Solid state physics 05-the effect of periodic potential

Solid State PhysicsUNIST, Jungwoo Yoo

1. What holds atoms together - interatomic forces (Ch. 1.6)2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET)---------------------------------------------------------------------------------------------------------------(Final)

All about atoms

backstage

All about electrons

Main character

Main applications


The effect of periodic potentials

1. Nearly free electron theory

2. Block theorem and exactly soluble model

2. Insulator, semiconductor, or metal

3. Band structure and optical properties

4. The tight binding approach


The Free Electron Model

In free electron model: the positive ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential. No details of crystal structure No el-ions interaction, No el-el interaction


Nearly Free Electron Theory

Nearly free electron model: Start with free electron model and consider the periodic potential as a small perturbation on the free electron potential. In reality, the potential is quiet large and this perturbation calculation are only qualitatively correct


For 3D


H 22

2ˆ

mH

From the periodic boundary condition,

),,(),,( zyxLzLyLx

1ikLe pL

k2

a

2xk p

L

2yk q

L

2zk r

L

m

k

2

22 )(

2222

2

zyx kkkm

Solution of the schrödinger equation

)(

2/12/1

11),,( zkykxkirki zyxe

Ve

Vzyx

Represent running wave with carrying momentum kp



rkie

~

a

,sin2 an ....3,2,1n

For 1 dimension, ,2an

a

nk


For 1D ,1

),,( ikxeL

zyx m

k

2

22 Represent running wave with car-

rying momentum kp

The dispersion relation gives,

0 a

a

2

a

a

3

a

2a

3

k

The Bragg’s law for the 1D crystal:

sin2d n

ank


)/sin(2)/exp()/exp()(

)/cos(2)/exp()/exp()(

axiaxiaxi

axaxiaxi

For wave number of , We can form two different standing

waves from the two traveling waves )/exp( axia

x

ank

For wavenumber of

Electronic wave become standing waveinstead of running wave



In quantum mechanics, the probability density of a particle in a position x given

by wavefunction as *2 x

Potential energy of is lower than that of the traveling wave,

whereas the potential energy of is higher than that of the traveling

wave.

)()(

),(cos)()( 22

a

xx

)(sin)()( 22

a

xx

The probability density of traveling wave

.const ikxikxx ee



At (Brillion zone boundary), the energy difference between and

introduce energy gap .

ak

)( )(

gE

,cos2

)(

a

x

L

a

x

L

sin2

)(At , a

k

We express the potential energy of an electron in the crystal at point x as

a Fourier series

1

2cos)(

nn a

nxVxV

In mathematics, a Fourier se-ries decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating func-tions,

Normalization constant from

1)()()()(00 LL

dxdx



L

nn

L

g a

x

a

x

a

nxVdx

LxdxVE

0

22

00

22sincos

2cos

2)()()(

100

2cos

2cos

2V

a

x

a

nxVdx

L

L

nn

Only n=1 survive, all other terms are orthogonal to n=1

Forbidden band

First allowed band

Second allowed band

0a

a

k

gE

0a

a

k

Free electron



0a

a

k

Energy gapEnergy band

1VEg

a

2

a

3

a

3

a

2

2VEg Energy gap

Energy band

Energy band

First BrillouinZone

SecondZone

SecondZone

ThirdZone

ThirdZone


Block Theorem

Block have proved the important theorem that the solutions of the Schrödinger equation for a periodic potential must be of a special form:

)exp()()( rkirur kk

Where has the period of the crystal lattice with )(ruk

)()( ruTru kk

T

: lattice translation vec-tor

: Block function

Block theorem: The eigenfunctions of the wave equation for a periodic poten-

tial are the product of a plane wave times a function with the

periodicity of the crystal lattice.

)exp( rki

)(ruk


An exactly-soluble model

)(xV

xb)( ba 0 a ba

0V

Kronig-Penny model of 1D periodic potential

,ˆ H )(2

ˆ2

2

xVxm

H

Periodic potential

Block showed the solution of this type of equation has the following form:

ikxexux )()( )(xu is a periodic function of the lattice

ikAexAx )()( )()0()( baikexbbaxa

III III II II II II



I :

at x = 0, boundary condition gives

xixiI BeAex )(

xxII DeCex )(II :

from Block theorem: )()0()( baikexbbaxa

boundary condition gives

02)(

22

Em

dx

xd

0)(2)(

22

VEm

dx

xd

2

2

mE

2

)(2

EVm

)()()( baikII eba a

at x = a,

)()( xx III

00

)()(

x

II

x

I

dx

xd

dx

xd DCBA

)()( DCBAi a

ax

II

ax

I

dx

xd

dx

xd

)()(

)()( xx III )()()( baikbbaiaiI eDeCeBeAea

)()()( baikbbaiai eDeCeBeAei a

DCBA

)()( DCBAi

)()( baikbbaiai eDeCeBeAe

)()()( baikbbaiai eDeCeBeAei



0

0

0

01111

)()(

)()(

D

C

B

A

eeeeeiei

eeeeee

ii

baikbbaikbaiai

baikbbaikbaiai

Secular Equation: it has non-trivial solution (A,B,C,D) only when the determinant is 0

0

1111

)()(

)()(

baikbbaikbaiai

baikbbaikbaiai

eeeeeiei

eeeeee

ii

)(cos)cos()cosh()sin()sinh(2

22

bakabab




22

bakabab

If we represent the potential by the periodic delta functionthen V0 is very large and b is very small, such a way that V0b is finite.

In this limit, and k 1b bb )sinh( 1)cosh( b

kaaaa

Pcos)cos()sin(

2

2

2 mVabab

P

1cos1 ka

a



a) If P (or Vb) is large, the curve proceeds more steeply, the allowed bands are narrow

b) If P is smaller, the allowed bands become wider.

c) If P g 0,

a free electron

d) If P g ∞,

infinite potential well

kaa coscos

2

2

mE

m

kE

2

22

0sin

a

a

0sin a

nmE

aa 2

2

22

2

a

n

mE

E

(a) (b) (c) (d)


Classification of Crystalline Solids into Metals, Insulators, and Semiconductors

We now consider how the occupation of the electron states in k-space is af-fected by the changes in the relation due to periodic potentials)(k

0a

a

k

Forbidden band

First allowed band

Second allowed band

gE

From periodic boundary condition,

pNa

k2

N possible k values

Each states for the allowed k value can accommodate 2 electrons

One up-spin, the other down-spin



For a solid with monovalent atoms: each atoms contribute one conduction electron

For a ground state (T = 0), electrons are filled up to Fermi energy corresponds to a

k2

Therefore, there is allowed state right above the Fermi energy

0a

a

k

Forbidden band

First allowed band

Second allowed band

gE

F

a2

a2

If electric field applied, the wavenumbers shift with no additional energy we expect metallic behavior for the solids with monovalent atoms a relation near the Fermi energy similar to the free electrons)(k



For a solid with divalent atoms: each atoms contribute two conduction electron

For a ground state (T = 0), electrons are filled up to first Brillouin zonea

k

Therefore, there is no allowed state right above the Fermi energy

At finite T, if is sufficiently small, some electrons can thermally excited into the second Brillouin zone a semiconductor behaviorFor a larger , the materials would continue to act as an insulator at finite T.

a Insulator at T = 0

0a

a

k

Forbidden band

First allowed band

Second allowed band

gEF

gE

gE



For a solid with trivalent atoms: each atoms contribute three conduction electronFor a ground state (T = 0), first Brillouin zone filled up, second Brillouin zone half-filled

a Metal at T = 0Therefore, there is allowed state right above the Fermi energy

In general, for odd-valence atoms, a metallic behavior for even-valence atoms, a insulating or semiconductor behaviorIn fact, number of valence electrons per primitive unit cellPosition of Brillouin zone determined by the periodicity of the lattice poten-tial size of primitive unit cell

0a

a

k

Forbidden band

First allowed band

Second allowed band

gE

F



Now, let’s consider 2D crystal: a square crystal of side L by L with a atomic spacing a in square lattice

The energy gap occur when the electron waves are synchronized with the period-icity of the lattice a case of a boundary of the first Brillouin zone in 2D

akak yx /,/

The constant energy of electron can be de-picted as a concentric circlessince

)(22

22222

yx kkmm

k

xk

yk

ak y

akx



For a solid of monovalent atoms at T = 0

xk

ykFermi surface (line)

ak y

akx



With the crystal potential, the energy inside the first Brillouin zone is lower close to the zone boundary. So the Fermi surface is extended towards the zone boundary as it gets close.

0a

a

k

1

For 1D




xk

yk

ak y

akx




xk

yk

ak y

akx




xk

yk

ak y

akx



With the crystal potential, the energy inside the first Brillouin zone is lower close to the zone boundary. So the energy contour is extended towards the zone boundary as it gets close.

xk

yk

ak y

akx




xk

yk

ak y

akx

Similarly the increased energy outside the zone boundary moves a constant energy contours in towards the boundary




xk

yk

ak y

akx





xk

yk

ak y

akx


Perturbed energy con-tour

2)/2( aThe area of the first Brillouin zone is and the den-sity of running wave states in k-space is

22 )2/()( Lkk

Therefore the first Brillouin zone con-tains

NaLakk 22 )/()/2)((

k-statesa can accommodate 2N elec-

trons



To determine which states are actually occupied for a lattice of divalent atomsWe must satisfy i) we must fill an area of k-space equal to that of the first zone ii) we must fill all levels below some fixed energy

F

xk

yk

ak y

akx

Fermi surface

In this case, the electric field can shift Fermi surface, so the material is metallic



xk

yk

ak y

akx

Fermi surface

E

Effect of electric fields on electrons in bands



Effect of electric fields on electrons in bands

xk

yk

ak y

akx

Fermi surface

E



Effect of magnetic fields on electrons in bands

xk

yk

ak y

akx

B

v

Bve

Electron orbit in k-space



xk

yk

Effect of magnetic fields on hole

B

Near the top of the band

hole orbit in k-space



Typical Fermi surface in 3D

Simple cubic

Face centered cubic

Fermi surface of copper



xk

yk

ak y

akx

For the free electron in 2D, states near the edge of the first Brillouin zone have larger energy than the some of states in second Brillouin zone.

1k

2k21 kk


1D free electron bands

First Brillouin zone

Second zone

Third zone


Second zone

Third zone


Second zone

Third zone


2D free electron bands


Second zone

Third zone

As the effect of periodic potential become larger


Second zone

Third zone


Second zone

Third zone

E


For 2D divalent atoms, if the effect of periodic potential get strong a all electrons are occupied in first Brillouin zone a material become insulator


xk

yk

ak y

akx

Fermi surface


0a

a

k


1VEg

a

2

a

3

a

3

a

2

2VEg Energy gap

Energy band

Energy band

Extended zone scheme



0a

a

k

1VEg

a

2

a

3

a

3

a

2

2VEg

Repeated zone scheme



0a

a

k

1VEg

a

2

a

3

a

3

a

2

2VEg

Reduced zone scheme




Insulator Metal, or semimetal Metal



Direct and indirect transitions

Absorption of light can be direct and indirect transitions and need to satisfy conservation of energy and momen-tum

Rarely happen since the pho-ton momentum is very small

cEk /



Optical properties of metal, insulator, and semiconductor

If bandgap is greater than 3.2 eV a insulator (no visible light can be ab-sorbed) bandgap is less than ~ 2.5 eV a semiconductor

For metal, they can absorb from low energy up to bandwidth


Second zone

Third zone

Bandwidth determine limit of en-ergy of photon they can absorb

Silver has larger bandwidth than the gold and copper


Tight binding model

Consider ion consists of two protons bound together by a single electron2H

R1

1

R1

2

R1 R2

21

bonding

R1 R2

21

Anti-bonding

I) Potential energy: electron density a lower for bonding

2

I) Kinetic energy: a lower for bonding

2

What’s the main origin of bonding ?

electron tends to be delocalizedin order to reduce its kinetic energy

The total wavefunction can be described as a linear combina-tion of each atomic orbital.


Tight binding model

Now, let’s think about one electron in a periodic array of atoms.

We can take wavefunction of electron as a linear combination of electron orbital in each atom in whole crystal

n

nnkk rrar )()( ,

If we take , the wavefunction become a block function)exp(1

, nnk rkiN

a

Then, the expectation value for en-ergy

nmn m

mnkk HrrkiN

rHr )](exp[1

)()(

m

mkkk kitH )exp(

Putting, nmm rr

n

mkk dVrHrkiH )()()exp(

,)()( dVrHr

tdVrHr )()(

If we ignore all integral, except for the case of and nearest neighbor

nm rr


Tight binding model

Tight binding model energy for 1D monoatomic chain

m

mkkk kitH )exp(

two nearest neighbor, with am

)cos(2)( kateetH ikaikakkk

tk 2

tk 2

-a a

Energy Bandwidth:

tW

tdVrHr )()(

The weaker the overlapa the narrower the energy band


Feynman’s coupled mode approach

The coupled mode approach is concerned with the properties of coupled oscil-lating systems like mechanical oscillators (e.g. pendulums), electric circuits, acoustic systems, molecular vibrations, etc

Divide the system up into its components, investigate the properties of the individual components in isolation, and then reach conclusions about the whole system by assuming that the components are weakly coupled to each other

Then, the time dependent Schrödinger eqn. becomes

jj

jj

jj dt

dwiHw ˆ

),()(),( twrtr By separating vari-

ables, The generalized solution can be written as

j

jj twrtr )()(),(

)(2

ˆ 22

rVm

H

Time dependent Schrödinger eqn.

),(ˆ),( trHtrt

i



Then multiply both sides by and integrate over the volume

dVdt

dwidVHw

jjk

jjk

jj ˆ

k

0kj

jk

CdV

jk

jk

if

if

Let’s consider system with two basic states, j=1, 2, then the differential eqn. becomes

2121111 wHwH

dt

dwi

2221212 wHwH

dt

dwi

Therefore, we get for each k j

jkjk wH

dt

dwi

Here, iskjH dVH jk ˆ

If the two states are not coupled. Then the solution becomes02112 HH

,exp 1111

t

HiDw

,exp 2222

t

HiDw

The probability of being state 1 and 2 respec-tively

,2

1

2

1 Dw ,2

2

2

2 Dw



Coupled ? Uncoupled ?

Consider hydrogen molecular ion (two protons + one electron), then we can think about two ground states (1) the state when the electron is in the vicinity of proton1 and occu-pying the lowest energy level, and proton 2 is just alone with no electron of its own. (2) vice versa

+ +

++

(1)

(2)

If protons are brought closer to each other, electron can tunnel through the barrier and jump over from proton 1 to proton 2 and vice versa. This is what we mean by coupling. The two states are not entirely separate.

If two states are weakly coupled, even in the presence of coupling, it is still meaningful to talk about one or the other state. The states influence each other but may preserve their separate entities.

Putting coupling constant as A

AHH 2112

This leads

2101 AwwE

dt

dwi 201

2 wEAwdt

dwi



Finally, we get AEE 0

Assuming weak coupling limit, the solution is of the form

,exp11

t

EiDw

,exp22

t

EiDw

Then we get2101 ADDEED

2012 DEADED

Which have a solution only if 00

0

EEA

AEE

A is tunneling probability, which vary exponentially with distance

E0 consists of the potential and kinetic energies of the electron and of the potential ener-gies of the protons

Ene

rgy

E

d

E0

A a

Ene

rgy

E

d

E0 + A

E0 - A



From time-dependent Schrödinger eqn.

j

jkjk wH

dt

dwi Here, iskjH dVH jk ˆ

111 jjjj AwAwwE

dt

dwi

From j-th atoms, where dVHE jj ˆ1

dVHdVHA jjjj ˆˆ11

Trial solution, /iEtjj eKw

)( 111 jjjj KKAKEEK

j 1j 2j1j2j

a

Trial solution, jkx

j exK )(

)( )()(1

axkaxkkxkx jjjj eeAeEEe

a kaAEE cos21


Tight binding model

Tight binding model energy for 2D monoatomic chain

m

mkkk kitH )exp(

four nearest neighbors ),0(),0,( aam

)cos(2)cos(2 aktaktH yxkkk

For divalent metal in 2D

xk

yk

Fermi surface plotted in a repeated zone scheme


Tight binding model energy for a simple cubic

Tight binding model

m

mkkk kitH )exp(

6 nearest neighbor);0,0,( am

);0,,0( a ),0,0( a

))cos()cos()(cos( akakakt zyxk

Constant energy surface of simple cubic lattice


Tight binding model

Tight binding model energy for bcc

8 nearest neighbor

)2

1cos()

2

1cos()

2

1cos(8 akakakt zyxk

Tight binding model energy for fcc

12 nearest neighbor

))2

1cos()

2

1cos()

2

1cos()

2

1cos()

2

1cos()

2

1(cos(4 akakakakakakt xzzyyxk

Constant energy surface of fcc lattice

m

mkkk kitH )exp(


Effective MassThe effect of periodic potential on the dynamics of the conduction electron wavepackets can be taken into account by using an effective mass in the equations of motion rather than the bare mass .

The deviation of from the free electron mass can be easily appreciated by stating the ratio , which has values slightly above or below 1. This deviation mainly from the interaction between the drifting electrons and the atoms in a crystal. For example, an electron which is accelerated in an electric field might be slowed down slightly because of “collisions” with some atoms. The ratio of is then larger than 1. On the other hand, the electron wave in another crystal might have just the right phase in order that the response to an external electric field is enhanced. In this case, is smaller than 1.

mm

m mmm /

mm /

mm /

dk

d

dk

dvg

1

Fdk

d

dt

dp

dk

d

dt

dp

dk

d

dt

dk

dk

d

dt

dva g

2

2

22

2

22

2

2

2 1111

1

2

22

dk

daF

1

2

22

dk

dm

a

a

For free electron, m

k

2

22

a mm


0a

a

k

Effective Mass

0a

a

k

0a

a

k

dt

dvg

k

1

2

22

dk

dm

Negative represent that states near the top of an energy band is equivalent to the unoc-cupied states behaving like pos-itively charged particles with positive masses.

a this fictitious particles are called holes, which play an im-portant role in the properties of semiconductors

m


SummaryNearly free electron model: Start with free electron model and consider the periodic potential as a small perturbation on the free electron potential. In reality, the potential is quiet large and this perturbation calculation are only qualitatively correct

For 1D ,1

),,( ikxeL

zyx m

k

2

22

The Bragg’s law for the 1D crystal:

sin2d n

ank

ank

For wavenumber of

Electronic wave become standing waveinstead of running wave

0a

a

k

The two standing wave can be repre-sented as

)/sin()(

)/cos()(

ax

ax

),(cos)()( 22

a

xx

)(sin)()( 22

a

xx

.const ikxikxx ee

The probability density of traveling wave

*2 x


Summary

Nearly free electron model

1

2cos)(

nn a

nxVxV

L

nn

L

g a

x

a

x

a

nxVdx

LxdxVE

0

22

00

22sincos

2cos

2)()()(

100

2cos

2cos

2V

a

x

a

nxVdx

L

L

nn

Only n=1 survive, all other terms are orthogonal to n=1

0a

a

k

Forbidden band

First allowed band

Second allowed band

gE

0a

a

k

Free electron


Summary

Block theorem: The eigenfunctions of the wave equation for a periodic po-

tential are the product of a plane wave times a function with

the periodicity of the crystal lattice.

)exp( rki

)(ruk

)exp()()( rkirur kk

)()( ruTru kk

T

: lattice translation vec-tor

Block func-tion:

Where

)(xV

xb)( ba 0 a ba

0V

Kronig-Penny model of 1D periodic potential

III III II II II II


Summary

boundary condition at x = 0, a gives

I : xixiI BeAex )(

xxII DeCex )(II :

02)(

22

Em

dx

xd

0)(2)(

22

VEm

dx

xd

2

2

mE

2

)(2

EVm

)()( xx III

00

)()(

x

II

x

I

dx

xd

dx

xd

ax

II

ax

I

dx

xd

dx

xd

)()(

)()( xx III

DCBA

)()( DCBAi )()( baikbbaiai eDeCeBeAe

)()()( baikbbaiai eDeCeBeAei


22

bakabab


If we represent the potential by the periodic delta functionthen V0 is very large and b is very small, such a way that V0b is finite.

In this limit, and k 1b bb )sinh( 1)cosh( b

kaaaa

Pcos)cos()sin(

2

2

2 mVabab

P

1cos1 ka

Summary


a) If P (or Vb) is large, the curve proceeds more steeply, the allowed bands are narrow

b) If P is smaller, the allowed bands become wider.

c) If P g 0,

a free electron

d) If P g ∞,

infinite potential well

kaa coscos

2

2

mE

m

kE

2

22

0sin

a

a

0sin a

nmE

aa 2

2

22

2

a

n

mE

E

(a) (b) (c) (d)

Summary


Tight binding model

Consider ion consists of two protons bound together by a single electron2H

R1

1

R1

2

R1 R2

21

bonding

R1 R2

21

Anti-bonding

I) Potential energy: electron density a lower for bonding

2

I) Kinetic energy: a lower for bonding

2

What’s the main origin of bonding ?

electron tends to be delocalizedin order to reduce its kinetic energy

The total wavefunction can by described as a linear combina-tion of each atomic orbital.

Summary


For a electron in a periodic potentials, we can take wavefunction of elec-tron as a linear combination of electron orbital in each atom in whole crystal

nnnkk rrar )()( ,

)exp(1

, nnk rkiN

a

If we take , the wavefunction become a block function)exp(1

, nnk rkiN

a

Then, the expectation value for en-ergy

nmn m

mnkk HrrkiN

rHr )](exp[1

)()(

m

mkkk kitH )exp(

Putting, nmm rr

n

mkk dVrHrkiH )()()exp(

,)()( dVrHr

tdVrHr )()(

If we ignore all integral, except for the case of and nearest neighbor

nm rr

Summary


m

mkkk kitH )exp(

Summary

For 1D monoatomic chain

two nearest neighbor, with am

)cos(2)( kateetH ikaikakkk

tk 2

tk 2

-a a

Energy Bandwidth:

tW

tdVrHr )()(

The weaker the overlapa the narrower the energy band


Summary

m

mkkk kitH )exp(

For 2D monoatomic chain

four nearest neighbors ),0(),0,( aam

)cos(2)cos(2 aktaktH yxkkk

For a simple cubic 6 nearest neighbor

);0,0,( am

);0,,0( a ),0,0( a

))cos()cos()(cos( akakakt zyxk For bcc

For fcc

8 nearest neighbor

)2

1cos()

2

1cos()

2

1cos(8 akakakt zyxk

12 nearest neighbor

))2

1cos()

2

1cos()

2

1cos()

2

1cos()

2

1cos()

2

1(cos(4 akakakakakakt xzzyyxk


0a

a

k


1VEg

a

2

a

3

a

3

a

2

2VEg Energy gap

Energy band

Energy band

Extended zone scheme

Summary


0a

a

k

1VEg

a

2

a

3

a

3

a

2

2VEg

Repeated zone scheme

Summary


0a

a

k

1VEg

a

2

a

3

a

3

a

2

2VEg

Reduced zone scheme

Summary


0 a

a

k

gE

F

a2

a2

0 a

a

k

gEF

0 a

a

k

gE

F

Summary

For a solid with trivalent atoms

For a solid with divalent atoms

For a solid with monovalent atoms


Summary

Energy contour in 2D crystal:

xk

yk

ak y

akx


)(22

22222

yx kkmm

k


Summary

Energy contour in 2D crystal:

xk

yk

ak y

akx


)(22

22222

yx kkmm

k

Fermi surface (line)For monovalent atoms at T = 0


xk

yk

ak y

akx

Summary

Effect of periodic potential


Summary

Constant energy surface in 3D

Simple cubic

Face centered cubic

Fermi surface of copper


The effect of periodic potential on the dynamics of the conduction electron wavepackets can be taken into account by using an effective mass in the equations of motion rather than the bare mass .

mm

0a

a

k

0a

a

k

0a

a

k

dt

dvg

k

1

2

22

dk

dm

Negative represent that states near the top of an energy band is equivalnet to the unoc-cupied states behaving like pos-itively charged particles with positive masses.

a this fictitious particles are called holes, which play an im-portant role in the properties of semiconductors

m

Summary

Science

Solid state physics 05-the effect of periodic potential