Measurements of a composite fermion split-gate

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Surface Science 361/362 (1996) 71-74

Measurements of a composite fermion split-gate C.-T. L i a n g *, C.G. Smith, D.R. M a c e , J.T. Nicholls, J.E.F. Frost, M.Y. S i m m o n s , A.R. H a m i l t o n , D.A. Ritchie, M. P e p p e r Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK Received 21 June 1995; accepted for publication 2 August 1995

Abstract Recent theoretical and experimental work demonstrates that at filling factor v=1/2, a two-dimensional electron gas can be described in terms of composite fermions (CFs), for which the effective magnetic field vanishes. We present low-temperature measurements on a wide split-gate (SG) device investigating CFs in the quantum point contact (QPC) defined by such a structure. Negative magnetoresistance due to suppression of backscattering in the QPC was observed both around zero magnetic field and v = 1/2. We have also measured the ~ c e of a CF QPC at zero effective magnetic field as a function of gate voltage, with a compensating field to maintain v = 1/2 in the QPC. Using a simple model, we have determined the channel widths for CFs, which are narrower than those for electrons.

Keywords: Electrical transport; Electrical transport measurements; Gallium arsenide; Hall effect; Heterojunctions; Quantum effects; Semiconductor-semiconductor fieterostructures

1. Introduction

The fractional quantum Hall effect (FQHE) [1] can be observed in high-mobility two-dimensional (2D) electron systems in the low-temperature, highmagnetic field re#me. It is believed that the FQHE arises from strong electron-electron interactions, causing the 2D electrons to condense into a fractional quantum Hall liquid. Jaill [2] proposed a new approach to the FQHE in terms of"composite fermions', consisting of electrons bound to an even number of magnetic flux quanta. The FQHE can then be understood as a manifestation of the integer quantum Hail effect of weakly interacting composite fermions. It has been shown [3] that at * Corresponding author. Fax: + 4 4 1223 337271; ~mail: ctl 100 lam~as.cam.ac.uk.

filling factor v = 1/2, a 2D electron system can be transformed into a composite fermion (CF) system interacting with a Chern-Simons gauge field. At v = 1/2 the effective magnetic field Bofr acting on the CFs is zero, and a variety of experiments [4-6] support this picture. Using the split-gate (SG) technique [7], a onedimensional (1D) channel [81 (or a quantum point contact) [91 can be formed in a two-dimensional electron gas (2DEG). Subsequent investigation of such a channel reveals negative four-terminal magnetoresistance R ~ given by [10]

Rx~ = (h/2e 2) (1/N.(B) )-- R,~ ,

where N°(B) is the number of occupied magnetoelectric sub-bands in the channel, R~ is the Hall resistance of the bulk 2DEG region, and B is the magnetic field. Assuming spin degeneracy and

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C.-T. Lianget aZ/SurfaceScience361/362(1996)71-74

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ignoring the discreteness of No(B), square well confinement is [10]

No(B) in a

N°(B)= ( kr-z?l¢){ arcsin(Wo/21o)+ (w°/21o) xlx/1-(w,/2l~)21},

for

w,<21o,

(2)

where kF is the Fermi wave-vector in the channel, lo=hkF/eB is the cyclotron radius, and w, is the channel width for electrons. Increasing the magnetic field causes electrons to start to execute skipping orbits towards the channel boundary, suppressing backscattering near the entrance to the channel, giving rise to negative magnetoresistance (NMR) centred around B=0. In this paper we present measurements on a wide SG device fabricated on a GaAs/AIGaAs heterostructure. Over the measurement range, the channel defined by the SG is very wide (see later) and there are at least twelve occupied 1D subbands occupied. In this case the confining potential can be described as a square well, and the carder density is uniform across the channel. This ensures a uniform filling factor in the channel, and thus we are able to measure a composite fermion quantum point contact (QPC) at B,er=0 as a function of gate voltage Vg with a compensating applied magnetic field to maintain v= 1/2 in the channel. Recently, Khaetskii and Bauer [ 11 ] have considered CF transport in a bar-gated sample. In this system, applying a small negative gate voltage reduces the carrier density underneath the gate, inducing an effective magnetic field in those regions. Injected CFs approaching those areas are bent back to the source contact by such an effective field. We shall show that our results agree qualitatively with this theory.

2. Experimental The SG device (0.3 lan long, 1.2 lan wide) was defined by electron-beam lithography on the surface of a GaAs/A1GaAs heterostructure, 297 nm above the 2DEG. It has been demonstrated that clean 1D channels showing good ballistic conductance quantisation can be defined in such deep

2DEGs [12]. After brief illumination with a red LED, the carrier concentration of the 2DEG was 1.385 x 1015 m-2, with a mobility of 350 m2/V's. Experiments were performed in a 3He cryostat at 0.3 IC The magnetoresistance was measured using standard four-terminal ac phase-sensitive techniques with a current of 10 nA. Similar results were obtained from experiments performed on seven devices fabricated on three different wafers. Measurements taken from one of the seven devices are presented in this paper.

3. Results and discussion When - 0.7 V was applied to the SG, the channel was first defined and NMR was observed. Note that a background resistance R ~ (V~=0) due to the bulk 2DEG was measured in series with the resistance of the channel [10]. After subtracting the background resistance (N100£~) we have a good fit to Eq. (1) and obtain the width of the channel to be 0.39-0.88gm over a range of - 3 V < Vs <~- 0 . 7 V. In this range of Vgthe channel width is very wide, and we assume that the carrier density n, is uniform across the channel, n,(Vs) can be varied by changing Vs on the SG, whereas the carder density of the bulk 2DEG remains unchanged. We can obtain n,(V~) by two measuremeats. Firstly, taking the magnetic field value of the middle point of the v = 1 two-terminal quantum Hall plateau, we determine n,(V~) from the relation eB/h=n,. Secondly, at Vg<-2.1 V, we observed extra minima in R ~ around B = 8 T, corresponding to v = 2/3 in the channel as shown in Fig. 1. Values of n,(Vs) obtained from the two methods are very similar and the difference in the deduced positions of v = 1/2 is within 0.06 T. For consistency we use n,(Vs) determined from the first measurement. Fig. 1 shows the high-field magnetoresistance measurements at different gate voltages. The minima in R ~ around B = 8.6, 9.5 and 10 T correspond to v = 2/3, 3/5 and 4/7 in the bulk 2DEG measured in series with the QPC. When the channel is defined, NMR around v= 1/2 is observed, as indicated by arrows in Fig. 1 for Vg=-2.3, - 2 . 5 and - 2 . 7 V. After subtraction of the background resistance, the resistance A R ~ - R ~ ( V s )

C.-T. Liang et al./Surface Science 361/362 (1996)

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--2.5 V is vertically offset by 0.3 klZ Insets (a) and (b) show the subtracted resistance AR,= centred a r o u n d v = 1/2 a n d B = 0, respectively. The positions of v = 1/2 in the channel at various Vs are indicated by arrows.

- R ~ ( V g = 0 ) also shows NMR centred around v= 1/2 and B =0, as displayed in the insets to Figs. la and lb, respectively. We believe the NMR in AR= around v--- 1/2 is evidence of suppression of backscattering of CFs in an effective magnetic field. If we consider that there is a factor of ~/2 between the Fermi wave vector of electrons and that of CFs and there is no spin degeneracy for CFs, the fit to Eq. (1) is poor. Thus it is not a reliable method to determine the channel widths for CFs. As shown later, we can determine the channel widths for CFs using a simple alternative model. The inset to Fig. 2 shows the measured resistance when v = l / 2 in the channel (V) and the background resistance R~(Vg=0) due to the bulk 2DEG at the corresponding magnetic field (O) as a function of V~. Thus the deduced resistance of the CF QPC at Boer=0 is given by • = • - O, as shown in Fig. 2. The resistance of the electron QPC at B = 0 is displayed for comparison. Two features can be readily seen in Fig. 2: (i) at equal Vs, the resistance of the CF QPC is higher than that of the electron QPC, and (ii) the definition voltage of the CF QPC is ~ -0.3 V, larger than that of the electron QPC (-0.7 V). The latter can be explained by the recent theory [ 11 ]. At v = 1/2, applying a small negative Vs induces a strong Boer under the SG. Thus CFs cannot transverse those regions. In this case, CFs can only go through the

Fig. 2. Inset: the measured resistance (V) a n d the background resistance (O) as a function of Vr M a i n figure: deduced resistance of the C F Q P C given by • = • - O as a function of Vr The resistance of the electron Q P C after subtraction of a series resistance of 93.3 fl due to the bulk 2 D E G is plotted as a solid line.

channel causing the channel to be defined at a larger Vg when electrons are not fully depleted underneath the SG. Finally we use a simple model to calculate the channel widths for CFs. At B = 0, the resistance of an electron QPC can be written as R,=h/2e2N°, where No is the number of 1D sub-bands occupied in the channel and the factor of two is due to spin degeneracy. Since No=2Wo/2~, we have / ~ = hL,/4e2Wo, where 2, is the Fermi wavelength in the channel. Assuming that 1D sub-bands also exist in a CF QPC, the resistance of such a device is RCF= h/e2Ncr, where Ncr is the number of occupied sub-bands, and there is no spin degeneracy. Following the same argument, we have Rcv= hAo,/2a/2e2wcF, where ACTis the Fermi wavelength of CFs and 2t/ZACF=2~ [3]. Thus the ratio of RcF to P~ for a given

= ,fiw./w

V,.

(3)

Using Eq. (1) to fit AR** centred around B = 0, we determine Wo(~). Rcv//~ can be deduced from the data shown in Fig. 2. Thus we can obtain WcF(Vs) from Eq. (3). As shown in Fig. 3, wcv(Vs) is smaller than wo(Vs) at equal Vs, in agreement with recent experimental results I-6]. This is due to lateral partial depletion from the SG. In this case, around v= 1/2, a slight reduction in the carrier density near the boundary of the channel induces a strong

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C- T. Liang et al. /Surface Science 361/362 (1996) 71-74

Boer,bending the CF trajectories back to the source contact. Thus CFs cannot go through those regions, consequently reducing the channel width. As shown in Fig. 3, we have a good linear fit of Wo(~). This predicts a pinch-off voltage of -4.89 V, in close agreement with the measured value of -5.18 V. The slope of the fit Wcr(~) for -2.9V
We have observed suppression of backscattering in a quantum point contact, both around B = 0 and v = 1/2. We have also measured the resistance of a composite fermion quantum point contact as a function of gate voltage at Boer= 0. Using a simple model, we have determined the channel widths for composite fermions at various gate voltages.

Acknowledgements This work was funded by the Engineering and Physical Sciences Research Council. We thank C.J.B. Ford and G. Kirczenow for helpful discussions. C.-T.L. acknowledges support from the Committee of Vice-Chancellors and Principals, United Kingdom.

References [1] D.C. Tsui, H.L. St6rmer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559.

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Fig. 3. The deduced channel widths for electrons wo(Vs) (111) and for CFs wcr(Vs) (A) from Eqs. (1) and (3). The straight line fits are also shown.

[2] J.K. Jain, Phys. Rev. Lett. 63 (1989) 199. [3] B.I. Halperm, P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312. [4] R.R. Du, H.L. St~rmer, D.C. Tsui, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 70 (1993) 2944. [5] D.R. Leadley, R.J. Nicholas, C.T. Foxon and J3. Harris, Phys. Re,/. Lett. 72 (1994) 1906. [6] V.I. Goldman, B. Su and J.K. Jain, Phys. Rev. Lett. 72 (1994) 2065. [7] TJ. Thornton, M. Pepper, H. Ahmexl, D. Andrew and GJ. Davies, Phys. Rev. Lett. 56 (1986) 1198. [8] D.A. Wharam, TJ. Thornton, L Newbury, M. Pepper, H. Ahmcd, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J. Phys. C 21 (1988) L209. [9] B.J. van Wee*, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel and C.T. Foxon, Phys. Rev. Lett. 60 (1988) 848. [10] H. van Houten, C.W3. Beenakker, P.H.M. van Loosdreeht, T.J. Thornton, H. Ahmed, M. Pepper, C.T. Foxon and J3. Harris, Phys. Rev. B 37 (1988) 8534. [11] A. Khaetskii and G.E.W. Bauer, Phys. Rev. B 51 (1995) 7369. [12] K.J. Thomas, M.Y. Simmons, J.T. Nicholls, D.R. Mace, M. Pepper and D.A. Ritchie, Appl. Phys. Lett. 67 (1995) 109.