Comment on ‘‘Quantum capacitance of resonant tunneling diodes’’

In a recent letter’ Hu and Stapleton (HS) introduced a quantity they call the quantum capacitance of resonance tunneling (RT) diodes. This quantity, which I shall denote by CHs, describes the modulation of charge SQ stored in the quantum well (QW) in response to a variation of external voltage 6V applied to the diode, SQ= C&S V. HS assert that it is CHs rather the geometric barrier capacitances C, =.cS/d, and C2=cY/d2 that determines the time constant of an RT oscillator. Proceeding to calculate Cns in a simple model of a RT diode, HS find that Cns.@Z1,C,, and in this context they criticize my earlier work.2 My problem with the HS letter is not that they had calculated their “quantum capacitance” incorrectly but that this quantity is unrelated to the oscillator time constant. Consequently, the use of Cn, by HS is based on a misunderstanding and their results are devoid of a definite physical significance. Consider the general circuit in Fig. 1, where the resistances R, and R2 are arbitrary functions of the voltages V and V,. For the purpose of calculating either the currentvoltage characteristics or the charge distribution inside the device, this circuit is equivalent to the model used by HS. Indeed, it has been rigorously shown by Payne3 that the following two calculational procedures produce identical results for the RT diode: One method is to first calculate the energy-dependent tunneling probability for a singlestep tunneling process between the emitter and the collector, and then calculate the current by integrating over the energies of the emitter electrons. The other method is to use an effective Hamiltonian for calculating the tunneling probabilities for two sequential steps and then calculate the current by a kinetic equation for the probability density. Even though it has become customary to refer to the former method as “coherent ” and the latter as “sequential,” it should be emphasized that the use of either method has nothing to do with the question of whether the tunneling process itself is coherent or sequential, which is an issue that can be decided upon only by including scattering processes. Within the “sequential ” framework, the equivalent circuit of Fig. 1 is reasonable. In general, the state of the circuit (e.g., the amount of charge Q stored in the central node) is not uniquely determined by the applied voltage V, cf. the well-known intrinsic bistability effect.4 Therefore, let us choose an operating regime away from any singular switching point and discuss the response of the circuit to a shock excitation SV u(t), where u(t) is a step function and 6 V is sufficiently small so as not to force switching into a different regime. Let us further assume that the current-voltage characteristics are sufficiently smooth so that variations in RI and R2 can be neglected within the range SV. Both assumptions are valid in the situation considered by HS. The time-dependent response SQ is easy to evaluate by the standard methods. For an arbitrary t > 0 the result is given by