Solid-State Electronics Vol. 42, No. 7±8, pp. 1179±1182, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98 $19.00 + 0.00 S0038-1101(97)00325-0
DETECTION OF FRACTIONAL EDGE CHANNEL BY QUANTUM POINT CONTACTS MASATO ANDO1, AKIRA ENDO1, SHINGO KATSUMOTO1,2 and YASUHIRO IYE1,2 1
Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan 2 CREST, Japan Science and Technology Corporation (JST), 1-4-25 Mejiro, Toshima-ku, Tokyo 171, Japan (Received 23 October 1997; in revised form 23 October 1997)
AbstractÐConductance spectrum through a quantum point contact in quantum Hall liquid is investigated by its dependence on the excitation voltage. The occurrence of conductance plateaus dierent from the bulk Hall conductance indicates a gradual con®nement potential at the sample edge. In the low excitation limit, the point contact is pinched o at smaller negative gate voltage for higher magnetic ®eld. This trend is interpreted in terms of the magnetic ®eld induced shift in the energy spectrum. # 1998 Elsevier Science Ltd. All rights reserved
Quantum point contact (QPC) in a two dimensional electron gas (2DEG) is a simple yet remarkable device that the conductance through a narrowed QPC can take quantized plateaus dierent from that corresponding to the bulk of the system. When the 2DEG is in an integer quantum Hall (IQH) state with ®lling factor n, the two-terminal conductance through the QPC often shows a series of plateaus at G = ne2/h; (n = 1, 2, . . . , n) as the QPC is narrowed. This phenomenon is given a natural explanation in terms of transmission and re¯ection of edge channels, each of which carries a conductance of e2/h. Conductance quantization takes place when the chemical potential lies in the gap between the energy levels of two channels. However, the situation is more complicated in the case where the 2DEG is in a fractional quantum Hall (FQH) state, where the electron correlation plays a crucial role. If one attempts to understand the transport properties of a FQH liquid in terms of edge channels, one inevitably has to consider an edge channel that carries a fractional conductance in units of e2/h. Also, the structure of the edge state, i.e. the number of channels and the conductance associated with each channel, cannot be derived straightforwardly as in the case of the IQH eect where the edge channels have one-to-one correspondence with the bulk Landau level. A wide range of studies have addressed the issue of edge channels in FQH liquids. Beenakker has argued that the edge of the 2DEG consists of alternating strips of compressible and incompressible liquids. The edge channels are identi®ed with those compressible strips, which respond to the elec-
tric ®eld and actually convey the current. In this picture, the edge con®nement potential must be smooth enough for the incompressible strips to form at the edge, and thus the structure of the edge channels depends on the smoothness of the con®ning potential as well as the energy gap of incompressible states. On the other hand, there is another picture put forward by MacDonald based on an assumption of a sharp boundary in which the topological order of the quantum Hall state directly shows up in the structure of the edge[3,4]. There are some numerical studies which indicate a transition between the above two pictures according to the smoothness of the con®ning potential[5,6]. In this paper, we present experimental results of two-terminal conductance measurement through a QPC in quantum Hall liquids that indicate the existence of fractional edge channels and are in accordance with the smooth con®nement model. 2. EXPERIMENT
The structure of the device used in this study is shown in the inset of Fig. 1. The device is fabricated from an MBE-grown GaAs/AlGaAs single heterojunction with electron density n = 1.23 1015 mÿ2 and mobility m = 97 m2/Vs at 4.2 K. A standard Hall bar is formed by photolithography and wet etching, and a pair of gold split gates with an opening of 400 nm is deposited on the top using electron-beam lithography and lift-o process. A negative bias voltage Vg applied to the split gate de®nes a QPC in the 2DEG. The sample is set in the mixing chamber of a dilution refrigerator placed in a 15 T superconducting solenoid and is cooled down to 30 mK. Figure 1 shows the mag-
M. Ando et al.
Fig. 1. Magnetoresistance and Hall resistance of the sample when the split gate is open (Vg=0). Inset: schematic view of the sample structure.
netoresistance and Hall resistance of the sample when the split gate is connected to the ground level. The Hall resistance shows clear plateaus at integral and fractional ®llings such as n = 2/3, 3/5, 2/5 and 1/3. The magnetoresistance falls to zero at ®llings n = 2/3 and 1/3, and also shows a distinct minima at n = 3/5 and 2/5. Transport through the QPC is investigated by measuring the two-terminal conductance and the voltage±current (V±I) characteristics in a d.c. voltage driven mode.
3. RESULTS AND DISCUSSION
Figure 2 shows the conductance through the QPC at ®ve dierent QHE states n = 1, 2/3, 3/5, 2/
Fig. 2. Conductance G as a function of gate voltage Vg at ®ve dierent quantum Hall states. The data for n = 3/5 and 2/5 are adjusted so that it gives the correct quantized conductance at Vg=0.
5 and 1/3. The source-drain excitation voltage is set to VSD1100 mV. For n = 3/5 and 2/5, a constant resistance is subtracted from the original data to compensate for the non-zero Rxx. It seems that the QPC is pinched o at smaller negative Vg for lower ®lling states. All ®ve curves have characteristic shapes such as peak structures near the pinch-o threshold and a few wider dips at intermediate conductance values. Two wide dips occur in the n = 1 state, one in the n = 2/3 and 3/5 states, and none in the n = 1/3 state. It is noted that the dip in the n = 2/3 and 3/5 states forms a clear plateau at conductance value of G = 1/3 (the conductance will be written in units of e2/h throughout this paper.). Figure 3 shows the results of similar measurements performed with various source-drain voltages ranging from 10 to 1000 mV for the n = 1, 2/3 and 1/3 states. The lower Vg side of the conductance curves depends sensitively on the excitation voltage VSD. This means that the V±I characteristics are highly non-linear in this region. This non-linearity sets in at smaller vVgv for lower ®lling states, or higher magnetic ®elds. The behavior of the dip structures mentioned above is particularly noteworthy. For n = 2/3, the minimum of the dip develops to a well-de®ned plateau at exactly G = 1/3. For n = 1, the two minima seem to approach G = 1/3 and 2/3, respectively. As the quantized conductance in a QPC is a signature of well-de®ned current carrying channels, the clear G = 1/3 plateau in the n = 2/3 state strongly suggests that the edge channel consists of two channels, each with conductance 1/3. This is in accordance with the smooth con®nement model, but not with the sharp con®nement model because the latter asserts the edge of n = 2/3 state consists of two channels with conductance 1 and ÿ1/3 to realize the total conductance of 2/3. Also the dips in the n = 1 case, whose minima are approaching 2/3 and 1/3 imply that the corresponding incompressible states, which are not related with the bulk quantum Hall state n = 1, are formed near the edge. The situation is similar for the G = 1/3 plateaus in the n = 3/5 and n = 2/5 states. Therefore, the con®nement potential is, at least in the vicinity of the metal-gated QPC, suciently smooth to allow the formation of strips of incompressible liquid dierent from the bulk state. The absence of such plateaus at conductance smaller than 1/3 may indicate that there is no stable FQH state with ®lling smaller than 1/3 in the present sample. The reason why the conductance does not create a simple quantized plateau but a wide dip with a ¯at bottom in some cases is not clear at present. Though the V±I characteristics are generally nonlinear, the pinch-o voltage, or the gate voltage when the conductance falls to zero is roughly constant for suciently small excitations. In order to illustrate this, the pinch-o voltage is plotted against shows VSD in Fig. 4(a). The pinch-o voltage V p.o. g
Detection of fractional edge channel
Fig. 3. G vs Vg for various source-drain voltage VSD for (a) n = 1, (b) n = 2/3 and (c) n = 1/3. Note the dierent scale of the y-axis in these ®gures.
an upward shift with decreasing VSD until it reaches a constant value at around Vg=50 mV. The overall shape of the conductance curves near the pinch-o threshold suggests that the tail part of the energy spectrum in the area near the QPC consists of a range of discrete levels which are fairly well separated from the continuum of the lowest 1D subband. Although there is some ambiguity in de®ning the pinch-of threshold, V p.o. in Fig. 4(a) is g determined by a smooth extrapolation of the falling conductance curve to zero. It therefore gives a measure of the energy that separates the continuum in the from the discrete levels. In Fig. 4(b), V p.o. g low-excitation limit is plotted as a function of the
applied magnetic ®eld. If the QPC near the pincho is approximated by a one-dimensional channel with a parabolic con®nement potential V
m*o 20 2 y 2
the bottom of the continuum in magnetic ®eld is given by q h eB epc V0 o 20 o 2c ; o c :
2 2 m* is given by the conThe pinch-o threshold, V p.o. g dition
M. Ando et al.
depend on the magnetic ®eld, the pinch-o voltage provides a probe of epc. The solid and dashed curves represent smooth and sharp con®nement, respectively. The factor a is taken to be a = 2 10ÿ2 to ®t the data. That this is a reasonable value can be inferred from our recent experimental analysis of the Weiss oscillation. The fact that the plot is almost linear to the magnetic ®eld suggests that the energy level in the QPC is dominated by cyclotron energy oc more than the curvature of the con®ning potential o0, and that the con®nement potential is smooth. To conclude, we have studied the two-terminal conductance of a quantum point contact in quantum Hall liquids. The conductance quantization at fractional values dierent from the bulk state suggests that the smooth con®nement model is more appropriate. The magnetic ®eld dependence of the pinch-o voltage also suggests that the con®nement is smooth. AcknowledgementsÐWe thank M. Hirasawa and K. Imura for valuable discussion. This work is supported by Grant-in-Aid for Scienti®c Research from the Ministry of Education, Science, Sports and Culture. Fig. 4. (a) Pinch-o gate voltage V p.o. plotted against g source-drain voltage at quantum Hall states n = 1/3, 2/5, p.o. 3/5, 2/3, 1, 2 and B = 0. (b) V g at lowest excitation plotted against magnetic ®eld. The vertical axis is common to (a). (c) Cross-sectional view of split gate and possible potential pro®le when the QPC is close to the pinch-o.
where a is a reducing factor for the gate bias that determines the potential at the 2DEG plain. Supposing that the chemical potential m does not
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