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Chem 253, UC, Berkeley
sp3 Semiconductor
Chem 253, UC, Berkeley
2
Chem 253, UC, Berkeley
2
Four-band Model (Parabolic Approximation)
Chem 253, UC, Berkeley
GaAs
3
Chem 253, UC, Berkeley
Diamond
Chem 253, UC, Berkeley
InP
4
Chem 253, UC, Berkeley
Si
Chem 253, UC, Berkeley
GaP
5
Chem 253, UC, Berkeley
InAs
Chem 253, UC, Berkeley
InSb
6
Chem 253, UC, Berkeley
GaN (Zinc Blende)
Chem 253, UC, Berkeley
GaN (Wurtzite)
7
Chem 253, UC, Berkeley
Bulk semiconductor:
)(2
222
*
2
)(
zyxe
zkykxki
kkkm
E
Ae zyx
2/12/322
)2
(2
)( EmV
dE
dNED
3/12 )3( nkF
Chem 253, UC, Berkeley
Independent Electrons
Free Electron Approximation
8
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
9
Chem 253, UC, Berkeley
])()()[(2
)( 2222
z
z
y
y
x
x
L
n
L
n
L
n
mkE
Bulk semiconductor:
)(2
222
*
2
)(
zyxe
zkykxki
kkkm
E
Ae zyx
2/12/322
)2
(2
)( EmV
dE
dNED
3/12 )3( nkF
Chem 253, UC, Berkeley
Periodic Potentials and Bloch's Theorem
Bloch Wavefunction:
R
RrVrVL )()(
Lattice vector
)()( ruV
er
rik
)()( Rruru periodic part of Bloch function
Bloch’s theorem: the eigenstates of the Hamiltonian above can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais Lattice.
10
Chem 253, UC, Berkeley
Quantum well, 1D confinement, 2D electronic system
d=Lz
zL
mED
2
*)(
dF nk 22
)(28
sin
22*
2
2*
22
)(
yxeze
z
ykxkiz
kkmLm
nhE
ezkA yx
Chem 253, UC, Berkeley
Quantum well, 1D confinement, 2D electronic system
2222
2
2
)(*2*2
/
)2
sin(2
)()(
*2
d
n
mm
kE
dnk
nzk
d
zEdz
zd
m
nn
n
nn
Infinite Potential Well
11
Chem 253, UC, Berkeley
Quantum well, 1D confinement, 2D electronic system
Infinite Potential Well
2222
2
2
)(*2*2
/
)2
sin(2
)()(
*2
d
n
mm
kE
dnk
nzk
d
zEdz
zd
m
nn
n
nn
Chem 253, UC, Berkeley
E
k
d=Lz
)(28
sin
22*
2
2*
22
)(
yxeze
c
ykxkin
kkmLm
nhE
ezkA yx
Quantum No.Integer No.
)11
(2 **2
2
..hez
ogwellg mmL
hEE
8)(28
22*
2
2*
22
yxhzh
v kkmLm
nhE
12
Chem 253, UC, Berkeley
Density of StatesThe number of orbitals/states per unit energy range
dE
dNED )(
3/22222
)3
(22 V
N
mm
kE
2/322
)2
(3
EVN
2/12/322
)2
(2
)( EmV
dE
dNED
Chem 253, UC, Berkeley
2D density of states:
22 )
2(
LV D
Lz
zz L
dkk
LLL
dkk
21
)2
(
22
2
zz L
dEmmEmE
L
dkk
22/1
22)
2(
2
For each subband, in term of energy states per unit volume
zL
mED
2
*)(
dE
dNED )(
m
kkE
2)(
22
13
Chem 253, UC, Berkeley
Quantum Confinement and Dimensionality
Chem 253, UC, Berkeley
2D density of states:
22 )
2(
LV D
Lz
dF
F
nL
Nk
N
L
k
22
2
2
2
22
2)
2(
1
14
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Single-sheet Single layer Few-layers
MoS2, C, BN, TMS2
15
Chem 253, UC, Berkeley
Infinite Potential Well
2222
2
2
)(*2*2
/
)2
sin(2
)()(
*2
d
n
mm
kE
dnk
nzk
d
zEdz
zd
m
nn
n
nn
Definite parity for each state
Chem 253, UC, Berkeley
Optical Transition: quantum well
Light polarized in x-y plane for normal incidence
xyxy
xyxy
rik
enc
rikhnv
ezruV
f
ezruV
i
izfiyfixf
ixfM
'
' )()(1
)()(1
Considering a general transition from thenth hole state to the n’th electron state, wecan write the initial and final quantum wellwave functions in Bloch function form:
16
Chem 253, UC, Berkeley
Selection Rule: Quantum well
Momentum conservation: kxy=kxy’
dzzzhnenM
uxuM
MMM
hnennn
vccv
nncv
)()(*'
0
''
'
Infinite well: dzd
zn
d
zndM
d
dnn )'
sin()sin(2
2/
2/'
Dipole allowed
MCV is the valence- conduction band dipole momentum;
Mnn’ is the electron-hole overlap.
Mnn’ = 1 if n = n’, Mnn’ = 0 otherwise.
Selection rules for infinite quantum well:n = 0
Chem 253, UC, Berkeley
17
Chem 253, UC, Berkeley
h
h
dm 2/
)11
(8 **2
2
..he
ogwellg mmd
hEE
Chem 253, UC, Berkeley
Exciton: 12nm
18
Chem 253, UC, Berkeley
)11
(8 **2
2
..he
ogwellg mmd
hEE
Chem 253, UC, Berkeley
2222
2
2
)(*2*2
/
)2
sin(2
)()(
*2
d
n
mm
kE
dnk
nzk
d
zEdz
zd
m
nn
n
nn
19
Chem 253, UC, Berkeley
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