15. Phonons

Institute of Solid State PhysicsTechnische Universität Graz

15. Phonons

May 15, 2018

Normal modes

The motion of the atoms can be described in terms of normal modes.

In a normal mode, all atoms oscillate at the same frequency .

The energy in a normal mode is quantized,

n is the number of phonons in that normal mode.

12( ).E n

Linear Chain

2

1 1 1 12 ( ) ( ) ( 2 )ss s s s s s s

d um C u u C u u C u u udt

21

22

23

24

25

26

2 0 0 02 0 0 0

0 2 0 0=0

0 0 2 00 0 0 2

0 0 0 2

AC m C CAC C m CAC C m CAC C m CAC C m CAC C C m

a

s = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3

Assume every atom oscillates with the same frequencyi t

s su A e

2 12 I T T 0.C m C A

Normal modes are eigen functions of T

solutions are eigenfunctions of the translation operator

a

( 1 ) ( )i k s a t ika i ksa t ikas k k sTu A e e A e e u

s = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3

N atoms, N normal modes, N eigenvectors of the translation operator, N allowed values of k in the first Brillouin zone.

( )iksa i t i ksa ts k ku A e e A e

solutions:

4 sin2

C kam

2

1 12 ( 2 )ss s s

d um C u u udt

( )i ksa ts ku A e

1 12

2

2

( 2 )( 2 )

2 (1 cos( ))

i k s a t i k s a ti ksa t i ksa t

ika ika

me C e e em C e em C ka

2 1sin 1 cos2 2ka ka

a

Linear Chains = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3

Linear Chain - dispersion relation

4 sin2

C kam

2

1 12 ( 2 )ss s s

d um C u u udt

( )i ksa ts ku A e

C am

speed of sound =

Max. freq.

Linear Chain - density of states

Determine the density of states numerically 4 sin2

C kam

Linear Chain - density of states

for every k calculate the frequency

4 sin2

C kam

4 sin2

C kam

1( )D k

cos2

C kad a dkm

( ) ( ) dkD D kd

2

1

14

DC mam C

This case is an exception where the density of states can be determined analytically.

density of states

D()

van Hove singularity

2

1

14

DC mam C

4 sin2

C kam

1( )D k

cos2

C kad a dkm

( ) ( )D k dk D d

maximum frequency

Linear chain M1 and M2

2

1 12 ( 2 )ss s s

d uM C v u vdt

( )i ksa ts ku u e

Newton's law:

2N modes

assume harmonic solutions

2

2 12 ( 2 )ss s s

d vM C u v udt

( )i ksa ts kv v e

21

22

(1 exp( )) 2

(1 exp( )) 2k k k

k k k

M u Cv ika Cu

M v Cu ika Cv

a

Linear chain M1 and M2

21

22

(1 exp( )) 2

(1 exp( )) 2k k k

k k k

M u Cv ika Cu

M v Cu ika Cv

a

21

22

2 (1 exp( ))0

(1 exp( )) 2k

k

uM C C ikavC ika M C

4 2 21 2 1 22 2 1 cos( ) 0M M C M M C ka

dispersion relation 2

22

1 2 1 2 1 2

4sin1 1 1 1 2

ka

C CM M M M M M

Optical phonon branch

Acoustic phonon branch

http://lampx.tugraz.at/~hadley/ss1/phonons/1d/1d2m.php

normal modes

density of states

2 22

1 2 1 2 1 2

1 1 1 1 4sin kaC CM M M M M M

Linear chain M1 and M2

2a

The branches of the dispersion curves can be translated by a reciprocal lattice vector G.

fcc

2

1 1 1 12

1 1 1 1 1 1 1 1

1 1 1 1 1 1

2

xx x x x x x x xlmnl mn lmn l mn lmn lm n lmn lm n lmn

x x x x x x x xl mn lmn l mn lmn lm n lmn lm n lmn

y y y y y y y yl mn lmn l mn lmn lm n lmn lm n lmn

lm

d u Cm u u u u u u u udt

u u u u u u u u

u u u u u u u u

u

1 1 1 1 1 1z z z z z z z z

n lmn lm n lmn l mn lmn l mn lmnu u u u u u u

and similar expressions for the y and z motion

1 2 3expx xlmn ku u i lk a mk a nk a t

These are eigenfunctions of T.

Normal modes are eigenfunctions of T

1 2 3expy ylmn ku u i lk a mk a nk a t

1 2 3expz zlmn ku u i lk a mk a nk a t

1 1 2 2 3 3

1 2 3 1 2 3

1 2 3

exp

exp exp

exp

x xpqr lmn k

xk

xlmn

T u u i lk a pa mk a qa nk a ra t

i lpk a qmk a rnk a u i lk a mk a nk a t

i lpk a qmk a rnk a u

fcc

1 2 3exp exp .

2 2 2yx zx x x

lmn k k

l n k al m k a m n k au u i lk a mk a nk a u i

Substitute the eigenfunctions of T into Newton's laws.

http://lamp.tu-graz.ac.at/~hadley/ss1/phonons/fcc/fcc.html

For every k there are 3 solutions for .

Phonon dispersion AuFr

om S

prin

ger M

ater

ials

: Lan

dhol

t Boe

rnst

ein

Dat

abas

e

Materials with the same crystal structure will have similar phonon dispersion relations

Cu

Au

Phonon DOS fcc

D()

From

Spr

inge

r Mat

eria

ls: L

andh

olt B

oern

stei

n D

atab

ase

Phonon dispersion Fe

From Springer Materials: Landholt Boernstein Database

H P N

Phonon DOS Fe

D()

From Springer Materials: Landholt Boernstein Database

Next nearest neighbors (bcc)

D() C2/C1 = 0 C2/C1 = 1/6 C2/C1 = 1/3

C2/C1 = 1/2

D()C2/C1 = 1

The normal modes remain the same (the translational symmetry is the same).

Phonon DOS Fe

D()

From Springer Materials: Landholt Boernstein Database

C2/C1 = 0

C2/C1 = 1/6

D()