In this chapter, we investigate the spin moiré engineering in two dimensions. Among many parameters in spin moirés, we specifically focus on the phase degree of freedom in the skyrmion lattice (SkL) composed of three waves. The phase degree of freedom plays a prominent role when the number of superposed waves is larger than the spatial dimension; this is the case for the 3Q-SkLs in two dimensions. Using the hyperspace representation developed in Sect. 4.6, we systematically investigate the effect of phase shifts on two types of SkLs composed of three helices and three sinusoidal waves. We clarify the topological phase diagram while changing the phases and the uniform magnetization, distinguished by the skyrmion number \(N_\mathrm{{sk}}\) ranging from –2 to 2. In the superposition of three helices, we find that the SkLs with \(N_\mathrm{{sk}}={\pm } 1\) are dominant, and the SkLs with higher topological numbers \(N_\mathrm{{sk}}={\pm } 2\) appear with nonzero magnetization. In contrast, in the superposition of three sinusoidal waves, the SkLs with \(N_\mathrm{{sk}}={\pm } 2\) are dominant and \(N_\mathrm{{sk}}={\pm } 1\) phases appear with nonzero magnetization. In addition, by analyzing the numerical data obtained in the previous study, we show that the phase shift is induced by the magnetic field, resulting in a topological transition with a change in \(N_\mathrm{{sk}}\) from 2 to 1.

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Spin Moiré Engineering in Two Dimensions

  • Kotaro Shimizu

摘要

In this chapter, we investigate the spin moiré engineering in two dimensions. Among many parameters in spin moirés, we specifically focus on the phase degree of freedom in the skyrmion lattice (SkL) composed of three waves. The phase degree of freedom plays a prominent role when the number of superposed waves is larger than the spatial dimension; this is the case for the 3Q-SkLs in two dimensions. Using the hyperspace representation developed in Sect. 4.6, we systematically investigate the effect of phase shifts on two types of SkLs composed of three helices and three sinusoidal waves. We clarify the topological phase diagram while changing the phases and the uniform magnetization, distinguished by the skyrmion number \(N_\mathrm{{sk}}\) ranging from –2 to 2. In the superposition of three helices, we find that the SkLs with \(N_\mathrm{{sk}}={\pm } 1\) are dominant, and the SkLs with higher topological numbers \(N_\mathrm{{sk}}={\pm } 2\) appear with nonzero magnetization. In contrast, in the superposition of three sinusoidal waves, the SkLs with \(N_\mathrm{{sk}}={\pm } 2\) are dominant and \(N_\mathrm{{sk}}={\pm } 1\) phases appear with nonzero magnetization. In addition, by analyzing the numerical data obtained in the previous study, we show that the phase shift is induced by the magnetic field, resulting in a topological transition with a change in \(N_\mathrm{{sk}}\) from 2 to 1.