Crystal Lattice Vibrations:
Phonons
Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
1
Lattice dynamics above T=0
•Crystal lattices at zero temperature posses long range order – translational
symmetry (e.g., generates sharp diffraction pattern, Bloch states, …).
•At T>0 ions vibrate with an amplitude that depends on temperature – because
of lattice symmetries, thermal vibrations can be analyzed in terms of collective
motion of ions which can be populated and excited just like electrons – unlike
electrons, phonons are bosons (no Pauli principle, phonon number is not
conserved). Thermal lattice vibrations are responsible for:
→ Thermal conductivity of insulators is due to dispersive
lattice vibrations (e.g., thermal conductivity of diamond is 6 times larger
than that of metallic copper).
→ They reduce intensities of diffraction spots and allow for
inellastic scattering where the energy of the scatter (e.g., neutron)
changes due to absorption or creation of a phonon in the target.
→ Electron-phonon interactions renormalize the properties of
electrons (electrons become heavier).
→ Superconductivity (conventional BCS) arises from multiple
electron-phonon scattering between time-reversed electrons.
PHYS 624: Crystal Lattice Vibrations: Phonons
2
Vibrations of small amplitude: 1D chain
Classical Theory: Normal Modes
 K 4  K1

d2 U
  K1


 0
dt 2

 K4
 K1
K1  K 2
K2
0
0
K2
K 2  K3
 K3
K4 

0 
U
 K3 

K 3  K 4 
2
3
1
K1  K 2  K 3  K 4
 u1 

 
u
2
V
2
 1
1
0
1
U   , U K U 
 u3 
 
 
 
 
m
1  1
1 0
1 1
1  1
 
0 
,1 
, 2 
, 3 
2  1
2 1 
2  1
2 0 
 u4 
 
 1
 
0
 
 1
4
 
 1
0  0, 1  2, 2  2, 3  2
A  cos( t   )
Quantum Theory: Linear Harmonic
Oscillator for each Normal Mode
2
2
x  xˆ , p  pˆ  i , H  Hˆ  

1
 m 2 x 2
2m
2
1

Hˆ  n ( x)   n  n ( x),  n    n   ,  n ( x) 
2

PHYS 624: Crystal Lattice Vibrations: Phonons
1
2n n !
e

m x 2
2

hn 

m 
x 

3
Normal modes of 4-atom chain in pictures
0 (u0  v0t )
1
2
PHYS 624: Crystal Lattice Vibrations: Phonons
3
4
Adiabatic theory of thermal lattice vibrations
•Born-Oppenheimer adiabatic approximation:
melectron
M ion
1;  electron
F
 ion 
1
Debye
•Electrons react instantaneously to slow motion of lattice, while remaining in
essentially electronic ground state → small electron-phonon interaction can
be treated as a perturbation with small parameter: melectron M ion or D  F
1. ions : R1 , , R N  Eelectron (R1 , , R N )
2. potential for ions :  (R1, , R N )  Eelectron (R1, , R N )  ion-ion interaction
Pi 2
3. Hamiltonian for ions : H  
  (R1 ,
i 1 2M
N
PHYS 624: Crystal Lattice Vibrations: Phonons
, RN )
5
Adiabatic formalism: Two Schrödinger
equations (for electrons and ions)
0
n
n
ˆ
 Hˆ electron


H

(
r
,
,
r
;
R
,
,
R
)

E
(
R
,
,
R
)

electron ion  electron 1
Ne
1
N
n
1
N
electron (r1 , , rNe ; R1 , , R N )

n
n
 crystal (r, R)   ion
(R) electron
(r; R)
n
0
p
p
 Hˆ ion


E
(
R
,
,
R
)

(
R
,
,
R
)

E

, R N )  Qp
p
1
N  ion
1
N
ion ( R1 ,

Q p  
n ,i
2
2M
p*
n
2 n*
n
n*


d
r

(
r
;
R
)

(
R
)


(
r
;
R
)

2


(
R
)


i ion
i electron (r; R ) 
 electron  ion i electron
The non-adiabatic term
Qp
0
T<100K!
PHYS 624: Crystal Lattice Vibrations: Phonons
can be neglected at
6
Newton (classical) equations of motion
•Lattice vibrations involve small displacement from the equilibrium ion
position: 0.1Å and smaller → harmonic (linear) approximation

Fn i  
m j
n i
 2 (rn i  sn i )

rn i rm j
 (rn i  sn i )
1
1
 H   M  sn2 i    nmi j sn i sm j
sn i
2 n i
2 n i ,m j
M  sn i     nmi j sm j
m j
0
 (rn i  sn i )
 2 (rn i  sn i )
1
 (rn i  sn i )   (rn i  sn i )  
sn i  
sn i sm j
rn i
2 n i ,m j rn i rm j
n i
0
•N unit cells, each with r atoms → 3Nr Newton’s equations of motion
PHYS 624: Crystal Lattice Vibrations: Phonons
7
Properties of quasielastic force coefficients
M  sn i     mni j sm j  mx  kx
m j
 mni j are analogous to elastic coefficients k
 mni j   nm ji
 mni j   (0m i n )  j from translational invariance
m j

 n i  0
m
PHYS 624: Crystal Lattice Vibrations: Phonons
8
Solving equations of motion: Fourier Series
1
u i (q)ei ( qrn t )  Ta sn i (q)  e  iqa sn i (q)
M
1
 2u i (q)  
 nmi j eiq ( rn rm )u j (q)
M M 
m j
sn i 
Di j 
1
M M 
m j iq ( rn rm )


 n i e
m
1
M M 
p j

 0 i e
iq ( r p )
p
Di j  dynamical matrix (does not depend on rn )
 2u i (q)   Di j u j (q)    Di j   2i j  u j (q)  0
j

j


det D(q)   I  0  for each q : d  r eigenvalues  (q)
2
PHYS 624: Crystal Lattice Vibrations: Phonons
2
s
9
Example: 1D chain with 2 atoms per unit cell
2
1
2
  f   sn1  sn 2    sn 2  sn 1,1 
2 n
 nn11   nn 22  2 f ;  nn12   nn12   nn11,2   nn 21,1   f
D 
1
M M 
   e
p
0
p
iq ( rp )




 

2f
M1

f
1  eiqa
M 1M 2


f

1  e  iqa 
M 1M 2


2f

M1

 1
1 
4f2
2  qa 
  2 f 


sin


0
 2 
 M 1 M 2  M 1M 2
4


2
PHYS 624: Crystal Lattice Vibrations: Phonons
10
1D Example: Eigenfrequencies of chain
optical mode
2 f ( M1  M 2 )
lim  (q) 
q 0
M 1M 2
+ (0)
acoustic mode
 1
1 
 (q)  f 

 f
 M1 M 2 
2
BvK: sn  N
f
lim  (q)  qa
q 0
2( M1  M 2 )
2
 1
1 
4
2  qa 

sin 

 

M
M
M
M
2


2 
1
2
 1
2 m
 sn  u (q)ei ( qna t )  q(n  N )a  qna  2 m  q 
PHYS 624: Crystal Lattice Vibrations: Phonons
Na
11
1D Example: Eigenmodes of chain at q=0
Optical Mode: These
atoms, if oppositely
charged, would form
an oscillating dipole
which would couple
to optical fields with
Center of the unit cell is not moving!

a
2f
 2f

 M

M
M
1
2 f (M1  M 2 )
1
2 
 (q  0) 
, D( q  0)  
 2f

2f
M 1M 2


M
2
 M 1M 2

2 fM 1M 2
2f
2f
  u1 

 M (M  M )
 
M
M
1
2
1
2
 1
    0  u   M 2 u , sn1   M 2
1
2

2 fM 1M 2   
2f
2f
M1
sn 2
M1


  
M
(
M

M
)
M 1M 2
2
1
2   u2 

PHYS 624: Crystal Lattice Vibrations: Phonons
12
2D Example: Normal modes of chain in 2D
space
•Constant force model (analog of TBH) : bond stretching and bond bending
0(  )  0
1(  )
0(  )
1(  )

 r  
r
2
2
1

    r    (s j  si )  rˆij    s j  si 

2 ij 
1 
 (q) 
 r      r2  2  2 r cos(q  a) 


M 
2
PHYS 624: Crystal Lattice Vibrations: Phonons
13
3D Example: Normal modes of Silicon
r
M Si

M Si
 8.828THz
 2.245THz
L — longitudinal
T
— transverse
O
— optical
A
— acoustic
PHYS 624: Crystal Lattice Vibrations: Phonons
14
Symmetry constraints
→Relevant symmetries: Translational invariance of the lattice and its
reciprocal lattice, Point group symmetry of the lattice and its reciprocal
lattice, Time-reversal invariance.
D*i j 
D*i j 
1
M M 
1
M M 
 0pi j e
 iq ( r p )
 0pi j e
iq ( r p )


1
M M 
1
M M 
 0pi j e
 0p ij e
iq ( r p )
iq ( rp )
 D ij
Di j (q  G )  Di j (q)
 (q  G )   (q  G )
u j (q  G )  u j (q + G )

D*T  D†  D is Hermitian matrix   2 
sn i 



1
u i (q)ei (qrn t )  t  t , q  q   (q)   (q), D ij (q)  D*j i (q)
M
Di j (q)   2 (q)i j u j (q)   Di j ( q)   2 ( q)i j u* j (q)  0

j
j


u* j (q)  u j (q)
PHYS 624: Crystal Lattice Vibrations: Phonons
15
Acoustic vs. Optical crystal lattice normal modes
→All harmonic lattices, in which the energy is invariant under a rigid translation
of the entire lattice, must have at least one acoustic mode  (q) q (sound waves)
1
m j
q  0  Di j 


n i 
M  M  m j
 (0)u i  
2
j
u


i
M
m
1
M M 

m j
n i  j
m
u
   (0)u i M   
2

j
u j
M
  
m j
n i
m
 2
 (0) u x M   0, 1 (0)  0



 0   2 (0) u y M   0, 2 (0)  0 ←3 acoustic modes (in 3D crystal)


 2 (0) u
M   0, 3 (0)  0

z



3r  3 optical modes which at q  0 behave as :
u (q  0)


i
PHYS 624: Crystal Lattice Vibrations: Phonons
M    M  s i (q  0) M   0

16
Normal coordinates
Qs (q, t )
→The most general solution for displacement is a sum over the eigenvectors of
1
the dynamical matrix: s 
iqrn
s
*
Q
(
q
,
t
)

(
q
)
e
,
Q

n i
s
i
s (q, t )  Qs ( q, t )
M  N qBZ , s
Ekinetic
1
1
2
  M  sn i    Qr (q, t ) r i (q)eiqrn Qs (k , t ) s i (k )eikrn
2 n i
2 n i q ,kBZ ,r , s
2
1
1
i ( k  q ) rn
r
s
e
  k , q ;    i (q)  i (k )   rs  Ekinetic   Qr (q, t )

N n
2 q ,r
i
1
2
E potential   s2 (q) Qs (q, t )
2
1
2
2 q,s
LE
E
   Q (q, t )   2 (q) Q (q, t ) 
kinetic
potential
2
q,s

s
s

s
•In normal coordinates
L
Ps* (q) 
 Qs* (q)
Newton equations describe
Qs (q)
dynamics of 3rN independent
harmonic oscillators!
d  L 
L

 0  Qs (q)  s2 (q)Qs (q)  0
 *

*
dt  Qs (q)  Qs (q)
PHYS 624: Crystal Lattice Vibrations: Phonons
17
Quantum theory of small amplitude lattice
vibrations: First quantization of LHO
→First Quantization:
Qs (q)  Qˆ s (q), Ps (q)  Pˆs (q)
Qr (k ), Ps (q)Poisson  kq rs  Qˆ r (k ), Pˆs (q)   i
H (Qs (q), Ps (q))  Ekinetic  E potential
 kq rs
2
2
1  ˆ
2
ˆ
ˆ
 H   Ps (q)  s (q) Qs (q) 

2 q , s 
Hˆ   E  E   Eq , s ,    Sym  q , s
q,s
PHYS 624: Crystal Lattice Vibrations: Phonons
q,s
18
Second quantization representation:
Fock-Dirac formalism
 (r, t )   ak (t ) k (r )
k
*

a

a

(
r
,
t
)
k
k
Hˆ  (r, t )  i
i
  H kk  ak  ;  i
  H kk  ak*
t
t
t
k
k
H   * (r ) Hˆ  (r )dr
kk

k
k
 H
ak
ak*  H
*
ˆ  (r, t )dr   a* H  a 
i

,

i

,
H


(
r
,
t
)
H
k
k
kk k
 k
t
i ak*
t
ak
kk 
i ak*  "generalized coordinate"; ak  "generalized momentum"
ak*  aˆk† , ak  aˆk ;
 aˆk , aˆk   0, ak , aˆk†   kk
ˆ (r, t )   aˆ (t ) (r), 
ˆ † (r, t )   aˆ † (t ) (r); 
ˆ (r, t ), 
ˆ † (r, t )    (r - r)

k
k
k
k


k
k
Ĥ   H kk  ak† ak   Oˆ   Okk  ak† ak 
kk 
PHYS 624: Crystal Lattice Vibrations: Phonons
kk 
19
Quantum theory of small amplitude lattice
vibrations: Second quantization of LHO
→Second Quantization applied to system of Linear Harmonic Oscillators:
Qˆ s (q), Pˆs (q)  aˆs (q), aˆs† (q)


2s (q)


s (q)

†
ˆ
ˆ
ˆ
Ps (q)  i
as (q)  as (q) 
2

Qˆ s (q) 

aˆs (q)  aˆs† (q)



1
 †
ˆ
H   s (q)  aˆs (q)aˆs (q)  
2

q,s
canonical transformation:  aˆs (k ), aˆr† (q)    sr kq ,  aˆ s (k ), aˆr (q)   aˆ s† (k ), aˆr† (q)   0
→Hamiltonian is a sum of 3rN independent LHO – each of which is a
refered to as a phonon mode! The number of phonons in state (q, s) is
described by an operator:
†
s
s
s
nˆ (q)  aˆ (q)aˆ (q)
PHYS 624: Crystal Lattice Vibrations: Phonons
20
Phonons: Example of quantized collective
excitations
→Creating and destroying phonons:
aˆs† (q) ns (q)  ns (q)  1 ns (q)  1
aˆs (q) ns (q)  ns (q) ns (q)  1
→Arbitrary number of phonons can be excited in each mode → phonons
are bosons:

ns ( q )
1 
†
ns (q)   
0
   aˆs (q) 
 q , s ns (q)!  q , s
→Lattice displacement expressed via phonon excitations – zero point motion!
sn i
1

M N

q,s

2 (q)

aˆs (q)  aˆs† (q) s i (q)eiqrn  s 2
s
PHYS 624: Crystal Lattice Vibrations: Phonons
T 0
0
21
Quasiparticles in solids
•Electron: Quasiparticle consisting of a real electron and the exchangecorrelation hole (a cloud of effective charge of opposite sign due to exchange
and correlation effects arising from interaction with all other electrons).
F
5eV, vF  kF me
106 ms 1
•Hole: Quasiparticle like electron, but of opposite charge; it corresponds to the absence
of an electron from a single-particle state which lies just below the Fermi level. The
notion of a hole is particularly convenient when the reference state consists of
quasiparticle states that are fully occupied and are separated by an energy gap from the
unoccupied states. Perturbations with respective to this reference state, such as missing
electrons, are conveniently discussed in terms of holes (e.g., p-doped semiconductor
crystals).
•Polaron: In polar crystals motion of negatively charged electron distorts the
lattice of positive and negative ions around it. Electron + Polarization cloud
(electron excites longitudinal EM modes, while pushing the charges out of its
way) = Polaron (has different mass than electron).
PHYS 624: Crystal Lattice Vibrations: Phonons
22
Collective excitation in solids
In contrast to quasiparticles, collective excitations are bosons, and they bear no
resemblance to constituent particles of real system. They involve collective (i.e.,
coherent) motion of many physical particles.
•Phonon: Corresponds to coherent motion of all the atoms in a solid — quantized
lattice vibrations with typical energy scale of   0.1 eV
•Exciton: Bound state of an electron and a hole with binding energye2  a  0.1eV
•Plasmon: Collective excitation of an entire electron gas relative to the lattice of
ions; its existence is a manifestation of the long-range nature of the Coulomb
interaction. The energy scale of plasmons is
ne2 m
5  20eV
e
•Magnon: Collective excitation of the spin degrees of freedom on the crystalline
lattice. It corresponds to a spin wave, with an energy scale of

PHYS 624: Crystal Lattice Vibrations: Phonons
0.001  0.1eV
23
Classical theory of neutron scattering
I   (K , ) , K = k 0 - k ,   0  
2
1
 (r (t ))    (r  rn (t )), rn (t )  rn 
u(q)ei ( qrn  ( q ) t )
M
n
 (K , )    dt ei ( K (r s
n
n
small amplitude: s n (t )
a, K
n ( t )) t )
2
  (K , )    dt ei ( K (rn t ) (1  iKs(t )  )
a
n
first term non-zero:K = k 0 - k  G,   0    0
Bragg or Laue
conditions for
elastic scattering!
second term non-zero:K  q = k 0 - k  q;   s (q)  0    s (q)  0
PHYS 624: Crystal Lattice Vibrations: Phonons
24
Classical vs. quantum inelastic neutron
scattering in pictures
•Lattice vibrations are inherently quantum in nature → quantum theory is needed to
account for correct temperature dependence and zero-point motion effects.
Phonon absorption is
allowed only at finite
temperatures where a
real phonon be excited:
T  0  ns (K )  0
PHYS 624: Crystal Lattice Vibrations: Phonons
25