Odd and even magnetic resonant modes in highly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$

We present inelastic neutron scattering data on highly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ single crystals with superconducting transition temperature $T_C=70\mathrm{K}$ and, for comparison, a nearly optimally doped crystal with $T_C=87\mathrm{K}$. In both samples, magnetic resonant modes with odd and even symmetry under exchange of the two $\mathrm{Cu}{\mathrm{O}}_2$ layers in the unit cell are observed. In the overdoped sample, the linewidth of the odd mode is reduced compared with the optimally doped sample. This finding is discussed in conjunction with recent evidence for intrinsic inhomogeneities in this compound. The data on odd and even resonant excitations are otherwise fully consistent with trends established on the basis of data on $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$. This confirms the universality of these findings and extends them into the highly overdoped regime of the phase diagram.

Figure 1

Resonant magnetic modes in nearly optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ $(T_C=87\mathrm{K})$ (OD87) measured at $\mathbf{Q}=(0.5,0.5,L)$ in the (a) odd channel $(L=14)$ and (b) even channel $(L=18)$. The signal is obtained by subtracting the intensity at $T=5\mathrm{K}$ from the intensity above $T_C$. The full lines are the result of a fit by a Gaussian profile on top of a background (dotted lines). The shape of this background is given by the difference between the phonon populations at the two temperatures, yielding negative values at low energy. The diamond symbols correspond to the background level obtained by constant energy scans.

Figure 2

Same as Fig. 1 in OD70, highly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ $(T_C=70\mathrm{K})$, in (a) odd channel $(L=13.2)$ and (b) even channel $(L=9.6)$. Note that the magnitude of the odd mode in absolute units is actually $\sim 3$ times weaker in OD70 than in OD87. An accurate value cannot be given due to uncertainties in the calibration procedure.

Figure 3

Temperature dependence of the peak intensity of both odd (left panel) and even (right panel) resonant modes (empty symbols) in OD70. The solid symbols show the temperature evolution of the intensity at a background point away from the magnetic wave vector. The solid lines are the results of fits to the data obtained using a power law in the superconducting state on top of a phononic background. Mn stands for monitor.

Figure 4

(Color online) Evolution of the magnetic resonant mode energy as a function of the hole concentration $\delta$ in the Bi2212 system (see Ref. 8 for the definition of $\delta$). The full symbols represent the odd $\left(E_r^o\right)$ and even $\left(E_r^e\right)$ modes, and the threshold energy $\left({\omega }_c\right)$ as explained in the legend. The open symbols are the energy gap measured by ARPES (Ref. 17) and by Electronic Raman Scattering (ERS) (Ref. 18). The inset shows the equivalent diagram for Y123 (Ref. 8)