NSM Archive - Silicon Germanium (SiGe)

Band structure and carrier concentration

Basic Parameters
Band structure
Intrinsic carrier concentration
Effective Density of States in the Conduction and Valence Band
Temperature Dependences
Dependences on Hydrostatic Pressure
Strain-Dependent Band Discontinuity
Effective Masses and Density of States
Dislocation Glide

Basic Parameters

SiGe			Remarks	Referens
Energy gaps, Eg_indirect (Δ conduction band min)	Si_1-xGe_x	1.12-0.41x + 0.008x² eV	300 K, x < 0.85
	Si (x=0)	1.12 eV	300 K, x = 0.	see Si. Band structure

Energy gaps, Eg_indirect (L conduction band min)	Si_1-xGe_x	1.86 - 1.2x eV	300 K, x > 0.85
	Ge (x=1)	0.66 eV	300 K, x = 1.	see Ge. Band structure

*Conduction band*
Energy separation E_Γ1	Si_1-xGe_x	4175 - (2814 ± 55)x meV	70 K; 0 < x < 0.3, Linear Fitting Si_1-xGe_x films on Si substrates	Ebner et al. (1998)
	Si (x=0)	3.4 eV	300 K	see Si. Band structure
	Ge (x=1)	0.8 eV	300 K	see Ge. Band structure
E_g(Γ-X)	Si_1-xGe_x	0.8941 + 0.0421x+ 0.1691x²	calculated	Krishnamurti et al. (1983)
E_g(Γ-L)	Si_1-xGe_x	0.7596 + 1.0860x+ 0.3306x²	calculated	Krishnamurti et al. (1983)
Energy separation E_Γ2	Si_1-xGe_x	(3400 ± 7) - (300 ± 40)x meV	70 K; 0 < x < 0.3, Linear Fitting Si_1-xGe_x films on Si substrates	Ebner et al. (1998)
	Si (x=0)	4.2 eV	300 K	see Si. Band structure
	Ge (x=1)	3.2 eV	300 K	see Ge. Band structure

*Valence band*
Energy of spin-orbital splitting E_so	Si_1-xGe_x	0.044 + 0.246x eV	300 K
	Si (x=0)	0.044 eV	300 K, x = 0.	see Si. Band structure
	Ge (x=1)	0.29 eV	300 K, x = 1.	see Ge. Band structure

Effective conduction band density of states	Si_1-xGe_x	~ 2.8 x 10¹⁹cm^-3	300 K, x < 0.85
	Si_1-xGe_x	~ 1.0 x 10¹⁹cm^-3	300 K, x > 0.85
	Si (x=0)	2.8 x 10¹⁹cm^-3	300 K, x = 0.	see Si. Band structure
	Ge (x=1)	1.0 x 10¹⁹cm^-3	300 K, x = 1.	see Ge. Band structure

Effective valence band density of states	Si (x=0)	1.8 x 10¹⁹cm^-3	300 K, x = 0.	see Si. Band structure
	Ge (x=1)	0.5 x 10¹⁹cm^-3	300 K, x = 1.	see Ge. Band structure

Intrinsic carrier concentration	Si_1-xGe_x		see Si_1-xGe_x. Intrinsic carrier concentration
	Si (x=0)	1 x 10¹⁰ cm^-3	300 K, x = 0.	see Si. Band structure
	Ge (x=1)	2 x 10¹³ cm^-3	300 K, x = 1.	see Ge. Band structure

Energy Gaps vs. Composition
E₁	Si_1-xGe_x	3452 - (1345 ± 25)x meV	70 K; 0 < x < 0.3, Linear Fitting Si_1-xGe_x films on Si substrates	Ebner et al. (1998)
E'₁	Si_1-xGe_x	5402 ± 25 + (280 ± 120)x meV
E₂ (X)	Si_1-xGe_x	4351 ± 38 + (210 ± 180)x meV
E₂ (Σ)	Si_1-xGe_x	4518 ± 129 + (880 ± 600)x meV
E_g(Γ-X)	Si_1-xGe_x	0.8941 + 0.0421x+ 0.1691x²	calculated	Krishnamurti et al. (1983)
E_g(Γ-L)	Si_1-xGe_x	0.7596 + 1.0860x+ 0.3306x²	calculated	Krishnamurti et al. (1983)

Band structure

Compositional dependence of band gaps (calculated): Ł,(F-X) 0.8941 + 0.042 lx+ 0.1691 ÷2 85Ę Łg(r-L) 0.7596+1.0860 ÷+0.3306 ÷2 Krishnamurti et al. (1983)

Band structure of Si at 300 K.

Energy gap E_g	1.12 eV
E_L	2.0 eV
E_X	1.2 eV
E_Γ1	3.4 eV
Energy separation (E_ΓL or E_Γ2 )	4.2 eV
Energy spin-orbital splitting E_so	0.044 eV
Intrinsic carrier concentration	1·10¹⁰ cm^-3
Intrinsic resistivity	3.2·10⁵Ω·cm
Effective conduction band density of states	3.2·10¹⁹ cm^-3
Effective valence band density of states	1.8·10¹⁹ cm^-3

Band structures of Ge. see also

Energy gap E_g	0.661 eV
E_x	1.2 eV
Energy separation (E_Γ1)	0.8 eV
E_Γ2	3.22 eV
Energy separation (ΔE>)	0.85 eV
Energy spin-orbital splitting E_so	0.29 eV
Intrinsic carrier concentration	2.0·10¹³ cm^-3
Intrinsic resistivity	46 Ω·cm
Effective conduction band density of states	1.0·10¹⁹ cm^-3
Effective valence band density of states	5.0·10¹⁸ cm^-3

The band structure shows a cross-over in the lowest conduction band edge from Ge-like [111] symmetry to Si-like [ 100] symmetry at x

0.15 . According to Kustov et al. (1983) this value should lie a little bit higher (x

0.25).

	Si_xGe_1-x. Indirect Energy gap vs. composition at 296 K One-phonone model of the absorption edge. At about x=0.15 a crossover occurs of the Ge-like [111] conduction band minima and the Si-like [100] conduction band minima Braunstein et al. (1958)
	Si_1-xGe_x. Fundamental (indirect) band gap & excitonic band gap at 4.2 K Squares - band gap of Si_1-xGe_x at 4.2 K (absorption measurements) ; Dots - excitonic band gap of Si_1-xGe_x at 4.2 K (photoluminescence measurements) Braunstein et al. (1958) and Weber & Alonso (1989)
	Si_1-xGe_x. Composition dependences of several important direct transitions observed. Kline et al. (1968) and Pickering et al. (1993)
	Si_1-xGe_x alloys.Fundamental indirect band gap vs. x of pseudomorphic Si_1-xGe_x (001) alloys: (a) on Si substrate; (b) on Si_0.5Ge_0.5 substrate; (c) on Ge substrate. Dashed lines - unstrained bulk band gap. Experimental points are taken from Lang et al. (1985) and Dutartre et al. (1991). Solid lines - calculated curves. People (1985, 1986) and Van de Walle & Martin (1986).

Energy Gaps vs. Composition (Linear Fitting)

Electroreflectance experiments on strained Si_1-xGe_x films on Si substrates at 70 K :

0 < x < 0.3 Remarks Referens

E_Γ1 = 4175 - (2814 ± 55)x meV 70 K;
Si_1-xGe_x films on Si substrates Ebner et al. (1998)

E_Γ2 = (3400 ± 7) - (300 ± 40)x meV

E₁ = 3452 - (1345 ± 25)x meV

E'₁ = 5402 ± 25 + (280 ± 120)x meV

E₂ (X) = 4351 ± 38 + (210 ± 180)x; meV

E₂ (Σ) = 4518 ± 129 + (880 ± 600)x meV

Si_1-xGe_x Compositional dependence of band gaps Remarks Referens

E_g(Γ-X) = 0.8941 + 0.0421x+ 0.1691x² calculated Krishnamurti et al. (1983)

E_g(Γ-L) = 0.7596 + 1.0860x+ 0.3306x²

Temperature Dependences

Temperature dependence of energy gap:

x=0 (Si)	E_g = 1.17 -4.73 x 10^-4 x T²/(T + 636)	(eV)	see also Si. Band structure and carrier concentration
x=1 (Ge)	E_g = 0.742- 4.8x 10^-4·T²/(T+235)	(eV)	see also Ge. Band structure and carrier concentration

where T is temperature in degrees K.

Temperature dependence of the direct band gap E_Γ

x=0 (Si)	E_Γ2 = 4.34 - 3.91·10^-4·T²/(T+125)	(eV)	see also Si. Band structure and carrier concentration
x=1 (Ge)	E_Γ₁ = 0.89 - 5.82·10^-4·T²/(T+296)	(eV)	see also Ge. Band structure and carrier concentration

where T is temperature in degrees K.

Si_1-xGe_x alloys. Fundamental indirect band gaps vs. temperature at different x.
Braunstein et al.(1958)

Intrinsic carrier concentration:

n_i = (N_c·N_v)^1/2exp(-E_g/(2k_BT))

Si_1-xGe_x alloys. Intrinsic carrier concentration vs. temperature at different x.
1 - x = 0 (Si);
2 - x = 0.4;
3 - x = 0.8;
4 - x = 1.0 (Ge);
Schaffler F.(2001)

Effective density of states in the conduction band N_c

At x < 0.85, Si_1-xGe_x- alloys are considered as "Si-like material:

N_c 4.82 x 10¹⁵ · M · (m_c/m₀)^3/2·T^3/2 (cm^-3) 4.82 x 10¹⁵ (m_cd/m₀)^{3/2
·} T^3/2 5.3 ^· 10¹⁵ x T^3/2(cm^-3) ,
where M=6 is the number of equivalent valleys in the conduction band.
m_c = 0.32m₀ is the effective mass of the density of states in one valley of conduction band.
m_cd = 1.06m₀ is the effective mass of density of states.

At0.85 < x < 1.0, Si_1-xGe_x- alloys are considered as "Si-like material:

N_c 4.82 x 10¹⁵ · M · (m_c/m₀)^3/2·T^3/2 (cm^-3) 4.82 x 10¹⁵ (m_cd/m₀)^3/2· T^3/2 2 ^· 10¹⁵ x T^3/2 (cm^-3) ,
where M = 4 is the number of equivalent valleys in the conduction band.
m_c = 0.22m₀ is the effective mass of the density of states in one valley of conduction band.
m_cd = 0.55m₀ is the effective mass of density of states.

Effective density of states in the valence band N_v

x=0 (Si)	N_c 4.82·10¹⁵ ^· (m_v/m₀)^3/2·T^3/2 3.5·10¹⁵·T^3/2 (cm^-3), m_v = 0.81m₀ is the hole effective mass of the density of states.	see also Si. Band structure and carrier concentration
x=1 (Ge)	N_c 4.82·10¹⁵ ^· (m_v/m₀)^3/2·T^3/2 9.6·10¹⁴·T^3/2 (cm^-3), m_v = 0.34m₀ is the hole effective mass of the density of states.	see also Ge. Band structure and carrier concentration

There is a large uncertainty in the available data on the composition dependence of hole effective mass of density of states m_v. For crude estimations one can use the simplest linear approximation:
m_v (x) = (0.81 - 0.47x)m₀
see also Effective Masses and Density of States

Dependence on Hydrostatic Pressure

x=0 (Si)	E_g=E_g(0)-1.4·10^-3P	(eV)	see also Si. Band structure and carrier concentration
x=1 (Ge)	E_g = E_g(0) + 5.1·10^-3P	(eV)	see also Ge. Band structure and carrier concentration

here P is pressure in kbar.

Strain-Dependent Band Discontinuity

Band discontinuities at an Si_1-xGe_x/Si_1-yGe_y are only defined, if the interface is coherent (that is, if the in-plane lattice constant is preserved across the interface). Generally, both layers are biaxially strained in the interface plane. The biaxial strain status can be converted into a hydrostatic and a uniaxial component, whose sign is opposite to that of the in-plane strain and directed ortogonal to the interface. Both the interface chemistry and the biaxial strain status are relevant for the determination of the band discontinuities at the interface, which will be given in the following for growth on a (001) surface.
The chemical contribution gives a linear variation of the weighted average valence band discontinuity ΔE_v between a strained Si_1-xGe_x film and an unstrained cubic Si_1-xsGe_xs substrate [Rieger & Vogi (1993)]:
ΔE_v (x,xs) = (0.47 - 0.06x)(x - xs) (eV)
The hydrostatic strain components leads to a change in the band gap of the strained Si_1-xGe_x film according to
ΔE_g = (&Teta;_d + 1/3&Teta;_u - a)(

+ 2

)
where (&Teta;_d + 1/3&Teta;_u - a) is the deformation potential difference relevant for the fundamental band gap

= (a_ort - a_o)/a_o are the perpendicular strain components of the strained Si_1-xGe_x film;

= (a_|| - a_o)/a_o - in-plane strain components of the strained Si_1-xGe_x film;.
a_o - undistorted lattice constant of the film;
a_ort - ortogonal components in the strained film;
a_|| - in-plane components in the strained film.

Deformation Potentials for the Calculation of the Band Discontinuities

	(&Teta;_d + 1/3&Teta;_u - a) (for A valley)	(&Teta;_d + 1/3&Teta;_u - a) (for L valley)	b	&Teta;_u	&Teta;_u	Remarks	Referens
Si	1.72 eV	-3.12 eV	-2.35 eV	9.16 eV	16.14 eV	Theory	Van de Walle & Martin (1986)
Si	1.5 ±0.3 eV		-2.10 ±0.1 eV	-4.85 ±0.15 eV		Experiment	Laude et al. (1971), Chandrasekar & Pollak (1977), Balslev (1966)
Ge	1.31 eV	-2.78 eV	-2.55 eV	9.42 eV	15.13 eV	Theory	Van de Walle & Martin (1986)
Ge		-2.0 ± 0.5 eV	-2.86 ±0.15 eV		16.2 ± 0.4 eV	Experiment	Laude et al. (1971), Chandrasekar & Pollak (1977), Balslev (1966)

Lacking predictions or measurements for Si_1-xGe_x alloys, a linear interpolation is suggested, which is certainly a compromise, whenever the conduction band changes from Si-like to Ge-like.

Both the valence and conduction band degeneracy are lifted by the uniaxial [001] strain component, which leads to the following splittings (Van de Walle and Martin, (1986)):

Valence Band

The valence band degeneracy are lifted by the uniaxial [001] strain component, which leads to the following splittings (Van de Walle and Martin, (1986)):
Heavy hole: E_v = 1/3Δ₀ - 1/2δE₀₀₁
Light hole: E_v1 = -1/6Δ₀ + 1/4δE₀₀₁ + 1/2(Δ₀² + Δ₀ δE₀₀₁+ 9/4δE²₀₀₁)^1/2
Spin-split hole: E_v3 = -1/6Δ₀ +1/4δE₀₀₁ - 1/2(Δ₀² + Δ₀ δE₀₀₁+ 9/4δE²₀₀₁)^1/2
were δE₀₀₁ =2b(

), b being a valence band deformation potential (see Deformation Potentials)

Conduction Band

The conduction band degeneracy are lifted by the uniaxial [001] strain component, which leads to the following splittings (Van de Walle and Martin, (1986)):
ΔE_v (Δ₂ ) = 2/3&Teta;_u^Δ (

)
ΔE_v (Δ₄ ) = -1/3&Teta;_u^Δ(

)
where Δ₂ are the two electron valleys along the [001] growth direction.
Δ₄ are the four in-plane electron valleys.
&Teta;_u^Δ is the relevant conduction band deformation potential for the Δ electrons that define the conduction band minimum in Si-like Si_1-xGe_x (x

0.85).

For higher Ge contents, the conduction band becomes Ge-like with electrons being located at the L minimum. With the uniaxial strain component being directed along [001], no splitting of the L minimum occurs for reasons of symmetry.

Si_1-xGe_x. Schematic diagram of the relevant band edges of Si subjected to hydrostatic and uniaxial strain as described in equations.
Energy values apply to a tensely strained Si quantum well on an Si_1-xGe_x substrate with x = 30%
Schaffler(1997)

Si_1-xGe_x. Contour plots of the conduction ΔE_c and valence ΔE_v band offsets of pseudomorphic Si_1-xGe_x layers on cubic Si_1-xsGe_xs substrates over the complete range of x and xs.
The signs correspond to an electronic energy scale, where the active layer (x) is referred to the cubic substrate of composition xs. Exciton-corrected experimental results indicate that for x > xs and x < 0.8, the conduction band offset is 0<ΔE_c<+40 meV [Penn et al. (1999)];
that is, for most of the (x,xs) combinations the band alignment is staggered (Type II) with the valence band offset being always in favor of the material with the higher Ge content. The theoretically predicted Type I region for x and xs being larger than about 80% has not been confirmed experimentally Schaffler(1997)

Si_1-xGe_x.
Solid lines - Variation of the relevant band edges of a strained Si layer on a cubic Si_1-xsGe_xs substrate .
The dashed lines correspond to the substrate bands.
LH, light holes;
HH, heavy holes;
SO, spin-orbit split holes
Schaffler(1997)

Si_1-xGe_x. Solid lines - Variation of the relevant band edges of a strained Ge layer on a cubic Si_1-xsGe_xs substrate .
The dashed lines correspond to the substrate bands. .
LH, light holes;
HH, heavy holes;
SO, spin-orbit split holes
Schaffler(1997)

Effective Masses and Density of States:

Electrons

Both the Δ and L electron masses are almost unaffected by either composition or biaxial (001) strain [Rieger & Vogl (1993)]. Hence, the electron effective masses are close to the Si bulk value for x

0.85 and are close to the Ge bulk values for x > 0.85:

At x < 0.85, Si_1-xGe_x alloys are considered as "Si-like" material: Remarks Referens

Effective electron mass
(longitudinal)m_l 0.92m_o Schaffler F.(2001)

Effective electron mass
(transverse)m_t 0.19m_o Schaffler F.(2001)

Effective mass of density of states m_cd=M^2/3 m_c
(for all valleys of conduction band) 1.06m_o Son et al. (1994);
Son et al. (1995)

Effective mass of the density of states m_c=(m_l+m_t²)^1/3
(in one valley of conduction band) 0.32m_o

Effective mass of conductivity m_cc= 3/(1/m_l+2/m_t) 0.26m_o

There are M=6 equivalent valleys in conduction band

At 0.85< x <1, Si_1-xGe_x alloys are considered as "Si-like" material: Remarks Referens

Effective electron mass
(longitudinal)m_l 0.159m_o Schaffler F.(2001)

Effective electron mass
(transverse)m_t 0.08m_o Schaffler F.(2001)

Effective mass of density of states m_cd=M^2/3 m_c
(for all valleys of conduction band) 1.55m_o Son et al. (1994);
Son et al. (1995)

Effective mass of the density of states m_c=(m_l+m_t²)^1/3
(in one valley of conduction band) 0.22m_o

Effective mass of conductivity m_cc= 3/(1/m_l+2/m_t) 0.12m_o

There are M=4 equivalent valleys in conduction band

Si_1-xGe_x. Variation of the conduction band effective masses vs. composition
Rieger and Vogl (1993)

Holes

Due to the spin-orbit interaction, hole effective masses depend strongly on composition and crystal direction (warping), and also on strain.
For unstrained bulk, the dispersion can be expressed by three valence band parameters A, B, C [or equivalent representations, Landoldt-Bornstein (1982)] according to
E_hh,lh = E_v -(0.5 h²k²/m_o) {A ± [B² +(C²/k⁴)(k²_x·k²_y+k²_x·k²_z+k²_y·k²_z)]^1/2 }
E_so = E_v -Δ - (0.5 h²k²/m_o) A,
where Δ is the spin-orbit splitting (44 meV in Si, 290 meV in Ge), and k = (k_x, k_y, k_z) is the direction in reciprocal space.

			Remarks	Referens
Effective hole masses (heavy) m_hh	Si (x=0)	0.537 m_o	4.2 K	see also Si. Effective Masses
	Ge (x=1)	0.33 m_o		see also Ge. Effective Masses

Effective hole masses (light) m_lh	Si (x=0)	0.153 m_o		see also Si. Effective Masses
	Ge (x=1)	0.0430 m_o		see also Ge. Effective Masses

Effective hole masses (spin-orbit-split ) m_so	Si_1-xGe_x	(0.23-0.135x) m_o	300 K	Schaffler F.(2001)
	Si (x=0)	0.234 m_o		see also Si. Effective Masses
	Ge (x=1)	0.095(7) m_o	30 K	see also Ge. Effective Masses

	Si_1-xGe_x. Valence band dispersion along [100] and [110] for Si_0.5Ge_0.5 on Si(001). (schematic view) Schaffler F.(2001)
	Si_1-xGe_x. Valence band parameters A, B, and \|C\| vs. composition x Schaffler F.(1997)
	Si_1-xGe_x. Heavy hole effective mass density of states m_hd vs. energy at different x . Manku & Nathan (1991)
	Si_1-xGe_x. Light hole effective mass density of states m_hl vs. energy at different x . Manku & Nathan (1991)
	Si_1-xGe_x. Experimental heavy hole cyclotron masses in strained Si_1-xGe_x quantum wells. Dots - Cheng et al. (1994), squares - Wong et al. (1995). Dashed line corresponds to unstrained bulk, Bottom solid line is a prediction for strained Si_1-xGe_x

	0 < x < 0.3		Remarks	Referens
E_Γ1	= 4175 - (2814 ± 55)x	meV	70 K; Si_1-xGe_x films on Si substrates	Ebner et al. (1998)
E_Γ2	= (3400 ± 7) - (300 ± 40)x	meV
E₁	= 3452 - (1345 ± 25)x	meV
E'₁	= 5402 ± 25 + (280 ± 120)x	meV
E₂ (X)	= 4351 ± 38 + (210 ± 180)x;	meV
E₂ (Σ)	= 4518 ± 129 + (880 ± 600)x	meV

Si_1-xGe_x	Compositional dependence of band gaps	Remarks	Referens
E_g(Γ-X)	= 0.8941 + 0.0421x+ 0.1691x²	calculated	Krishnamurti et al. (1983)
E_g(Γ-L)	= 0.7596 + 1.0860x+ 0.3306x²

	Si_1-xGe_x. Schematic diagram of the relevant band edges of Si subjected to hydrostatic and uniaxial strain as described in equations. Energy values apply to a tensely strained Si quantum well on an Si_1-xGe_x substrate with x = 30% Schaffler(1997)
	Si_1-xGe_x. Contour plots of the conduction ΔE_c and valence ΔE_v band offsets of pseudomorphic Si_1-xGe_x layers on cubic Si_1-xsGe_xs substrates over the complete range of x and xs. The signs correspond to an electronic energy scale, where the active layer (x) is referred to the cubic substrate of composition xs. Exciton-corrected experimental results indicate that for x > xs and x < 0.8, the conduction band offset is 0<ΔE_c<+40 meV [Penn et al. (1999)]; that is, for most of the (x,xs) combinations the band alignment is staggered (Type II) with the valence band offset being always in favor of the material with the higher Ge content. The theoretically predicted Type I region for x and xs being larger than about 80% has not been confirmed experimentally Schaffler(1997)
	Si_1-xGe_x. Solid lines - Variation of the relevant band edges of a strained Si layer on a cubic Si_1-xsGe_xs substrate . The dashed lines correspond to the substrate bands. LH, light holes; HH, heavy holes; SO, spin-orbit split holes Schaffler(1997)
	Si_1-xGe_x. Solid lines - Variation of the relevant band edges of a strained Ge layer on a cubic Si_1-xsGe_xs substrate . The dashed lines correspond to the substrate bands. . LH, light holes; HH, heavy holes; SO, spin-orbit split holes Schaffler(1997)