Physlib.CondensedMatter.Thermoelectric.Basic

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.

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

8 declarations

definition

Thermoelectric power factor $PF = \sigma S^2$

#powerFactor

The power factor $PF$ of a thermoelectric material $M$ is defined as the product of its electrical conductivity $\sigma$ and the square of its Seebeck coefficient $S$ , expressed as $PF = \sigma S^2$ .

theorem

The power factor $PF$ is positive if $S \neq 0$

#powerFactor_pos

For a thermoelectric material $M$ , if the Seebeck coefficient $S$ is non-zero ( $S \neq 0$ ), then the power factor $PF = \sigma S^2$ is strictly positive ( $PF > 0$ ).

definition

Total thermal conductivity $\kappa_l + \kappa_e$

#totalThermalConductivity

For a thermoelectric material $M$ , the total thermal conductivity is the sum of its lattice thermal conductivity $\kappa_l$ and its electronic thermal conductivity $\kappa_e$ , expressed as $\kappa_l + \kappa_e$ .

theorem

Total thermal conductivity is positive ( $\kappa > 0$ )

#totalThermalConductivity_pos

For any thermoelectric material $M$ , its total thermal conductivity $\kappa$ (the sum of the lattice thermal conductivity $\kappa_l$ and the electronic thermal conductivity $\kappa_e$ ) is strictly positive, i.e., $\kappa > 0$ .

definition

Thermoelectric figure of merit $zT = \frac{PF \cdot T}{\kappa_l + \kappa_e}$

#figureOfMerit

For a thermoelectric material $M$ at absolute temperature $T$ , the dimensionless thermoelectric figure of merit $zT$ is defined as the product of the power factor $PF$ and the temperature $T$ divided by the total thermal conductivity $\kappa_l + \kappa_e$ . Mathematically, this is expressed as: $zT = \frac{PF \cdot T}{\kappa_l + \kappa_e}$ where $PF = \sigma S^2$ is the power factor (the product of electrical conductivity $\sigma$ and the square of the Seebeck coefficient $S$ ), and the denominator is the sum of the lattice thermal conductivity $\kappa_l$ and the electronic thermal conductivity $\kappa_e$ .

theorem

Standard Formula for the Thermoelectric Figure of Merit $zT = \frac{\sigma S^2 T}{\kappa_l + \kappa_e}$

#figureOfMerit_eq

For a thermoelectric material $M$ at absolute temperature $T$ , the dimensionless figure of merit $zT$ is given by the expression: $zT = \frac{\sigma S^2 T}{\kappa_l + \kappa_e}$ where $\sigma$ is the electrical conductivity, $S$ is the Seebeck coefficient, $\kappa_l$ is the lattice thermal conductivity, and $\kappa_e$ is the electronic thermal conductivity of the material.

theorem

$zT > 0$ for $S \neq 0$ and $T > 0$

#figureOfMerit_pos

For a thermoelectric material $M$ with Seebeck coefficient $S \neq 0$ at a positive absolute temperature $T > 0$ , the dimensionless thermoelectric figure of merit $zT$ is strictly positive: $zT > 0$ The figure of merit is defined as $zT = \frac{\sigma S^2 T}{\kappa_l + \kappa_e}$ , where $\sigma$ is the electrical conductivity, $\kappa_l$ is the lattice thermal conductivity, and $\kappa_e$ is the electronic thermal conductivity.

theorem

Monotonicity of the figure of merit $zT$ with respect to lattice thermal conductivity $\kappa_l$

#figureOfMerit_le_of_le

For a thermoelectric material $M$ characterized by electrical conductivity $\sigma$ , Seebeck coefficient $S$ , lattice thermal conductivity $\kappa_l$ , and electronic thermal conductivity $\kappa_e$ , let $zT$ be the dimensionless figure of merit at absolute temperature $T$ . If $T \geq 0$ and the lattice thermal conductivity is increased to a value $\kappa_l' \geq \kappa_l$ (where $\kappa_l' > 0$ ), then the figure of merit of the resulting material is less than or equal to that of the original material. Mathematically, this states that $\frac{\sigma S^2 T}{\kappa_l' + \kappa_e} \leq \frac{\sigma S^2 T}{\kappa_l + \kappa_e}$ This result expresses the "phonon-glass electron-crystal" design principle: reducing the lattice thermal conductivity $\kappa_l$ (for example, by scattering phonons) while maintaining other transport coefficients increases the figure of merit $zT$ .

Physlib.CondensedMatter.Thermoelectric.Basic

Thermoelectric figure of merit

i. Overview

ii. Key results

iii. Table of contents

iv. References

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

Thermoelectric power factor PF=σS2PF = \sigma S^2PF=σS2

The power factor PFPFPF is positive if S≠0S \neq 0S=0

Total thermal conductivity κl+κe\kappa_l + \kappa_eκl​+κe​

Total thermal conductivity is positive (κ>0\kappa > 0κ>0)

Thermoelectric figure of merit zT=PF⋅Tκl+κezT = \frac{PF \cdot T}{\kappa_l + \kappa_e}zT=κl​+κe​PF⋅T​

Standard Formula for the Thermoelectric Figure of Merit zT=σS2Tκl+κezT = \frac{\sigma S^2 T}{\kappa_l + \kappa_e}zT=κl​+κe​σS2T​

zT>0zT > 0zT>0 for S≠0S \neq 0S=0 and T>0T > 0T>0

Monotonicity of the figure of merit zTzTzT with respect to lattice thermal conductivity κl\kappa_lκl​

Thermoelectric power factor $PF = \sigma S^2$

The power factor $PF$ is positive if $S \neq 0$

Total thermal conductivity $\kappa_l + \kappa_e$

Total thermal conductivity is positive ( $\kappa > 0$ )

Thermoelectric figure of merit $zT = \frac{PF \cdot T}{\kappa_l + \kappa_e}$

Standard Formula for the Thermoelectric Figure of Merit $zT = \frac{\sigma S^2 T}{\kappa_l + \kappa_e}$

$zT > 0$ for $S \neq 0$ and $T > 0$

Monotonicity of the figure of merit $zT$ with respect to lattice thermal conductivity $\kappa_l$