Phonon-assisted hopping due to interaction with both acoustical and optical phonons

Abstract

Previous calculations of the nonadiabatic phonon-assisted jump rate have included electron coupling to either optical or acoustical phonons. Such calculations have yielded expressions for the jump rate which are essentially identical in the high-temperature ($T\ensuremath{\gtrsim}{T}_{\mathrm{Debye}}$) regime, in which the lattice motion can be treated classically, but which differ qualitatively from one another in the complementary low-temperature ($T\ensuremath{\ll}{T}_{\mathrm{Debye}}$) regime. In this article we have extended the calculations to include coupling of the electron to both phonon modes. The theory is not limited by temperature or by the strength of the electron-phonon coupling. Results are obtained which exhibit the essential characteristics of the jump rate for numerous values of the relative coupling of the electron to the optical and acoustical modes. It is found that when the electron's coupling to acoustical modes is comparable to or stronger than its coupling to optical phonon modes and when the difference in energy between the final and initial electronic state, $\ensuremath{\Delta}$, is of the same order as $\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{Debye}}$, the jump rate is qualitatively similar to that obtained when only the coupling to acoustic phonons is included. For instance, in the strong-coupling (small-polaron) situation the jump rate exhibits a high-temperature activated form above some fraction of the Debye temperature (typically $\frac{{T}_{\mathrm{Debye}}}{2}$). However, below this temperature its temperature dependence decreases sharply, exhibiting a nonactivated behavior. Finally in the limit of sufficiently low temperatures the jump rate is found to be simply the product of the acoustic-phonon jump rate (electrons coupling to acoustical modes only) and an additional factor related to the extra binding of the electron due to its coupling with optical phonons. Only when the coupling of the electron to optical modes is much stronger than its coupling to acoustical modes does the jump rate exhibit deviations from the temperature-dependent behavior described above. Additionally, the dependence of the jump rate on $\ensuremath{\Delta}$ is discussed in detail. For instance, in the high-temperature case, it is found that only when $\ensuremath{\Delta}$ is small compared with the polaronic binding energy the jump rate varies with $\ensuremath{\Delta}$ as $\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\Delta}}{2kT})$. For sufficiently large positive or negative values of $\ensuremath{\Delta}$ the jump rate will decrease with increasing $|\ensuremath{\Delta}|$. Thus even in the case for $\ensuremath{\Delta}<0$ (a hop downward in energy) the jump rate will ultimately decrease to zero as $|\ensuremath{\Delta}|$ increases.