Atomic Vibrations in Solids: phonons

Photons* and Planck’s black body radiation law

phonons with properties in close analogy to photons

vibrational modes quantized

Goal: understanding the temperature dependence of the lattice contribution to the heat capacity CV

concept of the harmonic solid

when introducing

The harmonic approximation

Consider the interaction potential 1 3( , ..., )Nq q

Let’s perform a Taylor series expansion around the equilibrium positions:2

,

12st j k

j k j k

q qq q

jkA force constant matrix

,

12st jk j k

j k

A q q

j j jq q m

jkjk

j k

AA

m m

2 2

j k k jq q q q

Since

jk kjA A and jk kjjk kj

j k k j

A AA A

m m m m

A

such that 1 TT AT T AT

We can find an orthogonal matrix which diagonalizes 1 TT T

real and symmetric

where

21

22

23

0 ... 0

0

0 N

2,

,

Tj j j k k k j j kjk j k

T AT T A T

With normal coordinates k kjjk

q q T

we diagonalize the quadratic form,

12st jk j k

j k

A q q

j j jjj

q q T

From k k k kk

q T q

,

12st j k j kj k

A q q

,

12st j k j j k kj kj k j k

A T q T q

, , ,

12st j k j j k kj kj k j k

A T q T q

2

,

12st j jk j k

j k

q q

2 212st j j

j

q

Hamiltonian in harmonic approximation can always be transformed into diagonal structure

32 2 2

1

12

N

jj jj

H p q

harmonic oscillator problem with energy eigenvalues3

1

12

N

j jj

E n

problem in complete analogy to the photon gas in a cavity

0j jj

E n E

EZ e

0j j

j

nEe e

1 1 2 20

1 2

...

, ,..

n nE

n n

e e

0 0

31 2

1 1 1 1...1 1 1 1 j

E E

j

e ee e e e

0 ln 1 j

j

U E e

0 1

j

j

j

j

eE

e

0 1j

j

j

Ee

ln ZU

With

up to this point no difference to the photon gas

Difference appears when executing the j-sum over the phonon modes by taking into account phonon dispersion relation

The Einstein model

In the Einstein model j E for all oscillators

23

13 E

TBk/E N

eNU

zero point energy

vTU

vC

Heat capacity:

2

2

13

TBk/E

TBk/EBE

Bve

eTk/kNC

Classical limit

2

2

13

TBk/E

TBk/EBE

Bve

eTk/kNC

1 for

EBTk TBk/E

B

E eTk

2

for

EBTk

0 1 2 30.0

0.5

1.0

CV /3

Nk B

T/TE

•good news: Einstein model explains decrease of Cv for T->0

•bad news: Experiments show

3TCv for T->0

Assumption that all modes have the same frequency E unrealistic

refinement

The Debye model

Some facts about phonon dispersion relations:For details see solid state physics lecture

)k(

wave vector k labels particular phonon mode

1)

2)

3) total # of modes = # of translational degrees of freedom

3Nmodes in 3 dimensions N modes in 1 dimension

.constE

Example: Phonon dispersion of GaAs

data from D. Strauch and B. Dorner, J. Phys.: Condens. Matter 2 ,1457,(1990)

kfor selected high symmetry directions

00

Ud)T,(n)(DUmax

# of modes in , d

Energy of a mode # of excited phonons )T,(n

temperature independentzero point energy

= phonon energy

01j

j

j

U Ue

We evaluate the sum in the general result

via an integration using the concept of density of states:

In contrast to photons here finite # of modes=3N

d)(Dmax

0

total # of phonon modes In a 3D crystal

max

0

3( )D d N

k

vL

vT,1=vT,2=vT

Let us consider dispersion of elastic isotropic medium

Particular branch i: kv i

kd)()(

V)(D k3

32

dkkkd 23 4here

kv)k()k( i

22

i

k

vk

kid

vdk

1

k

ii

kk d

vv)(

)(V)(D 142

2

3 3

2

22 ivV

Taking into account all 3 acoustic branches

33

22

212 TL vvV)(D

D(ω)

00

Ud)T,(n)(DUmax

00

2

332 121

2Ud

evvVU

max

TL

How to determine the cutoff frequency max ?also called Debye frequency DDensity of states of Cu

determined from neutron scattering

2)(D

Nd)(DD

30

choose D such that both curves enclose the same area

withvT

UvC

de

eTk

NCmax

TBk/

TBk/

BDv

02

2

231

9

maxD

energy

BD k/

temperature

D:

Let’s define the Debye temperature D

Tkx

B

Substitution:

dxTkd B

dx

e

exTNkCT/D

x

x

DBv

0

2

43

19

dxe

exTNkCT/D

x

x

DBv

0

2

43

19

Discussion of:

0T dxe

exdxe

exx

xT/D

x

x

02

4

02

4

11

34125v B

D

TC Nk

DT

3/ /4

22

0 0

131

D DT TxD

x

x e dx x dxTe

3v BC Nk

Application of Debye theory for various metals with single fit parameter D