Griffiths phase and temporal effects in phase separated manganites

Author’s Accepted Manuscript Griffiths phase and temporal effects in phase separated manganites V.N. Krivoruchko, M.A. Marchenko

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To appear in: Journal of Magnetism and Magnetic Materials Received date: 16 December 2015 Revised date: 14 February 2016 Accepted date: 25 February 2016 Cite this article as: V.N. Krivoruchko and M.A. Marchenko, Griffiths phase and temporal effects in phase separated manganites, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.02.086 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Griffiths phase and temporal effects in phase separated manganites V. N. Krivoruchko and M. A. Marchenko Donetsk Institute for Physics and Engineering the NAS of Ukraine, 46, Nauki Avenue, 03680, Kyiv, Ukraine (Dated: February 26, 2016)

Abstract Phenomenological description of relaxation phenomena in magnetic and transport properties of perovskite manganites has been presented. The approach is based on generalization of some hypotheses appropriate to the Preisach picture of magnetization process for half-metallic ferromagnets and on an assumption that in doped manganites the phase separated state exists near the magnetic ordering temperature. For systems with the percolation type of a ferromagnet – paramagnet transition, distinctive features in relaxation of magnetization and resistivity have been found. The relaxation is shown to be most pronounced near the transition temperature, and to be an approximately logarithmic function of time. The theoretical results replicate a broad spectrum of behavior observed experimentally on time dependence of magnetization and resistivity of CMR systems and allow a direct comparison with available experimental data. We propose an additional experimental test to distinguish between the percolation scenario of magnetic and transport transitions in doped manganites, and the ferromagnetic polaron picture. In particular, an anomalously slow relaxation to zero of the order parameter can be considered as a key feature of the Griffiths-like phase transition in doped manganites. It is also shown that a system with the Griffiths-like state will exhibit nonequilibrium aging and rejuvenation phenomena, which in many aspects resemble that of a spin glass. We hope experimental observation of a set of time decay properties will provide a settlement of apparently conflicting results obtained for different characteristics of phase-separated manganites. Keywords: manganites, phase-separated sate, Griffiths phase, temporal effects

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I.

INTRODUCTION

Despite intense effort towards understanding the physics of manganites, many macroscopic features of their phase separated (PS) state, including its thermodynamic properties and dynamic behavior, still remain to be studied in more detail. One of the most interesting features of the PS state is the entwining between its dynamic and static properties. Numerous experimental data directly point to the fact that magnetization M(H,T) and resistivity ρ(H,T) are rather sensitive to the magnetic field (H) and temperature (T) sweeping rate. The mixed-valence manganites often display slow relaxation features, such glass like dynamic effects as aging and rejuvenation that hide experimentally the real equilibrium thermodynamic state of the system [1–10]. Whereas the phase separation scenario is generally accepted (an extensive review of all aspects of phase separation in manganites can be found in [11]), the origin of colossal magnetoresistance (CMR) in manganites is still under discussion. At present there are two competing opinions. Salamon et al. were the first who argued that the Griffiths-like phase and percolation type of metal-insulator (MI) transition could be a true theoretical ground for physics of manganites [12]. According to the opposite point of view, the Griffiths phase itself is insufficient for the appearance of CMR, and the formation of ferromagnetic polarons just above TC causes unique physical characteristics of doped manganites [13, 14]. Let us recall that the Griffiths phase (GP) was first proposed by Griffiths to explain the effects of quenched randomness on the magnetization of a diluted Ising ferromagnet [15]. The GP means the existence of short-range ordering of ferromagnetic clusters in paramagnetic matrix in the temperature region TC (p) < T < TG . Here TG stands for the ordering temperature of undiluted or homogeneous system TG = TC (p = 1) and the latter is known as the Griffiths temperature, while TC (p) is the Curie temperature of a diluted system; p is the probability of the nearest-neighbor exchange bonds to occur. For p < p c (p c stands for percolation threshold) no long-range ferromagnetic order is established in the system. In the GP, though the system does not hold spontaneous magnetization, there exist spatially distributed regions that are devoid of disorder forming finite size clusters having ferromagnetically correlated spins. As a direct consequence of limitation of the magnetic correlation length ξ to the finite size of the clusters (ξ does not diverge at the critical point) the magnetization fails to be an analytical function at the critical point. Bray [16]

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generalized this argument for any bond distribution (instead of bonds having strength of only J and 0, as discussed by Griffiths [15]) which reduces the long-range ordering transition temperature. It is generally accepted at present that quenched disorder is a prerequisite for the formation of the GP (for more details, see review [17] and references therein). Concerning doped manganites, in pioneering works of Salamon et al. [12], based on their results of magnetic and heat capacitance measurements, the ‘Griffiths phase’ paradigm in physics of these compounds was introduced. Indeed, in a manganite with a generic formula R1−x Ax MnO3 (where R is a trivalent rare earth and A is a divalent alkaline-earth ion) among the various forms of inhomogeneous phases, preformation of ferromagnetic (FM) clusters much above the FM long-range ordering temperature (TC ) was detected in different experiments: neutron scattering [18], ferromagnetic resonance [19], magnetic susceptibility measurements [12, 20], etc. (see also recent review on this topic [21]). At present, it is widely accepted that a system which demonstrates CMR is intrinsically inhomogeneous. Even in the best crystals, a competition between different phases results in the electronic PS state [22]. Authors [12] attributed these to the existence of the Griffiths-like phase in doped manganites. Accordingly, the MI transition is achieved due to percolation, i.e., following the Griffiths-like scenario, and theoretical model of the percolated MI transition was suggested to explain the CMR effect [23]. Namely, above TC , the FM metal clusters are isolated and a system is in insulating state. As temperature decreases, these isolated FM clusters increase in size and at TC the percolated channel forms. After that, the relevance of the Griffiths-like phase to colossal magnetoresistance of manganites was addressed in a number of reports [24]. Yet, applicability of the percolate phase transition scenario to CMR systems was addressed in a number of reports [13, 14], too. It is argued, in particular, that the unique physical characteristics of doped manganites in a temperature region above the Curie temperature TC are, most likely, due to ferromagnetic polarons, concentration of which increases upon cooling towards the Curie temperature [25]. Indeed, a strong electron-phonon coupling due to the Jahn-Teller effect of Mn3+ ion was shown to play a key role in CMR behavior and, as expected, is able to facilitate the formation of polarons [26]. The existence of polarons has been extensively confirmed experimentally [27]. It was argued that correlated polarons [28] may contribute to the nanometre-scale phase separation (see also [10] and references therein). Transition from itinerant large polarons to localized small polarons was proposed 3

to interpret the MI transition associated with the ferromagnet – paramagnet (FM – PM) transition in the CMR materials. (Models of polaron transport are discussed in detail in review [29].) In this report, we address the question whether these two scenarios of FM – PM phase transition, the percolation type (Griffiths-like) and the second-order-like, can be distinguished by their temporal behavior. The available data point out that temporal effects in the phase-segregated state can fundamentally differ from those in the state of large (or small) localized polarons. In particular, the Griffiths-like state is characterized by anomalously slow temporal evolution (relaxation) of the order parameter [30] and, thus, appears to be particularly favorable for the existence of out-of-equilibrium features. The competition between the coexisting phases opens the possibility for the presence of locally metastable states, giving rise to such interesting time dependent effects as cooling rate dependence [9], relaxation [1–8, 31] giant 1/f noise [32], two-level fluctuations [33], etc [34]. The similarity between phase-segregated manganites and glassy systems was also suggested [35]. The extremely slow relaxation observed in all the above-mentioned experiments opens an interesting question about temporal phenomena in a system in the GP. As far as the authors know, a clear understanding of the system’s slow dynamic behavior in the GP is still lacking, and in this report, we try to address some open questions. Systematic studies of the resistivity magnetization relation in doped manganites below and above the Curie temperature reveal a strong interplay between transport and magnetism in this system. This experimental fact was used as a physical basis of the model [36] which considered direct relationships between magnetization M(H,T), a function of magnetic field H and temperature T, and magnetoresistivity ρ(H,T). Mathematically the model is based on utilization of some hypotheses appropriate to the Preisach picture of the magnetization process [37] (see also textbook [38]). One of the remarkable features of the Preisach-based approach is that it yields joint description of hysteresis and thermal relaxation based on a few simple assumptions common to both aspects of the phenomenon [38–40]. We embraced this opportunity to address the temporal effects for a system in the PS state. The present paper discusses modeling of the temporal effects that leads to time dependence of magnetic and transport properties within the percolation scenario (Griffiths-like phase), and within the (conventional) FM – PM second-order-like transition. We try to replicate a broad spectrum of behavior observed experimentally on time dependence of mag4

netization and resistivity of CMR systems. The approach is based on generalization of some hypotheses appropriate to the Preisach picture of the magnetization process for half-metallic ferromagnets [36] and on the assumption that in doped manganites the PS state exists near the magnetic ordering temperature. For the systems under consideration, the noticeable relaxation of magnetization and resistivity has been found. The relaxation is shown to be most pronounced near the transition temperature, and to be approximately a logarithmic function of time. The theoretical results allow a direct comparison with the experimental data. We propose an additional experimental test to distinguish between the percolation scenario of magnetic and transport transition in doped manganites, and the ferromagnetic polaron picture. In particular, we show that a system with Griffiths-like state will exhibit nonequilibrium aging and rejuvenation phenomena, which in many aspects resemble those of a spin glass. We believe the experimental observation of a set of time decay properties will provide a settlement of apparently conflicting results obtained for different characteristics of PS manganites. The main part of the paper is organized as follows. In Sec. II the phenomenological model of the MI transition and magneto-resistivity of doped manganites is briefly outlined. For the sake of simplicity, we will consider the case when the PS state is formed by two phases, the FM phase with metal type conductivity and the PM one with polaron transport. A key feature introduced here is the interplay between the external magnetic field changes and the thermal relaxation effects that determines the time evolution of the system. In Sec. III the results of numerical simulations for a system with a second-order-like FM – PM phase transition and for a system with the Griffiths-like scenario FM – PM transition are presented. Our main goal is to reveal the effects related to the field or temperature sweeping time parameter t exp . In the next section, we discuss the results obtained and the available experimental data on temporal phenomena in doped manganites, and summarize our main results.

II.

THE MODEL

Let us briefly outline the model used (for details, see, Refs. [36]).

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A.

Magnetization

The conventional (scalar) Preisach model [37, 38] assumes that the system with an irreversible magnetic response consists of a large number of physically infinitely small interacting units (hysterons). Each unit is described with an elementary rectangular hysteresis loop, which has two field parameters, i.e. switching fields, hA and hB , hA ≥ hB . Under varying external magnetic field H, the unit will switch its magnetization µ to the ‘up’ state, if external field increases to H ≥ hA ; the unit will switch to the ‘down’ state, if external field decreases to H ≤ hB . However, hysterons magnetization will depend on the previous history of changes, if the external magnetic field is in the region hA ≥ H ≥ hB . The correspondence between a real magnetic system and a system of hysterons is achieved with Preisach function P(hA ,hB ). Function P(hA ,hB ) describes the density (number) of hysterons with switching fields (hA , hB ), is non-negative and normalized. A probability function P(hc ,hu ) is a given characteristic of the sample, which is attributed to the sample solely [38]. This function can be independently restored from magnetization measurements following some definite protocols (see, e.g., Ref. [41]). The plane with coordinates (hA ,hB ) or (hc ,hu ), where hu = (hA + hB )/2 and hc = (hA – hB )/2, is called the Preisach plane. With this plane, we can clearly illustrate the action of an external field on a system of hysterons. It is possible to assume, from the physical point of view, that hc may be non-negative only hc ≥ 0, i.e. the half of the plane, hA ≥ hB , will be meaningful only. Assume that the probability distribution function P(hA ,hB ) of the elementary units with switching fields (hA ,hB ) is known. Then the overall magnetization of the system may be calculated knowing the history of the magnetic field variation, which separates the Preisach plane (hA ,hB ) into two regions with hysterons being only in ‘up’ (+µ) or ‘down’ (- µ) states. It is possible to show (see Ref. [38]) that the half of the plane hA ≥ hB is always divided into two parts, S(+) and S(-), by a staircase line b(hc ). All hysterons in the part S(+) are switched ‘up’, all elemental switching volumes (hysterons) in the part S(-) have the magnetization ‘down’. As the result, the expression for the magnetization of the system has the form Z M (H) = 2µ

∞

Z dhc

0

b(hc )

P (hc , hu )dhu 0

Here the boundary line b(hc ) represents the history of the changes of the external magnetic field. 6

One of the remarkable features of the Preisach-based approach is that it yields joint description of hysteresis and thermal-fluctuation effects. Indeed, transitions between the two states of hysteron may also be induced by thermal fluctuations if the system is at a finite temperature T. For the experiment with a characteristic time parameter t exp (the field/temperature sweeping time), thermal transitions are bounded to those barriers which are less than or equal to the effective thermal fluctuation energy Wf l (T) = kB T ln(t exp /τ0 ), or the thermal fluctuation field Hf l = Wf l /µ, [38, 42]. Here kB is the Boltzmann’s constant; τ0 is a typical attempt time of the order of 10−9 – 10−10 s. For conventional static experimental measurements with experimental time t exp ∼ 100s the time parameter is ln(t exp /τ0 ) ∼ 20 25. As was demonstrated by Song et al. [39] (see also Ref. [43]) the field and thermal excitation conditions can be represented graphically in the Preisach plane (hc ,hu ), where each hysteron is located with respect to its characteristic fields hc and hu . Figure 1 (a) shows the Preisach plane for a system which was brought to a positive applied field H > 0 anhysteretically and which has been kept at finite temperature for some time. The quadrant enclosed between the boundaries hu = H + hc and hu = H - hc contains the bistable subsystems, which can potentially occupy either state ±µ(T). For the given temperature, there are two thermal excitation boundaries; one of which, hT min , identifies those subsystems whose lower energy barrier matches WT l (T) = µ(hc - |hu + H|), and the other one, hT max , is the location of subsystems whose higher barrier matches WT h (T) = µ(hc + |hu + H|). The subsystems that lie to the left or above the hT min boundaries have a thermally active lower barrier and a thermally inactive higher barrier, and consequently occupy their lower energy state exclusively. The subsystems within the shaded region in Fig. 1 (a) have two thermally active barriers, the equilibrium Boltzmann level populations and the superparamagnetic response function µsp (H,T) = µ(T)tanh[µ(T)(H - hu )/kB T]. The magnetization is obtained by evaluating the weighting state ϕ(H,T,hc ,hu ) of each hysteron by the Preisach function P(hc ,hu ) and integrating over the entire Preisach plane: Z ∞ Z ∞ M (H, T ) = dhc [ϕ(H, T, hc , hu )P (hc , hu )]dhu 0

−∞

For the regions I, II, III, and IV in the Preisach plane the hysterons state ϕ(H,T,hc ,hu ) is {+µ(T),-µ(T),µ(T)tanh[µ(T)(H - hu )/kB T], ±µ(T)}, respectively. Note that in the region IV, where hysterons can potentially occupy either of ±µ(T) state, the response is history 7

dependent and is determined by the ‘history’ line b(hc ). (As an example of the ‘history’ line b(hc ) a straight line d-e is shown in Fig. 1 (a) for simplicity). In order to make quantitative predictions about, e.g., the magnetization M(H,T) one must know the system state, given by the line b(hc ), and the Preisach distribution function P(hc ,hu ). The state line b(hc ) can be found, given the external field protocol. We will assume that the density of the Preisach characteristic fields is    

h2u (hc − hsw0 )2 2 −1/2 2 −1/2 exp − 2 , 2πΛu exp P (hc , hu ) = 2πΛc 2Λ2c 2Λu

(1)

where hsw0 , Λc and Λu stand for (related) distribution parameters. As already mentioned, for a given sample the function P(hc ,hu ) can be experimentally restored from magnetization measurements. The temperature dependence of the energy barriers is described by introducing the temperature dependence of the Preisach distribution parameters. The main physics can be captured by using a power law behavior defined as  γh  γc T T hsw0 (T ) = hc0 1 − , Λc (T ) = Λc0 1 − , TC/G TC/G γu γM   T T . Λu (T ) = Λu0 1 − , µ(T ) = µ0 1 − TC/G TC/G

(2) (3)

Here TC/G stands for the Curie temperature TC for the system with second-order-like FM – PM phase transition, or for the Griffiths temperature TG if above the magnetic ordering temperature the GP is realized. Thus, hc and hu distributions collapse into δ-function as T → TC/G from below, as expected on physical grounds.

B.

Magnetoresistivity

Consider the system under study as a system of magnetic hysterons distributed in real space [36]. In accordance with half-metallic properties of manganites, at zero temperature, the itinerant electron spin direction is parallel to the magnetic moment of the initial hysteron and may be parallel or antiparallel to the direction of the magnetic moment of the nearest neighbor hysterons. If parallel, the electron experiences weak scattering, and hence that is a ‘metallic path’. If antiparallel, the electron experiences strong scattering, and hence high resistance occurs (a ‘resistive path’). Therefore, the system can be represented as three-dimensional resistor networks. 8

At finite temperature, however, these networks are composed of hysterons which can potentially occupy three states. Figure 2 schematically illustrates a system of magnetic hysterons distributed in real space at finite temperature. Here the squares of different color represent hysterons with the magnetization ‘up’ (black), magnetization ‘down’ (white), and hysterons in superparamagnetic state (gray). The physics sketched in Fig. 2 enables to mimic the real situation observed experimentally for mixed-phase ferromagnetic manganites (cf. with Fig. 3 in Ref. [22]). The main concept introduced is that in the system under consideration the PS state exists near the magnetic ordering temperature and the percolation process can be not only due to the magnetic field variation, but due to temperature changes as well. Our approach implies that: (i) the metallic state reached from the insulator state with the magnetic field increase is not homogenous but has a substantial fraction of insulating clusters, and (ii) the metallic clusters exist in the range TM I ≤ T ≤ TG . Accordingly, this also suggests that the metallic state reached from the insulating state when temperature decreases is not homogeneous and even below TM I there is a substantial fraction of the insulating (paramagnetic) clusters. The metal-to-insulator transition is associated with the formation of a metallic cluster that spans the entire sample. Further increase in conductivity is likewise understood as the growth of this cluster. Thus, at finite temperature, as in the zero temperature case, if neighboring hysterons orientation is parallel, their conductivity is a ‘metallic type’. If antiparallel, the conductivity is an ‘insulating type’. An intermediate case for conductivity is observed if one or two of the neighboring hysterons are in the superparamagnetic state (we will attribute it to a ‘polaron type’ of conductivity). Overall, our model system can be imagined as a three-dimensional resistor networks. The system’s conductivity is obtained by weighting the conductivity of each resistor network path by the Preisach conductivity function, P(hAi ,hBi ,. . . , hAi+k ,hBi+k ). As a result, we obtain the following expression for an irreversible conductivity [36]: σirr (H, T ) =

∞ Z Z X k=1

b(hc )

Z Z dhAi dhBi ... dhAi+k dhBi+k b(hc )

P (hAi , hBi , . . . , hAi+k , hBi+k )σ(H, T ; hAi , hBi , . . . , hAi+k , hBi+k ) .

(4)

Here b(hc ) is the so-called Preisach memory function, the same as for magnetic hysteresis. Expression (4) contains all possible percolation channels including, in particular, the one shown in Fig. 2. (In Fig. 2, a metallic channel from the upper right corner to the bottom 9

left corner is formed by ‘white’ hysterons.) The correlation between magnetic and resistivity responses has now become a correlation between the Preisach magnetic function, P(hA ,hB ), and the Preisach conductivity function, P(hAi ,hBi ,. . . , hAi+k ,hBi+k ). For doped manganites, due to the short spin scattering length, the charge carrier probes the magnetization on a very short length scale. Therefore, as a good approximation one can assume that the Preisach conductivity function P(hAi ,hBi ,. . . , hAi+k ,hBi+k ) may be represented as a multiple function of the probability of each hysteron being involved in a conducting path. The irreversible part of the conductivity is obtained by evaluating the weighting state of the conductivity of hysteron pairs by the Preisach conductivity function with temperature-dependent parameters, and integrating over the entire Preisach plane. In the general case, all possible current trajectories through magnetic hysterons should be taken into account. Nevertheless, the analysis [36] shows that even the first approximation captures the main physics and reproduces qualitatively the main part of the experimental data. (This approximation suggests a qualitative relation between resistivity and normalized magnetization m of the form ρ(m) ∝ f (m2 ), which indeed is observed in numerous experiments on typical CMR manganites, see, e.g., [44].) Thus, in general, the system is considered as random three-dimensional resistor networks were a self-consistent formation of bonds with metal and polaron types of conductivity is not only due to magnetic field variation, but due to temperature changes, too. Both mechanisms of intrinsic percolation transition are considered on one basis. For comparison with real systems, it is necessary to supplement an irreversible response by a purely reversible term σrev (T ) that represents processes independent of the magnetic state of the system. We will assume, for simplicity, that the reversible term is due to a polaron mechanism of conductivity, and that above and below TM I the characteristic parameters of the polaron transport do not change, i.e., σrev (T ) = σsp (T ). The total conductivity then reads σtot (H, T ) = σm (T )Vm(R) (H, T ) + σsp (T )[1 − Vm(R) (H, T )].

(5)

(R)

Here Vm (H, T ) stands for the volume fraction with metallic type transport and is proportional to that of the ferromagnetic region on the entire Preisach plane. Within the first

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(R)

approximation, as follows from Eq. (4), the expression for Vm (H, T ) reads [36]: 2 2  bZ 1 (hc ) Z∞ Z∞ Z∞   Vm(R) (H, T ) = µ(T )/µ0 dhc dhu P (hc , hu ) + −µ(T )/µ0 dhc dhu P (hc , hu ) . 

0

−∞

0

b2 (hc )

(6) Here the boundary b1 (hc ,t exp ), line H-a-d-e in Fig. 1(a), separates the region where hysterons occupy the state +µ(T), while the boundary b2 (hc ,t exp ), line H-c-d-e in Fig. 1(a), separates the region were hysterons occupy the state –µ(T). Thus, the transport properties of this effective medium, Eq. (5), are characterized by a mixing of band-type and polaron-type conductivities. The results above could be easily rewritten for the system with the conventional (second order) FM – PM phase transition. In this case we will assume that only a polaron mechanism (R)

of conductivity exists above TC , i.e. Vm (H, T ) = 0 in Eq. (5) if T ≥ TC . At T ≤ TC the general Eq. (5) remains to be valid. To proceed further, we need expressions for the conductivities as functions of temperature. In the metallic regime, for such half-metals as manganites, electrical resistivity is found to obey the following relation (see, e.g., [45]): −1 ρdc (T ) = ρ0 + ρee T 2 + ρem T 9/2 = σm (T ),

(7)

where ρ0 = 1/σ0 is the residual resistivity arising from temperature independent processes such as impurities, vacancies, etc. The term ∼ T2 represents the electron-electron scattering, whereas the term ∼ T9/2 stands for the two-magnon scattering process in the ferromagnetic phase [46]. We take into account here that, in contrast to a conventional clean metal, for a ferromagnetic half-metal the one-magnon scattering process is forbidden. For manganites in particular, numerous experiments show that in the metallic phase of these materials electrical resistivity is mainly governed by the electron-electron scattering at low temperature while the contribution of the two-magnon scattering process is more important at high temperatures. Expression (7) reasonably describes experimental resistivity temperature dependence in manganites from low temperatures up to TC [45]. As already mentioned, a number of experimental results have provided strong evidence to the polaron-type conductivity in the paramagnetic region. In the framework of the small polarons model the electrical resistivity can be expressed as [47]: ρ(T) = ρ0 Texp(Ep /kB T), 11

where Ep is a sum of the activation energy required for creation and activation of the carriers hopping (for manganites, typically Ep ∼ 2÷3 kB TM I ). Experimental data also exhibit essentially the magnetic field independence of resistivity in the paramagnetic state. Within a comparative approach, we will approximate the conductivity of neighboring hysterons which are both in the superparamagnetic state, σsp−sp , as well as if only one of the neighboring hysterons is in the superparamagnetic state, σsp−m , by the form: σsp−sp (T ) = σsp−m (T ) = σsp (T ) = σ0 T −1 exp(−Ep /kB T ).

(8)

Expressions (1) - (8) are the basis for modeling the irreversible magneto-transport properties of the system for all experimental protocols. Let us give a brief comment concerning the approximations made (the reader can find the extended discussion of these questions in Ref. [36]b). Looking for intrinsic transport properties of phase-separated manganites we suggested that for an idealized system - a prefect single crystal - the contribution into magnetoresistivity of electrons scattering on a boundary between regions with different magnetization orientation (or on a boundary between ferromagnetic and paramagnetic regions) does not essentially change the results discussed here. We certainly simplify the picture. The contribution of these effects into magnetoresistivity can be accounted for within the models used for describing magnetotransport properties of granular ferromagnetic systems and polycrystalline manganites (see, e.g., [49] and references therein). Yet, as is well known, in manganites the itinerant charge carriers provide both the magnetic interaction between the nearest Mn3+ -Mn4+ ions (the so-called double-exchange interaction) and the system’s electrical conductivity. That is, the charge carrier probes the ions magnetization orientation on a lattice parameter distance. If the Mn3+ and Mn4+ core moments are not ferromagnetically ordered the double-exchange interaction is strongly suppressed [11]. For this reason, we believe that ignoring the above-mentioned contributions into magnetoresistivity is not fundamental for the results discussed here.

C.

Time dependent effects

It was demonstrated [40, 43] that the Preisach-based approach yields possibility to describe thermal relaxation dynamics, e.g., such as aging and memory effects, and in this subsection we briefly discuss how the model presented above can be generalized to include 12

the effects related to the experimental field or temperature sweeping rate, i.e. effects connected with the time constant t exp . To proceed in this direction note that the model contains two characteristic energies. One of these is the critical thermal fluctuation energy WCf l = kB TC/G ln(t exp /τ0 ), which quantifies the highest energy barrier that can be thermally activated for a system with critical temperature TC/G . The other is the mean zero-temperature anisotropy barrier WA (0) = µ0 hsw0 , which characterizes the ‘ability’ of the system to resist thermal fluctuations. As shown in Ref. [39], it is reasonable to introduce the dimensionless parameter η = WA (0)/WCf l . As was shown [36], a ratio of the characteristic energies WA (0)/WCf l is a natural parameter for classification of CMR systems in terms of their magnetoresistive properties, too. In order to make quantitative predictions about, e.g., the magnetization M(H,T,t exp ), one must know the system state, given by the line b(hc ,t exp ). The state line b(hc ,t exp ) can be found if external magnetic field protocol is known. On the other hand, in the presence of thermal activation, each unit (hc , hu ) relaxes to its energy minimum, with the transition rates described by the Arrhenius law. The competition between the external field changes and thermal relaxation, once averaged over the entire collection of units, now determines the time evolution of the system. If temperature is not too high, the relevant contributions to the relaxation process are concentrated mainly around the boundary of the superparamagnetic response region in the Preisach plane [40], see Fig. 1 (b) dashed region. With decreasing experimental sweeping rate the thermal fluctuation field Hf l = (kB T/µ0 )ln(t exp /τ0 ) increased. Thus, for the given system, the location of the superparamagnetic region in the Preisach plane is determined not only by temperature but by the experimental sweeping/observation time texp , too. As far as 1/η ∼ ln(t exp /τ0 ), by changing the experimental parameter t exp the related system’s response can be changed from reversible behavior, if ln(t exp /τ0 ) >> 1, to irreversible one, when ln(t exp /τ0 ) << 1. As will be shown in the next section, the magnetic and magnetoresistive responses, being functions of ln(t exp /τ0 ), qualitatively differ if the FM – PM transition is associated with the formation of a percolate ferromagnetic cluster, or if it follows a second order phase transition.

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III.

RESULTS OF NUMERICAL SIMULATIONS

Our particular interest is magneto-transport response of the system, which has been prepared following the field-cooling (FC) and the zero-field-cooling (ZFC) protocols (see, e.g., [38]). In the case of the FC protocol, the system is first heated in field H = 0 to some reference temperature Tref which is sufficiently high (Tref > TC/G ), for a time t ref which is sufficiently long, so that WT = kB Tref ln(t exp /τ0 ) exceeds all of the subsystem’s energy barriers. Thus, all of the individual subsystems are in thermal equilibrium, and the response of the ensemble to field changes is completely reversible. The sample is then cooled in non-zero field to the lowest measurement temperature. In the case of the ZFC protocol, after heating in zero field the system is cooled at H = 0 to the lowest measurement temperature, and then heated in non-zero field. In both cases, the system’s response depends on experimental field/temperature sweeping rate. Below we present the results of our numerical simulations of the system’s magnetotransport response for these two protocols. Here it is appropriate to note that while the electronic phase separation was believed to govern the magneto-transport in manganites close to phase transition, the energy scale of characteristic magnetic parameters (in particular those, which are included in the Preisach distribution function) still remains unclear. Within our qualitative analysis, we suggest the symmetric Preisach function; to be specific the parameters used in Eq. (1) are: Λc0 = Λu0 = 0.35, γc = γu = γh = 0.3, γM = 0.5. For other employed parameters, we choose: ρem /ρ0 = 2, ρee /ρ0 = 50, σm0 = 1, σsp0 = 0.2, and Ep /kB TG = 3 Oe/emu K. Magnetic field, resistivity, and magnetization are normalized to hc0 , ρ0 , and M0 , respectively. As mentioned above, a character of the system’s response to temperature is determined by the relation WA (0)/WCf l = µ0 hc0 /[kB TC/G ln(t exp /τ0 )]. Being interested in the temporal phenomena in the system’s properties, we choose the sweeping time t exp as a free parameter while the ratio µ0 hc0 /kB TC/G is fixed: µ0 hc0 /kB TC/G = 0.08 which corresponds to a system with η = 0.5 at static experimental measurements ln(t exp /τ0 ) = 25.

14

A.

Systems with polaron conductivity

Before considering systems with the GP, it is instructive to discuss the temporal effects in the systems with conventional second-order-like FM – PM phase transition and polaron type conductivity above the Curie temperature.

1.

Relaxation at constant magnetic field

The temperature dependence of the FC and ZFC magnetization response upon warming in magnetic field H/hc0 = 0.2 with different sweeping rate is illustrated in Fig.3. One can see that with changing t exp the system’s response drastically changes. Namely, with increasing t exp a bifurcation point at which the FC and ZFC magnetization follows different branches (we will refer to this temperature as the blocking one TB ), and near which the ZFC M(T) exhibits a peak as a function of temperature, is decreasing. While the FC response does not change with a sweeping rate (an exception is only a very fast sweeping when ln(t exp /τ0 ) = 5, 10), the ZFC response strongly depends on this parameter. Namely, with increasing t exp the response changes from irreversible to reversible one. This is a consequence of an already noted dependence of the parameter η on the sweeping rate. Note that the magnitude of the ferromagnetic ordering temperature, TC , does not changes with the ratio ln(t exp /τ0 ). The data for TB as function of t exc is summarized in Table I. The temperature dependence of the (normalized) resistivity in magnetic field H/hc0 = 0.2 for different cooling rate is shown in Fig.4. The ρ/ρ0 versus T/TC data in Fig.4 demonstrates that temperature at which the MI transition occurs, TM I , heavily depends on the parameter ln(t exp /τ0 ). The lower the cooling rate is, the lower the temperature where the MI transition is observed; simultaneously, the resistance drop increases by order. The data for TM I as function of ln(t exp /τ0 ) is collected in Table I, too. Comparing the results for TB and TM I we conclude that, within the accuracy of our calculations, in a system with the second-order-like FM – PM phase transition these temperatures coincide. A principal difference of the system under consideration from spin-glasses systems is the existence of a true equilibrium state that establishes a direction of the relaxation process. If at a given temperature and magnetic field the system is in a state where the ferromagnetic phase fraction is below its thermodynamic equilibrium value, there has to be an increase of

15

the magnetization as function of t exp . Accordingly, a decrease of the resistivity as function of time has been observed. In the opposite case, when the system is in a state where the ferromagnetic phase fraction is above its thermodynamic equilibrium value, the relaxation determines a decrease of this phase with increasing t exp . Simultaneously, the resistivity of the system increases. Figs. 5 and 6 illustrate these rejuvenation and aging effects. In accordance with the outlined physics, in the ZFC state, Fig. 5(a), the ferromagnetic phase volume is below its thermodynamic equilibrium value and the M(T,H,t exp ) grows with time. This is due to the relaxation process concentrated mainly around the superparamagnetic region boundary in the Preisach plane, which transforms the hysterons here into +µ(T) sate. In the FC state, Fig. 5(b), the ferromagnetic phase volume is above its thermodynamic equilibrium value and the M(T,H,t exp ) decreases with time. This is again due to the relaxation but now the hysterons mainly relax into −µ(T) sate (demagnetization processes). In both cases, the higher the temperature, the faster the thermodynamic equilibrium magnetization is achieved. At T > TC the ferromagnetic phase volume is independent on time (remember, for the system considered in this section at T > TC all hysterons are in the superparamagnetic state). Figure 6 illustrates the time dependence of the resistivity magnitude at different temperatures (the FC protocol). In accordance with the magnetization behavior, Fig. 5(b), the ρ(T,H,t exp ) grows with time as the volume of the phase with metallic type conductivity Vm (T,H,t exp ) decreases. In compliance with the approximation used, Eq. (6), the resistivity dependence on ln(t exp /τ0 ) is approximately two times slower than the magnetization dependence on ln(t exp /τ0 ). As follows from the data in Figs. 5 and 6, with the exception of a very short time, the magnetization and resistivity relaxation is seen to be described by logarithmic-like law. It is also clearly observed in the figures that due to the relaxation processes above some values of t exp (T) dependent on temperature the system is close to the equilibrium, i.e., the decay rate reaches zero at long time. Within this interval, both M and ρ can be fitted to a linear function of a logarithm of time, i.e., MZF C = M0 ln(t exp /τ0 ) + const, ρ = ρ0 ln(t exp /τ0 ) + const. Data in the figures enables us to estimate the time scale of the growth mechanism of the FM region, ranging from several seconds up to a few minutes. The relaxations are to be most pronounced at transition temperature; above TC the resistivity is independent on t exp . 16

2.

Relaxation at constant temperature, hysteretic response

Let us now consider the temporal effects at fixed temperature with changing magnetic field. There are two characteristic regions: below and above the blocking temperature. For the data shown in Figs. 7 - 9 the system’s parameters give static TB = 0.51, see Table I. Figure 7 illustrates a typical M(H) hysteretic response at T < TB for different sweeping time. One can see that the M(H,t exp ) behavior is conventional, i.e., fast sweeping field results in a large part of hysterons being in a quenched disorder that induces a broad hysteresis loop. On the other hand, slow sweeping rate leads to relaxation of hysterons due to thermal fluctuation field, so that magnetizations hysteresis loop shrinks. For the parameters used, magnetization response is reversible starting with ln(t exp /τ0 ) = 50 (at T = 0.3TC ). The results obtained for the resistivity dependence on magnetic field at T < TB and with different sweeping rate are shown in Fig. 8. The ρ(H,t exp ) behavior reproduces that for M(H,t exp ): the shorter the sweeping time, the larger the hysteresis loop. At slow field variation, the hysteresis loop shrinks but the magneto-resistive response is much more pronounced. Above the blocking temperature, T > TB , the static response is reversible. Yet, if the sweeping rate is fast enough, the hysteretic response is again observed in both magnetization and resistivity. The M(H,t exp ) and ρ(H,t exp ) dependences on magnetic field at T = 0.6TC > TB for different sweeping rate are shown in Fig. 9 (a) and (b), respectively. For comparison, in Fig. 9(b) the resistivity in paramagnetic phase with polaron type of conductivity is also shown (dashed line).

B.

Systems with the GP

Now we consider the relaxation effects in the system with the Griffiths phase.

1.

Relaxation at constant magnetic field

The temperature dependence of the ZFC and FC magnetization response upon warming in magnetic field H/hc0 = 0.2 with different temperature sweeping time t exp is shown in Fig. 10. Comparing the data in figures 3 and 10, we see that the system’s response differs qualitatively if the GP near the Curie temperature is present. As in the conventional case, 17

with the parameter ln(t exp /τ0 ) increasing a bifurcation temperature TB decreases. However, now the Curie temperature also strongly depends on the temperature sweeping time: it decreases with increasing ln(t exp /τ0 ). As a result, in contrast to a system with the secondorder-like FM – PM phase transition, the region with reversible response does not change. In addition, both the ZFC and FC magnetization responses are dependent on the time parameter. In particular, in contrast to the conventional case (see Fig. 3), the maximum of the ZFC magnetization response does not depend on the sweeping rate. Thus, for the system with the FM – Griffiths-like phase transition, not only the blocking temperature TGP B but also the actual transition temperature TGP C depends on the cooling rate (see Table II). As for the transport properties, we found that for a system with the GP resistivity the ρ(T,t exp ) dependence on temperature sweeping rate is qualitatively similar to that of a system with the conventional (second order) FM – PM phase transition, shown in Fig. 4. The data obtained for TGP M I as a function of ln(t exp /τ0 ) are presented in Table II. Analysis of Table II yields that in the system with the Griffiths-like phase the metal-to-insulator GP GP transition occurs within the interval TGP B < TM I < TC .

Fig. 11 illustrates rejuvenation and aging effects for a system with the GP: the ZFC and FC magnetization responses at field H/hc0 = 0.2 as a function of waiting time are shown in Fig. 11 (a), and (b), respectively. (For the data shown in Fig. 11 the system’s parameters give the static Curie temperature TGP C = 0.57 TG , see Table II.) Comparing the data in this figure and in Fig. 5, we see that the system’s magnetization response qualitatively differs if the GP exists above the Curie temperature. Namely, at low magnetic field a system with the GP behaves similar to a spin-glasses system. For MZF C , Fig. 11(a), thermal activation in applied magnetic field firstly switches hysterons with hA ≤ H to the ‘up’ state with the transition rates given by the Arrhenius law. Then, at large values of ln(t exp /τ0 ) the system relaxes to a demagnetized state which is an equilibrium one for the phase-separated system. In the case of the FC protocol, the starting magnetization already is finite, yet, in contrast to the system with the second order phase transition, presented in Fig.5 (b), the system relaxes to the equilibrium phase-separated state with M = 0, too. The dependence MF C = const - M0 ln(t exp /τ0 ) indeed was detected in some experiments on manganites [2, 8, 31, 48]. This behavior does not qualitatively change both below and above TGP C < TG (but at T < TGP C the relaxation is very slow, see the case T/TG = 0.2 in Fig. 11). The results in Fig. 11 obtained here within the phenomenological approach reproduce a more strict consideration 18

proving that the GP is characterized by an anomalously slow relaxation to zero of the order parameter (for more details see Refs. [16, 17, 30]). Figure 12 illustrates the time dependence of the metallic phase fraction, Eq. (6), and the resistivity ρ(T,t exp ) at different temperatures. In accordance to the magnetization behavior, presented in Fig. 11, the ρ(T,t exp ) grows with time as the volume fraction with metallic type conductivity Vm (T,t exp ) decreases. Data in the figures allows us to estimate the time scale of rejuvenation and aging effects from several seconds up to a few minutes. With the exception of a very short time, the resistivity relaxation becomes close to a logarithmic law and follows the dependence ρ = ρ0 ln(t exp /τ0 ) + const. When the system is close to the equilibrium-demagnetized state, the resistivity reaches saturation and relaxation rate is zero. This result agrees with the experimental observations [1, 2, 31, 35, 50, 51].

2.

Relaxation at constant temperature, hysteretic response

Let us now consider the temporal effects at constant temperature for a system with the GP phase when magnetic field is changing. Figure 13 (a) and (b) illustrates a typical M(H,t exp ) hysteretic response at T < TB and at T > TGP C , respectively. (For the data shown in Fig. 13 the system’s parameters give static TGP C = 0.57TG at t exp /τ0 = 25, H = 0.2 hc0 , see Table II.) One can see that now the M(H,t exp ) behavior differs from the conventional one (compare with Figs. 7 and 9(a), respectively) and at low magnetic field the M(H) behavior is substantially nonlinear. These peculiarities most distinctly reveal themselves in the GP, Fig. 13(b). For comparison, in Fig. 13(b) the conventional magnetization dependence in a paramagnetic phase, the Langeven function, is also shown (dashed line). In general, the magnetization dependence on t exp is as expected: the shorter the sweeping time is, the larger hysteresis loop is obtained. At T > TGP the magnetization response is reversible starting C with ln(t exp /τ0 ) = 25. We found that for the system with the GP the ρ(H,t exp ) dependence on magnetic field sweeping rate is qualitatively similar to that of a system with a second-order-like FM – PM phase transition. Namely, at T < TB the resistivity dependences on magnetic field are similar to those shown in Fig. 8. At T > TGP the ρ(H,t exp ) behavior qualitatively C reproduces that shown in Fig. 9 (b). Thus, the resistivity response qualitatively coincides for the systems with and without the GP. 19

IV.

DISCUSSION

Despite intense efforts towards microscopic understanding of the manganites’ physics many macroscopic features of the PS state, including its thermodynamic properties and slow dynamic behavior, still require further detailed research. In particular, since in PS materials the relative fraction of the phases can be altered by application of an external stimulus, one might expect to observe time dependence in ‘static’ physical characteristics. Time relaxation effects unveil the dynamic nature of the phase separated state. Thus, additional information about the physical properties of the system can be obtained by measuring temporal phenomena in these materials. In fact, slow dynamics of the PS state is the main factor determining the specifics of magneto-transport response of the system in different experiments. In this report, a phenomenological model of temporal effects in the PS state of manganites is developed. The PS state is considered within a general theoretical framework, based on the scalar Preisach model of hysteresis for describing irreversible phenomena in magnetically ordered materials. The approach includes the effects of the relaxation dynamics of collections of interacting, thermally activated two level subsystems [40, 43]. The theory identifies certain fundamental characteristics that play a primary role in determining the magnetic behavior characterized by nonequilibrium ageing and rejuvenation effects. Numerical simulations of nonequilibrium relaxation effects are presented and compared for two model systems that are actively discussed in connection with CMR of manganites. That is (i) a system with the percolation type of FM – PM phase transition and (ii) another one with the second-order-like FM – PM phase transition. Relaxation of magnetization and resistivity take place in both systems and low dynamics is displayed in the temperature range where out of equilibrium features are present. We found that the sweeping rate dependence of resistivity, as well as the magnetoresistive effect, in both systems is very similar, and the systems hardly can be distinguished by this parameter. An opposite situation is for magnetization: the temporal effects displayed in magnetization response for these systems were found to be principally different. For the system with the second-order-like magnetic transition the MF C response does not change with a sweeping rate, while the MZF C response strongly depends on this parameter. In the case of a system with the percolation type magnetic transition, both the ZFC and FC magnetization responses are heavily dependent on this parameter. In addition,

20

the magnetic transition temperature TC also strongly depends on the sweeping rate. Thus, the observation of the temporal effects, like those in Figs. 10 and 11, can be considered as a key feature of the Griffiths-like phase existence. There is a debate launched recently on whether the Griffiths phase is always a precursor to CMR in manganites or not [13]. Our results suggested that CMR effect is not always accompanied by the Griffiths phase near TC . A comment is needed here with respect to description of the PS as comprised out of two phases (FM and PM phases). It is obviously an oversimplified picture of the PS state in doped manganites. It is well known that the phase diagram of mixed-valence manganites is richer and the non-FM region could be formed by more than one insulating phase. e.g., by different charge-ordered antiferromagnetic (CO-AFM) phases [11]. As a result, temporal effects in real systems can be more complex. For example, in La5/8−y Pry Ca3/8 MnO3 (with y near 0.35) magnetization M(t) increases in time, while in La1−x Cax MnO3 (with x near 0.5) M(t) decreases with time, due to development of the CO phase (see Refs. [3, 9, 52] for more details). In conclusion, we carried out a phenomenological description of relaxation phenomena in magnetic and transport properties of perovskite manganites. The model is based on generalization of some hypotheses appropriate to the Preisach picture of the magnetization process on half-metallic ferromagnet and includes the effects of slow dynamics that lead to time relaxation and sweeping rate dependence. The temporal effects are considered for a system with the conventional (second order) ferromagnet – paramagnet phase transition and for a system with the Griffiths-like (percolation) transition at the Curie temperature. We found that the sweeping rate dependence of resistivity, as well as the magnetoresistive effect, in both systems is very similar, and by this parameter, the systems can hardly be distinguished. Yet, the temporal effects displayed in magnetization response for these systems are qualitatively different. In the case of a system with the percolation magnetic transition, magnetization response is heavily dependent on the sweeping rate, and the magnetic transition temperature itself strongly depends on this parameter. We point that observation of the temporal effects, such as an anomalously slow relaxation to zero of the order parameter, can be considered as a key feature of the Griffiths-like phase transition in doped manganites. The process of relaxation of phase fractions is unique to the phase-separated nature of the manganites and may constitute a new sort of magnetic glassiness - based on the macroscopic 21

yet dynamic coexistence of two (or more) dissimilar phases. This can add an additional aspect to the many unique properties of the perovskite manganites. Further studies of the detailed dynamics of phase separation are needed to fully characterize the nature of these processes. Fruitful discussions with S.M. Ryabchenko are gratefully acknowledged. The authors would like to thank Y.V. Melikhov for critical reading of the manuscript.

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TABLES Table I. The blocking TB and the metal-to-insulator transition TM I temperatures found at different temperature sweeping time t exp for a system with the second-order-like FM – PM phase transition; H =0.2 hc0 ; (see text for the magnetic parameters used). ln(t exp /τ0 ) 5 TB /TC

10

15

20

25

30

35

45

50

0.86 0.73 0.65 0.58 0.51 0.47 0.43 0.34 0.31

TM I /TC 0.88 0.75 0.65 0.58 0.52 0.49 0.44 0.38 0.36 GP GP Table II. The blocking TGP B , the Curie TC , and the metal-to-insulator transition TM I

temperatures, found at different temperature sweeping times t exp for a system with the Griffiths-like phase state; H = 0.2 hc0 . ln(t exp /τ0 ) 5

10

15

20

25

30

35

45

50

TGP B /TG 0.85 0.71 0.61 0.53 0.47 0.42 0.38 0.32 0.29 TGP C /TG 0.91 0.81 0.71 0.63 0.57 0.52 0.48 0.43 0.39 TGP M I /TG 0.88 0.75 0.65 0.58 0.52 0.49 0.44 0.38 0.36 Figure Captions FIG. 1. The Preisach plane of a system which was brought to a positive applied field H anhysteretically and which has been kept at finite temperature T < TC for some time. (a) Hysterons from the shaded region have two thermally active barriers, the equilibrium Boltzmann level populations and the superparamagnetic response function. (b) The dashed boundary describes the subsequent relaxation of hysterons to equilibrium population as a function of the observation time t exp . FIG. 2. An example of magnetic hysterons distribution in real space at finite temperature. Here the squares of different colors represent the hysterons in spin up (black), spin down (white), or superparamagnetic (gray) state. The picture imitates a local variation in the electronic properties of mixed-phase manganites. FIG. 3. (Color online) Temperature dependence of the FC and ZFC magnetization response upon warming in magnetic field H/hc0 = 0.2 at different sweeping rates.

26

FIG. 4. (Color online) Temperature dependence of the FC normalized resistivity in a magnetic field H/hc0 = 0.2 at different temperature sweeping rates. FIG. 5. (Color online) Time dependence of (a) the ZFC and (b) FC magnetization isotherms; H/hc0 = 0.2. FIG. 6.

(Color online) Time dependence (field cooling protocol) of the resistivity

isotherms; H/hc0 = 0.2. FIG. 7. (Color online) Magnetization dependence on magnetic field at T < TB (= 0.51) and at different sweeping rates. 1 - ln(t exp /τ0 ) ) = 15, 2 - ln(t exp /τ0 ) = 20, 3 - ln(t exp /τ0 ) = 25, 4 - ln(t exp /τ0 ) = 30, 5 - ln(t exp /τ0 ) = 35, 6 - ln(t exp /τ0 ) = 50. FIG. 8. (Color online) Resistivity dependence on magnetic field at given temperature below the blocking temperature (TB = 0.51) at different sweeping rates. 1 - ln(t exp /τ0 ) ) = 15, 2 - ln(t exp /τ0 ) = 20, 3 - ln(t exp /τ0 ) = 25, 4 - ln(t exp /τ0 ) = 30, 5 - ln(t exp /τ0 ) = 35. FIG. 9. (Color online) The magnetization (a) and resistivity (b) dependences on magnetic field at T > TB and at different sweeping field rates. A dashed line shows the resistivity in paramagnetic phase with polaron type of conductivity. FIG. 10.(Color online) Temperature dependence of the FC and ZFC magnetization response of a system with the Griffiths-like phase above TC upon warming in a magnetic field H/hc0 = 0.2 at different sweeping rates. FIG. 11. (Color online) Time dependence of the (a) ZFC and (b) FC magnetization < TG ; H/hc0 = isotherm for a system with the Griffiths-like phase below and above TGP C 0.2. FIG. 12. (Color online) Time dependence of (a) the metallic phase fraction and (b) the resistivity for a system with the Griffiths-like phase at different temperatures; solid lines – ZC, dashed – lines ZFC; H/hc0 = 0.2. FIG. 13. (Color online) The magnetization dependences on magnetic field at (a) T < TB and (b) T > TGP C at different sweeping rates. 1 - ln(t exp /τ0 ) = 15, 2 - ln(t exp /τ0 ) = 20, 3 ln(t exp /τ0 ) = 25, 4 - ln(t exp /τ0 ) = 30, 5 - ln(t exp /τ0 ) = 35, 6 - ln(t exp /τ0 ) = 50. A dashed line shows the Langeven dependence. 27