NSM Archive - Silicon Carbide (SiC) - Band structure

Band structure and carrier concentration
Basic ParametersBand structure
Intrinsic carrier concentration
Effective Density of States in the Conduction and Valence Band
Temperature Dependences
Dependences on Hydrostatic Pressure
Energy Gap Narrowing at High Doping Levels
Effective Masses and Density of States
Donors and Acceptors
More than 200 different polytypes of SiC are known. However, about 95% of all publications deal with three main polytypes: 3C, 4H, and 6H. In all main polytypes of SiC, some atoms have been observed in association both with cubic (C), with hexagonal (H) and with rombohedral (R) lattice sites.
Basic Parameters
3C-SiC: cubic unit cell (Zincblende) | Remarks | Referens | |
Energy gaps, Egind(Γ15v -X1c) | 2.416(1) eV | 2 K, wevelength modulated absorption | Bimberg et al.(1981) |
Energy gaps, Eg | 2.36 eV | 300 K | Goldberg et al.(2001) |
Energy gaps, Egdir(Γ15v -X1c) | 6.0 eV | 300 K, optical absorption | Dalven (1965) |
Excitonic Energy gaps, Egx | 2.38807(3) eV | 1.4 K, wevelength modulated absorption see also Temperature Dependences |
Gorban' et al.(1984) |
Conduction band | Remarks | Referens | |
Energy separation between Γ15v valley and L1c valleys EL | 4.6 eV | 300 K | Goldberg et al.(2001) |
Energy separation between Γ15v valley and Γ1c valleys EΓ | 6.0 eV | 300 K | |
Valence band | |||
Energy of spin-orbital splitting Eso | 0.01 eV | 300 K | Goldberg et al.(2001) |
Energy of crystal-field splitting Ecr |
--- | ||
Effective conduction band density of states |
1.5 x 1019cm-3 | 300 K | |
Effective valence band density of states |
1.2 x 1019 cm-3 | 300 K |
2H-SiC: Hexagonal unit cell (Wurtzite) | Remarks | Referens | |
Excitonic Energy gaps, Eg | 3.330 eV | optical absorption see also Temperature Dependences |
Patrick et al.(1966) |
4H-SiC: Hexagonal unit cell (Wurtzite) | Remarks | Referens | |
Energy gaps, Eg | 3.23 eV | 300 K | Goldberg et al.(2001) |
Excitonic Energy gaps, Egx | 3.20 eV | see also Temperature Dependences | Dubrovskii & Lepneva (1977) |
Conduction band | Remarks | Referens | |
Energy separation between Γ15v valley and L valleys EL | ~=4. eV | 300 K | Goldberg et al.(2001) |
Energy separation between Γ15v valley and Γ1c valleys EΓ | 5-6.0 eV | 300 K | |
Valence band | |||
Energy of spin-orbital splitting Eso | 0.007 eV | 300 K | Goldberg et al.(2001) |
Energy of crystal-field splitting Ecr |
0.008 eV | ||
Effective conduction band density of states |
1.7 x 1019cm-3 | 300 K | |
Effective valence band density of states |
2.5 x 1019 cm-3 | 300 K |
6H-SiC: Hexagonal unit cell (Wurtzite) | Remarks | Referens | |
Energy gaps, Eg | 3.0 eV | 300 K | Goldberg et al.(2001) |
Energy gaps, Egind | 2.86 eV | 300 K, optical absorption see also Temperature Dependences |
Philipp & Taft (1960) |
Excitonic Energy gaps, Eg | 3.0230 eV | wevelength modulated absorption see also Temperature Dependences |
Humphreys et al.(1981) |
Conduction band | Remarks | Referens | |
Energy separation between Γ15v valley and Γ1c valleys EΓ | 5-6.0 eV | 300 K | Goldberg et al.(2001) |
Valence band | |||
Energy of spin-orbital splitting Eso | 0.007 eV | 300 K | Goldberg et al.(2001) |
Energy of crystal-field splitting Ecr |
0.05 eV | ||
Effective conduction band density of states |
8.9 x 1019cm-3 | 300 K | |
Effective valence band density of states |
2.5 x 1019 cm-3 | 300 K |
8H-SiC: Hexagonal unit cell (Wurtzite) | Remarks | Referens | |
Excitonic Energy gaps, Eg | 2.86 eV | see also Temperature Dependences | Dubrovskii & Lepneva (1977) |
15R-SiC: Rhombohedral unit cell | Remarks | Referens | |
Excitonic Energy gaps, Eg | 2.9863 eV | wevelength modulated absorption see also Temperature Dependences |
Humphreys et al.(1981) |
21R-SiC: Rhombohedral unit cell | Remarks | Referens | |
Excitonic Energy gaps, Eg | 2.92 eV | see also Temperature Dependences | Dubrovskii & Lepneva (1977) |
24R-SiC: Rhombohedral unit cell | Remarks | Referens | |
Excitonic Energy gaps, Eg | 2.80 eV | see also Temperature Dependences | Dubrovskii & Lepneva (1977) |
Band structure
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SiC, 3C. Band structure. Important minima of the conduction
band and maxima of the valence band. . 300K; Eg = 2.36 eV; EΓ = 6.0 eV; EL = 4.6 eV; Eso = 0.01 eV. For details see Persson & Lindefelt (1997) |
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SiC, 3C. Band structure Hemstreet & Fong (1974) |
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SiC, 2H. Band structure Hemstreet & Fong (1974) |
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SiC, 4H. Band structure. Important minima of the conduction band
and maxima of the valence band. . 300K; Eg = 3.23 eV; EΓ = 5-6.0 eV; EL ~= 4.0 eV; EsM ~= 0.1 eV. Ecr = 0.08 eV; Eso = 0.007 eV. For details see Persson & Lindefelt (1997) |
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SiC, 4H. Band structure. Important minima of the conduction band
and maxima of the valence band. . 300K; Eg = 3.0 eV; EΓ = 5 - 6.0 eV; Ecr = 0.05 eV; Eso = 0.007 eV. For details see Persson & Lindefelt (1997) |
In all polytypcs except 3C- and riH-Sif atomic hiyers wilh cubic (C) and hexagonal (H) symmetry follow in a regular alternation in the direct ion of the c axis. This can be thought of as anutural one-dimensional superkmice imposed on the "pure" i.e. h-layer free 3C-SiC [Dean et al. (1977)], the period of the superlattice being different for different modifications.
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Brillouin zone of the cubic lattice. |
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Brillouin zone of the hexagonal lattice. |
Temperature Dependences
Temperature dependence of energy gap:3C-SiC | Eg = Eg(0) - 6.0 x 10-4 x T2/(T + 1200) | (eV) | Goldberg et al.(2001) |
Egx = 3.024-0.3055x10-4 +T2/(311K - T) | (eV) | Ravindra & Srivastava (1979) | |
4H-SiC | Eg = Eg(0) - 6.5 x 10-4 x T2/(T + 1300) | Goldberg et al.(2001) | |
6H-SiC | Eg = Eg(0) - 6.5 x 10-4 x T2/(T + 1200) |
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SiC, 3C, 4H, 6H. Energy gap vs. temperature. Choyke(1969) |
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SiC, 3C, 15R, 21R, 2H, 4H, 6H, 8H. Excitonic energy gap vs. temperature
Choyke(1969) |
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SiC, 4H. Excitonic energy gap vs. temperature Choyke et al.(1964) |
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SiC, 6H. Energy gap Egind vs. temperature
Philipp & Taft(1960) |
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SiC, 15R. Excitonic energy gap vs. temperature Patric et al. (1963) |
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SiC, 24R. Excitonic energy gap vs. temperature Zanmarchi (1964) |
Intrinsic carrier concentration:
ni = (Nc·Nv)1/2exp(-Eg/(2kBT))![]() |
SiC, 3C, 4H, 6H. Intrinsic carrier concentration vs. temperature
Goldberg et al.(2001) |
Effective density of states in the conduction band Nc
3C-SiC
Nc ~= 4.82 x 1015 · M · (mc/m0)3/2·T3/2
(cm-3) ~= 4.82 x 1015 (mcd/m0)3/2·x
T3/2 ~= 3 x 1015 x T3/2(cm-3)
,
where M=3 is the number of equivalent valleys in the conduction band.
mc = 0.35m0 is the
effective mass of the density of states in one valley of conduction band.
mcd = 0.72 is the effective mass of
density of states.
4H-SiC
Nc ~= 4.82 x 1015 · M · (mc/m0)3/2·T3/2 (cm-3) ~= 4.82 x 1015 (mcd/m0)3/2·x T3/2 ~= 3.25 x 1015 x T3/2(cm-3) ,where M=3 is the number of equivalent valleys in the conduction band.
mc = 0.37m0 is the effective mass of the density of states in one valley of conduction band.
mcd = 0.77 is the effective mass of density of states.
6H-SiC
Nc ~= 4.82 x 1015 · M · (mc/m0)3/2·T3/2
(cm-3) ~= 4.82 x 1015 (mcd/m0)3/2·x
T3/2 ~= 1.73 x 1015 x T3/2(cm-3)
,
where M=6 is the number of equivalent valleys in the conduction band.
mc = 0.71m0 is the
effective mass of the density of states in one valley of conduction band.
mcd = 2.34 is the effective mass of density
of states.
Effective density of states in the valence band Nv
3C-SiC | Nv = 2.23x1015 x T3/2 (cm-3) | Ruff et al. (1994), Casady and Johnson (1996) |
4H-SiC | Nv ~= 4.8x1015 x T3/2 (cm-3) | |
6H-SiC | Nv ~= 4.8x1015 x T3/2 (cm-3) |
Dependence on Hydrostatic Pressure
3C-SiC | Eg = Eg(0) - 0.34 x 10-3P | (eV) | Park et al. (1994) |
EL = EL(0) + 3.92 x 10-3P | (eV) | ||
EΓ = EΓ(0) + 5.11 x 10-3P | (eV) | ||
4H-SiC | Eg = Eg(0) + 0.8 x 10-3P | (eV) | |
EΓ = EΓ(0) + 3.7 x 10-3P | (eV) | ||
6H-SiC | Eg = Eg(0) - 0.03 x 10-3P | (eV) | |
EΓ = EΓ(0) + 4.03 x 10-3P | (eV) |
Energy Gap Narrowing at High Doping Levels:
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3C-SiC. Conduction and valence band displacements vs. ionized shallow
impurity. n-type material. For comparison, the band-edge displacements for Si are shown Lindefelt (1998) |
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3C-SiC. Conduction and valence band displacements vs. ionized shallow
impurity. p-type material. For comparison, the band-edge displacements for Si are shown Lindefelt (1998) |
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4H-SiC, 6H-SiC. Conduction and valence band displacements vs. ionized
shallow impurity. n-type material. For comparison, the band-edge displacements for Si are shown Lindefelt (1998) |
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4H-SiC, 6H-SiC. Conduction and valence band displacements vs. ionized
shallow impurity. p-type material. For comparison, the band-edge displacements for Si are shown Lindefelt (1998) |
The band-edge displacements for n-type material Lindefelt
(1998) :
ΔEnc=Anc(ND+)1/3
x 10-6 + Bnc(ND+)1/2
x 10-9 (eV)
ΔEnv=Anv(ND+)1/3
x 10-6 + Bnv(ND+)1/2
x 10-9 (eV)
where
n-type | Anc | Bnc | Anv | Bnv |
Si | -9.74x10-3 | -1.39x10-3 | -1.27x10-2 | -1.40x10-3 |
3C-SiC | -1.48x10-2 | -3.06x10-3 | -1.75x10-2 | -6.85x10-3 |
4C-SiC | -1.50x10-2 | -2.93x10-3 | -1.90x10-2 | -8.74x10-3 |
6C-SiC | -1.12x10-2 | -1.01x10-3 | -2.11x10-2 | -1.73x10-3 |
The band-edge displacements for p-type material Lindefelt
(1998) :
ΔEpc=Apc(ND+)1/3
x 10-6 + Bpc(ND+)1/2
x 10-9 (eV)
ΔEpv=Apv(ND+)1/3
x 10-6 + Bpv(ND+)1/2
x 10-9 (eV)
where
p-type | Apc | Bpc | Apv | Bpv |
Si | -1.14x10-3 | -2.05x10-3 | -1.11x10-2 | -2.06x10-3 |
3C-SiC | -1.50x10-2 | -6.41x10-4 | -1.30x10-2 | -1.43x10-3 |
4C-SiC | -1.57x10-2 | -3.87x10-4 | -1.30x10-2 | -1.15x10-3 |
6C-SiC | -1.74x10-2 | -6.64x10-4 | -1.30x10-2 | -1.14x10-3 |
Effective Masses and Density of States:
Electrons
3C-SiC. The surfaces of equal energy are ellipsoids:3C-SiC | Remarks | Referens | |
Effective electron mass (longitudinal)ml |
0.68mo | Son et al.
(1994); Son et al. (1995) |
|
0.677(15)mo | 45K, Cyclotrone resonance | Kaplan et al. (1985) | |
Effective electron mass (transverse)mt |
0.25mo | Son et al. (1994); Son et al. (1995) |
|
0.247(11)mo | 45K, Cyclotrone resonance | Kaplan et al. (1985) | |
Effective mass of density of states mcd | 0.72mo | Son et al. (1994); Son et al. (1995) |
|
Effective mass of the density of states in one valley of conduction band mc | 0.35mo | ||
Effective mass of conductivity mcc | 0.32mo |
4H-SiC. The surfaces of equal energy are ellipsoids:
4H-SiC | Remarks | Referens | |
Effective electron mass (longitudinal)ml |
0.29mo | Son et al.
(1994); Son et al. (1995) |
|
Effective electron mass (transverse)mt |
0.42mo | Son et al. (1994); Son et al. (1995) |
|
Effective mass of density of states mcd | 0.77mo | Son et al. (1994); Son et al. (1995) |
|
Effective mass of the density of states in one valley of conduction band mc | 0.37mo | ||
Effective mass of conductivity mcc | 0.36mo |
6H-SiC. The surfaces of equal energy are ellipsoids:
6H-SiC | Remarks | Referens | |
Effective electron mass (longitudinal)ml |
0.20mo | Son et al. (1994); Son et al. (1995) | |
Effective electron mass (transverse)mt |
0.42mo | Son et al. (1994); Son et al. (1995) | |
Effective mass of density of states mcd | 2.34mo | Son et al. (1994); Son et al. (1995) | |
Effective mass of the density of states in one valley of conduction band mc | 0.71mo | ||
Effective mass of conductivity mcc | 0.57mo |
Holes
Due to the small spin-orbit splitting the valence bands are highly non-parabolic, i.e. effective hole masses become strongly k-dependent.SiC | Remarks | Referens | ||
3C | Effective mass of density of state mv
|