( n 5c conduction bands, n 5v valence bands! and a wave vector k in the first Brillouin zone ~BZ! is related to a Kohn-Sham eigenvalue « n ( k) of the DFT-LDA. V denotes the crystal volume. The velocity operator v in the optical transition matrix elements is defined by the commutator of single-particle Hamiltonian and space operator. Its replace- ment by the momentum operator p/ m leads to the neglect of the influence of the nonlocal contributions due to the nonlo- cal pseudopotentials. For a detailed discussion the reader is referred to Ref. 13, where the effect is discussed for cubic semiconductors. The neglect of the nonlocality effects in- creases the average oscillator strength by 15%. A similar overestimation arises for the dielectric constants and the low- frequency reflectivity spectra. The collective plasmon peak in the energy loss spectra is shifted to higher energies by about 2 . . . 3 eV. For that reason all calculations are per- formed including nonlocal contributions. The real parts of the components of the dielectric tensor Re« ab ( v ) are calcu- lated from expression ~1! by means of a Kramers-Kronig transformation. Then the reflectivity and energy loss function follow using Fresnel’s formula for normal light incidence or definition. Expression ~1! is taken within the independent- particle approximation. Local-field effects have been ne- glected. Consequently, the exchange-correlation kernel due to the partial treatment of these effects in the DFT-LDA band structure does not influence the spectrum. PHYSICAL REVIEW B 15 JANUARY 1997-I VOLUME 55, NUMBER 3 55 0163-1829/97/55~3!/1422~8!/$10.00 1422 © 1997 The American Physical Society