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Chem 253, UC, Berkeley
Chem 253B
Crystal Structure
Chem 253C
Electronic Structure
Chem 253, UC, Berkeley
2
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
3
Chem 253, UC, Berkeley
Si NW
Chem 253, UC, Berkeley
4
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
6
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
L
nk
222
)(2 L
n
mEn
7
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.
10
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
15
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
16
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
17
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
18
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
19
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
20
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
21
Chem 253, UC, Berkeley
1st Brillouin Zone
Chem 253, UC, Berkeley
Wigner-Seitz cell
22
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
23
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
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