SiGe - Silicon Germanium

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  RemarksReferens
Energy gaps, Egindirect
      (Δ conduction band min)
Si1-xGex   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, Egindirect
      (L conduction band min)
Si1-xGex   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   EΓ1 Si1-xGex  4175 - (2814 ± 55)x meV70 K; 0 < x < 0.3, Linear Fitting
Si1-xGex films on Si substrates
Ebner et al. (1998)
  Si (x=0)   3.4 eV 300 Ksee Si. Band structure
 Ge (x=1)  0.8 eV 300 Ksee Ge. Band structure
Eg(Γ-X)Si1-xGex  0.8941 + 0.0421x+ 0.1691x2calculatedKrishnamurti et al. (1983)
Eg(Γ-L)Si1-xGex  0.7596 + 1.0860x+ 0.3306x2calculatedKrishnamurti et al. (1983)
Energy separation EΓ2 Si1-xGex  (3400 ± 7) - (300 ± 40)x meV70 K; 0 < x < 0.3, Linear Fitting
Si1-xGex films on Si substrates
Ebner et al. (1998)
  Si (x=0)   4.2 eV 300 Ksee Si. Band structure
 Ge (x=1)  3.2 eV 300 Ksee Ge. Band structure

Valence band     
Energy of spin-orbital splitting Eso Si1-xGex   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

Si1-xGex ~ 2.8 x 1019cm-3300 K, x < 0.85 
  Si1-xGex ~ 1.0 x 1019cm-3300 K, x > 0.85 
  Si (x=0)   2.8 x 1019cm-3300 K, x = 0.see Si. Band structure
 Ge (x=1)   1.0 x 1019cm-3300 K, x = 1.see Ge. Band structure
     
Effective valence band density of states Si (x=0)   1.8 x 1019cm-3300 K, x = 0.see Si. Band structure

 

Ge (x=1)   0.5 x 1019cm-3300 K, x = 1.see Ge. Band structure

Intrinsic carrier concentration Si1-xGex   see Si1-xGex. Intrinsic carrier concentration
  Si (x=0)  1 x 1010 cm-3300 K, x = 0.see Si. Band structure
 Ge (x=1)  2 x 1013 cm-3300 K, x = 1.see Ge. Band structure
 
Energy Gaps vs. Composition     
E1 Si1-xGex  3452 - (1345 ± 25)x meV70 K; 0 < x < 0.3,
Linear Fitting
Si1-xGex films on Si substrates
Ebner et al. (1998)
E'1 Si1-xGex  5402 ± 25 + (280 ± 120)x meV
E2 (X)Si1-xGex  4351 ± 38 + (210 ± 180)x meV
E2 (Σ)Si1-xGex   4518 ± 129 + (880 ± 600)x meV
Eg(Γ-X)Si1-xGex  0.8941 + 0.0421x+ 0.1691x2calculatedKrishnamurti et al. (1983)
Eg(Γ-L)Si1-xGex  0.7596 + 1.0860x+ 0.3306x2calculatedKrishnamurti et al. (1983)


Band structure

Compositional dependence of band gaps (calculated): ,(F-X) 0.8941 + 0.042 lx+ 0.1691 2 85 g(r-L) 0.7596+1.0860 +0.3306 2 Krishnamurti et al. (1983)
Band structure of Si at 300 K.
Energy gap Eg 1.12 eV
EL 2.0 eV
EX 1.2 eV
EΓ1 3.4 eV
Energy separation (EΓL or EΓ2 )4.2 eV
Energy spin-orbital splitting Eso0.044 eV
Intrinsic carrier concentration1·1010 cm-3
Intrinsic resistivity3.2·105Ω·cm
Effective conduction band density of states3.2·1019 cm-3
Effective valence band density of states1.8·1019 cm-3
see also Silicon. Band structure and carrier concentration
Band structures of Ge. see also
Energy gap Eg0.661 eV
Ex 1.2 eV
Energy separation (EΓ1)0.8 eV
EΓ2 3.22 eV
Energy separation (ΔE>)0.85 eV
Energy spin-orbital splitting Eso0.29 eV
Intrinsic carrier concentration2.0·1013 cm-3
Intrinsic resistivity46 Ω·cm
Effective conduction band density of states1.0·1019 cm-3
Effective valence band density of states5.0·1018 cm-3
see also Germanium. Band structure and carrier concentration
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).
SixGe1-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)
Si1-xGex. Fundamental (indirect) band gap & excitonic band gap at 4.2 K
Squares - band gap of Si1-xGex at 4.2 K (absorption measurements) ;
Dots - excitonic band gap of Si1-xGex at 4.2 K (photoluminescence measurements)
Braunstein et al. (1958) and Weber & Alonso (1989)
Si1-xGex. Composition dependences of several important direct transitions observed.
Kline et al. (1968) and Pickering et al. (1993)
Si1-xGex alloys.Fundamental indirect band gap vs. x of pseudomorphic Si1-xGex (001) alloys:
(a) on Si substrate;
(b) on Si0.5Ge0.5 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 Si1-xGex films on Si substrates at 70 K :

 0 < x < 0.3 RemarksReferens
EΓ1 = 4175 - (2814 ± 55)x meV70 K;
Si1-xGex films on Si substrates
Ebner et al. (1998)
EΓ2 = (3400 ± 7) - (300 ± 40)xmeV 
E1 = 3452 - (1345 ± 25)xmeV
E'1 = 5402 ± 25 + (280 ± 120)xmeV  
E2 (X)= 4351 ± 38 + (210 ± 180)x;meV  
E2 (Σ)= 4518 ± 129 + (880 ± 600)x meV 

Si1-xGexCompositional dependence of band gaps RemarksReferens
Eg(Γ-X)= 0.8941 + 0.0421x+ 0.1691x2  calculatedKrishnamurti et al. (1983)
Eg(Γ-L)= 0.7596 + 1.0860x+ 0.3306x2      


Temperature Dependences

Temperature dependence of energy gap:

x=0 (Si)  Eg = 1.17 -4.73 x 10-4 x T2/(T + 636)  (eV)   see also Si. Band structure and carrier concentration
x=1 (Ge)  Eg = 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 EΓ

x=0 (Si)  EΓ2 = 4.34 - 3.91·10-4·T2/(T+125)  (eV) see also Si. Band structure and carrier concentration
x=1 (Ge)  EΓ1 = 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.
Si1-xGex alloys. Fundamental indirect band gaps vs. temperature at different x.
Braunstein et al.(1958)

Intrinsic carrier concentration:

ni = (Nc·Nv)1/2exp(-Eg/(2kBT))

Si1-xGex 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 Nc

At x < 0.85, Si1-xGex- alloys are considered as "Si-like material:

Nc 4.82 x 1015 · M · (mc/m0)3/2·T3/2 (cm-3) 4.82 x 1015 (mcd/m0)3/2 · T3/2 5.3 · 1015 x T3/2 (cm-3) ,
where M=6 is the number of equivalent valleys in the conduction band.
      mc = 0.32m0 is the effective mass of the density of states in one valley of conduction band.
      mcd = 1.06m0 is the effective mass of density of states.

At0.85 < x < 1.0, Si1-xGex- alloys are considered as "Si-like material:

Nc 4.82 x 1015 · M · (mc/m0)3/2·T3/2 (cm-3) 4.82 x 1015 (mcd/m0)3/2 · T3/2 2 · 1015 x T3/2 (cm-3) ,
where M = 4 is the number of equivalent valleys in the conduction band.
      mc = 0.22m0 is the effective mass of the density of states in one valley of conduction band.
      mcd = 0.55m0 is the effective mass of density of states.

Effective density of states in the valence band Nv

x=0 (Si) Nc 4.82·1015 · (mv/m0)3/2·T3/2      3.5·1015·T3/2 (cm-3),
mv = 0.81m0 is the hole effective mass of the density of states.
see also Si. Band structure and carrier concentration
x=1 (Ge) Nc 4.82·1015 · (mv/m0)3/2·T3/2      9.6·1014·T3/2 (cm-3),
mv = 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 mv. For crude estimations one can use the simplest linear approximation:
mv (x) = (0.81 - 0.47x)m0
see also Effective Masses and Density of States


Dependence on Hydrostatic Pressure

x=0 (Si) Eg=Eg(0)-1.4·10-3P  (eV)  see also Si. Band structure and carrier concentration
x=1 (Ge) Eg = Eg(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 Si1-xGex/Si1-yGey 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 ΔEv between a strained Si1-xGex film and an unstrained cubic Si1-xsGexs substrate [Rieger & Vogi (1993)]:
ΔEv (x,xs) = (0.47 - 0.06x)(x - xs) (eV)
The hydrostatic strain components leads to a change in the band gap of the strained Si1-xGex film according to
ΔEg = (&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
      = (aort - ao)/ao are the perpendicular strain components of the strained Si1-xGex film;
      = (a|| - ao)/ao - in-plane strain components of the strained Si1-xGex 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 RemarksReferens
Si1.72 eV-3.12 eV -2.35 eV 9.16 eV16.14 eVTheoryVan de Walle & Martin (1986)
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 eVTheoryVan de Walle & Martin (1986)
  -2.0 ± 0.5 eV -2.86 ±0.15 eV  16.2 ± 0.4 eVExperiment Laude et al. (1971),
Chandrasekar & Pollak (1977),
Balslev (1966)
Lacking predictions or measurements for Si1-xGex 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: Ev = 1/3Δ0 - 1/2δE001
Light hole: Ev1 = -1/6Δ0 + 1/4δE001 + 1/2(Δ02 + Δ0 δE001+ 9/4δE2001)1/2
Spin-split hole: Ev3 = -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)):
ΔEv2 ) = 2/3&Teta;uΔ ( - )
ΔEv4 ) = -1/3&Teta;uΔ ( - )
where Δ2 are the two electron valleys along the [001] growth direction.
    Δ4 are the four in-plane electron valleys.
    &Teta;uΔ is the relevant conduction band deformation potential for the Δ electrons that define the conduction band minimum in Si-like Si1-xGex (x0.85).

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.
Si1-xGex. 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 Si1-xGex substrate with x = 30%
Schaffler(1997)
Si1-xGex. Contour plots of the conduction ΔEc and valence ΔEv band offsets of pseudomorphic Si1-xGex layers on cubic Si1-xsGexs 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<ΔEc<+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)
Si1-xGex.
Solid lines - Variation of the relevant band edges of a strained Si layer on a cubic Si1-xsGexs substrate .
The dashed lines correspond to the substrate bands.
LH, light holes;
HH, heavy holes;
SO, spin-orbit split holes
Schaffler(1997)
Si1-xGex. Solid lines - Variation of the relevant band edges of a strained Ge layer on a cubic Si1-xsGexs 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, Si1-xGex alloys are considered as "Si-like" material:  RemarksReferens
Effective electron mass
      (longitudinal)ml
0.92mo  Schaffler F.(2001)
Effective electron mass
      (transverse)mt
0.19mo Schaffler F.(2001)
Effective mass of density of states  mcd=M2/3 mc
      (for all valleys of conduction band)
1.06mo Son et al. (1994);
Son et al. (1995)
Effective mass of the density of states mc=(ml+mt2)1/3
      (in one valley of conduction band)
0.32mo  
Effective mass of conductivity  mcc= 3/(1/ml+2/mt)0.26mo  
There are M=6 equivalent valleys in conduction band

At 0.85< x <1, Si1-xGex alloys are considered as "Si-like" material:  RemarksReferens
Effective electron mass
      (longitudinal)ml
0.159mo  Schaffler F.(2001)
Effective electron mass
      (transverse)mt
0.08mo Schaffler F.(2001)
Effective mass of density of states  mcd=M2/3 mc
      (for all valleys of conduction band)
1.55mo Son et al. (1994);
Son et al. (1995)
Effective mass of the density of states mc=(ml+mt2)1/3
      (in one valley of conduction band)
0.22mo  
Effective mass of conductivity  mcc= 3/(1/ml+2/mt)0.12mo  
There are M=4 equivalent valleys in conduction band
Si1-xGex. 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
Ehh,lh = Ev -(0.5 h2k2/mo) {A ± [B2 +(C2/k4)(k2x·k2y+k2x·k2z+k2y·k2z)]1/2 }
Eso = Ev -Δ - (0.5 h2k2/mo) A,
where Δ is the spin-orbit splitting (44 meV in Si, 290 meV in Ge), and k = (kx, ky, kz) is the direction in reciprocal space.
 RemarksReferens
Effective hole masses (heavy) mhh Si (x=0) 0.537 mo 4.2 K see also Si. Effective Masses
 Ge (x=1) 0.33 mo  see also Ge. Effective Masses
 
Effective hole masses (light) mlh Si (x=0)0.153 mo   see also Si. Effective Masses
 Ge (x=1)0.0430 mo  see also Ge. Effective Masses
 
Effective hole masses (spin-orbit-split ) mso Si1-xGex(0.23-0.135x) mo 300 K Schaffler F.(2001)
  Si (x=0)0.234 mo   see also Si. Effective Masses
 Ge (x=1)0.095(7) mo 30 Ksee also Ge. Effective Masses
Si1-xGex. Valence band dispersion along [100] and [110] for Si0.5Ge0.5 on Si(001).
(schematic view)
Schaffler F.(2001)
Si1-xGex. Valence band parameters A, B, and |C| vs. composition x
Schaffler F.(1997)
Si1-xGex. Heavy hole effective mass density of states mhd vs. energy at different x .
Manku & Nathan (1991)
Si1-xGex. Light hole effective mass density of states mhl vs. energy at different x .
Manku & Nathan (1991)
Si1-xGex. Experimental heavy hole cyclotron masses in strained Si1-xGex 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 Si1-xGex