Introduction

Hidden-order phases have been observed in a variety of compounds containing 4f and 5f elements, for example, URu2Si2 (ref. 1), NpO2 (ref. 2), skutterudites3 and YbRu2Ge2 (ref. 4). They are characterized by a rich low-temperature phase diagram and it is assumed that the multipolar moments of the f-electrons in their specific crystal field environment play a decisive role5,6.

CeB6 is a heavy-fermion material showing a mass enhancement of the order of 100 (ref. 7), which is due to the hybridization of the localized f-electrons with the itinerant conduction electrons. Magnetism of heavy fermion materials is determined by the competition of Kondo screening and the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, the former quenching the local moments and favouring paramagnetic behaviour, the latter promoting magnetic order mediated by the conduction electrons8,9. In CeB6 the usual paramagnetic response is found before antiferromagnetic order with a double-Q commensurate structure characterized by the propagation vectors QAFM1=(π/2, π/2, 0) and QAFM2=(π/2, π/2, π) sets in below TN=2.3 K. However, the phase diagram is more complex: the antiferromagnetism is preceded by a famous hidden order state at TQ=3.2 K, the so called antiferroquadrupolar phase (AFQ), which has been explained by the ordering of quadrupole moments with QAFQ=(π, π, π)10,11. The latter has long been elusive to neutron diffraction experiments and was first directly visualized by X-ray scattering12.

Attempts were made to describe these observations by theories emphasizing the local character of the magnetic moments13,14. However, recently a magnetic resonance mode has been discovered at QAFQ below TN in close resemblance to the ones in unconventional superconductors15, underlining the importance of itinerant spin dynamics for this compound.

In an itinerant picture, the strength of the magnetic interactions is mediated by the conduction electrons and depends on the low-energy electronic structure. It can be expressed within linear response theory by the Lindhard function16. The latter quantifies the propensity of a given electronic structure towards nesting instabilities of the Fermi surface and the subsequent formation of a new, in our case magnetically ordered, state17. An interesting question in this context is whether or not the AFQ state also is directly linked to the electronic structure in a similar way. It has been theoretically suggested that the interactions between multipolar moments in CeB6 is driven by a RKKY-type interaction18. However, although CeB6 has been studied for more than 50 years19, the three-dimensional (3D) electronic structure of CeB6 was not known so far neither from experiment nor from theory with sufficient accuracy to test this hypothesis. This deficiency calls for a detailed investigation of the band structure and the Fermi surface of CeB6.

Here we implement a rigorous and innovative approach: We measured samples cleaved along all high-symmetry crystallographic planes (100), (110), (111). This probes different planes of k-space, resembling a tomographic type of measurement that yields complete 3D information about the electronic structure in contrast to conventional angle-resolved photoemission spectroscopy (ARPES), in which one direction orthogonal to the surface is always inferior to two others. We conducted photon-energy dependent measurements in the soft X-ray regime spanning a wide kz interval20. In comparison with conventional low-energy ARPES, this increases the photoelectron mean free path λ, which in turn increases the signal of the bulk states21 and, crucial for 3D materials like CeB6, enhances the intrinsic kz resolution22. From the consistency of the results obtained in this way, we can infer the absence of surface-related effects. Moreover, such a full data set offers a very precise view on the details of the electronic structure and increases the accuracy of model descriptions. We use the latter to calculate the Lindhard function and compare it with neutron scattering data. From the consistency of both, we conclude that the magnetic excitations and the AFQ propagation vector in CeB6 are dictated by the Fermi-surface geometry.

Results

Electronic structure from photoemission tomography

The method of choice to map the electronic structure in general is ARPES. But CeB6 holds several obstacles in store for this technique: it has a cubic crystal structure, that is, is fully 3D. This requires photon energy-dependent measurements to capture the kz dispersion perpendicular to the surface. Moreover the material is hard and difficult to cleave, which is the usual procedure to measure single crystals. Third, hexaborides are subject to surface reconstructions and possess surface states, which might mask the bulk electronic structure23,24,25,26. Previous ARPES studies on CeB6 are therefore sparse27,28,29. The Fermi surface has also been studied by de-Haas–van Alphen measurements. The results are consistent with ellipsoids centred around the X points30.

CeB6 crystallizes in a CsCl-type simple cubic crystal structure shown in Fig. 1a. The B6 octahedron is situated in a cubic environment of Ce atoms. In Fig. 1c–e, we present Fermi surface maps taken for different cleavage planes as indicated in the figure. The symmetry of the Fermi surface contours mirrors one of the cleavage plane. The (100) direction has a fourfold rotation axis, (110) only twofold and three- or sixfold for (111) depending on the used photon energy. The Fermi surface consists of ellipsoids centred at the X point. Figure 1f–h present 3D visualizations including the experimental cuts shown above. This Fermi surface agrees with previous measurements for CeB6 (refs 27, 28, 29, 30). The ellipsoids are typical for the hexaborides in general28,29. Their orbital character is composed of extended Ce 5d states with admixtures of localized Ce 4f near the Fermi energy (EF; ref. 28), similar to other 4f systems exhibiting a resonance mode31,32.

Figure 1: Fermi surface of CeB6 from different cleavage planes.
figure 1

(a) Crystal structure of CeB6. (b) Brillouin zone with high symmetry points. (c–e) Fermi surfaces and representations of the different cleavage planes. (c) (100), taken with a photon energy of hv=700 eV; (d) (110), hv=609 eV; (e) (111), hv=700 eV; measured at 12 K. (f–h) 3D representation of the measured Fermi surface and the measurement plane. The different colours of the ellipsoids are for clarity.

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The ellipsoid bands are electron-like. Away from EF, their size shrinks (see Fig. 2). Far below the bottom of the ellipsoid band at E=−8.5 eV, the constant energy contour consists of straight sections reflecting the symmetry of the Brillouin zone (Fig. 2d–f).

Figure 2: Constant energy cuts for different cleavage planes.
figure 2

(a–c) at E=−1 eV and (d–f) at E=−8.5 eV for (100) plane in (a,d), (110) in (b,e) and (111) in (c,f)

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Figure 3a,b show the energy distribution maps along the two mirror axes of the ellipsoid in the (100) plane. Along the ΓX direction, the band has a parabola-like or U shape, whereas the bottom of the band appears more cusp-like or V shaped along MX. Figure 3c presents the k-integrated spectrum of panel 3a featuring the typical shape of Ce-based materials: Around E=−2.5 eV, the f0 ionization peak is situated, which overlaps with the bottom of the ellipsoidal band. Near EF, the screened f1 states are found, which split due to the spin–orbit coupling in a J=5/2 and 7/2 component. The 5/2 state at EF is relevant here and splits further into crystal field levels, namely a Γ7 doublet and a Γ8 quartet. One of the Γ8 levels is occupied, whereas the Γ7 intensity seen in the spectrum is a satellite. The energy separation of the Γ7 and Γ8 levels (ΔE≈50 meV) is in agreement with previous reports33,34. Note that the large ground state degeneracy distinguishes CeB6 from many other Ce-based heavy fermion materials.

Figure 3: Band structure of CeB6.
figure 3

Energy distribution maps along (a) ΓX and (b) MX measured at 12 K. (c) k-integrated version of a with assignment of the f-levels. (d) The region near EF measured separately with increased resolution at 1 K. Crystal field levels are denoted.

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The electronic structure derived here by soft X-ray photoemission from various cleavage planes is consistent with previous low photon energy studies from the (100) plane28,29. Neupane et al. reported a strong k-dependent renormalization around the Γ point concluded from the deviation of the experimental data and the bandstructure calculation in this region and enhanced quasiparticle intensity. Our data also deviate from this calculation in a similar fashion but it is difficult to compare the intensity at the lowest energies due to the different integration windows imposed by the different energy resolution.

Lindhard function and model description

To investigate the connection of the electronic structure with the magnetism and the AFQ phase, the experimental band structure has to be fitted to a suitable model which is then used to calculate the Lindhard function. In two-dimensional systems, nesting instabilities can be identified sometimes just by visual inspection of the Fermi surface. However, in three dimensions, this is not a viable procedure anymore and a rigorous treatment is required. For this purpose, we used a tight-binding-like model fitted to the experimental ARPES data. Figure 4a–f show the results of the fitting. Figure 4a,b compares the (100) and (111) Fermi surfaces to the model. Figure 4c presents the (100) Fermi surface at kz=π/a, where the ellipsoid is cut at 90° and gives almost a circular contour. The latter is superimposed by shadow-like structures arising from the nearby parts of the ellipsoids parallel to the cutting plane by remnant final state kz broadening. Figure 4d shows a (100) kz scan, that is, a photon energy-dependent measurement. In all cases, the experimental contours are well reproduced by the model. The same holds true for the comparison with the near EF energy distribution maps in Fig. 4e,f. Note that the model electronic structure starts to deviate for E<−0.3 eV from the data. However, the Lindhard function falls off rapidly away from EF.

Figure 4: Parametrization of the low-energy electronic structure.
figure 4

Fermi surface for (a) (100), Γ plane; (b) (111), R plane; (c) (100), R plane; (d) (100), kz scan; energy distribution maps for (e) (100), ΓX direction; (f) (100), MX direction.

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The condition for the formation of a spin-density wave (SDW) is approximately given by16:

where Vq is the exchange interaction in the local approximation and χq is the real part of the Lindhard function in the static limit:

Here is the Fermi distribution. The system may become unstable against the formation of a SDW in the vicinity of maxima of χq, that is, where the Fermi surface nesting is large.

Inelastic neutron scattering

The relevance of the calculated Lindhard function is confirmed by the experimental inelastic neutron scattering (INS) data as shown in Fig. 5a. The two pictures are remarkably similar. The calculation reproduces not only the maxima but also the qualitative shape of the magnetic diffuse scattering at R (1/2, 1/2, 1/2) [=(π, π, π)] and X (0, 0, 1/2) [=(0, 0, π)]. Moreover, line structures appear in the calculation (for example, from (1/2, 1/2, 0) to (1/4, 1/4, 1/2)), which resemble the oval-shaped streaks connecting the R points in the neutron data, resulting in a weak local maximum around the propagation vector of the AFM2 order, (1/4, 1/4, 1/2) [=(π/2, π/2, π)]. In Fig. 5b, χq is extracted along certain high symmetry directions. The maxima at QAFQ and QAFM1 are clearly visible.

Figure 5: Comparison of Lindhard function and inelastic neutron scattering (INS) data.
figure 5

(a) Two-dimensional representation of the Lindhard function in the (HHL) plane compared with the distribution of magnetic quasielastic scattering intensity measured by INS at T=2.6 K>TN in reciprocal lattice units. (b) Lindhard function extracted for certain high symmetry directions with the indication of peaks coinciding with the propagation vectors of low-temperature ordered phases.

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There is an additional broad maximum around X (0, 0, 1/2), which has a significant spectral weight but does not correspond to any static ordering35. This possibly indicates proximity to another AFM instability with a propagation vector (0, 0, 1/2), which loses the energy competition with multiple other competing order parameters (possibly due to its broad width in Q). In fact, this type of order is realized in Ce1−x NdxB6 for x>0.4 (ref. 36). But here also, both neutrons and ARPES are in good agreement.

Discussion

The agreement between the Lindhard susceptibility derived from the measured electronic structure and the INS data proves the itinerant character of the magnetic excitations in CeB6 and suggests that the propagation vector of the AFQ order is dictated by the Fermi-surface geometry. It is important to emphasize that this result does not contradict the AFQ nature of the hidden-order phase, as the itinerant electrons determine the RKKY interactions between the Ce 4f multipolar moments that can be still considered as local. To visualize the nesting condition, we show in Fig. 6 the 3D Fermi surface together with shifted replicas. Figure 6a presents the nesting by (π, π, π). This vector effectively shifts a given ellipsoid into the void formed in between the other four. This maximizes the overlap among the ellipsoids although nesting in the strict sense is not apparent. Figure 6b shows the (π/2, π/2, 0) vector. Here, clear nesting between parallel segments of the Fermi surface is revealed.

Figure 6: 3D representations of the nesting conditions.
figure 6

(a,c) The original Fermi surface and their replicas by the respective nesting vector are shown. (b,d) Ellipsoids are skipped or added to emphasize the nesting condition. (π, π, π) in a,b; (π/2, π/2, 0) in c,d. The arrow highlights the place of increased nesting.

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Primordial signatures of the complex magnon spectrum of CeB6 are already observed above TN in the quasielastic neutron scattering response35. We show here that these signatures have their natural explanation in the low-energy electronic structure, establishing the importance of itinerant electrons for the spin dynamics. For CeB6, a picture emerges where the propensity towards specific magnetic order roots in favourable nesting conditions of the Fermi surface. The exact way by which this is achieved could be quite complex. For example, a weak magnetic Bragg peak has been observed at QAFQ above TN (refs 15, 37). It has been ascribed to a SDW-type order associated with the conduction electrons primarily of Ce 5d character37. (Note that the AFQ order itself is hidden to neutron diffraction10.) Although orders of magnitude weaker than the antiferromagnetic order observed below TN, this itinerant SDW could break the degeneracy of the Γ8 crystal field ground state and promote the multipolar moments to order in the AFQ phase with the same propagation vector. The value of TQ should then be associated with the low temperature evolution of the electronic structure, for example, the mass enhancement due to the formation of heavy quasiparticles. The latter could lead to an enhancement of the nesting-related energy gain. Intriguingly, this is indeed observed, as TQ coincides in zero field with the formation of the heavy fermion liquid indicated by the drop in resistivity38. Another plausible scenario would involve an enhancement of the RKKY interaction between the multipolar moments18 at the R point due to the Fermi-surface nesting, analogous to that seen previously in rare-earth silicides17.

The ellipsoidal Fermi surface observed here for CeB6 is also typical for other rare earth hexaborides, for example, the presumed topological Kondo insulator SmB6 (refs 39, 40, 41, 42). Interestingly, recent INS data of SmB6 bear out similarities to CeB6 as well, namely intensity maxima at the X and R points, for which Fermi surface nesting was discussed43. Our explicit observation of the relevance of nesting in CeB6 strongly indicates that the same mechanism is operative in SmB6 too.

Multipolar ordering phenomena have been suggested for other hidden order compounds, most prominently URu2Si2, which has so far even resisted proper characterization by conventional solid state probes1. CeB6, on the other hand, is well characterized with regard to its low-temperature ordered phases and by now also to its electronic structure and the connection between both. Therefore, it may serve as a model compound for many heavy-fermion metals and thus help to elucidate the interplay between the electronic structure, itinerant magnetism and complex order in correlated electron systems in general.

Moreover our methodology, namely the precise modelling of the 3D electronic dispersion based on experimental ARPES data and subsequent usage of this model to calculate complex response functions, is applicable to a wide class of 3D materials, not restricted to f-systems.

Methods

Sample preparation

Single crystal samples of CeB6 were prepared by floating zone method as described in ref. 15. Before the measurement, the crystals were oriented by Laue diffraction, and on the outside of the sample small notches were cut to create a predetermined cleavage plane along the desired direction. After that, the samples were cooled in ultrahigh vacuum with a base pressure of 1 × 10−10 mbar by a He flow cryostat to 12 K and cleaved for the measurement.

ARPES measurements

ARPES measurements have been done at the SX-ARPES end station of the Swiss Light Source44, ADRESS beamline45 in the photon energy range of 500–900 eV. The energy resolution was set to ΔE=100 meV. Angle resolution was better than 0.07°. The high resolution measurement of the crystal field splitting in Fig. 3d has been carried out at the 13 beamline at BESSY with ΔE=10 meV at a temperature of T=1 K with a photon energy of hv=90 eV.

INS measurements

INS data were collected using the cold-neutron time-of-flight spectrometer IN5 at the high-flux research reactor of the Institute Laue-Langevin35,46. A large single crystal specially synthesized from isotope-enriched 11B to minimize neutron absorption was mounted in a standard cryostat with its (110) and (001) directions in the horizontal plane. The incident neutron wavelength was fixed at 5 Å (3.27 meV), yielding an energy resolution (full width at half maximum) of 0.08 meV at zero energy transfer. The measurements were taken while rotating the sample about the vertical axis and then combined and transformed into energy-momentum space using HORACE analysis software. The quasielastic intensity distribution shown in Fig. 5a was obtained by integrating the four-dimensional data set in a narrow slab parallel to the horizontal plane along the vertical direction of the momentum and in a broad energy window between 0.15 meV (immediately above the elastic line) and 0.4 meV, thus providing an estimate of the integral quasielastic spectral weight within the (HHL) plane.

Model description

The low energy electronic structure has been parametrized by the following tight-binding-like model:

Where ti are hopping parameters, Ai tight-binding expansions of the i-th order and c is a correction term with

with =0.215 eV, =−0.091 eV, =0.118 eV, =0.0085, eV, =−0.011 eV, =−0.089 eV, =0.077 eV, =−0.0039, eV, =−0.011 eV; and

with , , , , , , , , . The model is only valid for E≥−0.3 eV.

Software availability

The open-source MATLAB-based HORACE software package is available from http://horace.isis.rl.ac.uk.

Additional information

How to cite this article: Koitzsch, A. et al. Nesting driven multipolar order in CeB6 from photoemission tomography. Nat. Commun. 7:10876 doi: 10.1038/ncomms10876 (2016).