chem 253b crystal structure chem 253c electronic...

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Chem 253, UC, Berkeley

Chem 253B

Crystal Structure

Chem 253C

Electronic Structure


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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


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Si NW


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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

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Independent Electrons

Free Electron Approximation

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L

nk

222

)(2 L

n

mEn

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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


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

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3D Periodic Boundary Condition

)exp( rkiAn Normalization:

VAdrrkirkiAdrnn22* )exp()exp(1

V: unit cell volume

)exp(2/1 rkiVn

2

k

de Broglie wavelength


)(22

)( 222222

zyx kkkmm

kkE

Energy Eigenvalue:

EF

KF

K

Moment Operator:ri

p

ˆ

)()()(ˆ rkrri

rp

Momentum Eigenvalue:

kp

)exp( rkiAn

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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


Reciprocal Lattice

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

Reciprocal lattice is always one of 14 Bravais Lattice.

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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


# 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

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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


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

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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


Quantum Confinement and Dimensionality


Fermi-Dirac distribution:

1]/)exp[(

1)(

TkEEEf

BF

14



15



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Nearly Free Electron Model

Adding small perturbation by the periodic potential of the ionic cores

EF

KF

K

m

kkE x

2)(

22


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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3D Periodic Boundary Condition

)exp( rkiAn Normalization:

VAdrrkirkiAdrnn22* )exp()exp(1

V: unit cell volume

)exp(2/1 rkiVn

2

k

de Broglie wavelength


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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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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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For 1D Lattice:Reciprocal lattice vector: n

aK

2

'k

k

Diffraction Condition: na

Kk

2

1

Can be extended to 3D

kkK'


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


Wigner-Seitz cell

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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


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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Pile electron on the core ionslower energy

Pile electron between the core ionshigher energy


Extended zone scheme reduced zone scheme

chem 253b crystal structure chem 253c electronic...

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