Exchange Boson Dynamics in Cuprates: Optical Conductivity of ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$

The electron-boson spectral density function $I^2\chi (\Omega )$ responsible for carrier scattering of the high temperature superconductor ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ ($T_c=90\mathrm{K}$) is calculated from new data on the optical scattering rate. A maximum entropy technique is used. Published data on ${\mathrm{HgBa}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{8+\delta }$ ($T_c=130\mathrm{K}$) are also inverted and these new results are put in the context of other known cases. All spectra (with two notable exceptions) show a peak at an energy (${\Omega }_r$) proportional to the superconducting transition temperature ${\Omega }_r\approx 6.3k_B{T}_c$. This charge channel relationship follows closely the magnetic resonance seen by polarized neutron scattering, ${\Omega }_r^{\mathrm{neutron}}\approx 5.4k_B{T}_c$. The amplitudes of both peaks decrease strongly with increasing temperature. In some cases, the peak at ${\Omega }_r$ is weak and the spectrum can have additional maxima and a background extending up to several hundred meV.

Figure 1

Experimental results for the real part of the optical self-energy $-2{\Sigma }_1^{\mathrm{op}}(T,\omega )$ as a function of photon energy $\omega$ for eight temperatures. Inset, theoretical results based on numerical solutions of the generalized Eliashberg equations. (Solid blue superconducting and dashed black normal state.) The electron-boson exchange spectral density used is shown as the dash-dotted blue curve. The superconducting gap value is $\Delta =22.4\mathrm{meV}$.

Figure 2

Top frame, the optical scattering rate $1/{\tau }^{\mathrm{op}}(T,\omega )$ for Hg1201 vs $\omega$ for 8 temperatures (light curves). The wider curves are our maximum entropy reconstructions. Bottom frame, the electron-boson spectral function $I^2\chi (\Omega )$ vs $\Omega$. The inset gives the peak position (blue triangles) left scales as a function of temperature and the red squares give the corresponding peak amplitude.

Figure 3

Same as Fig. 2 but for Hg1223 ($T_c=130\mathrm{K}$).

Figure 4

The optical resonance frequency ${\Omega }_r$ as a function of $T_c$. Bi2212 [6, 8], Tl2201 [5], YBCO [5, 6, 7], LSCO [9], and PCCO [10].