Kelvin Relations in Thermoelectric Materials: The Thermodynamic Connection between Seebeck, Peltier, and Thomson Effects
Kelvin Relations in Thermoelectric Materials: The Thermodynamic Connection between Seebeck, Peltier, and Thomson Effects
The Seebeck, Peltier, and Thomson effects are three manifestations of thermoelectric coupling between heat and electrical charge transport. The Seebeck effect describes the voltage generated by a temperature difference; the Peltier effect describes heat absorbed or released when electrical current crosses a junction; and the Thomson effect describes reversible heating or cooling when current flows through a material with a temperature gradient.
These effects are not independent. The Kelvin relations connect their coefficients and show that they arise from the same underlying thermoelectric transport processes. These relations are important in thermoelectric material characterization, device modelling, and the separation of reversible thermoelectric heat from irreversible Joule heating.
This article derives the Kelvin relations, explains their connection to Onsager reciprocity and thermodynamic consistency, and discusses how they are used in functional materials. Particular attention is given to sign conventions, temperature-dependent Seebeck coefficients, and the distinction between reversible and irreversible heat.
1. Three thermoelectric effects
1.1 Seebeck effect
A temperature gradient in a conductor or semiconductor can generate an electrical potential difference under open-circuit conditions. The Seebeck coefficient is conventionally defined as
[ S=-.|_{I=0} ]
where the sign depends on the specified voltage and temperature-gradient directions. The SI unit is V K(^{-1}), commonly reported as ().
1.2 Peltier effect
When an electrical current passes through the junction of two dissimilar materials, heat is absorbed or released at the junction. The Peltier coefficient of a material is denoted by (), with units of V or W A(^{-1}).
For a junction between materials A and B, the net junction heat rate is proportional to the current and to the difference between their Peltier coefficients. The sign indicates whether the junction absorbs or releases heat under the chosen current direction.
1.3 Thomson effect
When current flows through a homogeneous material that has a temperature gradient, reversible heat is exchanged continuously along the material. The Thomson coefficient is denoted by (), with units V K(^{-1}).
The Thomson effect differs from Joule heating. Joule heating is irreversible and proportional to (I^2R); Thomson heating is reversible, changes sign when the current direction is reversed, and depends on the temperature gradient.
2. The Kelvin relations
The classical Kelvin relations are
[ ]
and
[ ]
where (S) is the Seebeck coefficient, () is the Peltier coefficient, () is the Thomson coefficient, and (T) is absolute temperature.
The first relation connects the heat transported per unit charge at a junction to the Seebeck coefficient. The second relates the reversible heat generated in a temperature gradient to the temperature dependence of thermopower.
These equations apply when the coefficients are defined consistently and the material is described by local-equilibrium thermoelectric transport. Their signs must be interpreted using the same current, voltage, and heat-flow conventions throughout the analysis.
3. Deriving the Peltier–Seebeck relation from reversible transport
Consider a material carrying an electrical current (I) at temperature (T). The Peltier heat transported per unit time is
[ Q_{}=I ]
For a small reversible transfer of charge (dq=I,dt), the associated heat transfer is
[ dQ_{}=,dq ]
The Seebeck coefficient describes the reversible electrical response to a temperature difference. Thermodynamic consistency of the coupled heat and charge transport processes requires the heat transported by a charge current to be related to the voltage generated by a temperature gradient.
For a material in local equilibrium, this gives
[ =ST ]
The factor (T) converts the entropy transported per unit charge into heat transported per unit charge. In this interpretation, (S) is closely related to the transported entropy per charge, while () is the corresponding transported heat per charge.
This is a thermodynamic statement about coupled transport, not an assertion that all heat carried by charge carriers is directly measured as Peltier heat at every interface. The experimentally relevant junction heat depends on the difference in the Peltier coefficients of the two adjoining materials.
4. Peltier heat at a junction
For two materials A and B joined at a junction, the Peltier heat rate is
[ Q_{}=(_A-_B)I ]
with the sign determined by the chosen direction for positive heat flow and current.
Using the Kelvin relation,
[ Q_{} = T(S_A-S_B)I ]
This equation explains why a junction between two materials with different Seebeck coefficients can act as a localized heating or cooling element.
If the two materials have identical thermopower at the junction temperature, the ideal reversible Peltier heat associated with their coefficient difference vanishes. In practical devices, contact resistance, interfacial reactions, and thermal losses can still produce heating.
5. Deriving the Thomson relation
The Thomson effect occurs in a homogeneous material when current flows through a temperature gradient. The reversible heat generated per unit volume and time is conventionally written
[ q_{}=,T ]
where () is the current density. The sign of () depends on the convention used for positive heat generation.
The Thomson coefficient is related to the temperature derivative of the Peltier coefficient. With the conventional Kelvin definitions,
[ =-S ]
Substituting (=ST),
[ =-S ]
Applying the product rule,
[ =S+T-S ]
Therefore,
[ ]
The result shows that a material with a temperature-independent Seebeck coefficient has zero Thomson coefficient within this description, even though it may still exhibit Seebeck and Peltier effects.
6. What the Thomson coefficient tells us
The Thomson coefficient measures how strongly the thermopower changes with temperature. It is not determined by the absolute value of (S), but by its temperature derivative.
Three cases are useful:
- If (dS/dT>0), then () is positive under the stated convention.
- If (dS/dT<0), then () is negative.
- If (dS/dT=0), then the Thomson coefficient is zero.
A large Seebeck coefficient does not necessarily imply a large Thomson coefficient. A material can have a substantial but nearly temperature-independent (S), giving a small ().
Conversely, a material with a moderate Seebeck coefficient can have a significant Thomson response if (S) changes rapidly with temperature.
7. Connection to Onsager reciprocity
The Kelvin relations can also be understood through linear irreversible thermodynamics. Near equilibrium, charge and heat fluxes respond to electrical and thermal driving forces.
A simplified one-dimensional representation is
[ \[\begin{pmatrix} J_e\\ J_q \end{pmatrix}\] = \[\begin{pmatrix} L_{11}&L_{12}\\ L_{21}&L_{22} \end{pmatrix} \begin{pmatrix} X_e\\ X_T \end{pmatrix}\]]
where (J_e) is an electrical flux, (J_q) is a heat flux, and (X_e) and (X_T) are appropriately defined thermodynamic forces.
For a system obeying microscopic reversibility in the absence of a time-reversal-symmetry-breaking field, Onsager reciprocity gives
[ L_{12}=L_{21} ]
when the fluxes and forces are defined in conjugate pairs with consistent normalization.
The cross-coefficients describe reciprocal coupling: a thermal driving force can generate an electrical current, and an electrical driving force can transport heat. Translating this reciprocity into the conventional measurable coefficients yields the Kelvin relation (=ST).
In magnetic fields or systems with broken time-reversal symmetry, the more general Onsager–Casimir relations must be used. Tensor components may be related at opposite magnetic fields rather than being simply equal at the same field.
8. Reversible thermoelectric heat versus Joule heating
A central practical issue is separating reversible thermoelectric heat from irreversible dissipation.
For an ohmic conductor, the local Joule-heating term is
[ q_{}=J^2 ]
where () is electrical resistivity. It is non-negative and does not change sign when the current direction is reversed.
The Thomson contribution is proportional to the current and temperature gradient:
[ q_{}=,T ]
It reverses sign when the current is reversed while the temperature gradient is held fixed.
At a junction, the Peltier contribution is proportional to (I), and it also reverses sign with current direction. This current-reversal behaviour provides an experimental way to distinguish reversible thermoelectric contributions from Joule heating, although parasitic effects and transient thermal responses must be controlled.
9. The thermoelectric heat equation
A simplified one-dimensional steady-state heat equation for a conductor can be written as
[ () +J^2 -J=0 ]
for a chosen sign convention, constant current density (J), and with the Thomson term written as a heat source with the corresponding convention.
Here () is thermal conductivity and () is electrical resistivity. Depending on the definition of heat flux and the sign of the Thomson coefficient, the sign of the last term may be written differently. The governing equation must be derived consistently from the selected constitutive relations.
The equation illustrates that temperature profiles in a thermoelectric leg can be influenced by:
- Fourier heat conduction.
- Joule heating.
- Thomson heating or cooling.
- Peltier heat at material interfaces and electrical contacts.
- External heat exchange and radiation.
For high-temperature oxide thermoelectrics, radiative heat loss can also be significant and should not be mistaken for an intrinsic Thomson contribution.
10. Calculating the Peltier coefficient from Seebeck data
The Kelvin relation provides a direct route to estimate the Peltier coefficient when the Seebeck coefficient is known at the same temperature:
[ (T)=S(T)T ]
For example, suppose a material has
[ S=150 ]
at
[ T=800 ]
Then
[ =(150^{-6})(800) ]
[ ]
Since (1 =1 ), this corresponds to 0.12 W of reversible Peltier heat transport per ampere under the stated coefficient convention.
This is a material Peltier coefficient. The net heat at a junction depends on the difference between the Peltier coefficients of the two materials.
11. Calculating the Thomson coefficient from experimental data
The Thomson coefficient can be estimated by differentiating measured Seebeck data:
[ (T)=T ]
Suppose the Seebeck coefficient changes approximately linearly between 700 K and 900 K:
[ S(700 )=120 ]
[ S(900 )=160 ]
The approximate derivative is
[ =0.20 ]
At 800 K,
[ (0.20) ]
[ ]
This estimate assumes a locally linear dependence of (S) on temperature. Experimental differentiation amplifies noise, so a suitable smooth fit or physically justified interpolation should be used rather than taking differences between noisy adjacent points.
12. Implications for thermoelectric device operation
The Kelvin relations are useful in modelling both power-generation and cooling devices.
In a thermoelectric generator:
- A temperature difference produces a Seebeck voltage.
- Current through an external load produces Peltier heat transport at interfaces.
- A temperature gradient and current can produce Thomson heating or cooling within each leg.
- Joule heating opposes useful heat conversion.
In a thermoelectric cooler:
- Current direction determines whether a junction absorbs or releases heat.
- Peltier heat transport can move heat against a temperature gradient.
- Joule heating increases with current squared.
- Thomson heat can modify the temperature profile along a leg.
The optimal current is therefore not determined by the Peltier effect alone. It depends on electrical resistance, thermal conductivity, geometry, boundary conditions, and the temperature dependence of the thermoelectric coefficients.
13. Temperature-dependent properties and material design
In real thermoelectric materials, (S), (), and () are functions of temperature. A constant-property approximation may be acceptable over a narrow interval, but broad operating ranges require temperature-dependent data.
For oxide thermoelectrics, temperature-dependent thermopower can reflect:
- Changes in carrier concentration.
- Thermally activated transport.
- Changes in dominant scattering mechanisms.
- Contributions from multiple electronic bands.
- Oxygen nonstoichiometry and defect equilibration.
- Structural or magnetic transitions.
The Thomson coefficient is particularly sensitive to these variations because it depends on (dS/dT). Abrupt features in measured (S(T)) can produce large apparent derivatives, but they should be checked for phase transitions, measurement artefacts, and non-equilibrium changes in sample composition.
14. Experimental strategy for validating Kelvin relations
The Peltier and Thomson effects are more difficult to measure directly than the Seebeck coefficient. Nevertheless, the Kelvin relations allow consistency checks using accurate thermopower data.
A practical strategy is:
- Measure (S(T)) over the intended operating-temperature range.
- Ensure the sample atmosphere and equilibration protocol are controlled.
- Fit (S(T)) using a smooth function that does not suppress real physical transitions.
- Calculate ((T)=S(T)T).
- Calculate ((T)=T,dS/dT).
- Compare predicted reversible heat terms with measurements or device simulations.
- Report the sign convention and uncertainty introduced by differentiation.
Direct Peltier measurements can provide an independent check, but contact thermal resistance, parasitic heat flow, and interfacial effects must be considered. Direct Thomson measurements require careful separation of reversible heat from Joule heating and ordinary heat conduction.
15. Assumptions and limitations
The Kelvin relations are powerful, but their use requires a consistent thermodynamic and transport framework.
Important considerations include:
- Local equilibrium: transport coefficients are defined locally and vary smoothly enough for the continuum description.
- Linear response: the constitutive relations are used near equilibrium, with sufficiently small driving forces.
- Consistent conventions: current, heat flux, voltage, and temperature-gradient directions must be defined.
- Compatible coefficients: the Seebeck and Peltier coefficients refer to the same material state, temperature, and relevant transport conditions.
- Magnetic-field effects: Onsager–Casimir reciprocity may replace the simple zero-field reciprocity relation.
- Inhomogeneity: composition gradients and interfaces can introduce additional thermoelectric voltages and heat sources.
- Additional carriers: in mixed ionic-electronic conductors, ionic and electronic fluxes may require a multicomponent transport description.
The relations should not be used to infer an intrinsic material coefficient from a device measurement without accounting for contacts, interfaces, and parasitic heat paths.
16. Common mistakes
- Using (=S) instead of (=ST): the absolute temperature factor is essential.
- Using Celsius in Kelvin relations: all temperatures must be absolute temperatures in kelvin.
- Confusing material and junction Peltier coefficients: junction heat depends on the difference between the two materials’ coefficients.
- Treating Thomson heat as Joule heating: Thomson heat is reversible and odd in current; Joule heat is irreversible and even in current.
- Assuming a large (S) means a large (): Thomson heat depends on the temperature derivative of (S).
- Differentiating noisy data directly: numerical differentiation can amplify measurement noise.
- Ignoring sign conventions: inconsistent definitions can produce apparent sign disagreements between equations and experiments.
- Applying simple reciprocity in a magnetic field without checking symmetry: the appropriate Onsager–Casimir form may be required.
17. Further reading
These articles provide related thermodynamic and experimental foundations:
- Thermodynamics of the Seebeck Effect: Chemical Potential, Carrier Statistics, and Thermoelectric Transport
- Thermoelectric Transport: Seebeck Coefficient, Electrical Conductivity, and Power Factor
- Thermodynamic Potentials in Materials Science
- Chemical Potential and Activity in Solid Materials
- Thermodynamic Factors in Ionic Transport
- Electrical Resistance and Conductivity Measurements
18. Final perspective
The Kelvin relations express the thermodynamic consistency of the three classical thermoelectric effects:
[ ]
They connect the voltage generated by a temperature gradient to the heat transported by electrical current and to reversible heat exchange within a temperature gradient.
For functional oxides and other thermoelectric materials, these relations make it possible to estimate Peltier and Thomson coefficients from temperature-dependent Seebeck measurements. Their practical value depends on careful sign conventions, reliable (S(T)) data, and a clear separation of reversible thermoelectric effects from Joule heating and parasitic thermal processes.