1
3 Electrons and Holes in Semiconductors
3.1. Introduction
Semiconductors : optimum bandgap
Excited carriers → thermalisation
Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium
electron and hole currents
Chap. 4 : Charge carrier generation, recombination, transport equation
3.2. Basic Concepts
3.2.1. Bonds and bands in crystals
Free electron approximation to band
Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973
Atomic orbitals → molecular orbitals → bands
Valence band : HOMO
Conduction band : LUMO
2
Semiconductors : eVEg 35.0
Semi-metals : eVEg 5.00
Si : sp3 orbital, diamond-like structure
3.2.2. Electrons, holes and conductivity
Semiconductors
At KT 0 , all electrons are in valence band – no conductivity
At elevated temperature,
some electrons in conduction band, and some holes in valence band
3
3.3. Electron States in Semiconductors
3.3.1. Band structure
Electronic states in crystalline
Solving Schrödinger equation in periodic potential (infinite)
Bloch wavefunction
rkk rrk
ii eu, (3.1)
for crystal band i and a wavevector k, which is a good quantum number.
The energy E against wavevector k is called by crystal band structure.
Typical band diagram plotted along the 3 major directions in the reciprocal space.
111,011,001,000 LMX
a)
Conduction Band
Valence Band
Electrons
a)’
Electron in CB
b) b)’
Hole in VB
Fig. 3.4. Electron in CB and Hole in VB
4
Point
a
k
is called the Brillouin zone boundary.
For zinc blende structure,
aaa
L
aa
K
a
X
,0
2
3
2
3
,00
2
,000
3.3.2. Conduction band
At the vicinity of the minimum kE in the conduction band 0ckk , approximation
*
2
0
2
0
2 c
c
c
m
EE
kk
k
(3.2)
where 00 cc EE k and
*
cm is a parameter with dimensions of mass, and defined by
2
2
2*
11
k
kE
m
c
c
(3.3)
is called the parabolic band approximation.
eg.
*
cm for GaAs 0
* 067.0 mmc , for Si 0
*
// 19.0 mmc 0
* 92.0 mmc ,
where 0m is the static electron mass.
*
cm is determined by the atomic potentials, should actually be a tensor,
but is usually treated as a scalar, and is called the effective mass of electron.
Momentum of an electron and the applied force are related as
dt
d
dt
d kp
F (3.4)
Following eq. (3.2), the velocity and momentum can be obtained as
*
01
c
c
m
E
kk
kv k
(3.5)
0
*
ccm kkvp (3.6)
3.3.3. Valence band
At the vicinity of the maximum kE in the valence band 0vkk
*
2
0
2
0
2 v
v
v
m
EE
kk
k
(3.7)
5
where 00 vv EE k and
*
vm is the effective mass of hole.
*
01
v
v
m
E
kk
kv k
(3.8)
0
*
vvm kkvp (3.9)
2
2
2*
11
k
kE
m
v
v
(3.10)
eg.
*
vm for GaAs 0
* 5.0 mmv , for Si 0
* 54.0 mmv ,
*
cm is the curvature of the band.
3.3.4. Direct and indirect band gaps
Whether or not minimum of conduction band and maximum of valence band occur at the same
wavevector 00 vc kkk
Reciprocal lattices for (a) fcc and (b) bcc crystals
6
When the band gap is indirect, transition requires involvement of phonons for conservation of
momentum.
(Sze)
3.3.5. Density of states
According to Pauli’s exclusion principle, each quantum state only supports one electron (two,
considering spins). Also, from Heisenberg’s uncertainty principle,
hxp ~ ∴ hxk ~ ∴ 2~xk
Therefore, for a crystal of volume LLL
Lx
k
22
~
∴
3
2
L
states in this unit volume.
Hence, the density of electron states per unit crystal volume considering spin is
kkk
3
3
3
2
2
ddg
(3.11)
If the band structure kE is spherically symmetric and isotropic around 0k
dkkkgdg 23 4kk (3.12)
Therefore,
dE
dE
dk
kkgdEEg 24 (3.13)
∴
dE
dk
kEg 2
3
4
2
2
(3.14)
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From eq. (3.2)
*
2
0
2
0
2 c
c
c
m
EE
kk
k
∴ 02
*
2 2
c
c EE
m
k
(3.15)
Similarly, for the valence band (hole)
∴ EE
m
k v
v 02
*
2 2
(3.15’)
Differentiating (3.15)
210
21
2
*2
c
c EE
m
k
∴ 210
21
2
*2
2
1
c
c EE
m
dE
dk
Therefore, from (3.14)
210
23
2
*
2
21
0
23
2
*
3
21
0
21
2
*
02
*
3
21
0
21
2
*
2
3
2
3
2
2
12
2
4
2
2
12
4
2
2
2
2
1
4
2
2
4
2
2
c
c
c
c
c
c
c
c
c
c
c
EE
m
EE
m
EE
m
EE
m
EE
m
k
dE
dk
kEg
(3.16)
Similarly,
210
23
2
*
2
2
2
1
EE
m
Eg v
v
v
(3.17)
Question : Derive the DOS for 2D and 1D materials. (Box 3.3)
Excitons
A bound state of a pair of electron and hole of same k .
The wavefunction can be described as
rkrk ,, heex
Satisfying a hydrogenic effective mass equation
exexex
s
ex E
q
rr
42
2
2
*
2
* : reduced effective mass of e-h pair
exE : binding energy of the exciton
8
22
2
0
0
*
l
Ryd
m
E
s
ex
3.3.6 Electron distribution function
Fermi function in semiconductors
1exp
1
Tk
EE
Ef
B
F
This can be expressed in terms of rk,f although the energy is only dependent on the
wavevector.
krkkkr 33 , dfgdn
The total electron density is given by
k krkkr CB c dfgn
3, (3.20)
k krkkr VB v dfgp
3,1 (3.21)
