Abstract
Electrostatic superlattices have been known to significantly modify the electronic structure of lowdimensional materials. Studies of graphene superlattices were triggered by the discovery of moiré patterns in van der Waals stacks of graphene and hexagonal boron nitride (hBN) layers a few years ago. Very recently, gatecontrollable superlattices using spatially modulated gate oxides have been achieved, allowing for Dirac band structure engineering of graphene. Despite these rapid experimental progresses, technical advances in quantum transport simulations for largescale graphene superlattices have been relatively limited. Here, we show that transport experiments for both graphene/hBN moiré superlattices and gatecontrollable superlattices can be well reproduced by transport simulations based on a scalable tightbinding model. Our finding paves the way to tuningparameterfree quantum transport simulations for graphene superlattices, providing reliable guides for understanding and predicting novel electric properties of complex graphene superlattice devices.
Introduction
Hexagonal boron nitride (hBN), one of the most popular dielectric materials due to its atomic flatness^{1}, has been playing a crucial role in studying twodimensional materials using socalled van der Waals heterostructures^{2}. The graphene/hBN moiré pattern^{3} arising from the largescale lattice interference led to the discovery of graphene superlattices. First experiments revealing new transport phenomena, such as the emergence of the Hofstadter butterfly, were reported in 2013^{4,5,6}. In the following years, other exciting transport experiments have been reported^{7,8,9,10,11,12,13}, as well as a dynamic band structure tuning^{14,15}. More recently, another approach for inducing a superlattice potential in graphene has been demonstrated by using patterned dielectrics^{16}, allowing for miniband structure engineering. On the theory side, most works related to graphene superlattices focus either on calculations for the superlatticeinduced miniband structures^{17,18,19,20,21,22,23}, or on predicting transport properties by solving the Dirac equation with simplified superlattice model potential^{24,25}. Quantum transport simulations considering realistic experimental conditions have been relatively rare in the literature^{26,27}, not to mention a theory work combining transport simulations and miniband structure calculations, together with transport experiments.
This article aims at providing a straightforward method to perform reliable quantum transport simulations for graphene superlattices. As shown in the following, our transport simulations based on the realspace Green’s function method for twoterminal structures with the superlattice potential arising either from the graphene/hBN moiré pattern or from periodically modulated gating are consistent with transport experiments as well as miniband structures based on the continuum model. Our method is applicable equally well to multiterminal structures for simulating, for example, fourprobe measurements using the Landauer–Büttiker approach^{28}.
Results
Superlattice on a scaled graphene lattice
To perform quantum transport simulations for graphene working in realspace, the scalable tightbinding model^{29} has been proved to be a very convenient numerical tool^{30,31,32,33,34}: the physics of a real graphene system can be captured by a graphene lattice scaled by a factor of s such that the lattice spacing and nearestneighbor hopping parameter are given by a = sa_{0} and t = t_{0}/s, respectively, where a_{0} ≈ 0.142 nm and t_{0} ≈ 3 eV are the tightbinding parameters for a genuine graphene lattice. The scaling is justified as long as a remains much shorter than all important physical length scales in the considered graphene system.
In dealing with graphene superlattices (such as the one sketched in Fig. 1a), the newly introduced physical length scale not previously considered^{29} is the periodicity λ of the superlattice. The advantage of the scaling can be easily appreciated by comparing Fig. 1b, c: The former considers a genuine graphene lattice involving lots of carbon atoms, while the latter depicts a scaled graphene lattice (here s = 2 for illustrative purposes) involving a strongly reduced number of lattice sites that downscales with s^{2}. As long as a ≪ λ is satisfied, a reasonably large area covering enough superlattice periods can be implemented in realspace quantum transport simulations (see Methods) to reveal transport properties arising from the superlattice effects.
In the following, we demonstrate the reliability of transport simulations for graphene superlattices based on scaled graphene lattices, considering both graphene/hBN moiré superlattice and gatecontrolled superlattice.
Graphene/hBN moiré superlattice model potential
Formation of the moiré pattern due to the stacking of hBN and graphene lattices has been understood in one of the earliest experiments^{3}. Following their model, the moiré pattern results in a triangular periodic scalar potential described by
where V = 0.06 eV is the amplitude of the model potential and \({\mathbf{G}}_1(\tilde \lambda ,\tilde \theta )\) is the reciprocal primitive vector of the moiré pattern corresponding to the primitive vector L_{1}(λ, θ) = λ(cos θ, sin θ) in real space. The orientation angle \(\tilde \theta\) and wavelength \(\tilde \lambda\) are related to those in realspace through \(\tilde \theta = \theta + \pi {\mathrm{/}}2\) and \(\tilde \lambda = 4\pi {\mathrm{/}}\sqrt 3 \lambda\). The other two reciprocal vectors are given by \({\mathbf{G}}_2(\tilde \lambda ,\tilde \theta ) = {\mathbf{G}}_1(\tilde \lambda ,\tilde \theta + \pi {\mathrm{/}}3)\) and \({\mathbf{G}}_3(\tilde \lambda ,\tilde \theta ) = {\mathbf{G}}_1(\tilde \lambda ,\tilde \theta + 2\pi {\mathrm{/}}3)\). Following Moon and Koshino^{23} with the zigzag lattice direction arranged along the x axis, the moiré wavelength λ and the orientation angle θ of the pattern are given by
and
respectively, where \(a_{\mathrm{g}} = \sqrt 3 a_0 \approx 0.246\) nm is the graphene lattice constant, ϵ = (a_{hBN} − a_{g})/a_{g} ≈ 1.81% is the lattice constant mismatch with a_{hBN} ≈ 0.2504 nm the hBN lattice constant, and ϕ is the twist angle of the hBN lattice relative to the graphene lattice. An illustrative example with ϕ = 5° is sketched in Fig. 2a, where an overlay of U_{s}(x, y) given by Eq. (1) is shown to match perfectly the lattice structure of the resulting graphene/hBN moiré pattern. For selfcontainment, λ and θ as functions of the twist angle ϕ are plotted in Fig. 2b, c, respectively, where the hollow squares mark the ϕ = 5° example of Fig. 2a and the hollow circles mark the ϕ = 0.9° case corresponding to our transport experiments and simulations to be elaborated below.
Transport experiment
To confirm the validity of our simulation scheme illustrated above, the most convincing way is to fabricate a real device, perform a transport measurement, and finally simulate the experiment. Our singlegated twoterminal device was fabricated with a graphene/hBN stack (see the inset of Fig. 2d and Methods) and measured at low temperature (≈4.1 K), using standard lowfrequency (≈13 Hz) lockin technique. Figure 2d shows the twoterminal differential conductance of our sample as a function of the back gate voltage V_{bg}. Conductance dips at around −60 and +58 V are basic characteristics of the moiré superlattice potential^{4,5,6}, showing that our graphene and hBN lattices are nearly aligned. By analyzing from the Brown–Zak oscillation^{12,13}, the moiré wavelength λ ≈ 10.9 nm of our graphene/hBN sample was deduced. Note that although our graphene sample was encapsulated by two hBN layers, the single dip structure (in each of the electron and hole branches) of our conductance measurement (Fig. 2d) suggests that only one hBN layer exhibits a measurable moiré superlattice effect arising from a small twist angle, contrary to a recent report on doubly aligned hBN/graphene/hBN heterostructures^{35}. In the following simulations, therefore, we model our experiment with graphene/hBN, instead of hBN/graphene/hBN.
Transport simulation
To simulate such a conductance measurement, we have calculated the transmission T(E) as a function of Fermi energy E at zero temperature, and hence the conductance G(E) = (2e^{2}/h)T(E), based on an s = 4 tightbinding model Hamiltonian, for a twoterminal device similar to Fig. 1a with L = W = 500 nm (see Supplementary Fig. 1 and Supplementary Note 1 for sample size dependence), implementing the moiré model potential (1) with a twist angle ϕ. To compare with the experiment on the same voltage axis, we adopt the parallelplate capacitor formula to convert from the Fermi energy to the gate voltages; see Supplementary Note 2 for the conversion. With the chosen ϕ = 0.9° for the moiré model potential Eq. (1) corresponding to λ ≈ 10.4 nm consistent with the experimentally deduced value, the simulated conductance G(V_{bg}) transformed from G(E) and reported in Fig. 2e is found to show excellent agreement with the experiment, Fig. 2d, in the positions of the conductance dips. Note that the red curve reported in Fig. 2e considers leads with onsite and Fermi energies identical to those at the lattice sites attached to the leads; see Fig. 1a. In this way, the Fermi energy in the leads “floats” with that in the scattering region such that the interface between the lead and the scattering region is as transparent as possible. Compared to Fig. 2d, the electron–hole asymmetry is less pronounced due to the simple model of Eq. (1) from Yankowitz et al.^{3}. Although accounting for an electron doping from the metal contacts simply by fixing the Fermi energy in the leads with a positive value can make the conductance curve (black dashed curve in Fig. 2e with Fermi energy 0.32 eV in the leads) even more similar to the experiment, the nature of the electron–hole asymmetry observed in Fig. 2d comes from more subtle interactions between graphene and hBN lattices which require more advanced model Hamiltonians^{21,22,23} beyond the simple model of Eq. (1). We will not further address the electron–hole asymmetry but continue our discussions with calculations based on leads with “floating” Fermi energies.
Density of states and miniband structures
Without transforming to the gate voltage axis, the original conductance data of Fig. 2e as a function of energy is reported in Fig. 2f with a wider energy range up to ±0.4 eV. Compared to the density of states (Fig. 2g) and the band structure (Fig. 2h) which are calculated based on the same moiré superlattice model potential but within the continuum model (see Methods), consistent features in the energy spectrum can be seen. In view of Fig. 2d–h, our calculations significantly capture basic properties of the graphene/hBN moiré superlattice, at least at zero magnetic field.
Dependence on moiré orientation angle
In all the transport simulations for the moiré superlattice shown in the main text of this article, we have considered realistic moiré orientation angle θ prescribed by Eq. (2b), relative to the zigzag direction. We have further tested that the simulation results are insensitive to θ. See Supplementary Fig. 2 and Supplementary Note 3.
Magnetotransport experiment and simulation
We continue our comparison of the experimentally measured and theoretically calculated conductance G(V_{bg}) with finite magnetic field B perpendicular to the graphene plane, which can be modeled by associating the Peierls phase^{28,36} to the hopping t → te^{iΦ} in Eq. (3), where \({\mathrm{\Phi }} = (e{\mathrm{/}}\hbar ){\int}_{{\mathbf{r}}_j}^{{\mathbf{r}}_i} {{\mathbf{A}} \cdot d{\mathbf{r}}}\), choosing the Landau gauge A = (−yB, 0, 0) for the vector potential A; see the axes shown in Fig. 1. Conductance maps of G(V_{bg}, B) are reported in Fig. 3a (b) from the experiment (theory) showing magnetic field up to 5 T and the same gate voltage range as Fig. 2d (e), which corresponds exactly to the horizontal line cuts at B = 0 in Fig. 3a (b). The original data of the calculated conductance as a function of Fermi energy and magnetic field, G(E, B), is shown in Supplementary Fig. 3 with the full range. Within the gate voltage range of about −45 V ≲ V_{bg} ≲ +45 V, typical relativistic Landau fans can be seen in both experiment and theory. To compare further, we magnify the regions marked by the black dashed box in Fig. 3a (b) in Fig. 3c (d) painted by a different color code to highlight the quantized conductance plateaus. Numbers −6, −10, ⋯, −38 on Fig. 3d label the filling factor ν on the corresponding plateau with the expected conductance νe^{2}/h. Good agreement between experiment (Fig. 3a, c) and theory (Fig. 3b, d) within the main Dirac cone can be seen.
Unobserved 12fold quantum Hall plateaus
At gate voltages V_{bg} ≳ 45 V, transport properties are dominated by the extra Dirac cones arising from the modulating moiré superlattice. Discrepancies between the experiment and theory are evident. This suggests that the neglected higherorder terms of a more complete model Hamiltonian^{21,22,23} become important when the magnetic field is strong. Interestingly, we note that in the theory map of Fig. 3b, some unusual plateaus in the energy range around the electronbranch secondary Dirac point can be observed. We magnify this region in Fig. 3e with a horizontal line cut shown in Fig. 3f, where the gate voltage range showing quantized conductance plateaus is highlighted by a yellow background.
This V_{bg} range, transformed back to the energy through Eq. (S1) of Supplementary Note 2, corresponds to an energy window where part of the electron branch of the secondary Dirac cones at M points of the superlattice miniBrillouin zone are completely isolated. The respective energy window is highlighted also by yellow in Fig. 2h. Since there are effectively three such Dirac cones (six cones on six M points within each miniBrillouin zone but each cone shared by two neighboring miniBrillouin zones), the degeneracy factor is expected to be 3 × 2 × 2 = 12 with ×2 accounting for spin and another ×2 for valley. Indeed, in the quantum Hall regime, the calculated conductance is quantized to 6, 18, 30, 42 e^{2}/h as shown in Fig. 3f. Outside this energy (and hence back gate voltage) range, the higherorder Dirac cones are always mixed with background bands, so that no quantized conductance is observed. However, such a special energy window leading to the 12folddegeneracy of the Landau levels at the secondary DP is never observed in transport experiments with graphene/hBN moiré superlattices^{4,5,10,11,12}, including ours shown in Fig. 3a, indicating once again that the simplified model^{3} of Eq. (1) containing only the electrostatic scalar potential term is not sufficient to capture transport properties of graphene/hBN moiré superlattices at higher magnetic fields.
At this stage, we comment that the periodic scalar potential enters the tightbinding model Hamiltonian through the onsite energy term (see Methods), so that it is readily compatible with the scaling method^{29}. Including higherorder terms^{21,22,23} of the graphene/hBN moiré superlattice should be possible but is beyond the scope of the present study. As we will see below, when the graphene superlattice potential arises solely from the electrostatic gating, our method becomes even more precise because in such systems the scalar potential is the only term comprising the superlattice.
Gatecontrollable superlattices
To observe any superlattice effects in graphene, the mean free path must exceed enough periods of the superlattice potential. This means that either the sample quality must be extraordinary, or the superlattice periodicity must be short enough. When the periodicity is too short, however, the resulting extra Dirac cones appear at too high energy, exceeding the experimentally reachable range. This is why the discovery of the graphene/hBN moiré pattern^{3} led to studies on the graphene superlattice of its first kind—the periodicity corresponding to small twist angles turns out to be naturally in a suitable range for experiments; see Fig. 2b. A more flexible approach to design artificial graphene superlattice structures for band structure engineering was pursued with the realization of electrostatic gating schemes^{16}. To create an externally controllable periodic potential, the most intuitive way is to pattern an array of periodic fine metal gates on top of the graphene sample^{37,38}. However, due to technical difficulties such as instabilities of nanometerscale local gates, the low adhesion between metal gates and the inert hBN, etc., such superlattice graphene devices often suffer the problem of very low sample yield^{39}. The basic idea of the new technical breakthrough is to keep the hBN/graphene/hBN sandwich intact, while periodically modulating the gate capacitance. This can be achieved either by using fewlayer graphene as a local gate which is subsequently etched with a periodic pattern^{39,40}, or by etching the dielectric layer with a periodic pattern using a standard uniform back gate underneath the modulated substrate^{16}. In the following, we revisit the experiment of the latter by Forsythe et al.^{16} and reproduce the observed transport features by our transport simulations.
Electrostatic simulation
Following the geometry of the device subject to a square superlattice potential with periodicity λ = 35 nm presented in the work by Forsythe et al.^{16}, we have performed our own electrostatic simulation to obtain the back gate capacitance showing periodic spatial modulation. We consider an hBN/graphene/hBN sandwich (showing no measurable moiré superlattice effects) gated by a global top gate contributing a uniform carrier density n_{tg}, and a bottom gate at voltage V_{bg} with a prepatterned SiO_{2} substrate in between. See Figure 1 of Forsythe et al.^{16}. The bottom gate capacitance therefore shows a spatial modulation with a square lattice symmetry, as shown in the lower left inset of Fig. 4a.
With the electrostatically simulated positiondependent back gate capacitance per unit area C_{bg}(x, y), contributing carrier density n_{bg}(x, y) = [C_{bg}(x, y)/e]V_{bg}, together with the uniform n_{tg}, the resulting superlattice potential is given by \(U_{\mathrm{s}}(x,y) =  {\mathrm{sgn}}[n(x,y)]\hbar v_{\mathrm{F}}\sqrt {\pi \left {n(x,y)} \right}\) with n = n_{bg} + n_{tg}, in order to set the global transport Fermi level at zero^{41}. Slightly different from the case of the graphene/hBN moiré superlattice where the model potential U_{s}(x, y) given by Eq. (1) is independent of the gating, we consider U(x, y) = U_{s}(x, y) for the onsite energy term (4), and implement it in the tightbinding Hamiltonian (3) with s = 6 (such that a = sa_{0} ≈ 0.85 nm ≪ λ = 35 nm) to perform quantum transport simulations over a twoterminal structure with L = 420 nm and W = 385 nm; see the upper right inset of Fig. 4a.
Transport simulation
To compare with the resistance measurements reported by Forsythe et al.^{16}, we plot the inverse transmission 1/T as a function of n_{tg} and V_{bg} in the main panel of Fig. 4a, where most areas show high transmission (white regions correspond to low 1/T). Along the diagonal dark thick line showing high 1/T values due to the main Dirac point, multiple satellite peaks can be seen when increasing V_{bg} and hence the magnitude of the square superlattice potential, signifying the emerging multiple extra Dirac points due to the gatecontrolled square superlattice potential. Exemplary line cuts are plotted in Fig. 4b–d to show clearly the single and multiplepeak structures, in excellent agreement with the experiment (see Figure 2b of Forsythe et al.^{16}). Note that the peak positions of the secondary Dirac points at the electron and hole side seen in Fig. 4b, d are not symmetric due to the asymmetric superlattice potential shape obtained from the electrostatic simulation. When implementing a model periodic function symmetric in its potential profile, the density spacing from the main Dirac point to the secondary Dirac points at electron and hole sides becomes identical (not shown).
Gatedependent miniband structures
We have also checked the consistency between the calculated miniband structures and the simulated inverse transmission. Overall, we obtain band structures similar to those reported in the work by Forsythe et al.^{16}, but since each (n_{tg}, V_{bg}) point corresponds to a different U_{s}(x, y) profile and hence a different miniband structure, an overview consistencycheck like in Fig. 2f–h is technically not possible. Instead, the consistency can be checked by comparing the 1/T peaks and their corresponding miniband structure around E = 0. We have chosen three particular (n_{tg}, V_{bg}) configurations corresponding to the three black arrows in Fig. 4b marking three of the 1/T peaks, at which the E = 0 Fermi level is expected to hit either the main or the extra Dirac points.
These miniband structures, along with the actual U_{s}(x, y) profiles implemented individually in the continuum model (see Methods) are shown in Fig. 4e–g. Going from low to high n_{tg} (left, middle, and right black arrow in Fig. 4b), the highest filled energy rises relative to the main Dirac point, corresponding to the sinking of the whole band structure due to our choice of fixing the Fermi level at E = 0 (Fig. 4e–g). As expected, the highest peak in Fig. 4b marked by the middle black arrow corresponds to Fig. 4f, where the main Dirac point is nearly hit; see the lower subpanel therein and the relevant caption. From the E = 0 Fermi contours of Fig. 4e, g, the two satellite 1/T peaks seen in Fig. 4b are mainly contributed by the secondary Dirac points at X, labeling the midpoints on the edges of the square miniBrillouin zone. Note that the miniband structures shown in Fig. 4e–g, though corresponding to an increasing uniform n_{tg}, do not exhibit simply an energy shift without changing the band shape. Compare, for example, the shapes of the lowest shown subbands (purple).
Discussion
We have shown that quantum transport simulations based on the scalable tightbinding model^{29} correctly capture transport properties of electrostatic graphene superlattices. In the case of graphene/hBN moiré superlattice, the consistency of our simulation and experiment at zero and low magnetic field is rather satisfactory but breaks down at too strong magnetic field due to the adopted simple moiré model potential^{3} that neglects higher order terms. In the other case of gated superlattices, without such higher order terms the simulations are expected to be precise for all magnetic field range. Compared to the recent transport experiment on a gatecontrolled square superlattice device reported by Forsythe et al.^{16}, our simulations show an excellent agreement in revealing the emergence of multiple extra Dirac cones at zero magnetic field. Transport simulations at finite magnetic field for the gated superlattices are expected to reveal also consistent behaviors compared to the experiment, but are left as future work.
Our finding significantly lowers the computation burdens and hence paves the way to tuningparameterfree quantum transport simulations for graphene superlattices, providing reliable guides for understanding and predicting novel electric properties of complex graphene superlattice devices. We note some recent studies working on developing numerical techniques that allow largescale efficient transport simulations^{42,43,44}, but properly scaling the graphene lattice seems to be of least technical complexity and is readily applicable to anyone who is familiar with quantum transport using, for example, realspace Green’s function methods^{28} or the opensource python package KWANT^{45}.
Methods
Realspace Green’s function method
The model Hamiltonian including the superlattice potential U_{s}(x, y) using the scaled graphene lattice can be written as
where the operator c_{j} (\(c_j^\dagger\)) annihilates (creates) an electron at site r_{j} = (x_{j}, y_{j}). The first term in Eq. (3) represents the clean part of the Hamiltonian which contains nearest neighbor hoppings summing over site indices i and j with 〈i, j〉 standing for r_{i} − r_{j} = a, and the second term is the onsite energy
containing the superlattice potential U_{s}(x, y) smoothed by a model function F_{s}(x, y) = f_{s}(x, −L/2 + d)f_{s}(−x, L/2 − d)f_{s}(y, −W/2 + d)f_{s}(−y, W/2 − d) with \(f_{\mathrm{s}}(z,z_0) = \{ 1 + {\mathrm{tanh}}[(z  z_0){\mathrm{/}}\ell _{\mathrm{s}}]\} {\mathrm{/}}2\), where \(\ell _{\mathrm{s}}\) is a smoothing parameter typically taken as \(\ell _{\mathrm{s}} = \lambda {\mathrm{/}}4\). The purpose of smearing off the superlattice potential function U_{s} to zero at a distance d (typically taken as λ) away from the edges and the leads (see Fig. 1a and its inset) is to avoid any spurious effects due to the combination of the superlattice potential and the physical edges of the graphene lattice, as well as to avoid oversized unit cells for the lead selfenergies. Any contributions to the onsite energy term other than the superlattice potential are collected in the U_{0} term in Eq. (4).
With the model Hamiltonian Eq. (3) constructed, together with selfenergies Σ_{1} and Σ_{2} describing the attached two leads (following, for example, Wimmer^{46}), the retarded Green’s function at energy E is given by
leading to the transmission function
where \({\mathrm{\Gamma }}_j = i({\mathrm{\Sigma }}_j  {\mathrm{\Sigma }}_j^\dagger )\) with j = 1, 2 is the broadening function. In the lowtemperature lowbias limit, the conductance across the modeled scattering region is given by the Landauer formula G = (2e^{2}/h)T, where the factor of 2 accounts for the spin degeneracy. For a pedagogical introduction to the above outlined realspace Green’s function, see, for example, Datta^{28}. Note that in most simulations, the full matrix of Eq. (5) is not needed, suggesting that a partial inversion should be implemented in the numerics to avoid wasting computer memories and CPU time. On the other hand, the matrix version of the Fisher–Lee relation (6) can be implemented as the way it reads.
Device fabrication
Our twoterminal device contains a hBN/graphene/hBN stack on a Si/SiO_{2} substrate, where the crystallographic axis of the graphene flake is aligned with respect to one of the hBN flakes. Electric contact to the graphene is made from the edge of the mesa^{47} with selfaligned Ti/Al electrodes. We use the Si wafer as an overall back gate with a twolayer dielectric consisting of SiO_{2} with thickness \(d_{{\mathrm{SiO}}_{\mathrm{2}}}\) = 300 nm and the bottom hBN flake with thickness d_{hBN} = 20 nm. A typical exemplary junction similar to the measured device is shown by the atomicforce microscope (AFM) image in the inset of Fig. 2d and marked by the white dashed box.
Continuum model for minibands and density of states
To calculate the miniband structure of graphene in the presence of a superlattice potential U_{s}(r), we consider an infinitely large twodimensional pristine graphene described by \({\cal{H}}_0\) in kspace. Following Park et al.^{17}, we start with the continuum model Hamiltonian near the K valley:
where the first term is \({\cal{H}}_0\) and superlattice potential in the second term is treated as a perturbation. In Eq. (7), the product of the reduced Planck constant ħ and Fermi velocity v_{F} is related to the tightbinding parameters through ħv_{F} = (3/2)ta, and the twodimensional wave vector (k_{x}, k_{y}) = k is small relative to the K point. Using the eigenstates of \({\cal{H}}_0({\mathbf{k}})\) as a new basis, we solve the eigenvalue problem of Eq. (7) to obtain a set of linear equations:
where E is the energy eigenvalue of the graphene superlattice, ε_{s}(k) = sħv_{F}k is the eigenenergy of \({\cal{H}}_0({\mathbf{k}})\) associated with the s branch (s = 1 for electron above the Dirac point and s = −1 for hole below the Dirac point), \({\cal{U}}({\mathbf{G}})\) is the Fourier component of U_{s}(r) with the reciprocal lattice vector G = m_{1}G_{1} + m_{2}G_{2} of the superlattice potential, θ_{k,k−G} is the angle from k − G to k, and c_{s}(k) are the expansion coefficients of the pristine graphene eigenstates.
The infinitedimensional matrix spanned by the states with wave vectors \(\mathop {\sum}\nolimits_{m_1,m_2} {\mathbf{k}} + m_1{\mathbf{G}}_1 + m_2{\mathbf{G}}_2\) in Eq. (8) allows for solving for E and hence calculating the band structure. Since we focus on the lowenergy region, a matrix involving states with m_{1} ≤ 3 and m_{2} ≤ 3 is found to be sufficient to attain the convergence of the band structure. With the eigenenergies obtained, the density of states D as a function of energy can be calculated by
where the integration is taken over the first Brillouin zone.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank R. Huber and J. Eroms for valuable and stimulating discussions, R. Rückner for technical support on using the Athene Cluster of Rechenzentrum of Universität Regensburg, where most of the computations were performed, and J. Bundesmann for building and maintaining the Mordor Cluster of NCKU Physics, where part of the transport simulations were performed. Financial supports from Taiwan Ministry of Science and Technology (1072112M006004MY3 and 1072627E006001) and Ministry of Education (Higher Education Sprout Project), Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—ProjectID 10314695032—CRC 1277 (Subproject A07) and Project No. Ri681/131, and Helmholtz Society (Program STN) are gratefully acknowledged.
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S.C.C. calculated the miniband structures and density of states within the continuum model. R.K. performed experiment with the support of R.D. on the graphene/hBN moiré device. M.H.L. performed transport and electrostatic simulations, and wrote the draft of the manuscript. R.D., K.R., and M.H.L. guided the project. All authors contributed to the manuscript.
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Chen, SC., Kraft, R., Danneau, R. et al. Electrostatic superlattices on scaled graphene lattices. Commun Phys 3, 71 (2020). https://doi.org/10.1038/s4200502003351
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