Topological Winding Number Change and Broken Inversion Symmetry in a Hofstadter’s Butterfly

Abstract

Graphene's quantum Hall features are associated with a π Berry's phase due to its odd topological pseudospin winding number. In nearly aligned graphene-hexagonal BN heterostructures, the lattice and orientation mismatch produce a superlattice potential, yielding secondary Dirac points in graphene's electronic spectrum, and under a magnetic field, a Hofstadter butterfly-like energy spectrum. Here we report an additional π Berry's phase shift when tuning the Fermi level past the secondary Dirac points, originating from a change in topological winding number from odd to even when the Fermi-surface electron orbit begins to enclose the secondary Dirac points. At large hole doping inversion symmetry breaking generates a distinct hexagonal pattern in the longitudinal resistivity versus magnetic field and charge density. Major Hofstadter butterfly features persist up to ∼100 K, demonstrating the robustness of the fractal energy spectrum in these systems.

Twisted Bilayer Graphene: Interlayer Configuration and Magnetotransport Signatures

“…Ref. [] observes a particularly fascinating mixed case, qualitatively lying in between the above described contrasting scenarios in TBG: Magnetotransport in graphene lying on top of hBN (not encapsulated) reveals magnetic breakthrough at the mBz boundaries, while carriers on the according cyclotron orbits acquire additional phase attributed to superlattice Dirac points forming at the borders of the very same mBz.…”

Enhanced electron-phonon coupling in doubly aligned hexagonal boron nitride bilayer graphene heterostructure

“…One well established example is graphene on hexagonal boron nitride (hBN). The weak periodic potential due to underlying hBN gives rise to modulation of graphene electronic band structure [2][3][4] , with the emergence of clone Dirac cones 5 , wherein, the Fermi velocity could be controlled by the relative twist angle between graphene and hBN, showing interesting physics like the Hofstader butterfly 6 , resonant tunneling 7 , change of topological winding number 8 , topological valley current 9 , Brown-Zak oscillations 10 etc. Moreover, twisting individual layers leads to flat bands 11 , where emergent phenomenon like correlated insulating state 12 , superconductivity 13 , quantum anomalous Hall effect 14 , fractional Chern insulating states 15,16 , moiré excitons 17 , and ferromagnetism 14 has been observed.…”

Emergent quantum materials

“…Additionally, the 1.8% lattice mismatch between graphene and hexagonal BN (hBN) leads to a Moiré superlattice with a unit cell that is as large as 200 nm; 2 therein the disproportion between flux quantum and the large unit cell of the Moiré superlattice [46][47][48][49] leads to the celebrated Hofstadter butterfly fractal spectrum. 50 An important effect in heterostructures is the proximity effect, that is, the "endowment" of new properties onto a thin overlayer material by the proximity of a second material (Figure 3b).…”