Dispersion of the odd magnetic resonant mode in near-optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\mathit{\delta }}$

We report a neutron scattering study of the spin excitation spectrum in the superconducting state of the slightly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ system $(T_c=87\mathrm{K})$. We focus on the dispersion of the resonance peak in the superconducting state that is due to a $S=1$ collective mode. The measured spin excitation spectrum bears a strong similarity to the spectrum of the ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ system for a similar doping level (i.e., $x\sim 0.95-1$), which can be described as intersecting upward- and downward-dispersing branches. A close comparison of the threshold of the electron-hole spin flip continuum, known from angle resolved photoemission measurements in the same system, indicates that the magnetic response in the superconducting state is confined, in both energy and momentum, below the gapped Stoner continuum. In contrast to ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, the spin excitation spectrum is broader than the experimental resolution. In the framework of an itinerant-electron model, we quantitatively relate this intrinsic energy width to the superconducting gap distribution observed in scanning tunneling microscopy experiments. Our study further suggests a significant in-plane anisotropy of the magnetic response.

Figure 1

Difference between constant-$Q$ scans performed at 10 and $100\mathrm{K}$. INS measurements were carried out on spectrometer $2T$. The differential spectra are shown at different wave vectors of the form (a) $\mathbf{Q}=(H,H,L=14)$, (b) $\mathbf{Q}=(K∕3,K,L=4.2)$. In addition to the energy scan data (open circles) are also reported the magnetic intensities (open squares) and the location of the negative background (crosses), deduced from the analysis of constant energy scans (Fig. 2). Those data, obtained on spectrometer IN8, have been rescaled. The lines indicate the energy dependence of the negative background (see the text) and the enhancement of the magnetic intensity is described by a set of Gaussian functions on top of the negative background.

Figure 2

(Color online) Differences between constant-energy scans performed at 10 and $100\mathrm{K}$: (a) scans along the (110) direction and (b) scans along the (130) direction. All measurements were carried out on the spectrometer IN8. The position of the negative background used in the analysis of the energy scans, reported in Fig. 1, is also shown at the AF wave vector [red (dark gray) circles]. The solid lines correspond to the fit of the data by a single or double Gaussian functional form, on top of a sloping background.

Figure 3

(Color online) Dispersion of the resonant magnetic excitations deduced from constant-energy scans: (a) along the (130) direction (the full lines are described in the text) and (b) along the (110) direction. Horizontal and vertical error bars stand for the half width at half maximum of the signal and the energy resolution, respectively.

Figure 4

(a) Temperature dependencies at different wave vectors and energies. (b) Differences between constant-energy scans at 10 and $100\mathrm{K}$. Scans are performed at $40\mathrm{meV}$ around the AF wave vector along three different directions: (100),(130),(110). The label $\Theta$ corresponds to the angle between the scanning direction of the (100) direction [see Fig. 6f]. Data are normalized so that the intensities at the AF wave vector are the same.

Figure 5

Differences between constant-energy scans at 46 meV performed between (a) 10 and $300\mathrm{K}$ and (b) 100 and $300\mathrm{K}$. Measurements were carried out on the spectrometer IN8. The solid line corresponds to the fit of the data by a double Gaussian functional form, on top of a sloping background (dashed line). The background at the AF wave vector is set to 0 to get rid of the variation of the thermal enhancement of the nuclear background and to allow a direct comparison of both results.

Figure 6

(Color online) (a) Fermi surfaces in nearly optimally doped Bi2212 deduced from ARPES measurements.[50, 62] [red (dark gray)=bonding band, black=antibonding band]. Threshold of the electron-hole spin flip continuum in the SC state along different directions: (b) (130) and (c) (110). Below the threshold of the continuum, one distinguishes two distinct areas labeled, respectively I, and II as in (Ref. 14) (see the text). The vertical dashed lines indicate the location of the energy scans that were performed. The projection of the resolution ellipsoid at $40\mathrm{meV}$ is also represented. [(d)and (e)] Color (gray scale) maps showing the enhanced magnetic response in the SC state as deduced from the fit of the constant-energy scan, in addition to the threshold of the continuum. (f) Area occupied by the continuum at $40\mathrm{meV}$ (black). The red (dark gray) points indicate the extension of the magnetic response as deduced from Fig. 4b (see the text) and the white crosses show where the temperature dependencies of the scattering intensity were measured.

Figure 7

(Color online) Resonant magnetic modes in nearly optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ $(T_c=87\mathrm{K})$ measured at $\mathbf{Q}=(0.5,0.5,14)$ [as in Fig. 1a]. The solid lines correspond to the computed imaginary part of the susceptibility convoluted with the energy resolution function: the green (light gray) line was obtained using a single $d$-wave SC gap (RPA 1) and the red (dark gray) one with a distribution of $d$-wave SC-gaps (RPA 2). In the latter case, the SC- gap average is ${\Delta }_m=35\mathrm{meV}$ with a full width at maximum of $14\mathrm{meV}$.

Figure 8

(Color online) Spatial gap distribution considered for the calculation of $\chi (\mathbf{q},\omega )$. Using a Gaussian fit of this distribution, one obtains a spatial average gap of $35\mathrm{meV}$ and a FWHM of ${\sigma }_{\Delta }=15\mathrm{meV}$ corresponding to the sample with $T_c=87\mathrm{K}$.

Figure 9

(Color online) Width (FWHM) of the SC-gap distribution $\left({\sigma }_{\Delta }\right)$ measured by STM (Refs. 52, 53, 54) and the width of the magnetic resonance peak in INS data (see Table ) (Refs. 10, 11, 59) as a function of the hole doping level per ${\mathrm{CuO}}_2$ plane, $p$.

Figure 10

(Color online) Mappings of the imaginary part of the dynamical spin susceptibility $\mathfrak{I}\chi (\mathbf{q},\omega )$: [(a)–(c)] along the (130) direction and [(d)–(f)] along the (110) direction. $\mathrm{Im}\chi (\mathbf{q},\omega )$ was computed using a spin itinerant model (see the text) assuming [(a),(d)] a unique SC gap for the entire system or [(b),(e)] the spatial distribution of SC gaps reported in Fig. 8. In order to enable an immediate comparison of the computed mappings of $\mathrm{Im}\chi (\mathbf{q},\omega )$ and the INS measured ones, the experimental mappings of Figs. 6d, 6e are also reported [(c) and (f)]. In the absence of a calibration of the INS data in absolute units, mappings are scaled so that the maximum intensity corresponds to 100 counts.