How to calculate Seebeck coefficient thermoelectric ZT

Calculate Seebeck coefficient S = -V/dT thermoelectric thermocouples generators NASA RTG. Figure of merit ZT = S^2*sigma*T/kappa Bi2Te3 PbTe SiGe. Thomas Seebeck Estonia-Germany 1770-1831 1821. NASA JPL + ORNL + MIT.

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How to calculate the Seebeck coefficient (S) fundamental thermoelectric property measuring thermoelectric voltage generated per unit temperature difference between two points of a conducting or semiconductor material: S = -V / dT. V thermoelectric voltage V (microvolts to millivolts typical); dT temperature difference K (10-1000 K typical thermocouples; 1-100 K thermoelectric generators); minus sign convention: S positive for holes p-type carriers + S negative for electrons n-type.

Seebeck effect: discovered by Thomas Johann Seebeck 1821 + voltage generated by temperature gradient in conductor + basis of thermocouples + thermoelectric generators TEG + Peltier coolers + classical reference.

Typical coefficients: copper Cu S=-1.3 uV/K @ 300 K + Chromel S=22 uV/K + Alumel S=-18 uV/K + thermocouple K = Chromel-Alumel S~41 uV/K; bismuth telluride Bi2Te3 S=-220 uV/K @ 300 K p-type; lead telluride PbTe S=-180 uV/K @ 600 K; silicon-germanium SiGe S=-250 uV/K @ 1000 K NASA RTG; oxides Ca3Co4O9 S=120 uV/K + perovskites SrTiO3 S=-1000 uV/K.

Figure of merit ZT: ZT = S^2 * sigma * T / kappa + sigma electrical conductivity + kappa thermal conductivity + ZT > 1 good thermoelectric + ZT > 2 state-of-art + ZT_Carnot = (T_h - T_c)/T_h Carnot efficiency limit + classical reference; ZT_Bi2Te3 = 1.0 @ 300 K; ZT_PbTe = 1.5 @ 600 K; ZT_SiGe = 0.7 @ 1000 K; ZT_skutterudite = 1.5 @ 800 K.

Inverse Peltier effect: Pi = S * T Peltier coefficient + heat absorption or release when current crosses junction between two different materials + basis Peltier coolers solid-state refrigeration + classical reference.

Thomson effect: tau = T * dS/dT Thomson coefficient + heat absorption/release when current crosses temperature gradient in single material + classical reference Lord Kelvin 1854.

Mott formula semiconductors: S = -(pi^2 * k_B^2 * T) / (3 * e * E_F) + linear in T + degenerate + classical reference Sommerfeld.

General Cutler-Mott formula: S = (pi^2 * k_B^2 * T)/(3*e) * (d ln(sigma)/dE)|_E_F + classical reference electron energy dependence + Mott 1969.

Related numbers: electrical conductivity sigma; thermal conductivity kappa; Wiedemann-Franz law kappa_e/sigma = L*T = 2.44e-8*T W*ohm/K; power factor PF = S^2*sigma; thermoelectric efficiency eta = ((1+ZT)^(1/2)-1)/((1+ZT)^(1/2)+T_c/T_h).

Standard thermocouples IEC 60584: K type Chromel-Alumel S~41 uV/K + range -200 to 1372 °C + classical industrial; J type Iron-Constantan S~52 uV/K + -210 to 1200 °C; T type Copper-Constantan S~43 uV/K + -270 to 400 °C high precision; E type Chromel-Constantan S~68 uV/K + cryogenic; R type Pt-PtRh13% S~10 uV/K + 0 to 1768 °C noble; S type Pt-PtRh10% + IPTS-90 reference + classical reference.

Thermoelectric generators TEG: P = S^2 * sigma * (dT)^2 / R_load + waste heat recovery + automotive Tesla EV + steel furnace BMW; NASA Radioisotope Thermoelectric Generator RTG + SiGe + 238Pu heat source + Voyager 1977 Cassini 1997 New Horizons 2006 Curiosity Mars 2011 Perseverance Mars 2020 + classical reference space exploration. History: Thomas Johann Seebeck (Reval Estonia 1770 - Berlin 1831) Berlin Univ + Akademie Wissenschaften 1821 member; 1821 paper Akademie der Wissenschaften zu Berlin 'Magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz' classic Seebeck effect discovery + originally interpreted as 'thermomagnetism' + Hans Christian Oersted 1823 corrected to thermoelectric + classical reference + Seebeck coefficient eponym + Member US National Academy + classical reference Russian-Estonian scientist + Berlin Akademie.

Jean Charles Athanase Peltier (Ham France 1785-1845) French watchmaker physicist; 1834 paper Annales de Chimie et de Physique 56:371 'Nouvelles experiences sur la caloricite des courans electriques' classic Peltier effect inverse Seebeck + heating cooling at junction + classical reference + Peltier cooler eponym.

William Thomson Lord Kelvin (Belfast Ireland 1824 - Largs Scotland 1907) Glasgow University 1846-1899; 1854 paper Trans Roy Soc Edin 21:123 'On the Dynamical Theory of Heat' classic Thomson effect + Thomson relations Seebeck Peltier + 1851 absolute temperature scale Kelvin + classical reference + Lord Kelvin eponym + Member Royal Society + Knighted 1866 + Baron 1892.

Edmund Altenkirch (Germany 1880-1953) German engineer; 1909 paper Phys Z 10:560 'Uber den Nutzeffekt der Thermosaule' classic Altenkirch criterion ZT figure of merit thermoelectric efficiency + classical reference modern thermoelectrics + Altenkirch eponym Z dimensionless thermoelectric figure of merit.

Abraham Ioffe (Romny Russia 1880 - Leningrad 1960) Soviet physicist Ioffe Institute Leningrad founder 1918; 1957 textbook 'Semiconductor Thermoelements and Thermoelectric Cooling' Infosearch London 184 pages classic + classical reference modern thermoelectrics + Ioffe Institute eponym + classical reference Russian-Soviet + Bi2Te3 thermoelectric. Legacy: Goldsmid 'Thermoelectric Refrigeration' 1964 Plenum + 'Introduction to Thermoelectricity' 2nd 2016 Springer 282 pages classical + Rowe 'CRC Handbook of Thermoelectrics' 1995 CRC 701 pages + 'Thermoelectrics Handbook Macro to Nano' 2006 CRC 1014 pages + Snyder-Toberer 2008 Nature Materials 7:105 classical review modern + Tritt-Subramanian 2006 MRS Bulletin classical + Kanatzidis 2010 Chem Mater 22:648 classical advanced.

USA research thermoelectric + thermocouples: NIST Gaithersburg Boulder 4800 + thermocouple SRM standards IPTS-90 ITS-90; NSF Materials 12000; DOE national labs Argonne ORNL LBNL LANL Sandia 35000 + thermoelectric energy harvesting; NASA Jet Propulsion Lab JPL Pasadena 6000 + RTG SiGe Voyager Cassini Curiosity Perseverance world-leading + Glenn Research Center; NASA Glenn Cleveland OH 1500 + advanced thermoelectric materials; NREL Boulder 3700 + thermoelectric energy harvesting; Argonne National Lab 3500; ORNL Oak Ridge 6000 + skutterudites SiGe; NIH 18000.

USA universities thermoelectric: MIT 11500 + Mildred Dresselhaus thermoelectric nanowires legacy world-leading + Gang Chen heat transfer; Caltech 8800 + Sossina Haile high-temp; Stanford 11000; UC Berkeley 8500 + Peidong Yang nanowires; Northwestern 7100 + Mercouri Kanatzidis thermoelectric world-leading + chalcogenides PbTe SnSe; Univ Michigan 9000 + Pierre Ferdinand Poudeu; Penn State 8500; UCLA 31000; Boston College 9700 + Zhifeng Ren thermoelectric; Wake Forest 5000 + David Carroll polymer thermoelectric; Georgia Tech 36000; Texas Austin 50000.

USA industry thermocouples + temperature sensors: OMEGA Engineering Norwalk CT 2500 + thermocouples K J T E R S types world-leading; Honeywell Charlotte NC 95000 + thermocouples industrial; Watlow St Louis MO 3000; Pyromation Fort Wayne IN 1500; Conax Buffalo NY 250; Texas Instruments TI Dallas 33000 + thermocouple amplifiers ADS1118 + classical reference; Analog Devices Wilmington MA 25000 + AD8497 thermocouple; Maxim Integrated 7000 + MAX31855 K-type.

USA industry thermoelectric generators TEG: Marlow Industries Dallas 250 + Bi2Te3 modules world-leading; Laird Performance Materials Akron OH 4000 + Peltier modules; Ferrotec Bedford NH 2000 + thermoelectric modules; Phononic Durham NC 200 + solid-state cooling thermoelectric; II-VI Coherent 22000 + thermoelectric modules; Komatsu thermoelectric 200.

USA industry NASA RTG + space: NASA JPL 6000 + RTG SiGe MMRTG ASRG eMMRTG; Aerojet Rocketdyne Sacramento 5000 + RTG components; Teledyne Energy Systems 2500 + RTG; Westinghouse Cranberry 9000 + RTG nuclear; Idaho National Laboratory INL 5000 + 238Pu fuel; Oak Ridge ORNL 6000; Los Alamos LANL 13000.

USA industry HVAC Peltier coolers: Carrier Palm Beach 53000; Trane Davidson 36000; Lennox Richardson 14000; Daikin USA 1500; Marlow Industries 250 Peltier modules; Phononic 200.

USA industry steel thermocouples: Nucor Charlotte 32000 + R S type Pt-Rh thermocouples; U.S. Steel 22000; Steel Dynamics 11000; Cleveland-Cliffs 26000.

Seebeck formula

Definition: S = -V / dT. V thermoelectric voltage V + dT temperature difference K + minus sign convention.

Figure of merit ZT: ZT = S^2 * sigma * T / kappa + ZT > 1 good thermoelectric + ZT > 2 state-of-art.

Inverse Peltier effect: Pi = S * T Peltier coefficient + basis solid-state coolers.

Thomson effect: tau = T * dS/dT Thomson coefficient + Lord Kelvin 1854.

Mott degenerate formula: S = -(pi^2*k_B^2*T)/(3*e*E_F) + linear T + Sommerfeld.

Typical coefficients: copper Cu S=-1.3 uV/K; Chromel 22 + Alumel -18 + K-type 41 uV/K; Bi2Te3 S=-220 uV/K @ 300 K ZT=1; PbTe S=-180 @ 600 K ZT=1.5; SiGe S=-250 @ 1000 K ZT=0.7 NASA RTG; oxides Ca3Co4O9 S=120; perovskites SrTiO3 S=-1000.

History and context

Thomas Johann Seebeck (Reval Estonia 1770 - Berlin 1831) Berlin Univ + Akademie Wissenschaften 1821 member; 1821 paper 'Magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz' classic Seebeck effect discovery + originally interpreted as 'thermomagnetism' + Hans Christian Oersted 1823 corrected to thermoelectric + classical reference + Seebeck coefficient eponym + Member US National Academy + classical reference Russian-Estonian scientist.

Jean Charles Peltier (1785-1845); 1834 Annales de Chimie 56:371 classic Peltier effect inverse Seebeck + Peltier cooler eponym.

Lord Kelvin (1824-1907) Glasgow 1846-1899; 1854 Trans Roy Soc Edin 21:123 classic Thomson effect + Thomson relations + 1851 absolute temperature Kelvin + Lord Kelvin eponym + Royal Society + Knighted 1866.

Edmund Altenkirch (1880-1953); 1909 Phys Z 10:560 classic Altenkirch criterion ZT figure of merit + Altenkirch eponym.

Abraham Ioffe (1880-1960) Ioffe Institute Leningrad founder; 1957 'Semiconductor Thermoelements and Thermoelectric Cooling' Infosearch 184 + classical reference + Bi2Te3 thermoelectric + Ioffe Institute eponym. Legacy: Goldsmid 1964 Plenum + 2016 Springer 282 + Rowe 1995 CRC 701 + 2006 CRC 1014 + Snyder-Toberer 2008 Nature Materials 7:105 + Tritt-Subramanian 2006 + Kanatzidis 2010.

USA research: NIST 4800 + thermocouple SRM ITS-90; NSF Materials 12000; DOE labs 35000; NASA JPL 6000 + RTG SiGe Voyager Cassini Curiosity Perseverance world-leading; NASA Glenn 1500; NREL 3700; Argonne 3500; ORNL 6000.

Universities: MIT 11500 + Dresselhaus legacy world-leading + Gang Chen; Caltech 8800; Stanford 11000; UC Berkeley 8500 + Peidong Yang; Northwestern 7100 + Kanatzidis world-leading thermoelectric + chalcogenides; Michigan 9000; Penn State 8500; UCLA 31000; Boston College 9700 + Zhifeng Ren; Wake Forest 5000; Georgia Tech 36000; Texas Austin 50000.

Industry thermocouples: OMEGA Engineering 2500 world-leading; Honeywell 95000; Watlow 3000; Pyromation 1500; Texas Instruments 33000; Analog Devices 25000; Maxim Integrated 7000.

TEG modules: Marlow Industries 250 world-leading; Laird 4000; Ferrotec 2000; Phononic 200; II-VI Coherent 22000.

NASA RTG: NASA JPL 6000; Aerojet Rocketdyne 5000; Teledyne 2500; Westinghouse 9000; Idaho INL 5000; ORNL 6000.

HVAC Peltier: Carrier 53000; Trane 36000; Lennox 14000; Daikin USA 1500; Marlow 250; Phononic 200.

Steel: Nucor 32000; U.S. Steel 22000; Steel Dynamics 11000; Cleveland-Cliffs 26000.

Seebeck values table

  • S = -V/dT: definition
  • ZT = S^2*sigma*T/kappa: figure of merit
  • Pi = S*T: Peltier coefficient
  • tau = T*dS/dT: Thomson coefficient
  • Cu: S=-1.3 uV/K
  • Thermocouple K: 41 uV/K
  • Bi2Te3: -220 uV/K ZT=1
  • PbTe: -180 uV/K ZT=1.5
  • SiGe RTG: -250 uV/K ZT=0.7
  • SrTiO3: -1000 uV/K oxide

5 numerical examples

  1. Copper Cu metal: V=-13 uV + dT=10 K + S=1.3 uV/K small classical metal.
  2. K-type Chromel-Alumel thermocouple: V=-410 uV + dT=10 K + S=41 uV/K thermocouple K type IEC 60584 classical.
  3. Bismuth telluride Bi2Te3 thermoelectric: V=-2.2 mV + dT=10 K + S=220 uV/K + sigma=1e5 + kappa=1.5 + ZT=1.0 @ 300 K Peltier cooler classical Marlow.
  4. PbTe medium temperature: V=-1.8 mV + dT=10 K + S=180 uV/K + sigma=5e4 + kappa=2 + ZT=0.97 @ 600 K classical reference.
  5. SiGe NASA RTG Voyager: V=-2.5 mV + dT=10 K + S=250 uV/K + sigma=1e4 + kappa=4 + ZT=0.16 raw + classical reference + Voyager Cassini Curiosity Perseverance.

Limitations

Linear response: small dT relative T + nonlinear effects high dT + Goldsmid 2016.

Single carrier band: real semiconductor multi-band + heavy/light hole + Snyder-Toberer 2008.

Constant transport: sigma kappa S vs T + integration needed + Rowe 1995.

Thomson cross terms: irreversible coupling + Onsager reciprocity + De Groot 1962.

Quantum nano effects: sub-100 nm phonon glass electron crystal PGEC + classical reference modern.

Common mistakes

  • Sign convention: S = -V/dT minus sign + holes p-type S>0 + electrons n-type S<0 + Goldsmid 2016.
  • Confuse S and Pi: S = V/dT Seebeck + Pi = S*T Peltier + Thomson tau = T*dS/dT + Lord Kelvin 1854 distinct.
  • ZT vs efficiency: eta = (T_h-T_c)/T_h * ((1+ZT)^(1/2)-1)/((1+ZT)^(1/2)+T_c/T_h) + ZT high not sufficient + Carnot limit.
  • Single material vs couple: thermocouples need junction of two different materials + S_couple = S_A - S_B + IEC 60584.
  • Cold junction compensation: thermocouples need cold junction reference compensation + classical reference industrial.

Applications

  1. Industrial thermocouples: OMEGA + Honeywell + Watlow + K J T E R S types + Texas Instruments + Analog Devices + Maxim MAX31855.
  2. Thermoelectric generators TEG: Marlow Industries + Laird + Ferrotec + automotive Tesla EV exhaust waste heat + Marlow world-leading.
  3. Peltier coolers solid-state refrigeration: Carrier + Trane + Lennox + Daikin + Marlow + Phononic + small refrigerators.
  4. NASA Radioisotope Thermoelectric Generator RTG: NASA JPL world-leading + Voyager 1977 + Cassini 1997 + New Horizons 2006 + Curiosity Mars 2011 + Perseverance 2020.
  5. Steel industry furnaces: Nucor + U.S. Steel + Steel Dynamics + Cleveland-Cliffs + R S type Pt-Rh thermocouples.
  6. HVAC Peltier modules: small portable refrigerators + minibar + medical sample storage + Phononic.
  7. Spin Seebeck effect SSE: classical reference + spintronic energy harvesting.
  8. EV battery thermal management: Tesla 4680 + Peltier control.
  9. Wearable energy harvesting: flexible polymer thermoelectric.
  10. Cryogenic thermometry: NIST + ITS-90 standard + classical reference.

Updated July 2026

Frequently Asked Questions

What does the Seebeck coefficient S measure?

Seebeck coefficient S = -V/dT measures thermoelectric voltage V generated per unit temperature difference dT in a conducting or semiconductor material. S0 holes p-type. Thomas Johann Seebeck Estonia-Germany 1770-1831 1821 Akademie Berlin classic + Oersted 1823 thermoelectricity.

What is the figure of merit ZT?

Figure of merit ZT = S^2*sigma*T/kappa dimensionless measure of thermoelectric efficiency of a material. ZT > 1 good thermoelectric (Bi2Te3 ZT=1.0 @ 300 K Peltier; PbTe ZT=1.5 @ 600 K; SiGe ZT=0.7 @ 1000 K NASA RTG); ZT > 2 state-of-art (skutterudites SnSe). Edmund Altenkirch Germany 1909 Phys Z 10:560 introduced Z criterion + Abraham Ioffe Ioffe Institute Leningrad 1957 textbook + classical reference Goldsmid 2016 Springer 282 pages.

What are Peltier and Thomson effects?

Peltier effect 1834 Jean Peltier inverse of Seebeck: Pi = S*T Peltier coefficient + heat absorption or release when current crosses junction between two different materials + basis Peltier coolers solid-state refrigeration. Thomson effect 1854 Lord Kelvin Trans Roy Soc Edin 21:123: tau = T*dS/dT Thomson coefficient + heat absorption/release when current crosses temperature gradient in single material + classical reference Kelvin.

What are IEC 60584 standard thermocouples?

Standard thermocouples IEC 60584: K type Chromel-Alumel S~41 uV/K range -200 to 1372 °C most common industrial; J type Iron-Constantan S~52 uV/K -210 to 1200 °C; T type Copper-Constantan S~43 uV/K -270 to 400 °C high precision; E type Chromel-Constantan S~68 uV/K cryogenic; R type Pt-PtRh13% S~10 uV/K 0 to 1768 °C noble; S type Pt-PtRh10% IPTS-90 reference + classical industrial.

What is the NASA RTG?

RTG Radioisotope Thermoelectric Generator NASA: space thermoelectric generator that converts decay heat from 238Pu into electricity via SiGe ZT=0.7 @ 1000 K + classic SiGe alloy. Missions: Voyager 1 and 2 1977 + Galileo Jupiter 1989 + Cassini Saturn 1997 + New Horizons Pluto 2006 + Curiosity Mars 2011 + Perseverance Mars 2020. NASA JPL world-leading. Classical reference space exploration + Goldsmid 2016.

What does Wiedemann-Franz law state?

Wiedemann-Franz law 1853: ratio of electronic thermal conductivity kappa_e to electrical conductivity sigma is proportional to temperature T: L = kappa_e/(sigma*T) = pi^2/3*(k_B/e)^2 = 2.44e-8 W*ohm/K^2 = Lorenz constant. Consequence: hard to increase ZT = S^2*sigma*T/kappa because increasing sigma also increases kappa_e. Phonon glass electron crystal PGEC strategy + nanostructures classical Snyder-Toberer 2008 + Kanatzidis 2010.

Seebeck industrial applications?

Applications: industrial thermocouples OMEGA + Honeywell + Watlow + K J T types + Texas Instruments IC; thermoelectric generators TEG Marlow Industries world-leading + automotive Tesla EV exhaust waste heat; Peltier coolers solid-state Carrier + Trane + Phononic + medical sample storage; NASA RTG NASA JPL Voyager Cassini Curiosity Perseverance world-leading; steel furnaces Nucor + U.S. Steel + Cleveland-Cliffs R S type Pt-Rh; spin Seebeck classical.

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