Nanoscale Phase Separation of Incommensurate and Quasi-Commensurate Spin Stripes in Low Temperature Spin Glass of La_2−xSr_xNiO₄

Abstract

While spin striped phases in La_2−xSr_xNiO_4+y for 0.25 < x < 0.33 are the archetypal case of a 1D spin density wave (SDW) phase in doped antiferromagnetic strongly correlated perovskites, few information is available on the SDW spatial organization. In this context, we have measured the spatial variation of the wave vector of the SDW reflection profile by scanning micro X-ray diffractions with a coherent beam. We obtained evidence of a SDW order–disorder transition by lowering a high temperature phase (T > 50 K) to a low temperature phase (T < 50 K). We have identified quasi-commensurate spin stripe puddles in the ordered phase at 50 < T < 70 K, while the low temperature spin glassy phase presents a nanoscale phase separation of T = 30 K, with the coexistence of quasi-commensurate and incommensurate spin stripe puddles assigned to the interplay of quantum frustration and strong electronic correlations.

1. Introduction

Commensurate–incommensurate (C–IC) transitions [1] and the related complex multiscale distribution of dislocations with the formation of striped phases are of high interest in the quantum complex matter research field. Recently, this phenomenon has been observed in two-dimensional atomic layers with strongly correlated electronic matter forming stacks held together by a week Van der Waals force. For example, stripes appear in Ca-Intercalated bilayer graphene, induced by commensurate lattice matching [2] in graphene on hexagonal boron nitride [3] and in the monolayers on a honeycomb substrate [4].

The commensurate (C) phase in a crystalline lattice occurs when the ratio between a superlattice oscillation period (λ = 1/ε) (where ε is the wave vector) and the number of lattice units is a rational number. On the contrary, the incommensurate (IC) phase occurs if this ratio is an irrational number. In the C phase, the wave modulation is locked or pinned to the periodicity of the crystal while, in the incommensurate state, the wave is unlocked with the crystal periodicity. In this case the wave can be easily unpinned by a weak perturbation and dislocations, as bosonic strings can wander and sweep an increasingly greater volume at low temperatures driven by quantum fluctuations. As a consequence, a crystalline phase with real space ordering in a single state in the k-space can become disordered at lower temperatures.

Between any two commensurate phases, there is a sequence of quasi-commensurate (QC) locked-in and incommensurate (IC) phases forming a complex landscape called the Devil’s Staircase [1]. It has been observed in several systems, namely the Kr monolayer on graphite [5]; atomic Pb layer on Si(111) surface [6]; charge density wave (CDW) and spin density wave (SDW) [7]; ferroelectric smectic phases in liquid crystalline binary mixtures [8]; ferroelectrics [9]; atomic concentration waves in alloys [10]; periodic atomic displacements in NaV₂O₅ [11]; colloidal particles on a periodic substrate [12]; and field-induced SDW states in anisotropic two-dimensional conductors [13].

Commensurate or incommensurate stripes modulations can be formed by doping two dimensional (2D) strongly correlated magnetic quantum materials. Indeed, doping introduces charged solitons that can be viewed as a generalized Wigner charge density wave, as proposed by Hubbard [14], or as lines of domain walls in doped nickelates, which are the archetypal case of a multi orbital Hubbard model in a strained 2D atomic lattice [15,16,17,18,19]. Charge–spin stripes modulations are accompanied by orbital modulations and lattice distortions, like those that occur in cuprates [20,21,22,23,24,25], providing the archetypal case of quantum complex matter composed from a nanoscale phase separation [26,27,28,29]. Moreover, evidence now exists that proves this nanoscale phase separation is controlled by (i) the misfit strain between atomic layers in hetero-structures at the atomic limit [30,31,32,33], which is great in La_2−xNiO₄, giving a compressed [NiO₂] plane and an expanded [La₂O₂] layer; and (ii) by the proximity of the chemical potential to a Lifshitz transition [30,34,35] in multiband multiorbital Hubbard systems.

The layered La_2−xSr_xNiO_4+y nickelate is an archetypal doped Mott insulator exhibiting the universal features of the inhomogeneous spatial distribution of spin stripes phases in complex quantum materials [36,37,38], as lightly doped La_2−xSr_xCuO₄ [39,40], and photo-induced stripes in La₂CuO_4+y [41]. The nanoscale phase separation [42,43] can be generated by oxygen interstitial ordering [24,25,44,45,46,47], CDW [24,25] strain driven superstripes in cuprates [21,22,31,32,33], in iron-based superconductors [48,49], and diborides [50,51,52,53].

We have used scanning micro X-ray diffraction with a high coherent flux of X-ray, to visualize the spin stripes phase separation at different temperatures in La_1−xSr_xNiO₄. Nowadays, scanning micro X-ray diffraction is recognized as a general and powerful tool to investigate the emergence of complex geometry in several fields, such as complex biological systems [54,55,56,57,58,59,60], using time-resolved and space-resolved synchrotron radiation techniques and space–time dependent photo-induced structure evolution [61]. In this paper, we report the study of the SDW order–disorder transition in La_1−xSr_xNiO₄, which was observed as transitioning from the spin ordered phase at 50 < T < 70 K to the low temperature disordered phase at T = 30 K [62]. The formation of a glassy electronic phase at a low temperature was observed in other nickelates and related materials [63,64,65,66,67,68]. Moreover, it was found that the spin stripe phase in La_1−xSr_xNiO₄ below 120 K presented a strong spatially inhomogeneous landscape, with the formations of the spin stripe rich regions and spin stripes poor regions [43] providing evidence of nanoscale electronic phase separation [69,70] observed in several strongly correlate systems [71,72,73].

The formation of the inhomogeneous stripes landscape has been assigned to long-range interactions in combination with the confinement to periodic potentials, which generate a proliferation of metastable low-energy states, resulting in Devil’s Staircase-type structures [74,75,76,77]. Quantum frustration becomes relevant at low temperatures [78,79], in which stripe behavior called the superstripes phase can be generated, while preserving super fluidity [80,81,82] or superconductivity [83,84,85] or favoring competing glassy spin stripes landscape realizations. In the present study, we have identified a nanoscale phase separation at T = 30 K between quasi-commensurate and incommensurate glassy spin stripes, assigned to the interplay of quantum frustration and strong electronic correlations.

2. Results

2.1. Iincommensurate and Quasi-Commensurate Spin Stripes in La_2−xSr_xNiO₄

The temperature–doping phase diagram of La_2−xSr_xNiO₄ [36] presents the antiferromagnetic order (AFM) in the lower Sr-doping region with x < 0.2. In the higher Sr-doping region with x > 0.2, charge and spin stripes appear, extending diagonally to the Ni–O bond directions. For samples with tetragonal symmetry, such as the one analyzed here, the stripes order itself breaks the rotational symmetry of the ab-plane and, therefore, stripes with 2 different orientations show up, related by a 90-degree rotation around the c-axis. It has been found that the incommensurate superlattices of spin stripes occur at wave vectors (1−ε, 0, 0), where ε is a temperature dependent incommensurability value. A strong enhancement of the intensity peak can be obtained by resonant soft X-ray diffraction at the Ni 2p→3d L3-edge (852 eV), as reported in [62]. The temperature evolution of the SDW profile projected on the H and L directions is summarized in Figure 1a. The SDW rises for temperatures below T_SDW = 120 K. On further cooling, its intensity increases, reaching a maximum value between 50 and 70 K. Subsequently, quite a dramatic intensity reduction is reached when the temperature is decreased further to approximately 30 K (light blue area). Moreover, the peak position changes with the temperature. The commensurate periods 1/ε = (nx + my)/(n + m) intermediate between two main commensurate wave vectors (ε₁ = 1/n) and (ε₂ = 1/m) with periods of m and n lattice units, respectively, occuring at integer numbers of lattice unit cells η = (n + m)/ε. Each step corresponds to a modulation wave with a number of (m + n) periods λ = 1/ε locked with the underlying lattice onto a rational number. The temperature evolution of η value, named here as the commensurability value, indicates that the system tends to reach the quasi-commensurate state, namely QC, at η = 24 when the intensity reaches its maximum value of approximately T = 50 K. As the system moves to lower and higher temperatures, η moves away from the QC state, and the SDW puddles density presents an anomalous decrease.

A scheme of the superlattice of spin ordering is presented in Figure 1b. The typical SDW peak generated by the spin ordering, with cuts along the tH and L directions of he reciprocal space, is shown in Figure 1c. Fitting profiles with a Lorentzian lineshape, provided us with the in-plane correlation length of approximately 20 nm (36 unit cells) and the out-of-plane correlation length of 2 nm (2 unit cells), which is significantly smaller, reflecting the quasi two-dimensionality of the stripes phase [35]. For the present investigation, we used a sample with the following doping level: x = 0.29. The incommensurate SDW, with ε = 0.29 ordering, consists of the mixture of the ε = 1/4 stripe order and the ε = 1/3 spin order, i.e., alternating n = 3 stripes portions and m = 4 stripes portions as schematized in Figure 1d. The average modulation period is given by 1/ε = (3x + 4y)/(3 + 4), where x + y = 3 + 4. Therefore, η = 7/ε = 3x + 4y varies in the range 21 < η < 28 and, at each integer value of η, the system reaches a quasi-commensurate phase at the step of a Devil’s Staircase. The x = 0.29 doping is close to the staircase step of the quasi-commensurate spin modulation with 7/ε = 24 i.e., (x = 3, y = 4) 3 × 4+4 × 3 = 12 + 12 = 24 and, by changing the temperature and position, it fluctuates in the range 23 < η < 24.

2.2. Maps of Incommensurate and Quasi-Commensurate Spin Stripes in La_1.71Sr_0.29NiO₄

In order to shed light on the counterintuitive behavior of SDW disordering at low temperatures, we have investigated the spatial evolution of the SDW in real space. We used scanning micro X-ray diffraction (SµXRD), probing the local stripes order via the mapping of the superlattice peak at (1−ε, 0, 0) with ε = 0.29. We performed our measurements on the P10 beamline of the Petra III synchrotron. The key result obtained from this experiment has been the mapping of the spatial distribution of the (1−ε, 0, 0) stripes peak incommensurability over different areas of 40 × 80 μm² in steps of 1 μm in both directions, resulting in 3321 diffraction images.

The variability of the incommensurability, η, has already be observed as a function of temperature (Figure 1a). In Figure 2a, we present the maps of η at T = 30 K, 50 K, and 70 K. In Figure 2b, we show the probability density function of η and its multiple value η* = 5η, indicating a higher degree of commensurability. Focusing on the maps measured at the temperature T = 50 K, we can observe that microscale puddles with a narrow distribution occur in the range 23.9 < η < 24 (119.5 < η* < 120), which is associated with the QC long range SDW phase with the η = 24 (η* = 120) phase. By increasing the temperature to T = 70 K, we can observe that the average value of η remains at the QC phase η = 24 (η* = 120), but we can also observe the increasing number of incommensurate micro puddles with 24.0 < η < 24.2 (120 < η* < 121) induced by thermal noise. In the phase probing the spin glass at T = 30 K, which is characterized by the small coherence length of the SDW reflection [62], we can observe a shorter average SDW incommensurability η = 23.85. This is due to the appearance of a relevant number of spots with incommensurate values 23.6 < η < 23.8 (118 < η* < 119). Thus, at low temperatures, T < 50 K, we observe the coexistence of a double set of spots: the first set with the quasi-commensurate η* = 120 spots, and the second set of spots associated with an incommensurate SDW phase, namely IC.

2.3. Low Temperature Phase Separation between Incommensurate and Quasi-Commensurate Spin Stripes in La_1.71Sr_0.29NiO₄

The tendency of the system to assume a quasi-commensurate SDW conformation, when the intensity reaches a maximum at 50 K, is illustrated by plotting the SDW intensity as a function of η (η*), in Figure 3. Here, we can observe that the η distribution in the quasi-commensurate QC phase at T = 50 K is centered around the value of η = 24 and drops as soon as η overcomes this value.

The QC value of η is maintained at T = 70 K, even if the distribution appears larger due to the thermal noise. On the other hand, the incommensurability values decrease to approximately η = 23.6 in the IC phase at T = 30 K.

We can observe a continuous spread of the observed superlattice periods that broadens upon cooling below 50 K. At these low temperatures, a phase separation between the incommensurate and quasi-commensurate spin stripes phase occurs, as summarized in Figure 4. We indicate, here, the quasi-commensurate (QC) phase with η = 24 (η* = 120) and the incommensurate phase, at around η = 23.75 (η* = 118), with orange and blue light triangles, respectively. These phases correspond to slightly different x doping values, which can be estimated by fitting the experimental data of η (η*) as a function of Sr content, x [36]. These results indicate that the crystalline state at a low temperature is not a perfect periodic lattice but an intrinsic incommensurate lattice with expected dynamical lattice dislocations.

3. Discussion

We have investigated the spatial distribution of spin stripes order in La_1.71Sr_0.29NiO₄. The results obtained regarding the temperature dependence of SDW intensity are similar to those reported for the nickelates of different doping levels. Thus, we can assume the observed behavior to be a general characteristic of stripe order. This behavior resembles what has been predicted for incommensurate cuprate stripes at low temperatures, when the freezing of the lattice potentially disturbs the long-range order. The temperature where the stripes peak intensity is maximum coincides with a reorganization of more isotropic and quasi-commensurate SDW puddles at T = T* = 50 K. By decreasing the temperature, T < T*, the SDW shows a nanoscale phase separation with the coexistence of quasi-commensurate (QC) and incommensurate (IC) SDW puddles. These results shed light on the intrinsic inhomogeneous spatial organization of magnetic stripes forming domains organized in micrometric puddles, and their different spatial arrangement transitioning from the so-called ordered high temperature phase (T > 50 K) to the so-called disordered glassy SDW phase (T < 50 K). This inhomogeneous aspect could be a key issue for theoretically understanding the basic mechanisms for the formation of superstripes on particular lattice geometries in the regime of strong interactions.

The inhomogeneous spatial distribution of the spin ordering wave vector can be related to the spatial distribution of chemical doping in [La_2−xSr_xO_2+y] spacer layers, which are intercalated between [CuO₂] planes. Therefore, doping can either be controlled by the x substitutions of Sr for La, or by the additional oxygen interstitials. Indeed, it has been shown that in HgBa₂CuO_4+y [25], the oxygen interstitial ions form atomic stripes in the spacer layer [HgO_yBa₂O2] between [CuO₂] planes, which are anticorrelated with the spatial distribution of CDW order. Moreover, in the isostructural cuprate La₂CuO_4+y [24], in which oxygen interstitials form large La₂CuO_4.12 clusters, the CDW order appears in the poorly oxygen-doped domains anticorrelated with oxygen-interstitials-rich domains. In nickelates, the spatial distribution of dopants and spin/charge ordering is an open issue to be addressed by using new experiments with space resolved probes [86,87,88].

Our results show a disordered landscape of the spin stripes made of large striped puddles of about 20 nm, which are significantly larger than the CDW puddles observed in the cuprates [25]. The disordered striped landscape in manganites and cobaltites [89] has been described by the Anisotropic Random Field Ising Model (ARFIM), which predicts that any small amount of random field could stabilize long-range order, leading to a phase diagram with large critical regions corresponding to short-range quasi-2D spin/charge order. Furthermore, the ARFIM description has been applied to the nematic electronic phases in orthorhombic cuprates [90].

Finally, we observe that the spin stripes scenario in nickelates is similar to the spin stripes in isostructural multiband cobaltites. Indeed, the scattering pattern of the spin stripes shown in Figure 1c is similar to the spin order scattering found in cobaltites [91], which also presents nanoscale phase separation [42]. In these two perovskite families, the stripes are the result of the polaron ordering driven by lattice elastic interactions, as it has been proposed for the stripe formation in cuprates using X-ray spectroscopy [21,30].

4. Materials and Methods

Single-crystalline La_1.71Sr_0.29NiO₄ was grown via the floating zone technique. The seed and feed rods were prepared from polycrystalline powder obtained by the solid state reaction of La₂O₃, SrCO₃, and NiO with an excess of NiO. The reaction was performed at 1200 °C for 20 h with intermediate grinding. The rods were densified at 1500 °C for 5 h. All of the syntheses were carried out in air. The scanning micro X-ray diffraction μXRD experiments were carried out at the Coherence Beamline P10 of PETRA III synchrotron in Hamburg. The synchrotron radiation source was a 5 m long undulator (U29). The X-ray beam was monochromatized by a cooled Si(111) double-crystal monochromator with a bandwidth of ΔE/E ∼ 1.4 × 10⁻⁴ in the 8 KeV photon energy range. The collimated coherent X-ray beam was focused, using the beryllium refractive lens (CRL) transfocator, to a size of about 1 μm² on the sample positioned 1.6 m downstream of the transfocator center. The incident flux on the sample was about 10¹¹ photons/s. The exit window of the heating chamber and He cryostat, as well as the entrance window of the detector flight path, was covered by a 25 μm thick Kapton sheet. The scattered signal was detected using a large horizontal scattering setup with a sample-to-detector distance of 5 m with an evacuated flight path. A PILATUS 300 K detector was used to record the X-rays scattered by the sample. For the measurements, the sample was cooled to the lowest temperature (30 K) and the measurements were performed in a heating cycle. We aligned the crystal to detect the spin ordering satellite of the (100) reflection, which appeared only below the spin ordering temperature, Ts_DW = 120 K. The spin ordering satellite appeared at q_h = 0.71, where the incommensurate stripes wave vector was ε = |1 − q_h| = 0.29. The sample scanning was obtained by translating the sample over different areas of about 40 × 80 μm² in steps of 1 μm in both directions, resulting in more the 3000 diffraction images. Data analysis was performed by using customized and homemade written MATLAB routines [61].