Calculation of overdamped $c$-axis charge dynamics and the coupling to polar phonons in cuprate superconductors

In our recent paper we presented empirical evidence suggesting that electrons in cuprate superconductors are strongly coupled to unscreened $c$-axis polar phonons. In the overdoped regime the $c$ axis metallizes and we present here simple theoretical arguments demonstrating that the observed effect of the metallic $c$-axis screening on the polar electron-phonon coupling is consistent with a strongly overdamped $c$-axis charge dynamics in the optimally doped system, becoming less dissipative in the overdoped regime.

Figure 1

(Color online) The experimental $c$-axis optical conductivity in optimally doped (OP) Bi2201 from Ref. 16 (h). Various models for the optical conductivity of overdoped (OD) Bi2201 (e)–(g). The corresponding Eliashberg functions (solid line) and electronic self-energies (dashed line) assuming that the predominant electron-phonon coupling is due to the unscreened $c$-axis phonons (a)–(d); experimental data in symbols are from Ref. 4. Parameters: (a)–(c) ${\omega }_{\mathit{scr},c}\approx 60\mathrm{meV}$; (a) $\Gamma <{\omega }_{p,c}$, $\Gamma =45\mathrm{meV}$, and ${\omega }_{p,c}=190\mathrm{meV}$; (b) $\Gamma >{\omega }_{p,c}$, $\Gamma =430\mathrm{meV}$, and ${\omega }_{p,c}=350\mathrm{meV}$; and (c) $\Gamma \approx {\omega }_{p,c}$, $\Gamma =180\mathrm{meV}$, and ${\omega }_{p,c}=250\mathrm{meV}$. (d) OP case, $\Gamma =700\mathrm{meV}$ and ${\omega }_{p,c}=250\mathrm{meV}$.