Introduction

In recent years, the development of novel synthetic routes, allowing the delicate control of the morphology of nanoparticles, has triggered the interest in more complex magnetic structures such as bi-magnetic core/shell nanoparticles1. Among them, ferromagnetic (FM)/antiferromagnetic (AFM) systems are the most studied due to their exchange bias properties (e.g., loops shifts or coercivity enhancement)2. These systems not only combine the properties of the different constituents, but the core/shell interactions can provide an additional degree of freedom to improve the properties, thus opening new avenues for innovative applications of magnetic nanoparticles, ranging from energy storage3,4,5,6,7,8 to biomedicine9,10,11. Interestingly, currently, there is an increasing interest in, so-called, inverted structures (see Fig. 1), where the shell is FM or ferrimagnetic (FiM) and the core is AFM, containing for example Mn oxides12,13,14,15,16,17,18, Fe oxides19,20,21,22,23,24,25,26,27,28,29, Co oxides30,31,32,33,34,35,36, Cr oxides37,38,39, metallic FePt40 or even multiferroic BiFeO3 (Refs. 41,42). This type of inverted structures overcomes some of the limitations of conventional systems, since the AFM structure (and thus its magnetic properties) can be much better controlled in the core than in the shell (where usually it is forced to grow in non-ideal conditions). It has been demonstrated, experimentally and theoretically, that the poor crystallinity of the AFM counterpart can result in considerably inferior exchange bias properties43,44. In fact, inverted structures have already demonstrated very large coercivities and loop shifts, tunable blocking temperatures, enhanced Néel temperatures or proximity effects12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42 and have been proposed as potential magnetoelectric random access memories41. However, despite their potential, systematic studies of size effects (i.e., core diameter or shell thickness) are still rather scarce12,16,22,25,33,34,35. Remarkably, similar effects of the role of the position of the different magnetic phases (core vs shell) also arise in other types of bi-magnetic core/shell nanoparticles such as hard-FM/soft-FM vs. soft-FM/hard-FM nanoparticles, where systems with the hard counterpart in the core can have enhanced or different properties with respect to the ones with soft-FM cores45,46. Thus, understanding the role of the position of the diverse magnetic phases (core vs. shell) is of chief importance in the development of novel applications of bi-magnetic core/shell nanoparticles.

Figure 1
figure 1

Schematic representation and core size dependence of the hysteresis loops.

(left) Schematic representation of the inverse AFM/FiM core/shell structure and (right) hysteresis loops of the doubly inverted nanoparticles, for different AFM core sizes (Dcore) and constant FiM shell thickness of four lattice spacings.

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From the magnetic point of view, one can identify two main types of inverted structures depending on the transition temperature of the materials. Thus, “single inverted” systems are those where the Curie temperature of the FM, TC, is larger than the Néel temperature of the AFM, TN, i.e., TC > TN, e.g., FeO/Fe3O4. On the other hand, if TN > TC, the systems are usually denoted “doubly inverted”, e.g., MnO/Mn3O4. Although this type of structure is seldom studied in thin film systems, the available results evidence rather interesting properties47,48. Similarly, doubly inverted core/shell nanoparticles exhibit some novel properties such as a non-monotonic dependence of the coercivity and the loop shift on the core size12.

From the numerical simulation point of view, the mechanisms and the factors that influence the exchange bias behavior in conventional core/shell nanoparticles, i.e., soft FM core/hard AFM49,50 or FiM shell nanoparticles46,51,52 have been fairly investigated53,54. These studies support that the exchange bias field depends mainly on the structure of the interface (uncompensated spins) and the coercive field on the particle size. Studies of inverted, AFM/FM, structures (hard AFM/soft FM) are far scarcer55,56 and, in fact, there are no reports on doubly inverted structures. Notably, the factors responsible for the observed enhanced magnetic behavior of the inverse structure are largely unknown.

Herein, we investigate the core and shell size dependence of the magnetic properties of hard FiM nanoparticles in doubly inverted (TN > TC) AFM/FiM core/shell structures using Monte Carlo (MC) simulations. We consider a broad range of core sizes and shell thicknesses to elucidate the optimum conditions for improved properties for diverse possible applications. The results clearly replicate the non-monotonic dependence of the exchange bias properties observed experimentally. Namely, for very small core sizes both coercive (HC) and exchange bias (Hex) sharply increase as the core size becomes larger. However above a certain size HC and Hex start to decrease, thus exhibiting a maximum. Further, they show that the dependence of the magnetic properties on the FiM thickness depends critically on the size of the AFM core, leading to HC and Hex proportional to the FiM thickness for large AFM cores. The origin of these unusual effects is shown to arise from the competition between the AFM core spins and the hard FiM shell.

Results

Shown in the right panel of Fig. 1 are the hysteresis loops for doubly inverted nanoparticles for various core diameters, Dcore and a constant shell thickness, tSH of four lattice spacings, tSH = 4. The saturation magnetization of the loops decreases for larger Dcore as expected from the zero net magnetic moment of the AFM core. Moreover, it can be seen that the loops exhibit both HC and Hex, which have different behavior for the diverse Dcore. The dependence of HC and Hex on Dcore for tSH = 4, 6, 8 is shown in Fig. 2.

Figure 2
figure 2

Core size dependence of the coercivity and the exchange bias.

Dependence of the coercivity, HC (a) and exchange bias shift, Hex (b) on the AFM core size, Dcore for the doubly inverted (TC < TN – Jcore > Jshell) structures for tSH = 4, 6 and 8.

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The results clearly show that inverted structures can result in sizable loop shifts and coercivity enhancements, similar to conventional FM/AFM structures2. Nevertheless, contrary to conventional systems, a strong non-monotonic behavior is observed. Interestingly, both HC and Hex exhibit maximum values for rather small Dcore. Moreover, the maximum HC is obtained for very small Dcore (e.g., Dcore = 4.2 for tSH = 4) whereas the maximum Hex is observed for slightly larger Dcore (e.g., Dcore = 6 for tSH = 4). These results are in qualitative agreement with the experimental doubly inverse MnO/Mn3O4 nanoparticles case, which also shows an analogous non-monotonic dependence of HC and Hex on Dcore (Ref. 12). Remarkably, both HC and Hex decrease considerably slow with increasing Dcore. In fact, HC and Hex stabilize for large Dcore values and Hex does not vanish even for Dcore = 70. The overall behavior of both HC and Hex is similar for the three tSH. Explicitly, they all exhibit non-monotonic behavior with maxima both in HC and Hex at small Dcore values. However, a more careful analysis of the data reveals several appealing features. Concerning HC, it decreases slower as Dcore increases for thicker tSH, leading to a crossover behavior for large Dcore. Interestingly, although for small Dcore HC decreases for thicker tSH, the coercivity enhancement, i.e., ΔHC/HC, becomes larger for thicker shells, reaching almost 10% for tSH = 8. Regarding Hex, two remarkable features are worth mentioning. First, the maximum Hex is obtained for smaller Dcore as tSH increases. Secondly, similar to HC, Hex also exhibits a cross over behavior. As expected from the inverse FM thickness dependence of exchange bias observed in thin films or conventional core/shell particle2, for small Dcore Hex decreases for larger tSH. However, this trend is reversed for large Dcore and, counterintuitively, Hex increases for larger tSH. This is summarized in Fig. 3, where it can be clearly seen that while for small core sizes HC and Hex exhibit the conventional inversely proportional to tSH behavior (i.e., ~∝1/tSH), for large cores the behavior is against what is expected for an interface effect and is proportional to tSH (i.e., ~∝ tSH). This trend is somewhat broken for Hex for exceedingly small tSH in the case of small Dcore and for large tSH in the case of large Dcore.

Figure 3
figure 3

Shell size dependence of the coercivity and the exchange bias.

Dependence of the normalized HC and Hex on tSH for Dcore = 6 and 50.

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The simulations have revealed a number of notable results which, in some cases, are in clear contrast with conventional exchange bias wisdom in thin film and conventional FM/AFM core/shell nanoparticles: (i) both HC and Hex exhibit a strong non-monotonic behavior with Dcore; (ii) the maximum HC and Hex is obtained for rather small Dcore, with sizes comparable to those of the shell; (iii) the largest HC is obtained for smaller Dcore than for Hex; (iv) although Hex and HC increase for thinner tSH for small Dcore, they show an opposite behavior at large Dcore.

Discussion

To unveil the origin of these novel effects, the number of total and uncompensated spins in the core, interface, shell and surface have been evaluated. Note that uncompensated spins are those spins that due to their local environment of reduced number of neighbors feel a smaller mean field and can thus act more independently. As can be seen in the Fig. 4, the relative number of spins in the different positions with respect to the total number of spins depends strongly on the Dcore and tSH dimensions. Plotted in Fig. 5 is the absolute number of uncompensated spins Nup-Ndw as a function of the core size, for tSH = 4, normalized to the total number of spins, Ntot, for the core interface (IF), the shell IF and the surface, taking into account the fact that at the surface and the shell IF the spin magnitude is 1 and 1.5 respectively in the two sublattices. Hence, for the core IF region the normalized number of uncompensated spins is Abs[(NAFMup-NAFMdw)/(NAFMup + NAFMdw + NFiMup + NFiMdw*1.5)] while in the other regions is Abs[(NFiMup-NFiMdw*1.5)/(NAFMup + NAFMdw + NFiMup + NFiMdw*1.5)]. Notably, the notation up and down spins, introduced firstly by Néel57, is just schematic to indicate the spins in the two sublattices. Consequently, in the simulations, where the anisotropy parameters are not particularly strong anywhere in the nanoparticle, these spins do not actually act collinearly. Moreover, it should be emphasized that the number of spins and uncompensated spins is solely governed by geometry. Thus, it is important to emphasize that observed fluctuations in the number of uncompensated spins, especially for small core sizes are due to geometrical effects.

Figure 4
figure 4

Size dependence of the total number of spins.

Number of spins (N) in each region for various core diameters and shell thickness 2 (a), 4 (b), 6(c), 8(d), 10(e) and 12(f) lattice spacings. Shown in the insets are enlarged views of N vs. Dcore for low Dcore.

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Figure 5
figure 5

Core size dependence of the number of uncompensated spins.

Absolute number of uncompensated spins of the core IF, shell IF and surface normalized to the total number of spins, as a function of the core diameter, Dcore (for tSH = 4).

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For very small core sizes, in the range between 2.5–5.35 lattice spacings, the number of uncompensated spins at the core IF and shell IF is very small, while the number of uncompensated spins at the surface of the shell is considerably larger (Fig. 5). Thus, for very small Dcore Hex is dominated by the uncompensated spins from the surface of the FiM shell. Although the number of surface spins is relatively small their radial character keeps them pinned on the surface, resulting in finite exchange bias. As Dcore increases, in the range between 5.35 and 12.4 lattice spacings, the number of uncompensated spins in both the core IF and the shell IF increases (Fig. 5). The maximum of the Hex corresponds to the core size that gives the maximum value of the number of uncompensated spins at the shell interface and a sizable contribution of the core IF and surface uncompensated spins. Thus, the shell IF and the surface mainly contribute to the maximum Hex. As Dcore becomes even larger, for core sizes in the range 12.4–25 lattice spacings, the number of uncompensated spins of the core IF becomes almost negligible and finally vanishes for Dcore >25 lattice spacings, while the average number of uncompensated spins at the shell IF and the surface slowly decreases, leveling off for very large Dcore. This behavior is similar to one observed for Hex (Fig. 2). Consequently, for large Dcore Hex is controlled mainly by the shell. Thus, the evolution of the uncompensated spins can qualitatively explain the overall dependence of Hex on Dcore. Interestingly, the large exchange Jcore of the doubly inverted structure results in an extra anisotropy in the core spins. This gives rise to a stronger resistance of the core spins to be dragged by the reversal of the shell spins, leading to a non-vanishing Hex even for large Dcore (in the range 30–60.2 lattice spacings). For this Dcore range, although the number of core spins dominates, the shell spins still contribute to the exchange bias properties, acting as a pining center and therefore competing with the core spins over a wide range of core sizes, where the shell still has a sizeable contribution.

These arguments can also account for the non-monotonic behavior of Hex(tSH) for the different Dcores observed in Fig. 3. Namely, in the case of small Dcore as tSH becomes exceedingly small (e.g. tSH = 2) the number of surface spins and shell interface spins (see Fig. 4) dominate over core spins, leading to a Hex reduction. On the other hand for large Dcores, as tSH becomes sufficiently large the competition between core spins and shell spins increases resulting in an increase of Hex.

Concerning HC, for very small core sizes we have contribution mainly from the shell and the maximum occurs for the core size where the total number of spins from the shell plays the dominant role (see inset in Fig. 5). For larger core sizes, the extra anisotropy induced by the shell acting as pinning center (as Ncore starts to increase) results in a slow decrease HC as Dcore is increased.

Regarding the role of the shell thickness, given the larger shells in the tSH = 6 and 8 (compared to tSH = 4) the competition between the core and shell contributions (i.e., when then number of core spins become larger than the number of shell spins; see Fig. 4) occurs at different Dcore. The boundaries are roughly Dcore ~ 25, 30 and 50 for tSH = 4, 6 and 8, respectively. Consequently, the decrease of Hex and HC for large Dcore is pushed to larger Dcore as tSH increases. This gives rise to the crossover from the conventional FM thickness dependence of exchange bias systems (i.e., tFM ↑ → Hex, HC ↓) at small Dcore to the counterintuitive reversed behavior (tFM ↑ →Hex, HC ↑) for large Dcore.

Notably, this unusual behavior is to some extent different in single inverted AFM core/FiM shell nanoparticles shown in Fig. 6 (i.e., TN < TC – Jcore < Jshell). Namely, while HC exhibits a HC ∝ tSH dependence for large Dcores similar to the doubly inverted case, Hex shows this inverse behavior only in a very narrow range of Dcores since Hex vanishes at large Dcores. In this case since Jcore is weaker there is no competition between the core and the shell. Thus, the shell drags the core spins with the consequent decrease of Hex and HC. Consequently, the doubly inverted structures present improved properties compared to the single inverted ones, especially for large Dcores.

Figure 6
figure 6

Core size dependence of the coercivity and the exchange bias.

Dependence of the coercivity, HC (a) and exchange bias shift, Hex (b) on the AFM core size, Dcore for the single inverted (TC > TN – Jcore < Jshell) structures for tSH = 4, 6 and 8.

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From the applications point of view some of the features unveiled from the simulations are rather attractive. One possible use of these structures could be to utilize the coercivity enhancement for small Dcore to improve the performance of permanent magnets58. Given the higher HC for thin shells, probably thin shells with small cores would be ideal for this type of purposes. Note that despite the loss of saturation magnetization, MS (due to the zero MS of the AFM core), since the core is so small the energy product (i.e., the figure of merit of a permanent magnet) may actually be improved (similar to what has been observed in AFM/FM composites59) in these structures. For recording applications3,4, where, for example, enhancement of the blocking temperature of very small nanoparticles may be pursued, perhaps larger coercivity enhancement, ΔHC (which implies an increased effective anisotropy) may be more appealing. Thus, thicker shells with small cores would be more appropriate. Note that for this type of applications inverse AFM/FM structures may be more suitable than conventional FM/AFM, since the stray field of the nanoparticles (required for easy detection) would be less attenuated. For applications based on HC, doubly inverted structures may be more attractive, especially if large cores are needed. Concerning applications requiring exchange bias, e.g., miniaturized magnetotransport devices6,41, the optimum Dcore/tSH configuration would depend on the exact HC/Hex required. For example, for large HC and Hex, then small tSH with moderate Dcore may be best. Nevertheless for cases where moderate HC and Hex are needed, large tSH and Dcore would be better.

Conclusion

In conclusion, the behavior of the exchange bias and the coercive field in doubly inverted AFM core/hard FiM shell nanoparticle systems have been shown to depend on the core size in a different way at various core size ranges. For very small core sizes there is contribution on Hex from the surface uncompensated spins. For moderate core sizes the uncompensated spins of the core and the shell interface also contribute to the exchange bias, resulting in a maximum Hex value. For even larger core diameters the exchange in the core, Jcore and the AFM character of the core determine Hex. For large Dcore the whole shell plays the role of the shell IF, thus the exchange bias effect for these core sizes increases with shell thickness, in contrast to conventional systems. The study of the role of the shell thickness indicates that a sizable shell contribution is needed to ensure enhancement of the exchange bias properties. The improved magnetic properties can satisfy a range of technological demands.

Methods

For the study of the magnetic behavior of the nanoparticles we use the Monte Carlo simulation technique and the Metropolis algorithm53. We consider spherical nanoparticles of diameter D, expressed in lattice spacings, on a simple cubic lattice with inverted structure, consisting of an AFM core and a FiM shell (see Fig. 1-left panel). We take into account explicitly the microstructure of the system in an atomic scale. The spins of the nanoparticles are located at each lattice site of the core, the interface, the shell and the surface. They interact with nearest neighbours through Heisenberg exchange interactions and at each crystal site they experience a uniaxial anisotropy. At the surface of the particles, the crystal symmetry is reduced and consequently the anisotropy is stronger than in the bulk60,61. The FiM shell is considered as a layer surrounding the core.

In the presence of an external magnetic field, the total energy of the system is

Here is the atomic spin at site i and is the unit vector in the direction of the easy axis at site i. We consider the magnitude of the atomic spins in the two AFM sublattices equal to 1 and in the two FiM sublattices of the shell to be equal to 1 and 1.5, respectively. The first term in Eq. 1 gives the exchange interaction between the spins in the AFM core, while the second term gives the exchange interaction between the spins in the FiM shell. We consider two different cases, (i) single inverted with TC(shell) > TN(core) and (ii) doubly inverted with TN(core) > TC(shell). To take into account the difference in transition temperatures, for the first case we consider the exchange coupling constant of the core as Jcore = −0.5 JFM and that of the shell as Jshell = −1.5 JFM where JFM is the exchange coupling constant of a pure ferromagnet JFM = 1 as reference value. In the doubly inverted structures, Jshell is maintained the same, but Jcore in increased to account for the larger TN. Thus, the exchange coupling constants are set to Jcore = −3.0 JFM and Jshell = −1.5 JFM. The third term gives the exchange interaction at the interface between the core and the shell. The interface includes the last layer of the AFM core and the first layer of the FiM shell. The exchange coupling constant of the interface JIF is taken equal to that of the shell JSH. The forth term gives the anisotropy energy of the AFM core, KC. If the site i lies in the outer layer of the AFM core Kicore = KIF and Kicore = KC elsewhere (with KC = 0.05 JFM -i.e., a soft AFM- and KIF = 0.5 JFM). The fifth term gives the anisotropy energy of the FiM shell, which is taken as KSH = 0.5 JFM. If i lies in the outer layer (i.e., the surface) of the shell then the anisotropy is taken as KS = 1.4 JFM, which is assumed to be radial (rather than uniaxial). The last term in Eq. 1 is the Zeeman energy.

The aim of our model is to qualitatively reproduce the magnetic behavior of this new class of advanced nanomaterials, therefore the parameters used in the simulations for the exchange and the anisotropy constants were chosen from a careful analysis of the experimental magnetic behavior of the nanoparticle system MnO (AFM)/Mn3O4 (FiM) (i.e., the most studied doubly-inverse system)12. Note that since ab initio electronic calculations cannot be performed for this type of complex nanoparticles the parameters were estimated by modifying the bulk values Mn3O4 and MnO taking into account the lower crystal symmetry at the interface and at the surface, which results in higher effective anisotropy energy in these regions. Nevertheless, the K parameters are weighted by their corresponding volume49. The exchange coupling parameters for the core, the interface, the shell and the surface are considered here to be the same for all the different core and shell sizes since the exchange coupling parameters are short ranged and hence the influence of the second or third neighbors is almost negligible62. Moreover, it is worth noting that experimental studies of the detailed chemical characterization of MnO/Mn3O4 core/shell nanoparticles indicate that the interface between MnO and Mn3O4 is rather sharp63, thus justifying our model.

Importantly, the chosen parameters roughly maintain the main experimental characteristics (i.e., TC(Mn3O4)/TN(MnO) ~ 1/3 and KFiM(Mn3O4)/KAFM(MnO) ~ 50) with simulated values of TC(Mn3O4)/TN(MnO) ~ 1/3 and KFiM(Mn3O4)/KAFM(MnO) ~ 10. Note that the smaller than bulk ratio KFiM(Mn3O4)/KAFM(MnO) used in the simulations was chosen to take into account the possible worsening of the magnetic properties of the shell material due to structural deterioration of shell. Note, for example, that although the Mn3O4 shell in MnO/Mn3O4 core/shell nanoparticles exhibits a TC close to bulk values, their HC is usually smaller (i.e., lower K(Mn3O4))12,16,17,18.

We perform our simulations of the hysteresis loops on isolated nanoparticles using the Monte Carlo (MC) simulation technique with the implementation of the Metropolis algorithm64. The hysteresis loops are calculated after a field cooling procedure starting at temperature T = 7.0 JFM/kB down to Tf = 0.01 JFM/kB, at a constant rate under a static magnetic field Hcool = 4.0 JFM/gμB directed along the z-axis. The value of Hcool was selected as the optimum value to observe maximum Hex and HC as Fig. 7 shows. The hysteresis loop shift on the field axis gives the exchange field Hex = -(Hright+Hleft)/2. The coercive field is defined as HC = (Hright- Hleft)/2. Hright and Hleft are the points where the loop intersects the field axis. The coercive HC and exchange Hex fields are given in dimensionless JFM/gμB units and the temperature T in JFM/kB.

Figure 7
figure 7

Effect of the cooling field.

Dependence of coercivity, HC (left) and exchange bias, Hex (right) on the cooling field, Hcool, for Dcore = 8.2 (squares) and 26 (circles), with tSH = 4.

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In the Monte Carlo method, at each Monte Carlo step we select, at random, an atomic spin from the N spins of the nanoparticle and we make a small change in its orientation. This attempted change is accepted if it leads to the lowering of the system's energy or with a certain probability that corresponds to the Boltzmann probability. The process is repeated until equilibrium is reached. We have used 104 MC steps per spin (MCSS) at each field step and the results were averaged over 50–200 different samples (namely random numbers) depending on the size of the nanoparticle and the fluctuations in the values of the calculated fields. The standard deviation of the averages is depicted in the plots as error bars.