Introduction

In copper-oxide, iron-pnictide and heavy-fermion superconductors, antiferromagnetic (AFM) order can be induced by changes in pressure, doping level or magnetic field. Theories of magnetically mediated Cooper pairing inspired by this observation have received strong support from neutron-scattering experiments that revealed resonant magnetic excitations in the superconducting states of all of these compounds1,2,3,4,5,6. The resonant modes share a common set of characteristics. They are sharp in both energy and momentum, unlike the incoherent particle-hole excitations in a metal, and are located below the superconducting gap, 2Δ. As model calculations indicate that the spectral weight and momentum-space structure of the resonant modes contain essential information about the pairing interaction1,3,6, pairing symmetry7,8,9,10 and condensation energy11 of the superconducting state, research on their origin remains at the frontier of the quest for a comprehensive description of unconventional superconductivity.

Here we report inelastic neutron-scattering (INS) experiments that reveal a resonant mode with identical characteristics in the non-superconducting heavy-fermion metal CeB6, which has been intensely investigated because of its intriguing phase behaviour12,13,14 on the background of an exceptionally simple crystal structure and chemical composition. The electronic states near the Fermi level of CeB6 are composed of localised cerium-4f1 levels hybridised with itinerant cerium-5d and boron-2p electrons15,16, closely analogous to those of the superconductors CeCoIn5 and CeCu2Si2, where prominent resonant modes have been reported4,6. In contrast to these superconducting compounds, however, CeB6 exhibits AFM order below TN=2.3 K17. It is preceded by another phase transition at TQ=3.2 K, whose order parameter has long remained hidden to standard experimental probes such as neutron diffraction12,13. Theoretical work18,19 has attributed both transitions to interactions between the multipolar moments of the Ce-4f electrons mediated by the itinerant conduction electrons, which break the large ground-state degeneracy of the cerium ions in their cubic crystal field and stabilise an antiferro-quadrupolar (AFQ) order20,21,22.

Recent transport, thermodynamic and neutron diffraction experiments23,24,25,26,16 have revealed deviations from predictions of this interacting multipole model of magnetism in CeB6. In particular, weak Bragg intensity has been detected by spin-polarised neutron scattering at the AFQ wave vector, although the concept of quadrupolar ordering forbids such an intensity in the absence of external magnetic fields12,13, so that it is hidden to the elastic neutron scattering. This observation has been ascribed to the magnetic response of the itinerant Ce-5d electrons16. Our discovery of a magnetic resonant mode for T<TN confirms a more pronounced influence of itinerant electrons on the magnetic properties of CeB6 than anticipated by the prevailing theory, and establishes a surprising correspondence to unconventional superconductivity.

Results

Elastic neutron scattering

We start the presentation of our data by comparing the AFM and AFQ magnetic Bragg intensities, measured at and , respectively, in Fig. 1. From the different onset temperatures of the elastic signals, it is clear that they represent distinct order parameters. The elastic intensity at QAFQ is more than an order of magnitude weaker than the AFM Bragg intensity, and exhibits only a weak suppression below TN, indicating that both order parameters coexist in the AFM phase.

Figure 1: Bragg peaks associated with the AFQ and AFM order parameters.
figure 1

(a) Comparison of the elastic signals measured at (left) and (right) at T=1.6 K. The AFQ intensity has been multiplied by a factor of 5 for clarity. (b) Temperature dependence of the corresponding peak intensities that vanish at TQ and TN, respectively. The solid lines are guides to the eyes. The error bars in this and all the following figures correspond to one s.d. of the count rate.

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Manifestation of the resonant magnetic exciton mode

To give an overview of the inelastic-scattering signal, we present in Fig. 2 unprocessed INS intensity maps of the (hhl) plane in reciprocal space, measured in the AFM low-temperature phase (T=1.6 K) at fixed energy transfers of ħω=0.25, 0.5 and 0.9 meV. An intense peak centered at QAFQ dominates the spectrum at ħω=0.5 meV (Fig. 2a). It can be observed at the two equivalent R points and covered by the measurement, which are additionally connected by weaker streaks of intensity that surround the M point. Intensity maps at both lower (Fig. 2b) and higher (Fig. 2c) energies reveal greatly reduced intensities.

Figure 2: Reciprocal space maps.
figure 2

(a) Inelastic signal along the (hhl) plane in reciprocal space, measured in the low-temperature AFM phase (T=1.6 K) at 0.5 meV energy transfer. The labels represent different high-symmetry points, and the dashed lines mark Brillouin zone boundaries. (b,c) The same at lower (0.25 meV) and higher (0.9 meV) energies, respectively. (d,e) The same at two elevated temperatures (2.6 K and 10 K). All panels share the same colour scale.

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The strong concentration of magnetic spectral weight around isolated points in energy-momentum space is markedly different from the continuously dispersive spin waves in conventional antiferromagnets, but closely similar to the behaviour of the resonant modes observed in superconducting compounds below their respective transition temperatures, Tc1,2,3,4,5,6. Moreover, the mode is not centred at the S or Σ points characterising the AFM order, where the magnetic intensity is much weaker, but at the R point that reflects the 'hidden' AFQ phase. In Fig. 2b, one can also observe weak intensity near the S point that originates from conventional spin-wave excitations associated with the AFM phase. At 0.25 meV, their intensity is comparable in magnitude to the signal at the R point, whereas at the resonance energy of 0.5 meV (Fig. 2a), the latter is enhanced more than two-fold, although the spin-wave intensity at the S point can no longer be observed within the statistical noise.

Temperature dependence

As the temperature is increased above TN, the strong intensity at the R point is first sharply suppressed already in the AFQ phase (Fig. 2d) and then continues to diminish gradually with a further temperature increase far into the paramagnetic phase (Fig. 2e). At T=1.6 K, the resonant mode is sharp in energy (Fig. 3a) and is characterised by the intrinsic (resolution-corrected) full width at half maximum of ~0.2 meV. It is followed by a weaker second peak centred around 1.1 meV, with the full width at half maximum of ~0.6 meV. For T > TN, the 0.5 meV peak transforms into a quasi-elastic response27,28 that is typical for the paramagnetic state of heavy-fermion metals5,6. A similar behaviour, but with an exciton mode that is weaker and not as sharp in energy, can also be observed at other points throughout the Brillouin zone (Fig. 3b,c). We can therefore associate the formation of the resonant peak with the opening of a spin gap in the low-energy part of the quasielastic spectrum upon cooling and the consequent spectral-weight transfer from below ~0.35 meV to higher energies. In contrast, the second peak at 1.1 meV at the R point is nearly insensitive to TN and possibly arises from a low-energy crystal-field excitation similar to those previously studied in the AFQ phase.29,28

Figure 3: Unprocessed INS data.
figure 3

(a) Energy dependence of the INS signal in the AFQ (empty symbols) and AFM (filled symbols) states, measured at the point, where the intensity is maximised. Exciton peaks are marked by arrows. The horizontal black bar inside the peak shows the experimental energy resolution, the hatched region represents background intensity, and the lines are guides to the eyes. The inset shows constant energy scans across the peak maximum (ħω=0.5 meV) along the MRM′ line at different temperatures; the dashed line is the background level suggested by the intensity minima at and within the spin gap (ħω=0.25 meV). (b) Low-temperature (T=1.6 K) energy scans for the X, M, T and Λ points. The location of the scattering plane and the naming of high-symmetry points in the cubic Brillouin zone are shown in the inset. (c) Comparison of the energy scans measured at the X and Λ points above and below TN.

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The detailed temperature dependence of the resonant mode is illustrated in Fig. 4, where we plot the imaginary part of the dynamical spin susceptibility, χ″(QAFQ,ω), obtained by correcting the background-subtracted INS signal for the thermal population factor. In Fig. 4a, one sees that the mode gradually broadens and shifts towards lower energies upon warming, following the order-parameter-like trend shown in the inset of Fig. 4b. This behaviour is again strikingly similar to the one of the resonant mode in unconventional superconductors (Fig. 4c). For T > TN, the quasielastic signal continues to diminish in intensity with increasing temperature (Fig. 4b). Constant-energy scans in the inset of Fig. 3a demonstrate that it always remains peaked at the commensurate wave vector QAFQ, where the resonant mode develops below TN.

Figure 4: Temperature dependence.
figure 4

(a) Imaginary part of the dynamic spin susceptibility, χ″(QAFQ,ω), at various temperatures, illustrating the gradual development of the spin-exciton peak below the AFM transition. The lines are visual guides. (b) Temperature dependence of χ″(QAFQ,ω) at the peak maximum (ħω=0.5 meV) and within the spin gap (ħω=0.25 meV). The inset shows the exciton energy versus temperature, determined from the position of the peak maximum, ħωmax, in panel (a). (c) Comparison of the normalised resonance energies, ħωres/kBTN (or Tc), plotted versus reduced temperature, T/TN (or Tc), in CeB6, in several unconventional superconductors3,4,6, and in the low-temperature phase of CeRu2Al10 (ref. 35). The lines are guides to the eyes.

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Spin gap and the Q-space structure of the INS signal

Constant-Q scans at various high-symmetry points (Fig. 3b,c) further show a weakly energy-dependent magnetic response throughout the Brillouin zone. For T<TN, a spin gap of magnitude ~0.5 meV opens up nearly isotropically in momentum space, including the M point, where the magnetic INS signal is minimal, and the R point, where the resonant mode develops in the AFM state. The appearance of the spin gap is consistent with the depletion in the electronic density of states below TN, previously observed by point-contact spectroscopy30. The comparison in Fig. 4b of the temperature scans measured at the peak maximum (0.5 meV) and below it (0.25 meV) emphasises that the spectral-weight gain at the exciton energy is compensated by its depletion within the spin-gap region. Both effects show a simultaneous sharp onset at TN, closely similar to the behaviour of the resonant mode and the spin gap in an unconventional superconductor3. An analogous T-evolution is observed at the point (Fig. 3c), where a quasielastic signal of comparable intensity is found in the AFQ state. The redistribution of spectral weight at this point leads to an intensity enhancement below TN at ħω ≈ 0.9 meV, but the resulting feature is much broader than the one at the R point.

Although the resonant mode is most intense at , the intensity map in Fig. 2a suggests the presence of weaker tails extending well into the Brillouin zone. This observation is substantiated by Fig. 5, where we show momentum scans along the ΛTΛ′ (Fig. 5a) and ΣMΣ′ (Fig. 5b) directions that cut the two lobes of intensity extending from R towards the M point. The comparison of the low-temperature signals measured at 0.5 and 0.25 meV together with the energy scan at the Λ point (Fig. 3b,c) let us conclude that the mode exhibits at most a weak dispersion, so that even away from the R point, its intensity remains maximal near ħωmax ≈ 0.5 meV.

Figure 5: Anisotropic shape of the resonant mode in momentum space.
figure 5

(a,b) Low-temperature (T=1.6 K) Q-scans along the and (hh0) directions (left and right panels, respectively) cross the two intensity lobes that connect the resonant peaks at equivalent R points, resulting in two incommensurate peaks. The intensity is maximised at the resonance energy (ħω=0.5 meV) and is much reduced both within the spin gap (ħω=0.25 meV) and at a higher energy (ħω=0.8 meV). The letters on the horizontal axes mark points in the Brillouin zone according to the notation in Fig. 3b (inset).

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Although the quasielastic intensity in the paramagnetic state of CeB6 is similar to the one observed in other heavy-fermion compounds5,6, the behaviour of the resonant mode we have discovered for T<TN is quite different from that of low-energy magnons in other heavy-fermion antiferromagnets, including the isostructural PrB6 (ref. 31), as well as CeIn3 (ref. 32) and CePd2Si2 (ref. 33). These excitations exhibit the conventional dispersion with minimum energy and maximum intensity at the AFM-ordering wave vector. The resonant mode in CeB6, on the other hand, is much more intense, at most weakly dispersive, and centred at QAFQ rather than QAFM. The behaviour we have observed disagrees with calculations in the framework of the interacting-multipole model of CeB6, which predict magnon-like low-energy modes19, in addition to more weakly dispersive higher-energy crystal-field excitations34.

Discussion

The striking similarity of the resonant mode in CeB6 to the low-energy spin excitations in unconventional superconductors can be qualitatively understood in the framework of a two-component model that includes strong interactions of the localised magnetic moments with the itinerant electrons5. The formation of the AFM state accounts for the opening of a gap in the electronic density of states near the Fermi energy, which is similar to that found in spin-density-wave systems. As a consequence, it leads to the nearly isotropic spin gap in the continuous magnetic spectrum originating from strongly hybridised crystal-field excitations of the Ce 4f-electrons and the itinerant conduction electrons. The resonant mode can then be regarded as a collective mode below the onset of the particle-hole continuum, in close analogy to current theoretical models of the resonant modes in the superconductors. We note that this scenario is quite different from the standard interacting-multipole model of CeB6, according to which the magnetic response in the AFM phase is dominated by the magnetic dipole moments of the localised Ce electrons, but qualitatively consistent with other recent observations that emphasised the importance of itinerant magnetism in this system16,23,24,25,26.

Although there is general agreement on the collective-mode nature of the resonant modes in unconventional superconductors, the mechanisms leading to their formation are still controversial9,10. It is interesting to consider our data on CeB6 in the light of two classes of theories that describe the mode as a magnon-like excitation of localised electrons close to a magnetically ordered state, and as an excitonic bound state of itinerant electrons, respectively. A recent calculation in the framework of the former approach9 has reproduced the temperature evolution of the magnetic response of superconducting CeCoIn5 from an overdamped mode above Tc to a resonant collective mode below Tc by assuming that the super-conducting gap removes damping channels for a low-energy Ce crystal-field excitation. An analogous model in which the superconducting gap is replaced by a spin-density-wave-type gap may explain the resonant mode in CeB6. However, the appearance of the resonant mode at the R point rather points towards a relationship to the AFQ order, which is expected to support a soft multipolar mode at this wave vector. On the other hand, the close proximity of the resonant mode to the gap energy, which would have to be regarded as a coincidence in this approach, can be naturally explained within a scenario in which the mode is described as a weakly bound triplet exciton. The complex anisotropic shape of the collective mode in CeB6 also suggests an influence of the fermiology of the conduction-electron system, in analogy to itinerant-electron descriptions of the resonant mode in CeCoIn5 (ref. 10), and the hourglass shape of the resonant mode in the copper oxides1. Although such an influence is generally expected for an exciton within an itinerant-electron description, it will be interesting to explore theoretically the impact of the coherence factors on the expression for χ″(Q,ω) in the AFM state. This is important because the relationship between the superconducting coherence factors and the resonant mode has figured prominently in the ongoing debate about the Cooper pairing symmetry in iron pnictides3,8,7 and heavy-fermion compounds9,10. Answers to these questions will have to await calculations that take full account of the strong coupling between the localised Ce crystal-field states and the itinerant electrons, qualitatively beyond the current interacting-multipole model of CeB6. The first step in this direction has been already undertaken in a recent work35 that gives a qualitative theoretical description of the spin-exciton mode reported here in the framework of a fourfold degenerate Anderson lattice model, considering both order parameters in CeB6 as particle-hole condensates of itinerant heavy quasiparticles.

We end our discussion by noting that the resonant mode in CeB6 apparently results from the coexistence of the AFM state with the unusual AFQ order that had remained hidden to experimental probes in zero magnetic field until the recent advent of resonant X-ray diffraction methods20,21,22. This unique situation explains its unusual behaviour in comparison with other heavy-fermion antiferromagnets. Presumably, as a consequence of AFQ-order fluctuations, the dynamical susceptibility at the R point of CeB6 remains substantial even for T > TN, where the resonant mode is overdamped, and only decreases gradually upon further heating. This behaviour is reminiscent of the gradual onset of magnetic intensity in the pseudogap state of the underdoped cuprates2, for which various hidden order parameters have also been proposed. The mode we have observed may also be related to the higher-energy excitonic modes that have been reported in other materials with thus-far unidentified hidden order parameters, such as CeRu2Al10 (ref. 36), and in small-gap Kondo insulators SmB6 (ref. 37) and YbB12 (ref. 38), whose ground states remain poorly understood. We also note that multipolar ordering phenomena have been suggested responsible for the mysterious hidden-order phase in URu2Si2, which has so far avoided direct detection by conventional solid-state probes39. As the 'hidden' order in CeB6 has actually been extensively characterised, the comprehensive description of the magnetic dynamics of this compound we have initiated here will have model character for a broad class of correlated-electron materials.

Methods

Sample preparation

Two single-crystalline rods of Ce11B6 [110] with a total mass of 8 g were prepared in a multistage process: (i) synthesis of the source powder by borothermal reduction of cerium oxide at 1,900 K in vacuum according to the routine reaction: CeO2+8B → CeB6+2BO; (ii) cold compaction of the obtained powder into rods with dimensions Ø8×60 mm and their subsequent sintering at 2,000 K; (iii) growth of the single crystals by the crucible-free inductive floating-zone melting under argon pressure in the specialised 'Crystal-111' equipment of the Institute for Problems of Materials Science. Taking into account high metallic conductivity of CeB6, its high melting point (≥ 2,823 K), and its high reactivity in the molten state, this method has proven to be the most effective for purification and growth of sufficiently large single crystals of this boride. To reduce the absorption of neutrons by the 10B isotope, the synthesis was performed using 99.6% isotopically enriched polycrystalline 11B powder (Ceradyne Inc., USA) with a purity of 99.96%. The purity of the initial CeO2 (the Federal State Research and Design Institute of Rare Metal Industry, Russia) was 99.996%. The main technological parameters were: (i) crystallisation rate of 1 mm min-1; (ii) feed-rod rotation speed of 10 r.p.m.; (iii) gas pressure of 0.1 MPa. The desired [110] orientation of the Ce11B6 single crystals was achieved by using a correspondingly oriented seed that was prepared from a primary spontaneously grown crystal. The resulting samples had a diameter of 4–6 mm and a length of up to 50 mm. According to the spectral optical method, the concentration of impurities in the grown Ce11B6 crystals did not exceed 10−3 mass %.

Data acquisition

The INS data were collected using the cold triple-axis IN14 spectrometer (ILL, Grenoble, France) with a pyrolytic graphite monochromator. The sample was mounted into a standard cryostat with the (110) and (001) directions in the scattering plane. To acquire the overview maps of the reciprocal space (Fig. 2), we used the Flatcone detector, equipped with Si (111) analysers that select scattered neutrons with the final momentum kf=1.40 Å−1. In this configuration, a cold beryllium filter was installed in the path of the incident beam to eliminate the contamination from higher-order neutrons. As a consequence, the accessible energy-transfer range was restricted to 0.9 meV by the Be-filter cutoff energy. The rest of the measurements were done in the conventional triple-axis configuration with a pyrolytic graphite analyser and the Be filter installed on kf. The same fixed value of kf=1.40 Å−1 was selected. The data were corrected for the energy-dependent fraction of higher-order neutrons on the monitor. The imaginary part of the dynamical spin susceptibility, χ″(Q,ω), was obtained from the background-subtracted INS intensity by the fluctuation-dissipation relation , where S(Q,ω) is the scattering function.

Reciprocal-space notation

Throughout the paper, we index the reciprocal-lattice vectors on the simple-cubic unit cell (space group Pm3m, lattice constant a=4.1367 Å) and follow the conventional notation in labelling the high-symmetry points (Fig. 3b, inset). The wave-vector coordinates are given in reciprocal lattice units (1 r.l.u.=2π/a).

Additional information

How to cite this article: Friemel, G. et al. Resonant magnetic exciton mode in the heavy-fermion antiferromagnet CeB6. Nat. Commun. 3:830 doi: 10.1038/ncomms1821 (2012).