# Transport Properties¶

Electronic transport properties — namely, conductivity, Seebeck coefficient, and electronic component of thermal conductivity — are calculated through the Onsager coefficients. The spectral conductivity is calculated as

$\Sigma_{\alpha\beta}(\varepsilon) = \sum_n \int \frac{\mathrm{d}{\mathbf{k}}}{8\pi^3} v_{n\mathbf{k},\alpha}v_{n\mathbf{k},\beta}\tau_{n\mathbf{k}} \delta{\left(\varepsilon - \varepsilon_{n\mathbf{k}} \right )},$

where $\alpha$ and $\beta$ denote Cartesian coordinates, $\varepsilon_{n\mathbf{k}}$ and $v_{n\mathbf{k},\alpha}$ are the energy and group velocity of band index $n$ and wave vector $\mathbf{k}$, respectively. The spectral conductivity can be used to compute the moments of the generalized transport coefficients

$\mathcal{L}^n_{\alpha\beta} = e^2 \int \Sigma_{\alpha\beta}(\varepsilon) (\varepsilon - \varepsilon_\mathrm{F})^n \left [ -\frac{\partial f^0}{\partial \varepsilon} \right ] \mathrm{d}{\varepsilon},$

where $e$ is the electron charge and $\varepsilon_\mathrm{F}$ is the Fermi level at a certain doping concentration and temperature $T$. The Fermi–Dirac distribution is given by

$f^0_{n\mathbf{k}} = \frac{1}{\exp\left[{(\varepsilon_{n\mathbf{k}}-\varepsilon_\mathrm{F})/k_\mathrm{B}T} \right] + 1},$

where $k_\mathrm{B}$ is the Boltzmann constant. Electrical conductivity ($\sigma$), Seebeck coefficient ($S$), and the charge carrier contribution to thermal conductivity ($\kappa$) are obtained as

\begin{aligned} \sigma_{\alpha\beta} ={}& \mathcal{L}_{\alpha\beta}^0, \\ S_{\alpha\beta} ={}& \frac{1}{eT} \frac{\mathcal{L}_{\alpha\beta}^1}{\mathcal{L}_{\alpha\beta}^0}, \\ \kappa_{\alpha\beta} = {}& \frac{1}{e^2T} \left [ \frac{(\mathcal{L}_{\alpha\beta}^1)^2}{\mathcal{L}_{\alpha\beta}^0} - \mathcal{L}_{\alpha\beta}^2 \right ] . \end{aligned}