Post on 03-Jan-2016
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
Electrical conductivities based on the BTE
No temperature gradient, current density can be written as,
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
Electrical conductivities based on the BTE
No temperature gradient, current density can be written as,
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
Transport phenomena in the quantum or ballistic regimes
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
Transport phenomena in the quantum or ballistic regimes
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
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