VOLUME 69, NUMBER 20 P H YSICAL R EV I EW LETTERS 16 NOVEMBER 1992 Quantum Confinement in Laterally Squeezed Resonant Tunneling Devices In a recent Letter, Gueret et al. [1] investigated the current-voltage characteristics, I (V), of a double-barrier resonant-tunneling device in which the lateral dimensions, and hence the current, are controlled by a Schottky gate. They observed a series of peaks in the conductance dI/dV which moved to higher source-drain voltage (V) with in- creasing negative gate voltage VG. They claim that this behavior provides direct proof of a quantum size effect which is governed by Vg. In this Comment, we show that the overall dependence of the peak positions on Vo is qualitatively inconsistent with such a model. We base our argument on an analysis of the resonant-tunneling transitions of electrons from the occupied states of the negatively biased emitter into the quantum well under conditions of lateral confinement. It is claimed that the confinement provided by the gate potential quantizes the lateral motion, and hence the en- ergy, of the electron states in the emitter and quantum well. The exact form of this quantization depends on the detailed shape of the confining potential. However, the following considerations apply in general. Tunneling transitions from emitter to quantum well can be classified into two types: (1) those in which the lateral quantum numbers are conserved, and (2) those in which they are changed. In both cases, energy is conserved. If the de- gree of lateral confinement in the emitter and well is the same, the resonance condition for the features due to type-1 transitions is independent of the confinement and therefore they all occur at the same source-drain voltage, independent of VG. If the length scale (d ) which defines the lateral confining potential in the quantum well is smaller than that in the emitter (d, ), the features due to the various type-1 transitions will occur at different source-drain voltages, with their separation increasing as the confinement lengths are reduced. Features due to type-2 transitions will exhibit a strong dependence on gate voltage, even if d, =d„, since they involve a change in lateral quantum number(s). Their shifts, 5V, in source-drain voltage with increasing negative gate voltage are determined by the energy-level spacing of the quan- tized levels and by the change in quantum number(s) [2]. This variation is therefore a sensitive test of gate- controlled lateral confinement. Let us now compare this analysis with the data report- ed by Gueret et al. [1]. They find that the lowest voltage peak in conductance shifts to higher voltages with in- creasing negative Vg. However, most of the other prom- inent peaks in the dI/dV shift at a similar rate. This can be seen in Fig. 3 of Ref. [1] and also in Fig. 4 of Ref. [2] which plots the various peak positions for the data shown in Ref. [1] over the full range of gate voltage. In particu- lar, the variation of voltage separation of the lowest two peaks, h, Vi 2 = 5 V2 — h, V], is much less than the shift in the voltage of the lowest peak, h, Vi. From Fig. 4 of Ref. [3] we see AV|=15 meV and AVt q=4 meV as Vg is varied from — 1. 7 to — 2.4 V, although the estimate of AVi 2 is complicated by the appearance of an extra peak over this gate voltage range. This is also inconsistent with the model proposed by Gueret et al. [1]. We pro- pose that these observations are more consistent with an increase in the potential in the quantum well due to the electrostatic effect of the gate voltage than with a gate- controlled quantum confinement effect. An electrostatic effect would shift both type-1 and type-2 transitions equally, as is observed. In such a description, the peaks in the conductance are more likely to be due to transi- tions between states localized by random variations in the lateral potential rather than by the gate potential. For example, these might be due to the presence of ionized donors which have diffused or segregated from the doped contact layers. This explanation would also be more con- sistent with the voltage position of the lowest peak that Gueret et al. observe, which is (i) below the threshold for resonant tunneling, V, h=60 meV [4], and (ii) found to vary widely (30-50 mV) between different devices [5]. This type of effect has been proposed to explain the sharp peaks in the I(V) curves of similar [6] and related [7] de- vices. It is clear that the electrical properties of small devices of this type are sensitive to a number of phenomena such as quantum confinement, potential fluctuations, and Coulomb blockade. An unambiguous observation of a quantum confinement effect control!ed by an external gate potential would require evidence of an energy-level spectrum in which the level separation increased sys- tematically with increasing confinement. The experiment described by Gueret et al. shows only an overall shift of the conductance peaks with increasing negative bias. P. H. Beton, L. Eaves, and P. C. Main Department of Physics University of Nottingham NG7 2RD, United Kingdom Received 14 April 1992 PACS numbers: 73. 40.Gk, 72. 20. — i [I] P. Gueret et al. , Phys. Rev. Lett. 68, 1896 (1992). [2] Let us assume an infinite potential well of square cross section, and side length d . Gueret et al. claim the lateral potential is indeed sharp edged. The energy levels of the quantum well are then given by Eow =h (p +q )/ 8md + E„where (p, q) are the quantum numbers for la- 2 terai confinement and E, is the confinement energy pro- vided by the tunnel barriers. The variation with lateral confinement (and VG) in the source-drain voltage of the resonances is AV=h [(p +q )/d~ — (p' +q' )/d ]/ em* f, where (p', q') are the emitter state quantum num- bers and f= —, ' is the fraction of the source-drain voltage dropped across the emitter region. [3] P. Gueret et al. , Semicond. Sci. Technol. B 7, 462 (1992). [4] P. Gueret et al. , J. Appl. Phys. 66, 278 (1989). [5] P. Gueret et al. , Surf. Sci. 263, 212 (1992). [6] M. W. Dellow et al. , Phys. Rev. Lett. 68, 1754 (1992). [7] R. C. Ashoori et al. , Phys. Rev. Lett. 68, 3088 (1992). 2995