## Band structure and carrier concentration

Basic Parameters
Temperature Dependencies
Dependence of the Energy Gap on Hydrostatic Pressure
Energy Gap Narrowing at High Doping Levels
Effective Masses
Donors and Acceptors
Most Important Deep Levels Impurities

### Basic Parameters

 Energy gap 1.12 eV Energy separation (EΓL) 4.2 eV Energy spin-orbital splitting 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 structure of Si at 300 K. Eg = 1.12 eV EL = 2.0 eV EX = 1.2 eV Eso = 0.044 eV EΓ1 = 3.4 eV EΓ2 = 4.2 eV

### Temperature Dependences

#### Temperature dependence of the energy gap

Eg = 1.17 - 4.73·10-4·T2/(T+636) (eV),
where T is temperature in degrees K.

#### Temperature dependence of the direct band gap EΓ2

EΓ2 = 4.34 - 3.91·10-4·T2/(T+125) (eV)

#### Intrinsic carrier concentration

ni=(Nc·Nv )1/2·exp(-Eg/(2kBT])

#### Effective density of states in the conduction band

Nc=4.82·1015·M·(mc/mo)3/2·T3/2 = 4.82·1015·M·(mcd/mo)3/2·T3/2 (cm-3),
or
Nc=6.2·1015·T3/2 (cm-3),
M = 6 is the number of equivalent valleys in the conduction band.
mc = 0.36mo is the effective mass of the density of states in one valley of conduction band.
mcd = 1.18mo is the effective mass of the density of states.

#### Effective density of states in the valence band

Nv = 3.5·1015·T3/2 (cm-3).
 The temperature dependence of the intrinsic carrier concentration. (Shur [1990]). Fermi level versus temperature for different concentrations of shallow donors and acceptors. (Grove [1967]).

### Dependence of the Energy Gap on Hydrostatic Pressure

Eg=Eg(0)-1.4·10-3P (eV)

### Energy Gap Narrowing at High Doping Levels

 Electrical and optical energy gap narrowing versus donor doping density. (Van Overstaeten and Mertens [1987]).
For 1017 ≤ N ≤ 3·1017 cm-3
ΔEgel ~ 3.5·10-8·Nd1/3 (eV)
(Nd in cm-3).

### Effective Masses

 Electrons: The surfaces of equal energy are ellipsoids. ml= 0.98mo mt= 0.19mo Effective mass of density of states mc = 0.36mo There are 6 equivalent valleys in the conduction band. mcc= 0.26mo Holes: Heavy mh = 0.49mo Light mlp = 0.16mo Split-off band mso = 0.24mo Effective mass of density of states mv = 0.81mo

### Donors and Acceptors

#### Ionization energies of shallow donors (eV):

 As P Sb 0.054 0.045 0.043

#### Ionization energies of shallow acceptors (eV):

 Al B Ga In 0.072 0.045 0.074 0.157

### Most Important Deep Levels Impurities

 Impurity Type Position in theForbidden group σn (cm2) σp(cm2) Au d Ev+ 0.35 eV 10-15 3.5·10-15 a Ec- 0.55 eV 8·10-17 9.0·10-15 Cu d Ev+ 0.24 eV 3.5·10-20 a Ev+ 0.37 eV a Ev+ 0.52 eV Fe d Ev+ 0.39 eV 2.0·10-17 Ni a Ec- 0.35 eV 7·10-12 a Ev+ 0.23 eV Pt d Ev+ 0.32 eV 5·10-14 ~ 10-15 a Ev+ 0.36 eV a Ec- 0.25 eV Zn a Ev+ 0.32 eV 10-15 10-13 a Ec- 0.5 eV 10-19 10-13
a - acceptor, d - donor.