mat-edi-mobility

mat-edi-mobility

Goal

To compute the point-defect-limited carrier mobility $\mu(T)$ of a semiconductor and the underlying electron-defect scattering matrix elements $M(\mathbf{k}i, \mathbf{k}f) = \langle \psi{\mathbf{k}i} | \Delta V | \psi{\mathbf{k}f} \rangle$ from first principles, using the supercell difference-potential method of the Quantum ESPRESSO + EDI pipeline. Here $\Delta V = V{\text{defect}} - V{\text{pristine}}$ is the Kohn-Sham perturbation potential of an isolated point defect, extracted as the difference between the potentials of a defect-containing supercell and a pristine supercell. The matrix elements are computed on a coarse k-grid, transformed to a maximally localized Wannier basis $M(\mathbf{R}, \mathbf{R}')$, and interpolated onto dense fine grids. The state-resolved scattering rate follows from Fermi's golden rule,

$$\frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar}, n_{\text{d}}, \frac{1}{N_{\mathbf{k}}} \sum_{m,\mathbf{k}'} |M_{n\mathbf{k}, m\mathbf{k}'}|^2, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}'}),$$

and the mobility from the linearized Boltzmann transport equation in the self-energy (SERTA) and momentum (MRTA) relaxation-time approximations. The three first-class outputs are the direct matrix elements $M$ on the coarse grid, the Wannier-interpolated $M$ on a fine k-path, and the final mobility $\mu(T)$.

Background

Defect-limited transport requires $M(\mathbf{k}i, \mathbf{k}f)$ on k-grids far denser than any tractable supercell calculation can supply. EDI solves this the way EPW solves the electron-phonon problem: compute $M$ directly on a coarse grid commensurate with the supercell, rotate into a Wannier basis where $M(\mathbf{R}, \mathbf{R}')$ decays rapidly with $|\mathbf{R}|$ and $|\mathbf{R}'|$, and interpolate to $300\times300$ or denser at negligible cost. The scattering rate scales linearly with the defect concentration $n{\text{d}}$ (dilute, incoherent-defect limit), so **the mobility scales as $1/n{\text{d}}$ and the concentration must always be reported alongside $\mu$**. For 2D materials the pristine and defect supercells are computed independently, so their electrostatic potentials carry an arbitrary offset that must be removed by vacuum-level alignment (pot_align = 'vacuum').