Griffiths phase and spontaneous exchange bias in La_1.5Sr_0.5CoMn_0.5Fe_0.5O₆

The exchange bias (EB) effect is a remarkable phenomenon discovered at the 1950 decade in materials presenting ferromagnetic (FM)–antiferromagnetic (AFM) interfaces [1], and is manifested as a shift in the hysteresis loop measurement [M(H)]. For the recently discovered spontaneous EB (SEB) effect, the unidirectional anisotropy (UA) sets in spontaneously during the cycling of magnetic field (H) at the M(H) measurement even when the material is cooled from an unmagnetized state down to low temperature (T) in zero H. This results from a field-induced symmetry breaking formed across different magnetic phases [2, 3]. Conversely, for the conventional EB (CEB) effect discovered more than 60 years ago the asymmetry in the hysteresis loop is only manifested after the system is cooled in the presence of H [4]. This phenomena is a key functional ingredient in the sensing elements of read heads, as well as in the magnetic media of hard disks. Therefore, the fact that no external cooling H is needed to set the UA has raised to a great interest in the recently discovered SEB materials, where the presence of a spin glass (SG)-like phase seems to be a necessary condition for the appearance of SEB effect [5, 6].

In this context, the double-perovskite (DP) compounds are very appealing since intrinsic structural and magnetic inhomogeneity usually lead to competing magnetic interactions and frustration, both key ingredients to achieve glassy magnetic behavior. This makes such materials promising candidates to exhibit SEB effect [7, 8]. For instance, the La_2−x A_x CoMnO₆ (A = Ba, Ca, Sr) DP family presents unequivocal SEB, with the x = 0.5 Ba- and Sr-members showing a robust effect whilst for the Ca-one the shift in the M(H) loop is subtle [9–11]. Such differences are believed to be related to distinct amounts of low-spin (LS) Co³⁺ present in each sample, which is strongly correlated to the structural particularities of the materials, leading to distinct uncompensation in the AFM coupling between Co and Mn [12].

Recently, it was shown that a Fe to Co partial substitution in La_1.5Sr_0.5CoMnO₆ yields a large SEB effect which changes sign depending on the concentrations of Fe and Co, such changes being ascribed to Co and Fe-dependent spin state transitions [13]. In this work we show that the Fe to Mn substitution also leads to unambiguous SEB effect. The structural, electronic and magnetic properties of La_1.5Sr_0.5CoMn_0.5Fe_0.5O₆ (LSCMFO) were thoroughly investigated by means of x-ray powder diffraction (XRD), Mössbauer spectroscopy and magnetization as a function of H and T measurements, evidencing the presence of FM and AFM phases. The magnetic inhomogeneity leads to a Griffiths-like phase at high T and to an SG-like phase at low T. Both SEB and CEB effects are observed below 10 K in this material. The magnetic properties of LSCMFO are compared to those of similar DP compounds and explained in terms of its structural and electronic properties.

A polycrystalline sample of LSCMFO was synthesized by conventional solid-state reaction method. Stoichiometric amounts of La₂O₃, SrCO₃, Co₃O₄, MnO and Fe₂O₃ were mixed and heated at 750 °C for 10 h in air atmosphere. Later the sample was re-grinded before a second step at 1400 °C for 36 h, with intermediate grinding. Finally, the material was grinded, pressed into pellet and heated at 1400 °C for 24 h. After this procedure it was obtained a dark-black material in the form of a 10 mm diameter disk. High resolution XRD data was collected at room T using a Bruker D8 Discover diffractometer, operating with Cu K_α radiation operating at 40 kV and 40 mA. The XRD data was investigated over the angular range 10 ⩽ θ ⩽ 90°, with a 2θ step size of 0.01°. Rietveld refinement was performed with GSAS software and its graphical interface program [14]. ⁵⁷Fe-Mössbauer measurement was performed in transmission geometry at room temperature (RT), with a ⁵⁷Co:Rh source moving in a sinusoidal motion. Experimental data were fitted with the Normos program.

Magnetization as a function of T [M(T)] and M(H) measurements were carried out in both zero field cooled (ZFC) and field cooled (FC) modes using a commercial SQUID-VSM magnetometer. The M(H) measurements were performed at several T and, from one experiment to another, a systematic protocol of sending the sample to the paramagnetic (PM) state and demagnetize the system in the oscillating mode was adopted in order to prevent the presence of remanent magnetization. This procedure is particularly important for the ZFC M(H) measurements, since any trapped current in the magnet would prevent the correct observation of the SEB effect.

The room-T XRD pattern of LSCMFO is shown in figure 1. It revealed the formation of a single phase perovskite compound and could be successfully refined in the rhombohedral $R\bar{3}c$ space group, as for La_1.5Sr_0.5(Co_1−x Fe_x )MnO₆ and other resembling compounds [9, 12, 13]. The main parameters obtained from the Rietveld refinement are displayed in table 1.

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Comparing the lattice parameters of LSCMFO with those reported for La_1.5Sr_0.5CoMnO₆ (LSCMO) [15] it is observed a subtle increase in the unit cell volume of the former (344.09 ų) in relation to the later compound (343.58 ų). This was expected since our Mössbauer measurements presented below indicate trivalent state for Fe ions and previous works have shown a nearly tetravalent state for the Mn ions in LSCMO [9, 12, 15]. Since the Mn⁴⁺ ionic radius is smaller than that of Fe³⁺ [16], the incorporation of Fe in the system is expected to expand the lattice, as observed. On the other hand, previous studies have also indicated a Co²⁺/Co³⁺ mixed valence state in LSCMO and resembling compounds [12, 15, 17]. Therefore, the introduction of Fe³⁺ in place of Mn⁴⁺ induced the Co ions to a trivalent state in order to ensure the charge balance, which prevents a larger increase of the unit cell volume. For this scenario, we are assuming a near stoichiometry of oxygen ions, which may not be precisely the case since excess or vacancy of oxygen ions usually occurs in perovskites and should be verified by means of thermogravimetric analysis, x-ray photoelectron spectroscopy or x-ray absorption spectroscopy. Nevertheless, the Co, Fe and Mn formal valences here described are in agreement with the magnetization results.

The B–O bond length (B = Co/Mn/Fe) is slightly smaller for LSCMFO (1.941 Å) than for LSCMO (1.946 Å) [15], which may be related to a more pronounced tilt of the oxygen octahedra for the later compound. Although care must be taken in interpreting these data since they depend on the oxygen positions taken from XRD, the trends here described are highly expected for perovskite compounds [7] and are supported by our magnetization data, being thus very plausible. The structural and electronic changes observed for LSCMFO with respect to LSCMO will remarkably affect the compound magnetic properties, as will be discussed below.

⁵⁷Fe Mössbauer spectrum is displayed in figure 2. To fit this spectrum, a model with two PM doublets gives the best least-square fitting. The obtained hyperfine parameters are the isomer shift (δ), the quadrupole splitting (ΔE_Q), the spectral relative area (A) and the line width (Γ). Table 1 presents the hyperfine parameters, δ values are reported relative to the α-Fe. For these subspectra, the values of δ and ΔE_Q are consistent with Fe³⁺ at octahedral sites. The electric field gradient produced by the surrounding atoms at iron probe sites gives rise to the quadrupole splitting [18, 19]. In a perfect cubic symmetry, ΔE_Q = 0, other values indicate the deviation of the system from a cubic symmetry. Despite a signature of disorder is showed in the linewidth, the subspectra of two doublets have values of δ and ΔE_Q consistent with Fe³⁺ at octahedral sites. Then, the presence of two PM components could be related to the statistical distribution of the atomic environment around Fe³⁺, being the main part more symmetric. This was expected since in this disordered system one may have regions with Fe surrounded by Co and other ones where Mn is its nearest neighbor. More information could be obtained from the Mössbauer spectra below the magnetic order as observed for other perovskite compounds [20, 21].

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Figure 3(a) shows the ZFC and FC M(T) curves measured with H = 500 Oe, from which is observed an FM-like transition at T_C ≃ 150 K. The fit of the inverse of magnetic susceptibility (χ⁻¹) with the Curie–Weiss (CW) law at the PM region yielded a CW temperature θ_CW = 59 K and an effective magnetic moment μ_eff = 5.2 μ_B/f.u. The positive θ_CW confirms the presence of FM coupling. However, the fact that θ_CW lies far below T_C, together with the small magnetization values observed at low-T, rule out the possibility of a fully long-ranged FM state with the presence of Fe, Co and Mn magnetic ions. According to the Goodenough–Kanamory–Anderson (GKA) rules [22], the Fe³⁺–O–Mn⁴⁺ coupling is expected to be FM, which could explain the positive θ_CW and the FM-like curve depicted in figure 3(a). On the other hand, for the $R\bar{3}c$ space group all Co, Fe and Mn ions share the same crystallographic site, which is intrinsically related to the proximity of Co/Mn/Fe ionic radii and scattering factors. This results in a complex and disordered system with a high degree of permutation between the transition–metal (TM) ions, for which other nearest neighbor couplings as Fe³⁺–O–Fe³⁺, Mn⁴⁺–O–Mn⁴⁺ are expected. These are predicted by the GKA rules to be of AFM type, and in such condition the superexchange-driven FM contribution is interrupted by a large proportion of AFM phases, which explain the small magnetization observed for LSCMFO.

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As aforementioned, the Fe³⁺ to Mn⁴⁺ partial substitution drives the Co toward a trivalent state. This ion is particularly interesting due its multiple spin states resulting from a delicate balance between between the crystal field splitting and the intra-atomic Hund's exchange energy [23]. In octahedral coordination, Co³⁺ can be found in high spin (HS) [24, 25], low spin (LS) [12, 13, 26, 27] or even in intermediate spin (IS) configuration [28]. As a first attempt to reproduce the experimentally observed μ_eff, one can consider the HS configuration for Co³⁺. Using the spin only approximation for the TM ions, i.e. ${\mu }_{{\mathrm{F}\mathrm{e}}^{3+}}$ = 5.9 μ_B, ${\mu }_{{\mathrm{M}\mathrm{n}}^{4+}}$ = 3.9 μ_B and ${\mu }_{{\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{o}}^{3+}}$ = 4.9 μ_B, the effective magnetic moment can be theoretically calculated with the following equation for systems with two or more different magnetic ions [29, 30]

$$\begin{equation}\mu =\sqrt{{{\mu }_{{\mathrm{H}\mathrm{S}\mathrm{C}\mathrm{o}}^{3+}}}^{2}+0.5{{\mu }_{{\mathrm{F}\mathrm{e}}^{3+}}}^{2}+0.5{{\mu }_{{\mathrm{M}\mathrm{n}}^{4+}}}^{2}},\end{equation} \tag{ 1 }$$

yielding μ = 7.0 μ_B/f.u., far above the experimentally observed value. Now assuming LS Co³⁺ (S = 0) while keeping Mn⁴⁺ and Fe³⁺, the calculated magnetic moment becomes μ = 5.0 μ_B/f.u., much closer to the experiment.

Zhang et al reported the presence of a second FM transition for intermediate Co/Fe concentrations in the La_1.5Sr_0.5(Co_1−x Fe_x )MnO₆ series [13], finding that was not confirmed in the M(T) curves of the La_1.5Sr_0.5Co(Fe_0.5Mn_0.5)O₆ sample here investigated. Since both systems belong to $R\bar{3}c$ space group, for which the Co/Fe/Mn ions share the same crystallographic site, this different magnetic behavior is most likely related to changes in valence and spin states, which in turn are related to the molar ratios among these ions. The Co:Fe:Mn ratio for the latter system is (1−x):x:1, leading to mixed valence states for Mn (Mn³⁺/Mn⁴⁺) while Co presents both mixed valence and spin states (Co²⁺/Co³⁺ in LS and HS configurations) for intermediate x. The second transition was attributed to FM exchange couplings as (HS)Co³⁺–O–Mn⁴⁺, (LS)Co²⁺–O–(HS)Co³⁺, (LS)Co²⁺–O–Fe³⁺ and (HS)Co³⁺–O–(LS)Co²⁺ [13]. Contrastingly, the Co:Fe:Mn ratio for La_1.5Sr_0.5Co(Fe_0.5Mn_0.5)O₆ is 1:0.5:0.5, with this compound being better described as a single valence and single spin state system [Mn⁴⁺, (LS)Co³⁺], and naturally the second FM transition is not observed.

Figure 3(a) shows a split of the ZFC and the FC M(T) curves below 150 K. This is characteristic of disordered magnetic systems, where the presence of competing magnetic phases usually leads to glassy magnetic behavior [8]. Due to this, an interesting anomaly is observed in the M(T) curves carried with H ⩾ 2000 Oe, figure 3(b). A peak appears in the low-T region, whose intensity decreases with increasing H while it moves toward lower T. These are characteristic features of SG-like states, here resulting from the structural disorder together with competition between FM and AFM phases. The fact that the anomaly does not manifest for small H suggests that larger fields are necessary for one of the competing magnetic interactions to overmatch the other, thus indicating that the cluster glass (CG) state is most probable in LSCMFO [31–34]. The inset of figure 3(b) shows the evolution of the peak position (T*) as a function of the measured H. The T*, obtained from the first derivative of the ZFC M(T) curves, reasonably follows the H^2/3 Almeida–Thouless (AT) relation usually observed for SG-like systems [35], further indication that this peak may be associated to the onset of glassy magnetic behavior. From the extrapolation of the linear fit shown in the inset, the freezing T of the possibly CG phase in zero H would be T_g ≃ 73 K. The formation of the clusters, as well as their sizes, should be confirmed by other techniques as AC magnetic susceptibility, neutron powder diffraction and electron spin resonance.

In order to further investigate the SG-like state in LSCMFO, and to confirm that the glassy behavior is present even when the anomaly is not apparent in the M(T) curves (i.e. for H < 2000 Oe), we have performed a memory effect measurement in the H = 500 Oe ZFC susceptibility by adopting the following protocol: first, a standard ZFC curve was captured by cooling the system down to 5 K in zero field for a subsequent application of H = 500 Oe and the measurement of χ vs T in the heating mode. In the second step a similar measurement was performed, but with the difference that during the ZFC protocol the cooling procedure was paused for a time-interval t_w = 10⁴ s at T = 30 K. The inset of figure 3(a) shows the difference between these curves, where a dip is observed precisely at 30 K. This is a hallmark of glassy magnetic systems, thus confirming the emergence of SG-like state at low-T.

Figure 4 shows χ⁻¹ of the H = 500 Oe ZFC curve. One can note a rapid downward deviation from the CW law occurring well-above T_C, which is a signature of Griffiths phase (GP). This phenomena is related to the nucleation of small and weakly correlated clusters embedded in a PM matrix, resulting in a peculiar magnetic phase where the system neither behave like a PM nor shows long-range ordering [36, 37]. The inset of figure 4 shows that as H increases the downturn deviation gets softened. This is also a characteristic feature of GP, related to the fact the PM matrix magnetization increases linearly with increasing H [38, 39], and for a sufficiently high field its susceptibility dominates over the weakly correlated clusters. This scenario of short-ranged FM clusters may explain the small magnetization values observed in the M(T) and M(H) curves.

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The characteristics of the GP are usually described by the following power law equation [38]

$$\begin{equation}{\chi }^{-1}\propto {\left(T-{T}_{\mathrm{C}}^{\mathrm{R}}\right)}^{1-\lambda },\end{equation} \tag{ 2 }$$

which is simply a modified version of the CW law where the value of λ (0 ⩽ λ ⩽ 1) reflects the strength of the GP, i.e. it is a measure of the deviation from the CW behavior. ${T}_{\mathrm{C}}^{\mathrm{R}}$ defines the GP interval, ${T}_{\mathrm{C}}^{\mathrm{R}}{< }T{< }{T}_{\mathrm{G}}$, with T_G being the temperature above which the system behaves as a simple PM. The λ value is highly sensitive to ${T}_{\mathrm{C}}^{\mathrm{R}}$ [26, 40], which must be carefully computed in order to obtain a reliable description of the GP. In the case of LSCMFO, since the θ_CW obtained from the CW fitting of the PM region lies far below the FM ordering T, we have chosen ${T}_{\mathrm{C}}^{\mathrm{R}}={T}_{\mathrm{C}}$, yielding λ = 0.64. This value lies within the range found for resembling DP compounds [13, 17, 26, 41]. The lattice distortions and the cationic disorder at the TM ion-site, intrinsic to the $R\bar{3}c$ space group, certainly play a part in the GP. But the abundant non-magnetic LS Co³⁺ is probably the key for the appearance of such phenomena in LSCMFO as it gives rise to the dilution of the magnetic ions, thus hindering the formation of long-range order and aiding the nucleation of correlated clusters.

It is well known that multi-magnetic systems usually exhibit the EB effect [4], and recently it was shown that glassy magnetism is paramount for the emergence of SEB [5, 6]. Thereby such phenomenon was verified in LSCMFO. Figure 5(a) shows the M(H) curve measured after ZFC the sample down to 3 K. It is clearly a closed loop, nearly symmetric with respect to the M axis and shifted to the left along the H axis. It may also be noticed that the virgin magnetization curve falls outside the hysteresis loop up to a certain critical H (∼4.5 T), which is a common ground of several SEB compounds, being related to H induced magnetic transitions [2, 3, 9, 11]. Since the 3 K M(H) curve of LSCMFO results from the contributions of short-ranged FM clusters embedded in an AFM matrix, there is no magnetic saturation up to 70 kOe. Furthermore, by comparing this curve with the one taken at 100 K [bottom inset of figure 5(a)] one can see that for the lower T-curve the loop loses it squareness, indicating that the system has entered in the glassy state.

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The shift along the H axis is a measure of the EB field, herein defined as H_EB = |H₂ + H₁|/2, where H₁ and H₂ are respectively the left and right coercive fields (keeping the signs). The average coercive field is defined as H_C = (H₂ − H₁)/2. For the 3 K curve shown in figure 5(a) it was found a maximum shift of 4241 Oe, which is comparable to those of giant SEB materials [9, 42]. Several ZFC M(H) measurements were performed at different T in order to investigate the evolution of the SEB effect. It is shown in figure 5(b) that the SEB effect is negligible for T > 10 K, the drop in H_SEB being accompanied by a local increase of H_C. This is a signature of the EB effect [4], where the gain in thermal energy favors some AFM spins to be 'dragged' by the rotating FM clusters, thus increasing H_C.

There is a great increase in the EB effect of LSCMFO when the system is cooled in the presence of an external field. For a cooling field H_FC = 30 kOe the shift in the 3 K M(H) curve is ∼40% larger than that of the ZFC loop measured at the same T, as can be seen in figure 6. The qualitative evolution of both H_CEB and H_C as a function of T are very similar to those observed after the ZFC procedure. Once again, the emergence of EB at low T indicates that the glassy magnetic phase plays a key role on the EB effect of DP compounds. The overall CEB behavior of LSCMFO resembles the results observed for LSCMO and related compounds, for which H_FC induces the growth and the correlation between the FM clusters, favoring the pinning of spins and thus increasing the EB effect [9, 11, 43].

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In EB systems, H_EB generally depends on the number n of consecutive M(H) measurements, a property often called training effect [4]. This is usually related to the relaxation of uncompensated spins after successive M(H) cycles, and for practical applications in spintronics it is required of an EB material to be fairly stable under successive H cycles. Here we investigated the changes in the SEB effect for seven consecutive hysteresis loops. Figure 7 shows that the consecutive cycles lead to very small changes, evidencing the stability of the SEB effect. A closer inspection of the curves near the M = 0 region (upper inset of figure 7) indicates a monotonic decrease of the left coercive field H₁ with increasing the number of cycles n, suggesting that some spin rearrangements at the magnetic interfaces lead to the reduction of H_SEB. The decrease is more pronounced from the first to the second cycle, resulting in a larger reduction of the EB effect from n = 1 to n = 2. Conversely, the right coercive field H₂ systematically shifts to the left with increasing n. This acts in favor of increasing the EB effect, but since the changes are more pronounced in H₁ than in H₂ the consecutive M(H) loops act to slowly reduce H_SEB with increasing n. The decrement from n = 1 to n = 2 is ∼3% and to n = 7 is ∼4%, these very small values indicating that the metastable spin configuration at the magnetic interfaces is reasonably stable against the magnetic field cycles.

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The H_SEB evolution with n could be well fitted by a model proposed for FM/SG systems which considers that the SG state has both frozen and uncompensated rotatable spins [43, 44]

$$\begin{equation}{H}_{\text{SEB}}\left(n\right)={H}_{\text{SEB}}^{\infty }+{A}_{\mathrm{f}}\enspace {\mathrm{e}}^{\left(-n/{P}_{\mathrm{f}}\right)}+{A}_{\mathrm{r}}\enspace {\mathrm{e}}^{\left(-n/{P}_{\mathrm{r}}\right)},\end{equation} \tag{ 3 }$$

where ${H}_{\text{SEB}}^{\infty }$ is the value of H_SEB for n = ∞, A_f and P_f are parameters related to the change of the frozen spins, A_r and P_r are evolving parameters of the rotatable spins. The fitting with equation (3) yields ${H}_{\text{SEB}}^{\infty }\simeq$ 2981 Oe, A_f ≃ 1422 Oe, P_f ≃ 0.35, A_r ≃ 66 Oe, P_r ≃ 4.82. The precise fitting of the training effect with equation (3) gives further evidence of the presence of SG-like phase in LSCMFO. As observed for other DPs, the glassy magnetism seems to play a key role on the SEB of LSCMFO.

In summary, the structural, electronic and magnetic properties of LSCMFO compound were systematically studied. The polycrystalline sample forms in rhombohedral $R\bar{3}c$ space group, as observed for similar compounds. The Mössbauer spectroscopy has revealed that Fe is in trivalent oxidation state. The M(T) data indicate an FM transition at 150 K related to the Fe³⁺–O–Mn⁴⁺ coupling, whilst several other possible nearest neighbor interactions as Fe³⁺–O–Fe³⁺ and Mn⁴⁺–O–Mn⁴⁺ are predicted to be AFM. Magnetization results also indicate that the presence of non-magnetic LS Co³⁺ prevents the formation of long-range order, aiding the nucleation of FM clusters that, when embedded in a PM matrix at high T, drives the system to exhibit a GP. At low T, the onset of AFM couplings leads to the emergence of a CG state. The material exhibit a large SEB effect below 10 K, which is greatly enhanced when the system is cooled in the presence of an applied magnetic field.

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [No. 400134/2016-0, 425936/2016-3 and 309250/2011-0], Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)and Fundação de Amparo à Pesquisa do Estado de Goiás (FAPEG). EBS acknowledges several Grants from Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), including Professor Emeritus fellowship. CPCM acknowledges FAPERJ for the PDR Nota10 fellowship and KLSR thanks CNPq for PDJ fellowship.