Polaronic band tails in disordered solids: Combined effects of static randomness and electron-phonon interactions.

Abstract

The band-tail density of states at zero temperature for an electron coupled to a quantized elastic continuum and static disorder is derived from first principles with use of a variational method based on the Feynman path-integral representation of the one-electron propagator. In the absence of static disorder, nonperturbative effects in the electron-phonon interaction associated with small-polaron formation give rise to an exponentially decaying band tail below the free-electron continuum. The density of states projected onto the phonon vacuum exhibits three physically distinct regimes when the electron-phonon coupling is above small-polaron threshold: (i) at shallow energies, there is a shift of the square-root continuum band edge arising from the perturbative emission and reabsorption of virtual phonons; (ii) at intermediate energies, there is a linear exponential band tail of localized states analogous to an Urbach tail arising from quantum fluctuations of the lattice; (iii) at deeper energies, there are strongly localized or self-trapped states associated with the termination of this band tail at the polaron ground state. In the infinite-effective-mass approximation for polaron formation, these results are in close agreement with a simple physical argument based on an optimum potential-well method. In the presence of weak static disorder and electron-phonon coupling near polaron threshold we find substantial synergetic interplay between phonons and disorder. Static potential fluctuations provide nucleation centers for small-polaron formation and an exponential tail appears for any value of the coupling constant. The resulting density of states can be considerably larger than that arising from static disorder and electron-phonon interaction acting individually.