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Characteristics and Properties

NSM Archive - Band structure and carrier concentration of Silicon (Si)

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

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

Temperature dependence of the direct band gap E_Γ2

E_Γ2 = 4.34 - 3.91·10^-4·T²/(T+125) (eV)

Intrinsic carrier concentration

n_i=(N_c·N_v )^1/2·exp(-Eg/(2k_BT])

Effective density of states in the conduction band

N_c=4.82·10¹⁵·M·(m_c/m_o)^3/2·T^3/2 = 4.82·10¹⁵·M·(m_cd/m_o)^3/2·T^3/2 (cm^-3),
or
N_c=6.2·10¹⁵·T^3/2 (cm^-3),
M = 6 is the number of equivalent valleys in the conduction band.
m_c = 0.36m_o is the effective mass of the density of states in one valley of conduction band.
m_cd = 1.18m_o is the effective mass of the density of states.

Effective density of states in the valence band

N_v = 3.5·10¹⁵·T^3/2 (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

E_g=E_g(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
ΔE_g^el ~ 3.5·10^-8·N_d^1/3 (eV)
(N_d 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 the Forbidden 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.

 

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