Electron-electron scattering between parallel two-dimensional electron gases

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Electron-electron electron gases

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Electron-electron measured

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1. Introduction The importance of electron-electron interactions in two-dimensional electron systems is strikingly demonstrated by the existence of the fractional quantum Hall effect. Direct measurement of the electron-electron (e-e) scattering rate, however, remains challenging. In the diffusive regime, weak l~)calization analysis of magneto-resistance measurements has been employed to determine the electron dephasing time. This time is dominated by electron-electron scattering in some cases. Very recently, interference effects [I] in the quasi-ballistic regime have also been used to assess this dephasing time. Our approach, by contrast, is to directly measure the momentum relaxation rate between closely spaced parallel two-dimensional electron systems. We describe. in this work, quantitative measurements of electron-electron interactions in such a double layer system. In these experiments, access to the interlayer c-e interaction is provided by the momentum transferred from a current carrying two-dimensional electron gas (2DEG) to a nearby, parallel 2DEG. The interlayer spacing is such that tunncl-

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ing between the two electron layers is negligible: momentum is transferred by c-c scattering. The contribution of the Coulomb intcractioI1 to such ;I process has been considcrcd theoretically [3.3] in the past. Recent measurements [4] of the coupling between a 2DEG and a nearby th~c~r-dimensional electron system have been interpreted in terms of Coulomb scattering. The cxpcrimcnts discussed in this paper probe the intcr~~ct~~)I~s between two strictly two-dimensional electron systems, Our technique allows a direct dctermination of the interlayer electron scattering rate. In a previous paper [S], we focused on the contribution of Coulomb scattering to the coupling. In this work, we further explore the dependence 01 the coupling on the spacing between the two electron layers. As discussed earlier [S]. WC find that for closely spaced 2DEG layers, the magnitude and temperature dependence of the observed scattering rates arc in reasonable agrccment with CalcuI~ltions of interlaycr Coulomb scattering. For larger layer separations, howcvcr. there is evidence for an additional interaction mechanism. Both the spacing and temperature dependence of the scattering rates in this regime point to the existence of a second. long-ranpc

T.J. Gramila et al. / Electron-electron

scattering between parallel 2DEGs

441

interaction. This suggests that phonons may also contribute to the interlayer etectron-electron interaction

2. Experiment The samples which are used for this investigation are modulation-doped GaAs/AlGaAs double quantum well (DQW) structures grown by molec$ar beam epitaxy. Two GaAs layers, each 200 A thick, are embedded in the alloy Al,,Ga,,As. Si delta-doping layers are placed above and below the quantum we&s; the barrier separating the layers is undoped. These dopants create nearly equivalent [6] 2DEGs in the lowest electric subband in each well. A simplified band diagram for this geometry is shown in the inset in fig. 2. Samples with three different barrier thicknesses have been studied: 175, 225, and 500 A. These samples are otherwise nearly identical; in all samples, each 2DEG has a density and a sheet mobility 161 near 1.5 x 10” cm-’ and greater than 2 x 20h cm-‘/V. s, respectively. Standard photo-lithographic techniques are used to pattern a bar-shaped mesa on the sample front surface. Details of the sample geometry have been described elsewhere [.5]. A representation of the measurement scheme is given in fig. 1. A constant current (typically 200 nA) is driven through the 2DEG in one of the quantum wells (the drive). The drift of those electrons transfers momentum via interwell scattering to the 2DEG in the second (output) well. This “drag” on the electrons is balanced by the development of a voltage in the output well, which we measure with a high-impedance detection circuit. Two key elements are essential for making this measurement. First are separate ohmic contacts to the individual 2DEGs in the DQW. These are obtained using a selective depletion technique [7]. Second is the accurate detection of the small drag voltage, which requires that the measured signal, typicaily nanovolts, be free from undesired effects. This is accomplished with a transformer coupled detection circuit [S], enabling both the

Fig. 1. The measurement scheme used. A 25 Hz current is driven through one LZDEG,and the voltage that develops in the second 2DEG because of interlayer interactions is measured with a high-impedance detection circuit,

reduction and in situ detection of possible spurious signals. Verification that leakage and tunneling between the two 2DEGs is negligible is thus provided during the measurement.

3. Results Th,e measured drag signal for a sample with a 175 A thick barrier is shown in fig. 2 as an equivalent resistance (drag-voltage/ drive-current) versus temperature, scaled by the aspect ratio of the mesa in which the two 2DEGs interact. The signal is identical, within experimenta uncertainty, upon interchanging the roles of the

loo I

0

60

2 4 Temperature

6 (K)

Fig. 2. Temperature dependence of observed coupling between two 2DEG systems separated by a 17.5A barrier. Data is plotted as an equivalent resistance and a momentum transfer rate. The dotted line is the calculated rate for Coulomb scattering (see text). Inset is a simplified band diagram for a DQW structure indicating both the ground subband energy and the Fermi energy.

current carrying and output wells, and after thcrmally cycling the sample. Low excitation currents were used throughout the measurements to avoid non-equilibrium and heating effects [5]. In all cases, the measured signal was linear in the applied current. It is important to note that the sign of the measured drag voltage is consistent with momentum transfer between the two 2DEGs; this sign is opposite to the resistive voltage drop in the drive well. The voltage results from the accumulation of charge swept along in the direction of the drift velocity in the drive well. Because no current is allowed to flow in the output well, the electric field that develops precisely balances the drag due to the interlayer interactions. We have characterized these interactions [5] by means of an interwell scattering time, T”, by analogy with standard Drude transport. The drag force is simply the rate of momentum transfer from the drive well, which can be written as m * z’JrD, where l“, is the drift velocity in the drive well. The force balance then determines the relationship of the drag voltage Vp to the drive current I as simply V&I

= -( L/W)m*/Ne’r,,

(1)

where N is the electron density in the drive well, L and W are the length and width of the region in which the drag occurs. Our measurements. then, determine the scattering rate r;‘. This is shown in the right vertical axis of fig. 2. The overall temperature dependence of the scattering rate shown in fig. 2 is close to quadratic. This temperature dependence might be expected for direct electron-electron scattering; each electron system contributes one power of T from the thermal broadening of its Fermi surface. We have shown [5] that the magnitude and temperature dependence observed in this data are largely consistent with calculations based on simple Coulomb scattering. Comparison of these data with theory can be seen in fig. 2. The dotted line is the result of numerical calculations done in the low-temperature limit. The theoretical temperature dependence is quadratic, and agrees with the overall behavior of the measured rates. The small deviations from quadratic behavior, most prominent at intermediate temperatures, are discussed in de-

tail below. The amplitude of the calculated scattering rate is determined using known sample parameters and includes the effects of finite well widths, as well as vertex corrections to RPA screening [S]. This amplitude is expected to be somewhat uncertain. nevertheless, the agreement between theory and experiment evident in fig. 2 strongly supports the identification of Coulomb scattering as the predominant coupling mcchanism. There are several important aspects of interlayer Coulomb scattering. For the barriers and well widths used in this experiment, the process is dominated by small angle scattering. The finite spacing between the electron layers limits the magnitude of the bare Coulomb interaction, and thus limits large angle scattering. This is evident in the Fourier transformed interlayer Coulomb interaction, which includes a factor e-mqf’, strongly suppressing scattering events with wavevectors cl greater than &‘. For the 175 A barrier saomple, with a well center-to-center spacing of 3750A and an inverse Fermi wavevector of - 100 A, this factor limits the scattering angle to less than approximately 15 O. In addition, screening in the double layer system is substantially enhanced as compared to a single layer. In the random-phase approximation the double-layer screening wavevector [5,9] q,, = 29&l + yrrd). For large separation the term proportional to rl dominates q,,. In our samples q,, = SOL-/,, so the enhanced screening reduces r,>’ by roughly two orders of magnitudes. Calculations using Boltzmann transport theory [5] result in a scattering rate that varies as T’. Logarithmic corrections to this behavior are absent for the experimental parameters appropriate to our measurements. Such corrections [ 101 result from a divergence in the density of phase space available for scattering near 0 and 180 O. Because no momentum is transferred by forward scattering, the divergence at 0” is canceled in the scattering rate. Backscattering processes could potentially contribute to logarithmic corrections. however, as discussed above, these scatterings arc strongly suppressed by the finite spacing between the layers, and in these samples can become important only at extremely low temperatures.

T.J. Gramila et al. / Electron-electron 2.5

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00 0

4 2 Temperature

6 (K)

Fig. 3. Temperature dependence of the interlayer scattering rate divideed by T* for the three barrier thicknesses, 175, 225, and 500 A. All the rates show nearly identical deviations from a quadratic temperature dependence.

Both the dominance of small-angle scattering and the enhanced screening contribute to the spacing dependence of rD1. For sufficiently large spacing, and neglecting the finite well widths, the low temperature scattering rate varies as dp4 [5]. This spacing dependence is characteristic of Coulomb scattering between parallel 2DEGs, and provides an additional test for the coupling mechanism. We have measured the0 scattering rate for a sample with a larger 225 A barrier. For this sample, the well center-to-centez distance is only 12% greater than for the 175 A barrier sample discussed earlier, however, the dm4 dependence of the Coulomb coupling model will effect a 40% reduction in t,he scattering rate. Data for both the 175 and 225 A barrier samples are shown in fig. 3 as 7;‘/T2. The reduction in the scattering rate observed for the larger barrier sample is in good agreement with the dp4 dependence. Detailed calculation appropriate to these samples, including the effects of finite well widths as well as vertex corrections to RPA screening, predicts a similar reduction. The observed spacing dependence, as well as the magnitude and overall temperature dependence of the interwell scattering rate, all support the identification of Coulomb coupling as the dominant interlayer momentum transfer mechanism. As is evident in fig. 3, however, there are significant deviations from a strictly quadratic

scattering between parallel 2DEGs

449

temperature dependence. While these deviations, for the 175 A barrier sample, represent a variation of only 20 to 30% from an overall temperature dependence in which the scattering rate varies by two orders of magnitude, they find no adequate explanation in the Coulomb model. These deviations could result from inadequacies in this model, but they may also point to the existence of an additional coupling mechanism. Examination of the data for the 175 and 225 A barrier samples lend support to the second proposal. Both sets of data display strikingly similar deviations from a quadratic temperature dependence. This similarity could easily be explained by the presence of two coupling mechanisms: Coulomb scattering, which is proportional to T2 and has a strong spacing dependence, plus a second mechanism with the temperature dependence of the deviations, but a weak dependence on the layer seperation. To test for such a mechanism& the scattering rates of a sample with a 500 A barrier were measured. Those measurements are shown in the lowest trace in fig. 3. For this layer separation, the dp4 behavior of the Coulomb process would result in a contribution to the scattering that is less than 10% of that for the 175 A barrier sample. There remains substantial momentum transfer, however, and the shape of the data is dramatically similar to that of the other samples. This curve is striking evidence for the presence of an additional interaction. The long-range nature of this interaction suggests that phonons may be involved in the momentum transfer process. At these temperatures phonon mean free paths are extremely long compared to the interlayer spacings. Furthermore, phonons may afford a temperature dependence similar to the deviations from T2 observed in the data. The inverse of the phonon limited mobility, F;~‘, which is proportional to the rate of momentum transfer to the phonon bath, varies linearly with the temperature at high temperatures. At lower temperatures, however, phonons with wavevector of order 2k, are no longer thermally excited, and phase space restriction of the BlochGriineisen regime alter the temperature dependence to T” or T’ (for piezoelectric and defor-

mation potential electron-phonon coupling, rcspectively). This crossover occurs at a temperature of order of a few kelvin for the electron density of our samples. Thus, the rate at which momentum is delivered to the phonon bath has a temperature dependence very similar to the deviations from quadratic behavior measured for the scattering rates. The magnitude of the interwell momentum transfer rate estimated for a phonon emission and absorption process is, however, substantially smaller than necessary to account for the temperature dependence observed. From calculations [Ill of pph appropriate to our samples, we find that the fraction of excess phonon momentum generated which must be absorbed by the second 2DEG to account for the observed deviations from a T’ behavior is of order 0.5 to 1%. Estimates of thermal phonon absorption probabilities for the 2DEG in similar structures is in the lo-’ range [12], fully two orders of magnitude too small. Identification of the this second coupling mechanism thus remains uncertain. It remains possible that the absorption probabilities for phonons emitted by the drive well are substantially enhanced as compared to thermal phonons, or that the coupling is a virtual phonon process. These questions are currently being investigated.

however. implies the existence of a second. longer-range process. This process may involve phonons, but difficulties concerning the magnitude of the rate for such a mechanism remain. This technique allows the study of e-e interactions under a wide range of experimental conditions. Measurements of the scattering as a function of electron density, for example. may help in determining the nature of the long-range COLIpling mechanism. Many other aspects of the c-c interaction could also be studied. The effects of a magnetic field are one such area that can be probed using this technique. One particularly interesting direction would be to extend our measuremcnts to the dirty limit. The technique stems an ideal tool for studying the modification of c-c interactions in the strongly diffusive regime. Futurc experiments have been planned to explore these possibilities.

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