Doping Dependence of Bilayer Resonant Spin Excitations in $(\mathrm{Y},\mathrm{Ca}){\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$

Resonant magnetic modes with odd and even symmetries were studied by inelastic neutron scattering experiments in the bilayer high-$T_c$ superconductor ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$ over a wide doping range. The threshold of the spin excitation continuum in the superconducting state, deduced from the energies and spectral weights of both modes, is compared with the superconducting $d$-wave gap, deduced from electronic Raman scattering in the $B_{1g}$ symmetry on the same samples. Above a critical doping level of $\delta \simeq 0.19$, both mode energies and the continuum threshold coincide. We find a simple scaling relationship between the characteristic energies and spectral weights of both modes, which indicates that the resonant modes are bound states in the superconducting energy gap, as predicted by the spin-exciton model of the resonant mode.

Figure 1

(a) Differences between INS energy scans measured at the antiferromagnetic wave vector below $T_c$ in ${\mathrm{Y}}_{0.85}{\mathrm{Ca}}_{0.15}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ (OD75). Open and solid symbols correspond to the odd and even channels, respectively. The stars indicate the measured reference level of the magnetic scattering determined by constant-energy scans. (b) $L$ dependence at $E=34\mathrm{meV}$ for OD75. The lines describe the intensity of each mode following Eq. (1) multiplied by the Cu magnetic form factor (upper line). (c) Temperature dependence of the neutron intensity measured at characteristic energies of the even and odd modes for OD75. (d) Same as (a) but for sample ${\mathrm{YBCO}}_{6.6}$ (UD63).

Figure 2

Electronic Raman spectra in the $B_{1g}$ channel. The difference between spectra measured at 15 K in the SC state and at 90 K in the normal state is displayed. In addition to sharp features due to phonon renormalization, the difference indicates a broad peak due to the enhancement of the electronic response in the SC state. The electronic peaks were fitted with Gaussian profiles. The inset shows the peak energy of the 5 samples we have studied (see Fig. 3 caption for the determination of the doping levels).

Figure 3

Doping dependence of: (a) $E_r^o$, $E_r^e$, ${\omega }_c$, and $2{\Delta }_{\max }$ (ERS). ERS data were obtained from Ref. 21 (▵), in Ref. 20 (⋄), in Ref. 22 (□), and in the present study (▿). (b) Ratio and sum of the energy-integrated SW. The hole doping is estimated from the phenomenological relation $T_c(\delta )=T_c^{\mathrm{opt}}(1-82.6(\delta -{\delta }_{\mathrm{opt}}{)}^2)$ [15], ${\delta }_{\mathrm{opt}}$ is generally estimated to be 0.16 [15]. $T_c^{\mathrm{opt}}$ is 93 K for pure YBCO. Ca substitution reduces $T_c$ recording to $T_c^{\mathrm{opt}}=90.6-39.2x_{\mathrm{Ca}}$ obtained for our single crystals.

Figure 4

(a) Resonant modes SW vs the reduced binding energy. Sketch of two approaches for the resonant modes: (b) bound states of the electronic continuum; (c) preexisting modes whose intensities are enhanced by the disappearance of the fermionic damping below ${\omega }_c$.