Magnetic charge and moment dynamics in artificial kagome spin ice

Spin ice materials represent an intriguing class of frustrated magnetic systems which, through their geometry, admit an exponential number of approximately degenerate configurations. In this paper, the relaxation properties of a thermally active artificial kagome spin ice system are studied. Through application of an external magnetic field, an out-of-equilibrium vertex charge ordered configuration is selected and relaxed under approximate zero-field conditions. Using x-ray photo-emission electron microscopy, the magnetic moment and vertex charge degrees of freedom are followed in space and time, revealing different dynamics to that seen in past athermal equilibration protocols, and a relaxation which is well described by a point-dipolar model system. Furthermore the charge correlations are found to relax with a time scale several times smaller than that of the moment correlations.

Figure 1

(a) Scanning electron microscope image of part of an artificial kagome spin ice consisting of nanomagnets with a length $L$ = 470 nm, width $W$ = 170 nm, and thickness $d$ = 3 nm arranged on a lattice with nearest-neighbor center-to-center distance $a$ = 500 nm. Due to their elongated shape and the small value of $d$, each nanomagnet is well described by an Ising moment, which points parallel or antiparallel to its long axis. Superimposed on the image are two nearest-neighbor triplets of Ising moments that adhere to the ice rule of two in(out) moments and one out(in) moments. The net local flux may be identified as magnetic vertex charges of either $Q=q$ or $-q$ (red or blue circles) existing on the parent honeycomb lattice. (b) Upper schematic: At high enough temperatures these vertex charges, like the underlying Ising moments, are spatially disordered—a phase referred to as spin ice I. Middle schematic: As the temperature is lowered, the vertex charges order and the spin ice II phase is formed. Lower schematic: At even lower temperatures, the long-range nature of the underlying dipolar interaction gives rise to ordered moments, resulting in a long-range ordered phase where a partial tiling of microvortices defines the zero-temperature ground state.

Figure 2

(a–c) XMCD snapshots of artificial kagome spin ice starting from an initial out-of-equilibrium spin ice II phase, which then relaxes towards the equilibrium spin ice I phase. The insets show the corresponding smeared charge-density maps indicating the creation of emergent magnetic monopoles (see caption of Fig. 3), which remain confined due to a strict obedience to the local ice rule at each vertex. In these insets, the red and blue points reflect charge defects that can be seen as emergent magnetic monopoles connected by Dirac strings consisting of chains of magnetic moments connecting the charge defect pair (black lines). (d–f) The corresponding $Q=+q/-q$ red and blue vertex charge configurations, which show how the initially global charge order of the prepared state decays into regions of local charge order characteristic of spin ice I. The two shades of gray refer to the two possible local (central vertex plus nearest neighbors) charge order configurations. (g–j) XMCD image sequence highlighting a typical example of thermally induced Dirac-string evolution in artificial kagome spin ice under a strict obedience to the local ice rule.

Figure 3

(a–f) Thermal behavior of emergent magnetic monopoles in artificial kagome spin ice during relaxation to the spin ice I phase. XMCD images (upper panels), obtained as a function of time at constant temperature $T=420$ K, are shown together with smeared charge-density maps (lower panels) as well as the emergent monopole/antimonopole pairs ($\mathrm{\Delta }Q=+2q,-2q,0$ indicated, respectively, by red, blue, and white dots), where the back lines indicate the connecting Dirac string. It is seen that emergent monopole defects give rise to a nonzero smeared charge density, as in the case of athermal field protocols (see Ref. [10]). However, in contrast to field-driven experiments, each nucleated emergent monopole/antimonopole pair remains confined and immobile due to strict ice rule compliance.

Figure 4

Three local moment configurations differing by a single moment reversal. The vertex charges are given by the colored balls where blue corresponds to a vertex charge of $-q$, red corresponds to a vertex charge of $+q$, and black corresponds to a vertex charge of $-3q$. Due to the moment reorientation in going from (a) to (b), there is an exchange of vertex charges and the creation of an emergent monopole/antimonopole pair with charges $\mathrm{\Delta }Q=\pm 2q$. This involves a total change in energy equal to 0.045 eV. The small value reflects the fact that the configuration remains within the ice sector. The moment reorientation associated with going from (b) to (c) does, however, violate the ice rule, giving a much larger total change in energy equal to 0.302 eV.

Figure 5

Experimentally obtained moment relaxation at 420 K, characterized by instantaneous ($\tau =0$) course-grained correlation functions [Eq. (3)] for the (a) $\alpha \beta$, (b) $\alpha \gamma$, (c) $\alpha \delta$, and (d) $\alpha \nu$ neighbor shells [see Fig. 1], plotted as a function of time $t$. At $t=0$ the artificial kagome spin ice is in the out-of-equilibrium moment configuration where all moments are aligned in the direction of the applied saturating field shown in Figs. 2 and 2 (corresponding to the initial out-of-equilibrium charge ordered state). In all figures, the corresponding connected curves are four KMC simulated relaxation realizations. The dashed lines correspond to the equilibrium values expected for the spin ice I phase derived by Wills et al. [7] and also the KMC asymptotes obtained by extending the KMC simulations to up to 2000 min.

Figure 6

Experimentally observed vertex charge relaxation characterized by instantaneous ($\tau =0$) correlation functions [Eq. (2)] plotted as a function of time ($t$), starting from the initial out-of-equilibrium charge ordered state shown in Figs. 2 and 2. The connected curves of the same color are four corresponding KMC simulated relaxation realizations. The dashed line associated with the nearest-neighbor vertex charge correlation indicates the expected asymptotic behavior of one-third reflecting local charge neutrality.

Figure 7

Experimentally obtained (a) magnetic moment and (b) vertex charge correlations [Eqs. (1) and (2)] plotted as a function of time, $\tau$, with $t=0$, starting from the initial out-of-equilibrium charge ordered state (at $\tau =0$) shown in Figs. 2 and 2. In both (a) and (b), the connected curves are four corresponding KMC simulated relaxation realizations.