Abstract
Magnetic fluctuations is the leading candidate for pairing in cuprate, ironbased, and heavy fermion superconductors. This view is challenged by the recent discovery of nodeless superconductivity in CeCu_{2}Si_{2}, and calls for a detailed understanding of the corresponding magnetic fluctuations. Here, we mapped out the magnetic excitations in superconducting (Stype) CeCu_{2}Si_{2} using inelastic neutron scattering, finding a strongly asymmetric dispersion for E ≲ 1.5 meV, which at higher energies evolves into broad columnar magnetic excitations that extend to E ≳ 5 meV. While lowenergy magnetic excitations exhibit marked threedimensional characteristics, the highenergy magnetic excitations in CeCu_{2}Si_{2} are almost twodimensional, reminiscent of paramagnons found in cuprate and ironbased superconductors. By comparing our experimental findings with calculations in the randomphase approximation,we find that the magnetic excitations in CeCu_{2}Si_{2} arise from quasiparticles associated with its heavy electron band, which are also responsible for superconductivity. Our results provide a basis for understanding magnetism and superconductivity in CeCu_{2}Si_{2}, and demonstrate the utility of neutron scattering in probing band renormalization in heavy fermion metals.
Introduction
The discovery of superconductivity in CeCu_{2}Si_{2}^{1} marked the beginning of decadeslong intense research into unconventional superconductivity^{2,3}, encompassing cuprate^{4,5,6}, ironbased^{7,8,9,10}, and heavy fermion superconductors^{11,12,13}. The proximity of superconductivity to antiferromagnetic (AF) quantum critical points (QCP) in these systems implicate AF fluctuations that proliferate at the AF QCP as the pairing glue, leading to unconventional superconductivity with a signchanging superconducting order parameter^{2,3,14}.
Experimental evidence for magnetically driven superconductivity in these systems include: (i) reduction of the magnetic exchange energy is much larger than the superconducting condensation energy^{15,16,17,18,19,20,21,22}; (ii) the observation of a spin resonance mode, which in the spinexciton scenario indicates a signchanging superconducting order parameter^{2,23,24,25}; and (iii) the persistence of twodimensional (2D) highenergy AF fluctuations that resemble spin waves in magnetically ordered parent compounds^{19,26,27,28,29,30,31,32,33}.
From an empirical perspective, it is important to identify whether these features are common in different unconventional superconductors, so that ingredients for a unified pairing mechanism may be established. Of the above observations, (i) is modelindependent and has been verified for cuprate, ironbased and heavy fermion superconductors^{15,16,17,18,19,20,21,22}; (ii) the spin resonance mode has been found in cuprate, ironbased, and heavy fermion superconductors, but their spinexcitonic nature needs to be separately tested and requires a quasiparticle origin of the magnetic excitations^{2,23,24}; while (iii) has been established for cuprate and ironbased superconductors^{19,26,27,28,29,30,31,32,33}, magnetic excitations in heavy fermion superconductors such as CeCu_{2}Si_{2} are strongly threedimensional (3D) at low energies^{18}, and it is unclear whether they become 2D at higher energies.
CeCu_{2}Si_{2} (Stype) is an archetypal heavy fermion unconventional superconductor, and is naturally located near a 3D AF QCP^{34,35}. Upon the introduction of slight Cu deficiencies the system can be tuned to AF order (Atype), with an ordering vector τ ≈ (0.22, 0.22, 0.53) [Fig. 1a]^{36}. Similar dispersive paramagnons up to E ≈ 1 meV were found to stem from τ in both Atype and Stype CeCu_{2}Si_{2}^{18,37}. While these dispersive AF fluctuations were discussed in terms of an effective Heisenberg model with shortrange magnetic couplings^{18}, magnetic order in Atype CeCu_{2}Si_{2} was suggested to result from Fermi surface nesting^{36}, and AF fluctuations in CeCu_{2}Si_{2} were found to exhibit critical slowing down consistent with being near a spindensitywave QCP^{35}. In the superconducting state of CeCu_{2}Si_{2}, a spin gap forms with spectral weight built up just above it^{18}, consistent with the formation of a spin resonance mode with an energy E_{r}≈0.2 meV, which in the spinexcitonic scenario suggests magnetic pairing^{38,39}. The recent discovery of fullygapped superconductivity in CeCu_{2}Si_{2}^{40,41,42,43,44,45,46,47,48} challenges the role of magnetic excitations in its superconducting state and calls for a more detailed understanding of its magnetism, including the origin and highenergy properties of its magnetic excitations.
In this work, by carrying out detailed inelastic neutron scattering measurements over large energy and momentum ranges, we uncover magnetic fluctuations up to E ≳ 5 meV in CeCu_{2}Si_{2}. While magnetic fluctuations below E ≈ 1.5 meV are strongly 3D and dispersive^{18,35}, they become increasingly 2D with increasing energy and form an almost dispersionless column in energy. By comparing with theoretical calculations, we find magnetic excitations in CeCu_{2}Si_{2} can be accounted for by intraband scattering of quasiparticles associated with the heavy electron band [Fig. 1d], and therefore allowing us to estimate the band renormalization by matching our calculations with experimental data. We expect this method to be broadly applicable in heavy fermion metals near magnetic criticality. The agreement between our experimental and theoretical results suggests that despite signatures of nonFermiliquid behavior^{34,35}, the magnetic excitations in the normal state of CeCu_{2}Si_{2} are reasonably captured by a LDA+U band structure with additional mass renormalization. Our discovery of almost 2D highenergy magnetic excitations in CeCu_{2}Si_{2} is reminiscent of similar findings in cuprate and ironbased superconductors, and favors magnetic pairing with a signchanging superconducting order parameter.
Results
Inelastic neutron scattering
Large single crystals of Stype CeCu_{2}Si_{2} with T_{c} ≈ 0.5 K were grown using a vertical floating zone method^{49}. Multiple crystals with a total mass of ≈12 g were coaligned in the [H, H, L] scattering plane using the E3 fourcircle neutron diffractometer at the Chalk River Laboratory (see Methods for details). Inelastic neutron scattering measurements were carried out using the MultiAxis Crystal Spectrometer (MACS)^{50} at the NIST Center for Neutron Research, with fixed outgoing neutron energies E_{f} = 3 or 5 meV. We reference momentum transfer Q = (H, K, L) in reciprocal lattice units, with H = aQ_{x}/(2π), K = bQ_{y}/(2π), and L = cQ_{z}/(2π) (a = b ≈ 4.1 Å and c ≈ 9.9 Å). Ce^{3+} ions in CeCu_{2}Si_{2} form a bodycentered tetragonal lattice [Fig. 1b], and the corresponding Brillouin zone in the [H, H, L] plane is shown in Fig. 1c. Isotropic background intensities were estimated from regions with H = 0 and 1, and have been subtracted from our data.
Maps of the [H, H, L] plane at T = 1.6 K are compared in the left column of Fig. 2 for different energies, with the corresponding cuts along (H, H, 1.48) and (0.22, 0.22, L) shown in the middle and right columns. For E = 0.5 meV, magnetic excitations are relatively sharp, with spectral weight asymmetrically located around τ [Figs. 1c, 2a–c]. With increasing energy, the magnetic excitations gradually broaden, while maintaining the asymmetric distribution of spectral weight around τ [Fig. 2d–o]. Given the same asymmetric distribution is observed for multiple momentum and energy transfers, it is an intrinsic effect rather than a result of instrumental resolution. Two dispersive branches were previously observed at low energies in CeCu_{2}Si_{2}^{18}, with the branch closer to the zone boundary being increasingly dominant in intensity with increasing energy^{18}. In our results, a single branch is resolved and can be identified as the dominant branch in previous work, while the weaker branch is unresolved and likely shows up as a shoulder in intensity. Clear modulation of magnetic intensity along (H, H, 1.48) persists up to the highest measured energy (E = 5.5 meV), with little or no magnetic intensity at the (0, 0, L) and (1, 1, L) positions; on the other hand, intensity along (0.22, 0.22, L) becomes weakly Ldependent for E ≥ 2 meV. These observations suggest that although CeCu_{2}Si_{2} is close to a 3D AF QCP^{34,35}, its highenergy magnetic excitations are almost 2D (see Supplementary Note 1 and Supplementary Fig. 1 for theoretical evidence of quasi2D magnetic excitations), similar to cuprate and ironbased superconductors.
Dispersion of the magnetic excitations can be directly visualized in the energy(H, H, 1.48) map in Fig. 3a. By fitting scans along (H, H, 1.48) using Gaussian peaks symmetrically positioned around (0.5,0.5,1.48) as shown in the middle column of Fig. 2, the magnetic dispersion along (H, H, 1.48) can be quantitatively extracted from E = 0.5 meV to 5.5 meV [Fig. 3a] (see Methods section for details). Consistent with previous observations^{18,35,37}, the magnetic excitations are dispersive for E ≲ 1.5 meV, but at higher energies they form a column in energy away from the zone boundary. Such an evolution from dispersive to columnar magnetic excitations is unexpected for a localmoment magnetic system, but has been observed in itinerant magnetic systems, including heavily holedoped iron pnictides^{51}, Fedoped MnSi_{3}^{52}, and MnSi^{53}.
Theoretical calculations
To understand the origin of the magnetic excitations in CeCu_{2}Si_{2} which extend up to at least E = 5.5 meV, we calculated magnetic excitations within the randomphase approximation (RPA) [Fig. 3b] using a LDA+U band structure (see Methods for details). As can be seen in Fig. 3b, despite extending over a larger energy scale, the calculated magnetic excitations are in good agreement with our experimental results, with the extracted dispersion consisting of a dispersive part at low energies and a columnar part at high energies (see Methods for details). By introducing an overall band renormalization factor r that scales our LDA+U band structure, excellent agreement between experimental and theoretical dispersions can be achieved [Fig. 3c]. We fit the dispersions with a lineardispersing part that intersects a columnar part at E_{cross}, with the experimental dispersion further constrained to stem from τ = (0.22, 0.22, 0.53). Scaling the theoretical dispersion so that E_{cross} is identical for theoretical and experimental dispersions leads to r ≈ 40. For comparison, the normal statespecific heat coefficient from our LDA+U band structure is C/T ≈ 50 mJ/mol K^{2}, which requires r ≈ 20 to match the experimental normal state value of ≈1.0 J/mol K^{2} for T → 0 K^{35}.
The Ce f electrons in CeCu_{2}Si_{2} participate in the formation of an electron and a hole Fermi sheet, both indispensable for its superconducting state^{46}. Compared to specific heat measurements which contain contributions from all Fermi sheets, magnetic excitations measured by neutron scattering are sensitive to bands that exhibit good nesting properties. Therefore, the larger value of r inferred from our neutron scattering results suggests a larger renormalization factor for the wellnested heavy electron band, relative to the hole band. Such banddependent renormalization effects have been discussed in the context of ironbased superconductors^{54,55,56}, with bands exhibiting markedly different renormalization factors for different Fe 3d orbitals. In addition, while specific heat measurements are only sensitive to states within ~k_{B}T (~0.1 meV for T = 1.6 K) of the Fermi level, magnetic fluctuations probed in our experiments involve states on the order of several meVs within the Fermi level. Therefore, stronger renormalization effects compared to our LDA+U calculations for states away from the Fermi level (relative to those within k_{B}T of the Fermi level) can also contribute to the larger r values extracted from our inelastic neutron scattering measurements. Our findings illustrate the utility of neutron scattering measurements in extracting renormalization factors with bandspecificity, complementing specific heat measurements. It should be noted that such a bandspecificity is limited to the wellnested band, which dominates the magnetic excitation spectra, while other bands are effectively not probed. This method hinges on the fact that magnetic excitations arising from quasiparticles encode information on the band structure^{57,58,59}, and can be especially useful in heavy fermion metals, for which angleresolved photoemission spectroscopy measurements are challenging due to the small energy scales involved.
The structure of the heavy electron band along the (k_{x}, k_{y} = k_{x}, 2π/c) direction [Fig. 3d] offers an intuitive understanding of the unusual magnetic dispersion in CeCu_{2}Si_{2} [Fig. 3a, b]. The excellent agreement between our experiment and calculations demonstrates that the AF fluctuations in CeCu_{2}Si_{2} are welldescribed as particlehole excitations, in which quasiparticles below the Fermi level are excited to unoccupied states above the Fermi level. In our LDA+U calculations (without the renormalization by r), the band bottom is around −15 meV. Therefore, the crossover from dispersive to columnar behavior that occurs at around 60 meV is dominated by states above the Fermi level. Comparing states just above the Fermi level (E ≲ 30 meV) with those well above the Fermi level (E > 50 meV) reveals that the latter are much lighter, characteristic of conduction bands from nonef orbitals [Fig. 3d]. Therefore, the crossover from dispersive excitations at lowenergies to columnar excitations at highenergies in CeCu_{2}Si_{2} reflects the electron band’s reduction of forbital content above the Fermi level (see Supplementary Fig. 2 for the partial density states of CeCu_{2}Si_{2} from our LDA+U calculations). For other values of k_{z}, the band experiences a similar loss of forbital content for E > 50 meV, although more complex behaviors are seen closer to the Fermi level.
Temperature evolution of lowenergy magnetic excitations
Maps of the [H, H, L] plane for E = 0.3 meV at T = 0.3 K (T < T_{c}) and 1.6 K (T > T_{c}) are compared in Fig. 4a, b, with their difference shown in Fig. 4c. Since E = 0.3 meV is above the energy window of the spin resonance mode in CeCu_{2}Si_{2} (E_{r} ≈ 0.2 meV)^{18}, magnetic excitations in the superconducting and normal states are similar. Examining the difference of excitations measured at T = 0.3 K and 1.6 K nonetheless reveals a subtle shift of magnetic spectral weight toward the Brillouin zone center along the (H, H) direction upon cooling. The systematic presence of such behavior across multiple Brillouin zones [Fig. 4c] demonstrates this behavior to be an intrinsic property of CeCu_{2}Si_{2}. Cuts along the (H, H, 1.5) direction are compared in Fig. 4d for T = 0.3 and 1.6 K, and their difference is shown in Fig. 4e. Such a temperaturedependent shift is similar to the shift of ordering vector^{36} and magnetic excitations^{35} in Atype CeCu_{2}Si_{2}, which also move towards the Brillouin zone center upon cooling. These observations can be naturally understood now we have shown that magnetic excitations in CeCu_{2}Si_{2} arise from heavy quasiparticles, and result from a combination of the intrinsic asymmetry of the magnetic dispersion and depletion of the electronic density of states near the Fermi level upon cooling (see Supplementary Note 2 and Supplementary Fig. 3 for details). Such a depletion occurs in Atype CeCu_{2}Si_{2} due to a spindensitywave gap, and in Stype CeCu_{2}Si_{2} due to a superconducting gap.
Discussion
Our experimental observation of magnetic excitations extending up to at least E = 5.5 meV in CeCu_{2}Si_{2} demonstrates quasi2D magnetic fluctuations with an energy scale much larger than the superconducting pairing energy to be a common feature in unconventional superconductors. As the bandwidth of magnetic excitations is captured by effective magnetic interactions, which in turn determine the saving of magnetic exchange energy in the superconducting state ΔE_{mag}, the highenergy magnetic excitations observed in our work suggest a ΔE_{mag} that is at least as large as previously reported^{18}. The commonality of a much larger ΔE_{mag} compared to the superconducting condensation energy ΔE_{SC} [Table 1] and the presence of a spin resonance mode^{2,23,24} across different families of unconventional superconductors, suggest a common pairing mechanism and favors a signchanging superconducting order parameter such as d + d^{44} or s^{±}^{45,46} for CeCu_{2}Si_{2}, rather than s^{++}^{47}. To conclusively distinguish between these scenarios, it is important to study the dispersion of the spin resonance mode in comparison with theoretical results under different pairing symmetries to test its spinexcitonic nature^{60,61,62}. Our work presents a model that captures the normal state magnetic excitations of CeCu_{2}Si_{2}, and is consistent with magnetically driven superconductivity in CeCu_{2}Si_{2}.
Given that the magnetic excitations in Atype CeCu_{2}Si_{2} and the superconducting and normal states of Stype CeCu_{2}Si_{2} are similar for E ≳ 0.4 meV^{35,37}, we expect the observed high energy excitations in our Stype CeCu_{2}Si_{2} to also be present in Atype CeCu_{2}Si_{2}, as well as compositions in between. In addition, columnar spin excitations near τ are also seen in CeNi_{2}Ge_{2}, although compared to CeCu_{2}Si_{2} no lowenergy dispersive features were reported^{63}. Since CeNi_{2}Ge_{2} is paramagnetic and relatively far away from an AF QCP, this suggests that compared to the lowenergy dispersive excitations, the columnar excitations at high energies are more robust upon tuning away from the QCP. In both cuprate and ironbased superconductors, the quasi2D high energy magnetic excitations remain robust when tuning towards the superconducting state, while the lowenergy excitations may change dramatically. Therefore, the prevalence of robust highenergy magnetic excitations suggests shortrange 2D magnetic correlations provide a backdrop from which unconventional superconductivity emerges, while lowenergy AF fluctuations and electronic structure can range from strongly 2D to having significant 3D features and are more tunable. This is analogous to conventional superconductors, in which a large Debye cutoff energy provides a backdrop for potential hightemperature superconductivity.
Methods
Sample preparation and inelastic neutron scattering measurements
Several rodshaped Stype CeCu_{2}Si_{2} single crystal samples were grown using a vertical optical heating floating zone method [Supplementary Fig. 4a]^{49}. To avoid Si excess which results in Atype CeCu_{2}Si_{2}, we used a highpressure Ar atmosphere and a relatively small overheating of the floating zone beyond the melting temperature, which effectively reduces Cu evaporation. Using inductively coupled plasma atomicemission spectroscopy, we determined the atomic percentage of our samples to be Ce:20.1(1)%, Cu:40.3(1)%, and Si:39.6(1)%. This stoichiometry is found to be consistent across several pieces of our samples, indicating they are dominantly Stype^{64}, in agreement with previous transport measurements^{49}. Specific heat measurements were carried out for several pieces of our CeCu_{2}Si_{2} samples [Supplementary Fig. 4c], all exhibiting a specific heat jump below T_{c} ≈ 0.5 K, different from Atype and A/Stype CeCu_{2}Si_{2} samples^{36,65}. The magnitude of the specific heat jump exhibits some sample dependence, possibly due to parts of the samples being nonsuperconducting. The specific heat measurements evidence our CeCu_{2}Si_{2} samples are dominantly Stype, without prominent signatures of antiferromagnetism. While the presence of a minority phase of Atype or A/Stype CeCu_{2}Si_{2} is difficult to rule out, the highenergy magnetic excitations uncovered in our work should also be present in Atype and A/Stype CeCu_{2}Si_{2}, as discussed above. Therefore, the possible presence of such minority phases will not affect the conclusions of our work.
We cut the rodlike samples into segments of a few centimeters and used the E3 fourcircle neutron diffractometer to identify singlegrain pieces by mapping the ϕ and χ rotation angles, with the scattering angle 2θ adjusted to the scattering angle of an intense structural Bragg peak. We then coaligned four such segments in the [H, H, L] plane, as shown in Supplementary Fig. 4b.
Inelastic neutron scattering measurements were carried out using the MACS^{50} at the NIST center for neutron research in Gaithersburg, MD. Our measurements were carried out using fixed E_{f} = 3 or 5 meV, with Be filters placed after the sample for both E_{f} and before the sample for E_{f} = 3 meV. MACS consists of 20 spectroscopic detectors, and by rotating the sample and shifting the detectors, a map of the scattering plane at a fixed energy transfer can be efficiently constructed. The doublebounce analyzers are vertically focused, while the monochromator is doubly focused. Instrumental energy resolutions at the elastic line are ΔE ≈ 0.14 meV for E_{f} = 3 meV, and ΔE ≈ 0.35 meV for E_{f} = 5 meV. Sample alignment is confirmed on MACS for the (110) and (002) structural Bragg peaks. As shown in Supplementary Fig. 5, our samples are reasonably wellaligned for sample arrays used in inelastic neutron scattering measurements.
Extraction of experimental and calculated dispersions
To extract the experimental dispersion of magnetic excitations in CeCu_{2}Si_{2}, cuts along (H, H, 1.48) were obtained by binning data with 1.38 < L < 1.58 and fit using two Gaussian peaks
The same expression with an additional constant term is used to extract the calculated dispersion. In these fittings, x = 0 corresponds to (0, 0), x = 1 corresponds to (1, 1) and δ is the fit peak center position. Representative fits to our experimental data are shown in the middle column of Fig. 3, and representative fits to our theoretical results are shown in Supplementary Fig. 6.
LDA+U band structure
The LDA+U band structure calculations were performed using the fullpotential augmented planewave plus local orbital method as implemented in WIEN2k^{66}. The Perdew–Burke–Ernzerhof exchangecorrelation energy^{67} was used with spinorbit coupling and an effective onsite Coulomb interaction U = 5 eV^{68}. The orbital characters were obtained using WANNIER90 code^{69} via WIEN2WANNIER interface^{70}. Our LDA+U band structure was used previously to study the pairing symmetry of CeCu_{2}Si_{2}^{46}, and is similar to band structures in previous LDA+U calculations^{40,45} and from the renormalized band approach^{71}. As the felectrons are itinerant in our LDA+U calculations, the obtained Fermi surfaces are “large”.
We note that there is a subtle difference in the band structures of refs. ^{45} and^{46}, with the latter used in the calculations of this work. Comparing the heavy electron Fermi surfaces in these two works, there is an extra ringlike Fermi surface in ref. ^{46} around the Γ point (Supplementary Fig. 7). This difference results from details in implementing the calculations, and affects neither key features of the band structure nor the expected physics, which is dominated by the cylindrical heavy electron Fermi surface common to refs. ^{45} and^{46}.
The partial density of states of CeCu_{2}Si_{2} from our LDA+U calculations is shown in Supplementary Fig. 2. As can be seen, the Cef_{5/2} density of states is mainly located just above the Fermi level, and decreases rapidly above 50 meV, becoming increasingly small around 100 meV. Such an evolution of the partial density of states is consistent with the notion that a change of character of the band states causes the crossover from dispersive to columnar magnetic excitations. However, as the partial density of states contains contributions from all the electronic states, and not just the wellnested regions that give rise to the magnetic excitations, the signatures for such a change is not as clear in the partial density of states compared to Fig. 3d.
Calculation of magnetic excitations in CeCu_{2}Si_{2}
The bare magnetic susceptibility with four indices is:
where \({a}_{n}^{\mu }({\bf{k}})\), ϵ_{n}(k) are the unitary matrices diagonalizing H_{0} and the energy dispersion, respectively. The sum over n is taken over the entire band index. Using the Matsubara frequency sum rule and the FermiDirac function \({n}_{F}(\epsilon )=\frac{1}{{e}^{\beta \epsilon }+1}\), we get:
using iω → ω + iη, we have
The transverse RPA susceptibility for a multiband system is:
This is in fact a BetheSalpeter equation where \(\hat{U}\) is a n^{2} × n^{2} matrix with orbital number n.
Onsite interactions \(U=U^{\prime}\)=0.25 eV, \(J=J^{\prime} =0\) eV, and a 20 × 20 × 20 kmesh were used in our RPA calculations, similar to previous work^{45}. The U values used in our RPA calculations are smaller than those in our LDA + U calculations to avoid the divergence of RPA magnetic susceptibility. The key features of the RPA susceptibility are mostly determined by the bare susceptibility χ_{0} (Supplementary Fig. 8), which already contains the essential features of the magnetic susceptibility, and are not strongly affected by the interaction term U.
Disclaimer: The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The computer codes used for the LDA+U band structure calculations in this study are available from the corresponding authors upon reasonable request.
References
Steglich, F. et al. Superconductivity in the presence of strong Pauli paramagnetism: CeCu_{2} Si_{2}. Phys. Rev. Lett. 43, 1892–1986 (1979).
Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).
Stewart, G. R. Unconventional superconductivity. Adv. Phys. 66, 75–196 (2017).
Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: physics of hightemperature superconductivity. Rev. Mod. Phys. 78, 17–84 (2006).
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to hightemperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Sidis, Y. et al. Inelastic neutron scattering study of spin excitations in the superconducting state of high temperature superconductors. Comptes Rendus Phys. 8, 745–762 (2007).
Stewart, G. R. Superconductivity in iron compounds. Rev. Mod. Phys. 83, 1589–1652 (2011).
Dai, P. C. Antiferromagnetic order and spin dynamics in ironbased superconductors. Rev. Mod. Phys. 87, 855–896 (2015).
Fernandes, R. M., Chubukov, A. V. & Schmalian, J. What drives nematic order in ironbased superconductors? Nat. Phys. 10, 97–104 (2014).
Si, Q., Yu, R. & Abrahams, E. Hightemperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. 1, 16017 (2016).
Pfleiderer, C. Superconducting phases of f electron compounds. Rev. Mod. Phys. 81, 1551–1624 (2009).
Stockert, O., Kirchner, S., Steglich, F. & Si, Q. Superconductivity in Ce and UBased 122 heavyfermion compounds. J. Phys. Soc. Jpn. 81, 011001 (2012).
White, B. D., Thompson, J. D. & Maple, M. B. Unconventional superconductivity in heavyfermion compounds. Phys. C. 514, 246–278 (2015).
Moriya, T. & Ueda, K. Antiferromagnetic spin fluctuation and superconductivity. Rep. Prog. Phys. 66, 1299–1341 (2003).
Demler, E. & Zhang, S. C. Quantitative test of a microscopic mechanism of hightemperature superconductivity. Nature 396, 733–735 (1998).
Woo, H. et al. Magnetic energy change available to superconducting condensation in optimally doped YBa_{2} Cu_{3} O_{6.95}. Nat. Phys. 2, 600–604 (2006).
Stock, C., Broholm, C., Hudis, J., Kang, H. J. & Petrovic, C. Spin resonance in the d wave superconductor CeCoIn_{5}. Phys. Rev. Lett. 100, 087001 (2008).
Stockert, O. et al. Magnetically driven superconductivity in CeCu_{2} Si_{2}. Nat. Phys. 7, 119–124 (2011).
Wang, M. et al. Doping dependence of spin excitations and its correlations with hightemperature superconductivity in iron pnictides. Nat. Commun. 4, 2874 (2013).
Leiner, J. et al. Modified magnetism within the coherence volume of superconducting Fe_{1+δ} Se_{x} Te_{1−x}. Phys. Rev. B 90, 100501(R) (2014).
Carr, S. V. et al. Electron doping evolution of the magnetic excitations in NaFe_{1−x} Co_{x} As. Phys. Rev. B 93, 214506 (2016).
Dahm, T. et al. Strength of the spinfluctuationmediated pairing interaction in a hightemperature superconductor. Nat. Phys. 5, 217–221 (2009).
Eschrig, E. The effect of collective spin1 excitations on electronic spectra in highT_{c} superconductors. Adv. Phys. 55, 47–183 (2006).
Yu, G., Li, Y., Motoyama, E. M. & Greven, M. A universal relationship between magnetic resonance and superconducting gap in unconventional superconductors. Nat. Phys. 5, 873 (2009).
Waßer, F. et al. Strong spin resonance mode associated with suppression of soft magnetic ordering in holedoped Ba_{1−x} Na_{x} Fe_{2} As_{2}. npj Quantum Mater. 4, 59 (2019).
Birgeneau, R. J., Stock, C., Tranquada, J. M. & Yamada, K. Magnetic neutron scattering in holedoped cuprate superconductors. J. Phys. Soc. Jpn. 75, 111003 (2006).
Lipscombe, O. J., Hayden, S. M., Vignolle, B., McMorrow, D. F. & Perring, T. G. Persistence of highfrequency spin fluctuations in overdoped superconducting La_{2−x} Sr_{x} CuO_{4} (x = 0.22). Phys. Rev. Lett. 99, 067002 (2007).
Fujita, M. et al. Progress in neutron scattering studies of spin excitations in highT_{c} cuprates. J. Phys. Soc. Jpn. 81, 011007 (2012).
Le Tacon, M. et al. Intense paramagnon excitations in a large family of hightemperature superconductors. Nat. Phys. 7, 725–730 (2011).
Le Tacon, M. et al. Dispersive spin excitations in highly overdoped cuprates revealed by resonant inelastic xray scattering. Phys. Rev. B 88, 020501(R) (2013).
Dean, M. P. M. et al. Persistence of magnetic excitations in La_{2−x} Sr_{x} CuO_{4} from the undoped insulator to the heavily overdoped nonsuperconducting metal. Nat. Mater. 12, 1019–1023 (2013).
Liu, M. et al. Nature of magnetic excitations in superconducting BaFe_{1.9} Ni_{0.1} As_{2}. Nat. Phys. 8, 376–381 (2012).
Zhou, K. J. et al. Persistent highenergy spin excitations in ironpnictide superconductors. Nat. Commun. 4, 1470 (2013).
Gegenwart, P. et al. Breakup of heavy fermions on the brink of phase A in CeCu_{2} Si_{2}. Phys. Rev. Lett. 81, 1501 (1998).
Arndt, J. et al. Spin fluctuations in normal state CeCu_{2} Si_{2} on approaching the quantum critical point. Phys. Rev. Lett. 106, 246401 (2011).
Stockert, O. et al. Nature of the a phase in CeCu_{2} Si_{2}. Phys. Rev. Lett. 92, 136401 (2004).
Huesges, Z. et al. Robustness of magnons near the quantum critical point in the heavyfermion superconductor CeCu_{2} Si_{2}. Phys. Rev. B 98, 134425 (2018).
Eremin, I., Zwicknagl, G., Thalmeier, P. & Fulde, P. Feedback spin rsonance in superconducting CeCu_{2} Si_{2} and CeCoIn_{5}. Phys. Rev. Lett. 101, 187001 (2008).
Akbari, A. & Thalmeier, P. Spin excitations in the fully gapped hybridized two band superconductor. Annals of Physics 428, 168433 (2021).
Kittaka, S. et al. Multiband superconductivity with unexpected deficiency of nodal quasiparticles in CeCu_{2} Si_{2}. Phys. Rev. Lett. 112, 067002 (2014).
Yamashita, T. et al. Fully gapped superconductivity with no sign change in the prototypical heavyfermion CeCu_{2} Si_{2}. Sci. Adv. 3, e1601667 (2017).
Takenaka, T. et al. Fullgap superconductivity robust against disorder in heavyfermion CeCu_{2} Si_{2}. Phys. Rev. Lett. 119, 077001 (2017).
Kittaka, S. et al. Thermodynamic study of gap structure and pairbreaking effect by magnetic field in the heavyfermion superconductor CeCu_{2} Si_{2}. Phys. Rev. B 94, 054514 (2016).
Pang, G. et al. Fully gapped dwave superconductivity in CeCu_{2} Si_{2}. Proc. Natl Acad. Sci. U.S.A. 115, 5343–5347 (2017).
Ikeda, H., Suzuki, M. T. & Arita, R. Emergent loopnodal s^{±} wave superconductivity in CeCu_{2} Si_{2}: similarities to the ironbased superconductors. Phys. Rev. Lett. 114, 147003 (2015).
Li, Y. et al. Gap symmetry of the heavyfermion superconductor CeCu_{2} Si_{2} at ambient pressure. Phys. Rev. Lett. 120, 217001 (2018).
Tazai, R. & Kontani, H. Fully gapped swave superconductivity enhanced by magnetic criticality in heavyfermion systems. Phys. Rev. B 98, 205107 (2018).
Tazai, R. & Kontani, H. Hexadecapole fluctuation mechanism for swave heavy fermion superconductor CeCu_{2} Si_{2} : interplay between intra and interorbital cooper pairs. J. Phys. Soc. Jpn. 88, 063701 (2019).
Cao, C. D. et al. Single crystal growth of the CeCu_{2} Si_{2} intermetallic compound by a vertical floating zone method. Cryst. Growth Des. 11, 431–435 (2011).
Rodriguez, J. A. et al. Measurement science and technology MACSa new high intensity cold neutron spectrometer at NIST. Meas. Sci. Technol. 19, 034023 (2008).
Horigane, K. et al. Spin excitations in holeoverdoped ironbased superconductors. Sci. Rep. 6, 33303 (2016).
Tomiyoshi, S., Yamaguchi, Y., Ohashi, M., Cowley, E. R. & Shirane, G. Magnetic excitations in the itinerant antiferromagnets Mn_{3} Si and Fedoped Mn_{3} Si. Phys. Rev. B 36, 2181–2189 (1987).
Chen, X. et al. Unconventional Hund metal in a weak itinerant ferromagnet. Nat. Commun. 11, 3076 (2020).
de’ Medici, L., Giovannetti, G. & Capone, M. Selective Mott physics as a key to iron superconductors. Phys. Rev. Lett. 112, 177001 (2014).
Yi, M., Zhang, Y., Shen, Z. X. & Lu, D. Role of the orbital degree of freedom in ironbased superconductors. npj Quant. Mater. 2, 57 (2017).
Zhang, C. et al. Effect of pnictogen height on spin waves in iron pnictides. Phys. Rev. Lett. 112, 217202 (2014).
Goremychkin, E. A. et al. Coherent band excitations in CePd_{3} : a comparison of neutron scattering and ab initio theory. Science 359, 186–191 (2018).
Butch, N. P. et al. Symmetry and correlations underlying hidden order in URu_{2} Si_{2}. Phys. Rev. B 91, 035128 (2015).
Portnichenko, P. Y., Cameron, A. S. & Inosov, D. S., NeutronScattering Studies of Spin Dynamics in Pure and Doped CeB_{6} ", Rare Earth Borides, edited by D. S. Inosov, Jenny Stanford Publishing (2021).
Song, Y. et al. Robust upward dispersion of the neutron spin resonance in the heavy fermion superconductor Ce_{1−x} Yb_{x} CoIn_{5}. Nat. Commun. 7, 12774 (2016).
Song, Y. et al. Nature of the spin resonance mode in CeCoIn_{5}. Commun. Phys. 3, 98 (2020).
Hong, W. et al. Neutron spin resonance in a quasitwodimensional ironbased superconductor. Phys. Rev. Lett. 125, 117002 (2020).
FÅk, B., Flouquet, J., Lapertot, G., Fukuhara, T. & Kadowaki, H. Magnetic correlations in singlecrystalline CeNi_{2} Ge_{2}. J. Phys.: Condens. Matter 12, 5423–5435 (2000).
Steglich, F. et al. New observations concerning magnetism and superconductivity in heavyfermion metals. Physica B 223224, 1–8 (1996).
Lengyel, E., Nicklas, M., Jeevan, H. S., Geibel, C. & Steglich, F. Pressure tuning of the interplay of magnetism and superconductivity in CeCu_{2} Si_{2}. Phys. Rev. Lett. 107, 057001 (2011).
Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J., WIEN2k: An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Technische Universität Wien, 2014).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Anisimov, V. I., Aryasetiawa, F. & Lichtenstein, A. I. Firstprinciples calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method. J. Phys.: Condens. Matter 9, 767–808 (1997).
Mostofi, A. A. et al. wannier90: a tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).
Kuneš, J. et al. Wien2wannier: from linearized augmented plane waves to maximally localized Wannier functions. Comput. Phys. Commun. 181, 1888–1895 (2010).
Zwicknagl, G. & Pulst, W. CeCu_{2} Si_{2} : renormalized band structure, quasiparticles and cooperative phenomena. Physica B 186–188, 895–898 (1993).
Acknowledgements
We would like to thank Binod K. Rai and Emilia Morosan for help with specific heat measurements. Neutron scattering work at Rice is supported by the U.S. Department of Energy, BES under Grant No. DESC0012311. The singlecrystal characterization work at Rice is supported by Robert A. Welch Foundation Grant No. C1839. The work at NWPU is supported by National Key Research and Development Program of China (2016YFB1100101), National Natural Science Foundation of China (51971180, 51971179), Key Research and Development Program of Shaanxi (2021KWZ13), and Guangdong Science and Technology program (2019B090905009). The work at Lawrence Berkeley National Laboratory and the University of California, Berkeley is supported by the Office of Science, Office of Basic Energy Sciences (BES), Materials Sciences and Engineering Division of the U.S. Department of Energy (DOE) under Contract No. DEAC0205CH11231 within the Quantum Materials Program (KC2202) and BES, U.S. DOE Grant No. DEAC0376SF008. The work at IOP, Beijing is supported by the National Key R&D Program of China (2017YFA0303103) and the National Natural Science Foundation of China (11974397, 11774401). Access to MACS was provided by the Center for HighResolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No. DMR1508249.
Author information
Authors and Affiliations
Contributions
P.D., Y.Song, and C.C. conceived the project. C.C. and W.L. prepared the samples. Z.Y. and Y.Song coaligned the samples. W.W., Y.Song, C.C., and Y.Q. carried out the experiments. W.W. and Y.Song analyzed the data. Y.X., Y.Sheng, and Y.Y. carried out the theoretical analyses. Y.Song, P.D., and Y.Y. wrote the paper with input from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Song, Y., Wang, W., Cao, C. et al. Highenergy magnetic excitations from heavy quasiparticles in CeCu_{2}Si_{2}. npj Quantum Mater. 6, 60 (2021). https://doi.org/10.1038/s4153502100358x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4153502100358x
This article is cited by

Fractional and composite excitations of antiferromagnetic quantum spin trimer chains
npj Quantum Materials (2022)