Two types of equally sized (6.8 ± 0.6 nm) particles were prepared: γ-Fe2O3 nanoparticles (hereon described as “low anisotropy” or “soft”) and cobalt doped γ-Fe2O3 nanoparticles (“high anisotropy” or “hard”). See Fig. 1a,b and Methods.
Although at low temperatures the isolated (non-interacting) particles of both systems show a roughly similar saturation magnetization (MS), they exhibit very different blocking temperature, TB, and low temperature coercivity, HC, as can be seen in Supplementary Section 3. In this reference samples, the isolation/dilution of the magnetic particles was efficiently guaranteed by coating with a thick SiO2 shell (serving as spacer in pressed powders) a fraction of the particles from the corresponding batches39,44. The difference in TB and HC is due to the difference in anisotropy between the two types of particles. Crucially for this work, while the γ-Fe2O3 nanoparticles exhibit zero loop shift (i.e., no exchange bias), the Co-doped particles show a sizable exchange-bias field (HEB = 1.20 kOe at 5 K). The precise origin of the exchange-bias in the hard particles (an intraparticle effect) is irrelevant for this study, nonetheless we have included a preliminary discussion in the Supplementary Section 435,39,45–48. Dense composites made of a single type of particles (with no silica coating) produce a large increase of TB due to interparticle dipolar interactions (Supplementary Section 3)49, which, however, barely affect the exchange-bias of the individual particles (from 1.20 in the reference system to 1.28 kOe in the disc made of bare Co-doped particles, see Supplementary Section 3). In fact, this small variation may be driven by subtle changes in the nanoparticle surface upon growing the SiO2 layer39.
To prepare the dense binary nanocomposites, the pure and Co-doped γ-Fe2O3 nanoparticles were first mixed in liquid dispersion to ensure homogeneous mixing. The mixture was then dried and, subsequently, the oleic acid particle coatings were removed by repeated washing in acetone. Finally, disc-like compacts of the nanoparticle mixtures were formed by uniaxial die pressing. These highly dense binary assemblies (packing fraction close to 60%40, see Fig. 1c), hereon denoted as Mx where x is the percentage of soft particles, show a uniform distribution of the soft/hard particles down to the nanoscale (Methods). In the following, to avoid any confusion, we will call “exchange bias”, HEB, to the loop shifts that unambiguously stem from interface exchange coupling, an intraparticle effect, whereas the more general term “bias”, HB, will be used for loop shifts caused by a combination of the different mechanisms discussed here. All hysteresis loops were measured at T = 5 K after cooling in 50 kOe.
As the soft particle content in the mixtures increases several effects take place, the most prominent being a change in the shape of the hysteresis loop, from a simple form in the pure samples (M0 and M100) to a “double loop” shape in the mixed systems (Fig. 2 and Supplementary Section 5). This reflects a relatively weak hard-soft interparticle coupling (given the high anisotropy contrast between them) despite the relatively strong dipolar interactions at play50,51, which, as expected, leads to a strong monotonic reduction in the global HC of the mixtures with increasing x (Fig. 3a). Remarkably, in contrast to HC, the horizontal shift of the loop has a non-monotonic behavior, exhibiting a maximum at x = 15% before decreasing for larger contents of soft particles (Fig. 3b). Thus, for moderate amounts of soft particles, the global bias HB of the composites is surprisingly larger than the exchange bias HEB of the single-phased Co-doped sample (M0), despite the soft particles exhibiting perfectly centered loops (no bias).
In order to safely establish this unexpected enhancement in bias (shaped by just one experimental datapoint in Fig. 3a), and given the weak soft-hard coupling evidenced by the measured double-loops, “superposition loops” were obtained by simply adding the experimental loops of the single-phase samples M100 (soft) and M0 (hard), i.e. as \(\left(soft loop\right)\left(x\right)+\left(hard loop\right)(1-x)\), in the whole soft-hard proportion range. The coercivity and bias extracted from these superpositions was added to Fig. 3a,b (red circles), where it can be seen that the overall trends with x of both HC and HB in the experimental and superposition loops are similar. In particular, the superposition loops evidence a significant bias enhancement of ∼25% for x = 23% (Fig. 3b), demonstrating a ‘mathematical’ origin for this effect, namely, that the addition of certain differently-shaped loops may give rise to a sizable horizontal shift (\({H}_{B})\). A close inspection of the loop of the hard particles reveals that their magnetization reversal is not completely symmetric, namely the squareness of the descending (Shard,left) and ascending (Shard,right) branches of the loop are different (see Supplementary Section 6)52. To quantify the asymmetry in the reversal of the hard particles, each of the branches of the was fitted to a Stearns and Cheng functions (a relatively simple empirical model42),
$${M}_{\pm }\left(H\right)= \frac{2}{\pi }{M}_{S,hard}\text{atan}\left[\right(\frac{H + {H}_{EB,hard }\pm { H}_{C,hard}}{{H}_{C,hard}})\text{tan}(\frac{\pi {S}_{hard, \pm }}{2}\left)\right]+\chi H$$
1
where the ± symbol indicates the different signs used in the ascending (right) and descending (left) branches of the loops, \(\chi\) is a high-field susceptibility (see Methods), and \({H}_{EB,hard}\pm {H}_{C,hard}\) were fixed to the measured values. The resulting squareness parameters of the hard phase are \({S}_{hard, left}=0.65\), and \({S}_{hard, right}=0.56\), i.e., yielding an “asymmetry ratio” \({S}_{hard, right}/{S}_{hard, left}=0.86\) ). It is this asymmetry in the hard loop which literally shifts the overall loop upon adding a soft component, even when superposing two unbiased loops (see below), as schematized in Fig. 4a.
Once the asymmetric reversal of the hard loop was stablished as the origin of the bias enhancement in Fig. 3b, we investigated the optimization/sensitivity of the effect to the coercivity and asymmetry ratio of the hard component in a given soft-hard mixture. To this purpose, a full array of superposition loops was obtained using the experimental loop for the soft particles (unshifted and symmetric, i.e., with Ssoft,right = Ssoft,left = Ssoft = 0.32) and hard loops simulated with different degrees of asymmetric reversal (i.e., different Shard,right/Shard,left ratios) and coercivity (HC,hard). For clarity and to highlight the role of the reversal asymmetry in the hard component, its exchange-bias was set to zero [HEB, hard = 0 in Eq. (1)]. For each combination of Shard,right/Shard,left and HC,hard, superposition loops were calculated across the whole range of soft particles concentration, x (thus obtaining a “3D array” of loops), and the loop bias HB was extracted. Shown in the inset of Fig. 4b is an example for the case x = 28%, Shard,right/Shard,left = 0.86 and HC,hard = 12 kOe (the two latter parameters corresponding to the experimental Co-doped maghemite loop). It can be clearly seen that despite the absence of exchange bias in both these soft and hard loops, their superposition generates a strong shift of HB = 860 Oe. As can be seen in Fig. 4b, the asymmetry-driven bias of the superposition loops peaks at \({H}_{B}^{max}\)at different soft particle proportions (xmax) depending on the asymmetry ratio (Shard,right/Shard,left). The results for \({H}_{B}^{max}\) over a large Shard,right/Shard,left and \({H}_{C,hard}\) range, \({H}_{B}^{max}\left(\frac{{S}_{hard, right}}{{S}_{hard, left}};{H}_{C,hard}\right)\), are conveniently plotted as a contour plot in Fig. 4c. The plot, with its white vertical ridge at Shard,right/Shard,left = 1, serves to remark that the asymmetry-generated bias appears in all superpositions (see Supplementary Section 7), even if the constituent soft and hard loops are unbiased on their own. Note that loops shifts of several kOe can be obtained for realistically large Shard,right/Shard,left and \({H}_{C,hard}\) values. As expected, for a given asymmetry ratio the bias enhancement increases with the hard loop coercivity (see also Supplementary Section 7). It is therefore the combination of both parameters that generates bias upon the introduction of a soft unbiased loop. In short, the results clearly indicate that the enhanced bias observed experimentally in the binary mixtures (Fig. 3b) arises from the shape asymmetry of the magnetization curves of the hard phase (\({S}_{hard, right}/{S}_{hard, left}\ne 1\)), and not from its horizontal shift. This effect could be universally exploited in any soft-hard weakly-coupled binary system (e.g., in thin film multilayers) to increase or adjust the overall bias of the ensemble as long as one of the moieties exhibits an asymmetric loop.
Back to Fig. 3, although the parameters extracted from the experimental and the superposition loops follow similar trends with the soft-hard nanoparticle proportion x, marked differences exist between them for the central samples of the series (see difference plots of Fig. 9 in Supplementary Section 5). Consequently, besides the asymmetry effect described above (an “additive” effect which do not require interparticle interactions), there must be an additional mechanism arising from the dipolar interactions between the particles in our uniform dense mixtures. It is unlikely that the dipolar interactions in our system, quantified by the temperature \({\text{T}}_{\text{d}\text{d}}= \frac{{{\mu }}_{0}}{4{\pi }{\text{k}}_{\text{B}}}{\text{M}}_{\text{S}}^{2}\text{V}{\phi } 50\text{K}\) (with \({\text{k}}_{\text{B}}\) the Boltzmann constant, \({M}_{S}\) the saturation magnetization, \(\text{V}\) the particle volume and \({\phi }\) the particle packing fraction) for either soft-soft, hard-hard and soft-hard combination49, may affect significantly the strong intraparticle exchange-coupling (and therefore the exchange bias) of the Co-doped particles, the latter having a much larger associated energy26. In other words, the ratio \({\text{T}}_{\text{e}\text{x}}/{\text{T}}_{\text{d}\text{d}}\) (where the temperature \({\text{T}}_{\text{e}\text{x}}\)is proportional to the exchange-coupling energy at the interface, \({E}_{ex}={H}_{EB}{V}_{FiM}{M}_{S}\) 2) is rather large for small particles such as those studied here. The exchange stiffness constant Aex for the corresponding ferrites (∼10−12)53 is two orders of magnitude larger than the dipolar stiffness Adip extracted from a random anisotropy model (∼10− 14) of the present systems35, confirming that the exchange energies involved are much higher54,55. Indeed, a bibliographic study presented in the SI (Supplementary Section 8) shows that \(\frac{{\text{T}}_{\text{e}\text{x}}}{{\text{T}}_{\text{d}\text{d}}}\gg 1\) in all the nanocomposites considered except those exhibiting very low exchange bias fields26,49,56,57.
Thus, it is more likely that the strong dipolar interactions between the two types of particles are mutually influencing the magnetic response of each population, and therefore the overall magnetization loop. In this way, the hard particles will “pin” or delay the switching of the softer particles, and vice versa, which in turn determine the coercivity and loop shift of each population. Since the reversal of the hard particles is strongly biased, the pinning effect in the reversal of the soft particles will be correspondingly biased, leading to a dipolar-induced loop shift in the soft particles as illustrated in Figs. 5 and 6. In Fig. 5a we analyse the largest horizontal separation between the experimental and superposition loops in the low field region (i.e., near the constriction caused by the soft particles reversal). Such distance is larger in the left than in the right branch, which suggests a biased response of the soft particles. Figure 5b plots the difference between the “left” and “right” separations as a function of the soft particle concentration, which decreases upon reducing the content of hard particles, thus lending support to the previous statement.
To further test the idea of a dipolar transfer of bias to the soft particles, the experimental hysteresis loops measured in the mixtures were fitted to a double Stearn-Cheng function comprising soft and hard components and allowing individual bias for both of them,
$${M}_{\pm }\left(H\right)=\frac{2}{\pi }{M}_{S,soft}\text{atan}\left[\right(\frac{H+{H}_{B,soft}\pm {H}_{C,soft}}{{H}_{C,soft}})\text{tan}(\frac{\pi {S}_{soft}}{2}\left)\right]+ \frac{2}{\pi }{M}_{S,hard}\text{atan}\left[\right(\frac{H+{H}_{B,hard}\pm {H}_{C,hard}}{{H}_{C,hard}})\text{tan}(\frac{\pi {S}_{hard, \pm }}{2}\left)\right]+\chi H$$
2
This model factors in the asymmetric reversal of the hard loop, thus decoupling the mathematical effect described above from a possible dipolar-induced bias effect. An example of such a fit is given in Supplementary Section 2. The HB values obtained for the hard and soft particles (\({H}_{B,hard}\) and \({H}_{B,soft}\), respectively) as a function of x confirm the mutual influence between the two particle populations (Fig. 6a). Focusing on the initially unbiased soft particles, the results indicate the appearance of a “dipolar bias” in their magnetization reversal which increases, as expected, with the concentration of hard particles. This trend is analogous to that plotted in Fig. 5b, hence establishing the appearance of “dipolar bias” in the magnetization reversal of the soft nanoparticles in the mixtures. Conversely, the soft particles represent a dragging force favouring the field-alignment of the hard particles moments, which leads to the reduction of both their coercivity (see Supplementary Section 2 and ref.35) and loop shift for soft particle concentration x > 50% (Fig. 6a).
This finding, which constitutes the foremost demonstration of “dipolar bias” in a nanoparticle system, prompted a Monte Carlo study to further emphasize the role of dipolar interactions, since these simulations allow a straightforward separation of the magnetization reversal of the two populations. The results show precisely the same trends in both the soft and hard particles as those observed experimentally (Fig. 6b, filled symbols). Importantly, the simulations show that the results are similar independently of the chosen parameters, indicative of a robust effect (see Supplementary Section 9). Moreover, in the Monte Carlo simulations, dipolar interactions can be deactivated ad hoc. As can be seen in Fig. 6b (empty symbols), when dipolar interactions between all particles are switched off (g1 = g2 = g12 = 0), the bias of the soft particles remains zero for all concentrations, while HB for the hard particles remains constant for all x, thus providing further confirmation of dipolar interactions as the origin of the bias induced in the soft particles derived above from the fitting of the experimental loops.
It is important to emphasize that the novel phenomena we describe in the binary mixtures do not depend on the origin of the exchange bias (or of the asymmetry) of the hard particles. They are robust mechanisms relying solely on the existence of asymmetry or (in the case of dipolar transfer of bias to the soft particles) exchange bias in the hard nanoparticles, whatever their origin. Additionally, note that while the asymmetry-induced bias is independent of the particle size and magnetization or the packing fraction, the dipolar bias will be strongly influenced by these parameters, since they govern the magnetostatic interactions between the particles.
To conclude, using binary dense nanocomposites has been shown to be instrumental to uncover two new sources of bias (i.e., hysteresis loop shift) different from the usual interface exchange-coupling. Dipolar interactions induce a hysteresis bias (labelled “dipolar bias”) in the originally unbiased component (pure soft γ-Fe2O3 nanoparticles) due to their coupling with their exchange-biased hard Co-doped γ-Fe2O3 neighbours, which act as pinning centres. Importantly, the uniform mixing of nanoparticles in dense composites has been shown to double the effective bias of the ensemble with respect to the simple addition (superposition) of the same components. The second source of bias in soft-hard mixtures, responsible for an unexpected increase in the binary system bias, results from the addition of soft particles (not necessarily interspersed) to hard particles with an asymmetric magnetization reversal. Remarkably, it has been shown that neither interparticle interactions nor individual exchange bias (in the constituent particles) are necessary for the creation of bias in such soft-hard binary nanocomposites; a certain reversal asymmetry in the hard loop is sufficient to induce this novel effect when combined with a soft component. Such asymmetry is typically observed in exchange-coupled nanoparticles, whose bias field may therefore be readily and significantly increased by adding a fraction of soft particles.