PHYSICAL REVIEW B VOLUME 50, NUMBER 20 15 NOVEMBER 1994-II Description of resonant tunneling near threshold Gaston Garcia-Calderon Instituto de Fisica, Universidad Nacional Autonoma de Mezico, Apartado Postal 80-M4, Mexico 01000, Distrito Federal, Mexico Roberto Rorno* Centro de Investigacion Cientifica y Educacion Superior de Ensenada, Apartado Postal 87M, Ensenada, Baja California, Mezico Alberto Rubio Facultad de Ciencias, Universidad Autonoma de Baja California, Apartado Postal 1880, Ensenada, Baja California, Mezico {Received 15 July 1994) The behavior of a sharp resonance in a resonant tunneling structure as it approaches and passes through the emitter threshold due to an applied voltage is analyzed using a resonance formalism. The transmission coefBcient T(E) near resonance energy is written as a threshold factor times a Breit-Wigner formula. A typical example is used to show that the threshold factor and the resonance decay width are essential to reproduce the negative differential resistance region obtained from exact numerical calculations. For the valley contribution the resonance energy is below threshold and the crucial contribution comes froxn the part of the decay width that still lies above it. I. INTRODUCTION An important analytical expression for the description of electronic transmission in resonant tunneling struc- tures has been the Breit-Wigner formula for the trans- mission coefficient near resonance energy. This expres- sion has been widely used to study coherent processes in resonant tunneling structures. However, the standard Breit-Wigner formula is derived by assuming that the corresponding resonance is sharp, isolated, and far from the emitter energy threshold. The relevant point is, how- ever, that this is not the situation in the region of great- est physical interest, namely, the region where the peak value of the tunneling current and negative difFerential resistance take place. These e8ects are particularly im- portant in device applications. The peak current and the negative differential resistance occur when a resonance situated just above the energy threshold E = 0 moves down, as the voltage is increased, to end up just below this energy value, see Fig. 1. Although the threshold region has been considered in other fields of physics in the past, as for example, in nuclear or atomic physics, this problem has deserved almost no attention in resonant tunneling structures. In Refs. 2 and 3 the authors impose the threshold behavior on the transmission coefFicient by introducing an explicit energy dependence on the partial decay widths and hence on the total width. However, in this approach the decay widths are considered as adjustable parameters and the procedure becomes cumbersome in the presence of an applied voltage. In this work, we consider a formalism that provides a simple relation between the transmission coeKcient and the resonant properties of the system that incorporates in a natural way the description of resonant tunneling near threshold. We consider a purely coherent process and derive an expression for the transmission coefficient T(E) near resonance energy as a threshold factor times a Breit-Wigner formula. The threshold factor describes correctly the behavior of T(E) near and far from the emitter energy threshold. We also consider the implica- tions of the above expression for the tunneling current, in particular for the negative differential resistance re- gion and the description of the valley contribution to the tunneling current. A relevant feature of our approach is that the resonance parameters of the problem, the decay widths and the resonance energy, are not adjustable, as Emitter Collector FIG. 1. Potential pro6le of a double-barrier resonant struc- ture in the step approximation for the applied voltage. As the voltage increases the resonance state approaches the emitter threshold to eventually cross it. 0163-1829/94/59, '20)/15142(6)/$06. 00 50 15 142 1994 The American Physical Society