Tuning a resonance in Fock space: Optimization of phonon emission in a resonant-tunneling device

Abstract

Phonon-assisted tunneling in a double barrier resonant tunneling device can be seen as a resonance in the electron-phonon Fock space which is tuned by the applied voltage. We show that the geometrical parameters can induce a symmetry condition in this space that can strongly enhance the emission of longitudinal optical phonons. For devices with thin emitter barriers this is achieved by a wider collector barrier.Progress in mesoscopic semiconductor devices [1] and molecular electronics [2] is driven by the need of miniaturization and the wealth of new physics provided by coherent quantum phenomena. A fundamental idea behind these advances was Landauer's view that conductance is transmittance [3,4]. Hence, the typical conductance peaks and valleys, observed when some control parameter is changed, are seen as fringes in an interferometer. However, the many-body electron-electron (ee) and electron-phonon (e-ph) interactions restrict the use of this picture. The e-e effects received much attention in different contexts [1]. While interest on e-ph interaction remained mainly focused on double barrier Resonant Tunneling Devices (RTD) [5], where phononassisted tunneling shows up as a satellite peak in a valley of the current-voltage (I-V) curve, recent observation of electro-mechanical effects in molecular electronics [6] requires a reconsideration of the e-ph problem. Theory evolved from a many-body Green's function in a simplified model for the polaronic states [7] to quantum and classical rate equations approach [8]. The latter uses an incoherent description of the e-ph interaction by adopting an imaginary self-energy correction to the electronic states [9,10].In this article, we analyze a quantum coherent solution of transport with e-ph interaction. We resort to a mapping of the many-body problem into a one-body scattering system where each phonon mode adds a new dimension to the electronic variable [11,12]. Transmission of electrons between incoming and outgoing channels with different number of phonons are then used in a Landauer's picture where the only incoherent processes occur inside the electrodes. This allows to develop the concept of resonance in the e-ph Fock space and the identification of the control parameters that optimize the coherent processes leading to a maximized phonon emission. It also gives a clue as to how "decoherence" arise within an exact many-body description. As an application, we consider a RTD phonon emitter where the relevant parameters are best known. There, the first polaronic excitation serves as an "intermediate" state for the phonon emission. An electron with kinetic energy ε ≤ ε F and potential energy eV in the emitter decays through tunneling into an electron with energy ε + eV−hω 0 in the collector plus a longitudinal optical (LO) phonon. The tuning parameter is the applied voltage while the optimization of phonon emission requires the tailoring of the tunneling rates.Let us consider a minimal Hamiltonian:where c + j and c j are electron creation and annihilation operators at s...

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