Atomic Vibrations in Solids: phonons
Goal: understanding the temperature dependence of the lattice contribution to the
heat capacity CV
concept of the harmonic solid
Photons* and Planck’s black body radiation law
vibrational modes quantized
phonons with properties in close analogy to photons
The harmonic approximation
Consider the interaction potential   (q1 , ..., q3 N )
Let’s perform a Taylor series expansion around the equilibrium positions:
1
 2
1
  st  
q j qk   st   Ajk q j qk
2 j ,k q j qk
2 j ,k
when introducing q j  q j m j
Ajk  force constant matrix
A jk 
Since
 2
 2

q j qk qk q j
Ajk  Akj and
Ajk
Ajk 
m j mk
real and symmetric
We can find an orthogonal matrix T
such that
T
1
1
T
T

Akj
mk m j
A jk
m j mk
 Akj
which diagonalizes A
A T  T A T   where
T
 12 0

2
 0 2


 0

...
0 




2
3 N 

T
T AT

  T j j A j k Tk k   j  j ,k
2
jk
j , k 
With normal coordinates
q j   qkTkj
k
we diagonalize the quadratic form
From
q j   q jTjj
j
   st 
   st 
1
Ajk q j qk

2 j ,k
qk   Tk k q k
k
1
1
A
q
q



Ajk   T jj q j  Tk k q k


st
j k  j  k 
2 j , k 
2 j , k 
j
k
  st 
1
2
  st 
1
2
2

q
 j j
2 j

j , k , j , k
Ajk T jj q jTk k q k   st 
1
2


j  jk q j q k
2 j ,k
Hamiltonian in harmonic approximation can always be transformed into
diagonal structure
1 3N 2
2
2
H   p j j q j
2 j 1
harmonic oscillator problem with energy eigenvalues E 
3N

j 1


problem in complete analogy to the photon gas in a cavity
Z  e

  E
 e  E0  e


jnj
 e   E0
j


e
   1n1  2 n2 ...
n1 , n2 ,..
E    j n j  E0
j
1

 e  E0 

 1 e
1
1



 1  e
2
1



 1  e
3
1
 j  nj  
2
1


  E0
...

e





j  1 e
j




With U  
 ln Z

 

U

E

ln
1

e
 0 
 
j

j

  j
 je

  E0  

1

e
j

j
 E0  
j
j
e
 j
1
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
 j  E for all oscillators
In the Einstein model
3NE
3NE
U   / k T

B
2
e
1
zero point energy
 U 

Heat capacity: C v  
 T  v
Cv  3 N k B
Cv  3 N k B
Classical limit
E / kBT 2 eE / kBT
e
E / kBT

1
2
E / kBT 2 eE / kBT
e
E / kBT
1 for
kBT  E

1
2
 E 


 kBT 
2
e E / kBT
for
kBT  E
•good news: Einstein model
explains decrease of Cv for T->0
CV /3NkB
1.0
•bad news: Experiments
show
Cv  T3 for T->0
0.5
0.0
0
1
2
3
T/TE
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
1)
  (k )  E  const.
2) wave vector k labels particular phonon mode
3) total # of modes = # of translational degrees of freedom
3Nmodes in 3 dimensions
N modes in 1 dimension
Example: Phonon dispersion of GaAs
k
for selected high symmetry directions
data from D. Strauch and B. Dorner, J. Phys.: Condens. Matter 2 ,1457,(1990)
We evaluate the sum in the general result
U 
j
j
e
 j
1
 U0
via an integration using the concept of density of states:
# of modes in
U
max
  D()
,  d
n(, T ) d  U0
0
Energy of a mode
= phonon energy
temperature independent
zero point energy
 
# of excited phonons
n(, T)
In contrast to photons here finite # of modes=3N
max
 D()d 
max
total # of phonon modes
In a 3D crystal

D ( )d   3 N
0
0
Let us consider dispersion of elastic isotropic medium
Particular branch i:  

vik
vL
V
3
D() 

(



)
d
k
k
3 
( 2)
here
(k )  (k )  v ik
vT,1=vT,2=vT
d3 k  4k 2dk
1
dk

dk
2
vi
 k 
k 2   
 vi 
2
 k  1
V
D() 
4 (  k ) 
dk
3
( 2 )
 vi  vi
V 2
 2 3
2 v i
k
Taking into account all 3 acoustic branches
U
max
  D()
n(, T ) d  U0
V 2 1
2 
D()  2   3  3 
2
v T 
 v L
0
max


V
1
2
 2
U  2  3  3   
d  U0
2  v L
v T  0 e  1
How to determine the cutoff frequency max
D(ω)
Density of states of Cu
determined from neutron scattering
?
also called Debye frequency D
D
 D() d  3N
0
choose D such that both curves
enclose the same area
D()  2
with
 U 


Cv 
 T  v
Cv 
9N
D
max
3

0
  2e / kBT
d
2
2
k BT e / kBT  1

Let’s define the Debye temperature

D
D  max
energy
Substitution:

kBT
d 
kBT
dx

 T 
Cv  9Nk B  
 D 
3 D / T
x
D / kB : D
temperature
x 4e x
 ex  12 dx
0
Discussion of:
 T 
Cv  9Nk B  
 D 
D / T
T0 
x 4e x
e
0
x
3 D / T
x 4e x
 ex  12 dx
0


1
2
dx  
0
x 4e x
e
x

1
2
T 
12
Cv 
Nk B  
5
 D 
4
dx
3
T   D
D / T

0
x 4e x
e
x
 1
2
dx 
D / T

0
1  D 
2
x dx   
3 T 
3
Cv  3NkB
Application of Debye theory for
various metals with single fit
parameter D