Magnetic and Electronic Properties of URu2Si2 Revealed by Comparison with Nonmagnetic References ThRu2Si2 and LaRu2Si2

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The uranium heavy fermion URu 2 Si 2 has been attracting much interest since its discovery almost 30 years ago [1][2][3] because of its fascinating properties, including the mysterious phase transition at T HO ¼ 17:5 K, whose order parameter has not been identified [so-called "hidden order (HO)"], and its unconventional superconductivity below T c ¼ 1:4 K.Among a wide variety of experimental reports on this compound, studies using nuclear magnetic resonance (NMR) have played an important role in providing microscopic information on the issues of the HO and superconducting phases: for example, a decrease in the density of states at Fermi energy 4,5) and spurious antiferromagnetic ordering 6) in the HO phase, and the existence of line nodes in superconducting energy gap.4,5) Moreover, symmetry changes across the HO have recently been discussed from the magnetic [7][8][9] and electronic 10) points of view.
However, we have noticed that some pieces of fundamental information on magnetic fluctuations and U-valence have been missing owing to the lack of comparisons between the NMR data of URu 2 Si 2 and its nonmagnetic references so far.In this paper, we report on the results of NMR and nuclear quadrupole resonance (NQR) measurements performed on the two isostructural non-f compounds ThRu 2 Si 2 and LaRu 2 Si 2 .The comparisons of already reported NMR and NQR data on URu 2 Si 2 4,5) with the present results allow us to analyze them semiqualitatively and to extract unique information on the magnetic and electronic properties of URu 2 Si 2 .First, we discuss whether ThRu 2 Si 2 is suitable as the nonmagnetic reference of URu 2 Si 2 on the basis of the NQR data measured at the Ru site.The NQR frequency Q sensitively reflects local charge distribution.Then, by comparing the nuclear spin lattice relaxation rate 1=T 1 data of URu 2 Si 2 with those of ThRu 2 Si 2 , we conclude that there exist only spin fluctuations along the crystal c-axis in the HO phase and these Ising-like fluctuations may be important for the appearance of the unconventional superconductivity in URu 2 Si 2 .
The single crystals of ThRu 2 Si 2 were grown in a tetraarc furnace under high-purity Ar gas atmosphere by the Czochralski method.The details of the sample preparation are described elsewhere. 11)A single crystal sample was used for the estimation of the temperature dependence of Q at the Ru site by 99 Ru-NMR measurement.Most of other the 29 Si-NMR (nuclear spin 29 I ¼ 1=2), 99;101 Ru-NMR ( 99;101 I ¼ 5=2), and 101 Ru-NQR measurements on ThRu 2 Si 2 were performed using a powdered sample.A polycrystalline sample of LaRu 2 Si 2 was also synthesized by arc-melting constituent elements in Ar atmosphere, and we measured the 101 Ru-NQR spectra of this sample.All the NMR and NQR experiments were carried out by the spin-echo technique with a phase-coherent pulsed spectrometer.T 1 was measured by a single rf-pulse saturation method.
The U ion in URu 2 Si 2 has been regarded to be between U 3+ (5f 3 ) and U 4+ (5f 2 ) configurations.Actually, many theoretical models to explain the various properties of URu 2 Si 2 seem based on the regime starting from the 5f 2 configuration. 12)However, it is difficult to distinguish experimentally these two valences.One of the reasons is that the two 5f electron states have quite close values of effective moment (3.62 and 3.58 B for U 3+ and U 4+ , respectively), whose magnetic responses are indistinguishable.Therefore, we focused on the data on Q , which is sensitive to local charge distribution.
Figure 1(a) shows a field-swept 99 Ru-NMR line of ThRu 2 Si 2 .The spectrum consists of two peaks around 12.0 T arising from a transition between the nuclear spin states of À1=2 $ 1=2 and the two quadrupole-split first satellite lines (AE1=2 $ AE3=2) of 99 Ru-NMR.An additional resonance peak indicated by an arrow is assigned to one of the two first satellites of 101 Ru-NMR.From the obtained spectrum, we were able to extract 99 Q ¼ 1:046 MHz for the 99 Ru nucleus, yielding 101 Q ¼ 6:05 MHz for the 101 Ru nucleus using the ratio of nuclear quadrupole moments, 101 Q= 99 Q ¼ 457=79. 13)This estimation was verified by the observation of a 101 Ru-NQR signal at a frequency of 12:1 MHz ¼ 2 Â 101 Q , as shown in the inset of Fig. 1(a).At the Ru site with fourfold symmetry along the c-axis, the ratio of NQR frequencies Q : Q ð2Þ ¼ 1 : 2, where Q and Q ð2Þ are resonance frequencies for the transitions between AE1=2 $ AE3=2 and AE3=2 $ AE5=2, respectively.
Note that the 10,14) ] is close to that of ThRu 2 Si 2 [ 101 Q ðThRu 2 Si 2 Þ ¼ 6:05 MHz] in which the Th ion is tetravalent.Actually, it is difficult to estimate ionic valence quantitatively only from the Q data.However, if we see how Q changes in valence transition or valence-fluctuating systems (see, for example, Ref. 15 and the case of YbPd 2 Si 2 described below), the present small difference in 101 Q ($ 0:3 MHz) between URu 2 Si 2 and ThRu 2 Si 2 suggests the similarity of electronic configuration, except for 5f electrons, between the two compounds.
101 Q (ThRu 2 Si 2 ) shows weak temperature dependence, as shown in Fig. 2(a).A more precise temperature dependence of 101 Q was measured for LaRu 2 Si 2 in order to obtain further information on electronic states in URu 2 Si 2 . 101Q (LaRu 2 Si 2 ) estimated as 10.65 MHz at 4.2 K [see Fig. 1(b)] is close to 101 Q ¼ 10:52 MHz for CeRu 2 Si 2 , 16) both of which are trivalent or quasitrivalent compounds.On the other hand, 101 Q (LaRu 2 Si 2 ) is ∼1.9 times larger than 101 Q (ThRu 2 Si 2 ).The difference is predominantly attributed to the difference in the charge distribution between the two systems.This result also supports the fact that the U valence is closer to 4+ rather than 3+, so that we found that ThRu 2 Si 2 is applicable as the nonmagnetic reference of URu 2 Si 2 .
One should also notice that the temperature dependence of 101 Q (URu 2 Si 2 ) contrasts with that of 101 Q (LaRu 2 Si 2 ). 101Q (LaRu 2 Si 2 ) shows a gradual decrease with increasing temperature, as seen in the upper panel of Fig. 2(a), which is a characteristic behavior reflecting a normal thermal expansion of the lattice.The result suggests that no significant rearrangement of charge distribution occurs with temperature, namely, the La ion is in a stable trivalent state.In this context, we may expect a somewhat similar temperature dependence for 101 Q (ThRu 2 Si 2 ) as well.
On the other hand, 101 Q (URu 2 Si 2 ) increases with temperature above T HO .Therefore, the difference between 101 Q (URu 2 Si 2 ) and 101 Q (ThRu 2 Si 2 ) (∼0.3 MHz at 4.2 K as mentioned above) is even smaller at higher temperatures, suggesting a closer electronic configuration between these two compounds at high temperatures.The decrease in Q in the low-temperature region is generally observed in valence-fluctuating lanthanide compounds, where the positive valence of lanthanide ions simultaneously decreases upon cooling.We found that the temperature dependence of 101 Q (URu 2 Si 2 ) is analogous to that of the isostructural Yb compound YbPd 2 Si 2 .Here, the Yb valence was studied by X-ray absorption spectroscopy, 17) as shown in Fig. 2(b).As temperature decreases, the delocalization of 4f electrons is induced owing to the evolution of exchange interactions between conduction and 4f electrons, resulting in the decrease in the Yb valence (valence-fluctuating state) and simultaneously in Q (YbPd 2 Si 2 ) [see Fig. 2(c)].Similarly, the temperature dependence of 101 Q (URu 2 Si 2 ) revealed in Fig. 2(a) suggests the delocalization of U-5f electrons at low temperatures.For a more quantitative discussion, reliable theoretical calculations are required.
Next, we show the results of 29 Si-NMR measurements.The 29 Si-NMR line exhibits a slightly asymmetric powder pattern, as seen in Fig. 3, implying that it consists of small anisotropic components of the Knight shift K.However, the evaluated anisotropic component of the shift is quite small: K aniso ¼ 8 Â 10 À3 %.Therefore, compared with the strongly anisotropic Knight shift of URu 2 Si 2 , 4) the shift is regarded to be almost isotropic.As discussed in Ref. 4, the hyperfine coupling constant of URu 2 Si 2 is isotropic, and the present results are consistent with this.
By measuring the intensities of the 29 Si-NMR signal near the frequencies indicated by arrows in Fig. 3, we could estimate 1=T 1 of ThRu 2 Si 2 [hereafter denoted as Th ð1=T 1 Þ]  063702-2 © 2015 The Physical Society of Japan for H k c and H ? c, i.e., Th ð1=T 1 Þ k and Th ð1=T 1 Þ ?, respectively.Their temperature dependences are shown in Fig. 4, along with the previously reported data on URu which is a characteristic behavior of a Fermi liquid.Although we were able to measure Th ð1=T 1 Þ k only up to 10 K owing to a poor signal-to-noise ratio, the result indicates that Th ð1=T 1 Þ is isotropic within experimental accuracy.For T > T HO , we assume that Eq. ( 1) corresponds to a contribution from conduction electrons without 5f electrons in URu 2 Si 2 .As indicated in Ref. 4, U ð1=T 1 Þ shows the Korringa relation above T HO , as shown by dotted lines in Fig. 4. Since 1=T 1 T is generally proportional to Dð" F Þ 2 , with Dð" F Þ being the density of states at Fermi energy, the enhancement of U ð1=T 1 Þ compared with Th ð1=T 1 Þ above T HO implies the formation of a heavy Fermi liquid owing to the Kondo effect.A remarkable feature shown in Fig. 4 is that U ð1=T 1 Þ k abruptly decreases to the line expressed by Eq. (1) below T HO .ð1=T 1 Þ k and ð1=T 1 Þ ?are related to the dynamical susceptibilities perpendicular and parallel to the c-axis, ?ðq; !0 Þ and k ðq; !0 Þ, respectively, as follows: 1 hf;?

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© 2015 The Physical Society of Japan estimate it correctly in the HO phase, because one needs to consider the reduction in carrier density below T HO .Here, if we assume which should give a lower limit of U ð1=T 1 Þ f;i , we obtain KðÞ ( 1 for H k c.This result does not vary even if we assume U ð1=T 1 Þ f;i $ U ð1=T 1 Þ i , which should give an upper limit. According to the modified Korringa relation, the result indicates FM correlations in the HO phase.On the other hand, one expects antiferromagnetic (AFM) correlations originating from the neighboring AFM order phase under pressure.Indeed, a magnetic excitation with the AFM wave vector Q 0 ¼ ð1; 0; 0Þ was detected by an inelastic neutron measurement. 24)Here, to understand the present NMR result, we need to take into account the crystallographical surroundings of the Si site: it is located near an ab basal plane with four nearest-neighbor U ions.As expected from the Q 0 vector, the U magnetic moments tend to align ferromagnetically within the ab basal plane, so that they give rise to FM fluctuations that cannot be canceled out at the Si site.Thus, the T 1 relaxation is predominantly affected by the FM correlations owing to the geometrical factor; thus, the present result is not in disagreement with the AFM correlations between the planes.
In summary, the comparisons of already reported NMR and NQR data on URu 2 Si 2 with the data on isostructural nonmagnetic references (ThRu 2 Si 2 and LaRu 2 Si 2 ) and a valence-fluctuating compound (YbPd 2 Si 2 ) have uncovered some electronic and magnetic properties of URu 2 Si 2 .The data on 101 Q and its temperature dependence suggest that the U electronic configuration is close to 5f 2 at high temperatures, while the 5f electrons gradually delocalize with decreasing temperature.From the 1=T 1 data of 29 Si-NMR, we found Ising-like spin fluctuations along the c-axis in the HO phase.The unique magnetic property is plausibly associated with the mechanism of unconventional superconductivity.

Fig. 1 .
Fig. 1. (Color online) (a) Ru-NMR spectrum of ThRu 2 Si 2 measured at 4.2 K and a frequency of 23.56 MHz.The solid line shows a calculated spectrum with 99 Q ¼ 1:046 MHz.The peak indicated by the arrow is assigned to one of the two first satellites of 101 Ru-NMR.Inset: 101 Ru-NQR spectrum measured at 4.2 K and a frequency corresponding to a AE3=2 $ AE5=2 transition, i.e., 2 Â 101 Q .(b) 101 Ru-NQR spectrum of LaRu 2 Si 2 measured at 4.2 K and 101 Q .The measurement of the 101 Ru-NQR spectrum was performed at 101 Q because the spectrum at 2 Â 101 Q overlaps with the tail of the 3 Â 139 Q line of 139 La-NQR with I ¼ 7=2 and 139 Q ¼ 7:01 MHz.

2 Fig. 2 .
Fig. 2. (a) Temperature dependences of 2 Â 101 Q for URu 2 Si 2 , 10) LaRu 2 Si 2 , and ThRu 2 Si 2 .The ThRu 2 Si 2 data for T !20 K were estimated from 99 Ru-NMR measurement of a single crystal owing to the poor signal intensity of 101 Ru-NQR.Other data were obtained from 101 Ru-NQR measurements.(b) Temperature dependence of the Yb valence of the valence-fluctuating compound YbPd 2 Si 2 evaluated from X-ray absorption spectroscopy measurement. 17)(c) Temperature dependence of 2 Â 105 Q obtained from the 105 Pd-NQR measurement.

2 Fig. 4 .
Fig. 4. (Color online) Temperature dependence of 1=T 1 measured by 29 Si-NMR.The previous data of URu 2 Si 2 4) are compared with the present results for ThRu 2 Si 2 .The solid line is a linear fit to the data for H ? c, and the broken lines indicate Korringa-like behavior for URu 2 Si 2 above T HO .