PVMD Miro Zeman Delft University of Technology Equilibrium - - PowerPoint PPT Presentation
Carrier concentrations PVMD Miro Zeman Delft University of Technology Equilibrium Equilibrium The unperturbed state of a system, to which no external voltage, magnetic field, illumination, mechanical stress, or other perturbing forces are
Delft University of Technology
Carrier concentrations
Miro Zeman
Equilibrium
Equilibrium
The unperturbed state of a system, to which no external voltage, magnetic field, illumination, mechanical stress, or other perturbing forces are applied. ▪ At thermal equilibrium the observable parameters of a semiconductor do not change with time
Density of energy states function, g(E)
Describes the number of allowed energy states for electrons per unit volume and energy.
Conduction band Valence band
Arno H.M. Smets et al., Solar energy, 1st edition, page 53
Band gap energy
Density of energy states (DOS)
EC EV EG
E
EC EV EG
E
g(E) gC gV
EC EV EG
E
g(E) gC gV
Density of energy states function
Arno H.M. Smets et al., Solar energy, 1st edition, page 52
Conduction band Valence band
C n C
E E h m E g
2 3 2 *
2 4 ) (
3 * 2 2
2 ( ) 4
p V V
m g E E E h
gC DOS in conduction band E Electron energy EC Lowest energy level of CB mn Electron’s mass h Planck’s constant gV DOS in valence band E Electron energy EV Highest energy level of VB mp Hole’s mass h Planck’s constant
Occupation function, f (E)
Fermi-Dirac distribution function, f (E)
Gives the probability that a given available electron-energy state will be occupied at a given temperature.
Fermi-Dirac distribution function, f(E)
0.0 0.1 0.2 0.3 0.0 0.2 0.4 0.6 0.8 1.0
T= 400 K T= 300 K T= 200 K T= 100 K f(E) E-EF (eV) T= 0 K
▪ Temperature dependence
0.0 0.1 0.2 0.3 0.0 0.2 0.4 0.6 0.8 1.0
T= 100 K f(E) E-EF (eV) T= 0 K
0.0 0.1 0.2 0.3 0.0 0.2 0.4 0.6 0.8 1.0
f(E) E-EF (eV) T= 0 K
Fermi Level
Fermi level, EF
The total average energy of an electron related to its electro- chemical potential in a material. ▪ Energy level for which f(E) =
1 2
▪ EF in intrinsic semiconductor is approximately in the middle of the band gap
EC EV
EF = EFi
Boltzmann Approximation
When
EC EV
EF
Carrier concentration
Band diagram
EC EV EG
E
EC EV EG
E
g(E) gC gV
Density of states
f(E)
Fermi-Dirac distribution
EC EV
E
1/2
f(EF)=1/2
EF
Charge carrier densities
n (E ) and p (E )
EC EV
E
EF
Charge carrier concentration
Arno H.M. Smets et al., Solar energy, 1st edition, page 54
▪ Holes in the valence band ▪ Electrons in the conduction band
E EC n (E ) and p (E ) EF EV
Charge carrier concentration
Electrons in the CB Holes in the VB
Arno H.M. Smets et al., Solar energy, 1st edition, page 54
Effective density of states
Effective densities of valence band states Effective densities of conduction band states
Arno H.M. Smets et al., Solar energy, 1st edition, page 54
Intrinsic concentration of charge carriers ni
For an intrinsic semiconductor in equilibrium conditions
Nc =3.22×1019 cm-3 (effective density of CB states) Nv= 1.83×1019 cm-3 (effective density VB states) Eg= 1.12 eV (bandgap energy for c-Si) kB= 8.6×10-5 eV/K (Boltzman constant) T = temperature in K
ni (300K) ≈ 1×1010 cm-3
Arno H.M. Smets et al., Solar energy, 1st edition, page 54
n = p = ni
np = ni
2 = NCNV exp EV - EC
kBT æ è ç ö ø ÷ = NCNV exp - Eg kBT æ è ç ö ø ÷
Intrinsic carrier concentration temperature dependence