Observation of isotropic-dipolar to isotropic-Heisenberg crossover in Co- and Ni-substituted manganites

High-precision ac susceptibility data have been taken on the La0.7Pb0.3Mn1−y(Co, Ni)yO3 (y=0, 0.1, 0.2 and 0.3) manganite system over a wide range of amplitudes and frequencies of the ac driving field in a temperature range that embraces the critical region near the ferromagnetic (FM)–paramagnetic (PM) phase transition (occurring at the Curie point TC). Elaborate data analysis was performed that (i) enabled the first observation of a crossover from a three-dimensional (3D; d=3) isotropic long-range dipolar asymptotic critical behavior to a d=3 isotropic short-range Heisenberg critical regime as the temperature is raised from TC in the compositions y≠0 (no such crossover is observed in the parent compound, y=0) and (ii) brought out clearly the importance of dipole–dipole interactions between the eg electron spins and/or between eg–t2g electron spins in establishing long-range FM order in the insulating state. The final charge and spin states of Co and Ni ions, substituting for the Mn3+ and/or Mn4+ ions, are arrived at by using a scenario of substitution that is consistent not only with the present results but also with the previously published structural, thermo-gravimetric, bulk magnetization, dc magnetic susceptibility and electrical resistivity data on the same system. The marked similarity seen between the magnetic behavior of the manganite system in question and the quenched random-exchange ferromagnets, within and outside the critical region, suggests that the percolation model forms an adequate description of the FM metal-to-PM insulator transition.


Introduction
The rich diversity and complex nature of the physical phenomena occurring in hole-doped manganite perovskites R 3+ 1−x A 2+ x Mn 3+ 1−x Mn 4+ x O 2− 3 (R = La, Sm, Pr, Nd; A = Ca, Sr, Ba, Pb, Cd) have evoked a great deal of interest in the study of such systems (for comprehensive reviews, see [1]- [7]). A quantitative, or in some cases even qualitative, understanding of these physical phenomena necessitated a theoretical approach that goes beyond the double exchange (DE) mechanism, initially proposed for metallic ferromagnetism, in recognizing the importance of (i) the superexchange (SE) between the nearest-neighbor localized t 2g spins, which can be antiferromagnetic (AF), or even ferromagnetic (FM), depending on the relevant e g orbital configurations, (ii) the strong coupling of the twofold degenerate e g orbitals to the octahedral symmetry-breaking Jahn-Teller (JT) lattice modes and (iii) the strong on-site Mott-Hubbard Coulomb repulsion between two e g electrons in different orbitals. By contrast, the DE formalism relies on a large Hund coupling between the e g electron spin and the localized t 2g spin on the same site to optimize the kinetic energy, or the inter-site hopping amplitude, of the e g electrons. However, no single theoretical model is able to adequately handle the interplay between the charge, spin, orbital and lattice degrees of freedom and thereby account for the existence of a variety of phases in the magnetic phase diagram of manganites.
A partial substitution of Mn by TM (TM = Fe, Co, Ni) ions in hole-doped La 1−x A x Mn 1−y TM y O 3 manganites alters the magnetic and transport properties of the host (LAMO) in a specific fashion, depending on the charge and spin states of the TM solute ions, and thereby provides a fertile testing ground for the theoretical models proposed hitherto. The bulk of such studies have been carried out on the La 0.7 Pb 0.3 Mn 1−y (Fe, Co, Ni) y O 3 manganite system [8]- [14] rather than on the A = Ca or Sr counterparts [15]- [18]. While there seems to be broad consensus [8,9,11], [15]- [18] that the Fe ion exists in the Fe 3+ high-spin (HS) t 3 2g e 2 g S = 5/2 state, opinions about the charge and spin states of Co and Ni ions are sharply divided. Configurations such as Co 3+ low-spin (LS) [11,13] t 6 2g e 0 g S = 0, Co 3+ covalent intermediate-spin (IS) [13,19,20] t 5 2g e 1 g S = 1, Co 3+ HS [9] t 4 2g e 2 g S = 2, Co 4+ HS ( [13] and references therein) t 3 2g e 2 g S = 5/2, Ni 2+ HS [10] t 6 2g e 2 g S = 1, Ni 3+ LS ( [14] and references therein) t 6 2g e 1 g S = 1/2 3 and Ni 3+ HS [9] t 5 2g e 2 g S = 3/2 have been invoked in the case of Co and Ni ions. In all cases, the solute Fe, Co and Ni ions are supposed to partially substitute for the Mn 3+ HS t 3 2g e 1 g S = 2 ion.
Following the realization that the critical behavior of a spin system in the vicinity of a magnetic order-disorder phase transition is solely governed by the nature of the magnetic ordering present, critical phenomena have been investigated in a large number of manganite systems. However, conflicting reports about the critical exponents that characterize the FM-paramagnetic (PM) phase transition have rendered such studies inconclusive. For instance, the susceptibility critical exponent γ is in the range 1.03-1. 45 [39]. Since the reported exponent values either range between the values theoretically predicted for different universality classes (e.g. three-dimensional (3D) isotropic short-range Heisenberg d = 3, n = 3 with [40] γ = 1.386(4), 3D isotropic short-range Ising d = 3, n = 1 with [40] γ = 1.241(2), mean-field (MF) with γ = 1.0, where d and n are the space/lattice dimensionality and spin dimensionality, respectively) or are completely anomalous, the nature of the interactions that sustain FM order below T C remains obscure. Consequently, the basic issue of whether the percolation picture or the two-fluid model forms a correct description of both the FM-PM and the associated metal-insulator transition cannot be unambiguously resolved.
The unresolved issues mentioned above and the possibility that non-asymptotic data could be responsible for the chaotic dispersion in the γ values prompted us to undertake an exhaustive study of the critical behavior near the FM-PM phase transition in the La 0.7 Pb 0.3 Mn 1−y (Co, Ni) y O 3 (y = 0, 0.1, 0.2, 0.3) system. The critical behavior of y = 0 (the LPMO host) turns out to be markedly different from that of the Co-and Ni-doped (i.e. y = 0) samples. Barring a systematic trend with y, the compositions y = 0.1, 0.2, 0.3 exhibit essentially the same behavior in the critical region. Therefore, only the representative results (obtained on the compositions y = 0 and y = 0.2) are presented and the observed trend with y is highlighted as and when required.

Experimental details
Extensive ac magnetic susceptibility measurements have been carried out on polycrystalline samples of nominal composition La 0.7 Pb 0.3 Mn 1−y (Co, Ni) y O 3 (y = 0, 0.1, 0.2, 0.3) at an ac driving field of rms amplitude ranging from 0.1 to 10 Oe and frequency ranging from 100 Hz to 10 kHz over a wide temperature range 4 K T 355 K that embraces the critical region. These samples were prepared by the sol-gel method [8], thoroughly characterized by scanning electron microscopy, energy-dispersive absorption of x-rays and neutron diffraction [13,14], and are the same as those used previously for bulk magnetization, dc magnetic susceptibility and electrical resistivity measurements [12]- [14]. In the Ni-and Co-containing compounds, the presence of Ni 2+ , Ni 3+ , Ni 4+ and Co 2+ , Co 3+ , Co 4+ at different temperatures [13,14] was revealed by thermo-gravimetric analysis. The redox titration yielded the Mn 3+ /Mn 4+ ratio as 0. 35 Figure 1 displays the temperature dependence of the real (reactive/dispersion), χ (ν), and imaginary (dissipative/absorption), χ (ν), components of ac susceptibility (ACS) for LPMO measured at an ac driving magnetic field of rms amplitude h ac = 1 Oe and frequency ν = 300 Hz, and for Ni 0.2 and Co 0.2 measured either at a fixed frequency ν = 1 kHz and rms amplitudes h ac = 0.1, 1, 5, 10 Oe or at h ac = 1 Oe in the frequency range 100 Hz ν 10 kHz 5 over a wide temperature range (which embraces the critical region near the FM-PM phase transition). Recognizing that the critical fluctuations of the order parameter (spontaneous magnetization) have maximum amplitude as well as range (divergent spin fluctuation-spin fluctuation correlation length) at the critical point, T C , and that the fluctuations are directly related to dissipation (the fluctuation-dissipation theorem), the temperature T p at which a sharp peak in χ (T ) occurs provides a reasonably accurate estimate of T C . No discernible shift in T p , or equivalently in the Curie temperature T C , over a frequency range spanning two decades testifies to a true thermodynamic phase transition at T C . Since χ is at least two orders of magnitude smaller (figure 1) than χ at any temperature, χ decides the magnitude of ACS. In the PM state (T T C ), the inverse intrinsic susceptibility, χ −1 (T ), varies with temperature as
The 'range-of-fit' (ROF) analysis (detailed elsewhere [41]- [44]) of the χ −1 (T ) data, based on equations (1) and (2), permits an accurate determination of T C and N and hence of A eff (T ) and γ eff (T ), as elucidated below. Since T p ∼ = T C , T C is held constant at one of the temperatures (≈80 mK apart), in the range T p − 10 K T C T p + 10 K, at which χ has been measured. The parameters N , A eff and γ eff are varied to optimize agreement between the experimentally observed and theoretically predicted (by equations (1) and (2)) χ −1 (T ) over the temperature range min max , where min corresponds to the first data point above T C . In the ROF analysis, the temperature range min max of the fit is varied by keeping min fixed and progressively raising ≡ max (by including one more data point above T C for successive fits), and the variations of N , A eff and γ eff with are monitored. The correct choice of T C (and hence of N , A and γ ) is the one that yields constant values for N , A eff and γ eff in the asymptotic critical region (ACR), i.e. N ( ) → χ −1 (T C )/4π , A eff ( ) → A and γ eff ( ) → γ as → 0. Figure 2 illustrates this behavior of N ( ), A eff ( ) and γ eff ( ), as an outcome of the ROF analysis, by treating the χ −1 (T ) data taken at h ac = 1 Oe and ν = 1 kHz on Ni 0.2 as an example. Out of the temperature variations of N , A eff and γ eff yielded by the ROF analysis for the three specified choices of T C , the criterion that N ( ), A eff ( ) and γ eff ( ) should attain their limiting but constant values as → 0 is satisfied only for the (correct) choice T C = 247.43 K. The values of T C determined in this way are displayed in table 1. Figure 3 demonstrates that the value for T C , so obtained, exactly coincides with the temperature corresponding to the inflection point (where dχ /dT goes through a sharp well-defined minimum) in the χ (T ) curve (compare the T C values given in figures 2 and 3 and table 1) but lies just above (within 1 K) T p , the temperature at which χ peaks. Another important observation is that the T C of the parent compound gets depressed progressively as the Ni or Co concentration increases but the Co substitution depresses T C (of the host) at a much faster rate (see also [12]- [14]). The values of T C , so determined, are in much closer agreement with the numerical estimates directly obtained earlier from the 'zerofield' neutron diffraction data [13,14] than with those (less reliable values) deduced from the 'in-field' bulk magnetization data through an extrapolation to zero magnetic field. Having determined the T C correctly, the inverse intrinsic susceptibility, χ −1 ( ), is obtained by subtracting 4π N ( = 0) from χ −1 ( ). The behavior of χ −1 ( ) over a wide temperature range above T C is shown in the insets of figure 4. χ −1 ( ), in the critical region, is depicted in the upper panels of figure 4 and compared with the theoretical fits that use the 3D Heisenberg value of γ H = 1.386 for the asymptotic critical exponent γ in equation (1). The bottom panels display the corresponding percentage deviation of the observed χ −1 ( ) data from the theoretical fits plotted against the reduced temperature . These deviation plots demonstrate that the 3D Heisenberg value of γ adequately describes χ −1 ( ) in the ACR, more so in the LPMO compound. The physical significance of such deviations, particularly at intermediate temperatures ≈ 0.03 within the critical region, for Ni 0.2 and Co 0.2 , will become clear at a later stage when a theoretical explanation is provided for the γ eff versus plots shown in figure 5. The salient features of γ eff ( ) (figure 5) are as follows. (I) γ eff in LPMO assumes the 3D isotropic Heisenberg (1H) value of γ H = 1.386(4) at < H and goes through a peak at p ∼ = 0.01 with γ p eff ∼ = 2.45 before approaching the MF value of γ = 1 at > 0.06. (II) By contrast, irrespective of the Co or Ni concentration in the range 0.1 y 0.3, a dip in γ eff ( ) occurs at dip , a peak at p and γ eff attains the values γ D ∼ = 1.39, γ dip = 1.280(4) and γ H = 1.386(4) at * , dip and * * , respectively. Note that the values of dip , γ dip , γ H ( = * * ) and * * for different data sets taken on Co 0.2  (3) 1.389 (4) 1.389 (3) 1.389 (2) 1.389 (4) 1.389 (2 (2) 0.085 (4) 0.077 (3) 0.077 (2) 0.092 (2) 0.080 (6) [0.125 (6) (3) 1.281 (3) 1.280 (2) 1.280 (2) 1.279 (2) 1.283 (4 (2) 1.386 (3) 1.386 (2) 1.386 (5) 1.386 (3) 1.386 (5 (2) 0.041 (2) 0.041 (2) 0.040 (3) 0.039 (3) 0.041 (2) [0.060(2)]   and Ni 0.2 are listed in table 1. (IIIa) As y increases, * , dip , * * and p shift to higher temperatures, while the peak value γ p eff increases. (IIIb) For a given y, e.g. Co 0.2 and Ni 0.2 , * , dip , * * and p are higher, while γ p eff is lower in the case of Co. γ eff ( ) of type (I) has been observed [44,45] previously in several spin systems with quenched random-exchange disorder, which behave as normal ferromagnets down to the lowest temperature. In sharp contrast, γ eff ( ) of the type (II) + (III) occurs [44] in amorphous ferromagnets (a-FMs) that have composition close to, but above, the percolation threshold for the appearance of long-range FM order and exhibit re-entrant behavior at low temperatures. Consistent with these observations, the composition La 0.7 Pb 0.3 MnO 3 , corresponding to an Mn 3+ /Mn 4+ ratio that maximizes the DE interaction (and hence T C ), has a collinear FM ground state, whereas in the composition range 0.1 y 0.3, the Co-or Ni-substituted LPMO undergoes a transition to the re-entrant-like state (where long-range FM order coexists with cluster spin-glass-like order) at T T C [8]- [14]. In a-FMs, the type (I) γ eff ( ) has found the following interpretation in terms of the percolation model [45] in which finite FM clusters coexist with an infinite FM network for T T C . The spin-spin correlation length, ξ(T ), of the d = 3 Heisenberg-like FM network diverges at T = T C so that the presence of the spin clusters is not felt in the ACR, H , where 11 the d = 3 Heisenberg-like critical behavior is observed. With temperature increasing above T C , ξ(T ) declines rapidly and the infinite FM network breaks up into finite FM clusters, so that at T > T C , finite FM clusters are embedded in a PM matrix. At = H , ξ( ) equals the caliper dimension of the largest spin cluster, (d c ) max , with the result that the magnetic inhomogeneity in the spin system is now no longer averaged out and γ eff begins to increase as exceeds H till it attains a maximum value γ p eff at = max when ξ( ) ∼ = (d c ) min , the smallest cluster size. It follows that the smaller the average cluster size and the narrower the cluster size distribution, the higher the values of H and p , the narrower the peak in γ eff ( ) and the lower the peak value γ p eff (for details, see [45]).
In the case of manganites, a percolation picture (strikingly similar to that described above), in which metallic FM clusters of various sizes coexist with the insulating PM matrix at T > T C , has gained wide acceptance [3,5] over the years. As the temperature is lowered towards T C , the localized nearest-neighbor t 2g spins (coupled by Heisenberg-like interactions) in the insulating PM phase order better progressively due to the growing spin-spin correlations between them. Ordering of t 2g spins, in turn, facilitates hopping of e g electrons and hence the metallic FM clusters grow in size. For H , ξ(T ) is much larger than the size of the largest FM cluster and a d = 3 Heisenberg-like critical behavior is expected, as observed in figure 5 for LPMO. However, at T = T C and for temperatures below T C , metallic FM clusters coalesce to form an infinite percolating network, in which the finite insulating AF clusters are embedded, and a transition from the insulating (T > T C ) to metallic (T < T C ) state occurs. Generally, at constant hole density, the transition between the electronic phase-separated metallic FM and insulating AF/CO (charge ordered) phases in manganites is of the first order. However, the disorder introduced by a partial but random replacement of R 3+ ions by A 2+ ions of larger/smaller ionic radius, and in the present case by the substitution at the Mn site, results in randomly distributed coexisting insulating and metallic percolative clusters. The temperature-induced growth of clusters and the percolation of such clusters render the insulating PM-to-metallic FM transition continuous (second order).
Although the mechanisms responsible for the formation of finite FM clusters and FM network/matrix are radically different in the percolation pictures for a-FMs and manganites, the type of interplay between ξ(T ) and the temperature-dependent length scale introduced by the finite FM spin clusters at T > T C [45] permit the conclusion that with respect to the parent compound y = 0, the average FM cluster size reduces while the cluster size distribution narrows down as one goes from Ni to Co for a given y. Monte Carlo (MC) simulations [3] on hole-doped manganites have shown that the weaker the disorder, the larger the cluster size. In view of this MC result, the above inference about the cluster size and cluster size distribution asserts that the disorder increases in the sequence LPMO → Ni-→ Co-substituted LPMO. Figure 6 shows an enlarged view of the steep minimum (dip) in γ eff ( ), witnessed in the case of Ni 0.2 and Co 0.2 at temperatures close to T C in figure 5, for the χ (T ) data taken at h ac = 1 Oe for different frequencies (ν) and at ν = 1 kHz for different values of h ac . With a view to bringing out clearly the physical significance of the dip in γ eff ( ), and hence of the type (II) + (III) behavior of γ eff ( ), a brief account of the relevant theoretical results is given below. Motivated by the fact that long-range dipole-dipole interaction between spins (magnetic dipole moments) is invariably present in all real magnetic materials, the renormalization group (RG) calculations [46] on an isotropic spin system with d = 3 and n = 3 revealed that dipolar perturbations render the short-range IH fixed point of RG unstable and give rise to a new stable 'isotropic dipolar' (ID) fixed point. However, the critical exponents characterizing the ID fixed point, calculated to two-loop order, are very close [47] to (within 0.5%) those associated with a d = 3 pure IH ferromagnet. Subsequent theoretical investigations, using different RG methods [48]- [51], dealt exclusively with the thermally induced crossover from a critical behavior at T > T C , governed by the short-range isotropic exchange (Heisenberg) interaction, to the asymptotic ( → 0) critical behavior dictated by the long-range dipolar interaction. Irrespective of the RG method used, the most remarkable outcome of these theoretical approaches is the occurrence of a deep minimum in γ eff ( ) in the crossover region ( > 0), which turns out to be a universal feature of the isotropic d = 3, n = 3 spin systems with weak dipolar interactions (compared to the Heisenberg exchange interactions). The numerical results of these RG calculations are summarized in table 2.
The prediction of a dip in γ eff ( ) prompted us to attempt a quantitative comparison between the theory and experiment. These RG calculations [48]- [51] yield the final expressions for susceptibility and its effective critical exponent, defined as [52] γ eff ( ) = ∂[ln χ −1 ( )]/∂(ln ), as where is a non-universal critical amplitude, τ = ( g / H ) φ , H = [T − T C (0)]/T C (0), T C (0) ≡ T C (g = 0) is the transition temperature and γ H the susceptibility critical exponent of the pure (g = 0) IH d = 3, n = 3 spin system, g = [T C (g) − T C (0)]/T C (0) is the shift in T C caused by ID interactions of relative strength (i.e. the ratio of the ID energy to isotropic exchange energy) g, γ D is the susceptibility critical exponent of the d = 3 ID fixed point, φ = γ H is the crossover exponent, p(τ ) is the 'correction-to-scaling' function, τ −1/φ − 1 = = [(1 + g )/ g ] and = (T − T C (g))/T C (g). The explicit forms of p(τ ) and its derivative with respect to τ , p (τ ), are given in [50]. Equations (3) and (4) assert the following. (a) The ID interactions become important below a crossover temperature co ≈ dip ≡ g 1/φ such that, for co , the asymptotic critical behavior is that of a d = 3 ID ferromagnet, whereas for co the spin system behaves as a pure d = 3, n = 3 system. (b) γ eff ( ) goes through a minimum in the crossover region such that, at = dip , γ eff has the universal (independent of the RG coupling parameter [51]) value [50] γ dip = 1.28(1), for a weak dipolar system with g 10 −4 . (c) γ eff → γ D and γ H in the ID (τ → 1) and IH (τ 1) limits. The results for Ni 0.2 and Co 0.2 , presented in figure 6, testify to the validity of the predictions (a)-(c). Considering that the dipolar interactions between the localized t 2g electron spins are extremely weak compared to the Heisenberg-like SE interactions (which, in turn, are swamped by DE interactions) in hole-doped manganites such as LAMO (A = Ca, Sr, Ba, Pb, Cd), a crossover to the ID critical behavior is expected to occur at temperatures, < co ≈ 10 −6 , which are inaccessible to experiments. Thus, it is not surprising that a dip in γ eff ( ) is completely absent in LPMO. In the ACR (0 < * co ), the expansion of the scaling function p(τ ) in equations (3) and (4) yields the result [42,53] and where a χ and D are the leading 'correction-to-scaling' amplitude and exponent, and˙ is the reduced temperature at which χ −1 = 4π . The γ eff ( D ) data (symbols) and the best leastsquares linear fits (straight lines), based on equation (6), are shown in figure 7 (while the corresponding critical-point parameters are displayed in table 1 for different sets of data taken on Ni 0.2 and Co 0.2 ). It is evident from figure 7 that equation (6) with D = 0.55 [44] is valid in the ACR ( * ) and yields the true asymptotic susceptibility critical exponent γ D = 1.390(1), irrespective of Co or Ni concentration (and hence γ D is universal, as expected) and (nonuniversal) composition-dependent a χ . Like a χ , the width of the ACR increases with y and is larger for Co, e.g. a χ = 0.08(1), ACR width: 5.0 × 10 −4 * = 7.0 × 10 −3 for Ni 0.2 and a χ = 0.12(1); 5.0 × 10 −4 * = 9.0 × 10 −3 for Co 0.2 (table 1). This finding implies that among Ni-and Co-doped systems, the dipolar interactions are stronger in Co and grow in strength as y increases in both Ni-and Co-containing samples. Inserting the experimental value of˙ in equations (7) and (8) with D = 0.55 yields a χ = 0.973(6) for Ni 0.2 and a χ = 0.966(3) for Co 0.2 . Compared to the observed values displayed in table 1, these estimates are an order of magnitude higher. In order to make a direct comparison with the predictions of the RG calculations [50], γ eff is plotted against log 10 in figure 8. These γ eff -log 10 plots make use of the values, estimated from the relation = [(1 + g )/ g ] and equation (8) using the observed values of˙ . A glance at figure 8 reveals that the theory overestimates the width of the crossover region, underestimates γ D and γ H but estimates dip with reasonable accuracy and yields the same value γ dip = 1.28 as observed for γ eff at = dip . Considering that the most accurate theoretical value [40] γ H = 1.386(4) is in perfect agreement with the currently determined value of γ H , more refined RG calculations for the IH-ID crossover are needed to reproduce the experimental values of γ D and γ H . That dip is higher in Co than in Ni basically reflects that the strength of the dipolar interactions, g, is greater in Co.
A comparison between the deviation plots, shown in the lower panels of figures 4(b) and (c), and the γ eff ( ) plots, displayed in figures 6(a) and (b), reveals the following. Firstly, the inferences drawn from a direct fit to the χ −1 (T ) data, based on equation (1) and using the 3D Heisenberg value γ H = 1.386 for γ eff , are consistent with the results of the ROF data analysis in that γ eff ∼ = γ H for * 0.01 and at * * 0.04 for Ni 0.2 ( * * 0.06 for Co 0.2 ). Secondly, the significantly large deviations of the χ −1 (T ) data from the fit, yielded by equation (1) when γ eff = γ H , at intermediate temperatures (figures 4(b) and (c)) basically reflect the reduced magnitude (figures 6(a) and (b)) of γ eff , compared to γ H = 1.386, in the crossover region. Thirdly, a wide dispersion in the values of the critical exponent γ reported in the literature for a given series of the hole-doped manganites with or without substitution at the Mn site can be attributed to one or more of the following factors [45]. (a) T C not fixed to an accuracy demanded by a precise determination of the critical exponents. (b) The determinations are based on data that either fall completely within the crossover region or well outside the crossover region or partly overlap the ACR. (c) Uncertainties associated with the extrapolation of the magnetization data, taken in finite magnetic fields, to zero field to arrive at the 'zero-field' quantities, such as spontaneous magnetization and initial susceptibility.
Recognizing that the resolution of the controversy surrounding the charge and spin states of Co and Ni in Co-or Ni-substituted hole-doped manganites is crucial to understanding the origin of dipolar interactions and the process of electron localization/charge ordering, an attempt is made to arrive at the most probable scenario of substitution at Mn 3+ and/or Mn 4+ sites that is consistent with the following main observations. (i) Reduction in the unit-cell volume (v)/lattice parameters (a and c), the critical temperature T C and the magnetic moment per 3d transition metal atom (µ) in the FM and PM states, with the Ni or Co concentration, y.
(ii) Robustness of the tolerance factor against substitution. (iii) Localization of the conduction electrons as y increases, as inferred from the increase in electrical resistivity with y irrespective of temperature [12]- [14]. The substitution of Mn 3+ HS ions (with S = 2 and ionic radius r ion = 0.645 Å) by (a) Ni 3+ LS (S = 1/2, r ion = 0.56 Å) or Co 3+ LS (S = 0, r ion = 0.545 Å) reduces both v and µ at a rate that is much faster than the observed one; (b) Ni 2+ HS (S = 1, r ion = 0.69 Å) increases v but decreases µ; (c) Co 3+ HS (S = 2, r ion = 0.61 Å) reduces v but keeps µ constant; (d) Co 4+ HS (S = 5/2, r ion = 0.53 Å) reduces v drastically but increases µ. Thus, the above possibilities (a)-(d) invoked in the literature [9]- [14], [19,20], are in direct conflict with the consistency conditions (i) and (ii), the latter due to an appreciable change in the tolerance factor. After exhausting other possibilities involving substitution of Mn 4+ HS (S = 3/2, r ion = 0.53 Å) by Ni or Co ions of different charge and spin states, we propose the substitution scheme, sketched in figure 9, which is compatible with observations (i)-(iii). In this scheme, Ni 2+ HS (t 6 2g e 2 g , S = 1, r ion = 0.69 Å) [Co 2+ LS (t 6 2g e 1 g , S = 1/2, r ion = 0.65 Å)] and Ni 4+ LS (t 6 2g e 0 g , S = 0, r ion = 0.48 Å) [Co 4+ HS (t 3 2g e 2 g , S = 5/2, r ion = 0.53 Å)] substitute for Mn 3+ HS (t 3 2g e 1 g , S = 2, r ion = 0.645 Å) and Mn 4+ HS (t 3 2g e 0 g , S = 3/2, r ion = 0.53 Å) ions, respectively. Such a substitution introduces the least mismatch between the ionic radii of the solute and the substituted host ions with the result that only a minute reduction in v and essentially no change in the tolerance factor and the bond lengths Mn-O occurs, in accordance with the experimental observation. By virtue of the substitution at the Mn 3+ and Mn 4+ sites, Ni 2+ HS/Co 2+ LS and Ni 4+ LS/Co 4+ HS ions find themselves respectively in the neighborhood of Mn 4+ and Mn 3+ ions. In order to minimize the total energy of the system, Ni 2+ HS and Co 2+ LS ions assume their stable configurations, Ni 3+ LS and Co 3+ HS ions, by donating one e g electron per ion to Mn 4+ ions (via the intervening oxygen 2p orbitals) and converting them into Mn 3+ ions. By contrast, Ni 4+ LS and Co 4+ HS ions achieve the same goal (energy minimization) by accepting one e g electron per ion from the Mn 3+ HS neighbor (Co 4+ HS state lowers its energy through the JT distortion and changes over to the Co 4+ IS configuration before accepting the e g electron) and transform themselves into the stable Ni 3+ LS and Co 3+ HS ions (Mn 3+ HS ions get converted to Mn 4+ HS ions, in the process), as depicted in figure 9. Since the Within the framework of the above picture (figure 9), the Co 3+ HS ion has two localized e g electrons as against one per Ni 3+ LS ion. It immediately follows that the Co-substitution should be more effective in weakening DE (and hence in reducing T C ), as corroborated by experiments ( figure 10). Had the role of such e g -electron localization been merely to reduce the DE, T C would have dropped in proportion to the Ni/Co concentration or Mn 3+ /Mn 4+ ratio (with a faster drop in the Co case); SE interactions, favoring AF coupling between the t 2g electron spins of the ion pairs Ni 3+ /Co 3+ -O 2− -Ni 3+ /Co 3+ or Mn 3+ /Mn 4+ -O 2− -Mn 3+ /Mn 4+ , would make the decline of T C with increasing Ni/Co concentration (or decreasing Mn 3+ /Mn 4+ ratio) even faster. As noted from figure 10, this expectation is in direct conflict with the observed slowing down of the rate of decrease of T C in both cases as the concentration of Ni/Co solute ions increases or the Mn 3+ /Mn 4+ ratio decreases. We propose that the dipolar interactions, which couple the e g -e g (e g -t 2g ) spins on the Ni 3+   pairs ferromagnetically, slow down the rate of decline of T C caused by the reduced DE and/or enhanced SE. This is so because the dipolar interactions, although extremely weak compared to the DE and SE interactions, tilt the balance in favor of FM order when the competing DE-FM interactions between e g electron spins and SE-AF interactions between t 2g electron spins are both large but of similar magnitude. The dipolar interactions are stronger in Co-substituted LPMO because of the greater number of localized e g electrons and larger magnetic moments on the Co 3+ HS ion, and this also explains the higher electrical resistivity [12]- [14] for the Cocontaining manganites. Assuming that the substitution of Mn 3+ HS and Mn 4+ HS ions by Ni or Co solute ions is completely biased by the 35/65::Mn 3+ /Mn 4+ ratio prevalent in the LPMO host, we arrive at the values for the low-temperature (saturation) magnetic moment, µ LT , and hightemperature (in the PM state) magnetic moment, µ HT , for all of the compositions in the Niand Co-substituted series that are in very good agreement (figure 11) with the corresponding experimental values [13,14]. In the case of Co-substituted LPMO, the agreement between the calculated and observed values of µ LT and µ HT can be improved further by considering that at low (high) temperatures, almost all of the Co 3+ ions are in the LS (HS) state, as the energy difference between the two spin states is as low as [54] ≈0.03 eV.
Before assessing the impact of the present findings on the current understanding of the underlying physics of the novel physical phenomena exhibited by manganites, the salient features of this work are summarized below. (A) In the optimally hole-doped parent LPMO compound, the SE-AF interactions between the localized t 2g electron spins are completely dominated by the DE-FM interactions between e g electron spins and the dipolar interactions between the localized t 2g electron spins are extremely weak compared to the SE interactions.
Thus, dipolar interactions have essentially no role to play in establishing the FM state in LPMO. (B) Strong evidence for the presence of finite metallic FM clusters within the insulating PM matrix for T > T C , and hence for the phase separation, is provided by the peak in γ eff ( ) at p in the parent LPMO compound as well as in the Ni-or Co-substituted LPMO. (C) In LPMO and weakly Ni-or Co-doped (y < 0.1) LPMO, finite metallic FM clusters grow in size as the temperature is decreased towards T C , so much so that at T T C , metallic FM clusters coalesce to form an infinite FMM percolating network in which finite AF insulating clusters are embedded, and thus a percolative transition from the insulating (T > T C ) to metallic (T < T C ) state occurs. (D) As more and more of the Mn ions in LPMO are replaced by Ni or Co ions (i.e. as y increases), an increased number of e g electrons become localized via the mechanism illustrated in figure 9. Consequently, the DE-FM interactions are rapidly suppressed and become similar in magnitude to the SE-AF interactions, so that even weak dipolar interactions between the localized e g electron spins and/or between e g -t 2g electron spins are able to establish the FM insulating (FI) phase, which grows at the expense of the FMM phase with increasing y. Thus, a transition from the PI phase (T > T C ) to the FI phase (T < T C ) occurs at T = T C when the Ni or Co solute concentration exceeds a certain threshold value (y > 0.2 in the present case). That the dipolar interactions play a decisive role in establishing FM order in the vicinity of T C when the fraction of localized e g electrons becomes substantial (y 0.2) is clearly borne out by the increase in the ID-to-IH crossover temperature with y. Since the currently studied manganite system has the same underlying physics as other optimally doped manganites, e.g. La 0.7 A 0.3 Mn 1−y TM y O 3 where A = Ca, Sr, Ba, Cd and TM = Fe, Co, Ni, the above observations (A)-(D) should be applicable to these systems as well and provide a means of resolving the controversy surrounding the nature of the FM-PM phase transition as well as the spin and charge states of Co and Ni in La 0.7 A 0.3 Mn 1−y TM y O 3 manganites.
In recent years, substantial progress in understanding the pronounced 'electron-hole asymmetry' and the occurrence of a variety of phases in the phase diagram of La 1−x A x MnO 3 manganites has been made by recognizing the importance of the competition between DE interactions, Hund coupling between e g electron spin and localized t 2g electron spin, JT electron-phonon coupling between e g electrons and the local distortions of the MnO 6 octahedron, Coulomb interactions among the e g electrons and AF Heisenberg coupling between nearest-neighbor t 2g spins. However, many aspects of the physics of manganites are still poorly understood. For instance, no general consensus on how the FI state at low doping levels x ≈ 1/8 evolves from the AF insulating state at x = 0 has emerged so far. Taking cognizance of the inference (D), we suggest that the dipole-dipole interaction (completely ignored in the previous considerations) between the localized e g electron spins (regardless of the actual localization mechanism) may hold the key to the formation of the FI phase at low doping levels, as elucidated below. As the hole concentration (x) increases from zero, the DE interaction grows at the expense of the SE interaction so that at a certain value of x, the DE FM interactions between mobile e g electron spins, become similar in magnitude to, and compete with, the SE AF interactions between localized t 2g electron spins, with the result that even extremely weak dipolar interactions suffice to tilt the balance in favour of FM order.

Summary and conclusion
An extensive investigation of the magnetic behavior of the La 0.7 Pb 0.3 Mn 1−y (Co, Ni) y O 3 (y = 0, 0.1, 0.2, 0.3) hole-doped manganite system over a wide range of temperatures, which covers the critical region near the FM-PM phase transition and the PM region, has been carried out by measuring ACS at the ac driving field whose rms amplitudes (frequencies) ranged from 0.1 to 10 Oe (from 100 Hz to 10 kHz). A detailed ROF analysis of the ACS data enabled a precise determination of the Curie temperature, T C , and hence of the effective critical exponent, γ eff , for susceptibility as a function of reduced temperature, = (T − T C )/T C . The main observations are as follows. In the parent LPMO compound, (I) γ eff ( ) → γ H = 1.386 (the 3D IH value) in the ACR and goes through a peak before approaching the MF value of γ = 1 at T T C . (II) By contrast, regardless of the Ni or Co concentration in the range 0.1 y 0.3, as the temperature is raised from T C , γ eff starts with the value γ D = 1.390(1) in the ACR ( * ), goes through a steep minimum (dip) at dip with γ dip ( = dip ) = 1.280(4), rises to the value γ H = 1.386(4) at * * and then peaks at p assuming a value γ eff ( = p ), which depends on y. (III) As y increases, * , dip , * * and p shift to higher temperatures, while γ p eff increases. (IV) For a given y, * , dip , * * and p are higher, while γ p eff is lower in the case of Co. (V) A progressive drop in T C (from its value in the LPMO host) occurs as the Ni or Co concentration increases but the Co-substitution depresses T C at a much faster rate. However, the rate of decline of T C slows down with increasing y, i.e. T C (y) is concave upwards. The RG calculations predict that (a) a 3D isotropic spin system, in which short-range Heisenberg interactions coexist with weak long-range dipolar interactions, exhibits a thermally induced crossover from the 3D IH critical behavior to the 3D ID asymptotic critical behavior when the temperature is lowered towards T C from high temperatures, and (b) the characteristic feature of this IH to ID crossover is a dip in γ eff ( ) in the crossover region with γ eff attaining the values γ D in the ACR, γ dip = 1.28(1) at dip and γ H at * * , respectively. In view of these RG predictions, the observations (II)-(IV) permit us to conclude that an ID-to-IH crossover occurs in the Ni-and Co-substituted LPMO, and the ID interactions, not discernable in the parent LPMO compound, grow in strength with y and, among Ni and Co solutes, are stronger in the Co-substituted LPMO.
Within the framework of a percolation model, the peak in γ eff ( ) at p in the LPMO host and the Ni-or Co-substituted LPMO is shown to result from an interplay between the diverging spin-spin correlation length of the insulating PM matrix as → 0 and the temperaturedependent length scale due to the finite metallic FM clusters, while the observed upward shift in p and the decrease in γ p eff as one follows the sequence y = 0 → Ni → Co (for a given Ni or Co concentration, e.g. y = 0.2 in figure 5) imply that the average FM cluster size decreases and the cluster size distribution narrows down. The peak in γ eff for > 0 thus constitutes a new experimental signature for the phenomenon of phase separation in hole-doped manganites. To address the issues connected with the localization of e g electrons (and hence weakening of DE) and the origin of dipolar interactions, we have proposed the charge and spin states for Ni and Co ions in the Ni-and Co-substituted LPMO that are consistent not only with the present results but also with the previously published structural (neutron diffraction), thermo-gravimetric, bulk magnetization, dc magnetic susceptibility and electrical resistivity data on the same system. Finally, the physical implications of our results, so far as the physics of manganites is concerned, are discussed in the preceding section.
Department of Science and Technology, New Delhi) for permission to use the Quantum Design Physical Property Measurement System (PPMS).