Substrate-induced topological minibands in graphene

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 minibands 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 minibands, 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.

Figure 1

(a) High- and low-energy description of Dirac-like particles subject to a substrate potential. The large Brillouin zone (BZ) of the host material (left) generates Dirac cones at the $K$ and $K^{\prime }$ points (dark orange). Selecting one of the latter (here a $K$ point) defines the new $\gamma$ 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 $\kappa$ and ${\kappa }^{\prime }$ of the mini-BZ produce threefold degeneracies and the hybridization of the associated bands through the substrate generates bands with tunable topological properties. (b) Electronic dispersion along the lines $\gamma \to \kappa \to \mu \to {\kappa }^{\prime }\to \gamma \to \mu$, 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 $V\left(\mathit{x}\right)$ that lifts the degeneracies and opens gaps (we have chosen parameters $\mathrm{\Delta }=0.1{\epsilon }_0$ and $m_{\mathrm{S}}=0.13{\epsilon }_0,m_{\mathrm{A}}=0.03{\epsilon }_0$; the red square delimits the region for the dispersions shown in Figs. 2 and 5). (c) The band energies near the $\kappa$ and ${\kappa }^{\prime }$ points appear as projections of three equidistant points (shaded red triangle) on a circle of radius $\delta \epsilon$ centered at $\overline{\epsilon }$ and rotated by $\varphi$; see Eq. (8).

Figure 2

Singlet-doublet gaps at the $\kappa$ and ${\kappa }^{\prime }$ points depending on the TIS parameters $u_{\mathrm{S}},m_{\mathrm{S}},a_{\mathrm{S}}$ with vanishing $\mathrm{\Delta },u_{\mathrm{A}},m_{\mathrm{A}},a_{\mathrm{A}}$. The doublets locally define (warped) cones around the $\kappa$ and ${\kappa }^{\prime }$ points in the 2D BZ, while the singlet is the projection of a (warped) paraboloid; see Fig. 3. 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 threefold degeneracy at the $\kappa$ and ${\kappa }^{\prime }$ points where gap closures allow for band rearrangements. The surrounding insets show the different types of band splittings along the line $1\to \kappa \to \mu \to {\kappa }^{\prime }\to 2$ 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 $(\zeta ,s)=(+,+)$, blue for $(\zeta ,s)=(-,+)$, green for $(\zeta ,s)=(+,-)$, and orange for $(\zeta ,s)=(-,-)$. Crossing a plane in parameter space inverts the bands ($▾\leftrightarrow ▴$) 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 $(\zeta ,s)=(+,+)$. The evolution of the band structure along the line B to G in parameter space is shown in Fig. 4.

Figure 3

Band structure for the lowest three conduction bands ($s=+$) near the BZ edge ($\kappa -\mu -{\kappa }^{\prime }$) 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 ${\varphi }_{\zeta }^s\left(\mathit{0}\right)$, see Eq. (9), along the path $\kappa \to \mu \to {\kappa }^{\prime }$ in reciprocal space. Two inequivalent situations are shown, with the same arrangement $▾▾$ between $\kappa$ and ${\kappa }^{\prime }$ as featured in B and with an opposite arrangement $▴▾$ as in G. In the former case, the triangle undergoes a rotation from $\pi /3\to 0\to \pi /3$ along $\kappa \to \mu \to {\kappa }^{\prime }$, while in the latter case it evolves unidirectionally $0\to \pi /3$. (b) 3D illustration of the band structure near the BZ edge showing the nontrivial geometries of the band touchings at $\kappa$ and ${\kappa }^{\prime }$ (light red open circles) and at $\mu$ (black open circle) for parameters as in the maps B and G of Fig. 2.

Figure 4

Evolution of the lowest three conduction bands ($s=+$) near the BZ edge ($\kappa -\mu -{\kappa }^{\prime }$) 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 $\kappa$ (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 $\mu$ in B (blue open circle) moves towards the $\kappa$ point (see cyan arrows). Passing through the triple degeneracy at X, the band crossing moves further to higher bands as the parameters approach G.

Figure 5

Doublet splittings at the $\kappa$ and ${\kappa }^{\prime }$ points depending on the parameters $u_{\mathrm{A}},m_{\mathrm{A}},a_{\mathrm{A}}$ for the massless TI-antisymmetric situation and with vanishing parameters $\mathrm{\Delta },u_{\mathrm{S}},m_{\mathrm{S}},a_{\mathrm{S}}$. Colored segments describe the local Berry curvature ${\mathrm{\Omega }}^{sn}\left(\mathit{k}\right)$ [see Sec. 3d] of the $n\mathrm{th}$ electron and hole bands, with red (blue) denoting positive (negative) values; see ${\mathrm{\Omega }}^{sn}$ color scale. The four planes in the central figure mark the points of threefold degeneracy at the $\kappa$ and ${\kappa }^{\prime }$ points, where gap closures allow for band rearrangements and exchange of Berry curvature [see color code in A relating the four $(\zeta ,s)$ sectors with the relevant planes for gap closure]. The same color code as in Fig. 2 has been chosen, e.g., blue for $(\zeta ,s)=(-,+)$, implying a corresponding exchange of the Berry curvature when going from B to D. The fractions $\delta C^s$ denote the integrated Berry curvatures for the lowest ($n=1$) electron and hole bands arising from the vicinity of the $\kappa -\mu -{\kappa }^{\prime }$ points; adding the contribution $\delta C_{\gamma }^s=\pm 1/2$ from the vicinity of the $\gamma$ point (not shown) provides the Chern number $C^s=\delta C_{\gamma }^s+\delta C^s$ for the lowest bands. In B, the bands cross at $\mu$ and the Abelian Berry curvature is not well defined.

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 three cutting cones at the $\kappa$ or ${\kappa }^{\prime }$ point (see Fig. 1), contribute with $\delta C_{\zeta }^{sn}=({\xi }^{sn}/4)\mathrm{sgn}\left({{\mathrm{\Delta }}_{\zeta }^{\prime \prime }}^s\right)$ to the Berry curvature of the miniband; see also Ref. [39] for equivalent results found numerically. (b) The two bands (deriving from two cutting cones; see Fig. 1) approaching one another near the $\mu$ point contribute with $\delta C_{\mu }^{sn}=({\iota }^{sn}/2)\mathrm{sgn}\left(\tilde{\mathrm{\Delta }}\right)$ to the Berry curvature of the Bloch band.

Figure 7

Band structures and Berry curvature maps realizing different topological insulator phases with valley Chern numbers $C^{+}=0,1,2$ for the lowest electronic band (light gray shading). The local Berry curvatures are indicated in red and blue for positive and negative contributions, respectively. The Berry curvature is calculated using numerical diagonalization involving 62 bands and the Chern number is obtained with the help of Fukui's method [55]. 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 $C^{+}=0$) or commensurate grain boundaries ($C^{+}=1$) [39]. Case (c) highlights a situation where the Berry curvature has been pushed from a symmetric distribution to the ${\kappa }^{\prime }$ point. Case (d) shows a Berry curvature network connecting all $\kappa$ and ${\kappa }^{\prime }$ points and accumulating a total valley Chern number $C^{+}=2$.