Absence of a static in-plane magnetic moment in the 'hidden-order' phase of URu₂Si₂

In compounds that contain transition metal, lanthanide or actinide ions with partially filled d- or f-electron shells, the strong electronic correlations originating in the hybridization of localized d- or f-electron and conduction electron states often leads to the emergence of new electronic ground states such as heavy fermion metals, complex magnetic order, quadrupolar order, non-Fermi-liquid behavior and unconventional superconductivity (SC) [1]. The search for and the understanding of novel electronic ground states thus is an important research direction in the study of strongly correlated electron phenomena.

A prime example of such emergent behavior is the 'hidden-order' (HO) phase in the heavy fermion compound URu₂Si₂ that occurs below T₀ = 17.5 K and coexists with SC below T_c = 1.5 K [2–4]. Neutron scattering experiments demonstrate the presence of a small antiferromagnetic moment S_⊥ ≈ 0.01–0.04 μ_B/U perpendicular to the tetragonal basal plane in the HO phase that is, however, very much small to account for the entropy of ≈ 0.2R ln(2) associated with the observed specific-heat anomaly [5–7]. Recently, more detailed neutron scattering studies have, however, put forth the view that the magnetic structure in the HO phase is induced by strain [6, 7] and is due to a small amount of the neighboring large moment antiferromagnetic (LMAFM) phase that emerges at critical pressures P_c ⩾ 0.5–1.5 GPa [8].

This led to the terminology HO [9] to allude to the unknown identity of the corresponding order parameter (OP) that has eluded identification for almost three decades. Notably, the search for the OP of the HO phase has attracted an enormous amount of attention, and over the last few decades the full arsenal of experimental methods has been employed in the effort to unravel this notorious phase. This concentrated experimental effort has established that the presence of HO is reflected in many details of the complex electronic structure of URu₂Si₂ [10].

As originally inferred from the specific heat a charge gap of Δ ≈ 11 meV opens over about 40% in the Fermi surface [2, 3]. This is also evident from the large jump observed in the Hall coefficient R_H at T₀ that indicates a reduction of charge carrier concentration n from 0.10 holes per U atom in the paramagnetic state to 0.02 in the HO phase [11–13] and measurements of the optical conductivity [14]. Inelastic neutron scattering has further revealed that a spin gap opens simultaneously with the charge gap. Here a spin gap is observed both at the commensurate wave vector Q₀ = (1,0,0), as well as the incommensurate wave vector Q₁ = (0.4,0,0), where the values of the corresponding gaps are 2 and 4 meV, respectively [15, 16]. Above T₀ the Q₀ mode transforms to weak quasielastic spin fluctuations. In contrast, the Q₁ mode is due to itinerant-like spin excitations that are related to the heavy electronic quasiparticles that form below a coherence temperature T* ≈ 70 K. Here, Wiebe et al have shown that the gapping of these incommensurate spin fluctuations accounts for the loss of entropy at the HO transition [16].

Further details have been revealed by investigations of the Fermi surface (FS) of URu₂Si₂. Here quantum oscillation measurements [17, 18] have demonstrated that the FS in the HO phase of URu₂Si₂ is mostly dominated by small closed pockets which is again in agreement with the partial gapping of the FS. Moreover, measurements including angle-resolved photo emission spectroscopy [19], scanning-tunneling microscopy [20, 21] and point-contact spectroscopy [22] have revealed that the electronic structure of URu₂Si₂ is reorganized below T₀ where a heavy quasiparticle band shifts below the Fermi level, and the crossing with a light hole-like band at Q* = ± 0.3π/a leads to the formation of a hybridization gap Δ_Q* = 5 meV.

In parallel with these extensive experimental efforts the HO problem has also motivated a wide range of theoretical studies, in particular because many of the electronic signatures of the HO phase are also relevant more broadly for strongly correlated electron systems in general. In turn a multitude of models to explain the HO phase and its elusive nature have been proposed (see [10] and references therein). However, to date none of the models could be confirmed, often because the nature of the corresponding complex OP cannot be easily verified by means of current experimental methods.

Recent magnetic torque experiments suggest that the HO phase spontaneously breaks the rotational symmetry of the crystal in the [110] direction [23]. In combination with the absence of lattice distortions through the HO transition [24, 25], this suggests that the broken symmetry is solely of electronic origin. Interestingly, this new experimental constraint rules out many previously proposed models for the HO phase (see e.g. [10]), and has induced a flurry of new theoretical proposals, such as dynamical symmetry breaking [26, 27], staggered spin–orbit coupling order [28], spin–nematic states [29], a rank-5 superspin [30] and hastatic order [31]. Here we focus on the latter two, which both propose a non-zero magnetic moment S_∥ in the tetragonal basal plane—a prediction that can be experimentally tested via magnetic neutron diffraction.

The OP proposed by Rau and Kee [30] is a rank-5 E type spin density wave between the 5f crystal field doublets Γ⁽¹⁾₇ and Γ⁽²⁾₇. This candidate for the HO OP is based on a tight-binding model for the itinerant heavy quasiparticles in URu₂Si₂. The proposed OP breaks both time-reversal symmetry and the lattice point group symmetry D_4h, consistent with the torque magnetometry results. For this OP a small antiferromagnetic magnetic moment S_∥ in the tetragonal basal plane oriented along the [110] direction ordered at a wave vector (0, 0, 1) is expected due to second-order correlations.

Chandra et al [31] have based their proposal for the HO OP on the Ising-like nature of local 5f moments in URu₂Si₂ [2], as well as on the recent observation that the quasiparticles in the HO phase exhibit a giant Ising anisotropy [18, 32]. They demonstrate that such Ising quasiparticles result from a spinor OP that breaks double time-reversal symmetry, mixing states of integer and half-integer spin by hybridizing the conduction electrons with Ising 5f² states of the uranium atoms. This OP accounts for the large specific-heat jump and the torque magnetometry results. Similar to the proposal by Rau and Kee [30] a small in-plane magnetic moment S_∥ is predicted. Estimates from this theory suggest that S_∥ is of the order of 0.015 μ_B/U, which would be detectable in a sensitive neutron diffraction experiment.

To date such an in-plane magnetic moment in the HO phase of URu₂Si₂ has not been observed. Establishing an experimental limit for the size of S_∥ is therefore highly desirable in order to guide current and future theoretical efforts in disentangling the nature of the HO phase. Here we show by means of a carefully designed neutron diffraction study that within the detection limit of state-of-the-art neutron scattering no static in-plane magnetic moment exists within the HO phase.

The sample used for this work is a 7 g single crystal of URu₂Si₂ synthesized via the Czochralski technique using a tetra-arc furnace. The sample has been characterized by both x-ray and neutron diffraction measurements. High sample quality is demonstrated by a residual resistance ratio of ≈10 within the tetragonal plane that is identical to other neutron scattering studies of high-quality single crystals [7].

Our measurements were carried out on the BT-7 thermal triple-axis spectrometer at the NIST Center for Neutron Research [33]. BT-7 was operated in elastic mode with the wavelength λ = 2.36 Å. To reduce contamination due to higher-order Bragg scattering at the monochromator PG filters, both the up and down stream of the sample positions were employed. The instrument was optimized for high intensity in order to search for the postulated in-plane magnetic moment and used a position-sensitive detector (PSD) without the analyzer. BT-7 was equipped with an 80' Soller collimator in front of the sample position and an 80' radial collimator in front of the PSD.

In order to separate the postulated in-plane (S_∥) from the well-known out-of-plane (S_⊥) component of the weak magnetic moment we have exploited the condition that the magnetic scattering intensity depends on the mutual direction between the magnetic moments and the scattering vector Q. The magnetic neutron scattering cross-section is proportional to $\sum _{\alpha ,\beta }(\delta _{\alpha \beta }-\hat {Q}_{\alpha }\hat {Q}_{\beta })M_{\boldsymbol {Q}}^{\alpha \dagger }M_{\boldsymbol {Q}}^{\beta }$, where $\hat {Q}$ and M_Q are a unit vector parallel to the scattering vector Q and the magnetic structure factor, respectively. α and β describe their components with α,β = x,y,z. The magnetic structure factor is the Fourier transform of the local magnetization density M(r) and can be written in terms of magnetic Fourier components [34] that for URu₂Si₂ take the form $\boldsymbol{M}_{\!\boldsymbol{Q}}=\sum _d g_d F_d(\boldsymbol {Q})\exp (-W_d(\boldsymbol {Q}))\exp (\mathrm {i}\boldsymbol {Q}\cdot \boldsymbol {d})\boldsymbol {S}^{\boldsymbol {Q}}_d$. Here d describes the position of the dth magnetic ion in the unit cell. g_d, F_d(Q) and $\exp (-W_d(\boldsymbol {Q}))$ are the Landé g-factor, the magnetic form factor and the Debye–Waller factor for the dth ion, respectively.

URu₂Si₂ crystallizes in the space group I4/mmm, and the antiferromagnetic order found in the HO state is described by the two uranium ions (0, 0, 0) and (1/2, 1/2 1/2) that exhibit antiparallel magnetic moments S^Q₁ = (S_∥,x,S_∥,y,S_⊥) and S^Q₂ = (−S_∥,x,−S_∥,y,−S_⊥), where $S_{\parallel } = \sqrt {S_{\parallel ,x}^2+S_{\parallel ,y}^2}$. We denote the known component perpendicular to the tetragonal basal plane with S_⊥ (figure 1(a)), and the postulated component within the plane as S_∥ ( figure 1(b)).

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Inspecting the $(\delta _{\alpha \beta }-\hat {Q}_{\alpha }\hat {Q}_{\beta })$ term in the magnetic cross-section it is clear that only components of the magnetic structure factor that are perpendicular to Q are visible for the current scattering geometry as illustrated in figure 1. Our sample was oriented in such a way that only [H0L] Bragg reflections were accessible. In these experiments, we probed the components of the magnetic moments in two configurations. By measuring the intensity of the [100] magnetic Bragg reflection, only components parallel to the [0KL] plane (equivalent to the real space ac-plane) may be observed thus allowing us to probe the known component S_⊥ (figure 1(c)). Similarly, the existence of S_∥ was investigated by measuring at the [001] position, which only generates magnetic intensity from spin components within the tetragonal basal plane (figure 1(d)). Furthermore, we have also measured the [003] reciprocal position, for which the geometry is identical to the [001] position, and therefore was also used to probe S_∥. We note that both the [100], [001], [003] reflections are forbidden structural reflections and all scattering observed at these positions will be purely magnetic.

Figure 2 shows maps of the observed neutron intensity as measured around the [100], [001] and [003] Bragg reflections, respectively. The maps were obtained by rotating the sample around the axis perpendicular to the [H0K] scattering plane used in our experiment. For the [100] reflection, which probes mostly S_⊥, there is a difference in neutron intensity for the data taken at T = 2.5 K in the HO phase (figure 2(a)) and T = 20 K above T₀ (figure 2(b)). Even in the 20 K data set a clear peak is visible which is due to higher-order scattering from the [002] reflection which is the most intense nuclear Bragg reflection. This was verified at the end of our experiment by inserting a third PG filter after the sample position, which in turn eliminated the higher-order scattering completely. To isolate the magnetic scattering from the temperature-independent higher-order scattering, we additionally show a map that was obtained by subtracting the 20 K from the 2.5 K data set (figure 2(c)). As seen in this difference plot, there is a clear magnetic response corresponding to the previously observed magnetic structure with the magnetic moment parallel to the c-axis arises below T₀.

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In contrast, for the [001] reciprocal space position that purely probes S_∥, there is no visible difference between measurements at T = 2.5 K (figure 2(d)) and 20 K (figure 2(e)). This is also borne out in the difference data (figure 2(f)) indicating that within the detection limit of neutron diffraction S_∥ ≈ 0. Similar to the [100] position the temperature-independent intensity is due to higher order nuclear scattering. We note that the [001] reciprocal space position in our setup with λ = 2.36 Å is near to the unscattered neutron beam, e.g. the scattering angle for [001] is 2θ = 14.2°, and thus is affected by an increased background. Therefore, we have carried out additional measurements around the [003] reciprocal space position that also is purely sensitive to S_∥. While measurements at the [003] position are less sensitive to magnetic scattering compared to [001] because the magnetic form factor for U⁴⁺ ions is almost 30% reduced, the background is significantly improved at [003]. The corresponding maps for [003] for T = 2.5 and 20 K are shown in figures 2(g) and (h), respectively, and the difference map is shown in figure 2(i). Just as for the [001] position there is no additional magnetic intensity observed at the [003] reflection, confirming that S_∥ ≈ 0.

In order to provide an experimental upper limit for the size of S_∥, we plot Q-scans through the [100], [001] and [003] positions shown in figure 3 that have been extracted from the maps such as the ones in figure 2. In addition, we have measured more maps in the HO phase at [100] at T = 10 and 15 K, and at [001] and [003] at T = 10 K, and the corresponding extracted Q-scans are also plotted in figure 3. For all shown Q-scans the data recorded at T = 20 K were subtracted to remove the higher-order nuclear scattering.

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Q-scans through the [100] (see figure 3(a)) position show clear magnetic peaks arising in the HO phase. The integrated intensity of the [100] magnetic Bragg reflection was determined by means of fits with Gaussian profiles. The corresponding magnetic moment perpendicular to the tetragonal basal plane was calculated from the integrated intensity by calibrating with the integrated intensity of the [006] nuclear Bragg reflection. This reflection was the weakest nuclear Bragg reflection accessible in our experiment and was chosen to avoid the problem of extinction. Using this method we found that S_⊥ = 0.016(1) μ_B/U in agreement with the values reported in literature previously [5–7]. The temperature dependence of S_⊥ is shown in figure 4.

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As demonstrated in figures 3(b) and (c) the Q-scans through the [001] and [003] positions show no peak at all, and no temperature dependence, clearly indicating the absence of any moment in the tetragonal basal plane. We note that all measurements on the [100] position have been performed with count rates of 1 min per point, whereas the measurements around [001] and [003] have been carried out with 12 min per point, in turn resulting in 3.5 times better statistics. This allows us to define a simple experimental upper limit for S_∥. As illustrated in figure 4 the magnitude of the size of magnetic moment perpendicular to the basal plane decreases to S_⊥ = 0.007(3) μ_B/U at T = 15 K, which is just below T₀. Notably, from the Q-scans through the [100] magnetic Bragg reflection at T = 15 K shown in figure 3(a) it is clear that S_⊥(15 K) is at the detection limit of our experiments for scans performed with count rates of 1 min per point.

Using the fact that the Q-scans around the [001] and [003] reciprocal space positions have been carried out with 3.5 better statistics, we thus can define a conservative detection limit for the in-plane component S^conservative_∥ ⩽ S_⊥(15 K)/3.5 = 2 × 10⁻³ μ_B/U. However, using both the magnetic form factor dependence for the reciprocal space positions investigated and the known temperature dependence of the known component of the magnetic moment S_⊥, we may use the full statistics of the four Q-scans shown in figures 3(b) and (c) to define a more accurate detection limit.

Considering the magnetic form factor for U⁴⁺ ions on all three measured reciprocal space positions [100], [001] and [003], we find that F_d([001])/F_d([100]) = 1.20 and F_d([003])/F_d([100]) = 0.86. Furthermore, inspecting figure 4 we obtain S_⊥(10 K)/S_⊥(2.5 K) = 0.8. Therefore, the full statistics of the four Q-scans performed at the [001] and [003] positions is a factor $\sqrt{12 \{ [(F_d([001])/F_d([100]))^2+(F_d([003])/F_d([100]))^2 ] [1+(\frac {S_\perp (10\,\mathrm {K})}{S_\perp (2.5\,\mathrm {K})})^2 ] \}}=6.5$ better than for the Q-scan carried out at the [100] position. Here we have used that the magnetic neutron intensity is proportional to the square of both the magnetic moment and the magnetic form factor. Consequently, Q-scans carried out around the [001] and [003] reciprocal space positions with 6.5 better statistics would be able to detect a magnetic moment S^max_∥ ⩽ S_⊥(15 K)/6.5 = 1 × 10⁻³ μ_B/U.

Finally, our results additionally enable us to establish an upper limit for the magnetic correlation length ξ between magnetic moments S_∥ in the tetragonal basal plane. The width of a magnetic Bragg in a neutron scattering experiment is a direct measure for the inverse magnetic correlation length κ = 1/ξ. The region of the reciprocal space that we have investigated around the [001] and [003] directions (see figure 2) allows us to probe maximum inverse correlation lengths κ_a = 0.2 Å⁻¹ and κ_c = 0.12 Å⁻¹ along the crystallographic a- and c-directions, respectively. Because we have not observed magnetic Bragg peaks over these inverse length scales any magnetic correlations in URu₂Si₂ would have larger inverse correlation length, i.e. the associated magnetic peak would be much broader than the large portion of reciprocal space investigated in this experiment. This suggests that any magnetic correlations between in-plane magnetic moments S_∥ must develop over correlation lengths smaller than $\xi _a^{\max }=1/\kappa _a=8.2\, \mathring {\rm A}$ and $\xi _{ c}^{\max }=1/\kappa _{ c}=5.1\, \mathring {\rm A}$, which corresponds to approximately 2 and 0.5 lattice spacings for the a- and c-axis, respectively. This clearly establishes that any sort of magnetic correlations for magnetic moments parallel to the tetragonal basal plane must be very short range in nature, and that no long-range magnetic order with in-plane moments develops in the HO phase of URu₂Si₂.

We note that the absence of a static in-plane component of the magnetic moment S_∥ may be explained in terms of a fluctuating in-plane component. This possibility was not investigated in detail in this work, because current models suggest a static magnetic moment [30, 31]. However, we note that because our experiment was carried out in the diffraction mode of BT-7 where the intensities are integrated over all final neutron energies after the scattering process at the sample, a fluctuating moment S_∥ should have still led to some increased intensity at the [003] reciprocal space position if both the moment and the lifetime of the fluctuations were large enough. If the lifetime of the fluctuations were short or the fluctuating moment was very weak, only an energy-resolved neutron scattering experiment would be able to detect an in-plane fluctuating moment. This possibilities will be investigated in future work.

We briefly discuss the possibility of the parasitic character of the well-known out-of-plane component S_⊥ of the magnetic moment. There are reports that the tiny magnetic moment S_⊥ is induced by strain that causes a small amount of the neighboring LMAFM phase to coexist with the HO phase [7, 8]. The presence of this parasitic phase would decrease the volume fraction of the HO phase, and therefore the size of a possible in-plane component S_∥. However, comparing the measured moment in the LMAFM phase of 0.5 μ_B/U with the known size of S_⊥ ≈ 0.01–0.04 μ_B/U in the HO phase, one estimates that the parasitic LMAFM phase would be less than 8%. For high-quality samples such as the one used for this study it should be even smaller. Therefore, we conclude that a possible parasitic character of S_⊥ does not influence the upper bound for S_∥ significantly. Similarly, we note that our result is not influenced by the mosaic of our sample. In fact, a sample exhibiting mosaicity increases the acceptance of a typically divergent neutron beam. We therefore argue that a small sample mosaic typical for a high-quality sample is useful for detecting small magnetic moments.

We also consider the influence of magnetic domains on the limit provided for S_∥. URu₂Si₂ crystallizes in the tetragonal space group I4/mmm that does not contain any symmetry operations which may mix in-plane and out-of-plane components of the magnetic moment, and therefore, we limit our discussion to symmetry operations that transform the in-plane components (S_∥,x,S_∥,y,0) of the magnetic moment. Here, the four-fold rotation symmetry of the tetragonal basal plane will lead to four configuration domains for which the magnetic moments will be directed along (S_∥,x,S_∥,y,0), (−S_∥,x,S_∥,y,0), (S_∥,x,−S_∥,y,0), and (−S_∥,x,−S_∥,y,0), respectively. However, the selected measurement geometry (see figure 1(d)) probes all four domains simultaneously, and the total magnetic intensity at the probed reciprocal space positions [001] and [003] is the sum of the individual intensities for each domain. Thus, the presence of magnetic domains does not influence our result.

In conclusion our measurements provide an experimental upper limit for the size of the theoretically predicted component S_∥ of the antiferromagnetically ordered static magnetic moments in the HO phase of the heavy fermion system URu₂Si₂. The established experimental limit S^max_∥ ⩽ 1 × 10⁻³ μ_B/U is more than an order of magnitude smaller than the value S^theo_∥ = 0.015 μ_B/U suggested by theory [31]. This new limit is therefore an important constraint for present [30, 31] and future theories that aim to model the OP of the notorious HO phase of URu₂Si₂.

In particular, it is interesting to discuss our results with respect to the magnetic torque experiments that have been interpreted in such a way that the HO phase spontaneously breaks the rotational symmetry of the crystal in the basal plane [23]. Here it has been suggested that the line broadening of the NMR signal observed in URu₂Si₂ for magnetic fields parallel to the basal tetragonal plane [35] is due to a non-zero but tiny susceptibility χ_[110] that also breaks time-reversal symmetry. Our measurements clearly demonstrate that long-range magnetic order with magnetic moments parallel to the basal tetragonal plane is absent in URu₂Si₂ and that any static contribution to χ_[110] is zero. This suggest that both the proposed broken four-fold symmetry in the tetragonal plane and the putative broken time-reversal symmetry are not due to the presence of a magnetic dipole moment in the basal plane. Based on more recent NMR measurements and symmetry analysis it has been argued that the line-broadening and the associated broken time-reversal symmetry may be explained by the presence of magnetic multipoles such as octupoles or triakontadipoles in combination with domains or disorder [36]. In agreement with our study, this NMR study by Takagi et al [36] also concludes that the in-plane magnetic dipole moment S_∥ is zero. Finally, the broken rotational symmetry of the tetragonal basal plane may be explained via an electronic nematic state as proposed in [23]; however, more detailed investigations of the electronic structure of URu₂Si₂ are required to resolve this issue.

We thank Filip Ronning and Premala Chandra for useful discussions. Sample synthesis and characterization at UCSD were funded by the US DOE under grant no. DE FG02-04ER46105. Work at Los Alamos National Laboratory (LANL) was performed under the auspices of the US DOE, OBES, Division of Materials Sciences and Engineering and funded in part by the LANL Directed Research and Development program. MJ acknowledges financial support from the Alexander von Humboldt foundation.