Charge carrier mobility in systems with local electron-phonon interaction

We present a method for calculation of charge carrier mobility in systems with local electron-phonon interaction. The method is based on unitary transformation of the Hamiltonian to the form where the nondiagonal part can be treated perturbatively. The Green's functions of the transformed Hamiltonian were then evaluated using the Matsubara Green's functions technique. The mobility at low carrier concentration was subsequently evaluated from Kubo's linear response formula. The methodology was applied to investigate the carrier mobility within the one-dimensional Holstein model for a wide range of electron-phonon coupling strengths and temperatures. The results indicated that for low electron-phonon coupling strengths the mobility decreases with increasing temperature, while for large electron-phonon coupling the temperature dependence can exhibit one or two extremal points, depending on the phonon energy. Analytical formulas that describe such behavior were derived. Within a single framework, our approach correctly reproduces the results for mobility in known limiting cases, such as band transport at low temperatures and weak electron-phonon coupling and hopping at high temperatures and strong electron-phonon coupling.

Figure 1

Dependence of band dispersion narrowing factor $\frac{J_{\mathrm{eff}}}{J}$ on electron-phonon coupling strength $\frac{G}{J}$ and temperature $\frac{k_{B}T}{J}$ for different values of phonon energies $\frac{\hslash \mathrm{\Omega }}{J}$.

Figure 2

Spectral function at phonon energy of $\frac{\hslash \mathrm{\Omega }}{J}=1$ and different values of electron-phonon coupling strength $\frac{G}{J}$ and temperature $\frac{k_{B}T}{J}$: (a) $\frac{k_{B}T}{J}=3.16,\frac{G}{J}=\frac{\sqrt{2}}{10}$; (b) $\frac{k_{B}T}{J}=10,\frac{G}{J}=\frac{\sqrt{2}}{10}$; (c) $\frac{k_{B}T}{J}=0.631,\frac{G}{J}=1$; (d) $\frac{k_{B}T}{J}=6.31,\frac{G}{J}=1$; (e) $\frac{k_{B}T}{J}=1,\frac{G}{J}=\sqrt{6}$; (f) $\frac{k_{B}T}{J}=6.31,\frac{G}{J}=\sqrt{6}$.

Figure 3

Spectral function at phonon energy of $\frac{\hslash \mathrm{\Omega }}{J}=3$ and different values of electron-phonon coupling strength $\frac{G}{J}$ and temperature $\frac{k_{B}T}{J}$: (a) $\frac{k_{B}T}{J}=1.58,\frac{G}{J}=1$; (b) $\frac{k_{B}T}{J}=3.98,\frac{G}{J}=1$; (c) $\frac{k_{B}T}{J}=10,\frac{G}{J}=1$; (d) $\frac{k_{B}T}{J}=0.794,\frac{G}{J}=\sqrt{8}$; (e) $\frac{k_{B}T}{J}=2.00,\frac{G}{J}=\sqrt{8}$; (f) $\frac{k_{B}T}{J}=5.01,\frac{G}{J}=\sqrt{8}$.

Figure 4

Temperature dependence of mobility for different electron-phonon coupling strengths $\frac{G}{J}$ and phonon energies: (a) $\frac{\hslash \mathrm{\Omega }}{J}=0.2$, (b) $\frac{\hslash \mathrm{\Omega }}{J}=1$, and (c) $\frac{\hslash \mathrm{\Omega }}{J}=3$. The label “fc” denotes full calculation using the spectral function obtained from the self-consistent solution of Eqs. (21, 22, 23, 24) and (27) and mobility obtained from Eqs. (39)–(43). The label “wl” denotes the weak electron-phonon coupling limit results obtained from Eq. (49). The label “fc-fb” denotes full calculation where momentum dependence of Green's functions and self-energies was neglected, i.e., Eqs. (53), (54), (56), and (27) were used. The label “sl” denotes the strong electron-phonon coupling limit results obtained from Eq. (65). The label “nl” denotes the mobility obtained using the narrow linewidth approximation with spectral function obtained from Eqs. (70) and (69) and the mobility calculated as described in Appendix pp4. The label “m” denotes the mobility obtained using the Marcus formula from Eqs. (75) and (76).

Figure 5

Temperature dependence of the $\gamma$ and the ${\mathrm{\Theta }}_0$ term that determine the mobility for $\frac{\hslash \mathrm{\Omega }}{J}=1$ and $\frac{G}{J}=\sqrt{6}$ (solid line), as well as for $\frac{\hslash \mathrm{\Omega }}{J}=0.2$ and $\frac{G}{J}=1$ (dashed line).

Figure 6

Time dependence of the real part of the quantity ${\mu }^{\prime }\left(t\right)$, which is proportional to current-current correlation function and whose integral over time is equal to mobility. The dependence is shown for $\frac{\hslash \mathrm{\Omega }}{J}=1$ and (a) $\frac{G}{J}=\frac{\sqrt{2}}{10}$, (b) $\frac{G}{J}=1$, and (c) $\frac{G}{J}=\sqrt{6}$. Since the relation $\mathrm{Re}{\mu }^{\prime }\left(t\right)=\mathrm{Re}{\mu }^{\prime }(-t)$ holds, only the part with $t>0$ is shown.

Figure 7

Temperature dependence of the mean free path for different values of system parameters.

Figure 8

Comparison of temperature dependence of mobility obtained in this work (unlabeled data sets) with quantum Monte Carlo results of Ref. [32] (data sets labeled as “qmc”). The straight line labeling the $\sim \frac{1}{T}$ dependence is given as a guide to the eye.