Orbital Magnetization of Quantum Spin Hall Insulator Nanoparticles

Abstract

Both spin and orbital degrees of freedom contribute to the magnetic moment of isolated atoms. However, when inserted in crystals, atomic orbital moments are quenched because of the lack of rotational symmetry that protects them when isolated. Thus, the dominant contribution to the magnetization of magnetic materials comes from electronic spin. Here we show that nanoislands of quantum spin Hall insulators can host robust orbital edge magnetism whenever their highest occupied Kramers doublet is singly occupied, upgrading the spin edge current into a charge current. The resulting orbital magnetization scales linearly with size, outweighing the spin contribution for islands of a few nm in size. This linear scaling is specific of the Dirac edge states and very different from Schrodinger electrons in quantum rings. Modelling Bi(111) flakes, whose edge states have been recently observed, we show that orbital magnetization is robust with respect to disorder, thermal agitation, shape of the island and crystallographic direction of the edges, reflecting its topological protection.A central notion in magnetism is the fact that orbital moments associated to circulating currents are fragile. They naturally occur in open-shell isolated atoms [1], but these atomic orbital moments quench as soon as the atom is placed in a crystal. Circulating currents in artificially patterned mesoscopic quantum rings [2], studied in the last three decades [3, 4], require very special conditions to survive, such as very low temperatures so that the electrons keep their phase coherence around the entire ring, and small disorder, so that electrons do not localize. In contrast, robust spin currents occur naturally at the edge of quantum Spin Hall insulators (QSHI) [5][6][7] and are topologically protected. These spin currents are associated to Kramers doublets, where each state has a net charge current flowing with opposite chirality. In a finite sample, these counter-propagating currents can be associated to magnetic moments with opposite sign for each state in the Kramers doublet. Since these states are equally occupied, the resulting net orbital moment vanishes. Having an insulating bulk and robust spin currents at the edges, QSHI are natural quantum rings [8] for spin currents. The central idea of this paper is that, in the case of QSHI nanoislads [9] (or flakes) with a discrete edge state spectrum, it is possible to turn these robust spin currents into robust charge currents that result in very large orbital moments. To do so, two conditions are sufficient: a magnetic field has to split the Kramers doublets and, using electrical gating or chemical doping, only one electron has to occupy the highest occupied Kramers pair, providing thereby a net edge current, and a large orbital magnetization.Several systems have been predicted to be QSHI [10,11] and strong experimental evidence exists that CdTe/HgTe quantum wells [12], InSb/GaAs quantum wells [13], and with Bismuth (111) monolayers [14][15][16][17] host spin filtered edge states essential f...

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