# Excitonic Tunneling in the AB-bilayer Graphene Josephson Junctions

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## 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, 2, 3, 4, 5, 6], and a prediction about the superfluidity has been given for a condensed excitonic state. Ulteriorly, it has been the subject of an intense theoretical studies [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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, 26, 27], in the systems of magnons [28] and cavity exciton polaritons [29, 30, 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, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 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, 47, 48].

The excitonic gap formation and condensation has been examined also in the bilayer graphene structures [49, 50, 51, 52, 53, 54, 55, 56, 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, 50, 51, 52, 53, 54, 55, 56, 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, 62, 63, 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 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 junction, which causes the appearance of the ac Josephson current in the junction. The adiabatic changes of the external gate potential are important in many contexts concerning the tunnel junctions. First of all, here the adiabaticity of the external potential is important for the non-perturbative influence of the gate potential on electron densities in the individual bilayer structures, on both sides of the junction. Furthermore, this leads to the possibility of examination of the permitted values of interlayer Coulomb interaction parameters and excitonic gaps for a fixed value of the external gate potential. The half-filling condition in each layer of the BLGs is directly related to the adiabaticity of the external gate potential in the sense that the electronic reconfiguration time \(\tau _\mathrm{rec}\) in the BLG system is much faster than the variation time of the external potential \(\Delta {\tau }_\mathrm{Field}\): \(\tau _\mathrm{rec}<<\Delta {\tau }_\mathrm{Field}\). Such instantaneous changes of the electron densities in the system lead to the well-defined Coulomb potentials, for each value of the external field configuration. On the other hand, we consider the explicit dependence of the condensates phase difference parameter on the external gate potential and we show that the excitonic dc Josephson current appears in the junction array at the zero value of the external gate potential. We show how a finite difference between the static phases of the coherent excitonic condensates in the BLG subsystems leads to the excitonic Josephson dc current through the tunnel junction even in the case of the non-interacting regime in the BLG systems. The adiabatic switching of the external gate potential could be very important in the context of the adiabatic graphene logic constructions similar to the superconductor logic constructions, which are actually under the strong theoretical and experimental investigations [68, 69, 70]. In such devices, the intrinsic switching speed of the junctions increases rapidly with the increase of the damping resistance, and the phase differences change more adiabatically, which results in a considerable reduction in the energy dissipation of the single logical gates.

Furthermore, we show also that any finite voltage leads to the ac Josephson current irrespective of the phases of coherent condensates in both sides of the junction and we calculate the ac Josephson current as a function of external gate potential, for different values of the interlayer Coulomb interactions in different sides of the junctions. We study the amplitude of the zero voltage Josephson dc current as a function of the interlayer Coulomb interaction parameters in the subsystems. The symmetric and asymmetric interaction cases have been considered straightforwardly. Also, we calculate the normal quasiparticle tunneling current and we show that normal tunneling in the BLG/I/BLG heterostructure is an interaction-protected process and the threshold frequency of the normal tunneling current strongly depends on the values of the Coulomb interaction parameters in the BLGs. We analyze the role of the charge neutrality point (CNP) on the behavior of the normal and excitonic tunneling currents.

## 2 The Bilayer Graphene Josephson Junction

### 2.1 Description of the Hubbard Interactions

*a*,

*b*and \(\tilde{a}, \tilde{b}\) (and their conjugates \(a^{\dag }, b^{\dag }\) and \(\tilde{a}^{\dag }, \tilde{b}^{\dag }\)) the annihilation (creation) fermionic operators corresponding to different sublattice sites

*A*,

*B*in the left bottom, and \(\tilde{A}\), \(\tilde{B}\) in the left top layer in the left BLG. Similarly, we denote by

*c*,

*d*and \(\tilde{c}, \tilde{d}\) (and their conjugates \(c^{\dag }, d^{\dag }\) and \(\tilde{c}^{\dag }, \tilde{d}^{\dag }\)) the annihilation (creation) fermionic operators corresponding to different sublattice sites

*C*,

*D*in the right bottom and \(\tilde{C}\), \(\tilde{D}\) in the right top layer in the right BLG. In Fig. 1, we have presented the schematic setup of our BLG/I/BLG junction (here

*I*represents the dielectric layer between the BLGs). We assume here that the layers, which have lattice sites

*A*,

*B*and

*C*,

*D*(bottom layer) are biased-\(V_{\ell }/2\) (with \(\ell =L,R\)), and the layers with the lattice sites \(\tilde{A}\), \(\tilde{B}\) and \(\tilde{C}\), \(\tilde{D}\) (top layers) \(V_{\ell }/2\), so that the potential difference between the two layers is \(V_{\ell }\). For the first treatment of such a junction, we suppose the half-filling condition satisfied in both BLG systems, i.e., we suppose that \(\langle n^{\ell '}_{\ell } \rangle =1\), where \(n^{\ell '}_{\ell }\) is the total particle number operator in each layer with \(\ell '=1,2\) of each BLG with \(\ell = L,R\). Note, also that we attached the number \(\ell '=1\) to the bottom layers and \(\ell '=2\) to the top layers, in the heterostructure. The non-interacting tight-binding part of the total tunnel junction Hamiltonian could be written in the usual form

*A*(in the bottom layer \(\ell =1\), in the left BLG) and \(\tilde{B}\) (in the upper layer \(\ell =2\), in the right BLG), and, similarly, it is not defined for the electrons on the lattice sites

*C*(in the bottom layer \(\ell =1\), in the right BLG) and \(\tilde{D}\) (in the upper layer \(\ell =2\), in the right BLG). Indeed, these lattice sites have not their local nearest neighbors sites on the opposite layer in the given BLG. Both interaction parameters \(U_{\ell }\) and \(W_{\ell }\) are positive because we consider, initially, the electronic layers in both BLG systems. Then, we will use the interaction representation for the fermions [71], in which the time dependence of the fermionic operators is given by the unperturbed Hamiltonian \({H}_{L 0}(V)+{H}_{R 0}+H_{iL}+H_{iR}\), where \({H}_{L 0}(V)\) is the tight-binding Hamiltonian of the left side of the junction in the presence of the external gate voltage applied to the BLG/I/BLG (see). We assume that the gate voltage

*V*(

*t*) drops across the barrier and, in general, the Fermi levels in the left- and right-BLG structures will relatively shift by an amount proportional to the potential drop across the junction, i.e., \(\bar{\mu }_{L}-\bar{\mu }_{R}=-eV(t)\). We should emphasize here that the Fermi levels \(\bar{\mu }_{L}\) and \(\bar{\mu }_{R}\) are the functions of the shifted effective chemical potentials in the system which depend on the intralayer and interlayer Coulomb interaction parameters introduced in Eq. (5). For the reasons that will be clear in the following sections, we have denoted by \(\bar{\mu }_{l}, l=L, R\) the exact Fermi levels in different sides of the tunnel junction. We suppose here the half-filling conditions for the total electron densities in each layer of the separate BLG structure and we assume that the influence of the external gate voltage on the charge densities in the layers is infinitesimally small, and the excitonic gap parameters \(\Delta _{L}=W_{L}\left\langle \tilde{a}^{\dag }_{i}b_{i} \right\rangle \) (in the left bilayer) and \(\Delta _{R}=W_{R}\left\langle \tilde{c}^{\dag }_{i}d_{i} \right\rangle \) will not get modified by the external perturbation, i.e., \(\delta {V}\nrightarrow \delta {n^{\ell '}_{\ell }}\nrightarrow \delta {\Delta }_{\ell }\), and consequently \(\delta {n^{\ell '}_{\ell }}=0\), \(\delta {\Delta }_{\ell }=0\). Such a non-perturbative effect on the electron densities and on the excitonic gap parameter permits to include properly the effect of the applied gate voltage on the excitonic properties in the system and do not affect the excitonic condensate state in the system. For a more sophisticated case that evolves the variation of the excitonic condensates states in the junction, one should include the influence of the gate voltage on the excitonic gap parameter and the half-filling assumption will be failed in this case. Such a treatment is out of the scope of the present paper. We will consider the hopping parameter \(\gamma _{0}=1\) as the unit of energy measure in the system, and we set \(e=1\), \(k_{B}=1\), \(\hbar =1\) in the paper.

### 2.2 The Tunneling Hamiltonian

*A*,

*C*) and (

*B*,

*D*), where the first letter in the pair (

*x*,

*y*) is the lattice site in the left BLG and the second letter is the lattice site which belongs to the right BLG) will be written in the form

*B*in the bottom layer (or similarly between the sites \(\tilde{C}\) and

*D*in the right BLG system). There are no excitons between the electrons at the sites \(\tilde{B}\) (\(\tilde{D}\)) in the top layer and the holes at the sites

*A*(

*D*) in the bottom layer of the left (right) BLG. Thus, considering the zig-zag outermost sites as the sites which are close to the junction contacts with the insulating barrier, the tunneling of the electrons is coming only from the top layer of the right BLG system. Another equivalent geometry for the local excitons in the right BLG is also possible when the electrons at the sites

*D*in the right bottom layer \(\ell =1\) are coupled with the holes at the sites \(\tilde{C}\) in the top layer \(\ell =2\). In this case, we should consider the rate of change of the electron number operator \(N^{\ell =1}_{R}(t)\) in the right bottom layer. Those two descriptions are equivalent. We can consider also the tunneling current coming from the rate of change of the hole number operator in the right bottom layer, but, in fact, this contribution is exactly the same as that given by Eq. (8) except the sign in the right-hand side. The sign changing occurs because of the hole type of particles in the right bottom layer. Thus, the rate of total current between the layers will be twice as higher as the tunneling current given in Eq. (8) above. Let us mention also that the consideration of both the electron and hole particle number operator changes in the right BLG has nothing to do with the true excitonic tunneling process, and it defines only the amplitudes of the normal single-particle and excitonic tunneling currents. In reality, the only electron tunneling from the right top layer is sufficient to consider the single-particle and excitonic tunnelings through the junction because the electron-hole recombination times is much large than the single-particle tunneling time \(\tau _\mathrm{rec}>>\tau _{T}\) (Here \(\tau _{T}\) is the single-particle tunneling time.) The electron tunneling time is of order of \(\tau _{T}\sim 10^{-16}\) sec [72], while the time of the electron and hole recombination processes (Auger recombination time for Coulomb scattering) in graphene is of order of \(\tau _\mathrm{rec}\sim 1.1 \times 10^{-12}\)\(\mathrm{sec}\) for electron-hole densities smaller than \(10^{12} \mathrm{cm}^{-2}\) [73]. In order to not double counting the single-particle and excitonic tunneling effects, we have not considered the rate of change of the left bottom electron number operator \(N^{\ell =1}_{L}(t)=\sum _{\mathbf{{k}},\sigma }(a^{\dag }_{\mathbf{{k}},\sigma }a_{\mathbf{{k}},\sigma }+b^{\dag }_{\mathbf{{k}},\sigma }b_{\mathbf{{k}},\sigma })\). Thus, as a result, we have the single exciton tunneling through the tunnel junction, from left to right. The expectation value \(\langle \dot{N}_{R}\rangle \) is given by \(\langle \dot{N}_{R}\rangle =Tr(e^{-\beta {H}}\dot{N}_{R})/Tr(e^{-\beta {H}})\), where \(\beta =1/k_{B}T\), and

*H*is the total Hamiltonian of the system

*H*. The Heisenberg equation of motion for \({N}^{\ell =2}_{R}\) is

*t*, we obtain \(\Delta {\Phi }_\mathrm{Upper}(t)\) and \(\Delta {\Phi }_\mathrm{Lower}(t)\). Furthermore, we will define the time-dependent phase difference parameter \({\Delta \varphi }(t)\) as

*A*(

*t*),

*B*(

*t*),

*C*(

*t*) and

*D*(

*t*). Namely, we write

### 2.3 The Normal Quasiparticle and Excitonic Tunneling

*N*(

*t*) and

*F*(

*t*) for the normal and excitonic counterparts, which are defined with the help of the Kadanoff–Baym Green’s functions [80]. For the normal function

*N*(

*t*), we have

*F*(

*t*), we get

*N*(

*t*) and

*F*(

*t*), given in Eqs. (22) and (23), and, also, we have introduced the single-particle spectral functions \(A^{\ell }_{X_{\ell }}(\mathbf{{k}},\omega )\) (\(X=\tilde{a}, \tilde{b}\) for \(\ell =L\) and \(X=\tilde{c}, \tilde{d}\) for \(\ell =R\)) and excitonic spectral functions \(A^{L}_{\tilde{a}b}(\mathbf{{k}},\omega )\), \(A^{R}_{\tilde{c}d}(\mathbf{{k}},\omega )\). We have supposed in Eqs. (22) and (23) a simple form of the tunneling probability \(t_{\mathbf{{k}},\mathbf{{p}}}=\delta _{\mathbf{{k}},\mathbf{{p}}}t\). For a simple treatment, this approximation is sufficient to consider the excitonic effects.

*K*-point in the Brillouin zone. From the form of the total fermionic action, derived there, it follows that the normal spectral functions in different layers in the left BLG are interconnected:

*X*(

*x*,

*y*,

*z*) is defined as \(X(x,y,z)=(4\gamma _{0}|x|)^{-1}\sqrt{\left( x^{2}-y^{2}-z^{2}\right) ^{2}-4y^{2}z^{2}}\). The density of states (DOS) function \(\rho (x)\) in Eq. (40) appears after transforming the \(\mathbf{{k}}\)-summation in Eq. (22) into the integration over the continuous variable, i.e., \(\sum _{\mathbf {k}}\ldots =\int dx\rho (x)\ldots \). The DOS, in the non-interacting graphene layer, is defined as

*d*, in Eq. (48), refers to the carbon–carbon distance in the graphene layers. Beyond the Dirac’s approximation, the DOS can be analytically expressed [58, 82, 83] as

*f*(

*x*,

*y*,

*z*) in the denominators in Eq. (40) read as:

## 3 The Numerical Results and Discussions

### 3.1 The Normal Quasiparticle Tunneling

*V*and for two different limits of the right-BLG interlayer Coulomb interaction parameter \(W_R\): the weak and intermediate regime, starting from \(W_R=0\) (solid black curve), \(W_R=\gamma _0\) (solid blue curve), \(W^{*}_R=1.25\gamma _0\) (bold-dashed darker green curve) and \(W_R=1.5\gamma _0\) (dashed darker yellow curve) (the value \(W_R=1.5\gamma _0\) is chosen very close to the CNP value \(W^{C}_{R}=1.48999\gamma _0\)) and the high interaction limit, when \(W_R=1.8\gamma _0\) (dashed darker blue curve), \(W_R=2\gamma _0\) (solid green curve), \(W_R=3\gamma _0\) (dot-dashed darker red curve) and \(W_R=5\gamma _0\) (solid red curve). In all plots, in Fig. 2, we have set the intralayer Coulomb interaction parameters \(U^{\eta }_{\ell }\) equal and \(U^{\eta }_{\ell }=2\gamma _0\), for both BLGs \(\ell =L,R\) and for all layers in the individual BLGs, i.e., \(\eta =1,2\). We denoted by \(W^{*}_{R}\) the value of the interlayer Coulomb interaction parameter at which the excitonic gap parameter is maximal: \(\Delta _{R}=\Delta ^\mathrm{max}_{R}\). The interlayer interaction parameter in the left BLG is fixed at the value \(W_L=0.5\gamma _0\), for all curves in Fig. 2. We see first of all, in Fig. 2, that the normal quasiparticle tunneling through the BLG/I/BLG heterojunction, accompanied with the excitonic pair formations in the system, is a threshold process, and the threshold frequency \(\omega _0=(e/\hbar )V\) of the external field depends on the relative values of the Coulomb interaction parameters \(W_L\) and \(W_R\) at different sides of the construction. We also see that when augmenting the parameter \(W_R\) in the interval \(W_R\in [0,1.5\gamma _0]\), the curves, corresponding to the positive part of the current function \(I_{n}(V,T)\), are shifting into left and the intensity of curves is increased. In turn, the threshold values of the normal tunneling current are also shifting left. Starting from the upper bound (UB) critical CNP value of \(W_R\) (i.e., \(W_R\ge W^{C}_{R}(UB)\)), related to the upper bound solution of the chemical potential in the BLG (see in Ref. [56, 57]), the normal tunneling current is shifting right, on the

*V*-axis. Nevertheless, the same is not true for the negative part of the tunnel current. We see, in Fig. 2, that all curves of the negative part of the quasiparticle tunneling current are displacing to the right when increasing the interlayer Coulomb interaction parameter in the interval \(W_R\in [0, 5\gamma _0]\). Only a large jump of the threshold voltages occurs when passing across the lower bound (LB) CNP value \(W^{C}_{R}(LB)\), related to the lower bound chemical potential at the CNP. We observe also that for a very large disbalance between the values of the parameters \(W_L\) and \(W_R\), the additional low-frequency peaks appear in the positive part of the current spectrum, and the threshold frequency values of

*V*are gradually decreasing in these cases. We see also that the amplitudes of the low-frequency peaks are increasing with \(W_R\). It is interesting to note that for \(W_R=5\gamma _0\) (see the solid red curve in Fig. 2), the threshold frequency \(\omega _0\) in the positive part of the normal current is of the order of \(\omega _0\sim 2\gamma _1=0.256\gamma _0=0.76\) eV.

### 3.2 The Excitonic Josephson Tunneling

*V*(\(V=0.5\gamma _0\) and \(V=\gamma _0\), in the picture), even for the case \(\Delta \varphi _{0}=0\) (see the blue dotted lines in Fig. 6 with the square plot-markers). The additional phase difference \(\Delta {\varphi }_{0}=\pi /2\) only amplifies the excitonic tunneling current amplitude and leads to a phase shift (see the black dashed lines in Fig. 6). We observe also that the amplitude of the tunnel current decreases when increasing the parameter \(W_R\) (see the top panel with \(W_R=0\) and the middle panel with \(W_R=0.5\gamma _0\), in Fig. 6). In the bottom panel in Fig. 6, we have chosen the larger value of the applied gate potential \(V=1.5\gamma _0\), and \(W_R=\gamma _0\). We see that the oscillations of the tunnel current are multiplied in this case within the same time interval. This effect is clearly seen in Fig. 7, where different values of the applied gate voltage are considered straightforwardly, and the interaction parameter \(W_R\) is fixed at the value \(W_R=0\), in correspondence with the plots of the excitonic Josephson current in the upper panel in Fig. 6. We see that when multiplying the value of

*V*by an integer number \(V'=nV\), where \(n=1,2,3,4\), we have for the current wavelength \(\lambda '=\lambda /n\), thus a relation of type \(\lambda {V}=\text {const}\), between the applied gate potential and the current wavelength, emerges naturally. In Fig. 8, the same function \(I_\mathrm{Exc}(t)\) is presented for the case of the fixed applied gate potential \(V=0.5\gamma _0\), and for different values of the right-BLG interlayer Coulomb interaction parameter \(W_R\), below the critical CNP value \(W^{C}_{R}=1.48999\gamma _0\). The condensates phase difference is fixed at the value \(\Delta {\varphi }_{0}=\pi /2\). We see, particularly, that when augmenting the interaction parameter \(W_R\) (and keeping at the same time \(W_L\) fixed at \(W_L=0.5\gamma _0\)) the amplitude of the excitonic tunneling current is decreasing considerably. The zeros of the current function do not shift in their positions, contrary to the case presented in Fig. 7, where the shift of the zeros is caused by the reduction of the current wavelength in the junction. Thus, we realize that the interlayer Coulomb interaction in the right BLG affects only the current amplitudes, while the changes of the applied gate potential modify principally the frequency of the excitonic current and have not a significant effect on the current amplitudes. Next, in Fig. 9, we have shown the time dependence of the excitonic tunneling current for the values of \(W_R\) above the UB charge neutrality point \(W^{C}_{R}(UB)\). Contrary to the case, given in Fig. 8, the amplitude of the Josephson tunneling current is increasing with \(W_R\). This result is related again to the behavior of the chemical potential and Fermi energies in the BLGs (see in Ref. [56, 57]). In order to compare the results in Figs. 8 and 9, we kept the curve for \(W_R=0\), in both cases.

*V*at the zero phase difference (see the solid black curve in the lower panel, in Fig. 10). Contrary, for \(\Delta {\varphi }_{0}=\pi /2\) and at \(t=0\) the excitonic current has been developed in the system (see the solid black curve, in the upper panel, in Fig. 10).

The I–V characteristics of the excitonic Josephson current, for the case of the equal values of the interlayer interaction parameters, i.e., \(W_L=W_R=W\), is shown in Fig. 11. Different values of *W* are considered in the picture: \(W=0.5\gamma _0\) (solid black curve), \(W=0.8\gamma _0\) (solid blue curve), \(W=\gamma _0\) (solid yellow curve), \(W=W^{*}=1.25\gamma _0\) (solid red curve), \(W=1.3\gamma _0\) (dot-dashed darker green curve), \(W=1.4\gamma _0\) (large-dashed green curve), and \(W=W^{C}(LB)=1.48999\gamma _0\) (bold-dashed darker yellow curve). We observe that up to the value \(W^{*}=1.25\gamma _0\), which corresponds to the maximum of the excitonic gap parameters in the BLGs, the amplitude of the excitonic Josephson current is increasing (including the dc values at \(V=0\)). Furthermore, for \(W>W^{*}\), the amplitudes are continuously decreasing, for *W* up to the LB CNP value \(W=W^{C}(LB)=1.48999\gamma _0\). The further increase of *W*, above the LB CNP, leads to a drastic decrease (of about of one order of magnitude) of the excitonic tunnel current amplitude, and this is shown in Fig. 12. It is important to concentrate on another principal difference between the results presented in Figs. 11 and 12. This concerns the first deepest minima of the excitonic Josephson current. In Fig. 11, those minima appear for relatively small voltages, and the positions of the positive (negative) minima are shifting to right (left), for \(W\le W^{*}=1.25\gamma _0\). Then, with a further increase of *W* in the interval \(W^{*}<W<W^{C}(LB)=1.48999\gamma _0\), they are slightly shifting to the left (right). On the other hand, the first minima for the curves in Fig. 12, for \(W^{C}(UB)\le W\le 3\gamma _0\), appear for very large values of *V*, and the positions of positive (negative) minima are continuously shifting to right (left) in this case. Therefore, the very large values of *V* are very promising to observe the ac excitonic Josephson current in the system. Experimentally, this could be achieved with the appropriate choice of the left and right gate voltages, which will change the interlayer Coulomb interactions in the BLGs, until the expected effect takes place. In Fig. 13, we have presented the excitonic tunneling current for the asymmetric values of the interaction parameters \(W_L\) and \(W_R\) and we see how the increase of \(W_R\) in the right BLG, above the symmetric value \(W_{R}=0.5\gamma _0\) (the parameter \(W_L\) is fixed at the value \(W_L=0.5\gamma _0\) for all values of \(W_{R}\)), leads to the tunneling current transfer into right, on the positive axis of *V*. It is interesting and straightforward to consider the special case of the non-interacting BLGs and the excitonic Josephson current through the junction in this particular case. This result is shown in Fig. 14. We see that although the zero interaction limit, i.e., when \(U_{\ell }=0\) and \(W_{\ell }=0\), the excitonic Josephson current still present in the BLG/I/BLG junction, and the values of it are comparable to the case when \(W>W^{C}(UB)\), shown in Fig. 12, above. The reason of such excitonic effect is related to the presence of the excitonic condensates states, and the principal mechanism for the formation of the excitonic condensates phase is due to the interlayer hopping between the layer in the individual BLG systems (see in Ref. [56, 57]). In the context of the excitonic Josephson current, discussed here, the finite value of the excitonic current follows from the expressions of the functions \(I_{J_1}(V,T)\) and \(I_{J_2}(V,T)\), given in Eqs. (50) and (51), in Sect. 2.3. Indeed, the expression of the current amplitude \(I_{J_{1}}(V,T)\), in Eq. (52) is proportional to the product of the anomalous spectral functions \(A^{L}_{\tilde{a}b}(\mathbf{{k}},\omega )\) and \(A^{R}_{\tilde{c}d}\), given in Eq. (34); in Sect. 2.3, thus, the function \(I_{J_{1}}(V,T)\) is proportional to the product \((\gamma _{1}+\Delta _{L})(\gamma _{1}+\Delta _{R})\). The same is true for the function \(I_{J_{2}}(V,T)\), given in Eq. (53). At the zero interlayer coupling limit \(\Delta _{L}=\Delta _{R}=0\) while the functions \(I_{J_{1}}(V,T)\) and \(I_{J_{2}}(V,T)\) remain finite, due to the finite interlayer hopping amplitude \(\gamma _1\). Thus, in the non-interacting regime, the fundamental state of BLG systems in both sides of the junction is represented as the phase coherent macroscopic state of the excitonic condensate and the principal mechanism for such a state is attributed to the finite interlayer hopping between the layers in the BLGs. Another spectacular effect, related to the non-interacting regime in the BLGs, is the presence of a finite solution of the chemical potential and the Fermi energy in the separate BLGs (see in Ref. [56, 57], for more details).

## 4 Conclusion

We have calculated the normal quasiparticle and excitonic Josephson tunneling currents in the BLG/I/BLG heterostructure with the half-filled, AB-stacked bilayer graphene structures at different sides of the junction. By using the *S*-matrix approximation, we have derived the analytical expressions of both types of tunneling currents: normal quasiparticle and excitonic. The role of charge neutrality point has been discussed in details. Particularly, we have shown how the current spectrum is changing when passing through the CNP point of the interlayer Coulomb interaction parameters, in different sides of the junction. It has been shown that the normal quasiparticle tunneling, in the junction, is a threshold process, independently of the values of the interlayer Coulomb interactions in the BLGs and the threshold value of field frequency could be modified by changing the interaction parameter in one BLG, while keeping it fixed, at the same time, in another BLG. The formation of the resonant tunneling peaks in the normal tunneling spectrum has been analyzed in details by considering the vicinity of the charge neutrality point in the right BLG. The very large values of the threshold frequency have been obtained for the case of the interaction-symmetric junction \(W_L=W_R\), below and above the charge neutrality point. Particularly, from the form of I–V spectrum, in the case of the interaction-asymmetric junction, it is clear that the excitonic insulator state and the excitonic condensate states are two different states of matter in the BLGs. We have shown that the low-frequency peaks appear from the excitonic tunneling in the condensates regime in BLGs, and at the large values of the Coulomb interaction parameter in the right BLG (when the excitonic insulator state breaks down). The temperature dependence of the threshold values of field frequency has been derived for a very large interval of temperature: from zero up to very high temperatures.

Furthermore, the time dependence of the excitonic Josephson tunnel current has been studied in details for different limits of interlayer Coulomb interaction parameter in the right BLG, and the excitonic Josephson dc effect has been found for the case \(\Delta {\varphi }_{0}\ne 0\) and \(V=0\). Ulteriorly, an ac excitonic Josephson effect has been shown in the junction for nonzero values of the applied gate potential. It has been shown that the phase difference between the excitonic condensates in the BLGs, only amplifies the tunneling current when keeping *V* constant. It has been shown that a relation of type \(\lambda {V}=\text {const}\) is valid in the junction when changing the applied gate potential by the integer values. The time dependence of the excitonic Josephson tunneling current has been analyzed by considering different values of the right-BLG interaction parameter. The \(I-V\) characteristics of the excitonic Josephson effect has been found for both interaction-symmetric and interaction-asymmetric cases, and the effects of the charge neutrality point have been discussed in details. It has been shown that the very large values of *V* are promising to observe the ac excitonic Josephson current in the system. Finally, a particular case of the non-interacting BLGs junction has been considered separately, and the excitonic Josephson tunneling current has been calculated for this case. As a result, we have shown that the nonzero excitonic current is present in this case with the amplitudes comparable to that of the amplitudes of I–V spectrum at the high symmetric values of *W* . The principal mechanism for the excitonic condensate states and the nonzero excitonic tunneling current, in this case, is attributed to the interlayer hopping between the layers in the individual BLG systems.

We have presented a complete theory of the bilayer graphene-based tunnel junction, and the results, presented here, could represent a veritable framework on which the experimental setup could be made and the results could be compared straightforwardly. The theoretical results in the paper are especially important in the context of the long-standing problem about the excitonic condensation and pair formation in the BLGs and double-layer graphene systems. This is especially important for the theoretical understanding of the nature of the EI state and the coherent excitonic condensate states in such systems: their similarities and differences. The Josephson effect studied in the considered system could be furthermore an important construction in order to build the graphene-based quantum interference devices, which will bring a new idea about the ultra-sensitive magnetometers and voltmeters and will be more sensitive than the usual superconductors-based Quantum Interferometer Devices SQUIDs, due to the exceptional mobility of the electrons in graphene. The excitonic Josephson junctions would be also promising in the context of engineering a new type of ultrafast electronic circuits blocks, which could form a digital logic unit, in the modern ultrafast computers and fast electronics. The results, presented here, could be also important for the building of the new type of adiabatic graphene logic constructions based on the Graphene Quantum Interferometer Devices (Graphene-QUIDs) analog to the Josephson flux transfer logic devices such as the Adiabatic Quantum Flux Parametron (AQFP).

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