Intertwined Magnetic Sub-Lattices in the Double Perovskite Compound LaSrNiReO6

We report a muon spin rotation ($\mu^{+}$SR) study of the magnetic properties of the double perovskite compound LaSrNiReO$_{6}$. Using the unique length and time scales of the $\mu^{+}$SR technique, we successfully clarify the magnetic ground state of LaSrNiReO$_{6}$, which was previously deemed as a spin glass state. Instead, our $\mu^{+}$SR results point towards a long-range dynamically ordered ground state below $T_{\rm C}= 23$ K, for which a static limit is foreseen at $T=0$. Furthermore, between 23 K$<T\leq$300 K, three different magnetic phases are identified: a dense ($23$ K$<T\leq75$ K), a dilute ($75$ K$<T\leq250$ K), and a paramagnetic ($T>250$ K) state. Our results reveal how two separate, yet intertwined magnetic lattices interact within the unique double perovskite structure and the importance of using complementary experimental techniques to obtain a complete understanding of the microscopic magnetic properties of complex materials.

We report a muon spin rotation (µ + SR) study of the magnetic properties of the double perovskite compound LaSrNiReO6. Using the unique length and time scales of the µ + SR technique, we successfully clarify the magnetic ground state of LaSrNiReO6, which was previously deemed as a spin glass state. Instead, our µ + SR results point towards a long-range dynamically ordered ground state below TC = 23 K, for which a static limit is foreseen at T = 0. Furthermore, between 23 K< T ≤300 K, three different magnetic phases are identified: a dense (23 K< T ≤ 75 K), a dilute (75 K< T ≤ 250 K), and a paramagnetic (T > 250 K) state. Our results reveal how two separate, yet intertwined magnetic lattices interact within the unique double perovskite structure and the importance of using complementary experimental techniques to obtain a complete understanding of the microscopic magnetic properties of complex materials.

I. Introduction
Materials with perovskite crystal structure have for several decades been in the centre of attention across a wide scientific scope [1,2]. These compounds exhibit many interesting physical properties, such as various magnetic orders [3] and/or electronic states-metallic [4,5], insulator [6], and superconductivity [7]. Moreover, some perovskites also display multiferroicity [8][9][10][11], which is an area that during recent years has received an increasing attention due to both fundamental interests as well as applications in sensors, actuators and memory devices [12].
The smörgåsbord of properties for the perovskites originate from its ABX 3 -type crystal structure, where A and B are cations and X an anion that bond with B to form BX 6 octahedra. The significance of the crystal structure is that the octahedra is flexible and can contract/expand/distort to accommodate almost all elements in the periodic table [13]. A noteworthy and most common subgroup of perovskites is the oxide perovskite, which is simply achieved for systems where the X anion is an oxygen ion.
Lately, a new type of perovsikte has raised the interest of both experimental and theoretical physicists, the so-called double perovskite [14]. In this case, half of the B cations is substituted with another cation forming a * okfo@kth.se † yasmine.sassa@chalmers.se A 2 BB'O 6 structure. The B cation in these system may order, where the most common pattern being a rock-salt type (like NaCl, but column or layered also exits), consisting of corner shared BO 6 and B'O 6 octahedra. [15,16] The degrees of freedom for designing these material has opened up a wide door from application point of view. Lately, diverse combinations of compounds [17][18][19] and physical properties for the double perovskite have been reported [20,21]. In such systems, the magnetic and electronic properties is governed by B and B' superexchange interaction through the O atom. Much like conventional perovskite systems, the double perovskites have been reported to exhibit metallic [22], insulator [23], superconductivity [24], colossal magnetoresistance [14], magnetic order [25], frustrated magnetism [26] as well as multiferroicity [27].
Here, LaSrNiReO 6 is a double perovskite with two different magnetic ions, Ni (B) and Re (B') [see Figure 1 (a)]. In this particular system, an overlap between orbital symmetry is missing for an effective superexchange interaction. Instead, the ground state is determined by none or weakly interacting magnetic sub-lattices. Previous study by Ref. 28 observed a frequency dependent shift in the cusp of a.c.-susceptibility, from where a spin glass ground state was suggested. Moreover, neutron diffraction studies indicated absence of long-range ordering [28,29], in line with a spin glass scenario. For the current sample, the absence of magnetic Bragg peaks in the neutron diffraction pattern is confirmed, as shown in Fig. 1(b). In order to further clarify the ground state, we initiated a muon spin rotation, relaxation and res-arXiv:2007.04616v1 [cond-mat.str-el] 9 Jul 2020 onance (µ + SR) study. Being a local probe and highly sensitive to magnetism, µ + SR is the ideal tool to detect any weak magnetic interactions. Moreover, µ + SR allows for measurements in zero field (ZF) and/or weakly applied fields meaning any influence from the measurements itself can be considered minimal, in comparison to e.g. susceptibility measurement. In this study, a spin precession frequency observed at the lowest temperature of T = 2 K, clearly excluding a spin glass scenario. Instead, an incommensurate long-range dynamically ordered ground state below T C = 23 K is proposed. Such state is still in line with previouse neutron diffraction [28,29] and χ AC [28] results, as it will be shown. Furthermore, between 23 K and 300 K, our µ + SR results distinguish three other magnetic regimes, including dense, dilute, and paramagnetic states. Our findings demonstrate how two separate, yet intertwined magnetic lattices interact over a wide temperature range within the unique double perovskite structure. This study also establishes the unique capabilities of the µ + SR technique for investigating static and dynamic spins on the microscopic (local) length scale, and underlines the importance of combining complementary techniques to get insight on true physical properties of complex materials.

II. Experimental Methods
Polycrystalline sample was prepared using a solid state reaction based on pure La 2 O 3 , SrCO 3 , NiO, Re 2 O 7 and Re metal as starting materials. Stoichiometric mixtures of the starting materials in different steps at high temperatures resulted in single phased LaSrNiReO 6 . Details about the synthesis and basic characterization of the sample are found in Ref. 29.
The crystal structure of LaSrNiReO 6 was generated using the Visualization for Electronic and STructural Analysis (VESTA) [30] software. The magnetic susceptibility measurement was performed using both a Physical Property Measurement System (PPMS) and SQUID magnetometer (MPMS) from Quantum Design. The d.c. and a.c. magnetic susceptibility was performed under a field H= 200 Oe and H = 10 Oe, respectively, within a temperature range of T = 5 − 300 K.
The neutron powder diffraction (NPD) was conducted at the High-Resolution Powder Diffractometer for Thermal Neutrons (HRPT) [31] instrument at the Swiss Spallation Neutron Source (SINQ), Paul Scherrer Institute (PSI), Switzerland. About 1 g of sample was filled in a Vanadium can and measured with a neutron wavelength of λ = 1.8 and 2.95Å at T = 1 and 50 K. The difference plot was done using the FullProf suite [32].
The µ + SR measurements were performed at the surface muon beamlines M20 and DOLLY instruments at TRIUMF and PSI, respectively. Approximately, 1 g of sample was prepared inside a thin (∼ 50 µm) aluminium coated mylar envelope, mounted on a Cu fork sample stick. A standrad 4 He flow cryostat was used in order to reach T base = 1.8 K for PSI/DOLLY and T base = 2.4 K for TRIUMF/M20. Finally, the software package musrfit was used in order to analyze the data [33].

A. Neutron Powder Diffraction and Magnetic Susceptibility
Magnetisation and neutron powder diffraction measurements was first performed [ Fig. 1 (b-c)] before the µ + SR experiments. Starting with the χ(T), two clear magnetic transitions are observed around T = 250 K and T = 25 K, which agrees with previous reports [28 and 29]. An additional transition around T = 75 K is made clear in the ZFC (inset of Fig. 1 (c)). Figure 1 (b) displays the neutron diffraction patterns measured above and below the transition and their difference plot. No magnetic Bragg peaks nor reducing of paramagnetic diffuse scattering [see inset of Fig. 1 (b)] are observed (only a slight thermal expansion is seen). These results suggest an absence of long range magnetic order, in line with previouse NPD meausrements performed between T = 2 − 300 K [28,29]. Detailed NPD and magnetisation analysis can be found in Ref. 28 and 29. In order to clarify the nature of the ambiguous ground state, temperature dependent µ + SR measurements were performed.

B. Muon Spin Rotation
µ + SR measurements in zero field (ZF) and weak transverse field (wTF) configurations were performed. Here, the field in transverse directions refers to the applied field direction with respect to initial t 0 muon spin polarisation, whereas the term weak is to signify that the applied field is significantly weaker than the internal field at low temperatures.

Weak transverse field (wTF)
The obtained time spectra for selected temperatures under wTF = 50 G are presented in Fig. 2. While the phase and the frequency remains constant through the all temperature range, the oscillatory amplitude change drastically. The wTF time spectra were fitted using an oscillatory component and two none oscillatory depolarizing components according to where A 0 is the initial asymmetry and P TF is the muon spin polarisation function in a wTF configuration. A TF , ω TF , φ TF and λ TF are the asymmetry, frequency, relative phase and depolarisation rate respectively, originating from the applied wTF. Further, A S , λ S , A F and λ F are the asymmetry and the respective depolarisation rates originating from internal magnetic fields. The indices S and F are conventions used to represent slow and fast components. The obtained fit parameters for the oscillatory components are summarized in Fig. 3. As already implied in Fig. 2, A TF increases with increasing temperature. A TF corresponds roughly to the paramagnetic fraction and the abrupt increase represents a transition from a magnetically ordered to a disordered state. The transition temperature is then defined as the middle point of a sigmoid fit in which T TF C = 27.3(3) K is obtained. The full asymmetry is only recovered above 250 K, confirming a magnetic contribution to be present up to about 250 K as also suggested from the susceptibility data shown in Fig.  1(c). The nature of this transition will be further discuss in section IV. As expected, the slow component (A S ) is non-zero (not shown) until the full asymmetry of A TF is recovered. Moreover, A TF is low, but non-zero below the transition temperature (A TF 0.02 corresponding to ∼ 9% of the signal), suggesting that part of the muon beam partially hits the sample holder and/or beamline, creating a small background contribution.
The transverse field depolarisation rate, λ TF , approaches zero at higher temperatures as expected in the extreme motional narrowing limit of fluctuating magnetic moments. As the temperature is lowered, a critical behavior is displayed close to the transition with a sharp maximum around T TF C . Such behavior is consistent with critical slowing down of the magnetic moments and broadening of the internal field distribution. Given that the applied field is weak with respect to the internal field, the λ TF values below T TF C are not considered due to the demagnetisation field of the sample (λ TF 0.2 µs −1 originates from the background signal). Figure 4 displays a ZF measurement at T 2 K. The shorter time domain exhibits a highly damped oscillation, which originates from perpendicular field components. The longer time domain (inset Fig. 4) reflects the parallel components, resulting to a stretched exponentiallike depolarisation (the so-called tail component). The time spectrum was then fitted using a combination of a stretched exponential and a Gaussian depolarising oscillating function

Zero field (ZF)
where A 0 is the initial asymmetry, P ZF is the muon spin polarisation function in ZF configuration and A IC , f IC , β IC and λ IC are the asymmetry, frequency, stretched exponent and depolarisation rate for the oscillatory component, respectively. J 0 is the zero order Bessel function of its first kind, while A S , λ S and β S are the tail components originating from the fact that 1/3 of the field components inside the sample is parallel to the initial muon spin polarisation. The tail component exhibits a stretched exponential polarisation with a temperature dependent stretched exponent, β S . Physical interpretation of the stretched exponent is not trivial, but the function is derived by assuming a distribution of depolarization rates [34,35]. However, a microscopic origin for β = 1, 2, 0.5 exists: β = 1 denote an exponential depolarization channel (magnetically homogenous), while β = 2 is a Gaussian depolarization channel (quasi-static field distribution) and β = 0.5 is a so-called root exponential (e.g. a distribution of spin correlation times, also seen above the transition temperature in spin glasses in the motional narrowing limit). A complete interpretation behind Eq. 2 is further discussed in Sec. IV.
a. Below the transition temperature T = 27 K The ZF time spectra from T 2 K up to T = 27 K were analysed using Eq. 2 and the obtained fit parameters are displayed in Fig. 5. At the base temperature, A S A 0 /3 and A IC 2A 0 /3, consistent with being the tail and perpendicular components. While the asymmetries [ Fig. 5(a)] are more or less constant up to the transition, drastic changes can be seen around the transition temperature for the other parameters. The temperature dependent muon precession frequency confirms an order parameter like dependence and displays a clear transition from an ordered to a disordered state [ Fig. 5(b)]. The value of the frequency is directly proportional to the internal field at the muon site. From mean field theory, a fit according to f (T ) = f (0)( TC−T TC ) α results in T ZF C = 23.0(1) K, α = 0.348(47) and f (0) = 49.57(2.49). The low temperature part is not well fitted due to thermal magnon excitation, resulting into a reduction of local spontaneous magnetization given by the Bloch 3/2 law [36]. Note that the slightly higher transition temperature extracted from the wTF measurement may be due to the applied field and in general, only the ZF data reflect the intrinsic magnetic properties of materials.
Both depolarisation rates [ Fig. 5(c) plotted in log scale] seem to have similar temperature dependence, where the values are increasing as T approaches T ZF C , consistent with an increase of dynamics and broadening of the field distributions close to the phase transition. The parallel component, λ S , corresponds rougly to the spin-lattice relaxation rate, while λ IC is rougly the spin-spin relaxation rate. The parameter λ S contains information about the dynamics in the system and λ IC includes a mixture of both the field distribution and the dynamics. Therefore, the relatively high value of λ IC = 32.4(1.2) µs −1 at T base relates to a high field distribution and dynamics at the muon sites. However, the behavior of λ S → 0 as T→ 0, is consistent with the sample going towards the static limit. Since λ IC seems to level off at lower temperature, the high value of λ IC originates mostly from a broad field distribution. Based on the temperature dependence of λ S , a completely static mag-netic ground state is expected below T = 2 K. Nevertheless, the value λ S (2 K) = 0.012(1) µs −1 is still observed, meaning that the spins are dynamic even at lowest measured temperature. Finally, the stretched exponents are almost constant up to the transition. Stretched exponent of β = 2 is a Gaussian depolarization channel suggesting that the magnetic phase is quasi-static. β = 1 3 corresponds to the magnetic impurity limit and has already been reported in several spin glass systems [37][38][39].
b. Above the transition temperature T = 27 K The sample was also studied in ZF configuration for temperatures above T ZF C up to T = 250 K. The data are also well fitted using Eq. 2 in this temperature range (note that A IC = 0 above the transition). The obtained fit parameters as a function of temperature are shown in Fig. 6. A clear increase in asymmetry is shown until the maximum asymmetry A 0 0.24 is recovered. The depolarization rate is exhibiting a critical behavior just above T ZF C . The narrow temperature range of this critical slowdown of electronic moments suggest an exchange coupling J ∼ k B T C , agreeing with the wTF measurement. As for the stretched exponent, a recovery of β → 1 is observed at highest temperature, as expected. Moreover, β → 1 3 is seen around the transition, typical for many glassy like transitions [37][38][39]. The obtained value of β ∼ 0.7 above T ZF C (between 75 K and 250 K) can be further expanded upon. In fact, this temperature range is nicely fitted with a static Lorentzian Kubo-Toyabe (L-KT) function. For these temperatures, the following fitting function was utilized: where A 0 is the initial asymmetry, P ZF is the muon spin polarisation function in ZF configuration, A F and A KT are the asymmetries for their respective contribution, where L SLKT represents a static Lorentzian Kubo-Toyabe. λ F is the depolarisation rate for an initial fast depolarising signal, while ∆ is related to the internal Lorentzian half width half maximum. Here, A F is attributed to the fraction of muons coupled to fast fluctuating moments, whereas the L-KT is the dilute limit of the KT, commonly observed in dilute magnetic/paramagnetic state. Indeed, it is only for T > 250 K that a Gaussian KT could be used to fit the data, meaning that dilute electronic magnetic moments are present up to this temperature. Further, the data below T = 75 K could not be fitted to any Kubo-Toyabe function. This suggests the presence of a distribution of relaxation times for 23 K< T ≤ 75 K. Eq. 3 is further discussed in Sec. IV.
The obtained fit parameters using Eq. 3 are displayed in Fig. 7. As expected, the KT asymmetry slowly recovers the full asymmetry as the temperature increases, while the fast relaxing component decreases. Moreover, the field distribution width decreases with temperature, a typical behavior in the motional narrowing limit. Strangely, the depolarisation rate seem to exhibit a maximum around 150 K. Indeed, a small anomaly can also be observed around this temperature in the stretched exponent, shown in Fig. 6(c). The origin of this anomaly is currently unknown and further investigations are required. Although, such anomaly does not affect the main results and conclusion drawn in this report.

IV. Discussion
Based on the results presented in Sec. III, four magnetic states are identified: a paramagnetic state above T >250 K, a dilute magnetic phase between 75 K < T ≤ 250 K, a glassy like transition/dense state between 23 K < T ≤ 75 K, and a magnetically ordered state below T ZF C = 23 K [Fig. 8]. The pressence of several magnetic phases is further supported by the magnetic susceptibility measurements shown in Fig. 1 (c) and Ref. 29, where both the transition at T ≈ 23 K and the bifurcation between ZFC and FC curves at T ≈ 250 K were reported. The latter has also been observed in Sr 2 CaReO 6 [40] and Sr 2 InReO 6 [41] double perovskite compounds and attributed to the non-magnetic ions located at the B-site, causing a geometrical frustration of the Re site on the FCC sub-lattice. Such situation would lead to a dilute magnetic system and a L-KT like depolarisation would manifest the ZF time spectrum in µ + SR, as presented here. Typically, a L-KT fit is appropriate for dilute electronic moments existing in a non-magnetic matrix. Moreover, the temperature independent depolarisation rate from ZF measurement supports such situation in which dilute moments fluctuate [ Fig. 6(b)].
In same temperature region, a stretched exponent of β ∼ 0.7 was obtained. While the stretched exponential is somewhat phenomenological, such value is perhaps indicative of an intermediate case between dilute and dense motionally narrowed source of magnetic fields. If the magnetic distribution in real space is dense, the field distribution can be approximated by a Gaussian shape for which the muon spin depolarization follows an exponential in the motional narrowing limit. In the dilute limit, the muon spin depolarizes according to a root exponential in the narrowing limit. Therefore, a β ∼ 0.7 for 75 K< T ≤250 K could perhaps be explained as an intermediate case of dense and dilute source of magnetic fields.
A stretched exponential like depolarization is observed in a wide temperature range (23 K< T ≤ 75 K). Such a situation is usually explained by the presence of a distribution of muon depolarization channels (i.e. spatially disordered systems). The magnetically similar compounds Sr 2 NiWO 6 and Sr 2 NiTeO 6 poses none magnetic W 6+ and Te 6+ ions at B site. The Ni in these compounds orders at T N = 35 K and T N = 54 K, respectively [42]. Therefore, it seems that some localized moments of Ni 2+ becomes prominent at lower temperature for the title compound, effectively destroying the dilute limit and resulting into a distribution of relaxation rates. Therefore, we suggest that an independent Ni sub-lattice feature is realized below T ZF C = 23 K, resulting into magnetic order. In other words, the Re 5+ interactions/fluctuations dominates at higher temperature, while the Ni 2+ interactions becomes significant at lower temperature. Admittedly, the presented data cannot distinguish the Ni 2+ moments from Re 5+ moments, meaning that the opposite case in which the Re 5+ orders at low T instead of Ni 2+ is also probable. Nevertheless, a sigmoid function cannot fit the all temperature range of the wTF asymmetry plot [ Fig. 3(a)], but a continuous transition is seen up to 250 K.
Transport measurements [29] suggested the transition at T ∼ 30 K due to a weak ferromagnetic interaction, predicted by Goodenough-Kanamori rules. However, in this study, a Bessel function was used instead of a simple cosine function to fit the time spectrum. The usage of a Bessel function instead of a cosine function would suggest an incommensurate order. This is justified because using a cosine function results in a large offset in the initial phase φ ≈ −50 • . For an incommensurate single-k collinear magnetic structure, where the muon site is centre of symmetry, the polarisation function is given by: A phase offset may also be realized in commensurate orders, e.g. in cases where the local field fluctuates from parallel to anti-parallel direction at a rate v c < 2γ µ B fluc . However, the spin-lattice relaxation rate [λ S in Fig. 5(c)] indicates a decrease in dynamics with temperature, which in turn would change the phase offset as a function of temperature (not this case). Another case was reported by Ref. 43, where several muon sites were present within a larger magnetic unit cell. While the presence of several muon sites could explain the stretched exponential behavior in this study, only one clear oscillating signal was observed meaning only one muon site is expected for this compound. Consequently, LaSrNiReO 6 most likely display an incommensurate magnetic spin order below T ZF C . It should also be noted that the muon spin precession frequency at lowest temperature was effectively lowered due to the reduction in spontaneous magnetisation ( Fig. 5(b)). However, such situation is only realised for ferro and ferri magnets. Indeed, the transition at T ∼ 30 K was predicted to be due to a weak ferromagnetic interaction [29] according to Goodenough-Kanamori rules. Consequently, given the fact that we observe a incommensurate order, the ground state of LaSrNiReO 6 point towards an incommensurate ferrimagnetic state.
The presence of an oscillatory component in the µ + SR time spectrum for T ≤ 23 K (Fig. 4) suggest a magnetically ordered ground state, where χ AC measurements also predicted a magnetic transition occuring arond 27 K. However, any magnetic bragg peaks were not observed in the neutron diffraction measurements [28,29] and such discrepancy should be addressed. First off, lets point out that the internal field distribution is wide, as implied by the highly damped oscillation (λ IC ). As mentioned, the parallel component of the ZF measurement, λ S , at base temperature indicated a dynamic state. While it is possible for spins to be dynamic in only specific directions, it is not likely in this case considering the crystal structure. More importantly, the measurement was performed on a powder sample meaning any spatial direction should be averaged out. Furthermore, λ IC seem to level off at lower temperatures while λ S continues to decrease [ Fig. 6 (c)]. Therefore, the high damping of the oscillatory component, λ IC , observed in Fig. 5 should mainly be due to an wide field distribution width and supports the fact that the sample is magnetically in-homogeneous. Indeed, a stretched exponent β = 0.33 is seen for the parallel component below T C suggesting a very wide distribution of relaxation rates. Naturally, magnetically inhomogeneous systems do not yield any clear magnetic Bragg peaks, which together with the observation of oscillation in ZF µ + SR time spectrum suggest that the sample displays a magnetic order with a correlation length from the view point of NPD is short-range while from µ + SR is a long range.
A previous report based on a.c.-susceptibility measurements suggested a spin glass state below T ∼ 25 K [28]. Indeed, just like in this case, many spin glasses exhibits β → 1 3 around the transition temperature. However, presence of a muon spin precession clearly excludes such scenario. Instead, we suggest that an incommensurate order is stabilised with a distinct magnetic correlation length (ξ), that is long enough for detection for µ + SR and χ AC techniques, but not for neutron diffraction (hence magnetically in-homogeneous). On a macroscopic scale, the system may look like a spin glass, where small magnetic domains freezes randomly. However, microscopically, each of these domain are in fact ordered on a shorter length scale [ Fig. 8(a)]. Overall, this kind of microscopic picture would yield a frequency dependent shift in χ AC while also yielding muon spin precession and absence of magnetic Bragg peaks in neutron diffraction is explained. From our knowledge, this is a very rare case where such a situation is present and so clearly revealed.

V. Conclusions
We have utilised muon spin rotation (µ + SR) to elucidate the magnetic properties of the double perovskite compound LaSrNiReO 6 . Using the unique length and time scales of the µ + SR technique, we have successfully identified four magnetic states: a paramagnetic (T > 250 K), a dilute (75 K< T ≤ 250 K), a dense (23 K< T ≤ 75 K), and an incommensurate magnetically ordered state (2 K< T ≤ 23 K). The dilute state is established by weakly interacting and fluctuating Re 5+ ions sitting on the B site, which develops into an in-commensurate order around 23 K, driven by the FCC sublattice of Ni ions on the B sites. This state consist of weakly interacting domains/islands, established by a ferri interacting spins, forming an incommensurate spin wave. This study reveal in great detail how two separate, yet intertwined magnetic lattices interact over a wide temperature range within the unique double perovskite structure. It also show the unique capabilities of the µ + SR technique for studying static and dynamic spins on the microscopic (local) length scale. We also emphasize the importance of applying a set of complementary experimental techniques in order to obtain a complete and correct understanding of the microscopic magnetic properties in complex materials.