Hierarchy of gaps and magnetic minibands in graphene in the presence of the Abrikosov vortex lattice

Abstract

We determine bands and gaps in graphene subjected to the magnetic field of Abrikosov lattice of vortices in the underlying superconducting film. The spectrum features one non-dispersive magnetic miniband at zero energy, separated by the largest gaps in the miniband spectrum from a pair of minibands resembling slightly broadened first Landau level in graphene, suggesting the persistence of ν = ±2 and ±6 quantum Hall effect states. Also, we identify occasional merging point of magnetic minibands with a Dirac-type dispersion at the miniband edges. PACS numbers: 73.22.Pr, 73.21.Cd,Studies of superlattices in two-dimensional (2D) electron systems have, recently, been boosted by the development of van der Waals heterostructures of graphene with hexagonal boron nitride (hBN). In such systems the superlattice effects, observed in STM spectra [1-3], magneto-transport chracteristics [4][5][6] and quantum capacitance [7], are produced by a periodic moire pattern, with the period a determined by slight incommensurability and misalignment between graphene and hBN crystals [1,8,9] and reflect the formation of superlattice minibands for graphene's Dirac electrons [4,7,9]. To a large extent, the possibility to observe the superlattice effects in graphene-hBN heterostructures owes to the high mobility of electrons in such systems, where graphene is encapsulated between hBN sheets both protecting from contamination and permitting to vary electrons' density over a broad range using electrostatic gates. When subjected to a strong external magnetic field, the superlattice leads to the formation of a 'Hofstadter butterfly', a sparse spectrum of minibands [10][11][12] formed at magnetic field values corresponding to the magnetic flux, Φ = p q φ 0 (through the area S = √ 3a 2 /2 of the superlattice unit cell) commensurate with the flux quantum, φ 0 = h/e.Here, we consider a magnetic superlattice [13-18] that can be realised in a ballistic hBN-graphene-hBN stack by placing it over a high-H c2 superconductor film (e.g., Nb, W, or MoRe alloy). In such a system, where no alignment control of graphene and hBN lattices is required, longrange periodic structure is caused by the Abrikosov lattice of vortices [19,20] formed in a superconductor subjected to an external magnetic field H < H c2 , sketched in the inset in Fig. 1. In contrast to the earlier theories developed for spatially alternating magnetic fields with a zero average [21][22][23][24][25][26], the Abrikosov lattice produces magnetic induction with spatial average, B = φ 0 /( √ 3a 2 ), linked to the magnetic lattice period, a. As each vortex carries the flux h/2e, the vortex lattice realises the simplest fundamental fraction, p q = 1 2 , in the Brown-Zak commensurability condition for magnetic field flux in a 2D periodic system [10,11]. Figure 1 shows the hierarchy of bands and gaps in the corresponding spectrum of Dirac electrons calculated FIG. 1: Spectrum of Dirac electrons in graphene in a magnetic field of Abrikosov vortex lattice, with one degenerate band at E = 0. Energy is s...

Majorana lattices from the quantized Hall limit of a proximitized spin-orbit coupled electron gas

“…We remark that most treatments of Abrikosov vortex lattices in related contexts (see, for example, Refs. [14,15,50]) typically work in the "London" limit appropriate for intermediate flux densities, H c1 B H c2 . In this regime, the superconducting coherence length ξ is much less than the intervortex separation, i.e., ξ a, so that |∆(r)| = const.…”

Zero-energy modes and valley asymmetry in the Hofstadter spectrum of bilayer graphene van der Waals heterostructures with hBN

“…As the u 1 only perturbation can be considered to be a periodic pseudomagnetic field [10], it could alternatively be created using an Abrikosov lattice, i.e., a vortex structure generated from a type-II superconductor in a magnetic field. Such a system is also expected to possess degenerate zero energy eigenvalues [25].…”