G. Blanke, A. Lorke, and J. P. Kotthaus
Sektion Physik, Universität München, Geschwister-Scholl-Pl. 1, 80539 München, Germany.
J. H. English, A. C. Gossard, and P. M. Petroff
QUEST, University of California, Santa Barbara, CA 93106, USA.
Using different techniques to individually contact two closely spaced electron gases, we study the tunneling characteristics between weakly coupled GaAs quantum wells, with and without resonant far-infrared excitation. We find that for barriers as thick as 300 Å, the alignment between the subbands in the wells can be observed as an increased tunneling conductivity. To study photon-assisted tunneling in our samples, we use the cyclotron resonance as a strong, tunable electronic excitation in the far-infrared. When the Landau-level spacing h[[omega]]c corresponds to the laser energy h[[omega]]L, carriers are effectively pumped to higher Landau levels with an increased tunneling probability, and hence a reduced resistance across the tunnel barrier is observed. This photo-conductive signal is "doubly resonant" in that it is at maximum when [[omega]]c coincides with [[omega]]L and at the same time the subbands of the two wells are aligned.
Coupled double quantum wells (CDQWs) are standard text book examples for tunneling between bound states . Furthermore, they can be viewed as the constituting building blocks of semiconductor superlattices and as a first step in the transition from purely two-dimensional to three-dimensional transport in layered semiconductor structures. The interplay between quantum mechanical and electrostatic interactions in narrowly spaced, two-dimensional electron gases (2DEGs) leads to intriguing transport properties in the quantum Hall and fractional quantum Hall regime , where the coupling reveals itself in the in-plane transport properties. The vertical transport (between the wells) is dominated by resonant tunneling effects [3-5]. Compared to three-dimensional devices, resonant tunneling between two-dimensional electron gases is expected to have a sharper bias dependence, because of the reduced density of stateses. This, together with the possibility to realize mobility-modulated transistor structures [6,7], makes CDQWs also interesting from a technological point of view.
In addition to the dc transport properties, the dynamic behavior of electrons in CDQWs has recently been studied in far-infrared transmission [8,9] and time-resolved inter-band excitation experiments [10,11]. A combination of infrared excitation and time-resolved luminescence probing was used to observe electron transfer between asymmetric CDQWs .
Here, we report on the tunneling of electrons between double wells with and without resonant far-infrared (FIR) excitation. In order to study the vertical transport properties between CDQWs, it is necessary to make separate contacts to each well. Recently, a number of selective contacting schemes have been proposed and realized [3,4,13]. In the present study, we employ two different techniques: Masked growth  and selective pinch-off, using front and back gates .
The samples are grown by molecular-beam epitaxy. The active layers consist of two symmetric, modulation-doped GaAs quantum wells, separated by a 300 Å AlAs/GaAs short-period superlattice (sample A) and by 100 Å Al0.3Ga0.7As (sample B), respectively. These are relatively thick barriers compared to those commonly used in resonant tunneling devices. However, it has been shown that barriers of a few hundred Å still allow for the observation of resonant tunneling . The width of the wells are 150 Å in sample A and 82 Å in sample B. Details of the masked growth procedure and the layer sequence of sample A can be found in Ref. ; a schematic of the contact geometry is shown in Fig. 1 (a). Sample B is thinned down to 25 um and front and back gates are used to make separate contacts to the wells , as schematically shown in Fig. 1(b). A large area semi-transparent front gate ("tunneling gate" ) allows us to tune the carrier density in the upper well and bring the subband energies of both wells into resonance. Despite the large tunneling gate areas necessary for FIR studies (up to 25 mm2), leakage currents below a few 10-9 A are observed for a typical bias of 5 mV.
The samples are mounted in a liquid He cryostat in the center of a superconducting solenoid, with the direction of the magnetic field normal to the plane of the CDQWs. Far-infrared radiation from an optically pumped molecular laser is coupled into the cryostat through an oversized waveguide. The transmitted intensity is recorded using a custom made bolometer to determine the magnetic field BCR at which cyclotron resonance occurs.
Fig. 1. Schematic representation of the different techniques used to separately contact the narrowly spaced two-dimensional electron gases. In (a), the electron gases are shifted with respect to each other by directional growth through shadow masks . (b) illustrates the use of front and back gates to deplete the upper and lower electron gas, respectively.
Fig. 2. Tunneling conductance G as a function of tunneling gate voltage VTG for samples A and B. Both samples display multiple peaks which indicate small carrier density inhomogeneities. The dotted line in (a) gives the tunneling conductance for a different cool-down cycle. At zero bias, sample B has a much higher carrier density in the upper than in the lower well, so that large negative gate biases are necessary to achieve alignment of the subbands. The carrier densities at resonance are 5.7x1011 cm-2 for sample A and 1.7x1011 cm-2 for sample B.
Figure 2 (a) shows the vertical conductance of sample A as a function of the tunneling gate voltage. Because of the thick barrier in this sample, currents of only 2.5 nA are used in order to keep the bias between the wells low enough to not affect the subband structure or the carrier densities. A number of peaks are observed, indicating increased tunneling caused by the alignment of the energy states in the wells. Shubnikov-de Haas and capacitance measurements show that the lowest subbands in the wells should be aligned at moderate positive gate voltages, in agreement with the increased tunneling around VTG = +0.25 V. Because of leakage currents at high positive gate biases, the tunneling current could not reliably be measured beyond VTG = +0.3 V. Additional peaks, e.g. at -0.1 V, can be seen in Fig. 2. These are not very reproducible for different cool-down cycles (see dotted line in Fig. 2 (a)), and we attribute them to inhomogeneities in the carrier density ns which lead to small areas of the sample where the alignment of the subbands takes place at lower tunneling gate voltages (the subband spacing in this sample is approximately 40 meV, so that tunneling between different subbands  can be ruled out).
Similar tunneling characteristics are observed for sample B. Fig. 2 (b) shows that here, again, inhomogeneities of the carrier density give rise to additional peaks in the tunneling current. From the double peak in Fig. 2 (b), we estimate a [[Delta]]ns of ~ 14 %. The presence of distinct domains with slightly different carrier densities in molecular-beam epitaxially grown heterostructures has also been reported by, e.g., Wixforth et al. .
To study photon-assisted tunneling in sample A, we use the 118.8 um laser line, which is the strongest line available to us. From transmission experiments we determine the magnetic field at cyclotron resonance BCR = 6.5 T. The application of a high vertical magnetic field has only a small effect on the tunneling behavior and the peaks shift only marginally.
When the sample is illuminated by resonant FIR radiation, the conductance increases by approximately 2 %. This change can clearly be observed in the dc conductance. To obtain a better signal-to-noise ratio, however, the FIR radiation is chopped with ~ 30 Hz and the change in conductance is recorded using standard lock-in amplification.
Fig. 3 displays the change in conductance with laser excitation as a function of the tunnel-gate bias VTG. The magnetic field is BCR = 6.5 T. Two maxima are observed at ~ -0.1 V and ~ +0.25 V, the same positions where increased tunneling can be seen in Fig. 2 (a). We attribute these peaks to photon-assisted tunneling, as schematically depicted in the inset in Fig. 3. Electrons are pumped by the resonant FIR radiation into higher energy levels, where the tunneling probability is higher than in the ground level. This leads to the increased conductance observed in Fig. 3.
Fig. 3. Relative change in tunneling conductance with resonant far-infrared excitation for sample A. Superimposed upon a background signal of approximately 2 % are two peaks at the same positions as the maxima in Fig. 2 (a).
The tunneling through a rectangular barrier is a standard text book quantum mechanical problem. In the most simple treatment of an incoming plane wave with an energy E, the transmission probability for a thick barrier of height Vo >> E is given by 
where a is the thickness of the barrier and m the mass of the particle. The relative change of the transmission probability T for a small change in energy, [[Delta]]E, then follows to
Here, we assume [[Delta]] << E, Vo - E . For the parameters of the present sample, Eq. (2) gives an increase of the transmission probability of approximately 100 %, even though the change in energy, [[Delta]] = h[[omega]]Laser ~ 10 meV, is much smaller than the estimated confinement energy Vo - E ~ 400 meV. Note that [[Delta]]T/T increases exponentially with increasing barrier thickness a. We therefore expect [[Delta]]T/T to be much smaller in sample B than in sample A, which might explain why we have so far not been able to clearly observe photon-assisted tunneling in sample B. Equation (2) can only give a very crude estimation for photon-assisted tunneling, especially for cyclotron resonance excitation. It illustrates, however, that even for small excitation energies, strong changes in the transmission probability can be obtained. A more quantitative comparison between experiment and theory requires a detailed knowledge of the excitation and relaxation mechanisms, which is beyond the scope of this study.
An important question to address is whether the FIR induced conductance change can be attributed to heating effects. It is known that a 2DEG can effectively be heated by cyclotron resonance excitation [16,17], and it is likely that an increased electron or lattice temperature will result in an increased leakage or tunneling current. Photo-voltage measurements have shown that, for a similar experimental setup, the maximum temperature change of the electron gas is around T = 10 K . This corresponds to a thermal energy of kT = 1 meV, one order of magnitude smaller than the laser energy, and thus a much smaller change in transmission is expected from Eq. (2). Furthermore, heating effects are strongly dependent on the filling factor n, and the temperature increase of 10 K is only observed when the magnetic field at cyclotron resonance corresponds to an integer n. This is not the case in the present sample, and we therefore estimate the real temperature change to be much smaller than 10 K. When the magnetic field is adjusted so that the filling factor of the lower well is an integer, we also observe a strong photo-signal, even 0.5 T away from BCR. This signal, which might be caused by heating effects, however, does not show the resonant behavior observed in Fig. 3.[+]
Figure 3 shows that the transmission signal is largely independent of the tunneling gate bias (the slight increase is caused by the increase of the carrier density with increasing VTG). We can therefore rule out that the peaks in Fig. 3 are related to a possible change in absorption caused by the alignment of the subbands.
In conclusion, we study the tunneling behavior between narrowly spaced, two-dimensional electron gases in GaAs coupled double quantum wells. Resonant tunneling reveals itself in an increased vertical conductance when the subbands of both wells are aligned. Cyclotron resonance excitation of the carriers in the wells leads to a further increase in conductance which we attribute to photon-assisted tunneling through higher Landau levels. Maximum photo-response is observed when the subbands are aligned and the excitation energy matches the Landau level spacing.
Acknowledgment - We gratefully acknowledge financial support by QUEST, a NSF Science and Technology Center, under Grant No. DMR 20007 and would like to thank M. Hartung for valuable help with the sample preparation.
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