Electronic transport coefficients
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.
Electron–phonon coupling matrix elements
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. The scattering with phonons 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.
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 one of two ways determined by ELPH_TRANSPORT_DRIVER
- Linear energy grids and Simpson rule
The integrand is computed on a linear energy grid, and the Simpson rule is used for integration. The discretized Onsager coefficient is evaluated as
- [math]\displaystyle{ L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \; \sigma(\epsilon_k)\; (\epsilon_k - u)^{\,i+j-2}\; \left( -\frac{\partial f^0}{\partial \epsilon} \right). }[/math]
with [math]\displaystyle{ \epsilon_k = \epsilon_\text{min}+(k-1)\Delta \epsilon,\;\; k=1,\dots,N }[/math] and [math]\displaystyle{ \Delta \epsilon = \tfrac{\epsilon_\text{max}-\epsilon_\text{min}}{N-1} }[/math] and [math]\displaystyle{ \epsilon_\text{min} }[/math]=ELPH_TRANSPORT_EMIN and [math]\displaystyle{ \epsilon_\text{max} }[/math]=ELPH_TRANSPORT_EMAX or alternatively both [math]\displaystyle{ \epsilon_\text{min} }[/math] and [math]\displaystyle{ \epsilon_\text{max} }[/math] are set by ELPH_TRANSPORT_DFERMI_TOL and [math]\displaystyle{ w_k }[/math] the weights due to the Simpron integration rule.
- 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 w_k \, \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]
with [math]\displaystyle{ w_k }[/math] and [math]\displaystyle{ x_k }[/math] the weights and abcissae of the Gauss-Legendre quadrature rule.
The Gauss–Legendre approach has the advantage that the integration grid adapts naturally to the width of the Fermi window, making it numerically efficient without having define manually the energy window through ELPH_TRANSPORT_DFERMI_TOL or ELPH_TRANSPORT_EMIN and ELPH_TRANSPORT_EMAX. Instead, only the number of points [math]\displaystyle{ N }[/math] in the sum above needs to be defined through TRANSPORT_NEDOS.
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 |
| Peltier coefficient [math]\displaystyle{ \Pi }[/math] | [math]\displaystyle{ \Pi = T S = L_{11}^{-1} L_{12} }[/math] | Heat carried per unit charge current |
| 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 |
Electron and hole mobilities in semiconductors
In semiconductors, the electrical conductivity can be separated into contributions from conduction-band electrons and valence-band holes. This is only meaningful in materials with a finite band gap, where carriers can be clearly identified as either electrons in the conduction band (CB) or holes in the valence band (VB).
| Quantity | Definition | Carrier density |
|---|---|---|
| Electron mobility [math]\displaystyle{ \mu_e }[/math] | [math]\displaystyle{ \mu_e = \tfrac{\sigma_{n \in \text{CB}}}{n_e} }[/math] | [math]\displaystyle{ n_e = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{CB}} f(\varepsilon_{\mathbf{k}n}, T_\sigma, \eta) }[/math] |
| Hole mobility [math]\displaystyle{ \mu_h }[/math] | [math]\displaystyle{ \mu_h = \tfrac{\sigma_{n \in \text{VB}}}{n_h} }[/math] | [math]\displaystyle{ n_h = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{VB}} \big[1 - f(\varepsilon_{\mathbf{k}n}, T, \eta)\big] }[/math] |
Here:
- [math]\displaystyle{ \sigma_{n \in \text{CB}} }[/math] and [math]\displaystyle{ \sigma_{n \in \text{VB}} }[/math] denote the conductivity restricted to states in the conduction and valence bands, respectively.
- [math]\displaystyle{ f_{n\mathbf{k}} }[/math] is the Fermi–Dirac distribution.
- [math]\displaystyle{ \Omega }[/math] is the volume of the unit cell.
- [math]\displaystyle{ N_\mathbf{k} }[/math] is the total number of k-points.
- [math]\displaystyle{ \eta }[/math] is the chemical potential at the given temperature
| Important: The chemical potential is usually written as [math]\displaystyle{ \mu }[/math] or [math]\displaystyle{ \varepsilon_F }[/math]. To avoid confusion with the mobility, we use the notation [math]\displaystyle{ \eta }[/math]. |
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.