Band structure and carrier concentration of Germanium (Ge)

Ge - Germanium

Band structure and carrier concentration

Basic Parameters
Temperature Dependences
Dependences on Hydrostatic Pressure
Energy Gap Narrowing at High Doping Levels
Effective Masses and Density of States
Donors and Acceptors

Basic Parameters

Energy gap 0.661 eV
Energy separation (EΓ1) 0.8 eV
Energy separation (ΔE>) 0.85 eV
Energy spin-orbital splitting 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

Band structures of Ge.
Eg= 0.66 eV
Ex= 1.2 eV
EΓ1 = 0.8 eV
EΓ2 = 3.22 eV
ΔE = 0.85 eV
Eso = 0.29 eV

Temperature Dependences

Temperature dependences of the energy gap:

Eg = 0.742- 4.8·10-4·T2/(T+235) (eV),
where T is temperature in degrees K.

Temperature dependence of the direct band gap EΓ1:

EΓ1 = 0.89 - 5.82·10-4·T2/(T+296) (eV),

Effective density of states in the conduction band:

Nc = 4.82·1015·M·[mc/mo]3/2·T3/2 (cm-3), or Nc = 1.98·1015·T3/2 (cm-3)
M = 4 is the number of equivalent valleys in the conduction band,
mc = 0.22mo is the effective mass of the density of states in one valley of the conduction band.

Effective density of states in the valence band:

Nν = 9.6·1014T3/2 (cm-3)
The temperature dependence of the intrinsic carrier concentration ni.
Fermi level versus temperature for different concentrations of shallow donors and acceptors.
Dashed line shows Fermi level dependence versus temperature for intrinsic Ge.

Dependences on Hydrostatic Pressure

Eg = Eg(0) + 5.1·10-3P (eV)
EΓ1 = EΓ1(0) + 1.53·10-2P (eV),
where P is pressure in kbar.

Energy Gap Narrowing at High Doping Levels

Energy gap narrowing versus donor (curve 1) and acceptor (curve 2) doping density. Open curves are experimental values. (Jain and Roulston [1991])

For n-type Ge:

ΔEg = 8.67·(Nd·10-18)1/3 + 8.14·(Nd·10-18)1/4 + 4.31·(Nd·10-18)1/2 (eV)

For p-type Ge:

ΔEg = 8.21·(Na·10-18)1/3 + 9.18·(Na·10-18)1/4 + 5.77·(Nd·10-18)1/2 (eV)
(Jain and Roulston [1991]).

Effective Masses

The surfaces of equal energy are ellipsoids
  ml = 1.59mo
  mt = 0.0815mo
Effective mass of density of states
mc=(9mlmt2)1/3 mc=0.22mo
Effective mass of conductivity mcc=0.12mo
Heavy mh = 0.33mo
Light mlp = 0.043mo
Split-off band mso = 0.084mo
Effective mass of density of states mv = 0.34mo

Donors and Acceptors

Ionization energies of shallow donors (eV):

As P Sb Bi Li
0.014 0.013 0.010 0.013 0.093

Ionization energies of shallow acceptors (eV):

Al B Ga In Tl
0.011 0.011 0.011 0.012 0.013