Polariton-induced superconductivity in two-dimensional metals

Abstract

The electronic properties of two-dimensional (2D) metals are altered by changes in their three-dimensional dielectric environment. In this Letter we propose that superconductivity can be induced in a 2D metal by resonant coupling between its plasmonic collective modes and optical phonons in a nearby polar dielectric. Specifically, we predict that relatively high-temperature superconductivity can be induced in bilayer graphene twisted to an angle somewhat larger than the magic value by surrounding it with a THz polar dielectric. Our conclusions are based on numerical solutions of Eliashberg equations for massless Dirac fermions with tunable Fermi velocities and Fermi energies, and can be understood qualitatively in terms of a generalized McMillan formula.

Figures (9)

• Figure 1: Polariton enhancement of superconductivity in a MDF 2DES. The inset shows a sketch of the device analyzed in this work. The MDF 2DES is encapsulated between two slabs of polar material with thicknesses $d_{\rm t}$ and $d_{\rm b}$. In the main panel we plot the ratio $T_{\rm c}/ T_{{\rm F}}$, where $T_{\rm F}$ is the MDF 2DES Fermi temperature at velocity $v$, as a function of the longitudinal optical phonon frequency $\hbar\omega_{\rm L}$. The results in this plot have been obtained by setting $d_{\text{t}}=d_{\text{b}}\to\infty$, $\epsilon_\infty=3$, carrier density $n = e11cm^{-2}$, and Fermi velocity $v=v_0/2$ ($\theta=2.2^\circ$ and $T_{\rm F} = 150~{\rm K}$ in TBG). The two curves refer to different values of the phonon oscillator strength $\zeta \in [0,1]$ and hence to different values of $\epsilon_0$. We argue in the main text that these $T_{\rm c}$'s estimates are most reliable near the curve maxima.
• Figure 2: Resonantly enhanced electron-electron interactions. The $s$-wave (i.e. $\ell=0$) averaged interaction $I(k_{\text{F}},1.1k_{\text{F}},\mathrm{i} \Omega)$, defined in Eq. (\ref{['eq:avg-int-l']}), is plotted as a function of the frequency $\Omega$ on the imaginary-frequency axis. Results in this figure have been calculated for the same parameters as in Figs. \ref{['fig:one']}. In particular, $\epsilon_\infty=3$, $\epsilon_0=24$, and the energy of the longitudinal optical phonon mode $\hbar\omega_{\text{L}}$ is reported in the legend. The thin vertical lines are guides to the eye, representing the energy scales discussed in the main text, $\hbar\omega_{\text{L}}=2meV$, $0.2meV$, $200meV$, and the Fermi energy $\varepsilon_{\text{F}}$. The (green) dotted line is the averaged interaction in the absence of phonons, i.e. calculated with permittivity $\epsilon(\omega)\equiv\epsilon_\infty$.
• Figure 3: Critical temperature $T_{\text{c},\infty}$ (in units of the Fermi temperature $T_{\rm F}$) for plasmon-mediated superconductivity in MDF 2DESs. The results in this figure have been obtained by encapsulating the MDF 2DES between two semi-infinite dielectric slabs of permittivity $\epsilon_\infty$. Black circles (red squares) represent numerical results for a MDF 2DES with velocity $v=v_0/2$ ($v=v_0$) and $N_{\text{f}}=8$ ($N_{\text{f}}=4$) flavors, where $v_0=e6m/s$ is the single-layer graphene Fermi velocity. ($N_{\text{f}}$ enters the problem via the bare polarization function $\chi_0(q,\omega)$.) Thin solid lines illustrate exponential depedence on $\epsilon_\infty$. Our numerical results depart from simple exponential dependence at weak coupling, i.e. for large $\epsilon_\infty$.
• Figure 4: Plasmon-phonon polaritons. The spectral function, $- W(q,\omega)$, of the dynamically-screened electron-electron interaction (in units of $2\pi e^2/k_{\text{F}}$) is plotted as a function of wave vector $q$ and real frequency $\omega$. The blue and red solid lines represent the upper and lower polaritons $\omega_{\pm}(q)$ outside the intra-band electron-hole continuum of the 2DES. The solid black line represents the dispersion of the bare plasmon $\omega_{\text{p}}(q)$ of the 2DES. The dotted black lines represent the longitudinal and transverse optical phonon modes of the dielectric slabs ($\omega_{\text{L}}>\omega_{\text{T}}$). In this plot the energy of the longitudinal optical phonon mode has been set at $\hbar\omega_{\text{L}}=2meV$. All the other parameters are as in Fig. \ref{['fig:one']}. Panel (a) refers to $\epsilon_0=6$ ($\zeta=\sqrt{2}/2$)---see data labelled by red circles in Fig. \ref{['fig:one']}. Panel (b) to $\epsilon_0=24$ ($\zeta=\sqrt{7/8}$)---see data labelled by blue squares in Fig. \ref{['fig:one']}.
• Figure S1: Averaged interaction $I(k_{\text{F}},k',\mathrm{i}\Omega)$ as a function of the frequency $\Omega$, for different values of $k'$. Solid lines represent the average interaction in Eq. \ref{['eq:avg-int-l']}, while dashed lines represent the piecewise constant approximation. The system is sketched in Fig. \ref{['fig:one']}. The dielectric has $\epsilon_0=24$, $\epsilon_\infty=3$, and the frequency of the transverse optical phonon mode is $\hbar\omega_{\text{T}}=10^{-2}\varepsilon_{\text{F}}$. The parameters are: $d_{\text{b}}=d_{\text{t}}\to\infty$, $n = e11cm^{-2}$, $v=v_0/2$ ($\theta=2.2^\circ$).