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Probing the edge of a 2DEG by time-resolved transport measurements
Physica E 1 (1997) 95–100
Probing the edge of a 2DEG by time-resolved transport measurements G. Ernst a;∗ , N.B. Zhitenev a;b , R.J. Haug a;c , K. von Klitzing a a Max-Planck-Institut
fur Festkorperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany Institute of Technology, Cambridge, MA 02139, USA c Institut f ur Festkorperphysik, Appelstr. 2, 30167 Hannover, Germany
b Massachussettes
Abstract We have studied time-resolved transport in a two-dimensional electron gas in the integer and fractional quantum Hall regime. In samples with a smooth edge and an additional screening electrode, the propagation velocity of high-frequency signals depends on the number and width of the involved edge channels, and thus can be used to obtain an approximate electron density pro le. The amplitude of the transmitted signals oscillates with respect to the applied magnetic eld with maxima appearing close to integer and fractional bulk lling factors. While the data around lling factor 13 are qualitatively similar to those around lling factors 1 and 2, deviations appear around lling factor 23 . At odd lling factors, signal propagation in partially decoupled edge channels is observed. ? 1997 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional electron gases; Time-resolved transport measurement
Two-dimensional electron gases (2DEGs) in high magnetic elds have remarkable transport properties both in the DC and the high-frequency regime. Many of them are closely related to the electronic structure near the edge of the 2DEG, where the electron density decreases to zero. Under quantum Hall e ect conditions, this boundary region is characterized by the formation of alternating compressible and incompressible strips [1–5]. In the compressible regions, the so-called edge channels (EC) [6–8], a Landau level is pinned at the Fermi energy and therefore only partially occu∗ Correspondence address: Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA.
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pied. This leads to a metallic behaviour with a constant electrostatic potential. In the incompressible regions, all occupied Landau levels are completely lled, and the Fermi energy is located in an energy gap, which is E = ˜!c , if we ignore the spin splitting. Hence, an electrostatic potential di erence can exist across an incompressible strip even in equilibrium. The coupling between adjacent ECs as well as the coupling between ECs and the bulk of the 2DEG may be weak enough for a non-equilibrium potential distribution to be established on a macroscopic length scale. As a result, ECs can be selectively populated and detected, a phenomenon observed as adiabatic transport in DC magnetotransport experiments [6–8]. But even
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though the EC picture is now commonly accepted for the integer quantum Hall (IQH) regime [3–5, 9–11], the electronic structure in the vicinity of the edge of a 2DEG under conditions of the fractional Quantum Hall (FQH) e ect is still under discussion [1, 2, 12– 19]. Theoretical calculations have shown that, in the FQH regime, the charge density pro le for a smooth boundary exhibits plateaus at certain fractional lling factors [16–19] similar to the plateaus at integer lling factors, while the edge reconstruction for a steep edge leads to counter owing edge states for certain lling factors [12–17], i.e. the electron density in the vicinity of the edge increases beyond its bulk value, before decreasing to zero. Time-resolved measurements of a voltage pulse in a 2DEG in strong magnetic elds are a promising experimental technique for obtaining information about the electronic structure near the edge of a 2DEG [20–22]. A high-frequency voltage pulse applied to a contact at the edge of a 2DEG generates edge magnetoplasmons (EMPs) [23], which are collective electronic excitations localized near the edge of the 2DEG. The voltage pulse is thus transformed into a wave packet of EMPs, and is transmitted along the edge adjacent to the injection contact. The EMP propagation direction is determined by the direction of the Lorentz force acting on the electrons. The EMP dispersion !(k) and group velocity vg = 9!=9k depends on the electron density n(x) near the edge, and on the electromagnetic environment of the 2DEG. Therefore, in measuring the pulse propagation velocity, one can study the electron density pro le near the edge. Here, we report on time-resolved magnetotransport measurements of a 2DEG with a smooth edge potential in order to study the dynamic edge excitations in the IQH and FQH regime. For the measurements, standard Hall-bar geometries made from AlGaAs=GaAs heterostructures are used. The 2DEG in sample A has a carrier concentration n0 = 2:2 × 1015 m−2 , a mobility = 50 m2 =Vs, and is located d = 90 nm below the surface. The data for sample B are n0 = 1:0 × 1015 m−2 , = 75 m2 =Vs, and d = 130 nm. In Fig. 1a, the sample geometry is shown together with a sketch of the experimental setup for time-resolved transport measurements. The largest part of the sample is covered with a metallic top gate. A long voltage pulse with a rise time of 150 ps and an amplitude Vin of several mV reaches contact 1 at time t = 0 with contact
Fig. 1. (a) Sketch of the sample geometry and measurement setup. Gated areas are shaded. A rectangular input pulse Vinput is applied to contact 1. The wave packet runs along the shorter boundary to contact 3. The transient potential at the intermediate contact 2 is detected using gate G2 as a sampling switch which is opened and closed by means of the applied strobe pulse. (b) Dependence of the transmission delay t0 on the applied magnetic eld (sample A). A sawtooth behaviour is superimposed onto an overall increase, with maxima or shoulders appearing close to the indicated integer and fractional bulk lling factors. (c) Calculated electron density of the innermost incompressible strip versus measured width of the compressible region at the edge for the data shown in (b). The density pro le extrapolated from these data (dashed line) is compared with a theoretical prediction (solid line), see text.
3 being grounded. The transient signal is detected at the intermediate contact 2 using the box-car technique described in Ref. [24]. All measurements were made in a 3 He= 4 He dilution refrigerator at a temperature of 70 mK with magnetic elds up to 13 T. The direction of the eld is chosen such that the applied pulse
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propagates along the shortest connection between the injection (1) and the detection (2) contacts. This distance is 160 m in sample A and 60 m in sample B. Fig. 1b shows the delay time t0 as a function of the applied magnetic eld for the magnetic eld range between 1 and 9 T, and an input amplitude of Vin = + 5 mV (sample A). The magnetic eld dependence can be described by a monotonic increase onto which sawtooth-like oscillations are superimposed. Close to the bulk lling factors 4, 2, and 1, the delay time has a maximum with a distinct asymmetry. The delay time increases steadily at the low magnetic eld side of the local maximum and drops rapidly at the high magnetic eld side. For the quantitative analysis, we use a macroscopic hydrodynamic description which yields the EMP velocity [24] vg =
d 9! e2 = · ; 9k h 0 r l
(1)
where d is the distance between 2DEG and top gate, and 0 r is the dielectric constant of the material between them. Except for fundamental constants and these material parameters, the velocity depends on the lling factor of the innermost incompressible strip and on the width l of the compressible region at the edge. While the overall increase of the delay time with increasing magnetic eld stems from the decreasing Hall conductivity e2 =h, we attribute the superimposed sawtooth behaviour to the oscillating width l. The observed asymmetry indeed indicates that the origin of the oscillations lies in the magnetic eld dependence of the electronic edge structure. As the magnetic eld increases, all compressible and incompressible strips shift towards the centre of the hallbar. At the indicated lling factors the innermost incompressible strip, which con nes the compressible edge region, vanishes in the bulk of the 2DEG, and the nearest incompressible strip, which is positioned closer to the edge, becomes the boundary line between edge and bulk. Thus, the width of the compressible edge region de ned as the sum of all edge channels, exhibits the same sawtooth oscillations as the measured delay time. Next, we concentrate on the lling factors = 1; 2; 4. Eq. (1) allows us to obtain the width l of the compressible edge region from the measured delay times. Assuming that the width of the innermost
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incompressible strip is small compared to the sum of all other compressible and incompressible strips ( = compressible edge region), we identify the width l with the position of the innermost incompressible strip. Since l measures the total width of the compressible region, l = 0 is de ned as the point where the electron density goes to zero. Knowing the electron density n = eB=h and the position of the incompressible strips, we can replot the data in the form of an electron density pro le. In Fig. 1c we compare the data for the magnetic eld ranges 1.9 T ¡B¡2.15 T, 2.9 T¡B¡4.4 T, and 6.5 T¡B¡8.8 T, where the lling factor of the innermost incompressible strip is = 4; 2; 1, respectively. The three sets of data clearly fall onto the same curve, and approach the bulk density far away from the boundary. This curve (dashed line) can thus be regarded as the electron density pro le n(l) without magnetic eld. The width of the depletion layer which separates the boundary of the 2DEG from the physical edge of the sample cannot be deduced from these measurements. Although di erent models have been used in order to calculate this electron density pro le theoretically [3–5, 25–27], none of them applies to our sample geometry where the edge of the 2DEG is de ned by etching away the surrounding heterostructure, before covering the entire surface with a metallic top gate. This a ects, in particular, the screening length inside and outside of the 2DEG. Therefore, we do not expect to nd a perfect agreement when comparing our extracted electron density pro le with the existing theoretical prediction. The solid line in Fig. 1c shows the result of a self-consistent calculation for the case that the edge of the 2DEG is de ned by applying a negative voltage V = − Egap =2e to a top gate [5]. Without any tting parameter, the overall agreement is satisfactory, although the predicted pro le rises too fast close to the edge, and approaches the bulk value too slowly. Calculations by other authors show the same trend [3, 4, 25–27]. Not only the delay, but also the amplitude of the rst step-like increase reveals details about the electronic structure at the edge of the 2DEG [20–22]. For a small amplitude of the input pulse, the entire signal is transmitted in the edge channels, and the amplitude of the leading edge appearing at the detection contact is equal to the input amplitude. For larger amplitudes, a part of the signal penetrates into the bulk and
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Fig. 2. Amplitude of the rst step V0 as a function of the applied magnetic eld (sample B). Maxima or shoulders appear close to the indicated integer and fractional lling factors. The solid lines serve as a guide to the eye.
appears at the detection contact at a later time. The transmitted signal consists of two parts: a fast step-like increase up to a voltage V0 , followed by a slower increase until saturation. This non-linear dependence of the shape of the transmitted signal on the amplitude of the input signal re ects the break-down of non-local transport. Fig. 2 shows the amplitude V0 as a function of the applied magnetic eld for bulk lling factors down to 1 3 (sample B). For large magnetic eld values, the initial step is partially rounded at the top, while the onset, and thus the delay time, remains well de ned [28]. Therefore, we apply a suciently large voltage pulse, Vin = 5 mV, and use the amplitude of the transmitted signal 5 ns after the arrival of the wave packet at the detection contact as a measure of V0 . As the noise in the measurement was strongly enhanced around lling factor = 12 , no accurate determination of V0 was possible between 7 and 10 T. The most prominent feature apparent from Fig. 2 are the similarities that exist between integer and fractional lling factors, which both show up as maxima (or at least shoulders) of V0 which are located close to these lling factors. As the voltage V0 drops across the innermost incompressible strip, the width of this strip determines the measured values for V0 , and the largest values for V0 are found, when this width is largest.
The width of an incompressible strip, in turn, is determined by the value of the energy gap, which explains the relative amplitudes of the observed maxima in V0 . Of particular interest is the behaviour around = 23 , where the position of the corresponding maximum amplitude seems to be signi cantly shifted to higher magnetic elds. Since DC transport shows a larger energy gap for = 23 compared to = 35 , the maximum at 6.6 T has to be attributed to lling factor 23 . We interprete this shifted maximum position as an indication of a magnetic- eld-dependent charge density pro le with a maximum within the edge region, as is predicted for a suciently steep edge [12–17]. In this case, an incompressible strip with = 23 can exist even for a bulk lling factor ¡ 23 . The di erences between = 1, 13 and 23 can be explained if such a maximum in the electron density pro le appears only for certain lling factor ranges, e.g. around = 23 but not around = 1 or 13 . Further indications for this scenario are found in the magnetic eld dependence of the delay time [28]. Fig. 3a compares the transmitted signals for positive and negative input amplitude at bulk lling factor = 3 (sample A). For a positive input signal, the shape of the signal is characterized by an abrupt rise at time t0+ = 0:86 ns, while for a negative input signal, the initial step at t0− = 0:76 ns is followed by a second step-like increase at t1− = 2:8 ns. The di erence between positive and negative polarity arises from the fact that the width of an incompressible strip is in uenced by the potential di erence across it [29]. When the voltage at the edge is positive with respect to the bulk, the width of the incompressible strips is reduced compared to its equilibrium value and vice versa. Thus, for negative polarity, more incompressible strips can become suciently wide to decouple the adjacent compressible regions. Fig. 3b illustrates the situation encountered in Fig. 3a. For positive polarity the compressible regions and are coupled and carry one common EMP mode. Eq. (1) with = 3 yields l = 0:49 m for the total width of the compressible edge, i.e. for the sum of the regions and . For negative polarity the = 2, incompressible strip is suf ciently wide to decouple the and regions, allowing for the existence of an EMP mode in each of them. Since these modes propagate at di erent velocity, the transmitted signal consists of two step-like increases instead of only one. For the quantitative analysis of
G. Ernst et al. / Physica E 1 (1997) 95–100
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In conclusion, we have shown that time-resolved magnetotransport measurements are a powerful tool for the investigation of the electronic structure near the edge of a 2DEG. The measured transmission delay times and amplitudes allow us to clarify the role of the compressible and incompressible strips in magnetotransport experiments in general. We would like to thank B. Farid, R. Gerhardts, S. Mikhailov and D. Pfannkuche for useful discussions, A. Yacoby for critical reading of the manuscript, and K. Eberl for providing the heterostructures. N.B.Z. was supported by Alexander von Humboldt Foundation. References Fig. 3. (a) Leading edge of the signal transmitted through the 2DEG for bulk lling factors = 3 (sample A). The positive pulse is transmitted by one EMP mode, whereas two EMP modes contribute to the transmission of the negative pulse. (b) Sketch of the electronic structure at the edge of the sample for a lling factor = 3 + . Compressible areas (crosshatched) are separated by incompressible strips with integer lling factor. In the presence of a shielding metallic plane, the regions and are partially decoupled.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
the measured delay times, we have to adapt Eq. (1) by replacing by = r − l , the lling factor difference between the incompressible strips surrounding the compressible region, i.e. = 2 for the region and = 1 for the region. In addition, the geometrical factor d=0 r l in Eq. (1) has to be modi ed, since mixing of these EMP modes due to capacitive coupling between the regions and has to be taken into account. Due to the existence of the neighbouring compressible strip, the capacitance between each individual compressible strip and the top gate is increased, which leads to a reduction of the EMP propagation velocity. Treating the regions and as a coplanar stripline, we estimate the magnitude of the additional capacitance to be about half the value of the parallel plate capacitance to the top gate. The corresponding EMP velocities are reduced by about 60%, leading to a width of 18 m for region and 32 m for region , which is in good agreement with the entire width found for positive polarity.
[10] [11]
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
C.W.J. Beenakker, Phys. Rev. Lett. 64 (1990) 216. A. Chang, Solid State Commun. 74 (1990) 871. P.L. McEuen et al., Phys. Rev. B 45 (1992) 11 419. D.B. Chklovskii, B.I. Shklovskii, L.I. Glazman, Phys. Rev. B 46 (1992) 4026. K. Lier, R.R. Gerhardts, Phys. Rev. B 50 (1994) 7757. B.I. Halperin, Phys. Rev. B 25 (1982) 2185. For a review see: C.W.J. Beenakker, H. van Houten, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics, vol. 44, Academic Press, New York, 1991. R.J. Haug, Semicond. Sci. Technol. 8 (1993) 131. J. Dempsey, B.Y. Gelfand, B.I. Halperin, Phys. Rev. Lett. 70 (1993) 3639. M.M. Fogler, B.I. Shklovskii, Phys. Rev. B 52 (1995) 17 366. A. Manolescu, R.R. Gerhardts, in: G. Landwehr, W. Ossau (Eds.), Proc. 12th Int. Conf. High Magnetic Fields in the Physics of Semiconductors II, Wucrzburg, 1996, World Scienti c, Singapore, 1997, pp. 319. A.H. MacDonald, Phys. Rev. Lett. 64 (1990) 220. X.G. Wen, Phys. Rev. Lett. 64 (1990) 2206. M.D. Johnson, A.H. MacDonalds, Phys. Rev. Lett. 67 (1991) 2060. Y. Meir, Phys. Rev. Lett. 72 (1994) 2624. L. Brey, Phys. Rev. B 50 (1994) 11 861. D.B. Chklovskii, Phys. Rev. B 51 (1995) 9895. M. Ferconi, M.R. Geller, G. Vignale, Phys. Rev. B 52 (1995) 16 357. O. Heinonen, M.I. Lubin, M.D. Johnson, Phys. Rev. Lett. 75 (1995) 4110. N.B. Zhitenev et al., Phys. Rev. Lett. 71 (1993) 2292. N.B. Zhitenev et al., Phys. Rev. B 49 (1994) 7809. G. Ernst et al., Surf. Sci. 361=362 (1996) 102. For a review see: V.A. Volkov, S.A. Mikhailov, Sov. Phys. JETP 67 (1988) 1639, translated from Zh. Eksp. Teor. Fiz. 95 (1988) 217. N.B. Zhitenev et al., Phys. Rev. B 52 (1995) 11 277.
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[25] A.V. Khaetskii, V.I. Fal’ko, G.E.W. Bauer, Phys. Rev. B 50 (1994) 4571. [26] A.O. Govorov, Phys. Rev. B 51 (1995) 14 498.
[27] I.A. Larkin, J.H. Davies, Phys. Rev. B 52 (1995) R5535. [28] G. Ernst et al., Phys. Rev. Lett. 79 (1997) 3748. [29] N.B. Zhitenev et al., Europhys. Lett. 28 (1994) 121.
Probing the edge of a 2DEG by time-resolved transport measurements G. Ernst a;∗ , N.B. Zhitenev a;b , R.J. Haug a;c , K. von Klitzing a a Max-Planck-Institut
fur Festkorperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany Institute of Technology, Cambridge, MA 02139, USA c Institut f ur Festkorperphysik, Appelstr. 2, 30167 Hannover, Germany
b Massachussettes
Abstract We have studied time-resolved transport in a two-dimensional electron gas in the integer and fractional quantum Hall regime. In samples with a smooth edge and an additional screening electrode, the propagation velocity of high-frequency signals depends on the number and width of the involved edge channels, and thus can be used to obtain an approximate electron density pro le. The amplitude of the transmitted signals oscillates with respect to the applied magnetic eld with maxima appearing close to integer and fractional bulk lling factors. While the data around lling factor 13 are qualitatively similar to those around lling factors 1 and 2, deviations appear around lling factor 23 . At odd lling factors, signal propagation in partially decoupled edge channels is observed. ? 1997 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional electron gases; Time-resolved transport measurement
Two-dimensional electron gases (2DEGs) in high magnetic elds have remarkable transport properties both in the DC and the high-frequency regime. Many of them are closely related to the electronic structure near the edge of the 2DEG, where the electron density decreases to zero. Under quantum Hall e ect conditions, this boundary region is characterized by the formation of alternating compressible and incompressible strips [1–5]. In the compressible regions, the so-called edge channels (EC) [6–8], a Landau level is pinned at the Fermi energy and therefore only partially occu∗ Correspondence address: Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA.
1386-9477/97/$17.00 ? 1997 Elsevier Science B.V. All rights reserved PII S 1 3 8 6 - 9 4 7 7 ( 9 7 ) 0 0 0 2 1 - 0
pied. This leads to a metallic behaviour with a constant electrostatic potential. In the incompressible regions, all occupied Landau levels are completely lled, and the Fermi energy is located in an energy gap, which is E = ˜!c , if we ignore the spin splitting. Hence, an electrostatic potential di erence can exist across an incompressible strip even in equilibrium. The coupling between adjacent ECs as well as the coupling between ECs and the bulk of the 2DEG may be weak enough for a non-equilibrium potential distribution to be established on a macroscopic length scale. As a result, ECs can be selectively populated and detected, a phenomenon observed as adiabatic transport in DC magnetotransport experiments [6–8]. But even
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though the EC picture is now commonly accepted for the integer quantum Hall (IQH) regime [3–5, 9–11], the electronic structure in the vicinity of the edge of a 2DEG under conditions of the fractional Quantum Hall (FQH) e ect is still under discussion [1, 2, 12– 19]. Theoretical calculations have shown that, in the FQH regime, the charge density pro le for a smooth boundary exhibits plateaus at certain fractional lling factors [16–19] similar to the plateaus at integer lling factors, while the edge reconstruction for a steep edge leads to counter owing edge states for certain lling factors [12–17], i.e. the electron density in the vicinity of the edge increases beyond its bulk value, before decreasing to zero. Time-resolved measurements of a voltage pulse in a 2DEG in strong magnetic elds are a promising experimental technique for obtaining information about the electronic structure near the edge of a 2DEG [20–22]. A high-frequency voltage pulse applied to a contact at the edge of a 2DEG generates edge magnetoplasmons (EMPs) [23], which are collective electronic excitations localized near the edge of the 2DEG. The voltage pulse is thus transformed into a wave packet of EMPs, and is transmitted along the edge adjacent to the injection contact. The EMP propagation direction is determined by the direction of the Lorentz force acting on the electrons. The EMP dispersion !(k) and group velocity vg = 9!=9k depends on the electron density n(x) near the edge, and on the electromagnetic environment of the 2DEG. Therefore, in measuring the pulse propagation velocity, one can study the electron density pro le near the edge. Here, we report on time-resolved magnetotransport measurements of a 2DEG with a smooth edge potential in order to study the dynamic edge excitations in the IQH and FQH regime. For the measurements, standard Hall-bar geometries made from AlGaAs=GaAs heterostructures are used. The 2DEG in sample A has a carrier concentration n0 = 2:2 × 1015 m−2 , a mobility = 50 m2 =Vs, and is located d = 90 nm below the surface. The data for sample B are n0 = 1:0 × 1015 m−2 , = 75 m2 =Vs, and d = 130 nm. In Fig. 1a, the sample geometry is shown together with a sketch of the experimental setup for time-resolved transport measurements. The largest part of the sample is covered with a metallic top gate. A long voltage pulse with a rise time of 150 ps and an amplitude Vin of several mV reaches contact 1 at time t = 0 with contact
Fig. 1. (a) Sketch of the sample geometry and measurement setup. Gated areas are shaded. A rectangular input pulse Vinput is applied to contact 1. The wave packet runs along the shorter boundary to contact 3. The transient potential at the intermediate contact 2 is detected using gate G2 as a sampling switch which is opened and closed by means of the applied strobe pulse. (b) Dependence of the transmission delay t0 on the applied magnetic eld (sample A). A sawtooth behaviour is superimposed onto an overall increase, with maxima or shoulders appearing close to the indicated integer and fractional bulk lling factors. (c) Calculated electron density of the innermost incompressible strip versus measured width of the compressible region at the edge for the data shown in (b). The density pro le extrapolated from these data (dashed line) is compared with a theoretical prediction (solid line), see text.
3 being grounded. The transient signal is detected at the intermediate contact 2 using the box-car technique described in Ref. [24]. All measurements were made in a 3 He= 4 He dilution refrigerator at a temperature of 70 mK with magnetic elds up to 13 T. The direction of the eld is chosen such that the applied pulse
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propagates along the shortest connection between the injection (1) and the detection (2) contacts. This distance is 160 m in sample A and 60 m in sample B. Fig. 1b shows the delay time t0 as a function of the applied magnetic eld for the magnetic eld range between 1 and 9 T, and an input amplitude of Vin = + 5 mV (sample A). The magnetic eld dependence can be described by a monotonic increase onto which sawtooth-like oscillations are superimposed. Close to the bulk lling factors 4, 2, and 1, the delay time has a maximum with a distinct asymmetry. The delay time increases steadily at the low magnetic eld side of the local maximum and drops rapidly at the high magnetic eld side. For the quantitative analysis, we use a macroscopic hydrodynamic description which yields the EMP velocity [24] vg =
d 9! e2 = · ; 9k h 0 r l
(1)
where d is the distance between 2DEG and top gate, and 0 r is the dielectric constant of the material between them. Except for fundamental constants and these material parameters, the velocity depends on the lling factor of the innermost incompressible strip and on the width l of the compressible region at the edge. While the overall increase of the delay time with increasing magnetic eld stems from the decreasing Hall conductivity e2 =h, we attribute the superimposed sawtooth behaviour to the oscillating width l. The observed asymmetry indeed indicates that the origin of the oscillations lies in the magnetic eld dependence of the electronic edge structure. As the magnetic eld increases, all compressible and incompressible strips shift towards the centre of the hallbar. At the indicated lling factors the innermost incompressible strip, which con nes the compressible edge region, vanishes in the bulk of the 2DEG, and the nearest incompressible strip, which is positioned closer to the edge, becomes the boundary line between edge and bulk. Thus, the width of the compressible edge region de ned as the sum of all edge channels, exhibits the same sawtooth oscillations as the measured delay time. Next, we concentrate on the lling factors = 1; 2; 4. Eq. (1) allows us to obtain the width l of the compressible edge region from the measured delay times. Assuming that the width of the innermost
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incompressible strip is small compared to the sum of all other compressible and incompressible strips ( = compressible edge region), we identify the width l with the position of the innermost incompressible strip. Since l measures the total width of the compressible region, l = 0 is de ned as the point where the electron density goes to zero. Knowing the electron density n = eB=h and the position of the incompressible strips, we can replot the data in the form of an electron density pro le. In Fig. 1c we compare the data for the magnetic eld ranges 1.9 T ¡B¡2.15 T, 2.9 T¡B¡4.4 T, and 6.5 T¡B¡8.8 T, where the lling factor of the innermost incompressible strip is = 4; 2; 1, respectively. The three sets of data clearly fall onto the same curve, and approach the bulk density far away from the boundary. This curve (dashed line) can thus be regarded as the electron density pro le n(l) without magnetic eld. The width of the depletion layer which separates the boundary of the 2DEG from the physical edge of the sample cannot be deduced from these measurements. Although di erent models have been used in order to calculate this electron density pro le theoretically [3–5, 25–27], none of them applies to our sample geometry where the edge of the 2DEG is de ned by etching away the surrounding heterostructure, before covering the entire surface with a metallic top gate. This a ects, in particular, the screening length inside and outside of the 2DEG. Therefore, we do not expect to nd a perfect agreement when comparing our extracted electron density pro le with the existing theoretical prediction. The solid line in Fig. 1c shows the result of a self-consistent calculation for the case that the edge of the 2DEG is de ned by applying a negative voltage V = − Egap =2e to a top gate [5]. Without any tting parameter, the overall agreement is satisfactory, although the predicted pro le rises too fast close to the edge, and approaches the bulk value too slowly. Calculations by other authors show the same trend [3, 4, 25–27]. Not only the delay, but also the amplitude of the rst step-like increase reveals details about the electronic structure at the edge of the 2DEG [20–22]. For a small amplitude of the input pulse, the entire signal is transmitted in the edge channels, and the amplitude of the leading edge appearing at the detection contact is equal to the input amplitude. For larger amplitudes, a part of the signal penetrates into the bulk and
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Fig. 2. Amplitude of the rst step V0 as a function of the applied magnetic eld (sample B). Maxima or shoulders appear close to the indicated integer and fractional lling factors. The solid lines serve as a guide to the eye.
appears at the detection contact at a later time. The transmitted signal consists of two parts: a fast step-like increase up to a voltage V0 , followed by a slower increase until saturation. This non-linear dependence of the shape of the transmitted signal on the amplitude of the input signal re ects the break-down of non-local transport. Fig. 2 shows the amplitude V0 as a function of the applied magnetic eld for bulk lling factors down to 1 3 (sample B). For large magnetic eld values, the initial step is partially rounded at the top, while the onset, and thus the delay time, remains well de ned [28]. Therefore, we apply a suciently large voltage pulse, Vin = 5 mV, and use the amplitude of the transmitted signal 5 ns after the arrival of the wave packet at the detection contact as a measure of V0 . As the noise in the measurement was strongly enhanced around lling factor = 12 , no accurate determination of V0 was possible between 7 and 10 T. The most prominent feature apparent from Fig. 2 are the similarities that exist between integer and fractional lling factors, which both show up as maxima (or at least shoulders) of V0 which are located close to these lling factors. As the voltage V0 drops across the innermost incompressible strip, the width of this strip determines the measured values for V0 , and the largest values for V0 are found, when this width is largest.
The width of an incompressible strip, in turn, is determined by the value of the energy gap, which explains the relative amplitudes of the observed maxima in V0 . Of particular interest is the behaviour around = 23 , where the position of the corresponding maximum amplitude seems to be signi cantly shifted to higher magnetic elds. Since DC transport shows a larger energy gap for = 23 compared to = 35 , the maximum at 6.6 T has to be attributed to lling factor 23 . We interprete this shifted maximum position as an indication of a magnetic- eld-dependent charge density pro le with a maximum within the edge region, as is predicted for a suciently steep edge [12–17]. In this case, an incompressible strip with = 23 can exist even for a bulk lling factor ¡ 23 . The di erences between = 1, 13 and 23 can be explained if such a maximum in the electron density pro le appears only for certain lling factor ranges, e.g. around = 23 but not around = 1 or 13 . Further indications for this scenario are found in the magnetic eld dependence of the delay time [28]. Fig. 3a compares the transmitted signals for positive and negative input amplitude at bulk lling factor = 3 (sample A). For a positive input signal, the shape of the signal is characterized by an abrupt rise at time t0+ = 0:86 ns, while for a negative input signal, the initial step at t0− = 0:76 ns is followed by a second step-like increase at t1− = 2:8 ns. The di erence between positive and negative polarity arises from the fact that the width of an incompressible strip is in uenced by the potential di erence across it [29]. When the voltage at the edge is positive with respect to the bulk, the width of the incompressible strips is reduced compared to its equilibrium value and vice versa. Thus, for negative polarity, more incompressible strips can become suciently wide to decouple the adjacent compressible regions. Fig. 3b illustrates the situation encountered in Fig. 3a. For positive polarity the compressible regions and are coupled and carry one common EMP mode. Eq. (1) with = 3 yields l = 0:49 m for the total width of the compressible edge, i.e. for the sum of the regions and . For negative polarity the = 2, incompressible strip is suf ciently wide to decouple the and regions, allowing for the existence of an EMP mode in each of them. Since these modes propagate at di erent velocity, the transmitted signal consists of two step-like increases instead of only one. For the quantitative analysis of
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In conclusion, we have shown that time-resolved magnetotransport measurements are a powerful tool for the investigation of the electronic structure near the edge of a 2DEG. The measured transmission delay times and amplitudes allow us to clarify the role of the compressible and incompressible strips in magnetotransport experiments in general. We would like to thank B. Farid, R. Gerhardts, S. Mikhailov and D. Pfannkuche for useful discussions, A. Yacoby for critical reading of the manuscript, and K. Eberl for providing the heterostructures. N.B.Z. was supported by Alexander von Humboldt Foundation. References Fig. 3. (a) Leading edge of the signal transmitted through the 2DEG for bulk lling factors = 3 (sample A). The positive pulse is transmitted by one EMP mode, whereas two EMP modes contribute to the transmission of the negative pulse. (b) Sketch of the electronic structure at the edge of the sample for a lling factor = 3 + . Compressible areas (crosshatched) are separated by incompressible strips with integer lling factor. In the presence of a shielding metallic plane, the regions and are partially decoupled.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
the measured delay times, we have to adapt Eq. (1) by replacing by = r − l , the lling factor difference between the incompressible strips surrounding the compressible region, i.e. = 2 for the region and = 1 for the region. In addition, the geometrical factor d=0 r l in Eq. (1) has to be modi ed, since mixing of these EMP modes due to capacitive coupling between the regions and has to be taken into account. Due to the existence of the neighbouring compressible strip, the capacitance between each individual compressible strip and the top gate is increased, which leads to a reduction of the EMP propagation velocity. Treating the regions and as a coplanar stripline, we estimate the magnitude of the additional capacitance to be about half the value of the parallel plate capacitance to the top gate. The corresponding EMP velocities are reduced by about 60%, leading to a width of 18 m for region and 32 m for region , which is in good agreement with the entire width found for positive polarity.
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