Thermoelectricity Thermoelectricity / thermoelectric effect electric field and a temperature...

Thermoelectricity Thermoelectricity / thermoelectric effect

electric field and a temperature gradient along the z direction of a conductor

e 11 12J L E L Te

21 22q L E L Te

211 0 12 1 21 12 1 22 2

1, , ,

eL e L L TL e L

T T

2 FD

0

1( ) ( )

3n

n

fv D d

E

T

Let

: electrochemical potential (electrostatic + chemical)

Ee

0 0T if and

e 11 12 11J L E L T L Ee

22 2 FD

11 0 0= ( )

3

feL e v D d

ee 11

JJ E L

E

e 0 0E J if and

11 120 L L Te

21 22 21 22q L E L T q L L Te e

12

11

L T

e L

12 12 2121 22 22

11 11

L T L Lq L L T L T

L L

12 2122

11

L LL

L

The Seebeck Effect and Thermoelectric Power

Seebeck effect: used to produce an electrical power directly from a temperature difference. Seebeck coefficient: induced thermoelectric voltage across the material of unit length per unit temperature difference. (thermopower or thermoelectric power)

e 11 12 11 12J L E L T L L Te

21 22 21 22q L E L T L L Te

e 11 12 0J L L T

When no current flow,

21

11S

L

T L

Seebeck coefficient [W/K]

121

S 211 0

eL T

T L e

2 FD

0

1( ) ( )

3n

n

fv D d

2 2 21 2 FD

1 0

( )1( ) ( ) ( )

3 3Bf v k T

v D d D

Appendix B.8

2 2 2B

2 21B

S 2220

( )( ) ( )3

( ) ( )3

v k TeeD k T DTT

v De e De

0 2 FD0 0

1( ) ( )

3

fv D d

22 FD

0

1 ( )( )

3 3

f v Dv D d

2 2B

F F F

( )( ) ( ) ( ) 0

6

k TD D

(5.21b)

F2 2

B

6( )( )

( ) ( )

D

D k T

2 2 2B F B

F F 2F F

( )11

3 2 3 4

k T k T

(5.21a)

F2 2

B F

6( )( ) 1

( ) ( ) 2

D

D k T

2 2 2B B B

SF F

1

2 2

k T k k T

e e

2B B

SF2

k k T

e

F 7 eV for copper

at T = 300 K and 600 K

2 5 2

S

(8.617 10 eV/K) 300 K1.6 V/K

2 7 eVe

2 5 2

S

(8.617 10 eV/K) 600 K3.2 V/K

2 7 eVe

Experimental values are positive with 1.83 mV/K and 3.33 mV/K. The sign error is due to simplification used to evaluate 1.

Seebeck coefficientpositive for p-type semiconductorsnegative for n-type semiconductors

T2 V2

T1 V1

2

12 1 S ( )

T

TV V T dT

Thermoelectric voltage cannot be measured with the same type of wires because the electrostatic potentials would cancel each other.

T2

T1

∆V

T1Type I (+)

Type II (-)

2

1S,I S,II I,II( ) ( )

T

TV T T dT T

The Peltier Effect and the Thomson Effect Peltier effect: reverse of the Seebeck effect.a creation of a heat difference from an electric voltage.

e 11 12 11 12J L E L T L L Te

21 22 21 22q L E L T L L Te

e 12

11

J L T

L

21 12

11S

L TLT

L

11 21 12 12 11 S, , L L TL L L

e 12 2121 22 e

11

=J L T L

q L L T J TL

: Peltier coefficient

Thomson effect: Heat can be released or absorbed when current flows in a material with a temperature gradient. It describes the heating or cooling of a current-carrying conductor with a temperature gradient.

2e

e e= ( ) SdJJ q T

dJ

TTT

2eJ

( )T

eSd

TdT

J T

SdK T

dT

: Joule heating

Energy received by a volume element

: heat transfer due to temperature gradient

: Thomson effect

Thomson coefficient:

Thermoelectric Generation and Refrigeration

TL

TH

LAC

∆V

x

Assumptionsnegligible contact resistances same length and cross-sectional area of all thermoelectric elements heat transfer by conduction only through thermoelectric elementsJoule heating due to resistance of the thermoelectric element onlythermal, electrical conductivities and Seebeck coefficient: independent of temperature.very small temperature difference btw. the two heat reservoirs Steady-state temperature distribution

2e

e e= ( ) 0SJ dJ q T T J T

dT

2e ( ) 0

JT

2eJT

x x

H L0, (0) ; , ( )x T T x L T L T boundary conditions

2e

1 ,JT

x Cx

22e

1 2( ) 2

JT x x C x C

2H Le

H( )2

T TJT x L x x x T

L

eC

IJ

A

2 2H L

2 2C C2

T TdT I L Ix

dx A A L

2H L

20 C

,2x

T TdT I L

dx A L

2H L

2L C2x

T TdT I

dx A L

21e= S

C

L I dTq J T q T

A dx

C S C( )dT

q T A q T I Adx

0 H H S C0

( )x

dTq T q T I A

dx

2

H SC2C

T I LT I A

L A

2

L L L C L S CL C

( )2S

x

dT T I Lq T q T I A T I A

dx L A

H LP I V q q 2 2

H S L S C CC C2 2

T T I L I LT I T I A A

L L A A

22

H L S S 0C

( )I L

T T I I T I RA

output power

0C

LR

A

2S 0 ,P I T I R

S

0 L 0 L

TVI

R R R R

SV T

S 0 LT R R I

2 2 2 2S 0 0 L 0 LP I T I R I R R I R I R

L

0 H2

H L L*

H 0 0 H

11 1

2

R TR TP

q R R TZ T R R T

2 2* S S

C 0

LZ

A R

figure of merit

thermal efficiency

Onsager’s Theorem and Irreversible Thermodynamics

In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

T : heat diffusion and current

: current and heat flow

bothT & E → driving force Fj

current and heat flow→ flux of physical quantity Ji

i ij jj

J F

ij: Onsager kinetic coefficient

Onsager reciprocity relation ij ji

4th law of thermodynamics

Classical Size Effect on Conductivities and Quantum Conductance

• Classical Size Effect Based on Geometric Consideration

• Classical Size Effect Based on BTE

• Quantum Conductance

Classical Size Effect Based on Geometric consideration

Free-path reduction due to boundary scattering

Size dependence of the mean free path

Knudsen number and thermal conductivity relations

in the ballistic transport limit : d << b

f = d

f f

b b

1d

d Kn Kn

bKnd

b

1

3 vc v

conductivity ratio :

Matthiessen’s ruleeff b f

1 1 1

bf Kn

b f b

eff eff f b f

bb b b f bb

1

1Kn

KnKn

Consideration of free path duistribution

When d << eff and all energy carriers originate from the boundary

0

b 0

/ cos , 0( )

, / 2

d

2 / 2

0 0f2 / 2

b b0 0

( )sin

sin

d d

d d

0

0

/ 2 / 2

b0 0( )sin sin sin

cos

dd d d

0 bcos /d

0

0

/ 2

b 0 b 00ln cos cos ln cos cosd d

b

1 ln ln 1d

d d Kn

1

eff eff f

b b f b

1ln 1

Kn

Kn

Applicable for Kn > 5 Cannot be applied for small values of Kn since ln(Kn) becomes negative.

for Kn < 11

eff

b

1Kn

m

for thin films3m

2 / 2

0 0f2 / 2

b bb0 0

( )sin ln 1ln 1

sin

d d KndKn

Knd d

For the z-direction consideration

2 / 2

0 02 / 2

b, b0 0

( )cos sin

cos sin

z

z

d d

d d

0 02

b0 0 0

2 / 2 2

b0 0

cos sin cos sin2 1cos

cos sin

dd d d

Kn Knd d

1

ff , eff,

1b b, b,

12

e z z z

z z z

Kn

Kn

applicable for Kn > 5

0 bcos /d

b, b( )cos , cosz z

When d << eff and all energy carriers originate from the center

1

b 1

/ cos , 0( )

, / 2

d

11

b

cos2

d

2 / 2 2

0 02 / 2 2

b, b0 0

( )sin cos

sin cos

x

x

d d

d d

1

1

/ 2 / 22 2 2b0 0

( )sin sin sincos

dd d d

1

0

ln cos sin ln cos sin sin2 2 2 2

d

1

/ 2

b

1( cos sin

2

1 bcos / 2d

11 b 1 1 1

1

b

1 sin 1ln sin ( cos sin

cos 2 2

4

d

eff, eff ,

b b, b,

x x x

x x x

1

11 b 1 1 1

1

b

1 sin 1ln sin ( cos sin

cos 2 21

4

d

1 11

2 sin2ln 2 1 sin 1Kn

Kn Kn

b b2 1

2 2cosKn

d d

For circular wires, the conduction along a thin wire

when d << eff and all energy carriers originate from the center

b 2

2

cos , 0( )

cot / 2, / 2z d

12

b

sin2

d

2 / 2

w, 0 02 / 2

b, b0 0

( )sin

cos sin

zz

z

d d

d d

2

2

/ 2 / 2

b0 0( )sin cos sin cot sin

2z

dd d d

2

2

/ 22b

0

1cos sin

2d

2b 2 21 cos 1 sind

eff,w eff,w w,

b, b, w, b,

z

z z z z

1

2b2 21 sin 1 sin

4d

1

2

1 11

4Kn Kn

b b2 1

2 2sinKn

d d

1

111 4

Kn

Kn

As Kn bigger, effect of boundary scattering more significantPaths with larger polar angles are more important for parallel conduction, whereas paths with smaller polar angles are more important for normal conduction.

Reduction in thermal conductivity due to boundary scattering

eff

b

1

1 Kn

1

eff 1ln( ) 1b

Kn

Kn

1

eff ,

112

z

b

Kn

Kn

1

eff,w1

b,

11 4z

Kn

Kn

Classical Size Effect Based on the BTE

0

( )

f f f f fv a

t r v v

z

d

x

E

T

steady state

( )T T x

x e xF eE m a

0,y za a

0f

t

x z

f dT fv v

T dx z

xx

f fa a

v v

( , , )f r v t

( , ( ), )f T x z

xe

eEa

m

x y z

f f f fv v v v

r x y z

e x

eE f

m v

e x

eE f

m v

electron movement in x, z directionstemperature and electric field in x direction

0

( )x z

e x

eE f f dT f f fv v

m v T dx z

under the assumption that f is not far from f0

0 0 1 0 1

( )x z

e x

eE f f dT f f fv v

m v T dx z

1 0 0 01

1z x x

f f f dT fv f eEv v

z T dx

1 0 0 01

1 x x

z z z z

f eEv f v f dT ff

z v v v T dx v

electron movement in a particular direction

C

2 2 2 21 1

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

x

m vv

d

x

Upward motion

0zv

z1

1

1

z

ff C

z v

1 ,z z

z z

v vf e Cez

10

0 0

z z z

z zz

v v vzf e C e d C v e

1 1( , ,0) 1z z z

z z z

v v vz z zf e f T C v e v v C e

1 ,z z

z z

v vd f e Ce dz

10 0

z z

z zv vd f e C e d

1 11 ( , ,0)z z z

z z z

v v vzf Cv e e f T e

11 exp ( , ,0)expzz z

z zCv f T

v v

1( , ,0)expz

zf T

v

0 0 0 1 expx xz

z z z z

eEv f v f dT f zv

v v T dx v v

0 0 1 expx xz

z z z

eEv f v f dT zv

v v T dx v

0 11 exp ( , ,0)expz z

z zf f T

v v

0 01 01 expx

z

f f dT zf v eE f

T dx v

1 0( , ,0) expz

zf T f

v

0 00 1 ( )expx

z

f f dT zf v eE v

T dx v

( )v : arbitrary function that accounts for

the accommodation and scattering

for perfect accommodation with inelastic and diffuse scattering

( ) 1v 1 0( , ,0)f T f

d

x

Downward motion 0zv

z

11

1

z

ff C

z v

d z

1 ,z zv vf e Ce

, d z d dz

11

1,

z

ff C

v

similarly

0 01 0 1 ( )expx

z

f f dT d zf f v eE v

T dx v

1 ,z zv vd f e Ce d

1

0 0

z z

d z d zv vd f e C e d

current density

for electric conduction without any temperature gradient and with( ) 1v

01 0 1 exp , 0x z

z

f zf f v eE v

v

01 0 1 exp , 0x z

z

f d zf f v eE v

v

10

( ) ( )e N xJ z eJ z e v f d d

2 21

0 0 0sinxe v f v d d d

d

x

d

xcoszv v sin cosxv v

x

y

z

v

2 / 2 2 21, 1,

0 0 0 / 2( ) sin sine x u x dJ z e v f v d v f v d d

01, 0 1 exp , 0u x z

z

f zf f v eE v

v

01, 0 1 exp , 0d x z

z

f d zf f v eE v

v

( )eJ z

/ 2 FD 2

02

0 0FD 2

/ 2

1 exp sin

1 exp sin

x xz

x xz

f zv v eE v d

ve d d

f d zv v eE v d

v

2 / 2FD2 2 2

0 0 01 exp sin

cosx

f ze E d d v v d

v

2 2

/ 21 exp sin

cosx

d zv v d

v

sin cosxv v

2 / 2FD2 2 4 3

0 0 0

4 3

/ 2

( )

cos 1 exp sincos

1 exp sincos

eJ z

f ze E d d v d

v

d zv d

v

average current flux 0

1( )

d

e eJ J z dzd

energy integral at the Fermi surface

F F( ), v

effective electrical conductivity feJ E

f

b

( ),F Kn

b ,Knd

b F F( )v

3

/ 2 0 b

3sin 1 exp

4 cos

d d zdzd

d

/ 2f 3

0 0b b

3( ) sin 1 exp

4 cos

d zF Kn dzd

d

/ 23

b0 b

3sin cos 1 exp

2 cos

zd d

d

3 51

3 3 1 11 exp

8 2

Kn Kn tdt

t t Kn

1

cost

exponential integral function1

2 /

1 0( ) xt m m x

mE x e t dt e d

1( ) ( )x

m m

e xE x E x

m m

f3 5

b

3 3 1 1( ) 1

8 2

Kn KnF Kn E E

Kn Kn

asymptotic relations

f

b

31 , 1

8

KnKn

f

b

3ln, 1

4

KnKn

Kn

Thermal conductivity

f

b

( )F Kn

Electrical and thermal conductivities based on the BTE Assumption – Relaxation time approximation , under the local equilibrium conditions

Temperature gradient & electric field in the x-direction only

Finite thickness in the z-direction, the distribution function

z

fv

x

T

T

fv

v

f

m

eETfzTf zx

xe

10001 )(),(),,(

Temperature gradient & electric field, x-direction >> z-directionx-direction >> z-direction

)(01100

ff

z

fv

x

T

T

fv

f

m

eEzx

xe

01 ff

Replace T

f

T

f

01,

0,exp)(10001

zz

x vv

zv

dx

dT

T

ffeEvff

0,exp)(10001

zz

x vv

zdv

dx

dT

T

ffeEvff

General solution of the steady-state BTE under the relaxation time approximation

)( :arbitrary function

Electrical conductivities based on the BTE Assumption – Relaxation time approximation , under the local equilibrium conditions

:arbitrary function that accounts for the accommodation and scattering.)(

If perfect accommodation is assumed with inelastic and diffuse scattering,

1)(

0,exp1001

zz

x vv

zfeEvff

0,exp1001

zz

x vv

zdfeEvff

and electrical conduction without any temperature gradient,

0,exp)(10001

zz

x vv

zv

dx

dT

T

ffeEvff

0,exp)(10001

zz

x vv

zdv

dx

dT

T

ffeEvff

Electrical conductivities based on the BTE

No temperature gradient, current density can be written as,

dDf

eEvfveeJzJ FDzFDzNe

0

)()( (5.61a)

2

0

/2 22/2

0

22

0

2 sincos

exp1sincos

exp1 dvv

zdvdv

v

zvdd

fEe xx

FD

0,exp1001

zz

x vv

zfeEvff

0,exp10

01

zz

x vv

zdfeEvff

EdzzJd

zJ f

d

ee 0 )(1

)( FFb )(,

f : the effective electrical conductivity of the film

FF ),( : Properties at the Fermi surface



dDf

eEvfveeJzJ FDzFDzNe

0

)()( (5.61a)

)(3

2 2

FFFe

Dm

e (5.62)

)(KnFb

f

dtKn

t

tt

KnKn

dKn

KnKn

dzdd

dd b

b

exp11

2

3

8

31

cos/1expcoscossectan

2

3

8

31

cosexp1cossin

2

3

1 53

1

53

/2

0

3

dzd

zd

ddzd

z

d

d

b

d

b

0/2

3

0

/2

0

3

cosexp1sin

4

3

cosexp1sin

4

3

)cos/1( t



dtKn

t

tt

KnKnKnF

b

f

exp

11

2

3

8

31)(

1 53

KnE

KnE

KnKn 11

2

3

8

31 53

dttexE mxtm

1

)(

)()/(/)(1 xEmxmexE mx

m

mth exponential integral

8

31

Kn

b

f

Kn

Kn

b

f

4

)ln(3

for Kn << 1

for Kn >> 1

Asymptotic relations

Similar to the result of “electron originates from the center of the film”for Kn >> 1, (thinner film

case)

Kn

Kn

KnKn

Knxb

xeff

)4ln(2sin2

1)]sin1(2ln[2 11

1,

,

Because the derivation using the BTE presented earlier inherently assumed that the electrons are originated from the film rather than from the boundaries.

Thermal conductivities based on the BTE Assumption – Relaxation time approximation , under the local equilibrium conditions

If perfect accommodation is assumed with inelastic and diffuse scattering,

1)(

0,exp1001

zz

x vv

z

dx

dT

T

fvff

0,exp1001

zz

x vv

zd

dx

dT

T

fvff

and temperature gradient without any electric field

0,exp)(10001

zz

x vv

zv

dx

dT

T

ffeEvff

0,exp)(10001

zz

x vv

zdv

dx

dT

T

ffeEvff

Thermal conductivities based on the BTE

No electric field, thermal conductivity can be written as,

dDT

f

mFD

e

)())((3

20

(5.65a)

/2 22/2

0

22

0sin

cosexp1sin

cosexp1

3

2dv

v

zdvdv

v

zvdT

T

f

m xxFD

e

0,exp1001

zz

x vv

z

dx

dT

T

fvff

0,exp10

01

zz

x vv

zd

dx

dT

T

fvff

3

)()()(

3

2)())((

3

2 22

0

2 TkD

TmdD

T

f

TmB

FFFe

FD

e

)(KnFb

f

dtKn

t

tt

KnKn

exp

11

2

3

8

31

1 53)cos/1( t

Thermoelectricity based on the BTE Assumption – Relaxation time approximation , under the local equilibrium conditions

First-order approximation , L12 and L21 subjected to boundary scattering Seebeck coefficient along the film remains the same regardless of

boundary scattering

:specularity, represent the probability of scattering being elastic and specular

p

)/exp(1

1)(

zvdp

p

dtKntp

Knt

tt

KnppKnF

)/exp(1

)/exp(111

2

)1(31),(

1 53

dtKn

t

tt

KnKnKnF

exp

11

2

3

8

31)(

1 53

Thermoelectricity based on the BTE

<thermal conductivity as predicted by the BTE with different specularity>

For electronic transport, wavelength < 1nm, p=0 (as diffuse)

For phonons, wavelength may vary,

21

1

p /4 rms

rms : Surface roughness

In reality, p depends on the angle of incidence

Kn > 0.1, the size effect may be significant.

Kn > 10, boundary scattering dominate.

size effect importantboundary scatteringdominate

As temperature is lowered, size effect more significant.

Conduction along a thin wire based on the BTE

from reference 41,42

dtdt

t

Kn

tor

b

w

b

w4

2

1

1

0

2 1exp1

121

The asymptotic approximations, with about 1% accuracy

3

8

3

4

31 KnKnor

b

w

b

w

for Kn < 0.6

32 15

2

8

)1(ln31

KnKn

Kn

Knor

b

w

b

w

for Kn > 1

If p=1,

If p≠1,

1

12

),()1(12

1m

m

b

w

b

w mKnGmpp

or

dtdt

t

Kn

tmmKnG

1 4

21

0

2 1exp1),(where

,

Quantum conductance (1)

When the quantum confinement becomes significant, the relaxation time approximation used to solve the BTE is not applicable.

Electrical conductance of metallic materials and thermal conductance of dielectric materials.

0Bulk solid, 3-D

2/3

2

24)(

h

mD e

Quantum well, 2-D

2

*

)(h

nmD

for n = 1,2,3,…

Quantum well, 1-D

nl

m

h

nlD

,

*

2

2)(

For 3-D confined quantum dots, the energy levels are completely discrete; subsequently, the density of states becomes isolated delta functions.

Transport phenomena in the quantum or ballistic regimes

Landauer treated electrical current flow as transmission probability

a) Electrical current flow through a narrow metallic channel due to different electrochemical potentials

b) Heat transfer between two heat reservoirs through a narrow dielectric channel

Ballistic transmission, absence of losses by scattering and reflection )()( 211221 DevJJJ Fe The net current

flow: chemical potential


There is no resistance or voltage drop associated with the channel itself.The voltage drops are associated with the perturbation at each end of the channel as it interacts with the reservoir.

Transmission coefficient ξ12, the actual distribution function

)()( 21 DevJ Fe ,the electronic spin degeneracy

1)()( FvD

2

21

121

210 )(

))(( e

e

vev

VV

Jg FFee

dDffevJ FDFDFe )()],(),()[()( 21120


dDffevJ FDFDFe )()],(),()[()( 21120

For small potential difference, using the following approximation

),(),(),(),(

12

21 FDFDFDFD ffff

e

dDffev

VV

Jg

FDFDFee )(

)()],(),()[()(

21

21120

21

e

dvffev FFDFDF

)(

))](,(),()[()(

21

121120

dfe FD

),(

)(120

2

i

ie

eg

2 given by scattering matrix based on SchrÖÖdinger’s equation transmission coefficient between 0 and 1 i

Ballistic thermal transport process

Resembles electromagnetic radiation between two blackbodies separated by vacuum

For a 1-D photon gas,

2'' Tq Stefan-Boltzman law

P

BEp

D

p

dTfq

),()(

2

1121

PBEp

D

p

dTfq

),()(

2

1212

PBEpBEp

D

p

D

p

dTfdTfqqq

),()(),()(

2

121122112

1 2

1( ) ( , ) ( , )

2

D

pp BE BE

P

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Thermoelectricity Thermoelectricity / thermoelectric effect electric field and a temperature...

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Transcript of Thermoelectricity Thermoelectricity / thermoelectric effect electric field and a temperature...