Electrical Reservoirs for Bilayer Excitons

The ground state of two-dimensional (2D) electron systems with equal low densities of electrons and holes in nearby layers is an exciton fluid. We show that a reservoir for excitons can be established by contacting the two layers separately and maintaining the chemical potential difference at a value less than the spatially indirect band gap, thereby avoiding the presence of free carriers in either layer. Equilibration between the exciton fluid and the contacts proceeds via a process involving virtual intermediate states in which an unpaired electron or hole virtually occupies a free carrier state in one of the 2D layers. We derive an approximate relationship between the exciton-contact equilibration rate and the electrical conductances between the contacts and individual 2D layers when the contact chemical potentials align with the free-carrier bands, and explain how electrical measurements can be used to measure thermodynamic properties of the exciton fluids.

Figure 1

Schematic illustration of a 2D material heterojunction capable of supporting a spatially indirect exciton condensate, and of an electrode pair that can act as a reservoir for spatially indirect excitons.

Figure 2

Schematic band diagrams for the vertical vdW heterostructure systems of interest for the case of (a) zero applied bias and (b) a finite bias $V_b$ satisfying indirect gap $E_g>e{V}_b>{\mu }_{\mathrm{ex}}^0$ where ${\mu }_{\mathrm{ex}}^0$ is the energy of an isolated spatially indirect exciton. In case (b), the filled (empty) black circles represents electrons (holes) and the gray (pink) circle represents a virtual state in the $T(B)$ layer. 1 and 2 label two possible tunneling paths as discussed in detail in the text.

Figure 3

Equilibrium electrode densities (a) and exciton densities (b) at different values of $\beta =g_{XC}/g_H$, where $g_H$ is fixed with its value set by choosing $d_{\mathrm{DL}}=1\mathrm{nm}$. These results were calculated for $d_S=d_D=d_{\mathrm{DL}}$ spatially indirect gap $E_g(0)=1.1\mathrm{eV}$, and exciton binding energy $E_b=0.2\mathrm{eV}$. The extended (colored) dashed line represents electrode densities in the case in which the TMD double layer that hosts excitons is absent. The vertical gray dashed lines indicates the threshold voltage at which the dual electrodes establish a reservoir for excitons. Inset: Threshold voltage as a function of $d_S/d_{\mathrm{DL}}$. The dashed line indicates the zero electrode density limit of the isolated exciton energy ${\mu }_{\mathrm{ex}}^0$.

Figure 4

Differential conductance as a function of (a) $x=\omega C_{\mathrm{DL}}/{\overline{G}}_{\mathrm{ex}}$ and (b) bias voltage $V_b$. The dashed line and the dash-dotted line in (a) shows the linear relation at low and high frequency limits, respectively.