Ruderman-Kittel interaction between Si in URu2Si2

29Si nuclear magnetic resonance (NMR) has been studied in a 29Si–enriched single crystal sample of URu2Si2. The spin-echo decay for applied field H || [110] and [001] directions has been measured at 50 K. A clear spin-echo decay oscillation is observed for both cases, reflecting the Ruderman-Kittel interaction between Si nuclei. Since the observed oscillation frequency depends on the direction of applied magnetic field, the Ruderman-Kittel and anisotropic pseudo-dipolar interactions may not be negligible in this compound. The origin of spin-echo decay oscillations is discussed.


Introduction
The heavy fermion compound URu 2 Si 2 undergoes a phase transition [1,2,3] at T 0 ∼ 17.5 K. Since the order parameter of the transition has not ever been clearly identified, such order has been termed "hidden order" [4] and has been studied intensively. In addition, the superconducting transition appears only in the hidden ordered state, indicating that the superconducting pairing glue can be related to fluctuations that occur within the hidden order. Previously, the hidden order and superconducting states have been studied by means of NMR Knight shift and spin-lattice relaxation measurements [5,6,7,8]. Recently, we recognize that NMR spin-echo decay measurements are quite useful for investigating the electronic state of strongly correlated electron systems. It is well know that the spin-echo decay process oscillates due to the Ruderman-Kittel (RK) interaction [9,10] between nuclei mediated by the conduction electrons. Thus, the modification of conduction bands can be measured via the spin-echo decay. Indeed, a strong change of spin-echo oscillation frequency is observed at low temperatures in YbRh 2 Si 2 [11].
In the present study, we report data for the 29 Si nuclear spin-echo decay in URu 2 Si 2 . The observed spin-echo decay curves show a clear oscillation, which we attribute to the RK interaction. The anisotropic oscillation frequency indicates that the contributions of the pseudodipolar and dipolar (PDD) interactions are not negligible.

Experimental
Experimental data have been presented and discussed in a previous report [6]. Single crystals were grown using the Czochralski method in a tetra-arc furnace under an argon gas atmosphere, In the present study, high sample purity was confirmed by a residual resistivity of ∼ 5 µΩcm (RRR ∼ 70). The resistivity was measured in a piece cut (2 × 0.5 × 0.5 mm 3 ) from the same single crystal used for NMR measurements.
This has prevented highly accurate Si NMR measurements in URu 2 Si 2 in the past. For the present study, a single crystal sample with a 53% enriched 29 Si isotope has been prepared, improving the NMR sensitivity by a factor ∼ 11. As there are no quadrupolar interactions for I = 1/2 nuclei, 29 Si NMR spectra reflect only magnetic shift and broadening effects.
A single crystal specimen was mounted in a 4 He cryostat with an NMR pickup coil. Using a standard π/2 − π pulse sequence, the spin-echo intensity m(2τ ) so generated was measured as a function of the time τ between excitation pulses (see Fig. 1). Here, a typical π/2 pulse width is 4 ∼ 5 µsec. Since the resonance linewidth is quite narrow (e.g. ∼ 1 kHz at 2.5 T along the [110] axis in the paramagnetic state), all nuclear spins in the spectrum were quite uniformly excited by the radio-frequency pulses used. The pulse repetition time t rep was taken to be much longer than the previously determined T 1 [12].

NMR Spin-echo decay
The spin-echo intensity m decays with increasing time period τ between the first (π/2) and second (π) pluses due to several interactions between nuclei spins. Generally, the τ dependence of m may be expressed; where T 1L is the Lorentzian type relaxation. For the present case T 1L can be replaced by the spin-lattice relaxation time T 1 determined previously [12]; T 2G is the Gaussian spin-spin relaxation time. Ω(2τ ) is the modulation factor which is under investigation.

Spin-echo decay modulation due to RK and PDD interactions
It is well known that the RK and PDD interactions can induce oscillations in the spin-echo decay curve [13,14,15]. In metals the indirect RK interaction between nuclear spins I i , I j occurs through second-order scattering of conduction electrons [9], taking the scalar form In contrast to the scalar RK interaction (Eq. 2), the PDD interaction is a tensor quantity [10], where ⃗ r ij is the radius vector between the ith and the jth nuclei, δ ij is the angle between ⃗ r ij and the applied magnetic field, and b ij is the effective coefficient for a dipolar-type interaction B ij between the ith and jth nuclei. Thus, the PDD interaction depends on the direction of applied field. A complete formula for the spin-echo decay modulation factor Ω(2τ ) due to RK and PDD interactions is given by Alloul and Froidevaux [15]. As they have used polycrystalline samples, angle-integrated equations are given. Here, using their formulas, equations for the I = 1/2 case for a single crystal are presented. Considering the nearest neighbor (nn) sites among ions considered (i.e. Si for the present case), where J and B are the RK and PDD interactions between nn nuclear spins, respectively; N is the total number of nn sites; r corresponds to the number of occupied nn sites (i.e. with 29 Si nuclear spins); P r = N C r c r (1 − c) N −r is the binomial coefficient; c = 0.53 is the concentration of 29 Si isotope; n r is number of possible configurations of occupied nn sites for a given r, the configurations being indexed by k, ρ k,r is a distribution of situation k of r case; Θ α,k is the angle between the applied magnetic field H and the direction to the αth nn for the situation k. It should be noted that contributions of unlike spins are negligible in the present case.

URu 2 Si 2 case
For the case of 29 Si in URu 2 Si 2 , there is only one nn site, namely along the c axis, i.e N = 1 (Fig. 2), as is also the case for YbRh 2 Si 2 . The RK and PDD interactions (i.e. J and B) are proportional to R −3 (R is distance between nn sites). Since the distance to four second nn sites (3.75Å) is comparable with that of the nn site (2.35Å) in URu 2 Si 2 , both cases are included here. The definition of angles ϕ, θ and ψ is given in Fig. 2.

a) Ω 1st (2τ ) for the first nn case
The nn case is simple since there is only one nn site (N = 1, n r = 1), which was already discussed previously for YbRh 2 Si 2 [11] (note: definition of θ in Ref. [11] corresponds to present ϕ). There is a unique oscillation frequency G for Ω 1st (2τ ). Thus, first nn second nn where J 1 and B 1 are the RK and PDD interactions between the first nn sites, respectively. The double signs are coordinated. The signs in Eq. 9 cannot be determined by the present method; fortunately, they are not required in a fit to the data. For the nn site, Ω(2τ ) is independent of θ since the nn sites are located along the c ([001]) axis.

b) Ω 2nd (2τ ) for the second four nn case
The second four nn sites (N = 4) case is more complicated. All possible configurations n r for r = 0 − 4 cases should be considered. Based on Eq. 5, Ω 2nd (2τ ) may be expressed; G I.II are defined as; Generally, the contribution from the nn site is dominant. If the second nn contributions are not negligible, Ω(2τ ) may be expressed [15]; which is often a complicated function of 2τ .  Table 1. Figure 3 shows spin-echo decay data for H = 2.5 T ∥ [001] and [110] directions at 50 K. The data obtained can be well fitted using Eqs. 1 and 8, indicating that the contributions from the second nn sites are almost negligible. It should be noted that no better fitting can be obtained using Eq. 12, indicating that the first nn contribution dominates the spin-echo decay at the Si site in URu 2 Si 2 . As the G values obtained depend on the direction of field, certainly the PDD interaction should be considered in the present case. Based on Eq. 9 and G values from Table 1 Hz and |B 1 | = 148 Hz are obtained. If we consider the nuclear dipole-dipole contribution of 324 Hz in this enriched URu 2 Si 2 , the purely pseudo-dipolar interaction is estimated ±148+324 = 472 or 176 Hz (see [11]).

Summary
In the present study, spin-echo oscillations have been observed in the paramagnetic state of URu 2 Si 2 . The results are well accounted for by the Ruderman-Kittel interaction. As the Ruderman-Kittel interaction is due to conduction electrons, the spin-echo oscillation frequency is considered to be sensitive to modifications of the Fermi surface in URu 2 Si 2 . This is a possible topic for future investigation.