Lattice Dynamics

Physical properties of solids

determined by electronic structure

related to movement of atoms

about their equilibrium positions

•
Sound velocity

•
Thermal properties:
-
specific heat


-
thermal expansion


-
thermal conductivity


(for semiconductors)

•
Hardness of perfect single crystals
(without defects)






Lattice Dynamics



Reminder to the physics of oscillations and waves:

Harmonic oscillator in classical mechanics:

Example: spring pendulum

Hooke’s law

2
2
1
x
D
E
pot

x

spring
F
x
m



Equation of motion:

0


x
D
x
m


or

0


x
~
m
D
x
~


where

))
t
(
x
~
Re(
)
t
(
x

Solution with

t
i
e
A
~
)
t
(
x
~


)
t
cos(
A
)
t
(
x




where

m
D


X=A sin
ω
t



X

Dx

m
D


Traveling plane waves:

)
kx
t
(
cos
A
)
t
(
y





X

0

Y

X=0:

t
cos
A
)
t
(
y


t=0:

kx
cos
A
)
x
(
y

Particular state of oscillation Y=const

0


in particular

or

)
kx
t
(
i
e
A
~
)
t
(
y
~



)
kx
t
(
cos
A
)
t
(
y



travels according



0




.
const
dt
d
kx
t
dt
d
k
v
x








/
2
2
v



)
kx
t
(
i
e
A
~
)
t
(
y
~



2
2
2
2
2
1
x
y
t
y
v





solves wave equation

Transverse wave

Longitudinal wave

Standing wave

)
t
kx
(
i
e
A
~
y
~



1
)
t
kx
(
i
e
A
~
y
~



2


)
t
kx
(
i
)
t
kx
(
i
s
e
e
A
~
y
~
y
~
y
~








2
1


t
i
t
i
ikx
e
e
e
A
~





t
cos
e
A
~
ikx


2
Re( ) 2 cos cos
s s
y y A kx t
 

Large wavelength
λ


0
2




k
Crystal can be viewed as a continuous medium:

good for

m
8
10



λ
>10
-
8
m

10
-
10
m

Speed of longitudinal wave:





s
B
v
where B
s
:

bulk modulus

with

compressibility

B
s

determines elastic deformation energy density

2
2
1


s
B
U
dilation

V
V



(ignoring anisotropy of the crystal)

s
B
1




s
B
v
E.g.: Steel

B
s
=160 10
9
N/m
2

ρ
=7860kg/m
3

s
m
m
/
kg
m
/
N
v
4512
7860
10
160
3
2
9


(
click for details in thermodynamic context
)


>>

interatomic spacing


continuum approach fails

In addition:

phonons

vibrational modes quantized

Linear chain:

Remember: two coupled harmonic oscillators

Superposition of normal modes

Symmetric mode

Anti
-
symmetric mode

Vibrational Modes of a Monatomic Lattice

generalization

Infinite linear chain

How to derive the equation of motion in the harmonic approximation

?

n

n+1

n+2

n
-
1

n
-
2

u
n

u
n+1

u
n+2


u
n
-
1

u
n
-
2

u
n

u
n+1

u
n+2


u
n
-
1

u
n
-
2

fixed


D



1




n
n
l
n
u
u
D
F


1




n
n
r
n
u
u
D
F
a

Total force driving atom n back to equilibrium





1
1







n
n
n
n
n
u
u
D
u
u
D
F
n

n



n
n
n
u
u
u
D
2
1
1





equation of motion

n
n
F
u
m





n
n
n
n
u
u
u
m
D
u
2
1
1







Solution of continuous wave equation

)
t
kx
(
i
e
A
u



approach for linear chain

)
t
kna
(
i
n
e
A
u



)
t
kna
(
i
n
e
A
u





2


ika
)
t
kna
(
i
n
e
e
A
u




1
ika
)
t
kna
(
i
n
e
e
A
u





1
,

,

?

Let us try!



2
2






ika
ika
e
e
m
D


ka
cos
m
D



1
2
2
)
/
ka
sin(
m
D
2
2












2 2
2
1 1
1 1
1 1
1
2 2
0
2 2 0
2
2 0
n n n n n
n n
n n n n n
n n n n
D
L mu u u u u
d L L
dt u u
D
mu u u u u
D
u u u u
m
 
 
 
 
    
 
 
 
 
    
 
 
   
Alternative without thinking

Lagrange formalism

)
/
ka
sin(
m
D
2
2


Continuum limit of acoustic waves:

m
D
2
k

0
2




k
...
/
ka
/
ka
sin


2
2
k
a
m
D


a
m
D
v
k



Note: here pictures of transversal waves

although calculation for the longitudinal case

k

)
t
)
k
(
na
k
(
i
e
A
n
u





a
h
k
k




2
)
k
(
)
k
(




)
t
na
k
(
i
e
A




, here h=1

)
t
na
)
a
h
k
((
i
e
A





2
n
h
i
e
)
t
na
k
(
i
e
A




2
)
t
na
k
(
i
e
A



1
2


n
h
i
e
))
k
(
,
k
(
n
u
))
k
(
,
k
(
n
u





a
h
k
k




2
1
-
dim. reciprocal

lattice vector

G
h

a
k
a





Region

is called

first Brillouin zone

We saw: all required information contained in a particular volume in reciprocal space

first Brillouin zone

1d:

a

x
e
a
n
n
r

x
e
a
h
h
G


2
m
n
r
h
G



2
where m=hn integer

a

2
1st Brillouin zone

In general: first Brillouin zone

Wigner
-
Seitz cell of the reciprocal lattice

Brillouin

zones


Vibrational Spectrum for structures with 2 or more atoms/primitive basis

Linear diatomic chain:

2n

2n+1

2n+2

2n
-
1

2n
-
2

u
2n

u
2n+1

u
2n+2


u
2n
-
1

u
2n
-
2


D

a

2a



n
u
n
u
n
u
m
D
n
u
2
2
1
2
1
2
2







Equation of motion for atoms on even positions
:

Equation of motion for atoms on even positions
:



1
2
2
2
2
2
1
2






n
u
n
u
n
u
M
D
n
u


)
t
kna
(
i
e
A
n
u



2
2
Solution with:

)
t
ka
)
n
((
i
e
B
n
u





1
2
1
2
and













A
)
ika
e
ika
e
(
B
m
D
A
2
2












B
)
ika
e
ika
e
(
A
M
D
B
2
2
ka
cos
B
m
D
m
D
A
2
2
2









ka
cos
A
M
D
M
D
B
2
2
2


















2
2
2
m
D
ka
cos
B
m
D
A
ka
cos
Mm
D
M
D
m
D
2
2
4
2
2
2
2

















ka
cos
Mm
D
m
D
M
D
Mm
D
2
2
4
4
2
2
2
2
2
4







0
2
1
2
4
2
2
4

























ka
cos
Mm
D
M
D
m
D
ka
sin
2
Mm
ka
sin
M
m
D
M
m
D
2
4
2
1
1
1
1
2


















1 1
2D
m M
 
 
 
 
2
2
M
1
m
1
D
M
1
m
1
D

















m
D
2


,

M
D
2


m
D
2


M
D
2


2

2

•
Click on the picture to start the animation M
-
>m


note wrong axis in the movie

:
a
k
2


Atomic Displacement

Optic Mode

M
m
k
A
B



0
Atomic Displacement

Acoustic Mode

1
0


k
A
B
Click for animations

Dispersion curves of 3D crystals

•
Every additional atom of the primitive basis

•
3D crystal: clear separation into longitudinal and transverse mode only possible in


particular symmetry directions

•
Every crystal has 3 acoustic branches

sound waves

of elastic theory

1 longitudinal

2 transverse

acoustic

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

Intuitive picture:

1atom

3 translational degrees of freedom

3+3=6 degrees of freedom=3 translations+
2rotations

+1vibraton

Solid:
p
N atoms

no translations, no rotations

3p

N vibrations

x

y

z

# of primitive

unit cells

# atoms

in primitive

basis

diamond lattice: fcc lattice with basis

(0,0,0)

)
,
,
(
4
1
4
1
4
1
L
ongitudinal
A
coustic

L
ongitudinal
O
ptical

T
ransversal
A
coustic

degenerated

Part of the phonon dispersion relation of diamond

T
ransversal
O
ptical

degenerated

P=2

2x3=6 branches expected

2
fcc

sublattices

vibrate against one another

However, identical atoms

no dipole moment

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)

Inelastic interaction of light and particle waves with phonons

Constrains: conservation law of

momentum

energy

Condition for

elastic

scattering

hkl
G
k
k


0
in

±

q

incoming wave

phonon

wave

vector

hkl
G
q
k
k







0
0
0






)
q
(



elastic

sattering

in

“quasimomentum”

0
2
2
0
2
2
2
2




)
q
(
n
M
k
n
M
k



for neutrons

for photon

scattering

Phonon spectroscopy



0


)
q
(


0
k

k

q

Triple axis neutron spectrometer

@ ILL in Grenoble, France

Lonely scientist in the reactor hall

Very expensive and involved experiments


Table top alternatives

?

Yes, infra
-
red absorption and


inelastic light scattering (Raman and Brillouin)

However only

0

q
accessible

see homework #8