Inelastic Quantum Transport

We solve a Schrödinger equation for inelastic quantum transport that retains full quantum coherence, in contrast to previous rate or Boltzmann equation approaches. The model Hamiltonian is the zero temperature 1d Holstein model for an electron coupled to optical phonons (polaron), in a strong electric field. The Hilbert space grows exponentially with electron position, forming a non-standard Bethe lattice. We calculate nonperturbatively the transport current, electron-phonon correlations, and quantum diffusion. This system is a toy model for the constantly branching “wavefunction of the universe”. 71.38.+i, 72.20.Ht, 72.10.Di Typeset using REVTEX 1 This paper is a fundamental study of quantum transport in the presence of inelastic degrees of freedom. We retain coherent quantum effects in a way that has not to our knowledge been done previously. There are a number of relevant model systems, but for concreteness we consider an electron interacting with optical phonons (the Holstein model), driven by a strong electric field. Our calculation is variational. It is non-perturbative in the electron-phonon coupling, the electric field, and everything else. Most previous work on this problem has relied on rate or Boltzmann equations, which are valid only for weak electron-phonon coupling and ignore quantum coherence effects [1–4]. These theories calculate transition rate probabilities, rather than complex quantum amplitudes. It is not clear how accurate such a treatment is, since, for example, a polaron is a quantum coherent object. This work takes a different approach. We calculate the wavefunction |ψ〉 that is the scattering solution to the Schrödinger equation H|ψ〉 = E|ψ〉. The solution consists of the complex amplitude and phase for each basis state in the many-body Hilbert space. The results are relevant to high field transport experiments on semiconductors such as ZnS [5], tunnel cathode structures [6,7], and microchannels [8]. They may also be relevant to systems exhibiting polaron hopping conductivity, such as colossal magnetoresistance (CMR) materials [9]. There has been related theoretical work that takes a very different (path integral) approach to the Fröhlich polaron problem, which has long-range electron-phonon interactions and parabolic bands [10,11]. We consider the Holstein model describing a single electron in a one-dimensional tightbinding lattice, which interacts locally with dispersionless optical phonons [1]. (Other models of inelastic quantum transport can be treated by similar methods.) A constant electric field drives the electron. The Hamiltonian is