Chap 13 Phonons
• classical theory of vibration
• 1-dim, 3-dim
• quantum theory of vibration
• phonon specific heat
• Einstein model, Debye model
• thermal expansion
• neutron scattering
Dept of Phys
M.C. Chang
One dimensional vibration
• consider only longitudinal motion
• consider only NN coupling
d 2 un
M
  (un 1  un )   (un  un 1 )
2
dt
PBC:
Assume un  Aei ( kX n t ) , where X n  na ,
then we' ll get
M (  2 )eikna   2eikna  eik ( n 1) a  eik ( n 1) a ,
which leads to
 (k )=M sin(ka / 2) , M  2  / M
(k )=M sin(ka / 2)
Dispersion curve

  2a
  2a   2a
  2a (redundant)
k
-/a
/a
The waves with wave numbers k and k+2π/a describe the same
atomic displacement
Therefore, we can restrict k to within the first BZ [-π/a, π/a]
Displacement of the n-th atom
un (t )  Aei ( kX n t ) , X n  na
Pattern of vibration:
• k ～ 0, exp(ikXn) ～ 1.
Every atom move in unison. Little restoring force.
• k ～ π/a,exp(ikXn) ～ (-1)n.
Adjacent atoms move in opposite directions. Maximum restoring force.
Velocity of wave:
• k ～ 0, ω = (ωMa/2)k
Linear dispersion, phase velocity = group velocity
• k ～ π/a, group velocity ～ 0
PBC
Number of “normal modes”:
uN  u0  exp(ikNa)  1
m 2
△k=2π/Na
k 
N a
, m Z
Within the 1st BZ, there are N k-points.
Each k describes a normal mode of vibration
(i.e. a vibration with a specific frequency)
Vibration of a crystal with 2 atoms in a unit cell
a
d 2u2 n 1
M2
  (2u2 n 1  u2 n  u2 n  2 ),
dt 2
d 2 u2 n  2
M1
  (2u2 n  2  u2 n 1  u2 n 3 ).
dt 2
 u2 n1   A1eikX 2 n1  it X 2 n1  (2n  1)a / 2
Assume 
e ,

ikX 2 n2 
X 2 n2  (n  1)a
 u2 n 2   A2e


 2  M 2 2
2 cos(ka / 2)   A1 

    0,
2

2

cos(
ka
/
2)
2


M

1

  A2 

 2  M 2 2
2 cos(ka / 2) 
det 
  0.
2
2  M1 
 2 cos(ka / 2)
2
 1
 1
1 
1  4sin 2 (ka / 2)
2
    


.
  
 
M
M
M
M
M
M
 1
2 
 1
2 
1
2
Two branches of dispersion curves (assume M2 > M1)
a
d
c
b
Patterns of vibration
similar
See a nice demo at http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html
Three dimensional vibration
Along a given direction of propagation, there are
1 longitudinal wave and 2 transverse waves,
each may have different velocities
Sodium
(bcc)
3D crystal with atom basis
FCC lattice with 2-atom basis
cm-1
Rules of thumb
• For a 3-dim crystal, if each unit cell
has p atoms, then there are
3 acoustic branches,
3(p-1) optical branches
• If a crystal has N unit cells, then each branch
has N normal modes。
• As a result, the total number of normal modes
are 3pN (= total DOF of this system)
Quantum theory of vibration
Review: 1D simple harmonic oscillator (DOF=1)
p2  2
H
 x
2m 2
• Classically, it oscillates with a single freq ω=(α/m)1/2
• The energy ε can be continuously changed.
Quantization:
 x, p   i
1 
i

m

x

p


2 
m 
define
a
then
 a, a   1
†
1

H   a†a   
2

1

H n  n   n
2

Note:
a n  n n 1
a† n  n  1 n  1
a†a n  n n
1

• After quantization, the energy becomes discrete  n   n   
2

• The number n of energy quanta depends on the amplitude of the oscillator.
Quantization of a 1-dim vibrating lattice (DOF=N)
 p2 
H  
  u 1  u
2
m
2
1 
N

2



• Classically, for a given k, it vibrates with a single frequency ω(k).
The amplitude ( and hence energy ε) can be continuously changed.
Quantization:
u , p '   i
Fourier transf.
u 
then

'
1
eik a uk

N k
1
p 
eik a pk

N k
uk , pk† '   i  kk '
 1 †
mk2 † 
 H  
pk pk 
uk u k 
2
k  2m

Note:
u†  u ;
 uk†  u k ;
p†  p
pk†  p k
k=2πm/L, L=Na.
A collection of N
independent oscillators !
Similarly, define
1
ak 
2


i
pk 
 mk uk 


mk


  ak , ak†'    kk '
1

 H    ak† ak   k
2
k 
eigenstate
n1 , n2 ...nk ,...
1

H n1 , n2 ...nk ,...    nk   k n1 , n2 ...nk ,...
2
k 
• The number of energy quanta (called phonons) for the k-mode is nk.
• There are no interaction between phonons → “free” phonon gas.
• If there are p-atoms in a unit cell (p branches), then the total
vibrational energy of the lattice is
1

U    nk , s   k , s
2
k s 1 
p
Specific heat: experimental fact
Specific heat
approaches 3R
(per mole) at
high temperature
(Dulong-Petit law)
Specific heat
drops to zero at
low temperature
After rescaling the temperature by θ (Debye temperature), which
differs from material to material, a universal behavior emerges:
Debye temperature
In general, a harder material has a higher Debye temperature
Specific heat: theoretical framework
• Internal energy U of a crystal is the summation of vibrational energies
(consider an insulator so there’s no electronic energies)
U (T )   (nk , s  1/ 2) k , s , s =L/T, A/O...
k ,s
• For a crystal in thermal equilibrium, the average phonon number is
nk ,s 
• Therefore,
• Specific heat
1
e
k ,s / kT

U (T )= 
k ,s  e
1
1
k ,s / kT
Bose-Einstein
distribution
1
  k ,s
1 2 
CV   U / T V
• Density of states (similar to electron energy band)
f
k
k
V 
d 3k
 2 
 L 
D( )  

 2 
3

3
f k   d D( ) f
dS
k 
Ex: In 3D
if   vk , then D( )  V  2 / 2 2v3
Einstein model (1907)
Assume that each atom vibrates independently of each other,
and every atom has the same vibration frequency ω0
DOS
D()  3N   0 
3 dim  number of atoms
0

1

U  3N  n   0  3N
 3N 0
2
exp( 0 / kT )  1
2

e 0 / kT
 0 
CV  (U / T )V  3Nk 

 kT  e 0 / kT  1
2

 e
0 / kT
as T  0 K
(Activation behavior)

2
Debye model (1912)
Atoms vibrate collectively in a wave-like fashion.
U (T )   nk , s
k , s ( k ,s /2 neglected)
k ,s
=   d Ds ( )
s

e
 / kT
1
• Debye assumed a simple dispersion relation:
ω = vsk. Therefore, Ds ( )  V  2 / 2 2 vs3
Also, the 1st BZ is approximated by
a sphere with the same volume
3
  d D ( )  3N
s 1
s
V D3
  2 3  3N
s 1 6 vs
3
3
3
1


v3 s 1 vs3
 D  v(6 n) , n  N / V
2
1/3
If vg   k   0, then there is
"van Hove singularity".
3
V
s 1
2 2 vs3
U (T )  
=
D

d  2
0
3V
 kT 


2 2 v3 

T 
=9 NkT  
 
4 xD

e
 dx
0
 / kT
1
Debye temperature
D 
x
,
x

 , k  D
D
ex 1
kT
T
3
3 xD
x3
0 dx e x  1 = π4/15 as T→ 0
12 4
T 
 CV 
Nk    T 3 as T  0
5
 
3
solid Argon
(θ=92 K)
At low T, Debye’s curve drops
slowly because long wavelength
vibration can still be excited.
A simple explanation of the T3 behavior:
Suppose that
1. All the phonons with wave vector k<kT are excited
with thermal energy kT.
2. All the modes between kT and kD are not excited.
kD
kT
Then the fraction of excited modes
= (kT/kD)3 = (T/θ)3.
energy U ～ kT3N(T/θ)3, and the heat capacity
C ～ 12Nk(T/θ)3
Thermal expansion
Coeff. Of volume
expansion:
Bulk modulus:
1  V 

 ,
V  T  P
 P 
B  V 

 V T

 
1  P 


B  T V
 F 
P  

 V T
Next page we’ll show that,
P 

U (T )
V
1  P 




cV


B  T V
B
(cV  CV / V )
 T 3 at low T
use
 x 
1
and
   y

y


 z
 
 x  z
 x   y   z 
       1
 y  z  z  x  x  y
Partition function:
Z   e  Ei / kT
(Ei are the macroscopic eigen-energies)
i
 1
=  exp  
 kT
nks 
1


n



 ks
 ks 
2
ks 

 1
=  exp  
ks nks  0
 kT

Grüneisen parameter
1


n


 ks
 ks 
2


d ks
ks
F   kT ln Z

= kT  ln e
ks /2 kT
 e
ks /2 kT
 

ks
1   ks  e
 F 
P  





2 ks  V T e
 V T
P 
1
V

ks / kT
ks / kT
1
1
k s dk s V
k s
  k s
V k s dV
V
dV
V
1  P ' 


B  T V
1 1
 CV

C

 ks V ,ks B V
B V ks
 C
where  
C
ks
V , ks
ks
V , ks
ks
CV   CV ,ks
U ks (T ),
ks
ks

where U ks (T )  ks 
e

  ks
1
ks / kT
1
 
1 2 
ks
(γ～1-2 for most materials)
Neutron scattering
En
 k

Why neutron?
2
2mn
 2.07k 2 meVA 2
• Neutron has no charge (can probe bulk properties)
• Neutron wavelength comparable to interatomic spacings (1-5 Å)
• Neutron energy comparable to phonon’s (5-100 meV)
• Neutron has spin (can probe magnetic structure and magnetic excitations)
Measure phonon dispersions by neutron scattering
More than one phonon mode may be excited
nq.s  n 'q.s  nq.s
out E’,p’
in E, p
Conservation of energy
En '  q , s nq , s  En
q,s
Conservation of crystal momentum (for a proof. see App. M of A+M)
p '  qnq,s  p  G
q,s
(momentum of phonon with λ< a must be shifted by G)
One phonon scattering
En '  En  s (q )
p'  p  q  G
Neutron energy:
p '2
p2

 s (q ),
2mn 2mn
En=p2/2mn
Consider phonon absorption

p '2
p2

 s k ' k
2mn 2mn
 f k (q ) 
2
 p' 
k '; p  k
q  (k ' k )
k=0

2 2
k
| q  k |2

 s  q 
2mn
2mn
q and ωs can be determined from
the intersections in the figure.

2
q2
2M n
 (q )
q
k≠0
several solutions at a given direction.
→ a series of peaks in the data
Phonon dispersion curve for Si,
comparing experimental data and
ab initio calculation (Wei and Chou 1994)
• Width of one-phonon peaks due to anharmonic effect
• Multi-phonon scatterings give a continuous background
Neutron scattering: formal theory
2

 f V  i   E f  Ei 
• Fermi golden rule
(transition rate)
Ri  f  
• Neutron states scattered to
d3k’ (per unit time)
mn k ' d  d 
d 3k '
Rk k ' V

R

V
k k '
(2 )3
(2 )3
2
f
j  v
• Differential cross-section
 d 2

 d d 
 d 2

 d d 
Rk d  ,d  

 d d  
incident particle flux

initial:
final:
 i   k ( r ) i ,
 f   k ' ( r ) f ,
1 ik r
e
V
2
k2
Ei 
 Ei
2mn
Ef 
2
2
k 1 k

mn V mn
 k ' (Vmn ) 2
Rk k '

3
k
(2

)

• Quantum state of “neutron + crystal”
   k (r ) crystal ,  k ( r ) 
 k
2
k'
 Ef
2mn
Neutron Crystal
energy energy
• The energy gained by a neutron
due to a phonon in crystal
2
2 2
k '2
k

 Ei  E f  
2mn 2mn
Momentum transfer
q  k ' k
Neutron-ion interaction potential


V (r )   v r  r ( R) , where r ( R)  R  u ( R)
v(r) is the atomic
potential
R
Matrix element


let v r  r ( R) 
1
vq eiq r e  iq r ( R )

V q
 f V  i    k ' f v  k  i
R
=
1
vq  k ' eiq r  k  f e  iq r ( R )  i

V R ,q
 k  q ,k '
=
 d

 d d 
vk ' k
1
vk ' k  f e  i ( k ' k )r ( R )  i

V R
 k '  v0 mn 
 
2 
 k  2 
2
  E
f
v0
since k for phonon
the range of vq
 Ei    f eiqr ( R ) i
f
Sum over a complete set
of phonon states
R
2
108 cm 1
1013 cm 1
 d 
One can re-write  d d   using dynamical structure factor


1  E f  Ei
 1
  E f  Ei    
=

 2
use e
i ( E f  Ei ) t /
 d
then 
 d d 
it
 dt e e
i ( E f  Ei ) t /
for phonon absorption
(always from neutron’s viewpoint)
 f A  i   f A(t )  i
 k ' 1  mn v0 


2 
 k h  2 
2

 dt e 
i t
 i eiq r ( R ') e iq r ( R ,t )  i
R,R '
2
k ' 1  mn v0 
=
2 NSi (q ,  )

2 
k h  2 
Dynamical structure factor
Si ( q ,  ) 
Density operator (for ions)
1 dt it
iq r ( R ')  iq r ( R ,t )
e

e
e
i

i

N 2
R,R '
 (r )    (r  r ( R))
R
 q   d 3r eiq r  (r )   eiq r ( R )
dt it 1
=
e
 i  q (0)   q (t )  i
2
N
R
Density correlation function
For a crystal at finite temperature
 i  q (0)   q (t )  i

 q (0)   q (t )
T

e
i
 Ei / kT
 i  q   q (t )  i
e
i
 Ei / kT
Evaluation of the dynamical structure factor
eiqr ( R ') e iqr ( R ,t )
T
 eiq ( R ' R ) eiq u ( R ') e iq u ( R ,t )
It can be shown that, for A, B linear in a, a
e Ae B
T
e
1 2
A  2 AB  B 2
2
use  q  u ( R ') 
2
T
T
†
D. Mermin, J Math.Phys. 7,1038 (1966)
  q  u ( R, t ) 
and  q  u ( R ')   q  u ( R, t ) 
r ( R)  R  u ( R)
T
T
  q  u (0) 
2
 2W
2
T
T
  q  u (0)  q  u ( R  R ', t ) 
T
Translation symmetry
of the system
Debye-Waller factor
 S (q ,  )  e
e
T
2W
1 
dt it
 iq  R  q u (0) q u ( R ,t )  T
 2 e R e e
1 2

 ...
T
T
2
0-phonon 1-phonon process …
Exact so far
(for a harmonic crystal)
• a rough estimate (Kittel. App.A)
Zero-phonon process
2W 
S0 (q ,  )  e
2W
 ( )  e
R
DebyeWaller
factor
Elastic
scattering
 iq  R
N   q ,G
G
Laue’s diffraction
condition
T
T
cos 2 
T
1
= G2 u2
T
3
m
3
use ion  2 u 2  kT
T
2
2
2W
e

G2
mion 2
kT
2
 k ' 1  mn v0 
2 NS (q ,  )


2 
 k h  2 
k' N
or = a 2 S (q ,  )
k
 d

 d d 
2
Intensity I=I0 e-2W (I0 for a rigid lattice)
Differential cross-section
Cf:

= G2 u2
 e
 d

 d d 

G u
 N
 = S (q ,  )

2 2 a
v0 
mn
For X-ray scattering
(the same S)
Scattering length
For more discussion,
see A+M, App. N
use aks† aks
• A more accurate evaluation
1
aks 
2
a† ks 
1
2
uks 


i
ˆ
m

u

p

ks ks
ks   eks


mks




i
ˆ
m

u

p

ks ks
ks   eks


mks


(eˆ* ks =eˆks )
a
 (k )
2mion
ks
a
†
 ks
 eˆ
ks
s

2mions (k )
a
ks
 a† ks  eˆks

2

T
1
N
G  eˆks
 2m
ks
ion
2
s ( k )
(2nks  1)
In 3D Debye model (at T =0!)
3
G2
2W 
4 mion v k D
(Prob. 7)
In the calculation, one has

kD
0
pks  i

G u
 nks
T
k D 1dk
1 kD D  2
 k dk
k 0
• In 3D, W weakens the diffraction peaks.
• In 2D, W is finite at T=0 but infinite at finite-T.
1
uks eik R

N k ,s
1
p( R) 
pks eik R

N k ,s
u ( R) 
• In 1D, no long-range order even at T=0
Mermin-Wagner theo (Mermin PR1968) There
is no long-range crystalline order at finite-T in
2D.
One-phonon process
S1 (q ,  )  e2W 
dt it
e  eiqR  q  u (0)  q  u ( R, t ) 
2
R
T
1/2

 ik R
1
†
q  u ( R, t ) 

 e  aks (t )  a ks (t )  q  eˆks

N ks  2mionks 
 S1 (q ,  )  e 2W 
s
2mionqs
 q  eˆks 
2
aks (t )  e  iks t aks
(1  nqs ) (  qs )  nqs (  qs ) 
Phonon emission
absorption
Delta peaks are broadened only
if anharmonic effect (phononphonon interaction) is included.
One-phonon cross-section
 d

 d d 
 k' 2 N
S1 (q ,  )
= a
 k
Mössbauer effect (1958)
57Fe
nucleus
(radioactive)
1961
• Natural linewidth 4.65x10-9 (eV)
• Recoil energy 1.94x10-3 (eV)
Mössbauer found that, by placing emitting and
absorbing nuclei in a crystal, you could have
(almost) recoilless nuclei with resonant absorption
→ an extremely sensitive detector for energy shift.
The Pound-Rebka experiment (1959)
ν
note: precision measurement
using atom interferometer
ν0
Gravitational
freq shift
 gH 
v  v0 1  2 
c 

gH
c2
g  22.6 m
 14.4 keV
 3.5 10 11 eV
2
c
E  h(v  v0 )  hv0
Ref: hyperphysics
Müller H et al , Nature 2010
Quinn and Yi, SSP, Sec 2.3
Back to Mössbauer effect
Ri  f  
2

 f V  i   E f  Ei 
2
f
 f V i 
1
V
v
q
 k ' eiq r  k  f e  iq r ( R )  i
Recoilless fraction
q
v
 0  f e  iq r ( R )  i
V
f  i e
 iq r ( R )
i
Put the nucleus at the origin
1/2

1 
r (0)  u    ks  aks  a† ks  ,  ks 

 eˆks
N  2mionks 
ks
i e
 iq r ( R )
i  nks  e
iq u
nks   
nks e
consider
eˆks q

 iqks aks  a† ks
 n
ks
ks
nks e

 iq ks aks  a† ks

nks
(iq ks ) 2
 1
nks aks aks†  aks† aks nks 
2!
E (q) nks  1/ 2
=1 

ks
N
 i e  iq r ( R )  i
 E (q) nks  1/ 2 
  1 

ks
N
ks 

 E (q) nks  1/ 2 
 exp  


N

ks
ks


2
Using the Debye model
1
I
N

ks
nks  1/ 2
ks
3 V
 2 3
2 v N
9T
k 2
9

4k

D

0

d 
e
for T

for T

1
 / kT
1
 
1 2 
Cf: the specific
heat calculation
Recoilless fraction
f  i e
 iq r ( R )
i
2
 e2 E ( q ) I
Recoil energy E(q) ～2x10-3 eV, θ ～ 300 K
At low T,
f  e1/3  0.7
Put 57Fe impurities in a diamond (θ=2230 K)
f (295 K )  0.94
Mossbauer Effect in Lattice Dynamics. Chen and Yang 2007
Peierls instability in 1D metal (1955)
2
 k0 


 

 2a

2m
2
 
 
2

2a 




2m
2ma
  0  
2
2
increase elastic energy, decrease electron energy. Who wins?
Recall  k(1) 
 k0   k0 g
2
 /2 a
Eel  2 L 
0
 2L
c
0
  k0   k0 g
 

2

2

  U g

2
 0 
 
2
 Ug
2
(Chap 8)
dk 0
 k   k(1) 

2
dk
2

  

 
2
 Ug
2
 Ug

 
2
 1   U g 2 1 
 ln c  
  
 2 U g   c  4 
For small displacement δ, U～δ
Eel
 2   ln   but Eph  2
∴ distortion is always favored in 1D