IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 12, DECEMBER 1999 1887 Tunneling Time Asymmetry in Semiconductor Heterostructures Daniela Dragoman Abstract— Analytical expressions are given for the difference between left-to-right and right-to-left tunneling times in asymmet- ric single- and multiple-barrier heterostructures. This tunneling time asymmetry is related to the phase difference of the reflection coefficients of the electron wavefunction for the two tunneling directions. Examples for single- and double-barrier heterostruc- tures are given. The treatment in this paper can be used for designing devices with asymmetric frequency characteristics with respect to the electron tunneling direction. Index Terms—Berry phase, matrix theory, tunneling. I. INTRODUCTION T HERE IS still much controversy in the academic com- munity regarding the definition of tunneling time because time is not an operator in quantum mechanics; see, for exam- ple, the review papers [1] and [2]. One of the most intuitive definitions of tunneling time is based on the group velocity concept, namely (1) where the group velocity is defined as the ratio between the average probability current density Re and the probability density of the electron wavefunc- tion [3]. This definition, although apparently based on a semiclassical interpretation, was shown to be equivalent to the Bohm definition of the tunneling time, which takes explicitly into account the wave properties of the tunneling particle [1], as well as to a phase space definition in terms of the Wigner distribution function [4]. Numerical simulations have shown that this definition is in good agreement with experimental results [5]. Calculations of defined in (1) have shown that, for an asymmetric double-barrier heterostructure, the left-to-right and right-to-left tunneling times are different, although the reflection probabilities are equal. These tunneling times can differ by up to two orders of magnitude for the case of resonant tunneling [5]. To elucidate the origin of tunneling time asymmetry, the tunneling time through each layer in the heterostructure has been calculated; the results showed that there is no particular layer responsible for this asymmetry, but the left-to-right and right-to-left tunneling times through each layer are different [5]. An explanation of the tunneling time asymmetry was given in [6] where it was related to the fact Manuscript received June 25, 1999; revised August 30, 1999. The author is with the Physics Department, University of Bucharest, 76900 Bucharest, Romania. Publisher Item Identifier S 0018-9197(99)09443-9. that, although the reflection probabilities of the structure for the left-to-right and right-to-left tunneling directions are equal, the overall reflection coefficients differ by a phase factor. However, no explicit attempt to calculate the value of the tunneling time asymmetry was made in [6]. Due to the importance of tunneling time in designing the high-frequency behavior of optoelectronic devices [7], such as high-speed modulators, photodetectors, resonant tunneling and cascade lasers, etc., it is necessary to provide a quantitative estimate for the asymmetry of this parameter. The asymmetry of the tunneling time could also be responsible for instabilities in submicrometer devices [8]. In this paper, we provide not only a quantitative estimate, but also an analytical formula for the tunneling time asymmetry using (1) as the definition of tunneling time. The calculation of the tunneling time asymmetry was also addressed in [9], where it was expressed in terms of the different partial widths of the resonant energy level decay to the left and right of a resonant structure; it was, therefore, defined only for resonant tunneling conditions. The method of calculating the tunneling time asymmetry given in this paper is general, valid for either single- or multiple-barrier heterostructures, and is applicable for any conditions, not only for resonant tunneling or resonant tunneling heterostructures (for example, it accounts for the tunneling time asymmetry of a single barrier). Throughout this paper, we will consider a semiconductor heterostructure such as that shown in Fig. 1, with a step-like variation of the potential and a number of layers. A more detailed study will be made for a single-barrier ( ) and a double-barrier ( ) heterostructure. The electron wavefunction through this heterostructure satisfies the time- independent Schr¨ odinger equation (2) where is the effective mass of the particle, is its incident energy, and is the potential. Both and have a step-like variation, with their values in a layer being labeled with and , respectively. Then, as a solution of (2), we consider a plane-wave wavefunction, which takes in each layer the form with , and and the coefficients of the forward and backward components of the wavefunction, respectively. The regions situated to the left and right of the barrier/heterostructure are labeled with and , respectively, whereas the layers which form the heterostructure are labeled with numbers, in an increasing order starting from 0018–9197/99$10.00  1999 IEEE