# Boltzmann Transport Theory and Thermoelectric Performance

<img src="/files/CQ08k1kfL6oCQbcVKxj9" alt="" data-size="line"> *Written by Michael Y. Toriyama (MichaelToriyama2024 \[at] u \[dot] northwestern \[dot] edu)*

## Introduction

Thermoelectric properties in a material can often be described using Boltzmann transport theory under low applied field. The electrical and thermal current densities can be described in terms of the applied electric field and a thermal gradient. The relevant charge transport properties, namely the electrical conductivity, Seebeck coefficient, and the electronic contribution to the thermal conductivity, can be derived from electrical and thermal current densities.

## Basic Theory

### Thermoelectric Performance

The efficiency of a thermoelectric device, from a material engineering perspective, is parametrized by the figure-of-merit $$zT$$, which can be expressed as

$$
zT = \frac{S^2 \sigma}{\kappa\_e + \kappa\_L} T
$$

where $$\sigma$$ is the electrical conductivity, $$S$$ is the Seebeck coefficient, $$\kappa\_e$$ and $$\kappa\_L$$ are the electronic and lattice contributions to the thermal conductivity, and $$T$$ is the temperature.

### Single Parabolic Band Model

The single parabolic band assumption forms the simplest, first-order description of a material's electronic structure. Often, this is a reasonable approximation in regards to charge transport properties.&#x20;

We do not delve into the derivation of the transport parameters here. You can refer to the theory section of [this paper](https://chemrxiv.org/engage/chemrxiv/article-details/62fab7ef08a44b30fb7228cf) for more details. Rather, we list the final forms of the transport properties, namely the electrical conductivity ($$\sigma$$), Seebeck coefficient ($$S$$), and Lorenz number ($$L$$):

$$
\sigma(\eta) = \frac{8 \pi e}{3} \left(\frac{2 m\_e k\_B T}{h^2}\right)^{3/2} \mu\_w \left(r + \frac{3}{2}\right) F\_{r+\frac{1}{2}}(\eta)
$$

$$
S(\eta) = \frac{k\_B}{e} \left\[ -\eta + \left(\frac{r + \frac{5}{2}}{r + \frac{3}{2}} \right) \frac{F\_{r+\frac{3}{2}}(\eta)}{F\_{r+\frac{1}{2}}(\eta)} \right]
$$

$$
L(\eta) = \left(\frac{k\_B}{e}\right)^2 \left\[ \frac{r+\frac{7}{2}}{r + \frac{3}{2}}\~ \frac{F\_{r+\frac{5}{2}}(\eta)}{F\_{r+\frac{1}{2}}(\eta)} -  \left{\frac{r+\frac{5}{2}}{r + \frac{3}{2}}\frac{F\_{r+\frac{3}{2}}(\eta)}{F\_{r+\frac{1}{2}}(\eta)} \right}^2 \right]
$$

The electronic thermal conductivity can be expressed as

$$
\kappa\_e(\eta) = L(\eta) \sigma(\eta) T
$$

Fundamental constants are needed to describe transport; namely, $$e$$ is the electric charge, $$m\_e$$ is the free electron mass, $$k\_B$$ is the Boltzmann constant, and $$h$$ is Planck's constant.

The transport properties are all dependent on the reduced Fermi level ($$\eta$$) and the scattering parameter ($$r$$). The reduced Fermi level is given as

$$
\eta=\frac{E\_F}{k\_B T}
$$

where $$E\_F$$ is the Fermi level and $$T$$ is the temperature. $$E\_F$$is referenced to the band edge; in other words, $$E\_F=0$$ at the band edge, $$E\_F>0$$ when the Fermi level is in the band, and $$E\_F<0$$ when the Fermi level is in the band gap. This description holds regardless of whether the majority carriers are electrons (conduction band) or holes (valence band).

$$r$$ is a parameter representing the scattering mechanism of charge carriers. Some common scattering mechanisms are:

* Acoustic phonon scattering: $$r = -1/2$$
* Polar optical phonon scattering: $$r=1/2$$
* Ionized impurity scattering: $$r=3/2$$

The Fermi-Dirac integral $$F\_i(\eta)$$ is given as

$$
F\_i(\eta) = \int \frac{\varepsilon^i}{1 + \exp\left(\varepsilon - \eta\right)} d\varepsilon
$$

Using the Fermi-Dirac integral, the carrier concentration can also be expressed as

$$
n(\eta) = 4\pi \left( \frac{2 m\_{\rm DOS}^\* k\_B T}{h^2} \right)^{3/2} F\_{1/2}(\eta)
$$

The electrical conductivity $$\sigma$$ is also dependent on the weighted mobility $$\mu\_w$$, given as

$$
\mu\_w = \frac{e \tau\_0}{m\_b^*} N\_V \left(\frac{m\_b^*}{m\_e}\right)^{3/2}
$$

where $$m\_b^\*$$ is the effective band mass, and $$N\_V$$ is the valley degeneracy.

Given that the transport properties can be written in closed forms, we can express $$zT$$ in a more compartmentalized way:

$$
zT = \frac{S(\eta)^2}{L(\eta) + \frac{(k\_B/e)^2}{B \left( r+\frac{3}{2} \right)F\_{r+1/2}(\eta)}}
$$

where $$B$$ is the quality factor of the majority carriers defined as

$$
B = \left(\frac{k\_B}{e}\right)^2 \frac{8 \pi e}{3} \left(\frac{2 m\_e k\_B}{h^2}\right)^{3/2} \frac{\mu\_{w} T^{5/2}}{\kappa\_L}
$$

Notice that $$zT$$, as expressed this way, is determined by two principal quantities of the material: the intrinsic property $$B$$, and an extrinsic factor $$\eta$$.

### Two-Band Model

The two-band model described here involves the transport of both electrons and holes. In other words, information about the conduction and valence bands are necessary, as well as the band gap. It is convenient to represent the electron and hole contributions separately (using the single parabolic band model of transport as described above) and derive the total electrical conductivity, Seebeck coefficient, and electronic thermal conductivity from the two contributions.

If we reference the Fermi level $$E\_F$$ to the conduction band minimum (i.e. $$E\_F = 0$$ at the CBM), then the reduced Fermi levels of electrons ($$\eta\_n$$) and holes ($$\eta\_p$$) are given as

$$
\eta\_{n} = \frac{E\_F}{k\_B T}
$$

$$
\eta\_{p} = -\eta\_{n} - \frac{E\_g}{k\_B T}
$$

where $$E\_g$$ is the band gap. The form of $$\eta\_p$$ can be justified as follows: if $$E\_F$$ is in the conduction band (i.e. $$\eta\_n>0$$), then from the perspective of holes, $$E\_F$$ is far below the valence band edge (i.e. $$\eta\_p << 0$$). On the other hand, if $$E\_F$$ is $$E\_g$$ below the conduction band edge, then $$E\_F$$ is at the valence band edge, and therefore $$\eta\_p = 0$$.

In general, the reference energy for $$E\_F$$ is arbitrary, meaning that $$E\_F = 0$$ can be defined as the CBM, VBM, mid-gap, etc. It is more convenient to define this reference point by the majority carrier type in the material. For example, $$E\_F = 0$$ can be defined as the CBM when the material is *n*-type, whereas it can be defined as the VBM when the material is *p*-type. The mathematical formalism for the transport properties can be generalized by using the "majority" and "minority" carrier labels; in other words, we can express the reduced Fermi levels $$\eta$$ as

$$
\eta\_{maj} = \frac{E\_F}{k\_B T}
$$

$$
\eta\_{min} = -\eta\_{maj} - \frac{E\_g}{k\_B T}
$$

where $$E\_F = 0$$ at the edge of the majority carrier band. The thermoelectric transport properties can then be written as

$$
\sigma=\sigma(\eta\_{maj})+\sigma(\eta\_{min})
$$

$$
S = \frac{\sigma(\eta\_{maj})S(\eta\_{maj}) + \sigma(\eta\_{min})S(\eta\_{min})}{\sigma}
$$

$$
L = \frac{ \left\[ L\_{maj} + S\_{maj}^2 \right] \sigma\_{maj} + \left\[ L\_{min} + S\_{min}^2 \right] \sigma\_{min} }{\sigma} - S^2
$$

As before, the electronic thermal conductivity is given as $$\kappa\_e=L \sigma T$$.&#x20;

The important aspect of the two-band model to keep in mind is that the model can be built from the individual contributions of electrons and holes. Therefore, a rational way to write a script for the two-band model would be to write functions relevant for the single-band model and to subsequently use those functions to build the two-band model.

## Computational Details

### Necessary Modules

Scientific computing can be handled solely by `numpy`. The Fermi-Dirac integral can be evaluated using the [`fdint`](https://pypi.org/project/fdint/) module.

```
import numpy as np
from fdint import *
```

From `fdint`, we use the function `fdk(k=i, phi=eta)` to evaluate Fermi integrals of the form $$F\_i(\eta)$$.

### Fundamental Constants

We need to define fundamental constants (in SI units) to describe transport properties.

```
k = 1.38E-23      # Boltzmann's constant
e = 1.602E-19     # Charge
me = 9.109E-31    # Free electron mass
h = 6.626E-34     # Planck's constant
```

### Transport Properties for a Single Band

As suggested in the [two-band model section](#two-band-model), we build the transport model from two single-band transport models (one for the valence band, and the other for the conduction band). The transport properties ($$\sigma$$, $$S$$, and $$L$$) within a single-band model can be codified as follows:

```
# Electrical conductivity
def Sigma_SPB(mu_w, eta, T, r):
	return 8. * np.pi * e / 3 / h**3 * (2*me*k*T)**(3./2) * mu_w * (r+3./2) * fdk(k=r+1./2, phi=eta)

# Seebeck coefficient
def Seebeck_SPB(eta, r):
	return k/e * (-eta + (r+5./2)*fdk(k=r+3./2, phi=eta)/(r+3./2)/fdk(k=r+1./2, phi=eta))

# Lorenz number
def Lorenz_SPB(eta, r):
	return (k/e)**2 * ( (r+7./2)*fdk(k=r+5./2, phi=eta)/(r+3./2)/fdk(k=r+1./2, phi=eta) - ( (r+5./2)*fdk(k=r+3./2, phi=eta)/(r+3./2)/fdk(k=r+1./2, phi=eta) )**2 )
```

Here, `eta` is the reduced Fermi level and`r` is the scattering parameter. If needed, the electronic thermal conductivity ($$\kappa\_e$$) can be expressed straightforwardly as

```
# Electronic thermal conductivity
def Kappa_e_SPB(mu_w, eta, T, r):
        return Lorenz(eta, r) * sigma(mu_w, eta, T, r) * T
```

For example, if we set the following parameters for our model system:

```
temperature = 300 	# Temperature (K)
kappa_L = 1 		# Lattice thermal conductivity (W/mK)
band_mass = 0.5*me	# Effective electron mass (kg)
NV = 1 			# Valley degeneracy
r = -1/2 		# Scattering parameter (assumes acoustic phonon scattering)
tau0 = 1e-13		# Scattering prefactor (s)
```

we obtain the following plots:

<figure><img src="/files/xTVGVXSbNQ7VKi7UalZH" alt=""><figcaption><p>Transport properties as functions of the carrier concentration, within a single parabolic band assumption.</p></figcaption></figure>

### Transport Properties in a Two-Band Model

When both holes and electrons contribute to thermoelectric transport, the transport coefficients need to be evaluated using the two-band model.

The electrical conductivity is simply the sum of the conductivities of the electrons and holes:

```
# Electrical conductivity within the two-band model
def Sigma_TwoBand(mu_w_maj, mu_w_min, eta_maj, eta_min, T, r):
	sigma_maj = Sigma_SPB(mu_w_maj, eta_maj, T, r)
	sigma_min = Sigma_SPB(mu_w_min, eta_min, T, r)
	sigma_total = sigma_maj + sigma_min
	return sigma_total
```

Note that `Sigma_SPB` is the electrical conductivity from a single parabolic band.

The Seebeck coefficient is a sum of the Seebeck coefficients of the electrons and holes, weighted by their respective conductivities:

```
# Seebeck coefficient within the two-band model
# NOTE: Assumes n-type material (electrons = majority)
def Seebeck_TwoBand(mu_w_maj, mu_w_min, eta_maj, eta_min, T, r):
	sigma_maj = Sigma_SPB(mu_w_maj, eta_maj, T, r)
	sigma_min = Sigma_SPB(mu_w_min, eta_min, T, r)

	Seebeck_maj = Seebeck_SPB(eta_maj, r)
	Seebeck_min = Seebeck_SPB(eta_min, r)

	sigma_total = sigma_maj + sigma_min
	Seebeck_total = (-sigma_maj*Seebeck_maj + sigma_min*Seebeck_min) / sigma_total

	return Seebeck_total
```

Note that `Seebeck_SPB` is the Seebeck coefficient from a single parabolic band.

The electronic thermal conductivity can be codified as

```
def Kappa_e_TwoBand(mu_w_maj, mu_w_min, eta_maj, eta_min, T, r):
	
	sigma_maj = Sigma_SPB(mu_w_maj, eta_maj, T, r)
	sigma_min = Sigma_SPB(mu_w_min, eta_min, T, r)

	Seebeck_maj = Seebeck_SPB(eta_maj, r)
	Seebeck_min = Seebeck_SPB(eta_min, r)

	Lorenz_maj = Lorenz_SPB(eta_maj, r)
	Lorenz_min = Lorenz_SPB(eta_min, r)	

	sigma_ratio = sigma_maj / sigma_min

	sigma_total = sigma_maj + sigma_min
	Seebeck_total = (-sigma_maj*Seebeck_maj + sigma_min*Seebeck_min) / sigma_total
	kappa_e_total = (Lorenz_maj*sigma_maj + Lorenz_min*sigma_min)*T + sigma_total*T*(Seebeck_maj**2/(1.+1./sigma_ratio) + Seebeck_min**2/(1.+sigma_ratio) - Seebeck_total**2)

	return kappa_e_total
```

Note that `Lorenz_SPB` is the Lorenz number from a single parabolic band.

If we set the following parameters in our model system:

```
temperature = 300 	# Temperature (K)
kappa_L = 1 		# Lattice thermal conductivity (W/mK)
r = -1/2 		# Scattering parameter (assumes acoustic phonon scattering)
Eg = 0.4 		# Band gap (eV)

band_mass_e = 0.5*me    # Effective electron mass (kg)
band_mass_h = 0.5*me	# Effective hole mass (kg)
tau0_e = 1e-13		# Scattering prefactor for electrons (s)
tau0_h = 1e-13		# Scattering prefactor for holes (s)
NV_e = 1 		# Valley degeneracy for electrons
NV_h = 1 		# Valley degeneracy for holes
```

then we obtain the following plots:

<figure><img src="/files/7XVTkHqMgrwgbBObUZVW" alt=""><figcaption><p>Transport properties as functions of the Fermi energy, within a two-band assumption.</p></figcaption></figure>


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