1 Introduction

It is known that lanthanum manganite (LaMnO 3) is an anti-ferromagnetic (anti-FM) insulator. Substituting a part of La by a metallic element (usually an alkaline-earth or alkali element) makes the doped materials exhibiting ferromagnetic (FM) metallic behaviors below the Curie temperature (T C). The physical properties of manganites are arisen from two classes of competing effects: (i) supper-exchange interactions cause insulating and anti-FM-ordered ground states, and (ii) double-exchange interactions lead to metallic and FM ordered ground states. These interactions can be tuned by external and internal parameters, such as temperature, magnetic field, the crystalline size, concentration of carriers, crystal-structure type, the size of ions, and so forth.

It is known that for perovskite manganites, when their size is reduced to the nanometer scale, they exhibit a number of outstanding physical properties; for example, low-field magnetoresistance, surface spin-glass (SG) behavior, superparamagnetism (SPM), and low-field saturation magnetization [1–4]. These phenomena have promoted many research works on nanosized manganite materials. An exchange bias effect of magnetization and resistance relaxation at low temperatures for La 0.7Ca 0.3MnO 3 (LCMO) nano-ring network was found by Zhu et al. [2]. They suggested the existence of two SG types in the shell of LCMO grains, according to the core-shell model. Karmakar et al. [3] also observed the SG-like behavior in LCMO nanostructures at low temperatures. It was believed that the origin and nature of SG state in the nanostructured LCMO system were similar to the picture of hierarchical organization of metastable states. Evidences related to surface-spin freezing of nanopolycrystalline LCMO samples were observed by Dey and coworkers [4]. The glassy behavior at T S ≈ 40 K is attributed to competing magnetic interactions stabilized in a SG-like frozen state at the surface region of nanometric grains. However, physical mechanisms related to the SG phenomenon in LCMO nanoparticles are still a controversial issue. In this paper, we present a study on the deviation from the Bloch’s T 3/2 law and the SG-like behavior in LCMO nanoparticles.

2 Experiment

Three LCMO nanoparticle samples denoted as S8, S12, and S16 were prepared by a reactive milling method with the milling times t m = 8, 12, and 16 h, respectively. Detailed descriptions of this method can be found elsewhere [5]. The crystal structure and particle size of samples were examined by X-ray diffraction (XRD) performed on a SIEMENS-D5000 diffractometer. The size and shape of nanoparticles were determined by using scanning electron microscopy (SEM, Hitachi S-4800) and high-resolution transmission electron microscopy (HR-TEM, Philips CM20-FEG). Measurements of the dynamic magnetic relaxation, AC susceptibility, and magnetization were carried out on a physical properties measurement system (PPMS 6000) with T = 5–300 K, H = 0–50 kOe, and f= 297–1337 Hz.

3 Results and Discussion

Room-temperature XRD patterns, Fig. 1a, revealed the single phase in an orthorhombic structure (the space group Pnma) of the samples [6]. Averaged crystallite sizes calculated by using the XRD data and Warren-Averbach method are about 8, 7, and 6 nm for S8, S12, and S16, respectively. Figure 1b shows a representative SEM image for S8 (particle-size distributions are similar to the other samples), where the average particle size is about 15 nm. This size is quite larger than that obtained from the XRD and HR-TEM studies (notably, for the HR-TEM study, the averaged particle size of nanoparticles is about 7 nm, Fig. 1c). This could be explained by a high level of lattice disorders (marked as ↑), including dislocations (marked as ⊥) at the faceted grain surface of nanocrystals, as can be seen in Fig. 1c.

Fig. 1
figure 1

a XRD patterns of the samples S8, S12, and S16, where the XRD peaks are marked with the Miller indices, after ref. [6]. b SEM image and c and HR-TEM image for a representative sample 8. Arrows indicate lattice disorders and inverted letter Tindicates dislocations

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Following the structural analyses, we have investigated the magnetic properties of these nanoparticle samples. Figure 2a shows their temperature dependences of field-cooled (FC) and zero-field-cooled (ZFC) magnetizations, M ZFC/FC(T), for the field H = 10 Oe. These M ZFC(T) curves are similar to those recorded for SG and/or SPM systems. They exhibit a peak at the blocking temperature (T B), which is shifted towards lower temperatures with increasing t m (T B = 87, 76, and 63 K for t m = 8, 12, and 16 h, respectively). With decreasing temperature, meanwhile, M FC(T) firstly increases, and then saturates to a constant value. Such the features indicate that our LCMO nanoparticles have the SG-like behavior [7]. Furthermore, magnetization variations in the paramagnetic (PM) region for the samples can be analyzed by using the CurieWeiss law [8], χ(T) = C/(T − 𝜃), where C and 𝜃 are the Curie constant and Curie-Weiss temperature respectively. It appears in Fig. 2b that the χ −1(T) curves exhibit the linear behavior with two distinct slopes. The first one (denoted as R1) is related to the FM transition temperature while the other (denoted as R2) is located at a higher temperature. The C and 𝜃 values obtained from fitting the χ(T) data to the Curie-Weiss law are C 1 = 0.76, 1.31, and 1.34 emu ⋅K/mol and 𝜃 1 = 178, 168, and 137 K in the region R1; C 2 = 5.02, 9.92, and 10.90 emu ⋅K/mol and 𝜃 2 = −63, −241, and −323 K in the region R2 for S8, S12, and S16, respectively. Here, the C 1 values are comparable to the theoretical one with C =0.97 emu ⋅K/mol corresponding to PM Mn ions with an average valence of 3.35 (i.e., the mixture of Mn 3+ and Mn 4+ ions) [9]. The decrease of 𝜃 1 with increasing t m reflects the weakening of FM interactions in LCMO nanoparticles. The appearance of the region R2 with negative 𝜃 2 values suggests a strong anti-FM nature which is mainly related to superexchange interaction pairs Mn 4+Mn 4+ and Mn 3+Mn 3+. Therefore, magnetic moments become difficult to reach a saturation state, as can be seen in Fig. 3a.

Fig. 2
figure 2

a M(T) curves for H = 10 Oe and b χ −1(T) curves for S8, S12, and S16. In b, the solid and dashed lines are the fitting curves of the χ −1(T) data to the CurieWeiss law in the regions R1 and R2

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Quantitatively field dependences of magnetization, M(H) are further analyzed by using the SPM theory initiated by Langevin. According to Ningthoujam et al. [10], if there is a PM fraction in addition to SPM particles, temperature and field dependences of magnetization can be written as

$$M=M_{\mathrm{S}} \left[ {\coth \left( {\frac{\mu H}{k_{\mathrm{B}} T}} \right)-\left( {\frac{k_{\mathrm{B}} T}{\mu H}} \right)} \right]+\chi H, $$
(1)

where M S and μ are the saturation magnetization and magnetic moment of an individual SPM particle, respectively As shown in Fig. 3a, the M(H) data recorded at 5 K were fitted well with (2), see the solid lines Obtained M S values are 36.77, 19.42 and 15.16 emu/g for S8, S12, and S16, respectively. These values are much smaller than that of a bulk LCMO sample with M S = 97.5 emu/g [11]. The decrease of M S in LCMO nanoparticles is ascribed to the existence of a nonmagnetic or AFM layers on the surface that their thickness increases with increasing t m

Fig. 3
figure 3

a M(H) data at 5 K are fitted to (2) and b M S(T) data are the fitted to a function M S = M 0(1 − BT ε). The solid lines are the fitting curves

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Figure 3b shows the behavior of the M S versus temperature for the samples, where M S values obtained upon fitting the magnetization data at 5 K to (2). In general, the thermal dependence of magnetization for nanoparticles deviates from expected Bloch’s T 3/2 law when magnons with a wavelength longer than the particle diameter can not be excited, and a threshold of thermal energy is required to generate spin waves [12]. In our case, the temperature dependence of M S does not follow Bloch’s T 3/2 law, but obeys a T ε law with ε increasing from 1.69 to 1.76 if t m increases from 8 to 16 h, see the solid lines shown in Fig. 3b. A slightly greater value of ε compared to the bulk sample value (ε = 3/2) [13] is due to the effect related to small particle sizes.

To further understand the SG-like behavior existing in the samples, we have measured the AC susceptibility (χ AC) as the function of temperature under an AC applied field of 10 Oe at different frequencies (f= 297-1337 Hz). Figure 4a shows the AC susceptibility (consisting of real \(\chi ^{\prime } \) and imaginary \(\chi ^{\prime \prime } \) parts) for a representative frequency f = 897 Hz. One can see that the peak temperature (denoted as the freezing temperature T f) decreases with increasing t m. For each sample, T f shifts towards higher temperatures when frequency increases, see Fig. 4b–d. This is a typical feature of a conventional SG system [14] Additionally, to distinguish between SG and SPM systems, the c parameter is considered as an empirical criterion [15]. This parameter can be defined by

$$ c=\frac{\Delta T_{f} }{T_{f} {\Delta} \left( {\log_{10} f} \right)}, $$
(2)

where ΔT f and Δ(log 10 f) are the shifts of T f and log 10 f, respectively. Normally, c is about 10 −2 for a conventional SG system, whereas c ≥ 0.1 for SPM [16]. In our case, the obtained c values are 0.062, 0.055, and 0.051 for the samples S8, S12, and S16, respectively. Such the results prove that our LCMO nanoparticles at low temperatures exhibit the SG-like behavior rather than the SPM behavior. The existence of the SG-like state is considered as a consequence of competing magnetic interactions (between anti-FM and FM phases) and of spin frustration at surface layers of LCMO nanoparticles. A similar feature was observed in bilayer heterostructures composed of anti-FM SrMnO 3 and FM L 0.7Sr 0.3MnO 3 [1].

Fig. 4
figure 4

Temperature dependences of the real part of the AC susceptibility in an AC magnetic field H AC = 10 Oe for the samples measured at 897 Hz (a), and at different frequencies for the samples S8 (b), S12 (c), and S16 (d). The inset in a shows the image part of the AC susceptibility at 897 Hz for the samples. Arrow denotes the variation trend of T f

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4 Conclusion

LCMO nanoparticles with the averaged crystallite size of ∼7 nm prepared by a reaction milling method were carefully studied by measurements of the DC magnetization and AC susceptibility. Both the magnetization and T B values decrease with increasing t m. We believe that the magnetization decrease is attributed to the thickness of nonmagnetic or anti-FM layers on the surface of LCMO nanoparticles, which increases with increasing t m. Temperature dependences of saturation magnetization obeys a T ε law (with ε = 1.69–1.76). Interestingly, our studies indicate the existence of the SG-like behavior in the samples, which is originated from magnetic frustration at the surface of nanoparticles.