• Specific Heat of Solids

• Quantum Size Effect on the Specific Heat

• Electrical and Thermal Conductivities of

Solids

• Thermoelectricity

• Classical Size Effect on Conductivities and

Quantum Conductance

THERMAL PROPERTIES OF SOLIDS AMD THE SIZE EFFECT

Lattice Vibration in Solids: Phonon Gas

Specific Heat of Solids

Atoms in solids Inter-atomic forces keep them in position only move around by vibrations near their equilibrium positions

Specific heat of bulk solids: lattice vibration, free electronmacroscopic behavior from microscopic point of view

Lattice Periodical array of atoms Lattice vibration contribute to thermal energy storage and heat conduction

Einstein Specific Heat Model

3 vibrational degree of freedom high-temperature limit of the specific heat of elementary solidsgood prediction for specific heat of solid at hightemperature

3vc R Dulong-Petit law

Einstein Specific Heat Model

1, 1,2, ...

2i h i

simple harmonic oscillator model each atom : independent oscillator All atoms vibrate at the same frequency.

quantized energy level

specific heat as a function of temperature

1, 1,2, ...

2i i h i

B0 0B

1exp exp[ / ]

2i

V i ii i

Z g g i h k Tk T

Quantized energy of atoms

Vibrational partition function

E E/ 2 /E

0 0

1exp[ / ]

2T i T

i i

i T e e

E E E/ 2 / 2 /1T T Te e e

E E E/ 2 / / 21 ( )T T Te e e E

E

/ 2

/1

T

T

e

e

Einstein temperature

E B/h k

Internal energy

Specific heat

2 ln,

V

ZU NkT

T

E /Eln ln 1

2TZ e

T

EB E /

1 1

2 1TU Nke

E E

E E

2 / 2 /E E

B 2 22 2/ /( )

1 1

T T

vT T

v

U e ec T Nk R

T T Te e

vibration along single axis

E

E

2 /E

22 /( ) 3

1

T

vT

ec T R

T e

vibration along three axes

Limitation of Einstein model

2 /

22 /( ) 3

1

E

E

TE

vT

ec T R

T e

0 0, 3 v v Ec T c R T as as

Einstein specific heat is significantly lower than the experimental data in the intermediate range.

Force-spring interaction must be considered.

Debye Specific Heat Model

AssumptionVelocity of sound: same in all crystalline directions and for all frequenciesa high-frequency cutoff and no vibration beyond this frequency (shortest wavelength of lattice wave should be on the order of lattice constants)Upper bound m, determined by the total number

of vibration modes 3N (N : number of atoms)Vibrations inside the whole crystal just like standing waves

Equilibrium distributionPhonon : Bose-Einstein statistics

The total number of phonons is not conserved since it

depends on temperature.

p

, h

h pv

BE statistics

1

1i

i

i

N

g e e

0, 1/ Bk T

/

1

1i B

ik T

i

N

g e

The energy levels are so closely spaced that it can be regarded as a continuous function : BE distribution function

BE /

1( )

1Bh k T

dNf

dg e

total number of phonons is not conserved

D() : density of states of phonons

the number of quantum states per unit volume per unit frequency or energy (h) interval

va: weighted average speed

Degeneracy of phonons

The number of quantum states per unit volume in the phase spaceGiven volume V and within a spherical shell in the momentum space (from p to p + dp)

2 2

3 3p

4 4Vp dp V ddg

h v

2

3p

( ) 4( )

dg g dD d d

V V v

2 22

3 3 3 3p a

4 1 2 12( ) 4

l t

Dv v v v

p

hp

v

BE /

1( )

1Bh k T

dNf

dg e

BE BE( ) ( ) ( )dN f dg f g d

BE ( ) ( )f g ddNdn

V V

BE0( ) ( )n f D d

Relation between f() and fBE()

( )( )

dg g dD d

V V

BE ( ) ( ) ( )dn f g d f d

0( )f d

upper limit of the frequency (due to lattice constant)

Total number of quantum states must be equal to 3N

m2

30 0 0a

3 12( )

dg ND d d

V V v

3m

3a

43N

V v

1/ 3

am a

3

4

nv

Debye temperature1/ 3

amD a

B B

3

4

nh hv

k k

na = N/V : number density of atoms

( )( )

dg g dD d

V V

Vibration contribution to the internal energy

Distribution function for phonons

BE

1( ) ( ) ( )

dNf D f

V d

B B B

22 2a

m/ / /3 3 3a m m

912 9( ) ,

1 1 1h k T h k T h k T

n Nf

v e e V e

vibration contribution to internal energy

0 0( )U U f Vh d

m

0( )f Vh d

m

B

3

/30m

9

1h k T

N hd

e

1/ 32a

BE m a/3a

312 1( ) , ( ) ,

41Bh k T

nD f v

v e

B

hx

k T

Let

B B k Tx k Tdx

dh h

m D

B

33B

3B

3/30 0mm

99

11

x

xh k T

k TN x h

k TN h hd dx

hee

D

3 3B

0 B 0m

91

x

x

k T xU U Nk T dx

h e

D

3 3

B 0D

9 1

x

x

T xNk T dx

e

m DD

B

hx

k T T

Molar specific heat

D

3 3

0 B 0D

91

x

x

T xU U Nk T dx

e

A BR N k

D

3 3

0 0D

91

x

x

T xu u RT dx

e

D

3 3

0D

( ) 361

x

v xV

u T xc T R dx

T e

D

3 3

0D

91

x

x

T xRT dx

T e

D D3 3

D

0 0D1 1

x x

x x

xx xdx dx

T e x e T

m DD

B

hx

k T T

D

D D

3 43m D D D

2 0B D

1

1 1 1

x

x xx

h x x xxdx

k T x e T Te e

D

D D3 4 4

20 00

1 1

1 4 1 4 1

x xx x

x x x

x x x edx dx

e e e

D

D

44D

20

1 1

4 4 11

xx

xx

xx edx

ee

D

D

3 44D

20D

1 1( ) 36

4 4 11

xx

v xx

xT x ec T R dx

ee

D

3 4D

D

91x

xTR

e

D

3 4

20D

( ) 91

xx

vx

T x ec T R dx

e

DT

D D / 0,x T 1xe x

D D

D

44 3D

20 04

1 11

xx x

xxx

xx e xdx dx

e ee

D33D

0,

1 3

x

x

xxdx

e

When

D

3 4

20D

( ) 91

xx

vx

T x ec T R dx

e

3 3D

D

9 33

xTR R

D

43DD ,

1x

xx

e

D

34D

20 31

xx

x

xx edx

e

DT

Relative difference between calculated value and experimental data : about 5%

When

Riemann zeta function

1

20 0 11

n x n

xx

x e xdx n dx

ee

1

01 ! ( )

1

n

x

xdx n n

e

1 1 1

( ) 12 3 4n n nn

n 1 2 3 4 5 6 7 8

6

290

4...202.1 ...037.1 ...008.1

945

69450

8( )n

DT

D D

4 3 4 4

20 0

44 4 3! (4) 4 6

1 90 151

xx x

xx

x e xdx dx

ee

3 34 43

20

12( ) 9

( 1) 5D

xx

v xD D

T x e Tc T R dx R T

e

agrees with experiments within a few percent

/ 0.1DT

When

D D

D

44 3D

20 04

1 11

xx x

xxx

xx e xdx dx

e ee

m D

DB

hx

k T T

1

01 ! ( )

1

n

x

xdx n n

e

Debye Model vs. Einstein Model

Free Electron Gas in Metals

Free electron

Translational motion of free electrons within the solid: largely responsible to the electrical and thermal conductivities of metals

FD ( ) /

1( )

1Bk T

dNf

dg e

: Fermi energyF at 0 K

Fermi-Dirac distribution

due to positive and negative spins

in terms of the electron speed v

Degeneracy for electrons3 22 4 ( / )edg V m h v dv

3 2

( ) /

1 1( ) 8

1B

ek T

mdN dg dN vf v

V dv V dv dg h e

Distribution function

( )dN

dn f v dvV

in terms of the kinetic energy of the electron using

( ) ( )f v dv f d 2

e

2

m v

3/ 2

FD( ) /2

24 ( ) ( )

1B

ek T

mf D

h e

2e

e e e

2 2 1 1, ,

2 2 2

m v dv dv

m m m

( )dN

dn f dV

1 1 1 1( )

2 e

dN dN dv dNf

V d V dv d V dv m

3

( ) /

2 / 18

1 2B

e ek T

e

m m

h e m

3 2

( ) /

1( ) 8

1B

ek T

mdN vf v

V dv h e

FD( )dN

f vdg

FD FD( ) ( ) ( )dN f v dg f v g v dv

FD( ) ( )f v g v dvdNdn

V V

FD0( ) ( )n f v D v dv

( )( )

dg g v dD v dv

V V

FD( ) ( ) ( )dn f v D v dv f v dv

0( )f v dv

Density of states for free electrons3/ 2

2

2( ) 4 em

Dh

Fermi Level,F

The number density of electrons as

0T

B

3/ 2

e ee ( ) /20 00 0

2lim ( ) lim 4

1k TT T

N mn f d d

V h e

F

3/ 2 3 / 2

3 / 2e eF2 20

2 2 24 4

3

m md

h h

2/ 3 2 / 32 2e e

Fe e

3 3

8 2 8

n nh h

m m

3/ 2

FD( ) /2

2( ) 4 ( ) ( )

1B

ek T

mf f D

h e

Since – < 0,

Since the difference between (T) and F is small,

Sommerfeld expansion

Apply FD statistics to study free electrons in metalsResolve the difficulty in the classical theory for electron specific heat

e FD0( ) ( , )n D f T d

F

22B

F F F0( ) ( ) ( )

6

k TD d D D

F

e 0( )n D d

22B

F F F( ) ( ) 06

k TD D

Number of electrons does not depends on temperature

22B

F F F( ) ( ) 06

k TD D

3/ 2

2

2( ) 4 ,em

Dh

3/ 2

2

2 1( ) 4

2em

Dh

3/ 2 3 / 22 2

BF F2 2

F

2 2( ) 14 4 0

6 2e em mk T

h h

2 2

BF F

( )0,

12

k T

2 2B

FF

( )

12

k T

22 2 2 2B B B

F F F2F F F

( ) ( ) 11 1

12 12 3 2

k T k T k T

Internal energy

F

0( )

UD d

V

2 2 2 2

B BF F F F F F

( ) ( )( ) ( ) ( )

6 6

k T k TD D D

F2 2

BF0

( )( ) ( )

6

k TD d D

3/ 2 2 / 32e e e

F F2e F

2 3 3( ) 4 , ( )

8 2

m n nhD D

h m

22B

FF

3 51

5 12

k TU N

FD0 0( ) ( ) ( )U V f d V f D d

= Electronic contribution + Lattice contribution

Specific heat of free electrons

2 2B B B

, A FF F F

3 5

5 6 2v eV

k T k k Tuc N R

T

Electronic contribution to the specific heat of solids is negligible except at very low temperatures (a few kelvins or less)

3( )v sc T T BT

Quantum Size Effect on the Specific Heat

For nanoscale structures such as 2-D thin film and supperlattices, 1-D nanowires and nanotubes, or 0-D quantum dots and nanocrystals, substitution of summation by integration is no longer appropriate.

2-D thin film: confined in one

dimension

1-D nanowires: in two dimensions

0-D quantum dots: in three dimensions

1-D chain of N + 1 atoms in a solid with dimension Lwith end nodes being fixed in position

min

L0

0L

min 02L

max 2L

max

min 0

LN

L

N number of vibrational modes

eigenfunctions

2 3sin , sin , sin , , sin

x x x N x

L L L L

0

sinx

L

Periodic Boundary Conditions

Born-von Kármán lattice model

medium: an infinite extension with periodic boundary conditions

standing wave solutions for a solid with dimensions of Lx, Ly, Lz (see Appendix B.7) ( , ) exp i iu r t A k r t

k: lattice wavevector ˆ ˆ ˆ

x y zk k x k y k z

2 2 22x y zk k k k

total number of modes = total number of atoms along the 1-D chain

General Expressions of Lattice Specific Heat

lattice vibrational energy

B0 /

1 1( )

21k TP K

u T ue

0u: static energy at T = 0 K

B BBE / /

1 1

1 1h k T k Tfe e

1

2 : zero-point energy

K : wavevector index, dispersion relation ( )k

P : polarization index

specific heat

BE( )vP Kv

fuc T

T T

B0 /

1 1( )

21k TP K

u T ue

in terms of density of states

BE

0( ) ( )v

P

fc T D d

T

B

B

2 /

B 20 /B

( )1

k T

k TP

ek D d

k T e

density of states : the number of states (or modes) per unit volume and per unit energy interval

k

xk

yk

zk

3-D view

volume of one quantum state

Dimensionality: density of states

3-D reciprocal lattice space or k-space

number of quantum states per unit volume :

34 / 3

2 / 2 / 2 /x y z

kN

L L L

3 3

2 26 6

x y zL L L k Vk

3

2

1( )

6

dN d kD

V d d

2 2

2 2

3

6 2

k dk k dk

d d

For a linear dispersion relation av k

22

3 3 3a

1 2 12( ) 4

l t

Dv v v

22 2

2 2 2 3a a a

1 1( )

2 2 2

k dkD

d v v v

For a single polarization

22 2

3 3 2 3a a a

4 4( )

2 2 2

dD d d d

v v v

2

3a

4( )D

v

Eq.(5-7)

( ) ( )D d D d

2-D reciprocal lattice space or k-space

xk

yk

2-D projection

2 / xL

2

yL

k

volume of one unit cell

number of quantum states per unit area :

2 2

42 / 2 /x y

k AkN

L L

21( )

4

dN d kD

A d d

2

4 2

k dk k dk

d d

thin film, superlattice

a ,v k When 2a

( )2

Dv

1-D reciprocal lattice space or k-space

k xk

2 / xL

number of quantum states per unit length :

nanowires, nanotubes

2

2 / x

k LkN

L

1 1( )

dN d k dkD

L d d d

a ,v k When a

1( )D

v

independent of frequency

Thin Films Including Quantum Wells

Ex 5-4: specific heat of thin film

thin film made of a monatomic solidfilm thickness L with q monatomic layers, L = qL0

average acoustic speed va independent of temperature

.

.

.

L

0L

xkyk

zk

molar specific heat

,

BE

,B

( )3

x y z

vk k k

fVRc T

Nk T

BE( )vP K

fc T

T

total number of modes

.

.

L

0L

12 40, , , ,

220, , , ,

z

q

L L Lkq q

L L L

for q = 1, 3, 5, …

for q = 2, 4, 6, …

specific heat per unit volume

A A B

A B Bv v v v v v

VN VN kN VRVc c c c c c

N N Nk Nk

2zk

L

,

BE

,B

3( )

x y z

vk k k

fVRc T

Nk T

The lattice is infinitely extended in the x and y directions.

xk

yk

d

.

.

L

0L

xkyk

zk

Dk

yk

zk

2 2 2D zk k

xk

2 2

BE3 0

B

3( ) 2

2

D z

z

k k

v zk

fVRc T d k

TNk

2 2 2 , 2 /x y zk k k L

kD : cutoff value determined by setting the total number of modes equal to the number of atoms per unit area

2 2 2 22 2

4 4 4 4

x y D zA k k A k kAk AN

2 2

4z

D z

k

k kN

A

20

N q

A L

total number of modes

total number of atoms

.

.

L

0L

xkyk

zk

2 2

204

z

D z

k

k kN q

A L

2 2

2 2 2 220

1 1 1

4 4 4 4 4z z z z

D zD z D z

k k k k

k k q qk k k k

L

2

220

4,

z

zD

k

kk

L q

1/ 22

20

4

z

zD

k

kk

L q

For a single layer 1/ 2 1/ 2

22

2 20 0

4 4

z z

zD z

k k

kk k

L q L

1/ 2

20 0 0

4 2 3.540

L L L

12 40, , , ,

220, , , ,

z

q

L L Lkq q

L L L

for q = 1, 3, 5, …

for q = 2, 4, 6, …

As ,q 0

3.982Dk

L

In 3-D case,

3 2

0 0

6 3.90Dk

L L

2 2

BE3 0

B

3( ) 2

2

D z

z

k k

v zk

fVRc T d k

TNk

dispersion relation

2 2a a zv k v k

2 2 2zk transformat

ion

2 3B

2a

3( )

2 1

D

zz

xx

vxx

k

k TRA x ec T dx

N v e

a

B

v kx

k T

Electrical and Thermal Conductivities of Solids

Electrical Conductivity

e-

AVL

z

ec c

L LR r

A A

1er

VR

I

c c

L V I V

A I A L

J E

re: resistivity, : conductivity

Ac: cross-sectional area, J: current density

E: electric filed

3

Neweton’s 2nd law

J E

de e

udvF eE m m

dt

Current density J

- d

e

eEu

m

2

ee d e

e e

n eeEJ en u en E

m m

2e

e

n e

J E Em

2e

e

n e

m

: drift velocity

: relaxation time

ne: electron number

du

thermal conductivity (kinetic theory)

Thermal Conductivity of Metals

,

1

3 v e F ec v

2 2

, ,2

e Bv e

F

n k Tc

2 22 F

F e F F Fe e

v v v vm m

2

,

2 22

,

,

2

2 2

e e e B

Bv e

V F

e BBv e e e

F F

n m m R k

k Tuc R

T

n k Tk Tc n m R

2

2

1

22

e F

Fe

m v

vm

2 2 2 2

,

21 1

3 3 2 3e B e BF

v e F eF e e

n k T n k Tc v

m m

Wiedemann-Franz law

2 2 2e

e

, 3

e B

e

n e n k T

m m

2 2

22 2

2 2e

e

3 1 1

3 3

e B

e B B

n k T

m k kLz

n eT e eT

m

Lz: Lorentz number

T Lz

25

8 2 2 28.617 10 eV/K1

= 2.44 10 / /3

V K W Ke

Derivation of Conductivities from the BTE

Distribution functionlocal equilibrium, relaxation time

approximation( , , )f r v t

( , , ), ef r p t p m v

2( , , ), , , = ,

2 2

h h khf r k t p k h p k

2 2 2

( , ), = , ( , )2 2e e

p kf T T T r t

m m

1 1( , , ) ( , ) d ( , ) ( )dk

f r k t dk f T f T D dd

D() : density of states

steady state, relaxation time approximationcoll

f f f fv a

t r v t

electric filed E along with temperature gradient in z direction

0 11 1

( )z zz

f ff fv a

z v

e ee

zz z z z

dv eEF eE m m a a

dt m

0 11 1

e ( )zz

f ff feEv

z m v

0 11 1 ( , ) ( , )

( )ze z

f T f Tf feE Tv

z m T v

1 11 0( , ) ( , ) ( ) z

e z

f feE Tf T f T v

m v T z

0 FDf f

2 2 2 2e e

1 1

2 2 x y zm v m v v v e zz

m vv

01 ,ff

01 ff

T T

Assume under local equilibrium

In the case of no Temperature gradient 0T

z

1 11 0( , ) ( , ) ( ) z

e z

f feE Tf T f T v

m v T z

FD1 FD e

e

( , ) ( , ) ( ) z

feEf T f T m v

m

FD1 FD( , ) ( , ) ( ) z

ff T f T v eE

e FD0( )n f D d

: electron number density

FD0( )e z NJ e v f D d eJ

0( , ) ( ) ( )FD

e z FD z

fJ e v f T v eE D d

FD0( , ) ( ) 0zv f T D d

2 FD

0( ) ( )e z

fJ e v eE D d

2 2

e

1 2

3 3zv vm

e e /J E J E

2 2 FD

0( ) ( )z

fe v D d

2

0

2( ) ( )

3FD

e

feD d

m

FD ( )f

Note that

( ) : Dirac delta fucrtion

( ) ( ) ( )f x x a dx f a

F F( ) , ( )T T Assume that

2FD

0e

2( ) ( )

3

feD d

m

2

0e

2( ) ( ) ( )

3

eD d

m

2

F F Fe

2( )

3

eD

m

3/ 2

e2

2( ) 4

mD

h

(5.18)

2/ 32e

Fe

3

8

nh

m

(5.20)2/ 3

e e2

F

2 31

4

m n

h

3/ 23 / 2 2 / 3

1/2e e eF F F2

F F

2 3 31( ) 4 4

4 2

m n nD

h

22 2e e

F F F F F Fe e F e

32 2( )

3 3 2

n n ee eD

m m m

Drude-Lorentz expression2

e

e

n e

m

Thermal ConductivityIn the case of no electric field, E = 0

1 11 0( , ) ( , ) ( ) z

e z

f feE Tf T f T v

m v T z

for an open system of fixed volume,

dU Q dN

z E Nq J J

0

( )E z FDJ v f D d

energy flux

0

( )N z FDJ v f D d

particle flux

0 0 0

( ) ( ) ( ) ( )z z FD z FD z FDq v f D d v f D d v f D d

1 ( , ) ( )FD FD z

f Tf f T v

T z

0

( ) ( )z z FDq v f D d

( ) ( , )z FDv f T FD

0

( ) ( )z

f Tv D d

T z

z

dTq

dz

FD

0

( ) ( ) ( )z z z

fdz T dzq v v D d

dT T z dT

2 FD

0( ) ( ) ( )z

fv D d

T

2 2 3z ev m

FD FD

0 0

2 2( ) ( ) ( ) ( ) ( ) ( )

3 3e e

f fD d D d

m T m T

where

FD FD f f

T T

Eq.B.82 (Appendix B.8)

FD

0

2( ) ( ) ( )

3 e

fD d

m T

FD

0

2( ) ( ) ( )

3 e

fD d

m T

2 FD

0

2( ) ( ) ( )

3 e

fD d

m T

2 22 FD

0

( )( )( ) ( )

3Bf k T

G d G

Appendix B.8

2 2( )2( ) ( )

3 3B

F F Fe

k TD

m T

2 22 23 ( )2

3 2 3 3e e BB

F F Fe F e

n n k Tk T

m T m

same result as simple kinetic theory

ph

1

3 v gc v kinetic theory1

( ) ( ) ( )3 v g gc v v

Thermal Conductivity of Insulators

BE ( , )vP K

c f TT

2

20( )

( 1)

B

B

k T

B k TP B

ek D d

k T e

(5.31)

2

220

1( ) ( ) ( )

3 ( 1)

B

B

k T

B g k TP B

ek v D d

k T e

under local equilibrium

2 2 2

3 3 2 2 2

1 k 1 4 1( )

2 / 22 2 p g

d k dk kD

d d d dk v v

for isotropic distribution in k-space

2k = 4 , (k) , (k)g p

dd k dk v v

dk k

2 22

2 220

1( ) ( )

3 2( 1)

B

B

k T

B g k TP B p g

ek v d

k T v ve

2

2 2B B B

B

k T k T k Tx x x d dx

k T

2

2 2 22 2 20

1 1( ) ( )

3 2 ( 1)m

xx

B BB g x

P p g

k T k Tek x v x x x dx

e v v

3 4

2 2 20

( )( )

6 ( 1) ( )m

xx gB B

xP p

v xk k T x ex dx

e v x

for a large system with isotropic dispersion

xm: corresponds to maximum frequency of each phonon polarization