Solid state physics 06-waves in crystals

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


Waves in Crystals

- Elastic scattering of waves by a crystal- Wavelike normal modes - Block's theorem- Normal modes, reciprocal lattice, brillouin zone


The Bragg law identifies the angles of the incident radiation relative to the lat-tice planes for which diffraction peaks occur, but it gives no information on the intensities of the diffracted beams

Elastic Scattering of Waves by a Crystal

r

k

o

2

R

rR

'k

Crystal

Incident radiation

Detector

)](exp[0 trkiA

X-ray, electron, neutron

Elastic scattering a kinetic energy is conserveda direction of propagation is modified

kk

'

)'( kkK

For the weak elastic scattering, the contribution of the atom at to the scattered wave at a detector at a large distance from the atom can be written as the product of three factors

r

rR

Incident wave Atomic from factor

Amplitude decrease and phase change associated with a point source at the position of the atom

rR

efeAA

rRiktrki

r

)(

0



For the wave at a distant detector, wavevector is approximately parallel to both and

R

'k

rR

rkkRrkRkrRkrRk ''')('

rR

efeAA

rRiktrki

r

)(

0

rKitkRi

r efR

eAA

)(

0

kkK

'

Scattering vector

Same for all atoms

Amplitude of the scattered wave from whole atoms in the crystal is proportional to

n

rKin

nefA

Sum over all atoms in the crystal

nf is the atomic form factor of the nth atom



n

rKin

nefA

nrThe position of nth atom can be written

pln rrr

lr : the position of the lattice point for the atom n

pr: the position of the atom relative to the lattice point

sum over lattice point

sum over atoms in ba-

sisDetermine the di-rection for which diffraction occurs

Determine the rela-tive intensities of

the diffracted beams

(structure factor)

pr

\\

pln rrr

lrOrigin

Basis of atoms as-sociated with each

lattice point

p

rKip

l

rKi pl efeA



lrcan be represented with lattice vectors

cwbvaurl

w

wcKi

v

vbKi

u

uaKi

n

rKi eeee l

A large scattering amplitude is obtained when the contributions from all the lat-tice points are in phase and this is the case if

lcK

kbK

haK

2

2

2

lkh ,, are integers

Laue conditions for dif-fraction

321 NNNeeeew

wcKi

v

vbKi

u

uaKi

n

rKi l are the # of lattice spac-ings in the x,y,z directions re-spectively

321 NNN

,321 NNNN # of primitive unit cell

Direction of the diffracted beams are given by the set of vectors , which sat-isfy Laue conditions

K

The values of the scattering vector that satisfy the Laue conditions lie on a regular lattice in the space of reciprocal space.

K



Reciprocal lattice vector:

All the points of the reciprocal lattice can be generated from three primitive re-ciprocal lattice vectors and by following translation vectors ba

,

*,** clbkahGhkl

lkh ,, are integers

,)(

)(2*

cba

cba

,)(

)(2*

cba

acb

)(

)(2*

cba

bac

where

hklGK

Then, for , it satisfy Laue conditions

For example,

haclbkahaGaK hkl 2*)**(

,2* aa

0** acab

*,*,*, cba

are perpendiculareach other

Therefore, the scattering vector of each diffracted beam corresponds to a point in the reciprocal lattice , therefore we can use integers (hkl) to label that beam.

This diffracted beams of (hkl) are associated with lattice planes of Miller indices (hkl)

hklGK

*c

lcK

kbK

haK

2

2

2

lkh ,, are integers

Laue condi-tions

for diffraction


Laue methods

The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal.

The diffracted beams form arrays of spots, that lie on curves on the film

The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the val-ues of d and θ involved

The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident beam. Laue method is often used to determine the orientation of single crystals by means of illuminating the crystal with a continuos spectrum of X-rays

http://jcrystal.com/steffenweber/JAVA/jlaue/jlaue.html


X-RayFilmSingle

Crystal X-Ray Film

SingleCrystal

Transmission laue method

Back reflection laue method

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{



Reciprocal lattice vector:Now, let’s see the relationship between the reciprocal lattice vector and the set of lattice planes with Miller indices (hkl)

hklG

a

d

la /

ka /

ha /

bc

A vector perpendicular to the planes with length equal to the plane spacing satisfies

d

2/// dlcdkbdhad

since the component of along the direction is

ha /

d

d

2

2

2

ldcd

kdbd

hdad

similar to the scattering vec-tor

hklGK

hklGd

clbkahd

d

2*)**(

2

22

The reciprocal lattice vector is perpendicu-lar to the lattice planes with Miller indices (hkl) and has a length , where d is the spacing of the planes.

hklG

d/2



Reciprocal lattice vector:

(hkl) plane

k

'k

K

K

The scattering vector is perpendicular to the planes and thus parallel to the vector

The magnitude of is equal to that of if

K

hklG

hklG

Bragg law for first-order diffrac-tion

dk /2sin2

sin2da

Bragg and Laue formulations of the conditions for diffraction are thus com-pletely equivalent as is use of reciprocal lattice vectors and Miller indices to label the diffracted beams


Elastic Scattering of Waves by a CrystalExamples of Reciprocal lattice

For simple cubic: ,iaa

,jab

kac ˆ

,ˆ2

* ia

a

,ˆ2

* ja

b

ka

c ˆ2*

The reciprocal lattice is simple cubic with a unit cell of side a

2

For face-centered cubic:),ˆˆ(

2kj

aa

),ˆˆ(2

ika

b

),ˆˆ(2

jia

c

),ˆˆˆ(2

* kjia

a

),ˆˆˆ(2

* kjia

b

),ˆˆˆ(2

* kjia

c

The reciprocal lattice is body-centered cubic with a unit cell of side a

4

For body-centered cubic:),ˆˆˆ(

2* kji

aa

),ˆˆˆ(2

* kjia

b

),ˆˆˆ(2

* kjia

c

),ˆˆ(2

* kja

a

),ˆˆ(2

* ika

b

),ˆˆ(2

* jia

c

The reciprocal lattice is face-centered cubic with a unit cell of side a

4

For hexagonal close-packed structure:

,iaa

),ˆ2

3

2

ˆ( j

iab

,kac

),ˆ2

1ˆ2

3(3

4* ji

aa

,ˆ3

4* j

ab

,ˆ

2* k

cc

The reciprocal lattice is hexagonal lattice


Elastic Scattering of Waves by a CrystalThe Structure Factor



sis

p

rKip

l

rKi pl efeA

p

rKip

pefS

Structure fac-tor:

fS a

If basis consists of one atom on each lattice point 0pr

a intensity of the diffracted peaks depends only on the variation of the atomic form factor f, which is usually smooth and monotonic so that neighboring diffraction peaks have similar intensities.


The Structure Factor


For the hexagonal colse-packed structure,

basis of two identical atoms at

,01 r

cbar2

1

3

1

3

22

)]

2

1

3

1

3

2(*)**(exp[)0exp( cbaclbkahiifS

)]

3

2

3

4(exp[1 lkhif

))](

3

1)(exp[1 hklkhif

For the diffracted beam of (001),

The intensity of diffracted beam is propotional to 2

S

Likewise, some of the possible diffracted beams of are absent because of destructive interference of the scattering form the two atoms in the basis. This is based on the assumption that both atoms in the basis have identical form factors. Although the atoms are chemically identical, their environments within the crystal are different; electron state within the atoms are distorted slightly by the neighboring atoms and this distortion is reflected in the angular dependence of f. Thus the form factors of the two atoms differ slightly and very weak diffracted beams do occur in the ‘forbidden’ directions

hklGK

a intensity vanishes.a S = 0

2SThere are four possible value for for the case of

hcp.

P.11.3



For the face centered cubic structure, can be regarded as a simple cubic with a four basis

,01 r

),(2

12 bar

),(

2

12 cbr

),(

2

12 acr

]exp[]exp[]exp[]exp[ 4321 rGirGirGirGifS hklhklhklhkl

)()()(1 hlilkikhi eeef *,** clbkahGhkl

a/2

a/4

0

4 fS

when h, k and l are all odd or all evenotherwise

This is bcc reciprocal lattice of cube size obtained from primitive unit cell of the fcc lat-tice

a the same structure factor can be obtained using primitive unit cell

a/4



For the diamond structure, fcc with two basis ,01 r

)(

4

12 cbar

]exp[]exp[ 21 rGirGifS hklhkl

)](

2

1exp[1 lkhif

From fcc structure, only when h, k and l are all odd or all even, diffracted beam survive. 22

2 fS i) For h, k and l are all odd,

i) For h, k and l are all even,

0

4 22 f

Sif h+k+l is a multiple of 4,if h+k+l is not a multiple of 4,

But, there is weak diffraction beam from forbidden direction, such as (2 2 2) a two basis does not have identical atomic form factor, two atoms have dif-ferent orientation for tetrahedral covalent bonds

For the sodium chloride structure, fcc with two basis ,01 r

ar

2

12

hieffS

2

2

)(

)(

ff

ffS

for h, k, and l all odd

for h, k, and l all even

(111) diffracted beam is forbidden from KCla K+ and Cl- ions are identical

ff


Details of the structure can also be deduced from diffraction pattern

Ex. NaCl, KCl

NaCl shows weak but clear (111) peaksBut, absence of (111) peaks for KCl

K+ and Cl- both have argon electron shell a nearly same strength of x-ray scatteringBut, Na+ and Cl- have different shell structure

(111) plane: only Cl-

(222) plane: only Na+

For KCl: reflection from (222) plane induces destructive interference


Block Theorem

Block have 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


Periodicity of the Dispersion Relation

The normal modes of the waves in the crystal, which described by the wave vector k is equivalent to the waves with wavevector k’ related to k by

Where is any vector of the reciprocal lattice of the crystal as given by

Consider any lattice point, cwbvaurl

G

Then, )(2 wlvkuhrG l

lll rkirGirki eee ' lrkiwlvkuhi ee

)(2 lrkie

1

This result can be extended to any point r

*,** clbkahGhkl

)exp()()( rkirur kk

For Block wavefunction,

For any normal mode k values, normal mode with k’ is also equivalent waves

Gkk

' )()( ' rr kk

aFor

Gkk

'



1) For any branch of the dispersion relation, the frequency is periodic in k-space with the same periodicity as the reciprocal lattice

2) Any mode can be represented by a wavevector inside a single primitive unit cell of the reciprocal lattice

3) The number of normal modes associated with any branch of the dispersion relation is equal to the number Nc of the primitive unit cells in the crystal.

Possible k vectors are distributed uniformly in k-space with a density And all distinct modes can be represented by a k vector inside one primitive cell of the reciprocal lattice which has a volume cvcba /)2(*)*(* 3

ccc

Nv

V

v

V

3

3

)2(

)2(


Gkk

'

3)2/( V


0a

a

k

Energy gapEnergy band

1VEg

a

2a

3a

3

a

2

2VEg Energy gap

Energy band

Energy band

Extended zone scheme

Classification of Crystalline Solids into Metals, Insulators, and Semiconductors


0a

a

k

1VEg

a

2a

3a

3

a

2

2VEg

Repeated zone scheme



0a

a

k

1VEg

a

2a

3a

3

a

2

2VEg

Reduced zone scheme



Brillouin ZoneIt is useful to split k-space into Brillouin zones with boundaries at the k values for which Bragg diffraction of the electron wave occurs.

The Bragg diffraction produces standing waves rather than running waves at the zone boundaries, with the consequent vanishing of the group velocity of wavelike normal modes.

The energy gaps, produces by a periodic lattice potential, in the free electron dispersion curve also appear at the Brillouin zone boundaries

For 1-d, the nth Brillouin zone are given in terms of the lattice spacing a by

ankan //)1(

Here, the first Brillouin zone plays a special role in that it is a primitive unit cell of the re-ciprocal lattice and is the unit cell normally used when dispersion curves from different branches of the dispersion curve are plotted in the same cell.

For 2, and 3 d, the Brillouin zone boundaries to be given by the wavevectors k that satisfy the diffraction condition.

Gkk

'

a 222 2' GGkkk

a 22 GGk

22' kk since

a22 GGk

G

k2

G


Brillouin Zone


Brillouin Zone

1 22

2

2 33

3

3

3 3

3

3

4

4 4

4

4

4

4

4

4

4 4


Brillouin Zone

Contours of equal electron energy superimposed on the Brillouin zone boundaries (Extended zone scheme)

Reduced zone scheme for the dispersion relation in 2DThe second zone is remapped into the first zone by translation through the reciprocal lattice vectors

The energy contour in the first zone

The energy contour of the 2nd zone remapped into the 1st zone


Symbol Description

Γ Center of the Brillouin zone

Simple cube

M Center of an edge

R Corner point

X Center of a face

Face-centered cubic

KMiddle of an edge joining two hexagonal faces

L Center of a hexagonal face

UMiddle of an edge joining a hexagonal and a square face

W Corner point

X Center of a square face

Body-centered cubic

H Corner point joining four edges

N Center of a face

P Corner point joining three edges

Hexagonal

A Center of a hexagonal face

H Corner point

KMiddle of an edge joining two rectangu-lar faces

LMiddle of an edge joining a hexagonal and a rectangular face

M Center of a rectangular face

Brillouin Zone


Energy Band

The energy bands of the free electrons for the bcc structure

The energy bands of the free electrons for the fcc structure


Energy Band

The energy bands of copper (fcc) The energy bands of aluminum (fcc)


Energy Band

The energy bands of silicon (di-amond-cubic crystal structure) The energy bands of GaAs (zinc-

blende crystal structure)


Energy Band

Graphene

,ˆ2

1ˆ2

31

jiaa

,ˆ2

1ˆ2

32

jiaa

,ˆ2ˆ

3

21

j

ai

ab

,ˆ2ˆ

3

22

j

ai

ab

kaK

haK

2

2

2

1

a

,2

3cos4

2

3cos

2

3cos41)( 2

akakaktk yyx

From tight binding model


Summary



sis

p

rKip

l

rKi pl efeA


lcK

kbK

haK

2

2

2

lkh ,, are integers

Laue conditions for dif-fraction

hklGK

Then, for , it satisfy Laue conditions

*,** clbkahGhkl

lkh ,, are integers

,)(

)(2*

cba

cba

,)(

)(2*

cba

acb

)(

)(2*

cba

bac

Where, is reciprocal lattice vector ,hklG


Summary



sis

p

rKip

l

rKi pl efeA


p

rKip

pefS

Structure fac-tor:


Summary


1) For any branch of the dispersion relation, the frequency is periodic in k-space with the same periodicity as the reciprocal lattice

2) Any mode can be represented by a wavevector inside a single primitive unit cell of the reciprocal lattice

3) The number of normal modes associated with any branch of the dispersion relation is equal to the number Nc of the primitive unit cells in the crystal.


Gkk

'


Summary

Brillouin Zone

It is useful to split k-space into Brillouin zones with boundaries at the k values for which Bragg diffraction of the electron wave occurs.

The energy gaps, produces by a periodic lattice potential, in the free electron dispersion curve also appear at the Brillouin zone boundaries

Here, the first Brillouin zone plays a special role in that it is a primitive unit cell of the re-ciprocal lattice and is the unit cell normally used when dispersion curves from different branches of the dispersion curve are plotted in the same cell.

For 2, and 3 d, the Brillouin zone boundaries to be given by the wavevectors k that satisfy the diffraction condition.

Gkk

'

a22 GGk

G

k2

G


1 22

2

2 33

3

3

3 3

3

3

4

4 4

4

4

4

4

4

4

4 4

Summary

Brillouin Zone


Contours of equal electron energy superimposed on the Brillouin zone boundaries (Extended zone scheme)

Reduced zone scheme for the dispersion relation in 2DThe second zone is remapped into the first zone by translation through the reciprocal lattice vectors

The energy contour in the first zone

The energy contour of the 2nd zone remapped into the 1st zone

Summary

Science

Solid state physics 06-waves in crystals