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A P P L I E D P H Y S I C S

Abnormal conductivity in low-angle twisted bilayer grapheneShuai Zhang1,2*, Aisheng Song2*, Lingxiu Chen3*, Chengxin Jiang3, Chen Chen3, Lei Gao4†, Yuan Hou5, Luqi Liu5, Tianbao Ma2, Haomin Wang3†, Xi-Qiao Feng1,2, Qunyang Li1,2†

Controlling the interlayer twist angle offers a powerful means for tuning the electronic properties of two-dimensional (2D) van der Waals materials. Typically, the electrical conductivity would increase monotonically with decreasing twist angle owing to the enhanced coupling between adjacent layers. Here, we report a nonmonotonic angle- dependent vertical conductivity across the interface of bilayer graphene with low twist angles. More specifically, the vertical conductivity enhances gradually with decreasing twist angle up to a crossover angle at c ≈ 5°, and then it drops notably upon further decrease in the twist angle. Revealed by density functional theory calculations and scanning tunneling microscopy, the abnormal behavior is attributed to the unusual reduction in average carrier density originating from local atomic reconstruction. The impact of atomic reconstruction on vertical conductivity is unique for low-angle twisted 2D van der Waals materials and provides a strategy for designing and optimizing their electronic performance.

INTRODUCTIONVarying the interlayer twist angle provides an effective means for tuning the electronic properties of van der Waals structures (1–5). The incommensurate state of the twisted interface between adjacent layers typically leads to suppression of interlayer coupling, thereby decreased interlayer conductivity (6–9). For instance, recent experi-ments have shown that the interlayer conductivity of two-dimensional (2D) van der Waals structures, e.g., graphene/graphene (6) and graphene/graphite (10) junctions, typically decreases monotonically as the twist angle increases. Such monotonic angle-dependent inter-layer conductivity can be well explained using a phonon-mediated interlayer transport mechanism (11). Besides the interlayer conduc-tivity, the vertical conductivity probed by conductive atomic force microscopy (c-AFM), where the interlayer conductivity of 2D material is coupled with the probe-sample contact conductivity, also reveals a similar trend in MoS2/graphene (12), graphite/graphite (7), and graphene/graphene with a large twist (13) system. However, recent studies on low-angle twisted bilayer graphene (TBG) found that the competition between van der Waals interaction and in-plane elasticity might cause local atomic-scale reconstruction of graphene, resulting in unconventional electronic properties such as superconductivity, correlated insulators, and spontaneous ferro-magnetism (5, 14–20). How vertical conductivity of TBG evolves with the twist angle, especially at low twist angles when atomic re-constructions exist, remains scientifically intriguing and unexplored.

RESULTSTo systematically investigate the variation of vertical conductivity of TBG with a twist angle, we performed c-AFM measurements, which have been widely used in previous studies (7, 12, 13, 21). In the ex-periments, a thick h-BN flake was used as the substrate, on which bilayer graphene was grown by chemical vapor deposition (CVD) method (more details can be found in fig. S1 and in Materials and Methods). For our samples, the bottom graphene layer is a continuous polycrystalline film, while the top graphene layer is a single-crystal graphene island. This special structure can be confirmed by prescans (fig. S2). The continuous film with distinct grain boundaries and island in hexagon shape can be observed corresponding to the bottom polycrystalline and top single-crystal graphene, respectively. The unique sample structure enables a large number of TBG domains with a wide range of twist angles to be interrogated simultaneously, as illustrated in Fig. 1A. During a typical c-AFM measurement, a constant bias voltage was applied between the conductive probe and the bottom film, and the current was continuously monitored. Figure 1B shows a typical current image of a TBG sample, where four regions with different moiré patterns can be clearly observed. On the basis of correlation between the period of moiré pattern and the twist angle, the twist angles of these four regions can be estimated as indicated in Fig. 1B. It should be noted that when the twist angle of TBG is larger than 12°, the period of the moiré pattern is less than 1.2 nm, which is too dense to be clearly identified by our c-AFM. As shown by the 2D map in Fig. 1B and the line profiles in Fig. 1C, the TBG domain with a twist angle of 3.0° exhibits notable enhanced averaged conductivity compared with the TBG domain with twist angles larger than 12°. This trend is consistent with the previous reports, where vertical conductivity was found to gradually increase with decreasing twist angle (7, 10, 12, 13). However, when the twist angle continues to decrease to 1.1°, the overall conductivity of the TBG domain decreases unexpectedly as indicated by Fig. 1 (B and C). Such drop in vertical conductivity of TBG with decreasing twist angle is distinctively different from the monotonic angle-dependent conductivity observed in previous studies (7, 10, 12, 13).

To further explore the abnormal twist angle dependence, we carried out measurements on more TBG samples so that a wider range of

1AML, CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China. 2State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China. 3State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China. 4Beijing Advanced Innovation Center for Materials Genome Engineering, Institute for Advanced Materials and Technology, University of Science and Technology Beijing, Beijing 100083, China. 5CAS Key Laboratory of Nanosystem and Hierarchical Fabrication, CAS Center for Excel-lence in Nanoscience, National Center for Nanoscience and Technology, Beijing 100190, China.*These authors contributed equally to this work.†Corresponding author. Email: qunyang@tsinghua.edu.cn (Q.L.); gaolei@ustb.edu.cn (L.G.); hmwang@mail.sim.ac.cn (H.W.)

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).

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twist angles could be sampled. The averaged current versus twist angle data for a series of TBG samples are plotted in Fig. 2. Because several probes were used for different samples, to minimize the influence of the probe conditions on measured conductivity (12, 13), all the reported currents were normalized by the average current value ob-tained from the domain with a twist angle of >12° of each sample. As shown in Fig. 2, when the twist angle reduces from >12° to ~5°, the conductivity of the TBG increases gradually, which is consistent with the results shown in Fig. 1 (B and C) and previous reports (7, 10, 12, 13). However, an unusual reduction in conductivity is observed when the twist angle is decreased below ~5°. This non-monotonic angle dependence in vertical conductivity was repro-ducible for nine different samples as illustrated by different colors and shapes in Fig. 2. To rule out the influence of the h-BN substrate, we transferred monolayer graphene to the surface of graphite with a controllable low twist angle (more details can be found in Materials and Methods) and measured the vertical conductivity using c-AFM. As shown in the inset of Fig. 2, the conductivity of low-angle twisted graphene on graphite also shows a similar trend that the conductivity decreases notably with decreasing twist angle. Combining the trend that we observed for the low-angle twisted samples with the large-angle behavior reported previously (13), we expect that a similar crossover in vertical conductivity would also exist for TBG on a graphite surface.

Conductivity measurements with finer resolution were carried out to examine the origin of the abnormal decrease in conductivity when the twist angles are below ~5°. To make quantitative compar-isons, we carried out measurements by scanning across regions with both twist angle >12° and small twist angles simultaneously and normalized the current using the averaged current value from the region with the twist angle >12°. Figure 3A shows five normalized current profiles obtained from the TBG samples with twist angles of 2.9°, 1.5°, 0.9°, 0.8°, and 0.6°, respectively. In all current profiles, a periodic fluctuation can always be clearly observed in the small twist angle regions although their periods are different. As shown in Fig. 3 (B and C), the periodic fluctuation in the current profiles con-stitutes the 2D hexagonal patterns that are consistent with the moiré superlattice structures. In the meantime, no obvious contrast can be observed in topographic images, as shown in fig. S3, suggesting that height undulation is minimum for these samples. According to pre-vious studies (13, 22, 23), the regions with low and high conductivity

would correspond to approximate AB-/BA- and AA-stacked regions, respectively, as schematically illustrated in Fig. 3 (D to F). As indi-cated by Fig. 3A, when the twist angle decreases, the conductivity of the AA-stacked region remained nearly unchanged, but the con-ductivity of the AB-stacked region decreases notably. More-over, as the twist angle decreases from 1.5° to 0.6°, the areal ratio of the AB-stacked region within a moiré pattern period gradually in-creases, which leads to a notable decrease in averaged current. For example, by comparing Fig. 3B with Fig. 3C, we can clearly see that the proportion of the AA-stacked region decreases notably as the twist angle reduces from 2.9° to 0.6°. The large area of the AB-stacked region and the isolated AA-stacked region have been found to be a unique feature of TBG with low twist angles originating from local reconstruction, based on previous experiments and theoretical calculations (15–17).

0 10 20 30 40 500

5

10

15

Cur

rent

(nA

)

Distance (nm)

1.1° 3.0° >12°

0.0

9.0

B C

1.1º

3.0º>12º

20 nm

GBs

Cur

rent

(nA

)

+V

GBs

Tip

A

>12º

Fig. 1. Measurement of conductivity of TBG with varying twist angles. (A) Schematic of c-AFM for measuring vertical conductivity of bilayer graphene on h-BN sub-strate with different twist angles. A constant bias was applied between the conductive tip and the bottom graphene film. GBs, grain boundaries. (B) Typical current image measured on bilayer graphene showing domains with different twist angles (1.1°, 3.0°, and >12°) under a bias of 10 mV. Scale bar, 20 nm. (C) Typical current line profiles measured from different domains with twist angles of 1.1°, 3.0°, and >12°, respectively.

TBG/h-BN

0 2 4 60.5

1.0

1.5

2.0

2.5

Nor

mal

ized

cur

rent

(a.u

.)

Twisted angle (°)

TBG/graphite

0 5 10 15 200.6

1.0

1.4

1.8

2.2

Nor

mal

ized

cur

rent

(a.u

.)

Twisted angle ( )°

Fig. 2. Dependence of vertical conductivity on twist angle. The relationship be-tween the normalized current and the twist angle obtained on TBG/h-BN is shown. Data with the same symbol color and shape were obtained simultaneously from the same current image. The inset shows the relationship between current and twist angle obtained on twisted graphene on graphite, where the current values were normalized by the average current value of bilayer graphene with a twist an-gle of 0°. The error bar represents the SD of the current signal in each image. a.u., arbitrary units.

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To further characterize the moiré and sub-moiré scale structures with higher resolution, we have carried out scanning tunneling microscopy (STM) experiments on TBG with low twist angles. Figure 4A shows three typical 3D height images measured on TBG with twist angles of 0.6°, 1.1°, and 3.3°, where clear moiré superlattice structures with varying periods can be observed. As the twist angle decreases from 3.3° to 0.6°, the areal ratio of the AB-stacked region within a moiré pattern period gradually increases as illustrated in Fig. 4B, consistent with the c-AFM measurements. By comparing the height of the regions with a small twist angle and >12°, it can be clearly seen that as the twist angle decreases from 3.3° to 0.6°, the averaged height difference gradually reduces as illustrated in Fig. 4B and fig. S4. As the STM images were taken under a constant-current mode, the height signal would be a manifestation of both actual topographic height and local density of states (24–26). Considering that the variation of actual height is negligible in our experiments (fig. S3), the contrast in the STM height signal would primarily reflect the difference in local density of states (26). Therefore, the STM measurements suggest that the averaged local density of states on the surface of TBG decreases as the twist angle reduces from 3.3° to 0.6°. To verify the atomic reconstruction, we took the TBG with a twist angle of 1.1° as an example to characterize the sub-moiré scale structure. As shown in Fig. 4C, the zoom-in STM image shows clear AA-stacked (red color) and AB-stacked (blue color) zones together with a fine graphene lattice. By carefully examining the Fourier transform patterns and the atomically resolved images (Fig. 4D), we observed a slight orientation difference about 1° to 2° between AA- and AB-/BA-stacked regions. This suggests that the atoms at the AB-/BA-stacked region indeed locally rotate and re-construct to form commensurate stacking, as schematically shown in the lower panels of Fig. 4D.

To understand how the moiré superlattice structure and local reconstruction lead to the abnormal vertical conductivity, we per-

formed theoretical calculations on TBG with varying twist angles. The local conductivity of the junction of tip/TBG was calculated using the atomic-scale contact quality (ACQ) model (details can be found in Materials and Methods), which can capture the effect of carrier density and tunneling barrier on the electrical conductance of 2D interfaces (27), as schematically shown in Fig. 5A and fig. S5. The atomic configurations of TBG were relaxed firstly via density functional theory (DFT) calculations. Consist with the experimen-tal results and previous studies (13, 22, 23, 28, 29), the calculated real-space distribution maps of the tip/TBG junction conductivity (Fig. 5B) show clear moiré superlattice level modulation. In all cases, the AA-stacked regions indeed exhibit higher conductivity than the AB-stacked regions. To quantify the variation of conductivity with twist angle, we calculated the mean tip/TBG junction conductivity within a unit moiré superlattice structure as a function of the twist angle and plotted it in Fig. 5C. It can be clearly seen that the sim-ulation results well reproduce the experimental observations (Fig. 2): Mean conductivity increases with increasing before c ≈ 5.5° and reverses afterward.

To understand the origin of the crossover behavior, we calculated the graphene-graphene interlayer conductivity by considering the interlayer tunneling barrier using the ACQ model and DFT-relaxed configurations. As shown in Fig. 5D, in contrast to the vertical con-ductivity of the tip/TBG junction, the interlayer conductivity decreases monotonically with increasing twist angle, which is consistent with the previous experiments and the tunneling barrier calculations (7, 10, 12, 13, 27). As the STM results have suggested that local den-sity of states might decrease as the twist angle decreases (Fig. 4B and fig. S4), we speculated that the nonmonotonic behavior of the tip/TBG conductivity might result from the variation of the carrier density on the top graphene surface. To verify this idea, we further calculated the average carrier density (refers to the density of the elec-tron states that are near the Fermi level, i.e., EF − 0.5 eV < E < EF + 0.5 eV)

1.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

0 20 40 60 800.0

1.0

2.0

>12°

0.6°

0.8°

0.9°

1.5°

~

~

~

~2.9°

~

~

Nor

mal

ized

cur

rent

(a.u

.)

~

~

Distance (nm)

A

AA

AB

AA

AB

AA

B

AB

AA

AA

AA

AB

0.0

1.0

Cur

rent

(a.u

.)

C

2.9°

AAAB

BA

0.6°

0.0

1.0

Cur

rent

(a.u

.)

BA

D

E

F

AB

BAAA

AB

BAAA

Fig. 3. Conductivity and structure evolution with twist angles. (A) Typical current profiles measured on TBG across two domains (one domain with a twist angle of >12° and the other domain with twist angles of 2.9°, 1.5°, 0.9°, 0.8°, and 0.6°, respectively). (B and C) Typical current images obtained from TBG with twist angles of 2.9° and 0.6°, respectively. The AA-stacked regions are marked with black circles. Scale bar, 10 nm. (D to F) Schematics showing the atomic stacking in TBG with different twist angles and the atomic configurations for AA, AB, and BA stacking.

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of the top-layer graphene via DFT calculations. As shown in Fig. 5D and fig. S5, the averaged carrier density on the top graphene layer evolves in a similar trend as the tip/TBG junction conductivi-ty, indicating that the variation of the carrier density might account for the abnormal angular dependence of the tip/TBG vertical conductivity.

To better understand the nonmonotonic variation trend of the carrier density of graphene, we performed further DFT calculations and found that the presence of the AA-stacked regions would enhance the local carrier density (fig. S5) (13). Further calculations confirmed that such enhanced carrier density was barely affected by the electric field that we imposed regardless of the moiré pattern and reconstruction (fig. S6). According to previous studies (26, 30, 31), the AA-stacked region shows higher density of states near Fermi level to accommodate more carriers. From the distribution of inter-layer valence electron density obtained from our DFT calculations (fig. S6B), the overlap of interlayer electron states in the AA-stacked region is weaker than that in the AB-stacked region. This suggests that the electrons in the AA-stacked region tend to act as carriers rather than bonding electrons, resulting in higher local carrier den-sity in the AA-stacked region, as shown in fig. S6C. Consequently, the averaged carrier density on the graphene surface of TBG should roughly scale with the areal fraction of the AA-stacked region in moiré superlattice. To confirm this hypothesis, we calculated the areal fraction of the AA-stacked region (rAA/a)2 using DFT-relaxed atomic structures (details can be found in fig. S7), where rAA is the radius of the AA-stacked region and a is the period of moiré super-lattice. As shown in Fig. 5E, the areal fraction of the AA-stacked region of the relaxed structures exhibits a similarly nonmonotonic

trend as the carrier density. We believe that this nonmonotonic trend is primarily caused by the graphene reconstruction, which can be clearly seen in the DFT-relaxed configuration (inset of Fig. 5E and fig. S5). In contrast, when the graphene reconstruction is artificially suppressed by assuming an interface configuration with rigid rotation, the nonmonotonic trend is absent, as indicated by the blue crosses in Fig. 5E.

DISCUSSIONSince vertical transport property of the TBG is determined by two factors, surface carrier density and interlayer tunneling barrier, both high carrier density and low tunneling barrier are essential for high conductivity. When is near 0°, the overlap of interlayer elec-tron states is strong and the interlayer tunneling barrier is low, but the carrier density is low when is near 0° because of the low areal fraction of the AA-stacked region resulting from the atomic recon-struction. In contrast, when  > 12°, although the carrier density is higher than that of the AB-stacked graphene, the interlayer electron states is decoupled and the tunneling barrier is high. Therefore, the conductivities at nearly 0° and  > 12° were both lower than that at the crossover angle (around ~5°), resulting in the nonmonotonic dependence of vertical conductivity of TBG on the twist angle.

In conclusion, taking TBG as an example, we found that the vertical conductivity of van der Waals heterostructures exhibited a nonmonotonic dependence on the twist angle. In general, reducing the twist angle of TBG would enhance its vertical conductivity be-cause of increased interlayer conductivity and higher carrier density. However, when the twist angle is below a certain threshold (c ≈ 5°),

50 nm

0.0

1.0 1°–2° 1°–2°

AA AB BA

5 Å 5 Å 5 Å

D

AA

AA

AA

AB

BA

BA

2 nm

C 1.1º

A

B

Hei

ght (

a.u.

)

50 nm 50 nm

0.6º 1.1º 3.3º

Reconstruction

Reconstruction

0 10 20 30 40 50 60

Distance (nm)

Hei

ght

3 Å

AA

AA

AA

AB

AB

21.9 nm13.2 nm 4.3 nm

3.3°

2.3°

1.1°

0.6°

>12°

Fig. 4. STM characterizations of moiré and sub-moiré scale structures. (A) Three typical 3D height images measured on TBG with twist angles of 0.6°, 1.1°, and 3.3°, respectively. (B) Four typical height profiles measured on TBG across two regions (one region with a twist angle of >12° and the other region with twist angles of 3.3°, 2.3°, 1.1°, and 0.6°, respectively). (C) High-resolution characterization of sub-moiré scale structure measured on TBG with a twist angle of 1.1°. Scale bar, 2 nm. (D) Fourier trans-form patterns (top panels), Fourier-filtered atomically resolved images (middle panels), and the corresponding schematic diagram of atomic stacking structure (bottom panels) for AA-, AB-, and BA-stacked regions, respectively. Scale bar, 5 Å. The STM measurements were carried out under a constant-current mode with the same bias voltage of 50 mV.

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the vertical conductivity might decrease abnormally with reducing twist angle because of a notable drop in carrier density. Our find-ings emphasize the influence of atomic reconstruction on the vertical conductivity, a unique feature for 2D interfaces but not for bulk junctions, and offer guidance for the optimizing the electric perform-ances of TBG and other 2D van der Waals structures.

MATERIALS AND METHODSSample preparationThe TBG samples were grown on h-BN by CVD. First, h-BN flakes were mechanically exfoliated onto quartz and then annealed at 600°C in an oxygen flow to remove the residues. Second, the annealed samples were placed into a graphitic tube in the furnace chamber. The growth of graphene was carried out at 1300°C by applying a pulsed flow of C2H2/SiH4 mixture to form a polycrystalline layer of graphene on h-BN and then maintaining the flow of the mixture to grow the second layer of graphene. Last, the chamber was naturally cooling down to room temperature in a flow of Ar/H2 (fig. S1).

The twisted graphene on graphite samples were prepared by a positioning transfer method. First, multiple graphene layers were mechanically exfoliated on a SiO2/Si substrate, and the layer number of graphene was identified by optical contrast and Raman spectrum. Then, the monolayer graphene was lifted by polydimethylsiloxane film that was mounted on a high-precision translation stage. Assisted with the optical microscopy, the monolayer graphene was transferred on the nearby graphite to form the graphene/graphite structure with precisely controlled twist angles by a rotation stage. Last, the

conductive silver glue was deposited along the edge of bottom graphite for further characterization.

Sample measurementsAsylum Research Cypher atom force microscope (AFM) was used to perform the conductivity and STM measurements. The conductive probes for conductivity measurements have a nominal radius of ~25 nm (ASYELEC-01, Asylum Research); the Pt/Ir probes for STM measure-ments were prepared by mechanically cutting the Pt/Ir wire. Both measurements were conducted under ambient conditions (20° to 25°C; relative humidity, 20 to 30%). For the TBG/h-BN samples, the bias voltage was applied between the tip and the bottom graphene. For twisted graphene on graphite samples, the bias voltage was applied between the tip and the bottom AB-stacked graphite. According to the contact mechanics model (JKR model) (32) considering adhesion, the contact radius was estimated to be about 1.25 nm in the c-AFM experiments.

Ab initio calculationThe geometry optimization and carrier density calculations (partial charge density, i.e., electronic states within the energy range EF − 0.5 eV < E < EF + 0.5 eV) were implemented in the Vienna Ab initio Simulation Package (VASP). The calculation models are shown in fig. S5. The projector augmented-wave method was used to model the core electrons, and the optB88-vdW exchange-correlation func-tional was used to approximately describe the dispersion interaction (van der Waals forces). The plane wave basis kinetic energy cutoff was set to 400 eV. Both graphene layers were allowed to relax until the force on each atom was smaller than 0.01 eV/Å.

AA

AABA

AB

0 5 10 151.0

1.1

1.2

1.3

1.4 Interlayer conductivity (w/o tip)

Inte

rlaye

r con

duct

ivity

(a.u

.)

1.0

1.2

1.4

1.6

1.8

2.0 Carrier density of top graphene

Car

rier d

ensi

ty (a

.u.)

0.005

0.018

D

G

BTip

2 nm

AAAB AA

ABAAAB AA

ABAB

Con

duct

ivity

(a.u

.)

0 5 10 150.8

0.9

1.0

1.1

1.2

1.3

1.4

Con

duct

ivity

(a.u

.)

Twisted angle (°)

Conductivity of tip/TBG

C

Twisted angle (°)

0.0 0.017 Å

A 5.5º 11º4.7º3.5º0º

Rigid structure

0 3 6 9 120.6

0.7

0.8

0.9

1.0

Relaxed structure

Nor

mal

ized

(rA

A/a

)2

Twisted angle (°)

E

Fig. 5. Evolutions of conductivity, carrier density, and atomic configurations of TBG with twist angle. (A) Schematic showing the simulation model of c-AFM. (B) Simulated local conductivity maps of TBGs with twist angles of 0°, 3.5°, 4.7°, 5.5°, and 11°, respectively. (C and D) Averaged tip/TBG junction conductivity (C), TBG interlayer conductivity, and averaged carrier density of the top-layer graphene (D) calculated for different twist angles. (E) Normalized areal fraction of the AA-stacked region in moiré superlattice (rAA/a)2 calculated using relaxed and rigid atomic stacking structures. The inset shows the in-plane atomic displacements after relaxation for TBG with a twist angle of 3.5°. The dashed lines are schematically drawn to highlight the trend.

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ACQ model calculationsWhen calculating the tip/TBG junction conductivity map, the atomic configurations of TBG were first relaxed by DFT calculations. Then, the AFM tip apex is scanning over the top-layer graphene at a dis-tance of 3 Å. When the tip is at the position (x, y), the local conduc-tivity G(x, y) of the tip/TBG junction is calculated as (27)

G(x, y ) = j=1

n ( ( C topj (x, y ) ⋅ e −3.91 d C topj −tip (x,y)+10 )

−1

+ ( i=1

m e −4.20 d C topj − C boti (x,y)+10.93 )

−1 )

−1

where n represents the number of C atoms of top-layer graphene nearby the tip position (x, y) and m represents the number of C at-oms of bottom-layer graphene adjacent to each of the n carbon at-oms of top-layer graphene. Ctopj (x, y) is the surface carrier density at the jth C atom of top-layer graphene near the tip position (x, y). dCtopj-tip is the distance between the jth C atom of top-layer graphene and the tip apex, while dCtopj-Cboti is the distance between the jth C atom of top-layer graphene and the ith C atom of bottom-layer graphene. The averaged tip/TBG junction conductivity is calculated by averaging the local conductivity of the whole scan region.

The TBG interlayer conductivity is calculated as

G interlayer = Σ j=1

n′

Σ i=1

m′

e −4.20 d C topj ‐ C boti +10.93 / A

where m′ and n′ represent the C atom numbers of the top-layer graphene and the bottom-layer graphene, respectively. A is the area of the TBG supercell.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/47/eabc5555/DC1

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Acknowledgments Funding: We thank the support from the National Key Research and Development Program of China (no. 2017YFB0702100), National Natural Science Foundation of China (nos. 11772169, 11921002, 11890671, 51527901, 51772317, 91964102, and 51705017), the Initiative Program of State Key Laboratory of Tribology (SKLT2019B02), National Science and Technology Major Project (2017-VI-0003-0073), the National Key Research and Development Program of China (no. 2017YFF0206106), the Strategic Priority Research Program of Chinese Academy of Sciences (no. XDB30000000), and the China Postdoctoral Science Foundation (nos. 2019 T120366 and 2019 M651620). Computations were carried out on the “Explorer 100” cluster system of Tsinghua National Laboratory for Information Science and Technology. Author contributions: Q.L. and H.W. conceived the project. S.Z. performed the c-AFM and STM measurements.

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A.S., L.G., and T.M. carried out the first-principles and theoretical calculations. L.C., C.J., C.C., and H.W. prepared the TBG/h-BN samples. Y.H. and L.L. prepared the TBG/graphite samples. S.Z., A.S., L.C., L.G., T.M., H.W., and Q.L. wrote the paper. All authors analyzed and discussed the results and approved the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

Submitted 30 April 2020Accepted 7 October 2020Published 20 November 202010.1126/sciadv.abc5555

Citation: S. Zhang, A. Song, L. Chen, C. Jiang, C. Chen, L. Gao, Y. Hou, L. Liu, T. Ma, H. Wang, X.-Q. Feng, Q. Li, Abnormal conductivity in low-angle twisted bilayer graphene. Sci. Adv. 6, eabc5555 (2020).

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Abnormal conductivity in low-angle twisted bilayer graphene

Wang, Xi-Qiao Feng and Qunyang LiShuai Zhang, Aisheng Song, Lingxiu Chen, Chengxin Jiang, Chen Chen, Lei Gao, Yuan Hou, Luqi Liu, Tianbao Ma, Haomin

DOI: 10.1126/sciadv.abc5555 (47), eabc5555.6Sci Adv 

ARTICLE TOOLS http://advances.sciencemag.org/content/6/47/eabc5555

MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2020/11/16/6.47.eabc5555.DC1

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