We study the influence of inelastic processes on shot noise and the Fano factor for a one-dimensional double-barrier structure, where resonant tunneling takes place between two terminals. Most studies to date have found, by means of various approximate or phenomenological methods, that shot noise is insensitive to dephasing caused by inelastic scattering. In this paper, we explore the status of this statement by deriving a general Landauer-Büttiker-type formula that expresses the current noise and Fano factor in a one-dimensional conductor through inelastic scattering amplitudes. For a double-barrier structure, exact scattering amplitudes are calculated in the presence of a time-dependent potential that acts in the region between the barriers. This allows us to rigorously analyze the role of dephasing in the current noise generated by applying a finite bias voltage to the resonant level. As an example of the dephasing potential, we consider the one induced by equilibrium phonons. We show that for phonons propagating in one dimension, the random phase of the electron wave function, which is induced by the electron-phonon coupling, exhibits diffusionlike dynamics. At the same time, for higher-dimensional phonons, the electron phase dynamics turns out to be nondiffusive, such that the average square of the phase grows logarithmically with time. We calculate transmission coefficients of a double-barrier structure for these two types of phonon-induced dephasing. In the case of diffusive phase relaxation, the resonant level has a Lorentzian shape with the broadening determined by a sum of the elastic linewidth and the phase breaking rate. Logarithmic dephasing leads to an unusual shape of the size-quantized level: the transmission coefficient is characterized by the two energy scales, one governed by the transparency of barriers and the other by the phonon correlation time. We further calculate the Fano factor for these types of dephasing, using exact expressions for inelastic transmission and reflection amplitudes. It turns out that when an integer number of levels falls into the energy window of width eV, where V is the voltage applied to the structure, the Fano factor is really insensitive to inelastic processes inside the structure and coincides with the prediction of phenomenological models with an accuracy of small corrections depending on these processes. On the contrary, at low voltages, when the eV window is smaller than the level width, this dependence is particularly pronounced and the phenomenological formula does not work.

• Received 26 July 2022
• Revised 5 November 2022
• Accepted 9 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.245421

This article appears in the following collection:

Emmanuel Rashba: Breaking New Ground in Solid-State Exploration

Physical Review B is pleased to present the “Collection in Honor of Emmanuel I. Rashba and His Fundamental Contributions to Solid-State Physics” in the year of his 95th birthday, highlighting the many ways in which his work has changed the landscape of modern condensed matter physics. Papers belonging to this collection will be published through mid-2023. The contributed articles, and an editorial by Guest Editors Mark Dykman, Alexander Efros, Bertrand Halperin, Leonid Levitov, and Charles Marcus, are linked below.