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Crystallizing Kagome artificial spin ice

Artificial spin ices are engineered arrays of dipolarly coupled nanobar magnets. They enable direct investigations of fascinating collective phenomena from their diverse microstates. However, experimental access to ground states in the geometrically frustrated systems has proven difficult, limiting studies and applications of novel properties and functionalities from the low energy states. Here, we introduce a convenient approach to control the competing diploar interactions between the neighboring nanomagnets, allowing us to tailor the vertex degeneracy of the ground states. We achieve this by tuning the length of selected nanobar magnets in the spin ice lattice. We demonstrate the effectiveness of our method by realizing multiple low energy microstates in a Kagome artificial spin ice, particularly the hardly accessible long range ordered ground state - the spin crystal state. Our strategy can be directly applied to other artificial spin systems to achieve exotic phases and explore new emergent collective behaviors.

shown that direct experimental access of the spin crystal states of a Kagome ASI is challenging [12,31]. With the exception of the magnetic writing approach [31], no LRO spin crystal state has been unambiguously visualized in a Kagome ASI consisting of fully disconnected nanobar magnets, hindering investigations of emergent phenomena and phase transitions from its low energy manifolds.
The collective properties of ASIs are directly associated with their lattice geometries, and are intimately governed by the competing dipolar interactions between their constituent elements, i.e., the single-domain nanobar magnets [16,[32][33][34]. The difficulty of obtaining the LRO state of a Kagome ASI originates from the extensive degeneracy of the ground state, resulting from the high frustration of the tri-leg vertices. The ground state of a Kagome ASI exhibits neither charge nor spin order when only nearest-neighbor interactions are considered [21,35]. Further-neighbor interactions induce charge and/or spin ordering [13,21,28]. However, these longer-distance interactions are much weaker than that from the nearest-neighbor. As a result, the properties of artificial spin ices are mostly dominated by the nearest-neighbor interactions. Recently, utilizing the micromagnetic nature of connected vertices in the honeycomb structure of nanowire networks, the LRO spin crystal phase of Kagome ices was realized for the first time, in which notches were introduced to reduce the vertex degeneracy [36]. More recently, asymmetric bridges were introduced to break the six-fold symmetry of Kagome ASI's vertices, leading to a direct real-space imaging of the phase transitions [37]. However, these strategies are not applicable to Kagome ASIs consisting of disconnected nanobar magnets. In this letter, we develop a new method to tailor the vertex degeneracy and the ground states of a Kagome ASI of disconnected magnetic nanobars, and we directly present the phase transition from the SL1 liquid to LRO crystal state. Unlike the recently realized connected Kagome ASI structure [36,37], in which the coupling is dominated by short range exchange interaction, our disconnected ASIs maintain their long range dipolar interaction, therefore, allowing us to directly evaluate the critical role of the nearest and/or next nearest neighbor interactions, which is crucial for understanding the phase transitions between the various low energy manifolds of ASIs.
The low degeneracy of 2 in a square ASI is induced by the non-equal interactions between the nanobar magnets at each vertex. Following this notion, we developed a method to achieve the LRO spin ice state of a regular Kagome ice by inducing non-equal interactions between the three nanobar magnets at a vertex, which reduces the ground state degeneracy of a Kagome vertex from 6 to 2. As shown in Fig. 1(a), we increase the length of one of the three nanobar magnets (α) in each vertex, while maintaining the length of the other two magnets (β) and the lattice constant (see  [45]). This breaks the three-fold rotation symmetry of the vertex. The interactions between the three nanobar magnets at each vertex are no longer equivalent. As shown in Fig. 1(b), the original six-fold degenerate vertices are divided into two groups of Type K-I and K-II configurations with different energies. We denote the interaction energy of the frustrated magnet pair between two β nanomagnets in Type K-I vertices as J1, while that between α and β 5 nanomagnets in Type K-II vertices as J2 [ Fig. 1(b)]. Since each vertex satisfying the Kagome ice rule contains only one frustrated magnet pair, J1 and J2 also represent the energies of the Type K-I and K-II vertices, respectively. Because the endpoints of the lengthened α nanomagnet are closer to the vertex center, J1 is lower than J2, resulting in the two-fold degenerate Type K-I vertices to be the ground state configuration. As shown in Fig. 1(a), when all the vertices are satisfied to be in the Type K-I ground state configuration, a LRO spin crystal emerges. We can further tailor the energy difference between Type K-I and K-II vertices by varying the length (Lα) of the α nanomagnet. Figure 1(c) presents the vertex energy evolution of Type K-I and K-II vertices as a function of Lα, calculated from micromagnetic simulation using Mumax3 [46]. It shows that J2 increases with Lα while J1 remains constant. Thus, the energy difference J2-J1 between Type K-I and K-II vertices increases with Lα. Therefore, varying the length of the α nanomagnets allows us to regulate the effective temperature, similar to the effect of tuning the vertex notch in the connected honeycomb structures [36]. This enables us to tailor the phases in a fully disconnected Kagome ASI.
To experimentally validate this approach, we fabricated Kagome ASIs with Permalloy nanomagnets and with a series of Lα values (220 nm, 270 nm, 320 nm, 370 nm, 420 nm and 440 nm) for the α nanomagnets (see SEM images in Fig. S2 of Supplemental Material [45]). The length (Lβ) of the β magnets is fixed at 220 nm, and the lattice constant is a = 640 nm for all samples [ Fig.   1(a)]. The width and thickness of all the nanomagnets are 80 nm and 15 nm, respectively. Details of the sample fabrication process and parameters can be found in Supplemental Material [45]. A demagnetization procedure (see Methods) lasting 72 hours was performed to obtain the low-energy states [11,47].  We can further elucidate the spin and charge ordering using magnetic structure factors [47]. We have shown that the local interactions of an ASI have significant impact on their collective behavior and ultimately affect the properties of the entire system. The length of selective nanobar magnets can be used as a convenient knob to tune the local coupling strength, which enables us to access various low-energy manifolds and phase transitions in a fully disconnected ASI. We proved that although the next-nearest neighbor interaction plays a notable role through the magnetic charge structure factor maps, the crystallization of the Kagome ice structure with reduced vertex degeneracies is dominated only by the nearest neighbor interaction. This method could be used to manipulate the frustration in ASIs for attaining even more exotic ground state phases (see Figs. S8 of Supplemental Material [45]). It would also allow us to realize new magnetic configurations that are hard to access with magnetization method, e.g., to design novel ASIs with composite ground states in which different low energy states co-exist in the same sample, as illustrated by the hybrid ground state of both liquid and crystal in Fig. S9 of Supplemental Material [45]. This would allow us to investigate phase transitions between these new types of low energy states. Furthermore, kinetics becomes topologically protected in the SL2 phase and in the LRO state [48], and our structural modifications can be employed to fine-tune different kinetic regimes in thermal realizations or under field inversions. In addition, this method is also applicable to other types of artificial spin ices, leading to new opportunities to explore more exotic collective phenomena, such as novel phases and phase transitions. It could also be combined with the other structure 10 modification strategies, such as lattice transformation [49,50], to design new ASIs with controllable degeneracies. This method, maintaining the dipolar coupling between disconnected nanomagnets, could result in different spin dynamic properties and magnonic applications [51,52] than those in connected systems [36,37]. Moreover, our approach avoids complex analysis of the micromagnetic structures such as the domains/domain-walls in the vertices of connected systems, and thus offers a simpler Kagome model which can connect with a variety of systems outside of magnetism, such as metal organic frameworks [53] and mechanical metamaterials [54, 55]. Last but not least, the observed Bragg peaks' splitting induced by spin scattering between domains could lead to an alternative way to investigate domain/domain wall formation in ASIs.

Demagnetization protocol.
The sample was mounted on a motor rotating at 2000 RMP. An oscillating in-plane magnetic field (sine wave with period of 40 s) was applied to the sample. The amplitude of the oscillating magnetic field decreased slowly from 1000 Oe (well above the saturation field of our nanomagnets) to zero field in 72 hours.

Micromagnetic simulations.
The micromagnetic simulations were performed using Mumax3 [46], with the following material parameters for permalloy: the exchange constant of 1.3×10 -11 J/m, the saturation magnetization of 8.6 × 10 5 A/m, and the Gilbert damping of 0.01. The mesh size is 2 × 2 × 2 nm 3 . We consider a given pair of neighboring nanobar magnets. The ground state configuration is 'one-in, one out' or 'head to tail' configuration, and the frustrated high energy configuration is 'both in' or 'both out' configuration. We extract the energies of the ground state configuration and the frustrated high energy configuration from micromagnetic simulations. The interaction energy (J1 or J2) of the magnet pair is obtained by subtracting the frustrated high energy from the ground state energy. In this case, the ground state energy of the entire pair of two neighboring magnets is zero.

Monte Carlo simulated thermal annealing to the ground state.
Our Monte Carlo simulations were performed by using a single spin flip algorithm on 30×30×3 Kagome lattice sites with free boundary conditions. In the simulation, the Hamiltonian is defined as Magnetic spin structure factor.
The magnetic spin structure factor was calculated by following reference [47]. We define a perpendicular spin component S ⊥ as: q  is the unit scattering vector: q q q     So for every vector q = (qx, qy), the intensity I(q) scattered at location q in reciprocal space is written as: I is the magnetic spin structure factor that we calculate for the interval (qx, qy) = [-5.9π,-5.9π]−[5.9π, 5.9π] in 513×513 steps.
Magnetic charge structure factor.
The magnetic charge structure factor is calculated in a similar way as the spin structure factor. We define the direction of magnetic charge to be in the Z direction, perpendicular to the sample plane. For each vertex, we consider the total charge C  at the vertex center.
For every vector q = (qx, qy) the intensity I(q) is given by where N stands for the number of charges.
In order to simplify the calculation, the equation is divided into two parts, each containing site i and j respectively, and is rewritten as The above equation can also be written as )) sin( ) cos( ( )) sin( ) cos( ( 1 ) ( Then we obtain the equation

Comparisons to other recent bicomponent/hybrid ASIs.
Recently, bicomponent square ASIs constructed with nanobar magnets of either different materials [44] or different shapes [25,43] were introduced for reconfigurable magnonic applications. Although we also utilize bicomponent elements (nanobar magnets with different lengths) in the Kagome ASI, our tactic differs fundamentally from those reported.
There, the ground state configurations and multiple metastable configurations of square ASIs are realized through well-designed magnetization processes, which utilize the distinct magnetization properties, such as different magnetization reversal fields of different types of nanobar magnets [25,43,44]. The interactions between nanomagnets are not vital in those processes, therefore tuning the ASIs' configurations using polarizing magnetic fields does not directly reveal their collective behavior. In contrast, the interaction between nanomagnets in our modified Kagome ASIs is crucial for achieving their collective low energy manifolds. Varying the length and the arrangements of the selected nanobar magnets allows us to tailor their interaction and/or coupling strength, enabling us to obtain distinct ground states, e.g., the two exotic ground states in Fig. S8 and a hybrid ground state with both liquid and crystal states in Fig. S9.
In addition, recently site-specific exchange-bias fields were introduced to tune the ground state in a hybrid square ASI by introducing local pinning of nanomagnets through exchange bias effect [42]. In that work the magnetization state of selected nanomagnets were fixed or pinned by exchange bias. These fixed nanomagnets behave like controllable defects, therefore creating a new energy landscape for the rest of the nanomagnets. The ground state of the square ASI was tuned by introducing a designed pinning landscape. In our work, we modify the length of selected nanomagnets, which does not pin or fix any magnets. That is, we tailor the ground state configuration by tuning the interactions between the nanomagnets and not by producing a fixed new energy pinning landscape. On the other hand, as mentioned in Ref [42], high temperatures can destroy the interfacial exchange bias, and therefore the method of introducing pinning with exchange bias cannot be used to investigate the thermal properties, e.g., the sample cannot go through thermal annealing.
Although in our experiments we realize the ground state using a demagnetization procedure, our sample in principle can be used to explore the thermal properties at high temperatures.