Itinerant effects and enhanced magnetic interactions in Bi-based multilayer cuprates

The cuprate high temperature superconductors exhibit a pronounced trend in which the superconducting transition temperature $T_{\mathrm{c}}$ increases with the number of ${\mathrm{CuO}}_2$ planes $n$ in the crystal structure. We compare the magnetic excitation spectrum of ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CuO}}_{6+\delta }$ (Bi-2201) and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$ (Bi-2223), with $n=1$ and 3, respectively, using Cu $L_3$-edge resonant inelastic x-ray scattering. Near the antinodal zone boundary we find the paramagnon energy in Bi-2223 is substantially higher than that in Bi-2201, indicating that multilayer cuprates host stronger effective magnetic exchange interactions, providing a possible explanation for the $T_{\mathrm{c}}$ vs $n$ scaling. In contrast, the nodal direction exhibits very strongly damped, almost nondispersive excitations. We argue that this implies that the magnetism in the doped cuprates is partially itinerant in nature.

Figure 1

RIXS spectra for Bi-2201 and Bi-2223 taken at $\mathit{Q}=(0.39,0)$. Magnetic, orbital, and charge-transfer excitations are seen in energy ranges around $0\text{–}400$ meV, $1\text{–}3$ eV, and $2\text{–}8$ eV, respectively (high-energy scales $>6$ eV are not shown).

Figure 2

Low-energy RIXS spectra for Bi-2201 (left) and Bi-2223 (right) along $(\zeta ,0)$, showing the dispersion of the paramagnon excitation. Black points show the data and the solid gray line represents the results of the fitting, which is the sum of a Gaussian elastic scattering contribution (red), an antisymmetrized Lorentzian capturing the magnetic scattering (blue), and the smooth background (dashed black line).

Figure 3

The dispersion of the energy of the paramagnon mode in Bi-2201 and Bi-2223 along $(\zeta ,0)$, as obtained by the fitting analysis of the spectra in Fig. 2. The points for Bi-2212 are obtained by fitting the data in Ref. [27] in the same way. The lines are sinusoidal fits to the Bi-2201, Bi-2212, and Bi-2223 data, which serve as guides to the eyes. Error bars are determined by summing the estimated uncertainly in the elastic energy and the statistical error from the least-square-fitting routines in quadrature.

Figure 4

Low-energy RIXS spectra for Bi-2201 (left) and Bi-2223 (right) along $(\zeta ,\zeta )$, where a heavily overdamped, almost nondispersive excitation is observed. Black points show the data and the solid gray line represents the results of the fitting, which is the sum of a Gaussian elastic scattering contribution (red), an antisymmetrized Lorentzian (blue), and the smooth background (dashed black line).

Figure 5

Calculations of the imaginary part of the magnetic dynamical susceptibility ${\chi }^{\prime \prime }(\mathit{Q},E)$ in the superconducting state based on the renormalized itinerant quasiparticle approach used in Ref. [20]. A well-defined paramagnon mode is observed along the antinodal $(0,0)\to (0.5,0)$ direction whereas the intensity along the nodal $(0,0)\to (0.5,0.5)$ direction is more diffuse and poorly defined with the exception of the “resonance mode” around $(0.5,0.5)$. Apart from the “resonance mode,” calculations for the nonsuperconducting state are virtually identical.