Endotaxial α-Fe Nanoparticles in the High-Fluence Iron-Implanted Single-Crystal MgO

Abstract

In this work, we report on the endotaxial growth of α-Fe nanoparticles in the near-surface layer under high-fluence iron ion implantation of the single-crystal magnesium oxide substrate. Comprehensive Mössbauer effect and magnetometry studies show that the implanted sample reveals a pronounced ferromagnetic response even at room temperature, and the α-Fe nanoparticles serve as its main source. The broad band at ~1000 Oe in the X-band magnetic resonance spectra originates from the α-Fe fraction. It manifests the properties of the easy-plane system with the four-fold in-plane anisotropy. The last indicates that the α-Fe nanoparticles are coherently incorporated into the host MgO matrix.

1. Introduction

The last few decades have been marked by the search for and studies of dilute magnetic semiconductors. New materials were obtained by doping the semiconductor host with magnetic ions. Such materials exhibit simultaneously ferromagnetic and semiconductor properties. Interest in these materials was triggered by the discovery of ferromagnetism in manganese-doped indium and gallium arsenides [1,2] and the theoretical description of their magnetism [3]. Ferromagnetism development on doping with magnetic ions was discovered also in other semiconductors including oxides [3,4]. Various mechanisms responsible for magnetic ordering have been proposed. One of these mechanisms is the formation of secondary magnetic phase precipitates, which include the introduced magnetic elements [5].

The creation of materials where ferromagnetic inclusions grow within a host matrix indicates a transition from dilute to nanocomposite systems. An advantage of the nanocomposites that exhibit ferromagnetic properties is the possibility of a magnetic phase possessing a high Curie temperature and a large magnetization value. In such a case, these materials become promising for new nanoelectronic devices [6,7]. Another interesting effect observed in Co-MgO and Co-ZnO metal-insulator nanocomposites is the dependence of their resistance on an applied magnetic field (magnetoresistance) [8,9,10]. Here, magnetic anisotropy is one of the degrees of freedom that influence significantly composite properties, e.g., a system with a distribution of the magnetic anisotropy magnitudes would switch its magnetization gradually, thus making a resistance a function of a magnetic field and making a composite essentially a magnetic field sensor. In general, the magnetic anisotropy of a composite as a whole originates from the shape of individual magnetic particles (shape anisotropy), their magnetocrystalline anisotropy, and inter-particle coupling. The ability to control the shape and size of nanoparticles stipulates the high extent of applicability of ferromagnet-semiconductor nanocomposites [6]. Magnetocrystalline anisotropy provides a route to endow a system with higher (than second) order anisotropies. This can be achieved if the crystal structure of the embedded magnetic phase coherently follows the structure of the host (endotaxial particles). Thus, the observation of the anisotropic ferromagnetism of the composites was reported in [11,12,13]. In this context, nanocomposite systems exhibiting anisotropic magnetic properties are of particular interest.

Magnesium oxide is an insulator material with a bandgap value of 7.8 eV [14]. Ion implantation effects on MgO have been studied both theoretically and experimentally, especially for the dopant concentrations range below the limit of its solubility in the host matrix [15,16,17]. The valence states and site preferences of the implanted ions in MgO were revealed [18,19,20,21]. The formation of radiation defect centers upon irradiation of MgO and modifications of its micromechanical properties were also investigated [22,23].

Magnesium oxide is often used as a substrate for the deposition of magnetic thin films, and it is especially suitable for the deposition of metallic iron films due to a small crystal lattice mismatch. However, only a few works have dealt with the implantation of iron into the magnesium oxide at a high fluence sufficient for the precipitation of iron-containing particles. For example, in [24], the formation of coherent particles of magnesium ferrite was shown. In [25,26], in iron-implanted MgO, α-Fe and γ-Fe phases were found. The formation of superparamagnetic and ferromagnetic particles, as well as paramagnetic iron atoms, was also reported in [27]. Clustering of the implanted ions has been shown for other combinations of dopants and matrixes as well [5,13,28,29].

It should be noted that the formation of thin embedded ferromagnetic layers, consisting of nanoparticles, with anisotropic magnetic properties upon implantation of transition metals into magnesium oxide matrices is of interest in connection with the possibility of their use in spintronic memory cells. Since magnesium oxide is an insulator, it is expected that it can be used as a barrier layer in magnetic tunnel junction devices. Previously, it was reported that Fe/MgO and Fe/MgO/Fe epitaxial systems have a rather high tunnel magnetoresistance value, and they are promising systems for magnetic random-access memory [30]. However, in the case of implanted layers, the use of ferromagnetic nanoparticles in such devices requires their fixed crystallographic orientation with respect to the host matrix and knowledge of the features and characteristics of their magnetic anisotropy. This article is devoted to experimental studies of these features.

So, the aim of this paper is the study of a single-crystal magnesium oxide plate implanted with a high-dose of iron ions in order to produce magnetic nanoparticles embedded into a crystalline MgO. The main emphasis is given to the magnetic characterization of this nanocomposite system. The magnetic phase composition was studied by Mössbauer spectroscopy. The magnetic properties and their anisotropies were probed by magnetometry and ferromagnetic resonance techniques.

2. Experimental Methods

The single crystalline (100)-face-oriented MgO substrate (MTI Corp., Richmond, CA, USA, 10 × 10 × 0.5 mm³, 1sp) was implanted with iron ions (~40% enriched with ⁵⁷Fe isotope) on ILU-3 ion accelerator. The energy of ions was 40 keV and implantation was carried out to the fluence of 1.5 × 10¹⁷ ions/cm². The ionic current density was maintained at ~8 μA/cm² to avoid the substrate overheating.

The Mössbauer studies of the implanted sample were performed in the conversion electron counting geometry (CEMS) at 295 K and in the transmission geometry at 295 K and 80 K with the conventional WissEl spectrometer working in the constant acceleration mode. The low-temperature spectrum was obtained using a CFICEV flow cryostat (ICE Oxford). The velocity scale of the spectrometer was calibrated using thin metallic iron foil. The isomer shift values of the spectral components are given relative to the gravity center of the metallic iron spectrum at room temperature. The spectra were least-square fitted using the SpectrRelax software [31].

The static magnetic properties were studied using the vibrating sample magnetometer option of the PPMS-9 system (Quantum Design).

The magnetic resonance spectra of the sample were taken in the temperature range of 10–300 K with the cw X-band (9.6 GHz) spectrometer Bruker ESP300. The spectra were measured at different orientations of the substrate relative to the magnetic field, which makes it possible to study the anisotropy of the responses both in the sample plane (in-plane geometry) and in the plane containing the normal to the film (out-of-plane geometry). The spectra in both experimental geometries were obtained with step-in rotation angles of 5 degrees.

3. Results and Discussion

According to SRIM calculation [32], implanted ions’ depth profiles of iron ions have a Gaussian-like shape with a mean range (i.e., mean depth of impurity) of 23 nm and a variance of 8 nm (Figure 1). The radiation damage range depth is about 50 nm. It is clearly seen that impurity concentration reaches the maximal value of about 40%. Iron solubility limit in MgO was reported as about 3 at.% [33]. So, it is natural to expect the formation of a secondary phase from dopants since the region with a very high iron concentration exists.

Conversion electron Mössbauer spectroscopy (CEMS) was used to study the magnetic phase composition and valence states of implanted iron ions in the near-to-surface layer. Studying depth is limited by conversion electrons’ escape depth and typically is about 100 nm for oxides. Thus, this technique allows for the investigation of iron ions in the radiation-damaged region. The CEMS spectrum collected at room temperature (Figure 2a) was fitted into three components, namely, one sextet and two doublets. The sextet’s hyperfine parameters (see

Table 1. Hyperfine parameters, valence state, and spectral area of Mössbauer spectra partial components.

		Components	Hyperfine Parameters			Valence State	Spectral Area, %
			δ, mm/s	2ε, mm/s	HF, kOe
CEMS 300 K		Sextet	0	0	329	Fe0	43.5
		Doublet-I	0.28	0.64	-	Fe3+	22
		Doublet-II	0.95	1.43	-	Fe2+	34.5
Transmission Mössbauer spectra	300 K	Sextet	0.01	−0.08	329	Fe0	19
		Doublet-I	0.33	0.63	-	Fe3+	50
		Doublet-II	0.90	1.44	-	Fe2+	22
		Singlet	1.04	0	-	Fe2+	9
	80 K	Sextet-I	0.12	0.03	340	Fe0	11
		Sextet-II	0.44	0.06	491	Fe3+	22
		Doublet-I	0.44	0.73	-	Fe3+	38
		Doublet-II	1.11	1.68	-	Fe2+	25
		Singlet	1.16	0	-	Fe2+	4

δ–isomer shift relative α-Fe at RT (estimated error ± 0.01 mm/s), 2ε–quadrupole parameter (estimated error ± 0.01 mm/s), HF–average hyperfine magnetic field on ⁵⁷Fe nuclei (estimated error ± 1 kOe)

) are characteristic of metallic iron (α-Fe). Parameters of two other components, doublets, are typical for high spin ferrous and ferric ions, which are in a paramagnetic state at room temperature (RT).

Room temperature Mössbauer spectrum in a transmission geometry (Figure 2b) has a little deviation from the CEMS spectrum. First, another component, singlet, with hyperfine parameters specific to high-spin ferrous ions (see Table 1), appears. We suppose this singlet may be related to iron ions, which are diffused into the volume of MgO and substituted Mg ions on its cationic sites. Second, the spectral area of the ferric doublet increases, while the areas of the α-Fe sextet and ferrous doublet decrease by about two times.

It means that ferric ions are also diffused to the MgO volume, while part of implanted iron related to ferrous doublet and α-Fe sextet is near to surface. We may speculate that there are metallic iron nanoparticles at ~25 nm depth from the implanted surface of MgO.

With a temperature decrease, another sextet appears, while the spectral area of the ferric doublet decreases (see Figure 2b and Table 1). We suppose that this sextet and ferric doublet (partially or fully) are related to superparamagnetic MgFe₂O₄ particles. With the temperature decreasing, the relaxation time of the particle’s magnetic moment increases. When this characteristic time becomes larger than the measurement time (in the case of ⁵⁷Fe Mössbauer spectroscopy it is in the order of 10⁻⁷ s), the doublet disappears, and a resolved magnetic pattern (Zeeman’s magnetic splitting) appears. However, in real systems, the particles have a size distribution and, consequently, a relaxation time distribution as well. Therefore, with a temperature decrease, the ferric doublet “goes” into the sextet continuously, and these components may coexist at a wide temperature range.

The main feature, which should be noted from the results of Mössbauer studies, is that there is only one ferromagnetic contribution at room temperature. It is α-Fe nanoparticles near the implanted surface. Furthermore, the line intensities of the corresponding sextet are related to 3:4:1:1:4:3, and this shows the in-plane magnetization of the sample.

The in-plane magnetization curves reveal clear hysteretic behavior at room temperature and below (Figure 3a). However, the coercivity H_c and saturation magnetization M_s rise significantly with temperature decrease (Figure 3b). This is not expected for bulk metallic iron since its Curie temperature is very high, therefore it may be related to the small size of magnetic particles. The influence of superparamagnetic magnesium ferrite particles, that get blocked with cooling, should take place as well.

The magnetic resonance spectrum recorded at room temperature (Figure 4a) contains a broad ferromagnetic resonance (FMR) band (marked by the rectangle) and a number of sharp electron paramagnetic resonance (EPR) lines. Five EPR lines indicated by the arrows represent the fine structure components of the tetragonal-symmetry Cr³⁺-ion centers (spin S = 3/2) [34,35,36]. The angular dependencies of its EPR spectra are well known [36] and can be used to correctly relate the magnetic resonance spectra to the crystallographic directions of the MgO substrate. Intense EPR lines in the vicinity of g = 2 (~3400 Oe) originate from other 3d-metal impurities (such as V, Ni, Mn, etc.) in the native MgO substrate.

According to the Mössbauer effect measurements, the only ferromagnetic phase at room temperature is represented by α-Fe particles. It is natural to relate the FMR band to these particles. The large FMR linewidth may be attributed to the size and saturation magnetization distributions of the magnetic particles.

Magnetic resonance spectra recorded in the temperature range of 10–300 K are presented in Figure 4b. With the temperature lowering, the FMR band of the α-Fe particles experiences a significant broadening and can hardly be recognized below 50 K. Furthermore, below 100 K, another broad microwave absorption appears. The last is marked by the red dashed line in Figure 4b. Its resonance field decreases with the temperature lowering revealing thus a behavior typical for FMR. We suggest that this band originates from the clusters of magnesium ferrite that were found in the Mössbauer spectra of the sample.

The FMR resonance field at RT has a notable orientation dependence (Figure 5). With an increase in the out-of-plane angle (starting from the plane), the resonance field rises as well. However, on shifting towards the higher resonance fields, the FMR linewidth increases, and at angles larger than ~75 degrees, the resonance field values cannot be accurately determined. We suppose that the size distribution and related to its saturation magnetization dispersion of α-Fe particles and, therefore, the distribution of resonance fields may be a reason for this broadening. The in-plane orientation dependence of the FMR resonance field reveals clearly the fourfold (cubic) anisotropy with a minor admixture of the uniaxial anisotropy. Based on the abovementioned angle dependence of the tetragonal Cr³⁺-ion EPR spectra, the FMR hard axes coincide with the four-fold <100> axes of MgO, and the FMR easy axes match with the <110> axes of MgO. It is known that the α-Fe easy axis of magnetization corresponds to its <100> axes [37]. Therefore, the α-Fe particles are bonded to the MgO matrix in such a way that the <100>_α-Fe axes coincide with the <110>_MgO axes.

The FMR signal in the case of a granular magnetic film may be considered as the collective motion of particles’ magnetic moments. As a first approximation, such systems may be analyzed in terms of the macroscopic magnetization dynamics of a film as a whole. Qualitatively, this is the same approach as is commonly used for the studies of continuous magnetic films [38].

To analyze the orientation dependences of the resonance field, we introduce the Cartesian frame xyz (see Figure 6), with the z-axis coinciding with the normal of the sample and the x-axis–with the crystal [110]-direction of the MgO substrate (this choice of the x-axis is due to the fact that the FMR easy axis coincides with this direction). Let in the given coordinate system the magnetization $\overrightarrow{M}$ be directed relative to the z-axis at the angle ${\mathsf{\theta }}_M$, and its projection to the xy plane relative to the x-axis at an angle of ${\mathsf{\phi }}_M$. Let these angles for the applied magnetic field $\overrightarrow{H}$ be equal to ${\mathsf{\theta }}_H$ and ${\mathsf{\phi }}_H$, respectively.

The free energy of a magnetic system with magnetization $\overline{M}$ in an external magnetic field $\overline{H}$ can be written as the sum of the Zeeman energy, the demagnetization energy, and some anisotropy terms, including magnetocrystalline and in-plane easy axis anisotropies, as follows:

$$F=-\overline{M}\overline{H}+2\mathsf{\pi }{M}^2{\cos }^2{\mathsf{\theta }}_M+K_4^1({\cos }^2{\mathsf{\alpha }}_1\cdot {\cos }^2{\mathsf{\alpha }}_2+{\cos }^2{\mathsf{\alpha }}_1\cdot {\cos }^2{\mathsf{\alpha }}_3+{\cos }^2{\mathsf{\alpha }}_2\cdot {\cos }^2{\mathsf{\alpha }}_3)-K_u{\cos }^2\left({\mathsf{\alpha }}_1+\mathsf{\psi }\right),$$

(1)

where ${\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2,{\mathsf{\alpha }}_3$-angles formed by the magnetization $\overline{M}$ vector and x, y, z axes, respectively; $K_4^1$—the first-order fourfold (cubic) anisotropy constant; $K_u$—the in-plane uniaxial anisotropy constant, and ψ—rotation of easy axis contribution with respect to the x-axis. The FMR resonance condition can be expressed as follows [39]:

$$\omega =\frac{\gamma }{M\sin {\mathsf{\theta }}_M}{\left[\frac{{\partial }^{2}F}{\partial {\mathsf{\theta }}^2}\frac{{\partial }^{2}F}{\partial {\mathsf{\phi }}^2}-{\left(\frac{{\partial }^{2}F}{\partial \mathsf{\theta }\partial \mathsf{\phi }}\right)}^2\right]}^{\frac{1}{2}},$$

(2)

where ω = 2πf—resonance angular frequency, f—the spectrometer microwave radiation frequency, $\gamma =\frac{g{\mathsf{\mu }}_B}{\hslash }$-gyromagnetic ratio, g-the g-factor, ${\mathsf{\mu }}_B$-Bohr magneton, ħ-Plank’s constant. The equilibrium magnetization direction may be determined from the minimum of the free energy, i.e.,

$$\frac{\partial F}{\partial \mathsf{\theta }}=\frac{\partial F}{\partial \mathsf{\phi }}=0.$$

(3)

In addition, we may use following conditions:

$${\mathsf{\theta }}_M={\mathsf{\theta }}_H=\frac{\mathsf{\pi }}{2}$$

(4)

for in-plane FMR dependence, and

$${\mathsf{\phi }}_M={\mathsf{\phi }}_H=0$$

(5)

for out-of-plane geometry. As a result, we obtain the following systems of equations for the in-plane FMR orientation dependence:

$$\{\begin{array}{l}{\left(\frac{\omega }{\gamma }\right)}^2=\frac{\left(2K_u\cos 2\left({\mathsf{\phi }}_M-\mathsf{\psi }\right)+2K_4^1\sin 4{\mathsf{\phi }}_M+HM\cos ({\mathsf{\phi }}_M-{\mathsf{\phi }}_H)\right)\left(K_4^1(3+\cos 4{\mathsf{\phi }}_M)+8\mathsf{\pi }{M}^2+2HM\cos ({\mathsf{\phi }}_M-{\mathsf{\phi }}_H)\right)}{2M^2}\\ 2K_u\sin 2\left({\mathsf{\phi }}_M-\mathsf{\psi }\right)+K_4^1\sin 4{\mathsf{\phi }}_M+2HM\sin ({\mathsf{\phi }}_M-{\mathsf{\phi }}_H)=0\end{array}$$

(6)

and for the out-of-plane geometry (neglecting the in-plane easy axis term):

$$\{\begin{array}{l}{\left(\frac{\omega }{\gamma }\right)}^2=\frac{\left(2K_4^1\cos 4{\mathsf{\theta }}_M-4\mathsf{\pi }{M}^2\cos 2{\mathsf{\theta }}_M+HM\cos ({\mathsf{\theta }}_M-{\mathsf{\theta }}_H)\right)\left(2K_4^1{\sin }^4{\mathsf{\theta }}_M+HM\sin {\mathsf{\theta }}_M\sin {\mathsf{\theta }}_H\right)}{M^2{\sin }^2{\mathsf{\theta }}_M}\\ 4\mathsf{\pi }{M}^2\sin 2{\mathsf{\theta }}_M-K_4^1\sin 4{\mathsf{\theta }}_M-2HM\sin ({\mathsf{\theta }}_M-{\mathsf{\theta }}_H)=0\end{array}$$

(7)

Values of g = 2.02 and ψ = π/8 were fixed during the processing of experimental values. The parameters M, $K_4^1$, $K_u$ were adjusted for the best-fit achievement. It should be noted that, in the case of magnetic nanoparticles with a size distribution, derived parameters should be treated as effective values. A more complex analysis should be performed to account for the size distribution.

The least-square fit results of the FMR orientation dependences are shown by the red curves in Figure 5c,d for the out-of-plane and the in-plane rotations, respectively. The derived values of the fit parameters are as follows: M = 940 ± 40 G, $K_4^1$ = K_eff = 136,000 ± 12,000 erg/cm³, $K_u$ = 20,000 ± 700 erg/cm³.

The rough estimate of M using the Kittel’s formula for FMR field in the in-plane geometry as follows [37]:

$$f=\frac{\gamma }{2\mathsf{\pi }}\sqrt{H(H+4\mathsf{\pi }M)}$$

(8)

gives the value of M = 910 G, which falls within the experimental uncertainty of our fit result. The resonance field in the perpendicular direction can be estimated from the Kittel’s formula for the out-of-plane geometry [37] using the taken value of M as follows:

$$f=\frac{\gamma }{2\mathsf{\pi }}(H-4\mathsf{\pi }M).$$

(9)

So, the H value is ~14,750 G for the out-of-plane orientation.

Observation of the cubic anisotropy reflects the fact that, first, the α-Fe particles are crystalline and, second, they are not randomly oriented but rather coherently incorporated into the host MgO matrix. However, the extracted values of the magnetization M and magnetocrystalline anisotropy K_eff are smaller than for the bulk iron M(Fe) = 1707 G and $K_4^1(Fe)$ = 4.2 × 10⁵ erg/cm³ [37]. The reduced values of M and K_eff in our opinion are related to the small size of the nanoparticles. Indeed, it has been shown that magnetization and magnetocrystalline anisotropy are size-dependent [40,41].

The contribution of uniaxial anisotropy to the in-plane FMR orientation dependence was detected as well. The shape anisotropy of the particles may serve as the source of this term. Let us assume that particles have a spheroid shape with the length a along the symmetry axis (the last is rotated by π/8 about the fourfold axis of α-Fe) and equatorial radii b = c, as depicted in Figure 7. This fourfold axis matches with the x axis, as was introduced earlier. The energy of the shape anisotropy is expressed as follows [42]:

$$E_{shape}=\frac{1}{2}\overline{M}\widehat{N}\overline{M},$$

(10)

where $\widehat{N}$ is the demagnetization field tensor. The principal axes of the tensor match the axes of the spheroid. The demagnetization factor along the symmetry axis of a spheroid may be expressed as follows [43]:

$$N_a=\frac{4\mathsf{\pi }}{k^2-1}\left[\frac{k}{\sqrt{k^2-1}}\ln (k+\sqrt{k^2-1})-1\right],$$

(11)

where k = a/b. Then,

$$N_b=N_c=\frac{4\mathsf{\pi }-N_a}{2},$$

(12)

since [26]:

$$\mathrm{Tr}\left\{\widehat{N}\right\}=4\mathsf{\pi }.$$

(13)

In the frame based on the axes of spheroid

$$\overline{M}=(M\cos \mathsf{\beta },M\sin \mathsf{\beta },0),$$

(14)

where β = α₁ + ψ is an angle between the magnetization and the symmetry axis of the spheroid. Then, we can rewrite Equation (10) in the following way:

$$E_{shape}=\frac{1}{2}\left(N_a{M}^2{\cos }^2\mathsf{\beta }+N_b{M}^2{\sin }^2\mathsf{\beta }\right)=\frac{1}{2}{N}_b{M}^2+\frac{1}{2}{M}^2\left(N_a-N_b\right){\cos }^2\mathsf{\beta }.$$

(15)

The first term in the last expression is isotropic. We can omit this term since resonance conditions (2) and (3) are the derivatives with respect to angles. Then, using Equation (12),

$$E_{shape}=\frac{1}{4}{M}^2\left(3N_a-1\right){\cos }^2\mathsf{\beta }.$$

(16)

Comparing the last with the uniaxial anisotropy term in Equation (1), we obtain the following:

$$-K_u=\frac{1}{4}{M}^2\left(3N_a-1\right).$$

(17)

Using the estimated values of K_u, M, and Equation (11), one can solve Equation (17) with respect to k = a/b, i.e., we can estimate an effective value of the nanoparticles’ elongation. This value is ≈1.009. We suppose that a small deviation of accelerated Fe⁺ ions from the normal direction to the MgO surface in the course of implantation may stimulate such shape anisotropy of the nanoparticles.

Some estimations of sizes may be taken from Mössbauer spectroscopy results. The widths of the outermost lines of the Fe⁰ sextet are 0.47(2) mm/s. This value is higher than twice the natural width of S = 3/2 excited state of ⁵⁷Fe (≈0.19 mm/s) and the typical experimental value for thin α-Fe foil (~0.25 mm/s), which, for instance, we use for the spectrometer calibration. The sextet’s shape is not complicated by the superparamagnetic relaxation processes, but some broadening exists. This shows that superparamagnetic relaxation processes are just beginning to affect the shape of the spectrum. Consequently, nanoparticle sizes should be close to the critical value when the relaxation processes are beginning. In the Neel approximation, the characteristic time of relaxation may be expressed as follows:

$$\tau ={\tau }_0\cdot \exp \left[\raisebox{1ex}{$KV$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.\right],$$

(18)

where τ–characteristic relaxation time, τ₀ ≈ 10⁻¹⁰ s time constant, K–magnetocrystalline anisotropy, V–particle volume, k_B–Boltzmann constant, T–temperature. The particle is in the blocked state when the measurement time τ_meas (~100 ns in the case of ⁵⁷Fe Mössbauer spectroscopy) is less than τ. From the fitting of the angular dependence of the FMR field, K ≈ 1.36 × 10⁵ erg/cm³. Consequently, the critical size is ~16 nm. However, this estimation is far enough from the real case because dipolar interaction between particles was not considered. Indeed, this interaction may reduce the superparamagnetic relaxation frequency. This phenomenon is often called superferromagnetism.

In the simplest consideration, let us assume that identical α-Fe nanoparticles with magnetic moment μ are located equidistant from each other and form a cubic lattice. Since the dipolar interaction is inversely proportional to the cube of the distance, we will consider the effect only of the six nearest particles. In this case, characteristic relaxation time may be expressed as follows:

$$\tau ={\tau }_0\cdot \exp \left[\raisebox{1ex}{$KV+\mu H$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.\right].$$

(19)

Magnetic field H produced be the nearest particles may be shown as follows:

$$H=6\frac{\mu }{d^3},$$

(20)

where d–distance between particles. Magnetic moment of particle μ = M·V (M–magnetization). In the transmission Mössbauer spectrum, the relative area of Fe⁰ sextet $S_{F{e}^0}$ = 19%. If we assume close values of the Lamb–Mössbauer factors for various iron centers in the sample, the relative area may be used as the content of this phase. Then, using the known value of the fluence D = 1.5 × 10¹⁷ ions/cm² and the implanted ion range h ≈ 50 nm, we may calculate the concentration of Fe⁰ ions in the sample, $N_{F{e}^0}$. On the other hand, the number of iron atoms in one particle $n_{F{e}^0}$ may be expressed by the following formula:

$$n_{F{e}^0}=\raisebox{1ex}{$V\cdot \rho \cdot N_A$}\!\left/ \!\raisebox{-1ex}{$A_{Fe}$}\right.,$$

(21)

where $\rho$–density of α-Fe, A_Fe–atomic mass, N_A–Avogadro number. Then:

$$\frac{1}{d^3}=\frac{N_{F{e}^0}}{n_{F{e}^0}}=\frac{\raisebox{1ex}{$S_{F{e}^0}\cdot D$}\!\left/ \!\raisebox{-1ex}{$h$}\right.}{\raisebox{1ex}{$V\cdot \rho \cdot N_A$}\!\left/ \!\raisebox{-1ex}{$A_{Fe}$}\right.}.$$

(22)

Expression (19) may be rewritten as follows:

$$\tau ={\tau }_0\cdot \exp \left[\raisebox{1ex}{$\left(K\cdot V+6\frac{A_{Fe}\cdot S_{F{e}^0}\cdot D\cdot M^2}{h\cdot \rho \cdot N_A}V\right)$}\!\left/ \!\raisebox{-1ex}{$k_B\cdot T$}\right.\right].$$

(23)

Then, we obtain for the critical volume:

$$V_{crit}=\raisebox{1ex}{$k_B\cdot T\cdot \ln \left(\frac{{\tau }_{meas}}{{\tau }_0}\right)$}\!\left/ \!\raisebox{-1ex}{$\left(K+6\frac{A_{Fe}\cdot S_{F{e}^0}\cdot D\cdot M^2}{h\cdot \rho \cdot N_A}\right)$}\right.,$$

(24)

and for the critical size:

$$d_{crit}=\sqrt[3]{\raisebox{1ex}{$6\cdot V_{crit}$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}.$$

(25)

In such an approximation, using effective values of K and M from FMR studies, we can estimate the critical size as ~11 nm and the mean distance between particles (Expression (22)) as ~21 nm. This estimation matches well with the values shown above.

The size of the nanoparticles may be estimated from the value of the coercivity field also, since the only α-Fe phase is magnetically ordered at room temperature. At room temperature, the measured coercive field value is 160 Oe. According to [44], such a value is characteristic for α-Fe of ~10 nm in size. Moreover, in the close implantation conditions [45] where Co ions were implanted into TiO₂ (rutile), particles sized ~10 nm were precipitated in the irradiated layer. We believe that Fe nanoparticles in our Fe-MgO system are not far from this value since it matches well with the size values calculated from Mössbauer results and coercive field data.

4. Conclusions

High-fluence Fe-ion implanted MgO substrate was studied by using VSM magnetometry, Mössbauer spectroscopy, and electron magnetic resonance methods. The implanted sample reveals ferromagnetic properties at room temperature, whereas undoped MgO is diamagnetic. According to the Mössbauer spectroscopy results, the α-Fe nanoparticles are formed in the surface layer of the MgO substrate. The sizes of these particles are estimated to be about 10 nm. They are coherently incorporated into the MgO crystal matrix and display cubic magnetic anisotropy in the substrate plane. Effective values of the magnetization and magnetocrystalline anisotropy of the nanocomposite MgO:Fe layer were obtained from the analysis of the angular dependences of the FMR signal associated with crystallographically ordered and strongly interacting Fe nanoparticles. Small uniaxial anisotropy was detected from the FMR data as well and attributed to the shape anisotropy of Fe nanoparticles.