 # Thermoelectric Properties of Materials

### Thermoelectric Figure of Merit

The thermoelectric performance (for either power generation or cooling) depends on the efficiency of the thermoelectric material for transforming heat into electricity. The efficiency of a thermoelectric material depends primarily on the thermoelectric materials figure-of-merit, known as $$zT$$ .

$$zT= \frac{S^2 \sigma}{ \kappa}T\hspace{10 mm} \text{or} \hspace{10 mm} zT=\frac{\alpha ^2}{\rho \kappa}T$$

The voltage is produced by the Seebeck coefficient ($$S$$ or $$\alpha$$). In addition a low electrical resistivity $$\rho$$ (or high electrical conductivity $$\sigma$$) and low thermal conductivity $$\kappa$$ is required for high efficiency.

The $$zT$$ values of some common materials used commercially, on NASA missions or confirmed in our laboratories is shown below and provide a good baseline for improved materials . Numerous materials with higher $$zT$$ have been reported and the field is rapidly advancing, nevertheless one should be aware of stability and measurement issues .  zT for common thermoelectric materials

To achieve sufficient power a TE generator must be used efficiently across a large temperature difference $$\Delta T=T_{\text{h}}-T_{\text{c}}$$ and so the material $$zT$$ must be high across this temperatre range. The Device $$ZT$$ is a weighted average of the TE material $$zT$$ that gives the maximum efficiency $$\eta$$ across this finite $$\Delta T$$ and easily calculated from the material properties $$S$$, $$\rho$$ and $$\kappa$$.

$$\eta=\frac{\Delta T}{T_{\text{h}}}\frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+T_{\text{c}}/T_{\text{h}}}$$

### Thermoelectric Quality Factor

Nearly all good thermoelectric materials are heavily doped semiconductors: semiconductors which have so many free electrons that they have many properties similar to metals. The charge carrier concentration depends on intrinsic defects (such as atom vacancies) as well as extrinsic dopants (impurities). Because all the properties in $$zT$$, Seebeck coefficient, electrical resistivity and thermal conductivity depend on charge carrier concentration in a conflicting manner (see figure) achieving high $$zT$$ in a material typically requires optimizing the charge carrier concentration. Thus the search for (or comparison of) good thermoelectric materials is really a search for a material with the highest potential for high $$zT$$ presuming it can be optimally doped. This potential high $$zT$$ is determined by the thermoelectric quality factor $$B$$. The quality factor is determined by the weighted mobility of the electronic charge carriers $$\mu_\text{w}$$ and the phonon (or lattice) contribution to the thermal conductivity $$\kappa_\text{L}$$.[1, 7]

$$B= \frac{{8\pi k_\mathrm{B}{{\left( {2{m_e}} \right)}^{3/2}{\left(k_\mathrm{B} T\right)}^{5/2}}}}{{3e{h^3}}} \cdot \frac{\mu_\text{w}}{\kappa_\text{L}}$$

Thus the quality of a thermoelectric material can be divided into the quality of its electronic properties - the weighted mobility $$\mu_\text{w}$$ - and its thermal properties - the lattice thermal conductivity $$\kappa_\text{L}$$. The weighted mobility and lattice thermal conductivity can be easily evaluated (or at least estimated) from measurements of Seebeck coefficient, electrical conductivity and thermal conductivity, as described below, and therefore the potential of a new material can be estimated from a single sample.

# Electronic Properties of Thermoelectric Materials

### Electrical Conductivity and Resistivity

The thermoelectric effects are determined by the same transport function $$\sigma_\text{E}$$ that determines electrical resistivity ($$\rho$$) (or electrical conductivity ($$\sigma = 1/\rho$$) of a metal or semiconductor among other transport . The electrical conductivity is determined by density of charge carriers $$n$$, their drift mobility $$\mu$$ and the electronic charge $$e$$.

$$1/\rho = \sigma = ne\mu$$

#### Resistivity of a Metal or Heavily Doped Semiconductor

Resistivities from different electron scattering mechanisms typically add according to Mathiessen's Rule and therefore plotting experimental values as resistivities rather than conductivities is usually more helpful. For example in metals or heavily doped semiconductors such as thermoelectric materials the temperature dependent resistivity follows the approximate form

$$\rho=\rho_0 + \rho_1 T + \rho_\text{gb} \exp{\frac{-\Delta E}{k_\text{B} T}}$$

where $$\rho_0$$ is the resistivity from neutral defects (neutral impurities, dislocations and neutral grain boundaries), $$\rho_1 T$$ is the resistivity from the atom vibrations (phonon scattering), and the last term is a thermally activated resistivity often observed from charged grain boundaries .

### Hall Mobility and Carrier Concentration

The mobility and charge carrier concentration are often estimated using Hall Effect measurements. The Hall carrier concentration ($$n_\text{H}$$) and Hall mobility ($$\mu_\text{H}$$) are derived from the measured Hall Resistance ($$R_\text{H}$$). They are typically within 20% of the true $$n$$ and $$\mu$$ for semiconductors with electronic transport properties dominated by a single type (electron or hole) charge carrier.

$$n_\text{H} = \frac{1}{eR_\text{H}}$$ $$\mu_\text{H} = \frac{R_\text{H}}{\rho}$$

Mobility typically decreases with effective mass $$m^*$$ because heavier objects are harder to move with a force.

$$\mu=\frac{e\tau}{m^*_\text{c}}$$

Here $$\tau$$ is the scattering time and $$m^*_\text{c}$$ is the inertial or conductivity effective mass that in complex materials is different from but related to the density-of-states mass or Seebeck mass $$m^*_\text{S}$$ used below .

#### Mobility of a Metal or Heavily Doped Semiconductor

Electron scattering mechanisms are typically identified by their effect on electron mobility. At high temperatures (above 100K) phonon scattering often dominates, and is the only common mechanism that causes a strong decrease in mobility as a function of temperature, $$T^{-1}$$ ($$\rho_1$$ term above) or faster decrease. For non-degenerate (lightly doped semiconductors) the analytic form typically contains $$T^{-3/2}$$ that gradually changes to $$T^{-1}$$ for degenerate (heavily doped) semiconductors or metals. For example, in deformation potential scattering (deformation of the crystal changes the energy of electrons which scatters transporting electrons) . Although this scattering is often called Acoustic Phonon Scattering, in complex materials phonon scattering could be dominated by optical phonons.

### The Seebeck Coefficient

In a thermoelectric material there are free electrons or holes which carry both charge and heat. To a first approximation, the electrons and holes in a thermoelectric semiconductor behave like a gas of charged particles. If a normal (uncharged) gas is placed in a box within a temperature gradient, where one side is cold and the other is hot, the gas molecules at the hot end will move faster than those at the cold end (in addition to this thermodynamic difference between hot and cold, there can also be a scattering rate difference). The faster hot molecules will diffuse further than the cold molecules and so there will be a net build up of molecules (higher density) at the cold end. The density gradient will drive the molecules to diffuse back to the hot end. In the steady state, the effect of the density gradient will exactly counteract the effect of the temperature gradient so there is no net flow of molecules. If the molecules are charged, the buildup of charge at the cold end will also produce a repulsive electrostatic force (and therefore electric potential) to push the charges back to the hot end.

The electromotive force, measured as an electric potential (and electric field $$\vec{E}$$ ) produced by a temperature difference is known as the Seebeck effect and the proportionality constant is called the Seebeck coefficient ($$S$$ or $$\alpha$$). If the free charges are positive (the material is p-type), positive charge will build up on the cold which will have a positive potential. Similarly, negative free charges (n-type material) will produce a negative potential at the cold end. Here Thermopower refers to the magnitude of the Seebeck coefficient ($$|\alpha |$$).

$$V = S\Delta T \hspace{10 mm} \text{or} \hspace{10 mm} V = \alpha \Delta T$$ $$\vec{E} = S\nabla T \hspace{10 mm} \text{or} \hspace{10 mm} \vec{E} = \alpha \nabla T$$  Most of the Seebeck coefficient in electronic systems is related to the equilibrium thermodynamics can be described as the entropy transported per charge transported. Thus $$S$$ is often described as related to entropy and carrier concentration $$n$$ relative to the density-of-states $$g$$ which is characterized by the density of states effective mass $$g\propto (m^*)^{3/2}$$. Here we use the Seebeck effective mass $$m^*_\text{S}$$ measured by the Seebeck coefficient. There are additional, usually smaller, contributions to the Seebeck coefficient (such as electron scattering) that make this experimental effective $$m^*_\text{S}$$ different from other definitions of effective mass. At low temperature (typically less than 100K) and in materials with large phonon thermal conductivity, phonon drag thermopower can be large.

### Seebeck Effective Mass

The Seebeck coefficient and charge carrier concentration $$n$$ (or Hall $$n_\text{H}$$) can be used to calculate a density-of-state effective mass we call the Seebeck Effective Mass $$m^*_\text{S}$$ [11, 14]. Like the Hall mobility and weighted mobility the Seebeck coefficient must be dominated by a single charge carrier type for $$m^*_S$$ for easily interpreted. Nevertheless, $$m^*_\text{S}$$ can be simply calculated from experimental values of $$S$$ and $$n$$ (or $$n_\text{H}$$) .

When $$|S|> 75\mu V/K$$

$$m^*_\mathrm{S} = \frac{h^2}{2k_\mathrm{B}T} \left\{\frac{3n_\mathrm{H}}{16\sqrt{\pi}} \left(\exp\left[\frac{|S|}{(k_\mathrm{B}/e)}-2 \right] -0.17\right ) \right\}^{2/3}$$

When $$|S|< 75\mu V/K$$

$$m^*_\mathrm{S} = \frac{3h^2}{8\pi^2k_\mathrm{B}T} \frac{|S|}{(k_\mathrm{B}/e)} \left(\frac{3n_\mathrm{H}}{\pi}\right)^{2/3}$$

### Common Forms of the Seebeck Coefficient

The Seebeck coefficient has a number of simple, approximate forms in certain limiting cases - the intermediate cases are described in  and . The simple forms can all be related to special cases of the location of the Fermi Level and the transport function $$\sigma_\text{E}$$ that also determines electrical resistivity and other electronic transport properties . In systems with a transport edge (band edge or mobility edge) this makes $$\alpha$$ a measure of the Fermi Level ($$E_\text{F}$$) relative to the band edge ($$E_\text{B}$$) which is commonly set to zero (replace $$E_\text{F}$$ with $$E_\text{F}-E_\text{B}$$ in the equations below when $$E_\text{B}≠0$$). The convention when describing only valence bands is to consider the energy of the holes which results in the same equations as electrons for the thermopower ($$|S|$$)

#### Seebeck coefficient of a Metal

In metals or degenerate semiconductors (typically when $$S < 100 \mu V/K$$) where the Fermi level in inside the band ($$E_\text{F}>0$$) the thermopower increases with decreasing charge carrier concentration $$n$$ and also increases linearly with temperature $$T$$ with a slope proportional to the density of states effective mass $$m^*_S$$

$$|S |= \frac{\pi^2}{3} \frac{k_ \text{B}^2 T}{e} \frac{8m^*_S}{h^2}\left(\frac{\pi}{3n}\right) ^{\frac{2}{3}} \hspace{10 mm} |S |= \frac{\pi^2}{3} \frac{k_ \text{B}^2 T}{e} \frac{1}{E_ \text{F}}$$

where $$h$$ is Planck's constant.

#### Seebeck coefficient of an Insulator

The thermopower of an insulator often decreases with temperature. In this case the Fermi Level is pinned inside the band gap ($$E_\text{F}< 0$$ is fixed). Here the thermopower of a non-degenerate semiconductor ($$S > 200 \mu V/K$$) is used where the thermopower again depends on the Fermi Level.

$$|S |= \frac{k_ \text{B}}{e} \left( \frac{-E_ \text{F}}{k_ \text{B}T}+A \right)$$

where $$A\approx 2$$ depends on the scattering.

#### Heikes Formula

The Seebeck coefficient of a narrow band system, often used for insulating transition metal and rare earth oxides is given by the Heikes formula.

$$S = \frac{k_ \text{B}}{e} ln\left( \frac{c}{1-c} \right)$$

where $$c$$ is the fraction of states in the narrow band that are occupied by electrons. Here the Seebeck coefficient is entirely given by the entropy change due to adding an electron ($$S< 0$$ for small $$c$$) that becomes zero for a half filled band (($$c=\frac{1}{2}$$) and changes sign when $$c>\frac{1}{2}$$ when the system is better described by holes rather than electrons.

### Weighted Mobility and Power factor

The weighted mobility $${\mu_\text{w}}$$ is the best descriptor of the electronic part of the quality factor that makes a good thermoelectric material $$B \sim \frac{\mu_\text{w}}{\kappa_\text{L}}$$. Thus any strategy to improve a material for thermoelectric use by reducing $$\kappa_\text{L}$$ needs also to consider the effect on $$\mu_\text{w}$$.

The weighted mobility can be defined as a simple function of two measured properties, the Seebeck coefficient and the electrical resistivity.

When $$|S|> 100\mu V/K$$

$$\mu_\text{w}=331 \frac{\text{cm}^2}{\text{Vs}}\left(\frac{m\Omega\text{cm}}{\rho}\right)\left(\frac{T}{300\text{K}}\right)^{-3/2} \exp\left[\frac{|S|}{k_\mathrm{B} / e}-2\right]$$

When $$|S|< 100\mu V/K$$

$$\mu_\text{w}=331 \frac{\text{cm}^2}{\text{Vs}}\left(\frac{m\Omega\text{cm}}{\rho}\right)\left(\frac{T}{300\text{K}}\right)^{-3/2} \frac{3}{\pi^2}\frac{|S|}{k_\mathrm{B} / e}$$

In the simple free electron model, the weighted mobility $$\mu_\text{w}$$ is the drift mobility $$\mu$$ weighted by the density-of-states $$g\propto \left(m^*\right)^{3/2}$$.

$$\mu_\text{w} = \mu\left(\frac{m^*}{m_e}\right)^{3/2}$$

#### Power Factor

Historically, $$S^2\sigma$$, referred to as the thermoelectric "power factor" has been discussed to evaluate improvements in the electronic properties of thermoelectric materials or when thermal conductivity measurements are unavailable.

Although $$S^2\sigma$$, like $$zT$$ peaks at a charge carrier concentration between a metal and typical semiconductor, $$S^2\sigma$$ does not optimize where $$zT$$ does (figure above), overemphasizing more metallic doping concentrations because it ignores the impact of the electronic thermal conductivity $$\kappa_{\text{e}}$$ to $$zT$$. $$S^2\sigma$$ is frequently over analyzed to the point where it is incorrectly concluded that a high $$S^2\sigma$$ is preferable even at the expense of lower $$zT$$. An optimized thermoelectric design will produce more power when the material is optimized for $$zT$$ because it converts the heat flowing through it to electricity most efficiently . The weighted mobility, $${\mu_\text{w}}$$ is a better descriptor of the electronic properties for thermoelectric use than $$S^2\sigma$$, and with the use of the equation above is easy to calculate from the experimental $$S$$ and $$\rho$$ (or $$\sigma$$). For example, rather than report peak power factors, a more useful comparison would be to compare $${\mu_\text{w}}$$ at a given temperature.

### Electronic Contribution to the Thermal Conductivity

Electrons transport heat as well as charge and therefore contribute to the thermal conductivity that reduces $$zT$$. To a good approximation, the thermal conductivity of a semiconductor is the sum of the thermal conductivity from the free electrons (or holes) $$\kappa_{\text{e}}$$ and the thermal conductivity of the phonons (atomic vibrations) often called the lattice thermal conductivity $$\kappa_{\text{L}}$$.

$$\kappa = \kappa_{\text{e}} + \kappa_{\text{L}}$$

The electronic thermal conductivity derives from the same the transport function $$\sigma_\text{E}$$ that also determines the Seebeck and electrical resistivity. For a system with only one type of electron or hole charge carriers (otherwise see bipolar thermal conductivity) $$\kappa_{\text{e}}$$ is well approximated with the Wiedemann-Franz law $$\kappa_{\text{e}}=L\sigma T$$ where $$L$$ is the Lorenz factor. [Note that some older textbooks use the law differently as if ignoring $$\kappa_{\text{L}}$$]

Like the Seebeck coefficient $$S$$, the Lorenz factor $$L$$ depends primarily on the location of the Fermi Level (in a single band system) and so a good estimate for $$L$$ (at any temperature) is based only on the measured thermopower ($$|S|$$).

$$\frac{L}{10^{-8}\text{W}\Omega \text{K}^{-2}}=1.5+\exp\left(\frac{-|S|}{116\mu \text{V/K}}\right)$$

where $$L$$ is measured in $$10^{-8}\text{W}\Omega \text{K}^{-2}$$ and $$S$$ in $$\mu\text{V}/\text{K}$$.

Thus the most common way of reporting phonon thermal conductivity $$\kappa_{\text{L}}$$ is to calculate $$\kappa_{\text{L}}=\kappa-L\sigma T$$ where $$\kappa$$ is the measured total thermal conductivity and $$L$$ is estimated from the measured value of $$|S|$$. When $$\kappa_{\text{L}}$$ appears to rise at high temperatures due to the bipolar effect $$\kappa-L\sigma T$$ is perhaps best refereed to as $$\kappa_{\text{L}}+ \kappa_{\text{b}}$$.

### Bipolar Effects

When there are significant number of both electrons and holes contributing to charge transport (bipolar charge transport) the thermoelectric properties are greatly affected. This occurs when electrons are excited across the band gap producing minority charge carriers (e.g. holes in an n-type material) in addition to majority charge carriers (e.g. the electrons in an n-type material). Bipolar effects are observed in small band gap materials at high temperature ($$k_\text{B} T \sim E_\text{g}$$) and intrinsic (undoped) semiconductors where ($$n_\text{n} \sim n_\text{p}$$). The equations for two-band (n-type conduction band and p-type valence band) transport given below are special forms of the generalized multi-band result given in .

#### Bipolar Conductivity

The electrical resistivity is least affected because both electrons can holes contribute to the electrical conductivity with the same sign.

$$\sigma = n_\text{p} e\mu _\text{p} + n_\text{n} e\mu _\text{n}$$

#### Bipolar Seebeck

The Seebeck coefficient is dramatically affected because the minority charge carriers add a Seebeck voltage of opposite sign as the majority carriers greatly reducing the thermopower $$|S|$$. The contribution of each charge carrier to the total Seebeck coefficient is weighted by the electrical conductivity .

$$S = \frac{S_\text{p} \sigma _\text{p} + S_\text{n} \sigma _\text{n}}{ \sigma _\text{p} + \sigma _\text{n}}$$

Because $$S_p>0$$ but $$S_n<0$$ the resultant thermopower will be less when minority charge carriers have significant conductivity.

#### Bipolar Hall Effect

The Hall voltage, which is also opposite for electrons as holes, is also compensated when both electrons and holes are present making the apparent Hall carrier concentration $$n_{\text{H}} = \frac{1}{eR_{\text{H}}}$$ increase much faster than the real majority carrier concentration. The individual contributions are even more strongly weighted by the electrical conductivity (mobility) than is the Seebeck coefficient [1, 9] (note that in this notation $$\mu _\text{p} > 0$$ and $$\mu _\text{n} < 0$$).

$$R_{\text{H}}= \frac{1}{en_{\text{H}}} = \frac{\sigma_\text{p} \mu _\text{p} + \sigma_\text{n} \mu _\text{n}}{ \left(\sigma _\text{p} + \sigma _\text{n}\right)^2}$$

#### Bipolar Electronic Thermal Conductivity

The thermal conductivity is also affected by the bipolar effect but it is often not noticed because of the lattice contribution. Because there are more electron-hole pairs at high temperature then low temperature, in a temperature gradient there will be an effect of absorbing heat at the hot end by creating electron-hole pairs and releasing heat at the cold end when they recombine.

$$\kappa_{\text{e}}= L _\text{p}\sigma_{\text{p}}T + L _\text{n}\sigma_{\text{n}}T + \left(S_{\text{p}}- S_{\text{n}} \right)^2 \frac{\sigma_{\text{p}} \sigma_{\text{n}}T}{ \left(\sigma _\text{p} + \sigma _\text{n}\right)}$$

where the last term can be considered the bipolar thermal conductivity that increases exponentially with temperature as $$\exp{\frac{E_\text{g}}{k_\text{B} T}}$$

### Thermopower Peak and Band Gap

By having a band gap large enough, n-type and p-type carriers can be separated, and doping will produce only a single carrier type. Thus good thermoelectric materials have band gaps large enough to have only a single carrier type but small enough to sufficiently high doping and high mobility (which leads to high electrical conductivity).

In a semiconductor at high enough temperature electrons will have enough energy to excite across the band gap. When that happens there will be both n-type carriers in the conduction band and p-type carriers in the valence band such that the resultant thermopower will be compensated (reduced) because the two contributions subtract. In a heavily doped semiconductor, where the dopants produce many majority carriers (could be either n-type or p-type) the thermopower will be reduced at high temperature due to the excitation of minority carriers of opposite sign. Although there are fewer minority carriers than majority carriers, they have a larger thermopower. This leads to a peak in the thermopower as a function of temperature. The Figure above shows how the Seebeck coefficient changes when the doping changes from lightly doped (blue) to heavily doped (red).

The Temperature at which the thermopower peaks, $$T_{\text{max}}$$, and the thermopower of the peak, $$|S_{\text{max}}|$$, can be used to estimate the semiconducting band gap, $$E_{\text{g}}$$ .

$$E_{\text{g}}=2e |S_{\text{max}}|T_{\text{max}}$$

This equation can be reasonably accurate when both electrons and holes have similar weighted mobility, $$A=\frac{\mu _{\text{w,maj}}}{\mu _{\text{w,min}}}=\frac{\mu _{\text{maj}}}{\mu _{\text{min}}}\left(\frac{m^* _{\text{maj}}}{m^*_{\text{min}}} \right)^{3/2}$$ (here $$m^*$$ is the DOS mass including valley degeneracy), otherwise the carrier with higher weighted mobility will have a stronger influence and affect the estimation of band gap . The following figure can be used to estimate the error from the Band Gap equation. When the Seebeck peak is greater than 150µV/K this method for measuring Band Gap at high temperature is likely to be accurate.

# Thermal Properties of Thermoelectric Materials

Reducing the phonon or lattice thermal conductivity provides a good opportunity to enhance $$zT$$ because it directly appears in the thermoelectric quality factor $$B \sim \frac{\mu_\text{w}}{\kappa_\text{L}}$$. This can be done by increasing the phonon scattering by introducing heavy atoms, disorder, large unit cells, and grain boundaries or slowing the phonons so they transport less heat.

#### Callaway Model

Thinking of phonons as classical particles that can transport heat, the thermal conductivity can be considered as comprising the heat capacity, velocity and scattering time for each phonon. Considering the wide spectrum of heat carrying phonons, the Callaway model considers the contribution individually for each frequency $$\omega$$.

$$\kappa = \int c(\omega) v^2_\text{g}(\omega) \tau (\omega) d\omega$$

This is a good way to distinguish the importance of different phonon scattering mechanisms but requires numerical integration. Some simplified analytical forms for $$\kappa$$ are discussed below.

### Thermal Conductivity from Thermal Diffusivity Measurements

Measurements of thermal diffusivity is now the most common method to experimentally determine thermal conductivity in materials above 200K . The thermal conductivity $$\kappa$$ is determined from the measured thermal diffusivity $$D_\text{T}$$ and the heat capacity per volume which is the specific heat $$c_\text{P}$$ (heat capacity per mass) times density $$\unicode{x00FE}$$.

$$\kappa =D_\text{T} c_\text{P} \unicode{x00FE}$$

### Heat Capacity

Estimation of the heat capacity of a solid above 200K based on simple physical principles is often more accurate than measurements . Unless there are phase changes in the material, the Dulong-Petit heat capacity of $$3 k_\text{B}$$ per atom is usually accurate at high temperatures for the heat capacity at constant volume and to within 10% for the heat capacity at constant pressure for up to a few times the Debye temperature.

$$c_\text{DP}$$ is often the best estimate for $$c_\text{P}$$ of a new material if the Debye temperature (from elastic properties or speed-of-sound measurements) is not known. Minor compositional changes are easily accounted for by assuming the heat capacity per atom does not change. For example the specific heat capacity per mass can be calculated from the heat capacity per atom equations given here divided by the average atomic mass in the solid $$\overline{M}$$ (see Neumann-Kopp law).

$$c_\text{DP} = 3 k_\text{B}/\overline{M}$$

The heat capacity at constant pressure usually increases linearly with temperature above the Debye temperature due to the enthalpy to thermally expand the solid $$9\alpha^2_\text{L}BV_\text{a}T$$ where $$V_\text{a}$$ is the volume per atom, $$B$$ is the Bulk modulus, $$\alpha_\text{L}$$ is the coefficient of linear thermal expansion. If $$\alpha_\text{L}$$ and $$B$$ are known, for example from thermal expansion speed-of-sound measurements, then a good estimate for the heat capacity is 

$$c_\text{P} =\frac{3k_\text{B}}{\overline{M}}\left[1+\frac{1}{10^4}\frac{T}{\Theta_\text{D}}-\frac{1}{20}\left(\frac{T}{\Theta_\text{D}}\right)^{-2}\right] + \frac{9\alpha^2_\text{L}BT}{\unicode{x00FE}}$$

where $$\unicode{x00FE}=\overline{M}/V_\text{a}$$ is the density.

#### Phase Transformation Heat Capacity

When phase transformations occur on the timescale of measurements the enthalpy of the transformation is included in the heat capacity. For thermal conductivity this timescale is at least the timescale of atomic vibrations and thus significant phase transformation heat capacity has been found in the thermal diffusivity of many materials that are multi-phase or near a phase transformation . This additional heat capacity $$c_\text{p,PT}$$ can be related to the enthalpy of transformation $$\Delta H$$ and the rate that the phase fraction (order parameter $$\phi$$) changes with temperature which can be estimated with the equilibrium phase diagram .

$$c_\text{p,PT}=\frac{\Delta H}{\unicode{x00FE}} \left(\frac{\partial \phi}{\partial T}\right)_\text{p}$$

### Debye Model of Phonons

Atomic vibrations in a solid, called phonons, form collective (wave-like), low-energy acoustic modes. Very low energy acoustic modes make acoustic (sound) waves that can be measured as the speed-of-sound $$v_\text{s}$$ in a solid. The Debye model approximates all phonon modes to have the same velocity in the dispersion relation of frequency vs. wavevector $$\omega = vk$$. The Debye model works surprisingly well considering real solids have transverse and longitudinal modes with different velocities ($$v_\text{t}$$ and $$v_\text{l}$$) and have optical as well as acoustic modes that travel with group velocities $$v_\text{g}$$ slower than the speed of sound $$v_\text{s}$$. Nevertheless, the Debye model sufficiently describes the energy and number of phonon states to provide a good first-order description of atom vibrations in a solid that can be measured with acoustic properties such as the speed of sound.

The number of phonon modes is limited by the number of atoms and this leads to the maximum frequency in the Debye model $$\omega_\text{D}$$ that can be described as the Debye Temperature $$\Theta_\text{D}$$ and related to the average velocity (here the average speed of sound $$v_\text{s}$$) where $$V_\text{a}$$ is the volume per atom.

$$\hbar \omega_\text{D} = k_\text{B}\Theta_\text{D} = \hbar v_\text{s}\left( \frac{6\pi^2}{V_\text{a}}\right)^{1/3}$$

#### Speed of Sound

The speed of sound from pulse-echo experiments is an easy, accurate method to measure average phonon properties of an isotropic material at room temperature . The average speed of sound $$v_\text{s}$$ is an appropriate average of the longitudinal speed of sound $$v_\text{l}$$ and transverse speed of sound $$v_\text{t}$$.

Often the simple arithmatic mean is used

$$v_\text{s}=\frac{1}{3}\left(v_\text{l}+2 v_\text{t} \right)$$

although the average that preserves the phonon density-of-states (probably best for Debye Temperature) is

$$v_s=\left(\frac{1}{3}\left[\frac{1}{v_\text{l}^{3}}+\frac{2}{v_\text{t}^{3}}\right]\right)^{-1/3}$$

The speed of sound is directly related to the elastic properties of an isotropic material and so can be estimated from measurements or calculations of the bulk modulus $$B$$ and shear modulus $$G$$ and density $$\unicode{x00FE}$$.

$$v_\text{l}=\sqrt{\frac{B+\frac{4}{3}G}{\unicode{x00FE}}} \hspace{20 mm} v_\text{s}=\sqrt{\frac{G}{\unicode{x00FE}}}$$

Note that the speed of sound, phonon velocities, and elastic properties are not isotropic in crystals, even cubic crystals, which makes isotropic polycrystalline materials somewhat easier to work with. $$S$$, $$\rho$$, and $$\kappa$$ are second rank, symmetric tensors which makes them isotropic for cubic crystals.

#### Grüneisen Parameter

The thermodynamic Grüneisen parameter $$\gamma$$ is related to the coefficient of linear thermal expansion $$\alpha_\text{L}$$ in an isotropic material.

$$\gamma=\frac{3\alpha_\text{L}B}{c_\text{p}\unicode{x00FE}}$$

### Minimum Thermal Conductivity

In the extreme scattering limit where the vibrational modes no longer transport heat like propagating waves (propagons or classical phonons) but diffusively and are called diffusons. The minimum thermal conductivity from such heat diffusion can be estimated to be :

$$\kappa_\text{min} = 0.76 k_\text{B} \frac{ v_\text{s}}{V_\text{a}^{2/3}}$$

The Cahill minimum thermal conductivity (high temperature limit) is slightly higher $$1.21 k_\text{B} \frac{ v_\text{s}}{V_\text{a}^{2/3}}$$ is a good estimate for glasses.

### Umklapp Scattering Thermal Conductivity

In the least scattering limit, phonons are only scattered by other phonons, known as Umklapp scattering. Phonons interact with each other through the anharmonicity where the Grüneisen parameter $$\gamma$$ is a good measure. A reasonable estimate for the thermal conductivity of a crystal with no other scattering is 

$$\kappa_\text{u} = C_\text{u} \frac{\overline{M} v^3_\text{s}}{TV_\text{a}^{2/3}\gamma^2}$$

where the constant $$C_\text{u}\sim 0.4/N_\text{c}^{1/3}$$ is expected to scale with the number of atoms in the unit cell $$N_\text{c}$$ in this way to account only for the acoustic phonon branches .

The thermal conductivity of complex materials above the Debye temperature is often of the form $$\kappa=A/T + B$$ where the $$A$$ is from this umklapp thermal conductivity due to mostly the acoustic branches and $$B$$ is the $$\kappa_\text{min}$$ minimum or diffuson thermal conductivity from mostly the optical branches close to the value given above. Often $$C_\text{u}$$ is better fit to experimental $$A$$ value and used as a constant for similar materials.

### Point Defect Scattering of Phonons

The frequency dependence of point defect scattering and umklapp scattering combine fortuitously to give a relatively simple analytical form for the reduction of thermal conductivity due to point defects

$$\frac{\kappa_\text{L}}{\kappa_0} = \frac{\mathrm{tan}^{-1}u}{u} \hspace{20 mm} u^2 = \frac{(6\pi^5V^2_\text{a})^{1/3}}{2 k_\text{B} v_\text{s}}\kappa_0 \Gamma$$

Here $$\kappa_0$$ is the $$\kappa_\text{L}$$ without defects (umklapp $$\kappa_\text{u}$$) and $$\Gamma$$ is the average variance from the mass $$\Delta M$$ and strain $$\Delta R$$ disorder where $$\epsilon$$ is a fitting parameter .

$$\Gamma = \frac{\langle \overline{\Delta M^2} \rangle}{\langle \overline{M} \rangle^2} + \epsilon \frac{\langle \overline{\Delta R^2} \rangle}{\langle \overline{R}\rangle^2}$$

Examples of structures with complex anions are shown below in the Skutterudite and Yb14MnSb11 Phases that can be understood with Zintl electron counting rules. Crystal Structure of Skutterudite Crystal Structure of Yb14MnSb11 Thermal conductivity decrease in Skutterudites by the
introduction of various scattering mechanisms

Using these principles, a variety of high zT materials have been developed. Many materials have an upper temperature limit of operation, above which the material is unstable. Thus no single material is best for all temperature ranges, so different materials should be selected for different applications based on the temperature of operation. This leads to the use of a segmented thermoelectric generator.

The ideal thermoelectric material is then one which is a "Electron Crystal - Phonon Glass"  where high mobility electrons are free to transport charge and heat but the phonons are disrupted at the atomic scale from transporting heat.

 G. Jeffrey Snyder and Eric S. Toberer "Complex Thermoelectric Materials" Nature Materials 7, 105-114 (2008).

 Andrew F. May, G. Jeffery Snyder "Introduction to Modeling Thermoelectric Transport at High Temperatures" Chapter 11 in Thermoelectrics and its Energy Harvesting Vol 1, edited by D. M. Rowe. CRC Press (2012).

 G. Jeffrey Snyder, Tristan Ursell. "Thermoelectric efficiency and compatibility" Physical Review Letters, Vol 91 p. 148301 (2003)

 Zachary M. Gibbs , Hyun-Sik Kim , Heng Wang , and G. Jeffrey Snyder “Band gap estimation from temperature dependent Seebeck measurement—Deviations from the 2e|S|maxTmax relation” Applied Physics Letters 106, 022112 (2015)

 Hyun Sik Kim, Zachary Gibbs, Yinglu Tang, Heng Wang, and G. J. Snyder “Characterization of Lorenz number with Seebeck coefficient measurement” APL Materials 3 041506 (2015)

 Stephen D. Kang and G. J. Snyder "Charge-Transport Model for Conducting Polymers" Nature Materials 16, 252 (2017)

 G. D. Mahan "Good Thermoelectrics" in Solid State Physics Vol. 51 (Eds: H. Ehrenreich ,F. Spaepen ), Academic Press Inc , San Diego 1998 , p. 81.

 Yanzhong Pei, Heng Wang and G. Jeffrey Snyder “Band Engineering of Thermoelectric Materials” Advanced Materials 24, 6125 (2012)

 L. L. Baranowski, G. J. Snyder , Eric S. Toberer  “The Misconception of Maximum Power and Power Factor in Thermoelectrics” J. Applied Physics 115, 126102 (2013)

 Tristan Day, Wolfgang Zeier, David Brown, Brent Melot, G. J. Snyder “Determining Conductivity and Mobility Values of Individual Components in Multiphase Composite Cu1.97Ag0.03Se” Applied Physics Letters 105, 172103 (2014)

 Jimmy J. Kuo et al., “Grain boundary dominated charge transport in Mg3Sb2-based compounds” Energy & Environmental Science 11, 429 (2018)

 Z.M. Gibbs, Francesco Ricci, Guodong Li, Hong Zhu, K. Persson, G. Ceder, Geoffroy Hautier, Anubhav Jain, G.J. Snyder "Effective mass and Fermi surface complexity factor from ab initio band structure calculations" NPJ Computational Materials 3, 8 (2017)

 Heng Wang, Yanzhong Pei, Aaron LaLonde and G. Jeffrey Snyder “Weak Electron-Phonon Coupling Contributing to High Thermoelectric Performance in n-Type PbSe” PNAS 109, 9705 (2012)

 Kasper A. Borup, Johannes de Boor, Heng Wang, Fivos Drymiotis, Franck Gascoin, Xun Shi, Lidong Chen, Mikhail I. Fedorov, Eckhard Müller, Bo B. Iversen, G. Jeffrey Snyder “Measuring Thermoelectric Transport Properties of Materials” Energy and Environmental Science 8, 423 (2015)

 S. Kang and G.J. Snyder "Transport property analysis method for thermoelectric materials: material quality factor and the effective mass model" Chapter VI in

A. Zevalkink, S. D. Kang, G. J. Snyder, E. S. Toberer, et al. "A practical field guide to thermoelectrics: Fundamentals, synthesis, and characterization" Applied Physics Reviews 5, 021303 (2018)

 M. T. Agne, G.J. Snyder, et al “Heat Capacity of Mg3Sb2, Mg3Bi2, and their alloys at high temperature” Materials Today Physics 6, 83 (2018)

 Riley Hanus, G. J. Snyder et al “Lattice Softening Significantly Reduces Thermal Conductivity and Leads to High Thermoelectric Efficiency” Advanced Materials (2019)

 M. T. Agne, P.W. Voorhees, G.J. Snyder “Phase Transformation Contributions to Heat Capacity and Impact on Thermal Diffusivity, Thermal Conductivity, and Thermoelectric Performance” Advanced Materials (2019)

 M. T. Agne, R. Hanus, G. J. Snyder, “Minimum thermal conductivity in the context of diffuson-mediated thermal transport” Energy & Environmental Science 11, 609 (2018)

 Eric S. Toberer, Alexandra Zevalkink, and G. Jeffrey Snyder, “Phonon engineering through crystal chemistry” Journal of Materials Chemistry  21, 15843 (2011)(corrected version .pdf)

 Heng Wang, Yanzhong Pei, Aaron D. LaLonde and G. Jeffrey Snyder “The Criteria for Beneficial Disorder in Thermoelectric Solid Solutions” Advanced Functional Materials23, 1586 (2013)

[] R. Hanus, A. Garg, G. J. Snyder, “Phonon diffraction and dimensionality crossover in phonon-interface scattering” Communications Physics, 1, 78 (2018) 