thermal properties
TRANSCRIPT
Xing Sheng, EE@Tsinghua
1
Xing Sheng盛兴
Department of Electronic EngineeringTsinghua University
Thermal Properties
Fundamentals of Solid State Physics
Xing Sheng, EE@Tsinghua
Heat Capacity (Thermal Capacity) 热容
Thermal Conductivity 热导
Thermal Expansion 热膨胀
...
Thermal Properties
2
Xing Sheng, EE@Tsinghua
Thermal properties are the combinations of properties of lattice vibration (phonons) and free electrons
For insulators, there are no free electron. Thermal properties of lattice vibration (phonons) dominate.
For metals,
thermal properties = phonon part + free electron part
Thermal Properties
3p e
, ,V V p V eC C C Thermal capacity
Thermal conductivity
Xing Sheng, EE@Tsinghua
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Xing Sheng盛兴
Department of Electronic EngineeringTsinghua University
Thermal Properties- Phonons
Fundamentals of Solid State Physics
Xing Sheng, EE@Tsinghua
Vibration amplitude is continuous
Energy is continuous, and temperature dependent
Harmonic Oscillator: Classical Theory
5
potential energy V
atom distance
r
r0
Thermal vibrationaround r0
2 21 1 1 1
2 2 2 2B B BE Kx mv k T k T k T
Equipartition Theorem (能量均分定理)
x
energy of one spring + one atom= potential energy + kinetic energy
Xing Sheng, EE@Tsinghua
Total vibration energy of a crystal all the springs + all the atoms (3NL)
Heat capacity (Specific heat) 比热容 energy per unit of temperature
Internal Energy 内能
6
3 BU NLk T
3V B
V
UC NLk
T
Dulong–Petit Law
In the system, every atom contributes an energy of 3kBT
N - # of primitive cellsL - # of atoms in a primitive cell
Xing Sheng, EE@Tsinghua
Heat Capacity CV
7
3V B
V
UC NLk
T
Dulong–Petit Law
only valid at high temperature
experimental results
Xing Sheng, EE@Tsinghua
Harmonic Oscillator: Quantum Theory
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2 2 21 1( )
2 2V x Kx m x
x
2 2
2( ) ( ) ( ) ( )
2x V x x E x
m x
K
m
1
2nE n
0,1,2,...n
Vibration energy is quantized
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator
Xing Sheng, EE@Tsinghua
Quantum theory Vibration energy is quantized
At each state, the energy is the ground state energy (ħ/2)plus energy of n phonons (ħ)
average n in each state follows Bose-Einstein distribution
If there are N primitive cells, and L atoms in each cell, the total number of states is 3NL
Harmonic Oscillator: Quantum Theory
9
1( )
2E n
0,1,2,...n
/
1
1Bk Tn
e
Xing Sheng, EE@Tsinghua
Internal energy is the ground state energy plus the energy of all the phonons
Internal Energy 内能
10
max
3
0 /1
0 /0
1
( )1
i B
B
NLi
k Ti
k T
U Ue
U g de
ground state DOS average phonon numberin each state
phonon energy
V
V
UC
T
heat capacity
Xing Sheng, EE@Tsinghua
At high temperature, T >> 0 K
Heat Capacity CV
11
3
0 /1
3
01
0
1
/
3
i B
NLi
k Ti
NLi
i i B
B
U Ue
Uk T
U NLk T
3V B
V
UC NLk
T
constantheat capacity
Dulong–Petit Law
Xing Sheng, EE@Tsinghua
At low and medium temperatures
Heat Capacity CV
12
k/a0/aFor acoustic phonons:
assume linear -k relation The Debye Model (德拜模型)
2( )g B
For optical phonons: assume all the phonons have the same
The Einstein Model(爱因斯坦模型)
Xing Sheng, EE@Tsinghua
Phonon Density of States g()
13
DOS for copper
experiment
Debye2( )g B
Xing Sheng, EE@Tsinghua
At very low temperature, T --> 0 K
Most phonons have --> 0, just focus on the acoustic branch
Heat Capacity CV
14
max
max
0 /0
20 /0
( )1
1
B
B
k T
k T
U U g de
U B de
34
312
5V B
V D
U TC Nk T
T
heat capacity
Debye
T3 Law
The Debye Model
1/326gDD
B B
v N
k k V
Debye Temperature
Xing Sheng, EE@Tsinghua
At very low temperature, T --> 0 K
Most phonons have --> 0, just focus on the acoustic branch
Heat Capacity CV
15
1/326gDD
B B
v N
k k V
Debye Temperature is around room temperature for most materials
343
428
645
2230
D (K)
Cu
Al
Si
C (diamond)
http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/thrcn2.html
Xing Sheng, EE@Tsinghua
At median temperature, assume all the phonons in an optical branch have frequency 0
Heat Capacity CV
16
0
0
2 /0
/ 2
2 /
/ 2
( 1)
( 1)
B
B
E
E
k T
V B k TV B
TE
B T
U eC Nk
T k T e
eNk
T e
heat capacity(of one branch)
The Einstein Model0
00 / 1Bk T
U U Ne
0E
Bk
Einstein Temperature
Xing Sheng, EE@Tsinghua
Heat Capacity CV - Example
17
~T3
Dulong–Petit Law
Z. Guo, et al., Appl. Phys. Lett. 106, 111909 (2015)
Debye Model
Xing Sheng, EE@Tsinghua
Heat Capacity CV - Example
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Zhang Z.M. (2020) Thermal Properties of Solids and the Size Effect. In: Nano/Microscale Heat Transfer. Mechanical Engineering Series. Springer, Cham. https://doi.org/10.1007/978-3-030-45039-7_5
The Debye Model matches better with experimental results
Xing Sheng, EE@Tsinghua
Thermal expansion originates from the anharmonicnature of the potential
Vibration increases with temperature
Thermal Expansion 热膨胀
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r0T = 0 K
T increasesr1
r2
2 3 3( ) ...U r cu gu fu
potential energy U
atom distance
r
Xing Sheng, EE@Tsinghua
Fourier's Law heat flux is proportional to the temperature gradient
Thermal Conduction 热导
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dTQ
dx
https://www.khanacademy.org
Q - heat flux (W/m2) - thermal conductivity (W/m/K) T - temperature (K)
Xing Sheng, EE@Tsinghua
Thermal conductivity
Thermal Conductivity 热导率
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CV - thermal capacity vg - sound speed l - phonon mean free pathp - phonon relaxation time
l and p is dependent on crystal structure, defects, impurities, ...
21 1
3 3V g V g pC v l C v
Q: Which material has the highest thermal conductivity?
Ashcroft & Mermin, p20
Xing Sheng, EE@Tsinghua
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Xing Sheng盛兴
Department of Electronic EngineeringTsinghua University
Thermal Properties of Free Electrons
Fundamentals of Solid State Physics
Xing Sheng, EE@Tsinghua
Review
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Lecture 3.1, Sommerfeld Model
Ashcroft & Mermin, Chapter 2
Xing Sheng, EE@Tsinghua
Density of States (DOS) 态密度
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DOS - number of energy states/levels per unit energy in [E, E+dE], per unit volume
( )dn
g EdE
2 2
2 e
kE
m
3/2
3/2
2 2
1 2
3emn E
3/2
1/2
2 2
1 2( )
2edn m
g E EdE
2 1/3(3 )k n
n - free electron density
Xing Sheng, EE@Tsinghua
Density of States (DOS) 态密度
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3/2
1/2
2 2
1 2( )
2edn m
g E EdE
DOS g(E)
Energy
3D Free Electrons
Xing Sheng, EE@Tsinghua
Density of Electrons
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Density of electrons = DOS * probability ( ) ( )f E g E
DOS
( )/
1( )
1BE k Tf E
e
When T > 0 K, some electrons are excited to higher states (from 1 to 2)
f(E)
Energy
T = 0 KT > 0 K
~ EF
1
0
Xing Sheng, EE@Tsinghua
Internal energy is the energy of all the free electrons
Internal Energy 内能
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0( ) ( )U g E f E EdE
DOS average electron numberin each state
(Fermi-Dirac Distribution)
electron energy
The Sommerfeld Model
Xing Sheng, EE@Tsinghua
When T = 0 K
Internal Energy 内能
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( )/
1( )
1BE k Tf E
e
00
( )
3
5
FE
F
U g E EdE
E
f(E)
Energy
T = 0 KT > 0 K
~ EF
1
0
Homework 4.4
Xing Sheng, EE@Tsinghua
When T > 0 K
Heat Capacity CV
29
22
0 ( ) ( )6
B FU U k T g E
2
,2
V e B
V F
U TC nk T
T T
heat capacity
TF - Fermi temperature (~104 K)
Only a few electrons around EF contribute to CV, e.At room temperature, for free electrons CV, e << NkB
much smaller than CV from phonons
Xing Sheng, EE@Tsinghua
For metals at very low temperature T ~ 0 K Thermal properties = phonon part + free electron part
Heat Capacity CV
30
3, ,V V p V eC C C AT T 2/VC T AT
W. Lien and N. Phillips, Phys. Rev. 133, A1370 (1964)
Xing Sheng, EE@Tsinghua
Thermal conductivity for free electrons
Thermal Conductivity 热导率
31
CV - thermal capacity vF - Fermi velocity l - electron mean free pathe - electron relaxation time
21 1
3 3e V F V F eC v l C v
Ashcroft & Mermin, p20
Xing Sheng, EE@Tsinghua
Thermal conductivity for metals
Thermal Conductivity 热导率
32
2 2, ,
1 1
3 3p e V p g p V e F eC v C v
For conductive metals like Cu or Ag, vF >> vg
electron part dominates
Xing Sheng, EE@Tsinghua
Relationship of thermal conductivity and electron conductivity for certain metals Lorentz number L
Thermal Conductivity 热导率
33
228 22.44 10 W /K
3BkL
T e
Agree with experimental results (Wieddemann and Franz law in 1853)
2.31
Ag
2.30
Cu
2.23
Al
2.35
Au
2.45
Pb
2.47
Fe
L (10-8 W*/K2)
metals at 300 K
Q: Which metal has the highest thermal conductivity?
Homework 8.7
Xing Sheng, EE@Tsinghua
Thermal Conductivity 热导率
34
Q: High thermal conductivities of diamond and graphitehave different origins. Why?
400Cu
~200C (graphite)
0.05
1
130
2000
(W/m/K)
paper
glass
Si
C (diamond)
Xing Sheng, EE@Tsinghua
Thermal properties are the combinations of properties of lattice vibration (phonons) and free electrons
For insulators, there are no free electron. Thermal properties of lattice vibration (phonons) dominate.
For metals,
thermal properties = phonon part + free electron part
Summary
35p e
, ,V V p V eC C C Thermal capacity
Thermal conductivity
Xing Sheng, EE@Tsinghua
36
Thank you for your attention