Second-Order Josephson Effect in Excitonic Insulators

We show that in electron-hole bilayers with excitonic orders arising from conduction and valence bands formed by atomic orbitals that have different parities, nonzero interlayer tunneling leads to a second-order Josephson effect. This means the interlayer electrical current is related to the phase of the excitonic order parameter as $J=J_c\sin 2\theta$ instead of $J=J_c\sin \theta$ and that the system has two degenerate ground states at $\theta =0,\pi$ that can be switched by an interlayer voltage pulse. When generalized to a three dimensional stack of alternating electron-hole planes or a two dimensional stack of chains, the ac Josephson effect implies that electric field pulses perpendicular to the layers and chains can steer the order parameter phase between the two degenerate ground states, making these devices ultrafast memories. The order parameter steering also applies to the excitonic insulator candidate ${\mathrm{Ta}}_2{\mathrm{NiSe}}_5$.

Figure 1

(a) Schematics of the electron-hole bilayer showing electrons ($-$) and holes ($+$) and the interlayer current-phase relation in the excitonic insulating phase. (b) False color representation of the free energy on the plane of a complex order parameter where lower energy appears bluer. (c) Solid curve: time dependence of order parameter phase after a voltage pulse ${\varphi }_a=-{\varphi }_0{e}^{-(t-t_0{)}^2/T_0^2}$ (dashed curve) computed from Eq. (4) with ${\mathrm{\Delta }}_p=14\mathrm{meV}$, $T_0=0.3\mathrm{ps}$, $\gamma =0.3{\mathrm{\Delta }}_p$, $D=2$, and $C=1$.

Figure 2

(a) Schematic of the 3D stack of alternating electron (blue) and hole (unshaded) planes with pairing order parameters labeled. (b) Schematic of the 2D stack of alternating electron and hole chains. The orange and blue dots represent atomic orbitals forming the conduction and valence bands. Their different parities lead to asymmetric interchain hopping $t/-t$ [30]. Arrows represent the spontaneous circulating currents. (c) The ground state band dispersion of the 2D stack. (d) The order parameter phase dynamics (black curve) and the Josephson current (blue curve) induced by an electric field pulse $E_z(t)=E_{\max }{e}^{-(t-t_0{)}^2/T_0^2}$ (red curve) implied by Eq. (9), with $\mathrm{\Delta }=10{\mathrm{\Delta }}_p$, $E_{\max }=3.55{\mathrm{\Delta }}_p/d$, and $T_0=0.5/{\mathrm{\Delta }}_p$.