Spin glass behavior and enhanced but frustrated magnetization in Ho³⁺ substituted Co–Zn ferrite interacting nanoparticles

Abstract

Nanoparticles of Ho³⁺ substituted in Co–Zn ferrites were synthesised by sol–gel method. The phase formation of these samples has been confirmed by X-ray powder diffraction technique. XRD Rietveld refinement carried out using the FULLPROF program shows that the Co–Zn ferrite retains its single phase cubic structure with space group Fd3m for x ≤ 0.05. Occupancy of the cations is explained on the basis of site preference, size and valance of the substitution cations. The nanostructure and morphology of prepared samples were investigated by field emission scanning electron microscopy and transmission electron microscopy. The elemental percentage of the constituent ions was determined using energy dispersive spectroscopy. The magnetic interactions among the nanoparticles were analyzed by employing a temperature dependent vibration sample magnetometer, field cooled (FC)/zero field cooled (ZFC) measurements. The ZFC and FC curves diverge below the blocking temperature exhibiting a ZFC cusp at 195–225 K. The saturation magnetization of Co–Zn ferrite increased linearly with Ho³⁺ substitution for x ≤ 0.05 and almost remains constant thereafter. The frequency dependence of the AC susceptibility measurements was performed on the sample. It shows a peak at around spin freezing temperature, with the peak position shifting as a function of driving frequency, indicating a spin-glass-like transition of the sample.

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1. Introduction

The spinel ferrite material is one of the magnetic materials that shows novel properties when particle size is reduced to a nanometer scale. Spin glass has recently been observed in nanocrystalline magnetic materials owing to their inter-particle interactions (dipolar/exchange) and surface spin disorder.^1–5 A spin-glass is a magnetically disordered material exhibiting high magnetic frustration where each electron spin freezes in a random direction below the spin freezing temperature.^6,7 Frustration refers to the inability of the system to remain in a single lowest energy state that contains many metastable states separated by the potential barrier of ∼k_BT_f.⁸ Thus no conventional long-range order (of ferromagnetic or antiferromagnetic type) could be established. Nanoparticles of ferrite self-assemble in a 3-dimensional lattice with high packing densities due to inter-particle interaction where the dipolar coupling among the nanoparticles is comparable to anisotropy energy, which produces low temperature collective characteristics that are similar to atomic spin-glass behavior.^9–11 With the dilution of magnetic ions on the tetrahedral A-site of a ferrite, magnetic frustration and magnetic disorder on the octahedral B-site give rise to spin-glass-like ordering at low temperature. The physics of spin glass raises many fundamental questions and has become one of the main streams of research in condensed matter physics and material science researchers.

Special attention has been given to cobalt containing spinel ferrites due to their large magneto-crystalline anisotropy and high saturation magnetization which is promising for magnetic materials in recording media. CoFe₂O₄ is an inverse spinel oxide having structural formula (Fe³⁺)_A[Co²⁺Fe³⁺]_BO₄. All the Co²⁺ ions normally occupy the octahedral [B] sites due to its favorable atomic radius and charge distribution in an octahedral crystal field and the Fe³⁺ ions are equally distributed between the (A) and the [B] sites.¹² The Fe_A–O–Fe_B interaction is very strong and CoFe₂O₄ is ferrimagnetic below 790 K (T_N, Neel temperature). On the other hand, ZnFe₂O₄ is a normal spinel with structural formula (Zn²⁺)_A[Fe₂³⁺]_BO₄. The Fe_A–O–Fe_B interaction is not effective in ZnFe₂O₄ due to the absence of the magnetic ion at the tetrahedral (A) site and the T_N is only ∼9 K. In the case of Co–Zn ferrites the general distribution of cations among (A) and [B] sites is given by (Zn_1−xFe_x)_A[Co_xFe_2−x]_BO₄. On substitution of Zn²⁺ in Co ferrite the Fe³⁺ ions migrate from (A) to [B] site and consequently the Fe_A–O–Fe_B interaction becomes feeble.¹³ The magnetic measurements have revealed that the T_N of bulk Co_0.2Zn_0.8Fe₂O₄ is ∼200 K, which is much smaller than T_N of the CoFe₂O₄, but considerably higher than that of ZnFe₂O₄.¹⁴ Marked improvement in the saturation magnetization of Co ferrite has also been observed with the substitution of proper amount of Zn ions.^15–17 However the coercivity, remanent magnetization and resistivity found to decrease for these samples.

Rare earth lanthanides (Ln³⁺) substituted ferrites are nowadays under extensive investigations in order to enhance the saturation magnetization, permittivity and permeability. Experimental and theoretical investigations of the exchange interactions in the rare earth-transition metal (4f–3d) elements containing spinel oxides have received more attention in the recent years.^18–20 At the same time rare earth ions expected to affect spin-glass behavior.²¹ Large magneto-crystalline anisotropy in CoFe₂O₄ arises due to a strong spin (S)–orbit (L) coupling in Co²⁺ ions and similar strong spin–orbit (L–S) coupling has been noted in 4f grouped rare-earth (RE³⁺) ions. By introducing RE³⁺ Co–Zn, RE³⁺ ions are stabilized in B-sites of the spinel structure. Hence, one would expect a significant change of magnetic moment, ferromagnetic ordering temperature and anisotropy energy constant in the spinel structure of Co–Zn ferrite upon RE doping. RE³⁺ ions have a variety of magnetic properties, their magnetic moment vary from 0 to 10.6 μ_B, among them Ho³⁺ ions posses the highest magnetic moment of 10.6 μ_B.

One of the challenging issues for physics, chemistry and material science researchers with these compounds is their synthesis in single phase and understanding of their complex magnetism such as spin glass behavior at low temperature. Our aim is to probe the complex magnetic nature of Ho³⁺ ions in Co–Zn (Co_0.7Zn_0.3Ho_xFe_2−xO₄) compound using the probe of temperature, frequency and magnetic field dependent magnetic properties. We concentrated on some aspects, which we feel are important to unearth the complex magnetism of these systems.

2. Experimental

A series of Ho substituted Co–Zn ferrite with a formula Co_0.7Zn_0.3Ho_xFe_2−xO₄ was prepared by sol–gel auto combination method. Analytical grade cobalt nitrate (Co(NO₃)₂·6H₂O), zinc nitrate (Zn(NO₃)₂·6H₂O), holmium nitrate (Ho(NO₃)₃·5H₂O), iron nitrate (Fe(NO₃)₃·9H₂O) and citric acid (C₆H₈O₇·H₂O) were used as starting materials. The metal nitrate to citric acid ratio was kept as 1 : 3 and then the pH of the mixed solution was kept at 7 by adding ammonia solution. The mixed solution was evaporated to dryness by heating at 100 °C on a hot plate with continuous stirring and finally formed a very viscous brown gel. This viscous brown gel was ignited by increasing the temperature up to 100 °C and the loose powder of the samples was obtained. Finally, the as prepared powder was annealed at 700 °C for 4 h. Chemical reaction of the prepared sample is shown in Fig. 1.

Fig. 1 Chemical reaction of Co_0.7Zn_0.3Ho_xFe_2−xO₄ synthesized by sol–gel autocombustion method.

The phase formation identification of the prepared samples was characterized by X-ray diffraction (XRD, Philips X'Pert instrument) with Cu-Kα radiation (wavelength λ = 1.54056 Å) at room temperature. The grain morphology and particle size was investigated by field emission scanning electron microscopy (FE-SEM) (FEI NOVA NANOSEM 600) and transmission electron microscopy (TEM) (JEOL 3010) respectively. Magnetic hysteresis was measured at 10 and 300 K using vibrating sample magnetometer. Field cooled (FC) and zero field cooled (ZFC) measurements with an external applied magnetic field of 100 Oe were carried out in the temperature range of 50–350 K. AC susceptibility measurements were carried out with varying temperature from 50 to 350 K at three different frequencies; 100, 500 and 1000 Hz with an applied magnetic field of 10 Oe (with a commercial superconducting quantum interference device, Quantum Design, USA). ⁵⁷Fe Mossbauer measurements were carried out in transmission mode with ⁵⁷Co radioactive source in constant acceleration mode using standard PC-based Mossbauer spectrometer equipped with Wissel velocity drive. Velocity calibration of the spectrometer is done with natural iron absorber at room temperature. The spectra were analyzed with NORMOS program considering the distribution of hyperfine fields.

3. Results and discussion

3.1. Crystal structure and phase identification

XRD data were processed to software program FULLPROF for the Rietveld structure refinement, with goodness fit index value close to one. Rietveld refined discrepancy factors such as; unweighted profile R factor (R_p) weighted profile R-factor (R_wp), expected R factor (R_exp) and goodness fit factor (χ²) are given in ESI Table 1.† The Rietveld refined XRD patterns of all the sample are shown in Fig. 2. Crystal structure of the Co–Zn (x = 0.0) generated from the Rietveld refinement is presented in Fig. 3. It can be observed from Fig. 2 that the substitution of Ho³⁺ deviate Co–Zn ferrite from single phase cubic spinel (space group Fd3m) and becomes more prominent with the appearance of some un-indexed peak as secondary phase positioned at 2θ angle are; (25.98°), (33.14°), (33.94°) corresponding to the (111), (112) and (200) planes respectively which can be indexed to orthorhombic ferrites, HoFeO₃, (JCPDS-74-1479, space group Pbnm). Apart from the HoFeO₃ a co-existence unreacted Ho₂O₃ is also observed, similar orthorhombic distortions in rare earth Dy substituted Ni–Cu–Zn-²² and Ho substituted Co-²³ ferrite have also been reported in the literature. This weight% of secondary phase is calculated using the following relation:

Fig. 2 Rietveld refined X-ray diffraction patterns of x = 0.0 and 0.1 for Co_0.7Zn_0.3Ho_xFe_2−xO₄. Braggs peak positions are shown in green color at the bottom of XRD of x = 0.1. These Braggs peaks are referred to the space group Pbnm and Fd3m for HoFeO₃ and Co_0.7Zn_0.3Ho_xFe_2−xO₄ respectively.

Fig. 3 Cubic spinel crystal structure of Co_0.7Zn_0.3Fe₂O₄. The spinel structure belong to space group Fd3m. The cubic unit cell is formed by 56 atoms, 32 oxygen anions dispersed in a cubic close packed structure that form nearly fcc lattice with a number of interstitial positions partially occupied by tetrahedral-A and octahedral-B atoms. 24 cations occupying 8 of the 64 A-site and 16 of the 32 B-site. Yellow and pink shaded area corresponding to tetrahedral and octahedral sites respectively.

The obtained values of weight percent of spinel phase and second phase are listed in Table 1, that shows spinel phase for x > 0.05 slightly decreased with Ho substitution. Therefore, the small amount of Ho³⁺ ions in Co_0.7Zn_0.3Fe₂O₄ can affect not only the phase composition but also the size of the spinel matrix, which is due to orthoferrite (HoFeO₃) phase.^24,25 It has been reported that phase formation is affected by the concentration of rare earth element, Ho³⁺, in substituted samples of CoFe₂O₄.²⁶ The most important reason for the phase formation in the Ho³⁺ substituted Co–Zn ferrites is the electronic configuration, larger ionic radii of the Ho³⁺ rare earth element and diffusion of Ho³⁺ ions at the grain boundaries. K. Bharathi et al.²⁷ relate such distortion in the lattice to the changes in the O–Fe and O–Co bond lengths at octahedral B site. It is worth to mention here that, upto x = 0.05, Ho³⁺ is highly soluble in Co–Zn ferrite. This level of Ho³⁺ solubility is quite better as compared to previous reports (ref. 22–26) related to rare earth doping. This could be mainly related to the synthesis condition of the prepared samples. The lattice constant (a) increased from 8.401 to 8.419 Å with increase Ho ions. The increase in a is linear up to x = 0.05, whereas it almost remain constant for x > 0.05. The increase in a with Ho³⁺ substitution can be explained on the basis of difference in ionic radii between Ho (1.04 Å) and Fe (0.67 Å) ions, where smaller Fe ions at octahedral B site are replaced by the larger Ho ions. The rare earth Ho ions may induce root mean square (rms) lattice strain in Co–Zn ferrite due to its larger ionic radii. Therefore, using the XRD data; Williamson–Hall (W–H) extrapolations as a Lorentzian function were utilized to determine the lattice strain in all the samples (Fig. 4). The Williamson–Hall plots show that the slopes of all plots are negative. This negative slope indicates that strain could be very small. The negative slope of the plots of all samples also indicates compressive strain experienced in smaller grain size samples. As the Ho³⁺ substitution is increased up to x ≤ 0.5, slop is changing its sign from negative to positive which is an indication of increase in strain. Further, for x ≥ 0.075 the W–H plot almost remains flat confirming that Ho³⁺ ions have the limit to enter into Co–Zn spinel lattice. The average crystallite size (t_XRD) of the synthesized powders is calculated from the XRD line broadening of the (311) peak using Scherer's equation²⁸ value crystalline size are display in Table 1 crystalline size in order to few nanometer.

Table 1 Phase analysis (W%) and average crystallite size (t_XRD) of CoZnHo_xFe_2−xO₄

Comp. x	x = 0.0	x = 0.025	x = 0.05	x = 0.075	x = 0.1
W% of spinel phase	100	100	98.9	98.3	97.8
W% of secondary phase	0	0	1.1	1.7	2.2
tXRD (nm)	45	42	44	40	38

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Fig. 4 Williamson–Hall plot of Co_0.7Zn_0.3Ho_xFe_2−xO₄.

In spinel ferrites, metal ions occupy tetrahedral (A) and octahedral [B] site. In the present system, occupancy of atom is calculated using the FULLPROF program. The values of atomic coordinates, occupancy and inversion parameter are given in ESI Table 2.† It is observed from Table 2 that Zn ions preferred tetrahedral A site, the Ho³⁺ ions show a preference for octahedral B sites. It is to explain that the radius of the octahedral site is larger than that of tetrahedral site in the spinel lattice. As the ionic radius of the Ho³⁺ ion is 1.04 Å which could be large for tetrahedral A site and therefore these larger Ho³⁺ ions occupy octahedral site. The small amount of Ho³⁺ cations substituted for Fe³⁺ cations which enter into the octahedral sites by rearrangement of cations between the tetrahedral and octahedral sites to minimize the free energy of the system. It is also observed that Co²⁺ ion initially occupy only octahedral sites, but as the Ho³⁺ concentration increased Co²⁺ ions occupied both sites. Partial migration of Co²⁺ ions from B to A sites has been observed by increasing the Ho³⁺ concentration.

Table 2 Values of atomic occupancy (g) determined from Rietveld refinement of XRD pattern of CoZnHo_xFe_2−xO₄

Ions	x = 0.0	x = 0.025	x = 0.05	x = 0.075	x = 0.1
Zn	0.2998(2)	0.2998(3)	0.2997(3)	0.2998(2)	0.2997(3)
Fe	0.6998(2)	0.6999(1)	0.6998(2)	0.6998(2)	0.6999(1)
Co	0.6998(2)	0.7000(1)	0.6998(2)	0.6999(1)	0.6998(2)
Fe	1.2999(2)	1.2751(1)	1.2503(2)	1.2252(2)	1.2000(1)
Ho	0.0000	0.0249(1)	0.0498(2)	0.0749(1)	0.0999(1)
O	3.998(2)	3.997(4)	3.996(4)	3.999(2)	3.998(3)

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Energy dispersive spectroscopy (EDS) was carried out to determine the elemental composition of the prepared samples. Fig. 5 display the EDS pattern of a typical sample of Co_0.7Zn_0.3Fe₂O₄ (x = 0.0) and the results obtained on the elemental analysis are presented in ESI Table 3.† It is observed that the theoretical and observed elemental compositions are in good agreement with each other. The surface morphology and grain size were observed using FE-SEM (Fig. 6). It is evident that the grain sizes are uniformly distributed, however, there are some abnormal grain growth is also observed. The presence of smooth grain boundaries can also be seen in the FE-SEM images.

Fig. 5 Energy dispersive spectra of a typical sample (x = 0.0) of Co_0.7Zn_0.3Ho_xFe_2−xO₄.

Fig. 6 FE-SEM and TEM images of typical samples (x = 0.0 and 0.1) of Co_0.7Zn_0.3Ho_xFe_2−xO₄.

TEM images of typical samples (x = 0.0 and 0.1) are represented in Fig. 6. The TEM image for pure Co–Zn ferrite reveal that the nanoparticles are almost regular in shape, cubic in nature, partially agglomerated with uniform particle size. On the other side particle size is not uniform for Ho³⁺ substituted Co–Zn ferrite (x = 0.1). Some of the Ho³⁺ ions segregated on the grain boundaries of Co–Zn ferrite thereby minimizing the possibility of increasing the grain size by the process of agglomeration. Ho³⁺ with its larger ionic radii required greater energy to replace Fe³⁺ and to form the Ho–O bond. Energy is utilized for this process at the expense of crystallization and therefore the decrease in crystalline nature with smaller particle size is observed. Similar decrease in crystallinity is reported by L. Zhao et al.²⁹ with the inclusion of Nd³⁺ in cobalt.

3.2. Magnetic properties

Magnetization (M) as a function of applied magnetic field (H) was studied at room temperature (300 K) and at 10 K over the field range of +10 kOe to −10 kOe and +20 kOe to −20 kOe respectively. The magnetization curve at 300 K shows (Fig. 7) typical characters of soft magnetic materials; very small area surrounded by the closed curve is related to the soft magnetic properties of materials. It is observed from Fig. 7 that saturation magnetization (M_s) has the highest value of 67.55 emu g⁻¹ for x = 0.075 and it decreased for further increase in Ho³⁺ substitution (x = 0.1). On the other hand magnetization measured at 10 K; M_s has the highest value of 83.67 emu g⁻¹ for x = 0.1. The increase in M_s of Co–Zn ferrite with Ho³⁺ substitution is considered with the large magnetic moment of Ho³⁺ (10.6 μ_B) atoms and their preferential distribution in the octahedral lattice. In case of x > 0.75 for magnetization measured at 300 K, the magnetic properties do not follow the theoretical tendency suggesting a lattice distortion as revealed by XRD data. In addition, due to the higher substitution of Fe³⁺ with Ho³⁺ the M_s and coercivity (H_c) may decrease as a consequence of reduced exchange interaction between Fe^Td–Fe^Oh, which is substituted with the weaker Fe^Td–Ho^Oh interaction.

Fig. 7 Variation of magnetization (M) with applied magnetic field (H) of Co_0.7Zn_0.3Ho_xFe_2−xO₄, measured at 300 K and 10 K.

The increase in M_s at 10 K is attributed to frozen spin-glass-like state on the surfaces of the nano-particles where the decrease in thermal energy and magnetic moments get aligned along the external magnetic field direction. Since at high temperatures, the surface spins can experience many disorder states with similar energies in a short time, which weakens their response to the applied magnetic field and thus lowers the magnetization as compare to their bulk counterpart. However, below the freezing temperature, the surface spins may freeze to a certain state, which increases the projection of the moments in the direction of applied magnetic field. Magnetic hardness, associated with the sharp increase of coercivity is observed for Ho³⁺ substituted Co–Zn ferrite at 10 K. This may correspond to the appearance of the exchange anisotropy field (H_E). The observation of this exchange anisotropy field is a clear indication of the existence of a magnetically disordered surface layer that becomes frozen at low temperature.³⁰ Therefore, the observed increase of the coercivity can be attributed to the extra energy required for the switching of the core spins that are pinned by the exchange interactions with the frozen spin glass-like surface layer. The direct competition of exchange interactions between surface spins lies at the origin of the observed high field irreversibility and accounts for the large experimental values of the coercive fields.³¹

To inspect the existence of magnetic ordering in pure Co–Zn and Ho³⁺ substituted Co–Zn ferrite nanoparticles, these samples were subjected to FC-ZFC measurements with an applied magnetic field of 100 Oe (Fig. 8). The typical spin-glass cusp-like maximum in the ZFC curve is observed which is corresponding to the blocking temperature (T_B). To determine the T_B; the differentiate analysis is carried out and the temperature corresponding to the maximum temperature is determined as dM/dT becomes zero. It is observed that T_B shifts from 195 K (x = 0.0) to 225 K (x = 0.1). Below T_B, with decreasing temperature the B site antiferromagnetic (J_BB) interactions become stronger which compete with the (J_AB) ferrimagnetic interactions. The competition between the exchange interactions results in strong spin frustration and decrease of magnetization at low temperature. At temperatures above T_B, the thermal energy (i.e., k_BT) is higher than the magnetic energy barrier and thus the materials become superparamagnetic following the Curie–Weiss law. T_irr is the temperature, at which FC and ZFC curve bifurcate, also undergo changes with Ho³⁺ substitution. It is observed that T_irr for x = 0.0 is 229 K whereas it is 270 K for x = 0.1. In ZFC, the domain structure at low temperature state differs from the high temperature state and the domain structure gets adjusted as the temperature is increased. In FC, the domain structure retains its state at high temperature configuration from 50 K to 300 K. It is to be noted that bifurcation temperature, T_irr, is higher than that of the T_B, such behavior in FC-ZFC indicates the existence of strong dipole–dipole interaction among spins.³²

Fig. 8 Variation of magnetization (M) with temperature (T) measured in field cooled (FC) and zero field cooled (ZFC) mode at 100 Oe.

The large difference between the magnetizations in ZFC and FC indicate that Ho³⁺ substituted Co–Zn ferrite possess higher coercivity at lower temperature. This coercivity can be related to the anisotropy constant in Ho³⁺ substituted Co–Zn ferrite cubic spinel system. In this system Co²⁺ ion induces this anisotropy as the Fe³⁺ has no orbital momentum and Co²⁺ in the low symmetry coordination has an orbital contribution caused by the relevant spin–orbit coupling that gives rise to single ion anisotropy. The anisotropy constant was obtained from the fitted ZFC curve:³³

where M_ZFC, V, f(V) and V_B are the magnetization in the ZFC process, the particle volume, volume distribution function, and the blocking volume, respectively. The fitted curve show that (figure not shown here) the Co–Zn ferrite posses highest anisotropy constant of 4.6 × 10⁵ erg cm⁻³ for Ho³⁺ substitution level of x = 0.75.

It is observed from FC curves that magnetization continuously decreased with the increase in temperature. The temperature dependence of the M_s can be well fitted to Bloch's spin wave theory. Bloch's law describes the temperature-dependence of the M_s of ferri/ferromagnetic materials where M_s(T) is dominated by the excitation of long wavelength spin-wave fluctuations, called magnons. In case of contiguous distribution of spin-wave states as in the bulk, the temperature dependence of magnetization given by the Bloch law as:³⁴

where M_s(0) is the magnetization of the ground state at T = 0 K, (in our case it is T = 50 K), b is the Bloch's constant, n is the Bloch exponent, and T_c is the Curie temperature. The decrease in the value of b indicates an increase in the spin wave stiffness coefficient, thereby increasing the spin wave excitation energy required to switch spin states in the sample. Bloch' law indicates that the decrease in M_s with increasing temperature due to spin-wave excitations is described by a power law in T.³⁵

AC susceptibility measurements are useful to understand the dynamics of freezing of the spin-glass systems in which the temperature associated with the maxima of χ′ corresponding to spin freezing temperature (T_f), varies with the Ho³⁺ substitution. Fig. 9 shows the variation of χ′ vs. T at selected frequencies of 100, 500 and 1000 Hz for the end samples (x = 0.0 and x = 0.1) of the present study where the magnitude of ac magnetic field is kept constant at H_ac = 10 Oe. It is found that a small shift of the peak position of χ′ to higher temperature for higher driving frequency. The peak shift per frequency decade is quantified in the phenomenological parameter K = ΔT_f/T_f(Δlog f), where ΔT_f is the difference between T_f measure in Δlog f frequency interval.³⁶ The value of K represents the strength of inter-particle interaction among magnetic nanoparticles. This parameter lies in the range of 0.005 to 0.06 for interacting nanoparticles of spin glass systems.³⁷ In the present system it is 0.0097 for x = 0.0 and 0.011 for x = 0.1, supporting the claim of spin glass behavior. The intensity of the peak increases for lower measuring frequencies. Furthermore χ′(T) curves for different measuring frequencies overlap for temperatures higher than the peak temperature. This behavior is observed in many spin-glass and disordered magnetic system,^38,39 that confirmed the transition from paramagnetic to spin glass phase in pure Co–Zn (x = 0.0) and Ho³⁺ substituted Co–Zn (x = 0.1) ferrite. The spin freezing temperature T_f can be accurately determined by the position of the cusp of the real part of AC susceptibility, χ′, and it is observed for both the samples. In other words, the maximum relaxation time (τ) of the system is equal to 1/f at T_f. A shift in the T_f from 234 K to 251 K for f = 100 Hz, 241 K to 260 K for f = 500 Hz and 246 K to 268 for f = 1000 Hz is observed as Ho³⁺ substitution in Co–Zn ferrite changed from x = 0.0 to 0.1.

Fig. 9 (a and b) Temperature dependence of in-phase (χ′) components of the AC susceptibility measured at 100, 500 and 1000 Hz with an applied magnetic field H_ac = 10 Oe. (c and d) The frequency dependence of freezing temperature is fitted with Vogel–Fulcher law.

In order to gain clear insight into the spin glass behavior by considering the interaction nano-particles. The data is fitted to Vogel–Fulcher equation: τ = τ₀ exp[−E_a/k_B(T_f − T₀)], where τ₀ is characteristic relaxation time, E_a is activation energy, k_B is the Boltzmann constant and T₀ a characteristic temperature which is the measure interparticle interaction energy.⁴⁰ Fig. 9 shows the experimental and fitted curve for x = 0.0 and 0.1. The obtained parameters T₀ and τ₀ for E_a/k_B = 811 K are listed in Table 3. The obtained τ₀ from the fitted line of f = 100 Hz is 9.1 × 10⁻¹² s for Co–Zn ferrite (x = 0.0), whereas it is decreased to 4.5 × 10⁻¹³ s for Ho³⁺ substituted Co–Zn ferrite (x = 0.1). Values of τ₀ are in close proximity to spin glass system; the value of τ₀ ∼10⁻¹³ s is identical in magnitude with that for atomic spin-glasses, for which τ₀ denotes a spin flip time of individual magnetic moments belonging to atoms or ions. The decrease in τ₀ of Co–Zn ferrite with increase in Ho³⁺ substitution could be attributed to the exchange coupling strength between nanoparticles. Interparticle interaction energy, T₀, decreased from 195 to 217 K for Ho³⁺ substation and is in good agreement with the T_B obtained from the ZFC curve.

Table 3 Spin freezing temperature (T_f), interparticle interaction energy (T₀) characteristic relaxation time (τ₀) and transition temperature (T_g) obtained by Vogel–Fulcher model and spin glass power-law

Comp.	f (Hz)	Tf (K)	Vogel–Fulcher model		Spin glass power-law
			T0 (K)	τ0 (s)	Tg (K)	τ0 (s)
x = 0.0	100	234	195	9.1 × 10^−12	203	5.3 × 10^−12
	500	240	198	8.2 × 10^−12	207	4.3 × 10^−12
	1000	246	208	5.2 × 10^−13	210	4.9 × 10^−13
x = 0.1	100	251	217	4.5 × 10^−13	220	4.4 × 10^−13
	500	260	224	3.2 × 10^−13	229	2.2 × 10^−13
	1000	268	232	1.7 × 10^−13	238	1.2 × 10^−13

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To determine if the dynamics of a magnetic system exhibit critical slowing down, the ac susceptibility data can be compared to the predictions of dynamic scaling theory, which states that the relaxation time in a spin glass, related to the correlation length ξ as τ ∝ (T_f/T_g − 1)^−v as glass transition as T_g is approached. Fitting the data with spin glass power-law could provide concrete information about the spin glass state:⁴¹

where T_g is the transition temperature, T_f is the freezing temperature, τ the maximum relaxation time of the spin system at a temperature T_f, zv the dynamic critical exponent that falls within the range for glassy systems (zv ∼ 4–12)⁴² and τ₀ is related to the relaxation time of the individual particle magnetic moment in the system. The fitted parameters from spin glass power law are listed in Table 3. The obtained τ₀ and T_g values are in good agreement with τ₀ and T₀ values estimated using the Vogel–Fulcher model. It is a known fact that Vogel–Fulcher law gives the frequency dependence of the τ₀ in weakly interacting nanoparticles, whereas the spin glass power law describes the τ₀ in strongly interacting nanoparticles showing spin glass features. In the present study, both the models seem to describe the frequency dependence of T_f and also yield similar τ₀ values.

Fig. 10 shows the Mössbauer spectra of three typical samples x = 0.0, 0.05 and 0.1 recorded at room temperature. The data is analyzed with distribution of hyperfine fields. The observed sextet confirms the magnetic ordering in all the samples at room temperature. However, broadened nature six-line pattern may have arisen from the random distribution of magnetic (Fe, Co and Ho) and nonmagnetic (Zn) ions in the sublattices. It may also be due to the existence of magnetically ordered regions that are not coupled to each other by intersublattice, i.e. A–B superexchange interactions. Further, the center of the Zeeman pattern does not show any significant change for all the studied samples and could be related to the zero quadrupole splitting resulted from the presence of chemical disorder. The chemical disorder produces a distribution of electric field gradient (EFG) of varying magnitude, direction sign, and symmetry, that can be represented by the equation:⁴³

|ΔE_Q| = 1/2|ΔE_Q0|(3 cos² θ − 1),

where |ΔE_Q| is the magnitude of the quadruple shift when the magnetic interaction tends to be zero and θ is the angle between axially symmetric EFG and magnetic field direction. This distribution of field produces a noticeable broadening in individual lines of Zeeman pattern.

Fig. 10 Room temperature Mossbauer spectra and their corresponding variation of hyperfine field distribution for the typical samples (x = 0.0, 0.05 and 0.1) of Co_0.7Zn_0.3Ho_xFe_2−xO₄.

The obtained average isomer shift is consistent with Fe³⁺ state. The variation of hyperfine field distribution P(B_hf) as a function of field B_hf is also shown in Fig. 10. The shape of P(B_hf) shows a broad low field tail down to B_hf = 0 and a high field maximum. The hyperfine field distributions at high filed are broad for x = 0.0 and it becomes narrower as the substitution of Ho³⁺ increased to x = 0.1. This is connected with different local surroundings of the Fe atom in investigated compounds. The values of internal magnetic field B_hf depend upon the nearest neighbour distribution around Fe atoms, which increased as more Ho³⁺ atoms surround Fe atoms. The average hyperfine field of Co–Zn ferrite is increased from 45.55 T (x = 0.0) to 46.33 T (x = 0.1) with the substitution of Ho³⁺ substitution. According to Neel,⁴⁴ contribution to hyperfine magnetic field is due to the strongest A–B superexchange interactions and contributions from to A–A/B–B exchange interactions may be neglected. As the ratio of Fe_(B)³⁺/Fe_(A)³⁺ should be decreased with the incorporation of the Ho³⁺ ions at octahedral B-site that expected to decrease in hyperfine field. However, with the increase in Ho³⁺ substitution it seems to be that A–B interaction increased as evident by increase in hyperfine field. This also supports the increase in saturation magnetization as observed by M–H loops measured at room temperature and at 300 K.

4. Conclusions

Ho³⁺ substituted Co–Zn ferrite nanoparticle are successfully prepared by sol–gel method. Rietveld refined XRD pattern shows the existence of orthoferrite-HoFeO₃ and unreacted-Ho₂O₃ phase for higher Ho³⁺ substitution. The lattice constant increased from 8.401 to 8.419 Å with increase Ho³⁺ ions. XRD and TEM measurement confirmed the nanosize of crystallite and particle of all the prepared samples. RMS strain determined from the W–H plot revealed that strain changed the sign from negative to positive with the Ho³⁺ substitution which is an indication of increase in strain. Theoretical and observed elemental percentages are in good agreement as determined from EDS patterns. Cation distribution suggest that Co and Ho³⁺ ion prefer to occupy octahedral [B] site and Zn ion prefer to occupy tetrahedral (A) site where as Fe ion are randomly distributed among both sites. Importantly, M–H plots and Mossbauer spectra revealed that the saturation magnetization increased with the Ho³⁺ substitution. It can be concluded from the M–H loops, FC/ZFC and susceptibility measurements results that the Ho³⁺ substituted Co–Zn ferrite nanoparticles exhibits a spin-glass-like behavior at low temperatures. A frequency and field dependent sharp cusp is observed in AC susceptibility, indicating metastable magnetism in these systems. Blocking temperature, spin freezing temperature, interparticle interaction energy and transition temperature characteristic decreased whereas relaxation time increased with Ho³⁺ substitution.

Conflict of interest

The authors declare no competing financial interest.

Acknowledgements

Authors are thankful to Prof. P. L. Paulose (Tata Institute of Fundamental Research, Mumbai) for providing low temperature magnetization facilities. Thanks to Dr V. R. Reddy (UGC-DAE Consortium for Scientific Research, Indore) for providing the Mossbauer facility. They are also thankful to Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru for providing the FE-SEM and TEM facilities.

References

1. S. Mørup and E. Tronc, Superparamagnetic relaxation of weakly interacting particles, Phys. Rev. Lett., 1994, 72, 3278.
2. R. Mathieu, J. A. D. Toro, D. Salazar, S. S. Lee, J. L. Cheong and P. Nordblad, Phase transition in a super superspin glass, Europhys. Lett., 2013, 102, 67002.
3. R. H. Kodama, A. E. Berkowitz, E. McNiff Jr and S. Foner, Surface Spin Disorder in NiFe₂O₄ Nanoparticles, Phys. Rev. Lett., 1996, 77, 394.
4. K. Hiroi, K. Komatsu and T. Sato, Superspin glass originating from dipolar interaction with controlled interparticle distance among gamma-Fe₂O₃ nanoparticles with silica shells, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 224423.
5. A. C. Gandhi, P. M. Reddy, T.-S. Chan, Y.-P. Ho and S. Y. Wu, Memory effect in weakly-interacting Fe₃O₄ nanoparticles, RSC Adv., 2015, 5, 84782–84789.
6. G. S. Babu, M. Valant, K. Page, A. Llobet, T. Kolodiazhnyi and A.-K. Axelsson, New (Bi_1.88Fe_0.12)(Fe_1.42Te_0.58)O_6.87 Pyrochlore with Spin-Glass Transition, Chem. Mater., 2011, 23, 2619–2625.
7. V. Cannella and J. A. Mydosh, Magnetic Ordering in Gold-Iron Alloys, Phys. Rev. B: Condens. Matter Mater. Phys., 1972, 6, 4220.
8. S. Kirkpatrick and D. Sherrington, Infinite-ranged models of spin-glasses, Phys. Rev. B: Condens. Matter Mater. Phys., 1978, 17, 4384.
9. M. Rivero, A. del Campo, Á. Mayoral, E. Mazario, J. Sánchez-Marcos and A. Muñoz-Bonilla, Synthesis and structural characterization of Zn_xFe_3−xO₄ ferrite nanoparticles obtained by an electrochemical method, RSC Adv., 2016, 6, 40067–40076.
10. P. E. Jönsson, Superparamagnetism and spin glass dynamics of interacting magnetic nanoparticle systems, Adv. Chem. Phys., 2004, 128, 191–248.
11. R. Skomski, Are There Superspin Glasses?, J. Appl. Phys., 2011, 109, 07E149.
12. B. G. Toksha, S. E. Shirsath, M. L. Mane, S. M. Patange, S. S. Jadhav and K. M. Jadhav, Auto-combustion high-temperature synthesis, structural and magnetic properties of CoCr_xFe_2−xO₄ (0 ≤ x ≤ 1.0), J. Phys. Chem. C, 2011, 115, 20905–20912.
13. S. M. Patange, S. S. Desai, S. S. Meena, S. M. Yusuf and S. E. Shirsath, Random site occupancy induced disordered Néel-type collinear spin alignment in heterovalent Zn²⁺–Ti⁴⁺ ions substituted CoFe₂O₄, RSC Adv., 2015, 5, 91482–91492.
14. S. Dey, A. Roy, J. Ghosh, R. N. Bhowmik and R. Ranganathan, Size dependent magnetic phase of nanocrystalline Co_0.2Zn_0.8Fe₂O₄, J. Appl. Phys., 2001, 90, 4138–4142.
15. V. Mameli, A. Musinu, A. Ardu, G. Ennas, D. Peddis, D. Niznansky, C. Sangregorio, C. Innocenti, T. K. Nguyen and C. Thanh Cannas, Studying the effect of Zn-substitution on the magnetic and hyperthermic properties of cobalt ferrite nanoparticles, Nanoscale, 2016, 8, 10124–10137.
16. D. S. Nikam, S. V. Jadhav, V. M. Khot, R. A. Bohara, C. K. Hong, S. S. Mali and S. H. Pawar, Cation distribution, structural, morphological and magnetic properties of Co_1−xZn_xFe₂O₄ (x = 0–1) nanoparticles, RSC Adv., 2015, 5, 2338–2345.
17. S. S. Jadhav, S. E. Shirsath, B. G. Toksha, S. M. Patange, S. J. Shukla and K. M. Jadhav, Structural properties and cation distribution of Co–Zn nano-ferrites, Int. J. Mod. Phys. B: Condens. Matter Phys., Stat. Phys., Appl. Phys., 2009, 23, 5629–5638.
18. A. Sinha and A. Dutta, Microstructure evolution, dielectric relaxation and scaling behavior of Dy-for-Fe substituted Ni-nanoferrites, RSC Adv., 2015, 5, 100330–100338.
19. S. G. Kakade, Y.-R. Ma, R. S. Devan, Y. D. Kolekar and C. V. Ramana, Complex Impedance, and Electrical Transport Properties of Erbium (Er³⁺) Ion-Substituted Nanocrystalline, Cobalt-Rich Ferrite (Co_1.1Fe_1.9−xEr_xO₄), J. Phys. Chem. C, 2016, 120, 5682–5693.
20. M. Vucinic-Vasic, E. S. Bozin, L. Bessais, G. Stojanovic, U. Kozmidis-Luburic, M. Abeykoon, B. Jancar, A. Meden, A. Kremenovic and B. Antic, Thermal Evolution of Cation Distribution/Crystallite Size and Their Correlation with the Magnetic State of Yb-Substituted Zinc Ferrite Nanoparticles, J. Phys. Chem. C, 2013, 117, 12358–12365.
21. C. S. Hong, W. S. Kim, E. O. Chi, N. H. Hur and Y. N. Choi, Role of Rare Earth Ion in Spin Glass Behavior for R_0.7Sr_1.3MnO₄, Chem. Mater., 2002, 14, 1832–1838.
22. S. E. Shirsath, R. H. Kadam, S. M. Patange, M. L. Mane, A. Ghasemi and A. Morisako, Enhanced magnetic properties of Dy³⁺ substituted Ni–Cu–Zn ferrite nanoparticles, Appl. Phys. Lett., 2012, 100, 042407.
23. K. S. Lohar, A. M. Pachpinde, M. M. Langade, R. H. Kadam and S. E. Shirsath, Self-propagating high temperature synthesis, structural morphology and magnetic interactions in rare earth Ho³⁺ doped CoFe₂O₄ nanoparticles, J. Alloys Compd., 2014, 604, 204–210.
24. R. N. Bhowmik and R. Ranganathan, Magnetic properties in rare-earth substituted spinel Co_0.2Zn_0.8Fe_2−xRE_xO₄ (RE = Dy, Ho and Er, x = 0.05), J. Alloys Compd., 2001, 326, 128–131.
25. S. E. Shirsath, M. L. Mane, Y. Yasukawa and A. Liu Morisako, Self-ignited high temperature synthesis and enhanced super-exchange interactions of Ho³⁺–Mn²⁺–Fe³⁺–O²⁻ ferromagnetic nanoparticles, Phys. Chem. Chem. Phys., 2014, 16, 2347–2357.
26. E. Ateia, M. A. Ahmed and A. K. El-Aziz, Effect of rare earth radius and concentration on the structural and transport properties of doped Mn–Zn ferrite, J. Magn. Magn. Mater., 2007, 311, 545–554.
27. K. Kamala Bharathi, G. Markandeyulu and C. V. Ramana, Structural, Magnetic, Electrical, and Magnetoelectric Properties of Sm- and Ho-Substituted Nickel Ferrites, J. Phys. Chem. C, 2011, 115, 554–560.
28. S. M. Patange, S. E. Shirsath, G. S. Jangam, K. S. Lohar, S. S. Jadhav and K. M. Jadhav, Rietveld structure refinement, cation distribution and magnetic properties of Al³⁺ substituted NiFe₂O₄ nano particles, J. Appl. Phys., 2011, 109, 053909.
29. L. Zhao, H. Yang, X. Zhao, L. Yu, Y. Cui and S. Feng, Magnetic properties of CoFe₂O₄ ferrite doped with rare earth ion, Mater. Lett., 2006, 60, 1–6.
30. C. Caizer and M. Stefanescu, Magnetic characterization of nanocrystalline Ni–Zn ferrite powder prepared by the glyoxylate precursor method, J. Phys. D: Appl. Phys., 2002, 35, 3035.
31. S. Kumar, V. Singh, U. K. Mandal and R. K. Kotnala, Nanocrystalline Co_0.5Zn_0.5Fe₂O₄ ferrite: Synthesis, characterization and study of their magnetic behavior at different temperatures, Inorg. Chim. Acta, 2015, 428, 21–26.
32. M. Muro, R. Street, P. G. McCormick and J. Amighian, Magnetic properties of ultrafine MnFe₂O₄ powders prepared by mechanochemical processing, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 184414.
33. R. Sappey, E. Vincent, N. Hadacek, F. Chaput, J. P. Boilot and D. Zins, Nonmonotonic field dependence of the zero-field cooled magnetization peak in some systems of magnetic nanoparticles, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 56, 14551.
34. F. Bloch, Zur Theorie des Ferromagnetismus, Z. Phys., 1931, 61, 206–219.
35. S. Khanra, A. Bhaumik, Y. D. Kolekar, P. Kahol and K. Ghosh, Structural and magnetic studies of Y₃Fe_5−5xMo_5xO₁₂, J. Magn. Magn. Mater., 2014, 369, 14–22.
36. Z. Sabsabi, F. Vernay, O. Iglesias and H. Kachkachi, Interplay between surface anisotropy and dipolar interactions in an assembly of nanomagnets, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 104424.
37. H. Khurshid, P. Lampen-Kelley, Ò. Iglesias, J. Alonso, M.-H. Phan, C.-J. Sun, M.-L. Saboungi and H. Srikanth, Spin-glass-like freezing of inner and outer surface layers in hollow γ-Fe₂O₃ nanoparticles, Sci. Rep., 2015, 5, 15054,  DOI:10.1038/srep15054.
38. S. Dhar, O. Brandt, A. Trampert, K. J. Friedland, Y. J. Sun and K. H. Plorg, Observation of spin-glass behavior in homogeneous (Ga,Mn)N layers grown by reactive molecular-beam epitaxy, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 67, 165205.
39. M. A. Hakim, M. Manjurul Haque, M. Huq and P. Nordblad, Spin-glass-like ordering in the spinel ZnFe₂O₄ ferrite, Phys. B, 2011, 406, 48–51.
40. K. Parekh and R. V. Upadhyay, Static and dynamic magnetic properties of monodispersed Mn_0.5Zn_0.5Fe₂O₄ nanomagnetic particles, J. Appl. Phys., 2010, 107, 053907.
41. J. Mohapatra, A. Mitra, D. Bahadur and M. Aslam, Superspin glass behavior of self-interacting CoFe₂O₄ nanoparticles, J. Alloys Compd., 2015, 628, 416–423.
42. J. A. Mydosh, Spin Glasses: An Experimental Introduction, Taylor & Francis, 1993.
43. E. Matthias, W. Schneider and R. M. Steffen, Nuclear Level Splitting Caused by a Combined Electric Quadrupole and Magnetic Dipole Interaction, Phys. Rev., 1962, 125, 261.
44. L. Neel, C. R. Acad. Bulg. Sci., 1950, 230, 375.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra12541g

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