Exchange coupling effects in Co-Pt nanochessboards

Exchange-coupled nanocomposites consisting of a hard magnetic phase of high magnetic anisotropy and a soft magnetic phase of high saturation magnetization can potentially exhibit both large coercivity and high remanence magnetization,^1–3 providing an effective means to the development of improved permanent magnets with a high energy product. Since the remanence and the coercivity are extrinsic properties that sensitively depend on the composite microstructure, a key issue in developing such permanent magnets is microstructure optimization. Regular dispersion of the two phases in a fine microstructure is a necessary condition to effect particle-level exchange coupling between the hard and soft phase regions.^2,4 Coherent interfaces are desirable to aid exchange coupling across the interfaces between two magnetic phases of generally different crystal structures,¹ which are also necessary to facilitate the formation of regular and fine two-phase microstructures during solid-state phase transformations through long-range elastic interactions.⁵ Artificial compositing⁶ or processing methods⁷ that create phase separation on the nanoscale can be used to produce a crystallographically coherent and finely dispersed two-phase bulk system. As an interesting example, A1 → L1₀ + L1₂ eutectoid transformation in the binary Co-Pt system has been shown to produce a spectacular bulk nanocomposite with a quasi-periodic nanochessboard structure,⁸ which consists of aligned nanorods of the magnetically hard tetragonal L1₀ phase and the magnetically soft cubic L1₂ phase. The fascinating Co-Pt nanochessboard has fully coherent interfaces and is composed of regularly distributed nanorods that alternate between L1₂ and L1₀ phases, thus providing an excellent platform for the investigation of exchange coupling effects in magnetic nanocomposites relevant to permanent magnet applications.

A recent study⁹ on the nanochessboard structures in Co-Pt systems demonstrates that the periodicity (i.e., the characteristic dimension) can be controlled by varying the cooling rate of the bulk cast specimens through eutectoid A1 → L1₀ + L1₂. It shows that a reduction in the characteristic dimension to approximately 10–20 nm results in an increase in the coercivity and energy product, which is expected from an enhanced particle-level exchange coupling as a result of the reduced particle size. For this reduced characteristic dimension of 10–20 nm, the first-order reversal curve (FORC) diagram exhibits one peak confirming a single phase-like magnetic behavior due to the enhanced exchange coupling effects. These results agree with the theoretical findings by Kneller and Hawig¹ and by Leineweber and Kronmuller¹⁰ on exchange-coupled nanocomposites where magnetic hardening is achieved when the characteristic dimension is reduced below the critical correlation length. It raises natural questions whether further reduction of the characteristic structural dimension would provide additional magnetic hardening and how the strong exchange coupling impacts the domain processes in the nanochessboard structure. Furthermore, the microscopic magnetization states and magnetization reversal mechanisms in this system, which features competing easy-axis variants and interactions between shape and magnetocrystalline anisotropy, are not understood. To answer these questions, a micromagnetic simulation study of Co-Pt nanochessboards is performed and the simulation results are analyzed in this paper.

The simulation study shows that, below the critical correlation length, the two-phase composite system behaves like a single-phase magnet and exhibits peculiar magnetic anisotropy that is different from the magnetocrystalline anisotropy of the constituent phases. Magnetic hardening is observed, as expected, when the characteristic dimension decreases. However, upon further decrease in the characteristic dimension, an unexpected magnetic softening is observed. This unusual length scale dependency of the coercivity distinguishes the magnetic nanochessboards from other typical exchange-coupled nanocomposites explained by Kneller-Hawig theory. To understand the special magnetic behaviors of the nanochessboards, the detailed domain structures and evolution paths are analyzed. A simple analytical model is developed to further explain the simulation results and provide a clearer understanding of the magnetic behaviors, including how reducing the characteristic dimension of an exchange-coupled nanostructure can actually weaken its coercivity. The findings delineate the mechanisms responsible for the peculiar magnetic behaviors of exchange-coupled nanochessboard structures and provide insights into achieving novel magnetic properties in a variety of self-assembled regular nanostructures.

Micromagnetic modeling of two-phase magnetic composites⁹ is employed to simulate magnetic domain processes in nanochessboard structures of different tile sizes. In the model, the two-phase Co-Pt nanochessboard structure is characterized by a static phase field variable θ(r), which assumes the value 1 in the tetragonal L1₀ phase and 0 in the cubic L1₂ phase. The magnetic domain structure in the nanochessboard is described by a magnetization vector field M(r), which is a function of the temporally evolving directional unit vector field m(r) and saturation magnetization M_s(r) that depends on the two-phase microstructure characterized by θ(r),

where $MSL10$ and $MSL12$ are the saturation magnetization of the L1₀ phase and the L1₂ phase, respectively. The time evolution of the magnetic domain structure is described by the Landau-Lifshitz-Gilbert equation¹¹ 

where γ is the gyromagnetic ratio, α is the damping parameter, and H^eff is the effective magnetic field. The driving force H^eff is determined by the variational derivative of the system free energy with respect to the magnetization vector field, $Heff=−1/μ0(δF/δM)$⁠, where μ₀ is the permeability of a vacuum. The system free energy of the two-phase magnetic Co-Pt system is a sum of magnetocrystalline anisotropy energy, exchange energy, magnetostatic energy, and external magnetic energy (Zeeman energy)¹² 

where A is the exchange stiffness constant, H^ext the external magnetic field, $⨍$ the principal-value integral excluding the point k = 0, $M̃(k)=∫M(r)e−ik⋅rd3r$⁠, and $n=k/k$⁠.⁵ The magnetocrystalline anisotropy energy density $fan$ is a function of the magnetization direction and the underlying two-phase microstructure variable θ(r),

where $K1L10$ and $K1L12$ are, respectively, the magnetocrystalline anisotropy constants of the L1₀ phase and the L1₂ phase, and t(r) is the local orientation of tetragonal axis of the L1₀ phase which alternates between [100] and [010] in neighboring L1₀ tiles, as shown in Fig. 1(a). The material parameters used in the simulations are:⁹ $MSL10$ = 4.2× 10⁵ A/m, $K1L10$ = 1.5 × 10⁶ J/m³, $MSL12$ = 5.2 × 10⁵ A/m, $K1L12$ = 2 × 10⁴ J/m³, and A = 2.5 × 10⁻¹¹J/m (a constant exchange stiffness is assumed across L1₀ and L1₂ phases).

Figure 1(a) shows the 2D projected configuration of a nanochessboard oriented along the [001] axis of the A1 parent phase. The black and white color tiles indicate the L1₀ and L1₂ phases, respectively, that are actually nanorods extended in the out-of-plane direction. As such, the nanochessboard is a 2 + 1-D structure, with the 2-D in-plane heterostructure of the chessboard pattern extended uniformly in its out-of-plane direction. In the real Co-Pt bulk polycrystalline samples,^8,9 each parent A1 grain has chessboard colonies along all three ⟨100⟩ directions, with a typical rod length of hundreds of nanometers (much greater than the tile size of tens of nanometers). Since the out-of-plane dimension would not affect particle-level exchange coupling, it is the in-plane characteristic dimension (i.e., typical tile size, d) that determines the degree of particle-level exchange coupling. Using the above material parameters, the critical correlation length is estimated as $dcr∼2πAsoft/2Khard=$ 18 nm,¹ below which strong particle-level exchange coupling effects are expected. This paper investigates the exchange coupling effects within individual nanochessboard colonies by considering the ideal nanochessboard colony in Fig. 1(a). It should be noted that, without taking into account the magnetic interactions among nanochessboard colonies and grains which are beyond the scope of this paper, the simulated hysteresis curves cannot be directly compared with the experimentally measured hysteresis curves in Co-Pt bulk polycrystals. The simulation results rather reveal elementary mechanisms for the magnetic behaviors of the nanochessboard, and help explain the experimental measurements. Given the ideal nanochessboard structure in Fig. 1(a), Eq. (2) is numerically solved under a periodic boundary condition in the nanochessboard plane. Due to its special 2 + 1-D microstructure feature, the structural and magnetization variations in the out-of-plane direction would not significantly affect the magnetic exchange coupling phenomena in the chessboard; thus, we assume uniformity in both the microstructure and the magnetization distribution along the nanorods. This greatly reduces the computational burden of the micromagnetic simulations: it simplifies the quasi-3-D structure to a 2-D ideal nanochessboard with distributions of 3-D magnetization vectors and magnetic fields. However, the simulations do not capture domain walls in the out-of-plane direction. In the simulations, 256 × 256 computational cells are used, three tile sizes d = 7.2 nm (case A), 12.6 nm (case B), and 18.0 nm (case C) up to $dcr$ are considered, and an external magnetic field is applied along three different directions, namely, [100] and [110] in the nanochessboard plane and [001] along the nanorod direction, as shown in Fig. 1.

Consider the magnetic behaviors of cases A and B of tile sizes d = 7.2 nm and 12.6 nm, respectively, both of which are below the critical length. Figures 1(b) and 1(c) show the simulated hysteresis curves along three typical crystallographic directions of [100], [110], and [001] (the hysteresis curves for case B with d = 12.6 nm are reproduced from previous work⁹). Comparing the hysteresis curves among the three directions in cases A and B reveals that the chessboard structures exhibit magnetic anisotropy with a [110] (and $[1¯10]$⁠) easy axis (corresponding to the square hysteresis loops in green color) and a [001] hard axis (corresponding to the hysteresis-free curves in blue color). Having [001] as the hard axis is easy to understand since the easy directions of all L1₀ tiles lie in the plane, and the out-of-plane [001] direction is a common hard axis for all L1₀ tiles. The [001] magnetization curves are indeed reversible and anhysteretic with zero coercivity, featuring the typical magnetic response in the magnetic hard direction of a single-phase single-crystalline ferromagnetic material. It is observed that case A exhibits a linear M-H response along [001] up to the high field, while case B exhibits a nonlinear response in [001] at the high field. This difference is caused by the different characteristic dimensions of the nanochessboards. The smaller tile size in case A leads to a stronger exchange coupling effect producing a more uniform magnetization across neighboring tiles, and thus the nanochessboard behaves in a manner of single-domain magnetization rotation yielding the linear response. In contrast, the larger tile size in case B allows nonuniform magnetization across neighboring tiles resulting in a deviation from the single-domain rotation behavior. An increasing degree of magnetization uniformity in the nanochessboard with decreasing tile size can be seen by comparing the simulated domain structures B1 and A1 in Figs. 1(e) and 1(d) as the tile size is reduced from d = 12.6 nm in case B to d = 7.2 nm in case A. Comparing case A and case B, the changes in the in-plane hysteresis loops along [100] and [110] directions (red and green lines, respectively) are more notable than in the out-of-plane M-H curve (in blue lines), as seen in Figs. 1(b) and 1(c). In particular, the coercivity in both in-plane directions is significantly reduced, contrary to the expectation of higher coercivity for a smaller tile size according to Kneller-Hawig theory¹ and the experimental finding (which did not explore smaller characteristic size).⁹ To understand this unusual length scale dependence of the magnetization hysteretic behaviors, the magnetic domain processes are analyzed in the following.

In the chessboard structure, the magnetically hard L1₀ tiles (bridged by the in-between magnetically soft L1₂ tiles) alternate easy directions between two perpendicular [100] and [010] in-plane directions, as shown in Fig. 1(a). Assisted by the in-between bridging L1₂ tiles, the neighboring L1₀ tiles appear exchange-coupled when the tile size is so small that the exchange coupling effect extends over them. We refer to it as hard-hard exchange coupling between neighboring L1₀–L1₀ tiles as opposed to hard-soft coupling between adjacent L1₀–L1₂ tiles. The hard-hard exchange coupling would rotate the local magnetization vectors away from [100] or [010] magnetic easy direction in individual L1₀ tiles toward the intermediate [110] direction, as shown in the simulated domain structures A1 and B1 in Figs. 1(d) and 1(e). As a result, the hard-hard exchange coupling weakens the original individual [100] or [010] easy direction, while promotes the new effective [110] (⁠$or [1¯10]$⁠) easy direction in the nanocomposite system. Therefore, when the effect of the exchange coupling becomes stronger as the tile size decreases, weakening of the [100] (and [010]) easy direction while strengthening of the [110] easy direction are expected. Indeed, the [100] hysteresis loop of case A becomes narrower than that of case B (red lines) as shown in Figs. 1(b) and 1(c), manifesting a weakened [100] magnetic easy direction for the reduced tile size d = 7.2 nm compared to d = 12.6. However, contrary to the expectation of an enhanced [110] easy direction, the [110] hysteresis loop of case A becomes narrower than that of case B (green lines), indicating a weakened anisotropy in the effective [110] easy direction. To clarify this puzzling dependence of anisotropy in the [110] direction on the tile size, a different tile size d = 18 nm (case C) is also considered in our simulations, and the results are shown in Fig. 2. Comparing the three cases of A, B and C shown, respectively, in Figs. 1(b), 1(c) and 2(a), the hysteresis loop width (i.e., the coercivity) exhibits an expected monotonic increase with increasing tile size in the [100] direction, while the coercivity in the [110] direction increases from case A to case B and then decreases to case C. This implies that, under a magnetic field in the [110] effective magnetic easy direction, two different magnetization reversal mechanisms operate, respectively, above and below 12.6 nm tile size. Moreover, a change from the anhysteretic behavior of cases A and B to a notable hysteretic behavior in case C in the [001] magnetic hard direction is also observed, which indicates a change in the out-of-plane magnetization switching process when the tile size increases to the critical length d_cr, the onset of weakening in exchange coupling. To understand this interesting dependence of the magnetic behaviors in the effective easy direction [110] and the hard direction [001] of the nanocomposites on the characteristic microstructural dimension, i.e., the tile size, the details of the domain structures and evolution paths are examined further.

Consider first the effects of the tile size on the domain structure in the remanence states A1, B1 and C1 in Figs. 1(d), 1(e) and 2(b), by comparing their magnetization distributions in a representative unit of four neighboring L1₀ tiles surrounding one L1₂ tile. Due to the exchange coupling, the magnetization vectors in individual L1₀ tiles deviate from the underlying crystallographic easy directions illustrated in Fig. 3(a) and orient toward the intermediate [110] direction as shown in Fig. 3(b). The magnetization vector directions in the two L1₀ variants are represented, respectively, by m1 and m2 in Fig. 3(b) for the convenience of discussion. The deviation angle θ would depend on the tile size. As the tile size decreases, the particle-level exchange coupling effect increases, leading to a greater deviation angle θ and a smaller misalignment angle ϕ between m1 and m2, which correspondingly increases the degree of homogeneity in the magnetization distribution over the neighboring tiles. Based on the simulated magnetization distributions, the deviation angle θ (and misalignment angle ϕ) is determined from the magnetization vectors at the centers of the two neighboring L1₀ tiles: θ = 35.2° (ϕ = 19.6°) for case A in Fig. 1(d), θ = 21.4° (ϕ = 47.2°) for case B in Fig. 1(e), and θ = 11.8° (ϕ = 66.4°) for case C in 2(b). As such, the magnetization vectors deviate from their underlying magnetic easy directions due to the exchange coupling, and the deviation angle θ (misalignment angle ϕ) increases (decreases) as the tile size decreases because of the stronger exchange coupling effect at a smaller tile size.

Consider next the length scale effects on the domain switching processes under a reversal field in the [110] direction. Depending on the strength of exchange coupling, domain switching would follow different paths. As the tile size increases, four regimes of exchange coupling behavior are expected in the order of decreasing exchange coupling strength, namely strong hard-soft-hard, strong hard-soft, weak hard-soft, and decoupled, where the transition from the strong hard-soft to weak hard-soft coupled regime occurs near the critical length $dcr$⁠. Two different domain switching paths can be predicted below the critical length $dcr$ = 18 nm. In the strong hard-soft-hard coupled regime, m1 and m2 are bound and rotate together under the applied field, named Path I in Fig. 3(c). Otherwise, in the strong hard-soft coupled regime, the external field drives m1 and m2 to rotate separately in opposite directions, named Path II in Fig. 3(c). The simulation results do confirm both of these domains switching paths: Path I in the case A of d = 7.2 nm and case B of d = 12.6 nm, while Path II in the case C of d = 18 nm, as exemplified in Fig. 4. The domain structures in Fig. 4 correspond to A2, B2 and C2 points on the respective magnetization curves during domain switching for d = 7.2 nm in Fig. 1(b), d = 12.6 nm in Fig. 1(c), and d = 18 nm in Fig. 2(a). It is worth noting that when the tile size increases over the critical length $dcr$ = 18 nm (i.e., in the weak hard-soft coupled regime and in the decoupled regime), due to a weakened exchange coupling effect, the shape anisotropy of nanorods that dictates the [001] easy axis in the L1₂ phase becomes significant and the nanochessboard forms more complex domain structures and evolution paths involve out-of-plane domains. Although a detailed analysis in these two regimes is beyond the scope of this paper, the effects of shape anisotropy are discussed below. To better understand the length scale-dependent exchange coupling effects on the domain evolution path and the resultant coercivity in the nanochessboard of characteristic dimensions below the critical length, a simplified model is developed in the following to evaluate the misalignment angle ϕ and the switching field (coercivity) as a function of the tile size.

Consider a chessboard unit enclosed in the yellow dashed square in Fig. 3(b). The magnetization distribution in the unit is represented by m1 and m2 which, assuming that they stay in-plane, can be described by the misalignment angle ϕ and the rotating angle φ of the angle bisector in Fig. 3(c): $m1=cosφ−ϕ2+π4,sinφ−ϕ2+π4,0$ and $m2=cosφ+ϕ2+π4,sinφ+ϕ2+π4,0$⁠. Then, the free energy in Eq. (3) can be approximated in terms of ϕ and φ with the following simplifications. Firstly, the magnetocrystalline anisotropy energy is estimated based on the deviation of m1 and m2 vectors from the underlying magnetic easy directions in the L1₀ phase regions which occupy about one half of the unit volume. Secondly, the exchange energy is estimated using the change in the magnetization direction from m1 to m2 over the distance between two neighboring L1₀ tiles along both [100] and [010] directions (approximately $2d$⁠). Thirdly, the magnetostatic interaction energy is neglected, which is largely accommodated by the soft magnetic L1₂ phase. With these approximations and considering the reversal magnetic field $H=H2−1,−1,0$⁠, the free energy density becomes

where $Ms=(MSL10+MSL12)/2$ = 4.7 × 10⁵ A/m is the average saturation magnetization. In this analytical model, the free energy is simplified to a function of the misalignment angle ϕ, the rotating angle φ of the angle bisector, and the reversal magnetic field magnitude H. It allows us to solve the evolution of m1 and m2 as a function of H by determining the equilibrium values of ϕ and φ that minimize the free energy density in Eq. (5) at a given H value. The critical switching field H_c can be calculated accordingly, which is the stability limit as H increases. This simplified model is applied to three specific cases below.

The first case considers the equilibrium state under zero external field. Given H = 0 and the rotating angle φ = 0 correspondingly, the free energy density is a function of the misalignment angle ϕ. The equilibrium misalignment angle ϕ₀ is determined by minimizing the free energy density in Eq. (5): $df/dϕ=−14K1L10cosϕ+2Ad2sinϕ=0$⁠. This yields $ϕ0=arctanK1L10d28A$ and the equilibrium deviation angle $θ0=π4−ϕ02$ accordingly, which is plotted in Fig. 5(a). Compared with the micromagnetic simulation results for the three specific tile sizes (d = 7.2 nm, 12.6 nm and 18 nm in Figs. 1 and 2) shown by the three square dots in Fig. 5(a), the simplified model actually offers a very good approximation, accurately predicting the reduction in the misalignment angle as the tile size decreases, thus the strengthening of the particle-level exchange coupling effect.

The second case considers magnetic switching under an increasing reversal magnetic field $H=H2−1,−1,0$ along Path I, which is expected for the superstrong exchange coupling effect enabled by smaller tile sizes. Here, an in-plane rigid-body rotation of m1 and m2 is assumed, as illustrated in Fig. 3(c), where the angle $ϕ$ between m1 and m2 is set to the equilibrium misalignment angle $ϕ0$⁠. The magnetic domain evolution is then described merely by the rotating angle φ, and the free energy density becomes a function of φ and H. Using the free energy density in Eq. (5), the critical switching field $Hcr$ is obtained by the two conditions of stability limit: $∂f/∂φ=12K1L10sin2φsinϕ0−μ0MsHsinφcosϕ02=0$ and $∂2f/∂2φ=K1L10cos2φsinϕ0−μ0MsHcosφcosϕ02=0$⁠. The solution $Hcr=2K1L10μ0Mssinϕ02$ is plotted as the blue curve in Fig. 5(b). It predicts that the coercivity decreases as the tile size decreases, and the reason for this behavior is because the superstrong exchange coupling effect binds m1 and m2 together, and this binding enables their easy rigid-body rotation against their respective magnetocrystalline anisotropy energy.

The third case considers Path II that is expected for the relatively weaker exchange coupling effect as the tile size increases toward the critical length under the same reversal magnetic field $H=H2−1,−1,0$⁠. Here, separate rotations of m1 and m2 with an average rotating angle φ = 0 is assumed, as illustrated in Fig. 3(c). The magnetic domain evolution is now described by the misalignment angle $ϕ$⁠, the free energy density in Eq. (5) becomes a function of $ϕ$ and H, and the critical switching field $Hcr$ is determined by $∂f/∂ϕ=−14K1L10cosϕ+2Ad2sinϕ−12μ0MsHsinϕ2=0$ and $∂2f/∂2ϕ=14K1L10sinϕ+2Ad2cosϕ−14μ0MsHcosϕ2=0$⁠. As an analytical solution is unavailable, the numerical solution is obtained and plotted as the red curve in Fig. 5(b). Contrary to Path I, it predicts that the coercivity increases as the tile size decreases. The reason is that the exchange coupling effect suppresses the splitting of m1 and m2 driven by the reversal magnetic field.

The two curves (blue for Path I and red for Path II) in Fig. 5(b) together predict the non-monotonic dependence of coercivity on the tile size. It is the lower segments of the curves that give the coercivity and the magnetization path at a given tile size. As the tile size decreases from the critical length, the coercivity first increases because the domain evolution follows Path II; while the tile size is reduced below about 12 nm, the coercivity decreases because the domain evolution follows Path I. The transition between these two paths is determined by the strength of the exchange coupling effect, which in turn depends on the tile size. Since a smaller tile size is required to achieve sufficient strength of exchange coupling to bind m1 and m2 together during rotation, the magnetizations rotate along Path II at a larger tile size and along Path I at a smaller tile size, and consequently, the coercivity first increases and then decreases with decreasing tile size. To compare with the micromagnetic simulation results, the simulated coercivities in the [110] direction for the tile sizes of d = 7.2 nm, 12.6 nm and 18 nm are also plotted by three square dots in Fig. 5(b), which shows a non-monotonic dependence on tile size. It is noted that, for analytical tractability, the magnetization processes are significantly simplified for Path I and Path II, including neglecting variations of $ϕ$ in Path I and out-of-plane magnetization in Path II during the switching process, which are both observed in the simulations. Since any relaxation from these simplifying assumptions would provide an additional degree of freedom for magnetization processes, the simulated coercivity is expected to be lower than the analytical value, as confirmed in Fig. 5(b). Nevertheless, the simplified model for domain switching along Path I and Path II yields good agreement with the simulation results, and, most importantly, provides insights into the mechanism for the non-monotonic dependence of coercivity on the tile size.

Finally, consider the appearance of hysteresis in the magnetic hard direction [001] in case C in Fig. 2(a), which is associated with the weakening of the exchange coupling effect as the tile size approaches the critical length $dcr$ = 18 nm. When the exchange coupling effect weakens, the shape anisotropy of the nanorods starts to play a role, which dictates a magnetic easy direction along the rod axis [001] in the L1₂ phase. It is therefore expected that perpendicular domains can form in the L1₂ phase regions against the exchange coupling effect from neighboring L1₀ phase regions where the magnetizations stay in-plane to accommodate the underlying magnetic [100] and [010] easy directions. The simulated magnetic domain structure of the remanence state C3 in Fig. 2(a) is shown in Fig. 6(b), which confirms the existence of perpendicular domains in some L1₂ tiles. Such perpendicular domains are absent in the remanence state B3 in Fig. 1(c), as shown in Fig. 6(a). The shape anisotropy not only accounts for the hysteresis in the [001] direction, but also would contribute to the magnetization processes in other directions. As an example, during the domain switching under the [110] field, the significant involvement of out-of-plane magnetization in entire L1₂ tiles and portion of L1₀ tiles near the phase boundaries (as a result of exchange coupling) is observed in Fig. 4(c), which eases the energy cost of the domain switching, thus lowering the coercivity.

It is noted that there are many aspects of the real Co-Pt chessboard samples that have not been dealt with explicitly here. These include the macroscopic polycrystalline nature of the samples, the presence of three chessboard colonies in each grain (along each of the three ⟨100⟩ directions in the parent A1 phase), and dispersion in the size and the shape of tiles. Hence, the idealized simulations here are not expected to predict the actual coercivities in these samples, and will, in general, overestimate the coercivity. These simulations do, however, elucidate the detailed microscopic mechanisms associated with nanoscale exchange coupling in the chessboard, and the dependence on the length scale.

In summary, the effects of strong exchange coupling on the magnetic behaviors of Co-Pt nanochessboards are investigated using micromagnetic simulations, and a simplified analytical model is developed to provide a better understanding of the simulation results. The simulations reveal details of the magnetic domain structures and evolution processes in the nanochessboard microstructures of different characteristic dimensions and uncover an unusual non-monotonic length scale dependence of the coercivity, which is distinct from other exchange-coupled magnetic nanocomposites. The length scale-dependent domain structures and evolution paths are analyzed, and their impacts on the effective magnetic anisotropy and coercivity of the nanochessboard are discussed. It shows that due to the strong exchange coupling effect, the nanochessboard exhibits an effective magnetic anisotropy of the [110] easy direction different from the magnetocrystalline [100] and [010] easy directions of the hard L1₀ phase regions. It also shows that the nanorod shape anisotropy starts to play a role when the characteristic dimension increases toward the critical length, leading to the formation of perpendicular domains in the soft L1₂ phase regions and also the involvement of out-of-plane magnetization in the hard L1₀ phase regions during domain switching processes. Importantly, it reveals a change in the domain switching pathway that is responsible for the non-monotonic length scale dependence of the coercivity on the tile size, leading to magnetic hardening first followed by magnetic softening when the characteristic dimension of the nanochessboard structure is reduced below the critical length. This offers a guidance for experiments to explore other approaches rather than a further reduction of length scale to achieve higher coercivity in Co-Pt nanochessboards. The findings provide not only some new understanding of the special magnetic behaviors of exchange-coupled nanochessboard structures, but also insights into achieving novel magnetic properties in a variety of regular self-assembled nanostructures.

Funding from the NSF under Grant Nos. DMR-1409317 (L.D.G. and Y.M.J.) and DMR-1105336 (J.A.F. and W.A.S.) is gratefully acknowledged. The computer simulations were performed on XSEDE supercomputers.