In cuprate superconductors, a pronounced maximum of superconducting Tcmax\documentclass[12pt]{minimal}
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\begin{document}$${T}_{c}^{max}$$\end{document} is observed in compounds that have an in-plane Cu–O distance aCu-O\documentclass[12pt]{minimal}
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\begin{document}$${a}_{Cu-O}$$\end{document} close to ∼1.92\documentclass[12pt]{minimal}
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\begin{document}$$\sim 1.92$$\end{document} angstroms. On the other hand, direct measurements of the electron–phonon coupling λ⟨ω2⟩\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \langle {\omega }^{2}\rangle$$\end{document} as a function of aCu-O\documentclass[12pt]{minimal}
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\begin{document}$${a}_{Cu-O}$$\end{document} show a clear linear correlation, implying that Tcmax\documentclass[12pt]{minimal}
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\begin{document}$${T}_{c}^{max}$$\end{document} is a strongly non-linear function of λ⟨ω2⟩\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \langle {\omega }^{2}\rangle$$\end{document}. Conventional superconductivity theories based on the electron–phonon interaction predict a monotonic dependence of Tcmax\documentclass[12pt]{minimal}
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\begin{document}$${T}_{c}^{max}$$\end{document} on λ⟨ω2⟩\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \langle {\omega }^{2}\rangle$$\end{document}, which makes them incompatible with the observed behavior. The observed crossover behavior as a function of λ⟨ω2⟩\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \langle {\omega }^{2}\rangle$$\end{document} suggests that Tcmax\documentclass[12pt]{minimal}
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\begin{document}$${T}_{c}^{max}$$\end{document} occurs at the crossover from weak to strong coupling, which is also associated with the onset of carrier localization. A coexistence, with a dynamical exchange of localized and itinerant carriers in a two-component superconductivity scenario are in agreement with the observed anomalous behavior and are suggested to be the key to understanding the mechanism for achieving high Tcmax\documentclass[12pt]{minimal}
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\begin{document}$${T}_{c}^{max}$$\end{document}.
