Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices

Abstract

In graphene placed on hexagonal boron nitride, replicas of the original Dirac spectrum appear near edges of superlattice minibands. More such replicas develop in high magnetic fields, and their quantization gives rise to a fractal pattern of Landau levels, referred to as the Hofstadter butterfly. Some evidence for the butterfly has recently been reported by using transport measurements. Here we employ capacitance spectroscopy to probe directly the density of states and energy gaps in graphene superlattices. Without magnetic field, replica spectra are seen as pronounced minima in the density of states surrounded by van Hove singularities. The Hofstadter butterfly shows up in magnetocapacitance clearer than in transport measurements and, near one flux quantum per superlattice unit cell, we observe Landau fan diagrams related to quantization of Dirac replicas in a reduced magnetic field. Electron-electron interaction strongly modifies the superlattice spectrum. In particular, we find that graphene's quantum Hall ferromagnetism, due to lifted spin and valley degeneracies, exhibits a reverse Stoner transition at commensurable fluxes and that Landau levels of Dirac replicas support their own ferromagnetic states.2 When graphene is placed on top of atomically flat hexagonal boron nitride (hBN) and their crystallographic axes are carefully aligned, graphene's electron transport properties become strongly modified by a hexagonal periodic potential induced by the hBN substrate [1][2][3][4][5][6] . Replicas of the main Dirac spectrum appear [7][8][9][10][11][12] at the edges of superlattice Brillouin zones (SBZ) and, for the lowest SBZs, the second-generation Dirac cones can be reached using electric field doping [4][5][6] . Because the superlattice period, , for aligned graphene-hBN structures is relatively large (15 nm), magnetic fields B 10 T are sufficient to provide a magnetic flux  of about one flux quantum  0 per area A = √3 2 /2 of the superlattice unit cell. The commensurability between  and the magnetic length l B gives rise to a fractal energy spectrum, the Hofstadter butterfly [4][5][6][13][14][15][16][17][18][19] . An informative way to understand its structure is to consider the butterfly as a collection of Landau levels (LLs) that originate from numerous mini-replicas of the original spectrum, which appear at all rational flux values  = 0 (p/q) where p and q are integer 4,12 . At these fluxes, the electronic spectrum can be described [12][13][14][15] in terms of Zak's minibands 14 for an extended superlattice with a unit cell q times larger than the original one. In graphene, Zak's minibands are expected to be gapped cones (thirdgeneration Dirac fermions) 12 . Away from the rational flux values, these Dirac replicas experience Landau quantization in an effective field B eff =B -B p/q where B p/q = 0 (p/q)/A.In this work, we have employed capacitance measurements to examine the electronic spectrum of graphene superlattices and its evolution into the Hofstadter butterfly. In zero B, pronounced minima...

Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride

“…We note that around the right corner of the SBZ (labelled as κ ), no conical dispersion is observed, suggesting that the two superlattice valleys κ and κ are inequivalent. Therefore, both the constant-energy maps and the dispersion images presented above reveal directly that SDCs exist at the SBZ corners and only at one of the two superlattice valleys 23 , which is in agreement with the Landau level degeneracy implied from previous quantum Hall effect measurements 5,19 . Such direct information is critical in determining the parameters used to describe the generic band structure of SDCs 10,20 , and in further understanding other experimental results which probe the electronic structure indirectly.…”

“…The superlattice potential induced by the lattice mismatch and crystal orientation can significantly modify the electronic properties of graphene and lead to various novel quantum phenomena, for example, the emergence of second-generation Dirac cones (SDCs), which are crucial for the realization of Hofstadter butterfly states under an applied magnetic field 2-5 , renormalization of the Fermi velocity 8,[10][11][12] , gap opening at the Dirac point 4,13-16 , topological currents 15 and gate-dependent pseudospin mixing 17 . Hence, understanding the effects of the superlattice potential on the band structure of graphene is crucial for advancing its device applications, and for gaining new knowledge about the fundamental physics of Dirac fermions in a periodic potential.Previously, the existence of SDCs has been deduced from scanning tunnelling spectroscopy, resistivity and capacitance measurements 2,5,18,19 . However, such measurements are not capable of mapping out the electronic dispersion with momentum-resolved information, and the lack of direct experimental results has led to ambiguous and even conflicting results about the electronic spectra of SDCs and the existence of gaps.…”

“…Previously, the existence of SDCs has been deduced from scanning tunnelling spectroscopy, resistivity and capacitance measurements 2,5,18,19 . However, such measurements are not capable of mapping out the electronic dispersion with momentum-resolved information, and the lack of direct experimental results has led to ambiguous and even conflicting results about the electronic spectra of SDCs and the existence of gaps.…”

Spontaneous strains and gap in graphene on boron nitride

“…- 15 and experimentally. [16][17][18][19][20] The earliest experiments on different samples showed conflicting results on the existence of an insulating state at the neutrality point. Some experiments 21 suggested the existence of an electronic gap of about ∼ 30 meV, while others do not see any clear evidence of it.…”

In-plane chiral tunneling and out-of-plane valley-polarized quantum tunneling in twisted graphene trilayer

“…These crystals engineered with one-atomic-plane accuracy provide unprecedented opportunities to explore unusual properties and new phenomena [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. It was demonstrated that properties of the van der Waals structures depend not only on their building blocks with layer-specific attributes, but also on how the 2D atomic crystals are stacked [12,[16][17][18][19].…”