Substrate-induced topological mini-bands in graphene

Substrate-induced topological mini-bands in graphene

Tobias M. R. Wolf Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland    Oded Zilberberg Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland    Ivan Levkivskyi Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland    Gianni Blatter Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
July 26, 2019
Abstract

The honeycomb lattice sets the basic arena for numerous ideas to implement electronic, photonic, or phononic topological bands in (meta-)materials. Novel opportunities to manipulate Dirac electrons in graphene through band engineering arise from superlattice potentials as induced by a substrate such as hexagonal boron-nitride. Making use of the general form of a weak substrate potential as dictated by symmetry, we analytically derive the low-energy mini-bands of the superstructure, including a characteristic 1.5 Dirac cone deriving from a three-band crossing at the Brillouin zone edge. Assuming a large supercell, we focus on a single Dirac cone (or valley) and find all possible arrangements of the low-energy electron- and hole bands in a complete six-dimensional parameter space. We identify the various symmetry planes in parameter space inducing gap closures and find the sectors hosting topological mini-bands, including also complex band crossings that generate a Valley Chern number atypically larger than one. Our map provides a starting point for the systematic design of topological bands by substrate engineering.

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I introduction

The hunt for materials with topological properties, originally rooted in two-dimensional quantum Hall systems Von Klitzing (1986), has been fueled by numerous proposals for materials with electronic topological bands Kane and Mele (2005); Hasan and Kane (2010); Qi and Zhang (2011) and has recently sparked ideas for engineered meta-materials hosting topological bands for electromagnetic Haldane and Raghu (2008); Khanikaev et al. (2012); Ozawa et al. (2018) (photonic) or elastic Prodan and Prodan (2009); Süsstrunk and Huber (2016) (phononic) modes. Many of these proposals are based on the honeycomb lattice, which provides a natural host for topological phenomena through various types of engineering, from the (dynamical) Haldane model Haldane (1988); Oka and Aoki (2009), recently realized both in photonic Rechtsman et al. (2013) and cold atomic Jotzu et al. (2014) systems, to designer dielectrics holding topological photons Wu and Hu (2015). The topological properties in these systems arise from band crossings or Dirac cones. In time-reversal-symmetric systems, such cones appear in compensating pairs and topological features cancel out. Nevertheless, topological properties manifest in individual valleys or cones and are brought forward in the field of valleytronics Rycerz et al. (2007); Xiao et al. (2007); Schaibley et al. (2016)—as with topological materials, valleytronics can be engineered in non-electronic systems Ni et al. (2018).

In this paper, we investigate a generic valleytronic system where the Dirac electrons of the cone are engineered via a hexagonal substrate potential, with a well-known realization of such a system given by placing graphene on hexagonal boron nitride (G-hBN). Symmetry considerations on the substrate potential then define a six-dimensional parameter space that describes all possible arrangements of minibands and their topological properties. Focusing on the six lowest electron- and hole bands, the three-fold symmetry of the scattering potential leads to a characteristic ‘1.5’ Dirac cone deriving from three crossing bands as well as to strongly anisotropic two-band crossings. We discuss several pertinent examples for new topological band arrangements resulting from the atypical Berry curvatures generated by these crossings, including also situations with high Valley Chern number.

While depositing graphene on a substrate improves the electrical properties of the film Geim and Novoselov (2007); Castro Neto et al. (2009); Dean et al. (2010), such simple manipulation also allows for the deliberate tuning of its spectral properties Giovannetti et al. (2007), e.g., the gap-opening observed in graphene on SiC Zhou et al. (2007), or the band-flattening in twisted bilayer graphene Li et al. (2010); Suárez Morell et al. (2010); Bistritzer and MacDonald (2011). More complex reconstructions of a Dirac cone into minibands emerge when subjecting graphene to a hexagonal substrate Wallbank et al. (2015), such in the graphene on hexagonal boron-nitride (G-hBN) system. The scattering of the electrons on such three-fold symmetric substrate potentials naturally leads to the hybridization of backfolded cones that can result in secondary gap openings that give birth to conventional or even topological minibands Song et al. (2015)—such three-band hybridization of Dirac-like particles induced by the substrate at low energies is the subject of the present work.

Similar multi-band engineering has attracted quite some interest recently, starting with proposals to hybridize three (one Dirac cone plus a flat band) Liu et al. (2011); Huang et al. (2011) and four bands (a double-Dirac cone) Sakoda (2012) in photonic Wu and Hu (2015) or phononic Liu et al. (2012); Chen et al. (2014) metamaterials by exploiting properly tuned accidental degeneracies. While these degenerate multi-band configurationes reside at the -point, our three-band mixing occurs near the -points of a Dirac material and involves three linear bands, corresponding to what we call a 1.5 Dirac cone, in allusion to the double-Dirac cone of Refs. [Sakoda, 2012; Wu and Hu, 2015; Liu et al., 2012; Chen et al., 2014]. Alternatively, the formation of what we call the 1.5 Dirac cones by the scattering of a Dirac fermion on a hexagonal substrate potential can be understood in terms of the formation of the new ‘three-fermions’ of Ref. [Bradlyn et al., 2016].

Depositing graphene on a substrate with hexagonal symmetry generates both (incommensurate) Moiré Xue et al. (2011); Decker et al. (2011) or (commensurate) grain- boundary Yankowitz et al. (2012); Woods et al. (2014) superstructures. Below, we analyze how such a superstructure splits an individual Dirac cone into mini-bands, see Fig. 1(a). Exploiting that such a cone maps onto itself under the combined action of inversion (I) and time reversal (T), we can use symmetry arguments Wallbank et al. (2013) to characterize the scattering potential. The latter then is be described by six parameters that can be grouped into two sets of three TI symmetric (TIS) and three TI antisymmetric (TIAS) amplitudes, defining two three-dimensional parameter spaces. The three lowest electron- and hole bands derive from three backfolded cones that mix at the - and -points of the mini-Brillouin zone, see Fig. 1(a). A purely TI-symmetric potential then splits the three-fold degeneracies at the - and -points into combinations of a single cone and a parabolic band—the mutual arrangement of the latter depends on the chosen parameters. Turning on a TI-antisymmetric component of the substrate potential leads to a splitting of the remaining degeneracy of the cones and frees the Berry curvatures previously hidden in the degeneracy points Nielsen and Hedegård (1995). Interestingly, the Berry curvatures deriving from the 1.5 cone sum up to values for the top and bottom bands and averages to zero for the middle band, quite different from the usual weight characterizing a conventional Dirac-like cone. Finally, by proper tuning of parameters, we find values generating electron- or hole bands that are gapped away from other bands—appropriate placement of the chemical potential within the minigap then allows for realizing topological valley physics Xiao et al. (2007); Rycerz et al. (2007) with mini-bands. Furthermore, we find substrate configurations that generate such isolated bands with a network of Berry curvature with a higher-than-one Chern number.

In the following section II, we set up our phenomenological model Hamiltonian describing an isolated cone of Dirac-like particles subject to a substrate potential with TI-symmetric and more general symmetries. We solve the problem analytically for the six low-energy electron- and hole bands by folding back the neighboring unit cells in the Brillouin zone; more exact band-structure calculations are done numerically with 62 bands, i.e., including higher-order reciprocal vectors. In Sec. III, we analyse the miniband geometries for the and symmetric potentials, emphasizing the geometric arrangements of the bands with singlets and doublets at the and -points in the high symmetry case and the Berry-curvature maps characteristic of the low-symmetry (, TI-symmetry broken) situation. The latter derive from multiple band crossings and we present an analytic calculation for the curvatures associated with the various bands. In Sec. IV, we present specific examples where the substrate potential produces isolated minibands (with well defined gaps separating bands) characterized by non-trivial Chern numbers. We summarize our work and conclude in Sec. V.

Figure 1: (a) High- and low-energy description of Dirac-like particles subject to a substrate potential. The large Brilloulin zone (BZ) of the host material (left) generates Dirac cones at the - and -points (dark orange). Selecting one of the latter (here a -point) defines the new -point of the mini-BZ (right) generated by the triangular substrate potential. The scattering of the Dirac-like particle by the substrate produces a periodic mirroring (light gray cones) of the original Dirac cone and leads to band-hybridization at the boundary of the first mini-BZ (light shaded red). The high-symmetry points and of the mini-BZ produce 3-fold degeneracies and the hybridization of the associated bands through the substrate generates bands with tunable topological properties. (b) Electronic dispersion along the lines , with dotted orange lines referring to pristine graphene (dispersion backfolded to the mini-BZ, numbers indicate degeneracies) and black solid lines showing the dispersion for a finite scattering potential that lifts the degeneracies and opens gaps (we have chosen parameters and ; the red square delimits the region for the dispersions shown in Figs. 2 and 5). (c) The band energies near the - and -points appear as projections of three equidistant points (shaded red triangle) on a circle of radius centered at and rotated by , see Eq. (8).

Ii Dirac-like particles in and symmetric potentials

We study an effective model describing the low-energy physics of Dirac electrons subject to a weak hexagonal periodic potential. This situation is realized by the triangular Moiré pattern resulting when graphene is deposited on a hexagonal substrate, such as boron-nitride. To do so, we consider a spinless Dirac-like particle described by a pseudo-spinor with linear dispersion moving in two dimensions in the presence of a weak periodic potential ,

(1)

where denote Pauli iso-spin matrices, is the Fermi velocity, and we allow for a finite mass (or spectral gap) which constitutes a TIAS parameter. We assume a smooth, three-fold rotational-symmetric potential with only one set of long-wavelength amplitudes for the six reciprocal lattice vectors , . The reciprocal lattice constant , with the real-space periodicity, defines an energy scale via the minimal recoil momentum for elastic scattering,

(2)

We assume the latter to be much larger than the mass and the amplitudes of the potential. In building the Moiré pattern in the G-hBN system, the (approximate) periodicity is determined by the ratio of the lattice constants of graphene and the substrate, and their misfit angle Wallbank et al. (2013); Moon and Koshino (2014); note, that we deal with a slightly incommensurate situation where the lattices do not match exactly on the distance . An exact match can be obtained in twisted bilayer graphene and requires fine-tuning of the angle Moon and Koshino (2012). In the following, we ignore effects arising due to the quasi-periodicity appertaining to a Moiré pattern.

The eigenmodes of the kinetic part in the Hamiltonian (1) describe particles with dispersion

(3)

and associated momentum eigenstates

(4)

where is a plane wave state with wave vector . The phase and signs refer to particle () and hole () bands, in the following specified by the index .

The most general expression for the scattering amplitudes respecting three-fold rotational symmetry takes the form Wallbank et al. (2013)

(5)

where , , and define three complex parameters with the hat referring to normalized quantities of unit amplitude. The real (imaginary) parts of the parameters define a potential that is even (odd) under real space inversion. While quantifies the overall amplitude of the potential landscape, describes a periodic modulation of the Dirac mass . The parameters are associated with a spatially-periodic vector potential describing the action of an out-of-plane pseudo-magnetic field with the same spatial periodicity as the (substrate) potential; in graphene, such a term can arise due to non-uniform strain Castro Neto et al. (2009). Note that local symmetry in the Hamiltonian allows one to eliminate the longitudinal component of this vector potential via a proper transformation of the wavefunction (see Appendix A); the parameters then describe the transverse component of the vector potential after fixing the gauge.

The various components of the Hamiltonian can be grouped into two sets that are defined through their transformation properties under the combined action of time-reversal and spatial inversion, the TI-symmetric (TIS) parameters and the TI-antisymmetric (TIAS) parameters Wallbank et al. (2013), the latter picking up a minus sign under the action of TI. Dropping the TIAS parameters enhances the structural symmetry group from to . Note that T and I by themselves are not good symmetries of the Hamiltonian (1), unless we include the host material’s second Dirac cone (e.g., at the time-reversed point , with the parameters for the case of a time- reversal- symmetric mass and potential).

Including the scattering by the potential , the free Dirac-like spectrum is folded back in reciprocal space, defining the Brillouin zone (BZ) shown on the right of Fig. 1(a). The band structure is obtained from diagonalizing the Bloch Hamiltonian (6) with restricted to the first Brillouin zone and , integers, denoting the original position in reciprocal space, see top right in Fig. 1. Given a choice of scattering amplitudes, Eq. (6) can be diagonalized numerically Wallbank et al. (2013) including a sufficiently large set of bands , see Fig. 1(b).

Alternatively, focusing on the lowest bands, useful insights can be gained from an analytic solution involving only mixing of the three neighboring cells sharing the -point and the -point (equivalently, we denote the latter by -points, , with equivalent to ). Including scattering induced by the potential (5) between the unperturbed states , and , the many-band Bloch Hamiltonian (6) can be truncated to the lowest three electron () and hole () bands described by

(7)

with the unperturbed energies and matrix elements , where , , are -rotated -vectors.

Such a three-band degenerate perturbation theory provides a reliable analytical solution near the BZ boundary, while the band structure is given by the Dirac-like spectrum (3) near the -point. By diagonalizing (7), we find that the energies for electrons and holes can be written in the form of projections of three points on a circle of radius centered around the mean energy ,

(8)

for the three bands arranged in ascending order of excitation energy, see Fig. 1(c). Here, the mean derives from the unperturbed energies averaged over the -rotated -vectors, while the radius involves the energy disbalance . The offset-angle is given by

(9)

and the integer ensures the proper band ordering. While the radius defines the magnitude of the splittings, the phase determines their relative arrangements. The associated eigenfunctions can be found in a closed analytic form as well, see Appendix B.

The three-band mixing described by (7) determines the structure of the mini-bands near the corresponding edge of the Brillouin zone. In the absence of a scattering potential , the three energies in Eq. (8) collapse to a triplet at (i.e., ). Deviations away from are linear in and locally define three planes that derive from the cutting of the three original cones—these three planes define our 1.5 Dirac cone. A finite scattering potential lifts the 3-fold degeneracy near the -points; at the high-symmetry points, the splitting derives from the Hamiltonian (7) at

with

(10)

derived from the scattering amplitudes (we assume a vanishing Dirac mass , see Section III.3 for results with a finite ). Diagonalizing , we find the energy splittings

(11)

where the splittings and are associated with the TIS parameters , , and and the TIAS parameters , , and , respectively.

In a TI-symmetric situation, we have and the original triplet splits into a singlet and a doublet separated by . If (), the singlet will be higher (lower) in energy than the doublet, which is equivalent to an offset angle (); in the following, we will denote these arrangements by (), corresponding to the red shaded triangles in Fig. 1(c). In the opposite case with only finite TIAS parameters, we find that and the triplet fully splits into singlets in a symmetric fashion. This splitting is controlled by and comes with the offset phase and thus will be denoted with the symbol . A general TI-symmetry broken case will involve all parameters and leads to an interplay between the singlet-doublet splitting and the doublet splitting . In such a situation, the angle can assume any value. We summarize the above discussion in Table 1.

Parameters
TIS (), ()
TIAS ()
Table 1: Parameters driving the energy splittings and at the -points. The symbols , , indicate the arrangement of split energies, doublet below singlet when , doublet above singlet for , and full symmetric splitting for and , respectively. These arrangements are dictated by the offset angle [Eq. (9)].

A similar analysis can be done at the -point, where it is sufficient to consider two bands only; the Hamiltonian mixing the corresponding states and takes the form

(12)

where is the band gap at induced by the potential,

(13)

Diagonalizing (12), we obtain the energy splitting at the -point in the form

(14)

Iii Bloch bands along

As illustrated in Figs. 2 and 5, the TI symmetric parameters split the triplet at and into a doublet (with the doublet degeneracy protected by the TI symmetry) and an additional singlet, while the antisymmetric terms in additionally split the remaining doublet—it is the latter splitting that generates the topological properties of the mini-bands by breaking the TI symmetry.

iii.1 symmetry

We first focus on the TI-preserving situation with . The remaining TIS parameters define the 3D parameter space shown in Fig. 2. The four planes mark parameters for which residual triplet degeneracies remain at and , i.e., they signal singlet–doublet gap closures (). These planes111Including higher bands leads to a deformation of the planes. compartmentalize the three-dimensional parameter space into regions with different characteristic band arrangements, 7 of which are shown in the maps A to G in Fig. 2. Each of these maps is characterized by a different arrangement of singlets and doublets in the four sectors ( versus , electrons versus holes), with the singlet either above the doublet (, ) or vice versa (, ). The maps A to G show the situation with an arragement for positive energies at (); the configurations with a instead appear for . Note that the configurations and do not occur, since all bounding planes are meeting in the origin of the parameter space.

Figure 2: Singlet–doublet gaps at the and -points depending on the TIS parameters with vanishing . The doublets locally define (warped) cones around the and -points in the 2D BZ, while the singlet is the projection of a (warped) paraboloid, see Fig. 3(b). The relative band arrangement is indicated as () when the doublet is higher (lower) in energy than the singlet. The four planes in the central figure mark the points of three-fold degeneracy at the - and -points where gap closures allow for band rearrangements. The surrounding insets show the different types of band splittings along the line in the relevant energy window of the BZ [see Fig. 1] that occur for parameters away from the planes (dispersions are calculated numerically accounting for 64 Bloch bands). The color code for the bands is red for , blue for , green for , orange for . Crossing a plane in parameter space inverts the bands () of the respective color in the band structure, e.g., going from A to G the crossing of the red plane rearranges the bands in [color online]. The evolution of the bandstructure along the line B to G in parameters space is shown in Fig. 4.

For a given point in the TIS parameter space, the and configurations at and are smoothly related through the evolution of the angle (see Fig. 3). Configurations with the same singlet–doublet arrangement at and ( or ) as in case B have an intermediate level crossing with the phase continuously changing either from to and back to or from to and back to . Configurations that change the singlet–doublet arrangement as in case G ( or ) have no intermediate level crossing and the phase evolves unidirectionally from to or from to .

Figure 3: Band structure for the lowest three conduction bands () near the BZ edge () with TIS-symmetric parameters corresponding to the maps B and G in Fig. 2. (a) Band energies deriving from projections of three points on the circle (8); the corresponding triangle (light red shading) rotates with the offset angle , see Eq. (9), along the path in reciprocal space. Two inequivalent situations are shown, with the same arrangement between and as featured in B and with an opposite arrangement as in G. In the former case, the triangle undergoes a rotation from along , while in the latter case it evolves unidirectionally . (b) 3D illustration of the band structure near the BZ edge showing the non-trivial geometries of the band touchings at and (light red open circles) and at (black open circle) for parameters as in the maps B and G of Fig. 2.

Tuning the TIS parameters across the plane associated with a given , inverts the corresponding singlet–doublet configuration () by going through a gap closing and reopening. As the gap vanishes at , the radius of the circle in (8) goes through zero and the offset phase flips by . An example for for such a gap closure and reopening when going from the maps B to G in Fig. 2 is shown in Fig. 4. Note that the configuration of B features an intermediate band crossing along while the configuration to in G does not. Hence, while moving across the plane in parameter space, this intermediate crossing must continuously move into a higher band when passing through the triple degeneracy.

Figure 4: Evolution of the lowest three conduction bands () near the BZ edge () when going from the TIS-symmetric parameter map B to G via a straight line as shown in Fig. 2. The singlet–doublet splitting at (inverted red triangle) shrinks to zero upon approaching the point X and opens up in an inverted geometry upon continuing towards G (red triangle). At the same time, the band crossing at in B (blue open circle) moves towards the -point (see cyan arrows). Passing through the tripple degeneracy at X, the band crossing moves further to higher bands as the parameters approach G.

iii.2 symmetry

Next, we address the TI-antisymmetric case, i.e., with finite TIAS parameters and vanishing TIS parameters as well as . This defines a second 3D space of antisymmetric parameters describing spectra at - and -points where the singlet in Eq. (II) remains unchanged while the doublet is split symmetrically away, see Fig. 5. Most interestingly, this gap opening induces finite Berry curvatures [see section III.4 for a detailed analysis] in the 1.5 cone and generates the curvature maps A to G shown in Fig. 5. Shown are the configurations with equivalent Berry curvatures in the sector, i.e., for ; configurations with reverse Berry curvatures are realized for negative values . Again, the four planes crossing at the origin define the locations of triple-degenerate bands at the - and -points where bands rearrange when . As before, the evolution of the spectra when moving between and -points and when changing the antisymmetric parameters across one of the four triplet-degenerate planes can be understood in terms of the rotation and inflation/deflation of the circle defining the energies in (8).

Figure 5: Doublet splittings at the and -points depending on the parameters for the massless TI-antisymmetric situation and with vanishing parameters . Colored segments describe the local Berry curvature [see section III.4] of the -th electron- and hole bands, with red (blue) denoting positive (negative) values, see color scale. The four planes in the central figure mark the points of three-fold degeneracy at the - and -points where gap closures allow for band rearrangements and exchange of Berry curvature (see color-code in A relating the four -sectors with the relevant planes for gap closure). The same color code as in Fig. 2 has been chosen, e.g., blue for , implying a corresponding exchange of the Berry curvature when going from B to D. The fractions denote the integrated Berry curvatures for the lowest () electron- and hole bands arising from the vicinity of the points; adding the contribution from the vicinity of the -point (not shown) provides the Chern number for the lowest bands. In B, the bands cross at and the abelian Berry curvature is not well-defined.

In general, we deal with the situation that is neither purely symmetric nor purely antisymmetric w.r.t. TI symmetry and thus we have to cope with all six TIS and TIAS parameters assuming finite values. We then have to consider the interplay of the two cases. As we have seen, the TIS terms give rise to doublet–singlet gaps, while the TIAS terms control the splitting of the doublet. The magnitude of both splittings is determined by the distance of the configuration point to a particular plane in the respective parameter space. Knowing the two points in the parameter spaces of Fig. 2 and Fig. 5 allows one to quickly determine the band configurations at the - and -points as well as their associated Berry curvatures.

iii.3 Finite mass

In the above discussions of - and the -symmetric cases, we have assumed a vanishing Dirac mass . Relaxing this assumption to a situation where is finite and of similar or smaller magnitude as the other parameters, we find two effects: first, a gap opening at the -point, freeing additional Berry curvature and allowing to define an (abelian) Berry curvature for each band (see Section III.4). The periodic potential can perturbatively modify this mass gap, ; as the correction appears only in the third order of the scattering parameters,

(15)

the renormalization of the mass gap at is small and can usually be neglected. Second, a finite mass produces a (again small) correction in the band arrangement at the Brillouin zone edge. Interestingly, the previous classification of Figs. 2 and 5 remains quantitatively correct under proper replacement of the bare TIS and TIAS parameters with their (slightly) renormalized counterparts

(16)

where . This result can be obtained by evaluating Eq. (II) with a finite but small . It should be noted, however, that in such a case the -symmetry for states near the BZ boundary holds only approximately: while the doublet is no longer symmetry-protected, the -induced splitting near the BZ edge will be at least an order of magnitude smaller than any other discussed gap contribution.

iii.4 Berry curvatures and Chern numbers

The breaking of either inversion or time-reversal symmetry in our single-valley model (1) allows for a finite abelian 2D Berry curvature Xiao et al. (2010) (17) where is the band index with associated energy and wave function and denote derivatives along and ; we have suppressed the variable in the right- hand side of Eq. (17). When bands do not cross, their Berry curvature is well-defined and we can assign a Chern number

(18)

to each separate band . In the following, we determine the local Berry curvatures near the high-symmetry points , , , and in reciprocal space and integrate their contributions to the Chern number. For that matter, the contribution for the lowest electron- and hole bands at the -point involves a standard calculationXiao et al. (2010); the result is dominated by the Dirac mass and results in a local Berry curvature

(19)

where the wavevector denotes the deviation from the -point. The result (19) describes a smeared (by ) -function in 2D with weight 1/2 and hence contributes to the Chern number with

(20)

iii.4.1 Three-band crossings at and : Dirac cones

In order to determine the Berry curvatures near the - and -points, we study the three-band Hamiltonian (7) for a purely TIAS situation. A small-momentum expansion (with polar coordinates ) around or provides the Hamiltonian (21) where is the magnitude of the TIAS band splitting and we choose the zero of energy at the high-symmetry point , see Eq. (II). We restrict the sums in (17) to the three minibands constituting the 1.5 Dirac cone and make use of the eigenenergies and eigenfunctions of the Hamiltonian (21), see Appendix B. Proper arrangement of terms and using symmetries, one arrives at the Berry curvatures

(22)

for the bands with and defined above, see Eq. (9) (we suppress the indices and ). The factor

(23)

describes a three-fold -dependent modulation of the Berry curvature. Furthermore, given the simpler form of Eq. (21), the angle reduces to

(24)

The integration of over the angle generates the dependence of the Berry curvature on ,

(25)

with . As a result, the Berry curvature (22) assumes the form of a broadened (by ) and warped 2D -function of weight 1/4 (with the index reinstalled),

(26)

In Fig. 6, we show the dispersions of the three bands at the -point together with a Berry curvature map. The latter feature a symmetric angular dependence with the middle band exhibiting both signs and hence averaging to zero under angular integration. The distribution of the 1.5 Dirac cone Berry curvature can be understood as splitting the curvature at degeneracy into two parts and attributed to the top () and bottom () pairs of bands. The weight is again equally split between the pair constituents, such that the middle band assumes a vanishing contribution , while the top and bottom bands remain with a weight and .

Figure 6: Band structures and associated Berry curvatures calculated in a small-momentum expansion around the high-symmetry points. (a) The 1.5 Dirac cone (deriving from 3 cutting cones, see Fig. 1) at the - or -point contribute with to the Berry curvature of the miniband. (b) The two bands (deriving from two cutting cones, see Fig. 1) approaching one another near the -point contribute with to the Berry curvature of the Bloch band.

Above, we have discussed the novel situation of emergent Berry curvatures when splitting the 1.5 Dirac cones at the - and -points. A different situation arises when the three-bands crossing at these points is first split with a finite TIS parameter, see Fig. 3(b), leaving a singlet and a doublet at the - and -points. The doublet then is associated with a conventional (warped) Dirac cone where a finite TIAS parameter frees a conventional Berry curvature of integrated weight .

iii.4.2 Two-band crossings near : anisotropic Dirac cone

The second non-trivial contribution to the Berry curvature arises from the two-band splitting near the -point, see Fig. 6. While in Fig. 5 B the two bands (in ascending order of excitation energy) cross in , in a more general situation, these bands split and free a Berry curvature that resides a finite distance away from the -point, see Figs. 5 A, C, D, F as well as Fig. 6. In order to derive the Berry curvature for this situation, we again employ a small-momentum expansion with measured with respect to the point in reciprocal space. Including both TIS and TIAS parameters, we find the characteristic two-band anisotropic Dirac Hamiltonian

(27)

where is again the reciprocal lattice constant and we chose the zero of energy at . We see that the substrate potential defines the gap and the renormalized velocity along via Wallbank et al. (2013)

A straight-forward evaluation of Eq. (17) using the eigenenergies and eigenfunctions of (27), see Appendix B, results in the Berry curvatures (with )

(28)

where is the minimal gap displaced from the -point by along the -direction, i.e., towards the - or -point. The Berry curvature Eq. (28) assumes the form of an anisotropically broadened 2D -function with weight , see Fig. 6; in the limit , it approaches the 2D -function . The integration of (28) thus contributes a term

(29)

As becomes large, the curvatures at and start overlapping and our approximations break down. Nevertheless, the contributions still add, until a gap closure intervenes when (i.e., the minimal gap from passes through the -point) and the Berry curvatures originating from the -point get redistributed between the second and third band.

The Chern number of the individual mini-bands is obtained by adding the contributions from the -, -, -, and -points. Focusing on the lowest electron- and hole bands (with ), we first define the contribution arising from the vicinity of the -- points, with and and the factor 3 arising from the three -points in the first BZ. These contributions then add up to values and are shown in Fig. 5 for the various maps AG. Adding the contribution from the -point, we can reach values for the Chern numbers of the lowest excitation bands.

In the above discussion, we have focussed on a single Dirac cone deriving, e.g., from an original -point; adding a time-reversed cone at then adds the same bands but with opposite Berry curvatures such that the total Berry curvatures add up to zero (note that for a graphene derived system it is always the inversion symmetry that breaks the TI symmetry). In this situation, non-trivial topology manifests itself in valley physicsXiao et al. (2007); Rycerz et al. (2007); only upon breaking of time-reversal symmetry can the Berry curvatures of separate cones be decoupled and overall topological effects can be realized.

Iv Topological valley insulator from filled minibands

In order to realize topological (valley) physicsXiao et al. (2007); Rycerz et al. (2007) with surface-induced minibands, we have to tune the TIS, TIAS, and mass parameters in (1) such as to generate isolated bands with finite Chern numbers and place the Fermi level in the minigap. While numerous parameter settings provide access to such conditions, in Table  2 and Fig. 7 we present a few illustrative examples of the kind of interesting physics that can be brought forward.

(T+I)-symmetric (T+I)-antisymmetric
[]
(a) 0
(b) 1
(c) 1
(d) 2
Table 2: TIS and TIAS parameters for several topological band insulators involving the lowest electron- and hole-type mini-bands and resulting Chern numbers and . The -point is gapped with a mass . The associated Berry curvatures are shown in Fig. 7.

Cases (a) and (b) have been chosen with reference to the work of Song et al.Song et al. (2015) describing the emergence of (topological) minibands in the G-hBN system. It turns out that a smooth and incommensurate Moiré pattern with antisymmetric Dirac mass modulation generates a topologically trivial mini-band: the homogeneous mass parameter derived from perturbatively in third order assumes a sign different from , see Eq. (III.3), such that the Berry curvatures at cancel against those at the - and -points; in our phenomenological description this corresponds to the case (a) with . On the contrary, modulating the graphene with a commensurate grain boundary networkSong et al. (2015) produces equal-sign masses that corresponds to our case (b).

Figure 7: Band structures and Berry curvature maps realizing different topological insulator phases with valley Chern numbers for the lowest electronic band (light grey shading). The local Berry curvatures are indicated in red and blue (see online colors) for positive and negative contributions, respectively. The Berry curvature is calculated using numerical diagonalization involving bands and the Chern number is obtained with the help of Fukui’s method Fukui and Hatsugai (2007). The parameters describing the substrate potential for the cases (a)–(d) can be found in Table 2. Cases (a) and (b) have been discussed in the context of graphene on hexagonal boron-nitride involving either smooth incommensurate Moiré structures (with ) or commensurate grain boundaries () Song et al. (2015). Case (c) highlights a situation where the Berry curvature has been pushed from a symmetric distribution to the -point. Case (d) shows a Berry curvature network connecting all - and -points.

In case (c), we have chosen parameters such as to move the Berry curvature between - and -points. We start from a situation with very asymmetric band arrangement between the - and -points as driven by a large parameter (or alternatively ), see the configuration G (or E) in Fig. 2. The lowest two bands then have a single band touching at and vanishing Berry curvature due to TI symmetry. The gap opening at controlled by the doublet splitting frees a large Berry curvature at the -point. Note that with this procedure, we gap a conventional Dirac cone (providing a contribution from the -point) rather than the 1.5 Dirac cone discussed above and applying to case (b) (providing two contributions and from the - and -points).

Finally, in case (d), we start from a situation with symmetric band configurations at and and a band-crossing at as driven by the large TIS parameter , see the Fig. 2 B. Choosing a finite TIAS parameter frees the large Berry curvature near the -point. The flatness of the bands along the line spreads the Berry curvature to connect all and -points and thereby generates the curvature network illustrated in Fig. 7 (d). Note that with this choice of parameters, the Berry curvature near the - and -points derives from the -point.

V Summary and Conclusion

In the present paper, we have subjected a Dirac-like particle to a periodic substrate potential and have calculated the ensuing band structure as well as its topological properties. Within our phenomenological approach, the model Hamiltonian involves the Fermi-velocity of the Dirac-like particle, possibly a finite (TI-antisymmetric) mass opening a gap at the -point, and TI-symmetric (TIS) and TI-antisymmetric (TIAS) parameters opening up gaps at the , , and -points. While TIS parameters leave a cone at the and -points, these band touchings are lifted by the TIAS parameters and a finite Berry curvature emerges at the (due to a finite ) as well as at the , , and -points (due to the TI-antisymmetric potential part). Such a system opens the possibility for deliberate mini-band engineering and tuning of the Dirac material between different (valley) topological phases.

The phenomenological Dirac-like model described in this paper involves a single Dirac cone—in reality, such a model originates from a microscopic bandstructure where the microscopic lattice generates the effective low-energy Dirac-like dispersion while the periodic substrate potential defines a secondary or mini-bandstructure. Microscopic time-reversal symmetry then generates a partner cone that compensates the Berry curvatures of the original cone. The topological properties of the Dirac material then are reduced to valley-specific features that have to be brought to manifest through special measures, see, e.g., Refs. [Rycerz et al., 2007] and [Xiao et al., 2007].

The realization of such valley-topological physics through substrate-assisted mini-band engineering involves proper tuning in a high-dimensional parameter space. Not only does one require a proper set of TIAS parameters bringing forward Berry curvatures with finite Chern numbers, in addition, the topological minibands have to be isolated from the other bands through proper gaps. Our analysis of TIS and TIAS parameter-spaces summarized in Figs. 2 and 5 provides a systematic overview of possible arrangements of miniband structures for the lowest bands in the vicinity of the , , and -points. Our specific examples in Fig. 7 demonstrate that isolated mini- bands with non-trivial Chern numbers can be achieved in principle. The next important step in a program aiming at substrate-assisted topological mini- band engineering then has to establish the connection between the phenomenological and microscopic parameters describing real band electrons subject to a substrate potential. Inspiration for the solution of this task can be gained from several principles: the TIAS mass parameter derives from sublattice asymmetryZhou et al. (2007) breaking inversion symmetry, the potential parameters can be engineered with an electrostatic top-gate patternPark et al. (2008); Ye and Qi (2011), finite mass parameters derive from a modulated sublattice asymmetry Wallbank et al. (2013); Moon and Koshino (2014), and gauge field parameters are induced by bond modulations, e.g., through strain Castro Neto et al. (2009). More detailed analysis then involves realistic bandstructure calculations that pose a challenging problem given the large supercell of Moiré or grainboundary structures in real systems. Alternatively, optically engineered atomic crystalsUehlinger et al. (2013); Jotzu et al. (2014) may provide another arena for the implementation of topological minibands.

Acknowledgments

We thank M. Ferguson, J. Lado, M. Fischer, and I. Petrides for illuminating discussions and acknowledge financial support from the Swiss National Science Foundation, Division 2 and through the National Centre of Competence in Research “QSIT - Quantum Science and Technology”.

Appendix A Local symmetry and transverse gauge

For the reader’s convenience, we briefly elaborate on the role of gauge freedom and gauge fixing in the model of a Dirac-like particle elastically scattered on a static potentialWallbank et al. (2013). For this purpose, let us consider the Dirac particle minimally coupled to a vector potential , i.e.,

(30)

where , , and are defined in the main text, see Eq. (6). The vector potential can always be decomposed into longitudinal and transverse parts, i.e., with and . We can furthermore express these components through scalar functions, i.e.,

(31)

The vector potential can have different physical origins, such as the presence of electromagnetic fields. However, in G-hBN it arises due to the proximity of the substrate layer even in absence of external fields. The (pseudo) magnetic field associated with the vector potential is given by

(32)

We are allowed to make the unitary transformation without changing the physics of our system. This is our local gauge freedom. The identity

(33)

then readily implies that the transformation removes the longitudinal component of the vector potential from the Hamiltonian,