Physica E 6 (2000) 694–697
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Experimental observation of Coulomb drag in parallel ballistic quantum wires P. Debraya; b; ∗ , P. Vasilopoulosc , O. Raichevd , R. Perrina , M. Rahmane , W.C. Mitchela a Air Force Research Laboratory (AFRL=MLPO), Dayton, OH, USA de Physique de l’Etat Condense, CEA de Saclay, F-91191 Gif-sur-Yvette Cedex, France c Concordia University, Department of Physics, Montreal, Canada d Institute of Semiconductor Physics, National Academy of Sciences, Kiev, Ukraine e University of Glasgow, Department of Physics, Glasgow, UK
b Service
Abstract We report the experimental observation of Coulomb drag in two parallel, ballistic quantum wires fabricated from a high-mobility 2DEG at the interface of a AlGaAs=GaAs heterostructure and de ned by three independent Schottky gates. The Coulomb drag resistance RD in wire 1 due to a drive current in wire 2 was measured at 1.3 K and in a magnetic eld of 1 T, applied to completely suppress tunneling between the wires when their separation was small. As the widths of the wires are changed by appropriately biasing the Schottky gates, RD shows a structure with peaks whose magnitudes decrease rapidly with the distance d between the centers of the wires. These results can be understood qualitatively in the framework of recent theoretical treatments, which predict peaks in RD when the level splitting between the 1D subbands of the wires and the Fermi momentum kF are small. The peak is strongest for the lowest subbands and weakens rapidly for higher ones. c 2000 Published by Elsevier Science B.V. All rights reserved.
PACS: 73.40.Gk; 73.23.-b; 73.23.Ad Keywords: Coulomb drag; Ballistic transport; Drag resistance; Quantum wires
1. Introduction Momentum transfer, or Coulomb drag, between spatially separated electron layers has been studied experimentally [1– 4] and theoretically [5 –14] mostly ∗ Correspondence address: Service de Physique de l’Etat Condense, CEA de Saclay, F-91191 Gif-sur-Yvette Cedex, France. Tel.: 33-1-6908-7449; fax: 33-1-6908-8786. E-mail address:
[email protected] (P. Debray)
between 2D layers. Theoretically, this drag has also been studied between very long quantum wires in which the wire length L is much longer than the electron mean free path li (di ussive regime) [15,16] and recently between quantum wires of length L.li (ballistic regime) in which the electron motion along the wire at low temperatures is ballistic [17,18]. It has been predicted that the drag response should attain its maximum value when the subbands in the two
c 2000 Published by Elsevier Science B.V. All rights reserved. 1386-9477/00/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 1 7 5 - 7
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wires line up precisely and the Fermi wave vector is small. Interlayer tunneling has also been studied extensively and produced many interesting results [19 –22]. Motivated by these results and the absence, to our knowledge, of any experimental results on the 1D Coulomb drag, we undertook to study it experimentally in the ballistic regime. For reasons mentioned below this study was carried out in the presence of a magnetic eld applied perpendicular to the 1D layer plane of the wires. 2. Experimental setup and measurements The coupled quantum-wire devices, shown schematically in Fig. 1, were fabricated from a highmobility 2DEG ( = 100 m2 =Vs; EF = 10 meV, electron mean free path of 9 m) at the interface (80 nm below the wafer surface) of a AlGaAs=GaAs heterostructure, and lithographically de ned by three independent surface Schottky gates T, M, and B in order to have two identical, parallel quantum wires of lithographic length L = 0:4 m. By appropriately voltage biasing the gates, it was possible to create two parallel quantum wires of variable width and separation. Measurements of the conductance of the individual wires at 1.3 K showed nearly quantized conductance plateaus indicating ballistic transport. All measurements were carried out at 1.3 K using a standard low-bias (80 V), low-frequency (33 Hz), AC measurement technique. In all cases we chose the bottom wire as the drive wire and the top wire as the drag wire. Measurements were done in a perpendicular magnetic eld B = 1 T. This eld was chosen to suppress tunneling while at the same time having a reasonable separation d between the wires for which the drag e ect could be observed. If the eld had not been applied, d had to be substantially increased by strongly biasing negatively the middle gate to avoid tunneling. But for such d no drag could be observed. The Coulomb drag voltage was measured in two di erent ways at values of the middle gate voltage VMG for which no tunneling was observed between the drive and the drag wire. In the rst case, the voltage of the middle gate was kept at a xed value and the voltages of the top and bottom gates were swept together. The drive current I and the drag voltage VD were measured simultaneously as a function of the
Fig. 1. Schematic diagram of a coupled quantum wire device. Lithographic dimensions: L = 0:4 m, W = 0:25 m. Middle gate width is 50 nm.
sweep voltage. The measurements were repeated for di erent values of the middle gate voltage. These measurements were designed to study the e ect of wire separation on the drag voltage or equivalently the drag resistance RD = VD =I . In the second case, the voltage of the middle gate was kept at −0:550 V for all the measurements. A voltage of the top gate was chosen to be at the middle of a conductance step or rise of the drag wire. VD and I were then measured simultaneously as the voltage of the bottom gate was swept. The measurements were repeated for di erent xed values of the top gate voltage. These measurements were geared to study the e ect of wire width on the drag voltage. Figs. 2 and 3 show typical representative results for the two di erent ways of measuring the Coulomb drag e ect.
3. Results and discussion Fig. 2 shows the measured drive current I and the drag resistance RD as a function of the top and bottom gate voltage VTB changing simultaneously (VT = VB = VT; B ). In Fig. 3, I and RD are plotted as a function of the bottom gate voltage VB . The parameters used are indicated in the gures. In both cases, I shows the usual conductance plateaus for ballistic transport. In Fig. 2, RD shows a prominent peak at VB; T = −1:07 V, followed by a much smaller peak at −0:78 V. The peaks are thus centered at voltages just beyond the right ends of the respective conductance plateaus where the conductance rises into steps. In Fig. 3, I and
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Fig. 2. Drag resistance RD and drive current I as a function of the top and bottom gate voltage changing simultaneously (VT = VB = VT; B ).
Fig. 3. Drag resistance RD and drive current I as a function of the bottom gate voltage VB .
RD show a behavior similar to that in Fig. 2. RD has a strong peak at −1:01 V, and a weaker one at −0:71 V. The observed experimental results can be understood qualitatively in the framework of theories presented in Refs. [17,18] which dealt with Coulomb drag in the ballistic regime and considered only the interwire electron–electron coupling and neglected all other interactions. The theories cited above are valid for submicron wire lengths L that are considerably smaller than li , which is our case. More appropriate for the present results is the theory of Ref. [18], valid at nite magnetic elds B, whereas that of Ref. [17] is for B = 0. With parabolic con nement (along y-direction) of the same frequency for both the wires and transport along the lengths (x-direction) of the wires, the drag resistance is given by LT 3 z2 !c2 kF2 exp −C2 RD = C1 2 3 3
!2 ! kF sinh2 z "Z r ∞ C2 kF2 2 du exp(−u =2) × K0 × !2 −∞ 2 × |C3 1=2 d + u| :
(1)
The above expression corresponds to the case when only the lowest subbands, displaced in energy by (splitting energy), are occupied in each wire. C1 ; C2 ; C3 , are constants, and u is a dimensionless variable. K0 is the modi ed Bessel function, z = =2kB T , ! the frequency of the con ning potentials, 2 = !2 + !c2 , and !c = eB=m∗ . d is the distance between the centers of the wires, the dielectric constant of the wire material. The constant C’s are: C1 = e2 kB m3 =˜5 ; C2 = p 4˜=m; C3 = m∗ =˜, and m = m∗ 2 =!2 . Eq. (1) is valid when both and kB T are small compared to the Fermi energy F , de ned as F = ˜2 kF2 =2m, where kF is the x-component of the Fermi wave vector. This condition entails that the wave vectors of the electrons participating in Coulomb collisions in both the wires are close to kF . Eq. (1) shows that RD increases as z and kF decrease. Peaks in RD should therefore result when the highest occupied 1D subband bottoms of the drive and drag wire line up and sweep across the Fermi level. This dependence of RD on z and kF is also valid when the two wires are di erent and higher subbands are occupied [17]. Since K0 is a rapidly decreasing function of its argument, RD is expected to diminish rapidly as d increases. An
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increase in wire width corresponds to a decrease in ! and an increase in d. The combination of these two e ects as the wires widen should result in a signi cant suppression of the drag e ect. Eq. (1) demonstrates that for small values of B such that !c2 =!2 .1, the in uence of the magnetic eld is to signi cantly reduce RD , due to the exponential factor and also due to the increase with B of the argument of K0 . In the light of the above discussion we conclude that the observed peaks in RD shown in Figs. 2 and 3 result when the 1D quantized transverse energy levels of the drive and drag wire nearly line up and sweep across the Fermi level. Under these conditions and the Fermi momentum kF are small; the smallness of and kF results in a high RD . On a gate voltage scale, a peak in RD should therefore result as the conductance plateaus rise into steps. This is in agreement with our experimental observations. Assuming parabolic con nement for the wires, the con ning frequency decreases for the higher subbands since the wire width increases. The wire separation d increases at the same time. Both these changes should lower the drag e ect signi cantly. This explains why the second RD peaks in Figs. 2 and 3 are much weaker compared to the rst ones. The applied magnetic eld enhances this relative weakening e ect because of the presence of the exponential factor in Eq. (1). We have estimated the con nement strengths of the wires for gate voltages at which the two peaks in RD occur: ˜!1 = 6:94 meV and ˜!2 = 3:64 meV. Note that for these values of and a eld of 1 T, !c2 =!2 .1. The eld is thus too small to create magnetic edge states and an interpretation of our experimental results based on Eq. (1) is valid. To have an idea of the order of magnitude of the drag e ect, RD was computed using Eq. (1) with the following values for the wire parameters: L = 0:4 m, ˜! = 6:94 meV, F = 1meV, = 0; B = 1T; T = 1:3 K, and d = 50 nm. This value of d was chosen to correspond to the lithographic width (=50 nm) of the middle gate. We obtained a value of 423 for RD . Considering the uncertainties involved in the magnitudes of some of the above parameters, especially in d, we believe this value of RD is in reasonable agreement with experimental results.
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Acknowledgements One of us (P.D.) wishes to thank S.E. Ulloa for many helpful discussions and gratefully acknowledges the award of a Senior Research Associateship by the National Research Council, Washington, DC. This work was accomplished during the tenure of this award. References [1] N.P.R. Hill, J.T. Nichols, E.H. Linfeld, K.M. Brown, M. Pepper, D.A. Ritchie, G.A.C. Jones, B.Y. Hu, K. Flensberg, Phys. Rev. Lett. 78 (1997) 2204. [2] H. Rubel, A. Fisher, W. Dietche, K. von Klitzing, K. Eberl, Phys. Rev. Lett. 78 (1997) 1763. [3] U. Sivan, P.M. Solomon, H. Shtrikmann, Phys. Rev. Lett. 68 (1992) 1196. [4] T. Gramila, J. Eisenstein, A.H. MacDonald, L.N. Pfei er, K. West, Phys. Rev. Lett. 66 (1991) 1216. [5] H.C. Tso, P. Vasilopoulos, F.M. Peeters, Phys. Rev. Lett. 68 (1992) 2516. [6] H.C. Tso, P. Vasilopoulos, F.M. Peeters, Phys. Rev. Lett. 70 (1993) 2146. [7] K. Flensberg, B.Y. Hu, Phys. Rev. Lett. 73 (1994) 3572. [8] M. Bonsager, K. Flensberg, B.Y. Hu, A. Jauho, Phys. Rev. Lett. 77 (1996) 1366. [9] A. Rojo, G.D. Mahan, Phys. Rev. Lett. 68 (1992) 2074. [10] G. Vignale, A.H. MacDonald, Phys. Rev. Lett. 76 (1996) 2786. [11] L. Zheng, A.H. MacDonald, Phys. Rev. B 48 (1993) 8203. [12] A. Laikhtman, P. Solomon, Phys. Rev. Lett. 41 (1990) 9921. [13] E. Shimshoni, S.L. Sondhi, Phys. Rev. Lett. 49 (1994) 11 484. [14] O.E. Raichev, J. Appl. Phys. 81 (1997) 1302. [15] H.C. Tso, P. Vasilopoulos, Phys. Rev. B 45 (1992) 1333. [16] Yu.M. Sirenko, P. Vasilopoulos, Phys. Rev. B 46 (1992) 1611. [17] V.I. Gurevich, V.B. Pevzner, E.W. Fenton, J. Phys.: Condense. Matter 10 (1998) 2551. [18] O.E. Raichev, P. Vasilopoulos, Nano-99 Symposium, St. Petersburg, 1999. [19] J.A. del Alamo, C.C. Eugster, Appl. Phys. Lett. 56 (1990) 78. [20] I.M. Castleton, A.G. Davies, A.R. Hamilton, J.E.F. Frost, M.Y. Simmons, D.A. Ritchie, M. Pepper, Physica B 249 –245 (1998) 157. [21] N. Tsukada, A.D. Wieck, K. Ploog, Appl. Phys. Lett. 56 (1990) 2527. [22] M. Okuda, S.-I. Miyazawa, K. Fujii, A. Shimizu, Phys. Rev. B 47 (1993) 4103.
