Particle size dependence of the magnetic and magneto-caloric properties of HoCrO₃ Free

As a safe, efficient, and environmentally friendly technology, solid-state refrigeration is an alternative solution to conventional vapor compression technology.^1–4 Solid-sate refrigeration is based on three kinds of caloric effects: baro-caloric,¹ electro-caloric,² and magneto-caloric effect (MCE),^3,4 which are, respectively, driven by the application of isotropic stress, electric field, and magnetic field.^5–7 Nowadays, the interest has been extended to multi-caloric materials, which include more than one of these caloric effects. Multicaloric materials have been regarded as useful for improving the efficiency of refrigeration devices,^8–11 and strong candidates among them are multiferroic materials with two or more ferroic phases (ferroelectricity, ferromagnetism, and ferroelasticity). This is because cross-coupling effects between these ferroic phases lead to large entropy change.^12–17 Recently, rare-earth chromites (RCrO₃, R = Ho, Dy, Er, etc.) were proposed to be a new family of multiferroics showing large MCE,^18–20 thereby becoming suitable candidates for application in magnetic refrigeration (MR).^21–25 Among RCrO₃, HoCrO₃ with orthorhombically distorted perovskite structure has been a of great interest because of its multiferroic and MCE properties.^22,26–30 It has been reported that HoCrO₃ polycrystalline powder has MCE value of 7.2 J/kg K and refrigerant capacity (RC) ∼ 408 J/kg at 20 K and 7 T (70 kOe), and therefore deemed suitable for MR applications in low temperature range from 5 to 30 K.²² Tiwari et al. reported that, and Cr³⁺ ions in CrO₆ octahedra in HoCrO₃ shows splitting in spin-allowed d-d transition, induced by local magnetic field.²⁷ Ghosh et al., reported a high electric polarization ∼0.32 μCcm⁻² at 10 K, an ordering temperature of Cr³⁺ (⁠ $T N C r$⁠) at 142 K in HoCrO₃ polycrystalline sample, and polar ordering at a much higher temperature of 240 K indicating atypical multiferroic order.²⁶ 

In spite of the above-mentioned studies, an essential understanding of the size effect on the magnetic properties of HoCrO₃ polycrystalline samples is still lacking. This is an important issue to investigate in HoCrO₃ since prior studies in other perovskites have shown that particle size considerably affects the magnetic properties of polycrystalline powders.^31–34 For example, Wu et al. synthesized uniform ErMn₂O₅ nanorods of different lengths by a surfactant-templated hydrothermal process and the magnetization of these nanorods increases with their length, reaching a maximum value at the length of 175 nm because of the competition between surface strain and uncompensated spin.³¹ Park et al. synthesized BiFeO₃ nanoparticles of particle size ranging from 15 nm to 100 nm by controlling the annealing temperatures, and demonstrated that the nanoparticles with size smaller than 62 nm exhibited strong size-dependent magnetic properties.³² In the work of Cai et al., a modified polyacrylamide sol-gel method was employed to prepare DyMnO₃ nanoparticles of various sizes from 51 nm to 106 nm. With the reduction in particle size, the critical field where DyMnO₃ nanoparticles went through a field-induced metamagnetic transition at 4 K increases monotonically, whereas all the lattice constants and the antiferromagnetic interaction of Mn³⁺ ions decreased, because of reduction in Mn-O-Mn bond angle.³³ Likewise, the dependence of magnetic properties on particle size was also reported in DyCrO₃ nano-platelets synthesized using citric acid and oxalic acid as chelating agents. The average particle size were ∼90 nm and ∼50 nm, respectively, leading to different Néel temperatures (144 K and 146 K).³⁴ 

Particle size also influences the magnetic hysteresis behavior of a material.^35–37 This is important since in the refrigeration process, larger magnetic hysteresis leads to more energy loss, and thus abates MCE.³⁸ Since HoCrO₃ polycrystalline sample displays large magnetic hysteresis,²² variation of the particle size is expected to impact its MCE properties. As an example, the particle size of Gd₅Si₂Ge₂ alloys powders prepared via a milled processing condition in the micrometer range remarkably affected the MCE properties and powders with particle size 38.5–50 μm exhibited maximum entropy change.³⁹ Also, Pękała et al. revealed that MCE properties of La_0.7Ca_0.3MnO₃ nanocrystalline spreads over a broader temperature interval than its polycrystalline counterpart.⁴⁰ In addition, MCE properties of Ni₅₁Mn₃₄In₁₄Si₁ alloy of particle size ranging from 20 to 1000 μm were studied and samples with sizes 600–710 μm had high values of entropy change and RC.⁴¹ Similar result was reported by Toliński and Synoradzki in DyCo₃B₂ system.⁴² Likewise, the electro-caloric effect of BaTiO₃ nano-ceramics also depends on the particle size.⁴³ 

The particle size dependence of the MCE and electro-caloric effects observed in many systems noted above motivated us to undertake similar investigations in polycrystalline HoCrO₃ powder samples. The size-dependent structural, magnetic, and magneto-caloric properties for the 60 nm, 190 nm, 320 nm, and 425 nm size particles are reported in this work.

Powder samples of HoCrO₃ were synthesized using the citrate method. In the first step, equimolar quantities of Ho(NO₃)₃ and Cr(NO₃)₃ precursors were dissolved in water separately and then mixed together, followed by the addition of citric acid as the chelating agent. The mixed solution was then heated and dried on a hot plate followed by annealing at 700, 900, 1100, and 1300 °C, respectively for 2 h in an O₂ atmosphere to acquire the four HoCrO₃ powder samples with different particle sizes.

The structural properties and phase purity of the samples were determined by X-ray diffraction (XRD) measurements (using Bruker D2 Phaser X-ray diffractometer with Cu-Kα radiation λ = 1.54 Å). The scan mode was 2°/min with 0.02°/step from 20° to 90°. The experimental XRD data were Rietveld-refined via Fullprof Suite software in order to obtain the lattice constants of the samples. The Williamson-Hall analysis was adopted to estimate the particle size by Person 7 peak fitting of the XRD data (Fityk Software). Samples were imaged by field-emission scanning electron microscope (FE-SEM) to ascertain the microstructure and particle sizes, which were further confirmed by plotting the histogram of the measured particle diameters. Room temperature micro-Raman spectra were recorded using a 514 nm Argon laser in the Renishaw System 2000. The temperature dependent dc magnetization data {measured in zero-field cooled (ZFC) and field-cooled (FC) modes} and isothermal dc magnetization versus magnetic-field curves at diverse temperatures were evaluated using a vibrating sample magnetometer system attached to an Evercool Physical Property Measurement System from Quantum Design Inc.

The room temperature XRD patterns of the four prepared samples of HoCrO₃ are presented in Fig. 1(a), where the samples are labeled by their particle sizes obtained in the following discussion. The experimental data were fitted by Rietveld refinement that indicates that all the observed diffraction peaks in the four samples match with the orthorhombically distorted perovskite structure with space group Pbnm. Fig. 1(b) shows the representative refinement result of the HoCrO₃ sample annealed at 700 °C (labeled as 60 nm). It can be seen that no extra peak was identified, suggesting that the samples were phase pure. The lattice parameters (a, b, and c), Cr-O-Cr bond angles, and Cr-O bond lengths acquired from the Rietveld refinements are summarized in Table I. The values of the orthorhombic distortions from the cubic structure, defined by the orthorhombic strain factor S = 2(b−a)/(b+a),³³ are also summarized in Table I. These parameters for the HoCrO₃ samples annealed at the four different temperatures were found to be only slightly different from each other.

The crystallite sizes (D) and strain (η) of the HoCrO₃ samples were calculated using the Williamson-Hall analysis⁴⁴ 

where K is the shape parameter (0.89 was used in this study in accordance with earlier reports of synthesis of similar materials),⁴⁴ λ = 0.154 (nm) is the wavelength of the X-ray beam, β is the full width at half maximum of the Bragg diffraction peak, and η is the strain in the sample. For particle sizes above 50 nm, the instrumental width has to be considered by replacing β in Eq. (1) with $β ′ = β 2 − b 2$⁠, where b is the instrumental width (0.042° for the Bruker D2 Phaser system adopted here).

As listed in Table I, D was calculated to be 88, 182, 250, and 487 nm with the uncertainties using Eq. (1) for the HoCrO₃ samples annealed at 700, 900, 1100, and 1300 °C, respectively. Notably, D increases with an increase in the annealing temperature, because larger particles grow at the expense of smaller particles. This phenomenon is also known as Oswald ripening.⁴⁵ In the report of Ghosh et al., HoCrO₃ polycrystalline sample annealed at 1400 °C for 12 h by solid-state reaction method was examined to have even larger D of ∼1 μm,²⁶ which is consistent with our results. Similarly, Park et al.³² and Cai et al.³³ reported BiFeO₃ and DyMnO₃ nanoparticles synthesized with various D (14–245 nm and 51–106 nm, respectively) by tuning the annealing temperatures. Therefore, it can be inferred that the crystallite size of HoCrO₃ powders is tunable by simply controlling the annealing temperature (in addition to the sample synthesis method).

In Figs. 2(a)–2(d), the particle size (D′) of the HoCrO₃ samples was further explored by using the FESEM images, along with the particle diameter histograms accounting for about 100 particles (fitted to log-normal distribution). From these histograms, D′ are estimated to be 60 nm, 190 nm, 320 nm, and 425 nm, respectively, which are very close to the crystallite sizes (D) determined from the Williamson-Hall analysis and are within the experimental uncertainties (see Table I). This shows that the particle size rises with increase in the annealing temperature. In general, the crystallite size determined by XRD may be smaller than the particle size determined by SEM technique since a particle may contain more than one crystallite. However, as shown above, in the present case, the sizes determined by XRD and SEM are nearly equal within experimental uncertainties of the two methods. Therefore, in the rest of this paper, the average particle sizes obtained from the histograms as listed above have been used to label the HoCrO₃ samples. The shape of the particles was also measured in terms of the aspect ratio of the longest dimension to the shortest dimension, which were determined to be 1.45, 1.45, 1.33, and 1.21 for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively. Therefore, the aspect ratio decreases as the annealing temperature goes up.

Room temperature Raman spectra of the samples are presented in Fig. 3. Theoretical studies show that the orthorhombic Pbnm perovskite structure has 24 Raman active modes: 7A_g+5B_1g+7B_2g+5B_3g, 12 of which occur in the range of wave-number 100 cm⁻¹ to 600 cm⁻¹. These modes were assigned by following the recent room temperature Raman study on RCrO₃ system.^21,46 All of the vibration modes observed here are indicative of phase purity of the samples. There is slight shift in the Raman modes between these HoCrO₃ samples, attributed to the difference in the strain and lattice parameters (see Table I). It is noted that the width of the Raman modes becomes narrower with the rise in the particle size; this is interpreted in terms of the variation of lattice distortion, which is consistent with the trend observed in DyMnO₃ nanoparticles.³³ Because neither XRD nor Raman measurements show any impurity peaks (or modes) for all samples, it is inferred that even after annealing these samples at higher temperatures (1300 °C), no impurity phase was detected in the present samples.

The temperature dependent dc magnetization (M vs. T) data for the four HoCrO₃ samples under the field-cooled (FC) and zero-field-cooled (ZFC) protocols with applied field of H = 50 Oe are presented in Fig. 4. The FC data bifurcates from the ZFC data below ∼ 140 K which represents magnetic ordering as discussed below. Note that below 140 K, there is no evidence of a peak in the M vs. T data for the ZFC case. This implies that there is no blocking phenomenon characteristics of nanoparticle systems and that the sizes of these particles are larger than the superparamagnetic limit. This information is used later in the discussion of the size dependence of coercivity in these samples.

For the determination of the Néel temperature (⁠ $T N C r$⁠, where Cr³⁺ ions order), d(χT)/dT data (χ = M/H is susceptibility using dc field cooled magnetization data) are plotted as a function of temperature as shown in Fig. 5, revealing $T N C r$ of 139.1, 139.6, 140.1, and 139.7 K for the 60 nm, 190 nm, 320 nm, and 425 nm size particle samples, respectively. It is noted that the FC magnetization data were recorded with an interval of 0.5 K near $T N C r$ for better resolution. Theoretically and experimentally, it is well established that in antiferromagnets, the position of T_N from the temperature dependent magnetic susceptibility data is best determined by the peak position in d(χT)/dT vs T plot.^47,48 In the present samples with diverse particle size, there is only minor variations (<1 K) in the observed $T N C r$ values. However, in the report of Gupta et al., DyCrO₃ nano-platelets prepared by citric acid and oxalic acid methods with an average particle size of 90 nm and 50 nm, yielded slightly different $T N C r$ values (144 K and 146 K, respectively).³⁴ In addition to $T N C r$⁠, a second transition at ∼8 K, associated with the ordering of Ho³⁺ ions (⁠ $T N H o$⁠), was observed in the present samples as shown in the inset of Fig. 5.

Below 140 K (⁠ $T N C r$⁠), the magnetization of the present HoCrO₃ samples rises gradually with reduction in temperature. This is not typical of fully compensated collinear antiferromagnets, so that some canting of the moments is likely present in HoCrO₃. This is also verified by the temperature dependence of magnetization just above $T N C r$ in Sec. III B. In addition, there is clearly a size effect on the magnetization of HoCrO₃ samples measured with H = 50 Oe below $T N C r$ for both the ZFC and FC cases, as revealed in Fig. 4. The FC magnetization increases with an increase in particle size, whereas an opposite trend is evident for the ZFC magnetization. It is noted that such an increase of FC magnetization value with increasing particle size was also reported in multiferroic TbMnO₃ nanoparticles;⁴⁹ however, no such trend was observed for multiferroic DyMnO₃ nanoparticles in the size range of 51 to 106 nm.³³ 

In the paramagnetic region above 140 K, magnetization of the four HoCrO₃ samples with different particle sizes slowly decreases with increase in temperature, due to paramagnetism of Cr³⁺ and Ho³⁺ ions. Hence, the data of χ vs. T for T ≫ $T N C r$ were first fitted to the Curie-Weiss law

Here, C is the Curie constant and θ is the Weiss temperature. The fitting results of the present samples are shown in Figures 6(a)–6(d) and the fitted parameters (C and θ) are summarized in Table II. With the fitted C value, the effective magnetic moment $μ e f f$ was then calculated using

where k_B is Boltzmann constant and N is Avogadro constant.⁵⁰ The effective magnetic moment μ_eff^a was also calculated using the free ionic moments of Cr³⁺ (3.8 μ_B) and Ho³⁺ ions (10.4 μ_B)²² according to $μ e f f a = μ H o 2 + μ C r 2$⁠. These parameters are also summarized in Table II.

As evident in Figures 6(a)–6(d), the χ vs. T data just above $T N C r$ does not quite fit to the Curie-Weiss law of Eq. (2) due to the sharp drop in 1/χ with reduction in temperature. This is because HoCrO₃ is not a simple antiferromagnet, but a canted antiferromagnet giving rise to weak ferromagnetism, whose paramagnetic susceptibility versus temperature behavior was modeled by Moriya by including the antisymmetric exchange interaction, known as the Dzyaloshinsky-Moriya (DM) interaction.⁵¹ Only the susceptibility in the direction parallel to the easy magnetization axis of the crystal follows the Curie-Weiss law, while the susceptibility in the direction perpendicular to the easy axis must take the DM interaction into consideration. For the present polycrystalline powder samples, the parallel and perpendicular susceptibility could not be evaluated independently, and the contribution from the perpendicular part plays the dominant role. Thus, the susceptibility were fitted to the equation⁵¹ 

where T₀ and $T N C r ′$ are fitted parameters, expressed by Moriya⁵¹ as

where Z = 6 is the coordination of Cr³⁺ with respect to other Cr³⁺, S = 3/2 is the spin quantum number of Cr³⁺, J_e and D_e are the strength of the symmetric and antisymmetric Cr³⁺-Cr³⁺ exchange interactions, respectively. Equations (5) and (6) provide a semi-quantitative analysis of J_e and D_e.

The fitting of the data according to Eqs. (4)–(6) in Fig. 7 shows major improvements over the fitting to the Curie-Weiss law of Eq. (2). The fitted parameters, including C′, T₀, θ′, and $T N C r ′$⁠, are also given in Table II. Note that the fitted values of $T N C r ′$ agree well with those obtained by the peak in d(χT)/dT. The Cr³⁺-Cr³⁺ exchange interactions have been extracted from the above parameters, since other weaker exchange interactions (like Ho³⁺-Ho³⁺ and Ho³⁺-Cr³⁺ interactions) are ignorable at temperature well above the Ho³⁺ ordering temperature (∼8 K). Note that for the HoCrO₃ samples, D_e is an order of magnitude smaller than J_e as expected in most cases.⁵³ However, as evident from Table II, D_e increases with growth in particle size whereas J_e is essentially the same for the four present samples. Since the canting angle and hence magnetization is proportional to the ratio (D_e/J_e), the rise in magnetization (FC mode) with increase in particle size may be interpreted by this mechanism.

Fig. 8 shows M vs. H curve of the four HoCrO₃ samples measured at 5 K with H up to 40 kOe. As H is increased, M increases, but the low H variation of M is quite different among the four samples. As discussed later, this is related to significantly different coercivity (H_C) of the four samples. At H = 40 kOe, magnitudes of M for the 60 nm, 190 nm, 320 nm, and 425 nm particles are 80.1, 79.8, 77.4, and 75.8 emu/g, respectively. However, M does not saturate until applied field of 40 kOe for any of the four samples.

The isothermal hysteresis behavior of M vs. H in the four HoCrO₃ samples with H up to 40 kOe was recorded from 5 K to 160 K with the representative hysteresis loops at 5 K, 50 K, 100 K, and 145 K shown in Fig. 9. This M vs. H variation is interpreted in terms of two components: one is a ferromagnetic component M_FM resulting from canting of spins and responsible for the hysteresis, and the other is an antiferromagnetic component varying linearly with H or mathematically M = M_FM + χ_AF*H. Both M_FM and χ_AF are temperature dependent, and M_FM = 0 at temperature well above $T N C r$⁠, so that χ_AF changes to the paramagnetic susceptibility. The plots of the hysteresis loop in terms of M_FM = M − χ_AF*H at 5 K are shown in Fig. 10. From such plots M_FM vs. H at each temperature, coercivity H_C, remanence M_R, saturation magnetization M_S, and linear susceptibility χ_AF were determined and plotted in Fig. 11. As the temperature is increased, M_R, χ_AF and M_S decrease quite rapidly reaching zero at $T N C r$ except for χ_AF that remains non-zero at $T N C r$ as expected and noted above. In contrast, H_C shows insignificant variation versus temperature between 5 K and 120 K, except for the largest size 425 nm particles for which there is a steady decrease in H_C with increasing temperature. Above about 120 K, H_C for all four sizes reduces quite rapidly with the increase of temperature and approaches zero at $T N C r$ ∼ 140 K, because the system is in the paramagnetic state for T >  $T N C r .$

According to the plots in Fig. 11, although all these four parameters (M_R, χ_AF, M_S and H_C) exhibit some particle size dependence especially at the lower temperatures, the size dependence of H_C is strongest that shows several fold increase with growth in the particle size. As plotted in the inset of Fig. 11(a), H_C measured at 5 K is 1930 Oe, 2500 Oe, 4660 Oe, and 7790 Oe at 5 K for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively, including a data point on the 1 μm size particles of HoCrO₃ polycrystalline powder showing H_C = 15 kOe from prior report.²⁶ It is noted that the magnitude of H_C rises almost linearly with increment in particle size from 60 nm to about 1000 nm, suggesting a simple way to tune the magnetic hysteresis of HoCrO₃ samples. Such an increment in H_C at 5 K with growing particle size from 45 nm to 60 nm was also observed in multiferroic TbMnO₃ nanoparticles.⁴⁹ This near-linear relationship between H_C and the particle size is important for applications in magnetic storage, since increasing the coercivity of the magnetic materials utilized in write-heads is a key issue.⁵² Therefore, this study could prospectively strengthen the efforts towards synthesis and designing of new magnetic storage medium.

Another quantity of interest in connection with magnetic materials is the energy product M_S·H_C, which is often adopted to compare the strength of magnetic materials. The particle size dependence of this energy product for the four HoCrO₃ particles at 5 K is plotted in Fig. 12. The largest 425 nm particle-size sample has the largest magnitude of the energy product.

In an effort to understand the source of H_C and its particle size dependence in HoCrO₃, coercivity at 5 K was also measured in the FC mode by cooling the samples from room temperature to 5 K in H = 10 kOe, followed by collecting the hysteresis loop data. This comparison of observed hysteresis loop at 5 K between the ZFC and FC cases is shown in Fig. 13. Also, the measured H_C and the loop asymmetry - exchange bias (EB) between the FC and ZFC cases for the four particles are compared and plotted in Fig. 14, showing insignificant differences. The magnitude of EB is also quite small, perhaps well within the uncertainties of measuring large H_C values, thereby signifying the absence of interface effects between the ferromagnetic and antiferromagnetic components of the magnetic phases in these samples. This EB is much smaller than that reported for Dy_1-xNd_xCrO₃ solid solution.⁵³ 

As noted earlier, the absence of the blocking temperature in the M vs. T data of Fig. 4 shows that the particle sizes of HoCrO₃ investigated here are above the threshold of superparamagnetic limit. Nevertheless, it is still worthwhile to note that the observed size dependence of H_C in HoCrO₃ is quite different from that reported in nanoparticles of antiferromagnetic oxides such as CuO,⁵⁴ in which case H_C and EB of nanoparticles with sizes below about 50 nm show quite different particle size and cooling field dependence. In CuO nanoparticles, H_C increases with decrease in particle size which, along with the observed EB, is interpreted in terms of the interface effects between the disordered surface spins that yield the ferromagnetic component and the antiferromagnetically ordered spins in the core of particles. Therefore, the basic mechanism for the large H_C in HoCrO₃ and its particle size dependence in the range of 60–1000 nm reported here has to be quite different.

The weak ferromagnetism reported here in HoCrO₃ and interpreted in terms of the spin canting induced by the DM interaction is very similar to that known in hematite (α-Fe₂O₃), which is a well-recognized weak ferromagnet at room temperature due to canting produced by the DM interaction. In a recent paper, Ozdemir and Dunlop⁵⁵ have presented a thorough review of the particle size dependence of the magnetic parameters including H_C in hematite covering the particle size range of about 30 nm to 1000 μm. As the particle size rises, H_C increases until a maximum around D_o = 2 μm, and then decreases thereafter. It is argued that for D < D_o the particles have single domain (SD), whereas for D > Do the particles have multiple-domains (MD) with domain walls. In the SD state, the spin rotation with changing H must overcome exchange forces, making H_C grow with increasing particle size. On the other hand, in the MD state with domain walls present, relatively lower H is required to overcome the domain wall energy, making H_C decrease with increase in particle size. For the present HoCrO₃ samples with sizes in the range between 60 nm and 1 μm, H_C rises with increasing particle size. Therefore, these particles are in the SD state and the size D_o required for the SD to MD transition must be larger than 1 μm. It is noted that in a strong ferromagnet with large M_S such as Fe and Co, the SD to MD transition occurs at a much lower D_o ∼ 30 nm.⁵⁶ Since the temperature dependence of H_C is considerably more sluggish than that for M_R and M_S (Fig. 11), magnetocrystalline anisotropy alone cannot explain the large magnitudes of H_C in HoCrO₃. Because of the weak ferromagnetism and relatively small magnitude of M_S, demagnetization field and shape anisotropy are also less important. Other possible sources of H_C are magnetoelastic anisotropy and defects.^57,58 However, at present, a quantitative interpretation of H_C in HoCrO₃ samples is still lacking.

In order to further evaluate these samples for their applications in magnetic refrigeration, the MCE properties of the samples were acquired by collecting the isothermal magnetization data up to field of 7 T. Two parameters—magnetic entropy change ΔS_M(T,H) and refrigerant capacity (RC)—were used to characterize the MCE properties and are given by^4,38

where T₁ and T₂ are the low temperature and high temperature of the refrigeration circle, respectively. Fig. 15 shows the dependence of ΔS_M on temperature of the present HoCrO₃ samples. For all the four samples, ΔS_M increases from 1 T (10 kOe) to 7 T (70 kOe) because larger magnetic field generates larger magnetization, yielding larger ΔS_M according to Eq. (7). As temperature grows, ΔS_M rises from 5 to 15 K and declines thereafter. The maximum ΔS_M at ∼15 K is related to the ordering of Ho³⁺ ions and determined to be 8.73, 7.22, 7.77, and 6.70 J/kg K for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively. It can be seen that the HoCrO₃ sample with smallest particle size in this work shows the largest value of ΔS_M. The ΔS_M of HoCrO₃ samples is smaller than that of DyCrO₃ polycrystalline sample (max 5.13 J/kg K compared to 8.4 J/kg K at 4 T),²¹ but much larger than those of YCrO₃ (0.36 J/kg K at 5 T),⁵⁹ and LaCrO₃ (0.11 J/kg K at 8 T).⁶⁰ 

The field-dependent RC value of the present samples calculated by Eq. (8) (setting T₁ = 5 K, T₂ = 100 K) was shown in Fig. 16. With increasing magnetic field, RC value raises, because larger field results in larger ΔS_M and thus larger integration of ΔS_M. At 7 T, RC values were found to be 388, 354, 330, and 310 J/kg for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively. Alternatively, the refrigeration capacity can be characterized by the product of ΔS_M and the full width at half maximum δT_FWHM in the ΔS_M(T) curve, known as relative cooling power (RCP = −δT_FWHM × ΔS_M) and calculated to be 307, 265, 258, and 252 J/kg, respectively. Evidently, RC value of HoCrO₃ sample increases with the reduction of particle size. This, on the one hand, might be explained by the larger surface-to-volume ratio of the smaller particle size, which enhances the tangible contribution to the overall magnetization by the uncompensated surface spins.³² On the other hand, the HoCrO₃ sample with smaller particle size has smaller magnetic hysteresis, generating less energy lost in the thermal process, and thus the RC value is larger. As in the report of Phan and Yu, materials with nearly zero magnetic hysteresis were considered suitable for the application in MR because of better working efficiency.³⁸ In spite of larger RC value, the sample with 190 nm particle size has smaller ΔS_M than the sample with 320 nm particle size, because RC is the integration of sample's ΔS_M and it also depends on the width of ΔS_M vs. T curve. In addition, the RC values of the present samples are larger than those of polycrystalline YCrO₃ (7.1 J/kg at 5 T),⁵⁹ and Y-doped DyCrO₃, i.e., Dy_0.7Y_0.3CrO₃ (150 J/kg at 4 T),²⁴ but smaller than those of pure DyCrO₃, Dy_0.7Ho_0.3CrO₃, and Dy_0.7Er_0.3CrO₃.²⁴ It should be clarified here that some rare-earth based alloys and oxides are also promising candidates for low temperature MR application, and exhibit even larger MCE in the similar temperature range.^61–64 For example, Li et al. reported large ΔS_M = 17.1 J/kg K and 9.3 J/kg K in the temperature of 4–20 K and low field of 2 T for ErNiBC and GdNiBC compounds, respectively.⁶¹ A TmZn compound was also reported to have much larger ΔS_M = 26.9 J/kg K at 8.6 K and field of 7 T, but smaller RC value of 214 J/kg than the present HoCrO₃ samples.⁶⁵ Since RC value is the figure of merit for MCE property, the work presented here reveals important insights on a promising way to optimize the MCE property of an oxide material system via the control of particle size, such as by tuning the annealing temperature in the solution synthesis method.

In this paper, polycrystalline HoCrO₃ samples of various particle sizes were synthesized via a solution method and annealed at temperature from 700 °C to 1300 °C. Williamson-Hall analysis of the experimental XRD data and scanning electron microscopy images demonstrate that the samples are phase pure and have average particle size ∼60 nm, 190 nm, 320 nm, and 425 nm, respectively. Raman scattering data further certify the absence of impurity and there are frequency shift in the phonon modes between the HoCrO₃ samples with different sizes. The dc magnetization measurements indicate that the Néel temperature of Cr³⁺ ions are 139.1 K, 139.6 K, 140.1 K, and 139.7 K, and the max magnetization at 50 Oe are 19.73, 30.76, 30.85, and 33.14 emu/g for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively. The temperature dependent magnetization data was fitted to the Curie-Weiss law, and also the modified Curie-Weiss law including a correction from Dzyaloshinsky-Moriya interaction. The field dependent magnetization of the HoCrO₃ samples were collected at 5 K and the HoCrO₃ sample with smaller particle size show higher magnetization at the same field. Isothermal magnetization curves reveal that the HoCrO₃ sample with larger particle size has larger magnetic hysteresis loop that are characterized by the dependence of the coercive field and remnant magnetization on temperature. The max coercive field is 1930, 2500, 4660, and 7790 Oe for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively, which can be interpreted by a single domain model, but their remnant magnetization is very similar. The max magnetic entropy changes (ΔS_M) are 8.73, 7.22, 7.77, and 6.70 J/kg K and the refrigerant capacity values are 388, 354, 330, and 310 J/kg for the 60 nm, 190 nm, 320 nm, and 425 nm size particles, respectively. Therefore, the particle size of the HoCrO₃ samples affects the magnetic and magneto-caloric properties considerably and prospectively the MCE property can be optimized by controlling the particle size during processing of the materials.

This work was funded by the National Science Foundation Grant No. DMR-1310149. M.S.S. thanks D. J. Dunlop for bringing Ref. 57 to his attention.