Landau levels from neutral Bogoliubov particles in two-dimensional nodal superconductors under strain and doping gradients

Abstract

Motivated by recent work on strain-induced pseudo-magnetic fields in Dirac and Weyl semimetals, we analyze the possibility of analogous fields in two-dimensional nodal superconductors. We consider the prototypical case of a d-wave superconductor, a representative of the cuprate family, and find that the presence of weak strain leads to pseudo-magnetic fields and Landau quantization of Bogoliubov quasiparticles in the low-energy sector. A similar effect is induced by the presence of generic, weak doping gradients. In contrast to genuine magnetic fields in superconductors, the strain-and doping gradient-induced pseudo-magnetic fields couple in a way that preserves time-reversal symmetry and is not subject to the screening associated with the Meissner effect. These effects can be probed by tuning weak applied supercurrents which lead to shifts in the energies of the Landau levels and hence to quantum oscillations in thermodynamic and transport quantities.Introduction.-Elementary excitations in superconductors are comprised of coherent superpositions of electron and hole degrees of freedom [1][2][3]. These Bogoliubov quasiparticles are electrically neutral on average and therefore do not couple simply to the externally applied magnetic field. In addition, superconductors are known to expel magnetic field from their bulk either completely [4], or form a flux lattice [5], in which the quasiparticle dynamics is effectively zero-field [6]. For these reasons superconductors normally avoid formation of Landau levels which represent the canonical response of most other electron systems to magnetic field [7].In this work, we show that weak, in-plane strains and doping gradients generically lead to Landau quantization of Bogoliubov quasiparticles for a broad class of 2D nodal superconductors (SC). In these cases, the Dirac-like quasiparticles in the vicinity of point nodes are subject to emergent vector potentials which enter in a time-reversal invariant way. In contrast to genuine magnetic fields in a SC, there are no induced currents and no screening associated with the Meissner effect. Our work is motivated by interesting developments in graphene [8] where strain-induced pseudo-magnetic fields lead to Landau quantization and quantum oscillations, which were already observed in experiment [9], and more recent proposals in the context of Dirac and Weyl semimetals [10][11][12][13][14][15].A possible experimental setup is shown in Fig. 1. Since the most immediate realization of gapless, effectively twodimensional superconductivity is provided by the broad class of Cu-based materials, we study the case of a prototypical dwave SC. Strain can be induced in principle by allowing for the controlled deformation of an underlying substrate [16][17][18]. Controlled doping gradients [19,20] provide an alternate way of introducing unconventional vector potentials. As discussed in greater detail in the following, an additional application of a weak supercurrent provides a simple way to detect the underlying quantum oscillations.Vector po...

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