# Topological Bloch bands in graphene superlattices

^{a}Walter Burke Institute for Theoretical Physics, California Institute of Technology, CA 91125;^{b}Institute for Quantum Information and Matter, and Department of Physics, California Institute of Technology, CA 91125;^{c}Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

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Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved June 24, 2015 (received for review December 30, 2014)

## Significance

A family of designer topological materials is introduced, comprising stacks of two-dimensional materials which by themselves are not topological, such as graphene. Previously, topological bands in graphene were presumed either impossible or impractical. The designer approach turns graphene into a robust platform with which a host of topological behavior can be realized and explored.

## Abstract

We outline a designer approach to endow widely available plain materials with topological properties by stacking them atop other nontopological materials. The approach is illustrated with a model system comprising graphene stacked atop hexagonal boron nitride. In this case, the Berry curvature of the electron Bloch bands is highly sensitive to the stacking configuration. As a result, electron topology can be controlled by crystal axes alignment, granting a practical route to designer topological materials. Berry curvature manifests itself in transport via the valley Hall effect and long-range chargeless valley currents. The nonlocal electrical response mediated by such currents provides diagnostics for band topology.

Electronic states in topological materials possess unique properties including a Hall effect without an applied magnetic field (1⇓–3) and topologically protected edge states (4, 5). Accessing nontrivial electron topology depends on identifying materials in which symmetry and interactions produce topological Bloch bands. Such bands can only arise when multiple requirements, such as a multiband structure with a Berry phase and suitable symmetry, are fulfilled. As a result, topological bands are found in only a handful of exotic materials in which good transport properties are often lacking. Formulating practical methods for transforming widely available materials with a reasonably high carrier mobility (such as silicon or graphene) into a topological phase remains a grand challenge.

Here, we lay out an approach for engineering designer topological materials out of stacks of generic materials—“Chernburgers.” Our scheme naturally produces (*i*) topological bands with different Chern invariant values, and (*ii*) tunable topological transitions. As an illustration, we analyze graphene on hexagonal boron–nitride heterostructures (G/*h*BN), where broken inversion symmetry is expected to generate Berry curvature (6, 7), a key ingredient of topological materials. Indeed, recently valley currents have been demonstrated in a G/*h*BN system (8) signaling the presence of Berry curvature (6). As we will show, Berry curvature in G/*h*BN can be molded by stacking configuration, leading to a large variability in properties. Transitions between different topological states can be induced by a slight change in stacking angle.

Topological bands in G/*h*BN arise separately for valley *K* and valley *h*BN produces superlattice minibands (9⇓⇓⇓⇓–14), with Berry curvature *K* or *A*) *B*), the invariant [**1**] vanishes in these minibands,

Interestingly, the conditions for both topological and nontopological bands are met by currently available systems. Indeed, both commensurate and incommensurate stackings have been recently identified in G/*h*BN by scanning probe microscopy (15, 16). Further, the commensurate–incommensurate transition can be controlled by twist angle between G and *h*BN, providing a practical route in which to tailor electron topology via a tunable structural transition.

We note that time-reversal (TR) symmetry requires that *K* and

We also note that topological bands in graphene are sometimes presumed either impossible or impractical. Indeed, a connection between *K* and *h*BN superlattice to create energy gaps above and below the *K* and

## Minimal Model for Superlattice Bands

Modeling the superlattice bandstructure is greatly facilitated by several aspects of the G/*h*BN system. First is the long-wavelength character of superlattice periodicity, which results from nearly identical periods of graphene and *h*BN crystal structure. For commensurate stackings, the superlattice structure is defined by a periodic array of hexagonal domains (Fig. 1*A*). Its periodicity, which is set by the size of the domains, is on the order of *B*) the lattice mismatch and the twist angle between graphene and *h*BN produce long-period moiré patterns with wavelength *K* and *K* and

Another property of the G/*h*BN system that simplifies modeling is a relatively weak coupling strength. Indeed, the reported values for the *h*BN-induced energy gap at the Dirac point are on the order of 500 K (8, 16, 26), which is about 10 times smaller than the energy *b* is the superlattice wavevector and *h*BN. Our minimal model, given in Eq. **2**, is sufficient to understand the key features of the bandstructure for both stacking types. In particular, *K* and

We note parenthetically that a more general Hamiltonian can also include a scalar potential term modulated in the same way as the

## Global Gap and the Signs of Δ g and m 3

Turning to the analysis of the coupling in Eq. **2**, we first consider the commensurate case, where all of the hexagonal domains adopt the same lowest energy atomic configuration. The simplest arrangement to produce such a stacking is perfect crystal axes alignment when *G* and *h*BN lattices conform with each other as pictured in Fig. 1*A*. While we have used AB stacking where G and *h*BN crystal axes are aligned in our illustration in Fig. 1*A*, other stackings can also be used, yielding similar results. Other commensurate stackings in the absence of perfect crystal axes alignment may also occur and do not affect our main conclusions. The registration within each hexagonal cell is locked, producing an A/B sublattice asymmetry in graphene. Crucially, the sign of this asymmetry cannot change upon lateral sliding which is not accompanied by a rotation. Hence the asymmetry is of the same sign throughout the structure, leading to a global constant-sign gap.

To illustrate this important point, we present the argument in a form that does not depend on detailed knowledge of the registration within each of the domains. Of course, in practice the registration types (and hence the asymmetry signs) arise from general energetic and geometric constraints which can be easily accounted for (27). As an example, we consider three possible registrations: (*i*) site A in *h*BN aligned with site A in graphene and site B in *h*BN with site B in graphene; (*ii*) site A in *h*BN aligned with site B in graphene and site B in *h*BN with H (hollow) in graphene; and (*iii*) A in *h*BN aligned with site B in graphene whereas site B in *h*BN aligned with site A in graphene.

Configurations (*i*) and (*iii*) cost the same energy, but have a different energy than (*ii*). Importantly, lateral sliding of a cell with configuration (*i*) cannot generate configuration (*iii*) because it would require a lattice rotation. At the same time, whereas lateral sliding of a cell with configuration (*i*) can generate configuration (*ii*), it costs a different energy. As a result, stacking frustration between neighboring cells cannot occur, locking the registration between all hexagonal cells to yield a constant global gap,

Next, we note that imperfect registration around the domain boundaries yields a weaker coupling between G and *h*BN [strained graphene sheet buckles (16) increasing the G-to-*h*BN distance]. Reduction in sublattice-asymmetric potential *h*BN at the domain boundaries.

Because we are interested in bandstructure reconstruction in the lowest minibands, we expand **2** with*F* determines the sign of *A*, *δ*-functions along the hexagonal domain walls, we obtain the form factor*d* and *w* are the domain wall length and width, and

Evaluating Eq. **5** gives **2**, we find the relation between signs of *A*).

The incommensurate case (moiré superlattice) differs from the commensurate case in two important ways. One is that the G-to-*h*BN coupling is dominated by the modulational part *h*BN stackings at small twist angles, as shown in Fig. 1*B*.

Of course, one **2**] which we write as *V* yields a term describing a global gap at a third order in *V* via**6**. Whereas the gap size obtained at a third order of perturbation theory in a noninteracting system is small, electron interaction effects are expected to produce an enhancement and generate a large **9** lead to trivial topological classes for superlattice bands, *B*).

In addition to the difference in signs, the commensurate and incommensurate stackings differ in the relative magnitude of the

## Topological Classes

We proceed to explore how stacking types impact the band topology. The topological properties of G/*h*BN can be analyzed through the Berry curvature in the minibands. Even though the G/*h*BN Hamiltonian, Eq. **2**, possesses TR symmetry, its broken inversion symmetry allows for a finite Berry curvature to develop in the SBZ:*n* is the band index, **2**. In what follows, we concentrate on a single valley and the lowest conduction miniband (labeled “1” in Fig. 2*B*).

Using Eq. **10**, we evaluate **2**, and obtain Berry curvature maps in SBZ reciprocal space which are shown in Fig. 3 *A* and *B*. We adopted a numerical method similar to that outlined in ref. 28; see the *Supporting Information* for a full description. In Fig. 3 *A* and *B* we plot *B*); the lowest valence band exhibits the same behavior but with opposite sign. We find that

Integrating **1**, we identify two distinct cases. For the equal-sign case, Eq. **6**, which corresponds to commensurate stackings, we obtain *A*). For the opposite-sign case, Eq. **9**, which corresponds to incommensurate stackings, we obtain *B*). This gives topological and nontopological bands, respectively.

To gain more insight into band topology in Fig. 3, it is instructive to analyze the hot spots of *K* and *K*,

Berry curvature also features hot spots at SBZ corners **2** which mix the pseudospin textures; the energy spectrum and *Supporting Information*. We find that the net Berry flux in the conduction band,

We note that the “half-Dirac” flux *A* and *B*). Because there are two inequivalent

## Valley Currents and Berry Curvature Spectroscopy

Topological currents associated with each of the valleys can propagate over extended distances as long as the intervalley scattering is weak (17). Whereas TR symmetry requires no net charge Hall currents, the opposite signs of *K* and *h*BN,

The difference between topological bands and nontopological bands is reflected in the behavior of *C* and *D*. We note that sign-changing *C* and *D*.

Even though the currents *h*BN yield mean-free paths as large as several micrometers (17, 18). Nonlocal resistance measurements (Fig. 4) can therefore provide an all-electrical and robust way to probe the bulk valley-Hall conductivity.

Nonlocal resistance *A*. Transverse valley currents *I* can propagate over extended distances to induce a valley imbalance profile across the device, *A*, *A* is positive and depends on the longitudinal conductivity

Because *C*) changes sign as density is swept in a single band, we find that *B*. The sign-changing behavior of *C*. As a result, sign changes in

In summary, graphene superlattices provide a practical route to constructing topological bands out of generic materials, as illustrated via tunable electron band topology in commensurate–incommensurate stackings. Band topology can be inferred from nonlocal transport measurements. In addition, interesting behavior is expected in superlattice systems in which large-scale inhomogeneities give rise to topological and nontopological domains. In such systems, domain boundaries are expected to support topologically protected chiral edge states. Transitions between topological and nontopological states can be induced by temperature and strain. A number of different systems can be used, including SiC where superlattice stackings have been observed (31, 32), G/*h*BN (8, 16, 26), and twisted bilayer graphene (33⇓–35). The ease with which stacked G/*h*BN structures can be made (36) and the robust bulk transport signatures of their topological character open the door to accessing and probing electronic band topology in designer topological materials.

## Computing Berry Curvature

The topological properties of G/*h*-BN can be analyzed through the Berry curvature in the minibands. Even though the G/h-BN Hamiltonian, Eq. **2** of the main text, possesses TR symmetry, its broken inversion symmetry allows for a finite Berry curvature to develop in the SBZ*n* is the band index, **2** of the main text. In the main text, we concentrate on a single valley and the lowest conduction miniband (labeled “1” in Fig. 2*B* of the main text).

Here we comment on the procedure used to evaluate the Berry curvature. We use the eigenvectors, **2** of the main text [in doing so we use

k ⋅ p Theory for SBZ K ˜ , K ′ ˜ Points

The finite Berry curvature at *A*. The bandstructure near *A*. Evaluating the determinant in*B*, which mimics the behavior close to the *B* of the main text as expected.

We use eigenstates obtained from the above Hamiltonian, Eq. **S2**, to compute Berry curvature as shown in Fig. S1*C*. Here we adopted the same numerical method as outlined in the above section. The Berry curvature distribution shown in Fig. S1*C* is concentrated close to the avoided crossings as expected from Fig. 3 of the main text.

Summing up the Berry curvature in each of the bands, we directly verify that bands 1 and 3 contribute *C*; band 2’s net flux remains zero. As a result, we find that the net Berry flux in band 1 (the lowest conduction band) is described by Eq. 1 of the main text.

Importantly it is *h*-BN.

## Acknowledgments

This work was supported by National Science Foundation (NSF) Science and Technology Center for Integrated Quantum Materials, NSF Grant DMR-1231319 (to L.S.L.), and in part by the US Army Research Laboratory and the US Army Research Office through the Institute for Soldier Nanotechnologies, under Contract no. W911NF-13-D-0001 (L.S.L.). J.C.W.S. acknowledges support from a Burke Fellowship at Caltech, the Sherman Fairchild Foundation, and a National Science Scholarship.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: justin.song.cw{at}gmail.com or levitov{at}mit.edu.

Author contributions: J.C.W.S. and L.S.L. designed research; J.C.W.S., P.S., and L.S.L. performed research; and J.C.W.S. and L.S.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1424760112/-/DCSupplemental.

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