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In this contribution I will outline our recent measurements** of the tunneling conductanc e between high mobility 2DEG's in GaAs/AlGaAs double quantum well (DQW) structures . This discussion will include a brief description of the newly developed technique for producing independent , low-resistanc e contacts to the individual 2D layers, that is essentia l for direct tunneling studies. The tunneling results themselve s include the use of Schottky gates to locally induce resonan t tunneling, the observation of narrow resonance s attesting to a high degree of momentum conservation , and the use of 2D-2D RT as a novel Fermi surface mapping tool. Finally, brief mention will be made of interferometric configurations which may be realizable in DQW structures . 2,3

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Ð. SAMPLES AND CONTACTS Al l data to be presente d were obtained using modulation-dope d double quantum wells grown by molecular beam epitaxy. Most of the tunneling data were obtained with a sample consisting of two 140 A GaAs wells separate d by a 70 A pure AlA s barrier. Above and below the DQW are A l G a . 7 As layers in which Si ä-doping layers are placed. These dopants are disposed asymmetrically, one layer 700A above the DQW and one 900A below. This o

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Nanostructures and Mesoscopic Systems

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Copyright © 1992 by Academic Press, Inc. All right s of reproduction in any form reserved. ISBN 0-12-^09660-3

Figur e 1. Conductanceof DQW samplevs. top gate bias V depictssampleconfiguration. After Eisenstein et a/.

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asymmetry accommodate s for the natural tendency of the Si to diffuse upward during growth, and results in nearly equal 2D densities in the two quantum wells (nominally 1.5xlO c n T ) . Each 2DEG has a low temperatur e mobility around 8 x l 0 c m / V s. To determine these parameter s standard magneto-transpor t measurement s are performed on the individual layers. ( 5)

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Central to this work is a new method for making reliable contacts to the individual layers in the DQW. Each contact consists of a standard diffused In dot (which contacts both 2DEGs) and a pair of Schottky gates "surrounding" it, one on the sample top surface and one on its back side. (The sample is first thinned to ~50ìç é by etching the substrate. ) These gates are positioned so that any current ( 4)

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flowing in or out of the In dot must pass the gates. By applying an appropriate negative bias to one of the two gates we can fully deplete the closer 2DEG without significantly affecting the remote 2DEG. In this way a given In contact "sees" the rest of the sample through either one or the other 2DEG, but not both. These contacts are low resistanc e (typically 100Ù), are well isolated from the undesired 2DEG (by more than 50ÌÙ) , and can be switched between layers during an experiment. Figure 1 illustrates the sequentia l depletion technique , albeit with only top gates. The data represen t the two-terminal conductanc e (using a 50ìm-wide bar-shape d mesa) of a DQW sample with a 175 A AlGaAs barrier, qualitatively identical results are obtained with the thinner barrier tunnel sample. The conductance , measure d at T=1.5K using small amplitude ac excitation, is plotted as a function of the gate bias applied to the top gate associate d with one of the contacts. The two traces correspon d to different fixed biases applied to the top gate associate d with the other contact. In the upper trace, this second gate is unbiased and the conductanc e exhibits the two-step structure expected of a DQW carrying current in both its channels . The clear step near -0.9V signals the depletion of the upper well by the "analyzer" gate, and the subsequen t drop to zero conductanc e around -2.2V implies the lower well is being cut. (The unequal height of the steps is an artifact of the two-terminal measurement. ) In the lower trace the second top gate has been biased to deplete the upper well. Sweeping the analyzer gate reveals no current is flowing in the upper well, as expected . These data not only demonstrat e the validity of the sequentia l depletion technique, but the absenc e of any step around - 0 . 9V in the lower trace proves there are no unintentional "shorts" between the two layers. Analogous results are obtained using back-side gates to deplete the lower 2DEG first, only the required voltages are typically lOOx larger . (4)

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EQ. GATE-INDUCED RESONANT TUNNELING In the inset to Fig. 2 a longitudinal cross-sectio n through a bar-shape d mesa is shown. Each of the two In dots has a top and bottom gate associate d with it. For clarity, only one member of each gate pair is shown, the top gate for In dot 1 and the bottom gate for dot 2. These gates can be used to render dot 1 a bottom 2DEG contact and dot 2 a top 2DEG contact. So doing leaves the two contacts isolated except for tunneling through the barrier separatin g the quantum wells. As the figure indicates, two other gates are deposited across the central region of the mesa, one on top the sample and a larger one on the back-side. This pair of gates is usedto study the tunneling in the sample's central region. The two traces in Fig. 2 represen t the conductanc e G vs. V , the bias voltage applied to the central top gate, referred to as the tunnel gate. In the upper trace the gates used to produce separat e layer contacts are unbiased (as is the large back gate). This is the standard parallel configuration obtained when separat e contacts t

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Figure 2. DQW conductancevs. tunnel gate bias V . Upper trace taken with unbiased contact gates, lower trace with contact gates biased to produce tunneling configurationshownin inset. After Eisenstein,et a/. t

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are not available. The two-step gate characteristi c shows that both channels are conducting at V = 0. In the lower trace however, the two contact gates are biased to deplete 2DEG neares t to them, producing the "tunnel configuration" mentioned above. Again the conductanc e exhibits a two-step dependenc e on V plus a small peak near V = 0.1V. This peak is the primary subject of these experiments . t

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An important aspect of these measurement s is that the conductanc e is measure d at low temperature s (~1K) using small amplitude ac excitation (typically =0.1mV rms @ 27Hz); further reduction of the temperatur e or excitation amplitude has no observable consequences . In the tunnel configuration the excitation voltage creates a chemical potential difference Äì=âí â÷ between the two 2DEGs, thus driving the tunnel current. This Äì is much less than the typical Fermi energy of the 2DEGs (E ~5meV) and hence our data represent s equilibrium tunneling of electrons at the Fermi level. F

At zero magnetic field, conservatio n of energy and in-plane momentum allows 2D-2D tunneling only when the subband edges in the two quantum wells are closely aligned. In fact, the minimum energy width of 2D-2D tunneling resonance s is set by the symmetric/anti-symmetri c gap A at flat band. For our S A S

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samples, this energy is less than O.OlmeV justifying a weak tunneling approximation. As we shall see, the observed RT widths greatly exceed A owing to the presenc e of scattering. S A S

In order to scan the subband edges past one another we use the tunnel gate to locally change the density of the upper 2DEG. Since the tunneling is quasiequilibrium, changing the density of one well relative to the other must lead to band-bendin g and therefore to changes in the subband alignment. For negligible chemical potential difference Äì the difference between the two wells ground subband energies Ä Å is given by Ä Å = A N / D , with ÄÍ the density difference and D the 2D density of states. Thus, for equilibrium tunneling, matched subband energies imply matched densities. 0

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The following picture emerges for the data contained in the lower trace of Fig. 2. Since the sample, as grown, has nearly equal densities in the two wells, there is significant tunneling throughout the central part of the mesa. As expected , biasing the tunnel gate to deplete the upper quantum well ( V = - 0 . 9 V) roughly halves the background tunnel conductance . The small bump at V =0.1V, however, represent s the resonanc e in the tunneling under the tunnel gate expected at matched densities. To justify this assertion , we use the large back-side gate to alter the lower well density. This will alter the "background" tunneling conductanc e (by changing ÄÍ over a wide area) and should shift the position of the tunnel gate-induce d RT peak to a different bias voltage V . Figure 3a displays the tunnel gate bias position V of the RT peak vs. the voltage V applied to the large back gate. Since the gates act essentially as simple capacitors , one expects the linear dependenc e seen. The slope should be close to the ratio of the distance of the top and bottom surfaces of the sample to the quantum wells, about 1/120. The solid line indicates this slope. From this we can conclude that the RT peak is shifting in a way consisten t with the equal density criterion. As described earlier however, the data can also be used to estimate the actual densities of the two wells at resonance . This is done by utilizing the two depletion steps in conjunction with the peak position itself. Figure 3b shows the estimated ratio of the bottom to top well density n / n vs. the bottom well density n . The closenes s of these ratios to unity supports the general picture of the tunneling resonanc e outlined here. t

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The relative size of the RT peak in Fig. 2, compared to the background tunneling conductance , is determined in part by the ratio of the area under the tunnel gate to the total mesa area over which tunneling occurs, but also on the asgrown densities in the quantum wells. As already mentioned, this DQW is nearly balanced to begin with and so there is substantial tunneling throughout the structure. The relative strength of the gated RT peak can be enhance d either by increasing the gated-to-ungate d area ratio, or by suppressin g the background tunneling. The latter can be done using back-side gating or by designing an intentionally unbalance d sample. Figure 4 shows the result of the former

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Figure 3· Upperpanel: Top gate bias position of tunnel gatepeak vs. applied back gate bias. Lower panel: Estimateddensityratio at peak of tunnel resonance vs. lower well density. procedure using the same sample as in Fig. 2; even higher peak-to-valley ratios are sometimes obtained. Intentionally unbalance d samples should readily produce high contrast ratios for the RT peak; this may have a beneficial impact on possible device applicationsfor 2D-2D RT. IV . RESONANCE WIDTHS As stated in the introduction, the width of the tunneling resonanc e should be a measure of the scattering present during tunneling. To begin, we must convert the observed resonanc e width in terms of gate voltage to an actual electronic energy. This is easily done, again by using the depletion steps. The gate voltage interval between the RT peak and the upper well depletion edge is proportional to the well density at resonance . Since the density determines the Fermi energy ( E = N / D ) the required conversion factor can be obtained. We find the typical F

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Figure 4. OptimizedRTpeak obtainedby using back gates to suppressunwanted tunneling. AfterEisenstein, et al. {2)

RT full widths at half-maximum to be about r=0.5meV, only 10% the typical Fermi energy. Unlike earlier 2D-2D tunneling studies the barrier layer in our samples is undoped, thereby minimizing ionized impurity scattering. While these resonance s are narrow, they still greatly exceed the "natural" linewidth A <0.01meV. ( 1)

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What determines the linewidth Ã? The most obvious candidate is the scattering mechanis m that limits the mobility in the 2DEGs themselves . If we calculate the mobility scattering time ô and from the uncertainty relation calculate an energy width Ã = Ë/ô , we find Ã ~.02ðéâí, well less than the observed r=0.5meV. This, however, is not surprising since ô is the large angle scattering time. As the observed linewidth is only 10% the Fermi energy, the typical scattering wavevector is about 5% of the Fermi wavevector, k . This implies a small scattering angle. Furthermore , it is well known that in modulation-dope d heterostructure s the actual scattering time is often ten or more times less than ô owing to a great predominanc e of small angle scattering events. Zheng and MacDonald have suggeste d that the RT peak width presents a direct measure of the actual or quantum lifetime x . From approximate fits to our tunneling resonanc e curves, they conclude Ë / ô ~ Å / 15 and that this is ì

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reasonabl e assuming that the remote ionized donors dominate the scattering. Although not yet attempted , it would be interesting to estimate ô by the standard analysis of Shubnikov-deHaa s oscillations and compare that to the RT peak width. Q

Finally, we note the possibility of other broadening mechanisms , in particular those associate d with inhomogeneities . For example, fluctuations of the well widths owing to interfacial steps is also likely to create broadening . A simpleminded estimate assuming a single monolayer step leads to fluctuations in the bound state energy of around lmeV. This is probably an overestimate , as it ignores, for example, screening and the likely correlation of steps on lower interfaces with those on upper ones. V. MAPPING THE FERMI SURFACE WITH RESONANT TUNNELING Application of an in-plane magnetic field leads to a novel technique for mapping the 2D Fermi surface . As will become evident, the sensitivity of this technique relies on the high degree of momentum conservatio n on tunneling present in our samples . A number of earlier tunneling studies* ~ , including the purely 2D-2D work of Smoliner and co-workers , have examined the effect of in-plane magnetic fields. None, however, have discusse d the simple geometric picture of intersecting Fermi surfaces that explains the present 2D-2D equilibrium tunneling results . (3)

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For a single 2D layer, centered at the z=0 xy-plane, an in-plane magnetic field, to lowest order, simply adds a diamagnetic term to the energy of all electrons. The gauge can be chosen (e.g. A = - B z) so that the kinetic energy dispersion remains ft k /2m; i.e. a paraboloid centered at k=0. If a second 2D layer, centered at z=d, is present however, its kinetic energy dispersion becomes a paraboloid displaced along the x-axis by an amount Ak =eBd/fc. If tunneling between the two layers is to conserve in-plane canonical momentum, then this relative displacemen t of the kinetic energy paraboloids has fundamenta l importance. In particular, if the tunneling is equilibrium, i.e. at the Fermi level, then the conservatio n of energy and momentum implies that the available phase space is restricted to only those points where the initial and final Fermi surfaces intersect It is also obvious that RT can now occur without exact subband alignment, provided the field is large enough to create an intersection of the two Fermi surfaces (now of different area). x

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In Fig. 5 the total tunnel conductanc e (the entire region between the contact gates contributing) is plotted against applied in-plane magnetic field. As the field is first applied the conductanc e drops rapidly. It then levels out for a wide field range. By B=6T it rises to a small peak just before dropping to zero. The tunneling conductanc e remains quenche d to beyond 10T. This is completely consistent with the phase space restrictions just discussed . For this sample the two quantum wells are nearly equally populated ( N ~ 1 . 5 x l O c m " and so there n

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Figure 5. Upperpanel: Total measuredtunnel conductancevs. in-plane magnetic field. Lower panel: Calculated tunnel conductance. Insets depict various arrangementsof Fermi surfaces. is substantia l tunneling at B=0 since the Fermi circles are of equal diameter. As soon as the field is applied the circles displace and the phase space is reduced to two points; the conductanc e drops substantially. For a wide range of fields the conductanc e varies slowly as the intersection points move around the circles. Around 6T the circles are about to separat e altogether, the small bump reflecting the enhance d density of states expected when circles osculate. Beyond about 7T the circles are separate d by more than the scattering present can overcome and the tunneling shuts off. In the lower part of the figure a simple calculation, including only phase space limitations and a phenomenologica l broadening mechanis m (adjusted to match the RT linewidths observed at B=0), is presented . The auench 203

Figure 6. Twopossibleinterferometerconfigurations. field B is determined only by the DQW center-to-cente r spacing and the measure d 2D densities; the agreemen t with experiment is excellent. c

Although the Fermi surface for 2D electrons in GaAs should be circular, it is clear that by rotating the magnetic field in the plane, any anisotropy should be directly observable (in the quench field, for example). Unlike standard Fermi surface probes, e.g. the Shubnikov-deHaa s or deHaas-va n Alphen effects, the present technique does not merely measure the area of the Fermi surface but rather its extremal points. An obviouscandidate system for exploiting this are 2D holes in GaAs, which are expected to posses s highly anisotropic dispersions* \ 11

VL RESONANT TUNNELING INTERFEROMETERS To conclude this contribution, I would like to briefly mention some speculative possibilities for constructing novel interferometer s based on double quantum wells and 2D-2D tunneling . Figure 6 illustrates two simple configurations. In the upper diagram we have a DQW for which we assume separate connections to the individual quantum wells are provided, either with our technique** or some other. The figure represent s a longitudinal cross-sectio n through a thin bar-shape d mesa. Two tunnel gates are deposited on the front surface. We make the distance d between the gates short compared to the dephasing length 1 in the 2DEG*s. The gate lengths themselve s must be short compared to d . We further assume that although the DQW barrier is thin enough to permit tunneling, the individual well densities are imbalanced and so very littl e RT occurs in the absenc e of gating. As already discussed , this unbalancing can either be grown in to the sample or produced externally, e.g. with backside gates. (12)

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Without biasing the two gates to create RT under them, there is very littl e conductanc e between the two contacts. By adjusting the gate biases though, we can turn on the tunneling under each. Thus a loop with two tunnel junctions is created. The area of the loop is the product of the gate separatio n d and the center-to-cente r distance d between the quantum wells. This double-slit device should exhibit Bohm-Aharonov oscillations if a magnetic field is applied in the 2D plane and perpendicula r to the loop. A variation on this theme is depicted in the lower part of Fig. 6 wherein a third gate has been deposited between the other two. This middle gate is not used to induce tunneling, but rather to simply alter the Fermi wavevector in the top 2DEG inside the loop. Interference fringes as a function of voltage should appear in the tunnel conductanc e of the device, without any magnetic field. g

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These interferometric devices highlight the questionof quantum coherenc e in tunneling. The results already outlined in this paper already offer some insight into this question. For example, we may now envision an unbalanced , but tunnelable, DQW structure in which electrons remain localized in one or the other well. Using gates to locally produce balance, these electrons can tunnel. Without any scattering however, an electron entering the balanced region will oscillate back and forth between the wells. Which well the electron remains in on exiting the gated region will depend on the length of the gate and the Fermi velocity. The frequency of these oscillations is set by the symmetric/anti-symmetri c gap, A . For the samples used in these experiments A A S is much less than Ã, the RT linewidth. Consequently , our electrons scatter much more rapidly than they can oscillate between wells. In that sense , these samples exhibit incoherent tunneling. S A S

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ACKNOWLEDGEMENTS It is a sincere pleasure to acknowledge my collaborators in this work, T J. Gramila, L J i. Pfeiffer, and K.W. West I am also grateful to A.H. MacDonald and H i . Stormer for useful discussions . REFERENCES 1. The first truly 2D-2D tunneling experiments were reported by J. Smoliner, E.Gornik, and G. Weimann, Appl. Phys.Lett. 52,2136 (1988). 2. J P. Eisenstein , L.N. Pfeiffer, and K.W. West, Appl. Phys. Lett. 58, 1497 (1991). 3. J P. Eisenstein , TJ. Gramila, L.N. Pfeiffer, and K.W. West, submitted for publication. 4. J P. Eisenstein , L.N. Pfeiffer, and K.W. West, Appl. Phys. Lett. 57, 2324 (1990). 5. L.N. Pfeiffer, E.F. Schubert, K. West, and C. Magee, to be published. 6. S. Das Sarma and Frank Stern, Phys.Rev.532,8442 (1985).

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7. 8. 9. 10. 11. 12.

Lian Zheng and A.H. MacDonald, to be published. B.T. Snell, K.S. Chan, F.W. Sheard, L. Eaves, G.A. Toombs, D.K. Maude, J.C. Portal, SJ. Bass, P. Claxton, G. Hill , and M.A. Pate, Phys. Rev. Lett. 59,2806(1987) . J. Smoliner, W. Demmerle, G. Berthold, E. Gornik, G. Weimann, and W. Schlapp, Phys.Rev.Lett. 63,2116 (1989). J. Lebens, R.H. Silsbee, and S.L. Wright, Phys.Rev.537,10308 (1988). See for example, D.A. Broido and LJ. Sham, Phys.Rev.B31,888 (1985). Related configurations have been discusse d in the past. See for example, S. Datta, M. Cahay, and M. McLennan, Phys.Rev.£36,5655 (1987).

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