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Chem 253, UC, Berkeley
Chem 253B
Crystal Structure
Chem 253C
Electronic Structure
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Electronic Structures of Solid
References
Ashcroft/Mermin: Chapter 1-3, 8-10Kittel: chapter 6-9
Gersten: Chapter 7, 11
Burdett: chapter 1-3Hoffman: p1-20
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Si NW
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Carrier Mobility
Momentum gained during the mean free flight Momentum lost in a collision
*
*
m
eEv
vmeE
d
d
Drift velocity
Mobility: the ratio of the drift velocity over the applied electric field
*m
e
E
vd
cm 2 V -1 s -1
5
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Independent Electrons
Free Electron Approximation
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
L
nk
222
)(2 L
n
mEn
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Chem 253, UC, Berkeley
22
)(2 L
n
mEn
From 1D to 3D:
z
z
y
y
x
x
L
zn
L
yn
L
xnAr
sinsinsin)(
])()()[(2
)( 2222
z
z
y
y
x
x
L
n
L
n
L
n
mkE
Chem 253, UC, Berkeley
Periodic Boundary Condition
)()( Lxx
EF
KF
K
)()(2
22
rrm
Solution: traveling plane wave
)exp( xikAn Where: nLk 2
m
kkE x
2)(
22
L: periodicity, lattice constant
8
Chem 253, UC, Berkeley
3D Periodic Boundary Condition
)exp( rkiAn Normalization:
VAdrrkirkiAdrnn22* )exp()exp(1
V: unit cell volume
)exp(2/1 rkiVn
2
k
de Broglie wavelength
Chem 253, UC, Berkeley
)(22
)( 222222
zyx kkkmm
kkE
Energy Eigenvalue:
EF
KF
K
Moment Operator:ri
p
ˆ
)()()(ˆ rkrri
rp
Momentum Eigenvalue:
kp
)exp( rkiAn
9
Chem 253, UC, Berkeley With periodic boundary conditions:
1 zzyyxx LikLikLik eee
z
zz
y
yy
x
xx
L
nk
L
nk
L
nk
2
2
2
2D k space:Area per k point:
yx LL
22
3D k space:Area per k point: VLLL zyx
38222
A region of k space of volume will contain: allowedk values.
33 8
)8
(V
V
Chem 253, UC, Berkeley
Reciprocal Lattice
)(2
cba
cba
)(2
cba
acb
)(2
cba
bac
Reciprocal lattice is always one of 14 Bravais Lattice.
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Chem 253, UC, Berkeley
k space density of level: 38
V
Non-interacting electrons: Pauli exclusion principle
Each wave vector k two electronic level (spin up/down)
Fermi wave vector:
Volume enclosed by the Fermi surface:
Fk
3
3
4Fk
Chem 253, UC, Berkeley
# of allowed states within:
# of electrons N:
Electronic density:
VkV
k FF 2
3
3
3
683
4
Vk
N F2
3
3
2
3
3Fk
V
Nn
3/12 )3( nkF
11
Chem 253, UC, Berkeley
Free & independent electron ground state:
Fermi wave vector
Enclosed Fermi sphere
Fermi Surface
Fermi Momentum
Fermi energy
Fermi velocity
3/12 )3( nkF
FF kp
m
kE F
F 2
22
*/ mpv FF
Chem 253, UC, Berkeley
Estimation based on conduction electron density:
3
23 3
3
41
F
sk
rnN
V
Radius of sphere where volumeequals to the volume perconduction electron
ssF rr
k92.1)4/9( 3/1
20 )/(
1.50
ar
eVE
sF
2-3, for many metal
Fermi energy for metallic elements: 1.5 –15 eVFermi temperature:
Kark
ET
sB
FF
42
0
10)/(
2.58
m
kE F
F 2
22
12
Chem 253, UC, Berkeley
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
13
Chem 253, UC, Berkeley
Quantum Confinement and Dimensionality
Chem 253, UC, Berkeley
Fermi-Dirac distribution:
1]/)exp[(
1)(
TkEEEf
BF
14
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Nearly Free Electron Model
Adding small perturbation by the periodic potential of the ionic cores
EF
KF
K
m
kkE x
2)(
22
Chem 253, UC, Berkeley
Periodic Boundary Condition
)()( Lxx
EF
KF
K
)()(2
22
rrm
Solution: traveling plane wave
)exp( xikAn Where: nLk 2
m
kkE x
2)(
22
Dispersion Curve
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Chem 253, UC, Berkeley
3D Periodic Boundary Condition
)exp( rkiAn Normalization:
VAdrrkirkiAdrnn22* )exp()exp(1
V: unit cell volume
)exp(2/1 rkiVn
2
k
de Broglie wavelength
Chem 253, UC, Berkeley
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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.
Chem 253, UC, BerkeleyBragg reflection of electron waves in crystal is the cause of the energy gap.
First Bragg reflection:
Other gap:
a
a
n
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Reciprocal Lattice
n
'nd
R
cnbnanR 321
1)( '
kkRieLaue Condition
Reciprocal lattice vector
For all R in the Bravais Lattice
'k
k
kkK'
1 RiKe
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Chem 253, UC, Berkeley
For 1D Lattice:Reciprocal lattice vector: n
aK
2
'k
k
Diffraction Condition: na
Kk
2
1
Can be extended to 3D
kkK'
Chem 253, UC, Berkeley
Bragg reflection of electron waves in crystal is the cause of the energy gap.
First Bragg reflection:
Other gap:
a
a
n
First Brillouin Zone
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1st Brillouin Zone
Chem 253, UC, Berkeley
Wigner-Seitz cell
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Chem 253, UC, Berkeley
The wavefunction at are not traveling wave of free electrons:
Instead: equal parts of the waves traveling to the left and right
A wave travels neither to the left nor to the right is a standing wave.
a
)exp()exp( xa
iikx
Chem 253, UC, Berkeley
Two different standing waves:
xa
xa
ixa
i
xa
xa
ixa
i
sin2)exp()exp()(
cos2)exp()exp()(
Probability density: 2
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Chem 253, UC, Berkeley
Pile electron on the core ionslower energy
Pile electron between the core ionshigher energy
Chem 253, UC, Berkeley
Extended zone scheme reduced zone scheme