Transport properties with the Boltzmann equation

by Maria Troppenz, Patrick Dieu & Santiago Rigamonti for exciting nitrogen

0. Define relevant environment variables and download scripts

Read the following paragraphs before starting with the rest of this tutorial!

Before starting, be sure that relevant environment variables are already defined as specified in How to set environment variables for tutorials scripts.

From now on the symbol $ will indicate the shell prompt.

2. Theoretical background for calculating the transport coefficients by using the Boltzmann equation

The efficiency of a thermoelectric material is determined by the dimensionless figure of merit :

\begin{align} Z T= \frac { \sigma S^2}{\kappa} T ~, \end{align}

ZT depends on the temperature $T$ and the transport coefficients: Electrical conductivity $\sigma$, Seebeck coefficient $S$, and the thermal conductivity $\kappa$, which is given by the sum of electronic contributions $\kappa_e$ and the lattice contribution $\kappa_l$. The coefficients $\sigma$, $S$ and $\kappa_e$ are mainly related to the electronic structure of the material, whereas $\kappa_l$ is related to the lattice.

Solving the Boltzmann equation in the constant relaxation time approximation (see 1), the transport coefficients can be calculated as follows:

\begin{align} \sigma = e^2 \int \mathrm{d} \epsilon \left( - \frac{\partial f_{0}} { \partial \epsilon } \right) \Xi(\epsilon) \end{align}
\begin{align} S = \frac{ e\, k_B}{\sigma} \int \mathrm{d} \epsilon \left( - \frac{\partial f_{0}} { \partial \epsilon } \right) \Xi(\epsilon) \frac{\epsilon - \mu}{k_B T} \end{align}
\begin{align} \kappa_0 = k_B T \int \mathrm{d} \epsilon \left( - \frac{\partial f_{0}} { \partial \epsilon } \right) \Xi(\epsilon) \left(\frac{\epsilon - \mu}{k_B T} \right)^2 \end{align}

Here, $\mu$ is the chemical potential and $e$ the electron charge, and $- \partial f_0 / \partial \epsilon$ the derivative of the Fermi distribution function $f_0 = [ (\exp( (\epsilon-\mu) / k_B T) +1 ]^{-1}$.

Using the equation above, the electronic thermal conductivity $\kappa_e$ is obtained by the relation $\kappa_e = \kappa_0 - T \sigma S^1$. The lattice part of the thermal conductivity still remains to be calculated by other methods.
The kernal function $\Xi (\epsilon)$ of the three transport coefficients is the transport distribution function,

\begin{align} \Xi (\epsilon)= \sum_{ \pmb{k} } { \pmb{v} }_{ \pmb{k} } { \pmb{v} }_{ \pmb{k} } \, \tau_{ \pmb{k} }\, \delta(\epsilon-\epsilon_{ \pmb{k} }). \end{align}

It is calculated from the group velocities ${ \pmb{v} }_{ \pmb{k} }$ of the charge carriers with cystal momentum $\pmb{k}$ and the corresponding carrier's lifetime $\tau_{\pmb{k}}$, both summed over the first Brillouin zone. In the constant relaxation time approximation, the $\tau$ is independent from $\pmb{k}$.

3. Transport coefficients of Bi2Te3

   <title>Bulk Silicon: Plot example</title>
   <structure speciespath="$EXCITINGROOT/species">
      <crystal scale="10.26">
         <basevect>0.0   0.5   0.5</basevect>
         <basevect>0.5   0.0   0.5</basevect>
         <basevect>0.5   0.5   0.0</basevect>
      <species speciesfile="Si.xml">
         <atom coord="0.00  0.00  0.00"></atom>
         <atom coord="0.25  0.25  0.25"></atom>
          ngridk="5 5 5"
    <momentummatrix fastpmat="true"/>
        windtdf="-1.0 0.5" 
        windtemp="300 1200" 
        windmu="0.2026454363 0.2106454363"
              <condcomp>1 1</condcomp>
              <condcomp>2 2</condcomp>
              <condcomp>3 3</condcomp>
Attribute Element Description
windtdf boltzequ It determines the energy window in which the transport distribution function is calculated $\chi$ .
nwtdf boltzequ The quality xs calculation.
windtemp boltzequ It determines the size.
tgrid boltzequ This is the value of.
windmu boltzequ It determines the size of the basis set any calculation.
mugrid boltzequ This is the value of neglected.

3. Transport coefficients of Bi2Te3

1. T. J. Scheidemantel, C. Ambrosch-Draxl, T. Thonhauser, H. V. Badding, and J. O. Sofo, "Transport coefficients from first-principles calculations", Phys. Rev. B 68, 125210 (2003).
2. G. D. Mahan and J. O. Sofo, "The best thermoelectric", Proc. Natl. Aad. Sci. USA 93, 7436 (1996).
3. Method described in detail in: B. R. Nag, "Electron Transport in Compound Semiconductors" (Springer Verlag, Berlin, 1996).
4. S. Nakajima, J. Phys. Chem. Solids 24, 479 (1963).
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