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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

Σαβ(ε)=ndk8π3vnk,αvnk,βτnkδ(εεnk), \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 α\alpha and β\beta denote Cartesian coordinates, εnk\varepsilon_{n\mathbf{k}} and vnk,αv_{n\mathbf{k},\alpha} are the energy and group velocity of band index nn and wave vector k\mathbf{k}, respectively. The spectral conductivity can be used to compute the moments of the generalized transport coefficients

Lαβn=e2Σαβ(ε)(εεF)n[f0ε]dε, \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 ee is the electron charge and εF\varepsilon_\mathrm{F} is the Fermi level at a certain doping concentration and temperature TT. The Fermi–Dirac distribution is given by

fnk0=1exp[(εnkεF)/kBT]+1, f^0_{n\mathbf{k}} = \frac{1}{\exp\left[{(\varepsilon_{n\mathbf{k}}-\varepsilon_\mathrm{F})/k_\mathrm{B}T} \right] + 1},

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

σαβ=Lαβ0,Sαβ=1eTLαβ1Lαβ0,καβ=1e2T[(Lαβ1)2Lαβ0Lαβ2]. \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}