The modulations of the electronic band structures of hexagonal (2H, 4H, and 6H) and cubic (3C) SiC under biaxial (0001) and (111) in-plane strain are investigated by using first-principles calculations including spin–orbit coupling effects. We have clarified that the strain dependency of the valence bands is closely related to the crystal symmetry and hexagonality. Specifically, tensile strain induces hybridization and crossover between the heavy-hole and light-hole bands in the hexagonal polytypes. On the other hand, the degeneracy between the heavy-hole and light-hole bands breaks in the cubic polytype under tensile strain. Consequently, the hole effective masses change significantly under certain tensile strain in all four polytypes. The values of the critical tensile strain are approximately proportional to the energy differences between the heavy-hole and crystal-field splitting bands under no strain and, in turn, show a correlation with the hexagonality. In contrast to the case of the valence bands, the band structures around the conduction band minima and, therefore, the electron effective masses are insensitive to the strain, except for the ML direction in 6H–SiC. The present study provides principles for elucidating and designing the crystal structure and strain dependency of the electronic band structures and transport properties of SiC.

Most semiconducting materials entering our daily lives consist of sp3-bonding networks.1 The sp3-bonded crystals typically take cubic and hexagonal structures, where zincblende and wurtzite are well-known examples of the former and latter, respectively. It is closely related to the ionicity of each material whether the material takes a cubic or hexagonal structure. In general, the smaller the ionicity is, the more preferred the cubic structure is, and the larger the ionicity is, the more preferred the hexagonal structure is. In fact, the silicon (Si) crystal, which is one of the most familiar semiconducting materials, takes the diamond structure with a cubic symmetry, and the gallium nitride (GaN) crystal takes the wurtzite structure.

From the crystallographic point of view, silicon carbide (SiC) exhibits an interesting property. It has several hundred polytypes including cubic, rhombohedral, and hexagonal structures in nature,2 which is associated with a delicate balance of ionicity and covalency. Kobayashi and Komatsu reported that the ion–ion Coulomb repulsive energy from the third nearest neighbor atoms changes depending on the ionicity and affects the relative structural stability of cubic and hexagonal structures in SiC as well as BN and AlN.3 In order to distinguish SiC polytypes, a useful notation is introduced as follows. A building block for the SiC polytypes is an atomic Si–C bilayer in the planes spanned by hexagonal lattice vectors a1 and a2 with an angle of 120°. The structures of polytypes can be systematically described as the stacking sequences of the Si–C bilayers. For example, the stacking sequence of the zincblende structure contains three periodically repeated bilayers A, B, and C (ABC), and that of the wurtzite structure is two bilayers A and B (AB). Thus, each polytype can be denoted nC, nR, or nH, where n is the periodicity of the stacking sequences and C, R, or H represents the lattice type, i.e., cubic (C), rhombohedral (R), or hexagonal (H). By using this notation, the zincblende structure is described as 3C, and the wurtzite structure is 2H (Fig. 1). 4H (ABCB stacking) and 6H (ABCACB stacking) are also well-known SiC polytypes.

FIG. 1.

Crystal structures for 2H, 3C, 4H, and 6H–SiC. Blue and brown balls depict silicon and carbon atoms, respectively. The stacking sequence is indicated by symbols A, B, and C. For the hexagonal polytypes, primitive unit cells are shown. A hexagonal representation is used to illustrate 3C to facilitate the comparison with the hexagonal polytypes. Indexes h and k indicate a Si–C bilayer sandwiched between those in the same adjacent stacking, i.e., a hexagonal bilayer, and a bilayer sandwiched between those in different stacking, i.e., a cubic bilayer, respectively.

FIG. 1.

Crystal structures for 2H, 3C, 4H, and 6H–SiC. Blue and brown balls depict silicon and carbon atoms, respectively. The stacking sequence is indicated by symbols A, B, and C. For the hexagonal polytypes, primitive unit cells are shown. A hexagonal representation is used to illustrate 3C to facilitate the comparison with the hexagonal polytypes. Indexes h and k indicate a Si–C bilayer sandwiched between those in the same adjacent stacking, i.e., a hexagonal bilayer, and a bilayer sandwiched between those in different stacking, i.e., a cubic bilayer, respectively.

Close modal

Furthermore, we introduce an important structural indicator called hexagonality describing how the structure is similar to the wurtzite structure. A Si–C bilayer sandwiched between those in the same adjacent stacking is defined as a hexagonal (h) bilayer. On the other hand, a bilayer sandwiched between those in different stacking is defined as a cubic (k) bilayer. Each bilayer is indexed accordingly in Fig. 1. The ratio of the number of hexagonal bilayers to the total number of bilayers in the unit cell is called hexagonality. Therefore, the hexagonality is 100% for the 2H structure, 50% for 4H, 33% for 6H, and 0% for 3C.4 

We have previously reported the effect of biaxial strain on the band structure of 4H–SiC using first-principles calculations.5 It has been shown that the effective masses of holes manifest a large dependence on the strain, which is associated with its crystal symmetry, i.e., P63mc. Besides, there are some preceding works that investigated the effect of uniaxial and biaxial strain on the electron effective mass in 4H–SiC.6,7 However, it has not been studied well how the crystal symmetry affects the hole and electron effective masses in association with biaxial strain. In this study, we have investigated the role of crystal symmetry (P63mc for 2H, 4H, and 6H and F4¯3m for 3C) in determining effective masses in the four SiC polytypes using first-principles calculations. As a result, we have elucidated that the effective masses and their strain dependencies are closely related to both crystal symmetry and hexagonality.

This paper consists of the following sections. In Sec. II, we describe the computational methods. The results on the electronic properties for the unstrained SiC polytypes are provided in Sec. III A, and the strain effects are discussed in Sec. III B. We summarize our findings in Sec. IV.

Our first-principles calculations are based on the projector augmented-wave method8 and the generalized Kohn–Sham scheme,9,10 as implemented in the VASP (Vienna Ab initio Simulation Package) code.11–13 We used the HSE06 hybrid functional14,15 with the Fock-exchange mixing and screening parameter values of 0.25 Å−1 and 0.208 Å−1, respectively, which has been reported to well reproduce the band structures of covalent semiconductors.15–17 We employed a plane-wave basis set with a 520 eV energy cutoff and sampled the Brillouin zone with 8 × 8 × 4, 6 × 6 × 2, 6 × 6 × 2, and 6 × 6 × 6 k-point meshes for the primitive cells of 2H, 4H, 6H, and 3C, respectively. A hexagonal representation of 3C was also used with a 6 × 6 × 2 mesh to facilitate the comparison with the hexagonal polytypes. We obtained band structures including the spin–orbit interaction (SOI) after geometry optimization without the SOI.

In this study, we consider uniform biaxial strain within the basal (0001) plane for the hexagonal polytypes and the (111) plane for the 3C polytype, which corresponds to the (0001) plane in the hexagonal representation. The lattice parameters on these planes are changed from −1% (compressive strain) to +1.2% (tensile strain), allowing the lattice parameter perpendicular to these planes and the internal parameters to fully relax. The data about 4H–SiC, i.e., the crystal structure, band structure, and effective masses, are taken from Ref. 5, in which the same computational settings as the present ones are used.

First, we show the lattice constants of the four SiC polytypes. The calculated lattice constants a and c are 3.07 Å and 5.03 Å for 2H, 3.07 Å and 10.05 Å for 4H, 3.07 Å and 15.07 Å for 6H, and 3.07 Å and 7.53 Å for 3C (in the hexagonal representation used to facilitate the comparison with the hexagonal polytypes), which are in good agreement with the experimental values of 3.076 Å and 5.048 Å for 2H,18 3.073 Å and 10.053 Å for 4H,19 3.081 Å and 15.117 Å for 6H,20 and 3.082 Å and 7.549 Å for 3C,21 respectively; the calculated lattice constant a is 4.35 Å for 3C in the standard cubic representation, while the experimental value is 4.3596 Å.22 

Next, we report the band structures of the four SiC polytypes in Fig. 2. For these polytypes, the valence band maxima (VBMs) are located at the Γ point, and the conduction band minima (CBMs) are located at the K, M, U (on the ML path), and M (in the hexagonal representation) points, which are illustrated in Fig. 2(e), for 2H, 4H, 6H, and 3C–SiC, respectively. In the 6H polytype, the energy difference in the lowest conduction band between the U point, where the CBM is located, and the M point is less than 1 meV. The small energy difference is consistent with the previously reported values of ∼3 meV using the local density approximation24 and ∼12 meV using the generalized gradient approximation.4 The bandgaps are 3.16 eV, 3.15 eV, 2.91 eV, and 2.22 eV for 2H, 4H, 6H, and 3C–SiC, comparable to the respective experimental values of 3.30,25 3.25,26 3.023,26 and 2.417 eV.27 The mechanism of the large bandgap variation of the SiC polytypes is theoretically clarified in Refs. 28 and 29.

FIG. 2.

Band structures for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. (e) Brillouin zone for a hexagonal system. In (d), the left and right panels show the band structures in the cubic and hexagonal representations, respectively. The valence band maxima are aligned to the zero energy. The band paths conform to those in Ref. 23.

FIG. 2.

Band structures for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. (e) Brillouin zone for a hexagonal system. In (d), the left and right panels show the band structures in the cubic and hexagonal representations, respectively. The valence band maxima are aligned to the zero energy. The band paths conform to those in Ref. 23.

Close modal

We now discuss the band structure near the VBM in detail. The top of the valence bands in the SiC polytypes consists of three bands: the heavy-hole (HH) band, the light-hole (LH) band associated with spin-orbit (SO) splitting, and the crystal-field (CF) splitting band. They are non-degenerate for the hexagonal polytypes (Fig. 3). In contrast, the HH and LH bands are degenerate even with the SOI in 3C, where the LH band corresponds to the CF band in the hexagonal polytypes and is also denoted the CF band hereafter although there is no CF splitting in 3C. These differences are derived from the space symmetries: P63mc for the hexagonal polytypes and F4¯3m for 3C. The energy differences between the HH and SO bands at the Γ point are 9.8 meV, 9.1 meV, 8.1 meV, and 16.3 meV, comparable to the preceding theoretical values of 8.7 meV, 8.5 meV, 8.2 meV, and 14.5 meV24 for 2H, 4H, 6H, and 3C–SiC, respectively. The energy differences between the HH and CF bands are 110.1 meV, 48.2 meV, 33.0 meV, and 0 meV, comparable to the experimental values of 119 meV, 59.5 meV, 43.6 meV, and 0 meV24 for 2H, 4H, 6H, and 3C–SiC, respectively. Here, we note that the energy differences between the HH and CF bands are approximately proportional to the hexagonality, which are 100%, 50%, 33%, and 0% for 2H, 4H, 6H, and 3C–SiC, respectively. We assume that the nearly linear relation is derived from the hexagonal stacking because the smaller the hexagonality is, the closer the polytype is to the cubic in which the energy difference between the HH and CF bands is zero. In the hexagonal polytypes, the HH and LH (SO) bands consist mainly of the px and py orbitals of the carbon atoms, and the CF band corresponds to the pz orbitals. On the other hand, in 3C, these three bands consist of the px, py, and pz orbitals of the carbon atoms equally at the Γ point. The irreducible representations of the HH, LH (SO), and CF bands at the Γ point are Γ9, Γ7+, and Γ7− for the hexagonal polytypes, and those of the HH and LH degenerate bands and the SO band are Γ8 and Γ7 for 3C,30 respectively.

FIG. 3.

Band structures on the ΓA path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC in the hexagonal representation. The energies of the highest valence bands at the Γ point are set to zero. The data for 4H–SiC are taken from Ref. 5.

FIG. 3.

Band structures on the ΓA path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC in the hexagonal representation. The energies of the highest valence bands at the Γ point are set to zero. The data for 4H–SiC are taken from Ref. 5.

Close modal

Furthermore, we show the hole and electron effective masses for the SiC polytypes in Tables I–IV; the hole effective masses are those for the highest valence bands, i.e., the HH bands (see Figs. 2 and 3). For 3C, the standard band paths for the cubic primitive cell are used for calculating the effective masses to compare our results with those in previous studies. Our calculations well reproduce the preceding experimental and theoretical values for both holes and electrons. We note that the SOI is essential to reproduce the anisotropy of the hole effective masses, as reported in Refs. 4, 5, and 24.

TABLE I.

Hole and electron effective masses in unstrained 2H–SiC. For reference, reported theoretical4,24,30,31 values are also listed.

HoleElectron
Γ-MΓ-KΓ-AK-ΓK-MK-H
This work 0.53 0.53 1.38 0.42 0.41 0.25 
Calc. 0.6424  0.6224  1.52,4 1.64,24 1.5430  0.42,24 0.4331  0.42,24 0.4331  0.25,24,30 0.2631  
 0.61,4 0.5730(mh ⊥)  0.4330(me ⊥)  
HoleElectron
Γ-MΓ-KΓ-AK-ΓK-MK-H
This work 0.53 0.53 1.38 0.42 0.41 0.25 
Calc. 0.6424  0.6224  1.52,4 1.64,24 1.5430  0.42,24 0.4331  0.42,24 0.4331  0.25,24,30 0.2631  
 0.61,4 0.5730(mh ⊥)  0.4330(me ⊥)  
TABLE II.

Hole and electron effective masses in unstrained 4H–SiC. Reported theoretical4,5,24,30–32 and experimental33,34 values are listed; the theoretical values in Ref. 5 are obtained using the same computational settings as those in the present study.

HoleElectron
Γ-MΓ-KΓ-AM-ΓM-KM-L
Calc. 0.54,5 0.8624  0.54,5 0.9524  1.60,4 1.48,5 1.58,24 1.5630  0.58,4 0.54,5 0.53,24 0.57,30,31 0.6232  0.28,4,5,30,31 0.2724,32 0.31,4,5,30–32 0.3024  
 0.62,4 0.5930(mh ⊥)     
Expt.  0.5833  0.3133  0.3333  
 0.6634 (mh ⊥) 1.7534 (mh ∥) 0.4534 (me ⊥) 0.3034 (me ∥) 
HoleElectron
Γ-MΓ-KΓ-AM-ΓM-KM-L
Calc. 0.54,5 0.8624  0.54,5 0.9524  1.60,4 1.48,5 1.58,24 1.5630  0.58,4 0.54,5 0.53,24 0.57,30,31 0.6232  0.28,4,5,30,31 0.2724,32 0.31,4,5,30–32 0.3024  
 0.62,4 0.5930(mh ⊥)     
Expt.  0.5833  0.3133  0.3333  
 0.6634 (mh ⊥) 1.7534 (mh ∥) 0.4534 (me ⊥) 0.3034 (me ∥) 
TABLE III.

Hole and electron effective masses in unstrained 6H–SiC. For reference, reported theoretical4,24,30,31,35 and experimental36,37 values are also listed.

HoleElectron
Γ-MΓ-KΓ-AM-ΓM-KM-L
This work 0.54 0.54 1.51 0.68 0.25 3.34 
Calc. 0.62,4 0.58,30 0.6035 (mh1.67,4 1.54,30 1.6535  0.77,4 0.71,24 0.7530,31 0.244,24,30,31 1.42,4 1.95,24 1.8330,31 
Expt. 0.6636 (mh1.8536 (mh0.48,36 0.4237 (me3–6,36 2.037 (me
HoleElectron
Γ-MΓ-KΓ-AM-ΓM-KM-L
This work 0.54 0.54 1.51 0.68 0.25 3.34 
Calc. 0.62,4 0.58,30 0.6035 (mh1.67,4 1.54,30 1.6535  0.77,4 0.71,24 0.7530,31 0.244,24,30,31 1.42,4 1.95,24 1.8330,31 
Expt. 0.6636 (mh1.8536 (mh0.48,36 0.4237 (me3–6,36 2.037 (me
TABLE IV.

Hole and electron effective masses in unstrained 3C–SiC. For reference, reported theoretical24,31,32,38 and experimental39,40 values are also listed.

HoleElectron
Γ-XΓ-KΓ-LX-ΓX-UX-W
This work 0.54 1.18 1.52 0.62 0.23 0.23 
Calc. 0.5938  1.3538  1.6838  0.64,24 0.68,31 0.7232  0.22,24 0.2331  0.22,24,32 0.2331  
Expt.    0.6739,40 0.2539,40 0.2539,40 
HoleElectron
Γ-XΓ-KΓ-LX-ΓX-UX-W
This work 0.54 1.18 1.52 0.62 0.23 0.23 
Calc. 0.5938  1.3538  1.6838  0.64,24 0.68,31 0.7232  0.22,24 0.2331  0.22,24,32 0.2331  
Expt.    0.6739,40 0.2539,40 0.2539,40 

First, we show distortions along the [0001] direction for the hexagonal polytypes and the [111] direction for 3C under the biaxial (0001) and (111) in-plane strain, respectively, in Fig. 4. The lattice constants along the [0001] direction change almost linearly with the biaxial strain. The calculated slopes are −0.189, −0.193, −0.191, and −0.195 for 2H, 4H, 6H, and 3C–SiC, respectively. These values are close to −0.195,41 −0.194,41 −0.188,42 and −0.21543 from the preceding theoretical and experimental studies for the respective polytypes (estimated using values of 2C13C33, where C13 and C33 are elastic constants).

FIG. 4.

Dependence of distortions along the [0001] or [111] axis on biaxial (0001) or (111) in-plane strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

FIG. 4.

Dependence of distortions along the [0001] or [111] axis on biaxial (0001) or (111) in-plane strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

Close modal

Next, we discuss the effect of biaxial strain on the electronic properties. The variation of the bandgaps under biaxial strain is shown in Fig. 5. For all four polytypes, the bandgaps increase almost linearly with the strain up to a particular tensile strain and subsequently turn to decrease. These results are consistent with those in the recent non-relativistic theoretical studies of 2H, 4H, and 6H–SiC by Chokawa and Shiraishi.7 

FIG. 5.

Dependence of the bandgaps on biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

FIG. 5.

Dependence of the bandgaps on biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

Close modal

The volume and strain dependency of band energies in semiconductors is commonly described using deformation potential.44,45 We have calculated bandgap deformation potentials (dEg/d ln V) keeping the c/a ratio fixed, where Eg is the bandgap and V is the volume of the unstrained unit cell. The resultant values are 1.54 eV, 0.62 eV, and −0.51 eV for the 2H, 3C, and 4H polytypes, respectively. The bandgap of the 6H polytype hardly changed with the uniform strain, indicating that its bandgap deformation potential is nearly zero.

We found that at the critical strain values, i.e., 1.2%, 0.4%, 0.2%, and 0% for 2H, 4H, 6H, and 3C–SiC, respectively, where the bandgaps become largest, the order of three valence bands changes. For the hexagonal polytypes, the Γ7+ and Γ7− bands, which emanate from the LH (SO) and CF bands, move upward with increasing tensile strain, and then the Γ7+ band forms the VBM at the Γ point [Figs. 6(a)6(c)]. On the other hand, with compressive strain, the Γ9 and Γ7− bands, which originate from the HH and CF bands, remain the highest and lowest, respectively. For 3C, the irreducible representations of the HH and LH degenerate bands change from Γ8 to (Γ4 and Γ5) and Γ6+ and that of the SO band from Γ7 to Γ6−46 because the 3C structure loses the four-fold rotational axis under the biaxial (111) in-plane strain. Consequently, the degeneracy between the HH and LH bands is broken by the tensile strain, and the highest band becomes the Γ6+ band, which emanates from the LH band [Fig. 6(d)]. On the other hand, with the compressive strain, the Γ4 and Γ5 degenerate bands, which emanate from the HH band, remain the highest in a similar manner to the hexagonal polytypes despite degeneracy breaking. However, the Γ6+ band is lowered with an increase in compressive strain, and eventually, the Γ6+ and Γ6− bands are interchanged, and then the Γ6+ band becomes the lowest among the three relevant bands. In summary, in the hexagonal and cubic polytypes, the order of the three valence bands changes differently with the tensile strain. With compressive strain, the order of the three valence bands remains unchanged in the hexagonal polytypes, whereas the Γ6+ band is lowered and eventually becomes the lowest among the (Γ4 and Γ5), Γ6+, and Γ6− bands in the cubic polytype. The resultant band order in the cubic polytype is similar to those in the hexagonal polytypes.

FIG. 6.

Band structures on the ΓA path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC under 1.2%, 0.6%, 0.4%, and 0.2% tensile strain, respectively. The energy of the highest valence band at the Γ point is set to zero.

FIG. 6.

Band structures on the ΓA path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC under 1.2%, 0.6%, 0.4%, and 0.2% tensile strain, respectively. The energy of the highest valence band at the Γ point is set to zero.

Close modal

The difference and similarity of the band order change can be understood in view of the symmetries of the wavefunctions of the three valence bands. For the hexagonal polytypes, our detailed orbital analysis shows that the hybridization between the Γ7+ and Γ7− bands changes with the tensile strain. The orbital character of the Γ7+ band is changed to pz and that of the Γ7− band to px and py gradually. In other words, the Γ7+ band is changed to the Γ7− band, and the Γ7− band is changed to the Γ7+ band. Owing to the change in the hybridization between the Γ7+ and Γ7− bands, the Γ7+ band goes upward, leading to band crossing and reordering [Figs. 6(a)6(c)]. The irreducible representations of the three valence bands do not change with the biaxial (0001) in-plane strain because it does not change the crystal symmetry. Accordingly, the change in the hybridization between the Γ7+ and Γ7− bands is reasonable considering the same irreducible representation of them. In contrast, the hybridization between the Γ9 band and the other bands is not allowed by symmetry, and the orbital character of the Γ9 band remains px and py even with the strain. In the 3C polytype, the Γ4 and Γ5 degenerate bands emanating from the HH band remain the highest under compressive strain despite degeneracy breaking, and their orbital characters remain px and py.

In addition, we have investigated the strain dependency of the energy difference between the Γ9 and Γ7− bands (or that between the Γ9 and Γ7+ bands above critical tensile strain values where the band crossover occurs) in the hexagonal polytypes. Similarly, the energy difference between the Γ4 and Γ5 degenerate bands and the Γ6+ band was considered for the 3C polytype. These values, Esplit, decrease almost linearly with the increasing strain, and they become zero around the critical strains where band crossing or degeneracy occurs (Fig. 7). When the Γ7+ or Γ6+ bands are higher than the Γ9 or (Γ4 and Γ5) bands above the critical tensile strain, Esplit takes negative values. We have also calculated the CF splitting energies defined as the energy differences between the HH and CF bands without the SOI (Fig. 8). For the four polytypes we have considered here, the CF splitting energies are close to the relevant Esplit values. The values of the critical strains, which are 1.2%, 0.4%, 0.2%, and 0% for 2H, 4H, 6H, and 3C, respectively, are almost proportional to the values of the energy differences between the HH and CF bands under zero strain, which are 110.1 meV, 48.2 meV, 33 meV, and 0 meV, respectively. These energy differences are approximately proportional to the hexagonalities we noted in Sec. III A, and thus, we conclude that the values of critical strains are determined by the hexagonalities, which describe the stacking sequences of the Si–C bilayers.

FIG. 7.

Energy difference between the Γ9 and Γ7− bands (or that between the Γ9 and Γ7+ bands above critical tensile strain values where the band crossover occurs) for (a) 2H–SiC, (b) 4H–SiC, and (c) 6H–SiC. (d) Energy difference between the Γ4 and Γ5 degenerate bands and the Γ6+ band for 3C–SiC. These energy differences (Esplit) are shown as a function of biaxial strain; the positive and negative values in the abscissa indicate tensile and compressive strain, respectively.

FIG. 7.

Energy difference between the Γ9 and Γ7− bands (or that between the Γ9 and Γ7+ bands above critical tensile strain values where the band crossover occurs) for (a) 2H–SiC, (b) 4H–SiC, and (c) 6H–SiC. (d) Energy difference between the Γ4 and Γ5 degenerate bands and the Γ6+ band for 3C–SiC. These energy differences (Esplit) are shown as a function of biaxial strain; the positive and negative values in the abscissa indicate tensile and compressive strain, respectively.

Close modal
FIG. 8.

Dependence of the crystal-field splitting energies on biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively.

FIG. 8.

Dependence of the crystal-field splitting energies on biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively.

Close modal

Furthermore, we have analyzed the band structures along the ΓM and ΓK paths for the hexagonal polytypes and 3C (in the hexagonal representation) because we have observed curious shifts of the VBM with the tensile strain, as shown for the ΓM path in Fig. 9. Moving away from the Γ point, the irreducible representations of the three valence bands split into Σ3 ⊕ Σ4 on the ΓM path and Λ3 ⊕ Λ4 on the ΓK path.46 The Γ9 and Γ7+ bands in the hexagonal polytypes and the Γ4 and Γ5 degenerate bands and the Γ6+ band in the 3C polytype can therefore be hybridized on these paths under the biaxial strain, and the hybridization causes the shift of the VBM from the Γ point. The amount of the shift in both energy and wavevector spaces is correlated with the hexagonality, increasing in the order 3C, 6H, 4H, and 2H.

FIG. 9.

Strain dependence of the highest valence bands along the ΓM path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The energy of each band at the Γ point is set to zero. The data for 4H–SiC are taken from Ref. 5.

FIG. 9.

Strain dependence of the highest valence bands along the ΓM path for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The energy of each band at the Γ point is set to zero. The data for 4H–SiC are taken from Ref. 5.

Close modal

We have also calculated the strain-induced changes in the hole effective masses for the highest valence bands, which originate from the modulations of the band structures discussed above. The results are shown in Fig. 10. The effective masses along the ΓA direction drastically decrease at the critical tensile strain because of band crossing and reordering among the three valence bands. For the hexagonal polytypes, those with the ΓM and ΓK directions increase with tensile strain because of the hybridization between the Γ9 and Γ7+ bands at wavevectors other than the Γ point. In particular, we have observed that a delta-function-like increase in the effective masses occurs around these critical strains [Figs. 10(a)10(c)]; there should exist a magic strain where the second derivative of the highest band is exactly zero, indicating that an infinite value of the in-plane hole effective mass exists with the particular tensile strain. For 3C, the highest band becomes Γ6+ with tensile strain, and the effective masses also increase by a factor of about 4 in the ΓM and ΓK directions. The origin of the change in the effective masses in 3C is the breaking of the degeneracy between the HH and LH bands.

FIG. 10.

Hole effective masses under biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

FIG. 10.

Hole effective masses under biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

Close modal

Finally, we show the variations of the electron effective masses under strain in Fig. 11. For 2H, the effective masses in the KΓ and KM directions slightly increase with the strain. In the KH direction, the effective mass hardly changes. For 3C and 4H, the effective masses along MΓ and ML gradually increase with the strain, and in the MK direction, the effective mass decreases slightly. For 6H, the effective mass in MΓ hardly changes with the strain, and MK gradually decreases. On the other hand, the electron effective mass along the ML is much more sensitive to the strain than those in the other directions (Fig. 12). With the tensile strain, the effective mass gradually decreases from 3.34 to 1.76 (m0), and the location of the CBM moves from the U to the L point. In contrast, with the compressive strain, the effective mass increases drastically to about 23 (m0). The location of the CBM moves from the U to the M point with the compressive strain, and it becomes the M point at the −0.8% strain where the steep increase in the effective mass occurs.

FIG. 11.

Electron effective masses under biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

FIG. 11.

Electron effective masses under biaxial strain for (a) 2H–SiC, (b) 4H–SiC, (c) 6H–SiC, and (d) 3C–SiC. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. The data for 4H–SiC are taken from Ref. 5.

Close modal
FIG. 12.

(a) Electron effective mass along the ML path in 6H–SiC under biaxial strain. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. (b) Strain dependence of the lowest conduction band along the ML path in 6H–SiC. The energy and wavevector at the U point that gives the CBM are set to zero for each strain value.

FIG. 12.

(a) Electron effective mass along the ML path in 6H–SiC under biaxial strain. The positive and negative values in the abscissa indicate tensile and compressive strain, respectively. (b) Strain dependence of the lowest conduction band along the ML path in 6H–SiC. The energy and wavevector at the U point that gives the CBM are set to zero for each strain value.

Close modal

Our hybrid functional calculations including the SOI have clarified the strain effect on the electronic band structures in 2H, 4H, 6H, and 3C–SiC. In particular, we have shown that the crystal symmetry considerably affects the strain dependency of their valence bands. For the hexagonal and cubic polytypes, the orders of the three valence bands become similar to each other under compressive strain, and this behavior is related to the similarity of the valence orbital characters in the four polytypes. Tensile strain induces the hybridization between the HH and LH bands in the hexagonal polytypes, which leads to the crossing and reordering of these bands. For the cubic polytype, the degeneracy between the HH and LH bands breaks with tensile strain. The difference in the band order change with tensile strain is derived from the difference in the crystal symmetry between the hexagonal and cubic polytypes, and the consequent difference in the valence band structure.

Reflecting the strain dependency of the valence bands, the hole effective masses are insensitive to compressive strain in all four polytypes. Under moderate tensile strain, their hole effective masses increase by a factor of more than two in the (0001) or (111) in-plane directions and decrease by almost an order of magnitude in the out-of-plane direction. The values of the critical tensile strains are approximately proportional to the energy differences between the HH and CF bands under zero strain, which are closely related to the hexagonality. In contrast to the case of the holes, the electron effective masses are found to be insensitive to the strain, except for the ML direction in 6H–SiC. We have thus shown principles for elucidating and designing the crystal structure and strain dependency of the electronic band structures and transport properties of SiC.

The data that support the findings of this study are available within this article.

This work was supported by JSPS KAKENHI Grant Nos. JP18H03770 and JP20H00302. Computing resources of the Academic Center for Computing and Media Studies at Kyoto University were used.

1.
S. M.
Sze
,
Semiconductor Devices
(
Wiley
,
2002
).
2.
A. R.
Verma
and
P.
Krishna
,
Polymorphism and Polytypism in Crystals
(
Wiley
,
1966
).
3.
K.
Kobayashi
and
S.
Komatsu
,
J. Phys. Soc. Jpn.
77
,
084703
(
2008
).
4.
W. R. L.
Lambrecht
,
S.
Limpijumnong
,
S. N.
Rashkeev
, and
B.
Segall
,
Phys. Status Solidi B
202
,
5
(
1997
).
5.
Y.
Kuroiwa
,
Y.-i.
Matsushita
,
K.
Harada
, and
F.
Oba
,
Appl. Phys. Lett.
115
,
112102
(
2019
).
6.
F. M.
Steel
,
B. R.
Tuttle
,
X.
Shen
, and
S. T.
Pantelides
,
J. Appl. Phys.
114
,
013702
(
2013
).
7.
K.
Chokawa
and
K.
Shiraishi
,
Jpn. J. Appl. Phys., Part 1
57
,
071301
(
2018
).
8.
P. E.
Blöchl
,
Phys. Rev. B
50
,
17953
(
1994
).
9.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
10.
A.
Seidl
,
A.
Görling
,
P.
Vogl
,
J. A.
Majewski
, and
M.
Levy
,
Phys. Rev. B
53
,
3764
(
1996
).
11.
G.
Kresse
and
J.
Furthmüller
,
Phys. Rev. B
54
,
11169
(
1996
).
12.
G.
Kresse
and
D.
Joubert
,
Phys. Rev. B
59
,
1758
(
1999
).
13.
J.
Paier
,
M.
Marsman
,
K.
Hummer
,
G.
Kresse
,
I. C.
Gerber
, and
J. G.
Ángyán
,
J. Chem. Phys.
124
,
154709
(
2006
).
14.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
,
J. Chem. Phys.
118
,
8207
(
2003
).
15.
A. V.
Krukau
,
O. A.
Vydrov
,
A. F.
Izmaylov
, and
G. E.
Scuseria
,
J. Chem. Phys.
125
,
224106
(
2006
).
16.
F.
Oba
and
Y.
Kumagai
,
Appl. Phys. Express
11
,
060101
(
2018
).
17.
Y.
Hinuma
,
T.
Hatakeyama
,
Y.
Kumagai
,
L. A.
Burton
,
H.
Sato
,
Y.
Muraba
,
S.
Iimura
,
H.
Hiramatsu
,
I.
Tanaka
,
H.
Hosono
, and
F.
Oba
,
Nat. Commun.
7
,
11962
(
2016
).
18.
R. F.
Adamsky
and
K. M.
Merz
,
Z. Kristallogr.
111
,
350
(
1959
).
19.
N. W.
Thibault
,
Am. Mineral.
29
,
249
(
1944
).
20.
A. H.
Gomes de Mesquita
,
Acta Crystallogr.
23
,
610
(
1967
).
21.
O.
Madelung
,
Semiconductors—Basic Data
(
Springer
,
1996
).
22.
J. R.
O’Connor
and
J.
Smiltens
,
Silicon Carbide—A High Temperature Semiconductor
(
Pergamon Press
,
1960
), p.
147
.
23.
Y.
Hinuma
,
G.
Pizzi
,
Y.
Kumagai
,
F.
Oba
, and
I.
Tanaka
,
Comput. Mater. Sci.
128
,
140
(
2017
).
24.
Z. C.
Feng
,
SiC Power Materials
(
Springer
,
2010
), Vol. 73, pp.
72
75
.
25.
L.
Patrick
,
D. R.
Hamilton
, and
W. J.
Choyke
,
Phys. Rev.
143
,
526
(
1966
).
26.
W. J.
Choyke
,
D. R.
Hamilton
, and
L.
Patrick
,
Phys. Rev.
133
,
A1163
(
1964
).
27.
R. G.
Humphreys
,
D.
Bimberg
, and
W. J.
Choyke
,
Solid State Commun.
39
,
163
(
1981
).
28.
Y.-i.
Matsushita
,
S.
Furuya
, and
A.
Oshiyama
,
Phys. Rev. Lett.
108
,
246404
(
2012
).
29.
Y.-i.
Matsushita
and
A.
Oshiyama
,
Phys. Rev. Lett.
112
,
136403
(
2014
).
30.
C.
Persson
and
U.
Lindefelt
,
J. Appl. Phys.
82
,
5496
(
1997
).
31.
C.
Persson
and
U.
Lindefelt
,
Phys. Rev. B
54
,
10257
(
1996
).
32.
G. L.
Zhao
and
D.
Bagayoko
,
New J. Phys.
2
,
16
(
2000
).
33.
D.
Volm
,
B. K.
Meyer
,
D. M.
Hofmann
,
W. M.
Chen
,
N. T.
Son
,
C.
Persson
,
U.
Lindefelt
,
O.
Kordina
,
E.
Sörman
,
A. O.
Konstantinov
,
B.
Monemar
, and
E.
Janzén
,
Phys. Rev. B
53
,
15409
(
1996
).
34.
N. T.
Son
,
P. N.
Hai
,
W. M.
Chen
,
C.
Hallin
,
B.
Monemar
, and
E.
Janzén
,
Phys. Rev. B
61
,
R10544
(
2000
).
35.
C.
Persson
and
U.
Lindefelt
,
J. Appl. Phys.
86
,
5036
(
1999
).
36.
N. T.
Son
,
C.
Hallin
, and
E.
Janzén
,
Phys. Rev. B
66
,
045304
(
2002
).
37.
N. T.
Son
,
O.
Kordina
,
A. O.
Konstantinov
,
W. M.
Chen
,
E.
Sörman
,
B.
Monemar
, and
E.
Janzén
,
Appl. Phys. Lett.
65
,
3209
(
1994
).
38.
M.
Willatzen
,
M.
Cardona
, and
N. E.
Christensen
,
Phys. Rev. B
51
,
13150
(
1995
).
39.
R.
Kaplan
,
R. J.
Wagner
,
H. J.
Kim
, and
R. F.
Davis
,
Solid State Commun.
55
,
67
(
1985
).
40.
J.
Kono
,
S.
Takeyama
,
H.
Yokoi
,
N.
Miura
,
M.
Yamanaka
,
M.
Shinohara
, and
K.
Ikoma
,
Phys. Rev. B
48
,
10909
(
1993
).
41.
42.
K.
Kamitani
,
M.
Grimsditch
,
J. C.
Nipko
,
C.-K.
Loong
,
M.
Okada
, and
I.
Kimura
,
J. Appl. Phys.
82
,
3152
(
1997
).
43.
K. B.
Tolpygo
,
Sov. Phys. - Sol. State
2
,
2367
(
1961
).
44.
A.
Punya
and
W. R. L.
Lambrecht
,
Phys. Rev. B
85
,
195147
(
2012
).
45.
Q.
Yan
,
P.
Rinke
,
A.
Janotti
,
M.
Scheffler
, and
C. G.
Van de Walle
,
Phys. Rev. B
90
,
125118
(
2014
).
46.
L.
Elcoro
,
B.
Bradlyn
,
Z.
Wang
,
M. G.
Vergniory
,
J.
Cano
,
C.
Felser
,
B. A.
Bernevig
,
D.
Orobengoa
,
G.
de la Flor
, and
M. I.
Aroyo
,
J. Appl. Crystallogr.
50
,
1457
(
2017
).