Field-Induced Magnetic Monopole Plasma in Artificial Spin Ice

Artificial spin ices (ASIs) are interacting arrays of lithographically defined nanomagnets in which novel, frustrated magnetic phases can be intentionally designed. A key emergent description of fundamental excitations in ASIs is that of magnetic monopoles—mobile quasiparticles that carry an effective magnetic charge. Here, we demonstrate that the archetypal square ASI lattice can host, in specific regions of its magnetic phase diagram, plasmalike regimes containing a high density of mobile magnetic monopoles. These regimes result from the magnetic field-tunable tension on the Dirac strings connecting mobile monopoles. By passively “listening” to spontaneous monopole noise under conditions of strict thermal equilibrium, we reveal their intrinsic dynamics and show that monopole kinetics are most diffusive (that is, minimally correlated) in the plasma regime. These results open the door to on-demand monopole regimes having continuously field-tunable densities and dynamic properties, thereby providing a new paradigm for probing the physics of effective magnetic charges in synthetic matter.

Popular Summary

Magnetic monopoles are notional elementary particles that possess a magnetic “charge,” analogous to the electric charge carried by electrons. While the existence of elementary magnetic monopoles remains hypothetical, certain natural “spin-ice” materials exhibit excitations at low temperature that can be described as monopolelike quasiparticles. Monopolelike excitations can also occur at room temperature in engineered arrays of nanomagnets known as artificial spin ices (ASIs). We demonstrate that the archetypal square ASI lattice hosts, at certain magnetic fields, a high density of mobile magnetic monopoles.

ASIs were initially conceived as 2D analogs of natural spin-ice materials. Investigations of ASIs now extend well beyond these original goals and enable detailed studies of a vast range of possible interacting lattice arrangements, including exotic magnetic topologies not found in nature. Their monopolelike quasiparticles can move through the lattice in response to applied magnetic fields, enabling the study of a new phenomenon known as “magnetricity,” in analogy to electricity. By passively “listening” to spontaneous magnetic noise generated by these mobile monopoles, we probe the density of monopolelike excitations, reveal their intrinsic dynamics, and show that monopole kinetics are actually most diffusive in the high-density phase.

These results open the door to on-demand monopole regimes having tunable densities and dynamic properties, thereby providing a new paradigm for probing the physics and possible applications of effective magnetic charges in synthetic matter.

Figure 1

Field-dependent phase diagram of thermally active square ASI. (a) Scanning electron microscopy image of the sample. Each ${\mathrm{Ni}}_{0.8}{\mathrm{Fe}}_{0.2}$ island has lateral dimensions of $220\mathrm{nm}\times 80\mathrm{nm}$ and thicknesses of 3.5 nm, and behaves as a single superparamagnetic Ising moment that, in the absence of any biasing field, exhibits rapid thermodynamic fluctuations near room temperature. (b) Four vertex types in archetypal square ASI, in order of increasing energy at zero applied magnetic field. Type-I vertices have the lowest energy because the nearest-neighbor dipolar coupling $J_1$ exceeds the next-nearest-neighbor coupling $J_2$. Types-I and -II vertices have 2-in-2-out configurations and therefore obey ice rules (but only type-II vertices have a net polarization), while type-III vertices have 3-in-1-out or 3-out-1-in configurations and therefore have a monopole effective magnetic “charge.” (Type-IV vertices also have magnetic charge but are energetically very unfavorable and only rarely occur.) (c) Notional schematic of the anticipated field-dependent phase diagram of square ASI, showing the two well-defined ground states of the system: full tiling with type-I vertices at small applied fields $B_{x,y}\approx 0$, and polarized type-II vertex tiling when $|B_x|$ and $|B_y|$ are both large. Near the boundaries, the equilibrium dynamics are determined by the thermal creation, annihilation, and motion of type-III monopole vertices, which generate magnetization noise. (d) Schematic showing how a thermal fluctuation in the type-I phase creates a pair of type-III monopole vertices. (e) Subsequent fluctuations can cause the monopoles to diffuse, leaving behind a Dirac string of type-II (yellow) vertices. Red arrows show islands that have flipped; blue and red dots indicate the mobile monopole vertices.

Figure 2

Optical detection of broadband magnetization noise in ASI. (a) Experimental schematic: A weak probe laser is linearly polarized (LP) and focused on the ASI. Magnetization fluctuations of the horizontal islands, $\delta m_x(t)$, impart Kerr rotation fluctuations $\delta {\theta }_K(t)$ on the reflected laser, which are detected with a polarization beamsplitter (PBS) and balanced photodiodes. The frequency spectrum of the magnetization noise power, $P(\omega )$—equivalent to the Fourier transform of the temporal magnetization correlation function $⟨\delta m_x(0)\delta m_x(t)⟩$—is computed and averaged in real time. (b) Example of characteristic magnetization noise from square thermal ASI in the time domain (inset), and in the corresponding frequency domain spanning nearly 6 orders of magnitude from 1 Hz to 1 MHz. The dashed red line shows a fit to the model described in the main text, and the dashed green line shows that the noise decays as a power law at high frequencies, in this case as $P\sim {\omega }^{-1.2}$, which indicates correlated and subdiffusive kinetics.

Figure 3

Noise maps reveal the magnetic phase diagram of thermally active square ASI. (a) Map of the total magnetization noise power [$\int P(\omega )d\omega$, integrated from 250 Hz to 250 kHz] from a control sample that contains only horizontal islands, versus applied in-plane magnetic fields $B_x$ and $B_y$. Here, $T=-10.0°\mathrm{C}$. For $B_x\approx 0$, the islands fluctuate spontaneously, but fluctuations are suppressed at larger $\pm B_x$ where all the islands are polarized by the field. (b) Corresponding noise map from square ASI. The diamond-shaped boundary, where type-I and type-II vertices are degenerate in energy, is revealed by thermodynamic noise arising from the dynamics of type-III (monopole) vertices. The map lacks the fourfold symmetry of the square lattice because it measures only fluctuations along $\widehat{x}$ but not $\widehat{y}$. The very slight skewing of the diamond is likely due to imperfections in the lithography. Supplemental Material Fig. S2 shows raw time-domain noise signals at various applied fields [26]. (c) Corresponding map of the calculated density of monopole vertices from Monte Carlo spin dynamics simulations. Along the diamond-shaped boundary where the density is large, the majority of the monopoles are freely diffusing (see Fig. 4). (d) Map of the calculated rate of horizontal magnetization flips, which is directly related to the measured total noise. Measured and simulated noise maps at $T=6°\mathrm{C}$ and $23°\mathrm{C}$, and measured noise maps for square ASI with a different lattice constant are shown in Supplemental Material Figs. S3 and S4 [26].

Figure 4

Calculated density maps of monopoles both with and without an oppositely charged monopole on a neighboring vertex, vs in-plane magnetic field, from Monte Carlo simulations. (a) Map of the density of type-III monopole vertices that have a neighboring type-III vertex of opposite magnetic charge vs magnetic field, simulated at $T=1.2$ $J_2/k$. (b) Corresponding map of monopoles that do not have a neighboring monopole of opposite charge. These monopoles are more common when $B\approx B_c$ and the Dirac string tension vanishes because thermally generated monopoles can separate and diffuse freely. (c) Densities of these two classes of monopoles vs magnetic field applied along a lattice diagonal direction [i.e., along the dotted lines in panels (a) and (b)]. In the monopole plasma regime (i.e., near the crossover field $B_c$ where type-I and type-II vertices are energetically degenerate), the total monopole density increases mainly due to the huge increase in the number of free monopoles, which in turn is due to the absence of Dirac string tension at $B_c$. Note that away from $B_c$, the nonzero Dirac string tension favors keeping thermally generated monopole pairs close together, and therefore, there is a higher likelihood that monopoles will be found in close proximity to one another (compare blue and red curves). By comparing simulations and data, we estimate that a monopole density of approximately 10% is realized in our experiments.

Figure 5

Magnetization noise spectra through the monopole plasma regime. (a) Measured noise spectra along the indicated path (keeping $|B_x|=|B_y|$), which traverses between type-I and type-II order. Here, $T=-10.0°\mathrm{C}$. (b) Extracted relaxation rates ${\omega }_0$ (blue points), decay exponents $\beta$ (red points), and total noise power (black points). Note that $\beta$ exhibits a maximum and is closest to a value of 2 when $B=B_c$, indicating that the monopoles exhibit their most-uncorrelated (most-diffusive) dynamics in the plasmalike regime where their density is largest. (c,d) Corresponding computed noise spectra from Monte Carlo simulations and extracted parameters ${\omega }_0$, $\beta$, and total noise. Dashed lines are guides to the eye, highlighting power-law decays. The total integrated noise is peaked when type-I and type-II vertex energies are degenerate, i.e., in the monopole plasma regime that occurs at the crossover field $B_c$. By contrast, simulations show that ${\omega }_0$ exhibits a minimum when the ASI orders antiferromagnetically (spontaneous type-I ordering), which occurs at an applied field that is slightly less than $B_c$ when $T>0$ (see also Supplemental Material Fig. S5 [26]).