Ab Initio Magnetic Properties Simulation of Nanoparticles Based on Rare Earth Trifluorides REF₃ (RE = Tb, Dy, Ho)

Abstract

Recently, nanoparticles based on rare earth fluorides have been widely investigated due to their possible medical and spintronic applications, where it is desirable to know the magnetic moment volume distribution. However, this feature is hard to find out from the experiments, so we employ ab initio spin-polarized calculations for this purpose in order to find out the tendencies and common features in three different compounds. In the present work, the nanoparticles of dipole magnets with different sizes, namely TbF₃, DyF₃ and HoF₃, were simulated, and optimized structures were found. We present the optimized structures for the particles with various sizes, as well as for slabs constructed from same compounds. Lastly, for optimized geometries, the analysis of magnetic moments’ distribution over the nanoparticles and slabs volume was performed.

1. Introduction

In the recent years, great attention has been paid to various materials in the nanoparticle (NP) structure due to their unique properties, which arise thanks to their constrained geometry and which have many outstanding prospects both in science and technology [1,2,3,4]. Various materials were created and investigated in such a nano form and the study of rare earth fluoride NPs is gaining momentum thanks to the recent progress in technologies. The rare earth fluorides are known first of all for their magnetic properties. Apart from this, many of their practical applications are known today; for example, nanocrystalline fluoride powders are used as catalysts due to their significant specific area. Another important field is their use in medicine and biotechnology, namely the creation of nanoprobes with high biocompatibility [1], the development of radiation [2] and photodynamic [3] therapy, and nanothermometry [4].

More specifically, the trifluorides of rare earth elements are a class of materials with outstanding electronic, optical, and magnetic characteristics. For this reason, they are actively used in laser technology [5], as contrast agents in magnetic resonance imaging [6], and biological labels [7]. Nanocrystals of these compounds are used as model objects for both theoretical and practical research in various fields. The magnetic properties of rare earth trifluorides are of great importance when moving to the nanoscale due to the fact that their NPs exhibit a unique variety of magnetic properties, especially at low temperatures. In recent years, these properties have been actively studied by various methods, including computer simulation.

In order to underline the actuality of research objects, each material under investigation will be briefly discussed further separately in the introduction section.

1.1. Dipole Ferromagnet TbF₃

Terbium trifluoride has a structure of the β-YF₃ type, i.e., recrystallization is absent in it up to the melting point. The compound belongs to the orthorhombic symmetry space group Pnma (${\mathrm{D}}_{2h}^{16}$) [8] with lattice constants a = 0.6513 nm, b = 0.6949 nm and c = 0.4384 nm [9]. The corresponding crystal structure is shown in Figure 1.

According to the magnetization measurements and the behavior of the susceptibility, TbF₃ is a dipole ferromagnet [10], the phase transition from the paramagnetic to the ferromagnetic state occurs at temperatures below T_C = 3.95 K [10,11]. At temperatures T < T_C, the magnetic moments of Tb³⁺ ions (≈9 ${\mu }_B$ [12,13]) are ordered in two magnetically nonequivalent sublattices in the ac crystallographic plane at angles $\varphi$ = ±28° to the a axis. Due to the large values of the magnetic moments, the magnetic ordering is induced mainly by the classical dipole–dipole interaction between terbium ions [8,10].

In Ref. [14], TbF₃ NPs were used as contrast agents for magnetic resonance imaging under the conditions of an ultrahigh magnetic field (>3 T). The compound demonstrated an ability to attenuate X-rays and low toxicity compared to more popular contrast agents, which opens up great prospects for popularizing its use. In addition, TbF₃ crystals can be used as a material for the manufacture of magneto-optical instruments and devices. The magnetic properties of the TbF₃ compound are less studied than those of DyF₃.

1.2. Dipole Ferromagnet DyF₃

Dysprosium trifluoride DyF₃ has a orthorhombic symmetry space group Pnma (${\mathrm{D}}_{2h}^{16}$) and crystal lattice parameters a = 0.6456 nm, b = 0.6909 nm and c = 0.4380 nm [15]. From the comparison of the calculated and measured magnetization values it was obtained that the main contribution to the magnetization is due to the dipole–dipole interaction [15]. Its crystal structure is same as for TbF₃ and shown in Figure 1. The temperature of the phase transformation from paramagnetic to ferromagnetic state is T_C = 2.55 K [16]. Magnetic ordering is induced mainly by dipole–dipole interactions between Dy³⁺ ions due to large values of magnetic moments [15].

Dysprosium fluoride DyF₃ has unique magnetic properties at low temperatures. For instance, it was demonstrated that it can be used to improve the properties of Nd-Fe-B magnets [17]. Furthermore, dysprosium fluoride NPs have also been proposed for use as a contrast agent for high-field magnetic resonance imaging [6]. In a recent article [18], DyF₃ NPs with sizes from 16 to 225 nm were synthesized by hydrothermal treatment in an autoclave, and it was also found that the Curie temperature of NP shifts towards lower temperatures with decreasing the size. Moreover, when the NP size is less than 5 nm, the transition to the magnetically ordered state is not observed. Additionally, the size effect was attentively investigated in another recent paper [19], where it was found that saturation of magnetization decreases with decreasing the particle size (16 nm–7 µm). The authors emphasised the need to consider both the nanoparticle core and shell, as well as the clustering process to describe the magnetic properties. Thus, experimentally the size effect onto the magnetic properties was clearly established for DyF₃.

1.3. Dipole Ferromagnet HoF₃

HoF₃ belongs to the orthorhombic symmetry space group Pnma (${\mathrm{D}}_{2h}^{16}$) with lattice parameters a = 0.6404 nm, b = 0.6875 nm and c = 0.4379 nm [20]. The crystal structure is same as for previously described compounds and shown in Figure 1. It contains four Ho³⁺ and twelve F⁻ ions per unit cell. The compound exhibits a strong anisotropic susceptibility at low temperatures and it is close to the Ising system with two easy directions. The two lowest electronic states of HoF₃ ions are singlets, clearly separated from the other energy levels [21]. The primary interaction between ions is the classical dipole interaction, which induces magnetic ordering at the Curie temperature T_C = 0.53 K [22].

As for example of possible application, a facile strategy was proposed for preparing HoF₃ NPs with good water solubility, low cytotoxicity and biocompatibility as a dual-modal contrast agent [23]. Authors discussed the prospects of those NPs for cancer diagnosis. They claimed that, grown by a facile one-pot solvothermal approach, these NPs could be used as both computer tomography (CT) and magnetic resonance imaging (MRI) which is also confirms the potential of Ho-based nanomaterials for bioapplication research.

To sum up, one of the problems in the physics of rare earth compounds is the effect of the size of fluoride NPs on their magnetic properties. In particular, the magnetic moments distribution within the NP has not yet been fully understood. Moreover, surface and geometry effects in magnetic core–shell NPs are attention-grabbing and a great challenge for the scientific community [24]. Also, one of the fundamental questions of the physics of nanosized objects is the question, “what size of a NP is sufficient for a formation of a magnetic ordering inside the particle?”. Furthermore, how are domains formed inside a NP? The main difficulty in the theoretical point of view is related, on the one side, with the restrictions of the precise ab initio calculations for a systems with a large number of atoms, and on the other side with inability of methods based on force-fields to conduct the magnetic calculations. Therefore, the purpose of this work is to make an attempt for ab initio calculations of magnetic NPs based on TbF₃, DyF₃ and HoF₃ and establish the distribution of magnetic moments depending on the NP size and get close to understanding the issues raised.

2. Materials and Methods

Structural and magnetic properties calculations were realized as based on the density functional theory [25]. Exchange and correlation effects were accounted for using generalized gradient approximation (GGA-PBE) [26]. The Kohn–Sham equations [27] were solved using projectively extended wave potentials and wave functions [28]. All calculations were carried out using the VASP-6.3 (Vienna Ab-initio Simulation Package) program [29] built into the MedeA computational software [30]. The cutoff of the plane wave was taken to be 400 eV, the convergence criterion for atomic relaxation was 0.02 eV/Å, and the convergence condition for self-consistent calculations was the invariance of the total energy of the system with an accuracy of 10${}^{-5}$ eV. The Brillouin zones were sampled using Monkhorst–Pack grids [31,32,33], including 1 × 1 × 1 k-points for all NPs, and 5 × 1 × 7 for the slab. The k-mesh for bulk geometries was higher up to 5 × 5 × 7. The Gaussian smearing was 0.05 eV.

The following procedure was used for the calculation simplification: the geometry optimization were performed within the non-magnetic approach in order to simplify and fasten the calculations, and then the electronic and magnetic properties were extracted within the spin-polarized approach. Additionally, the optimization was performed with semi-core f electrons being treated as core states despite being higher in energy than other valence states. Even though the procedure is a brutal approximation, we checked for smaller particles to ensure that involving spin multiplicities does not affect the final result significantly.

Finally, in this work the modelling cells were created by spherical-shaped NPs based on an existing periodic model (shown in Figure 1) and surrounded by a thick-enough vacuum region to prevent the interactions with periodic copies. The analyzed surfaces were y-cut three and four unit cell slabs also surrounded by vacuum region in order to simulate the boundary of real particles with ≈2.0, 2.7 and 3.4 nm size.

3. Results

3.1. Bulk TbF₃, DyF₃ and HoF₃

The magnetic calculations were performed within the spin-polarized approach and will be presented for each considered size separately.But, at the fist step, the bulk components were fully optimized, lattice parameters and magnetic moments were extracted and collected in

Table 1. The calculated lattice parameters and magnetic moments for studied bulk TbF₃, DyF₃ and HoF₃ compounds along with available experimental data.

Structure	TbF3	DyF3	HoF3
Lattice parameter a b c, nm Exp. lattice parameter	0.651 0.698 0.441 0.651 0.695 0.438 [9]	0.646 0.691 0.438 0.646 0.691 0.438 [15]	0.641 0.687 0.437 0.640 0.687 0.438 [34]
Magnetic moment per Tb/Dy/Ho ion, ${\mu }_B$ Exp. magnetic moment	5.954 9.96 [12,13]	4.950 8 [15]	3.943 5.7 [35]

along with experimental data. The obtained lattice parameters agree well with reported experimental works even though the last were obtained at finite temperatures and at various conditions. However, the comparison of DFT and experimental magnetic moments is tricky, since the DFT assumes zero temperature. Additionally, DFT usually underestimates the magnetization values, since only the spin component is taken into account. That is why +U correction is usually used, or taking the spin-orbit coupling into account might be required. The first is associated with peculiarities while choosing the U term, the last one is time-consuming. However, the aim of the present research is a qualitative description of the magnetization distribution over the NPs, not quantitative so far. The spin-orbit calculations and the search for the best U term will be conducted in future.

In the further narration, the obtained magnetic moments of NPs will be compared with the bulk values.

3.2. Nanoparticle Geometry

In the following section the structural and magnetic properties will be presented for constructed stoichometric fully relaxed NPs. After the optimization, the RE-F distances were checked to ensure the reasonableness of the geometry relaxation. We found that RE-F distances do not change significantly while increasing the NP size for all investigated trifluorides.

3.2.1. 0.7 nm Particles

In order to keep the structures stoichiometric, the TbF₃, DyF₃ and HoF₃ NPs were constructed as based on the bulk structures being 0.7 nm spheres while maintaining the stoichiometry of the material. The resulting structures have a chemical formula, (REF₃)₄, and are presented in Figure 2. Such NP size corresponds to very small structures containing only four rare earth ions as shown. As a result of optimization, structures of 0.7 nm nanoparticles take on the most symmetrical shape possible for such an initial configuration. The shape reminds the ring for all investigated materials with the difference being negligible, only in hundredths of ion-to-ion distances. In Figure 2, various sides of particles are shown.

As for the magnetic structure, the distribution is rather uniform and similar for all particles. However, the magnitude is different compared with the bulk materials. Furthermore, since the systems have symmetrical shapes with respect to the center, it would expected for them to have the same magnetic moments for all rare earth ions. On the contrary, each system has a difference in the neighboring rare earth ions of 0.656, 0.434 and 0.684 ${\mu }_B$ for TbF₃, DyF₃ and HoF₃, respectively. At the same time, the mean values are close to the bulk magnetic moment (collected in

Table 2. The summarized collection of investigated structures and obtained magnetization. N is a number of rare earth ions, m is a magnetic moment. The N for the slabs correspond to the number of rare earth ions per computational unit cell (/u.c.).

Structure	Size, nm	N	Mean m, ${\mathit{\mu }}_{\mathit{B}}$
TbF${}_3$	bulk		5.95
	0.7	4	5.94
	0.8	6	5.92
	1.0	10	5.92
	1.6	44	5.90
	2.0	12/u.c.	5.94
	2.7	16/u.c.	5.93
	3.4	20/u.c.	5.91
DyF${}_3$	bulk		4.95
	0.7	4	4.93
	0.8	6	4.93
	1.0	10	2.78
	1.6	44	4.6
	2.0	12/u.c.	4.94
	2.7	16/u.c.	4.95
	3.4	20/u.c.	4.93
HoF${}_3$	bulk		3.94
	0.7	4	3.92
	0.8	6	3.9
	1.0	10	3.91
	1.6	44	2.85
	2.0	12/u.c.	3.95
	2.7	16/u.c.	3.92
	3.4	20/u.c.	3.94

), being ≈0.02 ${\mu }_B$ lower.

3.2.2. 0.8 nm Particles

The structures of 0.8 nm NPs of TbF₃, DyF₃ and HoF₃ are shown in Figure 2. The resulting structures have a chemical formula, (REF₃)₆. The final nanostructures exhibit more atoms with a complex and notable structure with six rare earth ions. In particular, the rare earth ions locate the octachedral positions by poking out the F ions. All the particles are slightly elongated in one direction; the shape is ellipsoidal.

The complex geometry led to a redistribution of magnetization in accordance with the optimized, energetically favorable position of atoms. For all structures the distribution of magnetization is symmetrical with respect to the center of the NP. For TbF₃, the distribution is more uniform (all ions have similar moments) and the mean value is close to the bulk. Moreover, two-side Tb ions have 6.0 ${\mu }_B$ as in the bulk. On the contrary, for DyF₃, the deviations for each ion from the bulk value are more prominent and equal up to 1.2 ${\mu }_B$. The magnetization ordering maintained as ferromagnetic for all structures here.

3.2.3. 1 nm Particles

The optimized structures of 1 nm NPs have a chemical formula, (REF₃)${}_{1}0$, and are collected in Figure 3. Obviously, it was found that the character of distribution is more complicated than in the previous case for smaller particles. The bigger particles contain 10 rare earth ions and have more complex structures and a more inhomogeneous distribution of magnetization.

Indeed, as depicted in Figure 4, the shape for all particles looks like an ellipsoid, but the TbF₃ is least elongated, the rare earth ions are uniformly distributed and approximately equidistant from each other. On the contrary, the DyF₃ particle has a distinct flattened grain shape. Still, as a result of optimization both Dy and F ions are uniformly distributed. The last HoF₃ is similar to the TbF₃ NP, also elongated but more flattened.

As for the magnetic moments distribution over the volume of TbF₃ we observed rather homogeneous magnetization with a mean value for Tb of 5.916 ${\mu }_B$ being close to the bulk value. However, the most distant-from-the-center ions received ≈4.95 ${\mu }_B$, whereas the center ions were more magnetized, up to ≈6.89 ${\mu }_B$.

The DyF₃ nanostructure was found to be most elongated as a result of relaxation. The distribution of magnetic moments is complex and not symmetrical even though the structure was fully optimized. The magnetic moments vary from negative values of −6.76 ${\mu }_B$ up to 6.82 ${\mu }_B$ making the mean value smaller than in bulk. That means that there is one ion being antiferromagnetically coupled to other moments. Lastly, a similarity to the TbF₃ case is that the center ions are most magnetized.

The last considered structure of HoF₃ is ferromagnetic as well with more homogeneous magnetization than in the case of DyF₃, but the values still vary significantly from 0 up to 6.83 ${\mu }_B$, however the mean magnetic moment is close to the bulk one. Again, it could be concluded that the most distant-from-the-center ions are less magnetized.

3.2.4. 1.6 nm Particles

The structures of constructed NP of TbF₃, DyF₃ and HoF₃ with 1.6 nm size contain 176 atoms, so the optimization of such large systems is complicated within the DFT approach and it does not converge. Nevertheless, we decided to calculate the magnetization without optimization for a overview. Indeed, for TbF₃, even for a non-optimized cell, we found the mean magnetic moment being close to the bulk value; in DyF₃, Dy ions received the mean magnetic moment of 0.3 ${\mu }_B$ which is less than the bulk value; whereas in the HoF₃, the mean magnetic moment of Ho is 1 ${\mu }_B$ is smaller than the bulk. Obviously, the optimization matters, so in order to overcome the restriction the slab geometries will be constructed and analyzed in the following section.

3.3. Slab Geometry

3.3.1. 2 nm Thick Slabs

As was pointed out in the previous subsection, the optimization process of particles with 1.6 nm size is associated with computational problems. However, in experiments, the synthesized NPs have an average dimension of ≈100–200 nm, and in order to approximate the real sizes we constructed the infinite slabs in order to investigate the distribution of magnetic moment within the thickness of it and to check the surface impact. Here, the crystals of TbF₃, DyF₃ and HoF₃ were cut along xz-planes. The other cuts might be checked as well but we expect that, qualitatively, the situation would be the same.

The constructed and optimized ≈2 nm thick slabs are collected in Figure 5. In terms of the structural properties as a result of optimization, the atom positions did not change significantly in comparison with bulk and all three TbF₃, DyF₃ and HoF₃ slabs look similar in terms of structure. At the same time the distribution of magnetic moments differs significantly. The TbF₃ slab exhibited a core-type distribution with higher-magnetized ions surrounding the less-magnetized inner ions. The DyF₃ slab is rather similar to TbF₃ in character, with ≈5.9 ${\mu }_B$/Dy at the surface and 3.9 and 4.9 ${\mu }_B$ for inner ions, respectively. Oppositely, the HoF₃ slab had a negligible surface magnetization, while the inner part was quite highly magnetized at the second atomic layer up to 6.6 ${\mu }_B$, and the middle layers had 4.9 ${\mu }_B$/Ho, so there was an alternation.

At the same time, the calculated mean magnetic value was close to the bulk one for all three structures. Since the conclusions concerning the character of magnetization are not obvious here and no similarities are found, the cell was increased in thickness.

3.3.2. 2.7 nm Thick Slabs

At the next step, the computational cell of the slabs were increased by one unit cell of trifluoride. In this way, each cell represents four unit cells in the y-direction surrounded by a vacuum region, as depicted in Figure 6.

Here, for the TbF₃ slab, the distribution slightly changed in comparison with the 2 nm thick slabs; no core-like distribution was observed here, it was rather uniform. On the contrary, the DyF₃ slab possessed the least-magnetized middle atomic layers and up to 5.9 ${\mu }_B$/Dy surface layers. The last HoF₃ slab, as in the previous case of 2 nm thick, had a very big alternation of up to 6.8 ${\mu }_B$/Ho and negligibly small ≈1 ${\mu }_B$/Ho atomic layers.

3.3.3. 3.4 nm Thick Slabs

Lastly, the cells were increased by one more unit cell, making the thickness of the slab 3.4 nm (Figure 7). This was a relatively thick slab, and the middle layers were supposed to be close to the bulk conditions. However, for the TbF₃ slab, the distribution was similar to the previous case, but there were two atomic layers with 2.9 ${\mu }_B$/Tb and the middle layers were more magnetized than bulk. Overall the layers’ magnetization alternated with values being both lower and higher than bulk.

The DyF₃ slab also possessed this alternation with slightly different character. As in the case of TbF₃ slab, the surface had a high magnetization, but the third layer was higher, so all considered thicknesses of DyF₃ slab demonstrated the alternation of higher and lower magnetization.

The HoF₃ slab’s distribution was similar to the previous case of 2 nm, when the surface layers were ≈1 ${\mu }_B$/Ho and then the alternation was present. Similarly to DyF₃, each investigated thickness of HoF₃ possessed the alternation of more- and less-magnetized atomic layers but in the opposite way, with a low-magnetized surface.

4. Discussion

In this section we provide some common features observed along with corresponding discussions.

In terms of a shape the following was observed:

For 0.7–0.8 nm size NPs, all three investigated materials (TbF₃, DyF₃ and HoF₃) have very similar atomic distributions with the only difference being in the atom-to-atom distance. When increasing the size of a NP, the shape changes from a sphere to a flattened elongated grain shape. This change becomes visible for a 1 nm size. The elongation was observed for all three compounds, but the DyF₃ is the most stretched. This finding agrees with the experimental work of Ref. [6], where the morphology and crystal structure of HoF₃ and DyF₃ were analyzed by microscopy imaging. There, the size of synthesized NP was ≈70–110 nm, which is much higher than considered in our study, but the conclusions concerning the shape are similar. The authors found that the NPs have elongated shapes. The difference with our findings is that in the experimental research, the HoF₃ and DyF₃ NPs are nearly same shape and size of ≈70–110 nm length and ≈30–50 nm width depending on the initial compound used for synthesis, whereas the calculations revealed a slight difference between them, i.e., DyF₃ is the more elongated. Finally, for the slabs, no structural difference was observed; the atom positions are similar to bulk ones.

In terms of magnetization the following features were found:

The distribution of magnetic moments within the NPs is mostly uniform and symmetrical with respect to the center of the particle. For the most symmetrical NP in terms of geometry, as for 0.8 nm HoF₃, all rare earth ions receive similar magnetic moments.

The same is true for 1 nm, with HoF₃ and TbF₃ being less stretched than DyF₃, and they have more homogeneous magnetization than DyF₃. Indeed, the DyF₃ NP being the most elongated has one Dy with zero magnetic moments and one with −6.76 ${\mu }_B$. That fact makes the average magnetization much less than in bulk, except for that case all the other mean magnetic moments for NPs of 0.7–1 nm size are close to the bulk value. The 1.6 nm NPs were not geometrically optimized, so we cannot take that into consideration. The summary of the obtained magnetic moments in comparison with bulk values is collected in Table 2.

As for the slab geometry, the distribution of magnetization is complex as well. Moreover, each of trifluoride slab demonstrates different behavior. The TbF₃ is mostly homogeneous among the considered structures. TbF₃ and DyF₃ slabs have the surface with the most magnetized atomic layers. Unlike this, the HoF₃ slabs of 2 and 3.4 nm thickness have negligible surface magnetization. Overall, each of the considered slab structures has an alternation of higher and lower magnetic moments. The biggest difference in the magnetization of neighboring layers was found for a DyF₃ slab of 2.7 nm thickness.

5. Conclusions

Using the VASP package and DFT, calculations of the magnetic moments of TbF₃, DyF₃ and HoF₃ NPs with sizes of 0.7–1.6 nm and 2–3.4 nm slabs were performed. An inhomogeneous distribution of magnetic moments and a different magnetic arrangements were observed depending on the NP size and slab thickness while maintaining the average value of the magnetic moment close to the bulk. Overall, each considered structure possesses the ferromagnetic arrangement. The slabs have a complex alternation of magnetized atomic layers.

The practical use of the rare earth trifluoride nanoparticles as contrast agents is limited by the toxicity of fluoride compounds to the human body. In this regard, they are coated with non-toxic polymers. The nature of the resulting coating depends on the distribution of magnetic moments and electric fields on the surface of nanoparticles. As a rule, the distribution of magnetic moments on the surface of nanoparticles is uneven, which affects the thickness of the coating and its quality. Simulations of the magnetic moments’ distribution over the nanoparticles volume and surface will allow experimentalists to assess the uniformity of the applied coating, and to propose control methods for uniform application of the polymer.

Further calculations might be conducted for the thicker slabs in order to understand the patterns of magnetization arrangement and approach the experimental sizes. Additionally, the spin–orbit approach might be used in order to investigate the spatial arrangement of each magnetic moment. Finally, other cuts of the crystals might be considered as well.