Surface Science 263 (1992) 234-239 North-Holland ,. .’ ” ,, . . zyxwvutsrqpo ‘ka.4rface science : ,. .:, .’ .‘,,. ,... Calculated contribution of localised LO phonon modes to the valley current of a double-barrier diode A. Nogaret and J.C. Portal zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA LPS-INSA, 31077 Toulouse. Fruncr untl C‘NRS-SNCI, 38042 Grenoble. France Received 3 June 1991: accepted for publication 26 August IYOI We have calculated the form factor and the current density assisted hy bulk and localised LO phonon cmisaion in an ideal caxc of a double-barrier diode (DBD). The transmission rate from emitter to collector is given by the transfer IIamiltonian approximation. We extend the formalism developped by Chevoir and Vinter by calculating the matrix elements using the appropriate electron-optical interaction Hamiltonians. We compute the relevant form factors and valley currents versus the applied voltage in the GaAs/Ga,,.,AI,,,,As system. We discuss the coupling dependence upon the phonon emission parameters and notably show the importance of zone center emission to explain the emergence and sharpening of the replica peaks in the current. Finally. evidence is given for a strong dependence of the amplitude of the second AlAs phonon satellite as a function of the emitter harrier width. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1. Introduction Recent experimental results [l] indicate that the valley current of a double-barrier diode (DBD) is due to non-coherent processes. In addition, several theoretical calculations based on this assumption were able to predict the experimental peak-to-valley ratios. The problem of weighting the different contribu- tions to the inelastic current has also been solved in ref. [2]. This work notably shows that in the GaAs/GaAlAs system the contribution to the inelastic current due to the interaction with the longitudi- nal optical (LO) phonons dominates. However, none of these calculations account for the coupling of an electron with lattice vibrations confined by the consecutive layers of the epitaxially grown structure. The interest in this study is doubly motivated by the experimental observations in two systems: GaAs/Ga,,.,Al,,,As [3] and (Galn)As/ (Alln)As [4] where, respectively, the GaAs, AlAs and InAs, AlAs modes were found to be active. In this paper, we calculate the transmission by extending the validity of the Bardeen transfer Hamiltonian approximation [5] to the case of inelastic tunneling. We make use of the corresponding electron-phonon interaction Hamiltonians calculated in the case of heterostructures composed of diatomic polar semiconductors [6] and the current is derived according to the method suggested zyxwvutsrqpon in ref. [2]. In particular, the non-coherent process is assumed to consist of two sequential steps. The first one allows the electron from the emitter to reach the quasi-bound level in the well by interacting with a phonon. The second one allows the electron in the well to tunnel through the collector barrier near resonance conditions. The first mechanism is the slowest and consequently will be the current-limiting factor. 2. The model Let us first clarify some preliminary points before setting up the theory. We have assumed a linear drop of potential in the active part of the DBD. In the flat-band approximation. self-energies and zyxwvuts 003Y-~ 02x/ ~ ~ 2/ $0s.00 c 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All right\ rcselvcd