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Skyrmion crystals in frustrated Shastry-Sutherland magnets

The phase diagrams of the frustrated classical spin model with Dzyaloshinskii-Moriya (DM) interaction on the Shastry-Sutherland (S-S) lattice are studied by means of Monte Carlo simulation. For ferromagnetic next-nearest-neighboring (J2) interactions, the introduced exchange frustration enhances the effect of the DM interaction, which enlarges the magnetic field-range with the skyrmion lattice phase and increases the skyrmion density. For antiferromagnetic J2 interactions, the so-called 2q phase (two-sublattice skyrmion crystal) and the spin-flop phase are observed in the simulated phase diagram, and their stabilizations are closely dependent on the DM interaction and J2 interaction, respectively. The simulated results are qualitatively explained from the energy landscape, which provides useful information for understanding the intriguing phases in S-S magnets.


I. Introduction
A skyrmion is a topological defect with a vortex-like spin structure where the spins point in all directions wrapping a sphere. [1,2] During the past a few years, skyrmions have drawn extensive attention for their potential applications in future memory technology, especially considering their outstanding merits such as the nanoscale sizes, [3,4] the topological protection, and the ultralow critical drive-currents [5] . Specifically, skyrmions have been discovered experimentally in chiral magnets including MnSi [1,6] , MnGe [7] and FeGe [8] , and have been proven theoretically to be stabilized by the competition among the ferromagnetic (FM) exchange, Dzyaloshinskii-Moriya (DM) [9,10] , and Zeeman couplings in the presence of thermal fluctuations. Subsequently, several effective methods for manipulating skyrmions using spinpolarized current [11,12] , magnetic field gradients [13][14][15] , microwave field [16] , and temperature gradient [17][18][19][20] have been revealed experimentally and/or theoretically.
Furthermore, strong interfacial DM interactions can be induced at heavy metal (HM)/FM interfaces in films such as Pt/Co/MgO [21] heterostructure and Pt/Co/Ta [22] thin film due to broken inversion symmetry and strong spin-orbit coupling, contributing to the stabilization of Néel-type magnetic skyrmions even at room temperature (T). More recently, in the IrMn/CoFeB/MgO heterostructure, it has been reported that the exchange bias at the antiferromagnetic (AFM)/FM interface can stabilize the skyrmions even in the absence of external magnetic field (B). [23] More interestingly, the DM interaction at the interface can be significantly enhanced by a factor of 7 through increasing the IrMn thickness, which is very meaningful for spintronic applications.
In addition, several earlier works on finding new magnetic systems with skyrmions have been reported. For example, several frustrated magnets such as the triangular antiferromagnets have been predicted to host skyrmion lattice phase. Instead of DM interaction, the competition between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions is essential for the stabilization of the skyrmions, as discussed in detail in the earlier works. [24][25][26][27][28][29][30] Most recently, the first observation of skyrmionic magnetic bubbles formed at room temperature has been reported experimentally in the frustrated kagome Fe3Sn2 magnet, [31] well confirming the earlier predictions.
It is well known that finding materials with skyrmions stabilized in a wide B range is essential both in basic physical research and for potential applications. In some extent, frustrated magnets such as Shastry-Sutherland (S-S) magnets may be promising candidates. On the one hand, there are real materials such as rare-earth tetraborides RB4 (R = Tb, Dy, Ho, Tm, etc.) are famous representatives of S-S magnets. In these materials, DM interactions could be induced by spin-orbit coupling between rare-earth ions and/or by structural inversion asymmetry through S-S magnet film design, and may play an important role in stabilizing noncollinear phases. For example, in the S-S Kondo lattice model, the DM interactions have been suggested to be necessary for the emergence of the chiral spin structures. [32] Thus, skyrmion lattice phase could be expected to be stabilized in S-S magnets with DM interactions. On the other hand, the exchange frustration has been proven to be important in understanding the intriguing magnetization behaviors observed in these materials. [33,34] More importantly, it is strongly expected that the DM effect could be enhanced by the effective reduction of the neighboring spin interactions due to the exchange frustration, and in turn further stabilizes skyrmions. As a matter of fact, similar phenomenon has been observed in the frustrated spin model on the square lattice, and the enlargement of the B range with the skyrmions by considering the interaction frustration has been reported. [35] Thus, there is still an urgent need to elucidate complex magnetic orders in frustrated S-S magnets with DM interactions, although the multi-step magnetization behaviors in these materials have been extensively investigated in recent years. [36][37][38] In this work, we study phase diagrams of a frustrated spin model with DM interaction on the Shastry-Sutherland lattice. For FM NNN interaction (J2), the introduced exchange frustration enhances the effect of the DM interaction, leading to an enlargement of the magnetic field-range with the skyrmion lattice phase and an increase of the skyrmion density. For AFM J2 interaction, the 2q phase (two-sublattice skyrmion crystal) and the spin-flop phase are suggested to be stabilized by the DM interaction and J2 interaction, respectively. The simulated results are qualitatively explained from the energy landscape.

II. Model and method
In the presence of additional DM interaction, the Hamiltonian is given by The first term is the exchange interactions where j = i + δn and δn connects site i and its n-th nearest neighbor sites with n = 1, 2, 3. Here, Jn are the exchange couplings and Si represents the Heisenberg spin with unit length on site i. The second term describes the interfacial DM interaction arises from structural inversion asymmetry along thin-film normal direction with the DM vectors Dij are shown in Fig. 1. The last term is the Zeeman coupling with B applied along the z direction.
Our simulation is performed on an N = 24 × 24 S-S lattice with period boundary conditions using the standard Metropolis algorithm and the parallel tempering algorithm. [39] We take an exchange sampling after every 10 standard Monte Carlo (MC) steps. Generally, the initial 5 × 10 5 MC steps are discarded for equilibrium consideration and another 5 × 10 5 MC steps are retained for statistical averaging of the simulation. To explore the phases in the system, besides the well-known magnetization M and magnetic susceptibility χ, we also characterize the spin structures by performing the Fourier transform where ri is the spatial coordinate of site i, and then calculating the intensity profile |Ak| 2 . In addition, we also calculate the topological winding number Qsk defined by to characterize the number of skyrmions in the system.  interpenetrating spirals on sub-lattice A and B are with a same ordering wave vector, as clearly demonstrated in the intensity of the spin structure factor (insert in Fig. 5(a)). When B is increased to a critical value, the phase transition from the AFM-spiral phase to the 2q phase is occurred. In the 2q phase of which typical configuration under B = 1.8 is given in Fig. 5 The simulated phase diagram can be well understood from the energy landscape. Fig. 6(a) gives the calculated B-dependence of H1, the J2 coupling energy H2, H3, the DM interaction energy HDM, and Hzee for J2 = 0.25. The corresponding magnetization and susceptibility curves are also presented in Fig. 6