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1 Epitaxial Growth of Single-Domain Graphene on Hexagonal Boron Nitride Wei Yang 1, Guorui Chen, Zhiwen Shi 1, Cheng-Cheng Liu 1,3, Lianchang Zhang 1,4, Guibai Xie 1, Meng Cheng 1, Duoming Wang 1, Rong Yang 1, Dongxia Shi 1, Kenji Watanabe 5, Taashi Taniguchi 5, Yugui Yao 3, Yuanbo Zhang, and Guangyu Zhang 1 * 1 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing , China. State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 00433, China. 3 School of Physics, Beijing Institute of Technology, Beijing , China. 4 Department of Physics, Kunming University, Kunming 65014, China. 5 Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namii, Tsuuba, , Japan. * gyzhang@aphy.iphy.ac.cn NATURE MATERIALS 1

2 I. More AFM images of G/h-BN Figure S1 shows various hexagonal graphene grains as well as etched hexagonal pits. It is clear that these hexagons are aligned with each other, regardless of grains or pits. These images reveal that the as-grown graphene is single crystalline, confirming the epitaxial nature of graphene grown on h-bn. a b c d e f 5 nm Figure S1. a-c, AFM images of various hexagonal MLG grains with same orientation. d-f, AFM images of various etched samples: quasi-continuous MLG (d), continuous MLG film, and continuous BLG film (f). After etching, all etched hexagonal pits have the same orientation. Scale bars in a-f are 00 nm. II. Raman spectra fittings for G/h-BN Zoomed-in plot of G and D Raman peas for h-mlg grains and BLG are shown in Fig. S. G-band of h-mlg grains (blac) shows split peas at 1563 cm -1 (blue) and 1581cm -1 (cyan), while that of BLG (red) contains only 1581 cm -1. D-band of h-mlg (blac) is fitted by single Lorentz pea, while that of BLG (red) is fitted by 4 Lorentz peas (green). NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION Figure S. a, G-band of h-mlg grains (blac) is split into two peas at cm -1 (blue) and cm -1 (cyan), while that of BLG (red) contains only cm -1. b, D-band of h-mlg (blac) is fitted by single Lorentz pea, while that of BLG (red) is fitted by 4 Lorentz peas (green). III. Graphene growth at h-bn step edges AFM images of graphene grown across h-bn step edges are shown in Fig. S3. Three types of step edges, i.e. monolayer, bilayer, and triple layer h-bn edges, were investigated. For a monolayer step, graphene can grow across the edges (Fig. S3a). While as to bilayer and triple layer steps, graphene grains would nucleate at the edges, and given enough growth time graphene film at up step would connect together with that at down step. Figure S3. a, Graphene grains nucleated across a single atomic h-bn step. b, Graphene nucleated along edges of bilayer h-bn step. c, Graphene growth at a step with three h-bn layers. Scale bars in a-c are 00 nm. IV. Temperature dependence of graphene growth NATURE MATERIALS 3

4 It is worthy of mentioning that growth temperature (T) also plays an important role. Figure S4 shows two AFM images of graphene nucleation at 500 o C and 530 o C. We can see that monolayer nucleation at 500 o C is suitable for graphene growing-up and coalescence, yielding large MLG grains (height of ~0.4 nm). While at higher temperatures of 530 o C, etching effect is weaened 1 and multilayer nucleation occurs (height ~1. nm). Figure S4. a, Graphene grains nucleated at 500 o C, and the lower height profile of the dashed line cut shows monolayer graphene nuclei. Scale bar in a is 00 nm. b, Graphene nucleated at 530 o C with nuclei of both monolayer and multilayer. Scale bar in b is 300 nm. V. Defects characterization of the as- grown graphene In order to characterize the defects in our graphene samples, we performed gas phase anisotropic hydrogen plasma etching 1. It has been proven that this technique, which enlarges the defects to mae them observable by AFM, is capable of identifying the structural defects in graphitic materials. We can count the number of the etched pits to estimate the defect density, and we can also chec if there is etched trenches or not to testify the existing of the grain boundaries. Experimentally, we first performed H-plasma etching for the as-grown samples (continuous first layer graphene with some second layer grains) for 30mins at 350 o C 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION and then do AFM imaging for the etched samples to visualize the defects and grains boundaries. There are three types of samples, #1-3 (Fig. S5b-d), and they have different second layer coverage varying from ~5%, 45%, to 75%, respectively. It is shown in Fig. S5b that there are some small etched hexagonal pits, which originated from in-plane defects, while the relative big ones originated from uncovered area (to be covered upon further growth) of the first layer graphene. For the well covered first layer (Fig. S5c and d), which has a relative high second layer coverage (usually >5%), there are only small etched pits with a reduced defect density. The average defect density is statistically plotted in Fig. S5a, yielding a value of around 10 /µm, which is comparable to that of Kish graphite. What is more, there is no trench observed in Fig. S5, indicating the as-grown graphene is free of grain boundaries. a b c d Figure S5. a, Average defect density (n d ) for three types of MLG samples. b-d, Typical AFM images corresponding to sample 1-3 in a with different second layer coverage. Scale bars in b-d are 300 nm. VI. Uniform moiré pattern and single crystalline nature of as-grown graphene The moiré pattern period depends strongly on the relative lattice rotation angle between graphene and h-bn underneath 3. The h-bn substrate is single crystal. If there is moiré pattern discrepancy, it should be relevant to the graphene lattice rotation. Thus the universal moiré pattern in large scale should provide us a hint of single NATURE MATERIALS 5

6 crystal nature of our epitaxial graphene. We firstly imaged a sample with near seamless first layer graphene and some second grains within a 10µm 10µm area by AFM (Fig. S6), and then did zoomed-in imaging (size<1µm) step by step to reveal the moiré pattern. It is clear that all moiré patterns in different location shares the same period (~15 nm) as well as the same pattern alignment. a b c f b c e d d e f Figure S6. a, A 10µm 10µm AFM image of a sample with near seamless first layer graphene and some second grains. Scale bar in a is µm. b-f, Zoomed-in image of the white squares (b-f) in a, showing universal moiré pattern. Scale bars in b-f are 100 nm. We did also chec the coalescence of the two independently nucleated grains with no evidence of dislocation or distortion of the moiré pattern (Fig. S7a and b). Note that all the moiré pattern images are original AFM data, not treated by FFTs. In order to experimentally confirm the single crystalline nature of our epitaxial graphene on h-bn, we carried out further STM imaging especially at those locations of connected grains in the Fig. S7c-e. STM images were taen in air and at room-temperature by a Multi-mode nanoscope SPM system (Veeco). Pt-Ir alloy tip was used and all atomic resolution images (Fig. S7d-f) are in constant-height mode. From these images, together with many others not shown, we found that the lattice of 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION two coalescent grains is continuous and the coalescence is good with few defects observed. Note that all imaging was taen from second layers of as-grown graphene since there are technical limits on imaging the first layer graphene on h-bn (STM tip will be crashed when scanning to the insulating BN region). However, it does not change the overall conclusion (supposing that the out-of-phase induced mismatch is present in first layer; it is the same thing for second layer). Thus our epitaxial graphene is indeed single crystal without rotation to h-bn lattice underneath. a c nm e G1 50 nm G 30 nm nm b d na f 50 nm 10 nm nm Figure S7. a-b, Moiré patterns of grains with different degree of coalescence. c-e are step-by-step zoom-in STM images on region of coalescence of two neighboring grains at room temperature in ambient atmosphere. e is the current image showing the atomic resolution; The typical STM image has artifacts such as random noise and scan lines, which can be removed by a Fourier transform filter. And f is the Fourier transform filtered image for (e). VII. Band structure calculation The well-developed moiré pattern in our graphene sample on an h-bn substrate indicates that the substrate exerts influence on the graphene sample by wea periodic potentials. The first Brillouin zone of graphene superlattice is shown in Fig. S8, and NATURE MATERIALS 7

8 the external periodic potentials, with moiré pattern periodicity, U U be performed the Flourier transformation U U e ig x G G x x L can x, (S-1) where L is the two-dimensional periodic vector for moiré pattern superlattice potential, and G α is the corresponding reciprocal vector. The effective low-energy Hamiltonian around Dirac points in unperturbed freestanding graphene reads H v, (S-) 0 f x x y y with σ a Pauli matrix for sublattice freedom and v f Fermi velocity for quasiparticle. The eigenvalues and eigenstates of this unperturbed Hamiltonian are s sv and 0, f x i se, (S-3) i, s e where Ω is the cell area, and θ is the angle of between and x. s=±1 are for the conduction band and valence band, respectively. As for the above wea superlattice periodic potentials, the coefficientu G decreases rapidly with the increase of G α. Especially, the naed coulomb potential, U declines in accordance with the G 1/G law. Therefore, the six shortest reciprocal vectors are taen into account for simplicity. Consequently, under the above wea superlattice periodic potentials, the total effective low-energy Hamiltonian around Dirac points can be written as H v 6 ig x UG e I 1 f xx y y, (S-4) where the wea periodic potential amplitude UG can be obtained by second perturbation theory 3. The superlattice periodic potential between the unperturbed eigenstates in perturbation theory 4 reads 6 ' 1 i ' 0, s Ux 0, s UG 1 e '. (S-5), G 1 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION The eigenstates of the total Hamiltonian are given by, s c, s 0. (S-6) After some algebraic calculations, one obtain the so-called central equation 6 UG i G 0, s, sc 1e cg 0. (S-7) 1 Similar to the nearly free electron model 5, the free Dirac electron under external periodic potential can result in Bragg scattering at the superlattice Brillouin zone G boundaries. The momentum can be again labeled by D near these boundaries, where D is a small offset with respect to the superlattice Brillouin zone boundaries. By solving the above central equation, the final energy dispersion is given by 1, s G G G G 0 D, s 0 D, s 0 D, s 0 D, s V and V in Fig. 3d. (S-8) U G 1 cos G G. And the energy dispersion is shown D D G M * G 1 * ( K ) K * Figure S8. The first Brillouin zone of graphene superlattice under an external periodic potential. The Dirac point K of original freestanding graphene, located in parentheses in the middle of the hexagon, is taen as a new Г * of the first Brillouin zone of the superlattice. VIII. Temperature dependence of superlattice Dirac point in transport NATURE MATERIALS 9

10 measurements The structure of the MLG device in Fig.3a is shown in the optical image in Fig. S9a, and the h-bn is purple colored (graphene in Hall bar geometry is discernable with darer purple) with a thicness of 10 nm. By zooming into the details of the T-dependent transfer curve in Fig. 3a, we found that the position ( Vg) of the superlattice Dirac point (SDP) is also T-dependent, which is shown by green squares in Fig. S9b. And similar trends are found in other devices with h-bn thicness of 55 nm and 40 nm (red circles and blue triangles in Fig. S9b). Taing Vg e /( Cg) (derived in the manuscript) into account, there are two factors mae contribution, superlattice period λ and capacitivity C g. However superlattice period λ is considered a constant in the following discussion, because of the negligible value of thermal expansion coefficients of graphene and h-bn (both with a value on the order of 10-6 /K) 6,7. One possible cause is the temperature dependence of capacitivity. QHE fan diagram of the device with 10 nm h-bn was performed at 1.5 K, and by fitting the filling factors in Fig. S11 we derived the capacitivity of the device, yielding a value of 7.6 nf/cm. Compared to the calculated 8.3 nf/ cm (10 nm h-bn/300 nm SiO ) at room temperature, the reduced value at lowered temperature is supportive to our observed shift of SDP. We also found that device with thicer h-bn behave stronger T-dependent Vg changes than that with thinner ones. In Fig. S9b for 10 nm h-bn Vg changes 47% from 8.6 V at 300K to 4 V at 50K while for 55 nm h-bn it changes 35% from 4.5 V to 3.5 V, and for both samples Vg were stabilized below 50 K, suggesting a phase transition happens. All these evidence lead to an assumption that the reduced capacitivity is somewhat induced by the decrease of dielectric constant of h-bn. As temperature is lowered, dipoles in the h-bn may freeze out gradually due to the increased rotation energy barrier for polarization, leading to a decreased dielectric constant and thus a decreased Cg, and when temperature is below some critical temperature dipoles become pinned, giving a stabilized T-independent Cg. Based on our observed data, we roughly estimate that the dielectric constant of 10 NATURE MATERIALS

11 SUPPLEMENTARY INFORMATION h-bn decrease from 4 at 300 K to 3 at below 50 K, and the critical temperature is around 50 K. However there are other possible factors playing a part, and we reluctant to mae other estimations. a b Figure S9. a, Optical image of a typical device (MLG device in Fig. 3a) with thicness of h-bn 10 nm. B, Temperature dependence of Vg with green squares, red circles, and blue triangles referred to h-bn thicness of 10 nm, 55 nm, and 40 nm, respectively. IX. Conductivity and mobility of MLG and BLG Conductivity (σ) versus applied gate voltage for MLG and BLG at different T is plotted in Fig. S10, and the superlattice generated new Dirac points are vividly depicted. By fitting constant mobility model as did in ref.8 we can estimate its value, which gives ~5,000 cm - V -1 s -1 for MLG at 1.5 K and ~4,500 cm - V -1 s -1 for BLG at 4 K. Our relative decreased mobility, compared to that of exfoliated graphene on h-bn in ref.9, is due to the scattering of the point defects and additional second layers. Note that the carrier mean free path (l) of our samples is l~70 nm. The average point defects density is around 10 /µm for different samples; that means that the average distance of adjacent defects is around 300 nm, which is much bigger than l. Besides we can see that the mobility tends to increase for samples with less secondary layers in Fig. S10c. For the MLG device in manuscript, the relative coverage for different layer are as follows, first layer >99%, second layer <60%, and multilayer <1%. Thus, scattering from additional second layer grains 10 is also an important factor. Our next NATURE MATERIALS 11

12 step wor will focus on decreasing these defects densities and suppressing the nucleation of additional second layers to increase the sample mobility. However, this mobility issues does not affect much on finding new physics on our sample. a b c Figure S10. a,b, Conductivities as a function of applied gate voltage for MLG (a) and BLG (b) at different temperature. c, Room temperature mobility of MLG samples with different second layer coverage. Value extracted for the sample in (a) is mared as red in (c). X. Additional magnetotransport measurements Figure S11a shows the longitudinal R xx and Hall resistance R xy of the MLG device against applied magnetic field, with a fixed V g = -40 V, at T = 1.5 K. Quantum Hall states with filling factors ν =, 6 as well as Shubniov-de Haas oscillation were observed, indicated by quantized hall resistance R xy and the corresponding R xx minimum. Figure S11b shows the magnetoresistance R xx and Hall resistance R xy of the BLG device against applied magnetic field, with a fixed V g = -35 V, at T = 1.6 K. Quantized Hall resistance with filling factor ν = 4 n, where n is landau level index, is clearly indicated in Fig. S11b. What is more, R xx versus applied gate voltage under different magnetic field for BLG was investigated in Fig. S11c. It was found that the resistance pea at DP begins to split into two as B > 6.8 T (Fig. S11c), which suggests emergence of n = 0 landau level for BLG. Providing comparable mobility with ref.8, a full sequence of lifted landau levels could be expected. 1 NATURE MATERIALS

13 SUPPLEMENTARY INFORMATION a R xy V 1.5 g = -40V T =1.5K b R xy V R g = -35V T=1.6K c xx MLG R xx R () =6 = R () 6 4 BLG =1 =8 =4 R xx () T 7.3 T 8.8 T 10 5 BLG T=1.6K B (T) B (T) Vg (V) Figure S11. a, Longitudinal (R xx, blue) and Hall resistance (R xy, red) of MLG versus magnetic field at V g = -7 V, T = 1. 5 K. b, Longitudinal (R xx, blue) and Hall resistance (R xy, red) versus magnetic field at V g = -35 V, T = 1. 5 K. c, Splitting of n = 1 LL of BLG for B > 6.8 T at T = 1.6 K. XI. QHE fan diagram of MLG The QHE contour plot of R xx and R xy are included in Fig. S1, where magnetic field B is swept from 0 to 14 T and the carrier density n is calculated by parallel capacitor model with capacitivity C g = 7.6 nf/cm (C g is extracted from the fitting of filling factors at landau levels). The white dashed lines along R xx valley and R xy plateau indicate the filling factors ν of landau level, while the blac and red dotted lines indicates the QHE of superlattice Dirac point. At around the prominent SDP at hole-branch, the blac dotted lines are attributed to QHE of SDP with filling factor ν = ±, whose slope equals to that of white line with ν = ±. Interference pattern due to QHE at SDP and QHE at DP is vividly revealed in the modulated regions where blac and white line crosses. Similar feature is also observed in SDP related QHE at electron-branch. Even though its magnitude is much reduced, the modulated interference pattern is discernable and lines with filling factors ν = ±, 6 is fitted in red dots. NATURE MATERIALS 13

14 a 0 b Figure S1. a, b Contour plot as a function of carrier density n and magnetic field B for R xx (a) and R xy (b). The negative sign of n is referred to holes. Reference 1. Yang, R. et al. An Anisotropic Etching Effect in the Graphene Basal Plane. Adv. Mater., (010).. Wu, S. et al. Identification of structural defects in graphitic materials by gas-phase anisotropic etching. Nanoscale, 4, (01). 3. Yanowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nature Phys. 8, 38 (01). 4. Par, C-H. et al. Anisotropic behaviors of massless Dirac fermions in graphene under periodic potential. Nature Phys. 4, (008). 5. Kittel, C. Introduction to Solid State Physics (John Wiley & Sons, Inc., 005). 6. Retajczy, T. F. & Sinha, A. K. Elastic stiffness and thermal expansion coefficient of BN films. Appl. Phys. Lett. 36, (1980). 7. Balandin, A. A. Thermal properties of graphene and nanostructured carbon materials. Nature Mater. 10, (011). 8. Shi, Z. W. et al. Zhang. Patterning graphene with zigzag edges by self-aligned anisotropic etching. Adv. Mater.3, (011). 9. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nature Nano. 5, 7-76 (010). 10. Ji, Shuai-Hua et al. Atomic-scale transport in epitaxial graphene. Nature Mater. 11, (01). 14 NATURE MATERIALS