Resonance mode in $B_{1g}$ Raman scattering: A way to distinguish between spin-fluctuation and phonon-mediated $d$-wave superconductivity

We argue that Raman scattering in $B_{1g}$ symmetry allows one to distinguish between phonon-mediated and magnetically mediated $d$-wave superconductivity. In spin-mediated superconductors, $B_{1g}$ Raman intensity develops a resonance at a frequency ${\Omega }_{res}<2{\Delta }_{max}$, whose origin is similar to a neutron resonance. In phonon-mediated $d$-wave superconductors, such a resonance does not develop. Several extensions of the argument are presented.

Figure 1

The real part of the $B_{1g}$ Raman response in the superconducting state ${\chi }_0^{\prime }$ for the case of small fermionic damping (solid line) and no fermionic damping (dashed line). The vertical dotted line marks $2{\Delta }_0$. A small change between $2{\Delta }_0$ and the frequency where ${\chi }_0^{\prime }\left(\Omega \right)$ is peaked is due to the fact that the maximum value of the gap $\Delta \left(\mathbf{k}\right)$ along the Fermi surface is slightly smaller than ${\Delta }_0$.

Figure 2

(Color online) The full Raman response plotted for $V{N}_0=0$ (black solid line), and $\pm 0.01,0.1,0.5$ (dotted, dashed, dashed-dotted lines, respectively) for phonon-mediated ($V=V_{ph}$, green lines) and spin-mediated ($V=V_{spin}$, red), respectively.

Figure 3

(Color online) The Raman data (open squares) from five different groups [Ref. 23 (red), Ref. 15 (magenta), Ref. 20 (orange), Ref. 21 (green), Ref. 16 (blue), Ref. 17 (cyan), respectively], together with the tunneling data (red triangles) from Ref. 13 and ARPES data from Ref. 14. (Black circles are derived from leading edge spectra while green circles are from the peak position of the $\pi ,0$ spectra.) The doping $p$ was determined by the formula $p=0.16\pm \sqrt{\frac{\mid T_c-T_c^{max}\mid }{82.6T_c^{max}}}$, where $T_c$ is the superconducting transition temperature. The solid line plots $70\mathrm{meV}(T_c∕T_c^{max})$.