Physica B 324 (2002) 15–33

Electronic structure of several polytypes of SiC: a studyof band dispersion from a semi-empirical approach

Maria Matos*

Departamento de F!ısica, Pontificia Universidade Catolica do Rio de Janeiro, CP 38071, G !avea, 22453-970 Rio de Janeiro, RJ, Brazil

Received 30 January 2001; received in revised form 28 March 2002

Abstract

A study of the most common polytypes of silicon carbide, namely 3C, 2H, 4H, 6H and 15R is performed, by using the

extended H .uckel tight binding approximation. Special account is taken in describing optical properties of the material.

New basis sets and empirical parameters are suggested which contain excited s* orbitals for 3C, 4H, 6H and 15R, and

p* orbitals for 2H. The new basis sets reproduce most important features of the topology of valence and conduction

bands given in the literature from ab initio, pseudopotential and local density approximation methods, in special the

appearance of an indirect gap of the correct order in each polytype. It is argued that the present results indicate that

one-electron methods are suitable for describing optical properties in more complex polar semiconductors, such as the

polytypes of SiC. r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Silicon carbide; Polytypes; Electronic structure; Band dispersion; Extended H .uckel approximation

1. Introduction

Silicon carbide is an important material for thedeveloping electronic industry [1,2]. Besides ex-cellent thermal and mechanical stability it presentspolytypism, a one-dimensional polymorphism.Different polytypes have different stacking se-quences of Si–C double layers. Starting from theperfect cubic structure—zincblende (3C-SiC)—hexagonal and rhombohedral structures areformed with the pure hexagonal (2H-SiC) at theother extreme. The most commonly used polytypesfor basic research and application are the purelycubic (3C) and purely hexagonal (2H), thehexagonal 4H and 6H and the rhombohedral

15R. Other tetrahedral compounds show polytyp-ism but SiC is the only one in which differentcrystal structures have significantly different elec-tronic properties [3]. Band gaps, for instance, varyin the range 2.4–3.3 eV if one goes from 3C-SiC to2H-SiC. Another important aspect for deviceengineering is the fact that different polytypeshave different electronic effective masses [4].In the last few years, theoretical studies have

addressed mainly the subject of understandingrelationships between the crystalline and electronicstructures of the distinct forms of SiC. Quantumchemical calculations within ab initio, pseudopo-tential and local density approximation haveshown to be powerful theoretical tools; muchprogress has been achieved in reproducing detailedexperimental data of structural, electronic, optical,dynamical and surface properties [4–6] of SiC.

*Fax: +55-021-512-3222.

E-mail address: mmatos@fis.puc-rio.br (M. Matos).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 2 0 3 - 6

Since the early work of Vogl et al. [7], semi-empirical tight binding methods have been used toinvestigate the electronic structure of semiconduc-tors. In spite of their mathematical simplicity, theone electron approach can be very useful toexplain qualitative features, basic mechanisms ofinteraction and their connection to the observedband structure, chemical trends, etc. [8].Semi-empirical hamiltonian depends on empiri-

cal parameters. It is believed that once a good,robust set of parameters is determined whichreproduces satisfactorily the band structure ofthe bulk, reliable and useful insight can be gainedin the study of the more complex systems. Someexamples of interest in the study of silicon carbideare doping [9,10], surface reaction [11–13], ohmic[14] and Schottky [15] contacts and the electronicstructure of heteroepitaxial films [16]. In the studyof dopants [17], the calculation of ionizationenergies of shalow impurities requires the use oflarge supercells, which becomes difficult if morepowerful theoretical methods are used.It was soon realized [7] that a basis set which

contains only valence atomic orbitals is notsuitable to describe the conduction bands ofsemiconductors. This comes from the well knowninadequacy of molecular orbital methods torepresent empty electronic levels. However, it hasbeen possible to cope with this limitation byincluding, in the basis set, excited s orbitals (s*).This has led to a considerable improvement ofresults, in particular, the description of the indirectgap appearing in most semiconductors of thezincblende structure [7].Recently [18,19], s* orbitals were used in tight

binding extended H .uckel [20] calculations ofelectronic properties of diamond [18], silicon andcubic SiC [19]. For diamond [18] and crystallinesilicon [19] s* parameters were found whichprovided excellent results for the band structure.The position of the indirect gap and the overalltopology of valence and conduction bands couldbe very satisfactorily reproduced.In the case of SiC [19] the situation was

different. Si and C parameters, good for the puresystems, diamond and crystalline silicon, could notprovide reasonable results for the calculation ofthe band structure of the heteronuclear material.

By using s* orbitals on silicon it was possible toreproduce an indirect gap. However, its positionand the overall shape of the conduction band wasin disagreement with experimental data and abinitio and pseudopotential results. Another dis-advantage of the use of the pure system para-meters concerns the calculation of the direct gapenergy, which was largely overestimated. In theextended H .uckel approach, two center electron–nuclei interaction terms are determined by the offdiagonal terms of the Hamiltonian, Hij : These aredefined by Hij ¼ KSijðHii þ HjjÞ; where Hii and Hjj

are valence state ionization potentials and theoverlap integrals are calculated from Slater typeorbitals (STO); K is a conveniently definedconstant. The definition of Hij in terms ofindependent atomic parameters satisfies a general-ity criterion for the parameters. The use of theoverlap integral in connection with atom-relateddata aims at describing local effects taking intoacount the geometry of molecule and crystal.However, as seen above, for crystalline SiC thisscheme does not lead to a good description ofband structure and consequently of importantoptical properties of the material.For highly ionic systems [21], a scaling trans-

formation of atomic orbitals was used to adjust thesizes of atomic orbitals to the lattice parameter ofthe crystal. In the case of SiC the transformationwas used [19] for the valence atomic orbital set (theorbitals 2s and 2p of C and 3s and 3p of Si), withan excellent improvement in the calculation of thedirect gap. Nevertheless, when applied to excitedorbitals, scaling transformation leaded to spuri-ous, unphysical results, such as the breakdown ofthe valence–conduction band energy separation.A more systematic study is therefore necessary

in order to obtain appropriate extended H .uckelparameters and atomic basis set for crystallineSiC. This is done in the present paper. Theextended H .uckel band structure of 3C, 2H, 4H,6H and 15R silicon carbide polytypes are investi-gated by taking special account of the topology ofvalence and conduction bands. The idea followedhere is similar to that employed in empiricalpseudopotential and other tight binding calcula-tions of crystalline materials. Band energies atspecial points of the Brillouin Zone (BZ), directly

M. Matos / Physica B 324 (2002) 15–3316

related to optical properties are used as criteria forthe definition of the crystal parameters [7,22].Some basic assumptions are made: (i) the newparametrization should present no essential differ-ences from the standard molecular one; (ii)parameters for SiC should be somewhat relatedto parameters of diamond and crystalline silicon;(iii) for a final adjustment it is the actual bandstructure of a given polytype which determines thebest parameters and basis set.One of the advantages of the extended H .uckel

tight binding approach is the fact that it takesexplicitly into account interactions between next-nearest neighbors without the introduction of newparameters. It could be expected to be suited tostudy polytypes since, in this case, it is theinteraction of next-nearest neighbors that distin-guishes one polytype from another. The presentstudy will help understanding the extent of thisassertion in the case of SiC.

2. Method

The use of the extended H .uckel theory in thestudy of infinite systems has been already dis-cussed in the literature [20]. It is briefly mentionedhere that, in a crystal, translational symmetryleads to the construction of Bloch functions, cðiÞ

k ðrÞ(i ¼ 1; 2;yn), infinite linear combinations ofatomic orbitals, which form the n � n hamiltonianmatrices HðkÞ: n is the number of atomic orbitalsin the crystalline unit cell and k a vector in thereciprocal lattice, taken in the first BZ. Theeigenvalues eiðkÞ; which give rise to the bandstructure of the crystal, are obtained through thesolution of the secular equation detðHijðkÞ �eSijðkÞÞ ¼ 0 for each reciprocal vector k of a givenchosen set. The matrix elements HijðkÞ depend onempirical parameters hii and hij ; defined in thesame way as for molecular calculations; hii aregiven by valence state ionization potentials and hij

are calculated according to the Wolfsberg–Helm-holz formula, hij ¼ Kðhii þ hjjÞSij ; where K isdefined elsewhere [19] and Sij are the overlapintegrals between the STO [23] of the atomic basisset. Density of states (DOS) analysis can be doneby adjusting gaussian curves to crystal energy

levels. In this case, the number of k points usedwas 408 for cubic and 450 for hexagonal struc-tures.STO exponents can be taken as varying para-

meters. A scaling transformation can be performedwhich consists of defining the exponent x0 of agiven STO in a crystal host of lattice parameter a0

according to x; the STO exponent in a crystal hostof lattice parameter a; through the relationshipxa ¼ x0a0: The transformation can be viewed aspartly describing Pauli repulsion effect on theelectronic density in a given crystal host. Acontraction (expansion) of the orbitals occurswhen the lattice parameter decreases (increases).In the present paper, scaled atomic orbitals of Siand C in SiC are determined, respectively, fromthe corresponding orbitals of crystalline siliconand diamond. The lattice parameters increase fromdiamond (3.567 (A) to silicon (5.429 (A), with, forinstance, cubic SiC in between both (4.360 (A) [5].Therefore scaled Si exponents for SiC—x0Si ¼ 1:72for cubic and x0Si ¼ 1:727 for the hexagonalstructure—lead to more contracted orbitals thanthose of crystalline silicon, whose exponents havethe standard value xSi ¼ 1:383: For carbon ex-ponents one gets x0C ¼ 1:33 in cubic SiC and x0C ¼1:331 in hexagonal SiC, leading to expandedorbitals if compared with those of diamond’s, forwhich xC ¼ 1:625:A rough estimate of the effect of scaling can be

given by comparing the scaled and standardaverage radii, /rS; for each orbital. Since, forSTO [23], /rS ¼ ð2n þ 1Þ=2x; the ratio betweenscaled and standard orbitals of Si and C can bereadily obtained from the lattice parameters ratio.One gets 0.80 for Si and 1.22 for C, showing about20% contraction or expansion of the STO,whichever should the case be.For the cubic polytype, a face-centered cubic

crystal (FCC) [24], the primitive (rhombohedral)unit cell will be used in all calculations. It containstwo atoms, a Si atom at its origin and a C atomlocated in the main diagonal of the cubic unit cell,at a distance equal to one fourth of its size. Thecrystal structure and corresponding BZ are shownin Fig. 1(a), with some special k points indicated.For the other polytypes, the hexagonal unit cellwill be used. It contains four (2H), eight (4H),

M. Matos / Physica B 324 (2002) 15–33 17

twelve (6H) and thirty (15R) atoms [25]. Thehexagonal crystal structure and corresponding BZ,also hexagonal, are shown in Fig. 1(b); special k

points are also indicated in the BZ.A few more definitions are necessary for the

foregoing discussion. The direct gap, Ed; is definedas the minimum energy difference between thelowest unoccupied and highest occupied bandlevels of a given k point. For SiC, as for mostsemiconductors, it occrus at the band center (seepoint G in Fig. 1). The indirect gap Ei; is obtainedas the difference between the minimum energy ofthe conduction band and the highest energy of thevalence band, occurring at different k points. Forcubic SiC, Ei comes from a G2X transition[5,8,22], while for the pure hexagonal polytypethe transition is of order G2K [5,8]. In the case of4H, 6H and 15R, an indirect gap of order G2Mwas found [5,6]. The crystal ionization potential,Ip; is equal to minus the energy of the top of thevalence band, which in this work is called thesystem’s Fermi level, eF:

3. Results and discussions

3.1. Cubic SiC

In this section, the 3C-SiC polytype is investi-gated. As a starting point, and in order to makethe discussion more clear, the extended H .uckelband structure and DOS curves when no s*orbitals are included in the basis set are repro-duced in Fig. 2 [19]. Two sets of parameters areused, the standard set (Fig. 2(a)), and parametri-zation I (Fig. 2(b)). Standard parameters are thosecompiled by Alvarez [26]; in set I, parameters weredetermined in order to give a correct description ofthe band structures of diamond and crystallinesilicon [18,19], with a scaling transformation ofSTO (see parameters in Table 1). Thus, inparametrization set I, silicon orbitals are con-tracted and carbon orbitals are expanded, ascompared to standard orbitals.The valence bands presented in Fig. 2, for both

sets of parameters, reproduce well results obtainedwith more sophisticated theoretical methods[5,8,22]. The main differences concern the width,

Fig. 1. The crystalline structure and corresponding BZ of

cubic (a) and hexagonal (b) SiC; special k-points are shown in

the BZ.

M. Matos / Physica B 324 (2002) 15–3318

d; and its internal gap, DEval; the energy splitting atX, characteristic of polar semiconductors of thezincblende structure [27]. For the standard setd ¼ 19:7 eV is smaller than for set I, in whichd ¼ 23:1 eV. Different pseudopotential calcula-tions give for this quantity calculated values inthe range 15–22 eV [5,8,22]. As for the internalgap, the standard parametrization value of 5.92 eVis larger than that given by the parameter set I, of0.71 eV. In pseudopotential calculations this quan-tity varies from 1.4 to 4.7 eV [5,8,22] (Table 1).The obtained conduction bands are, however,

poorly reproduced in either calculation of Fig. 2.

For the standard set (Fig. 2(a)), an indirect gapappears at point U and for set I a sharp minimumis found at point G (Fig. 2(b)). Both resultsdisagree with pseudopotential calculations whichshow that cubic SiC possesses a fundamentalindirect gap with a G2X transition, as mentionedabove. In addition, the overall shape of theconduction band in Figs. 2(a) and (b) do notreproduce the relative positioning of energy levelsEðkÞ; EXoEUoEDoELoEG; predicted for thismaterial. However, in spite of these disadvantages,parametrization I leads to an excellent estimate ofthe direct gap, Ed ¼ 5:77 eV, as compared to

Fig. 2. Extended H .uckel band structure and projected DOS of cubic SiC by using standard parametrization (a) and parametriza-

tion I (b). Parameters and calculated quantities are shown in Table 1. Energies are given in eV and DOS in arbitrary units.

M. Matos / Physica B 324 (2002) 15–33 19

experimental [22] and other calculated values,which appear in the range 5.14–6.30 eV [5,6,18,22,29]. From standard parameters one gets Ed ¼15:9 eV, which is overestimated by a factor ofthree. The projected density of states, also shownin Fig. 2, do not show relevant changes when onegoes from the standard set to set I. In both cases,little s–p mixing in the energy range around thesemiconductor gap is seen, with the main con-tributions coming from C-2p and Si-3p orbitals.Tight binding calculations in which only first

neighbors interactions are considered [27] predictno s–p mixing at point G; on the other hand, p–pas well as strong s–p mixing could be found atpoint X. Thus, in principle, one could expect thecalculated extended H .uckel direct gap (a G2Gtransition) to be mainly a result of C-2p–Si-3pinteractions, and the indirect gap to depend onboth, p–p and s–p C–Si interactions. This isconfirmed in Fig. 3, where the effects of p–p ands–p interactions are separately analyzed. Fig. 3(a)shows the result of a calculation in which thecontribution of C-2s and Si-3s orbitals are nottaken into account. In Figs. 3(b) and (c), 2s and 3sare separately included. Parameters are those ofset I. As will become clear, these reduced basis setcalculations were found to be useful to theunderstanding of the electronic interactions whichlead to the formation of the indirect gap. In Fig. 3,the top of the valence band (the system’s Fermi

level, eF) obtained with parametrization I isindicated by a dotted line.In all cases, the calculated value of the G2G

splitting which appears just above eF is the same asthe calculated direct gap energy provided byparametrization I, Ed ¼ 5:77 eV (at the bandcenter, the valence band maximum, �8.04 eV,and the conduction band minimum, �2.27 eV,remain constant in the several situations consid-ered in Fig. 3, leading to the obtained G2Gsplitting). This indicates that within the extendedH .uckel approximation the direct gap is a conse-quence of 2p–3p (C–Si) interactions, analogous toresults found within first-neighbor tight bindingcalculations, a symmetry condition. It is interest-ing to observe that the global conduction bandminimum and the smooth dispersion along the lineL2G2X2U; seen in Fig. 3(a) (represented by theband which appears above eF) is strikingly similarto the conduction band dispersion found bypseudopotential methods along the same line[5,8,22] and which gives rise to the appearance ofthe indirect gap. It has a dispersion characteristicof p-like bands. Since the calculation of Fig. 3(a)does not contain s orbitals it could be inferred thatthe presence of these orbitals is responsible for theloss of the indirect gap feature in the material. Inthe calculations shown in Figs. 3(b) and (c), sorbitals are present in the basis set. Their effect onthe conduction band is such as to spoil the

Table 1

Extended H .uckel parameters for cubic (3C) SiC and corresponding calculated quantities for different parameterizations considered in

the paper

SiC-3C

Standard I II Other works/exp.

C-2s ðHii=ziÞ �21.4/1.625 �21.4/1.33 �21.4/1.625 —

C-2p �11.4/1.625 �11.4/1.33 �13.6/1.33 —

C-3s* — — 3.25/7.014 —

Si-3s �17.3/1.383 �17.3/1.72 �17.3/1.72 —

Si-3p �9.2/1.383 �6.6/1.72 �6.6/1.72 —

Si-4s* — — �2.24/4.80 —

Ed 15.9 (G2G) 5.77 (G2G) 6.0 (G2G) 5.14–6.30 [5,6,18,22,29]

E0d — — 7.27 (G2G) 6.47 [8]–7.07 [7]

Ei 12.8 (G2U) None 2.42 (G2X) 2.42 [22]

d 19.7 23.1 19.6 15.0–22.0 [5,8,22]

DEval 5.92 0.71 1.53 1.4–4.7 [5,8,22]

All energies are given in eV.

M. Matos / Physica B 324 (2002) 15–3320

dispersion eðkÞ; by destabilizing the levels aroundX, thus destroying the indirect gap.The effect of C-2s is mainly manifested in the

bands above eF with negligible change in the bandsbelow eF (see Fig. 3(b)). This is a consequence ofthe fact that in the case of the conduction band,the relevant s–p interactions are mostly 2s–3p(first-neighbor interations) while for the valenceband the interactions are mostly 2s–2p (secondneighbor interactions). The same argument ex-plains why Si-3s (Fig. 3(c)) affects the bands beloweF more strikingly than C-2s does. Note that atpoint X, the lower lying energy level above eF hasthe same energy in calculations shown in Figs. 3(a)and (c), indicating that the destabilization of theminimum at X, in the band structure calculationobtained with parameterization I, is mainly due tothe interaction between C-2s and Si-3p atomicorbitals.The results above suggest that in 3C-SiC the

scaling transformation of C-2s should not beapplied, since the transformation (an expansionof C-2s) overestimates the interaction of thatorbital with Si-3p, thus enhancing the destabiliza-tion of the conduction band edge at X. Acalculation done with parameterization I exceptfor STO exponent of C-2s, which was given itsstandard value (x2s ¼ 1:625), has shown someadvantages; first, no changes were found in thecalculation of Ed; as compared with parameteriza-

tion I; this was expected, since Ed does not dependon s orbitals. In addition, the contraction of C-2sleads to an increse of the X–X splitting inside thevalence band, to 2.34 eV, and causes a widthreduction to d ¼ 20:8 eV; both quantities are thusmade to stay within the range of pseudopotentialresults. In the subsequent analysis the latter set ofparameters (set I except for the use of standard C-2s orbital) will be used. The inadequacy of scalingtransformation on C-2s is probably due to theweak dependence of these valence core orbitals onexternal lattice conditions.The destabilization of the minimum conduction

band energy at point X, caused by 2s–3p interac-tions, can be compensated by the inclusion ifexcited, C-3s* orbitals in the basis set, providedthat the atomic term H3s�3s� is positive. This can beseen in Fig. 3(d). The presence of a low-lyingconduction band with a local minimum at X canbe readily noted. Moreover, the overall dispersionis strikingly similar to that found by pseudopo-tential approaches (and in the reduced basis set—2p,3p-calculation of Fig. 3(a)). The parameters ofC-3s* are Hii ¼ 3:25 eV [18] and x3s� ¼ 7:0: InFig. 3(e) the effect of including a Si-4s* (withH4s�4s� ¼ 3:25 eV, x4s� ¼ 5:0) excited orbital in thebasis set is shown. It can be noted that nominimum appears at point X, while the overalldispersion disagrees with results predicted bypseudopotential calculations.

Fig. 3. Extended H .uckel electronic structure of cubic SiC obtained by using different sets of atomic orbitals in the basis: (a) only

carbon 2p and silicon 3p orbitals are considered; (b) carbon 2s and (c) silicon 3s orbitals are included; in (d) and (e) effects of carbon

3s* and silicon 4s* are shown, respectively; parameterization I is used; energies in eV; DOS in arbitrary units; dotted line: the top of the

valence band (Fermi level).

M. Matos / Physica B 324 (2002) 15–33 21

A scrutiny of individual band level energies at Xhas confirmed the basic mechanism of the appear-ance of the indirect gap through the inclusion ofC-3s*. The interaction of this orbital with Si-3popposes the destabilizing interaction with C-2s. Asimilar analysis has shown that, at point X, Si-4s*interacts with a low lying valence level, mostlyC-2p; the anti-bonding combination, mainly Si-4s*, is located above the conduction band mini-mum at G; causing the system to present noindirect gap, as seen in Fig. 3(e).Si-4s* is, nevertheless, suitable to describe other

features of the conduction band provided thatappropriate parameters are chosen. Fig. 4 showsthe band structure and DOS curves for calcula-tions in which both, C-3s* and Si-4s* are includedin the basis set. The final choice of 3s* and 4s*parameters was based in the following criteria:carbon 3s* STO exponent was fitted to reproducethe experimental indirect gap, Ei ¼ 2:42 eV [22];Si-4s* parameters were fitted in order to reproduceother features of the CB, such as the non-degenerate character of the conduction bandminimum at G [5,8,22]. The flat band whichappears around �3 eV, mainly Si-4s*, as seenfrom the projected DOS curves, is consistent withresults obtained within pseudopotential methods[5,8,22]. The parameters thus found (set II) aregiven in Table 1. Note that the atomic term H2p,2p

is 2.2 eV lower than the standard value. This wasdone in order to fit the direct gap of the material.The lowering of H2p2p could be understood as aresult of the long-range electrostatic interactionswith the neighboring charged ions. This feature,given by three center integrals, is not explicitlytaken into account in the extended H .uckel hamil-tonian. A rough estimate of that interaction, whichwas done by considering the first (Si) and second(C) neighbor shells, gives about 1 eV for the crystalelectrostatic potential. For that purpose, the totalMulliken atomic Si and C charges were used.The width of the valence band from parameter-

ization II was found to be 19.6 eV and the internalvalence band gap at X, 1.53 eV. The second G2Gvalence to conduction band transition was foundto have an energy of 7.27 eV. These values arewithin the range of pseudopotential results, as seenfrom Table 1.

3.2. The polytypes 2H, 4H, 6H and 15R

In the pure hexagonal 2H polytype, resultsfound for the extended H .uckel band structurecalculations, when standard parameters are used(Fig. 5(a), Table 2), present features similar tothose obtained in the cubic polytype. Valencebands are well reproduced, as compared to abinitio pseudopotential [5] and Hartree–Fock [28]

Fig. 4. Band structure and DOS analysis of cubic SiC by using parameterization II. The Fermi level is shown by a dotted line; energies

in eV. See parameters in Table 1.

M. Matos / Physica B 324 (2002) 15–3322

calculations, while the direct gap (Ed ¼ 15:7 eV) isoverestimated with respect to other calculatedvalues, which vary in the range 4.43–5.0 eV[5,8,29]. The conduction band minimum at thereciprocal space point K leads to a fundamentalindirect gap of the correct order, but its calculatedvalue is also considerably overestimated (10.4 eV)with respect to other calculations, found to be inthe range 2.0–3.3 eV [5,8,29].In the band structure shown in Fig. 5(b),

parameter set II was used for the valence orbitals2s, 2p, 3s and 3p. As a consequence of scaling, thedirect gap is considerably reduced (Ed ¼ 6:86 eV,from Table 1), a result already found in the cubicpolytype. This was to be expected since thetetrahedral vicinity is the same in both polytypes.On the other hand, the G2K energy differencedecreases by a much smaller amount (Ei ¼9:59 eV), the conduction band minimum appearingat G: the system is no longer described as anindirect gap material.

The decrease in Ed is mainly due to contractionof Si-3p atomic orbitals (which contributes about60% to the change) while expansion of C-2porbitals leads to a negligible increase (3%) of theconduction band minimum. A slight increase(B1 eV) is due to the change of H3p3p (�6.6 eVin set I). The strong Si-3p character of lower lyingconduction bands could be seen in a DOS analysis(not shown). Other features of the use ofparametrization II for the valence shell are changesin the valence band, as seen in Table 1.In the foregoing discussion, on the effects of

excited orbitals in the polytypes 2H, 4H, 6H and15R, parameterization II will be used for thevalence shell orbitals (carbon 2s and 2p, silicon 3sand 3p).The band structure obtained with parameteriza-

tion II, including 3s* and 4s* defined above forcubic SiC, is shown in Fig. 5(c). It can be seen thatthe K point minimum, with significant contribu-tion of C-3s* (as seen from a DOS analysis), turns

Fig. 5. The band structure and projected DOS of the hexagonal (2H) structure of SiC by using different sets of parameters:

(a) standard parameterization; (b) parameterization (II) for valence orbitals and (c) the effect of excited s* orbitals; energies in eV. See

parameters and calculated quantities in Table 2.

M. Matos / Physica B 324 (2002) 15–33 23

out to be two-fold degenerate; the global conduc-tion band minimum appears near but slightlyshifted to the left of K. Although the minimum atG is non-degenerate, it is not a local minimum.These features are in disagreement with resultsfound within other theoretical approaches[5,8,28,29]. Si-4s* orbitals were seen to play norelevant role in the gap region.Inspection of the energy levels at special k points

has shown that the conduction band minimum atG is a bonding combination of the two C-3s*orbitals centered on carbon atoms of the unit cell.On the other hand, the K-point two-fold degen-erate minimum comes from interactions of C-3s*with valence levels. Attempts to lower the mini-mum at the band center were done by changingC-3s* and Si-4s* parameters in a wide range (from�1.25 eV to 3.44 eV for H3s*3s* and from �2.24 eVto+3.25 eV for H4s*4s*). The corresponding ex-ponents were also modified in a significant range.They were, however, unsuccesful in improving thetopology of the conduction bands. The bandsgenerated by s* orbitals were found to be rigidwith respect to C-3s* and Si-4s* parameters, inwhat concerns the general aspect of the curvedispersions seen in Fig. 5(c). An inversion of G andK relative energies could be seen for negativevalues of H3s*3s*. But in that case the fundamentalgap was seen to arise from a G2G transition rather

than G2K; thus spoiling the order of gap. Theseresults indicate that (probably due to symmetryconditions of the hexagonal crystalline structure) agood description of the conduction bands of 2H-SiC cannot be obtained through the inclusion of s*orbitals in the basis set.Inclusion of p* orbitals has thus been consid-

ered. The behavior of Si-4p* is very similar to thatof Si-4s* in the energy region around the forbiddengap. For a significant range of parameters (H4p*4p*

varying from �2.24 to 3.25 eV and STO exponensvarying from 3.0 to 5.0) a group of six Si-4p*atomic orbitals was seen to form a very narrowband around the atomic term H4p*4p*. Mixturesinvolving these orbitals appear in the higher lyingconduction bands when the orbital exponent issmall.In Fig. 6(a), the results of including C-3p*

orbitals in the basis set are seen, through the bandstructure and projected density of states. The sameparameters, H3p*3p*=3.25 eV, x3p� ¼ 7:014; de-fined for the C-3s* orbital in 3C-SiC, were usedin this calculation. From the DOS curves, it can benoted that C-3p* gives rise to six conductionbands just above eF: These provide a non-degenerate local minimum at K but the globalconduction band minimum appears at G; with atwo-fold degenerate level. Aside from this dis-advantage, the dispersion of the two lower bands

Table 2

Extended H .uckel parameters for hexagonal (2H) SiC and corresponding calculated quantities for different parameterizations

considered in the paper

SiC-2H

Standard I val-II III Other works/exp.

C-2s (Hii=zi) �21.4/1.625 �21.4/1.331 �21.4/1.625 �21.4/1.625 —

C-2p �11.4/1.625 �11.4/1.331 �13.6/1.331 �13.6/1.331 —

C-3s* — — — — —

C1-3p* — — — 3.73/7.014

Si-3s �17.3/1.383 �17.3/1.727 �17.3/1.727 �17.3/1.727 —

Si-3p �9.2/1.383 �6.6/1.727 �6.6/1.727 �6.6/1.727 —

Si-4s* — — — — —

Ed 15.7 (G2G) 5.39 6.86 4.41 4.43–5.0 [5,8,29]

Ei 10.4 (G2K) None 12.10 3.30 2.0–3.3 [5,8,29]

d 19.6 22.8 19.53 19.4 15.0 [29]

DEval 5.24 1.09 1.83 1.2 1.7 [29]

All energies are given in eV.

M. Matos / Physica B 324 (2002) 15–3324

of the group is very similar to the one found bypseudopotential calculations. Attempts to lowerthe energy of these bands were done by includings* orbitals, either in Si or C. The more signi-ficant changes ocurred trough the inclusionof C-3s*. As seen in Fig. 6(b), the effect ofC-3s* is to introduce a band near eF; with similarfeatures of that observed in the calculation alreadyshown in Fig. 5(c). These results reinforcethe inadequacy of the use of C-3s* orbitals inthe hexagonal polytype. Variation of parametersof both orbitals in a wide range did not succeedin providing a global non-degenerate minimumat K.

An analysis is made in Fig. 7 on the way theexcited orbital C-3p* acts upon the band structureof 2H-SiC. The first three panels, (a)–(c), representC-2p–Si-3p interactions. It can be seen that thelower energy band (2p) is only slightly modified bythe interaction, while the higher lying bandsseparate in two groups, with a significant increasein the band width. This pattern is also seen inpanel (d), for the interaction between Si-3p andC-3p*. By including C-2p orbitals in the bandstructure of panel (d) two effects are seen(Fig. 7(e)): (i) higher lying bands appear, similarto those of panel (b), a result of 2p–3p interactionsand (ii) six intermediate bands appear, mainly

Fig. 6. Band structure and DOS analysis of 2H-SiC by using different sets of excited orbitals. Note the stabilization of two conduction

bands due to interaction with C-3s*; Energies in eV.

M. Matos / Physica B 324 (2002) 15–33 25

C-3p*. The 2p–3p–3p* interaction leads to thenon-degenerate local minimum at point K, givenby the higher lying intermediate bands. Inpanel (f), where C-2s is included, it can be seenthat C-2s interacts repulsively with the two lowerlying intermediate bands, exposing the localminimum at K; a DOS analysis has shownthe former to be 94% Si-3p. This strong first-neighbor s–p interaction has already been ob-served in Fig. 6(b) through the 3s*–3p* stabiliza-tions of the conduction band degenerate minimumat K. Si-3s plays no relevant role in the bandsabove eF:Since the intermediate C-3p* bands interact

repulsively with the p-bands above (mainly

through its Si-3p component) an increase of thediagonal hamiltonian term H3p*3p* would expect-edly stabilize it. In fact, this parameter can beadjusted to fit the indirect gap of 2H-SiC. Thisis shown in Fig. 8(a), where all parameters arethe same as those in Fig. 6(a), except forH3p*3p*=3.73 eV, giving a fundamental indirectgap of 3.3 eV. The basis set will be referred to asðsp3Þð3p�Þ2; meaning two excited carbon atoms inthe unit cell. In spite of providing an indirect gapof the correct order, the basis set does notreproduce the CB dispersion, as required. Thelatter becomes very similar to the one obtainedwith pseudopotential methods if only one carbonatom in the unit cell holds an excited C-3p*

Fig. 7. The calculated band structure of 2H-SiC by using different atomic orbitals in several reduced basis sets. See the interaction of

C-2p with Si-3p by comparing (a) and (c) with (b). This analysis is repeated in (d)–(f) which include C-3p*. H3p*3p*=3.25 eV,

x3p� ¼ 7:014: Valence shell parameters are taken from set II; energies in eV.

M. Matos / Physica B 324 (2002) 15–3326

orbital, constituting the ðsp3Þ3p� basis set. This isseen in the second panel of Fig. 8.The projected DOS in Fig. 8 consider atomic

contributions of C(2) and Si(1) in the unit cell (seelast panel of Fig. 8). These were seen to be therelevant contributions to the CB minimum at K.C(2) is the one which holds 3p*. The enhancementof the Si(1)–C(2) bond, as compared to the bondbetween C(2) and Si(3)—along c—and Si(3)–C(4),could be confirmed through a crystal orbitaloverlap population (COOP) analysis, which in-dicates that B1 eV around the global CB mini-mum, there is only the (antibonding) Si-3p–C-2pcombination from atoms 1 and 2. These resultssuggest that the minimum at K could be associatedwith the stabilization of a lateral Si(1)–C(2)interaction promoted by the C(2)-3p* orbital.Another COOP analysis was made for the casewhere C(2) and C(4) hold the 3p* orbital—theðsp3Þ3p2� basis. It could be seen that in this casethere is a mixing of lateral and vertical (along c)interactions at the CB minimum.The advantages of the ðsp3Þ3p� basis set, in

describing the topology of the lower lying conduc-tion bands around the minimum, poses somequestions. In principle, the addition of a single3p* orbital in the unit cell should carry no serious

theoretical objections since basis sets usuallyemployed in quantum chemical methods do notnecessarily satisfy completeness. The physicalreasoning presented above helps in the under-standing of the significance of this basis. Apartfrom the possibility of providing some insight intothe physical interaction mechanisms in the system,the ðsp3Þ3p� basis could be justified on the groundsof its descriptive quality. However, it should bepointed out that the more complete ðsp3Þð3p�Þ2basis set provides considerable improvement uponthe standard basis set. The choice between the twopossibilities might depend on the quantity whichone wishes to study. The single p*-excited carbonbasis, by providing a smooth dispersion around K,would likely be more convenient if energy deriva-tives were considered.For the polytypes 4H, 6H and 15R, the ðsp3Þ3p�

basis (single p*-excited carbon) gives an indirectG2K transition with energies 3.54, 3.54 and3.52 eV, respectively. For the three polytypes,dispersion around the conduction band minimumat K was found to be the same as that for 2H. Withincreasing number of p*-excited carbons, the bandstructures of these polytypes behave similarly tothat of 2H. That is, the G2K transition energyremains roughly constant while other bands

Fig. 8. Results of electronic structure calculations of 2H-SiC by using two different choices of excited carbon orbitals. In the first panel

(a), two 3p*-excited carbon atoms are considered in the unit cell, the ðsp3Þð3p�Þ2 basis; in (b) only one is considered, the ðsp3Þð3p�Þbasis. DOS analysis refers to the calculation of the second panel (b). In the last panel the unit cell is presented; energies in eV.

M. Matos / Physica B 324 (2002) 15–33 27

appear around K. A calculation in 3C SiC, byusing C-3p* in a hexagonal unit cell, has shownthe same behavior, the appearance of an indirectG2K transition, instead of the G2X transitionobtained with C-3s*. These results show that theeffects of C-3p* are equivalent in all the polytypessuggesting a strong dependence on the orbitaltype, rather than on the particularity of thecrystalline structure.Pseudopotential calculations [5,6] ascribe a

G2M fundamental gap to the 4H, 6H and 15Rstructures. Since point M in the hexagonal cell isequivalent to point X in the rhombohedral cell (seeFigs. 2 and 3 of Ref. [6], for example), improve-ments in the band structure of these polytypescould be expected if s* orbitals were used insteadof p*. In fact, very good results were found, whencalculations were performed by using the 3C SiCparameters of C-3s*. In all cases an indirect G2Mfundamental transition was found, as shown in theband structures of 4H (Fig. 9), 6H (Fig. 10) and15R (Fig. 11). In addition to the appearance of anindirect gap of the correct order, the overalldispersion of the lower lying conduction bands

are strikingly close to those obtained, for example,in Ref. [6]. The calculated energy of the indirecttransition, Ei; were found to be 2.31, 2.34 and2.77 eV, respectively for 4H, 6H and 15R. For thedirect gap, Ed; one obtains 4.60 eV in 4H and 6Hand 5.0 eV in 15R. Inclusion of 4s* silicon orbitalsdid not provide considerable changes in the bandstructure of the three polytypes.d orbitals have been proposed [30] for GaAs

through the use of an sp3 d2 basis set. In that case,the excited d orbitals led to considerable improve-ment in the understanding of surface spectroscopydata. Electronic band structure was seen to beaffected mainly in the dispersion of the secondconduction band. No changes were found forlower lying levels. In particular, excited d orbitalsdid not affect the order of the gap, preserving thedirect ðG2GÞ fundamental transition of GaAs. Inthe present work, calculations have been done byusing an sp3d5 basis set, considered to be moreappropriate for bulk.For carbon-centered d orbitals with parameters

in the range of those found for the s* and p*orbitals, the extended H .uckel band structure did

Fig. 9. Band structure of 4H SiC by using parametrization II, except for Si-4s* which was not included in the basis set; energies in eV.

M. Matos / Physica B 324 (2002) 15–3328

not show a tendency to form an indirect G2X gapin the cubic structure. Also, such orbitals were seento affect the valence band dispersion, whichdeviates considerably from that given by s*

orbitals. Several calculations have been performedby changing the Slater exponent of d functions,with H3d3d around +3.0 eV. By increasing thestrenght of d interactions it could be noted that a

Fig. 10. Band structure of 6H SiC by using the same basis set and parameterization of Fig. 9; energies in eV.

Fig. 11. Band structure of 15R SiC by using the same basis set and parameterization of Figs. 9 and 10; energies in eV.

M. Matos / Physica B 324 (2002) 15–33 29

prominent indirect gap appears along U–W, whilethe shape of valence bands visibly change.In the case of 2H SiC, the inclusion of C-3d in

the sp3d5 basis set does provide a conduction bandminimum at K. Nevertheless, the conduction banddispersion does not reproduce the expected shapeobserved previously in Fig. 8. The valence band isstrongly affected, in such a way as to destroy themaximum at the band center (see Fig. 12). Thelower CB levels are mainly due to C-2p and C-3d.A COOP calculated between 2p and 3d (secondneighbor carbon atoms) was seen to be 10�4 assmall as the one between 3p and 3d (first Si–Cneighbors)—seen in Fig. 10(e) to be an anti-bonding combination around the fundamentalgap. This indicates the indirect interaction 2p–3p–3d as being responsible for the formation of theminimum at K, analogously as for 3p*. If oneconsiders the presence of 3d orbitals in C(2) only,an increase in the energy of the VB levels could beobserved, with recovery of a band center max-imum, although the parabolic-like shape is lost,with considerable flattening in the band dispersionaroud that point. On the other hand, the minimumCB energy at K is raised, with loss of thefundamental G2K transition.

Si-3d orbitals in the sp3d5 basis produce anindirect gap of order G2K; as seen in Fig. 13, withCB dispersion around K characteristically differ-ent from that of Fig. 8. However, the interestingfeature here is the fact that the effects over thevalence bands are sensibly small, being almostnegligible. The main contribution to the conduc-tion bands comes from Si-3d, although C-2p andSi-3p are not negligible. A COOP analysis hasshown anti-bonding 2p–3d (first neighbors) com-bination, while that of 3p–3d (second neighbors) isnegligible. Wiggles in the conduction bands aredue to d–d interactions. This could be noted bycomparing these results with a calculation with asingle d-excited Si atom, in which case the wigglesin part disappeared.The effect of d orbitals in 4H, 6H and 15R gets

progressively more complicated, as the number ofsuch orbitals in the unit cell increases. In general,the overall features are similar to those found forthe pure hexagonal structure. That is, C-3dorbitals lead to a decrease in the energy of thevalence bands, with loss of the global VBmaximum at the band center. When less than thetotal number of C atoms in the unit cell is made tohold 3d orbitals, the same characteristic behavior

Fig. 12. The effect of carbon d orbitals on the electronic structure of 2H SiC; energies are given in eV.

M. Matos / Physica B 324 (2002) 15–3330

is found as that of 2H. The energy of thevalence band raises and the band becomes flatat G: Apart from the disadvantages in thevalence band description, C-3d are able to providelocal CB K-minima when apparently aleatorychoices of the number of d-excited carbons atomsin the unit cell are made. As an example, aparabolic-like minimum appears in 6H when fived-excited carbon atoms are considered and for15R the same happens with four d-excitedcarbons.For Si-3d orbitals, the valence bands of the three

polytypes, as in 2H, are not strongly affected. Theparabolic-like shape of the CB minimum at Kappears in an aleatory way also, depending on thenumber of d-excited Si atoms in the unit cell. Forexample, in the 6H structure a parabolic-likeminimum at K is found when the unit cell containstwo neighbor d-excited Si atoms, although thegreat number of d orbitals do produce wigglesaround the minimum. In this case also the valencebands become flattered at G: For 4H no parabolicminimum was found when one, two, three or fourd-excited Si atoms are considered. For 15R,parabolic CB bands at K are produced, forexample, with two and three neighbor d-excitedSi atoms.

From the discussion above it could be said thatthe results found for 3d excited orbitals, withparameters in the range analyzed in this paper,point to a tendency to reproduce an indirect gap oforder G2K in 2H, 4H, 6H and 15R, although thedispersion is not good. For 2H, whose indirect gapis of that order, it is not clear whether 3d orbitalspresent advantages as compared to 3p* orbitals.Nevertheless, due to the higher flexibility of dorbitals it is possible that improvements could befound, especially in the higher lying CB levels. For4H, 6H and 15R, as well as for 3C, it seems thatthe best choice remains to be the use of s* orbitals.However, also in these cases, it would be advisableto say that that d orbitals could provide someadvantages to the description of higher lyinglevels.Distortions on the ideal hexagonal structure

have been considered elsewere [4], in order toadjust the atom positions in the unit cell accordingto the experimental size of the c-axis in 2H SiC(5.048 (A, 0.5% bigger than the ideal value). Byusing density functional calculations to optimizethe geometry, a 0.04 (A stretching of the Si–C bondalong the c-axis was obtained. This suggesteddistorted geometry was used in the present paper,in order to investigate its effects upon the extended

Fig. 13. The effect of silicon d orbitals on the electronic structure of 2H SiC; energies in eV.

M. Matos / Physica B 324 (2002) 15–33 31

H .uckel band structure on the 2H polytype, but nodifferences were seen.

4. Conclusion

In this paper, a theoretical study was done onthe electronic structure of the polytypes 2H, 3C,4H, 6H and 15R of silicon carbide, by using thetight binding extended H .uckel method. Thevalence basis set was modified by the inclusion ofexcited s*, p* and d orbitals in carbon and siliconatoms. Empirical parameters were chosen in orderto reproduce experimental data on optical proper-ties, related to valence-to-conduction band transi-tions at special points of the reciprocal space, inthe pure cubic and hexagonal structures.For the valence shell, a transformation of 3s and

3p orbitals of silicon and of the 2p orbital ofcarbon was used through a lattice parameterscaling with pure silicon and diamond crystals.Changes in one center parameters, H2p2p andH3p3p, were proposed, which are in the correcttrend if one takes into account electrostatic long-range interactions in the crystal lattice. Consider-ing that scaling transformation is not a directfitting procedure, changes in the parameterizationof the valence shell could be thought to beminimal, with respect to parameters normally usedin the literature.In the cubic polytype, a 10-orbital basis set,

consisting of 2s, 2p and 3s* orbitals of carbon and3s, 3p and 4s* orbitals of silicon, was able toreproduce most important features of the bandstructure. The topology of the lower lying con-duction bands was found to be very good ascompared with results found in the literature byusing more sophisticated methods of calculation,based on ab initio, pseudopotential and densityfuncional approaches. In special, the order of thefundamental indirect gap of the material wascorrectly predicted to arise from a G2X transition.For 2H SiC, the pure hexagonal polytype, s*

orbitals were shown to be unsuitable to reproducethe topology of the conduction bands. A wide,exhaustive, range of parameters was attempted,with no success in the description of that property.By using instead excited p orbitals, 3p*, centered

on carbon atoms, dispersion of the conductionbands and the order, G2K; of the fundamentalindirect gap was correctly reproduced. The generalaspect of the band structure was seen to approachvery satisfactorily ab initio, pseudopotential anddensity funcional results when one carbon atom inthe unit cell holds the excited orbital C-3p*. It wasfound that the origin of the indirect gap is mainlyrelated to a lateral Si–C interaction. It remains tobe answered the question on why is a single excitedC atom in the unit cell enough to reproduce moresophisticated quantum chemical calculations onthe system.As for the polytypes 4H, 6H and 15R, a basis

set which contains C-3s* orbitals is the onewhich provided a correct description of the bandstructures, according with ab initio pseudopoten-tial calculations. I was pointed out that this resultis due to the equivalence between the hexagonal Mand cubic X special BZ points.The effect of 3d orbitals were also analyzed,

with no clear advantage being found with respectto s* and p* orbitals. It was pointed out that dorbitals could possibly bring improvements in thedescription of higher lying conduction band levelsin all polytypes.The results found in the present study indicate a

strong dependence of the band structure of theseveral polytypes of SiC on the type of additionalorbitals used in the basis set. Little effect comesfrom going from one polytype to another with thesame basis set. This is very likely and indicationthat second neighbor interactions are of littleimportance, as far as band dispersion is concerned.In general, it could be said that s* orbitals are wellsuited to reproduce an indirect gap of order G2X(or G2M) p* being more indicated to describe aG2K transition. The extended H .uckel method wasshown to be suitable to study such complex sytemsas the polytypes of SiC. A more detailed compara-tive study between extended H .uckel and ab initiocalculations on these systems could help under-stand why certain structures favor a given type ofexcited basis set orbital.Finally, the effects of lattice distortions were

analyzed for the pure hexagonal lattice but nosignificant changes were found in the calculationof the band structure of this polytype.

M. Matos / Physica B 324 (2002) 15–3332

Acknowledgements

All calculations were done with the computa-tional code yaehmop, developed by Dr. GregLandrum, in the Chemistry Department of theUniversity of Cornell.

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