Motivated by recent studies on ferroelectriclike order coexisting with metallicity, we investigate ferroelectric (FE) superconductivity in which a FE-like structural phase transition occurs in the superconducting state. We consider a two-dimensional $s$-wave superconductor with Rashba-type antisymmetric spin-orbit coupling (ASOC). Assuming a linear relationship between the polar lattice displacement and the strength of the ASOC, we treat the Rashba-type ASOC as a molecular field of FE-like order. It is shown that the FE-like order is induced by the magnetic field when the system is superconducting. Furthermore, we clarify the FE superconductivity in a low carrier density regime, which was recently discovered in doped ${\mathrm{SrTiO}}_3$. It is demonstrated that the FE superconducting state can be stable in this regime in the absence of the magnetic field. Our results open a way to control the electric polarization by superconductivity, that is, superconducting multiferroics.

• Received 17 March 2018
• Revised 23 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.024521