Long-range ferroelectric order in two dimensional excitonic insulators

Abstract

It is generally argued that Mermin-Wagner theorem excludes the possibility of long-range order in two dimensional bosonic systems at non-zero temperatures. In contrast, we show here that generic bilayer semiconductors could demonstrate true Bose-Einstein condensation of interlayer excitons. We show that the key requirements include (i) reduction of the interlayer band gap using an applied electric field so that excitons spontaneously appear in the ground state, (ii) band structure that allows for long-range electron-hole exchange interaction, and (iii) a finite magnetic field. Our results indicate that superfluidity and ferroelectric order can co-exist in two dimensional excitonic insulators.

Figures (6)

• Figure 1: (a) The van der Waals heterostructure that we analyze. The two Transition Metal Dichalcogenide (TMD) layers have type II band alignment and they are either in direct contact or are separated by monolayer hexagonal Boron Nitride (hBN). Voltages applied on top and bottom graphite gates are used to adjust the vertical electric field $E_z$. (b) Sketch of the band diagram around the K-valley. The spatially indirect band gap, $E_g$ can be tuned via electric field $E_z$ to reach $E_g < E_B$ to ensure that interlayer excitons appear spontaneously.
• Figure 2: Schematic illustration of the heterobilayer excitonic band structure in the presence of magnetic field induced Zeeman splitting $g\mu_B B$, where $g$ is the exciton $g$-factor, $\mu_B$ is the Bohr magneton and $B$ is the normal component of the magnetic field. Series of bound excitonic states are shown by the dashed ($1s$ state) and dotted ($2p$ and other higher-lying states) lines. Parabolas show the continuum states of electron-hole pairs. It is seen that the exciton insulator instability condition \ref{['cond:pairing']} is first met for $1s$ exciton in the $K_+$ valley.
• Figure 3: Spectrum of excitations. Only one bound state, $1s$, is shown. (a) Non-condensed system, $\Delta=0$. The excitations correspond to the exciton center of mass motion with parabolic dispersion (red parabola) and relative motion of electron-hole pairs in continuum (shaded region bounded by blue parabola). Here, we neglect the weak light-matter interaction. (b) Excitonic insulator state, $\Delta=0$. The relative motion continuum (shaded region bounded by blue parabola) is separated by the gap $\sqrt{E_g^2+4\Delta^2}$ from the condensate ground state. Red curve sketches the dispersion of the collective, Goldstone, mode. Orange region of small momenta is of interest for analysis of the condensate stability.
• Figure 4: Dispersion of elementary excitations of excitonic insulator with allowance for the light matter coupling. Open symbols show the dispersion calculated numerically, solid lines show analytical approximations; $x$-axis corresponds to the polarization of the condensate $\bm P \parallel x$, see text for details. (a) $\bm Q\parallel x$ ($\varphi=0$). Dashed curve with $\textcolor{blue}{c_s}Q$ asymptotics is shifted to allow for the $Q$-independent overall shift. (b) Small, but non-zero $\varphi=10^{-5}$. Dot-dashed and dashed curves show $Q$-linear asymptotics \ref{['light:like']} and \ref{['tr2']}, blue dot-dot-dashed line shows $Q^{3/2}$ law, Eq \ref{['tr1']}, and black solid line shows the interpolation \ref{['disper:interp']}. (c) $\bm Q\parallel y$ ($\varphi=\pi/2$). Inset shows the angular dependence of the energy with the curve calculated after Eq. \ref{['light:like']}.
• Figure 5: Detection of the exciton insulator state with the ferroelectric order. (a) Schematics of the TMD heterobilayer bandstructure in the $K_+$ valley that shows in addition to the nearest valence and conduction band the remote $c+2$ band of the opposite to the $c$-band chirality. The selection rules for the interband transitions are shown by arrows. (b) Interference of transitions for excitation or recombination of the high-lying exciton associated with the remote conduction band. In the presence of excitonic condensate, $a\ne 0$, the selection rules for such transition involve elliptical polarization.