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