Artificial Square Ice and Related Dipolar Nanoarrays

We study a frustrated dipolar array recently manufactured lithographically by Wang et al. [Nature (London) 439, 303 (2006)] in order to realize the square ice model in an artificial structure. We discuss models for thermodynamics and dynamics of this system. We show that an ice regime can be stabilized by small changes in the array geometry; a different magnetic state, kagome ice, can similarly be constructed. At low temperatures, the square ice regime is terminated by a thermodynamic ordering transition, which can be chosen to be ferro- or antiferromagnetic. We show that the arrays do not fully equilibrate experimentally, and identify a likely dynamical bottleneck.

Figure 1

Left: Atomic force microscope image of an array studied in Ref. 3. The islands have length $l=220\mathrm{nm}$, width 80 nm and thickness 25 nm. Right: Map of the ratio $J_2/J_1$ of the second to the first nearest-neighbor interactions (highlighted in the left part) for different values of lattice constant, $a$, and sublattice height offset, $h$. In the white zone, $|J_2/J_1-1|<5\%$. In the left (blue) region, the ordered state is antiferromagnetic, whereas it is ferromagnetic in the right (yellow-red) area.

Figure 2

Left: Spectrum for artificial ice (cut off after $10a$, for pointlike dipoles). Right: same for $l/a=0.7$ and $h/a=0.207$: the lower band becomes almost flat, (less than 1.5% of the total bandwidth).

Figure 3

Ising-ice vs dipolar arrays: frequency of vertex types (% deviation from random distribution for single vertex, which is $\frac{1}{8}$, $\frac{1}{4}$, $\frac{1}{2}$, and $\frac{1}{8}$ for Types I–IV, respectively), entropy ($S$), and heat capacity ($C$) from Monte Carlo simulations; $x$ axes were scaled for high-$T$ asymptotics to coincide. The fraction of non-ice-rule vertices is below 1% for $T<0.42J$ for all depicted systems. At low $T$, the ice regime, which widens with increasing $l/a$, is terminated by ferro- (dashed) or antiferromagnetic (dots) order for different $h$.

Figure 4

Frequency of vertex types (as in Fig. 3). Interaction energies scale approximately as $1/a^3$, a factor of 30 between the extremal points. Experiments (symbols) are shown against dynamics simulations for needle dipoles [Eq. (1), dotted line], and for increased $J_2$ (dashed line, see text).