Download Introduction to Lattice Dynamics

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)
Uniform Solid Material
There is energy
associated with the
vibrations of atoms.
They are tied together
with bonds, so they can't
vibrate independently.
(X-1)
The vibrations take the
form of collective modes
which propagate through
the material.
(X)
(X+1)
Wave-Particle
Duality
Just as light is a wave motion that is considered as composed of
particles called photons, we can think of the normal modes of
vibration in a solid as being particle-like.
Quantum of lattice vibration is called the phonon.
Phonon:
A Lump of Vibrational Energy
Propagating lattice vibrations can be considered
to be sound waves, and their propagation speed
is the speed of sound in the material.
Roughly how big is ?
Phonon:
Sound
Wavepackets
Reminder to the physics of oscillations and waves
Harmonic oscillator in classical mechanics
Equation of motion:
Hooke’s law
m x  Fspring
Example: vertical springs
x  K ~
m x  K x  0 or ~
x 0
m
where
x( t )  Re(~
x( t ))
~
x ( t )  A ei  t
Solution with ~
x(t )  A cos(t  )
where

K
m
Kx
X
E pot
X=A sin ωt
1 2
 Kx
2
x

K
m
Displacement as a function of time and k
Traveling plane waves:
Y
or
y(t )  A cos (t  kx  )
~ i( t kx )
~
y( t )  A e
  0 in particular
y(t )  A cos (t  kx )
0
X
(Phonon wave vector also often given as q instead of k)
Consider a particular state of oscillation
Y=const traveling along
~
~
y( t )  A e
solves wave equation
i( t kx )
1 2y 2y
v t
2
2

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
 2A 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
K
un-2
n-1 a
un-1
n
n+1
un
un+1
n+2
un+2
Fnr   K un  un 1 
Fnl   K un  u n 1 
un-2
un-1
un
fixed
un+1
un+2
?
Total force driving atom n back to equilibrium
Fn   K un  un1   K un  un1 
n
n
 K un1  un1  2un 
n  Fn
mu
equation of motion
K
un  un 1  un 1  2un 
m
Old solution for continuous wave equation was u  A ei(kx t ) . Use similar?
approach for linear chain un  A ei(kna t )
n  2A ei(kna t ) ,
u

K ika ika
 
e e 2
m
2

un1  A ei(kna t )eika
2  2
K
1  cos ka
m
?
,
Let us try!
un1  A ei(kna t )eika
K
2
sin( ka / 2)
m
2
K
sin( ka / 2)
m
2
K
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


K
ak
m

k
v
K
a
m
Technically, only have longitudinal modes in 1D
(but transverse easier to see what’s happening)
x = na
un
un
(a) Chain of atoms in the absence of vibrations.
(b) Coupled atomic vibrations generate a traveling longitudinal (L)
wave along x. Atomic displacements (un) are parallel to x.
(c) A transverse (T) wave traveling along x. Atomic displacements
(un) are perpendicular to the x axis.
(b) and (c) are snapshots at one instant.
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
Vibrational Spectrum for structures with 2 or more atoms/primitive basis
Linear diatomic chain:
2n-2
K
2n-1
a
2n
2n+1
2n+2
2a
u2n-2
u2n
u2n-1
u2n+1
K
u2n1  u2n1  2u2n 
m
Equation of motion for atoms on even positions:
u2 n 
Equation of motion for atoms on odd positions:
u2 n 1 
i( 2kna t )
Solution with: u2n  A e
and
u2n+2
K
u2 n 2  u2 n  2u2 n1 
M
u2n1  B ei(( 2n1)ka t )
 A 2 


K
 K

A2   2   2 B cos ka
m
 m

K
B(eika  e ika )  2 A
m

K
 B 
A(eika  e ika )  2 B
M
2

K2
 K
2  K
2
2
2 m    2 M     4 Mm cos ka
2
4
2
K B cos ka
m K
2
2 m   
K
 K

B 2   2   2
A cos ka
M
M


•Click on the picture to start the animation M->m
note wrong axis inKthe movie
1 
1
 2
  2K  

m
m M 
2
K
K
K
K
 2 2  2 2 4  4
cos 2 ka
Mm
M
m
Mm
 K K
   2 
 m M
4
A2
 2
K
M
K2


1  cos 2 ka  0
  4
Mm

2
2
sin2 ka

k
:
2a
 1 1
 1 1
  D    D   
 m M
 m M
2
2
2
1 1 
 1 1  4 sin ka
2
  K    K    
Mm
m M 
m M 
 2
K ,
m
 2
K
M
2
Transverse optical mode for
diatomic chain
Amplitude
s
of different
atoms
A/B=-m/M
Transverse acoustic mode for
diatomic chain
A/B=1
u2n  A ei( 2kna t )
u2n1  B ei(( 2n1)ka t )
Analogy with classical mechanical pendulums attached by spring
Amplitude of vibration is strongly
exaggerated!
Longitudinal Eigenmodes in 1D
What if the atoms were opposite charged?
Optical Mode: These
atoms, if oppositely
charged, would form an
oscillating dipole which
would couple to optical
fields with λ~a
Summary: What is phonon?
• Consider the regular lattice of atoms in a uniform
solid material.
• There should be energy associated with the
vibrations of these atoms.
• But they are tied together with bonds, so they
can't vibrate independently.
• The vibrations take the form of collective modes
which propagate through the material.
• Such propagating lattice vibrations can be
considered to be sound waves.
• And their propagation speed is the speed of
sound in the material.