  Citation: Berman, O.L.; Gumbs, G.; Martins, G.P.; Fekete, P. Superfluidity of Dipolar Excitons in a Double Layer of α − T 3 with a Mass Term. Nanomaterials 2022, 12, 1437. https://doi.org/10.3390/ nano12091437 Academic Editors: Yia-Chung Chang and Daniele Fazzi Received: 13 January 2022 Accepted: 12 April 2022 Published: 22 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). nanomaterials Article Superfluidity of Dipolar Excitons in a Double Layer of α - T 3 with a Mass Term Oleg L. Berman 1,2, *, Godfrey Gumbs 2,3,4 , Gabriel P. Martins 1,2,3 and Paula Fekete 5 1 Physics Department, New York City College of Technology, City University of New York, New York, NY 11201, USA; gpimentamartins@gradcenter.cuny.edu 2 The Graduate School and University Center, City University of New York, New York, NY 10016, USA; ggumbs@hunter.cuny.edu 3 Department of Physics and Astronomy, Hunter College, City University of New York, New York, NY 10065, USA 4 Donastia International Physics Center (DIPC), P de Manuel Lardizabal, 4, 20018 San Sebastian, Spain 5 US Military Academy at West Point, 606 Thayer Road, West Point, NY 10996, USA; paula.fekete@westpoint.edu * Correspondence: oberman@citytech.cuny.edu Abstract: We predict Bose-Einstein condensation and superfluidity of dipolar excitons, formed by electron-hole pairs in spatially separated gapped hexagonal α − T 3 (GHAT3) layers. In the α − T 3 model, the AB-honeycomb lattice structure is supplemented with C atoms located at the centers of the hexagons in the lattice. We considered the α − T 3 model in the presence of a mass term which opens a gap in the energy-dispersive spectrum. The gap opening mass term, caused by a weak magnetic field, plays the role of Zeeman splitting at low magnetic fields for this pseudospin-1 system. The band structure of GHAT3 monolayers leads to the formation of two distinct types of excitons in the GHAT3 double layer. We consider two types of dipolar excitons in double-layer GHAT3: (a) “A excitons”, which are bound states of electrons in the conduction band (CB) and holes in the intermediate band (IB), and (b) “B excitons”, which are bound states of electrons in the CB and holes in the valence band (VB). The binding energy of A and B dipolar excitons is calculated. For a two-component weakly interacting Bose gas of dipolar excitons in a GHAT3 double layer, we obtain the energy dispersion of collective excitations, the sound velocity, the superfluid density, and the mean-field critical temperature T c for superfluidity. Keywords: Bose-Einstein condensation; superfluidity; dipolar exitons 1. Introduction The many-particle systems of dipolar (indirect) excitons, formed by spatially separated electrons and holes, in semiconductor coupled quantum wells (CQWs) and novel two- dimensional (2D) materials have been the subject of numerous experimental and theoretical studies. These systems are attractive in large part due to the possibility of Bose-Einstein con- densation (BEC) and superfluidity of dipolar excitons, which can be observed as persistent electrical currents in each quantum well, and also through coherent optical properties [1–5]. Recent progress in theoretical and experimental studies of BEC and superfluidity of dipolar excitons in CQWs have been reviewed in [6]. Electron-hole superfluidity in double layers can occur not only in the BEC regime, but also in the Bardeen-Cooper-Schrieffer (BCS)-BEC crossover regime [7]. A number of experimental and theoretical investigations have been devoted to the BEC of electron-hole pairs, formed by spatially separated electrons and holes in a double layer formed by parallel graphene layers. These investigations were reported in [8–13]. Both BEC and superfluidity of dipolar excitons in double layers of transition-metal dichalcogenides (TMDCs) [14–18] and phosphorene [19,20] have been discussed, because the exciton binding energies in novel 2D semiconductors are quite large. Possible BEC in a long-lived dark Nanomaterials 2022, 12, 1437. https://doi.org/10.3390/nano12091437 https://www.mdpi.com/journal/nanomaterials