Efficient ab initio calculations of electron-defect scattering and defect-limited carrier mobility

Electron-defect ($e-\mathrm{d}$) interactions govern charge carrier dynamics at low temperature, where they limit the carrier mobility and give rise to phenomena of broad relevance in condensed matter physics. Ab initio calculations of $e-\mathrm{d}$ interactions are still in their infancy, mainly because they require large supercells and computationally expensive workflows. Here we develop an efficient ab initio approach for computing elastic $e-\mathrm{d}$ interactions, their associated $e-\mathrm{d}$ relaxation times (RTs), and the low-temperature defect-limited carrier mobility. The method is applied to silicon with simple neutral defects, such as vacancies and interstitials. Contrary to conventional wisdom, the computed $e-\mathrm{d}$ RTs depend strongly on carrier energy and defect type, and the defect-limited mobility is temperature dependent. These results highlight the shortcomings of widely employed heuristic models of $e-\mathrm{d}$ interactions in materials. Our method opens avenues for studying $e-\mathrm{d}$ scattering and low-temperature charge transport from first principles.

Figure 1

Workflow for computing the $e-\mathrm{d}$ matrix elements and relaxation times, and the defect-limited mobility. In the first step, several inputs are computed with DFT, including the KS wave functions and eigenvalues of a primitive cell, and the local and nonlocal parts of the KS potential, separately in a pristine and a defect-containing supercell. The local matrix elements are then computed by Fourier transforming and interpolating the local perturbation potential, $\mathrm{\Delta }{V}_{\mathrm{L}}\left(\mathit{r}\right)$, and combining it with the plane-wave matrix elements of the primitive cell. The nonlocal matrix elements are similarly computed by splitting the calculation into a primitive cell and a supercell part, using only the KS wave functions of the primitive cell. The total $e-\mathrm{d}$ matrix elements are then formed by adding the local and nonlocal parts, following which they are employed to compute as output the properties of interest, including the band and momentum-resolved $e-\mathrm{d}$ relaxation times and the defect-limited mobility.

Figure 2

Absolute value of the local $e-\mathrm{d}$ matrix elements, obtained from our approach and, for comparison, with the all-supercell method, for two cell sizes, a primitive cell and a $2\times 2\times 2$ supercell. The initial state is in the lowest valence band at $\mathrm{\Gamma }$, and the final states are in the same band with crystal momenta ${\mathit{k}}^{\prime }$ along the $\mathrm{\Gamma }-X$ high-symmetry line.

Figure 3

(a) Relaxation times and their inverse, the scattering rates, for electrons and holes due to $e-\mathrm{d}$ scattering with vacancy defects in silicon. The valence band maximum is labeled ${\varepsilon }_v$ and the conduction band minimum ${\varepsilon }_c$. The density of states (DOS) is also plotted, in arbitrary units. (b) Comparison between the relaxation times due to $e-\mathrm{d}$ scattering with vacancy and interstitial defects in silicon. The calculations use a ${200}^3$ BZ grid with a $5\mathrm{meV}$ broadening, and a supercell size of $6\times 6\times 6$ (432 atoms) for vacancy and $8\times 8\times 8$ (1024 atoms) for interstitial defects, where the number of atoms refers to the pristine cells.

Figure 4

Convergence of the $e-\mathrm{d}$ RTs, shown here for vacancy defects in silicon. We consider the effect of (a) supercell size, (b) structural relaxation, and (c) BZ grid and energy broadening in Eq. (1), where for each broadening value we use a converged BZ grid. Also shown in (c) is the function $(-\partial f/\partial E)\times \mathrm{\Sigma }\left(E\right)$, which is the integrand of the mobility formula in Eq. (15), at several temperatures in arbitrary units.

Figure 5

Defect-limited mobilities in silicon, as a function of temperature below 150 K. Shown are the results for electron-vacancy ($e-\mathrm{V}$), hole-vacancy ($\mathrm{h}-\mathrm{V}$), electron-interstitial ($e-\mathrm{I}$), and hole-interstitial ($\mathrm{h}-\mathrm{I}$) interactions. The electron (solid circles) and hole (empty squares) mobilities are given for vacancy (dotted line) and interstitial (solid line) defects. The temperature dependence of the mobility above 50 K follows approximately a power law, $\mu \propto T^{-\alpha }$, with coefficients $\alpha$ given in the figure. A defect concentration of one defect in ${10}^6$ atoms is assumed.