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
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