# IGBT Modules - Technologies, Driver and Application (Second Edition) - page 17

5
E
F
: Fermi energy [eV]
k:
Boltzman constant
K
J 10
38065 .1
23
The result of multiplying the distributions with the densities of states N
C
and N
V
is the
number of electrons andholes in the respective band:
Tk
EE
C
n C
F C
eN)E(FNn
= ⋅
=
Eq. 1.5
Tk
EE
V
p V
V F
eN)E(FNp
= ⋅
=
Eq. 1.6
n: Electron density [cm
-3
]
p: Hole density [cm
-3
]
N
C
: Density of states in the conductionband [cm
-3
]
N
V
: Density of states in the valence band [cm
-3
]
The densities of states themselves describe the number of possible states, expressed
by the quantum numbers, according to the Pauli Exclusion Principle within the energy
level under consideration. These in turn are functions of the temperature and the
followingequations apply:
2
3
2
*
n
C
h
Tkm 2 2 N
⋅ ⋅
⋅π⋅
⋅ =
Eq. 1.7
2
3
2
*
h
V
h
Tkm 2 2 N
⋅ ⋅
⋅π⋅
⋅ =
Eq. 1.8
m*: Effectivemass of electrons or holes [kg]
h: Planck constan
sJ 10
62607 .6
34
The electrons stimulated by the supply of energy and which were able to leave the
valence band, and thereby entered the conduction band leave a corresponding number
of holes in the valence band. This means that the number of electrons and holes
remains in equilibrium. Therefore:
np n
i
==
Eq. 1.9
n
i
: Intrinsic charge carrier concentration [cm
-3
]
can be reformulated by multiplying
by
and considering
The result is the so-called lawof mass action:
7
The effectivemassm
*
describes the apparent mass due to quantummechanic effects of an accelerated particle
within a specific material. For electrons in the conduction band the effective mass is larger and for holes in the
valence band smaller than their actual restmass.
8
Named after theGerman physicist MaxKarl Ernst LudwigPlanck (1858 - 1947).
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