Spectral Analysis of Topological Defects in an Artificial Spin-Ice Lattice

Arrays of suitably patterned and arranged magnetic elements may display artificial spin-ice structures with topological defects in the magnetization, such as Dirac monopoles and Dirac strings. It is known that these defects strongly influence the quasistatic and equilibrium behavior of the spin-ice lattice. Here, we study the eigenmode dynamics of such defects in a square lattice consisting of stadiumlike thin film elements using micromagnetic simulations. We find that the topological defects display distinct signatures in the mode spectrum, providing a means to qualitatively and quantitatively analyze monopoles and strings that can be measured experimentally.

Figure 1

(a) Simulated singly charged monopole pair ($G^{+}{G}^{-}$) linked by a Dirac string (DS). Vortices $V$ form along the string. The reference state (RS) of the lattice is shown in pale blue, while the string and monopoles are highlighted. (b) Doubly charged monopoles can also be observed experimentally, although relatively rarely [5]. In this case, each vertex with a doubly charged monopole leads to the formation of an adjacent oppositely charged monopole (for example, $G^{-}$ forms next to a $G^{*+}$ vertex). The color map corresponds to the projection of the magnetization along the (11) direction of the string; the magnetic charge is taken as positive when the magnetization points out of the vertex.

Figure 2

(a) Simulated lattice composed of 112 magnetic elements. It contains four monopole-antimonopole pairs connected by Dirac strings extending over 28 elements. (b) Evolution of the magnetization dynamics spectrum with increasing string length and number of monopole-antimonopole pairs compared to the reference state. The shaded labels correspond to the main, distinct signatures of the topological defects. The amplitude units are identical in all cases.

Figure 3

Examples of local oscillations of the magnetization ($\Delta m$) characteristic for monopoles (a), Dirac strings (b), and doubly charged monopoles (c). The first column shows the static magnetic configuration. The second and third columns respectively show the oscillations of the $x$ and $y$ components of the magnetization. The plots show the change in amplitude of the spectral signature of the different defects relative to the reference state. The last row (d) illustrates the splitting of the ferromagnetic resonance peak in the presence of a Dirac string. Labels are from Fig. 2.

Figure 4

Evolution of the spectral features with decreasing element separation for an array containing one monopole-antimonopole pair connected by a string of 6 elements, compared to a single stadium. As the lattice constant of the array is reduced, the spectral features are blueshifted while the amplitude of the ferromagnetic resonance peak decreases. $G$: peak characteristic of the monopole pair oscillations. DS: peak characteristic for the Dirac string oscillations.