Excitonic Tunneling in the AB-bilayer Graphene Josephson Junctions

Journal of Low Temperature Physics (2019) 194:325–359 https://doi.org/10.1007/s10909-018-2107-9 Excitonic Tunneling in the AB-bilayer Graphene Josephson Junctions V. Apinyan1 · T. K. Kopeć1 Received: 29 May 2018 / Accepted: 25 November 2018 / Published online: 1 December 2018 © The Author(s) 2018 Abstract We have considered the AB-stacked bilayer graphene tunnel junction construction. The bilayers are supposed to be in the charge equilibrium states and at the half-filling in each of the electronic layers of the construction and at each value of the external gate potential. By considering the interacting bilayers in both sides of the junction and by taking into account both intralayer and interlayer Coulomb interaction effects, we have calculated the normal and excitonic tunnel currents through the junction. The electronic band renormalizations have been taken into account, due to the excitonic pairing effects and condensation in the BLGs. The exact four-band energy dispersions, including the excitonic renormalizations, have been used for the bilayers without any low-energy approximation. The normal and excitonic tunneling currents have been calculated for different values of the gate potential and for different values of the interlayer interaction parameters in both sides of the tunnel junction. We demonstrate the existence of the excitonic Josephson current through the junction which persist even for the non-interacting bilayer graphene junction. For the non-interacting case, the mechanism of the excitonic condensates formation and tunneling between the condensates is attributed to the interlayer hopping between the layers. The role of the charge neutrality point has been discussed in details. Keywords Bilayer graphene · Josephson junction · Excitonic pairing · Exciton condensation 1 Introduction The existence of the electron-hole bound states for a semimetal with overlapping bands has been postulated long years ago by Keldysh, Kopaev and Kozlov [1–6], and a predic- B 1 V. Apinyan Institute for Low Temperature and Structure Research, Polish Academy of Sciences, PO. Box 1410, 50-950 Wrocław 2, Poland 123 326 Journal of Low Temperature Physics (2019) 194:325–359 tion about the superfluidity has been given for a condensed excitonic state. Ulteriorly, it has been the subject of an intense theoretical studies [7–22]. Experimentally, the strong evidence of an excitonic insulator (EI) and excitonic Bose–Einstein condensate (BEC) ground states have been determined only in the quantum Hall regime (in a large magnetic field) and under the high pressure, in a series of the experimental works on the rare-earth chalcogenide compounds, transition metal dichalcogenides and tantalum chalcogenides [23,24]. The exciton condensation was experimentally observed also in quantum Hall bilayers [25–27], in the systems of magnons [28] and cavity exciton polaritons [29–31]. Recently, other solid-state systems were proposed as possible candidates for the achievement of the BEC of excitons. It concerns the quantum well heterostructures with the excitons trapped in the cavities of the potential wells [32–45] (the structure, utilized in these works, were double-layer GaAs/AlGaAs or InAs/GaSb quantum-wells with the electric field applied perpendicularly to the structure), the double-layer heterostructures and bilayers [46–48]. The excitonic gap formation and condensation has been examined also in the bilayer graphene structures [49–57]. Namely, the bilayer graphene is very promising for the optoelectronic applications due to its unique gate-controllable band structure properties [58]. The imposition of the external electrical field can tune the bilayer graphene from the semimetal to the semiconducting state. Nevertheless, the excitonic condensation in the bilayer graphene structures remains controversial in the modern solid-state physics because of the complicated nature of the single-particle correlations in these systems [49–57]. It has been shown recently [59] that the critical temperature, which describes the transition from the condensate state to the normal state in graphene double-layer structures, can be very high due to the extremely small effective mass of excitons. The coherence in exciton BEC condensates survives at the very high temperatures. An analog conclusion has been drawn in Ref. [56,57], concerning the bilayer graphene, where the condensate evolution has been analyzed as a function of the interlayer Coulomb interaction parameter in the BLG. The excitonic condensation has been realized experimentally in the double bilayer graphene heterostructure, in the strong quantum Hall regime, with the help of a combination of the Coulomb drag and current counterflow measurements [60]. They have also found the evidence of strong interlayer coupling between graphene layers, thanks to the observation of quantized Hall “drag plateau”. The zero-valued longitudinal resistance, measured there, confirms the dissipationless (friction-free) nature of the electron-hole condensate state. A quite simple experimental way to observe the excitonic condensate states in the bilayer structures is related to the possibility of engineering of a spatially confined excitonic condensates in the potential traps, and the investigation of the Josephson tunneling effects for excitons [61–64], related to the tunneling between two trapped Bose condensates [65] that possess the macroscopic phase coherence. The excitonic Josephson tunneling effects and thermal transport properties in the electron-hole type double-layer graphene junctions, separated by a dielectric layer, have been recently considered in Refs. [66,67]. In the present paper, we study the excitonic tunneling effects in the tunnel junction based on the AB-stacked bilayer graphene structures. We suppose the presence of the macroscopic phase coherent regime with the well-defined condensates phases and amplitudes and we consider only the local, on-site, interlayer excitonic pairing in 123 Journal of Low Temperature Physics (2019) 194:325–359 327 each side of the junction. We treat the local intralayer and interlayer Coulomb interactions within the bilayer Hubbard model at the half-filling (in each layer of the BLG). Supposing the electronic bilayers, without the initial optical pumping mechanism, we study the normal and excitonic tunneling currents through the BLG/I/BLG junction for different values of the external gate voltage, applied to the heterostructure, including the dc limit (i.e., V = 0) when a dc excitonic Josephson current passes through the tunnel junction. We will assume the half-filling regime in each layer of the BLG structures, even in the presence of the applied gate potential, thus by supposing that not considerable changes occur in the electron density during the adiabatic switching of the external potential. Here, by the adiabatic switching, we mean the very slow changes of the external switching potential applied to the jun (...truncated)