Fig. 4: ℓ₀, T_c and k_F.

a Doping evolution of ℓ₀=\(\frac{{v}_{TO}}{{\omega }_{TO}}\) for Nb-doped (blue points), La-doped (red points), reduced SrTiO₃ (in green points estimated from ref. ³⁰, see the text) and \(\frac{a}{2}\) (in gray dot line) where a = 3.9 Å is the lattice parameter of SrTiO₃. b Doping evolution of the Fermi momentum (k_F) in the three sub-bands i = 1,2 and 3. c Doping evolution of the superconducting critical temperature (Tc) according to^11,12,46. The gray areas in (a, c) mark the end of the superconducting dome concomitant with the saturation of ℓ₀ to about \(\frac{a}{2}\). d \({k}_{F}^{i}{\ell }_{0}\)vs.n for each sub-band. e Comparison of the carrier dependence of ℓ₀ and T_c in linear scale. f Sketch of the electron doping evolution of 2ℓ₀ (blue disk) and 2k_F (black dot arrows). With electron doping ℓ₀ decreases and saturates to about \(\frac{a}{2}\), the Ti-O distance, while k_F increases. The summit of the superconducting dome occurs when \({k}_{F}^{3}{\ell }_{0}\) is the largest, see the text. Error bars on T_c are from^11,12,46.