Enhanced Magnetic Properties in Antiferromagnetic-Core/Ferrimagnetic-Shell Nanoparticles

Bi-magnetic core/shell nanoparticles are gaining increasing interest due to their foreseen applications. Inverse antiferromagnetic(AFM)/ferrimagnetic(FiM) core/shell nanoparticles are particularly appealing since they may overcome some of the limitations of conventional FiM/AFM systems. However, virtually no simulations exist on this type of morphology. Here we present systematic Metropolis Monte Carlo simulations of the exchange bias properties of such nanoparticles. The coercivity, HC, and loop shift, Hex, present a non-monotonic dependence with the core diameter and the shell thickness, in excellent agreement with the available experimental data. Additionally, we demonstrate novel unconventional behavior in FiM/AFM particles. Namely, while HC and Hex decrease upon increasing FiM thickness for small AFM cores (as expected), they show the opposite trend for large cores. This presents a counterintuitive FiM size dependence for large AFM cores that is attributed to the competition between core and shell contributions, which expands over a wider range of core diameters leading to non-vanishing Hex even for very large cores. Moreover, the results also hint different possible ways to enhance the experimental performance of inverse core/shell nanoparticles for diverse applications.

I n 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 nanoparticles 1 . 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 storage 3-8 to biomedicine [9][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 oxides [12][13][14][15][16][17][18] , Fe oxides [19][20][21][22][23][24][25][26][27][28][29] , Co oxides [30][31][32][33][34][35][36] , Cr oxides 37-39 , metallic FePt 40 or even multiferroic BiFeO 3 (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 properties 43,44 . In fact, inverted structures have already demonstrated very large coercivities and loop shifts, tunable blocking temperatures, enhanced Néel temperatures or proximity effects  and have been proposed as potential magnetoelectric random access memories 41 . However, despite their potential, systematic studies of size effects (i.e., core diameter or shell thickness) are still rather scarce 12,16,22,25,[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 cores 45,46 . 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.
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, T C , is larger than the Néel temperature of the AFM, T N , i.e., T C . T N , e.g., FeO/Fe 3  is seldom studied in thin film systems, the available results evidence rather interesting properties 47,48 . Similarly, doubly inverted core/shell nanoparticles exhibit some novel properties such as a nonmonotonic dependence of the coercivity and the loop shift on the core size 12 .
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 AFM 49,50 or FiM shell nanoparticles 46,51,52 have been fairly investigated 53,54 . 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 scarcer 55,56 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 (T N . T C ) 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 (H C ) and exchange bias (H ex ) sharply increase as the core size becomes larger. However above a certain size H C and H ex 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 H C and H ex 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, D core , and a constant shell thickness, t SH of four lattice spacings, t SH 5 4. The saturation magnetization of the loops decreases for larger D core as expected from the zero net magnetic moment of the AFM core. Moreover, it can be seen that the loops exhibit both H C and H ex , which have different behavior for the diverse D core . The dependence of H C and H ex on D core for t SH 5 4, 6, 8 is shown in Fig. 2.
The results clearly show that inverted structures can result in sizable loop shifts and coercivity enhancements, similar to conventional FM/AFM structures 2 . Nevertheless, contrary to conventional systems, a strong non-monotonic behavior is observed. Interestingly, both H C and H ex exhibit maximum values for rather small D core . Moreover, the maximum H C is obtained for very small D core (e.g., D core 5 4.2 for t SH 5 4) whereas the maximum H ex is observed for slightly larger D core (e.g., D core 5 6 for t SH 5 4). These results are in qualitative agreement with the experimental doubly inverse MnO/ Mn 3 O 4 nanoparticles case, which also shows an analogous non-monotonic dependence of H C and H ex on D core (Ref. 12).   Remarkably, both H C and H ex decrease considerably slow with increasing D core . In fact, H C and H ex stabilize for large D core values and H ex does not vanish even for D core 5 70. The overall behavior of both H C and H ex is similar for the three t SH . Explicitly, they all exhibit non-monotonic behavior with maxima both in H C and H ex at small D core values. However, a more careful analysis of the data reveals several appealing features. Concerning H C , it decreases slower as D core increases for thicker t SH , leading to a crossover behavior for large D core . Interestingly, although for small D core H C decreases for thicker t SH , the coercivity enhancement, i.e., DH C /H C , becomes larger for thicker shells, reaching almost 10% for t SH 5 8. Regarding H ex , two remarkable features are worth mentioning. First, the maximum H ex is obtained for smaller D core as t SH increases. Secondly, similar to H C , H ex 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 particle 2 , for small D core H ex decreases for larger t SH . However, this trend is reversed for large D core and, counterintuitively, H ex increases for larger t SH . This is summarized in Fig. 3, where it can be clearly seen that while for small core sizes H C and H ex exhibit the conventional inversely proportional to t SH behavior (i.e., , / 1/t SH ), for large cores the behavior is against what is expected for an interface effect and is proportional to t SH (i.e., , / t SH ). This trend is somewhat broken for H ex for exceedingly small t SH in the case of small D core and for large t SH in the case of large D core .
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 H C and H ex exhibit a strong non-monotonic behavior with D core ; (ii) the maximum H C and H ex is obtained for rather small D core , with sizes comparable to those of the shell; (iii) the largest H C is obtained for smaller D core than for H ex ; (iv) although H ex and H C increase for thinner t SH for small D core , they show an opposite behavior at large D core .

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 D core and t SH dimensions. Plotted in Fig. 5 is the absolute number of uncompensated spins N up -N dw as a function of the core size, for t SH 5 4, normalized to the total number of spins, N tot , 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 Notably, the notation up and down spins, introduced firstly by Néel 57 , 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.
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 D core H ex 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 D core 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 H ex 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 H ex . As D core 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 D core .25 lattice spacings, while the average number of uncompensated spins at the shell IF and the surface slowly decreases, leveling off for very large D core . This behavior is similar to one observed for H ex (Fig. 2). Consequently, for large D core H ex is controlled mainly by the shell. Thus, the evolution of the uncompensated spins can qualitatively explain the overall dependence of H ex on D core . Interestingly, the large exchange J core 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 H ex even for large D core (in the range 30-60.2 lattice spacings). For this D core 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 H ex (t SH ) for the different D cores observed in Fig. 3. Namely, in the case of small D core as t SH becomes exceedingly small (e.g. t SH 5 2) the number of surface spins and shell interface spins (see Fig. 4) dominate over core spins, leading to a H ex reduction. On the other hand for large D cores , as t SH becomes sufficiently large the competition between core spins and shell spins increases resulting in an increase of H ex .
Concerning H C , 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 N core starts to increase) results in a slow decrease H C as D core is increased.
Regarding the role of the shell thickness, given the larger shells in the t SH 5 6 and 8 (compared to t SH 5 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 D core . The boundaries are roughly D core , 25, 30 and 50 for t SH 5 4, 6 and 8, respectively. Consequently, the decrease of H ex and H C for large D core is pushed to larger D core as t SH increases. This gives rise to the crossover from the conventional FM thickness dependence of exchange bias systems (i.e., t FM " R H ex , H C #) at small D core to the counterintuitive reversed behavior (t FM " RH ex , H C ") for large D core .
Notably, this unusual behavior is to some extent different in single inverted AFM core/FiM shell nanoparticles shown in Fig. 6 (i.e., T N , T C -J core , J shell ). Namely, while H C exhibits a H C / t SH dependence for large D cores similar to the doubly inverted case, H ex shows this inverse behavior only in a very narrow range of D cores since H ex vanishes at large D cores . In this case since J core is weaker there is no competition between the core and the shell. Thus, the shell drags the core spins with the consequent decrease of H ex and H C . Consequently, the doubly inverted structures present improved properties compared to the single inverted ones, especially for large D cores .
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 D core to improve the performance of permanent magnets 58 . Given the higher H C 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, M S (due to the zero M S of the AFM core), since the core is so small the energy product (i.e., the figure of merit of   59 ) in these structures. For recording applications 3,4 , where, for example, enhancement of the blocking temperature of very small nanoparticles may be pursued, perhaps larger coercivity enhancement, DH C (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 H C , doubly inverted structures may be more attractive, especially if large cores are needed. Concerning applications requiring exchange bias, e.g., miniaturized magnetotransport devices 6,41 , the optimum D core /t SH configuration would depend on the exact H C /H ex required. For example, for large H C and H ex , then small t SH with moderate D core may be best. Nevertheless for cases where moderate H C and H ex are needed, large t SH and D core 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 H ex 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 H ex value. For even larger core diameters the exchange in the core, J core and the AFM character of the core determine H ex . For large D core 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 algorithm 53 . 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 bulk 60,61 . 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 s l ! is the atomic spin at site i and b e i 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 T C (shell) . T N (core) and (ii) doubly inverted with T N (core) . T C (shell). To take into account the difference in transition temperatures, for the first case we consider the exchange coupling constant of the core as J core 5 20.5 J FM and that of the shell as J shell 5 21.5 J FM where J FM is the exchange coupling constant of a pure ferromagnet J FM 5 1 as reference value. In the doubly inverted structures, J shell is maintained the same, but J core in increased to account for the larger T N . Thus, the exchange coupling constants are set to J core 5 23.0 J FM and J shell 5 21.5 J FM . 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 J IF is taken equal to that of the shell J SH . The forth term gives the anisotropy energy of the AFM core, K C . If the site i lies in the outer layer of the AFM core K icore 5 K IF and K icore 5 K C elsewhere (with K C 5 0.05 J FM -i.e., a soft AFM-and K IF 5 0.5 J FM ). The fifth term gives the anisotropy energy of the FiM shell, which is taken as K SH 5 0.5 J FM . If i lies in the outer layer (i.e., the surface) of the shell then the anisotropy is taken as K S 5 1.4 J FM , 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)/Mn 3 O 4 (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 Mn 3 O 4 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 volume 49 . The exchange coupling parameters for the core, the interface, the shell and the surface are 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, D core (for t SH 5 4).  We perform our simulations of the hysteresis loops on isolated nanoparticles using the Monte Carlo (MC) simulation technique with the implementation of the Metropolis algorithm 64 . The hysteresis loops are calculated after a field cooling procedure starting at temperature T 5 7.0 J FM /k B down to T f 5 0.01 J FM /k B , at a constant rate under a static magnetic field H cool 5 4.0 J FM /gm B directed along the z-axis. The value of H cool was selected as the optimum value to observe maximum H ex and H C as 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 10 4 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.