Airy-coordinate technique for nonequilibrium Green’s-function approach to high-field quantum transport

We develop a formalism for studying the motion of nondegenerate electrons in semiconductors, which allows a nonperturbative description of the effects of an external high electric field on electron-phonon scattering within the Kadanoff-Baym-Keldysh nonequilibrium Green’s-function approach. Based on the exact solutions of Schrödinger’s equation, we solve Dyson’s equation for the single-particle retarded Green’s function $G^r$ ${(\mathrm{k}}_{\perp }$,z,z’,ω). Recognizing that high fields break the translational symmetry of the system and that momentum is no longer a good quantum number, we use Airy transforms to handle the position dependence parallel to the applied high field. From this we are able to derive a model for the spectral density function A${(\mathrm{k}}_{\perp }$,s,ω) through which we can account for quantum effects such as collisional broadening and the intracollisional field effect in a simple and rigorous way. We demonstrate the theory by considering only weak scattering with nonpolar optical phonons and restrict ourselves to fields that are constant in space and time. Our formulation suggests the presence of a novel quantum effect induced by the simultaneous presence of scattering and electric field which produces a discontinuous trajectory of the electron path along the field direction.