Due to shape anisotropy, an elongated magnet with submi crometer dimensions (bar nanomagnet) is in a single-domain state with a bistable remnant magnetization pointing along its long axis. These single-domain bar nanomagnets have been exten sively used in a wide range of fundamental and applied meso scopic magnetic systems. For example, coupled bar nanomagnets were used to design majority logic gates for low dissipation digital computation in magnetic quantum-dot cellular automata systems. In artificial spin ices, the bistable magnetization of these bar nano magnets behaves like macro-Ising spins.
Specially arranged interacting single-domain bar magnets in artificial spin ices enable the investigation of geometric frustration, emergent magnetic monopoles, and phase transitions in a material-by-design approach. By coupling with the other functional materials, such as superconductors, these reconhgurable arrays of bar nanomag nets were used as reprogrammable magnetic potential landscapes to tailor the electronic properties of a hybrid device. They were also used to tailor spin wave transformation for reconfigurable magnonic applications.
Recently, thermally active bar nano magnets in artificial spin ice were adopted to demonstrate both deterministic and probabilistic computation. More recently, uti lizing single-domain bar nanomagnets with different magnetization reversal switching fields, a reprogrammable shape-morphing micro machine was designed, which showed great advances in controllabil ity and functionality. In of the above, the magnetization reversal switching field of the bistable remnant magnetization is a crucial parameter for tuning the physical properties and functionalities of the assembled artificial systems. Therefore, the ability to modulate the magnetization switching field of single-domain bar nanomag nets would enable new routes for enhanced control and creation of new functionalities in artificial hybrid systems.
The magnetization reversal process of a nanomagnet is strongly affected by its magnetic anisotropy, which not only depends on the Fermi surface structure of the material but also relies on the geometric shape of the magnet. Adjusting the aspect ratio of a magnet, such as tuning its length, width, and/or thickness, is a widely used method to control the magnetization reversal switching field, as recently demonstrated in magnetic shape-morphing microbots. Here, using micromagnetic simulations, we show the existence of an upper limit in the length of a bar nanomagnet, beyond which the magnetization reversal switching field cannot be tuned. This greatly limits the tuning of magnetization reversal by engineering aspect ratios, especially for a long nanomagnet. We circumvent this limitation by introducing a simple method to tailor the magnetiza tion reversal process of a single-domain bar nanomagnet by shaping its end geometry. Our work provides an easily accessible method to tune the properties of mesoscopic magnetic systems based on single-domain bar nanomagnets.
Micromagnetic simulations of the magnetization reversal pro cesses on a bar nanomagnet were carried out using MuMax3.24 The material parameters of Permalloy (NigoFezo) were used in our investigation because it is a soft magnetic alloy with very large permeability, and more importantly, it is one of the most widely used materials in related research. The material parameters were set as standard values widely used for Permalloy: the saturation magnetization (Ms) is 8.6 x 105 A/m, and the damping constant is 0.5 in order to quickly minimize the energy. The exchange stiffness (A) and the crystalline anisotropy constant (K) were 13 x 10~12 J/m and 0 j/m2, respectively. The simulation cell size was 2.5 x 2.5 x 2.5 nm3.
We first investigated the magnetization reversal process by varying the aspect ratios of a standard, stadium-shaped nanomagnet with a semicircular geometry at both ends [ :ig. l(a ]. The remanent magnetization spontaneously forms along the longitudinal direction of the bar under zero external magnetic field. The magnetic switch ing field Hs, i.e., the minimal magnetic field required to reverse the remanent magnetization, is determined from the sudden change in the magnetization hysteresis loop [ ig. l(d ]. The hysteresis loop is calculated by sweeping the magnetic field along the long-axis of the bar nanomagnet. The switching field Hs decreases when the width is increased from 25 to 100 nm [ ig. 2(b ] and increases when the thickness is changed from 5 to 30nm [Fig. 2(i ]. Fur ther tuning the switching field by width and/or thickness is not experimentally favorable as fabricating nano-bars with a width less than 25 nm requires very expensive ultrahigh resolution electronbeam lithography tools and a wider nanomagnet would transform from a single-domain state to a vortex state. Furthermore, very thin nanomagnets have a very low total magnetic moment, which is not favored in the application of magnetic microbots. In addi tion, very thin nanomagnets produce weak magnetic stray fields, thereby reducing the interactions between the nanomagnets and other proximal functional materials in hybrid devices. Fabricating nanomagnets using a thicker layer is challenging for the lift-off pro cess. Moreover, a thicker sample changes the in-plane anisotropy of Permalloy into an out-of-plane anisotropy, creating a complex domain and/or vortex structure in the bar nanomagnet (see Fig. SI). Due to all of the above reasons, tuning magnetization reversal by tailoring the nanomagnets' width and thickness has a variety of limitations.
Modulating the length of a bar nanomagnet with a moderate width and thickness seems to be an experimentally favorable way to control its magnetization reversal. To investigate the magnetiza tion reversal's dependence on the nanomagnet's length, we fixed the nanomagnet's thickness and width to experimentally favorable val ues of 25 and 80 nm, respectively, and varied its length L between 160 and 1000 nm. A shorter nanomagnet (L < 160 nm) does not maintain the single-domain state, and a vortex state emerges dur ing the magnetization reversal process, as demonstrated in Video 1. We show the length dependence of the switching fields in Fig. 1(e). Figure l(d displays the hysteresis loops of the bar nanomagnets with several selected lengths. One can see that the magnetization switch ing field increases when the length is increased from 160 to 480 nm. However, the switching field Hs stays unchanged at lengths beyond 480 nm [ ig. l(e ]. This significantly limits the modulation of mag netization reversal of a single-domain bar nanomagnet by aspect ratios. Therefore, an additional method is highly desired to further tune the magnetization reversal process, especially for a very long magnet.
The saturation of the switching field Hs for a nanomagnet with a length beyond 480 nm suggests that the magnetization rever sal process is not only controlled by the aspect ratios. Previous theoretical and micromagnetic simulation studies showed the anisotropy of the end of an elongated magnetic nanostructure plays a crucial role. We plot the microscopic magnetic structures of the nanomagnets immediately before their reversal, as shown in the insets of :ig. 1(d). Video 2 shows the reversal processes of several magnets with various lengths. The nucleation of the magnetization reversal begins at the two ends of the nanomagnet, and the magne tization at the central part of the nano-bar reverses instantly, follow ing the magnetization reversal at the two ends. This indicates the magnetization reversal of the entire nano-bar is intimately domi nated by its two ends. Previous investigations also showed that the Therefore, further tuning of the magnetization switching field of a bar nanomagnet could be realized by engineering the geometric shape of its two ends.
To demonstrate this approach, we investigated a simple nano bar with semi-elliptical shaped ends [ Tg. 2(a ]. We fixed the vertical axis of the semi-ellipse to the width of the nano-bar and varied the length of the horizontal axis, d, as shown in Fig. 2(a). The width and thickness of the nano-bar are 80 and 25 nm, respectively, while the total length was maintained at 800 nm, which is in the saturated magnetization reversal range L > 480 nm [ ig. 1(e)]. This allows us to examine the effects from the end geometries on the entire bar by adjusting the horizontal axis, d, of the ellipse. As shown in "ig. 2(b), hysteresis loops with different switching fields are obtained for bars with different axis lengths for the semi-ellipse. In Fig. 2(c), we plot the switching field Hs as a function of the axis length d. Increasing d from 0 to 80 nm reduces the switching field Hs to a minimum value of about 83.5 mT. On increasing d further, Hs increases rapidly up to a maximum value of 222.5 mT at d = 320 nm. Video 3 demonstrates the evolution of the magnetization reversal dynamics with different d values. We also simulate the d dependence of the switching field for nano-bars with various lengths ranging from 300 to 1000 nm (Fig. S2 of the supplementary materia ). All the results indicate the nano-bar with semi-circle ends has the smallest switching field and increasing the semi-ellipse axis length d enhances the switching field more effectively than changing only the aspect ratio, as shown in Fig. 1.
The above-mentioned results clearly indicate that the end geometry of a single-domain bar nanomagnet plays a critical role in the magnetization reversal process. While all the above-mentioned investigations were performed on symmetric end geometries, it would be interesting to examine the case for asymmetrical end geometries. Subsequently, we carried out simulations for nanomag nets with elliptical axis lengths of d\ and c/2 for its two ends, respec tively, as illustrated in the inset of Pig. 3(a). The overall size of the magnet bar was maintained at 800 x 80 x 25 nm3. In Fig. 3(a), we plot the switching field Hs as a function of both axis lengths rfi and c/2. Figure 3(b) displays the c/i dependent switching field for several selected c/2 values. The results clearly show that the switch ing field can be effectively tuned by modifying the geometry of one end. Comparing the result of ig. 3(b with that from a symmetric nanomagnet displayed in :ig. 2(c), we find the switching field of a nanomagnet with two different ends is determined by the end with the lower switching field. The inset of ;ig. 3(t displays the micro scopic magnetic structure right before the switching for the nano-bar with asymmetric ends. The corresponding magnetization reversal process can be found in Video 4. We can clearly see that the nucleation of the magnetization reversal is initiated at the end with the lower switching field. These simulation results indicate that effective and significant modulation of the magnetization reversal field can be realized by tuning the geometry of just one end of a single-domain bar nanomagnet.
In a real system consisting of bar nanomagnets, the applied external magnetic field may not always be directed along the long axes of all the bars. In all the aforementioned mesoscopic mag netic systems, such as in magnetic majority logic gates, artifi cial spin ices, and/or magnetic shape-morphing micromachines, the nanomagnets are patterned in various different orientations. Therefore, it is necessary to investigate the end geometry depen dence of the magnetization reversal process of a single-domain nanomagnet under magnetic fields applied in different orientations.
Figure 4(a demonstrates the switching field as a function of the field angle 8h and the axis lengths of the semi-ellipses at the ends. The size of the magnet bar was also maintained at 800 x 80 x 25 nm3.
The result shows that the end geometry introduces a notable effect on the field-angle dependency of the magnetization reversal. We plot the field angle dependence of the switching fields for several selected d values in Tg. 4(b). For the nano-bar with large values of d > 80 nm, e.g., at d = 200 nm, the switching field decreases rapidly with increasing 8h first, reaching a minimum at 8h ~ 45', and then becoming larger at a high field angle 8h [green curve in Fig. 4(b ].
That is, the field angle for the easiest switching (minimal switch ing field) is around 8h = 45'. This result is consistent with the AIP Advances ARTICLE scitation.org/journal/adv end geometries of a bar nanomagnet, the magnetization reversal switching field can be tuned over a wide field range. This method is more effective than tuning magnetization reversal by aspect ratios, especially for very long nano-bars in which the switching field is insensitive to the bar length beyond a certain span. We note that the "effective" end-geometry of the nano-bars is not limited to semi ellipse shapes and that other geometries, such as triangular shapes (as demonstrated in Fig. S3), can also modulate the magnetization reversal. The ability to control the magnetization reversal of a single domain bar magnet could lead to complex magnetization reversal behavior in coupled nanomagnet systems. Fabricating arrays of nanomagnets with tailored magnetization reversal switching fields would enable modulation of phase transition temperatures and/or magnetic fields in artificial spin ices, which could greatly affect the access to ground state configurations as well as realizing novel excited low energy states. Creating artificial spin ices using nano magnets with asymmetric end geometries may lead to novel phe nomena by introducing a local reversal barrier with designed defects. Nanomagnets with controllable magnetization reversals could also be used to modify phase coexistence and domain wall frustra tion in artificial magnetic systems. Our method would significantly simplify the design of magnetic shape-morphing micromachines, in which nanomagnets with different switching fields are required. Engineered end-geometries of nanomagnets could also be used to control the operation process in magnetic majority logic gates, as well as to manipulate the tuning parameters of the current-driven magnetic logic devices. Stoner-Wohlfarth model for a single-domain ferromagnet. The field angle for the minimal switching field decreases with the axis length d and approaches ~20 when the end geometry becomes a semi-circle (d = 80 nm). When reducing the axis length of the semi ellipses at the two ends further, the angle dependent switching curves become more complex with a peak emerging at small angles (8h < 45'), as shown in Fig. 4(b), which might be due to the competi tion between the shape anisotropy of the end segments and that of the center segment of the nano-bar. The difference in the switch ing fields for different d is only shown at small field angles (roughly <45 ). The switching field curves under large field angles 6h > 45 are nearly coincident. This indicates that the magnetization reversal process of the center segment of the nano-bar dominates the endsegments when the magnetic field is tilted away from the long-axis of the bar.