Nature of the spin resonance mode in CeCoIn₅

In the spin-exciton scenario, the SRM is a bound state residing below the PHCO with $E({\bf Q})<\rm{min}(\mid\Delta({\bf k})\mid+\mid\Delta({\bf k}+{\bf Q})\mid)$, resulting from a sign-change in the superconducting order parameter DScalapino ; Eschrig . For cuprates with a $d_{x^{2}-y^{2}}$-wave superconducting order parameter, the SRM peaks at the antiferromagnetic wavevector ${\bf Q}_{\rm AF}=(0.5,0.5)$, which connects hot spots that are close to the antinodal points of the $d_{x^{2}-y^{2}}$-wave superconducting order parameter [Fig. 1(a)]. As the SRM disperses away from ${\bf Q}_{\rm AF}$ towards ${\bf Q}_{\rm n}$, which connects the nodal points of the superconductivity order parameter, the PHCO is progressively pushed towards zero. The reduction of the PHCO away from ${\bf Q}_{\rm AF}$ requires a spin-excitonic SRM to exhibit a downward dispersion away from ${\bf Q}_{\rm AF}$, so that it stays below the PHCO. Inelastic neutron scattering measurements of the SRM in hole-doped cuprates demonstrated that it dominantly disperses downwards, consistent with expectations of the spin-exciton picture [Fig. 1(b)] PBourges2000 ; dai00 ; SPailhes2004 ; Reznik04 ; Hayden04 ; Sidis . For iron pnictide superconductors with isotropic $s^{\pm}$-wave superconducting gaps, the SRM is also consistent with being a spin-exciton DScalapino ; dai . Here, unlike the cuprates, the PHCO depends weakly on momentum ${\bf Q}$, allowing spin-excitonic SRMs to exhibit upward dispersions, as observed in electron- Kim13 and hole-doped compounds RZhang .

In the prototypical heavy fermion superconductor CeCoIn₅, like the cuprates, the superconducting order parameter is $d_{x^{2}-y^{2}}$-wave KIzawa ; BBZhou ; MPAllan13 and the SRM peaks around ${\bf Q}_{\rm AF}$ in momentum and $E_{\rm r}\approx 0.6$ meV in energy CStock08 ; thus also like the cuprates, the PHCO is gradually suppressed moving from ${\bf Q}_{\rm AF}$ towards ${\bf Q}_{\rm n}$ [Figs. 1(c) and (d)], resulting in a downward dispersion of the SRM in the spin-exciton scenario [Fig. 1(d), see Supplementary Note 1 for details] IEremin . Experimentally, however, the SRM is found to be dominated by a robust upward dispersion for $E\gtrsim E_{\rm r}$, contrary to expectations in the spin-exciton picture YSong2016 ; SRaymond2015 . These upward-dispersing features and the strong $L$-dependence of the SRM in CeCoIn₅ suggest it is a magnon-like mode, rather than a spin-exciton YSong2016 ; AVChubukov . While the SRM in CeCoIn₅ is dominated by an upward-dispersing branch for $E\gtrsim E_{\rm r}$, whether a downward-dispersing branch expected in the spin-exciton scenario also exists for $E<E_{\rm r}$, remains unclear.

To elucidate whether the SRM in CeCoIn₅ has any downward-dispersing features, we carried out detailed inelastic neutron scattering measurements of the SRM in CeCoIn₅ using PANDA, along the $(H,H,0.5)$ direction for $E\lesssim E_{\rm r}\approx 0.6$ meV, with results shown in Fig. 2. The magnetic scattering at $E=0.375$ meV is weaker in the superconducting state compared to the normal state [Fig. 2(a)], demonstrating a partial gapping of the magnetic fluctuations at this energy upon entering the superconducting state. With increasing energy, scattering in the superconducting state becomes more intense compared to the normal state [Figs. 2(b)-(f)], and the SRM can be clearly identified by such enhanced magnetic scattering. Constant-energy scans along $(H,H,0.5)$ for $E\geq 0.4$ meV [Figs. 2(b)-(f)] clearly reveal two peaks at ${\bf Q}=(0.5\pm\delta,0.5\pm\delta,0.5)$, in good agreement with previous work (see Supplementary Fig. 1 and Supplementary Note 2 for details) SRaymond2015 . While the magnetic scattering for $E=0.375$ meV appears to be a single peak, its broad width compared to higher energies suggests the magnetic scattering at this energy also consists of two peaks. By fitting the results in Figs. 2(b)-(f) using two Gaussian peaks at ${\bf Q}=(0.5\pm\delta,0.5\pm\delta,0.5)$, we find $\delta$ does not change significantly for $E\leq 0.45$ meV [Fig. 2(b)-(d)] and increases monotonically with increasing energy for $E>0.45$ meV [Fig. 3(a)], ruling out any downward-dispersing features. Combined with similar measurements for $E\gtrsim E_{\rm r}$ obtained using Multi-Axis Crystal Spectrometer (MACS) (see Supplementary Figs. 2-3 and Supplementary Note 3 for details), we find the SRM in CeCoIn₅ disperses only upward, inconsistent with calculations for the spin-exciton scenario, based on an electronic structure from scanning tunneling microscopy measurements [Figs. 1(d) and 3(a), see Supplementary Note 1 for details] MPAllan13 ; JVanDyke2014 . Instead, the dispersion of the SRM resemble spin waves in CeRhIn₅ [Fig. 3(b)] PDas2014 ; CStock2015 , suggesting it to be magnon-like (see Supplementary Fig. 4 and Supplementary Note 2 for additional comparisons).

For a spin-excitonic SRM that is isotropic in spin space, the application of a magnetic field should split it into a triplet in energy JPIsmer . In CeCoIn₅, the application of an in-plane magnetic field splits the SRM into a doublet, rather than a triplet CStock2012 ; SRaymond2012 , likely due to the presence of magnetic anisotropy AAkbari ; YSong2016 . The doublet splitting of the SRM under applied field, combined with the upward dispersion, raises the question of how the dispersive features of the SRM in CeCoIn₅ evolve with applied field, and whether the absence of a downward-dispersing branch is robust under applied field.

To address these questions, we studied the SRM in CeCoIn₅ using MACS, under an applied magnetic field perpendicular to the $[H,H,L]$ scattering plane, with results shown in Figs. 4 and 5. Constant-energy scans along $(H,H,0.5)$ and $(0.5,0.5,L)$ directions in Fig. 4 reveal dramatic changes to the SRM away from ${\bf Q}_{\rm AF}$ under applied magnetic field. For $E=0.5$ meV and $E=0.6$ meV, the SRM broadens upon increasing the magnetic field from $B=0$ T to $B=6$ T [Figs. 4(a)-(d)]. On the other hand, for $E=0.8$ meV and $E=1.0$ meV, two split peaks around ${\bf Q}_{\rm AF}$ are clearly seen at $B=0$ T, while increasing the magnetic field to $B=3$ T significantly reduces the splitting and only a single peak can be resolved at $B=6$ T [Figs. 4(e)-(h)]. We note that while the SRM is peaked slightly away from ${\bf Q}_{\rm AF}$ for $E\lesssim E_{\rm r}$ at zero-field, as demonstrated in Fig. 2, the resolution of our MACS measurements is insufficient to resolve such a small splitting, instead a single peak at ${\bf Q}_{\rm AF}$ is observed [Figs. 4(a) and (c)].

These disparate behaviors at different energies can be understood to result from the doublet splitting of the upward-dispersing SRM, as schematically depicted in Fig. 4(i). The broadening of the peaks along $(H,H,0.5)$ at $E=0.5$ and 0.6 meV under applied field is due to a downward shift of the lower branch of the SRM, and increased damping resulting from the PHCO also moving to lower energies. For higher energies $E=0.8$ and 1.0 meV, the intensity of magnetic scattering is dominated by the upper branch, and because the upper branch moves to higher energies under applied field, a reduction in peak splitting is observed. Our results indicate the dispersive SRM in CeCoIn₅ splits into two branches under an in-plane magnetic field, while maintaining its upward-dispersing character. This conclusion is also supported by the analysis of peak splittings for the data in Fig. 5 (See Supplementary Fig. 5 and Supplementary Note 4 for details).

In addition to splitting the SRM into two upward-dispersing branches, energy-$(H,H,0.5)$ and energy-$(0.5,0.5,L)$ maps in Fig. 5 and Supplementary Fig. 6 (obtained from data shown in Supplementary Figs. 3,7 and 8, see Supplementary Note 3 for details) suggest that applied magnetic field also results in significant damping to the SRM in CeCoIn₅ (see Supplementary Fig. 9 and Supplementary Note 4 for additional evidence from constant-${\bf Q}$ scans). While the dispersive features can be clearly observed in the $B=0$ T data [Figs. 5(a) and (b)], with applied field the dispersive features become less prominent for $B=4$ T [Figs. 5(c) and (d)] and for $B=6$ T no dispersive features can be resolved [Supplementary Figs. 6(d) and (h)]. These results suggest that with applied field, the SRM becomes progressively damped and its dispersive character smeared out, becoming similar to overdamped magnetic excitations in the normal state, as the applied field approaches the upper critical field (see Supplementary Fig. 10 and Supplementary Note 3 for details). The increase in damping is unexpected in the spin-exciton scenario. This is because the SRM and the PHCO are shifted in energy in unison by an applied magnetic field, the SRM should therefore remain undamped (see Supplementary Figure 11 and Supplementary Note 1 for details). Instead, the observed damping of the SRM with increasing field suggests that the SRM and the PHCO move independently with increasing magnetic field, consistent with the suggestion that the SRM in CeCoIn₅ results from the removal of damping to a pre-existing magnetic mode in the superconducting state AVChubukov ; YSong2016 , rather than being a spin-exciton.

Single crystals of CeCoIn₅ were prepared by the indium self-flux method VSZapf . Hundreds of CeCoIn₅ single crystals with a total mass $\sim$1 g were co-aligned in the $[H,H,L]$ scattering plane on aluminum plates using a hydrogen-free glue. Magnetic field is applied perpendicular to the scattering plane, along the $(1\bar{1}0)$ direction.

Neutron scattering experiments were carried out on the PANDA cold three-axes spectrometer at the Heinz Maier-Leibnitz Zentrum panda and the Multi-Axis Crystal Spectrometer (MACS) at the NIST Center for Neutron Research. The inelastic neutron scattering experiments on PANDA used fixed $k_{\rm f}=1.3$ Å^-1. A sapphire filter is used before the monochromator and a Be filter cooled to 40 K is used before the sample. The monochromator has horizontal and vertical variable focusing mechanics, vertical focusing of the analyzer is fixed (variable focusing is not needed because the detector is a vertically placed 1 inch ³He tube) and horizontal focusing is variable. In the focused mode, variable focusings are adjusted depending on the neutron wavelength based on empirically optimized values. The inelastic neutron scattering measurements at MACS used Be filters both before and after the sample with fixed $E_{\rm f}=3.0$ meV or $E_{\rm f}=3.7$ meV. Most of measurements on MACS were made using the 20 spectroscopic detectors simultaneously to efficiently obtain the magnetic scattering within the $[H,H,L]$ scattering plane. Constant-${\bf Q}$ scans at ${\bf Q}_{\rm AF}$ Supplementary Fig. 9 were carried out using MACS with a single detector. The analyzers are vertically focused, while the monochromator is doubly focused.

Data shown in Fig. 2 and Supplementary Fig. 1 are obtained using PANDA. The constant-energy scans were fit to a single Gaussian peak or two Gaussian peaks equally displaced from the center; scans at different energy transfers are fit globally with the same peak center. Constant-${\bf Q}$ scans in Supplementary Fig. 9 are measured on MACS using a single detector. All the rest of neutron scattering data are obtained using MACS by measuring maps of large portions of the $[H,H,L]$ scattering plane, simultaneously using the 20 detectors available at MACS. The maps of $[H,H,L]$-plane are folded into a single quadrant to improve statistics. Cuts along $(H,H,0.5)$ were obtained by binning data with $0.37\leq L\leq 0.63$ and a step size of 0.025; cuts along $(0.5,0.5,L)$ are obtained by binning data with $(0.42,0.42)\leq(H,H)\leq(0.58,0.58)$ and a step size of 0.05. Normal state magnetic excitations measured at $T=2.5$ K have been subtracted in all the MACS data except Supplementary Fig. 10. The cuts along $(H,H,0.5)$ are fit with a single Gaussian peak centered at ${\bf Q}=(0.5,0.5,0.5)$ or two Gaussian peaks at ${\bf Q}=(0.5\pm\delta,0.5\pm\delta,0.5)$. The cuts along $(0.5,0.5,L)$ are fit using a lattice sum of a single Lorentzian peak centered at ${\bf Q}=(0.5,0.5,0.5)$ or a lattice sum of two Lorentzian peaks at ${\bf Q}=(0.5,0.5,0.5\pm\delta)$. $B=4$ T data are collected using $E_{\rm f}=3.0$ meV, while measurements at other fields used $E_{\rm f}=3.7$ meV. Using MACS we collected high statistics data for selected energies (Fig. 4 and Supplementary Fig. 2) and lower statistics data with finer energy steps (Fig. 5 and Supplementary Figs. 3, 6-8). The zero-field data shown in Fig. 4 and Supplementary Fig. 2 are reproduced from Ref. YSong2016 , to compare with data under applied field.

DATA AVAILABILITY:

All relevant data are available from the corresponding authors upon reasonable request.

ACKNOWLEDGEMENTS:

We thank S. Raymond and C. Stock for helpful discussions. The neutron scattering work at Rice is supported by the U.S. DOE, BES under grant no. DE-SC0012311 (P.D.). Part of the material characterization efforts at Rice is supported by the Robert A. Welch Foundation Grant Nos. C-1839 (P.D.). Research at UC San Diego was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Grant No. DEFG02-04-ER46105 (single crystal growth) and US National Science Foundation under Grant No. DMR-1810310 (characterization of physical properties). Access to MACS was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No. DMR-1508249.

AUTHOR CONTRIBUTIONS:

Y. S. and P. D. led the project. The neutron scattering experiments were performed by Y. S., W. W., A. S., P. C. and Y. Q. The samples were prepared by N. P., S. R., D. Y. and M. B. M. Y. S. co-aligned the samples. Y. S. and W. W. analyzed the data. Theoretical calculations were carried out by J. V. and D. K. M. The manuscript was written by Y. S., D. K. M. and P. D. with input from all coauthors.

COMPETING INTERESTS:

The authors declare no competing interests.

Correspondence and requests for materials should be addressed to Y. S. ([email protected]), D. K. M. ([email protected]) or P. D. ([email protected]).