Study of Thermoelectric Materials Performance

This software allows to calculate properties of 2D thermoelectric materials depending on their band structure. This is useful to estimate the performance of thermoelectric materials and understand how they can be optimized.

1. Introduction

Thermoelectric materials are able to convert a temperature gradient into electrical energy, or vice-versa. Since the majority of primary energy is wasted as heat, the development of thermoelectric devices is essential to utilize this waste heat and convert it into useful electric power. To continue this research and to direct it towards the realization of novel thermoelectric materials, simulations that, based on the characteristics of the material, are able to compute the thermoelectric quantities of such materials are of crucial importance

This software allows to calculate the following quantities characterizing 2D thermoelectric materials:

• Seebeck coefficient $S$
• electric conductivity $\sigma$
• thermal electronic conductivity $\kappa_e$
• figure of merit $ZT$

The most important result is the figure of merit $ZT$, which characterizes thermoelectric materials: the higher the $ZT$ value, the better the performance of the material.

The software takes the band structure of the material as a starting point. The simulation can be done based on three different models describing the band structure:

• single-parabolic-band model
• double-parabolic-band model
• double-Dirac-band model

All three models are based on Boltzmann transport theory in the linear response regime in the relaxation-time approximation.

In the end, the computed thermoelectric quantities depend on three main parameters of the material: the energy gap, the chemical potential and on the thermal lattice conductivity.

2. Setup

Using Anaconda or Windows

Do the following steps, either in the Anaconda Prompt or in the Windows Prompt.

(a) Clone the repository

Write:

git clone https://github.com/ViolaFerretti/Thermoelectric_Materials.git

(b) Install packages

Install the necessary packages in python3 using the pip command:

• to manage calculations:

pip install numpy
pip install scipy
pip install mpmath

• to visualize data

pip install matplotlib

• to set parameters and perform the simulations

pip install PySimpleGUI

(c) Run the software

Navigate to the project directory with the cd command and then write:

python GUI.py

3. Start

3.0 Configuration

The user can load a set of data from the "load_data.txt" file to configure the software. This can be done by entering the path to the configuration file and by selecting "Load Configuration". In this way, the file will pre-fill all the parameters; then, the user can change those parameters by modifying the example data in the file or from the sofware windows.

Example: configuration window

By selecting "Next >", the user can start working with the software.

3.1 Select the model

Single-parabolic-band model

The single-parabolic-band model assumes that only one band participates in the charge transport. This assumption can be justied in materials with relatively large band gaps (with respect to the targeted operational temperature range). S-parabolic-band modeling has been employed successfully in many material systems like $Mg_2(Si,Sn)$, $ZnSb$, $Bi_2Te_3$ and $PbTe$. A single band is, of course, not enough to capture the complete picture of the material when more than one majority carrier band and/or minority carrier bands are contributing substantially to charge transport. However, in the case of a highly doped sample single-parabolic-band model is a correct description over the entire temperature range.

To apply this model, write "SBMP" (the first three letter distinguish between Single-Band Model and Double-Band Model, and the last one selects the approximation, being either Parabolic or of Dirac type) in the first window of the graphic user interface.

Double-parabolic-band model

The single-band model is not sufficient to describe moderately or lightly doped materials, for which a double-band model is required, being the simplest and, in most cases, sufficient improvement. A very promising class of materials, is that of members of $Mg_2X$ (with $X = Si$, $Ge$, $Sn$) and their solid solutions, which can be correctly described by the double-parabolic-band model.

To apply this model, write "DBMP" in the first window of the graphic user interface.

Double-Dirac-band model

A lot of research efforts and grants have especially been invested on the 2D materials whose electronic structure can be modeled by the Dirac Hamiltonian, or the so-called 2D Dirac materials. Dirac matter is any material where the low-energy excitation spectrum can be described by the Dirac equation ($E \propto k$) rather than the more usual quadratic dispersion ($E \propto k^2$ ), considered in the two previous models. The examples of 2D Dirac materials include graphene, silicene, germanene, transition metal dichalcogenides (TMDs), and hexagonal boron nitride. Some of them possess excellent electronic and thermal properties, and some recent findings may indicate the possibilities of 2D Dirac materials as good thermoelectrics.

To apply this model, write "DBMD" in the first window of the graphic user interface.

Example: select the Double-Parabolic-Band model

By solving the Boltzmann equations within the selected model and the relaxation-time approximation, one finds that the Seebeck coefficient $S$, the electric conductivity $\sigma$ and thermal electronic conductivity $\kappa_e$ are given by:

$$\begin{equation} \begin{cases} \sigma=q^2L_0\\\\\ S=\frac{1}{qT}\frac{L_1}{L_0}\\\\\ \kappa_e=\frac{1}{T}\left(L_2-\frac{L_1^2}{L_0}\right) \end{cases} \end{equation}$$

where $L_i$ are the transport integrals.

Then, the final figure of merit $ZT$ is computed through the folowing equation:

$$\begin{equation} ZT=\frac{S^2\sigma T}{\kappa_e+\kappa_L} \end{equation}$$

where $\kappa_L$ is the thermal lattice conductivity.

For details on the procedure see References.

3.2 First part: energy gap and chemical potential

In this first part, the focus is ont the dependency of thermoelectic quantities on energy gap and chemical potential, while the thermal lattice conductivity is kept fixed.

3.2.1 Set parameters

Single-parabolic-band model

If the chosen model is a double-band one, the next window allows the user to insert the following parameters:

• minimum value of the chemical potential
• maximum value of the chemical potential
• step to consider in the chemical potential range
• thermal lattice conductivity

Double-band models

If the chosen model is a double-band one, the next window allows the user to insert the following parameters:

• minimum value of the energy gap
• maximum value of the energy gap
• step to consider in the energy gap range
• minimum value of the chemical potential
• maximum value of the chemical potential
• step to consider in the chemical potential range
• thermal lattice conductivity

Example: first part of the study with Double-Parabolic-Band model

Note that the enetered parameters are defined as adimensional (see References for details).

3.2.2 Visualize data

If the selected model is the single-parabolic-band one, the user can visualize the 2D plot of the thermoelectric quantity with respect to the chemical potential by clicking on "Show Plots".

When working with double-band models, the user has two choices to visualize the computed thermoelectric quantities:

• by selecting "Show 2D plots", the graphs of each quantity will be shown on the y-axis with respect to the chemical potential on the x-axis, for the different values of energy gap
• by selecting "Show 3D plots, the graphs of each quantity will be shown on the z-axis with respect to both the chemical potential and the energy gap, which vary on the x-y plane; this visualization can be useful to have a better understanding of how the three parameters depend simultaneously on each other.

3.3 Second part: the role of thermal lattice conductivity

The expression of $ZT$ tells that a bigger value of the figure of merit can be obtained with higher thermopower and electrical conductivity and a small thermal conductivity. Thermal conductivity has two contributions, one from electrons, analyzed before, and the other from the lattice (phonons). $\kappa_e$ can vary but cannot be lowered much, because the electric conductivity $\sigma$ (to me maximized) depends on it through Franz-Wiedemann's law relation:

$$\begin{equation} \kappa_e=L_0T\sigma \end{equation}$$

where $L_0=2.45\times 10^4\mu V^2K^{-2}$ is the Lorenz number. Therefore, one must act on $\kappa_L$, which in general is defined through the following equation:

$$\begin{equation} \kappa_L=\frac{1}{3}\nu_sc_s\lambda_{ph} \end{equation}$$

where $\nu_s$ and $c_s$ are the frequency and the speed of sound respectively, and $\lambda_{ph}$ is the wavelength of phonons oscillations. In the end, ideal thermoelectric materials are those that have a low thermal lattice conductivity (phonon glass) and a low resistivity (electron crystal). Indeed, for a phonon glass the phonons have a short mean free path (lattice vibrations are not relevant) and for an electron crystal the electrons have high mean free path (current flows easily). The lowest value that can be reached by acting on $\lambda_{ph}$ is $\kappa_L^{min}\simeq0.25\ W/mK$ (phonon glass). To better understand how the thermoelectric quantities depend on this parameter, the thermal lattice conductivity can be written can be written in units of $\kappa_0$ as $\kappa_L=r_{\kappa}\kappa_0$, where $r_{\kappa}$ is a parameter that can be changed by the user during the simulation, and $\kappa_0$ is defined in each model (seen previously).

3.3.1 Set parameters

In this second part, the parameters to indicate are:

• minimum of $r_{\kappa}$
• maximum of $r_{\kappa}$
• step to consider in the $r_{\kappa}$ range

Example: second part of the study with Double-Parabolic-Band model

3.3.2 Visualize data

Single-parabolic-band model

In the single-parabolic-band model, $ZT$ depends now on two parameters: the chemical potential (whose role was analyzed in the first part) and the thermal lattice conductivity. Therefore, now the software computes $ZT$ for a fixed chemical potential range and the selected $r_{\kappa}$ range. Thus, the software directly plots a 3D graph on $ZT$ depending on these two variables.

Double-band models

In the case of double-band-models, the situation is more complex, because, as seen in the first part, $ZT$ already depends on chemical potential and energy gap; the addition of $\kappa_L$ makes $ZT$ a 3-variable function, which would be difficult to study. Therefore, the software gets rid of the chemical potential-dependency by only keeping the maximized value of $ZT$ with respect to it. In this way, one can analyze the 3D plot of $ZT$ with respect to energy gap and thermal lattice conductivity.

3.4 Save data

By clicking on "Save", the user can save the data by specifying:

• the path to save the data on
• the name of the file, indicating: sigle of the model + the part of the simulation ("1" for the first part and "2" for the second part) + additional specifications needed by the user

In this way, for the first part, the software will save chemical potential and thermoelectric quantities in the case of single-parabolic-band model, and also the energy gap in the case of double-band models.

For the second part, the software will save chemical potential, thermal lattice conductivity and $ZT$ in the case of single-parabolic band, or thermal lattice conductivity, energy gap and $ZT$ for double-band models.

Example: save data of the second part of the study with Double-Parabolic-Band model

4. References

• Microscopic Kinetics and Thermodynamics course by Professor L. Pasquini at University of Bologna
• Mitra, Sunanda. Chalcogenide of type $IV-VI_2$ for thermoelectric applications. Diss. Université Paris Saclay (COmUE), 2016.
• Hung, Nguyen T., Ahmad RT Nugraha, and Riichiro Saito. "Universal curve of optimum thermoelectric figures of merit for bulk and low-dimensional semiconductors." Physical Review Applied 9.2 (2018): 024019.
• Naithani, Harshita, Eckhard Müller, and Johannes de Boor. "Developing a two-parabolic band model for thermoelectric transport modelling using $Mg_2Sn$ as an example." Journal of Physics: Energy 4.4 (2022): 045002.
• Hasdeo, Eddwi H., et al. "Optimal band gap for improved thermoelectric performance of two-dimensional Dirac materials." Journal of Applied Physics 126.3 (2019).

For details about the calculations: Details of calculations.pdf