Carrier Density Dependence of Superconducting Transition Temperature in Electron-doped $\rm{SrTiO_3}$ Based on the First-principles Calculations
Electron-doped strontium titanate $\rm{SrTiO_3}$, known to be one of the most dilute superconductors, is investigated on the basis of the first-principles calculations. When the carrier density n decreases, the frequencies of the ferroelectric optical phonons near the $Γ$-point monotonically decreases in the overdoped regime with $n<10^{20}/\rm{cm}^{3}$, while unphysical imaginary phonon frequencies due to ferroelectric instabilities appear in the underdoped regime with $n>10^{20}/\rm{cm}^{3}$. We estimate the superconducting transition temperature $T_{\rm{c}}$ by using the McMillan equation in the overdoped regime and find that $T_{\rm{c}}$ increases with decreasing n as consistent with experiments in the overdoped regime. Detailed analysis of the Eliashberg function reveals that the increases in $T_{\rm{c}}$ with decreasing n in the overdoped regime is mainly due to the contributions from the ferroelectric soft-mode optical phonons.