Even-denominator fractional quantum Hall state in conventional triple-gated quantum point contact

The even-denominator states have attracted considerable attention owing to their possible applications in future quantum technologies. In this letter, we first report a 3/2 diagonal resistance, indicating the existence of a 3/2 state in a nanometer-sized triple-gated quantum point contact (QPC) fabricated on a high-mobility (not ultra-high-mobility) single-layer two-dimensional (2D) GaAs wafer. The center gate plays a crucial role in realizing the QPC's 3/2 state. Our observation of the 3/2 state using a conventional QPC device, which is a suitable building block for semiconductor quantum devices, paves a new path for the development of semiconductor-based quantum technologies.

The even-denominator states have attracted considerable attention owing to their possible applications in future quantum technologies. In this letter, we first report a 3/2 diagonal resistance, indicating the existence of a 3/2 state in a nanometer-sized triple-gated quantum point contact (QPC) fabricated on a high-mobility (not ultra-high-mobility) single-layer twodimensional (2D) GaAs wafer. The center gate plays a crucial role in realizing the QPC's 3/2 state. Our observation of the 3/2 state using a conventional QPC device, which is a suitable building block for semiconductor quantum devices, paves a new path for the development of semiconductor-based quantum technologies.
Template for APEX (Jan. 2014) 2 Fractional quantum Hall states with even denominators have attracted attention owing to their novel carrier interactions and possible applications to error-free quantum manipulations.
A well-known example is the ν = 5/2 state where a non-Abelian topological order is expected. [1][2][3] To understand the background physics of the even-denominator fractional quantum Hall state and to also manipulate such a state in future quantum technologies, it is important to study the possibility of even-denominator states at the most fundamental ground Landau level, such as the ν = 3/2 state. Despite the clear observation of the 5/2 state, the ν = 3/2 state was not observed in a single-layer two-dimensional (2D) GaAs system even with an ultra-high mobility of >4×10 7 cm 2 /Vs. 4) On the contrary, even-denominator fractional quantum Hall effects were observed at the ground Landau level (ν = 1/2, 3/2) in certain systems, such as GaAs wide quantum wells with an effective bilayer carrier distribution, 5,6) GaAs bilayer systems, 7,8) ZnO/MgZnObased quantum wells with a parallel field, 9) and bilayer graphene, 10) reflecting not only spin but also pseudospin and/or valley freedom characteristics. In a GaAs wide quantum well, both symmetric and antisymmetric states are used to reduce the Coulomb energy stabilizing the 331 state. 11,12) This state does not follow non-Abelian statistics; nevertheless, a change to the non-Abelian phase can be predicted by controlling parameters such as the tunnel coupling between the layers. 13,14) Another interesting and important study reported the formation of the ν = 3/2 state at the microscopic scale with an area of approximately 4 μm 2 . 15,16) These observations were made using a single-layer 2D GaAs system (and not a bilayer system) with an ultra-high mobility of >10 7 cm 2 /Vs, despite the lack of the ν = 3/2 state in wide 2D systems starting from the same wafer. In possible applications of even-denominator non-Abelian systems to faulttolerant quantum computation, the key issue is braiding; a small-area interference device may address this issue. [17][18][19] The clear observation of the ν = 3/2 state in the microscopic regime is an interesting finding that can further the development of quantum information technology based on semiconductor microstructures.
On the other hand, the confined condition possibly modifies the state characteristics.
Although thermal conductance 20) and NMR Knight-shift 21) measurements support the non-Abelian nature of the ν = 5/2 state for a large sample, tunneling experiments have shown contradicting results for microstructures with the ν = 5/2 state. Radu et al. 22) suggested a Template for APEX (Jan. 2014) 3 non-Abelian state; however, similar experiments conducted by Lin et al .23) and Baer et al. 24) support an Abelian 331 state. Notably, recent experiments conducted by Fu et al. 25) suggested that the potential shape of the confinement plays an important role. The obtained characteristics favor a non-Abelian state for a soft confinement at a relatively high temperature. 25) Moreover, in a microstructure, we can expect a complicated edge channel scheme, including counter-propagating channels, which are hidden in large-scale samples. [26][27][28][29] These results have motivated us to test the possibility of a 3/2 state formation in a conventional well-established quantum point contact (QPC) that can flexibly control the shape of one-dimensional (1D) confinement potential. Our experiments raise questions on the need for ultra-high-mobility wafers. QPCs are the most important building blocks of future semiconductor devices in quantum technology; therefore, our study has important implications for advancements in semiconductor-based quantum information technology.
In this study, we experimentally investigated the transport characteristics of a triple-gated QPC with a center gate in the fractional quantum Hall regime. The fabricated QPC structure, schematically shown in Fig. 1(a) along with its scanning electron microscopy (SEM) image ( Fig. 1(b)), has a nominal length L of 400 nm, a separation between the split Schottky gates W of 600 nm, and a 200-nm-wide center gate. In addition to the split Schottky gates and center gate, this device has a back gate. We can control the 2D electron density in the 20 nm GaAs quantum well sandwiched by Al0.33Ga0.67As barriers using the back gate voltage (Vbg).
The quantum well is located 175 nm below the surface. The low-temperature electron mobility is μ = 1.47 × 10 6 cm 2 /Vs at electron density of 1.8 × 10 11 cm −2 .
The transport characteristics of the triple-gated QPC were first measured under a zero magnetic field. Figure 1(c) shows the results. We observed quantized conductance at a temperature of 100 mK, particularly when a positive Vcg (center gate voltage) was applied, reflecting the successful fabrication of the QPC device. As expected for a triple-gated device, the conductance curve gradually shifts to the negative Vsg (voltage applied to the split Schottky gates) region with increasing Vcg. 30 Next, we applied a magnetic field B = 6 or 7 T and measured the characteristics in the fractional quantum Hall regime. We measured the diagonal resistance (Rdiag = VD/IAC) as schematically shown in Fig. 1(a). Although a magneto-depopulation of the 1D sub-band increases the plateau length, 31) there is a gradual shift in the Rdiag curve with the center gate voltage Vcg, similar to the zero-field characteristics shown in Fig. 1(c), when the filling factor of the bulk region, νbulk, is not 5/3. The situation changes when νbulk is set at 5/3, as shown in Fig. 2(a). When Vcg is increased from −0.1 to 0.1 V, the Rdiag curve gradually shifts, and the Rdiag value decreases with Vcg, as expected. However, with further increase in Vcg, Rdiag starts to increase and finally stabilizes at 17.2 kΩ, which corresponds to the νQPC = 3/2 value. This behavior can be clearly confirmed in Fig. 2(b), which represents Vcg dependence of Rdiag at Vsg = −1.05 V and B = 6 T. This is a unique behavior and the main experimental finding of this study. Once decreased, Rdiag starts to increase with Vcg despite the fact that a positive Vcg supplies electrons in the QPC channel. This suggests that the inside of the QPC is maintained at νQPC = 3/2. The bulk region on both sides of the QPC is maintained at νbulk = 5/3 in this experiment; hence, the diagonal resistance corresponding to νQPC = 3/2 suggests ν = 1/6 scattering channels at both edges of the QPC, as discussed by Fu et al.. 16) The expected edge channel schematic diagram near the QPC center region is discussed in the supplementary information S1.
We conducted similar measurements using conventional QPCs with the same nominal width of W = 600 nm but without the center gate to prove the importance of the center gate (see Supplementary Information S2). The obtained results indicate that the lateral spread of the depletion from the side gates is too high, and therefore, the electron density in the QPC becomes lower than that at νQPC = 3/2 for the rather narrow channel with a width of 600 nm.
This may explain why a wider (approximately 2 µm) confinement was required to observe the νQPC = 3/2 structure in previous experiments, 15,16) where there were no center gates.
A comparison between the two devices (with and without the center gate) confirms the essential role of the center gate in forming the ν = 3/2 structure in the conventional QPC whose width is <1 µm. A positive Vcg increases the electron density in the narrow channel, reaching νQPC = 3/2 even for a channel width of 600 nm. The center gate also contributes to Template for APEX (Jan. 2014) 5 effectively suppressing the disorder potential. In our previous QPC experiments using a medium-mobility wafer (µ = 3×10 5 cm 2 /Vs at n = 2×10 11 cm −2 ), clear quantized conductance could be observed under the application of positive Vcg; otherwise, there was no quantized conductance due to the influence of strong potential disorder. 32) This probably suggests another important role of the positive center gate voltage, i.e., effective screening of the disorder potentials. This can partly explain why the even-denominator fractional quantum Hall states were observed in QPCs fabricated on high-mobility 2D systems and did not require ultra-high-mobility systems. Furthermore, we experimentally verified how the plateau value varies with the experimental parameters, such as Vbg and magnetic field. Figure 3(b) shows the appearance of the plateau regions under varying Vbg. As shown in the inset of Fig. 2(a), Rxx becomes minimum almost at the center of the νbulk = 5/3 plateau in Rxy. On both sides, Rxx, i.e., the bulk resistance, gradually increases, particularly in the higher Vbg side. Notably, the Rdiag value reflects this series resistance change in the νbulk = 5/3 region, as shown in Fig. 3(c).
Despite such an effect of the series resistance, the Vsg range in which the νQPC = 3/2 plateau appears is almost constant in the Vbg region of 1.58-1.64 V (Fig. 3(b)), independent of the series resistance of the bulk, namely backscattering in the bulk region. This result suggests the robust feature of the νQPC = 3/2 plateau. A similar behavior can be observed when the magnetic field is changed, as shown in Figs. 3(d) and (e). The electron density corresponding to the νbulk = 5/3 decreases with the magnetic field, with the threshold Vsg value for the formation of the constricted region becoming less negative. Correspondingly, the QPC 3/2 plateau region also moves to less negative Vsg with decreasing magnetic field (Fig. 3(d)).
Template for APEX (Jan. 2014) 6 The νbulk = 5/3 fractional state becomes less pronounced with decreasing magnetic field, and the series resistance increases, as shown in Fig. 3(e). Finally, the plateau value largely deviates from the exact 3/2 value at B = 4 T. Nevertheless, the appearance of the wide plateau region in Fig. 3(d) once again demonstrates the robustness of the formation of the QPC 3/2 state. When we increase the measurement temperature, the bulk 5/3 state becomes obscure and the νQPC = 3/2 plateau disappears similar to the low magnetic field case.
Here, the accuracy of the plateau quantization should be considered. In previous reports, 15,16) the plateau resistance was reported to be within ±0.02% of the exact 2/3×(h/e 2 ) resistance for the approximately 4 μm 2 constricted region when using an ultra-high mobility 2D system at a very low temperature of approximately 20 mK. A precise quantization is the essence of the quantum Hall effect. However, as discussed, the plateau values are affected by the series resistance in the bulk areas. When not using ultra-high mobility 2D systems, the Rxx value of the bulk area is not completely zero; hence, we cannot expect extremely accurate quantization in our experimental setup. Although we cannot expect a good quantization accuracy, our ±0.8% accurate plateau value and experimentally obtained characteristics clearly suggest the existence of the νQPC = 3/2 state not in ultra-high-mobility but in high-mobility QPC with the conventional size of < 1 µm.
The origin of the νQPC = 3/2 state is unclear. Although the 20 nm quantum well used in the present experiment excludes the possibility of bilayer formation, unlike wide wells, 5,6) a complicated edge channel in the microscopic structure may allow the formation of the local hole edge channel and stabilize the νQPC = 3/2 state by the similar mechanism with the 331 state. Theoretical studies have indicated an important role of the finite quantum well thickness in producing the incompressible fractional quantum Hall state, particularly the ν = 5/2 state. 33,34) The modification of the vertical electron distribution by the front and back gates might play a similar role as the quantum well thickness. It also contributes to a change in the edge characteristics. Whether the νQPC = 3/2 state is Abelian or non-Abelian is a sensitive question; nevertheless, the flexible tunability of the confinement potential of the triple-gated QPC with a back gate may allow switching between the Abelian and non-Abelian states. 13,25) Further study is required to understand the role of spin freedom. 35) Tunneling experiments through the constriction [22][23][24][25] and resistively-detected nuclearmagnetic resonance 21,36,37) can help clarify the detailed characteristics of the νQPC = 3/2 state.
Template for APEX (Jan. 2014) 7 In conclusion, we have demonstrated appearance of the even-denominator fractional quantum state in the conventional triple-gated QPC device. We found the plateau feature corresponding to the νQPC = 3/2 state in the diagonal resistance when the bulk filling sets to νbulk = 5/3 and the center-gate voltage is positive enough to keep electron density in the center region of the QPC at the 3/2 filling. Another important point of our finding is that we can see the QPC 3/2 state by using conventional high-mobility but not ultra-high-mobility device thanks to the screening effect of the center gate. Although further studies are necessary, our finding suggests a possible formation of the unique even-denominator fractional-quantum-Hall state in the nanoscale local area by conventional high-quality twodimensional systems that can be easily achieved with modern crystal growth technology. It will pave a way to open new quantum information devices based on semiconductor quantum systems. It will also give us a hint to consider edge channel formation in the fractionalquantum-Hall regime.

S1. Schematic diagram near QPC at νQPC = 3/2
The physics behind the formation of νQPC = 3/2 structure is still unclear yet so that it is difficult to draw an exact edge channel diagram. Nevertheless, we show in Fig. S1 an expected edge channel schematic diagram near the QPC center region when νQPC = 3/2 is

S2. Characteristics of QPC without center gate
To prove the important role of the center gate in defining the νQPC = 3/2 state, we conducted similar measurements using conventional QPCs without the center gate. This device has a gap of 600 nm between the split gates, as schematically shown in Fig. S2 (a), and is fabricated on the same wafer. The successful fabrication of the QPC device is confirmed by the clear quantized conductance step observed at B = 0 T (Fig. S2 (b)). The quantized step becomes prominent with increasing Vbg, hence electron density of the two-dimensional system. In the next step, we set the device at 7 T, tune the bulk filling at νbulk = 5/3 by controlling Vbg, and then measure Rdiag as a function of Vsg. Figure S2 (c) shows the obtained result. There is no observation at Rdiag corresponding to νQPC = 3/2. From the change in the QPC conductance slope observed at B = 0 T (Fig. S2(b)), we can estimate the Vsg value at which the electrons under the split gates are fully depleted, and the center of the QPC changes to the constricted wire region. When Vbg = 3.54 V, the wire confinement is expected in the region of Vsg < −0.75 V. At this point, Rdiag increases beyond the value expected at νQPC = 3/2 (17.2 kΩ). This result indicates that the lateral spread of the depletion from the side gates is too high, and therefore, the electron density in the QPC becomes lower than that at ν = 3/2 for the rather narrow channel with a width of 600 nm. The situation was the same when we changed Vbg and realized νbulk = 5/3 at different magnetic field. The parameter, Vbg, changes the 2D bulk electron density. A clear quantized conductance is observed in the entire Vbg region, reflecting a high-quality QPC device. (c) Example of Rdiag as a function of Vsg at B = 7 T. The filling of the bulk region is set to νbulk = 5/3 by applying Vbg = 3.54 V to the back gate. The shaded area indicates the 2D region where depletion under the split gates is incomplete, judging from the 1D-to-2D transition estimated from (b). For the 600-nm-wide QPC without the center gate, the electron density in the center region of the QPC is always lower than the density corresponding to the 3/2 filling even when the magnetic field (then Vbg) is changed. All measurements were performed at 100 mK.