Attraction of indirect excitons in van der Waals heterostructures with three semiconducting layers

We study a capacitor made of three monolayers of transition metal dichalcogenide (TMD) separated by hexagonal Boron Nitride (hBN). We assume that the structure is symmetric with respect to the central layer plane. The symmetry includes the contacts: if the central layer is contacted by the negative electrode, both external layers are contacted by the positive one. As a result a strong enough voltage $V$ induces electron-hole dipoles (indirect excitons) pointing towards one of the external layers. Antiparallel dipoles attract each other at large distances. Thus, the dipoles alternate in the central plane forming a 2D antiferroelectric with negative binding energy per dipole. The charging of a three-layer device is a first order transition, and we show that if $V_1$ is the critical voltage required to create a single electron-hole pair and charge this capacitor by $e$, the macroscopic charge $Q_c = eSn_c$ ($S$ is the device area) enters the three-layer capacitor at a smaller critical voltage $V_{c} < V_{1}$. In other words, the differential capacitance $C(V)$ is infinite at $V = V_{c}$. We also show that in a contact-less three-layer device, where the chemically different central layer has lower conduction and valence bands, optical excitation creates indirect excitons which attract each other, and therefore form antiferroelectric exciton droplets. Thus, the indirect exciton luminescence is red shifted compared to a two-layer device.