1 Introduction

Recently, the effect of crystallite size reduction on the physical properties of hole-doped manganites has attracted lots of interest of scientists for both fundamental and application aspects [1–5]. Owing to finite-size effect and inter-particle interactions, remarkable new phenomena were observed in nano-materials [1]. The inter-particle interaction was found to be strongly modifying the magnetic response of nanoparticle systems; e.g., the dc susceptibility obeys the Curie-Weiss law rather than the Curie law at high temperatures. Moreover, the magnetic behavior of the particle surface differs from that of the core [2, 3]. Higher magnetic disorder that is usually present in the surface layer is known as a dead magnetic layer. Before one investigates the magnetic behavior of manganites in nanoscales, one has to prepare nanomaterials through a variety of synthesis methods. However, almost authors have used the bottom-up approach (e.g., sol-gel, mechanochemical milling, interactive milling, or sputtering methods) to prepare nanostructured samples. In this work, instead, we have used a simple preparative method that is the top-down approach to prepare nano-polycrystalline La0.7Ca0.3MnO3 (LCMO) samples and have investigated their magnetic properties.

2 Experiment

First, LCMO ceramic samples were prepared by conventional solid-state reaction. High-purity powders La2O3,CaCO3, and Mn were used as starting materials. These powders were ground and mixed well, and then heated in air at 1,200 °C for 24 h. After heating, the obtained powder was re-ground, pressed into pellets, and sintered at 1,400 °C for 24 h in air. These pellets were then used for mechanical ball milling (by using a Spex 8000D system) with the mass ratio of ball/powder = 4.1/1. The milling time (t m) was varied from 0 to 30 min (denoted as LC0, LC10, LC20, and LC30 for t m = 0, 10, 20, and 30 min, respectively). Their structure was checked by using an X-ray diffractometer (Siemens D5000, with λ = 1.5406 Å). The surface morphology of samples has been observed by scanning electron microscopy (SEM). The M(T) and M(H) curves were measured via a SQUID magnetometer.

3 Results and Discussions

Figure 1 shows the X-ray diffraction (XRD) patterns for the LCMO samples with various t m values. Our detailed analyses, i.e., identifying the Miller-indexed peaks, reveal that they are single phase in an orthorhombic structure, with the lattice parameters a ≈ 5.48 Å, b ≈ 7.75 Å, and c ≈ 5.46 Å. The lattice parameters were almost independent of t m. Using the Williamson-Hall (W-H) method [6], the average values of both the crystallite size (D) and strain (ε) parameters could be obtained from the relation βcosθ = (K λ / D) + 2ε sin θ where β is the full width at a half maximum of an XRD peak, θ is the Bragg angle, and K = 0.9 is the shape factor. The W-H analysis for LC10 is representatively shown in the inset of Fig. 1. The crystallite size obtained for t m = 0, 10, 20, and 30 min are D = 200, 72, 54, and 45 nm, and ε = 0.002, 0.003, 0.003, and 0.010, respectively. Notably, D = 200 nm is herein an estimated value for the as-prepared sample (t m = 0), and its real value can be larger because the W-H method is more accurate for the size of nanoparticles smaller than 100 nm. Figure 2 shows the SEM image for a representatively sample, LC10. In comparison with the SEM image, the D values deduced from W-H method are in good agreement.

Fig. 1
figure 1

XRD patterns for LCMO samples. The inset shows a detailed W-H analysis for the LC10 sample

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Fig. 2
figure 2

SEM image for LC10 sample

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Figure 3a shows the temperature dependence of field-cooled (FC) and zero-field-cooled (ZFC) magnetizations of the samples in the field of H = 100 Oe. The T C values determined from the minima of the dM/dT vs. Tcurves are 252, 252, 248, and 246 K for D = 200, 72, 54, and 45 nm, respectively. Here, the reduction in D did not change the T C significantly, but it broadened the phase transition region. The width of the FM-PM phase transition (ΔT C) determined from the full width at half maximum of the dM/dT vs. T curve increases from 6 to 34 K with decreasing D from 200 to 45 nm. In Fig. 3b, we showed the χ −1(T) curves deduced from M ZFC(T) data. One can see that in the high-temperature region, χ −1 varies almost linearly with temperature following the Curie-Weiss law [7], χ(T) = C/(T − θ), where C is the Curie constant, and θ is the ordering temperature. Here, θ decreases gradually in the temperature range of 251–240 K with decreasing D. This confirms that the strength of FM interaction decreases in LCMO nanoparticles with a smaller crystallite size. Notably, χ −1(T) data at high-temperature regions obey the Curie-Weiss law well for LC0 sample with D = 200 nm, whereas for samples with smaller crystallite sizes (D = 72, 54, and 45 nm), the Curie-Weiss law is not satisfied in the full PM temperature range. That is because there are obviously sharp downturns in χ −1(T) curves well above T C , which is the characteristic of the Griffiths singularity [8]. It means that the Griffiths phase exists in LCMO nanoparticles. This could be explained by higher degree of disorder in nanoparticle samples than that in as-prepared sample (LC0). According to Salamon et al. [8], the disorder due to the bending of Mn-O-Mn bond induces the formation of Griffiths phase in the La1−x Ca x MnO3 system.

Fig. 3
figure 3

a ZFC (solid symbols) and FC (open symbols) M(T) curves with H = 100 Oe. b χ −1(T) data fitted to the Curie-Weiss law (solid lines)

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Figure 4a shows hysteresis loops of M(H) curves measured at 5 K. One can see that the magnetization value was reduced in the smaller LCMO nanoparticles This reduction is related to the nonmagnetic layer or spin disorder on the surface; its thickness increases with decreasing D. The coercivity (H C) value substantially increases when the crystallite size decreases see the inset in Fig. 4a. A monotonic decrease of H C with increasing temperature in the range of 5–240 K becomes more rapid when temperature increases above 240 K see Fig. 4b. This is associated with the FM-PM phase transition, where magnetic moments located within FM domains/clusters become disordered due to thermal agitation. At temperatures above T C, the value of H C is thus almost 0, in good agreement with the above M(T) analyses. Interestingly, the H C observed below T C increases with decreasing D. The H C at 5 K as a function of the surface/volume ratio (D −1) is shown in the inset of Fig. 4b In general, when D is reduced, H C increases and reaches a maximum value at some critical diameter (D cr) of the single domain state, and then, it decreases with further decrease in the crystallite size [7]. The solid line in the inset of Fig. 4b is a fit of our experimental H C data to the function H C = m + n/D, where m and n are constants [7]. This indicates that our LCMO systems are close to a multidomain structure [9]. It means that the D values are larger than D cr values of the single domain state, which makes the anisotropy energy smaller. Hence, the H C value decreases monotonically with increasing D as shown in the inset of Fig. 4b

Fig. 4
figure 4

a M vs. H measured at 5 K. The inset shows the zoom-in view of the region between H = −500 and 500 Oe. b H C as a function of T. The inset shows D −1 dependence of H C at 5 K. Here, triangles, circles, diamonds, and squares indicate LC0, LC10, LC20, and LC30 samples, respectively

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We have also determined the saturation magnetization (M S ) for the samples at different temperatures. From M 2(H) curves, the linear extrapolation from high fields to the intercepts with the M 2-axis gives the values of \({M_{S}^{2}} (T,0)\). The obtained M S (T) data are plotted in Fig. 5a. A power law form of M S (T)=M S (0)[1 − BT ε] [7] is used to fit the data with M S (0) = 97.0, 86.4, 78.9, and 68.6 emu/g, B = 1.46 × 10−6, 4.05 × 10−4, 3.17 × 10−3, and 7.30 × 10−3 K−ε; and ε = 3.07, 2.13, 1.77, and 1.61 for D = 200, 72, 54, and 45 nm, respectively. One can see that the temperature dependence of the magnetization do not follow Bloch’s law, but obeys a T ε law with ε increasing from 1.61 to 3.07 when D increases from 45 to 200 nm (see the solid lines in Fig. 5a). The variation tendency of the exponent ε for our samples is quite similar to a previous report [10]. Additionally, the M S(0) value obtained is smaller than that of the LCMO bulk single crystal (M S(0) = 97.5 emu/g) [11]. The M S(0) values for the samples are reduced linearly with D −1 (see Fig. 5b). This result is consistent with that reported by Lopez-Quintela et al. [12], confirming that the magnetization is actually influenced by the particle surface. We used a model of the core-shell structure [13] to determine the core diameter (d C) for magnetic particles in the samples. The LCMO particles are assumed to have a spherical shape, and each of them is composed of an ideal single-crystalline core with the saturation magnetization of the core, M C = 97.5 emu/g [11], and the density of ρ C = 5.9 g/cm3 [14]. For a particle of radius r, consisting of a core surrounded by a shell of thickness d Shell ≪ r (with the corresponding ρ S value roughly assigned to be 4 g/cm3 [3]), the core diameter (d C) can be found from [13]

$$ d_{\mathrm{C}}=D\cdot\left[{\frac{\rho_{\textit{S}} /\rho_{\textit{C}}}{\rho_{\textit{S}} /\rho_{\textit{C}} + (M_{\textit{C}} -M_{\textit{S}})/(M_{\textit{S}}-M_{\textit{Shell}})}}\right]^{\frac{1}{3}}. $$
(1)

By using (1), while assuming a nonmagnetic shell (M Shell = 0), the d C values obtained are 199.5, 67.9, 48.9, and 38.3 nm for D = 200, 72, 54, and 45 nm, respectively Clearly, the nonmagnetic shell thickness, which can be estimated by d Shell = (D − d C)/2, increases from 0.25 to 3.35 nm with decreasing D. The increase in d Shell and decrease in D (or the increase in D −1) are the main reason for the reduction in the strength of FM inter-particle interactions and in the M S for LCMO nanoparticles, as mentioned above.

Fig. 5
figure 5

a M S vs. T for the samples (symbols) is fitted to a power law T ε (solid lines). b D −1 dependence of M S(0)

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4 Conclusion

The analyses of the magnetic properties were performed for LCMO nanoparticles with the crystallite size of D = 45-200 nm. Based on χ −1(T) curves, the existence of Griffiths phase in LCMO nanoparticles was observed, which implied the higher degree of disorder in nanoparticle samples. The results show that the D values of our LCMO samples are larger than the D cr of the single domain state. Therefore, H C value increases monotonically with decreasing D. Also, the temperature dependence of magnetization in this case obeys a power law T ε, with ε > 3 / 2. The reduction in M S vs. D is mainly due to the formation of a nonmagnetic surface shell surrounding the nanoparticles, which increases from 0.25 to 3.35 nm when D is decreased form 200 to 45 nm.