Edge-Limited Valley-Preserved Transport in Quasi-1D Constriction of

Aug 15, 2018 - We investigated the quantization of the conductance of quasi-one-dimensional (quasi-1D) constrictions in high-mobility bilayer graphene...
1 downloads 0 Views 2MB Size
Download PDF
Letter Cite This: Nano Lett. XXXX, XXX, XXX−XXX

pubs.acs.org/NanoLett

Edge-Limited Valley-Preserved Transport in Quasi-1D Constriction of Bilayer Graphene Hyunwoo Lee,† Geon-Hyoung Park,† Jinho Park,† Gil-Ho Lee,† Kenji Watanabe,‡ Takashi Taniguchi,‡ and Hu-Jong Lee*,† †

Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea National Institute for Material Science, Tsukuba, 305-0044, Japan

‡

Nano Lett. Downloaded from pubs.acs.org by UNIV OF SOUTH DAKOTA on 08/21/18. For personal use only.

S Supporting Information *

ABSTRACT: We investigated the quantization of the conductance of quasi-one-dimensional (quasi-1D) constrictions in high-mobility bilayer graphene (BLG) with different geometrical aspect ratios. Ultrashort (a few tens of nanometers long) constrictions were fabricated by applying an undercut etching technique. Conductance was quantized in steps of ∼4e2/h (∼2e2/h) in devices with aspect ratios smaller (larger) than 1. We argue that scattering at the edges of a quasi-1D BLG constriction limits the intervalley scattering length, which causes valley-preserved (valley-broken) quantum transport in devices with aspect ratios smaller (larger) than 1. The subband energy levels, analyzed in terms of the bias-voltage and temperature dependences of the quantized conductance, indicated that they corresponded well to the effective channel width of a physically defined conducting channel with a hard-wall confining potential. Our study in ultrashort high-mobility BLG nano constrictions with physically tailored edges clearly confirms that physical edges are the major source of intervalley scattering in graphene in the ballistic limit. KEYWORDS: Bilayer graphene, conductance quantization, intervalley scattering, undercut etching, valleytronics

B

degeneracy, depending on the conservation of the valley symmetry.11,12,15−18 Here, e is the electron charge and h is Planck’s constant. Therefore, the value of the quantized conductance steps can serve as an indicator of the effect of edge scattering on preservation of the valley symmetry. Taking account of λv according to the system size,19,20 we prepared five nanoscale BLG constrictions with different geometrical aspect ratios [i.e., the ratio of the length (L) of the constriction to its effective channel width (W*)]. We fabricated short nano constrictions beyond the resolution limit of electron-beam lithography using plasma reactive ion etching with an undercut electron-beam resist. The quantized conductance steps of the nano constrictions exhibited valley-preserved and valleybroken behavior when their aspect ratios (L/W*) were smaller and larger than 1, respectively. Devices with small aspect ratios (L/W*< 1) exhibited valley-preserved conductance quantization in steps of ∼4e2/h, while the steps were smaller than 2e2/h in devices with large aspect ratios (L/W* > 1). This indicates that λv was limited by the channel width due to short-ranged sharp scatterers located dominantly at the disordered edges, which caused large-momentum-transfer scattering. The results of the bias-voltage spectroscopy measurements of the quasi-1D

ilayer graphene (BLG) has been proposed as a promising platform for realizing valleytronics because it can exhibit in situ tunable valley polarization1−6 by opening a bandgap7 under applied vertical electric fields. It is essential to effectively suppress intervalley scattering in valleytronic applications. One suggested major source of intervalley scattering is short-ranged sharp defects, which enable a large momentum transfer that is required for mixing to occur between valleys K and K′.8,9 Thus, for ballistic graphene, the intervalley scattering at atomically disordered physical edges is believed to dominate over bulk scattering. In recent studies,1,2,5,10 it has been suggested that the intervalley scattering length (λv) in graphene is comparable to the size of micrometer-scale systems. Meanwhile, electrostatic gate-confined carrier guiding without physical edges has enabled valley-preserved transport in monolayer graphene11 and BLG12−14 with λv that is not limited by electrostatically defined confining-potential boundaries. Despite the predicted crucial role of the physical edges for intervalley scattering in graphene, systematic investigation of λv governed by physical edges is lacking, as are quantitative estimations of λv, especially in the case of high-mobility graphene systems. In this study, we investigated the conductance of physically tailored quasi-one-dimensional (quasi-1D) channels in ballistic BLG to gain insight into the intervalley scattering arising from disordered edges. A conducting channel, with a width in the range of the Fermi wavelength of carriers, exhibited quantized conductance in steps of ∼4e2/h or ∼2e2/h with spin© XXXX American Chemical Society

Received: July 6, 2018 Revised: August 8, 2018 Published: August 15, 2018 A

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters

Figure 1. Device fabrication process of an ultrashort bilayer graphene (BLG) constriction. (a) Deposition of a metallic mask on a boron nitride (BN)/BLG/BN/graphite heterostructure (left), and side view of the device along the black dashed line in the top view (right). (b) Electron-beam lithography and development (left), and undercut of the PMMA layer along the electron-beam patterning boundary (right). (c) O2/CF4 plasma etching after deposition of Al2O3 (left), where the heterostructure is etched only through the undercut, except for the narrow region below the metallic mask (right). (d) After the lift-off process, an ultrashort BLG constriction forms below the metallic mask. (e) Scanning electron microscopy (SEM) image of a device. (f) Transmission electron microscopy (TEM) image of the etched region corresponding to the white dotted line in part e. The length of the constriction is about 50 nm. The BLG sheet is marked by a broken orange dashed line.

Figure 2. Measurement configuration and conductance map for dual gating. (a) Device schematic showing the measurement configuration for constriction of encapsulated BLG. The metallic etching mask was used as the top gate electrode. (b) Color-coded conductance map of device A measured at 0.16 K as a function of top and bottom gate voltages (VTG and VBG, respectively). Line 1 (line 2) corresponds to changes in the carrier density (energy gap) with a fixed energy gap (carrier density).

in existing observations.21,22 This result can easily be extended to monolayer graphene. A novel fabrication method was used to prepare ultrashort constrictions, as illustrated in Figure 1. We started with a BLG sheet encapsulated between a pair of hexagonal boron nitride (BN) flakes using a multistage dry-transfer technique.23 In this heterostructure, BLG was protected from charged impurities introduced during device fabrication. Graphene-encapsulated by BN is known to have an enhanced mean free path, reaching tens of micrometers.23 Moreover, we used a graphite bottom

energy subbands formed in the BLG constrictions matched well with the hard-wall confinement model for the physical edges. This study gives systematic confirmation that the scattering at the physical edges of a BLG sheet in the ballistic limit is the major source of the intervalley scattering, with its scattering length limited by the spacing between the neighboring physical edges. Better understanding on valleyrelated transport in bilayer graphene can promote its valleytronic applications and can help resolve the controversies B

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters

Figure 3. Quantized conductance steps. (a) Conductance G of devices A and B with different widths (W) and lengths (L), determined by SEM images, as a function of the Fermi wavenumber kF. G was measured at 0.16 K for device A and at 4 K for device B. The gray lines indicate a linear relationship between G and kF. The magenta lines are to guide the eye. (b) Conductance G of devices A and B as a function of the total carrier density induced by gate voltages. The magenta curves represent the modified relationship between G and ntot, taking into account the trap states. (c) Quantized conductance steps of device A are denoted as a cyan trace in part b. The minima of transconductance dG/dntot versus G correspond to the conductance plateaus, with a modulation period of ∼4e2/h. (d) Quantized conductance steps of device B are denoted as a blue trace in part b. The minima of transconductance dG/dntot versus G correspond to the conductance plateaus, with a modulation period of ∼1.7e2/h. (e) Quantized conductance steps versus aspect ratios of the constrictions for the five devices used in this study.

gate24 to avoid charged impurities25 at the interface between the bottom BN and the bottom gate. This reduced the hysteresis with respect to the electrical bottom gating. Thus, the heterostructures in this study consisted of a stack of topBN/BLG/bottom-BN/graphite from top to bottom. To obtain an ultrashort constriction, we used conventional electron-beam lithography and reactive ion etching. Our fabrication method was unique because it used the undercut of an electron-beam resist (poly methyl methacrylate, PMMA) layer and the metallic top-gate layer26−28 as the etching stencil. First, we deposited a metal mask of Cr/Au (3 nm/12 nm) layers by electron-beam evaporation (Figure 1a) followed by spin-coating of PMMA. After electron-beam patterning and developing, an undercut was formed along the PMMA boundary due to backscattered electrons from the substrate (Figure 1b). The range of the undercut could be controlled depending on the type of PMMA, energy of the electron beam, and resistdevelopment time. After depositing a 30 nm-thick Al2O3 mask layer, the unmasked part of the heterostructure around the PMMA undercut was etched using O2/CF4 plasma. During the plasma etching, the heterostructure was protected by Al2O3 and Cr/Au mask layers (Figure 1c). As a result, the width of the metal mask and the range of the undercut determined the width and length of the constriction, respectively. Figure 1e shows a scanning electron microscopy (SEM) image of a device. A cross-sectional transmission electron microscopy (TEM) image along the white dotted line in Figure 1e shows

that the constriction was about 50 nm long (Figure 1f). The constriction length ranged from 20 to 110 nm for different devices. This novel undercut-etching technique enabled us to create an ultrashort constriction, which was shorter than or comparable to the electron-beam lithography limit. Figure 2a shows schematics of the measurement configuration of the BLG constriction, including the top and bottom gates. Electrical contacts were made of Cr/Au (10 nm/90 nm) layers. A metallic layer was used as an etching mask and top gate electrode, which allowed dual-gate operation in combination with the graphite bottom gate. Transport measurements were performed using low-frequency lock-in techniques with a root-mean-square bias current of 100 nA at a frequency of 17.7 Hz. A four-probe configuration was used to exclude the contact resistance. Five devices were examined in this study. The discussion is focused primarily on two devices of different widths (170 and 110 nm for devices A and B, respectively) and lengths (40 and 110 nm for devices A and B, respectively). The width (W) and length (L) of the constrictions were determined using SEM images. Each device consisted of two BLG reservoirs and a constriction. Wider BLG reservoirs made a negligible contribution to the series resistances; thus, the measured conductance solely represents the value of a constriction. Figure 2b shows the conductance of device A with varying top (VTG) and bottom (VBG) gate voltages. Using a parallel plate capacitor model,7 the total carrier density (ntot) of the constriction can be expressed as ntot = εBNε0(VBG − V0bot)/ C

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters

Figure 4. Energy dependence of the quantized conductance steps. (a) Variation of the transconductance dG/dntot for device A with increasing temperature for T = 0.16, 2, 6, 10, and 15 K from left to right. Quantized conductance steps survive even at T = 15 K. (b) Voltage-bias dependence of the differential conductance (Gdiff) measured at 4 K for device B. Each curve was taken at a fixed gate voltage, corresponding to the curve in Figure 3d. Emergence of Gdiff plateaus at a higher bias of Vdc = 7 mV leads to a subband energy spacing of 14 meV (i.e., the spacing between the dotted vertical lines).

both the spin and valley symmetry, ΔG should be (4e2/h)t, but it becomes (2e2/h)t for spin symmetry only. As t ≤ 1, a ΔG value near 4e2/h indicates that both the spin and valley symmetries are preserved with t ∼ 1. This suggests that the channel length of device A (∼40 nm) was shorter than λv. We also observed valley-symmetric conductance steps in two other devices with ultrashort lengths (L < 60 nm) and similar widths (170−190 nm). However, device B was longer (L ∼ 110 nm) and showed conductance steps of ΔG ∼ 1.7e2/h. We fit the conductance using the modified relationship that accounts for trap states near the edge disorders (magenta curves in Figure 3(b)) to extract an effective channel width of W* ∼ 130 nm for device A and W*t ∼ 42.5 nm for device B. We attribute the discrepancy between W* and W to overetching of the graphene edges16 by reactive ion etching and/or disorderinduced localization of carriers31 at the edges. When ΔG is less than 2e2/h, it is not clear whether valley symmetry is preserved or broken, where ΔG < 2e2/h can be either due to small t < 0.5 with preserved valley symmetry, or to broken valley symmetry. However, previous studies have suggested that λv is comparable to the width of graphene, because intervalley scattering occurs predominantly at the edges of graphene rather than the interior.1,2,5,10,19,20 This indicates that L should be shorter than W* for valley-symmetry preservation. Figure 3e shows the measured ΔG values versus the aspect ratios of the five devices used in this study (see Supporting Information). We observed conductance steps close to 4e2/h with L < W* (shaded area in Figure 3e), indicating preservation of the valley symmetry. In contrast, devices with L > W* showed conductance steps smaller than 2e2/h, suggesting broken valley symmetry. Thus, assuming broken valley symmetry for device B with L > W*, we used ΔG = (2e2/h)t to obtain the value of t ∼ 0.85 for the measured ΔG = 1.7e2/h. As shown in Figure 3e, devices with smaller aspect ratios have ΔG values close to 4e2/h for preserved valley symmetry, which confirms that λv is comparable to W*. We used the temperature dependence and bias voltage dependence of the conductance quantization steps to estimate the size of the energy spacing between subbands. Figure 4a shows transconductance dG/dntot for device A at temperatures ranging from 0.16 to 15 K. Increasing temperature diminishes

edbot + εBNε0(VTG − V0top)/edtop, where ε0 is the electric permittivity of the free space, εBN (=3.9), is the relative dielectric constant of BN,29 dbot (dtop) is the thickness of the bottom (top) BN sheet, and V0bot(V0top) is the bottom (top) gate voltage for the CNP. In addition, the displacement field, D = εBN(VBG − V0bot)/2dbot − εBN(VTG − V0top)/2dtop, opens the energy gap (Egap) by breaking the inversion symmetry of BLG.7 As a result, lines 1 and 2 in Figure 2b correspond to the variation in ntot and Egap, respectively. Egap and ntot are fixed in the respective cases. BLG reservoirs were only affected by the bottom gate. Therefore, npn or pnp junctions formed between the reservoir and the constriction in the gating regions α of the conductance map in Figure 2b, where the transmission probability was reduced by the presence of the pn potential barriers.30 We measured quantized conductance with the same carrier types in both the reservoir and constriction regions, corresponding to the gating regions β in Figure 2b. Figure 3a shows conductance (G) as a function of the Fermi wavenumber kF = πntot for the zero displacement field (D = 0). For device A, this corresponds to the gate sweep indicated by the red arrow in Figure 2b. Both conductance traces are horizontally shifted from the respective gray lines in Figure 3a, representing the linear relationship between G and kF as G = 4e2/πh·WtkF. Here, t is the transmission probability. The extent of the horizontal shift depends on the amount of additional trap states [nT(E)] that arise from the edge disorders.16 The shift modifies the relationship between the Fermi wavenumber and the carrier density as ntot=k′F2π−1 + nT(E) near the CNP, where k′F is the modified Fermi wavenumber. Because the trap states do not contribute to transport, the expression of G above should be rewritten in terms of the effective channel width W* and k′F as 2 2 * * G = 4e /πh·W tk′F = 4e /h π ·W t ntot − nT (E) (magenta curves in Figure 3b). The modified G value agrees well with the measured data, except near the CNP. Parts c and d of Figure 3 show zoomed-in views of G in Figure 3b as a function of ntot with the corresponding transconductance dG/dntot. The conductance steps are emphasized by the vanishing dG/dntot. For device A, we observed distinct conductance quantization steps, ΔG, close to 4e2/h (Figure 3c). With preservation of D

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters the conductance plateaus due to thermal broadening of kBT, where kB is the Boltzmann constant. The conductance plateaus survived at 15 K, which was the highest temperature reached in our measurements; thus, the energy spacing was larger than ∼1.3 meV. Voltage bias spectroscopy was carried out to extract the energy spacing of the subbands for device B at T ∼ 4 K (Figure 4b). Each trace shows differential conductance with respect to the voltage bias at a fixed gate voltage. Zero-bias differential conductance exhibits quantized conductance steps corresponding to the trace in Figure 3d. At a finite bias of eVdc∼ ± 7 meV, additional plateaus appear between quantized conductance steps at zero-bias, Vdc∼ 0. This indicates that the energy spacing between subbands of device B was ∼14 meV. In a single-particle configuration, each subband energy of BLG with a hard-wall potential is EN = ℏ2π2N2/2m*W2, where m* (=0.33me) is the electronic effective mass of BLG and N is a positive quantum number, yielding an increasing subband energy spacing with increasing quantum number N. However, voltage bias spectroscopy shows that the subband energy spacing was nearly equal for different N (Figure 4b). We attribute this to the dispersion relationship in the band structure of BLG, which becomes almost linear for large wavenumbers. In the tight-binding model, the low energy band of BLG is approximated by a quadratic equation (E ∼ p2/ 2m*) for small wavenumbers, but it becomes linear (E ∼ vp) for large wavenumbers,32 with v ∼ 106 m/s. Such a quadraticto-linear crossover takes place for k′F ∼ γ1/2vℏ = 260 × 106 m−1, corresponding to ntot∼ 3.5 × 1012 cm−2 for the device B, where γ1(=0.35 eV) is the interlayer hopping parameter.32 Because voltage bias spectroscopy was performed above the crossover point, the dispersion relationship should be approximately linear, which results in equally spaced subband energies with ΔE = vℏπ/W* = 42 meV in a hard-wall potential. In the presence of valley mixing, however, the valleydegenerate subbands are split and valley is no longer a good quantum number, giving rise to the quantized conductance steps of 2e2/h. This indicates that a modified subband energy spacing ΔEV is needed when considering the valley mixing. Because the measured subband energies are equally spaced for different N as shown in Figure 4b, the subband splitting energy is estimated as ΔEKK’= 0.5ΔE. Thus, the modified subband energy spacing is predicted to be ΔEV = 0.5ΔE = 21 meV, which corresponds well to the observed value of ∼14 meV. In summary, to understand the role of physically etched edges for intervalley scattering, we studied quasi-1D transport of ultrashort constrictions in BLG fabricated via PMMA undercut etching. For devices with small aspect ratios (1), quantized conductance steps smaller than 2e2/h were observed with broken valley-symmetry. This result indicates that L < W* is essential for valley-symmetrypreserved transport in physically etched BLG because λv is limited by W*. This study gives solid confirmation of the general belief that the scattering at the physical edges of a graphene layer in the ballistic limit is the major source of the intervalley scattering. In the presence of valley mixing, the voltage bias dependence of quasi-1D transport shows almost equally spaced confinement subbands of ∼14 meV. This agrees with the calculation for a hard-wall potential of width W*, assuming a linear dispersion of the band structure. The latter is

explained by the tight-binding model for large wavenumbers. In contrast to the gate-confined carrier guiding, a hard-wall confining potential of physically etched BLG is robust against the gate tuning, which provides the merits of generating valley polarized current near the CNP.1−5 Combining observed valley-preserved transport and gate-independent confining potential, dual-gated BLG with physically etched edges would create new opportunities for valleytronic devices with gate-tunable BLG.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.8b02750.

■

Ballistic transport of encapsulated bilayer graphene constriction, quantized conductance steps for different aspect ratios of the BLG constrictions, and quantized conductance steps for different displacement fields (PDF)

AUTHOR INFORMATION

Corresponding Author

*(H.-J.L.) E-mail: [email protected]. ORCID

Kenji Watanabe: 0000-0003-3701-8119 Hu-Jong Lee: 0000-0002-3208-0047 Author Contributions

H.L. and H.-J.L. conceived the idea and designed the experiments. H.L. prepared the devices and performed the measurements. All authors analyzed the data. H.L., G.-H.L., and H.-J.L. wrote the manuscript. H.-J.L. supervised the study. All authors contributed to the discussion and approved the final version of the manuscript. K.W. and T.T. provided highquality hexagonal boron nitride substrates. Funding

This work was supported by the National Research Foundation (NRF) through the SRC Center for Topological Matter, POSTECH, Korea (Grant No. 2011−0030046) and the SRC Center for Quantum Coherence in Condensed Matter, KAIST, Korea (Grant No. 2016R1A5A1008184 for GHL) and the Elemental Strategy Initiative conducted by the MEXT and JSPS KAKENHI, Japan (Grant Number 18K19136 for K.W. and T.T.). Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Sui, M.; Chen, G.; Ma, L.; Shan, W.-Y.; Tian, D.; Watanabe, K.; Taniguchi, T.; Jin, X.; Yao, W.; Xiao, D.; Zhang, Y. Nat. Phys. 2015, 11 (12), 1027−1031. (2) Shimazaki, Y.; Yamamoto, M.; Borzenets, I. V.; Watanabe, K.; Taniguchi, T.; Tarucha, S. Nat. Phys. 2015, 11 (12), 1032−1036. (3) Xiao, D.; Yao, W.; Niu, Q. Phys. Rev. Lett. 2007, 99 (23), 236809. (4) Lensky, Y. D.; Song, J. C. W.; Samutpraphoot, P.; Levitov, L. S. Phys. Rev. Lett. 2015, 114 (25), 256601. (5) Gorbachev, R. V.; Song, J. C. W.; Yu, G. L.; Kretinin, A. V.; Withers, F.; Cao, Y.; Mishchenko, A.; Grigorieva, I. V.; Novoselov, K. S.; Levitov, L. S.; Geim, A. K. Science 2014, 346 (6208), 448−451.

E

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters (6) Ju, L.; Shi, Z.; Nair, N.; Lv, Y.; Jin, C.; Velasco, J., Jr; OjedaAristizabal, C.; Bechtel, H. A.; Martin, M. C.; Zettl, A.; Analytis, J.; Wang, F. Nature 2015, 520 (7549), 650−655. (7) Zhang, Y.; Tang, T.-T.; Girit, C.; Hao, Z.; Martin, M. C.; Zettl, A.; Crommie, M. F.; Shen, Y. R.; Wang, F. Nature 2009, 459 (7248), 820−823. (8) Tikhonenko, F. V.; Kozikov, A. A.; Savchenko, A. K.; Gorbachev, R. V. Phys. Rev. Lett. 2009, 103 (22), 226801. (9) Ki, D.-K.; Jeong, D.; Choi, J.-H.; Lee, H.-J.; Park, K.-S. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78 (12), 125409. (10) Engels, S.; Terrés, B.; Epping, A.; Khodkov, T.; Watanabe, K.; Taniguchi, T.; Beschoten, B.; Stampfer, C. Phys. Rev. Lett. 2014, 113 (12), 126801. (11) Kim, M.; Choi, J.-H.; Lee, S.-H.; Watanabe, K.; Taniguchi, T.; Jhi, S.-H.; Lee, H.-J. Nat. Phys. 2016, 12, 1022. (12) Overweg, H.; Eggimann, H.; Chen, X.; Slizovskiy, S.; Eich, M.; Pisoni, R.; Lee, Y.; Rickhaus, P.; Watanabe, K.; Taniguchi, T.; Fal’ko, V.; Ihn, T.; Ensslin, K. Nano Lett. 2018, 18 (1), 553−559. (13) Lee, J.; Watanabe, K.; Taniguchi, T.; Lee, H.-J. Sci. Rep. 2017, 7 (1), 6466. (14) Li, J.; Wang, K.; McFaul, K. J.; Zern, Z.; Ren, Y.; Watanabe, K.; Taniguchi, T.; Qiao, Z.; Zhu, J. Nat. Nanotechnol. 2016, 11, 1060. (15) Tombros, N.; Veligura, A.; Junesch, J.; Guimarães, M. H. D.; Vera-Marun, I. J.; Jonkman, H. T.; van Wees, B. Nat. Phys. 2011, 7 (9), 697−700. (16) Terrés, B.; Chizhova, L. A.; Libisch, F.; Peiro, J.; Jörger, D.; Engels, S.; Girschik, A.; Watanabe, K.; Taniguchi, T.; Rotkin, S. V.; Burgdörfer, J.; Stampfer, C. Nat. Commun. 2016, 7, 11528. (17) Allen, M. T.; Martin, J.; Yacoby, A. Nat. Commun. 2012, 3, 934. (18) Goossens, A. M.; Driessen, S. C. M.; Baart, T. A.; Watanabe, K.; Taniguchi, T.; Vandersypen, L. M. K. Nano Lett. 2012, 12 (9), 4656− 4660. (19) Tikhonenko, F. V.; Horsell, D. W.; Gorbachev, R. V.; Savchenko, A. K. Phys. Rev. Lett. 2008, 100 (5), 056802. (20) Tikhonenko, F. V.; Horsell, D. W.; Wilkinson, B.; Gorbachev, R. V.; Savchenko, A. K. Phys. E 2008, 40 (5), 1364−1366. (21) Cresti, A.; Nikolic, B. K.; Garcia, J. H.; Roche, S. Riv. Nuovo Cim. 2016, 39 (12), 587−667. (22) Marmolejo-Tejada, J. M.; García, J. H.; Petrović, M.; Chang, P.H.; Sheng, X.-L.; Cresti, A.; Plechác,̌ P.; Roche, S.; Nikolic, B. K. arXiv:1706.09361, 2018. (23) Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.; Kim, P.; Hone, J.; Shepard, K. L.; Dean, C. R. Science 2013, 342 (6158), 614−617. (24) Zibrov, A. A.; Kometter, C.; Zhou, H.; Spanton, E. M.; Taniguchi, T.; Watanabe, K.; Zaletel, M. P.; Young, A. F. Nature 2017, 549, 360. (25) Wang, H.; Wu, Y.; Cong, C.; Shang, J.; Yu, T. ACS Nano 2010, 4 (12), 7221−7228. (26) Lian, C.; Tahy, K.; Fang, T.; Li, G.; Xing, H. G.; Jena, D. Appl. Phys. Lett. 2010, 96 (10), 103109. (27) Bai, J.; Duan, X.; Huang, Y. Nano Lett. 2009, 9 (5), 2083− 2087. (28) Yu, W. J.; Duan, X. Sci. Rep. 2013, 3, 1248. (29) Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. L.; Hone, J. Nat. Nanotechnol. 2010, 5, 722. (30) Huard, B.; Sulpizio, J. A.; Stander, N.; Todd, K.; Yang, B.; Goldhaber-Gordon, D. Phys. Rev. Lett. 2007, 98 (23), 236803. (31) Evaldsson, M.; Zozoulenko, I. V.; Xu, H.; Heinzel, T. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78 (16), 161407. (32) McCann, E.; Koshino, M. Rep. Prog. Phys. 2013, 76 (5), 056503.

F

DOI: 10.1021/acs.nanolett.8b02750 Nano Lett. XXXX, XXX, XXX−XXX