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The effect of the exchange interaction on the vibrational properties and onthe electron-phonon coupling were investigated in several recent works. In mostof the case, exchange tends to enhance the electron-phonon interaction,although the motivations for such behaviour are not completely understood. Herewe consider the class of weakly doped two-dimensional multivalleysemiconductors and we demonstrate that a more global picture emerges. Inparticular we show that in these systems, at low enough doping, even a moderateelectron-electron interaction enhances the response to any perturbationinducing a valley polarization. If the valley polarization is due to theelectron-phonon coupling, the electron-electron interaction results in anenhancement of the superconducting critical temperature. We demonstrate theapplicability of the theory by performing random phase approximation and firstprinciples calculations in transition metal chloronitrides. We find thatexchange is responsible for the enhancement of the superconducting criticaltemperature in Li$_x$ZrNCl and that much larger T$_c$s could be obtained inintercalated HfNCl if the synthesis of cleaner samples could remove theAnderson insulating state competing with superconductivity.

Exchange Enhancement of the Electron-Phonon Interaction: the Case of Weakly Doped Two-Dimensional Multivalley Semiconductors