Atomic Vibrations in Solids: phonons
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Transcript Atomic Vibrations in Solids: phonons
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 jTjj
j
st
st
1
Ajk q j qk
2 j ,k
qk Tk k q k
k
1
1
A
q
q
Ajk T jj 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
Ajk T jj 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
3NE
3NE
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 eE / kBT
e
E / kBT
1
2
E / kBT 2 eE / 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 4k 2dk
1
dk
dk
2
vi
k
k 2
vi
2
k 1
V
D()
4 ( k )
dk
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 12 dx
0
Discussion of:
T
Cv 9Nk B
D
D / T
T0
x 4e x
e
0
x
3 D / T
x 4e x
ex 12 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