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Mesoscopic transport properties of composite fermions
surface science ELSEVIER
Surface Science 361/362 (1996) 59--62
•
,
Mesoscopic transport properties of composite fermions A.S. Sachrajda a, y. Feng a, H.A. Carmona b, A.K. Geim b, P.C. Main b,,, L. Eaves b, C.T. Foxon b • Institute for Mtcrostrucmral Sciences, NRC, Ottawa, Canada KIA OR6 b Department of Physics, University of Nottingham, No~ngham NG7 2RD, UK Received 19 June 1995; accepted for publication 15 August 1995
Almraet We have measured the magnetoresistance of quantum wires, junctions and rings near Landau filling factor v = 1/2 to search for effects due to composite fermions (CF). We find evidence for semi-classical ballistic effects in the CF which are similar to those seen near zero magnetic field. However, we were not able to observe effects due to the phase coherence of the CF, even where their analogues were apparent in the electron transport. We attribute this to sample inhomogeneity.
Keywords: Electrical transport; Electrical transport measurements; Gallium arsenide; Heterojunctions; Many body and quasi-particle theories; Semiconductor-semiconductor heterostruetures
1. Introduction
There has been a great deal of recent interest in the modelling of the fractional quantum Hall effect (FQHE) in terms of a novel "composite fermion" (CF) particle 1-1]. The CF comprises an electron bound to an even number of flux quanta, ~bo= hie. Since the bound flux quanta are part of the CF, the composite particle does not experience the magnetic field due to these flux lines. For the simplest CF, where an electron is bound to two flux quanta, this implies that the CF experience an effective magnetic field given by B*=B-T-2n~bo, where B is the applied magnetic field and n is the electron (and CF) density. It follows that B*---0 when the Landau filling factor, v=n~o/B, is 1/2. Experimental evidence for this model was provided * Corresponding author. Fax: + 4 4 115 9515180;, e-mail: ppzpem@ppn 1.nott.ae.uk.
by two demonstrations of semi-classical ballistic behaviour close to v= 1/2 in periodic potentials 1-2] and magnetic focusing [3]. However, these effects are by no means easy to observe. For example, our attempts to repeat the focusing experiments in an equally high mobility device, which showed well developed "electron" magnetic focusing peaks, have proved unsuccessful. Two possible explanations for this difficulty in observing semi-classical effects in CF are: (i) a short CF mean free path, and (ii) fluctuations in B* resulting from inherent density fluctuations. In order to confirm the semi-classical behaviour at v = 1/2, it is clearly important to find other ballistic effects which are more easily observed. The two ballistic effects described above occur in open strt~'ctures and, as a result, the density fluctuations are expected to be smaller than in more confined geometries in which ballistic effects have been studied (e.g. quantum wires and junctions).
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved
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tL S. Sachrajdaet aL/Surface Science 361/362 (1996) 59-62
However, there are two properties of these latter geometries which make them suitable for establishing semi-classical CF behaviour. First, a small mean free path is sulticient, typically less than 1/an. Secondly, the ballistic effects are largely dependent on the randomiTJation of trajectories, a process which is likely to be more resilient to fluctuations in B*. In particular, the boundary scattering effect [4,5] observed in quantum wires is a result of non-specular reflection at the wire edges, and the quenching of the Hall effect in quantum wires [6,7] relies largely upon scattering at the junction walls randomizing the electron trajectories. We present evidence below that these ballistic effects have been observed in our samples in the CF regime.
2. Experimental details, results and discussion The experiments were performed on high mobility (~200-300 m2/V .s) devices with n = 0.7-2x l0 ts m -a) devices. Submicron quantum wire and ring devices were defined by a shallow wet-etching technique. After processing, the quality of material remained high. For example, Fig. I shows Aharanov-Bohm oscillations in the magnetoresistance near B = 0 of a 1.5/an diameter ring. An unexpected observation in nearly all the devices was a slight curvature in the Hall slope, indicating an apparent increase of n (a few per cent) with increasing B. This was accompanied by an equivalent shift in the minima of the longitudinal resistance. We have no explanation of this effect but we believe it is linked to inhomogeneities. Ballistic effects in wires and junctions have been studied experimentally and theoretically near B = 0 [4--7]. The quenching of the Hall effect is primarily due to the scrambling of trajectories [6] and the geometrical collimating properties of the junction [7]. The related effects in the longitudinal resistance are accompanied by an additional boundary scattering effect [4] due to diffuse scattering at the wire edges, which is responsible for a reduction in the electron longitudinal momentum and hence an increase in the wire resistance. As B is increased the electron collides more frequently with the wire wall, decreasing the mean free path and creating a
positive magnetoresistance. However, as the field is raised still further and the cyclotron diameter 2R~ becomes smaller than the wire width w, the electrons are "guided" along the wire and the effective mean free path increases again, leading to a negative magnetoresistance. In this way a resistance peak at low magnetic fields [4] (w//~=0.55) is formed in addition to the peak caused by junction scattering. Examples of these effects near B = 0 for wires of width 0.5/an are shown in the inset of Fig. 2 and Fig. 3b. A difference between our results and those described above [4,5] is that there is no dip in the broad magnetoresistance peak at B=0. We believe that this is due to an additional, narrow peak at B=0. The amplitude of this central peak decreased with increasing temperature at a much faster rate than the rest of the feature (for example the "bumps" labelled 2 and 3 in Fig. 3b), which we identify with the ballistic effects discussed above. This suggests that the central peak is not semi-classical in origin, but is a phase-coherent effect due to the interference of back-scattered trajectories, similar to weak localisation [8]. We note that some of the original work also had an unexplained narrow central peak around zero magnetic field [4]. Fig. 2 illustrates the Hall (Rxy) and longitudinal (R,=) resistance of a quantum wire junction around v = 1/2 at 1 K. At this high temperature, the feature at v = 1/2 is readily distinguished. A clear peak can be seen in R ~ , whereas the resistance of the bulk 2DEG has a dip at v = 1/2. The width of the feature is similar to that expected from scaling the width of the peak around zero field, as shown in Fig. 2 by the line below the trace. The observation of the quenching of the Hall effect is made difficult by the gentle curvature in R~, of these devices, as mentioned above. Nevertheless, an extremely weak "quenching" feature can be seen in the Hall resistance at v = 1/2. The width in magnetic field of this feature is slightly larger than that predicted from the equivalent zero-field effects. In some devices, the feature at v = l / 2 showed a more complex structure which could be directly compared with the zero-field feature. Figs. 3a and 3b show a direct comparison of the low field and v = 1/2 features (i.e. the low field range has been stretched by 2t/2). The features marked by 2 and 3, at the side of the
A.S. Sachrajda et aL /Surface Science 361/362 (1996) 59-62
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aft) Fig. 1. Magnetoresistance oscillations for a 1.5/an diameter ring, wire width 0.7/an. Before subtraction of the background, the relative amplitude of the oscillations was about 2%. Inset: Fourier transform of the oscillations.
2.5
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BCO Fig. 2. Magnetoresistance at 1 K of a junction with wire width 0.5/an. (i) R ~ indicating a weak quench at v = 1/2. The solid line is a guide to the eye. (ii) R ~ revealing a peak in the magnetoresistance at v = 1/2. The line underneath the peak is the predicted width of the feature based on the low-field ballistic feature. Inset: (iii) R ~ and (iv) R ~ around B=O.
central peak in Fig. 3b, can be related to those marked 2 and 3 in Fig. 3a. The overall width of the magnetoresistance peak in Fig. 3a (between points 1 and 4) corresponds to the scaled width of the low-field feature. The central peak, which we associate with coherent effects near B = 0, is absent in Fig. 3a, leaving a slight dip in R = as expected for the ballistic effect. The missing central peak is consistent with our lack of observation of any other CF coherent effects in these devices. For example, the ring which shows clear magnetoresistance oscillations in Fig. 1 has no trace of equivalent oscillations near v = 1/2. The absence of phase-coherent effects may be understood in terms of device inhomogeneity. Weak localisation is caused by interference between time-reversed trajectories. The characteristic peak at B = 0 is due to the exact equivalence of the path lengths of the two trajectories, regardless of the length of the interfering paths; the role of the magnetic field is to break the time-reversal symmetry. In the CF case, inhomogeneities mean that B * = 0 occurs at slightly different magnetic fields in different parts of the device. To observe weak localisation, this variation in B* must be less than
62
tl. S. Sachrajda et aZ /Surface Science 361/362 (1996) 59-62
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attribute the absence of weak localisation to sample inhomogeneity leading to spatial variation of B*. Further investigations are required, however, to understand the absence of the other phase-coherent effects in our measurements. In particular, the effects of (i) the divergence of the effective mass at B*=0, and (ii) the inevitable spatial fluctuations of B* on the phase-coherence length need to be understood. We note that in other experiments [2], the elastic mean free path was found to be much shorter for CF than for electrons. Similarly, we have not been able to reproduce the observation of magnetic focusing of CF. However, weak features in both Rxx and R,~ were observed around v= 1/2. We attribute these to ballistic effects and relate them to the corresponding electron effects around B=0. We argue that the apparent ease with which these effects can be observed in comparison with other ballistic effects is due both to the intrinsic nature of the effects and the short mean free path required for their observation. These experiments provide further support for the CF description of the FQHE.
Bfr) Fig, 3. R~, of a wire of width 0.5/an at 300 mK around (a) v = 1/2 and (b) B=0. The numbered arrows indicate equivalent points in the two curves. The B axes are sealed to allow direct comparison. Inset: R ~ over a wider field range with v=2/3 and v = 1/2 indicated.
the equivalent of one flux quantum threading the largest phase-coherent loop in the device. For a phase-coherence length of ~1 tan, this would require homogeneity of electron density considerably better than 0.1%, better than is routinely achieved.
3. Conclusion We have searched for both ballistic and coherent effects in quantum wires and rings. Although universal conductance fluctuations, weak localisation and Aharanov-Bohm oscillations were present in the magnetoresistance of our devices near B = 0, no equivalent CF effects have been found. We
Acknowledgements This work is supported by EPSRC (UK) and ECAMI, and H.A.C. wishes to thank CNPq (Brazil) for financial support.
References [11 B.I. Halperin, P.A. Lee and N. Read, Phys. Re,/. B 47 (1993) 7312. [2] W. Kang et aL, Phys. Rev. Lett. 71 (1993) 3850. [3] V.J. Goldman B. Stl and J.K. Jail], Phys. Rev. Lett. 72 (1994) 2065. [4] TJ. Thornton et al., Phys. Rev. Lett. 63 (1989) 2128; Proc. "Quantum Coherence in Mesoscopic Systems", NATO ASI, Savoy, France, 1990. [5] RJ. Blakie ct al. in: Quantum Effects: Devices and Applications, Ed. IC Ishmail (Adam H,ilger, Bristol, 1992). [6] M.L. Roukes et aL, Phys. Rev. Lett. 59 (1987) 3011. [-7] C.W.J. Beenakker and H. van Houten, Phys. Rev. Lett. 63 (1989) 1857. [8] For an excellent review of weak localisstion, see G. Bergmann, Phys. Rep. 107 (1984) 1.
Surface Science 361/362 (1996) 59--62
•
,
Mesoscopic transport properties of composite fermions A.S. Sachrajda a, y. Feng a, H.A. Carmona b, A.K. Geim b, P.C. Main b,,, L. Eaves b, C.T. Foxon b • Institute for Mtcrostrucmral Sciences, NRC, Ottawa, Canada KIA OR6 b Department of Physics, University of Nottingham, No~ngham NG7 2RD, UK Received 19 June 1995; accepted for publication 15 August 1995
Almraet We have measured the magnetoresistance of quantum wires, junctions and rings near Landau filling factor v = 1/2 to search for effects due to composite fermions (CF). We find evidence for semi-classical ballistic effects in the CF which are similar to those seen near zero magnetic field. However, we were not able to observe effects due to the phase coherence of the CF, even where their analogues were apparent in the electron transport. We attribute this to sample inhomogeneity.
Keywords: Electrical transport; Electrical transport measurements; Gallium arsenide; Heterojunctions; Many body and quasi-particle theories; Semiconductor-semiconductor heterostruetures
1. Introduction
There has been a great deal of recent interest in the modelling of the fractional quantum Hall effect (FQHE) in terms of a novel "composite fermion" (CF) particle 1-1]. The CF comprises an electron bound to an even number of flux quanta, ~bo= hie. Since the bound flux quanta are part of the CF, the composite particle does not experience the magnetic field due to these flux lines. For the simplest CF, where an electron is bound to two flux quanta, this implies that the CF experience an effective magnetic field given by B*=B-T-2n~bo, where B is the applied magnetic field and n is the electron (and CF) density. It follows that B*---0 when the Landau filling factor, v=n~o/B, is 1/2. Experimental evidence for this model was provided * Corresponding author. Fax: + 4 4 115 9515180;, e-mail: ppzpem@ppn 1.nott.ae.uk.
by two demonstrations of semi-classical ballistic behaviour close to v= 1/2 in periodic potentials 1-2] and magnetic focusing [3]. However, these effects are by no means easy to observe. For example, our attempts to repeat the focusing experiments in an equally high mobility device, which showed well developed "electron" magnetic focusing peaks, have proved unsuccessful. Two possible explanations for this difficulty in observing semi-classical effects in CF are: (i) a short CF mean free path, and (ii) fluctuations in B* resulting from inherent density fluctuations. In order to confirm the semi-classical behaviour at v = 1/2, it is clearly important to find other ballistic effects which are more easily observed. The two ballistic effects described above occur in open strt~'ctures and, as a result, the density fluctuations are expected to be smaller than in more confined geometries in which ballistic effects have been studied (e.g. quantum wires and junctions).
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved
PH S0039-6028 ( 9 6 ) 00352-4
60
tL S. Sachrajdaet aL/Surface Science 361/362 (1996) 59-62
However, there are two properties of these latter geometries which make them suitable for establishing semi-classical CF behaviour. First, a small mean free path is sulticient, typically less than 1/an. Secondly, the ballistic effects are largely dependent on the randomiTJation of trajectories, a process which is likely to be more resilient to fluctuations in B*. In particular, the boundary scattering effect [4,5] observed in quantum wires is a result of non-specular reflection at the wire edges, and the quenching of the Hall effect in quantum wires [6,7] relies largely upon scattering at the junction walls randomizing the electron trajectories. We present evidence below that these ballistic effects have been observed in our samples in the CF regime.
2. Experimental details, results and discussion The experiments were performed on high mobility (~200-300 m2/V .s) devices with n = 0.7-2x l0 ts m -a) devices. Submicron quantum wire and ring devices were defined by a shallow wet-etching technique. After processing, the quality of material remained high. For example, Fig. I shows Aharanov-Bohm oscillations in the magnetoresistance near B = 0 of a 1.5/an diameter ring. An unexpected observation in nearly all the devices was a slight curvature in the Hall slope, indicating an apparent increase of n (a few per cent) with increasing B. This was accompanied by an equivalent shift in the minima of the longitudinal resistance. We have no explanation of this effect but we believe it is linked to inhomogeneities. Ballistic effects in wires and junctions have been studied experimentally and theoretically near B = 0 [4--7]. The quenching of the Hall effect is primarily due to the scrambling of trajectories [6] and the geometrical collimating properties of the junction [7]. The related effects in the longitudinal resistance are accompanied by an additional boundary scattering effect [4] due to diffuse scattering at the wire edges, which is responsible for a reduction in the electron longitudinal momentum and hence an increase in the wire resistance. As B is increased the electron collides more frequently with the wire wall, decreasing the mean free path and creating a
positive magnetoresistance. However, as the field is raised still further and the cyclotron diameter 2R~ becomes smaller than the wire width w, the electrons are "guided" along the wire and the effective mean free path increases again, leading to a negative magnetoresistance. In this way a resistance peak at low magnetic fields [4] (w//~=0.55) is formed in addition to the peak caused by junction scattering. Examples of these effects near B = 0 for wires of width 0.5/an are shown in the inset of Fig. 2 and Fig. 3b. A difference between our results and those described above [4,5] is that there is no dip in the broad magnetoresistance peak at B=0. We believe that this is due to an additional, narrow peak at B=0. The amplitude of this central peak decreased with increasing temperature at a much faster rate than the rest of the feature (for example the "bumps" labelled 2 and 3 in Fig. 3b), which we identify with the ballistic effects discussed above. This suggests that the central peak is not semi-classical in origin, but is a phase-coherent effect due to the interference of back-scattered trajectories, similar to weak localisation [8]. We note that some of the original work also had an unexplained narrow central peak around zero magnetic field [4]. Fig. 2 illustrates the Hall (Rxy) and longitudinal (R,=) resistance of a quantum wire junction around v = 1/2 at 1 K. At this high temperature, the feature at v = 1/2 is readily distinguished. A clear peak can be seen in R ~ , whereas the resistance of the bulk 2DEG has a dip at v = 1/2. The width of the feature is similar to that expected from scaling the width of the peak around zero field, as shown in Fig. 2 by the line below the trace. The observation of the quenching of the Hall effect is made difficult by the gentle curvature in R~, of these devices, as mentioned above. Nevertheless, an extremely weak "quenching" feature can be seen in the Hall resistance at v = 1/2. The width in magnetic field of this feature is slightly larger than that predicted from the equivalent zero-field effects. In some devices, the feature at v = l / 2 showed a more complex structure which could be directly compared with the zero-field feature. Figs. 3a and 3b show a direct comparison of the low field and v = 1/2 features (i.e. the low field range has been stretched by 2t/2). The features marked by 2 and 3, at the side of the
A.S. Sachrajda et aL /Surface Science 361/362 (1996) 59-62
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5OO I000 1500 2OO0
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aft) Fig. 1. Magnetoresistance oscillations for a 1.5/an diameter ring, wire width 0.7/an. Before subtraction of the background, the relative amplitude of the oscillations was about 2%. Inset: Fourier transform of the oscillations.
2.5
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°
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BCO Fig. 2. Magnetoresistance at 1 K of a junction with wire width 0.5/an. (i) R ~ indicating a weak quench at v = 1/2. The solid line is a guide to the eye. (ii) R ~ revealing a peak in the magnetoresistance at v = 1/2. The line underneath the peak is the predicted width of the feature based on the low-field ballistic feature. Inset: (iii) R ~ and (iv) R ~ around B=O.
central peak in Fig. 3b, can be related to those marked 2 and 3 in Fig. 3a. The overall width of the magnetoresistance peak in Fig. 3a (between points 1 and 4) corresponds to the scaled width of the low-field feature. The central peak, which we associate with coherent effects near B = 0, is absent in Fig. 3a, leaving a slight dip in R = as expected for the ballistic effect. The missing central peak is consistent with our lack of observation of any other CF coherent effects in these devices. For example, the ring which shows clear magnetoresistance oscillations in Fig. 1 has no trace of equivalent oscillations near v = 1/2. The absence of phase-coherent effects may be understood in terms of device inhomogeneity. Weak localisation is caused by interference between time-reversed trajectories. The characteristic peak at B = 0 is due to the exact equivalence of the path lengths of the two trajectories, regardless of the length of the interfering paths; the role of the magnetic field is to break the time-reversal symmetry. In the CF case, inhomogeneities mean that B * = 0 occurs at slightly different magnetic fields in different parts of the device. To observe weak localisation, this variation in B* must be less than
62
tl. S. Sachrajda et aZ /Surface Science 361/362 (1996) 59-62
13.9
14.3
!
!
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-0.2
3
,
,
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attribute the absence of weak localisation to sample inhomogeneity leading to spatial variation of B*. Further investigations are required, however, to understand the absence of the other phase-coherent effects in our measurements. In particular, the effects of (i) the divergence of the effective mass at B*=0, and (ii) the inevitable spatial fluctuations of B* on the phase-coherence length need to be understood. We note that in other experiments [2], the elastic mean free path was found to be much shorter for CF than for electrons. Similarly, we have not been able to reproduce the observation of magnetic focusing of CF. However, weak features in both Rxx and R,~ were observed around v= 1/2. We attribute these to ballistic effects and relate them to the corresponding electron effects around B=0. We argue that the apparent ease with which these effects can be observed in comparison with other ballistic effects is due both to the intrinsic nature of the effects and the short mean free path required for their observation. These experiments provide further support for the CF description of the FQHE.
Bfr) Fig, 3. R~, of a wire of width 0.5/an at 300 mK around (a) v = 1/2 and (b) B=0. The numbered arrows indicate equivalent points in the two curves. The B axes are sealed to allow direct comparison. Inset: R ~ over a wider field range with v=2/3 and v = 1/2 indicated.
the equivalent of one flux quantum threading the largest phase-coherent loop in the device. For a phase-coherence length of ~1 tan, this would require homogeneity of electron density considerably better than 0.1%, better than is routinely achieved.
3. Conclusion We have searched for both ballistic and coherent effects in quantum wires and rings. Although universal conductance fluctuations, weak localisation and Aharanov-Bohm oscillations were present in the magnetoresistance of our devices near B = 0, no equivalent CF effects have been found. We
Acknowledgements This work is supported by EPSRC (UK) and ECAMI, and H.A.C. wishes to thank CNPq (Brazil) for financial support.
References [11 B.I. Halperin, P.A. Lee and N. Read, Phys. Re,/. B 47 (1993) 7312. [2] W. Kang et aL, Phys. Rev. Lett. 71 (1993) 3850. [3] V.J. Goldman B. Stl and J.K. Jail], Phys. Rev. Lett. 72 (1994) 2065. [4] TJ. Thornton et al., Phys. Rev. Lett. 63 (1989) 2128; Proc. "Quantum Coherence in Mesoscopic Systems", NATO ASI, Savoy, France, 1990. [5] RJ. Blakie ct al. in: Quantum Effects: Devices and Applications, Ed. IC Ishmail (Adam H,ilger, Bristol, 1992). [6] M.L. Roukes et aL, Phys. Rev. Lett. 59 (1987) 3011. [-7] C.W.J. Beenakker and H. van Houten, Phys. Rev. Lett. 63 (1989) 1857. [8] For an excellent review of weak localisstion, see G. Bergmann, Phys. Rep. 107 (1984) 1.