1 Introduction

Magnetocaloric effect (MCE) has attracted much research, both from the physics and application viewpoints, since its discovery in 1881 [1,2,3,4,5,6,7,8]. The discovery of the MCE in iron [9] has opened the door to extensive research on a variety of compounds, including the transition metal rare earth compounds, in an attempt to find the best MCE materials that operate near room temperature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

The fast development of nanometer-scale electronics demands the development of future cooling systems that can operate in complex geometries in the nanoscale range. An increase in the saturation magnetization in transition elements has been reported, when particle size is reduced to the nano-scale, at very low temperatures [15,16,17].

Research efforts, both experimental and theoretical, have been reported on the conventional MCE in the 3d transition elements at low and high fields, e.g., [14, 16, 18]. On the other hand, the anisotropic MCE (AMCE) is of interest because an increase in the entropy change could be achieved by rotating the crystal, along a specific crystalline direction, without changing the magnetic field intensity. This property was studied first by Kuzmin and Tishin in DyAlO3 [17].

The application of magnetocaloric materials, either ordinary or anisotropic, in magnetic refrigeration has been the subject of many papers [14, 18,19,20,21,22,23]; in particular, search for materials with magneto-volume coupling and first-order phase transition is currently an active research area. Other applications of MCE materials are in the domain of medicine, e.g., in treating hyperthermia and in drug delivery [15, 24,25,26,27,28], where nanoparticles could be used.

In the present work, we report on the anisotropic size-dependent magnetic properties and the anisotropic magnetocaloric effect of crystalline cubic Fe, for spherical particles of radii in the 25–150-Å range, and at temperatures up to Tc. A statistical mechanics-based model is used, in the current work, for calculating the size-dependent magnetic and magnetocaloric properties of Fe. This model has been successfully used in studying a host of magnetic properties of magnetic systems of different crystalline symmetries. Examples of those systems are hexagonal, tetragonal, and mixed-anisotropy systems [29,30,31,32,33]. The aforementioned magnetic properties of those systems include magnetization curves, magnetic susceptibility, energy, torque, and probability landscapes.

2 Model and Analysis

The interplay between different magnetic energies dictates the magnetic properties of a given magnetic system and the total energy is subsequently used in evaluating magneto-thermal and magnetocaloric properties using the well-known relations of statistical mechanics.

Many publications provide experimental values of the temperature dependence of the anisotropy constants: K1, K2, and K3 of Fe [34,35,36,37,38,39,40]. There is almost an agreement between these references on the sign and order of magnitude of K1; however, this trend is not found regarding the value and the sign of K2 and K3. Therefore, we preferred to use the Callen and Callen power law [41, 42] to calculate the temperature-dependent anisotropy constants. The Callen and Callen law has been checked and verified for elements, e.g., Fe and Co [41,42,43,44]. The values of K1 and K2, at very low temperatures, used in our model were obtained from the reference by Getzlaff [40]. Those values proved to simulate magnetization curves which are in fair agreement with the experimental magnetization curves along the [100], [110], and [111] directions.

The energy density ET is given by Eq. (1), where Ea and Ez are the anisotropy and Zeeman energy densities respectively:

$${E}_{T}={E}_{a}+{E}_{z}$$
(1)

where, for cubic systems, Ea can be expressed as [45]:

$${E}_{a}={K}_{0}+{K}_{1}f+{K}_{2}s+{K}_{3}w$$
(2)

where:

$$\begin{aligned}f&=a_x^2 \, a_y^2+{a_y^2 a}_z^2+a_x^2 a_z^2, \\ s&={a}_{x}^{2} \, {a}_{y}^{2} {a}_{z}^{2}, \\ w&={a}_{x}^{4} \, {a}_{y}^{4}+{a}_{y}^{4} {a}_{z}^{4}+{a}_{x}^{4} {a}_{z}^{4} \\ \text{and } {E}_{z}&=-H\cdot {M}_{s}\end{aligned}$$
(3)

We assume that the saturation magnetization vector is oriented at an angle θ with the z–axis and its projection on the x–y plane makes an angle φ with the x-axis which is chosen to be along the [100] direction of the cubic crystal. Hence, the Ms direction cosines are:

αx = sinθ cos φ, αy = sinθ sinφ, and αz = cosθ.

Using the dimensionless quantities:

εT = VβET, α = VβK1, γ = VβK2, ξ = VβK3, and εz = VβH MS, we rewrite Eq. (1) as:

$${\varepsilon }_{T}=af+ys+\xi w-{\varepsilon }_{z} \mathrm{cos}\psi$$
(4)

where V, β, and ψ are the volume, 1/kBT, and the angle between H and Ms respectively. The quantity εz is the ratio between the Zeeman and the thermal energies. As a result, the value of εz represents the relative strength of these two quantities [45]. Iron particles are assumed to be spherical in shape with radius r. The classical partition function is given by:

$$Z={\int }_{0}^{\pi }\mathrm{sin}\left(\theta \right)d\theta {\int }_{0}^{2\pi }\mathrm{exp}\left(-{\varepsilon }_{T}\right)d\varphi$$
(5)

and is subsequently used to evaluate magneto-thermal and magnetocaloric quantities.

The isothermal change in the magnetic entropy is given by:

$$\left|\Delta {S}_{m}\right|=\sum\nolimits_i\frac{\left({M}_{i+1}({H}_{i+1},{T}_{i+1})-{M}_{i}({H}_{i},{T}_{i}\right)}{{T}_{i+1}-{T}_{i}}\Delta {H}_{i}$$
(6)

where M(H,T) is the isothermal magnetization along a specific crystalline direction.

Equation (6) could be casted to take the following form [8, 44,45,46,47,48,49]:

$$\Delta {S}_{m}(Tav)=\frac{\delta H}{2\delta T}(\delta {M}_{1}+2\sum\nolimits_{k=2}^{n-1}\delta {M}_{k}+\delta {M}_{n})$$
(7)

In the reference by Pecharsky et al. [49], the authors discuss the limitations that should be imposed on \(\delta T \mathrm{and} \delta H\) in order for Eq. (7) to be valid. The same reference also discusses and comments on the analysis given in another reference [50] on the same subject.

The adiabatic change in temperature for a specific change in the external magnetic field is given by many references, e.g., [4, 7, 8, 12]:

$$\Delta T_{ad}(T,\Delta H)=-\int_{H_1}^{H_2}\frac{T}{C(T,H_{F})}(\frac{\partial M(T,H)}{\partial T})_{H}dH$$
(8)

However, one may write Eq. (8) in the following approximate form if the heat capacity is weakly field-dependent [8, 12]

$${\Delta \mathrm{T}}_{ad}(T, \Delta H)\approx \frac{-T}{C(T,H) }\Delta {S}_{m} \left(T, \Delta H\right)$$
(9)

A figure-of-merit called the relative cooling power (RCP) has been introduced to assess the performance of magnetocaloric materials. The RCP is determined by multiplying the magnitude of the maximum ∆Sm, at a definite field, times the difference in temperature, at full width at half maximum, of the ∆Sm vs. temperature curve as shown in the following equation [8] :

$$RCP=\Delta {S}_{m}^{\mathrm{max}}\times {\delta T}_{FWHM}$$
(10)

3 Results and Discussion

3.1 Size-Dependent Magnetization

Figure 1 shows the room temperature magnetization curves along the [100], [110], and [111] directions for a particle of radius r = 150 Å using our classical model. The saturation magnetization Ms is about 1713 emu/cm3 ( 220.17 emu/g), i.e., very close to the experimental value of 217.2 emu/g as reported by Coey [52] and Kittel [53].

Fig. 1
figure 1

The isothermal magnetization vs. magnetic field for Fe particles (r = 150 Å) at H = 800 Oe at room temperature. Our model (solid curves) experimental data (★ easy, · mid-hard, ○ hard directions)

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The sequence of the easy [100], mid-hard [110], and hard [111] directions is in agreement with the well-known data on cubic Fe crystals, e.g., [53, 54]. We have studied the effect of particle size on the magnetization curves for particles in the size range of 25–150 Å at room temperature. Figure 2A–E show the dependence of the magnetization curves on particle size in the range 25–125 Å. We noticed a relative decrease in the anisotropy between the [100] and [111] directions as the particle size is reduced (e.g., Fig. 2C). For a particle size of 25 Å, as shown in Fig. 2E, the magnetization curves along the three directions become identical and saturation (~ 1700 emu/cm3) is far from being achieved in a field up to 800 Oe. Figure 3 displays the dependence of magnetization on particle size (r = 25–150 Å), at 300 K and in 800 Oe field. The magnetization drops considerably as the particle size is reduced below about 50 Å. This behavior is consistent with the experimental results on Fe [55,56,57] and other particles like Gd [29]. Figure 4A–D shows the isothermal magnetization curves parallel to the easy [100] axis for particles with radii in the 25- to 150-Å range and at temperatures 600, 400, 150, and 4 K, respectively. The field required to achieve saturation increases as the particle size decreases, for these three temperatures 600, 400, and 150 K.

Fig. 2
figure 2

The isothermal magnetization curves at 300 K for Fe particle of different sizes at A 125 Å, B 100 Å, C 75 Å, D 50 Å, and E 25 Å

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Fig. 3
figure 3

Magnetization vs. particle size at 300 K and H = 800 Oe for particles of sizes in the 25–150-Å range

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Fig. 4
figure 4

The isothermal magnetization curves along [100] axis at A 600 K, B 400 K, C 150 K, and D 4 K for Fe particles of different sizes in the 25–150-Å range

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In addition, for particle radii in the range 25–75 Å and at 4 K, a significant increase in the saturation magnetization has been achieved, e.g., for a radius of 25 Å, the magnetization is raised to ~ 1700 emu/cm3 as shown in Fig. 4D. This behavior is expected because thermal energy is insufficient, at low temperatures, to reduce the magnetization of extremely small particles, whereas this effect becomes more significant at high temperatures. A common feature of super-paramagnetic single-domain particles is that at high temperatures, the thermal energy overcomes the magnetic anisotropy energy barrier [29]. The most remarkable feature of superparamagnetism is shown in Fig. 5, namely the overlap of the M/Ms vs. H/T curves at 77, 150, and 200 K for the critical particle radius of 25 Å. The same behavior was previously pointed out in Fe, iron oxide–based alloys [58], and Gd element [29]. In contrast, Fig. 6 displays curves at temperatures 100, 150, and 200 K for size radius of 100 Å. The behavior confirms the ferromagnetic state for Fe particles [57].

Fig. 5
figure 5

Fractional magnetization as function of H/T for a 25 Å particle at T = 77, 150, and 200 K in a magnetic field up to 2 kOe

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Fig. 6
figure 6

Fractional magnetization as function of H/T for a 100 Å particle at T = 100, 150 and 200 K in a magnetic field up to 1 kOe

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3.2 Anisotropic Magnetocaloric Effect (AMCE)

The anisotropic MCE effect is manifested in the difference between the isothermal entropy changes calculated along different crystallographic directions in a field of constant magnitude [59, 60]. Figure 7A–D display the AMCE, using the trapezoidal method, in the field range 50–500 Oe, and at temperatures between 900 and 1050 K, for Fe particles of size 150 Å. The anisotropic ΔSanis can be calculated by taking the difference between ΔSm along the easy and hard directions. A large ΔSanis is achieved in the low field region 50–500 Oe because the entropy of the spins is changed by application of a small field, in cluster and superparamagnetic materials more easily than in paramagnetic systems [61]. Table 1 shows the maximum value of ΔSm at different fields 50, 100, 300, and 500 Oe for a radius of 150 Å. Figure 8 displays the temperature dependence of ΔSm for r = 150 Å in a field of 30 kOe. The maximum value is 8.36 J/kg K, along the easy axis, in comparison with 7.95 J/kg K for the bulk Fe [16]. So, the anisotropic MCE is enhanced over the ordinary MCE. Figures 9 and 10 display the size-dependent ΔS, in a 1kOe field along the [100] direction for radii in the range 25–150 Å. A plateau feature [51] is found for 25-Å particles almost in the whole temperature range. Table 2 shows the RCP values for radii 50, 75, 100, and 150 Å in a field of 1 kOe. The highest value of RCP is 8.11 J/kg for a radius of 50 Å.

Fig. 7
figure 7

The isothermal AMCE ΔSanis for r = 150 Å, in the temperature range from 900 to 1050 K for applied fields of A 50 Oe, B 100 Oe, C 300 Oe, and D 500 Oe

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Table 1 The maximum values of anisotropic ΔSm which represents the difference between the anisotropic ΔS of the easy [100] and hard [111] directions, at different fields and for a radius of 150 Å
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Fig. 8
figure 8

The isothermal AMCE ΔSanis for r = 150 Å, in the temperature range from 900 to 1050 K for applied fields at 30 kOe along [100]

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Fig. 9
figure 9

ΔSanis vs. temperature at H = 1kOe, along [100], for particles of sizes in the 25–150-Å range

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Fig. 10
figure 10

The AMCE vs. temperature at H = 1kOe for particles of sizes of 25, 75, and 150 Å along [100] direction

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Table 2 The relative cooling power RCP for Fe particles of different radii along the easy axis [100] for a field of 1 kOe
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This behavior was reported previously in nano-sized Gd element by Miller [62] and Belo et al. [21]. However, the previous studies have not taken into account the calculation of anisotropic ΔTad and only the conventional ΔTad was presented experimentally as reported by [8, 15]. Based on Eq. (9), the anisotropic ΔTad is calculated along the easy direction [100] for applied field of 30 kOe. The heat capacity data was obtained from the experimental work presented in [8]. The anisotropic ΔTad has been calculated, using our model, with maximum value of 8 K as compared to the experimental value of 5 K as shown in Fig. 11. The increase in the value of anisotropic ΔTad relative to ordinary MCE confirms the importance of applying a field along a specific crystallographic direction for obtaining a high adiabatic temperature change at the same field intensity.

Fig. 11
figure 11

The adiabatic AMCE ΔTanis along [100] easy axis for r = 150 Å, in the temperature range from 1000 to 1050 K for applied fields of 30 kOe compared to the conventional experimental ΔT

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4 Conclusions

The size-dependent magnetization curves for Fe particles showed a ferromagnetic behavior at room temperature, for radii in the 50–150-Å range. Particles with a smaller radius, about 25 Å, exhibit a super-paramagnetic behavior. The isothermal change in entropy for field change in the 50–500-Oe range showed an anisotropy in the ΔSm behavior for field applied along the [100], [110], and [111] axes. The particles in the 25-Å size range exhibit a plateau in its temperature dependence of ΔSm. The adiabatic change in temperature ΔTad, along the [100] axis, in a 30-kOe field, is about 37.5% larger than the experimental value. The lack of experimental data on Ctotal(T, H) does not enable us to calculate ∆Tad without the use of the approximation form of Eq. (9).

The statistical mechanics model proved to be fairly suitable for calculating the magnetothermal, magnetic size-dependent, and anisotropic magnetocaloric properties of the iron element.