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Although conventional superconductivity (SC) is generally incompatible with magnetism, magnetic excitations seem to play an important role in the Cooper pair formation of unconventional superconductors such as the high-Tc cuprates or the low-Tc organic and heavy-fermion superconductors. Since the discovery of SC in CeCu2Si2 (ref. 1), antiferromagnetic (AF) spin excitations have been proposed as a viable mechanism for SC (refs 2, 3, 4). The discovery of SC at the boundary of AF order in CePd2Si2 (ref. 5) has pushed this notion into the framework of AF quantum criticality6. Unfortunately, such quantum critical points (QCPs) proximate to heavy-fermion superconductors typically arise under pressure, which makes it difficult to probe their magnetic excitation spectrum.

Here, we report a detailed study of the magnetic excitations in CeCu2Si2, which exhibits SC below Tc≈0.6 K. This prototypical heavy-fermion compound is ideally suited for our purpose, as SC here is in proximity to an AF QCP already at ambient pressure (see Fig. 1a). As shown in Fig. 1b, CeCu2Si2 crystallizes in a structure with body-centred tetragonal symmetry and is one of the best studied heavy-fermion superconductors and well characterized by low-temperature transport and thermodynamic measurements7. Moreover, those measurements in the field-induced normal state have already provided evidence that the QCP in this compound is of the three-dimensional (3D) spin-density-wave (SDW) type8. The spatial anisotropy of the spin fluctuations in superconducting CeCu2Si2 was measured at T=0.06 K and at an energy transfer ℏω=0.2 meV and is shown in Fig. 1c. These magnetic correlations exhibit only a small anisotropy (a factor of 1.5) in the correlation lengths between the [110] and the [001] direction. Therefore, these quite isotropic spin fluctuations are in line with thermodynamic and transport measurements, exhibiting or ρ−ρ0=A Tα, α=1–1.5 (refs 8, 9), and strongly support a 3D quantum-critical SDW scenario10. We are able to identify the magnetic excitations in the normal state of paramagnetic, superconducting CeCu2Si2, around the incommensurate wave vector11 of the SDW order nearby in the phase diagram (see Fig. 1a), and further establish the system’s proximity to the AF QCP through the observation of a considerable slowing down in the spin dynamics. Going into the superconducting state, a spin gap opens out of a broadened quasielastic response that extends to much higher frequencies (×10 the superconducting gap). These data allow us to establish a saving in the AF exchange energy that is considerably larger than the superconducting condensation energy, thereby providing the first demonstration of the nearly quantum-critical AF excitations as the main driving force for unconventional SC.

Figure 1: Schematic phase diagram around the QCP, crystal structure and nearly isotropic spin fluctuations of CeCu2Si2.
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a, Schematic T–g phase diagram of CeCu2Si2 in the vicinity of the quantum critical point (QCP) where the antiferromagnetic (AF) phase vanishes as a function of the effective coupling constant g. Superconductivity (SC) is observed around the QCP and extends far into the paramagnetic (PM) regime. The composition as well as hydrostatic pressure can be used to change the coupling constant g and to tune the system to the QCP. The positions of the A-type and the S-type single crystals in the phase diagram are marked. b, Tetragonal crystal structure (space group: I4/m m m) of CeCu2Si2. The nearest- and next-nearest-neighbour interactions between the cerium atoms are labelled I1 and I2. It should be noted that the distances between next-nearest-neighbour Ce atoms in the basal plane and out-of-plane are almost identical. c, The spin fluctuations at T=0.06 K and B=0 and at an energy transfer ℏω=0.2 meV. The anisotropy factor between the [110] and the [001] directions is about 1.5. Note that the correct aspect ratio [110]*:[001]* has been taken into account although the axes are labelled in reciprocal lattice units (rlu). Black dots mark the (Q,ω) positions at which data were taken.

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Superconductivity and antiferromagnetism in CeCu2Si2

The SC in CeCu2Si2 we consider is close to the AF QCP, and is to be contrasted with a second superconducting dome appearing at high pressure that is thought to be associated with a valence instability and the concomitant fluctuations9,12. This AF QCP is located within the narrow homogeneity range of the ‘122’ phase in the ternary chemical Ce–Cu–Si phase diagram of this tetragonal compound13. Correspondingly, we can prepare homogeneous samples (with slight Cu deficit) from the antiferromagnetically ordered side (A-type) and (with tiny Cu excess) from the paramagnetic, superconducting side of the QCP (S-type); in contrast, crystals very close to the 1:2:2 stoichiometry exhibit a ground state where SC and AF compete with each other without microscopic coexistence (A/S-type)13. The AF order was found to be an incommensurate SDW (ref. 11). At T=0.05 K, well below TN≈0.8 K, the A-type CeCu2Si2exhibits an ordered magnetic moment μord≈0.1 μB and an incommensurate propagation vector τ≈(0.215 0.215 0.53). The latter can be ascribed to a nesting wave vector of the renormalized Fermi surface. However, a full microscopic description of the magnetic order remains to be addressed.

To study the superconducting state in CeCu2Si2 in detail, neutron scattering results (see Supplementary Information) on an S-type single crystal are presented in this Letter. A previous experiment was severely hampered by a large experimental background and a low signal-to-background ratio14. Thermodynamic measurements confirmed that this crystal is superconducting with a Tc≈0.6 K and an upper critical field Bc2(T=0)<2 T (see Supplementary Information). Elastic neutron scattering measurements did not feature resolution-limited magnetic Bragg peaks in S-type CeCu2Si2 in accordance with thermodynamic measurements. However, at positions where magnetic satellite peaks are observed in A-type CeCu2Si2 (ref. 11), for example, at QAF=(0.215 0.215 1.458), relative to a nearby nuclear Bragg reflection G (QAF=G±τ), the S-type crystal exhibits quite weak correlation peaks at low temperatures14. They are still present above Tc and disappear at T≈0.8 K, very similar to the behaviour of the SDW order in A-type CeCu2Si2 (ref. 11). Although these peaks were found to be purely elastic within the energy resolution (≈57 μeV), their linewidth in Q space is considerably broadened, corresponding to a correlation length of 50–60 Å, comparable to the superconducting coherence length of order 100 Å (ref. 15). Thus, static magnetically ordered regions seem to exist in a fairly small part of the sample and are separated from the surrounding superconducting regions.

Spin dynamics in CeCu2Si2

We probe the magnetic response of CeCu2Si2 through extensive inelastic neutron scattering measurements around Q=QAF=(0.215 0.215 1.458), as no appreciable magnetic intensity has been detected elsewhere in the Brillouin zone. Figure 2a shows energy scans at this QAF position and at a general position, Q=Qarb=(0.1 0.1 1.6), where no correlation peaks emerge, but which has the same |Q|. Both data sets were recorded in the superconducting state at T=0.07 K. At Qarb only the incoherent elastic background contribution is seen with the instrumental resolution, and no magnetic intensity could be detected. In contrast, at QAF the response shows a strong inelastic signal with a long intensity tail extending beyond ℏω=2 meV (see inset of Fig. 2a). The missing spectral weight at low energies is an indication of a spin excitation gap in the superconducting state. The spectrum recovers the missing weight at the gap edge, thereby constituting an inelastic line. The data can be described by a quasielastic Lorentzian line with a spin excitation gap ℏωgap≈0.2 meV and with a density of states as for the electronic gap of a d-wave Bardeen–Cooper–Schrieffer (BCS) superconductor (solid lines in Figs 2a and 3a). ℏωgap≈3.9kBTc is found to be 10% smaller than the value predicted for a weak-coupling d-wave superconductor16 and falls 20% below 2Δ0/kBTc=5.0, as determined by copper nuclear quadrupole resonance for CeCu2Si2 (refs 17, 18). To unambiguously relate the inelastic magnetic excitation to the superconducting state, it was necessary to carry out additional measurements in the normal state.

Figure 2: Inelastic neutron scattering spectra in the normal and superconducting states of CeCu2Si2.
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a,b, Energy scans (neutron intensity S=Sela+Sqe/ine,mag versus energy transfer ℏω) in S-type CeCu2Si2 at Q=QAF=(0.215 0.215 1.458) in the superconducting state at T=0.07 K, B=0 (a) and in the normal state at T=0.8 and 1.7 K, B=0 and T=0.07 K, B=2 T (b). For comparison the magnetic response at an arbitrary, general Q position Q=Qarb=(0.1 0.1 1.6) at T=0.07 K, B=0 is also plotted in a. The inset in a shows the magnetic response at QAF (T=0.07 K, B=0) extending beyond ℏω=2 meV. The solid lines represent fits to the data comprising the incoherent and coherent elastic signal Sela and the quasielastic/gapped inelastic magnetic response Sqe/ine,mag (dashed lines) with Lorentzian line shape convolved with the resolution. The gapped magnetic response at T=0.07 K, B=0 has been modelled by a quasielastic Lorentzian line, taking into account a spin gap with a value ℏωgap and an enhanced density of states above the gap, as for the electronic gap of a BCS superconductor. The error bars represent the statistical error.

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Figure 3: Magnetic response, relaxation rate and spin gap at the AF wave vector of CeCu2Si2.
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a, Magnetic response Sqe/ine,mag at QAF and T=0.07 K in the superconducting and the normal state, that is, at B=0 and B=2 T, as extracted from the data shown in Fig. 2. The data have been put on an absolute intensity scale (see Supplementary Information). Below ℏω≈0.1 meV (approximately 1.6 times the instrumental resolution) the errors in Smag increase strongly (some data points fall even outside the plotted range), because the strong elastic scattering Sela is subtracted from the total scattering to obtain Smag and also because of small uncertainties in the resolution function. These uncertainties are the same for both data sets and are irrelevant, as only the difference is analysed for the estimation of the exchange energy saving. b, Linewidth Γ versus temperature T of the quasielastic magnetic response at QAF in the normal state, as yielded by fits to the data shown in Fig. 2. Plotted here is Γ(T)–Γ0 versus T, with Γ0=0.112 meV. The solid line Γ(T)–Γ0=a T3/2 (with a=0.061 meV K−1.5) is the expected behaviour near a 3D SDW QCP. c, Temperature dependence of the spin excitation gap ℏωgap at QAF together with the scaled d-wave BCS superconducting gap function (solid line). The error bars denote the statistical error.

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Energy scans recorded at QAF in the normal state are shown in Fig. 2b. Notably, independent of how the normal state is reached, that is, above Tc at T=0.8 K and B=0 or above Bc2 at T=0.07 K and B=2 T, the magnetic response is almost identical and seems to be quasielastic. The fits to the quasielastic magnetic response with a Lorentzian line shape give a good description of the data, as seen in Figs 2b and 3a. With increasing temperature in the normal state the magnetic response weakens in intensity and broadens considerably. Starting from Γ≈0.11 meV at T=0.07 K the linewidth of the quasielastic response at QAF increases to Γ≈0.235 meV at T=1.7 K (Fig. 3b). This considerable slowing down of the response when lowering the temperature indicates the proximity of S-type CeCu2Si2 to the AF QCP. Γ(T) extrapolates to a finite value at T0, as the S-type single crystal is located on the paramagnetic side of the QCP (see Fig. 1a). A related critical slowing down was observed in magnetically ordered A-type CeCu2Si2 (ref. 19).

The fact that the magnetic excitation gap disappears in the normal state, that is, above Tc, and also above Bc2 at low temperatures, where the magnetic short-range correlations still persist, gives direct evidence that the spin gap ℏωgap is related to the superconducting state. Its temperature variation is plotted in Fig. 3c and has been derived from fits to the data shown in Figs 2a and 3a and additional scans. As indicated by the solid line, ℏωgap follows, within the error bars, the BCS form for the superconducting gap amplitude 2Δ(T).

We now turn to the momentum dependence of the magnetic response around QAF in the superconducting state. Figure 4 shows Q scans along (h h 1.458) across QAF recorded at different energy transfers ℏω and at T=0.06 K. The single peak seen at ℏω≈0.2 meV splits on increasing energy transfer into two peaks, which move apart from each other accompanied by a marked decrease in intensity. Fits with two peaks of Gaussian line shape (solid lines) yield a good description of the data. The peak positions for different ℏω, shown in the inset of Fig. 4a, yield a linear dispersion relation. We conclude that the spin excitations are part of an overdamped dispersive mode. Its velocity, as read off the slope of the dispersion curve, vexc=(4.44±0.86) meV Å, is substantially smaller than the strongly renormalized Fermi velocity vF*≈57 meV Å (ref. 15) (1 meV Å=153 m s−1). This indicates a retardation of the coupling between the heavy quasiparticles and the quantum-critical spin excitations.

Figure 4: Dispersion of the magnetic response in superconducting CeCu2Si2.
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a, Wave vector Q dependence of the magnetic response around QAF in S-type CeCu2Si2in the superconducting state at T=0.06 K for different energy transfers ℏω. The scans are shifted by 100 counts with respect to each other. Solid lines denote fits of two peaks with Gaussian line shape to the data and dashed lines are only guides to the eye. From the linewidth at small energy transfers a dynamic correlation length ξ≈25 Å is inferred. Inset: Dispersion of the magnetic excitation around QAF at T=0.06 K as a result of the fits to the Q scans. The solid line indicates a fit to the data with a linear dispersion relation, yielding a velocity vexc=(4.44±0.86) meV Å (for comparison spin-wave velocities in other heavy-fermion metals, UPd2Al3: v=10–15 meV Å (ref. 36), URu2Si2: v≈45 meV Å; ref. 42). The error bars represent the statistical error. b, Colour-coded intensity plot of the data shown in a and additional data, clearly indicating the dispersion of the gapped spin excitation. The black dots mark the (Q,ω) positions at which data were taken.

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Superconducting condensation and exchange energies

The observed spin excitations both below and above Tc allow us to estimate the decrease of magnetic exchange energy in the superconducting state as compared with the putative normal state. This saving of exchange energy is determined as follows20,21:

where ExN and ExS are respectively the exchange energy in the normal and superconducting states, 〈 〉 indicates an average over the first Brillouin zone, and q=(qx,qy,qz) denotes momentum transfer in the first Brillouin zone, that is, Q=G+q. I(q) is the exchange interaction between the localized f-moments and contains nearest (I1) and next-nearest (I2) neighbour terms:

where a and c are the lattice constants and the precise form of f2 is given in the Supplementary Information. The inclusion of the next-nearest-neighbour terms is a consequence of the 3D nature of the spin excitations of CeCu2Si2 (see Fig. 1c). This is different from the cuprate superconductors and, for example, CeCoIn5, where the observed behaviour is predominantly 2D. As described in detail in the Supplementary Information, we find a magnetic exchange energy saving of ΔEx=η 4.8×10−3 meV per Ce (η≈1.25, η being a measure of the SC volume fraction (see Supplementary Information)). This energy saving stems primarily from the spectrum at low energies below the magnetic excitation gap. This follows from the fact that the spin excitations are peaked around the wave vector QAF at which I(q) is positive. Figure 5 illustrates which part of the spectrum of Imχ(QAF,ω) increases/decreases ΔEx. This energy gain must be compared with the superconducting condensation energy ΔEC, which is the difference in internal energy between the (putative) normal and the superconducting state at T=0 (refs 20, 21):

Using the specific-heat data shown in Supplementary Fig. S1a, we find the condensation energy to be η 2.27×10−4 meV per Ce. Compared with the high-Tc cuprates where similar analyses have been carried out22,23,24, the considerably lower energy scales in the heavy-fermion systems enable us to carry out a quantitative analysis of the data in terms of an accessible putative normal state. As noted above, extrapolating the spin excitations from above Tc is in good qualitative agreement with the excitations of the field-driven normal state at the lowest temperatures. Furthermore, the electronic specific heat of both the superconducting and normal state can be reliably determined as phonons do not contribute at such low temperatures. Despite this apparent advantage of heavy-fermion systems, ΔEC and ΔEx have not received much attention in the context of heavy-fermion SC. ΔEC has been determined for CeCoIn5 (ref. 25). The proximity of this compound to quantum criticality is not yet certain, as SC sets in before AF order can develop. Our study represents the first determination of the saving in both the exchange energy and the condensation energy for a superconductor near an AF QCP, as well as for any unconventional low-temperature superconductor.

Figure 5: Schematic plot of the imaginary part of the dynamic spin susceptibility Imχ(QAF,ω) in the normal and superconducting states.
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The dynamic correlation function S(QAF,ω) of Fig. 3a is related to Imχ(QAF,ω) through the fluctuation-dissipation theorem after deconvolving the data with the instrument’s energy resolution function. The blue area (marked with a ‘+’) leads to an increase in ΔEx, whereas the green area (marked with a ‘−’) leads to a decrease in ΔEx. The fact that the opening of the gap contributes to the saving of exchange energy is a consequence of I(QAF)>0 at the wave vector QAF, where ImχN/S(Q,ω) is peaked.

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Our observation that the magnetic exchange energy saving is more than one order of magnitude larger than the condensation energy implies that AF excitations are the primary driving force for SC. A comparable factor of exchange energy saving over condensation energy has recently been observed in the unconventional superconductor YbBa2Cu3O6.6 (ref. 24). As described above, the temperature dependence of ωgap(T) in CeCu2Si2 follows a rescaled BCS form. For a conventional BCS superconductor, where ΘD≫ωgap, the saving in potential energy is enhanced over the condensation energy by a factor that depends logarithmically on the ratio of Debye temperature ΘD and superconducting gap ωgap(T=0) (ref. 26). The corresponding enhancement factor over the condensation energy in CeCu2Si2, where the Kondo temperature TK≈15 K replaces the Debye temperature ΘD, turns out to be two. The fact that the observed magnetic energy saving is more than a factor 20 larger than the condensation energy indicates a large loss in kinetic energy. A natural origin for this loss lies in the Kondo effect, as the kinetic energy of the quasiparticles appears through the Kondo-interaction term. As superconducting pairing in CeCu2Si2 occurs in the spin-singlet channel, the opening of the superconducting gap therefore weakens the Kondo-singlet formation and, by extension, reduces the spectral weight of the Kondo resonance (see Supplementary Information).

Comparison with other unconventional superconductors

Our understanding of the magnetic exchange energy saving in the heavy-fermion superconductor CeCu2Si2 near its AF QCP naturally leads us to ask whether the effect is universal. SC-induced enhancement of the spin-fluctuation spectrum in some frequency range has also been observed in high-Tc cuprates such as YBa2Cu3O7−δ (ref. 27), iron pnictides such as K- or Co-doped BaFe2As2(refs 28, 29) or FeTe1−xSex (refs 30, 31), as well as two other heavy-fermion compounds, UPd2Al3(refs 32, 33) and CeCoIn5 (ref. 25). However, there are some striking differences between the spectrum observed in CeCu2Si2 and those seen in the other superconductors. In contrast to CeCu2Si2, where SC and long-range AF order exclude each other, SC in UPd2Al3 occurs inside the antiferromagnetically ordered part of its magnetic phase diagram, which is far away from any QCP (ref. 34). Whether a QCP underlies SC in the cuprates, the iron pnictides or CeCoIn5, is yet to be established. The normal-state magnetic response of S-type CeCu2Si2 at QAF slows down considerably when lowering the temperature, indicating its proximity to a QCP, and exhibits pronounced dispersion. CeCu2Si2 represents, therefore, the only system in which we can unambiguously establish the link between AF quantum criticality and unconventional SC, even though the effect may well prove to be broadly relevant. In comparison with other heavy-fermion superconductors, the inelastic spin response in CeCu2Si2 is broad in energy and extends beyond ten times the gap value, whereas in CeCoIn5 a rather sharp, resolution-limited spin resonance is found25. Furthermore, unlike CeCoIn5, the temperature dependencies of the spin excitation gap in CeCu2Si2 and UPd2Al3 (ref. 35) do follow the expected BCS form. In comparison with CeCu2Si2, UPd2Al3 also exhibits a dispersive spin excitation starting at the low-energy inelastic line (related to the edge of the spin gap33) with a slightly higher in-plane mode velocity36. However, the situation in the cuprate superconductors is more complex, with an hour-glass-like dispersion of the resonance mode37,38,39.

Experimentally, the most prominent difference between CeCu2Si2 and other unconventional superconductors is the Q position of the spin excitation gap, which is observed in all reported unconventional superconductors at or close to simple commensurate positions with half-integer indices. For example, in UPd2Al3 and CeCoIn5it occurs at commensurate positions, Q=(0 0 1/2) and (1/2 1/2 1/2) respectively25,36. In contrast, in S-type CeCu2Si2 the gapped spin excitations are restricted to the vicinity of the ordering wave vector of the system, τ≈(0.215 0.215 0.53), which is incommensurate, far away from a simple commensurate value. As a result, the opening of a spin gap becomes the main source of exchange energy saving. By extension, an additional excitonic resonance in χS(Q,ω) due to the superconducting state40 would reduce the energy saving. This is a striking difference between CeCu2Si2 on the one hand, and CeCoIn5(ref. 25) and high-Tc cuprate superconductors23 on the other.

Our inelastic neutron scattering experiments in CeCu2Si2 reveal spin excitations associated with the AF (3D-SDW) QCP. These spin excitations are overdamped, dispersive and gapped in the superconducting state. Our quantitative estimate of both the change in magnetic exchange energy and the superconducting condensation energy identifies the AF excitations as a main driving force for SC. At present, AF QCPs are being explored in a variety of strongly correlated electron systems, including the new Fe pnictide superconductors. Ba(Fe1−xCox)2As2 (ref. 41), for instance, exhibits a T–x phase diagram very similar to the T–p phase diagram of CePd2Si2, raising the prospect that AF quantum-critical excitations also drive the superconducting pairing in these new high-Tc superconductors.

Methods

High-resolution inelastic neutron scattering experiments were carried out on the cold-neutron triple-axis spectrometer IN12 at the high-flux reactor of the Institut Laue–Langevin in Grenoble/France. A vertical focusing graphite (002) monochromator and a doubly focused (vertical and horizontal) graphite (002) analyser were used. The horizontal collimation was provided by the neutron guide in front of the monochromator and 60′ before the sample, whereas no collimation was provided in the scattered beam. A liquid-nitrogen-cooled Be filter was placed in the incident neutron beam to reduce higher-order contamination. The measurements were carried out with a fixed final wave vector kf=1.15 Å−1, which corresponds to a final neutron energy Ef=2.74 meV and yields a high energy resolution ΔE≈57 μeV full-width at half-maximum. All experiments were carried out on an S-type CeCu2Si2 single crystal (m≈2 g). The crystal was mounted with the axis vertical on a copper pin attached to the mixing chamber of a dilution refrigerator. The set-up results in a [110]–[001] scattering plane. Data were taken at temperatures between T=0.06 and 1.7 K and in magnetic fields up to B=2.5 T applied along the vertical axis. The inelastic neutron scattering measurements were converted to units of μB2/(meV f.u.) by normalizing the intensities to the incoherent scattering of the sample.