Electronic transport coefficients

Introduction

The theoretical framework is based on the linearized Boltzmann transport equation (BTE) within the relaxation time approximation (RTA). The goal is to calculate electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity.

Electronic states and wavefunctions

The starting point is the set of Kohn–Sham eigenstates obtained from density functional theory (DFT). For a given Bloch state,

[math]\displaystyle{ H_{\mathbf{k}} |\psi_{n\mathbf{k}}\rangle = \epsilon_{n\mathbf{k}} S_{\mathbf{k}} |\psi_{n\mathbf{k}}\rangle, }[/math]

where [math]\displaystyle{ n }[/math] is the band index, [math]\displaystyle{ \mathbf{k} }[/math] is a crystal momentum, and [math]\displaystyle{ S_{\mathbf{k}} }[/math] is the overlap matrix.

Electron–phonon coupling matrix elements

Phonon scattering is described by the electron–phonon coupling matrix elements

[math]\displaystyle{ g_{n\mathbf{k},n'\mathbf{k}'}^{\nu\mathbf{q}} = \langle \psi_{n\mathbf{k}} | \partial_{\nu\mathbf{q}} V | \psi_{n'\mathbf{k}'} \rangle, }[/math]

where [math]\displaystyle{ \partial_{\nu\mathbf{q}} V }[/math] is the perturbation of the crystal potential due to a phonon of branch index [math]\displaystyle{ \nu }[/math] and wavevector [math]\displaystyle{ \mathbf{q} }[/math]. These matrix elements determine the scattering probability between states [math]\displaystyle{ (n,\mathbf{k}) }[/math] and [math]\displaystyle{ (n',\mathbf{k}') }[/math].

Scattering rates and lifetimes

Within Fermi’s golden rule, the inverse lifetime (scattering rate) of an electron in state [math]\displaystyle{ (n,\mathbf{k}) }[/math] is

[math]\displaystyle{ \frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar} \sum_{n'\nu\mathbf{k}'} w_{n\mathbf{k},n'\mathbf{k}'} \, |g^{\nu}_{n\mathbf{k},n'\mathbf{k}'}|^2 \left[ (n_{\nu\mathbf{q}} + 1 - f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} - \hbar\omega_{\nu\mathbf{q}}) + (n_{\nu\mathbf{q}} + f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} + \hbar\omega_{\nu\mathbf{q}}) \right] }[/math]

where:

• [math]\displaystyle{ f_{n\mathbf{k}} }[/math] is the Fermi–Dirac occupation,
• [math]\displaystyle{ n_{\nu\mathbf{q}} }[/math] is the Bose–Einstein phonon occupation,
• [math]\displaystyle{ \omega_{\nu\mathbf{q}} }[/math] is the phonon frequency.
• [math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} }[/math] weight determined by the ELPH_SCATTERING_APPROX

The two terms correspond to phonon emission and absorption, respectively.

Linearized Boltzmann transport equation

The distribution function of electrons under an applied electric field [math]\displaystyle{ \mathbf{E} }[/math] can be written as

[math]\displaystyle{ f_{n\mathbf{k}} = f^0_{n\mathbf{k}} + \delta f_{n\mathbf{k}}, }[/math]

where [math]\displaystyle{ f^0 }[/math] is the equilibrium Fermi–Dirac distribution. In the relaxation-time approximation,

[math]\displaystyle{ \delta f_{n\mathbf{k}} = - e \tau_{n\mathbf{k}} \, \mathbf{v}_{n\mathbf{k}} \cdot \mathbf{E} \left(-\frac{\partial f^0_{n\mathbf{k}}}{\partial \epsilon_{n\mathbf{k}}}\right). }[/math]

Here [math]\displaystyle{ \mathbf{v}_{n\mathbf{k}} = \nabla_{\mathbf{k}} \epsilon_{n\mathbf{k}} / \hbar }[/math] is the group velocity.

Transport distribution function

The energy-resolved transport distribution function is

[math]\displaystyle{ \sigma(\epsilon) = \frac{e^2}{N\Omega} \sum_{n\mathbf{k}} \tau_{n\mathbf{k}} \, \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}} \, \delta(\epsilon_{n\mathbf{k}}-\epsilon), }[/math]

where [math]\displaystyle{ \Omega }[/math] is the unit-cell volume and [math]\displaystyle{ N }[/math] the number of [math]\displaystyle{ \mathbf{k} }[/math]-points.

Onsager coefficients

The Onsager coefficients are defined as

[math]\displaystyle{ L_{ij} = \int d\epsilon \, \sigma(\epsilon) \, (\epsilon-\mu)^{i+j-2} \left( -\frac{\partial f^0}{\partial \epsilon} \right), }[/math]

where [math]\displaystyle{ \sigma(\epsilon) }[/math] is the transport distribution function, [math]\displaystyle{ \mu }[/math] the chemical potential, and [math]\displaystyle{ f^0 }[/math] the Fermi–Dirac distribution.

In practice, this integral can be evaluated in two ways:

Direct energy integration

The integrand is computed on an energy grid, and the derivative of the Fermi–Dirac distribution is explicitly included. This method is straightforward but may require careful broadening of δ–functions to converge smoothly.

Gauss–Legendre quadrature

A change of variables is introduced to avoid explicitly sampling the sharp derivative of the Fermi–Dirac function. Define

[math]\displaystyle{ x = 1-2f(\epsilon-\mu,T) }[/math]

so that [math]\displaystyle{ \epsilon = \mu + k_B T \ln\frac{1+x}{1-x} }[/math]. With this substitution, the derivative of the Fermi–Dirac distribution is absorbed into the Jacobian, and the Onsager coefficients take the form

[math]\displaystyle{ L_{ij} = \tfrac{1}{2} \sum_{k=1}^N d x \, \left( \frac{k_B T}{-e} \ln \frac{1+x_k}{1-x_k} \right)^{i+j-2} \sigma\!\left(\mu + k_B T \ln\frac{1+x_k}{1-x_k}\right), }[/math]

which corresponds to the Gauss–Legendre formula shown in the figure.

The Gauss–Legendre approach has the advantage that the integration grid adapts naturally to the width of the Fermi window, making it numerically efficient and avoiding the need for artificial broadening parameters.

Transport coefficients

Quantity	Formula	Physical meaning
Electrical conductivity [math]\displaystyle{ \sigma }[/math]	[math]\displaystyle{ \sigma = L_{11} }[/math]	Charge current response to an electric field
Seebeck coefficient [math]\displaystyle{ S }[/math]	[math]\displaystyle{ S = \tfrac{1}{T} L_{11}^{-1} L_{12} }[/math]	Voltage generated per temperature gradient
Electronic thermal conductivity [math]\displaystyle{ \kappa_e }[/math]	[math]\displaystyle{ \kappa_e = \tfrac{1}{T} ( L_{22} - L_{21} L_{11}^{-1} L_{12} ) }[/math]	Heat current carried by electrons in response to a thermal gradient

Approximations and methods

• Tetrahedron method: used for Brillouin-zone integration, avoiding the need for ad-hoc smearing parameters.
• Plane-wave Bloch states: ensure systematic convergence and avoid interpolation errors.
• Selection algorithms: restrict scattering processes to those allowed by energy conservation (delta functions), minimizing the number of matrix elements to compute.