Band structure and carrier concentration of Germanium (Ge)
Band structure and carrier concentration
Basic ParametersTemperature 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 (E_{Γ1})  0.8 eV 
Energy separation (ΔE>)  0.85 eV 
Energy spinorbital splitting  0.29 eV 
Intrinsic carrier concentration  2.0·10^{13} cm^{3} 
Intrinsic resistivity  46 Ω·cm 
Effective conduction band density of states  1.0·10^{19} cm^{3} 
Effective valence band density of states  5.0·10^{18} cm^{3} 
Band structures of Ge.

Temperature Dependences
Temperature dependences of the energy gap:
E_{g} = 0.742 4.8·10^{4}·T^{2}/(T+235) (eV),where T is temperature in degrees K.
Temperature dependence of the direct band gap E_{Γ1}:
E_{Γ}_{1} = 0.89  5.82·10^{4}·T^{2}/(T+296) (eV),Effective density of states in the conduction band:
N_{c} = 4.82·10^{15}·M·[m_{c}/m_{o}]^{3/2}·T^{3/2} (cm^{3}), or N_{c} = 1.98·10^{15}·T^{3/2} (cm^{3})M = 4 is the number of equivalent valleys in the conduction band,
m_{c} = 0.22m_{o} is the effective mass of the density of states in one valley of the conduction band.
Effective density of states in the valence band:
N_{ν} = 9.6·10^{14}T^{3/2} (cm^{3})Dependences on Hydrostatic Pressure
E_{g} = E_{g}(0) + 5.1·10^{3}P (eV)E_{Γ1} = E_{Γ1}(0) + 1.53·10^{2}P (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 ntype Ge:
ΔE_{g} = 8.67·(N_{d}·10^{18})^{1/3} + 8.14·(N_{d}·10^{18})^{1/4} + 4.31·(N_{d}·10^{18})^{1/2} (eV)For ptype Ge:
ΔE_{g} = 8.21·(N_{a}·10^{18})^{1/3} + 9.18·(N_{a}·10^{18})^{1/4} + 5.77·(N_{d}·10^{18})^{1/2} (eV)(Jain and Roulston [1991]).
Effective Masses
Electrons:  
The surfaces of equal energy are ellipsoids  
m_{l} = 1.59m_{o}  
m_{t} = 0.0815m_{o}  
Effective mass of density of states  
m_{c}=(9m_{l}m_{t}^{2})^{1/3}  m_{c}=0.22m_{o} 
Effective mass of conductivity  m_{cc}=0.12m_{o} 
Holes:  
Heavy  m_{h} = 0.33m_{o} 
Light  m_{lp} = 0.043m_{o} 
Splitoff band  m_{so} = 0.084m_{o} 
Effective mass of density of states  m_{v} = 0.34m_{o} 
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 