Thermoelectric figure of merit

i. Overview

A thermoelectric material converts a temperature difference into electrical power. In the linear-response (Onsager) regime, the coupled transport of charge and heat in an isotropic material is governed by the constitutive relations

J = σ (E − S ∇T), q = T S J − κ ∇T,

so a thermoelectric material is characterized by exactly four transport coefficients: the Seebeck coefficient S, the electrical conductivity σ, and the thermal conductivity κ = κl + κe, split into its lattice (phonon) and electronic contributions because the two channels respond independently to material design. These four coefficients are collected in the structure ThermoelectricMaterial, and everything in this file is stated on it.

The performance of the material at absolute temperature T is captured by the dimensionless figure of merit

zT = σ S² T / (κl + κe),

whose numerator groups into the power factor PF = σ S². Temperature is a state variable, not a material property, so T enters as an argument rather than a field. The four coefficients of a real material are themselves temperature-dependent, so a ThermoelectricMaterial should be read as a material at an operating point, its coefficients evaluated at the same temperature at which zT is evaluated.

In this file all quantities are real numbers in a fixed consistent system of units, following the convention of Physlib.Thermodynamics.IdealGas.Basic.

ii. Key results

ThermoelectricMaterial: the four linear-response transport coefficients of a thermoelectric material, with their physical sign constraints.

powerFactor: the thermoelectric power factor σ S².

totalThermalConductivity: the total thermal conductivity κl + κe.

figureOfMerit: the dimensionless figure of merit zT.

figureOfMerit_eq: the flat form zT = σ S² T / (κl + κe).

figureOfMerit_pos: positivity of zT for a material with a nonzero Seebeck coefficient at positive temperature.

figureOfMerit_le_of_le: lowering the lattice thermal conductivity raises zT, the phonon-glass electron-crystal design principle.

iii. Table of contents

A. The thermoelectric material

B. The power factor

C. The total thermal conductivity

D. The figure of merit

D.1. Equalities for the figure of merit

D.2. Positivity of the figure of merit

D.3. Monotonicity in the lattice thermal conductivity

iv. References

Ioffe, A.F., Semiconductor Thermoelements and Thermoelectric Cooling, Infosearch (1957).

Snyder, G.J., Toberer, E.S., Complex thermoelectric materials, Nature Materials 7, 105–114 (2008).

A. The thermoelectric material

The four fields are the complete set of linear-response coefficients for coupled charge and heat transport in an isotropic material: no further material parameter enters the steady-state thermoelectric equations, and zT is a function of exactly these four together with the temperature. The sign constraints are physical: a thermoelectric material conducts charge (0 < σ) and its lattice conducts heat (0 < κl), while the electronic heat channel can be negligible but never negative (0 ≤ κe). The Seebeck coefficient is unconstrained: its sign records the carrier type (negative for electrons, positive for holes), and it vanishes at compensation points.

A thermoelectric material in the linear-response regime, characterized by its four transport coefficients.

The Seebeck coefficient S: the voltage developed per unit temperature difference. Its sign records the dominant carrier type.

The electrical conductivity σ.

The lattice (phonon) contribution κl to the thermal conductivity.

The electronic contribution κe to the thermal conductivity.

B. The power factor

The thermoelectric power factor PF = σ S² of a material.

The power factor is positive when the Seebeck coefficient is nonzero.

C. The total thermal conductivity

The total thermal conductivity κl + κe of a material, the sum of the lattice (phonon) and electronic contributions.

The total thermal conductivity is positive.

D. The figure of merit

The dimensionless thermoelectric figure of merit zT = PF · T / (κl + κe) of a material at absolute temperature T.

D.1. Equalities for the figure of merit

The figure of merit in its standard flat form zT = σ S² T / (κl + κe).

D.2. Positivity of the figure of merit

The figure of merit is positive at positive temperature when the Seebeck coefficient is nonzero.

D.3. Monotonicity in the lattice thermal conductivity

Lowering the lattice thermal conductivity raises the figure of merit: if M.κl ≤ κl' then the material with lattice conductivity κl' (all other coefficients equal) has the smaller zT. This is the phonon-glass electron-crystal design principle: scatter phonons without degrading electronic transport. Note the hypotheses: no condition on the Seebeck coefficient is needed, since monotonicity only requires the numerator σ S² T to be nonnegative.