Exact solution of electronic transport in semiconductors dominated by scattering on polaronic impurities

The scattering of electrons on impurities with internal degrees of freedom is bound to produce the signatures of the scatterer's own dynamics and results in nontrivial electronic transport properties. Previous studies of polaronic impurities in low-dimensional structures, like molecular junctions and one-dimensional nanowire models, have shown that perturbative treatments cannot account for a complex energy dependence of the scattering cross section in such systems. Here we derive the exact solution of polaronic impurities shaping the electronic transport in bulk (3D) systems. In the model with a short-ranged electron-phonon interaction, we solve for and sum over all elastic and inelastic partial cross sections, abundant in resonant features. The temperature dependence of the charge mobility shows the power-law dependence, $\mu \left(T\right)\propto T^{-\nu }$, with $\nu$ being highly sensitive to impurity parameters. The latter may explain nonuniversal power-law exponents observed experimentally, e.g., in high-quality organic molecular semiconductors.

Figure 1

${\mathrm{\Gamma }}^{\alpha ,\alpha }$ diagram for $\alpha =0$, involving the static ${\varepsilon }_0\ne 0$ scattering (vertical dashed lines) and the dynamic scattering on phonons (wavy lines). Dotted lines represent the unperturbed electron propagator.

Figure 2

GF diagrams. Single lines represent the electron propagator with static scattering included; double lines represent the exact one. The square in the fourth diagram is the exact electron-phonon vertex function.

Figure 3

Exact LDOS at the impurity site for different impurity parameters, $\rho \left(\omega \right)=-{\pi }^{-1}\text{Im}{G}_{\mathbf{l},\mathbf{l}}^{0,0}$.

Figure 4

Partial cross sections (10) for few elastic and inelastic scattering channels as a function of the incoming electron energy ${\varepsilon }_{\alpha }$.

Figure 5

Mobility $\mu \left(T\right)$ as a function of temperature for different couplings $g$, shown in the log-log scale.

Figure 6

$\mu \left(T\right)$, shown in the log scale, as a function of $g$ (inset) and ${\varepsilon }_0$.