Charge-stripe order, antiferromagnetism, and spin dynamics in the cuprate-analog nickelate La$_4$Ni$_3$O$_8$
Oscar O. Bernal Douglas E. MacLaughlin Gerald D. Morris Pei-Chun Ho Lei Shu Cheng Tan Jian Zhang Zhaofeng Ding Kevin Huang Viktor V. Poltavets


rev. 2.0




Department of Physics and Astronomy, California State University, Los Angeles, California 90032, USA

Department of Physics and Astronomy, University of California, Riverside, California 92521, USA

Centre for Molecular and Materials Science, TRIUMF, Vancouver, British Columbia V6T 2A3, Canada

Department of Physics, California State University, Fresno, California 93740,USA





[Present address: ]Lawrence Livermore National Laboratory, Livermore, California 94550, USA.
State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China

Department of Chemistry and Advanced Materials Research Institute, University of New Orleans, New Orleans, Louisiana 70148, USA




We report results of a muon spin rotation (μSR) study of the cuprate-analog nickelate La_4Ni_3O_8, which undergoes a transition at 105 K to a low-temperature phase with charge-stripe and antiferromagnetic (AFM) order on square planar NiO_2 layers. Zero-field μSR shows that the AFM transition is abrupt, commensurate and has a quasi-2D character below ∼25 K. Comparison of observed muon precession frequencies with Ni dipolar field calculations yields Ni moments ≲ 0.5μ_B. Dynamic muon spin relaxation above 105 K suggests critical slowing of Ni spin fluctuations, but is inconsistent with corresponding ^139La NMR results. Critical slowing and an abrupt transition are also observed in the planar cuprate AFM La_2CuO_4+δ, where they are taken as evidence for weakly interplanar-coupled two-dimensional AFM spin fluctuations, but our μSR data do not agree quantitatively with theoretical predictions for this scenario when applied to the nickelate.



75.47.Lx,75.50.Ee,76.75.+i

Charge-stripe order, antiferromagnetism, and spin dynamics
 in the cuprate-analog nickelate La_4Ni_3O_8
    V. V. Poltavets
    December 30, 2023
=======================================================================================================






§ INTRODUCTION
 

Bednorz and Müller searched for superconductivity in nickel-based systems without success before they discovered superconducting La_2BaCuO_4+δ <cit.>. For many years after, high-T_c superconductivity was found only in cuprate compounds with CuO_2-layered structures. It is now well known that besides the cuprates other layered systems (e.g., Fe-pnictide/chalcogenide and BiS_2-based materials) also become superconductors, suggesting a potential universality in layered superconductivity. This in turn revives interest in the quest for Cu-analog, Ni-based, superconducting materials. The motivation for studying these materials lies in their potential to help answer current experimental and theoretical questions regarding superconductivity in layered compounds, to assess the universality of layer superconductivity, and to provide clues for where to search for new layered superconductors. 

The trilayer T^'-type nickelate La_4Ni_3O_8 <cit.> is one such material. The crystal structure of this compound involves square planar NiO_2 trilayers, isoelectronic to the Cu^2+O_2 layers of the cuprates if the Ni valence were 1+. Further interest in comparing these systems arises from an antiferromagnetic (AFM) phase transition in La_4Ni_3O_8 at a Néel temperature T_N = 105 K into a state with commensurate spin and charge stripes <cit.>, similar in some respects to the AFM transition occurring in the cuprates. A charge/spin stripe phase has been observed in the T-type nickelate La_5/3Sr_1/3NiO_4 <cit.> which, like La_4Ni_3O_8, is 1/3 hole-doped. 

This article reports a muon spin rotation and relaxation (μSR) study of the magnetic structure and spin dynamics of La_4Ni_3O_8. μSR is a magnetic resonance technique <cit.>, conceptually similar to NMR, that uses muon spins implanted into the sample as microscopic probes of their static and dynamic magnetic environment. We find that the 105-K transition is abrupt (discontinuous to within experimental resolution), and that the commensurate AFM configuration is consistent with both magnetic and charge stripe order as found in x-ray <cit.> and neutron <cit.> diffraction studies, respectively. Observed spectra of muon spin precession frequencies resemble histograms of calculated Ni dipolar fields at candidate muon sites for specific AFM stripe configurations, with reduced Ni moments 0.4–0.5μ_B. We present μSR evidence that the magnetic transition at T_N is quasi-2D in nature, in which the trilayers are ordered but fluctuate independently down to ∼25 K. Below this temperature three-dimensional (3D) static magnetism is observed, with disorder along the normal to the trilayer planes as found by neutron diffraction <cit.>.

Dynamic muon spin relaxation in the paramagnetic phase suggests critical slowing of Ni spin fluctuations with a divergence at ∼T_N. This is not expected for two-dimensional (2D) Heisenberg magnets, where the critical behavior is due to 2D spin fluctuations that do not order at finite T <cit.>. In that scenario the 3D transition is due to a weak interplanar coupling, and does not dominate the critical spin dynamics. Evidence for this 2D criticality in the planar cuprate AFM La_2CuO_4+δ is found from NMR studies <cit.>. Our μSR data in La_4Ni_3O_8 do not agree quantitatively with this scenario, and instead suggest the possibility of a second-order transition with a very sharp onset at T_N. This is in contrast with the T = 0 divergence of the ^139La NMR relaxation rate in La_4Ni_3O_8 <cit.>, which has been taken as evidence for 2D fluctuations. 

The article is organized as follows: The current understanding of La_4Ni_3O_8 and related systems is reviewed in Sec. <ref>, with emphasis on the charge/stripe nature of the 105-K transition. Section <ref> gives experimental details and a brief description of the μSR technique. Candidate muon stopping sites in the crystal are discussed in Sec. <ref>, and Sec. <ref> reports μSR results in the ordered phase and their interpretation. Section <ref> reports muon spin relaxation in the paramagnetic phase above 105 K. Our results are discussed in Sec. <ref> and summarized in Sec. <ref>. Appendix <ref> gives supplementary information on AFM configurations, and Appendix <ref> presents a simple model for zero-field μSR at an abrupt magnetic transition with an inhomogeneous distribution of transition temperatures.



§ BACKGROUND
 

In 1999 Anisimov, Bukhvalov, and Rice <cit.> concluded “Only if the Ni ions are forced into a planar coordination with the O ions can a S = 1/2 magnetic insulator be realized with the difficult Ni^1+ oxidation state and possibly doped with low spin (S = 0) Ni^2+ holes directly analogous to the superconducting cuprates.” I.e., a system with electronic configuration Ni^1+/Ni^2+ would have the same configuration as that of Cu^2+/Cu^3+ in the cuprates, but to be a true analog it would also need to be in a square planar coordination with O ions. The Ni^1+/Ni^2+ configuration is rare in oxides but it does exist, and conforms with the condition of Anisimov et al. in the T'-type nickelates Ln_n+1Ni_nO_2n+2, Ln = La, Nd, n = 2, 3 and ∞ <cit.>. In this formula n is the number of NiO_2 layers in a multilayer of square-coordinated Ni ions.

Poltavets et al. <cit.> reported a magnetic transition at 105 K for La_4Ni_3O_8 (n = 3), and attributed it to Ni-spin inter-trilayer interactions in analogy with undoped La_2CuO_4+δ. No transition was seen in the magnetization for applied fields lower than 0.1 T, but the observed change in slope of the resistivity ρ(T) and the lambda anomaly in the specific heat at 105 K were taken as evidence for a transition in zero field. At low temperatures ρ(T) exhibits semiconducting behavior <cit.>. ^139La NMR measurements indicated that the transition is to an AFM phase (i.e., 105 K is a Néel temperature T_N), further suggesting the possibility of a cuprate analog. 

The results of Ref. PLNC10 triggered a good deal of theoretical and experimental work <cit.>. Pardo and Pickett <cit.> predicted an insulating ground state of the Mott type, and related the transition to binding of dz^2 orbitals of Ni ions in outer and middle layers of a trilayer (designated Ni1 and Ni2 respectively). Checkerboard AFM ordering in NiO_2 planes was assumed, although powder neutron diffraction has not yielded specific indications for magnetic order <cit.>. A predicted metal-insulator transition under pressure <cit.> was not observed <cit.>.

Liu et al. <cit.> and Wu <cit.> predicted La_4Ni_3O_8 to be a C-type AFM using different theoretical methods. A correlation of the metal/insulator character of the system with dimensionality was found from first-principle calculations in La_n+1Ni_nO_2n+2 by Liu et al. <cit.>. The quasi-2D n = 2 and 3 varieties were predicted to be molecular insulators in contrast with 3D (n=∞) LaNiO_2, which according to their calculated electronic structure is metallic by virtue of Ni-La hybridization. The inclusion of electron correlation effects in first-principles calculations by Patra et al. <cit.> suggested that the two inequivalent Ni1 and Ni2 atoms in the La_4-xSr_xNi_3O_8 crystal structure are both in a high-spin state, with an average valence of +1.33 independent of x doping.

There were indications, however, that the physics of the nickelates is more complicated than expected. A fit to the temperature dependence of the ^139La NMR relaxation rate 1/T_1 in La_4Ni_3O_8 <cit.> required a Korringa term 1/T_1T = const., indicative of a Fermi liquid phase. This is in contrast to the AFM cuprate compounds, which are Mott insulators. In addition, 1/T_1 in the paramagnetic phase above T_N could be fit as diverging at T = 0 <cit.>, rather than at T_N as expected for a second-order (continuous) transition in 3D. A T = 0 divergence is characteristic of critical fluctuations of a 2D Heisenberg quantum antiferromagnet <cit.>, and is observed in the quasi-2D cuprate antiferromagnet La_2CuO_4+δ <cit.>. 


More recently, NMR measurements by ApRoberts-Warren et al. <cit.> in the paramagnetic phase of the bilayer (n=2) T'-type compound La_3Ni_2O_6 revealed very similar behavior of the ^139La relaxation rate to that found previously in trilayer La_4Ni_3O_8. Although they observed no magnetic transition in the bilayer down to 5 K, their results were ascribed to similarity in the electronic structure of the two materials, presumed to be of the 2D variety.

The discussion of the 105-K transition in La_4Ni_3O_8 shifted in 2016, when Zhang, Chen et al. <cit.> reported results from synchrotron x-ray diffraction measurements on single crystals. They found a quasi-2D stripe-ordered charge configuration below 105 K, with orientation at 45^∘ to the Ni-O-Ni bonds in NiO_2 squares and a 2D supercell 3√(2)a ×√(2)a, where a is the Ni-Ni nearest-neighbor distance. This is similar to the charge order found in La_5/3Sr_1/3NiO_4 <cit.>, a related 1/3-hole doped single-layer Ruddlesden-Popper nickelate, where the average Ni valence is 2.33+. The stripe pattern in La_4Ni_3O_8 is two Ni^1+ (S = 1/2) rows followed by one Ni^2+ (S = 0) row, oriented at 45^∘ to the planar Ni-O bonds, with staggered stacking along the c-axis. The 2:1 ratio preserves the formal Ni valence 1.33+. All three layers in a trilayer possess stripes. The magnetic structure was not obtained in this study, which sampled only charge degrees of freedom.

Ab initio calculations by Botana et al. <cit.> yielded stripe charge order from a combination of structural distortion and AFM order, in which Ni^2+ ions have spin S = 0, and Ni^1+ S = 1/2 ions order antiferromagnetically, similar to the spin-1/2 insulating antiferromagnetism of the cuprate parent materials. In this result the valences of corresponding Ni1 and Ni2 ions in a trilayer are the same. Two 2D supercells of the observed stripe pattern and the proposed 2D AFM order in a single Ni layer <cit.> are shown in Fig. <ref>. 


Very recently Zhang, Pajerowski et al. <cit.> reported results of neutron Bragg diffraction experiments on La_4Ni_3O_8. Their measurements confirm the magnetic ground state, with stripe AFM order commensurate with the charge stripes in individual trilayers but little or no correlation between trilayers. In contrast to La_5/3Sr_1/3NiO_4 and to most other transition metal oxides, the charge and spin stripes form simultaneously at 105 K.




§ EXPERIMENT
 



 §.§ Sample
  

Polycrystalline La_4Ni_3O_8 was prepared as described previously <cit.> and characterized by powder x-ray diffraction, x-ray absorption spectroscopy, magnetization, specific heat, resistivity, and NMR measurements <cit.>. For our μSR experiments we used enough material to fully cover a circular area of about 1 cm in diameter and 1 mm thickness.

The two-trilayer tetragonal unit cell of La_4Ni_3O_8 (space group I4/mmm, no. 139) is shown in Fig. <ref>. 

There are two inequivalent Ni sites (Ni1 and Ni2) and three inequivalent oxygen sites, two of which coordinate the Ni ions in the planar NiO_2 layers. The third oxygen sites form separate square-coordinated layers, with La ions above and below the square centers forming flourite La-O blocks that separate and isolate one NiO_2 trilayer from the next. NiO_2 layers within a trilayer are separated by La-only planes. The candidate muon sites shown in Fig. <ref> are discussed in Sec. <ref>.



 §.§ μSR experiments
 

In the μSR technique <cit.>, 100% spin-polarized muons are implanted into a sample, where they decay [μ^±→ e^± + ν_e(ν_e) + ν_μ(ν_μ)] with a mean lifetime 2.197 μs. Usually positive muons (μ^+) are used, which stop in interstitial sites [Negative muons go into tight Bohr orbits around host nuclei and are less sensitive to their magnetic environment than interstitial positive muons.]. The decay positron is preferentially emitted in the direction of the μ^+ spin at the time of decay. Thus the time-dependent positron count-rate asymmetry A(t) (the asymmetry spectrum), where t is the time after implantation, directly yields the time evolution of the ensemble μ^+ spin polarization G(t) [G(0) ≡ 1]: A(t) = A_0G(t), where the initial asymmetry A_0 = 0.2–0.3 is instrument-dependent. Information on local magnetic behavior is contained in G(t). Measurement of a single asymmetry spectrum typically requires ≳ 10^7 events.

μ^+SR experiments on La_4Ni_3O_8 were carried out with the LAMPF spectrometer at the M20D muon beam line, TRIUMF, Vancouver, Canada, over the temperature range 1.2–300 K, in zero magnetic field (ZF) and in longitudinal applied fields (LF) (field parallel to the initial μ^+ spin polarization) up to 4 kOe. Data were analyzed using the Paul Scherrer Institute  fitting program <cit.> and the TRIUMF physica programming environment [].



§ Μ^+ STOPPING SITES


μ^+ precession frequencies in ordered magnets yield static magnetic field values at the μ^+ stopping sites, which can help to characterize the magnetic order if these sites are known. Implanted positive muons in oxides are likely to stop near O^2- ions, and in La_4Ni_3O_8 with stripe order the large unit cell yields a considerable number of such candidate sites. To date there are no reported calculations of site locations or relative occupations in La_4Ni_3O_8 using, for example, DFT theory <cit.>. 

In lieu of such information, we consider candidate μ^+ sites that are symmetric with respect to oxygen near neighbors. Eight such sites that are crystallographically inequivalent are shown in Fig. <ref> and listed in Table <ref>.

It is customary <cit.> to describe magnetic and stripe structure in La_4Ni_3O_8 using the setting F4/mmm, in which basal-plane unit vectors are rotated 45^∘ to define a face-centered square lattice of Ni ions. The lattice constants in these directions are then √(2) times the Ni-Ni nearest neighbor distance a in the basal plane. The lattice constant c remains the same as in the I4/mmm setting, and the unit cell volume is doubled. The observed charge-stripe supercell <cit.> is 3√(2)a ×√(2)a × c, where the stripe direction is the b axis (cf. Fig. <ref>). Candidate μ^+ sites are given in Table <ref> for both settings. 
 
In the I4/mmm tetragonal unit cell there are a total of 58 such sites (the sum of the Wycoff multiplicities in Table <ref>). With two trilayers per I4/mmm unit cell, there are 58 sites per trilayer in the F4/mmm setting and 3 × 58 = 174 sites per trilayer in a stripe supercell. The two trilayers are crystallographically equivalent, so that it is sufficient to consider μ^+ sites for a single trilayer provided this equivalence extends to the ordered magnetic structure. Figure <ref> shows two F4/mmm supercells of La_4Ni_3O_8 viewed along the c axis, with the stripe AFM configuration of Fig. <ref> <cit.> and μ^+ sites from Table <ref>. 

The number of magnetically inequivalent sites in a full 3D AFM configuration is clearly greater in stripe models than for AFM order without stripes. 

The μ^+ sites of Table <ref> may not be realistic: they would be unstable if the O^2- ions were classical point charges, and our calculations do not take account of off-site ionic positions, intrinsic to the La_4Ni_3O_8 crystal structure <cit.> or due to the μ^+ charge. The candidate sites sample a large fraction of the unit cell, however (Fig. <ref>), so that calculated dipolar fields at these sites in the magnetically ordered phase can be taken as rough guides to the local field distribution. 




§ ANTIFERROMAGNETIC CONFIGURATIONS
 



 §.§ Experimental results
 

Figure <ref> shows Fourier transforms of ZF-μSR asymmetry data from La_4Ni_3O_8 from 2.2 K to 110 K. 

There are at least six discrete peaks with nonzero frequency, corresponding to oscillations in the asymmetry spectra due to μ^+ spin precession in well-defined magnetic fields. The peaks are clearly visible at 50 K; we label them from 1 to 6 on the abscissa of Fig. <ref>. In the temperature range 99–105 K the frequency of Peak 6 (∼19 MHz) hardly changes, but its amplitude decreases smoothly to nearly zero. Several of the peaks are broad, and probably contain unresolved frequencies from multiple sites. 

A representative series of ZF-μSR asymmetry spectra for temperatures between 1.85 K and 105 K (the AFM phase) is shown in Fig. <ref>.

The curves in Fig. <ref> are fits to the data using the exponentially-damped multi-frequency function

    G(t) = f_0exp(-λ_0t) + ∑_i=1^6 f_iexp(-λ_it)cos(ω_it+ϕ_i)
 
for the μ^+ spin polarization [For ZF measurements in a random powder with static local fields, 1/3 of the initial μ^+ polarization relaxes with zero frequency <cit.>. A ω = 0 component is also present in an antiferromagnet if a muon site is symmetric with respect to the sublattices, since this results in cancellation of local fields.]. The frequencies ω_i give the local magnetic fields ω_i/γ_μ, where γ_μ= 8.5156 × 10^4 s^-1 G^-1 is the gyromagnetic ratio of the muon. The rates λ_i characterize line broadening, dominated by inhomogeneity in these fields. The amplitudes f_i are related to μ^+ site occupation probabilities, and are not expected to depend on temperature. The polycrystalline nature of our sample is not involved in the ZF results shown in Fig. <ref>, except that the f_i are averaged over local-field orientations relative to the initial μ^+ polarization direction.

Frequencies obtained from fits of Eq. (<ref>) to asymmetry data are plotted vs temperature in Fig. <ref> (a) and similarly labeled from 1 to 6. 

The rather smooth differences in temperature dependence of the frequencies between T_N and ∼25 K 
is evidence for a gradual Ni spin reorientation below the transition.

The amplitudes and rates of Peak 1 (∼2 MHz) and Peak 6 are compared in Figs. <ref>(b) and <ref>(c). They behave quite differently: below ∼90 K the amplitude and rate of Peak 6 are nearly constant, whereas for Peak 1 they both increase substantially with decreasing temperature below ∼25 K. At lower temperatures Peak 1 is hard to see in the Fourier spectra (Fig. <ref>), because its increase in time-domain (asymmetry) amplitude by a factor of ∼2 [Fig. <ref>(b)] (probably an artifact due to overlapping adjacent peaks) is offset [Due to the general Fourier-transform proportionality between time-domain amplitudes and frequency-domain areas, increased broadening causes a loss of frequency-domain amplitude.] by a damping-rate (linewidth) increase by a factor of ∼30 [Fig. <ref>(c)]. This behavior is discussed in detail in Sec. <ref>.



 §.§ Models of AFM order: dipolar-field calculations
 

This section describes lattice-sum calculations of Ni-moment dipolar magnetic fields at the candidate μ^+ stopping sites of Table <ref> for a number of candidate AFM configurations, with and without charge stripes. The calculations assume the same moment on all moment-bearing Ni^1+ ions in a trilayer, consistent with the neutron diffraction results <cit.>. These dipolar fields are summed over a sphere of radius 200 Å centered on the μ^+ site, after determining that increasing this radius does not affect the results appreciably. The Ni-ion moment is scaled for rough agreement of the dipolar-field frequencies with a typical experimental spectrum (T = 50 K, cf. Fig. <ref>). 



  §.§.§ AFM configurations with charge stripes
 



  
Stripe models. For completeness, we consider the four charge stripe models discussed by Zhang, Chen et al. <cit.>, which differ in the positions of the stripes on the individual layers. These models are shown in Fig. <ref>, where the view is along the stripe direction (the b direction in the F4/mmm setting), and a supercell boundary is shown. 

The model numbering (1–4) of Ref. ZCPZ16 is used in the following. The stripes in a trilayer are stacked along the c axis in Models 1 and 4 and staggered in Models 2 and 3. The Coulomb repulsion energy would seem to be lower for the latter, but x-ray diffraction data favor the stripe structure of Model 1 <cit.>. The latter is also used for AFM configurations in the neutron diffraction study of Ref. ZPBZ19u. Model 4 is assumed in the ab initio treatment of charge and AFM order in La_4Ni_3O_8 of Botana et al. <cit.>, where it is argued that the weak coupling between trilayers does not play a significant role in the electronic structure. 

The calculations and neutron diffraction results cited above <cit.> both yield the 2D AFM structure shown in Fig. <ref>, with Ni moments aligned along the c axis. We have calculated dipolar fields at μ^+ sites for 16 3D AFM structures, obtained by placing 2D AFM layers with this structure on each of the four stripe models in four configurations, designated A–D in the following. In Configuration A the moment orientations are those of Fig. <ref> on all layers. In Configuration B all moments on alternate trilayers of Configuration A are inverted. In Configuration C the Ni2 (middle layer) moments of Configuration A are inverted for all layers, yielding intra-trilayer AFM ordering. Configuration D can be obtained in either of two ways: by inverting the Ni2 moments of Configuration B, or by inverting alternate trilayers of Configuration C.



We discuss Model 1 below. Models 2–4 are described in Appendix <ref>. 



  
Dipolar fields in stripe Model 1.  The four AFM moment configurations are shown in Fig. <ref> for charge stripe Model 1, and histograms of frequencies from μ^+-site dipolar-field calculations for these configurations are compared with the observed 50-K spectrum in Fig. <ref>. 
The calculated histograms assume equal stopping probabilities for all μ^+ sites, which may not be the case, and do not take into account possible transferred μ^+ hyperfine fields. A rough correspondence between the calculated histograms and observed spectra is obtained with Ni^1+ moments 0.4–0.5μ_B, somewhat smaller than the predicted values 0.6–0.7μ_B <cit.>. A high-frequency group of peaks is separated from a low-frequency group by a region with less spectral weight. The comparison suggests that the high-frequency peak is formed from a group of discrete frequencies. These are mainly from μ1 sites, which is not surprising as they are the closest to Ni ions (Fig. <ref>).

There are a number of general features of the calculated histograms (Fig. <ref> and Appendix <ref>). Histograms for stripe Models 1 and 4 are similar, as are those for stripe Models 2 and 3, due to stacked stripes in the former compared to offset stripes in the latter. For a given model, only small differences and no change of Ni moment are found between A and B histograms, and also between C and D histograms. Inverting alternate trilayers changes the μ^+-site fields considerably less than inverting Ni2 moments within each trilayer. 

Reference ZPBZ19u concludes that trilayers are essentially uncorrelated along the c axis, corresponding to a random mix of Configurations 1A and 1B (or 1C and 1D). From Fig. <ref>, such a random mix would have little effect on the higher frequencies, which arise from μ^+ sites in or near the trilayers. There are visible changes of the lower frequencies associated with sites far from trilayers. This behavior may have been observed, as described below in Sec. <ref>.

For all stripe models a calculated peak or group is found near the observed small peak at ∼12 MHz for AFM Configurations A and B but not for Configurations C and D; agreement with the observed spectrum is better for the former configurations. This is not strong evidence for them, however, due to the approximate nature of the treatment and the uncertainty in the μ^+ sites. Nevertheless, we note the discrepancy between this result and the neutron scattering data <cit.>, which favor the AFM intra-trilayer order of Configurations 1C and 1D (trilayer model 5 of Ref. ZPBZ19u, Supplementary Information) over the FM intra-trilayer order of Configurations 1A and 1B (their model 6).



  §.§.§ AFM order without stripes?
 


For completeness we also consider AFM configurations without stripes (although these are ruled out by the neutron diffraction results <cit.>). Using the F4/mmm setting, Fig. <ref> shows four such configurations in La_4Ni_3O_8: the ferromagnetic (FM) configuration (for completeness), and three AFM configurations <cit.>. Figure <ref> compares frequency histograms from dipolar field calculations for the configurations of Fig. <ref> with the observed 50-K spectrum. The subscripts 1 and 2 for the FM and A-AFM configurations designate alternate trilayer moments as pictured in Fig. <ref> or inverted, respectively. Alternate-trilayer inversion followed by 90^∘ rotation about the c axis is a symmetry operation for the C-AFM and G-AFM configurations, and therefore does not affect the dipolar-field histogram. 

These histograms have a rough correspondence with those for the stripe-model AFM configurations of the previous section and, similarly, have no clear agreement with details of the experimental spectrum. This is not surprising, since commensurate stripes are long-range order (i.e., points in reciprocal space) best probed by a diffraction technique, whereas μSR, like NMR, is a point probe in real space. The histograms of Fig. <ref> are not as dense as those from stripe configurations, and there does not seem to be enough spectral weight at low frequencies, but the evidence for a striped AFM configuration from neutron diffraction <cit.> is much more compelling.




§ PARAMAGNETIC PHASE
 

Two mechanisms for μ^+ spin relaxation are expected in the high-temperature paramagnetic phase of a local-moment magnet: dynamic relaxation due to local-moment fluctuations, and (quasi)static relaxation from nuclear dipolar fields. In zero or low applied fields the resulting Kubo-Toyabe (KT) relaxation <cit.> is due to the combined effects of these mechanisms. 

LF-μSR is a standard method for separation of the static and dynamic contributions <cit.>, since a longitudinal field H_L “decouples” the μ^+ spin polarization from the static local field distribution [For large enough H_L (≫ the local-field distribution width) the resultant field is parallel to the μ^+ spin polarization. Then there is no precession and thus no static relaxation, so that any observed relaxation is dynamic.]. Static μ^+ relaxation functions G_s(H_L,t) for arbitrary H_L (including zero field) have been determined for several local field distributions <cit.>. The combined effect of static and dynamic relaxation is often modeled by exponential damping of G_s(H_L,t) with rate λ:

    G(H_L,t) = e^-λ t G_s(H_L,t) .


In La_4Ni_3O_8 G(H_L,t) is complicated by the large number of inequivalent μ^+ sites. This results in a broad inhomogeneous distribution of μ^+ dipolar fields from Ni^1+ moments, as can be seen from the AFM-phase histograms discussed above. Similarly, the dipolar-field distribution from randomly-oriented ^139La nuclei is further broadened because of the multiple μ^+ sites. Obtaining G(H_L,t) is then a difficult numerical problem even if these distributions were known, which is not the case for La_4Ni_3O_8 because of uncertainty in the μ^+ sites.

We have carried out ZF- and LF-μSR experiments in the paramagnetic phase of La_4Ni_3O_8 between T_N and 275 K. For analysis of the data we have chosen a rough approximate form of Eq. (<ref>), using the static Lorentzian KT function G_s^Lor(H_L,α,t) <cit.> with “stretched exponential” damping:

    G(H_L,α,t) = exp[-(λ^∗ t)^K]  G_s^Lor(H_L,α,t)  .

We have taken a Lorentzian distribution of ^139La dipolar fields with half-width α/γ_μ as an approximation beyond the Gaussian that is appropriate for a single μ^+ site in a homogeneous system <cit.>. The LF static Lorentzian KT function is <cit.>


    G_s^Lor
        =     1 - α/ω_Lj_1(ω_L)exp(-α t) - (α/ω_L)^2[j_0(ω_L t)exp(-α t) - 1] 
           - α[1+(α/ω_L)^2] ∫_0^t j_0(ω_Lτ)exp(-ατ)  dτ ,


where ω_L = γ_μ H_L, and j_0 and j_1 are spherical Bessel functions. In the ZF limit Eq. (<ref>) simplifies to

    G_s^Lor(α,t) = 1/3 + 2/3 (1 - α t)exp(-α t) .

The stretched-exponential form of the damping in Eq. (<ref>) represents an inhomogeneous distribution of local dynamic relaxation rates λ_loc <cit.>, where λ^∗ is a characteristic rate (but not the average <cit.>) and the stretching power K < 1 characterizes the distribution function P(λ_loc) of the local rates. 

The stretched-exponential dynamic and Lorentzian static forms are crude approximations (both are strictly valid only for a r^-3 interaction with local moments in the dilute limit <cit.>). Their use is justified by their relative simplicity, and a posteriori by the fact that they give good fits to the data (see below). An alternative model for static relaxation in disordered systems, the “Gaussian broadened Gaussian” KT function <cit.>, does not fit well; for early times it varies as t^2 (i.e., as a Gaussian), whereas the data exhibit the linear early-time behavior of an exponential <cit.>.



 §.§ Zero field
 

Representative ZF asymmetry spectra above T_N are shown in Fig. <ref>, and Fig. <ref> shows the temperature dependence of λ^∗ and K in zero field from fits of Eqs. (<ref>) and (<ref>) to the data.


In the fits α was fixed at its 200-K value, but the results do not change qualitatively for ±10% variations of α. The stretching power K(T) is essentially constant, whereas λ^∗(T) increases markedly with decreasing temperature as T_N is approached.

The curves in Fig. <ref>(a) give fits of theoretical expectations for 2D AFM spin fluctuations <cit.>, as discussed in Ref. A-WDPG11 with respect to ^139La NMR relaxation data. The solid curve is a phenomenological Curie-Weiss-like fit: λ^∗(T) ∝ T/(T-T_0) <cit.>, which yields T_0 = 90(1) K. The dashed curve in Fig. <ref>(a) is the function

    λ^∗(T) ∝ x^3/2 e^1/x/(1+x)^3 ,   x = T/1.13J  ,

derived by Chakravarty and Orbach <cit.> (CO) from the theory of a 2D quantum Heisenberg antiferromagnet <cit.> mentioned in Sec. <ref>, in which the critical point is at T = 0. The weak inter-layer interaction that precipitates the 3D AFM transition in this picture is not expected to affect the critical behavior appreciably. Equation (<ref>) is only valid for T < 0.57J, however <cit.>, where J is the AFM exchange coupling. The fit value of J is 86(22) K, so that temperatures in the paramagnetic phase of La_4Ni_3O_8 violate this condition. Thus the CO fit is inconsistent.



 §.§ Longitudinal field, T = 200 K
  

Figure <ref> shows representative asymmetry spectra at 200 K for 0 ⩽ H_L ⩽ 4 kOe. 

The curves are fits of Eqs. (<ref>)–(<ref>) to the data. The observed spectra are quite smooth, and the parameters α, λ^∗, and K in Eqs. (<ref>)–(<ref>) are highly correlated statistically, making it hard to determine them separately. The following procedure was used: at high enough fields G_s^Lor(H_L,t) = 1 independent of α, so that stretched-exponential fits determine λ^∗ and K. Assuming these are roughly field-independent, they were fixed and the lowest-field data were fit to determine α. Then all three parameters were iteratively fixed and freed at low fields: α was fixed and λ^∗ and K freed, and vice versa, until convergence was obtained. Finally α was fixed, and λ^∗ and K were freed for fits at all fields. 

For low fields, the overall relaxation slows with increasing field but the early-time relaxation is not affected, which is a characteristic of static local-field decoupling. For higher fields the initial slope decreases, indicating a field dependence of λ^∗ and/or K.

The field dependences of λ^∗ and K at 200 K are shown in Fig. <ref>. 

The scatter in the parameters is obviously strongly correlated. The field dependences at low fields are probably artifacts of the assumed field independence of α; a field dependence (as discussed in Ref. HUIN79 for the Gaussian relaxation rate at a single μ^+ site) might be expected in La_4Ni_3O_8 because the μ^+ frequency γ_μ H_L is of the order of the lower of the two observed ^139La quadrupolar splittings <cit.> for H_L ∼ 100 Oe. Nevertheless, for H_L ≳ 50 Oe the fits no longer depend on α, and the parameters are only weakly field-dependent. The stretching power K is close to the value 1/2 expected from a Lorentzian distribution of fluctuating fields in the motional-narrowing limit (rapid fluctuations) <cit.>, which is perhaps further justification for the use of Lorentzian distributions to describe μ^+ relaxation in La_4Ni_3O_8.



 §.§ Longitudinal field, T = 110 K
 

A similar analysis was applied to LF data taken for T = 110 K, a temperature just above T_N. Representative asymmetry spectra are shown in Fig. <ref>, and the field dependences of λ^∗ and K, determined as described above for 200 K, are shown in Fig. <ref>. 


The values of K at the two temperatures are comparable, but at 110 K a clear increase of λ^∗ at low fields is observed. This increase and the temperature independence of λ^∗ at high fields are discussed further in Sec. <ref>.

Values of λ^∗ for H_L = 0.8 Oe at T = 200 K and 110 K (Figs. <ref> and <ref>) are larger than the ZF values at those temperatures (Fig. <ref>). This field dependence at low fields is not presently understood.




§ DISCUSSION
 



 §.§ AFM transition and spin configuration


The complex zero-field μSR spectra observed below 105 K in La_4Ni_3O_8 (Fig. <ref>) clearly indicate the magnetic nature of the transition. The multiple frequencies arise from a number of magnetically inequivalent μ^+ stopping sites in the crystal. The sharp peaks indicate commensurate magnetic order, since in an incommensurate structure crystallographically equivalent μ^+ stopping sites would be subject to a continuous distribution of frequencies. The smooth differences in temperature dependencies of the various peak frequencies between T_M and ∼25 K [Fig. <ref>(a)] suggest reorientation of the AFM magnetic structure with decreasing temperature below T_N.



  §.§.§ Abrupt onset at T_N
 

From Figs. <ref>–<ref> we conclude that the transition at T_N = 105 K has an abrupt onset, since the peak frequencies appear discontinuously at T_N. This is suggestive of a first-order transition. It is, however, also consistent with a continuous second-order transition in an inhomogeneous system <cit.>, if the intrinsic onset of the order parameter is sufficiently abrupt (i.e., the critical exponent β is sufficiently small) compared to the width of the inhomogeneous transition temperature distribution. A simple model for zero-field μSR (and NQR) spectra in this situation, described in Appendix <ref>, shows that near T_N the spectra become broadly distributed, with reduced-amplitude peaks at their maxima. Then the transition appears discontinuous (Fig. <ref>), as we observe in La_4Ni_3O_8 (Fig. <ref>). We cannot discriminate between this and a true first-order discontinuity on the basis of our data (the term “abrupt” describes both possibilities).



  
Comparison with neutron diffraction. A commensurate AFM phase is found in La_4Ni_3O_8 by neutron Bragg diffraction <cit.>, but with a smooth continuous transition. This discrepancy with the μSR result can be understood in the above scenario <cit.>. The effect of the distribution is quite different in the two techniques, due to the fact that magnetic resonance experiments probe real space whereas neutron diffraction probes reciprocal space. In neutron diffraction the AFM Bragg intensity is proportional to the volume fraction of magnetic order, and therefore increases smoothly as the temperature is decreased through an inhomogeneous distribution of transition temperatures.



  
Comparison with NMR. The μSR evidence for an abrupt transition in La_4Ni_3O_8 is also in conflict with the smooth transition observed in ^139La NMR spectra [Ref. PLNC10 (suppl.)]. The NMR spectra were obtained from a partially-oriented powder sample in an applied field of 42 kOe rather than zero field, and the conclusion of a smooth transition is mainly based on a single point at ∼95 K. It seems possible that partial random orientation and/or the applied field affected the results in these experiments.



  
Comparison with other materials. μSR studies of the doped nickelate La_2-xSr_xNiO_4 <cit.> yield an abrupt but continuous frequency onset at T_N with a small value of the critical exponent β, similar to our results in La_4Ni_3O_8.

Discontinuous transitions were observed in the layered AFM cuprate La_2CuO_4+δ in ^139La NQR <cit.> and other hyperfine experiments <cit.>, but not in neutron diffraction; the discrepancy can be understood as noted above in Sec. <ref>. Subsequent ^139La NQR experiments in a sufficiently homogeneous La_2CuO_4+δ crystal <cit.> resolved an “abrupt but continuous” change of frequency at the transition. Our results in La_4Ni_3O_8 might also be due to such an inhomogeneity-induced discontinuity, and there is a clear need for experiments on more homogeneous samples of this compound.



  §.§.§ AFM spin configuration
 

Histograms of calculated Ni-ion dipolar fields at candidate μ^+ sites for a number of charge stripe models and AFM configurations (Sec. <ref>, Appendix <ref>) capture the general features of frequency spectra and suggest reduced Ni moments (0.4–0.5μ_B), but do not uniquely determine the specific AFM configuration. Dipolar-field histograms for AFM configurations without stripes have somewhat less structure than the observed spectra. 

The broadening of Peak 1 below ∼25 K [Fig. <ref>(d)] is evidence for the onset of significant disorder at low temperatures. The low-frequency peaks arise from μ^+ sites far from Ni trilayers, suggesting that the disorder is associated with the lack of c-axis correlation reported in Ref. <cit.>. The absence of broadening for Peak 6, which is from μ^+ sites in or near the Ni trilayers, shows that the AFM state there is commensurate and not strongly disordered at low temperatures. Peak 1 becomes narrow above 25 K (Fig. <ref>), which indicates motional averaging of the corresponding local field. 

This behavior suggests that the 105-K transition is within the trilayers and not between them. We speculate that at intermediate temperatures, each trilayer fluctuates dynamically as a whole but independently of neighboring trilayers, maintaining nearby μ^+ local fields more or less constant. Such “stripe fluctuation” has been reported in La_2-xSr_xNiO_4 <cit.>. Below 25 K a weaker (possibly anisotropic) inter-trilayer interaction leads to disordered full 3D spin freezing, with a broad distribution of local fields at μ^+ sites distant from trilayers.



 §.§ Paramagnetic-phase relaxation; critical slowing
 

As discussed in Sec. <ref>, increase of the ZF μ^+ relaxation rate λ^∗(T) as T → T_N from the paramagnetic phase [Fig. <ref>(a)] suggests critical slowing of spin fluctuations associated with the transition. The data can be fit with Eq. (<ref>) from the CO theory <cit.>, in which the critical fluctuations are 2D in nature and diverge as T → 0. The value of the exchange constant J from this fit is lower than the temperatures of measurement, however, which invalidates the fit. A Curie-Weiss-like fit yields a divergence at 90(1) K, suggesting that it is associated with T_N rather than zero temperature.

Curie-Weiss and CO-theory fits [Fig. <ref>(a)] both yield characteristic temperatures (T_0 and J, respectively) of the order of T_N. This seems compatible with a connection between critical slowing and the 3D phase transition, although, as noted above, the CO fit is not valid.

Comparison of the field dependencies of λ^∗(T) at 200 K [Fig. <ref>(a)] and 110 K [Fig. <ref>(a)] suggests that the critical point is at zero field. The high-field data at both temperatures have similar magnitudes, but the low-field relaxation increases strongly with decreasing field at the lower temperature (which is close to T_N). 



  
Comparison with NMR. Like the μSR data, the paramagnetic-phase ^139La nuclear spin-lattice relaxation rate in La_4Ni_3O_8 <cit.> can be fit with a Curie-Weiss law, but with a divergence as T → 0 that disagrees strongly with the μSR result. The NMR data were also fit by Eq. (<ref>), which was taken as evidence for the CO theory as discussed above. However, the exchange constant 129(5) K from the NMR data is, like that from the μSR data, of the order of T_N, and therefore the temperatures of measurement (T > T_N) do not obey the condition T < 0.69J required for self-consistency of the theory <cit.>. Furthermore, the NMR experiments were carried out in a field of 42 kOe on very broad lines, which make the measurement difficult. For nonzero but much lower fields (0.1–4 kOe, Figs. <ref> and <ref>) μ^+ spin relaxation rates show no sign of critical slowing between 200 K and 110 K as noted above. 



  
Specific heat peak. A peak in the temperature dependence of the specific heat is observed at T_N <cit.>. We note that that a specific-heat “peak” could also be due to first-order latent heat in an inhomogeneous sample with a spread of transition temperatures. Furthermore, charge ordering in La_4Ni_3O_8 might contribute to the specific heat (μSR is not directly sensitive to charge ordering). Nevertheless, the peak has been taken as evidence for a second-order transition <cit.>, which is not in conflict with the μSR relaxation data.



  
Comparison with other materials. Critical slowing with a T → 0 divergence was observed in NQR experiments on La_2CuO_4+δ <cit.>, and taken as evidence for the CO theory <cit.>. Fits of Eq. (<ref>) to the NQR data yield T ≪ J <cit.>, consistent with applicability of the theory. 

La_4Ni_3O_8 is intrinsically overdoped, and a more appropriate comparison might be with results from doped La_2CuO_4 with charge ordering. Previous ^63,65Cu and ^139La NQR studies of underdoped La_2-xSr_xCuO_4, 0 ⩽ x ⩽ 0.15, <cit.> found that critical slowing for x = 0 at high temperatures is rapidly suppressed by Sr doping. Recent NMR experiments <cit.> using a single crystal of La_1.885Sr_0.115CuO_4 revealed inhomogeneity and considerable structure associated with staging and oxygen diffusion at low temperatures, but did not revise the earlier conclusions. A NMR study <cit.> of single-crystal La_2CuO_4+y, y ∼ 0.11, showed Curie-Weiss behavior of the ^63Cu relaxation rate for temperatures ≳ 60 K. This result is similar to both NMR and ZF-μSR data from La_4Ni_3O_8, but the doping level is higher in the nickelate. To our knowledge no μSR studies of critical slowing in the paramagnetic phases of La_2CuO_4-based systems, doped or undoped, have been reported.




§ CONCLUSIONS
 

Comparison of μSR spectra in the AFM phase of La_4Ni_3O_8 with calculations of dipolar fields at candidate μ^+ sites does not determine the AFM configuration uniquely, but is consistent with the stripe structure from neutron diffraction and suggests a reduced ordered Ni^1+ moment 0.4–0.5μ_B.

The nature of the 105-K transition is not well determined. Critical slowing is not expected for a first-order transition, however, and its observation is probably the best evidence that the transition in La_4Ni_3O_8 is “abrupt but continuous” <cit.>. 

The behavior of Peaks 1 and 6 in the AFM spectra (Figs. <ref> and <ref>) is strong evidence for commensurate AFM trilayers that fluctuate dynamically at temperatures between 25 K and T_N, and freeze with uncorrelated c-axis disorder below ∼25 K.

There is no evidence for 2D critical slowing with a divergence as T → 0 in μSR data from La_4Ni_3O_8. The low-frequency spin dynamics of La_4Ni_3O_8 and La_2CuO_4+δ are not comparable, in spite of some similarities. 

More work, both theoretical and experimental, is clearly necessary to understand the phase diagram of this enigmatic material.



We are grateful for the assistance of B. Hitti and D. Arsenau of the TRIUMF Centre for Molecular and Materials Research during these experiments. One of us (D.E.M.) thanks J. Spalding for useful comments. This research was supported in part by the National Science Foundation under the Cal State LA/Penn State PREM Program, NSF Grants DMR-1523588
, DMR-1809306
, and DMR-1506677
, by the National Natural Science Foundation of China No. 11774061
, and by the University of California, Riverside, Academic Senate
.


 



§ OTHER AFM STRIPE MODELS
 

AFM configurations 
and corresponding dipolar-field histograms for stripe Models 2–4 are shown in Figs. <ref>–<ref>. The trends seen in Model -1 histograms and discussed in Sec. <ref> (low Ni moments, low- and high-frequency groups, only small changes for inverted alternate trilayers) continue to be found.







[4] 


§ INHOMOGENEITY AND THE SPONTANEOUS FREQUENCY DISTRIBUTION FOR ABRUPT TRANSITIONS
 

We consider a simple model for a system with an abrupt second-order magnetic transition, i.e., a small value of the order-parameter critical exponent β. A first-order transition can be simulated by letting β→ 0. 

The temperature dependence of a spontaneous zero-field frequency f (proportional to the order parameter) is assumed to be of the critical-exponent form

    f(T,T_c) = f_0(1 - T/T_c)^β, 0 < T < T_c  ,

where f_0 is the zero-temperature value. At a temperature T the frequency distribution function P(f,T) (the spectrum) is given by

    P(f,T) = {[ P[T_c(f,T)] dT_c(f,T)/df , f_min < f < f_max ,;                                  0 otherwise , ].

where from Eq. (<ref>) 

    T_c(f,T) = T/1 - (f/f_0)^1/β ,

and P(T_c) is the inhomogeneous distribution of transition temperatures T_c, taken to be nonzero for T_c^min < T_c < T_c^max. At a given temperature T, P(f,T) lies between the limits f_min (max)(T) = f(T,T_c^min (max)). This yields

    P(f,T) = {[ P[T_c(f,T)] T(f/f_0)^1/β/β f[1 - (f/f_0)^1/β]^2 , f_min < f < f_max ,;                                                         0 otherwise . ].


In our simple model P(T_c) is taken to be uniform with mean T_c0 and half width Δ T_c:

    P(T_c) = {[ 1/2Δ T_c , T_c^min < T_c < T_c^max ,;                        0 otherwise , ].

 with T_c^min (max) = T_c0∓Δ T_c.  

Figure <ref> shows P(T_c), f(T,T_c) for T_c =  T_c^min , T_c0 , and T_c^max, and the normalized spectral weight

    w(T) = {[ 1 - ∫_T_c^min^T P(T_c) dT_c , 
     T_c^min < T < T_c^max ,;                                           1 otherwise . ].


The integral is the fraction 1 - w(T) of the sample volume with T_c < T, for which f = 0. 

Representative spectra from Eq. (<ref>) are shown in Fig. <ref>. Their behavior is qualitatively similar to that of Peak 6 of our experimental results in La_4Ni_3O_8 (Fig. <ref>): common features include a slight frequency decrease and loss of f > 0 spectral weight (transferred to f = 0) for T > T_c^min. The remaining weight for f > 0 becomes broadly distributed, with a peak at the maximum. The shapes of the observed and calculated spectra differ, but these depend strongly on the form of the T_c distribution and would not be expected to be reproduced in this simple model. The observed peaks are broader than those of the model, probably reflecting disorder that is not taken into account in the model.











56
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Bednorz and Müller(1987)]BNobel
  author author J. G. Bednorz and author K. A. Müller, @noop  title Perovskite-Type Oxides—the
  New Approach to High-T_c Super­con­duct­ivity, howpublished The Nobel Prize Lecture (year 1987), https://arxiv.org/abs/
   NoStop
[Lacorre(1992)]Laco92
  author author P. Lacorre, title title Passage from T-type to
  T^'-type arrangement by reducing R_4Ni_3O_10 to
  R_4Ni_3O_8 (R = La, Pr, Nd), https://doi.org/https://doi.org/10.1016/0022-4596(92)90061-Y journal journal J. Solid State Chem. volume 97, pages 495  (year
  1992)NoStop
[Poltavets et al.(2007)Poltavets, Lokshin, Croft, Mandal, Egami, and Greenblatt]PLCM07
  author author V. V. Poltavets, author K. A. Lokshin, author M. Croft,
  author T. K. Mandal, author T. Egami, and author
  M. Greenblatt, title
  title Crystal Structures of Ln_4Ni_3O_8 (Ln = La, Nd)
  Triple Layer T^'-type Nickelates, https://doi.org/10.1021/ic701480v journal journal Inorg. Chem. volume 46, pages 10887 (year 2007), note pMID:
  17985874, https://arxiv.org/abs/http://dx.doi.org/10.1021/ic701480v
  http://dx.doi.org/10.1021/ic701480v NoStop
[Poltavets et al.(2010)Poltavets, Lokshin, Nevidomskyy,
  Croft, Tyson, Hadermann,
  Van Tendeloo, Egami, Kotliar,
  ApRoberts-Warren, Dioguardi, Curro, and Greenblatt]PLNC10
  author author V. V. Poltavets, author K. A. Lokshin, author A. H. Nevidomskyy, author M. Croft,
  author T. A. Tyson, author J. Hadermann, author
  G. Van Tendeloo, author
  T. Egami, author G. Kotliar, author N. ApRoberts-Warren, author A. P. Dioguardi, author N. J.  Curro, and author M. Greenblatt, title title
  Bulk Magnetic Order in a Two-Dimensional
  Ni^1+/Ni^2+ (d^9/d^8) Nickelate,
  Isoelectronic with Superconducting Cuprates, https://doi.org/10.1103/PhysRevLett.104.206403 journal
  journal Phys. Rev. Lett. volume 104,  pages 206403 (year 2010)NoStop
[Zhang et al.(2016)Zhang,
  Chen, Phelan, Zheng,
  Norman, and Mitchell]ZCPZ16
  author author J. Zhang, author Y.-S. Chen,
  author D. Phelan, author H. Zheng, author
  M. R. Norman, and author
  J. F. Mitchell, title
  title Stacked charge stripes in the quasi-2D trilayer
  nickelate La_4Ni_3O_8, https://doi.org/10.1073/pnas.1606637113 journal journal Proc. Nat. Acad. Sci. volume 113,  pages 8945 (year 2016)NoStop
[Zhang et al.(2019)Zhang, Pajerowski, Botana,
  Zheng, Harriger, Rodriguez-Rivera, Ruff, Schreiber,
  Wang, and Chen]ZPBZ19u
  author author J. Zhang, author D. M. Pajerowski, author A. S. Botana, author H. Zheng,
  author L. Harriger, author J. Rodriguez-Rivera, author J. P. C. Ruff, author
  N. J. Schreiber, author
  B. Wang, and author
  Y.-S. Chen, title
  title Spin stripe order in a square planar trilayer
  nickelate, https://arxiv.org/abs/1903.03246 arXiv:1903.03246
  [cond-mat.str-el]  (year 2019)NoStop
[Lee and Cheong(1997)]LeCh97
  author author S.-H. Lee and author S.-W. Cheong, title title Melting of
  Quasi-Two-Dimensional Charge Stripes in
  La_5/3Sr_1/3NiO_4
, https://doi.org/10.1103/PhysRevLett.79.2514 journal
  journal Phys. Rev. Lett. volume 79,  pages 2514 (year 1997)NoStop
[Brewer(1994)]Brew94
  author author J. H. Brewer, title title Muon Spin
  Rotation/Relaxation/[2]Resonance, in @noop 
  booktitle Encyclopedia of Applied Physics,  Vol. volume 11, editor edited by editor
  G. L. Trigg, editor
  E. S. Vera, and editor
  W. Greulich (publisher VCH
  Publishers, address New York, year 1994) pp.  pages 23–53NoStop
[Blundell(1999)]Blun99
  author author S. J. Blundell, title title Spin-polarized muons in
  condensed matter physics, https://doi.org/10.1080/001075199181521
  journal journal Contemp. Phys. volume 40, pages 175 (year
  1999)NoStop
[Yaouanc and Dalmas de
  Réotier(2011)]YaDdR11
  author author A. Yaouanc and author P. Dalmas de Réotier, @noop  title Muon Spin
  Rotation, Relaxation, and Resonance: Applications to Condensed Matter,  International Series of Monographs on Physics (publisher
  Oxford University Press, address New York, year
  2011)NoStop
[Chakravarty et al.(1988)Chakravarty, Halperin, and Nelson]CHN88
  author author S. Chakravarty, author B. I. Halperin, and author D. R. Nelson, title title Low-Temperature Behavior
  of Two-Dimensional Quantum Antiferromagnets, https://doi.org/10.1103/PhysRevLett.60.1057 journal journal Phys. Rev. Lett. volume 60, pages 1057 (year 1988)NoStop
[Chakravarty et al.(1989)Chakravarty, Halperin, and Nelson]CHN89
  author author S. Chakravarty, author B. I. Halperin, and author D. R. Nelson, title title Two-dimensional quantum
  Heisenberg antiferromagnet at low temperatures, @noop  journal journal Phys. Rev. B volume
  39, pages 2344 (year 1989)NoStop
[Chakravarty and Orbach(1990)]ChOr90
  author author S. Chakravarty and author R. Orbach, title title Electron and Nuclear
  Magnetic Relaxation in La_2CuO_4 and Related
  Cuprates, https://doi.org/10.1103/PhysRevLett.64.224 journal journal Phys. Rev. Lett. volume 64, pages 224 (year 1990)NoStop
[Imai et al.(1993)Imai,
  Slichter, Yoshimura, and Kosuge]ISYK93
  author author T. Imai, author C. P. Slichter,
  author K. Yoshimura, and  author K. Kosuge, title title Low Frequency Spin Dynamics in Undoped and
  Sr-Doped La_2CuO_4, https://doi.org/10.1103/PhysRevLett.70.1002 journal journal Phys. Rev. Lett. volume 70, pages 1002 (year 1993)NoStop
[Imai et al.(1994)Imai,
  Slichter, Yoshimura, Katoh,  and Kosuge]ISYK94b
  author author T. Imai, author C. P. Slichter,
  author K. Yoshimura, author M. Katoh, and author
  K. Kosuge, title title High-temperature ^63,65Cu NQR and NMR study of the
  high-temperature superconductor La_2-xSr_xCuO_4 (0 ⩽ x
  ⩽ 0.15), https://doi.org/https://doi.org/10.1016/0921-4526(94)90262-3 journal journal Physica B volume
  197, pages 601  (year 1994)NoStop
[ApRoberts-Warren et al.(2011)ApRoberts-Warren, Dioguardi, Poltavets,
  Greenblatt, Klavins, and Curro]A-WDPG11
  author author N. ApRoberts-Warren, author A. P. Dioguardi, author V. V. Poltavets, author M. Greenblatt, author P. Klavins, and author N. J. Curro, title title Critical spin dynamics in
  the antiferromagnet La_4Ni_3O_8
  from ^139La nuclear magnetic resonance, https://doi.org/10.1103/PhysRevB.83.014402 journal journal Phys. Rev. B volume 83, pages 014402 (year 2011)NoStop
[Anisimov et al.(1999)Anisimov, Bukhvalov, and Rice]ABR99
  author author V. I. Anisimov, author D. Bukhvalov, and author T. M. Rice, title title Electronic structure of
  possible nickelate analogs to the cuprates, https://doi.org/10.1103/PhysRevB.59.7901 journal journal Phys. Rev. B volume 59, pages 7901 (year 1999)NoStop
[Poltavets et al.(2006)Poltavets, Lokshin, Dikmen, Croft, Egami, and Greenblatt]PLDC06
  author author V. V. Poltavets, author K. A. Lokshin, author S. Dikmen,
  author M. Croft, author T. Egami, and author
  M. Greenblatt, title
  title La_3Ni_2O_6: A New Double T^'-type Nickelate
  with Infinite Ni^1+/2+O_2 Layers, https://doi.org/10.1021/ja063031o journal journal J. Am. Chem. Soc. volume 128, pages 9050 (year 2006)NoStop
[Pardo and Pickett(2010)]PaPi10
  author author V. Pardo and author W. E. Pickett, title title Quantum Confinement
  Induced Molecular Correlated Insulating State in
  La_4Ni_3O_8, https://doi.org/10.1103/PhysRevLett.105.266402 journal
  journal Phys. Rev. Lett. volume 105,  pages 266402 (year 2010)NoStop
[Pardo and Pickett(2012)]PaPi12
  author author V. Pardo and author W. E. Pickett, title title Pressure-induced
  metal-insulator and spin-state transition in low-valence layered
  nickelates, https://doi.org/10.1103/PhysRevB.85.045111 journal journal Phys. Rev. B volume
  85, pages 045111 (year 2012)NoStop
[Sarkar et al.(2011)Sarkar,
  Dasgupta, Greenblatt, and Saha-Dasgupta]SDGS-D11
  author author S. Sarkar, author I. Dasgupta,
  author M. Greenblatt, and  author T. Saha-Dasgupta,  title title Electronic and magnetic structures of
  bilayer La_3Ni_2O_6 and trilayer
  La_4Ni_3O_8 nickelates from first principles, https://doi.org/10.1103/PhysRevB.84.180411 journal journal Phys. Rev. B volume 84, pages 180411 (year 2011)NoStop
[Cheng et al.(2012)Cheng,
  Zhou, Goodenough, Zhou,
  Matsubayashi, Uwatoko, Kong,
  Jin, Yang, and Shen]CZGZ12
  author author J.-G. Cheng, author J.-S. Zhou,
  author J. B. Goodenough,
  author H. D. Zhou, author K. Matsubayashi, author
  Y. Uwatoko, author P. P. Kong, author C. Q.  Jin, author W. G. Yang, and author G. Y. Shen, title title Pressure Effect on the
  Structural Transition and Suppression of the High-Spin State in the
  Triple-Layer
  T^-La_4Ni_3
O_8, https://doi.org/10.1103/PhysRevLett.108.236403
  journal journal Phys. Rev. Lett. volume 108, pages 236403 (year
  2012)NoStop
[Liu et al.(2012)Liu,
  Zhang, Zhang, Jia,
  Zeng, and Lin]LZZJ12
  author author T. Liu, author G. Zhang, author X. Zhang, author
  T. Jia, author Z. Zeng, and author H. Q.  Lin, title title Electronic
  structure and magnetism of La_4Ni_3O_8 from first principles,  http://stacks.iop.org/0953-8984/24/i=40/a=405502 journal journal J. Phys. Condens. Matter volume 24, pages 405502 (year
  2012)NoStop
[Wu(2013)]Wu13
  author author H. Wu, title title Charge-spin-orbital states in
  the tri-layered nickelate La_4Ni_3O_8: an ab initio study, http://stacks.iop.org/1367-2630/15/i=2/a=023038 journal
  journal New J. Phys. volume 15,  pages 023038 (year 2013)NoStop
[Patra et al.(2016)Patra,
  Ashwin Kishore, Vidya, Sjåstad, Fjellvåg, and Ravindran]PAKVS16
  author author L. Patra, author M. Ashwin Kishore, author R. Vidya, author A. O. Sjåstad, author H. Fjellvåg, and author P. Ravindran, title title Electronic and magnetic
  structures of hole doped trilayer La_4-xSr_xNi_3O_8 from
  first-principles calculations, https://doi.org/10.1021/acs.inorgchem.6b01938 journal
  journal Inorg. Chem. volume 55,  pages 11898 (year 2016)NoStop
[Liu et al.(2014)Liu,
  Wu, Jia, Zhang, Zeng, Lin, and Li]LWJZ14
  author author T. Liu, author H. Wu, author T. Jia, author
  X. Zhang, author Z. Zeng, author H. Q.  Lin, and author X. G.  Li, title title
  Dimensionality-induced insulator-metal crossover in layered nickelates
  La_n+1Ni_nO_2n+2 (n = 2, 3, and ∞), https://doi.org/10.1063/1.4873537 journal journal AIP Adv. volume 4, pages
  047132 (year 2014)NoStop
[ApRoberts-Warren et al.(2013)ApRoberts-Warren, Crocker, Dioguardi,
  Shirer, Poltavets, Greenblatt, Klavins, and Curro]A-WCDS13
  author author N. ApRoberts-Warren, author J. Crocker, author A. P. Dioguardi, author K. R. Shirer, author V. V. Poltavets, author M. Greenblatt, author P. Klavins, and author N. J. Curro, title title NMR evidence for spin
  fluctuations in the bilayer nickelate La_3Ni_2O_6,  https://doi.org/10.1103/PhysRevB.88.075124 journal
  journal Phys. Rev. B volume 88,  pages 075124 (year 2013)NoStop
[Botana et al.(2016)Botana,
  Pardo, Pickett, and Norman]BPPN16
  author author A. S. Botana, author V. Pardo,
  author W. E. Pickett, and  author M. R. Norman, title title Charge ordering in
  Ni^1+/Ni^2+ nickelates:
  La_4Ni_3O_8 and
  La_3Ni_2O_6, https://doi.org/10.1103/PhysRevB.94.081105 journal journal Phys. Rev. B volume 94, pages 081105 (year 2016)NoStop
[Note1()]Note1
  note Negative muons go into tight Bohr orbits around host nuclei
  and are less sensitive to their magnetic environment than interstitial
  positive muons.Stop
[Suter and Wojek(2012)]SuWo12
  author author A. Suter and author B. Wojek,  title title : A Free
  Platform-Inde­pendent Framework for μSR Data Analysis, http://www.sciencedirect.com/science/article/pii/S187538921201228X
  journal journal Phys. Procedia volume 30, pages 69  (year 2012),  note 12th International Conference on Muon Spin Rotation,
  Relaxation and Resonance (μSR2011)NoStop
[Note2()]Note2
  note NoStop
[Bonfà and De Renzi(2016)]BoDR16
  author author P. Bonfà and author R. De Renzi, title title Toward the
  Computational Prediction of Muon Sites and Interaction Parameters, https://doi.org/10.7566/JPSJ.85.091014 journal journal J. Phys. Soc. Jpn. volume 85, pages 091014 (year 2016)NoStop
[Note3()]Note3
  note For ZF measurements in a random powder with static local
  fields, 1/3 of the initial μ ^+ polarization relaxes with zero
  frequency <cit.>. A ω = 0 component is also
  present in an antiferromagnet if a muon site is symmetric with respect to the
  sublattices, since this results in cancellation of local fields.Stop
[Note4()]Note4
  note Due to the general Fourier-transform proportionality between
  time-domain amplitudes and frequency-domain areas, increased broadening
  causes a loss of frequency-domain amplitude.Stop
[Kubo and Toyabe(1967)]KuTo67
  author author R. Kubo and author T. Toyabe,  title title A stochastic model for low field
  resonance and relaxation, in @noop  booktitle
  Magnetic Resonance and Relaxation, editor edited by  editor R. Blinc (publisher North-Holland, address Amsterdam, year 1967) pp. pages 810–823NoStop
[Hayano et al.(1979)Hayano,
  Uemura, Imazato, Nishida,
  Yamazaki, and Kubo]HUIN79
  author author R. S. Hayano, author Y. J. Uemura,
  author J. Imazato, author N. Nishida, author
  T. Yamazaki, and author
  R. Kubo, title title Zero- and low-field spin relaxation studied by positive muons,  https://doi.org/10.1103/PhysRevB.20.850 journal
  journal Phys. Rev. B volume 20,  pages 850 (year 1979)NoStop
[Note5()]Note5
  note For large enough H_L (≫ the local-field distribution
  width) the resultant field is parallel to the μ ^+ spin polarization.
  Then there is no precession and thus no static relaxation, so that any
  observed relaxation is dynamic.Stop
[Uemura et al.(1981)Uemura,
  Nishiyama, Yamazaki, and Nakai]UNYN81
  author author Y. J. Uemura, author K. Nishiyama,
  author T. Yamazaki, and  author R. Nakai, title title Muon spin relaxation in a spin glass CuMn
  observed in finite longitudinal magnetic fields, https://doi.org/10.1016/0038-1098(81)90640-2 journal
  journal Solid State Commun. volume
  39, pages 461  (year 1981)NoStop
[Uemura et al.(1985)Uemura,
  Yamazaki, Harshman, Senba,  and Ansaldo]UYHS85
  author author Y. J. Uemura, author T. Yamazaki,
  author D. R. Harshman, author M. Senba, and author
  E. J. Ansaldo, title
  title Muon-spin relaxation in AuFe and CuMn spin
  glasses, https://doi.org/10.1103/PhysRevB.31.546 journal journal Phys. Rev. B volume
  31, pages 546 (year 1985)NoStop
[Noakes and Kalvius(1997)]NoKa97
  author author D. R. Noakes and author G. M. Kalvius, title title Anomalous zero-field muon
  spin relaxation in highly disordered magnets, https://doi.org/10.1103/PhysRevB.56.2352 journal journal Phys. Rev. B volume 56, pages 2352 (year 1997)NoStop
[Keren et al.(1996)Keren,
  Mendels, Campbell, and Lord]KMCL96
  author author A. Keren, author P. Mendels,
  author I. A. Campbell, and  author J. Lord, title title Probing the Spin-Dynamical Autocorrelation
  Function in a Spin Glass Above T_g Via Muon Spin Relaxation,  @noop  journal journal Phys. Rev. Lett.  volume 77, pages 1386 (year
  1996)NoStop
[Johnston(2006)]John06
  author author D. C. Johnston, title title Stretched exponential
  relaxation arising from a continuous sum of exponential decays, https://doi.org/10.1103/PhysRevB.74.184430 journal journal Phys. Rev. B volume 74, pages 184430 (year 2006)NoStop
[Tse and Hartmann(1968)]TsHa68
  author author D. Tse and author S. R. Hartmann, title title Nuclear spin-lattice
  relaxation via paramagnetic centers without spin diffusion, https://doi.org/10.1103/PhysRevLett.21.511 journal journal Phys. Rev. Lett. volume 21, pages 511 (year 1968)NoStop
[Moriya and Ueda(1974)]MoUe74
  author author T. Moriya and author K. Ueda,  title title Nuclear magnetic relaxation in weakly
  ferro- and antiferromagnetic metals, @noop  journal
  journal Solid State Commun. volume
  15, pages 169 (year 1974)NoStop
[Millis et al.(1990)Millis,
  Monien, and Pines]MMP90
  author author A. J. Millis, author H. Monien,  and author D. Pines, title title Phenomenological model of nuclear relaxation in
  the normal state of
  YBa_2Cu_3O_7, https://doi.org/10.1103/PhysRevB.42.167 journal journal Phys. Rev. B volume 42, pages 167 (year 1990)NoStop
[MacLaughlin et al.(1994)MacLaughlin, Vithayathil, Brom,
  de Rooy, Hammel, Canfield,
  Reyes, Fisk, Thompson, and  Cheong]MVBdR94
  author author D. E. MacLaughlin, author J. P. Vithayathil, author H. B. Brom, author J. C. J. M. de Rooy, author P. C. Hammel,
  author P. C. Canfield, author A. P. Reyes, author
  Z. Fisk, author J. D. Thompson, and author S.-W. Cheong, title title
  Abrupt but Continuous Antiferromagnetic Phase Transition in Nearly
  Stoichiometric La_2CuO_4+δ, https://doi.org/10.1103/PhysRevLett.72.760 journal journal Phys. Rev. Lett. volume 72, pages 760 (year 1994)NoStop
[Imbert et al.(1993)Imbert,
  Jéhanno, and Hodges]IJH93
  author author P. Imbert, author G. Jéhanno,  and author J. A. Hodges,  title title Comment on “Magnetization and
  Spin-Wave Velocities in La_2CuO_4”, https://doi.org/10.1103/PhysRevLett.71.654 journal journal Phys. Rev. Lett. volume 71, pages 654 (year 1993)NoStop
[Chow et al.(1996)Chow,
  Pattenden, Blundell, Hayes,
  Pratt, Jestädt, Green,
  Millburn, Rosseinsky, Hitti,
  Dunsiger, Kiefl, Chen, and  Chowdhury]CBPH96
  author author K. H. Chow, author P. A. Pattenden,
  author S. J. Blundell, author W. Hayes, author
  F. L. Pratt, author
  T. Jestädt, author
  M. A. Green, author
  J. E. Millburn, author
  M. J. Rosseinsky, author
  B. Hitti, author S. R. Dunsiger, author R. F.  Kiefl, author C. Chen, and author A. J. S. Chowdhury, title title Muon-spin-relaxation
  studies of magnetic order in heavily doped
  La_2xSr_xNiO_4
+, https://doi.org/10.1103/PhysRevB.53.R14725
  journal journal Phys. Rev. B volume 53, pages R14725 (year
  1996)NoStop
[Jestädt et al.(1999)Jestädt, Chow, Blundell, Hayes, Pratt, Lovett, Green, Millburn, and Rosseinsky]JCBH99
  author author T. Jestädt, author K. H. Chow,
  author S. J. Blundell, author W. Hayes, author
  F. L. Pratt, author
  B. W. Lovett, author
  M. A. Green, author
  J. E. Millburn, and author
  M. J. Rosseinsky, title
  title Temperature and doping-level dependence of magnetic order
  in
  La_2xSr_xNiO_4
+ studied by muon spin rotation, https://doi.org/10.1103/PhysRevB.59.3775 journal journal Phys. Rev. B volume 59, pages 3775 (year 1999)NoStop
[Klauss(2004)]Klau04
  author author H.-H. Klauss, title title Spin stripe order and
  superconductivity in layered transition metal oxides, https://doi.org/10.1088/0953-8984/16/40/004 journal journal J. Phys.: Condens. Matter volume 16,  pages S4457 (year 2004)NoStop
[Borodin et al.(1990)Borodin, Doroshev, Ivanchenko,
  Savosta, and Filippov]BDIS90
  author author V. A. Borodin, author V. D. Doroshev, author Y. M. Ivanchenko, author M. M. Savosta, and author A. E. Filippov, title title La_2CuO_4 in the
  fluctuation region of the magnetic transition, @noop  journal journal Pis'ma Zh. Eksp. Teor. Fiz. volume 52, pages 1073 (year
  1990), note [JETP Lett. 52, 469–475
  (1990)].Stop
[Uemura et al.(1987)Uemura,
  Kossler, Yu, Kempton,
  Schone, Opie, Stronach,
  Johnston, Alvarez, and Goshorn]UKYK87b
  author author Y. J. Uemura, author W. J. Kossler,
  author X. H. Yu, author J. R. Kempton, author
  H. E. Schone, author
  D. Opie, author C. E. Stronach, author D. C.  Johnston, author M. S.  Alvarez, and author D. P.  Goshorn, title title
  Antiferromagnetism of
  La_2CuO_4y
  Studied by Muon-Spin Rotation, https://doi.org/10.1103/PhysRevLett.59.1045 journal journal Phys. Rev. Lett. volume 59, pages 1045 (year 1987)NoStop
[Tang et al.(1990)Tang,
  Xiao, Singh, Tešanović,
  Chien, and Walker]TXST90
  author author H. Tang, author G. Xiao, author A. Singh, author
  Z. Tešanović, author
  C. L. Chien, and author
  J. C. Walker, title
  title Magnetic dynamics of La_2CuO_4 studied by Mssbauer
  spectroscopy, https://doi.org/10.1063/1.344855 journal journal J. Appl. Phys. volume
  67, pages 4518 (year 1990)NoStop
[Imai et al.(2017)Imai,
  Takahashi, Arsenault, Acton,
  Lee, He, Lee, and  Fujita]ITAA17
  author author T. Imai, author S. K. Takahashi,
  author A. Arsenault, author A. W. Acton, author
  D. Lee, author W. He, author Y. S.  Lee, and author M. Fujita, title title Revisiting
  ^63Cu NMR evidence for charge order in superconducting
  La_1.885Sr_0.115CuO_4, https://doi.org/10.1103/PhysRevB.96.224508 journal journal Phys. Rev. B volume 96, pages 224508 (year 2017)NoStop
[Arsenault et al.(2018)Arsenault, Takahashi, Imai, He, Lee, and Fujita]ATIH17
  author author A. Arsenault, author S. K. Takahashi, author T. Imai,
  author W. He, author
  Y. S. Lee, and author
  M. Fujita, title title ^139La NMR investigation of the charge and spin order
  in a La_1.885Sr_0.115CuO_4 single
  crystal, https://doi.org/10.1103/PhysRevB.97.064511 journal journal Phys. Rev. B volume
  97, pages 064511 (year 2018)NoStop
[Imai and Lee(2018)]ImLe18
  author author T. Imai and author Y. S. Lee,  title title ^139La and
  ^63Cu NMR investigation of charge order in
  La_2CuO_4+y (T_c=42 K), https://doi.org/10.1103/PhysRevB.97.104506 journal journal Phys. Rev. B volume 97, pages 104506 (year 2018)NoStop



