Moir\'{e} effects in graphene--hBN heterostructures

Encapsulating graphene in hexagonal Boron Nitride has several advantages: the highest mobilities reported to date are achieved in this way, and precise nanostructuring of graphene becomes feasible through the protective hBN layers. Nevertheless, subtle effects may arise due to the differing lattice constants of graphene and hBN, and due to the twist angle between the graphene and hBN lattices. Here, we use a recently developed model which allows us to perform band structure and magnetotransport calculations of such structures, and show that with a proper account of the moir\'e physics an excellent agreement with experiments can be achieved, even for complicated structures such as disordered graphene, or antidot lattices on a monolayer hBN with a relative twist angle. Calculations of this kind are essential to a quantitative modeling of twistronic devices.

Graphene, the first successfully isolated twodimensional material, has opened a new hot research area [1,2]. Due to the linear bands crossing the Fermi level, low-energy carriers in graphene behave like massless, relativistic Dirac fermions, allowing predictions from quantum electrodynamics to be tested in a solid-state system. The high Fermi velocity [3], Dirac-cone band structure [1] and ultra-strong mechanical properties [4] make graphene a promising material for next-generation electronic nanodevices and high-speed switching devices. However, the intrinsic zero energy gap of graphene has hampered its applications in modern electronics. In a practical nanoelectronic device semiconducting graphene is necessary.
A sizable band gap opening around the Fermi level in a graphene antidot lattice (GAL, a regular arrangement of antidots in a graphene lattice) has been predicted by several theoretical studies [5][6][7][8][9], and was recently realized in an experiment [10]. The band gap in GAL can be tuned by the size, shape, and symmetry of both the antidot and the superlattice cell [5][6][7][8][9]. The tunable band gap can be used to design quantum wells and channels for electronic devices [5][6][7][8][9]. Interestingly, transport under magnetic fields in an antidot lattice is predicted to show Hofstadter butterfly features arising from the competition between the antidot lattice periodicity and the magnetic length [11].
In this paper we consider two examples of recent experimental interest where the relative angle between graphene and hBN plays an important role [10,26] : (i) disordered graphene, and (ii) antidot lattices. We first summarize our most important physical findings, and discuss the technical details in subsequent paragraphs. By using an effective lattice model, we first calculate the electronic structure and conductance of G/hBN with and without a relaxation of the graphene and hBN monolayers comprising the system. Our results show that without relaxation the band structure is particle-hole symmetric, in disagreement with experimental data, while the fully relaxed graphene shows, correctly, a particlehole asymmetry emphasizing the importance of lattice relaxation of G/hBN. Next, we compute the conductance in the presence of a magnetic field perpendicular to the graphene sheet. The computed magnetoconductance shows a moiré potential induced secondary Dirac point which clones the Landau fan of the primary Dirac point. The Hofstadter butterfly features are also observed in our numerical results. The computed results are in excellent agreement with experimental data [10,[15][16][17]. Based on an Anderson model, we next investigate how disorder affects the electronic structure and conductance of G/hBN. arXiv:2011.13014v1 [cond-mat.mes-hall] 25 Nov 2020 We find that even though disorder can lift the degeneracy of the bands at high symmetry points, the main features of the band structure are similar to the clean case. While the magnitude of the conductance with disorder is reduced to almost a half of the one without disorder, the magnetoconductance stays unchanged. Finally, we systematically study the electronic structure and transport behavior of antidot lattice on twisted G/hBN. A major theoretical finding is that the secondary Dirac point will disappear once the distance between antidot edges ("the neck width", denoted by d n and see [28]) is smaller than the moiré wave length, a feature seen in experiments [10].
Our effective tight-binding model is proposed in Ref. [27] and we follow the procedures outlined there (Supplemental Material [28] summarizes the details pertinent to this work). Carbon atoms can be removed from the antidot regions by removing the associated rows and columns from the system Hamiltonian. Any dangling σ bonds for a carbon atom with only two neighboring carbon atoms are assumed to be passivated with Hydrogen atoms so that the π bands are unaffected. Transport quantities are calculated using recursive Green's function techniques, following Ref. [29]. The zero-temperature conductance is given by the Landauer formula G = 2e 2 h T , where T is transmission coefficient. A finite magnetic field B perpendicular to the graphene plane is modeled by associating a Peierls phase to the hopping amplitude t → te iϕij , where ϕ ij = (e/h) rj ri A · dr. Here A is the vector potential and r i is the position of atom i. We choose the Landau gauge A = (−By, 0, 0), and the Peierls phase becomes In the leads the magnetic field is set to zero.
Microscopic theoretical investigations on the G/hBN system are made cumbersome by the size of the unit cell, which for small relative rotations contains thousands of atoms making first-principles calculations very expensive. Effective continuum models [30] or several tight-binding models [14,31] with empirical parameters controlling the interlayer interaction between graphene and hBN have been applied to calculate the electronic properties of G/hBN. However, the effective continuum model cannot be used to simulate the transport for realistic experimental conditions, while results for the tight-binding models with empirical parameters must be carefully scrutinized to ascertain their reliability. For a large device transport simulation Chen et al. [32] applied a scaled graphene lattice with a triangular periodic scalar moiré potential, and successfully reproduced the main features of the secondary Dirac point. However, as we show below, this simple moiré potential does not lead to particle-hole asymmetry. In our effective lattice model, the Hamiltonian terms at any local part of a twisted and relaxed G/hBN only depend on the local relative shift and relaxationinduced strain and can be derived from a transparent set of parameters calculated by density functional theory. Moreover, our effective lattice model can be used to calculate the electronic structure of G/hBN with any twist angle, and does not require a recalculation of the parameters for a new twist angle. We next calculate the band structure and conductance for the twist angle θ=1.0047 • with and without lattice relaxation; the results are reported in Fig. 1. The band structure ( Fig.1(a)) and DOS ( Fig.1(b)) of fully relaxed G/hBN show a particle-hole asymmetry which is consistent with the experiments [10,[15][16][17][19][20][21][22]. To emphasize the importance of full relaxation, we plot in Fig.  1(d) the band structure of graphene with a scalar moiré potential [32]: the band degeneracy at high symmetry points is lifted and one finds secondary Dirac points at ±0.225eV. However, in this case the DOS around the two secondary Dirac points are equal and obey particle-hole symmetry ( Fig.1(e)), in contrast to experiment, and Fig.  1(a) and 1(b). The reason for the difference is that a fully relaxed graphene lattice has, in addition to a modified onsite energy, also a modified hopping between neighboring C atoms.
The Supplemental Material [28] reports the band structure and conductance for additional twist angles and edge orientations. The two main conclusions are: (i) the secondary Dirac points shift to larger energies as the twist angle increases, because the interaction between graphene and hBN decreases as the twist angle increases (see Fig. S1-S3), and (ii) the calculated conductance does not significantly depend on whether the edges of the simulated device are in the armchair or zigzag directions (see Fig. S4). In subsequent calculations we consider a device with zigzag edges. We next carry out magnetotransport simulations using the effective lattice model for a 300 nm×300 nm device ( 10 7 atoms). Our results are shown in Fig. 2 (some finer details in G, not visible in Fig. 2, are discussed in [28]). Both the primary (n/n 0 = 0) and the secondary Dirac points (n/n 0 = ±4) break into sequences of Landau levels upon application of a magnetic field, forming the so called Hofstadter butterfly spectrum. The high conductance areas (red, white) separate the gapped Landau levels (blue). As mentioned above, the moiré superlattice potential breaks the partical-hole symmetry, as also seen in transport experiments [10,[15][16][17][19][20][21][22]. As the magnetic field increases, the Landau levels will intersect when φ φ0 = 1 q , where q is an integer, indicated in Fig. 2 with black dashed lines. The intersection of the primary and secondary Landau levels leads to a closing of the magnetic band gap, and a resulting high conductivity, seen as bright spots in Fig. 2. Results for other twist angles, showing the same general trends, are given in [28].
Disorder is ubiquitous to all graphene samples, even for those synthesized with state-of-the-art technologies [33]. The properties of nanoribbons are known to be strongly affected by disorder [34], and recent studies suggest that the electronic and transport properties of graphene antidot lattice may also be strongly perturbed by relatively modest disorder [35][36][37][38][39]. An exploration of the effect of disorder on G/hBN is thus called for. Here, we introduce disorder as a site-diagonal random potential with matrix element H ij = δ ij v i , where v i are independent, uniformly distributed random variables in the range of [-V 0 ,V 0 ] (where V 0 is set to 0.5eV larger than the onsite energy ε i , maximum 0.14eV), and have zero mean and unit variance. Other details on the disorder model are provided in [28]. One would expect that disorder breaks certain symmetries with concomitant modifications in the band structure. Here, our simulations show that disorder leads to band degeneracy lifting at high symmetry points, especially it opens a band gap at the M point ( Fig. 3(a)). The band gap opening leads to a kink in DOS (indicated by horizontal green lines in Fig. 3 (b)). Even though the DOS is modified by disorder, the generic features in conductance stay qualitatively unchanged, except for an overall reduction of 50% in magnitude (compare Fig.  1(c) and 3(c)). In particular, the features in the conductivity at the secondary Dirac points still remain. Overall, we conclude that the transport properties of G/hBN at B = 0 are very robust against disorder.
To investigate whether the robustness persists for finite magnetic fields, we next calculate the magnetotransport properties of disordered G/hBN; the results are shown in Fig. 3(d). One observes that the main features of the magnetoconductivity are essentially the same as those without disorder, shown in Fig. 2. The main differences are the overall reduction of magnetoconductiviy, as already discussed above, and that many fine features are washed out by disorder, and thus the results for the disordered system appear significantly more regular than those for the pristine system. The conclusion is that the magnetotransport in G/hBN is indeed robust with respect to disorder, and its salient features should survive even a nonideal fabrication process.
The band structure of graphene antidot lattices (GAL) may differ qualitatively from that of graphene, as witnessed by the observation of a band gap in recent experiments [10]. We next describe how the techniques developed in this work can yield additional important information of transport in a GAL on a (twisted) hBN substrate. In addition to the magnetic length, there are now two (at least) other competing other length scales: the moiré length λ, which is maximally 14.4 nm [27], and the length scale(s) characterizing the GAL. A schematic geometry of the GAL system is shown in Fig. S7. We consider triangular antidot lattices, because they are known to lead to gapped systems [8] Our numerical results reveal an important relationship between λ and d n . The secondary Dirac point features in conductance are observed only when d n > λ, while they vanish if d n < λ as shown in Fig. 4(a). This finding explains an important detail in the recent experiments [10]. Namely, pristine graphene encapsulated in hBN shows a primary Landau fan and two cloned fans corresponding to moiré periods of 10.2nm and 17.2nm. The second moiré length is larger than the longest possible moiré length in single layer graphene on hBN. Recent experimental and theoretical studies show that the second moiré pattern is due to the simultaneous effects of top and bottom hBN [40][41][42][43]. However, after fabrication of the GAL the second peak related to the 17.2 nm moiré wavelength is lost. As the neck length in the fabricated GALs is 12-15nm, and thereby smaller than the second moiré wavelength 17.2 nm, one does not expect to see Landau fans related to this length scale, which indeed is the case in the experiment. In Fig. 4(b) we show the Landau fan diagram of G/hBN with an antidot lattice whose d n = 15 nm. Compared to the results shown in Fig. 2, there is a reduction in both the magnitude of magnetoconductance, and the number of Landau levels. Importantly, the secondary Dirac point survives, just as in experiments [10].
In summary, we have performed a systematic examination of the consequences of the lattice mismatch and relative orientation between graphene and hBN, and show that several experimental observations find a common explanation rooted in the interplay of the moiré length, and other relevant length scales. The method described in this work can be extended to many other systems of current interest, including twistronic devices.