Lattice Dynamics
related to movement of atoms
about their equilibrium positions
determined by electronic structure
Physical properties of solids
•Sound velocity
•Thermal properties: -specific heat
-thermal expansion
-thermal conductivity
(for semiconductors)
•Hardness of perfect single crystals
(without defects)
Reminder to the physics of oscillations and waves:
Harmonic oscillator in classical mechanics:
Equation of motion:
Hooke’s law
m x  Fspring
Example: spring pendulum
m x  D x  0 or
where
x  D ~
~
x0
m
x(t )  Re(~
x( t ))
~
x( t )  A ei  t
Solution with ~
x(t )  A cos(t  )
where

D
m
Dx
X=A sin ωt
1
Epot  D x 2
2
X
x

D
m
y(t )  A cos (t  kx  )
~
~
y( t )  A ei( t kx )
  0 in particular
Traveling plane waves:
or
Y
y(t )  A cos (t  kx )
0
X
X=0:
y(t )  A cos t
t=0:
y( x)  A cos kx
Particular state of oscillation Y=const
travels according
~
~
y( t )  A ei( t kx )
solves wave equation
2
2
1  y
v t
2
2

 y
x 2
d
t  kx   d const.  0
dt
dt

2
   v
x  v 

k
2 / 
Transverse wave
Longitudinal wave
Standing wave
~
~
y1  A ei(kx t )
~
~
y 2  A ei(kx  t )


~
~
ys  ~
y1  ~
y 2  A ei(kx t )  ei(kx t )
~
~
 A eikx eit  e it
 2 A eikx cos t


ys  Re( ys )  2 A cos kx cost
Large wavelength λ
k
2
0

λ>10-8m
10-10m
8
Crystal can be viewed as a continuous medium: good for   10 m
Speed of longitudinal wave: v   
(ignoring anisotropy of the crystal)
Bs

where Bs: bulk modulus with  
1
2
Bs determines elastic deformation energy density U  Bs2
(click for details in thermodynamic context)
dilation

compressibility
V
V
E.g.: Steel
v
Bs

Bs=160 109N/m2
ρ=7860kg/m3
v
160 109 N / m2
7860 kg / m3
1
Bs
 4512
m
s
>> interatomic spacing
continuum approach fails
In addition: vibrational modes quantized
phonons
Vibrational Modes of a Monatomic Lattice
Linear chain:
Remember: two coupled harmonic oscillators
Symmetric mode
Anti-symmetric mode
Superposition of normal modes
generalization
Infinite linear chain
How to derive the equation of motion in the harmonic approximation
n-2
D
un-2
n-1
a
un-1
n
n+1
un
un+1
n+2
un+2
Fnr  Dun  un1 
Fnl  Dun  un1 
un-2
un-1
un
fixed
un+1
un+2
?
Total force driving atom n back to equilibrium
Fn  Dun  un1   Dun  un1 
n
n
 Dun1  un1  2un 
Alternative without thinking
Lagrange formalism
1
D
2
2
mun2   un  un 1    un  un 1  

2
2
d L L

0
dt un un
equation of motion
L
n  Fn
mu
D
un  un1  un1  2un 
m
mun 
un 
D
 2  un  un 1   2  un  un 1    0
2
D
 2un  un 1  un 1   0
m
Solution of continuous wave equation u  A ei(kx t )
approach for linear chain un  A ei(kna t )
n  2 A ei(kna t ) ,
u
 2 

D ika
e  eika  2
m

un1  A ei(kna t )eika
2  2
D
1  cos ka 
m
?
,
Let us try!
un1  A ei(kna t )eika
D
2
sin(ka / 2)
m
D
2
sin(ka / 2)
m
2
D
m
Note: here pictures of transversal waves
although calculation for the longitudinal case
k
Continuum limit of acoustic waves:
sin ka / 2  ka / 2  ...
k
2
0

D

ak
m

D
v
a
k
m
k  k  h
2
, here h=1
a
k
2
i
((
k

h
) na t )
 na (k )t )

i
(
k
a
un  A e
 A ei(k na  t )  A e
 A ei(k na t )ei2 h n  A ei(k na  t )
(k )  (k )
k  k  h
2
a
un(k, (k))  un(k, (k))
1-dim. reciprocal
lattice vector Gh
Region



k
a
a
ei2 h n  1
is called
first Brillouin zone
Brillouin zones
We saw: all required information contained in a particular volume in reciprocal space
a
first Brillouin zone
1d:
rn  n a ex
Gh  r n  2m
2
a
In general: first Brillouin zone
2
Gh  h
e
a x
where m=hn integer
1st Brillouin zone
Wigner-Seitz cell of the reciprocal lattice
Vibrational Spectrum for structures with 2 or more atoms/primitive basis
Linear diatomic chain:
2n-2
D
2n-1
a
2n
2n+1
2n+2
2a
u2n-2
u2n
u2n-1
u2n+1
2n 
Equation of motion for atoms on even positions: u
u2n+2
D
u2n1  u2n1  2u2n 
m
2n1 
Equation of motion for atoms on even positions: u
i( 2kna t )
Solution with: u2n  A e
and
D
u2n 2  u2n  2u2n1 
M
u2n1  B ei(( 2n1)ka t )
D
 A2  B(eika  e ika )  2 A 

m 
D
 B2   A(eika  e ika )  2B

M 
D
 D

A 2  2   2 B cos ka
m
 m

A2
D B cos ka
m D
2
2 m   
D
 D

B 2  2   2 A cos ka
M
 M

•Click on the picture to start the animation M->m
D
note wrong
 1 1
 axis
2 in the movie
  2D   
m
m M
2
 D
2  2 D  2   4 D cos2 ka
2


 m
 M

Mm



D2
D 2
D 2
D2
4
4
2  2    4
cos 2 ka
Mm
M
m
Mm

2
D
M
  D D 
D2 
4
2
2
   2    4
1  cos ka   0

Mm 
  m M 
2
2
sin2 ka

k
:
2a
2
1
1
4 sin2 ka
1
1
2
  D    D    
Mm
m M
m M
 1 1
 1 1
  D    D   
 m M
 m M
2

2
D
m
,

2
D
M
2
Atomic Displacement
B
m


k 0
A
M
Optic Mode
B
k 0  1
A
Atomic Displacement
Click for animations
Acoustic Mode
Dispersion curves of 3D crystals
•3D crystal: clear separation into longitudinal and transverse mode only possible in
particular symmetry directions
•Every crystal has 3 acoustic branches
1 longitudinal
acoustic
2 transverse
sound waves
of elastic theory
•Every additional atom of the primitive basis
further 3 optical branches
again 2 transvers
1 longitudinal
p atoms/primitive unit cell (
primitive basis of p atoms):
3 acoustic branches + 3(p-1) optical branches = 3p branches
1LA +2TA
(p-1)LO +2(p-1)TO
z
y
Intuitive picture: 1atom
3 translational degrees of freedom
x
3+3=6 degrees of freedom=3 translations+2rotations
+1vibraton
# atoms
in primitive
basis
# of primitive
unit cells
Solid: p N atoms
3p N vibrations
no translations, no rotations
Part of the phonon dispersion relation of diamond
2 fcc sublattices vibrate against one another
However, identical atoms
no dipole moment
Longitudinal Optical
Transversal Optical
degenerated
Longitudinal Acoustic
(0,0,0)
1 1 1
( , , )
4 4 4
diamond lattice: fcc lattice with basis
Transversal Acoustic
degenerated
P=2
2x3=6 branches expected
Calculated phonon dispersion relation
of Ge (diamond structure)
Calculated phonon dispersion relation
of GaAs (zincblende structure)
Adapted from:
H. Montgomery, “ The symmetry of lattice vibrations in zincblende and diamond structures”, Proc. Roy. Soc. A. 309, 521-549 (1969)
Phonon spectroscopy
Inelastic interaction of light and particle waves with phonons
Constrains: conservation law of
energy
momentum
Condition for
inelastic scattering
k  k 0 ± q  Ghkl
phonon
wave
vector
in elastic sattering
  0  (q)  0
for photon
scattering
 2k 2  2k 0 2

 (q)  0
2Mn
2Mn
for neutrons
incoming wave
“quasimomentum”
k  k 0   q  Ghkl

0
k
( q)
k 0
q
Triple axis neutron spectrometer
@ ILL in Grenoble, France
Very expensive and involved experiments
Table top alternatives
?
Yes, infra-red absorption and
inelastic light scattering (Raman and Brillouin)
However only
q0
accessible
see homework #8
Lonely scientist in the reactor hall