Superfluidity of Dipolar Excitons in a Double Layer of α - T3 with a Mass Term

Abstract

We predict Bose-Einstein condensation and superfluidity of dipolar excitons, formed by electron-hole pairs in spatially separated gapped hexagonal α-T3 (GHAT3) layers. In the α-T3 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-T3 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 Tc for superfluidity.

Keywords: Bose-Einstein condensation; dipolar exitons; superfluidity.

Figure 1

Schematic illustration of a dipolar excitonin a pair of GHAT3 double layers embedded in an insulating material.

Figure 2

The exciton binding energy ${\mathcal{E}}_b(\alpha ,\Delta ,D)$ for CV and CI excitons as functions of the gap $\Delta$ for chosen parameter $\alpha =0.6$ and interlayer separations $D=25\mathrm{nm}$. The lattice constant of $\alpha -T_3$ is $a=2.46$.

Figure 3

The exciton binding energy ${\mathcal{E}}_b(\alpha ,\Delta ,D)$ for CV and CI excitons as functions of the parameter $\alpha$ for chosen gap $\Delta =0.5ħv_F/a$ and interlayer separations $D=25\mathrm{nm}$.

Figure 4

The exciton binding energy ${\mathcal{E}}_b(\alpha ,\Delta ,D)$ for CV and CI excitons as functions of the interlayer separation D for chosen parameter $\alpha =0.6$ and gap $\Delta =0.5ħv_F/a$.

Figure 5

The effective masses of a dipolar exciton for CV and CI excitons as functions of the gap $\Delta$ for chosen $\alpha =0.6$ for (a) center-of-mass exciton mass M on the left panel and (b) reduced exciton mass $\mu$, on the right.

Figure 6

The effective masses of dipolar excitons for CV and CI excitons as functions of the hopping parameter $\alpha$ for chosen gap $\Delta =0.5ħv_F/a$ for (a) center-of-mass exciton mass M in the left panel and (b) reduced exciton mass $\mu$, on the right.

Figure 7

Plot of the sound velocity $c\equiv c_1$ versus $\alpha$ for chosen gap $\Delta =ħv_F/a$, interlayer separations $D=25\mathrm{nm}$ at fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 8

The sound velocity $c\equiv c_1$ versus the gap $\Delta$ for chosen parameter $\alpha$ = 0.6, interlayer separations $D=25\mathrm{nm}$ at the fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 9

The sound velocity $c\equiv c_1$ versus the interlayer separation D for chosen parameter $\alpha =0.6$ and gap $\Delta =0.5ħv_F/a$, at fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 10

The sound velocity $c\equiv c_1$ versus the concentrations $n_A$ and $n_B$ of A and B excitons, respectively, for chosen parameter $\alpha =0.6$ and gap $\Delta =0.5ħv_F/a$, at the fixed interlayer separation $D=25\mathrm{nm}$.

Figure 11

The mean-field phase transition critical temperature $T_c(n_A,n_B,\alpha ,\Delta ,D)$ versus the parameter $\alpha$ for chosen gap $\Delta =0.5ħv_F/a$, interlayer separations $D=25\mathrm{nm}$ at the fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 12

The mean-field phase transition critical temperature $T_c(n_A,n_B,\alpha ,\Delta ,D)$ versus the gap $\Delta$ for chosen parameter $\alpha =0.6$, interlayer separations $D=25\mathrm{nm}$ at the fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 13

The mean-field phase transition critical temperature $T_c(n_A,n_B,\alpha ,\Delta ,D)$ versus the interlayer separation D for chosen parameter $\alpha =0.6$ and gap $\Delta =0.5ħv_F/a$, at fixed concentrations $n_A=50\times {10}^{11}{\mathrm{cm}}^{-2}$ and $n_B=50\times {10}^{11}{\mathrm{cm}}^{-2}$ of A and B excitons, respectively.

Figure 14

Density plot for the mean-field phase transition critical temperature $T_c(n_A,n_B,\alpha ,\Delta ,D)$ versus the concentrations $n_A$ and $n_B$ of A and B excitons, respectively, for chosen parameter $\alpha =0.6$ and gap $\Delta =0.5ħv_F/a$, at the fixed interlayer separation $D=25\mathrm{nm}$.