URu2Si2 is one of the most enigmatic strongly correlated electron systems and offers a fertile testing ground for new concepts in condensed matter science. In spite of >30 years of intense research, no consensus on the order parameter of its low-temperature hidden-order phase exists. A strong magnetic field transforms the hidden order into magnetically ordered phases, whose order parameter has also been defying experimental observation. Here, thanks to neutron diffraction under pulsed magnetic fields up to 40 T, we identify the field-induced phases of URu2Si2 as a spin-density-wave state. The transition to the spin-density wave represents a unique touchstone for understanding the hidden-order phase. An intimate relationship between this magnetic structure, the magnetic fluctuations and the Fermi surface is emphasized, calling for dedicated band-structure calculations.
For since three decades, a huge experimental and theoretical effort has been employed to try to determine the ground state of the heavy-fermion material URu2Si2, where a second-order phase transition occurs at the temperature T0=17.5 K and for which the order parameter has resisted experimental identification1. The case of URu2Si2—which is also superconducting2—is not isolated, since mysterious electronic phases, identified as ‘pseudo-gap’ and ‘nematic’ phases, have also been reported in high-temperature cuprate and iron-based superconducting materials3,4. Initially, a small magnetic moment of 0.02 Bohr magneton per uranium (μB/U) antiferromagnetically aligned along c and modulated with the wavevector k0=(0 0 1) has been considered as the order parameter of URu2Si2 below T0 (ref. 5). However, this moment is too small to account for the large entropy associated with the transition and is now suspected to result from sample inhomogeneities6. In the hidden-order phase, the low-energy magnetic fluctuations are peaked at the hot wavevectors k0 and k1=(0.6 0 0)7,8, and the Fermi surface presents nesting with these two wavevectors9,10,11,12,13,14,15, suggesting an itinerant origin of the magnetic properties. Recent experimental works have been interpreted as the signature of nematicity, that is, a correlated electronic state associated with a breaking of the fourfold symmetry, in the hidden-order phase16,17,18.
The application of extreme conditions permits to modify the ground state of a system and get new insights about it. In URu2Si2 under pressure, the hidden-order state is replaced by an antiferromagnetic phase19, whose order parameter is a moment of ≃0.3–0.4 μB/U aligned antiferromagnetically along c with the wavevector k0 (refs 20, 21). From the similar Fermi surfaces observed in the hidden-order and pressure-driven antiferromagnetic phases, it has been proposed that the hidden order, although of unknown nature, has a periodicity of wavevector k0 (ref. 19). Under a high magnetic field applied along the easy magnetic axis c, a cascade of first-order phase transitions at the fields μ0H1=35 T, μ0H2=36/37 T (rising/falling fields), and μ0H3=39 T, leads to a polarized paramagnetic regime above H3, where a magnetization of ≈1.4 μB/U is reached (Fig. 1b)22,23. Quantum oscillations probes have shown that a magnetic field along c induces Fermi surface instabilities deep inside the hidden-order phase, as well as at the magnetic transitions H1, H2 and H3 (refs 24, 25, 26, 27, 28). By comparison with 3d itinerant magnets, a novelty in heavy-fermion compounds is that a large magnetic polarization can be induced by a magnetic field, opening the path to new phases driven by combined changes of magnetic order and Fermi surface. Such interplay of metamagnetic and Lifshitz transitions has been evidenced in the heavy-fermion paramagnet CeRu2Si2 (ref. 29).
In Rh-doped U(Ru0.96Rh0.04)2Si2, no hidden-order phase is reported, but first-order transitions are also induced at magnetic fields of 26 and 37 T applied along c (refs 30, 31, 32). Between 26 and 37 T, a squared up–up–down ferrimagnetic phase was deduced from Bragg peaks (harmonics) at the wavevector k2=(2/3 0 0) and at the Brillouin zone centre kZC=(0 0 0)33. However, the extrapolation to the pure case of URu2Si2 is not straightforward, as doping may modify the low-carrier content of the pure lattice, and, thus, the Fermi surface topology, leading to different nesting effects. Here thanks to a new cryomagnet allowing neutron diffraction up to 40 T, we have determined the magnetic structure of URu2Si2 in fields between 35 and 39 T. A tilt by 4.2° of the field direction from c in the (c,a) plane allowed accessing a large number of momentum transfers Q, with almost no modification of the magnetic phase diagram expected for H || c (refs 26, 34). The relationship between the high-field magnetic structure, the low-energy magnetic fluctuations and the Fermi surface is emphasized, appealing for new band-structure calculations describing the electronic properties of URu2Si2.
Spin-density wave with wavevector k1
Figure 1a shows the time profile of a magnetic field pulsed up to 38 T, as well as the time dependence of the diffracted neutron intensities recorded during 38-T pulses at the temperatures T=2 and 18 K and at Q=(0.6 0 0) and (1.6 0 −1). These momentum transfers are satellites of wavevector k1=(0.6 0 0)=Q–τ around the structural Bragg positions τ=(0 0 0) and (1 0 −1), respectively. The high-field enhancement of the diffracted intensity at T=2 K, absent at T=18 K, shows that long-range ordering with wavevector k1=(0.6 0 0) is established at high field and low temperature. Figure 1c emphasizes in the window 32–41 T the field dependence of the diffracted neutron intensities measured up to 40.5 T. At T=2 K, the diffracted intensity with wavevector k1 is enhanced in fields higher than μ0H1=35 T and drops down to the background level in fields higher than μ0H3=39 T. No hysteresis is observed at H1 and H3 within the neutron counting statistics, and a slight field-induced decrease of intensity, suggesting a modification of magnetic ordering, is visible at μ0H2≈36 T for rising field and at μ0H2≈37 T for falling field, in agreement with the hysteresis observed by magnetization33 (Fig. 1b). Figure 2 presents the high-field neutron-diffracted intensity at momentum transfers Q in the vicinity of (0.6 0 0) at T=2 K. The field dependence of the intensity at momentum transfers (Qh 0 0) and (0.6 0 Ql) is shown in fields up to 36.2 T in Fig. 2a,b, respectively. Figure 2c,d presents Qh—and Ql—scans at μ0H=36 T, confirming that the field-induced magnetic intensity is peaked at Q=(0.6 0 0). From the intensity of the magnetic Bragg peaks induced at Q=(0.6 0 0) and Q=(1.6 0 −1) at μ0H=36 T>μ0H1, we extract the amplitude 2M(k1)≃0.5±0.05 μB/U of the sine modulation with wavevector k1 of the magnetic moment M(k1,r)=2M(k1)cos(2πk1·r+ϕ1), where r is an U-ion position and ϕ1 the associated phase (cf. Supplementary Note 1). This result is in agreement with the high-field magnetization22, since the amplitude 2M(k1) extracted here at 36 T is of same order than the variation of magnetization ΔM≃0.4 μB/U (estimated from 36–39 T) induced by the quench of the field-induced ordered moments into a polarized regime above μ0H3=39 T.
The field dependence of the diffracted intensity is presented in Fig. 3 for a large set of momentum transfers Q from (Qh 0 0), (Qh 0 −1) and (1 0 Ql) scans in the reciprocal space at T=2 K. Except at Q=(0.6 0 0) and (1.6 0 −1) corresponding to k1, no field-induced change of the neutron-diffracted intensity is observed at the different probed momentum transfers. In particular, no change of intensity occurs in high field at the harmonics nk1, where n≠1 is an integer, including the Brillouin zone centre kZC=(0 0 0), within an error of ∼5–10% of the induced intensity at k1 (cf. Supplementary Table 1). The absence of harmonics supports that the high-field magnetic structure is a spin-density wave, that is, a sine-modulated magnetic structure, in opposition with a squared-modulated structure, where harmonics of the main wavevector would lead to full moments, either up or down, on the U sites (cf. Supplementary Note 2). Spin-density-wave modulations of the magnetic moment are generally observed at low temperature in itinerant magnets at zero field, as chromium35, or, in a magnetic field, as the heavy-fermion superconductor CeCoIn5 (ref. 36) and the low-dimensional magnet Sr3Ru2O7 (ref. 37). However, in the present case, we cannot fully exclude the possibility of a ‘special’ sequence of squared Ising moments, for which high-order harmonics nk1 would all have intensities smaller than our error bar, that is, 5–10% of the magnetic intensity at k1. The absence of neutron Bragg peaks at k2=(2/3 0 0) and kZC is in contradiction with the up–up–down squared modulation proposed from nuclear magnetic resonance (NMR) measured on URu2Si2 in the field window 35.5–36.5 T (ref. 38). The NMR spectrum, composed of a central narrow peak and two broad satellites, is not compatible with a single-k spin-density wave, which would lead either to a few discrete and thin peaks (if k1 is commensurate) or to a double-horn line shape (if k1 is incommensurate)38. However, an incommensurability of k1 (that is, a slight deviation from (0.6 0 0) compatible with our neutron resolution Δk≃(±0.01±0.05±0.003)), combined with a multi-k character of the spin-density wave, could induce peculiar features, as a central peak, in the NMR spectrum39, and could possibly reconcile the NMR and neutron studies. Such multi-k structure would imply a conservation of the fourfold symmetry, that is, of the equivalence between x and y, in the high-field spin-density-wave phase, in contrast with the twofold symmetry proposed in refs 16, 17 for the low-field hidden-order phase.
Absence of a ferromagnetic Bragg peak
Surprisingly, the neutron-diffracted intensity at the Brillouin zone centre kZC is field invariant, while the magnetization M is strongly enhanced above 35 T (ref. 22). Usually, a magnetic Bragg peak develops at kZC and has an intensity ∝M2. From M≃0.5–1 μB/U in the field window 35–39 T, and assuming a magnetic form factor corresponding to localized f electrons (estimated from a low-field study of the magnetic form factor40), one expects a magnetic intensity of ≃2,000–8,000 counts per s, in addition to the non-magnetic Bragg peak of ≃19,000 counts per s, at Q=(1 0 −1) corresponding to kZC (cf. Supplementary Note 1). However, no such change of intensity is observed here at Q=(1 0 −1), within an error of 200 counts per s. Several scenarios could explain this unexpected feature. (i) The absence of a magnetic Bragg peak at kZC could result from a strongly-reduced magnetic form factor fM related to itinerant electrons. Since the observation of magnetic Bragg peaks at Q=(0.6 0 0) and (1.6 0 −1) demonstrates that nearly localized electrons with fM≲1 drive the high-field spin-density wave of wavevector k1, scenario (i) would imply that the high-field magnetism would be localized, or nearly localized, at k1 but itinerant at kZC. (ii) Alternatively, the absence of magnetic Bragg peak at kZC could be explained by field-induced magnetic excitations at kZC, which would contribute to the enhanced magnetization (cf. Kramers–Kronig relationship41), but not to the magnetic Bragg peak at kZC. (iii) Finally, this absence could also result from a high-field decrease of the nuclear Bragg peak at Q=(1 0 −1) (related via local stress effects to a change of some atomic positions), which would counteract the increase of magnetic intensity at kZC. The missing magnetic Bragg intensity at kZC constitutes an open question and will merit further studies.
Relationship with Fermi surface nesting
In an itinerant picture of magnetism, long-range magnetic ordering with a wavevector km can be related to a partial or complete nesting of two parts of the Fermi surface. The dual localized itinerant behaviour is a central issue in heavy-fermion metals, where the magnetic properties often result from itinerant f electrons. In URu2Si2, 5 f electrons hybridize with electrons from the conduction bands and modifications of the Fermi surface and carrier mobility accompany the establishment of the hidden-order phase at the temperature T0 (refs 26, 42, 43, 44, 45). The itinerancy of the f electrons has already been shown to be a key ingredient for the magnetic properties of this system. Angle-resolved photoemission spectroscopy and band-structure calculations showed that Fermi surfaces centred on the Г and Z points are nested with the wavevector k0 (refs 9, 10, 11, 12, 15). In refs 9, 15, a nesting with the wavevector k1 between Fermi surface ‘petals’ around the Г point has been identified. In refs 11, 12), a nesting with the wavevector (0.4 0 0) between these petals and a Fermi surface centred on the Г point has also been found. In ref. 14, another picture has been proposed, with a nesting with wavevector k0 between small Fermi pockets centred on the X points (defined in the first Brillouin zone) and a nesting with wavevector k1 between Fermi surfaces centred on the Г and Z points. Despite these nesting of the Fermi surface, no long-range magnetic order has been reported yet in URu2Si2 in its hidden-order phase. Instead of long-range magnetic order, inelastic magnetic excitations at the wavevectors k0 and k1 have been observed by neutron scattering in the hidden-order phase5,6,7,8. Their relationship with nesting effects and long-range magnetic ordering will be discussed below.
Assuming a high-rank multipolar order with wavevector k0, which cannot be seen directly by neutron diffraction, band-structure calculations resulted in a Fermi surface of the hidden-order state, which is very similar to that of the pressure-induced antiferromagnetic state9, in accordance with the experimental observations19. Under a high magnetic field applied along c, successive Fermi surface modifications occur inside the hidden-order phase and at the transition fields H1, H2 and H3 delimiting the high-field phases24,25,26,27,28. Several Fermi surface instabilities in fields smaller than μ0H1=35 T, that is, inside the hidden-order phase24,26,27,28, are due to Lifshitz transitions and suggest a progressive modification of Fermi surface nestings. Resulting from the Fermi surface reconstruction at H1, the Fermi surface nesting with k1 is probably reinforced, leading to the onset of the spin-density wave observed here with the wavevector k1.
Magnetic fluctuations as a precursor of magnetic ordering
Figure 4 presents the magnetic phase diagram of URu2Si2 under pressure and magnetic field, over a wide range of temperatures up to 75 K. At zero field and ambient pressure, URu2Si2 is in a correlated paramagnetic regime below Tχmax=55 K at which the static magnetic susceptibility χ is maximal22. The correlations are accompanied by a strong and broad magnetic-fluctuation spectrum peaked at the wavevector k1, which progressively sets at temperatures of the order of Tχmax (ref. 8). When the system enters in its hidden-order phase, the magnetic fluctuations give rise to sharp inelastic spectra at the wavevectors k0 and k1, with the gaps ΔE0≃1.7 meV and ΔE1≃4.5 meV, respectively, at low temperature. Below T0, the intensity of the magnetic fluctuations at k0 suddenly develops, while the intensity of the fluctuations at k1, as indicated by the static susceptibility χ′(k1, E=0), drops by 20% (refs 8, 46).
The magnetic fluctuations peaked at the hot wavevectors k0 and k1 indicate the proximity of quantum-phase transitions associated with long-range ordering with the same wavevectors. Such precursory role of the magnetic fluctuations has been emphasized in the heavy-fermion paramagnet CeRu2Si2, where doping (La, Rh or Ge) and magnetic field permit to tune competing quantum instabilities (cf. ref. 47 and refs therein). With pressure20,21 (as well as Rh doping30), URu2Si2 can be tuned to antiferromagnetic long-range order with wavevector k0. The disappearance of the hidden order in URu2Si2 under pressure coincides with a transfer of weight from magnetic fluctuations with wavevector k0 into antiferromagnetic order with the same wavevector48,49. Contrary to those with k0, the fluctuations with wavevector k1 survive under pressure48 and their progressive variation may be in relationship with the continuous evolution of Tχmax (ref. 50). With magnetic field, a spin-density wave is stabilized with the wavevector k1. Both Tχmax and T0 vanish in the quantum critical area μ0H≃35–39 T (ref. 22) where long-range magnetic ordering sets in. In the polarized regime above μ0H3=39 T, from the magnetization >1.4 μB/U, one can expect a loss of intersite magnetic correlations at k0 and k1, the magnetic excitations becoming wavevector-independent, that is, of local nature. This picture is compatible with the observation in fields up to 17 T that the gap in the excitations at k0 increases with H, while that in the excitations at k1 remains constant, their extrapolations converging at ≃35 T (ref. 51). In the critical field window (35 and 39 T), the enhancement of the Sommerfeld coefficient22 is the signature of critical fluctuations, presumably of magnetic origin (cf. CeRu2Si2, where critical magnetic fluctuations were reported at the La-doping- and magnetic-field-induced instabilities52,53). However, inelastic neutron scattering under steady fields up to 40 T, which is far beyond the present state of the art (26 T achieved recently at the neutron source in Berlin54), would be needed to unveil the high-field evolution of the magnetic fluctuations in URu2Si2.
We succeeded in detecting a high-magnetic-field switch from hidden-order to spin-density-wave phases in URu2Si2. This result illustrates the dual localized and itinerant facets of the 5f electrons in this system. The relationship between Fermi surface nesting and the magnetic wavevector k1 of the spin-density wave, as well as the cascade of Lifshitz transitions induced by the high-field magnetic polarization24,25,26,27,28, show the itinerant character of the 5f electrons. The neutron-diffracted intensity with the wavevector k1 is compatible with the magnetic form factor fM corresponding to U3+ or U4+ configurations40 and highlights the localized character of the 5f electrons. Knowing that a change of the Fermi surface in the polarized regime of URu2Si2 was detected indirectly by Hall effect55 and directly by quantum oscillations experiments25, a challenge is to determine whether the missing magnetic Bragg peak at the wavevector kZC=(0 0 0) in a high magnetic field could be related to a change in the itinerancy of the f-electrons.
A relationship between the long-range magnetic order, the Fermi surface and the magnetic fluctuations has been highlighted. In the hidden-order phase, one may suspect ‘inelastic’ nestings between electronic band surfaces of different energies, to be connected with the observed gapped magnetic excitations at the wavevectors k0 and k1 (refs 5, 6, 7, 8). These excitations are precursors of the long-range ordered states induced under magnetic field and pressure. Under a magnetic field, the transformation of the paramagnetic state with gapped magnetic excitations into a spin-density wave with wavevector k1 might be related to the stabilization of Fermi surface nesting with the wavevector k1. Under pressure, a modification of the Fermi surface nesting with the wavevector k0 might also lead to long-range magnetic order with the wavevector k0. Additional effects, as modifications of the crystal field, could also play a role in the stabilization of magnetically ordered phases.
Interestingly, the effect of Rh doping leads to a decrease of T0 and Tχmax (refs 30, 31) similar to the magnetic field effect, and to antiferromagnetic order with wavevector k0 (for Rh contents 0.02≤x≤0.03) similar to the pressure effect. The main wavevector k2=(2/3 0 0) of the high-field squared magnetic structure of U(Ru0.96Rh0.04)2Si2 (ref. 33) is rather close to the wavevector k1=(0.6 0 0) of the spin-density wave of URu2Si2, confirming a relationship between these two magnetic phases. The differences between these wavevectors might result from subtle differences in the band structures of the two compounds, due to the doping sensibility of their low-carrier density.
Our observation of a spin-density wave in magnetic fields between 35 and 39 T will certainly push to develop models incorporating on equal basis the Fermi surface topology and the magnetic interactions. With the objective to describe competing quantum-phase transitions between the hidden-order and long-range ordered phases, such models will be a basis to solve the hidden-order problem. From an instrumental point of view, this work paves the way for neutron magnetic diffraction on weak-moment systems under pulsed magnetic field up to 40 T.
The single crystal of mass m≃1 g and dimensions 6 × 4 × 4 mm3 used in this study has been grown by the Czochralski technique in a tetra-arc furnace. It has been characterized by X-ray Laue. A sample from the same batch was checked by electrical resistivity, showing a residual resistivity ratio of ∼50 and a superconducting temperature Tc of ∼1.5 K, indicating its high quality. A slit perpendicular to the basal plane was cut into the sample to reduce the surface exposed to the pulsed field and, thus, self-heating of the sample by eddy currents.
A new long-duration (rising time of 23 ms and total time of >100 ms) and high duty cycle (40, 36 and 31-T shots can be repeated every 10, 7 and 5 min, respectively) 40-T conical pulsed magnet developed by the LNCMI-Toulouse has been used. A cryostat specially designed by the ILL-Grenoble for this magnet allowed reaching temperatures down to 2 K, and the diffraction experiment was carried out on the triple-axis spectrometer IN22 (Collaborative Research Group-CEA at the ILL, Grenoble) operated in a double-axis mode with an incident neutron wavelength of 1.2 Å. At each momentum transfer, the field dependence of the neutron intensities has been extracted by summing the data accumulated over a few tens of pulsed-field shots, with either constant time- or constant field-integration windows (the rise and fall of the pulsed field were analysed separately).
The data that support the findings of this study are available from the corresponding author on request.
How to cite this article: Knafo, W. et al. Field-induced spin-density wave beyond hidden order in URu2Si2. Nat. Commun. 7, 13075 doi: 10.1038/ncomms13075 (2016).
We acknowledge R. Ballou, J. Béard, P. Dalmas de Réotier, H. Ikeda, M.-H. Julien, M. Horvatic, E. Ressouche and G Rikken for useful discussions. Part of this work was funded by the ANR grant Magfins N° ANR-10-0431. H.N. acknowledges KAKENHI 23224009. D.A. acknowledges KAKENHI, 15H05745, 15H05882, 15H05884, 15K21732, 25247055. H.N. and D.A. acknowledge support by ICC-IMR.
Supplementary Table 1, Supplementary Note 1-2 and Supplementary References
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