Phonon-mediated superconducting transitions in layered cuprate superconductors

A phonon-mediated $d$-wave BCS-like model is presented for a homologous series of layered cuprate superconductors. We show that the dependence of both the superconducting transition temperature $T_c$ and oxygen isotope exponent $\alpha$ on the doping level, number of $\mathrm{Cu}{\mathrm{O}}_2$ layers, pressure, and cation disorder in the model superconducting series $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ can be well reproduced within one single theoretical framework. We also find that the theoretical model accounts well for the superconducting transitions in other homologous series such as the bismuth- and thallium-based family. Our results suggest that the interlayer coupling plays an important role in the significant enhancement of $T_c$ and in the systematic reduction of $\alpha$ in a layered homologous series.

Figure 1

(Color online) The superconducting transition temperature $T_c$ versus the hole concentration $n_H$ in the homologous series of $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$.

Figure 2

(Color online) Variation of the superconducting transition temperature $T_c$ with the number of $\mathrm{Cu}{\mathrm{O}}_2$ layers in the optimally doped $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$. The experimental data points were taken from Ref. 11.

Figure 3

(Color online) The oxygen isotope exponent $\alpha$ versus the hole concentration $n_H$ in $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(n=1,2,3)$.

Figure 4

(Color online) Pressure dependence of the superconducting transition temperature $T_c$ in the optimally doped $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(n=1,2,3)$. Theoretical results are plotted by the curves. Circles, squares, and diamonds denote the experimental data points taken from Ref. 40 for the monolayer, bilayer, and trilayer compounds, respectively.

Figure 5

(Color online) Calculated oxygen isotope exponent $\alpha$ as a function of pressure in the optimally doped $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(n=1,2,3)$.

Figure 6

(Color online) Cutoff frequency ${\omega }_0$ dependence of both the superconducting transition temperature $T_c$ in the optimally doped $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(n=1,2,3)$.

Figure 7

(Color online) Cutoff frequency ${\omega }_0$ dependence of both the oxygen isotope exponent $\alpha$ in the optimally doped $\mathrm{Hg}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(n=1,2,3)$.

Figure 8

(Color online) The superconducting transition temperature $T_c$ and oxygen isotope exponent $\alpha$ versus the chemical potential $\mu$ relative to the optimal doping in layered cuprate superconductors ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ $\left[\mathrm{Bi}22(n-1)n\right]$, $\mathrm{Tl}{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+3+\delta }$ $\left[\mathrm{Tl}12(n-1)n\right]$, and ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ $\left[\mathrm{Tl}22(n-1)n\right]$. Our theoretical results are plotted by the curves. The experimental data for Bi2212 are denoted by triangles from Ref. 71, circles from Ref. 70, and stars from Ref. 37. The vertical dashed lines at $\mu ={\mu }_{\mathrm{opt}}$ denote the optimal doping level. UD and VD represent the underdoped and overdoped sides, respectively.