Superconducting dome in ferroelectric-type materials from soft mode instability

We present a minimal theory of superconductivity enhancement in ferroelectric-type materials. Simple expressions for the optical mode responsible for the soft mode transition are assumed. A key role is played by the anharmonic phonon damping which is modulated by an external control parameter (electron doping or mechanical strain) causing the appearance of the soft mode. It is shown that the enhancement in the superconducting critical temperature $T_c$ upon approaching the ferroelectric transition from either side is due to the Stokes electron-phonon scattering processes promoted by strong phonon damping effects.

Figure 1

Simplified model expressions for the optical mode energy (blue) and the inverse lifetime ${\tau }^{-1}$ (orange) upon approaching the soft mode instability as a function of a generic control parameter $n$. Here, ${\omega }_0=0.5,n_c=1$.

Figure 2

Plot of normalized critical temperature with damping $\mathrm{\Gamma }$ for different effective couplings $\overline{\lambda }$. For large values of $\overline{\mathrm{\Gamma }}$, there is no solution for superconductivity.

Figure 3

Plot of normalized critical temperature around the ferroelectric critical point $\overline{n}-{\overline{n}}_c$ for different effective couplings $\overline{\lambda }$.

Figure 4

Superconducting dome predicted by the anharmonic theoretical model with a dispersion of the soft phonon quadratic in momentum transfer, i.e., $\mathrm{\Omega }\left(q\right)={\omega }_0+\alpha q^2$, where coefficient $\alpha$ is the stiffness or “velocity.” $T_c$ is plotted as a function of the soft phonon control parameter $n$ upon approaching the soft mode instability at $n=n_c=1$. In each plot, curves are plotted for the following values of the optical energy ${\omega }_0$, from top to bottom: ${\omega }_0=0.3,0.4,0.5,0.6,0.7$. In each plot, the chemical potential $\mu$ is fixed to the value shown.