h-BN

Subscriber access provided by Kaohsiung Medical University

Communication

Accurate Gap Determination in Monolayer and Bilayer Graphene/h-BN Moiré Superlattices Hakseong Kim, Nicolas Leconte, Bheema Lingam Chittari, Kenji Watanabe, Takashi Taniguchi, Allan H MacDonald, Jeil Jung, and Suyong Jung Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b03423 • Publication Date (Web): 20 Nov 2018 Downloaded from http://pubs.acs.org on November 22, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Accurate Gap Determination in Monolayer and Bilayer Graphene/h-BN Moiré Superlattices Hakseong Kim,† ,* Nicolas Leconte,‡,* Bheema L. Chittari,‡ Kenji Watanabe,∥ Takashi Taniguchi,∥ Allan H. MacDonald,⊥ Jeil Jung‡,§ and Suyong Jung†,§

†

Korea Research Institute of Standards and Science, Daejeon 34113, Korea ‡

Department of Physics, University of Seoul, Seoul 02504, Korea

Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba

∥

305-0044, Japan Department of Physics, The University of Texas at Austin, Austin, TX 78712

⊥

KEYWORDS: graphene-h-BN moiré superlattice, bilayer graphene-h-BN superstructure, Landau level tunneling spectroscopy, graphene energy gap, high-precision measurement

ACS Paragon Plus Environment

1

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 38

ABSTRACT. High mobility single and few-layer graphene sheets are in many ways attractive as nanoelectronic circuit hosts but lack energy gaps, which are essential to the operation of fieldeffect transistors. One of the methods used to create gaps in the spectrum of graphene systems is to form long period moiré patterns by aligning the graphene and hexagonal boron nitride (h-BN) substrate lattices. Here, we use planar tunneling devices with thin h-BN barriers to obtain direct and accurate tunneling spectroscopy measurements of the energy gaps in single- and bi-layer graphene-h-BN superlattice structures at charge neutrality (first Dirac point) and at integer moiré band occupancies (second Dirac point, SDP) as a function of external electric and magnetic fields and the interface twist angle. In single-layer graphene we find, in agreement with previous work, that gaps are formed at neutrality and at the hole-doped SDP, but not at the electron-doped SDP. Both primary and secondary gaps can be determined accurately by extrapolating Landau fan patterns to zero magnetic field and are as large as ≈ 17 meV for devices in near perfect alignment. For bilayer graphene, we find that gaps occur only at charge neutrality where they can be modified by an external electric field.


2

Page 3 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

The sublattice symmetry of graphene forbids the energy gap needed for graphene-based electronic applications.1

Breaking sublattice symmetry would directly generate gaps, but

requires atomic scale structure control that is still out of reach. Several alternative approaches for engineering graphene energy gaps have therefore been proposed, including narrowing graphene channels to sub nanometer dimensions2 and subjecting graphene to periodic potentials to generate gaps at the superlattice-zone boundary.3–5 Here we explore gap engineering based on recent experimental advances in realizing two-dimensional vertical van der Waals heterostructures between graphene and h-BN lattices.

It is known that minimizing the

orientation difference between the two lattices maximizes the influence of h-BN on the graphene electronic structure.6–11 If the aligned lattices were commensurate, gaps near 50 meV would be expected at charge neutrality,12 which are yet too small for practical electronic device applications at room temperature. At the same time, the difference in lattice constants implies that moiré patterns are formed when the lattices are aligned,12–15 giving rise to Hofstadter butterfly effects in a magnetic field and inducing second-generation Dirac points (SDPs) at the corners of the graphene-h-BN superlattice Brillouin zones.16,17 The outstanding question that follows is whether graphene superlattices generate energy-gaps at the SDPs, as expected for graphene under a periodic potential,3,4,15,16 and, if so, how large these gaps can be. In single-layer graphene, the emergence of the energy gaps at the FDP and the SDPs has been

experimentally

confirmed

by

multi-probe

conductance,8,9,16,18

magneto-optical

spectroscopy19 and angle-resolved photoemission spectroscopy (ARPES) .20 However, the gap sizes inferred from different experiments have varied.

It has been argued that a delicate

interdependence between the atomic and electronic structures of graphene-h-BN heterostructures could affect energy-gap formation. For example, alteration of the gaps could take place upon


3

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 38

encapsulating devices with another h-BN layer,21 and application of external pressure in graphene on h-BN leads to a progressive increase of the gap at the FDP due to enhanced interlayer coupling.18 Indeed recent ARPES measurements have established that sub-Angstrom changes of the interlayer distance between graphene and h-BN layers can induce a one-order swing.20 Some of us, in an earlier theoretical work, had argued that relaxations of both graphene and h-BN lattices upon forming graphene-h-BN heterostructures and electron-electron interactions both play important roles in deciding the energy-gap size at charge neutrality.11 Compared with the rather rich experimental and theoretical reports on the gap states in singlelayer graphene, however, the experimental studies on the effect of moiré patterns in bilayer graphene are practically absent in the literature, and little is known about the influence of interactions with their underlying h-BN substrates. Electron tunneling spectroscopy is a reliable and accurate metrology for analyzing the electronic structure of solid-state systems, and is especially useful for those with energy gaps, for example, superconductivity, charge- and spin-density waves, or quantum Hall effects.22–24 By monitoring the tunneling current variation (differential conductance) with respect to energy (or tunneling bias voltage), the size of gaps and their positions with respect to the Fermi level (EF) can be acquired straightforwardly and more directly than in temperature-dependent multi-probe transport or magneto-optical spectroscopy measurements.

With a scanning tunneling

spectroscopy measurement, Yankowitz et al. explicitly demonstrated the formation of SDPs in graphene-h-BN superstructures and tracked their evolution as functions of an external gate voltage and the period of graphene superlattice.25 Despite the obvious advantages of tunneling spectroscopy for addressing gapped states, however, extracting accurate energy-gap sizes at the


4

Page 5 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

SDPs and even at the FDP has not been a focus, largely due to the ambiguity of locating the energy-gap boundaries in tunneling spectra,26 a problem we will discuss later in detail. In this report, we present tunneling spectroscopy measurements on single- and bi-layer graphene-h-BN superlattices with thin h-BN as a tunnel barrier fabricated using mechanical exfoliation and transfer approaches. The enhanced tunnel-junction stability of our graphene-hBN heterostructures27,28 allows us to identify the positions of SDPs, and to establish whether an energy gap is formed at both the FDP and the SDPs as functions of the twist angle; the size of graphene superlattices, external electric and magnetic fields, and the number of carbon layers. As a major advantage over previous reports addressing gaps,8,9,17–20,29 we accurately track band edges by monitoring the evolution of Landau levels (LLs) that originate from energy-gap boundaries when gaps exist, or at band-crossing points in the absence of an energy gap. In single-layer graphene superlattices, the energy gap at the FDP becomes as large as ≈ 17 meV at near-perfect alignment. The broken-inversion symmetry of graphene-h-BN superlattices causes electron-hole asymmetries in the electronic structures of single-layer graphene superlattices, opening an energy gap as large as ≈ 16 meV at the SDP in the hole-doped region. No gap opens at the SDP in the electron-doped area. We observe that the gap at the SDP closes quickly as the twist-angle increases, while the gap size at the FDP decreases more slowly, and interpret the difference as being due to different microscopic origins of energy-gap formations in the two cases, i.e. that inversion symmetry breaking is sufficient at the FDP whereas the details of pseudospin mixing are crucial at SDPs.30 In bilayer graphene superlattices, energy gaps fail to emerge at either the electron- and hole-doped SDPs because of a higher overall density of states and the presence of additional bands at mini-zone boundaries near the relevant energy. We establish that remote hopping contributions to the tight-binding Hamiltonian for intrinsic bilayer


5

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 38

graphene play an essential role near the SDPs, and must be incorporated in the moiré superlattice Hamiltonian.31 Near the FDP, we also observe tunneling signals that can be associated with the moiré potential induced midgap states near the band edges when the electric field opens a band gap, whereas the electric-field induced energy gap dominates tunneling spectra at the charge neutrality point. All the key experimental observations are qualitatively understood within a simple single-particle picture of single- and bi-layer graphene-h-BN superlattices. A better quantitative agreement of the gap sizes with experiment should be achievable through the inclusion of many-body Coulomb interactions. The optical viewgraph and the schematics of the tunneling and multi-probe devices are shown in Fig. 1(d). Firstly, we carefully select mechanically exfoliated graphene flakes with well-defined crystalline edges. For some devices, flakes with both single- and bi-layer regions are chosen in order to systematically investigate layer-dependence effects on graphene-h-BN superstructures with the same twist angle. Next, a thin h-BN flake with three to five layers and a thick graphite probe are sequentially transferred on top of the crystallographically aligned graphene-h-BN heterostructures to form planar graphene-h-BN tunnel junctions. To provide complementary information, we also fabricated edge-contacted multi-probe graphene devices, sharing both bottom h-BN/SiO2 back-gate insulators and the graphene layer with tunneling devices (Fig. 1(d)).

With edge-contacted devices, the presence of the graphene-h-BN

superlattices are confirmed through multi-probe conductance measurements with additional conductance (or resistance) dips (peaks) (Fig. S1). We obtain the back-gate capacitance (Cg) defined by bottom h-BN and SiO2 insulators from the Wannier-type fan diagram in high magnetic field measurements. With attained Cg and Vg-spacings between the SDPs and the FDP, basic parameters of graphene-h-BN superlattices can be extracted such as the location of the SDP


6

Page 7 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

in energy with respect to the FDP, the size of graphene-h-BN superlattice, and the twist angle defined between graphene and bottom h-BN flakes (See Supporting Information). The twist angle between graphene and h-BN layers can be estimated from Raman measurements as well.33 As displayed in Fig. S3, the Raman intensity and the width of the 2D peak are heavily influenced by the formation of a graphene-h-BN superlattice.33 In our devices, the twist angles estimated from the Raman measurements are consistent with the values from multi-probe measurements, implying that our graphene superlattices have little disarray from nearby graphene ripples or bubbles.34 Figures 1(e) and 1(f) display the series of tunneling spectra, differential conductance (G = dI/dVb) curves at B = 0 T from the single-layer (SL, Fig. 1(e), sample A) and bi-layer (BL, Fig. 1(f), sample B) graphene-h-BN superlattices. Both SL- and BL-graphene devices are fabricated with a flake where SL and BL regions coexist, and the misalignment angle with the underlying h-BN insulator θ is small (θ ≤ 0.1°).

With these small twist angles, the graphene-h-BN

superlattice defined in the single-layer region has SDPs located ≈ ± 0.17 eV away from the FDP and a moiré lattice constant estimated to be λM ≈ 13.9 nm. At zero Vg, a dI/dVb-dip feature relating to diminished density of states at the FDP is positioned at Vb ≈ 0 mV for un-doped SLand BL-graphene devices, as marked with orange triangles in Figs. 1(e) and 1(f). Similar to our previous report,28 the dI/dVb-dip feature at the FDP shifts away from Vb = 0 mV as Vg increases from Vg = 0 V to Vg = ± 10 V in magnitude. In addition, the FDP dI/dVb feature in the bilayer (Sample B, Fig.1(f)) evolves into a ‘U’-shaped gap that widens as the electric field-induced gap increases at non-zero Vg.28

Here, the most noticeable differences, compared with strongly

misaligned graphene-h-BN heterostructures28 are the presence of additional dI/dVb-dip structures, marked with red and blue triangles in Figs. 1(e) and 1(f). The pair of secondary dI/dVb dips are


7

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 38

equidistant at both sides of the FDP, and change their positions together with FDP for different Vg. The Vg-dependent characteristics confirm that these secondary dI/dVb features are linked to the electronic structures of the graphene-h-BN superlattices, particularly the SDPs at the minizone boundary.27,35 In the sample A (Fig. 1(e)), the dI/dVb dip positioned lower than the FDP in Vb, therefore formed in the hole-doped region, is conspicuous when compared to the dI/dVb dip located in the electron-doped area. Quite on the contrary, both dI/dVb dips in the sample B (Fig. 1(f)) for BLh-BN superlattices have similar dI/dVb magnitude for electrons and holes. We note that the electron-hole asymmetry in graphene-h-BN superstructures can be traced to the destructive and constructive combinations for electrons and holes of the moiré pattern matrix elements due to local mass and virtual strain patterns of comparable magnitudes.36 These obvious differences in the degree of electron-hole asymmetry between SL- and BL-h-BN superlattices are due to negligibly small virtual strains that couple the two low-energy carbon sites in bilayer graphene on h-BN. Figures 2(a) and 2(b) show the two-dimensional representation of the tunneling spectra in the window of Vg and Vb, which consist of 121-independent dI/dVb curves (Figs. 1(e) and 1(f)) from Vg = -30 V to Vg = 30 V with a spacing of ΔVg = 0.5 V, respectively for the SL (sample A) and BL (sample B)-h-BN superlattices. In the sample A (Fig. 2(a)), the suppressed dI/dVb at the FDP (marked with a dotted white arrow) and the secondary dI/dVb dip (marked with a dotted yellow arrow) for the SDP in the hole-doped region are well recognized. Along with those dark dI/dVb bands running diagonally across the gate map, an additional dI/dVb dip (marked with a dotted green arrow) at the SDP in the electron-doped area is shown distinctly with a significantly weakened visibility in the gate mapping (Fig. 2(a)). In contrast, as shown in Fig. 2(b), the


8

Page 9 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

electron-hole asymmetries become weaker in the BL-h-BN superlattice, featuring as a pair of dI/dVb dips equidistantly spaced in Vb around the electrostatically induced bilayer-graphene energy gap at the FDP. As discussed in our previous report,28 the bilayer energy gap at the FDP is proportional to the width of the suppressed dI/dVb band, which develops in the presence of an electric field between carbon layers.28 Here, note that the suppressed visibility of the dI/dVb dips at the SDPs in the bilayer is close to the dI/dVb at the SDP in the electron-doped region in the single layer. In contrast, the dI/dVb dips at the FDP and the SDP in the hole-doped region of the SL-h-BN superlattice (Fig. 2(a)) have dI/dVb signatures, just as clearly visible as the gap at the FDP in the BL-h-BN superlattice (Fig. 2(b)). We find that the observed visibility differences in dI/dVb are related to the presence of energy gaps either at the FDP or at the SDPs in SL- and BL-h-BN superlattices. Additionally, the tunneling features in the planar graphene-h-BN tunnel devices predominantly reflect the electronic structures at EF, which is tunable through the capacitive couplings across the thin h-BN tunnel insulator and the relatively thick bottom-gate h-BN layers. Thus, the absence of observable variations in both the linear scaling of the FDP and the SDPs and the equidistance of the SDPs from the FDP confirm that the planar capacitive model is sufficient to explain the tunneling spectra, guaranteeing little variations in the electronic structure of graphene superlattices in our measurement Vg–Vb window. We now discuss in detail the electronic-structure variations and plausible energy-gap formations at the FDP and the SDPs at the mini-zone boundaries in graphene-h-BN superlattices. Although the strongly suppressed dI/dVb spectra suggest the presence of energy gaps at the FDP and the SDP in the hole-doped region in the SL-h-BN superlattice, the dataset at B = 0 T is not sufficient either as indisputable evidence for the presence of a gap or for quantitative energy-gap


9

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

analyses.

Page 10 of 38

Indeed, a suppressed dI/dVb band can be observed at the FDP in completely

misaligned graphene-h-BN heterostructures where no gap exists.28 A low-valued dI/dVb band is also present at Vb = 0 mV (Figs. 2(a) and 2(b)), which is ascribed to the diminished phonon density of states and reduced electron tunneling rates in graphene and other low-dimensional systems.28,37,38 Moreover, tunneling signals are often delineated by dI/dVb features like the strong dI/dVb peaks marked with blue arrows in Figs. 1(e) and 2(a). Those resonance peaks are due to the relative crystalline-angle alignments of graphene and the top graphite probe, and are not directly related to the electronic structures of graphene-h-BN superlattices. Landau level spectroscopy, in turn, is proven to be the most efficient and accurate metrology for quantitative analyses of gapped states in graphene.26 In the presence of a perpendicular magnetic field, charged carriers in graphene collapse into well-separated LLs and the lowest LLs for electrons and holes coincide with the energy-gap boundaries. Therefore, the accurate assessment of energy gaps can be directly inferred from monitoring LL evolutions as the external magnetic field varies. Figure 2(c) displays the gate mapping of the sample A, SL-h-BN superlattice at B = 1 T, revealing the most direct and indisputable evidences for the energy-gap formations at the FDP and the SDP in the hole-doped region, and the absence of an energy gap at the SDP in the electron-doped area. First, we can easily identify suppressed dI/dVb bands where the FDP and the SDP in the hole-doped region exist at B = 0 T, as indicated with dotted white and yellow lines in Fig. 2(c). Around these suppressed dI/dVb bands, a pair of LLs, each denoted as LL0+ (LL0+,SDP) and LL0- (LL0-, SDP) exist with other LLs with higher LL indices, N. In comparison, a dI/dVb peak forms exactly at the SDP in the electron-doped region with a pair of dark lines with low dI/dVb in value. The dI/dVb peak, representing the lowest LL at the SDP in the electron-


10

Page 11 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

doped area is marked with a dotted green arrow and one of the gaps in the spectra is guided with a dotted white line. The developments of LLs at both Dirac points can be traced to magnetic fields as low as B = 0.1 T. Figure 2(d) shows the gate mapping of the sample B, BL-h-BN superlattice, measured at B = 1 T. In contrast with the SL-h-BN superlattice, aforementioned spectra features relating to energy-gap formations such as suppressed dI/dVb band paired with a couple of LLs are completely absent. The lowest-indexed LL, which is supposedly formed at Dirac points without a gap is not developed in BL-h-BN superlattices either. Instead, as marked with dotted yellow and green arrows in Fig. 2(d), dI/dVb dips are eminent at both the SDPs with far lower visibility than the ones for the gaps in the SL-h-BN superlattice. Around the dI/dVb dips, a secondary pair of LLs is observed at the lower Vb for the SDP in the hole-doping region and at the higher Vb for the SDP in the electron-doped area. All these observations point out that the energy gaps at both SDPs are not completely developed due to additional accessible bands around the BL-h-BN superlattice mini-zone boundaries, which will be further confirmed with detailed experimental results and theoretical calculations. We also defer the experimental and theoretical discussions on the electronic structures at the charge-neutrality point to a later part in this manuscript. We are able to investigate the energy-gap formations and the states around both Dirac points in more detail by utilizing LL spectroscopy through fine control of external magnetic fields at a fixed Vg. Figure 3(a) shows the LL-fan diagram from the SL-h-BN superlattice (sample A) with a spacing of B = 0.1 T.

The gate voltage is adjusted at Vg = -20 V to better

resolve the electronic structures at the FDP and both SDPs. The presence of energy gaps, represented by the bands of suppressed dI/dVb, is easily identified with the LLs that arise from the boundaries of the gaps at the FDP (delineated with a dotted red box) and the SDP in the hole-


11

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 38

doped region (marked with a dotted green box). Note that, at high magnetic fields, the tunneling associated to LLs and energy gaps often interfere with other features with different slopes as marked with dotted white lines in Fig. 3(a). Those resonances are due to the relative positions of LLs and electronic band structures of a graphite probe and graphene-h-BN superlattices in energy-momentum space.39 Additionally, the pre-split LLs at B ≈ 0 T become fully split into four well-separated LLs with newly developed gaps due to broken spin degeneracy and electronelectron interactions, allowing to quantitatively measure the gaps resulting from the broken inversion symmetry of graphene-h-BN superlattices. Figures 3(b)–3(d) show the high-resolution dI/dVb spectra from the sample A at the FDP (Fig. 3(b)), the SDP in the hole-doped region (Fig. 3(c)), and the SDP in the electron-doped area (Fig. 3(d)), respectively. The suppressed dI/dVb bands with a pair of LLs around evidence the energy gap at the FDP (Fig. 3(b)) and the SDP in the hole-doped region (Fig. 3(c)) in SL-h-BN superlattices.

Conversely, the LL (LL0,

SDP)

developed right at the SDP (Fig. 3(d)) proves that no energy gap is formed at the mini-zone boundary in the electron-doped area. In our planar graphene-h-BN tunnel devices, EF of graphene does not change its position in energy even with varying Vb, when electro-statically induced charged carriers are occupying the LLs. Accordingly, the high conductance dI/dVb bands representing each LL extend their width in Vb as external B increases. Thick yellow and orange ticks in Figs. 3(a)–3(d) indicate how much Vb is required to fill the LLs with the filling factor (FF) of either two or four depending on whether valley symmetry is broken or not. We consider that spin degeneracy is not lifted at lower B. Thus, accurate energy-gap assessment is obtained from extrapolating each LL-peak position to B = 0 T and extracting Vb spacing for an energy gap. The white circles in Figs. 3(b) and 3(c) indicate LL-peak positions at different B at the FDP and the SDP, and the


12

Page 13 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

energy gaps are estimated to be (16.2 ± 0.1) mV at the FDP and (15.6 ± 1.0) mV at the SDP in the hole-doped region for the SL-h-BN superlattice in near perfect alignments (θ ≤ 0.1°). Here, we point out that the Vb-spacings measured in our graphene-h-BN tunnel junctions can be directly related to the actual energy bandgaps because of the large geometric capacitance of planar tunnel junctions and the absence of in-gap states.28,40,41 The energy gaps induced from the broken inversion symmetries of SL-h-BN superlattices are expected to be sensitive to the twist angles of graphene and underlying h-BN substrates,30 and our measurements confirm the expectations. In total, we have measured eight SL-h-BN superlattice devices with different twist angles, and the extracted energy gaps are summarized in Fig. 4. There are a few intriguing points to note in the energy-gap evolutions at different twist angles. Firstly, energy gaps at the SDP are as big as those at the FDP in size at ≈ 16 meV for closely aligned SL-h-BN superlattices, but close out much faster than the FDP gap as the twist angle increases. For SL-h-BN superlattices with a twist angle larger than 0.6°, SDP energy gaps close out almost completely and dI/dVb features of a gap are practically absent. In contrast, the energy gap at the FDP shrinks with a rather moderate rate for increasing twist angle. Even for the SL-h-BN superlattice with a twist angle of 1.1°, the gap is estimated to be as large as (11.7 ± 0.2) meV. The obvious differences at the FDP and SDP highlight the microscopic-scale differences of the energy-gap formation mechanisms at the FDP and the SDP in SL-h-BN superlattices. The closed blue squares and red circles in the inset of Fig. 4 respectively represent theoretically expected energy gaps at the FDP and the SDP in hole-doped regions, with a consideration of fully relaxed graphene and h-BN lattices without electron-electron interactions. Here, we point out that simple single-particle considerations are sufficient enough to qualitatively explain most


13

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 38

of the experimentally observed characteristics at both Dirac points. Expected gaps at both the FDP and the SDP are close to ≈ 8 meV in size at a zero twist angle, and the fast closure of SDP gaps at 0.6° and larger twist angles is confirmed, as well as the modest decrease of the energygap sizes for the FDP. Note that the experimentally observed energy gaps are consistently larger than theoretically obtained ones by approximately a factor of two.

We attribute the

approximately two-fold increase of the experimental data to substantially enhanced electronelectron interaction effects, which are expected in high quality graphene-h-BN heterostructures.42 Within a single-particle picture, we can further calculate the LL evolutions near the gaps at both Dirac points. Numerically generated LL-fan diagrams are presented for the FDP in Fig. 3(f), the SDPs in the hole- and electron-doped regions in Figs. 3(g) and 3(h), and the agreement with the experimental data is more than satisfactory. The valley degeneracy at both SDPs is considered gv = 2, same as the degeneracy at the FDP, as experimentally confirmed with the orange (FF = 4) and the yellow (FF = 2) tick marks in Figs. 3(a)–3(d).43 Next, we switch our discussions to the electronic structure variations at the charge neutrality point and the mini-zone boundaries in BL-h-BN superlattices. As shown in Figs. 2(b) and 2(d), the most radical differences of BL-h-BN superlattice structures when compared with SL-h-BN devices are the lack of the fully-formed energy gaps with moderate electron-hole asymmetries at both SDPs.

Figure 5(a) shows the LL-fan diagram from the BL-h-BN

superlattice (sample B) with a spacing of B = 0.1 T at a fixed Vg = 10 V. It is clear that no fullyformed energy gaps develop at the SDPs, with the electrostatically induced gap at the charge neutrality point at Vb ≈ -120 mV and Vg = 10 V. The absence of the energy gaps at the SDPs is further confirmed in the high-resolution dI/dVb mappings taken at the SDP in the hole-doped region (Fig. 5(b)) at Vg = 10 V, and the SDP in the electron-doped area (Fig. 5(c)) at Vg = -20 V,


14

Page 15 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

respectively. Around the weakly suppressed dI/dVb bands at the SDPs, secondary sets of LLs are present at lower Vb for the SDP in the hole-doping region and higher Vb for the SDP in the electron-doped area. At B = 1 T, the secondary LLs start interfering with the gapped structures at the SDP and completely overtake them at higher B fields. We numerically calculate the electronic band structure of BL-h-BN superlattices (Fig. 1(c)), and the LL-fan diagrams around the FDP and the SDPs (Figs. 5(e)–5(h)). We find that the simplest model accounting only for interaction effects between the h-BN and the bottom graphene layer is sufficient to capture most of the experimental observations in the LL-fan diagrams. A moderately advanced model considering the interactions between the h-BN and the top graphene layer is found to slightly facilitate the electron-hole symmetries by enhancing tunnel-feature visibility at the SDP in the electron-doped region. A full DFT model accounting for the coupling with the substrate would likely improve the agreement even more, but a detailed analysis is beyond the scope of the current study. As for the tunneling features at the FDP, however, the electrostatic interactions with the moiré patterns in BL-h-BN superlattices seems to be critical to understand experimental observations, as discussed below. Probing the electronic structures at the charge neutrality point (FDP) in BL-h-BN superlattices is not as straightforward as the previously discussed tunneling features at the SDPs and those for single-layer graphene superstructures. As shown in the gate mappings (Figs. 2(b) and 2(d)) and the LL-fan diagram in Fig. 5(a), the electric-field induced energy gap dominates the tunneling features since there exists a single set of electron- and hole-energy bands around the charge neutrality point in bilayer systems. Quite interestingly, however, we observe a faintbut-clear pair of dI/dVb peaks developing inside the electric-field induced gap in both positive and negative Vb regions, as marked with red arrows in Figs. 2(b) and 2(d). These dI/dVb peaks


15

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 38

evolve with a different angle across the gate mappings when compared with the slopes of electric-field induced energy-gap boundaries, suggesting that the observed dI/dVb peaks inside the gap are not directly related to the layer-symmetry breaking in bilayer graphene. In addition, the mid-gap structures do not change their positions in Vb while varying B, as marked with a dotted red line in the high-resolution dI/dVb mapping in Fig. 5(d). In total, we have measured three BL-h-BN superlattices and all three devices reveal similar dI/dVb features inside electricfield induced energy gaps. Figure 6 shows the high-resolution dI/dVb spectra map from the additional BL-h-BN superlattice device with a twist angle of 0.65°. The mid-gap feature developed inside the electrostatically induced gaps is marked with a dotted red arrow. It is interesting to point out that the spectra are much clearer in the sample C (Fig. 6) when compared with those in the sample B (Fig. 2(b)), indicating that the planar graphite-tunnel h-BN-BL-h-BN superlattice tunnel junction in the sample C is less susceptible to momentum-relaxation tunneling events from mechanical deformations such as bubbles and structural defects in the junction. The strong resonance peaks, appearing as negative dI/dVb bands (blue arrows in Fig. 6), further support that the relative crystalline-angle alignments of graphene and the top graphite probe are well preserved in the sample C. These intriguing in-gap tunneling signals seem to be consistent with a picture of a moiré potential induced energy broadening in the edge states around the electric field induced band gap as shown in Fig. S8 for a bilayer graphene nanoribbon subject to moiré potentials. Additionally, we note that the simultaneous presence of moiré potentials and an electric field in BL-h-BN superlattice gives rise to either zero or finite valley resolved Chern number C = ±2 depending on the sign of interlayer potential difference, 32 as displayed in Fig. S9.


16

Page 17 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

In summary, we have carried out a careful analysis of the electronic structure modifications in graphene-h-BN superlattice structures with varying twist angle, external electric and magnetic fields, and the number of carbon layers. Benefited by the much improved tunneljunction stability with thin h-BN as a tunnel barrier, the energy gaps formed both at the FDP and the SDP in the hole-doped region in SL-h-BN superlattices are accurately assessed by highresolution LL spectroscopies.

The absence of the gap around the electron-doped SDP is

confirmed. In BL-h-BN superlattices, the electron-hole asymmetries are significantly moderated with an energy gap not fully developed at both SDPs. We also find out experimental signatures of the mid-gap modes at the FDP due to the simultaneous presence of an electric field and moiré patterns in BL-h-BN superlattices.

All the key experimental observations are qualitatively

understood within a single-particle picture.

Method. Device Fabrication Procedures. Our tunneling and multi-probe devices are fabricated with multiple steps of dry-transfer method. First, bottom thick h-BN (> 50 nm) is mechanically exfoliated from high-quality single crystals and transferred onto thermally grown 90 nm thick SiO2 on Si. We carefully examine the h-BN flakes with well-defined crystallographic edges under bright- and dark-filtered optical microscope and an atomic force microscope to ensure surface cleanness. Then, either single- or bi-layer graphene flakes are mechanically exfoliated on the stacks of PMMA (poly(methyl Methacrylate)) and water soluble PSS (poly-styrene sulfonic) layers, and transferred on top of to the pre-located h-BN flakes with a micromanipulating transfer module equipped with a rotation stage for accurate twist-angle alignments.


17

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 38

We remove the PMMA film in warm acetone for an hour and annealed the samples at 350 oC for 7 hours in a mixture of Ar : H2 = 9 :1 to guarantee residue-free graphene surfaces and reduce graphene bubbles formed at the graphene-h-BN interface. Transferred graphene flakes are then patterned into graphene-ribbon structures whose widths are in the range of a couple of microns, utilizing conventional electron-beam lithography and O2-based dry-etching procedures. Next, either thin (three to five layers) or thick (> 50 nm) h-BN flakes are respectively transferred on top of the tunneling or multi-probe graphene/h-BN heterostructures with the aforementioned drytransfer methods with PMMA/PSS polymer stacks.

During the top h-BN transfers, we

intentionally misalign the twist angles of the top h-BN flakes and underlying graphene/h-BN superstructures. After each transfer, we meticulously repeat the procedures of removing PMMA films in the solvent and annealing the samples in an Ar-H2 environment at elevated temperature. For planar tunneling devices, the graphite flake is transferred on top of the thin h-BN tunnel insulator, which covers the pre-patterned single- and bi-layer graphene/h-BN moiré superlattices, while final devices are completed through an electron-beam lithography and metal (Ti ≈ 5 nm / Au ≈ 60 nm) lift-off process. For edge-contacted multi-probe devices, separate electron-beam lithography and dry-etching procedures are performed to pattern graphene Hall-bar structures in a single-layer region, and electric contacts are made with metal evaporations of Cr ( ≈ 5 nm) and Au (≈ 60 nm) films.

FIGURE CAPTIONS Figure 1. (a) Schematic of single-layer graphene-h-BN superlattices at a twist angle θ with detailed atomic structure configurations of BA, AB, and AA with carbon, nitrogen and boron


18

Page 19 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

atoms. The hexagonal superlattice structure with a moiré wavelength λM is marked with a green hexagon. (b),(c) Energy band diagrams of single- and bi-layer graphene-h-BN superlattices at θ = 0°.

For bilayer graphene, no electric-field induced energy gap is considered for the

calculations. (d) Optical viewgraph and schematics of graphene-h-BN heterostructures for a planar tunnel device and an edge-contacted multi-probe device. Both devices share a graphene and a bottom h-BN insulator. Red arrows indicate current flows for tunneling and quantum Hall devices. Scale bar is 10 μm. (e),(f) dI/dVb spectra in zero magnetic field as a function of backgate voltage in steps of ΔVg = 2 V for the single (sample A, (e)) and bilayer (sample B, (f)) superstructures. Twist angle for both devices is θ < 0.1°. Inverted orange triangles mark the FDP at charge-neutrality point, and red and blue ones indicate the SDPs for hole- and electrondoped regions, respectively.

Figure 2. (a),(b) dI/dVb gate maps comprised with 121-independent spectrum measured at varying Vg in steps of ΔVg = 0.5 V at B = 0 T for the sample A (a) and the sample B (b). The positions for the FDP are marked with a dotted white arrow, and those for the SDPs in hole- and electron-doped regions are indicated with dotted yellow and green arrows. (c),(d) dI/dVb gate maps at B = 1 T for the sample A (c) and the sample B (d). Each Landau level (LL) is labeled with a LL-index N (LLN), and dotted arrows mark the same locations of the FDP and the SDPs as those in Figs. 2(a) and 2(b).


19

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 38

Figure 3. (a) Two-dimensional display of dI/dVb tunneling spectra from the sample A at a fixed Vg = -20 V and T = 4 K with a varying external magnetic field up to B = 5 T, revealing series of LLs originated from either the FDP or the SDP in the hole-doped region. (b–d) High resolution LL-fan diagrams focused on the FDP (b), the hole-doped SDP (c), and the electron-doped SDP (d) regions. Each graph is obtained from the zoomed-in areas, marked with dotted boxes in Fig. 3(a). Orange and yellow tick marks indicate how much Vb is required to fill the LLs with a filling factor FF = 4 and FF = 2, respectively. White circles mark LL-peak positions from Lorentzian fittings and energy gaps (marked with yellow arrows) are obtained from extrapolating LL-peak positions in Vb to B = 0 T.

(e–h) Numerically simulated LL-fan diagrams.

Our B-field

dependent tunneling spectroscopy shows the structure of the bands near the SDP. Qualitative agreement between the single-particle theory and experiments is found, while some differences in the slopes of the tunneling gaps can arise due to electrostatic charging effects of the LL.

Figure 4. Energy gaps developed at the FDP (blue squares) and the SDP (red circles) in the hole-doped region in SL-h-BN superlattices at different twist angles. (inset) Numerically calculated energy gaps at the primary Dirac point in SL-h-BN superlattices with a consideration of fully relaxed graphene and h-BN lattices within a single-particle picture. We observe a fast closure of the secondary Dirac point energy gap that is independent of strain effects due to lattice relaxations.


20

Page 21 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 5. (a) Two-dimensional display of dI/dVb tunneling spectra from the sample B at a fixed Vg = 10 V and T = 4 K with a varying external magnetic field up to B = 5 T. The electrostatistically induced energy gap at the FDP is shown as a vertical dark band at Vb ≈ -125 mV, accompanying with series of LLs originated from the charge neutrality point in the bilayer graphene. (b–d) High resolution LL-fan diagrams focused on the hole-doped SDP (b) at Vg = 10 V, the electron-doped SDP (c) at Vg = -20 V, and the FDP (d) at Vg = 30 V. Back-gate voltages are chosen for each region to better resolve the not-fully-formed energy gaps (marked with yellow arrows) and the LLs from nearby energy bands (marked with dotted white lines) at both the SDPs. The dotted red arrow (d) mark the additional dI/dVb peak inside the electric-field induced energy gap at the FDP. (e–h) Numerically simulated LL-fan diagrams. The simulated diagrams capture the SDP band features that are in qualitative agreement with experiments. Some discrepancies in the slopes for the gaps can root in electrostatic charging effects due to filling of Landau levels which are not considered in our theoretical model.

Figure 6. High resolution dI/dVb gate map for the sample C with a twist angle 0.65°, focused on the dI/dVb tunneling features (marked with red arrows) inside the electric-field induced energy gap in BL-h-BN superlattices. Blue arrows mark the resonance peaks, appearing as negative dI/dVb bands in the gate map.

ASSOCIATED CONTENT


21

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 38

Supporting Information The Supporting Information is available free of charge on the ACS Publication website at DOI:

Additional data for estimating the twist angles in single-layer graphene-h-BN superlattices from Raman spectroscopy and multi-probe transport measurements, and detailed method for determining energy gaps of graphene superstructures with Landau level spectroscopy. Models and formalisms of the theoretical calculations for moiré band models of single- and bi-layer graphene-h-BN superlattices.

AUTHOR INFORMATION Corresponding Authors §

E-mail: [email protected]

§

E-mail: [email protected]

Author contributions *H. K. and *N. L. contributed equally to this work.

Notes The authors declare no competing financial interests.


22

Page 23 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

ACKNOWLEDGMENTS This work was supported by the research grant funded by Korea Research Institute of Standards and Science. Support from the National Research Foundation (NRF) of Korea is acknowledged

for

N.L.

through

NRF-2018R1C1B6004437,

B.L.C.

through

NRF-

2017R1D1A1B03035932, and J.J. through NRF-2016R1A2B4010105. N.L. and J. J. also acknowledge the support by the Korea Research Fellowship Program through the NRF funded by the Ministry and Science and ICT (KRF-2016H1D3A1023826).

REFERENCES

(1)

Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E. Rev. Mod. Phys. 2011, 83, 407–470.

(2)

Han, M. Y.; Ozyilmaz, B.; Zhang, Y.; Kim, P. Phys. Rev. Lett. 2007, 98, 206805.

(3)

Park, C. H.; Yang, L.; Son, Y. W.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2008, 101, 126804.

(4)

Park, C. H.; Yang, L.; Son, Y. W.; Cohen, M. L.; Louie, S. G. Nat. Phys. 2008, 4, 213– 217.

(5)

De Jamblinne De Meux, A.; Leconte, N.; Charlier, J. C.; Lherbier, A. Phys. Rev. B 2015, 91, 235139.

(6)

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. Nano 2010, 5, 722–726.


23

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(7)

Page 24 of 38

Dean, C. R.; Wang, L.; Maher, P.; Forsythe, C.; Ghahari, F.; Gao, Y.; Katoch, J.; Ishigami, M.; Moon, P.; Koshino, M.; Taniguchi, T.; Watanabe, K.; Shepard, K. L.; Hone, J.; Kim, P. Nature 2013, 497, 598–602.

(8)

Hunt, B.; Sanchez-Yamagishi, J. D.; Young, A. F.; Yankowitz, M.; LeRoy, B. J.; Watanabe, K.; Taniguchi, T.; Moon, P.; Koshino, M.; Jarillo-Herrero, P.; Ashoori, R. C. Science 2013, 340, 1427–1430.

(9)

Ponomarenko, L. A.; Gorbachev, R. V.; Yu, G. L.; Elias, D. C.; Jalil, R.; Patel, A. A.; Mishchenko, A.; Mayorov, A. S.; Woods, C. R.; Wallbank, J. R.; Mucha-Kruczynski, M.; Piot, B. A.; Potemski, M.; Grigorieva, I. V.; Novoselov, K. S.; Guinea, F.; Fal’ko, V. I.; Geim, A. K. Nature 2013, 497, 594–597.

(10)

Wang, L.; Gao, Y.; Wen, B.; Han, Z.; Taniguchi, T.; Watanabe, K.; Hone, J.; Dean, C. R. Science 2015, 350, 1231–1234.

(11)

Jung, J.; DaSilva, A.; Adam, S.; MacDonald, A. H. Nat. Commun. 2015, 6, 7308.

(12)

Giovannetti, G.; Khomyakov, P. A.; Brocks, G.; Kelly, P. J.; Van Den Brink, J. Phys. Rev. B 2007, 76, 073103.

(13)

Wallbank, J. R.; Patel, A. A.; Mucha-Kruczyński, M.; Geim, A. K.; Fal’Ko, V. I. Phys. Rev. B 2013, 87, 245408.

(14)

Jung, J.; Raoux, A.; Qiao, Z.; Macdonald, A. H. Phys. Rev. B 2014, 89, 205414.

(15)

Moon, P.; Koshino, M. Phys. Rev. B 2014, 90, 155406.


24

Page 25 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(16)

Yang, W.; Chen, G.; Shi, Z.; Liu, C.-C.; Zhang, L.; Xie, G.; Cheng, M.; Wang, D.; Yang, R.; Shi, D.; Watanabe, K.; Taniguchi, T.; Yao, Y.; Zhang, Y.; Zhang, G. Nat. Mater. 2013, 12, 792–797.

(17)

Yu, G.; Gorbachev, R.; Tu, J. Nat. Phys. 2014, 10, 525–529.

(18)

Yankowitz, M.; Jung, J.; Laksono, E.; Leconte, N.; Chittari, B. L.; Watanabe, K.; Taniguchi, T.; Adam, S.; Graf, D.; Dean, C. R. Nature 2018, 557, 404–408.

(19)

Chen, Z.-G.; Shi, Z.; Yang, W.; Lu, X.; Lai, Y.; Yan, H.; Wang, F.; Zhang, G.; Li, Z. Nat. Commun. 2014, 5, 4461.

(20)

Wang, E.; Lu, X.; Ding, S.; Yao, W.; Yan, M.; Wan, G.; Deng, K.; Wang, S.; Chen, G.; Jung, J.; Fedorov, A. V.; Zhang, Y.; Zhang, G.; Zhou, S. Nat. Phys. 2016, 12, 1111–1115.

(21)

Woods, C. R.; Britnell, L.; Eckmann, A.; Ma, R. S.; Lu, J. C.; Guo, H. M.; Lin, X.; Yu, G. L.; Cao, Y.; Gorbachev, R. V; Kretinin, a V; Park, J.; Ponomarenko, L. A.; Katsnelson, M. I.; Gornostyrev, Y. N.; Watanabe, K.; Taniguchi, T.; Casiraghi, C.; Gao, H.; Geim, A. K.; Novoselov, K. S. Nat. Phys. 2014, 10, 451–456.

(22)

Giaever, I. Phys. Rev. Lett. 1960, 5, 147–148.

(23)

Giaever, I. Phys. Rev. Lett. 1960, 5, 464–466.

(24)

Wolf, E. L. Principles of Electron Tunneling Spectroscopy; Oxford University Press: New York, 1989.

(25)

Yankowitz, M.; Xue, J.; Cormode, D.; Sanchez-Yamagishi, J. D.; Watanabe, K.; Taniguchi, T.; Jarillo-Herrero, P.; Jacquod, P.; LeRoy, B. J. Nat. Phys. 2012, 8, 382–386.


25

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(26)

Page 26 of 38

Rutter, G. M.; Jung, S.; Klimov, N. N.; Newell, D. B.; Zhitenev, N. B.; Stroscio, J. A. Nat. Phys. 2011, 7, 649–655.

(27)

Jung, S.; Park, M.; Park, J.; Jeong, T.-Y.; Kim, H.-J.; Watanabe, K.; Taniguchi, T.; Ha, D. H.; Hwang, C.; Kim, Y.-S. Sci. Rep. 2015, 5, 16642.

(28) Jung, S.; Myoung, N.; Park, J.; Jeong, T. Y.; Kim, H.; Watanabe, K.; Taniguchi, T.; Ha, D. H.; Hwang, C.; Park, H. C. Nano Lett. 2017, 17, 206–213. (29)

Woods, C. R.; Withers, F.; Zhu, M. J.; Cao, Y.; Yu, G.; Kozikov, A.; Ben Shalom, M.; Morozov, S. V.; van Wijk, M. M.; Fasolino, A.; Katsnelson, M. I.; Watanabe, K.; Taniguchi, T.; Geim, A. K.; Mishchenko, A.; Novoselov, K. S. Nat. Commun. 2016, 7, 10800.

(30)

Jung, J.; Laksono, E.; DaSilva, A. M.; MacDonald, A. H.; Mucha-Kruczyński, M.; Adam, S. Phys. Rev. B 2017, 96, 085442.

(31)

Mucha-Kruczyński, M.; Wallbank, J. R.; Fal’Ko, V. I. Phys. Rev. B 2013, 88, 205418.

(32)

Chittari, B. L.; Chen, G.; Zhang, Y.; Wang, F.; Jung, J. arxiv:1806.00462, 2018, 1–8.

(33)

Eckmann, A.; Park, J.; Yang, H.; Elias, D.; Mayorov, A. S.; Yu, G.; Jalil, R.; Novoselov, K. S.; Gorbachev, R. V.; Lazzeri, M.; Geim, A. K.; Casiraghi, C. Nano Lett. 2013, 13, 5242–5246.

(34)

Leconte, N.; Kim, H.; Kim, H.; Ha, D. H.; Watanabe, K.; Taniguchi, T.; Jung, J.; Jung, S. Nanoscale 2017, 9, 6041–6047.


26

Page 27 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(35)

Brar, V. W.; Wickenburg, S.; Panlasigui, M.; Park, C.-H.; Wehling, T. O.; Zhang, Y.; Decker, R.; Girit, C.; Balatsky, A. V.; Louie, S. G.; Zettl, A.; Crommie, M. F.; Girit, Ç.; Balatsky, A. V.; Louie, S. G.; Zettl, A.; Crommie, M. F. Phys. Rev. Lett. 2010, 104, 036805.

(36)

Dasilva, A. M.; Jung, J.; Adam, S.; Macdonald, A. H. Phys. Rev. B 2015, 92, 245422.

(37)

Efros, A. L.; Shklovskii, B. I. J. Phys. C Solid State Phys. 1975, 8, L49–L51.

(38)

Rutter, G. M.; Crain, J. N.; Guisinger, N. P.; Li, T.; First, P. N.; Stroscio, J. A. Science 2007, 317, 219–222.

(39) Greenaway, M. T.; Vdovin, E. E.; Mishchenko, A.; Makarovsky, O.; Patanè, A.; Wallbank, J. R.; Cao, Y.; Kretinin, A. V; Zhu, M. J.; Morozov, S. V; Fal, V. I.; Novoselov, K. S.; Geim, A. K.; Fromhold, T. M.; Eaves, L. Nat. Phys. 2015, 11, 1057–1062. (40)

Braga, D.; Gutiérrez Lezama, I.; Berger, H.; Morpurgo, A. F. Nano Lett. 2012, 12, 5218– 5223.

(41)

Ye, J.; Craciun, M. F.; Koshino, M.; Russo, S.; Inoue, S.; Yuan, H. PNAS 2011, 108, 13002–13006.

(42)

Chae, J.; Jung, S.; Young, A. F.; Dean, C. R.; Wang, L.; Gao, Y.; Watanabe, K.; Taniguchi, T.; Hone, J.; Shepard, K. L.; Kim, P.; Zhitenev, N. B.; Stroscio, J. A. Phys. Rev. Lett. 2012, 109, 116802.

(43)

Yang, W.; Lu, X.; Chen, G.; Wu, S.; Xie, G.; Cheng, M.; Wang, D.; Yang, R.; Shi, D.; Watanabe, K.; Taniguchi, T.; Voisin, C.; Plaçais, B.; Zhang, Y.; Zhang, G. Nano Lett. 2016, 16, 2387–2392.


27

Nano Letters

BA

(b)

AB

AB

AA

E (eV)

BA

λM

AA

AB

carbon

BA

AB

0.2

0.2

0.1

0.1

FDP

0.0

-0.2

-0.2 M

tunneling device

(e)

tunnel h-BN -

Ti/Au

graphite

-

h-BN Si/SiO2

h-BN

K’

Sample A, SL

Ti/Au h-BN Si/SiO2

-

Ti/Au

0

M

0V

-600

SDP@hole

0

Vb (mV)

300

600

K

Γ

Vg = 10 V SDP@electron

2

0V

1

0 -300

K’

Sample B, BL

3

Vg = -10 V

B = 0.0 T T = 4.0 K

Γ

(f ) FDP

2

I

h-BN -

Γ

SDP@electron FDP

1 Vb

K

Vg = 10 V

3

I Ti/Au

quantum Hall device

Γ

SDP@hole

Vb

graphite

graphene

FDP

0.0

-0.1

-0.1

boron

(d)

Bilayer / h-BN (θ = 0°)

nitrogen

dI/dVb (µS)

BA

(c)

Single layer / h-BN (θ = 0°)

E (eV)

θ

(a)

dI/dVb (µS)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 38

Vg = -10 V

B = 0.0 T T = 4.0 K

-500

-250

0

Vb (mV)

250

500

Figure 1. (a) Schematic of single-layer graphene-h-BN superlattices at a twist angle θ with detailed atomic structure configurations of BA, AB, and AA with carbon, nitrogen and boron atoms. The hexagonal superlattice structure with a moiré wavelength λM is marked with a green hexagon. (b),(c) Energy band diagrams of single- and bi-layer graphene-h-BN superlattices at θ = 0°. For bilayer graphene, no electric-field induced energy gap is considered for calculations. (d) Optical viewgraph and schematics of graphene-h-BN heterostructures for a planar tunnel device and an edge-contacted multi-probe device. Both devices share a graphene and a bottom h-BN insulator. Red arrows indicate current flows for tunneling and quantum Hall devices. Scale bar is 10 µm. (e),(f) dI/dVb spectra in zero magnetic field as a function of back-gate voltage in steps of ∆Vg = 2 V for the single (sample A, (e)) and bilayer (sample B, (f)) superstructures. Twist angle for both devices θ < 0.1°. Inverted orange triangles mark the FDP at charge-neutrality point, and red and blue ones indicate the SDPs for hole- and electron-doped regions, respectively.


Page 29 of 38

-9

(a)

600

400

-9

1x10-5

5x10

Log(dI/dVb)

(b)

Sample A SL-h-BN superlattice

SDP@electron

Vb (mV)

Vb (mV)

600 Sample B BL-h-BN superlattice

200

SDP@hole

0 SDP@hole

Eg @ FDP

-200

-400

-400 B = 0.0 T T = 4.0 K

-600 -30

-20

-10

0

10

20

30

Vg (V)

-30

(d)

-20

-10

0

10

FDP

20

30

Vg (V) 600


SDP@electron

400

B = 0.0 T T = 4.0 K

-600

(c) 600

Sample B BL-h-BN superlattice

400

200

200 LL0+

SDP@hole LL0+, SDP

-200

LL-1 LL-2

LL1 LL2

Vb (mV)

0

1x10-5

5x10

SDP@electron

FDP

0

-200

Log(dI/dVb)

400

200

Vb (mV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

LL0-

0

-200

LL0-,SDP

-400

-400 B = 1.0 T T = 4.0 K

B = 1.0 T T = 4.0 K

-600

-600 -30

-20

-10

0

10

20

30

-30

Vg (V)

-20

-10

0

10

20

30

Vg (V)

Figure 2. (a),(b) dI/dVb gate maps comprised of 121-independent spectrum measured at varying Vg in steps of ∆Vg = 0.5 V at B = 0 T for the sample A (a) and the sample B (b). The positions for the FDP is marked with a dotted white arrow, and those for the SDPs in hole- and electron-doped regions are indicated with dotted yellow and green arrows. (c),(d) dI/dVb gate maps at B = 1 T for the sample A (c) and the sample B (d). Each Landau level (LL) is labeled with a LL-index N (LLN), and dotted arrows mark the same locations of the FDP and the SDPs as those in Figs. 2(a) and 2(b).


Nano Letters

4

2.5

Sample A

Vg = -20 V

FF:4

FF:2

-8

2x10

2.5

FF:2

FF:4

LL0+ LL1

LL-1

B (T)

LL-2

C

c

0

(e)

-300

b 0

d 300

0.0

600

Vb (mV)

5

(f )

0.0 210

250

Vb (mV)

290

LL0+, SDP LL1, SDP

330

(g)

0.5

hole-doped SDP -230

-190

0.0 -150

Vb (mV)

-110

2.5

Theory

-70

(h)

Theory

2.0

3

1.5

1.5

1.5

LL0-

LL0+ LL1

LL-1 LL-2

1.0

g

0 -250

f -125

h 0

E (meV)

125

0.0 250

LL0+, SDP

FDP -25

0

E (meV)

25

620

Vb (mV)

660

700

Theory

LL0, SDP 1.0

LL1, SDP

0.5

-50

580

LL2

0.5

1

540

B (T)

B (T)

B (T)

2.0

LL0-,SDP

electron-doped SDP

2.5

2.0

1.0

LL0, SDP

1.0

4

2

FF:4

1.5

LL0-,SDP 1.0

FDP 170

Vg = -20 V

2.0

0.5

2.5

Theory

FF:4

-6

2x10

LL2

0.5

1

2.5

FF:2

-7

5x10 Log(dI/dVb)

(d)

1.5

B (T)

LL0-

1.0

-6

1x10

Vg = -20 V

2.0

1.5

2

-8

2x10 Log(dI/dVb)

(c)

Vg = -20 V

2.0

3

-6

1x10 Log(dI/dVb)

(b)

B (T)

5

-5

1x10

B (T)

-9

5x10 Log(dI/dVb)

(a)

B (T)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 38

50

0.0

0.5

hole-doped SDP -230

-205

-180

E (meV)

-155

-130

0.0

electron-doped SDP 130

155

180

E (meV)

205

230

Figure 3. (a) Two-dimensional display of dI/dVb tunneling spectra from the sample A at a fixed Vg = -20 V and T = 4 K with a varying external magentic field up to B = 5 T, revealing series of LLs originated from either the FDP or the SDP in the hole-doped region. (b–d) High resolution LL-fan diagrams focused on the FDP (b), the hole-doped SDP (c), and the electron-doped SDP (d) regions. Each graph is obtained from the zoomed-in areas, marked with dotted boxes in Fig. 3(a). Orange and yellow tick marks indicate how much Vb is required to fill the LLs with a filling factor FF = 4 and FF = 2, respectively. White circles mark LL-peak positions from Lorentzian fittings and energy gaps (marked with yellow arrows) are obtained from extrapolating LL-peak positions in Vb to B = 0 T. (e–h) Numerically simulated LL-fan diagrams. Our B-field dependent tunneling spectroscopy shows the structure of the bands near the SDP. Qualitative agreement between the single-particle theory and experiments is found, while some differences in the slopes of the tunneling gaps can arise due to electrostatic charging effects of the LL.


Page 31 of 38

18

FDP (Experiment)

15

Energy gap (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

12 8

9

6

6

2

3 0

FDP (Theory)

4

SDP@hole (Experiment) 0.0

0.2

0.4

0.6

0.8

0

SDP@hole (Theory) 0.0

1.0

0.6

1.2

1.8

1.2

Angle θ (degree)

Figure 4. Energy gaps developed at the FDP (blue squares) and the SDP (red circles) in the hole-doped region in SL-h-BN superlattices at different twist angles. (inset) Numerically calculated energy gaps at the primary Dirac point in SL-h-BN superlattices with a consideration of fully relaxed graphene and h-BN lattices within a single-particle picture. We observe a fast closure of the secondary Dirac point energy gap that is independentr of stain effects due to lattice relaxations.


Nano Letters

-9

5

-6

5x10 Log(dI/dVb)

(a)

5x10

Sample B

Vg = 10 V

-7

4x10 Log(dI/dVb)

(b) 2.5

-6

2x10

2.5

Vg = 10 V

-7

6x10 Log(dI/dVb)

(c)

-6

2x10

2.5

Vg = -20 V

2.0

3

1.5

1.5

1.5

2

1.0

1.0

1.0

1

0.5

0.5

0.5

(b) -600

(c) -300

0

300

Vb (mV)

(e)

0.0

hole-doped SDP -510

-470

(f )

5

Theory

0.0 -430

Vb (mV)

2.5

-390

-350


(g) Theory

510

0.0 550

Vb (mV)

2.5

590

Theory

1.5

1.5

1.5

B (T)

2

1.0

1.0

1.0

1

0.5

0.5

0.5

-80

-40

0

E (meV)

40

80

hole-doped SDP -80

-60

E (meV)

-40

0.0

Theory

B (T)

3

B (T)

2.0

0.0

-400 -325 -250

Vb (mV)

2.5

2.0

(g)

FDP

(h)

2.0

(f)

Vg = 30 V

-550 -475

630

4

0

-6

2x10

B (T)

B (T)

B (T)

2.0

B (T)

2.0

0

-8

2x10 Log(dI/dVb)

(d)

4

B (T)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 38


FDP 70

E (meV)

90

0.0

-20

0

E (meV)

20

Figure 5. (a) Two-dimensional display of dI/dVb tunneling spectra from the sample B at a fixed Vg = 10 V and T = 4 K with a varying external magentic field up to B = 5 T. The electro-statistically induced energy gap at the FDP is shown as a vertical dark band at Vb ≈ 125 mV, accompanying with series of LLs orginated from the charge neutrality point in the bilayer graphene. (b–d) High resolution LL-fan diagrams focused on the hole-doped SDP (b) at Vg = 10 V, the electron-doped SDP (c) at Vg = -20 V, and the FDP (d) at Vg = 30. Back-gate voltages are chosen for each region to better resolve the not-fully-formed energy gaps (marked with yellow arrows) and the LLs from nearby energy bands (marked with dotted white lines) at both the SDPs. The dotted red arrow (d) mark the additional dI/dVb peak inside the electric-field induced energy gap at the FDP. (e–h) Numerically simulated LL-fan diagrams. The simulated diagrams capture the SDP band features that are in qualitative agreement with experiments. Some discrepancies in the slopes for the gaps can root in electrostatic charging effects due to filling of Landau levels which are not considered in our theoretical model.


Page 33 of 38

-9

Log(dI/dVb)

-100

3x10-7

1x10

-300

Vb (mV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

-500 Sample C BL-h-BN superlattice B = 0.0 T T = 4.0 K

-700 10

20

30

40

Vg (V)

Figure 6. High resolution dI/dVb gate map for the sample C with a twist angle 0.65°, focused on the dI/dVb tunneling features (marked with red arrows) inside the electric-field induced energy gap in BL-h-BN superlattices. Blue arrows mark the resonance peaks, appearing as negative dI/dVb bands in the gate map.


(a)

-9

600

Nano Letters (b)

Log(dI/dVb)

5x10

1x10-5

Page 34 of 38

600


SDP@electron FDP 1 2 3 200 4 5 0 6 7 -200 SDP@hole 8 9 -400 B = 0.0 T 10 T = 4.0 K -600 11 -30 -20 12 (c) 600 13 14 FDP 15 400 16 17 200 18 SDP@hole 19 0 20 21-200 LL 22 -400 23 B = 1.0 T 24 T = 4.0 K 25-600 -30 -20 26

-9

1x10-5

5x10

Log(dI/dVb)

400

Sample B BL-h-BN superlattice

SDP@electron

400

Vb (mV)

Vb (mV)

200

0 SDP@hole

Eg @ FDP

-200

-400 B = 0.0 T T = 4.0 K

-600 -10

0

10

20

30

Vg (V)

-30

(d)

-20

-10

0

10

30

600


SDP@electron

20

Vg (V) Sample B BL-h-BN superlattice

400

200

LL0+, SDP

LL-1 LL-2

LL1 LL2

Vb (mV)

Vb (mV)

LL0+ LL0-

0

-200

0-,SDP

-400 B = 1.0 T

T = 4.0 K ACS Paragon Plus Environment

-600

-10

0

Vg (V)

10

20

30

-30

-20

-10

0

Vg (V)

10

20

30

FF:4

FF:2

2.5

Vg = -20 V FF:2

2.0

FF:4

LL0+ LL1

LL-1 LL-2

-300

b 0

d 300

0.0

600

Vb (mV)

(f )

210

250

Vb (mV)

290

330

-125

0

E (meV)

125

-230

-190

0.0 -150

Vb (mV)

-110

2.5

-70

(h)

B (T)

LL0-,SDP 1.0

LL0+, SDP

-50

0

E (meV)

25

580

620

Vb (mV)

660

700

Theory

LL0, SDP

LL2 0.5

hole-doped ACS Paragon Plus Environment SDP -25

540

1.0

LL1, SDP

0.5

FDP

electron-doped SDP

2.5

Theory

1.5

0.0 250

hole-doped SDP

1.5

B (T)

B (T)

-250

(g)

0.5

1.5

LL0+ LL1

LL0, SDP

1.0

2.0

0.5

h

LL1, SDP

2.0

LL0-

FF:4

2.0

2.0

LL-2

f

LL0+, SDP

1.0

Theory

LL-1

Vg = -20 V

1.5

LL0-,SDP

0.0

1.0

g

FF:4

-6

2x10

LL2

FDP

2.5

Theory

2.5

0.5

170

-7

5x10 Log(dI/dVb)

(d)

1.5

0.5

c

-6

1x10

FF:2

B (T) LL0-

1.0

-8

2x10 Log(dI/dVb)

Vg = -20 V

2.0

1.5

B (T)

1 4 2 3 4 3 5 6 2 7 8 1 9 10 C 11 0 12 (e) 13 5 14 15 4 16 17 18 3 19 20 2 21 22 23 1 24 25 0 26 27

2.5

Sample A

Vg = -20 V

-6

2x10

Nano (c) Letters

B (T)

5

-8

1x10 Log(dI/dVb)

(b)

B (T)

-5

1x10

B (T)

-9

5x10 (a) 35 ofLog(dI/dV Page 38 b)

50

0.0

-230

-205

-180

E (meV)

-155

-130

0.0


155

180

E (meV)

205

230

Nano Letters

18

Energy gap (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13

Page 36 of 38

FDP (Experiment)

15 12

8

9

6

6

2

3 0

FDP (Theory)

4

SDP@hole (Experiment)

SDP@hole (Theory)

0

0.0 0.6 ACS Paragon Plus Environment

0.0

0.2

0.4

0.6

0.8

Angle θ (degree)

1.0

1.2

1.2

1.8

Vg = 10 V

b -600

c -300

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

-510

-470

(b)

0.0 -430

Vb (mV)

2.5

-390

-350

f -80

g -40

0

E (meV)

40

470

Theory

510

0.0 550

Vb (mV)

2.5

590

Theory

1.5

1.5

1.5

B (T)

1.0

1.0

1.0

0.5

0.5

0.5

-80

ACS Paragon Plus Environment electron-doped -60

E (meV)

-40

-400 -325 -250

Vb (mV)

2.5

2.0

0.0

FDP

(d)

2.0

hole-doped SDP

-6

2x10

Vg = 30 V

-550 -475

630

2.0

0.0 80

electron-doped SDP

(c)

B (T)

B (T)

Theory

2.5

2.0

hole-doped SDP

-8

2x10 Log(dI/dVb)

(d)

2.0

300

Vb (mV)

-6

2x10

2.0

0.0

0

-7

6x10 Log(dI/dVb)

Vg = -20 V

B (T)

B (T)

1 4 2 3 3 4 5 6 2 7 8 1 9 10 11 0 12 (a) 13 5 14 15 16 4 17 18 3 19 20 21 2 22 23 1 24 25 26 0 27

2.5

-6

2x10

Nano (c) Letters 2.5

B (T)

Sample B

Vg = 10 V

-7

4x10 Log(dI/dVb)

(b)

Theory

B (T)

-6

5x10

B (T)

-9

5x10

(a) 37 of Log(dI/dV Page 38 b) 5

FDP

SDP

50

70

E (meV)

90

0.0

-20

0

E (meV)

20

-9

3x10-7

1x10

-100

Page 38 of 38

Vb (mV)

1 2 3 -300 4 5 6 7 8 -500 9 10 11 12 -700 10 13 14

Log(dI/dV b) Nano Letters

Sample C BL-h-BN superlattice B = 0.0 T T = 4.0 K

ACS Paragon Plus Environment 20

30

Vg (V)

40

h-BN

Recommend Documents