thermal properties

Xing Sheng, EE@Tsinghua

1

Xing Sheng盛兴

Department of Electronic EngineeringTsinghua University

[email protected]

Thermal Properties

Fundamentals of Solid State Physics


Heat Capacity (Thermal Capacity) 热容

Thermal Conductivity 热导

Thermal Expansion 热膨胀

...

Thermal Properties

2


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


4

Xing Sheng盛兴


[email protected]

Thermal Properties- Phonons



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


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


Heat Capacity CV

7

3V B

V

UC NLk

T

Dulong–Petit Law

only valid at high temperature

experimental results


Harmonic Oscillator: Quantum Theory

8

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


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


Internal energy is the ground state energy plus the energy of all the phonons


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


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


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)


Phonon Density of States g()

13

DOS for copper

experiment

Debye2( )g B


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


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


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


Heat Capacity CV - Example

17

~T3

Dulong–Petit Law

Z. Guo, et al., Appl. Phys. Lett. 106, 111909 (2015)

Debye Model


Heat Capacity CV - Example

18

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


Thermal expansion originates from the anharmonicnature of the potential

Vibration increases with temperature

Thermal Expansion 热膨胀

19

r0T = 0 K

T increasesr1

r2

2 3 3( ) ...U r cu gu fu

potential energy U

atom distance

r


Fourier's Law heat flux is proportional to the temperature gradient

Thermal Conduction 热导

20

dTQ

dx

https://www.khanacademy.org

Q - heat flux (W/m2) - thermal conductivity (W/m/K) T - temperature (K)



Thermal Conductivity 热导率

21

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


22

Xing Sheng盛兴


[email protected]

Thermal Properties of Free Electrons



Review

23

Lecture 3.1, Sommerfeld Model

Ashcroft & Mermin, Chapter 2


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


Density of States (DOS) 态密度

25

3/2

1/2

2 2

1 2( )

2edn m

g E EdE

DOS g(E)

Energy

3D Free Electrons


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


Internal energy is the energy of all the free electrons


27

0( ) ( )U g E f E EdE

DOS average electron numberin each state

(Fermi-Dirac Distribution)

electron energy

The Sommerfeld Model


When T = 0 K


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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


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


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)


Thermal conductivity for free electrons


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


Thermal conductivity for metals


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


Relationship of thermal conductivity and electron conductivity for certain metals Lorentz number L


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



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)


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



36

Thank you for your attention

thermal properties

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