Solid State Physics
Lecture 8 – The Debye model
Professor Stephen Sweeney
Advanced Technology Institute and Department of Physics
University of Surrey, Guildford, GU2 7XH, UK
s.sweeney@surrey.ac.uk
Solid State Physics - Lecture 8
Recap from Lecture 7
• Concepts of “temperature” and thermal
equilibrium are based on the idea that
individual particles in a system have
some form of motion
• Heat capacity can be determined by
considering vibrational motion of
atoms
• We considered two models:
• Dulong-Petit (classical)
• Einstein (quantum mechanical)
• Both models assume atoms act
independently – this is made up for in
the Debye model (today)
Solid State Physics - Lecture 8
Dulong-Petit
Summary of Dulong-Petit and
Einstein models of heat capacity
Dulong-Petit model (1819)
Einstein model (1907)
•
•
•
•
•
Atoms on lattice vibrate
independently of each
other
Completely classical
Heat capacity
independent of
temperature (3NkB)
Poor agreement with
experiment, except at
high temperatures
•
•
Atoms on lattice vibrate
independently of each
other
Quantum mechanical
(vibrations are quantised)
Agreement with
experiment good at very
high (~3NkB) and very low
(~0) temperatures, but
not inbetween
Solid State Physics - Lecture 8
A more realistic model…
•
Both the Einstein and Dulong-Petit models treat each
atom independently. This is not generally true.
•
When an atom vibrates, the force on adjacent atoms
changes causing them to vibrate (and vice-versa)
•
Oscillations can be broken down into modes
1D case
Nice animations here: http://www.phonon.fc.pl/index.php
Java applet: http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html
Solid State Physics - Lecture 8
3D case
Debye model
•
Basic idea similar to Einstein model, with one key difference:
Einstein:
Energy of system = Phonon Energy x Average number of phonons
Debye:
Energy of system = Phonon Energy x Average number of phonons x number of modes
Einstein: number of modes = number of atoms
Debye: each mode has its own k value (and
hence frequency)
Solid State Physics - Lecture 8
The number and type of
modes are the key difference
Modes: lattice vibrations
Modes exist in various areas of physics/nature
Butterfly wing-beat
Water Molecules
Guitar modes
Solid State Physics - Lecture 8
Modes: Quantum mechanics
Modes are quantised in units of
mode is 

where the fundamental frequency of each
The Einstein model assumed that each oscillator has the same frequency
Debye theory accounts for different possible modes (and therefore different

)
Modes with low  will be excited at low temperatures and will contribute
to the heat capacity. Therefore heat capacity varies less abruptly at low T
compared with Einstein model
Low frequency modes
correspond to multiple
atoms vibrating
together (sound or
acoustic waves)
Solid State Physics - Lecture 8
Standing waves: revision
Consider a vibrating string
n=4
Lowest (fundamental) frequency
n=3
n=2
More generally
n=1
L
Other results follow:

2

2L
n L  λ
2
n
vn
vn
f  
 ω  2πf 
 2L
L
v
k
2π


n
L
Solid State Physics - Lecture 8
L
Standing waves in a 1D crystal
Consider solid as a continuous elastic medium:
N atoms, 3 degrees of freedom  3N standing modes
a
2π πn n
k


λ
L
Na
1D array of atoms:
L=Na
max  2 L  2 Na  kmin
Fundamental mode
(n=1)
π

Na
λmin  2a  kmax 
Highest order mode
(n=N)
Therefore we get N modes for N atoms
Solid State Physics - Lecture 8
 nmax  N

a
Standing waves in 2D crystals
Fundamental mode (2D)
Each component of the wave is quantised
separately and added in quadrature
k x  ky 
x
L
y
L
π
L
Magnitude of k-vector for mode
k
kx
2
π
 ky  2
L
2
Corresponding angular frequency
ω  2πf  2
v

 vk
ωv 2

L
Solid State Physics - Lecture 8
Standing waves in 2D crystals:
Degeneracy
ky 
π
kx 
L
2π
L
ky 
2π
kx 
L
π
L
x
y
L
L
L
L
π
    2 
k    
  5
L
L  L 
2
π
 2    
k 
    5
L
 L  L
2
2
In both cases
ωv 5

2
so these two modes are degenerate
L
As frequency increases, more and more states share the same frequency & energy
(called DEGENERACY)
Solid State Physics - Lecture 8
Back to reciprocal space… (2D)
•
We can represent each mode as a
point in reciprocal (k) space
Q. How many modes are available at a
particular k value?
A. Need three pieces of information:
1. How “big” is an individual k-state
2. How much of k-space is covered at a
particular k
3. Account for degeneracy
kl 2
g k dk 
dk
2
Solid State Physics - Lecture 8
Number of States in 3D
In 3D we consider the number of states
within a sphere of radius k
Sphere “volume” =
kz
4 3
k
3
k
ky
“volume” of k-state =
3
l3
kx
2
Vk
g k dk 
dk
2
2

l

l
k-state
Solid State Physics - Lecture 8

l
Number of States in 3D
Vk 2
g k dk 
dk
2
2
We know that
Hence
kz
k
ω  vk  dω  vdk
V 2
g  d 
d
2 3
2 v
ky
kx
i.e. the number of standing waves (modes)
increases as 2

l
Sound can propagate with 2 transverse and 1 longitudinal
wave in a solid  total no. of states = 3g()d 
Solid State Physics - Lecture 8

l
k-state

l
Debye frequency
For any one wavelength of oscillation there are shorter wavelength oscillations that
will also have the atoms in the same position on the lattice (c.f. aliasing in electronics)
There is a minimum wavelength which can
oscillate which corresponds to a maximum
frequency, max (Debye frequency)
We can calculate max since we know (from
earlier) that the maximum number of states
= 3N
 3N 
max
 3g  d
0

ωmax

0
3V
V
3
dω

ω
max
2 3
2 3
2π v
2π v
2
Solid State Physics - Lecture 8
So,
ωmax
 2 N
 v 6π

V

1
3
Some crystal modes of vibration
Phonon animations here: http://www.phonon.fc.pl/index.php
Solid State Physics - Lecture 8
Debye model: Total average energy of System
From earlier: Energy of =
system
E
max

0
Phonon
Average no.
x
energy
of phonons
1
 
 3g  d 
  
  1
exp 
 k BT 
Integrate over
all modes
(NB: Ignoring zero-point energy)
Solid State Physics - Lecture 8
x
max

0
No. of
modes
3g  
d
  
  1
exp 
 k BT 
Debye model: Total average energy of System
E
max

0
3g  
d
  
  1
exp 
 k BT 
3V
E
2 2 v 3
V 2
From before: g  d 
d
2 3
2 v
ω
x
k BT
Make substitution:
4
B
4
3Vk T
E 
2 2 v 3 3
D
T

0
max

0
 3
d
  
  1
exp 
 k BT 
and define Debye temperature:
x3
dx
exp x   1
ωmax
θD 
kB
From which (finally) we can
extract the heat capacity, C
Solid State Physics - Lecture 8
The Debye Temperature D
This is perhaps the most useful parameter in the Debye theory
• It allows us to predict the heat capacity at any temperature
• It provides an indication of the temperature at which we approach the
classical limit of the Dulong-Petit theory
ω
θ D  max
kB
From earlier, we know that
Therefore,
v
max
 2 N
 6

V

1

3
 DkB 
and
2 N 
6



 
V
N

ωmax  v 6π 2 
V

1
3
 13
So if we know N/V then we can predict the
speed of sound in a solid
Solid State Physics - Lecture 8
The Debye Temperature D: examples
High D corresponds to a large max
Large max implies large forces, low max implies weak bonds
Diamond D = 2230K
Dulong-Petit poor fit at room temperature. Strongly bonded
Iron
D = 457K
Dulong-Petit reasonable fit at room temperature.
Lead
D = 100K
Dulong-Petit good fit at room temperature. Weakly bonded
Solid State Physics - Lecture 8
Debye model: Heat Capacity
4
B
4
3Vk T
E
2 2v 3 3
D
T

0
x3
dx
exp x   1
where
ω
x
k BT
At high T:
exp( x)  1  x  ...
x is small 
D
T

0
3
x
dx 
exp x   1
Vk T
E 
2 v 
4
3
B
D
2 3 3
 E  3Nk BT
and since
D
T

0
so…
3
x
dx 
1  x  ...  1
v
 DkB 
2 N 
6



 
V
Heat capacity, C 
dE
 3Nk B
dT
Solid State Physics - Lecture 8
D
T

0
x 2 dx 
 D3
3T 3
 13
Dulong-Petit !
dE
 Cmolar 
 3N A k B
dT
Debye model: Heat Capacity
4
B
4
3Vk T
E
2 2v 3 3
D
T

0
x3
dx
exp x   1
where
At low T:

Take limit that D/T   and use identity
so…
x3
4
0 exp x   1dx  15
3Vk B4T 4  4 3 4 Nk BT 4
E

2 3 3
2 v  15
5 D3
Heat capacity, C 
v
ω
x
k BT
 DkB 
N
 6

 
V
 13
Debye T3 law
T 
dE 12

Nk B  
dT
5
 D 
2
Solid State Physics - Lecture 8
4
3
Debye T3 law
Heat Capacity (mJ mol-1 K-1)
Heat capacity for solid Argon (from Kittel)
T3 (K3)
Solid State Physics - Lecture 8
Comparison of Dulong-Petit, Einstein
and Debye models of heat capacity
Dulong-Petit
Solid State Physics - Lecture 8
Thermal Conductivity
•
Thermal conduction is a measure of how much heat energy is
transported through a material per unit time
•
In metals conduction is due to free electrons (a later lecture)
•
In non-metals conduction is largely due to phonons
• most hard insulators have a low thermal conductivity
•
Phonons have energy and can therefore conduct heat
• Scattering mechanisms limit the thermal conductivity of non-metals,
due to
• Imperfections (grain boundaries, point defects, dislocations)
• Phonons themselves can scatter other phonons (Umklapp
processes – we won’t cover that here)
Solid State Physics - Lecture 8
Thermal Conductivity
Thermal conductivity
Thigh
Tlow
1 dE
dT
Q
 
A dt
dx
Area, A
Energy flow along x
 
1 dE dx
1 dE dx

A dt dT
A dT dt
i.e. thermal conductivity scales with heat capacity
Solid State Physics - Lecture 8
C
v