Pergamon Solid State Communications, Vol. 97, No. 9, pp. 791-793, 1996 Copyright @ 1996 Published by Elscvicr Science Ltd Printed in Great Britain. All rights reserved 0038-1098196 $12.00 + 0.00 zyxwvuts 0038-1098(95)00694-X DWELL TIME FOR AN ASYMMETRIC ONE-DIMENSIONAL BARRIER V. Gasparian * Departamento de Fisica Aplicada, Universidad de Alicante, Alicante, Spain M. Ortuno, E. Cuevas and J. Ruiz Departamento de Fisica, Universidad de Murcia, Murcia, Spain zyxwvutsrqponmlkjihgfedcb (Received and accepted 23 September 1995 by A. Efros) We calculate the dwell times for incident particles comming from both the right and from the left of an asymmetric one-dimensional barrier. We prove that these times have a common contribution proportional to the density of states and an asymmetric contribution that depends on the reflection amplitudes from the right and from the left, which cancels in the-symmetric case. THERE HAS BEEN a great deal of interest in the study of the influence of the asymmetry of a potential barrier on the electronic transport properties and, in particular, on the resonant tunneling through a poten- tial barrier [l-4]. In this case the mean dwell time TD, which is defined as the average time that the particle spends within the barrier, independently of whether it is ultimately transmitted or reflected, is an important time scale. The dwell time was first introduced by Biittiker [5] as the ratio of the accumulated number of particles in the barrier to the incident flux: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA L (1) 0 where the integral extends over the barrier, and 2k is the incident flux (we will take e = c = A = 1, and mo = l/2 for the electron mass). q(x) is the steady- state scattering solution of the time-independent SchrGdinger equation, whose energy dependence is not written explicitly. As shown in papers [6,7] the Biittiker’s expression for a dwell time T@) is correct in all case and does not depend on the approaches, which is usually the case in the theory of tunneling problems (see i.g. [8,9]. On the other hand, with the expression of the dwell time de- fined as a weighted average between the transmission and reflection times TD = TTT -I- R~R (2) * Permanent address: Department of Physics, Yerevan State Uni- versity, 375049, Yerevan Armenia. we have a many problems (see, e.g. [7]). For example, the phase time for transmission and reflection of Bohm and Wigner do not satisfy Eq. (2). Our previous results [9], where we determine the dwell time for a symmetric barrier in terms of the Green function, and prove that it is proportional to the density of states, also are in contradiction with the dwell time defined by Eq. (2). The purpose of this work is to calculate directly the dwell time from Eq. (1) for the case of a general one- dimensional asymmetric barrier in terms of the density of states of system. Let us consider a particle moving along the x- direction in the presence of an arbitrary potential barrier V(x) in the interval (0, L) . The potential is zero outside the barrier. Our aim is to calculate the dwell time, given by Eq. (l), for particles coming both from the left and from the right. We evaluate Eq. (1) in three steps. First, we incorporate the fact that the wavefunction appearing in this equation is a solution of the Schr&dinger equation. Second, we rewrite the wavefunctions in terms of Green functions. And fi- nally, we express the Green functions in terms of the density of states and the reflection coefficients. First of all, we can trivially rewrite the wavefunction y(x), which is a solution of the Schrijdinger equation, as: cy(x) = (V(X) -El $4 - &V’(x) - E) ~4x1. (3) Then the modulus square I I&(X) I2 takes the following 791