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Introductory Nanotechnology
~ Basic Condensed Matter Physics ~
Atsufumi Hirohata
Department of Electronics
Quick Review over the Last Lecture
Waves :
Wave modes :
(
(
transverse wave
(
longitudinal wave )
Photon : (

h
), phonon : (

optical mode
)
)

(
)
acoustic mode )
Contents of Introductory Nanotechnology
First half of the course :
Basic condensed matter physics
1. Why solids are solid ?
2. What is the most common atom on the earth ?
3. How does an electron travel in a material ?
4. How does lattices vibrate thermally ?
5. What is a semi-conductor ?
6. How does an electron tunnel through a barrier ?
7. Why does a magnet attract / retract ?
8. What happens at interfaces ?
Second half of the course :
Introduction to nanotechnology (nano-fabrication / application)
How Does Lattices Vibrate Thermally ?
•
Specific heat
Einstein model
•
•
Debye model
Imagine You Are on a Beach in Summer …
When you walk on a beach ...
Comfortable in water !
Very hot on sand !
Water : difficult to be warmed up / cooled down.
Specific Heat Capacity
In order to compare :
Thermal energy required to increase temperature of a unit-volume material
(1 mole at constant volume)
Molecule
1-atom
2-atom
3-atom
Many-atom (>3)
Material
cV,mol [J/molK]
He
12.5
Ar
12.5
H2
20.4
CO
20.8
HCl
21.0
H2 S
26.9
SO2
32.3
CH4
27.0
C2H2
33.1
cV,mol increases with increasing numbers of atoms in a molecule.
 increases with increasing numbers of degree of freedom.
Thermal Fluctuation in a Molecule
At finite temperature :
Atoms in a molecule vibrates (translation and rotation).
* http://www.wikipedia.org
Amplitude of Lattice Vibration
Amplitude increases with increasing temperature :
 (r)
r
Higher vibration energy state
Thermal activation
Lower vibrtion energy state
Specific Heat 1 - Classical Model for Ideal Gas
Equal volume specific heat :
U 
cV   
T V
According to Dulong-Petit empirical law,

cV  6 [cal/molK] 25.1 [J/molK]
T
(  RT)
In order to explain this law, L. Boltzmann introduced classical thermodynamics :
Average kinetic energy for a particle in a vacuum is written as follows in 3D

EK  3
kBT
2
For 1 mole (N0 : Avogadro constant),

EK,mol 
3N 0kBT 3RT

2
2
where R = N0kB : gas constant, and EK, mol equals to internal energy of ideal gas.

* http://www.wikipedia.org
Specific Heat 1 - Classical Model for a Crystal Lattice
For a crystal lattice, each atom at a lattice point have potential energy of
EU, mol 
3N 0kBT 3RT

2
2
Therefore, internal energy of 1 mol solid crystal is written as


Umol  EK, mol EU, mol  3RT
By substituting R = 1.99 cal/molK,
Umol 
cV, mol  
  3R  5.96 [cal/mol K]
 T V
This agrees very well with Dulong-Petit empirical law.

However, about 1900,
J. Dewar found that specific heat approaches 0
at low temperature.
* http://www.wikipedia.org
Specific Heat 2 - Einstein Model 1
In a crystal lattice as a harmonic oscillator, energy is expressed as
En 
h E
 nh E
2
n  0,1,2, 
Einstein assumed that  is constant for all the same atoms in the oscillator.

Now, the numbers of the oscillators with energy of E0, E1, E2, ... are assumed
to be N0, N1, N2, ..., respectively, and these numbers to follow the MaxwellBoltzmann distribution.
 En  E0 
 nh E 
Nn
 exp
 exp

N0
k
T
k
T


 B 
B
In order to calculate average energy <E> of an oscillator, probability for En is

Nn
N
n
n

 En 

n




N
1
En  n  h E 
 N n  2


 n



n
 nh E 
nh E exp

 kBT 
 nh E 
exp

 kBT 
n

* http://www.wikipedia.org
Specific Heat 2 - Einstein Model 2
Substituting x = -hE / kBT, the second term in the right part becomes
h E
n expnx 
n
 h E

n
ex  2e2x 
expnx  h E
1 ex  e2x 
d
log 1 ex  e2x 
dx

 h E
d
1
1
log

h

E x
dx
1 ex
e 1
Therefore, average energy of an oscillator is

1
1
E  h E  h E
2
exph E kBT 1
 Energy of a phonon

En  nh E
1
n 
exph E kBT 
1
Also, assuming,

Planck distribution

1
h
2 E
 nh E  h E n
By neglecting zero point energy,
n  kBT h E
n  exph E
high temperatureh E  kBT 
kBT  low temperatureh E  kBT 
Specific Heat 2 - Einstein Model 3
Average energy of an oscillator can be modified as
E  h E 2 n h E
 E  h E 2 kBT  kBT


high temperatureh E  kBT 
 Energy only depends on T (= classical model).
 E  h E 2 h Eexp-h E kBT 
low temperatureh E  kBT 
For 1 mol 3D harmonic oscillator,

3
1
Umol  3N 0 E  N 0h E  3N 0h E
2
exph E kBT 1
Thus, equal volume specific heat is

h E 2 exph E kBT 
Umol 
cV, mol  
  3R

 T V
kBT  exph E kBT 12
Since E depends on materials, both cV, mol and hE depends on materials.

Accordingly, characteristic temperature E for hE is introduced as
h E  kBE
 Einstein temperature
Specific Heat 2 - Einstein Model 4
With using Einstein temperature, equal volume heat is rewritten as
E 2 expE T 
E 
cV, mol  3R 
 3RfE  
 T  exp T 12
 T 
E
 
f E  x 2ex ex 1


2
 fE (x) : Einstein function
For high temperature (T > E),
f E x 1
cV, mol  3R
 Agrees with Dulong-Petit empirical law

For low temperature (T << E),
cV, molexpE T 
With decreasing temperature,
Einstein model decrease

faster than measurement.
* N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, London, 1976).
Specific Heat 2 - Einstein Model 5
In Einstein model, discrete energy levels are assumed :
 1 
E3  3 h E
 2 
 1 
E2  2  h E
 2 
 1 
E1  1 h E
 2 
1
E0 
h E
2
n3
n2
n 1
n0
For low temperature (T << E), most of the atoms stay at the zero point energy
(E0).

With increasing temperature, very few atoms are excited to E1 as compared with
the theoretical prediction.
 Departure from experiment at low temperature
Specific Heat 3 - Debye Model 1
Debye introduced quantum harmonic oscillators :
Phonon can be produced by lattice vibration and can fill in one energy state.
 follows Planck distribution with energy of
E  h  
Numbers of particles occupying an energy level Ei, which is gi-fold degenerated
at angular frequency of i are calculated to be

ni  gi n 
gd
gi

exp i kBT 1 exp i kBT 1
  is treated as a continuous function

Here, the density of states for a phonon is written as
gd 
V
2 
3
4 k 2dk
For longitudinal / transverse waves,


l  vl k, t  vt k
Vl 2 1
Vt 2 2
gl  
, gt  
2
3
2 vl
2 2 vt 3
* http://www.wikipedia.org
Specific Heat 3 - Debye Model 2
By using average  and add both longitudinal and transverse waves :
V 2  1
2 
2
g 


C



3
2 2 
vt 3 
vl


V  1
2 
C  2 

 2 v 3  v 3 

 l

t 
For a N-atom 3D lattice, 3N modes are allowed :




CD 3
C d 
 3N
0
3
9N 2
g 
0    D
3
D

D
g ()
C 2
2
  D : Debye angular frequency
Therefore, Debye temperature is defined as
D 
D
kB
0
D

Specific Heat 3 - Debye Model 3
Now, numbers of states can be rewritten as
1
9N 2
n
d
exp  kBT 1 D3
By neglecting the zero point energy, total internal energy is

E  n  
U 




D
0
1
gd
exp  kBT 1

9N 2
exp  kBT 1 D
3
d
Therefore, equal volume specific heat is calculated to be
U 
cV, mol    
T V

D
0
 T 3
cV, mol  9R 
D 

  2 exp  kBT  9N 2
kB 
d
2
3
k
T
 B  exp  kBT 1 D
D T
0
x 4ex
e 1
x
2
dx



D
, D 
, R  NkB  N 0kB 
x 
k
T
k


B
B
Specific Heat 3 - Debye Model 4
For high temperature (D << T),
 T 3 1 D 3
cV, mol  9R     3R
D  3  T 

x 4ex


  
e 1
x
2
x 4 1 x

2
x  x 2 2
x4
 2  x2
x
 Agrees with Dulong-Petit empirical law

For low temperature (T << D),
3
3




12 R T
T
3
cV, mol 
   464.5  cal/mol K T
5 D 
D 
4




D T
0
x 4ex
e 1
x
2
dx 


0
4 4
dx 
2
x
15
e 1
x 4ex
 
 Agrees with experiment