Broken vertex symmetry and finite zero-point entropy in the artificial square ice ground state

We study degeneracy and entropy in the ground state of artificial square ice. In theoretical models, individual nanomagnets are typically treated as single spins with only two degrees of freedom, leading to a twofold degenerate ground state with intensive entropy and thus no zero-point entropy. Here, we show that the internal degrees of freedom of the nanostructures can result, through edge bending of the magnetization and breaking of local magnetic symmetry at the vertices, in a transition to a highly degenerate ground state with finite zero-point entropy, similar to that of the pyrochlore spin ices. We find that these additional degrees of freedom have observable consequences in the resonant spectrum of the lattice, and predict the occurrence of edge “melting” above a critical temperature at which the magnetic symmetry is restored.

Figure 1

(a) Twofold degenerate macroscopic ground state configuration in artificial square ice. The arrows represent the overall orientation of the magnetization within each island. The circle indicates the vertex enlarged in the subsequent images. (b) Microscopic structure of the magnetization at the edges of 4-nm-thick Permalloy nanoislands, in a symmetric state (labeled $e$-0). The small arrows represent the computed local magnetization, while the large arrows serve as a guide to the eye. (c) and (d) show two possible degenerate configurations of the magnetization in strongly magnetostatically coupled islands (labeled $e$-I and $e$-II). While the magnetic configuration $e$-0 has two magnetic symmetry axes, ${\mathcal{A}}_1$ and ${\mathcal{A}}_2,e$-I and $e$-II have a single axis, thus lowering the vertex symmetry.

Figure 2

(a) Simulated (normalized) edge mode amplitudes for the $e$-I and $e$-II vertex configurations after applying an external field pulse along the $(1,1)$ direction. (b) The ground state (GS) spectrum (gray) for a lattice only made of $e$-I-type vertices displays a peak at 3 GHz. Introducing an increasing number of $e$-II vertices in the system leads to a decrease of the original mode intensity and to the evolution of a peak at 3.6 GHz corresponding to the mode associated to the $e$-II vertices. (c) Peak amplitude evolution of the two modes following pulses along the $(1,1)$ (red curves) and $(1,-1)$ directions (blue curves) as a function of an increasing number of introduced $e$-II vertices. (d) Ground state spectrum (gray) for a lattice with $e$-0-type symmetric vertices following a pulse along the $(1,1)$ direction. The spectrum is identical if the pulse is along the $(1,-1)$ direction (blue). The inset shows the normalized edge mode amplitude.

Figure 3

Energy barrier, $\mathrm{\Delta }E/k_B$, separating the $e$-I and $e$-II states as a function of island thickness, $h$, and lattice constant, $a$, in a square lattice with $470\mathrm{nm}\times 170$ nm islands. The intersection point of the two curves represents the parameters of the system used in Fig. 4. The change in energy barrier is studied for variations of $a$ and $h$ away from that point. Although the barrier is computed up to the Curie temperature, the validity of the used assumptions breaks down earlier, due to changes in the value of $M_s$.

Figure 4

(a) Snapshot of the magnetic configuration of a square ice vertex made of $470\mathrm{nm}\times 170\mathrm{nm}\times 10\mathrm{nm}$ islands at 400 K. Thermal excitation was introduced at $t=0$ when the vertex was in the $e$-I state. (b) At $t=50$ ns, the vertex is in the $e$-II state. On longer time scales, the time-averaged state restores the local symmetry, indicating that the edges have melted. (c) Thermal magnetization dynamics spectra averaged over 1000 ns for a vertex with broken symmetry below ($T=200$ K) and at the edge melting temperature ($T=400$ K). The spectrum shows a dramatic increase in low-frequency noise as the temperature is increased, indicating thermal transitions across the energy barrier. (d) Low-frequency data below 0.2 GHz at 400 K and $1/f^{\alpha }$ fit with $\alpha \sim 0.7$. (e) Spectra for a vertex in a symmetric ground state ($e$-0). These display no significant temperature-dependent low-frequency noise.