Configurable Artificial Spin Ice with Site-Specific Local Magnetic Fields

We demonstrate ground state tunability for a hybrid artificial spin ice composed of Fe nanomagnets which are subject to site-specific exchange-bias fields, applied in integer multiples of the lattice along one sublattice of the classic square artificial spin ice. By varying this period, three distinct magnetic textures are identified: a striped ferromagnetic phase; an antiferromagnetic phase attainable through an external field protocol alone; and an unconventional ground state with magnetically charged pairs embedded in an antiferromagnetic matrix. Monte Carlo simulations support the results of field protocols and demonstrate that the pinning tunes relaxation timescales and their critical behavior.

Figure 1

(a) SEM images of (i) ASI ($a=400\mathrm{nm}$), (ii) $h_{2a}\text{−}\mathrm{ASI}$, and (iii) $h_{3a}\text{−}\mathrm{ASI}$. The exchange-biased sites are highlighted, and its direction indicated by arrows. The device edge is aligned to the direction of $H_{\mathrm{EB}}$ which is along Fe $⟨100⟩$ crystallographic axis. (b) Schematic cross section of $x$ sublattice and the local field due to $H_{\mathrm{EB}}$ as a function of position in $h_{2a}\text{−}\mathrm{ASI}$. (c) Hysteresis loop of Fe (7 nm) film with IrMn layer of different thicknesses measured using longitudinal magneto-optic Kerr effect. 8 nm of IrMn imparts an $H_{\mathrm{EB}}$ of 80 Oe. $H_{\mathrm{EB}}$ is almost nonexistent when IrMn layer is thinned down to around 1–2 nm by ion milling.

Figure 2

(a) The integrated correlation time (${\tau }_{\mathrm{int}}$) shows a peak in each case at the critical temperature; however, this is greatly suppressed by the presence of a nearby pinning site. The inset is a schematic showing three classes of nearest neighbors in ASI: first nearest neighbors (1NN), second nearest neighbors located in the same vertex ($2\mathrm{NN}\text{−}L$), and second nearest neighbors which span adjacent vertices ($2\mathrm{NN}\text{−}T$). Central spin is shown in red. (b) The variance of the $s_i$ ($t$) as a function of distance from a pinned spin. The dashed line denotes variance in case of no pinning. In both cases, the variance is evaluated at the critical temperature. (c) Four vertex types in square ASI with different spin topologies. (d) Two (i) and (ii) degenerate GSs/T1 phase domains of square ASI. We present these two phase domains using square and round symbols throughout.

Figure 3

(a) Vertex map and (b) population statistics of demagnetized $h_{pa}\text{−}\mathrm{ASI}$ ($p=0$ and 2). Vertices $T1$, $T2$, $T3$, and $T4$ are color coded in red, blue, green, and yellow, respectively. Arrows indicate direction of the net moment of a vertex and the pinned spins are indicated by black-white blocks. In (b) filled and open symbols represent results of experiments and simulations, respectively. (c) Schematic of one of the antiferromagnet order in the $x$ sublattice of $h_{2a}\text{−}\mathrm{ASI}$ with pinned sites highlighted in yellow. The bipartite lattices (i) and (ii) of the antiferromagnet order are indicated. All the pinned sites appear only in bipartite lattice (i). The bipartite lattice (ii) has no pinning centers. (d) Schematic shows the two $T1$ orderings (i) and (ii). Exchange bias at the site highlighted prefers (i) over (ii). The $T1$ domain (ii) with moment antiparallel to exchange bias (red arrow) has higher energy.

Figure 4

(a) The vertex maps of field demagnetized $h_{pa}\text{−}\mathrm{ASI}$ with odd $p$ values ($=1$, 3). The zigzag phase in $h_{1a}\text{−}\mathrm{ASI}$ is highlighted in cyan (b) vertex statistics of $h_{pa}\text{−}\mathrm{ASI}$ with odd $p$ values. Filled and open symbols represent results of experiments and simulations, respectively. (c) Schematic of one of the antiferromagnet order in the $x$ sublattice of $h_{3a}\text{−}\mathrm{ASI}$ with pinned sites highlighted in yellow. The bipartite lattices (i) and (ii) of the antiferromagnet order are indicated. Pinned sites appear in both Néel bipartite sublattices (i) and (ii), as shown. The bipartite sublattice (i) has all the spins aligned parallel to exchange bias but the (ii) has spins antiparallel to exchange bias (red arrows). Thus, a simple antiferromagnetic Néel ordering cannot simultaneously minimize both exchange-bias energy and dipolar energy in case of $h_{3a}\text{−}\mathrm{ASI}$.

Figure 5

(a) Ground states (GS1 and GS2, with pinning sites indicated) vertex maps of $h_{3a}\text{−}\mathrm{ASI}$ obtained in Monte Carlo simulations. (b) Heat capacities for no pinning $p=0$ and $p=1$, 2, 3. Even (odd) pinning raises (lowers) the critical temperature.