Cavity-Mediated Electron-Photon Superconductivity

We investigate electron paring in a two-dimensional electron system mediated by vacuum fluctuations inside a nanoplasmonic terahertz cavity. We show that the structured cavity vacuum can induce long-range attractive interactions between current fluctuations which lead to pairing in generic materials with critical temperatures in the low-kelvin regime for realistic parameters. The induced state is a pair-density wave superconductor which can show a transition from a fully gapped to a partially gapped phase—akin to the pseudogap phase in $\mathrm{high}\text{−}{T}_c$ superconductors. Our findings provide a promising tool for engineering intrinsic electron interactions in two-dimensional materials.

Figure 1

(a) A 2D electron gas depicted by yellow sites interacts with a cavity, whose electric field distribution is indicated in red. The electronic hopping amplitude $t$ is also sketched. (b) The exchange of virtual photons between electric intraband transitions creates an effective electron interaction. This is decomposed into two contributions as described in the main text: panel (b1) displays the effect of the lattice geometry on the cavity-mediated interaction across the first Brillouin zone as a function of the nesting vector $\overrightarrow{Q}$ and panel (b2) shows the interaction Eq. (4) in arbitrary units (arb. units). (c) Leading eigenvalues ${\nu }_i$ according to Eq. (5). (d) The eigenfunctions corresponding to eigenvalues ${\nu }_1$ and ${\nu }_2$ pertaining to a nesting vector $\overrightarrow{Q}=k_0(1,1)$ are shown along the positive quadrant of the Fermi surface, and positive and negative sectors of the wave functions are indicated. The color code is given on the right-hand side. ${\nu }_1$ corresponds to a singlet and ${\nu }_2$ to triplet symmetry. We used parameters appropriate for a 2D electron gas in GaAs: relative permittivity $\epsilon =13$, electron band mass $m^{*}=0.069m_e$, chemical potential $\mu =-3.98t$, lattice constant $a=5.6\AA$, and a cavity frequency ${\omega }_0=2\pi \times 5\mathrm{THz}$.

Figure 2

(a) The leading eigenvalue of Eq. (5) is plotted as a function of the electron density $n_e$, corresponding to chemical potentials $\mu \in [-3.999,-3.98]t$, and a cavity compression factor $A={10}^{-5}$. (b) Electron dispersion $E(k_x)$ along $k_x$, with $k_y=0$ (left) and $k_y=k_f$ (right) for low electron density $n_e\simeq 0.3\times {10}^{11}{\mathrm{cm}}^{-2}$, and ${\mathrm{\Delta }}_0={10}^{-3}t$. The axes $E(0)$ and $k_x=0$ are indicated for orientation. The system is fully gapped everywhere along the Fermi surface. (c) Same as (b) with larger electron density $n_e\simeq 2\times {10}^{12}{\mathrm{cm}}^{-2}$, and ${\mathrm{\Delta }}_0={10}^{-3}t$. It is gapped only along certain directions in reciprocal space.

Figure 3

(a) The leading positive eigenvalue of Eq. (5) is plotted as a function of the cavity enhancement $A^{-1}$, for $\mu =-3.99t$. (b) Absolute value of the difference between the leading ${\nu }_1$ and subleading eigenvalue ${\nu }_2$, normalized to the leading eigenvalue at $A={10}^{-5}$. At low enhancements, ${\nu }_1$ is negative (and the $s$ wave repulsive), resulting in large differences. The inset on the left sketches the effective interaction potential in this region in gray. The green wave function sketches the $s$-wave function which is strongly affected by the local repulsion, and thus suppressed energetically. Hence, the $p$ wave (red) is energetically favorable, and pairing occurs in the triplet channel. On the right, the repulsion is too weak to suppress the $s$ wave.