Dynamic behavior of electron tunneling and dark current in quantum-well systems under an electric field David M.-T. Kuo and Yia-Chung Chang Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ~Received 24 May 1999; revised manuscript received 29 July 1999! A time evolution method is used to study the time dependence of electron tunneling in a quantum well under an electric field. A comparison of phase-shift analysis and the full time-dependent calculation is presented. Good agreement between the two on a large time scale is found, but discrepancy exists on the small time scale especially at high fields. Our studies confirm that the direct tunneling rate (1/t t ) evaluated by the phase-shift analysis is reliable at low fields. The phonon-assisted tunneling rate (1/t p ) is also calculated. We found that 1/t p dominates at low fields, while 1/t t increases exponentially with increasing field and becomes dominant at high fields. Contributions due to the thermionic emission, direct tunneling as well as phonon-assisted tunneling to the dark current are also evaluated. @S0163-1829~99!12947-3# I. INTRODUCTION Understanding the dark current of semiconductor quan- tum wells ~QW’s! is important for assessing the usefulness of quantum wells as infrared detectors 1 or field-effect transistors. 2 Three different processes, thermionic emission, direct tunneling, and phonon-assisted tunneling, contribute to the dark current. The dark current due to the tunneling pro- cesses is related to the escape rate of carriers. The escape rate of electrons in QW’s has been studied by a number of authors. 3–5 At low fields, the phase-shift analysis 5 and the complex-energy method 4 both work well, and they give essentially identical results. In the phase-shift approach, the energy eigenfunctions of the Hamiltonian in the presence of a field is solved, and the lifetime of the quasibound state is related to the inverse of the width of the resonance profile. At low fields, this resonance profile is well approximated by a Lorentzian function, or a d function in complex energy space. Thus it is expected that a complex- energy solution to a time-independent Schro ¨ dinger equation with a proper boundary condition gives rise to the same life- time for the quasibound state. A more physical and direct way of calculating the escape time is to find the time it takes for the electron wave packet to leave the quantum well by solving the time-dependent solution to the system. The time-dependent wave function for this system was previously calculated by Juang et al. via an iteration method. 3 This numerical method does not pro- vide a direct comparison with the known results at low fields as given in Refs. 4 and 5, so its numerical accuracy cannot be assessed. In our approach, we express the exact time- dependent solution as the Fourier transform of the energy eigenfunctions which are solved analytically in terms of lin- ear combinations of the Airy functions. The advantage of this method is that the time dependence of the charge distri- bution can be evaluated very accurately and a direct com- parison with the phase-shift analysis can be made at low fields. We found that the normalized charge density @ Q ( t )/ Q (0) # confined in the quantum-well decays exponen- tially on a large time scale, and its decay rate (1/t t ) agrees well with the results of the phase-shift analysis. However, on a small time scale the behavior is quite different, especially at high fields. We found that Q ( t )/ Q (0) displays a sudden drop initially, then begins to oscillate in time for t ,t t , and finally decays exponentially ~for t .t t ). The oscillatory be- havior is attributed to the quantum interference effect. The phonon-assisted tunneling rate (1/t p ) is calculated by using Fermi’s golden rule. Only the polar-optical scattering process is considered here, since it is the dominant process for polar semiconductor heterostructures at temperatures above 77 K. We found that 1/t p is of importance only at low fields, since 1/t t increases exponentially with increasing field and becomes dominant at high fields. Finally, we give a rough estimate of the dark current due to tunneling processes and thermionic emission in a simple classical model. II. DIRECT TUNNELING We consider a one-dimensional quantum-well system whose time-dependent Schro ¨ dinger equation is given by i \ ] c ~ z , t ! ] t 5H c ~ z , t ! 52 \ 2 2 m * ] 2 c ~ z , t ! ] 2 z 2 1@ V ~ z ! 2eFz # c ~ z , t ! , ~1! where V ~ z ! 5 H 0 inside the well V 0 outside the well, and 2eFz is potential introduced by an electric field F ap- plied along the z direction. Initially the electric field is zero and the system is in the ground state with energy E 0 and wave function c 0 ( z ). We have c 0 ~ z ! 5De k z , z ,2d ~2! PHYSICAL REVIEW B 15 DECEMBER 1999-I VOLUME 60, NUMBER 23 PRB 60 0163-1829/99/60~23!/15957~8!/$15.00 15 957 ©1999 The American Physical Society