Hiroyuki KOIZUMI

1. Principle

Seebeck effect

Peltier effect

Thomson effect

Thermoelectric effect

Δ𝑇

𝐼

Δ𝑉

𝑄

𝐼 𝑄Heattransfer

Current

Voltagedifference

Temperaturedifference

Seebeck effect

Peltier effect

Thomson effect

Δ𝑉 = −𝑆Δ𝑇

𝑄 = Π𝐴 − Π𝐵 𝐼

𝑄 = −𝜅𝐼Δ𝑇

Thermoelectric effect

Peltier effect

Thomson effect

𝑄 = Π𝐴 − Π𝐵 𝐼

𝑄 = −𝜅𝐼Δ𝑇

Electricity Heat

Joule heating 𝑄 = 𝑅𝐼2Generation

Transfer (Q>0 = output)

Transfer

Peltier effect

Thomson effect

𝑄 = Π𝐵 − Π𝐴 𝐼

𝑄 = 𝜅𝐼Δ𝑇

Electricity Heat

Joule heating 𝑄 = 𝑅𝐼2Irreversible

Reversible

Reversible

Thermoelectric EMF(熱起電力)

Seebeck coefficient or Thermopower (熱電能)

Found by T.J. Seebeck

Seebeck effect (1821)ゼーペック効果

𝑇 𝑇 + Δ𝑇

Δ𝑉

Δ𝑉 = −𝑆 ΔT

𝑇𝐴 𝑇𝐵

𝑉𝐴𝐵

𝑉𝐴𝐵 = − 𝐴

𝐵

𝑆 𝑇 𝑑𝑇

Seebeck effect (1821)ゼーペック効果

8

Thermal equilibrium condition with Electron diffusion

No temperature gradient case

With temperature gradient case

heating

Same temperatures

Charge is carried by electron flow

MaterialSeebeck

coefficient/(μV/K)

Selenium 895

Tellurium 495

Silicon 435

Germanium 325

Antimony 42

Nichrome 20

Molybdenum 5.0

Cadmium, tungsten 2.5

Gold, silver, copper 1.5

Rhodium 1.0

Tantalum -0.5

Lead -1.0

Aluminium -1.5

Carbon -2.0

Mercury -4.4

Platinum -5.0

Sodium -7.0

Potassium -14

Nickel -20

Constantan -40

Bismuth -77

Wide variety

Dependency on 𝑇

P-type semiconductor

Carrier: positive hole

Δ𝑉 = −𝑆 Δ𝑇

High 𝑇

Lower hole density(stochastically, by random walk)

Negative potential

Low 𝑇

𝑆 > 0

N-type semiconductor

Carrier: negative electron

Δ𝑉 = −𝑆 Δ𝑇

High 𝑇

Lower electron density(stochastically, by random walk)

Positive potential

Low 𝑇

𝑆 < 0

P-N junctionPCarrier: positive hole

NCarrier: negative electron

Found by J.C.A. Peltier

Peltier effect (1844)ペルチェ効果

Q = Π𝐼

A

ΠA𝐼 Π𝐵𝐼

BQAB = (Π𝐴 − Π𝐵)𝐼

QAB

Π: Peltier coefficient

Metal N-type P-typeEnergy

Electron energy state in solids

Metal AEnergy

Electron energy state in solids

Metal Bcurrent

Energy gap

Metal AEnergyMetal Bcurrent

Energy gap

HeatHeating

Metal AEnergyMetal Bcurrent

Energy gap

HeatCooling

N-type

carrier: electron

P-type

carrier: hole

current

Heat

Energy release

N-type

carrier: electron

P-type

carrier: hole

current

Heat

Energy injection

20

𝜅: Thomson coefficient

(electric specific heat)

Predicted by William Thomson (Lord Kelvin)

Thomson effect (1854）トムソン効果

𝑇 𝑇 + Δ𝑇

Q = −𝜅𝐼Δ𝑇

𝐼

Current

Energy

𝑇 𝑇 + Δ𝑇

Low energycarrier

High energycarrier

Heat

Current

Energy

𝑇 𝑇 + Δ𝑇

Low energycarrier

High energycarrier

Heat

Seebeck effect

Peltier effect

Thomson effect

Δ𝑉 = 𝑆Δ𝑇

𝑄 = Π𝐵 − Π𝐴 𝐼

𝑄 = 𝜅𝐼Δ𝑇

Thermoelectric effectAll the phenomena are caused by the current carriers

They should be related each other

𝑇 𝑇 + Δ𝑇

Δ𝑉

𝑄in 𝑄out

𝑄ex

𝑄J

𝑄in = Π 𝑇 𝐼

Current𝐼

𝑄out = Π 𝑇 + Δ𝑇 𝐼

𝑄J = −𝐼Δ𝑉

Peltier effect

𝑄J + 𝑄in − 𝑄out − 𝑄ex = 0 Energy balance

Note, voltage drop with current is −Δ𝑉

Δ𝑉

𝑄in 𝑄out

𝑄ex

𝑄J

Current𝐼

Δ𝑉 = −𝜌Δ𝑥

𝐴𝐼 − 𝑆Δ𝑇

Resistance effect+Seebeck effect

𝜌 : resistivity

𝐴 : cross section

𝑇 𝑇 + Δ𝑇

𝑄ex = 𝜌Δ𝑥

𝐴𝐼2 −

dΠ

𝑑𝑇− 𝑆 Δ𝑇𝐼

𝑄ex = 𝜌Δ𝑥

𝐴𝐼2 −

dΠ

𝑑𝑇− 𝑆 Δ𝑇𝐼

Thomson effect

𝑄 = −𝜅𝐼Δ𝑇

Joule heating

𝑄 = 𝑅𝐼2

𝜅 =dΠ

𝑑𝑇− 𝑆

The first Thomson relation

Current𝐼

Two different materials

Temperature difference

Voltage differenceand current flow

Adjusting voltage to neglect 𝐼2 term

Voltage supply

to 𝐼2 ≅ 0

B A

𝑇H

𝑇C𝑉

Voltage supply

to 𝐼2 ≅ 0

B A

𝑇 + Δ𝑇

𝑇

𝑉 = 𝑆𝐵Δ𝑇 − 𝑆𝐴Δ𝑇 + 𝛿𝑉

𝑉

to flow a little current

to compensate the thermoelectric EMF

𝑄T,𝐵 𝑄T,𝐵

𝑄P,𝐵𝐴

𝑄P,𝐴𝐵

𝑄P,𝐵𝐴 = Π𝐵𝐴 𝑇 + Δ𝑇 𝐼

𝑄P,𝐴𝐵 = Π𝐴𝐵 𝑇 𝐼

𝑄T,𝐵 = −𝜅𝐵Δ𝑇𝐼

𝑄T,𝐴 = 𝜅𝐴Δ𝑇𝐼

Π𝐴𝐵 = Π𝐴 − Π𝐵

𝑉 ≅ −𝑆𝐴𝐵Δ𝑇

𝑉𝐼 = 𝑄P,𝐵𝐴 + 𝑄P,𝐴𝐵 + 𝑄T,𝐵 + 𝑄P,𝐴

𝑑Π𝐴𝐵𝑑𝑇

− 𝑆𝐴𝐵 = 𝜅𝐴𝐵

𝑆𝐴𝐵 = 𝑆𝐴 − 𝑆𝐵

𝜅𝐴𝐵 = 𝜅𝐴 − 𝜅𝐵

(The first Thomson relation)

Energy balance

Entropy balanceIrreversible process, Joule heating, is neglected by 𝐼2 ≅ 0

𝑄P,𝐵𝐴𝑇 + Δ𝑇

+𝑄P,𝐴𝐵𝑇

+𝑄T,𝐵

𝑇 + Δ𝑇/2+

𝑄T,𝐴𝑇 + Δ𝑇/2

= 0

Π𝐵𝐴 𝑇 + Δ𝑇

𝑇 + Δ𝑇+Π𝐴𝐵 𝑇

𝑇+

𝜅𝐴𝐵Δ𝑇

𝑇 + Δ𝑇/2= 0

𝑑Π𝐴𝐵𝑑𝑇

−Π𝐴𝐵𝑇= 𝜅𝐴𝐵

Π𝐵𝐴 𝑇 + Δ𝑇

𝑇 + Δ𝑇=Π𝐵𝐴𝑇+dΠ𝐵𝐴d𝑇

Δ𝑇

𝑇−Π𝐵𝐴𝑇2Δ𝑇 + 𝑂 Δ𝑇2

Δ𝑇 → 0

𝑑Π𝐴𝐵𝑑𝑇

−Π𝐴𝐵𝑇= 𝜅𝐴𝐵

𝑑Π𝐴𝐵𝑑𝑇

− 𝑆𝐴𝐵 = 𝜅𝐴𝐵

Energy balance(The first Thomson relation)Entropy balance

Π𝐴𝐵𝑇= 𝑆𝐴𝐵

The second Thomson relation

𝑑Π

𝑑𝑇− 𝑆 = 𝜅

Π

𝑇= 𝑆

Seebeck coefficient: 𝑆

Peltier coefficient: Π

Thomson coefficient: 𝜅

Three coefficients

Two relations

One of three coefficientsgives the other two coefficients

The only one directly measurable for individual materials

Onsager reciprocal relationsin Non-equilibrium thermodynamicsCheck it for more exact and more universal deviation.

Potential: 𝜙

Its conjugate: 𝑝

Its flow: 𝐽

𝐽1𝐽2⋮𝐽𝑁

=𝐿11 ⋯ 𝐿1𝑁⋮ ⋱ ⋮𝐿𝑁1 ⋯ 𝐿𝑁𝑁

∇𝜙1𝛻𝜙2⋮𝛻𝜙𝑁

𝐿𝑖𝑗 = 𝐿𝑗𝑖 Onsager reciprocal relations

𝑇, 𝜙𝑒 , 𝑃, 𝜇,⋯

𝑠, 𝑞, 𝑉,𝑚,⋯

(𝑝𝜙 has the unit of energy)

Intensive variables

Extensive variables

2. Thermocouple

Thermocouple thermometer

Thermocouple“very basic” temperature measurement way.Using Seebeck effect

𝑉

𝑉𝐴𝐵 = − 𝐵

𝐴

𝑆 𝑇 𝑑𝑇

Thermocouple“very basic” temperature measurement way.Using Seebeck effect

Unknown

𝑇𝐴

Known

𝑇𝐵

Unknown

𝑇𝐴

Known

𝑇𝐵

Thermocouple“very basic” temperature measurement way.Using Seebeck effect

𝑉 Meter

Wire

Connection is (usually) necessary

Thermocouple

𝑉 Meter

𝑉𝑀𝐴 = − 𝑀

𝐴

𝑆w 𝑇 𝑑𝑇

𝑉𝐵𝑀 = − 𝐵

𝑀

𝑆w 𝑇 𝑑𝑇

Unknown

𝑇𝐴

Known

𝑇𝐵

What you measure is 𝑉𝐵𝐴 − 𝑉𝑀𝐴 − 𝑉𝐵𝑀

Thermocouple

𝑉

𝑉 = 𝐵

𝐴

𝑆+ 𝑇 − 𝑆− 𝑇 𝑑𝑇

Unknown

𝑇𝐴

Known

𝑇𝐵

What you measure is

Uniform temperature

Material-

Material+

𝑉

Use two materials(no other way)

Thermocouple

𝑉 = 𝐵

𝐴

𝑆+ 𝑇 − 𝑆− 𝑇 𝑑𝑇

Coupled propertiesare important

Type Materials𝑆±/

(𝜇𝑉/℃)

K Chromel Alumel 41

J Iron Constantan 50

N Nicrosil Nisil 39

R 87%Pt/13%Rh

Platinum 10

T Copper Constantan 43

E Chromel Constantan 68

Thermocouple

T Range/℃ Remarks

-200 +1350High sensitivityHigh linearity

-40 +750High sensitivityEasily rusting

-270 +1300Wide range

stability

0 +1600High temperature

Expensive

-200 350Low temperature

Thermal noise

-110 +140Highest

sensitivity

Type Materials𝑆±/

(𝜇𝑉/℃)

K Chromel Alumel 41

J Iron Constantan 50

N Nicrosil Nisil 39

R 87%Pt/13%Rh

Platinum 10

T Copper Constantan 43

E Chromel Constantan 68

Thermocouple

Type Materials𝑆±/

(𝜇𝑉/℃)

K Chromel Alumel 41

J Iron Constantan 50

N Nicrosil Nisil 39

R 87%Pt/13%Rh

Platinum 10

T Copper Constantan 43

E Chromel Constantan 68

Color code

IEC BS

3. ThermoelectricPower Generation

44

Semiconductor thermoelectric circuit

Small heat engines Non-mechanical engine(Radioisotope generators) Recovery of waste heat (Energy Harvesting)

Thermoelectric power generation

Load

resistance: 𝑅

Heat input

𝑄𝑇H

𝑇C

Ptype

Ntype

45

Thermoelectric power generation

Load

resistance: 𝑅

Heat input

𝑄𝑇H

𝑇C

Generated power W

Excited current IPtype

Ntype

Current𝐼

𝐼 =𝑉

𝑅 + 𝑟=𝑆 𝑇𝐻 − 𝑇𝐶𝑟 𝑚 + 1

𝑚 =𝑅

𝑟

𝑊 = 𝐼2𝑅 =𝑆2 𝑇𝐻 − 𝑇𝐶

2

𝑟 𝑚 + 1 2

h : hightA : cross section ρ : resistivity λ : thermal conductance

𝑟 =ℎp𝜌p

𝐴p+ℎn𝜌n𝐴n

Thermoelectric power generation

Load

resistance: 𝑅

Heat input

𝑄𝑇H

𝑇C

Ptype

Ntype

Current𝐼

Ohmic heating

Heat conduction

Peltier heat

h : hightA : cross section ρ : resistivity λ : thermal conductance

𝑄𝑂 = 𝑟𝐼2 𝑟 =

ℎp𝜌p

𝐴p+ℎn𝜌n𝐴n

𝑄𝐻 = Λ(𝑇𝐻 − 𝑇𝐶)Λ =

𝜆p𝐴p

ℎp+𝜆n𝐴nℎn

𝑄𝑃 = 𝑆𝑇𝐻𝐼

Thermoelectric power generation

Load

resistance: 𝑅

Heat input

𝑄𝑇H

𝑇C

Ptype

Ntype

Current𝐼

Heat balance on hot side

𝑄 +1

2𝑄𝑂 − 𝑄𝐻 − 𝑄𝑃 = 0

𝑄 = 𝑆𝑇𝐻𝐼 + Λ 𝑇𝐻 − 𝑇𝐶 −1

2𝑟𝐼2

Thermoelectric power generation

Theoretical thermal efficiency

𝑚opt = 1 +𝑍

2𝑇𝐻 − 𝑇𝐶

𝜂 =𝑇𝐻 − 𝑇𝐶𝑇𝐻

𝑚opt − 1

𝑚opt + 𝑇𝐶/𝑇𝐻

𝜂 =𝑊

𝑄= 𝑓(𝑇𝐻 , 𝑇𝐶 , 𝑚, 𝑍)

Maximum efficiency (impedance matching)

𝑍opt = S2 𝜆𝑝𝜌𝑝 + 𝜆𝑛𝜌𝑛

−2

𝑍 =𝑆2

Λ𝑟Figure-of-merit (熱電素子対の性能指数 )

Thermoelectric materials

49

Temperature dependence of ZT (dimensionless parameter)

p-type (left) and n-type (right) semiconductors

Design example

50

Specifications

p n

e [mV/K] 210 ‐170

r [mWm] 18 14

l [W/mK] 1.1 1.5

h [cm] 1.0 1.0

S [cm2] 1.3 1.0

TH=1,000K and TC=400K（S has been optimized）

Thermal efficiency

Output =4.5[W]

6

p n 380 10 [V/K]e e e

2

2 -1

max p p n n 0.00177[K ]Z e l r l r

opt 1.5m R r

max

1000 400 1.5 10.6 0.26 0.16

1000 1.5 400 1000

2 2

opt opt

opt

0.2280.004127

0.006886

TW R

R r

e

2.8mr W

=

Radioisotope Generator: RTG 原子力電池

Energy from the decay of a radioactive isotope to generate electricity（different from nuclear reactor）

Nuclear ReactorUse of nuclear chain reaction

Natural decay Chain reaction

Control the rateby the material and environment

Chain reactionUse of nuclear chain reaction

Electron

Nucleus

= Protons+ neutrons

Atom

Chemical energyUse of electron energy states

Electron

Radioactive decayUse of nucleus energy

Plutonium 238

He

Uranium 234

x 94

x 144

x 94

x 92

x 142

x 92

x 2

x 2

x 2Half decayby 88 years

Radioactive decayUse of nucleus energy

Plutonium 238

He

Uranium 234

x 94

x 144

x 94

x 92

x 142

x 92

x 2

x 2

x 2Half decayby 88 years

540 W/kg

RTG～5 W/kg

SAP～50 W/kg(1 AU)

59

Radioisotope-Thermoelectric Generator

Electric output 290W/250W

Thermal Output 4,234Wt

TH 1000℃

Total mass 55kg

Pu mass 7.561kg

size 114cm×f42cm

Galileo RTG

Radioisotope Generator: RTG 原子力電池

Energy from the decay of a radioactive isotope to generate electricity（different from nuclear reactor）

VoyagerRTG was located with a distance from the main body.Power would be 73% of BOL after 39 years.

CuriosityRTG on the back (hip)

CassiniThree RTGswith a cover for each

New HorizonsThe latest RTG

Thank you