Two-dimensional electrons in a lateral magnetic superlattice

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1 surfaco soionce ELSEVER Surface Science 361/362 (1996) Two-dimensional electrons in a lateral magnetic superlattice H.A. Carmona *, A. Nogaret,, A.K. Geim a, P.C. Main a'*, T.J. Foster ~, M. Henini ~, S.P. Beaumont b, H. McLelland b, M.G. Blamjre c Department of Physics, University of Notttnghara, Nottingham NG7 2RD, UK b Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow G2 8QQ, UK * Department of Materials Science and Metallurgy, University of Cambridge, Cambridge CD2 3QZ, UK ' Received 21 June 1995; accepted for pubfication 30 August 1995 Abstract We have measured the msgnetoresistance measurements of a 2D electron gas subjected to a periodic magnetic field created by the presence of type 11 superconducting stripes on the surface of the heterostructure. We observe oscillatory behaviour due to a commensurability effect between the cyclotron radius and the period of the magnetic field. The results can be understood in terms of a simple semi-cla~cal theory. Keywords: Gallium arsenide; Semiconducting filma; Semiconductor-superconductor interfaces i Superconductor-semiconductor heterostructures 1. ntroduction There have been several recent investigations of the motion of electrons in a two-dimensional electron gas (2DEG) subjected to an inhomogeneous magnetic field created by patterned superconductors on the surface of the heterostructure. For example, the contribution of a single vortex to the quantum conductivity of a mesoscopic device, and the motion of ballistic electrons in a random magnetic field have been reported F1]. n this work, a periodic array of type 1 superconducting stripes is patterned on top of a heterojunction to create a periodic magnetic field. t has been predicted [2] that the electron mobility undergoes resonant enhancements whenever the cyclotron *Corresponding author. Fax: , ppepcm@plm 1.nott.ac.uk. diameter at the Fermi level is commensurate with the period of the magnetic potential within a constant phase factor. This effect can be understood in a semi-classical picture [3] analogous to the guiding-centre drift resonance in the case of periodic electric potentials [4]. The effects of a modulated magnetic field acting on a 2DEG have been reported recently by two groups [5,6]. 2. Experimental The structure used in the experiments is shown schematically in Fig. la. A high-mobility 2DEG was formed in a standard GaAs/(A1Ga)As heterojunction 300nm from the sample surface. The electron density was ~ 1.5 x 10 is m -2 and the elastic mean free path ~ 10/an. A metallic gate of thickness 150nm was deposited on the surface /96/$15.00 Copyright 1996 Elsevier Science B.V. All rights reserved PH S (96)

2 1t.,4. Carmona et al./ Surface Science 361/362 (1996) superconductor superlattico lamlator layer h,e 2DEG.,o.65t, m ~' -, -, Experiment 2 '! /'\... o i '1 i s -4 (a) o.oo o.os ", B(T) o.'lo oas Fig. 1. (a) Schematic diagram of the dcvices. (b) Micrograph showing 2 ~m imriod magnetic supcclattice lmtlcrnexi on top of the 2DEG. covering a standard Hall bar, followed by a 200 nm insulating Ge layer. An array of Pb or Nb stripes (200 nm thick) was fabricated on top of the insulator layer by electron beam lithography defining a grating of or 2/an period (Fig. lb). The magnetoresistance was measured sweeping the magnetic field from B < -- 1 T through zero to B> 1 T (sweep up) and vice versa (sweep down). Strong oscillations were observed, periodic in lb and which survive up to high temperature. These arise from a periodic electric potential at the 2DEG. We attribute this periodic electric potential to a strain effect causedby different thermal expansion coel~cients of the superconductor stripes and the insulating Ge [7]. 3. Results and discussion Fig. 2a shows the difference AR=, between the sweep-up and sweep-down traces at 0.3 K for a device with a Pb grating of 2/an period. The traces 7, 0 mv -'i~ ~t 6 " 100 mv ~-" )~.~ 200mV.-""..:)~/" l 5 300mY.../.:-~")':~ 4., 3,.~":':.. 21S.~"",'" ~, imo--l) (b) Fig. Z (a) AR~ versus B for a 2 ~m~period device. Above Bo, AR~=0. (b) nclices of the ma~imfi and minima of ARa~ versus /B for various gate voltages and hence electron densities. For both diagrams, theory lines have no adjustable parameter~ for sweep-up and sweep-down are identical at low magnetic fields and above some critical value Bc (AR~,=0). However, at intermediate magnetic fields, AR~, exhibits an oscillatory behaviour reflecting a difference between the magnetoresistance oscillations for sweep-up and sweep-down traces. As we increase the temperature-above the critical temperature of the. superconductor (~ 7.2 K), AR~ disappears indicating that the oscillations in AR~, are caused by the superconducting stripes. Further proof can be produced by investigating the temperature dependence of the magnetoresistance [5]. By plotting the index of the peaks and valleys versus 1/B, one can see in Fig. 2b that the oscillations in ARx= are periodic in lb with no phase shift. We now address the origin of the oscillations in AR~. Thin films of lead are type superconductors [8] with a low-temperature upper critical field Be2 ~0;12 T. Below Boz the flux penetrating the

3 330 H.tL Carmona et al /Surface Science 361/362 (1996) superconductor stripes is segregated into a distribution of vortices. The magnetoresistance hysteresis between sweep-up and sweep-down is caused by the pinning of these vortices. When the applied field is increased from 0 to B > Bo2 (sweep-up), flux penetrates from the edges of the stripes whereas the magnetic field at the centre of the stripes remains close to zero. n this situation the modulation in magnetic field at the plane of the 2DEG is out of phase with the electric potential modulation originating from strain [7]. When B is above B=2 the superconductivity is destroyed, leaving a homogeneous magnetic field at the plane of the 2DEG. When the applied field is decreased from a value larger than Be2 (sweep-down), the pinning mechanism prevents the vortices from easily leaving the superconductor. The field inside the stripes in this situation becomes larger than that between them, so the magnetic field modulation is in phase with the electric modulation. This change by x in the phase between magnetic field modulation and electric potential explains the hysteretic behaviour in Fig. 2a, although the situation is likely to be more complicated for B < 10 mt [1,5]. The low-field magnetoresistance oscillations observed when the 2DEG is subjected to both periodic electric and magnetic fields can be understood within a semi-classical picture. Fig. 3a shows computer simulations of electron trajectories at the Fermi energy for the cases where only electric modulation (dashed line) and when only magnetic modulation (solid line) are present. n each case, the magnetic field is chosen to correspond to a maximum in a magnetoresistance oscillation. For the calculations, the electrostatic modulation is described by V(x)= Vo sin(kx) and the magnetic one by B=(B+B o sin(kx))$. ("+" in phase, and "-" n out of phase), where K=2~/a with a the period of the modulation. For the electrostatic case the resonance condition occurs when the (l/b) V V(x) x B drift velocity at the two extremes x =Xo +P~ (here x0 is the orbit centre position and Re is the cyclotron radius at the Fermi energy) has the same sign, i.e. when 21~=(n+ 1/4)a [3]. when only magnetic modulation is present, the drift velocity is a maximum if the two extremes of the orbit lie at maximum and minimum magnetic fields, respectively. Because/~ oc l/b, the orbit size 'f 'fl 2t ',gg -.gg (a) Co) Fig. 3. (a) Semi-classical trajectories for electrons at the Fermi energy with an electric modulation (dashed curve) and with a spatially modulated magnetic field (solid curve). (b) Semiclassical trajectories for electrons at the Fermi energy in the presence of both electric and magnetic modulation n each case B is chosen to correspond to the resonance condition. The shadowed xegions correspond to the positions of the superconducting stripes. is slightly different on either side, leading to a finite drift velocity of the orbit centre in the direction of constant B (y direction). Following Beenaker [3], one can show that the resonance condition is 2R~ = (n-1/4)a. By averaging the mean square drift velocities over all orbit centres, we obtain the enhancement of the diffusion in the y direction and, therefore, a maximum in the resistivity in the x direction. n considering both electric and magnetic modulations, the drift velocities simply add together. n Fig. 3b, electron trajectories at the Fermi energy are plotted both for electric and magnetic modulations in phase (full line) and 7r out of phase (dashed line). The effect of adding a small magnetic modulation to the electric one is to change the phase angle of the resulting magnetoresistance oscillations. Resonance now occurs when 2R,--(n--1/4+~/70a with tan = 2, Vo/(ak t Oo), (1)

4 H.A. Carmona et,,l /Surface Science 361/362 (1996) where Ogo=eBo/m*, m* is the electron effective mass, and k F is the Fermi wave-vector. This simple semi-classical picture leads to expressions for the resistivity as a function of B identical to those obtained by Peeters and Vasilopoulos [2] for the case of magnetic modulation and combined electric and magnetic modulations. Using this theory we obtain an analytical expression for ARm [5] which has maxima strictly periodic in lb at R,=na, where n is an integer. n Fig. 2a, the experimental AR=, trace is compared with the theory, showing very good qualitative agreement with no adjustable parameters (see below). The periodicity of the maxima and minima in AR~, is shown in Fig. 2b. The lines are calculated by taking k F from the Shubnikov-de Haas oscillations, and there is an excellent quantitative agreement between theory and experiment. t is possible to determine both Vo and Bo from the experimental data. To this end, we measure the phase difference 2~ between the magnetoresistance maxima and minima for sweep-up and sweepdown, and use Eq. (1) to calculate values of Vo/Bo. Vo may be estimated using the cut-off magnetic field for the positive magnetoresistance at low B [9] and is ~ 1 mev. The maximum value for Bo, at the plane of the stripes, in the 2/an period samples with Pb grating was ~20 nat for the external field 35 mt [5]. With these measured parameters we are able to generate the theoretical curve shown in Fig. 2a. We did not observe hysteresis in the magnetoresistance of samples with 1 fan period. Assuming that the modulation amplitude decays exponentially [10] as exp(-2nz/a) at a distance z from the stripes, a reduction of about an order of magnitude relative to the 2/an period device is expected, but that should still be measured within our experimental resolution. From the maximum B o obtained for the 2 tan samples, we can estimate the separation between two vortices in the Pb to be about 200 nm. When this distance is comparable to the size of the stripes there is no longer any macroscopic gradient in the vortex concentration due to pinning, thus reducing the amplitude of the magnetic modulation in 1/an period samples. Similarly, we observe no modulation effect in samples with Nb stripes. This might reflect the larger value of the penetration length 2 in Nb [ 11] (>0.12/an) relative to Pb. Lead flms are known to have one of the smallest penetratior/ lengths that can be obtained for thin film,q [8]. n our structures we can estimate 2 for Pb from the cut, off field Bo to be ~ 0.07/an. The low-field magnetoresistance in our structures is dominated by the electric modulation. n order to have the magnetic modulation as the dominant effect in our samples, we estimate that the amplitude of the electric modulation would have to be as low as 0.2 mev for the 2/an period sample. These values are difficult to achieve in a lithographed structure and will require careful sample design. A 2DEG closer to the surface may make the magnetic modulation relatively stronger, since its amplitude is exponentially dependent on the distance from the surface, but strain-induced electrostatic modulation is not. 4. Summary We have observed novel magnetoresistance oscillations resulting from the comensurability of R~ with the period of a lateral magnetic superlattice. Our results can be understood within a semiclassical picture that takes into account both magnetic and electric modulations at the plane of the 2DEG. Acknowledgements This work is supported by EPSRC (UK), H.A.C. wishes to thank CNPq (Brazil) for financial support. References [1] A.K. Geim et al, Phys. Rev. B 46 (1992) 324. A.K. Gelm et al, Phys. Rev. B 49 (1994) [2] F.M. Peeters and P. Vasilopoulos, Phys. Rev. B 47 (1993) [3] C.WJ. Beenakker, Phys. Re','. Lett. 62 (1989) [4] D. Weiss et al, Europhya. Lett. 8 (1989) 179.

5 332 H.A. C.armona et al /Surface Science 361/362 (1996) R.R. Gehardts, D. Weiss and K. von Klitzing~ Phys. Rev. Lett. 62 (1989) [5] H.~L Carmona et aj, Phys. Rev. Lett. 74 (1995) "6] P J). Ye e~ al, Phys. Rev. Lett. 74 (1995) "7] J.Z Davies and.a. Larldn, Phys. Rev. B 49 (1994) [8] R.P. Huebener, Magnetic Flux Structures in Superconductors, Springer Series in Solid Satatv Physics, Vol. 6 (Springer, Berlin, 1979). [9] P.H. Bvton et al, Phys. Rw. B 42 (1990) [10"1 J. Rammer and A.L. Shelankov, Phys. Rev. B 36 (1987) [11] C.E. Cunn/tt~ham et al~ AppL PhyL Lett. 62 (1993) 2122.