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Single-Carrier Transport in Graphene/hBN Superlattices

Graphene/hexagonal boron nitride (hBN) moir\'e superlattices have attracted interest for use in the study of many-body effects and fractal physics in Dirac fermion systems. Many exotic transport properties have been intensively examined in such superlattices, but previous studies have not focused on single-carrier transport. The investigation of the single-carrier behavior in these superlattices would lead to an understanding of the transition of single-particle/correlated phenomena. Here, we show the single-carrier transport in a high-quality bilayer graphene/hBN superlattice-based quantum dot device. We demonstrate remarkable device controllability in the energy range near the charge neutrality point (CNP) and the hole-side satellite point. Under a perpendicular magnetic field, Coulomb oscillations disappear near the CNP, which could be a signature of the crossover between Coulomb blockade and quantum Hall regimes. Our results pave the way for exploring the relationship of single-electron transport and fractal quantum Hall effects with correlated phenomena in two-dimensional quantum materials.

Graphene stacked on hexagonal boron nitride (hBN) with the crystallographic axis alignment generates a long-period superlattice potential because of a lattice-constant mismatch of ~1.8%. [1][2][3][4][5][6][7] This moiré superlattice breaks inversion symmetry and modifies the electronic band structure of graphene, leading particularly to an energy gap at the charge neutrality point (CNP) and the emergence of satellites of the CNP. Previous studies have intensively examined the exotic transport properties in graphene/hBN superlattice systems, such as fractal quantum Hall effects, 1-3 topological valley Hall effects, [8][9][10] and correlated superconductivity. 11,12 On the other hand, single-carrier transport properties in graphene have been investigated by employing graphenebased quantum dot (QD) devices. [13][14][15][16][17][18][19][20][21][22][23][24][25] To form a potential barrier for graphene QDs, geometric patterning has been performed on nanostructures by etching [13][14][15][16][17][18][19] or applying a perpendicular electric field to bilayer graphene (BLG). [20][21][22][23][24][25] For etching, the graphene channel must be narrowed to a few tens of nanometers to obtain a sizable gap. [13][14][15][16][17][18]26 Consequently, the carrier transport property is dominated by the unintentional carrier localization induced by edge disorders, leading to an unstable dot configuration. [13][14][15][16] Additionally, the previous devices fabricated using graphene flakes on SiO2 substrates suffered from strong charge inhomogeneity due to surface roughness and substrate impurities, [13][14][15][16][17][18] which further disturbed the dot definition. On the other hand, with recent progress in fabrication technologies such as hBN encapsulation and the fabrication of a graphite back gate 3 through an all-dry transfer process, 27 high-quality BLG-based QD devices defined by electrostatically-induced potential barrier have been achieved under a perpendicular electric field. [22][23][24][25] In this work, to study the single-carrier transport in graphene superlattices, we fabricated a BLG/hBN moiré superlattice-based QD device (Figure 1c-e), in which the double dots are defined 3 by geometric patterning and local gating with the contribution of the superlattice potential. Note that widths of constrictions and diameters of the dots were slightly large compared to the previous graphene QD devices, [13][14][15][16][17][18] for avoiding strong potential fluctuation which leads to unstable dot configuration. We demonstrate highly controllable single/double dot operation in the range not only near the CNP, but also near the hole-side satellite. Furthermore, near the CNP, the Coulomb blockade effect is found to be suppressed under a perpendicular magnetic field, which could be a signature of the collapse of single-electron tunneling in the quantum Hall regime.
To elucidate the influence of the superlattice potential, we fabricated a Hall bar close to the double dots on the same heterostructure, as shown in Figure 1a,b, and measured them in the same environment. Note that the devices were fabricated in a region where interfacial bubbles were absent, which was achieved through a bubble-free transfer technique. 28 The devices consist of onedimensional edge-contacting electrodes 27 and the graphite back-gate. The device fabrication process is detailed in Supporting Information. All transport measurements were performed in a dilution refrigerator with the bath temperature of 40 mK. The effective electron temperature was ∼160 mK estimated by fitting of the Coulomb peak with the orthodox Coulomb blockade model 29 (shown in Supporting Information). The double-dot device was measured in two-terminal DC configuration, while the Hall bar device was measured using AC lock-in techniques with an excitation current of 10 nA and a frequency of 17 Hz. A superconducting magnet was used to apply a perpendicular magnetic field up to 6 T.
To confirm the superlattice property in the double dots, we first study the transport characteristics of the Hall bar. Figure 1f shows a map of the longitudinal resistivity ρ of the Hall bar as a function of the back-gate voltage VBG and a perpendicular magnetic field B. We observe well-resolved constant-filling-factor regions corresponding to the filling factor ν = ±4NLL (NLL is 4 the Landau level (LL) index). These regions fan out from VBG ~ 0 V where the carrier type is switched, indicating that the CNP is located at VBG ~ 0 V. In addition, lines fanning out from VBG ~ ±15 V are observed, which correspond to the second-generation Landau fan, and its origin at B = 0 T can be identified as the satellites. [1][2][3] From the distance between the CNP and the satellites, we estimate the moiré wavelength to be λ ~ 10.7 nm, 2 which corresponds to a BLG/hBN crystallographic-axis alignment angle of θ ~ 0.85°. 4 Moreover, as shown in Figure 1g  coupling between gate LDG and left dot is larger than that between gate RG and right dot, which is in good agreement with the device structure. By using the gate CG located between the dots (the 6 position of the gate CG is shown in Figure 1d), we tune C m and the dot configuration to the strongly coupled double-dot regime ( Figure 3c) and the single-dot-like regime ( Figure 3d).
Furthermore, each dot can be individually operated by tuning the gate voltage VLDG and VRG. consequence, the superlattice device shows well-defined closed double dots with remarkable controllability. Note that excited states are not observed, possibly because the single-particle level spacing is too small because of the large dot diameter of ~135 nm. 13,18 Next, we discuss the transport property at the dip near the hole-side satellite. Figure 4a shows the charge-stability diagram drawn by fixing VBG at −13.13 V. A honeycomb structure is observed, similar to the suppression region near the CNP, but with a large background current. As in the above discussion, the dot configuration can be switched between single and double dots. By tuning the gate voltages VLDG and VRG, the oscillation of conductance (G = ID/VSD) can be made more periodic (upper panel in Figure 4b). Under the same condition, non-lifted Coulomb diamonds are observed (lower panel in Figure 4b). These results indicate the interference effect in the open dot system (a Fabry-Pérot interferometer), in which the dot (cavity) is strongly coupled with source and drain leads. 30,31,33 From the upper panel in Figure 4b, the average peak spacing is estimated to be BG peak ~ 11.3 mV. In the Fabry-Pérot cavity, resonant peaks occur under the condition kFLC = qπ, where kF is the Fermi wave vector, LC is a cavity length, and q is an integer. 31 We extract the cavity length as LC = π/kF = �   Figure 5d. These behaviors resemble the Fock-Darwin state 14,15,29,37 or interdot repulsion in a QD system. 32 A detailed analysis is beyond the scope of the present paper and is a topic for future research.
In conclusion, we investigated the single-carrier transport in the BLG/hBN superlattice by realizing a highly controllable QD device. Single/double closed and open dot operations were demonstrated in the range near the CNP and the hole-side satellite, respectively. In particular, the dip induced by the superlattice Under a perpendicular magnetic field, we observed the suppression of Coulomb oscillations and the shifting/bundling of Coulomb peaks, which could be related to fractal quantum Hall and/or Fock-Darwin spectra in the massive Dirac fermion system. The Hofstadter's butterfly spectra were observed in both the Hall bar and the double-dot devices in the same heterostructure, suggesting that the observed results in the double-dot device could be due to the confined single carrier in the moiré superlattice system. For further characterization of these features in detail, the theoretical modeling/calculation and the investigation of shell-filling states are remained as future tasks. Although our device showed the high-quality and remarkable controllability, a contribution from the edge disorder due to plasma etching cannot be excluded completely, remained to be improved in future, i.e., by using electrostatic confinement induced by a perpendicular electric displacement field. [20][21][22][23][24][25] Our results show that superlattice single-electron devices could be a significant basis for investigating the interaction between exotic many-body phenomena and single-electron transport.

Notes
The authors declare no competing financial interest.      is estimated to be 110,000 cm 2 V -1 s -1 and 140,000 cm 2 V -1 s -1 for the electrons and the holes, respectively. From the width of the peak at the CNP, the residual carrier density is estimated to be ~2.0×10 10 cm -2 . 3 These indicate the high-quality in our BLG superlattice sample.
Using the distance between the CNP and the satellites, the moiré wavelength λ can be expressed by the formula, 4 where, e is the electron charge, CBG is the geometric capacitance for the back gate, VCNP is the position of the  To estimate the energy gap from the Arrhenius analysis, we measure the T dependence of the transport properties of the Hall bar device from 3 K to 15 K. Figure S2a shows ρ of the Hall bar device as a function of VBG near the CNP. The peak broadens and its height is lowered with increasing T. Near the hole-side satellite as shown in Figure S2b, the satellite peak is also lowered with increasing T. These results indicate semiconductor-like behavior. Figure S2c,d shows the Arrhenius plot for the CNP (ρCNP) and the hole-side satellite (ρsat). For the data for high T region, we fit the thermal activation transport model, where, Δ is the activation energy, kB is the Boltzmann constant. As a result, the activation energy at the CNP is estimated to be Δ ~ 0.44 meV. For the hole-side satellite peak, as a result of the same method at the CNP, the activation energy is Δ ~ 0.28 meV. These results are inconsistent with the theoretical prediction. 6 However, the prediction is for the superlattice with perfect angle alignment θ = 0°, and the energy gap is abruptly dropped when the angle alignment deviates from the θ = 0°. 23 4. Estimation of the effective electron temperature Figure S3a shows one of the normalized Coulomb peaks for T = 100 mK ~ 1 K at VSD = 10 μV. The peak width is broadened with increasing T. As shown in Figure S3b, we estimate the effective electron temperature Te from the peak at T = 40 mK by the orthodox Coulomb blockade model, 7 where, αBG ~ 0.0575 is the back-gate conversion factor estimated from the Coulomb diamonds by sweeping VBG, Vpeak is the position of the peak center. Then, we obtain Te ~ 160 mK, which almost agrees with the base temperature.

5. Estimation of the double-dot parameters
The double-dot parameters are derived by using the pure capacitance model. 8 Figure S4a,b shows the zoom up of the honeycomb structure in the charge-stability diagram (in Figure 3a in the main text), we extract the doubledot parameters. The results are summarized in Table S1. The left-dot and right dot have similar parameters, in good agreement in the symmetric device geometry (Figure 1c in the main text).   6. ID-VBG curves at the fixed perpendicular magnetic field near the charge neutrality point In the transport properties of the double-dot device under B, we observe the region where the Coulomb oscillation and current are suppressed. For clarity, the cutting-line-plot is shown in Figure S5. It is obvious that the suppressed region expands with increasing B. 26 7. Transport property in a parallel magnetic field To gain a better understanding of the current suppression region near the CNP under the perpendicular magnetic field, we measure the transport properties of the double-dot device in a parallel magnetic field B||. Note that this measurement was performed in a different cooling cycle due to the limitation of our equipment. As shown in Figure S6, neither the Hofstadter's butterfly spectrum for the wide VBG range nor the current suppression region near the CNP are not observed in contrast to the behavior under the perpendicular magnetic field. This result supports the discussion in the main text and suggests that the current suppression region could be due to the Landau level gap.