PHYSICAL REVIEW B VOLUME 29, NUMBER 4 15 FEBRUARY 1984 Physics of resonant tunneling. The one-dimensional double-barrier case B. Ricco* and M. Ya. Azbelt IBM Thomas J. 8'atson Research Center, Forktown Heights, 1Vew Fork 10598 (Received 21 September 1983) In this work we discuss how the occurrence of resonant tunneling through a one-dimensional (1D) double barrier involves some interesting phenomena which have so far been overlooked. The effect of an externally applied electric field is considered, and it is shown that with fully symmetrical bar- riers it leads to weaker resonances than otherwise possible. Furthermore, the time required for reso- nance to be fully established is discussed, and it is shown that, depending on the barrier transmis- sion coefficients and experimental conditions, it can be exceedingly long, thus contributing to a reduction of resonance effects on the usual experimental time scale. We alsa show that resonant tunneling under the usual experimental conditions implies carrier trapping, hence a buildup of space charge available for modifying the potential-energy barrier. Different current behaviors then result from the inherent feedback mechanism. The effects of temperature an the measured current are fi- nally discussed. I. INTRODUCTION In this paper, we discuss in some detail the problem of resonant tunneling through a one-dimensional (1D) double (potential energy) barrier. Pioneering work on this subject is primarily due to Esaki, Chang and Tsu' who used GaAs-Ga~ zAlzAs heterostructures, and to Hirose et al. who, instead, worked on the Si-Si02 system. From a theoretical point of view significant contributions came from J. C. Penley and Sandomirskii in the 60's while re- cently new experiments were performed by Sollner et al. and by Ricco et al. on double barrier grown by means of molecular-beam epitaxy. The purpose is primarily that of showing how, although the occurrence of resonances is a simple and well-known result of the usual treatment of tunneling, it involves some complicated physical effects which have so far been al- most completely overlooked. The failure to fully under- stand the importance of such effects and the consequent lack of attention to their regard might, in our opinion, ex- plain the only partial success of the pioneering experimen- tal work' aimed at showing resonant tunneling. Al- though truly remarkable these works share the rather un- pleasant characteristic of finding effects much weaker and less pronounced than expected from the theory. This im- plicitly suggested the idea that resonant tunneling is too critically dependent on experimental parameters to be really controllable and (re)producible, let alone exploited, in real life where samples' or devices conditions' (defect concentration, surface cleanliness, actual dimensions, etc. ) are of course different from the clear-cut exact pictures of theoretical models. Although differences between theory and experiments are certainly to be expected, we feel that the criticality of resonant tunneling has been, and still is, overestimated and that the somewhat disappointing experimental results might well be due to nonoptimal samples and rneasure- ment conditions. Recently, new experiments carried out on GaAs- Gai Al„As double barriers similar to those of Ref. 1 have produced I-V characteristics with well pronounced peaks. The key factor for the improvement seems here to be the better material quality available today. From a quantitative point of view, the improvement over previous experiments is relevant (a 6:1 ratio between peaks and val- leys is observed) but a large discrepancy still exists with theoretical predictions. As will be shown in this paper this cannot, at least in part, be considered surprising since in almost all respects, these experiments are similar to those of Ref. 1. Two points, in particular, are important and will be dealt with in this paper. First, while the measurements have been carried out with a nonnegligible electric field applied across the double barrier, the assumed underlying theory did not include it. ' The most important remark with respect to this is that the electric field destroyed the symmetry of the two realized barriers and, consequently, contributed significantly to reduce the effects of reso- nances. As will be shown later, better structures than those used so far can be designed to maximize resonance peaks but it is essential to take into account the effects of the field. This involves some nontrivial engineering. Futhermore, because the field cancels the intrinsic simple symmetry, the optimization can only be performed for one of the possible several peaks. A second important point to grasp about the experi- rnent is that because of the way they were realized the oc- currence of substantial effects is not an immediate phenomenon. Resonant tunneling is time dependent and, to be fully established, requires a non-negligible time (ex- ponentially long as will be explained later). Therefore, if the experiments feature ramping voltages and no care is taken to allow for enough time at the peaks, resonance ef- fects might well be almost completely lost. A further remark concerns the space-change buildup necessarily taking place because of the accumulation of 29 1970 1984 The American Physical Society