Incommensurate magnetism near quantum criticality in CeNiAsO

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Jul 30, 2017 - ... Huq,5 David W. Tam,6 Benjamin A. Frandsen,7, 8 Benjamin Trump,1, 9 ... 2Department of Physics, Columbia University, New York, New York ...
Incommensurate magnetism near quantum criticality in CeNiAsO Shan Wu,1, ∗ W. A. Phelan,1 L. Liu,2 J. R. Morey,1 J. A. Tutmaher,1 J. C. Neuefeind,3 M. Feygenson,3 Matthew B. Stone,4 Ashfia Huq,5 David W. Tam,6 Benjamin A. Frandsen,7, 8 Benjamin Trump,1, 9 Cheng Wan,1 S. R. Dunsiger,10 T. M. McQueen,1, 11 Y. J. Uemura,2 and C. L. Broholm1, 4, 11

arXiv:1707.09645v1 [cond-mat.str-el] 30 Jul 2017

1

Department of Physics and Astronomy and Institute for Quantum Matter, Johns Hopkins University, Baltimore, Maryland 21218, USA 2 Department of Physics, Columbia University, New York, New York 10027, USA 3 Oak Ridge National Laboratory, Chemical and Engineering Materials Division, Oak Ridge, Tennessee 37831, USA 4 Oak Ridge National Laboratory, Quantum Condensed Matter Division, Oak Ridge, Tennessee 37831, USA 5 Oak Ridge National Laboratory, Chemical and Engineering Materials Division, Oak Ridge, Tennessee 37831, USA 6 Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA 7 Department of Physics, University of California, Berkeley, California 94720, USA 8 Department of Physics, Columbia University, New York, 10027 USA 9 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 10 Department of Physics, Ohio State University, Columbus, Ohio 43210, USA 11 Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA (Dated: August 1, 2017) Two phase transitions in the tetragonal strongly correlated electron system CeNiAsO were probed by neutron scattering and zero field muon spin rotation. For T < TN 1 = 8.7(3) K, a second order phase transition yields an incommensurate spin density wave with wave vector k = (0.44(4), 0, 0). For T < TN 2 = 7.6(3) K, we find co-planar commensurate order with a moment of 0.37(5) µB , reduced to 30% of the saturation moment of the | ± 12 i Kramers doublet ground state, which we establish by inelastic neutron scattering. Muon spin rotation in CeNiAs1−x Px O shows the commensurate order only exists for x ≤ 0.1 so the transition at xc = 0.4(1) is from an incommensurate longitudinal spin density wave to a paramagnetic Fermi liquid.

The competing effects of intra-site Kondo screening and inter-site Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions in rare earth intermetallics epitomize the strongly correlated electron problem. The extreme Kondo lattice and N´eel limits are perhaps well understood [1]. By contrast, the transition regime not only involves magnetic symmetry breaking but also a profound alteration in the character of 4f electrons, which can undergo a band selective metal to insulator transition as they develop spin order [2–6]. Isostructural to the 1111 iron pnictides, CeNiAsO is an exciting new addition to the landscape of strongly correlated electron systems [7]. By substitution of P for As or by applying pressure, the material can be driven into a paramagnetic Fermi-liquid phase. Non-Fermi-liquid transport is found up to the critical pressure Pc = 6.5 kbar and the critical phosphorus concentration xc = 0.4(1) [8]. A sign change in the Hall coefficient at Pc indicates Fermi surface (FS) reconstruction. In this letter, we enrich this picture by determining the ordered spin structures through a combination of muon spin rotation (µSR) and neutron scattering. The first transition is to an incommensurate spin density wave (SDW) state polarized mostly along a with wave vector k = (0.44(4), 0, 0). The second transition yields co-planar commensurate order with moment 0.37(5)µB . Using inelastic neutron scattering we show the magnetism of Ce3+ arises from a | ± 21 i Kramers

(a)

Ce   O   As  

μ+    Site   (b)

CeNiAs1-­‐xPxO  

SDW    

X

Ni  

! k

Néel     AFM  

M

(e)

FIG. 1: (a) Crystallographic structure of CeNiAsO, and spin structure for (c) & (e) TN 2 < T < TN 1 and (d) T < TN 2 . Blue marks indicate the muon site, ( 14 , 34 , zCe ) and its three crystallographic equivalents inferred from simulation of µSR spectra. (b) Summary of the phase diagram reported here. The Fermi surface plot for the miller index l = 0 is based on a standard ab-initio band structure calculation. The arrow indicates the proposed intra-band nesting condition. The blue dashed line is the outer phase boundary for commensurate order inferred from µSR measurement on P-doped samples.

doublet ground state of the 4f 1 J =

5 2

multiplet subject

2 to a C4v crystal field. Polycrystalline CeNiAsO was synthesized as described in the supplementary information (SI). Neutron scattering employed the NOMAD and POWGEN diffractometers [9, 10] and the SEQUOIA [11] spectrometer at the ORNL spallation neutron source. The µSR experiment was conducted on the M15 beam line of the TRIUMF facility in Vancouver, Canada. Specific heat measurements were carried out by a 14 Tesla Quantum Design PPMS with a dilution fridge insert. We used standard density functional theory as implemented in VASP (LDA+U+SOC) for an approximate view of the Fermi surface. Fig. 1(a) shows the tetragonal structure of CeNiAsO where magnetism is associated with Ce3+ sandwiching a square lattice of oxygen. The structure and the single phase nature of the sample was ascertained by Rietveld refinement of high resolution neutron diffraction data (see SI). The specific heat data in Fig. 3(d) show two successive peaks indicating second order phase transitions at TN 1 = 9.3(3) K and TN 2 = 7.6(3) K consistent with previous results [7]. As the peaks shift towards lower temperature (T ), they also approach each other in a field of µ0 H = 14 T. This is consistent with two distinct antiferromagnetic phases. In the inset of Fig. 3 (d), an upturn of the Sommerfeld constant Cp /T , ascribed to the nuclear spin specific heat of 61 Ni and 75 As [12], was calculated and subtracted [13, 14]. A broad anomaly at T3 = 0.5 K is suppressed by a field µ0 H > 9 T. Electrical resistivity measurements on a polycrystalline sample showed no indications of superconductivity in this T range. To understand the nature of the two AFM phases, we conducted zero field µSR in the longitudinal configuration [15, 16]. Fig. 2 shows spontaneous muon spin precession indicative of a well defined static internal field for T < TN 1 . A marked change in the nature of the oscillation for T < TN 2 indicates two distinct magnetic phases. For TN 2 < T < TN 1 , the asymmetry pattern can be characterized by two distinct muon sites sampling the dipole field generated by an incommensurate SDW [17]. For T < 6 K, on the other hand, the signal is oscillatory (Fig. 2 (b)) with a beating pattern that can be associated with two distinct precession frequencies. This implies the loss of a sinusoidal amplitude modulation of spins and commensurate magnetism for T < TN 2 . We shall see the two patterns can be fitted directly by magnetic structures that are consistent with the neutron data and a single crystallographic muon stopping sites. To determine the detailed spin structures, we turn to time-of-flight neutron diffraction. Weak magnetic peaks are apparent at T = 2 and 8 K after subtracting data at T = 15 K (Fig. 2 (c-d)). At T = 2 K, the difference pattern shows several resolution limited peaks. The peak with the lowest wave vector transfer Q ≈ 0.77 ˚ A−1 can be indexed as Qm = (0.5, 0, 0).

k = (0.44(4), 0, 0) χ2

φ °  

k = (0.5, 0, 0)

φ °  

FIG. 2: Zero-field µSR spectra at T = (a) 7 K and (b) 0.05 K. The colored solid/dashed lines are fits to Fourier transformation of the density of field generated by magnetic structures in Fig. 1 at given muon sites. The inset is a contour plot of the reduced χ2 for µSR fits to T < TN 2 data, where the magnetic field was calculated for the indicated dipole moment m and rotation angle φ (tanφ = m⊥k /mkk ). (c-d) Diffraction patterns collected at T = 2 K and 8 K on NOMAD, subtracted T = 15 K data as a measure of nuclear diffraction. The data is normalized by nuclear Bragg intensity. Red and blue lines calculate the spin configurations in Fig. 1. The orange line describes a transverse polarized state with spins along b axis. The inset indicates the goodness of fit (χ2 ) versus φ. The grey dashed lines in (c) mark the nuclear Bragg positions, where thermal expansions gives rise to a peak-derivative anomaly. The horizontal green bar is the instrument resolution 0.04 ˚ A−1 from adjacent nuclear peaks.

Magnetic neutron diffraction probes spin polarization perpendicular to wave vector transfer so this indexing implies spin components along the b and/or c axes. Upon warming to 8 K < TN 1 , the absence of this first peak is indicative of a longitudinal spin density wave polarized along a. The width of the intensity maxima for T = 8 K and Q ≈ 1.1 ˚ A−1 in Fig. 2 (c) exceeds the instrumental Q-resolutions. The incommensurability indicated by the µSR can account for this. The magnetic signal at 8 K is however quite weak and since there is no energy resolution, inelastic magnetic scattering may also contribute to the broadened peaks, particularly near the putative polarization suppressed Qm peak. The diffraction data thus do not permit a unique determination of the spin structure for TN 2 < T < TN 1 .The combination of muon, specific heat, and neutron

3 IR Γ2

Γ1

BV ψ(1) ψ(2) ψ(3) ψ(4) ψ(5) ψ(6)

Atom1 mx my mz 000 010 001 000 000 100

Atom2 mx my mz 0 -1 0 000 000 100 0 0 -1 000

TABLE I: The 6 basis vectors associated with magnetic structures that transform according to irreducible representations with propagation vector k = ( 12 , 0, 0), using Kovalev notation [18] in SARAh [19].

FIG. 3: Temperature dependence of (a) the longitudinal (ma ) and (b) the transverse moments (mc for high T and mb for low T phase). Black dots were extracted from Rietveld fits to neutron diffraction data. The value given at 2 K and 8 K were averaged over the 30 Hz and 60 Hz chopper settings, and the rest are from the 60 Hz. Blue diamonds were inferred from µSR fits. The solid lines are guides to the eye. (c) Temperature dependence of the averaged static field. Inset: T < 1 K. (d) Specific heat Cp /T in zero field and for µ0 H = 14 T from 55 mK to 12 K. The upturn in Cp /T at 14 T is due to the nuclear spin contributions, and indicated by the solid red line. Inset: Cp /T below 2 K, after subtracting the nuclear spin contributions.

data, however, allow us to reach a firm conclusion about the nature of two ordered states in CeNiAsO. Using Kovalev notation [18] as implemented in SARAh [19], the reducible magnetic representation associated with k = (µ00) decomposes into three two-dimensional (2) (2) irreducible representation (IR): Γmag = 2Γ1 + Γ2 with a total of 6 Basis Vectors (BVs) as listed in Table I. The transitions are characterized as second-order based on the specific heat data. Landau theory requires that one IR should be associated with each of the two phase transitions. Below TN 1 , BVs ψ(4) and ψ(6) of Γ1 depict a candidate spin structure with moments oriented along a. Adding ψ(3) and ψ(5) are further allows for moments along c. Below TN 2 , we can account for the diffraction pattern in Fig. 2 (d) by adding ψ(1) and ψ(2) of Γ2 . The ratio between the a (ψ(4) and ψ(6)) and b (ψ(1) and ψ(2)) components of the low T order is determined by the relative intensities of magnetic diffraction peaks. The best fit (Fig. 2(d) ) corresponds to a reduced χ2

= 1.95 and a staggered moment hmi = 0.37(5) µB /Ce that is canted by φ ≈ 36(6)◦ with respect to the a axis (Fig. 1 (d)). While allowed by symmetry, the diffraction data place a limit of 0.06 µB on any c-component of the staggered magnetization. µSR, which probes magnetism in real space, offers an independent assessment of the proposed low T structure. We find a consistent description of the precession data with the muon stopping site ( 14 , 34 , zµ ) marked as blue stars in Fig. 1 (a). The fitting analysis described below yields zµ = 0.1471(3) (=zCe ), which places the muons 1.1 ˚ A from O2− , and close to the preferred distance of about 1˚ A [20]. This location is consistent with the electrostatic potential-energy map calculated for CeFeAsO [21]. The observation of two muon precession frequencies suggests two magnetically inequivalent muon sites marked as site 1 and 2 in Fig.1 (c-d). The asymmetry pattern can be fitted to the following expression: Z X 2 z −λi t Pµ (t) = (Atot − Abg ) ( e ρi (B)cos(γµ Bt)dB 3 i 1 + e−λs t ) + Abg e−λbg t 3

(1) Here Atot − Abg is the total asymmetry stopped in the sample at t = 0. We assume equal weight for the two muon sites in CeNiAsO (ratio = 1.0(2)). ρi (B) is the magnetic field distribution at the muon site i. λi and λs are the rapid transverse and slower longitudinal rate respectively, and γµ /2π = 135.53 MHz T−1 is the gyromagnetic ratio of the muon. The last term accounts for muons stopped beyond the sample. For the low T commensurate state, ρi (B) is described theoretically by a delta function at two distinct local fields. The contour plot in Fig. 2 (b) shows the reduced χ2 fit of muon spectra versus different spin moments m and rotation angles in the ab plane φ. The best fit is obtained with m = 0.37(2)µB and φ = 36(7)◦ , which is in excellent agreement with the Rietveld refinement of neutron diffraction. For the high T incommensurate state, the muon ensemble is subject to continuous set of internal fields proportional to modulated spins with moments mn,j = ma,j cos(Rn,j · k)ˆ a + mc,j sin(Rn,j · k)ˆ c. The subscripts denote j th Ce site in the nth unit

4 Energy (meV) 70(8) 18(3) 0

Doublet wave function 0.7518| ± 25 i + 0.6593| ∓ 32 i 0.6593| ± 25 i − 0.7518| ∓ 32 i | ± 12 i

TABLE II: Wave functions for three doublets associated with Ce3+ in CeNiAsO inferred from inelastic neutron at 200 K.

FIG. 4: Normalized inelastic spectrum with incident energy Ei = 100 meV for (a) CeNiAsO and (b) the non-magnetic reference LaNiAsO. (c) The T = 7 K difference spectrum: ˜ I(Q, E) = ICe − rILa where r = σCeNiAsO R /σLaNiAsO . R(d-e) ˜ ˜ Momentum-integrated scattering I(E) = I(Q, E)dQ/ dQ at T = 7 K and 200 K, including data collected with the indicated instrumental configurations. The inset in (d) shows the onset of low energy excitation at 2 meV in the ordered state. The blue and red solid lines were calculated for Crystalline Electric Field (CEF) model described in the text.

cell. The frequency density which is proportional to ρi (B) is shown in Fig. ??. The best fit leads to an incommensurate wave vector k=(0.44(4),0,0), ma = 0.27(6)µB , and mc = 0.08(3)µB . The corresponding calculated muon asymmetry and neutron diffraction are in Fig. 2 (a)&(c). The small component of mc implies this is a magnetic cycloid. The corresponding lack of inversion symmetry could have interesting consequences for electronic transport. However, since mc  ma we retain the terminology of a longitudinal SDW. The T -dependence of the transverse and longitudinal moment mk and m⊥ from the µSR analysis is consistent with the Rietveld refinement of neutron data (Fig. 3 (a-b)). To determine the specific nature of the lowest energy Kramers doublet, inelastic neutron scattering was performed using the SEQUOIA spectrometer at SNS. The intensity measured (Fig. 4 (a-b)) was normalized as a scattering cross section (I(Q, E) ≡ ki /kf d2 σ/dΩdE)) by nuclear Bragg diffraction for two configurations and two samples. At T = 7 K, the intensities of modes at E ≈ 10 meV, 30 meV and 40 meV are proportional to Q2 and observed in both CeNiAsO and non-magnetic LaNiAsO. Therefore we identify these modes as phonons [22]. ˜ ˜ The difference spectrum I(Q, E) and I(E) (Fig. 4 (c-e)) were obtained as described in the caption. Because their intensity decreases with Q, we associate the two broad modes at E1 ≈ 18(3) meV and E3 ≈ 70(8) meV with magnetic excitations. In the tetragonal environment of Ce3+ , the sixfold degenerate J = 52 multiplet splits into

three Kramer’s doublets. The two magnetic modes are assigned to crystal-field-like excitations from the ground state (GS) to two excited states. At T = 200 K, there is significant population of the excited state and this is reflected in the appearance of a broad mode at E2 ≈ 50 meV. Other than CEF levels En=1,2,3 one sharp mode at E0 ≈ 2 meV develops along with AFM order (inset, Fig.4 (d)). In the language of CEF theory, this is an intra-doublet transition driven to inelasticity by the molecular exchange field. The peak reflects the density of states for low energy excitations from the ordered state. To extract more specific information about the nature of the ground state doublet and the crystal field excitations, we develop a phenomenological model to fit the spectra as described in the SI. In the PM state the Hamiltonian is expressed in terms of Stevens operators [23] that are allowed by the point group symmetry and account for the observed CEF modes. In the ordered state, a mean field description of inter-site interactions adds a molecular field term. We carried out a global fit ˜ for normalized data I(E) at T = 7 K and 200 K varying the crystal field parameters and the molecular exchange field (Table ??). Fig. 4(d-e) shows good agreement between the model and the neutron data. The associated energy levels and wave functions for the three doublets in the PM phase are listed in Table II. Consistent with the easy plane character of the ordered states, the GS wave function is simply | ± 12 i (Γ7 ). As indicated in the DFT Fermi surface plot (Fig. 1 (b)), the ordering wave vector k = (0.44(4), 0, 0) appears to satisfy a nesting condition. This suggests the ordered state for TN 2 < T < TN 1 should be classified as a Spin Density Wave [24–28]. It is common for incommensurate (IC) magnets to undergo a longitudinal to transverse spin reorientation transition that reduces the modulation in the magnitude of the dipole moment per unit cell while sustaining the incommensurate modulation [26, 29]. The situation is different for CeNiAsO, which not only develops transverse magnetization but also becomes commensurate (C) for T < TN 2 . To arrive at the spin structure in Fig. 1 (d) from the commensurate version of Fig. 1 (c) involves counter-rotating the upper and lower AFM layers of a CeO sandwich (Fig. 1 (a)) by φ = 36◦ (5) around c. While inter-layer bi-linear interactions vanish at the mean field level for k = (0.5, 0, 0) type order, inter-layer bi-quadratic interactions[30, 31] give rise to a term in the free energy of the form (m2 cos 2φ)2 that can favor φ = 45o for a commensurate structure only.

5 As m grows upon cooling this term can be expected to induce both the IC to C transition and the symmetry breaking associated with appearance of the transverse magnetization at TN 2 . This brings us to the character of magnetism in CeNiAs1−x Px O. Upon cooling, CeNiAsO passes from Fermi liquid to incommensurate SDW to commensurate non-collinear order in two second order phase transitions. P doped samples CeNiAs1−x Px O that we examined (x > 0.1) all show the characteristic µSR oscillation associated with incommensurate magnetism (Fig.2 (a)) down to 50 mK. This indicates the commensurate state is limited to a low T , low x pocket (Fig. 1 (b)). If we may neglect the impacts of disorder, this suggests that all the way to the pressure driven quantum critical point the initial instability of the strongly correlated Fermi liquid is to an incommensurate SDW. Within the conventional Hertz-Millis framework this suggests the volume of the fermi-surface is sustained through the QCP and yet the anomalous transport properties are incompatible with this framework. The combined experimental picture thus does not presently have a satisfactory theoretical description. This research was funded by the US Department of Energy, office of Basic Energy Sciences, Division of Material Sciences and Engineering under grant DE-FG02-08ER46544. The research at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Activities at Columbia and TRIUMF have been supported by NSF DMR-1436095 (DMREF) and DMR-1610633, and the REIMEI project of Japan Atomic Energy Agency.

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175701 (2011). [8] Y. Luo, L. Pourovskii, S. Rowley, Y. Li, C. Feng, A. Georges, J. Dai, G. Cao, Z. Xu, Q. Si, et al., Nature Mat. 13, 777 (2014). [9] J. Neuefeind, M. Feygenson, J. Carruth, R. Hoffmann, and K. K. Chipley, Nuclear Instruments and Methods in Phys. Res. B: Beam Interactions with Materials and Atoms 287, 68 (2012), ISSN 0168-583X. [10] A. Huq, J. P. Hodges, O. Gourdon, and L. Heroux, Z. Kristallogr. Proc 1, 127 (2011). [11] G. E. Granroth, A. I. Kolesnikov, T. E. Sherline, J. P. Clancy, K. A. Ross, J. P. C. Ruff, B. D. Gaulin, and S. E. Nagler, J. Phys.: Conf. Series 251, 012058 (2010). [12] J. Emsley, The Elements, Oxford Chemistry Guides (Oxford University Press, New York, 1995). [13] C. Kittel, Introduction to Solid State Physics (Wiley, 2005). [14] E. Heltemes and C. Swenson, J. Chem. Phys. 35, 1264 (1961). [15] A. Schenck, Muon Spin Rotation Spectroscopy: Principles and Applications in Solid State Physics (Bristol: Adam Hilger, 1985). [16] S. Lee, S. Kilcoyne, and R. Cywinski, Chemistry and Materials, Proceedings of the 50th Scottish University Summer School in Physics 51 (1998). [17] Y.J.Uemura, Muon Science pp. 85–114 (1999). [18] O. V. Kovalev, Irreducible Representations of the Space Groups (Routledge, 1965). [19] A. Wills, Physica B: Cond. Matter 276, 680 (2000). [20] P. Meier, J. Chappert, and R. Grynszpan, Muons and pions in materials research (1984). [21] H. Maeter, H. Luetkens, Y. G. Pashkevich, A. Kwadrin, R. Khasanov, A. Amato, A. A. Gusev, K. V. Lamonova, D. A. Chervinskii, R. Klingeler, et al., Phys. Rev. B 80, 094524 (2009). [22] V. F. Sears, Neutron News 3, 26 (1992). [23] K. W. H. Stevens, Proc. Phys. Soc. A 65, 209 (1952). [24] W. Bao, P. Pagliuso, J. Sarrao, J. Thompson, Z. Fisk, J. Lynn, and R. Erwin, Phys. Rev. B 62, R14621 (2000). [25] C. Stock, J. A. Rodriguez-Rivera, K. Schmalzl, E. E. Rodriguez, A. Stunault, and C. Petrovic, Phys. Rev. Lett. 114, 247005 (2015). [26] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988). [27] Y. Sidis, M. Braden, P. Bourges, B. Hennion, S. NishiZaki, Y. Maeno, and Y. Mori, Phys. Rev. Lett. 83, 3320 (1999). [28] I. I. Mazin and D. J. Singh, Phys. Rev. Lett. 82, 4324 (1999). [29] G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, T. Yildirim, M. Kenzelmann, C. Broholm, et al., Phys. Rev. Lett. 95, 087205 (2005). [30] Y. Akagi, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 108, 096401 (2012). [31] S. Hayami and Y. Motome, Phys. Rev. B 90, 060402 (2014).

Supplementary Material for ‘Incommensurate magnetism near quantum criticality in the Kondo lattice CeNiAsO’ Shan Wu1,∗ , W. Adam Phelan 1 , L. Liu2 , J. R. Morey1 , J. A. Tutmaher1 , J. C. Neuefeind3 , M. Feygenson3 , Matthew B. Stone4 , Ashfia Huq5 , David W. Tam6 , Benjamin A. Frandsen7,8 , Benjamin Trump1,9 , Cheng Wan1 , S. R. Dunsiger9 , T. M. McQueen1,10 , Y. J. Uemura2 , and C. L. Broholm1,4,10

arXiv:1707.09645v1 [cond-mat.str-el] 30 Jul 2017

(Dated: August 1, 2017) In support of the main text, the following supplementary information is provided: (1) synthesis details and nuclear structure refinement from neutron diffraction; (2) details about the neutron measurements, µSR experiments, and DFT calculation; (3) details and discussion about the additional low temperature T3 anomaly in specific heat; (4) crystal fields analysis of inelastic neutron scattering data.

SYNTHESIS METHOD AND NUCLEAR STRUCTURE DETERMINATION

Synthesis method

Samples of CeNiAsO were synthesized by pelletizing CeO2 (Alfa Aesar, 99.99 %), NiAs (pre-fired), and Ce metal (Alfa Aesar, 99.8 %) stoichiometrically, sealing the pellet in an evacuated-fused silica tube. The reaction ampoule was placed in a furnace and heated to 1000 ◦ C at a rate of 100 ◦ C/hr. The temperature was held constant at 1000 ◦ C for 5 hours. The ampoule was then allowed to furnace cool to room temperature. Phosphorous doped samples were synthesized following the procedure described in Ref. [1]. For each sample, CeO2 , Ni5 P4 (pre-fired), Ce metal, purified and dried P, and NiAs (pre-fired) were mixed and ground thoroughly and stoichiometrically to CeNiAs1−x Px O0.98 . The pressed pellet was placed in a dry alumina crucible and sealed into an evacuated ampoule. The ampoule was heated from 750 ◦ C to 1050 ◦ C over 1 hour and held at 1050 ◦ C for 5 hours. After cooling to room temperature in the furnace, the sample was re-grounded and re-pressed with excess of 2 % P to account for vaporization during pre-reaction. The sample was then pressed into a pellet again and placed in the same alumina crucible into a new ampoule. The ampoule was sealed into a larger ampoule backfilled with 1/3 atm Ar (99.99%). This ampoule was placed directly into a furnace that was pre-heated to 1300 ◦ C and it remained there for 8-12 hours. After cooling on the bench top, such heating to 1300 ◦ C and bench-top cooling was repeated 2-4 times with intermediate grinding, until the starting materials were undetectable by laboratory powder X-ray diffractometer. The entire synthesis was conducted in an argon atmosphere. The time exposed to the air was less than 10 mins during each regrind/reheat/reseal cycle.

FIG. S1: Rietveld refinement of neutron diffraction for the nuclear structure of CeNiAsO at T = 300 K. Red dots are on POWGEN in the year of 2013 with a total proton charge of 3.3e-3 Ah. Black curve is the Rietveld profile based on the tetragonal P 4/nmm structure. Blue trace is difference between observation and calculation. Nuclear structure determination

In Fig. S1, crystallographical structure of CeNiAsO was determined by Rietveld refinement of T = 300 K data from POWGEN, using Fullprof [2] with the tetragonal space group P 4/nmm. The corresponding atomic positions and lattice parameters are listed in Table S1 (Bragg R-factor = 5.37). The inferred structure is consistent with powder x-ray diffraction (XRD) refinement. Laboratory powder x-ray diffraction data were collected using a Bruker D8 Focus diffractometer equipped with Cu Kα radiation and a LynxEye detector. Phase identification and phase purities checks were conducted by performing Rietveld refinements. A 4 hours XRD experiment found only 0.75% by volume of CeNi2 As2 as the only detectable impurity phase. A pair distribution analysis of the NOMAD data found no evidence of a local structural distortion as in CeFeAsO [3]. In CeNiAs1−x Px O samples with nominal values of

2

Ce Ni As O

2c 2b 2c 2a

1/4 3/4 1/4 3/4

1/4 1/4 1/4 1/4

0.1471(3) 1/2 0.6439(3) 0

0.0070(5) 0.0120(3) 0.0081(4) 0.0056(4)

T=7K Frequency density

TABLE S1: T = 300 K atomic positions for CeNiAsO with space group P 4/nmm determined by the Rietveld analysis of neutron powder diffraction data. The corresponding lattice parameters are a = 4.0621(1) ˚ Aand c = 8.1058(3) ˚ A. Atom Site x y z Uiso (˚ A)

Fourier transform of the μSR data Simulation

Rwp = 7.07, Rp = 10.3,Rexp = 2.06, χ2 = 10.9

x=0.2, 0.3, 0.4, and 0.5, we found x = 0.19(1), 0.29(1), 0.33(1) and 0.49(1) by XRD refinement, correspondingly for nominal 0.2, 0.3, 0.4 and 0.5 concentration. These values were found to be consistent with SEM/EDX measurements. EXPERIMENTAL DETAILS

Frequency (MHz)

FIG. S2: The frequency density f at T =7 K which is proportional to the internal field B as f = γµ /2πB. The solid pink line is the Fourier transformation of the experimental µSR data. Histogram shows the calculated frequency density for the incommensurate phase described in the main text.

elastic energy resolutions ∆E were 1 meV and 5.7 meV respectively.

Elastic scattering

Neutron diffraction was carried out using the Nanoscale Ordered Materials Diffractometer (NOMAD) at the Oak Ridge National Labratories (ORNL). Data were collected with a total proton charge of 4.5 mAh at T = 2 K, 8 K, and 15 K, with a 30 Hz bandwidth (BW) chopper, passing a wavelength band from 0.1 ˚ A to 3 ˚ A in the first frame. Temperature dependence measurements were counted with a proton charge of 2 mAh each from 2 K to 10 K in 2 K step, with 60 Hz pulse rate and passing a wavelength band from 3 ˚ A to 6 ˚ A. We used a vanadium sample can and corrected for background through separate measurements of the empty instrument and an empty sample can. Diamond scans were used for calibration purposes. Absolute unites for magnetic diffraction were obtained by scaling to nuclear Bragg intensities. Inelastic scattering

Inelastic neutron scattering was performed using the SEQUOIA Fermi chopper spectrometer at SNS, ORNL. We used a multi-sample exchange system to mount CeNiAsO (m ≈ 2.31 g) and its non-magnetic analogy LaNiAsO (m ≈ 2.32 g) in two thin cylindrical aluminum cans, sealed under 1 atm of 4 He. The sample was cooled to T = 7 K using a close-cycle cryostat. To optimize intensity and resolution, we used two configurations with different incident energy Ei : 1) Fermi fine chopper frequency ν2 = 360 Hz and T0 chopper frequency ν0 = 90 Hz for Ei = 50 meV; 2) Fermi sloppy chopper frequency ν1 = 240 Hz and ν0 = 60 Hz for Ei = 100 meV. The corresponding Full Width at Half Maximum (FWHM)

µSR experiment

We performed zero field (ZF) µSR with the detectors in the longitudinal configuration on the M15 beam line at Canada’s national laboratory for particle and nuclear physics (TRIUMF). The samples were mounted using Apiezon N grease, covered with Alfa Aesar 0.025 mm thick silver foil, and cooled in a dilution refrigerator to 0.04 K. We calculated the internal field distribution on the putative muon interstitial sites and spin configurations for comparison to the µSR data. The spatial extent for calculating the local magnetic field on one muon site was 61 times crystallographic unit cells, which is sufficient to reach convergence of the internal field. For the low-T commensurate phase, plugging the value of field into Eq. ?? and comparing to the oscillation pattern, we determined moment m and rotation angle φ. For the high-T phase, the frequency density is shown in Fig. S2. We performed an average over the phase of the incommensurate spin configuration. From the optimal fits to T = 7 K µSR data, we determined the wave vector k, the moment amplitude along the a-axis and the c-axis, ma and mc respectively.

DFT calculation

Electronic structure calculations were conducted using the Vienna Ab-Initio Simulation Package (VASP), which models valence electrons using projector augmented plane waves (PAW) [4–6]. The exchange-correlation effects were modeled using the local (spin) density

3 approximation (L(S)DA). An 11x11x11 gamma-centered Monkhurst-Pack k-point grid was used for all calculations along with an upper energy cut-off 434 eV, and a convergence of the total energy to better than 0.01%. The calculations proceeded in three steps. First, a structural and ionic relaxation was performed, with the ionic relaxation considered complete when the total free energy difference between steps was less than 0.1 %. Second, a non-magnetic static calculation using this structure was performed. Finally, a magnetic static calculation initiated from the converged (non-magnetic) charge density was performed. For the magnetic calculation, an initial 0.5 µB magnetic moment along ±c was applied to the Ce sites, converging to a simplified anti-ferromagnetic structure with a moment of 0.1 µB per Ce. Spin-orbit interactions and an onsite (Ce) Hubbard U of 7.5 eV with an exchange (J) parameter of 0.68 eV were included in all calculations. LOW TEMPERATURE SPECIFIC HEAT ANOMALY

Specific heat data were acquired using a quantum design Physical Properties Measurement System with a dilution refrigerator insert for measurements below 2 K. The single pulse method was employed near the ordering temperature and otherwise we used the adiabatic method. Two successive anomalies TN 1 = 9.3(3) K and TN 2 = 7.6(3) K were observed in the specific heat data as studied and discussed in the main text. The error bar for the two characteristic temperatures were estimated. A strong upturn in Cp /T below 0.1 K (Fig. ?? (d)) is identified as a hyperfine enhanced 61 Ni nuclear Schottky anomaly. It can be accounted by the following expression[7, 8]. 2 2 CN = EN β R

e−EN β (1 + e−EN β )4 + 4e−3EN β (S1) (1 + e−EN β + e−2EN β + e−3EN β )2

Here EN = −gN µN ∆IB is the level splitting, where µN is nuclear magneton and gN is the nuclear gyromagnetic ratio. The inset in Fig. ?? (d) shows a broad specific heat anomaly at T3 = 0.5 K after subtracting the nuclear Schottky anomaly. The corresponding change in entropy, S(µ0 H = 14 T)-S(µ0 H = 0 T) ≈ 80 mJ/K/mol/f.u. up to 2.2 K, is 1.4% of a full doublet entropy Rln2. Resistivity measurements on a polycrystalline sample shows no indication of a superconducting phase transition down to 55 mK. In addition, no clear change of oscillation patterns were observed in µSR spectra (inset in Fig. ?? (c) ). Possible explanations for the low T specific heat anomaly are (1) coherence effects in the Kondo lattice as for CeAl3 [9], (2) changes in the magnetic domain wall configuration and (3) partially gapped Fermi surface with sensitivity to an applied fields.

CRYSTAL FIELD ANALYSIS

The crystal field inelastic spectrum (Fig. ?? (c)-(e)) was obtained from inelastic neutron scattering data acquired on SEQUOIA after subtracting scaled data from the non-magnetic reference sample LaNiAsO and normalizing based on Bragg diffraction. Broad CEF excitations and one sharp mode that appears in the ordered state can be reproduced by the following phenomenological model. The Ce atoms are located at the 2c crystallographic site which has C4v point group symmetry. In the paramagnetic state, the CEF Hamiltonian that operates on the ground state J-multiplet can be written as: ˆ CEF = B20 O ˆ 20 + B40 O ˆ 40 + B44 O ˆ 44 . H

(S2)

Here Blm are the pre-factors of Stevens operators ˆ m are the Stevens’ (denoted CEF parameters here) and O l equivalent operators [10]. In the ordered state, the contribution of a molecular exchange field was added to ˆ CEF as follows: the CEF Hamiltonian H ˆ =H ˆ CEF − gj µB Beff · J. H

(S3)

The cross section for unpolarized neutron scattering from a polycrystalline sample arising from transitions from state |ii to |ji can be expressed as follows [11]: kf 1 d2 σ = N | gj r0 F (Q)|2 dΩdE ki 2 2X × pi |hi|Jα |ji|2 δ(Ei − Ej + ~ω) 3 i,j,α

(S4)

We replaced the Dirac delta function with a unity normalized Lorentzian function with Half Width at Half Maximum (HWHM) Γn = 13(2), 23(3), 24(1) meV for the three CEF modes and Γ0 = 2.0(2) meV for the sharp low energy mode. The molecular field term Beff · J can be further simplified as Beff Jk , considering in-plane effective field. Three CEF parameters and one molecular field parameter were obtained by fitting to the T -dependent magnetic neutron scattering spectra. The corresponding fitting parameters and wave functions are listed in Table S2 and S3. The difference in B20 and B44 between the two measurements is not significant. The sign change for B40 could result from a multipolar field. TABLE S2: CEF parameters B20 , B40 , B44 and molecular field parameters Beff for the order and paramagnetic state. T (K) B20 (meV) B40 (meV) B44 (meV) Beff (T) 7 200

2.84(3) 2.21(5)

0.068(8) -0.086(4)

1.04(1) 0.93(2)

1.5(1) -

4 TABLE S3: Wave functions considering effective molecular field at T = 7 K. The three doublets are divided into 6 non-degenerate singlets. The ± sign prior to the coefficient means the summation over each component of angular moment. ψ5 ∓0.641| ± 25 i ± 0.298| ± 32 i ∓ 0.009| ± 12 i

in-plane spin structure. However, the ordered moment size 0.37 µB is about half of the value of hµx i which is consistent with Kondo screening and proximity to a quantum critical point separating the ordered state from a Kondo lattice Fermi-liquid.

ψ4 −0.648| ± 25 i − 0.281| ± 32 i + 0.008| ± 12 i ψ3 ±0.035| ± 25 i ± 0.052| ± 32 i ∓ 0.704| ± 12 i ψ2 −0.043| ± 25 i + 0.079| ± 32 i − 0.701| ± 12 i ψ1 −0.278| ± 25 i + 0.643| ± 32 i + 0.091| ± 12 i ψ0 ±0.296| ± 25 i ± 0.639| ± 32 i ± 0.062| ± 12 i

We can extract the inelastic spectral weight for each CEF mode En at low temperature T = 7 K as following: RR 2 Q I˜n (E)(1 + e−Eβ )/|r0 F (Q)|2 dQdE R δm2n = 6µ2B Q2 dQ (S5) I˜n (E) is momentum-integrated scattering defined in the main text. The sum of the observed inelastic spectral weight δm2n and static moment hmi2 (= 0.372 µ2B ) spin correlation yields a total spectral weight of m2tot = 6.0(4) µ2B per Ce, consistent with the expected value gj2 J(J + 1) = 6.47 µ2B with gj = 67 . The magnitude of hµx i = hψ0 |gj Jx |ψ0 i with the ground state singlet listed in Table S3 is calculated as 0.9 µB , while the magnitude of hµz i (= hψ0 |gj Jz |ψ0 i ) ∼ 0. This is consistent with the proposed

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