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 transport coefficients are defined as

[math]\displaystyle{ L_{ij} = \frac{1}{2} \int d\epsilon \, \sigma(\epsilon) \, (\epsilon-\mu)^{i+j-2} \left( -\frac{\partial f^0}{\partial \epsilon} \right), }[/math]

where [math]\displaystyle{ \mu }[/math] is the chemical potential and [math]\displaystyle{ T }[/math] the temperature.

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.