Evidence for spiral magnetic order in the heavy fermion material CeRhIn5

We report 115 In nuclear quadrupolar resonance measurements of the heavy fermion compound CeRhIn 5 in the paramagnetic and antiferromagnetic states. The magnetic order is consistent with a model of helical modulation of the Ce moments that is incommensurate with the lattice, and the magnetic dynamics indicate possible two-dimensional behavior.

The interplay between magnetism and superconductivity in heavy fermion materials has attracted much interest since the discovery of heavy fermion superconductivity. 1 In several of the Ce based heavy fermion compounds, the behavior of the magnetic order can be understood qualitatively in terms of a simple picture, in which long-range Ruderman-Kittel-Kasuya-Yosida ͑RKKY͒ interactions between the Ce moments compete with short-range Kondo interactions. 2 In this picture, both the RKKY and the Kondo interactions depend on a coupling constant J that can be tuned with pressure or chemical substitution. Since each of these interactions evolves differently with J, the nature of the ground state can change drastically with pressure or chemical substitution. Recent experimental work indicates that when the manybody Kondo effect dominates ͑large J), the system undergoes a zero-temperature magnetic-nonmagnetic transition. In CeIn 3 and CePd 2 Si 2 , both of which have antiferromagnetic ground states at ambient pressure, T N decreases monotonically with pressure and superconductivity develops close to the magnetic instability where T N vanishes. 3 Transport measurements indicate that these systems exhibit unusual non-Fermi-liquid behavior near this point, supporting the idea that the physics of these materials is dominated by proximity to a quantum critical point, and it has been suggested that the superconductivity in these materials is mediated by spin fluctuations near this boundary. 3 Recently a very interesting class of heavy fermion materials with the formula Ce m M n In 3mϩ2n , where M is Rh or Ir, was studied. 4 CeRhIn 5 is an antiferromagnet at ambient pressure with T N ϭ3.8 K, and becomes a superconductor with T c ϭ2.1 K above a critical pressure P c ϳ16 kbar. CeRhIn 5 differs somewhat from other Ce based heavy fermion materials in that the magnetic-nonmagnetic transition where superconductivity develops is apparently a first-order transition. 4 It has been suggested that CeRhIn 5 exhibits twodimensional ͑2D͒ magnetic character, thus the magnetic behavior and the origin of the superconductivity may be related to that of the cuprates. A detailed understanding of the magnetism is crucial in order to understand how the antiferromagnetism evolves into superconductivity.
Here we report 115 In nuclear quadrupolar resonance ͑NQR͒ (Iϭ9/2) measurements at ambient pressure in a powder sample of CeRhIn 5 in the paramagnetic and antiferromagnetic states. NQR is an ideal probe of the local magnetic fields in the unit cell as well as the dynamic behavior of the electronic system. In the paramagnetic state we find metallic behavior. Just above T N , the magnetic correlations slow down, dominating the spin-lattice relaxation. Below T N we observe the rapid development of an internal field that indicates that the magnetism is localized on Ce sites with a moment Ce ϳ0.1Ϫ0.2 B . The NQR spectrum in the antiferromagnetic state indicates a spiral modulation of the Ce moments that is incommensurate with the lattice.
Crystals of CeRhIn 5 grown from an In flux were ground to a powder in order to facilitate the penetration of the rf fields necessary for NQR. Roughly speaking, CeRhIn 5 consists of alternating layers of CeIn 3 and RhIn 2 and so has two inequivalent In sites per unit cell. The In͑1͒ site, analogous to the single In site in cubic CeIn 3 , is located on the top and bottom faces of the tetragonal unit cell. There are four In͑2͒ sites per unit cell, two on each of the lateral faces of the unit cell, equidistant above and below the Rh layer. The NQR spectrum in the paramagnetic state at 4 K reveals two sets of resonances associated with the two In sites: site A is characterized by Q ϭ6.78Ϯ0.01 MHz and ϭ0.0, whereas site B is characterized by Q ϭ16.665Ϯ0.001 MHz and ϭ0.445 Ϯ0.001, where Q and are defined by the NQR Hamil- Here I is the nuclear spin, Q ϵeQV zz /6I(2Iϩ1), Q is the quadrupolar moment of the nucleus, V ␣␣ is the electric field gradient along the ␣ direction, and ϵ(V xx ϪV yy )/V zz following the convention V zz ϾV xx ϾV yy . 5 In zero field, there are four transitions per site; experimentally these are quite narrow ͓full width at half maximum ͑FWHM͒ ϳ75 kHz͔ indicating a small distribution of electric field gradients and thus a high-quality sample. By considering the symmetry of the two In sites in the crystal, we assign the axially symmetric A site to In͑1͒ and the lower symmetry site B to In͑2͒; this assignment is further supported by the intensity ratio. A simple point-charge calculation indicates that the electric-fieldgradient ͑EFG͒ vector q ͑the principal axis of the EFG tensor with the largest eigenvalue͒ points along the crystal c axis for In͑1͒ and points inward towards the center of the unit cell ͑along the crystal a or b axes͒ for In͑2͒. Therefore for In͑1͒ the z axis lies along the crystal c direction, and for In͑2͒ the z axis lies along crystal a or b directions. Furthermore, for the In͑2͒ the x axis lies along the crystal c direction.
In order to measure the internal field in the antiferromagnetic state, we consider the splitting of the In͑1͒ spectrum by We diagonalize the full Hamiltonian H nuc ϭH Q ϩH Z and determine H int as a function of temperature from the resonance positions shown in the inset to Fig. 1; the results are shown in Fig. 1. The internal field of 1.7 kG at the In͑1͒ site can be compared with an internal field of 5.0 kG at the equivalent In site in CeIn 3 , 6 where neutron-scattering measurements indicate a Ce moment of 0.4 B . 7 Assuming a similar hyperfine coupling, the Ce moment in CeRhIn 5 is ϳ0.1-0.2 B . The direction of the Ce moments within the ab plane cannot be determined because the field gradient at the In͑1͒ site is axially symmetric about the c axis, thus the splitting of the NQR transition is sensitive only to the magnitude of the internal field. Note, however, that the sharp lines in the antiferromagnetic state ͑see the inset of Fig. 1͒ indicate that there exists a single value of the magnitude of the internal field.
The growth of the sublattice magnetization below T N is quite rapid: a fit to H int versus (1ϪT/T N ) ␤ gives ␤ϭ0.25 Ϯ0.03, much faster than mean field (␤ϭ0.5). It is interesting to compare this result with a similar result found in the cuprate antiferromagnet La 2 CuO 4 , in which the transition to the antiferromagnetic state is almost first-order-like, with critical exponent ␤ϳ0.1. 8 Note that the reduced critical exponent in La 2 CuO 4 is related to the two-dimensional magnetism in this material; the reduced exponent in CeRhIn 5 may also indicate 2D magnetic behavior.
The spin lattice relaxation rate (T 1 Ϫ1 ) of the In͑1͒ ͑Fig. 1͒ was measured as a function of T by fitting the recovery of the magnetization at the 4 Q transition to equilibrium after an inversion pulse. This transition exhibits no splitting due to the internal field in the antiferromagnet state. The time dependence of the recovery was well fit to the expected multiexponential recovery for a spin 9/2 nucleus 9 using a single T 1 at all temperatures. T 1 Ϫ1 exhibits critical fluctuations within ϳ1 -2 K of T N ; this behavior contrasts with that of CeIn 3 , which shows no critical slowing down through T N ͑ϭ10 K͒. 6 The dotted line through the data in Fig. 1 where aϭ133 (s K) Ϫ1 , bϭ41.1 s Ϫ1 K Ϫ1/2 , and c/M 0 ϭ12.5 (s K) Ϫ1 are fitting parameters, and M (T)ϭM 0 (1 ϪT/T N ) 0.25 is the antiferromagnetic order parameter. The SCR expression for T 1 Ϫ1 has been used successfully to fit the T 1 Ϫ1 (T) in other heavy fermion materials which exhibit magnetic order, 11 however, it fails to capture all of the behavior below T N in CeRhIn 5 , perhaps reflecting enhanced fluctuations associated with the 2D character. The first term in the expression allows for a Korringa-type relaxation expected in a metal. Above 10 K CeRhIn 5 is metallic, and р10 ⍀ cm. Below T N , spin-lattice relaxation due to quasiparticles at the Fermi surface is suppressed as magnetism develops. Figure 2 shows the NQR spectrum of the In͑2͒ site in the paramagnetic and antiferromagnetic states. Since this site is not axially symmetric, the transition frequencies do not pos- The filled region is a fit as described in the text. Inset: The In͑2͒ spectrum in the paramagnetic state at 4 K. The small peaks to the left of the main peaks are signals from the low abundance 113 In isotope. A schematic In͑2͒ nuclear-spin energy-level diagram indicating the transitions used in the SEDOR experiment is shown.

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sess a simple relationship as do those of the In͑1͒ site. In the paramagnetic state the spectrum consists of four sharp transitions; however, as the internal field develops below T N the low-frequency lines broaden ͑by several MHz͒ and exhibit a pattern which indicates a distribution of internal fields. The electric field gradient at the In͑1͒ site is axially symmetric; in contrast, the spectrum of the In͑2͒ site is sensitive not only to the azimuthal angle between H int and the EFG ͑along a or b for this site͒, but to the polar angle as well. The fact that the transitions at 48 and 65 MHz are affected less than the low frequency transition demonstrates that H int has no component parallel to the EFG, for if it did then each transition would experience an equal splitting ␥H int . We find good agreement between the measured spectrum and a simulation assuming the internal field varies along the c axis as H int ϭH 0 cos(2z/). We numerically diagonalize the summed quadrupolar and Zeeman Hamiltonian to obtain the transition frequencies; these are then weighted by the probability of a given internal field H int to generate the simulated spectrum shown in Fig. 2 ͑shaded regions͒. Using a maximum field H 0 ϭ2.7 kG, all of the relevant features in the spectrum are accurately reproduced; this strongly indicates the broadening results from a distribution of internal fields.
The continuous nature of the broadened In͑2͒ spectrum implies a spatially modulated magnetic structure incommensurate with the lattice ͑or having a very long period͒. The In͑1͒ spectrum, however, implies a single magnitude of the internal field; a distribution of magnitudes would cause the In͑1͒ spectrum to also broaden into a powder patternlike distribution. Therefore we are led to consider a spiral modulation where the orientation of the Ce moment rotates in the ab plane. Without a detailed knowledge of the transferred hyperfine couplings one cannot completely determine H int at each site, however, we estimate H int for this type of structure by calculating the direct dipolar field at both the In͑1͒ and In͑2͒ sites. We assume that the magnetic moments are localized on Ce sites, and the spiral twists along the c axis: nml ϭ Ce (Ϫ1) nϩmϩl ͓acos(klc)ϩb sin(klc)͔, where nml is the moment on the Ce located at rϭnaϩmbϩlc, and k ϭ2/, where is the wavelength of the modulation. This structure gives rise to an internal field at the In͑1͒ site which lies in the ab plane, remains constant in magnitude, but varies in angle in the ab plane. This result is entirely consistent with the spectrum of the In͑1͒. This magnetic structure distinguishes half of the In͑2͒ sites from the others: two of the four sites experience a field H 2a ϭϮH 0 cos(2lc/) along the c axis and the other two sites experience a field H 2b ϭ ϮH 0 sin(2lc/), also along the c axis. Hence both types of In͑2͒ sites experience the same distribution of internal fields. For In͑2͒ the summed contributions from all Ce moments result in a net internal field along the c axis with no components in the ab plane.
Further insight into the magnetic structure can be obtained by performing spin echo double resonance ͑SEDOR͒ between nearest-neighbor In͑2͒ nuclei along the c axis. 5 Since only neighboring nuclei are correlated ͑via a short-range magnetic interaction͒, SEDOR can determine the difference in resonance frequencies between neighboring unlike nuclei. In the antiferromagnetic state of CeRhIn 5 neighboring In͑2͒ nuclei along the spiral direction experience different internal fields, rendering them unlike. By applying SEDOR inversion pulses at a set frequency and searching the spectrum for a SEDOR reduction at a different observe frequency, one can determine the frequency difference between neighbors and hence deduce the difference in internal field and the pitch of the spiral. One important aspect of such a scheme is that normally SEDOR is performed between two different nuclear species which resonate at different frequencies, whereas in this case we conduct SEDOR between nuclei of the same species. One can perform SEDOR between two like nuclei if they possess multiple nuclear transitions at different frequencies. However, if the two transitions are neighboring transitions, the SEDOR pulse will create a higher-order coherence, which is undesired. 5 To demonstrate the viability of such an experiment we measured the SEDOR reduction between different nuclear transitions of the In͑2͒ in the paramagnetic state. We employed the double-resonance probe design of Haase et al., which allows one to tune to close frequencies. 12 The inset of Fig. 3 shows the SEDOR reduction as a function of the pulse spacing. Here the 48 MHz transition is observed with a standard echo sequence, and a pulse is applied to the 34 MHz transition ͑see inset, Fig. 2͒. The echo reduction with increasing pulse spacing clearly indicates a strong coupling between In͑2͒ neighbors. Figure 3 presents the SEDOR data in the antiferromagnetic state. Here the SEDOR irradiation ͑IRR͒ pulse is applied slightly below the 37 MHz horn ͑see Fig. 2͒, and the observed frequency is swept from 45 to 50 MHz. The irradiation frequency corresponds to an internal field H 0 ϭ2.29 kG; if the coupled nucleus experienced the same internal field one would expect a correlation at 46.3 MHz at the higher nuclear transition. Instead, there is a clear decrease in the echo integral at 45.95 MHz corresponding to an internal field of 2.63Ϯ0.31 kG ͑obtained by diagonalization of H nuc ). This increased internal field experienced by the nearest In͑2͒ neighbor of the irradiated nucleus ͑separated along the c axis by a distance 0.38c where cϭ6.782 Å is the lattice spacing 13 ͒ implies a wavelength ϭ4.6cϮ0.8c assuming that the internal field varies as cos(2z/). Note that there is a component of the SEDOR fraction that is independent of frequency, i.e., the signal is reduced by approximately 10%. The origin of this offset is not understood, but may be related to higher-order coherences generated by the SEDOR pulse on the observed nuclei. We are currently investigating this effect in greater depth.
A well-known class of helical antiferromagnets includes the rare-earth metals Ho, Dy, and Tb, where the magnetic coupling is given by the RKKY interaction. A model Hamiltonian that can give rise to the magnetic structure seen in these materials as well as in CeRhIn 5 is one in which the couplings between all nearest-neighbor pairs, J 1 , is antiferromagnetic and along one particular direction the couplings between next-nearest-neighbor pairs, J 2 , is ferromagnetic. 14 The frustration between the two competing interactions gives rise to the spiral along the c direction when ͉J 2 /J 1 ͉ exceeds a critical value. It is not unreasonable to expect that such interactions may be present in CeRhIn 5 . Spiral magnetic order along the c axis can also explain the unusual anisotropy of the bulk magnetic susceptibility (T). In the paramagnetic state Ϫ1 ϰT for TϾ100 K, with Curie-Weiss constant c ϭϪ79 K for H along the c axis, and ab ϭϩ16 K for H in the ab plane. 4 Although anisotropic crystal-field effects cannot be excluded, such a result could arise if the average coupling along the c direction were ferromagnetic, and the average coupling in the plane were antiferromagnetic. Furthermore, c (T)ϳ2 ab (T), suggesting that the easy axis is along the spiral direction. This picture is further supported by de Haas-van Alphen and magnetization measurements, which reveal a magnetic transition in fields on the order of 2 T along the c axis. 15 The behavior of this transition is consistent with an anisotropic antiferromagnetic spin-density wave or helical magnetic order. Finally, recent elastic neutron-scattering results indicate an incommensurate spin modulation along the c direction with period 3.3c. 16 We thank N. Hasselmann, A. Suter, and C. P. Slichter for inspiring discussions, and especially C. P. Slichter for suggesting the SEDOR experiment. This work was performed under the auspices of the U.S. Department of Energy.