Nesting-driven multipolar order in CeB6 from photoemission tomography

Abstract

Some heavy fermion materials show so-called hidden-order phases which are invisible to many characterization techniques and whose microscopic origin remained controversial for decades. Among such hidden-order compounds, CeB₆ is of model character due to its simple electronic configuration and crystal structure. Apart from more conventional antiferromagnetism, it shows an elusive phase at low temperatures, which is commonly associated with multipolar order. Here we show that this phase roots in a Fermi surface instability. This conclusion is based on a full 3D tomographic sampling of the electronic structure by angle-resolved photoemission and comparison with inelastic neutron scattering data. The hidden order is mediated by itinerant electrons. Our measurements will serve as a paradigm for the investigation of hidden-order phases in f-electron systems, but also generally for situations where the itinerant electrons drive orbital or spin order.

ARTICLE

Received 12 Oct 2015 | Accepted 28 Jan 2016 | Published 15 Mar 2016

A. Koitzsch1, N. Heming1, M. Knupfer1, B. Bchner1, P.Y. Portnichenko2, A.V. Dukhnenko3, N.Y. Shitsevalova3,V.B. Filipov3, L.L. Lev4,5, V.N. Strocov4, J. Ollivier6 & D.S. Inosov2

Some heavy fermion materials show so-called hidden-order phases which are invisible to many characterization techniques and whose microscopic origin remained controversial for decades. Among such hidden-order compounds, CeB6 is of model character due to its simple electronic conguration and crystal structure. Apart from more conventional antiferromagnetism, it shows an elusive phase at low temperatures, which is commonly associated with multipolar order. Here we show that this phase roots in a Fermi surface instability. This conclusion is based on a full 3D tomographic sampling of the electronic structure by angle-resolved photoemission and comparison with inelastic neutron scattering data. The hidden order is mediated by itinerant electrons. Our measurements will serve as a paradigm for the investigation of hidden-order phases in f-electron systems, but also generally for situations where the itinerant electrons drive orbital or spin order.

DOI: 10.1038/ncomms10876 OPEN

Nesting-driven multipolar order in CeB6 from photoemission tomography

1 Institute for Solid State Research, IFW-Dresden, PO Box 270116, D-01171 Dresden, Germany. 2 Institut fr Festkrperphysik, TU Dresden, D-01069 Dresden, Germany. 3 I.M. Frantsevich Institute for Problems of Materials Science of NAS, 3 Krzhyzhanovsky str., Kiev 03680, Ukraine. 4 Paul Scherrer Institut, Swiss Light Source, CH-5232 Villigen PSI, Switzerland. 5 National Research Centre Kurchatov Institute, 123182 Moscow, Russia. 6 Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex, France. Correspondence and requests for materials should be addressed to A.K. (email:mailto:[email protected]

Web End [email protected] ).

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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 specic crystal eld 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 RudermanKittel KasuyaYosida (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 (p/2, p/2, 0) and QAFM2 (p/2, p/2, p) 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 (p, p, p)10,11. The latter has long been

elusive to neutron diffraction experiments and was rst 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 quanties 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 sufcient accuracy to test this hypothesis. This deciency 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 crystal-lographic 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 l, 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.

ResultsElectronic 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 difcult 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 structure2326. Previous ARPES studies on CeB6 are therefore sparse2729. The Fermi surface has also been studied by de-Haasvan 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. 1ce, we present Fermi surface maps taken for different cleavage planes as indicated in the gure. 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 1fh present 3D visualizations including the experimental cuts shown above. This Fermi surface agrees with previous measurements for CeB6 (refs 2730). 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.

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 reecting the symmetry of the Brillouin zone (Fig. 2df).

Figure 3a,b show the energy distribution maps along the two mirror axes of the ellipsoid in the (100) plane. Along the GX 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 f 0 ionization peak is situated, which overlaps

with the bottom of the ellipsoidal band. Near EF, the screened f 1 states are found, which split due to the spinorbit coupling in a J 5/2 and 7/2 component. The 5/2 state at EF is relevant here

and splits further into crystal eld levels, namely a G7 doublet and a G8 quartet. One of the G8 levels is occupied, whereas the G7 intensity seen in the spectrum is a satellite. The energy separation of the G7 and G8 levels (DEE50 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.

The electronic structure derived here by soft X-ray photo-emission 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 G 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 difcult to compare the intensity at the lowest energies due to the different integration windows imposed by the different energy resolution.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10876 ARTICLE

a b

R

X

M

k100

k001

k010

100 110 111

Ce

B

c d e

X R

X X

1

1

M

R

k y(/a)

k y(/a)

1

k y(2/a)

M

1 1

3

22/a

1 1

1 1

kx (/a)

kx (2/a)

1

1 kx (6/a)

f g h

z

z

z

x x x

Figure 1 | Fermi surface of CeB6 from different cleavage planes. (a) Crystal structure of CeB6. (b) Brillouin zone with high symmetry points.(ce) 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. (fh) 3D representation of the measured Fermi surface and the measurement plane. The different colours

of the ellipsoids are for clarity.

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 tted to a suitable model which is then used to calculate the Lindhard function. In two-dimensional systems, nesting instabilities can be identied 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 tted to the experimental ARPES data. Figure 4af show the results of the tting. Figure 4a,b compares the (100) and (111) Fermi surfaces to the model. Figure 4c presents the (100) Fermi surface at kz p/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 nal 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 Eo 0.3 eV from the data. However, the Lindhard function falls

off rapidly away from EF.

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

Vq4 1wq 1

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

wq X

k

nFEk

q

nFEk

Ek

2

Here nFE 1=expE=kBT 1 is the Fermi distribution.

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

Inelastic neutron scattering. The relevance of the calculated Lindhard function is conrmed 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) [ (p, p, p)] and X (0, 0, 1/2)

[ (0, 0, p)]. 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) [ (p/2, p/2, p)]. In

Fig. 5b, wq is extracted along certain high symmetry directions. The maxima at QAFQ and QAFM1 are clearly visible.

There is an additional broad maximum around X (0, 0, 1/2), which has a signicant 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

Ek

q

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a b c

100

110 111

E= 1 eV

k y(/a)

1

3

1

1

k y(2/a)

1

1

1

1

1

1

1

1

1

1

d e f

E= 8.5 eV

k y(/a)

k y(2/a)

1

1

1

3

1

1

1 1

kx (/a)

kx (2/a)

1 1

kx (6/a)

Figure 2 | Constant energy cuts for different cleavage planes. (ac) at E 1 eV and (df) at E 8.5 eV for (100) plane in (a,d), (110) in (b,e)

and (111) in (c,f)

EF

0.00

f 1

8

7

0.5

0.05

1.0

0.10

Energy (eV)

d

1.5

0.15

Intensity (a.u.)

2.0

f 0

2.5

3.0

a

b

c

X X

M M

5/2 7/2

Intensity (a.u.)

k

Figure 3 | Band structure of CeB6. Energy distribution maps along (a) GX 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 eld levels are denoted.

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 x40.4 (ref. 36). But

here also, both neutrons and ARPES are in good agreement.

DiscussionThe 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 (p, p, p). 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 (p/2, p/2, 0) vector. Here, clear nesting between parallel segments of the Fermi surface is revealed.

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 specic 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

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a

b

a

h = [0.15, 0.4] meV

T = 2.6 K (AFQ)

k y(/a)

k y(/a)

2

1

0

1.5

1.0

0.5

0.0

k y(1)

0.2

1

1

1 1

1

0

1

2

Lin (HHL)

X q(a.u.)

kx (/a)

kx (1)

c d

0.5

1.0

1.5

19

Max

Min

k z(/a)

17

INS intensity

1.0 1.0

0.5 0.5

0.0

1

5

3

1

H in (HHL)

1 kx (/a) kx (/a)

b

QAFQ

e f

0.1

QAFM1

0.0

Energy (eV)

0.0

0.1

Energy (eV)

0.1

0.2

0.3

1.2 0.8 0.4

0.2 0.1 0.0 0.1 0.2

15

k (1)

k (1)

Figure 4 | Parametrization of the low-energy electronic structure. Fermi surface for (a) (100), G plane; (b) (111), R plane; (c) (100), R plane;

(d) (100), kz scan; energy distribution maps for (e) (100), GX direction; (f) (100), MX direction.

X M R

Figure 5 | Comparison of Lindhard function and inelastic neutron scattering (INS) data. (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 K4TN 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.

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 G8 crystal eld 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 eld 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 3942).

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 oating 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 ow 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 500900 eV. The energy resolution was set to DE 100 meV. Angle

resolution was better than 0.07. The high resolution measurement of the crystal eld splitting in Fig. 3d has been carried out at the 13 beamline at BESSY with

DE 10 meV at a temperature of T 1 K with a photon energy of hv 90 eV.

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a b

Q = ([afii9843],[afii9843],[afii9843])

Q = ([afii9843]/2,[afii9843]/2,0)

c d

Figure 6 | 3D representations of the nesting conditions. (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. (p, p, p) in a,b; (p/2, p/2, 0) in c,d. The arrow highlights the place of increased nesting.

INS measurements. INS data were collected using the cold-neutron time-ofight spectrometer IN5 at the high-ux 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 xed at 5 (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 between0.15 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:

Ek c E0 X

A7 4 coskx 2ky 2kz coskx 2ky 2kz coskx 2ky 2kz coskx 2ky 2kz cos2kx ky 2kz cos2kx ky 2kz cos 2kx ky 2kz cos 2kx ky 2kz cos2kx 2ky kz cos2kx 2ky kz cos 2kx 2ky kz cos 2kx 2ky kz

10

A8 16 cos3kx cos3ky cos3kz 11

with E0 0.215 eV, t1 0.091 eV, t2 0.118 eV, t3 0.0085 eV, t4 0.011 eV,

t5 0.089 eV, t6 0.077 eV, t7 0.0039 eV, t8 0.011 eV; and

ck jEc0 X

8 tiAi 3

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

A1 16 coskx cosky coskz 4

A2 8 coskx ky coskx ky coskx kz coskx kz cosky kz cosky kz

5

8 tciAij 1 12

with Ec0 0:00604, tc1 0:038, tc2 0:047, tc3 0:0054, tc4 0:000049,

tc5 0:021, tc6 0:047, tc7 0:021, tc8 0:025. The model is only valid for

EZ 0.3 eV.

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

Web End =http://horace.isis.rl.ac.uk .

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A3 12 coskx ky kz cos kx ky kz coskx ky kz

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A4 16 cos2kx cos2ky cos2kz 7

A5 4 cos2kx ky cos2kx ky cos2ky kx cos2ky kx cos2kx kz cos2kx kz cos2kz kx cos2kz kx

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Acknowledgements

We thank C.G.F. Blum for help with the crystal preparation. A.K. acknowledges support by the German Research Foundation (DFG) under grant No. KO 3831/3-1 and D.S.I. under grant No. IN 209/3-1. D.S.I. is grateful to A. Akbari and P. Thalmeier for enlightening the discussions.

Author contributions

A.K., N.H. and D.S.I. conducted the ARPES experiments with assistance from L.L.L. andV.N.S. P.Y.P. and D.S.I. conducted the INS experiments with assistance from J.O. A.V.D., N.Y.S. and V.B.F. grew the samples. A.K. wrote the paper with assistance from N.H., M.K., B.B., P.Y.P., L.L.L., V.N.S., J.O. and D.S.I.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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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).

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