Pyramid Artificial Spin Ice: 3D Magnetic Frustration

• Pyramid artificial spin ice is a three-dimensional nanomagnetic system featuring square-pyramidal geometry and distinct vertex types that introduce controlled geometric frustration.
• Selective external field protocols enable decoupled control of tilted and in-plane nanomagnets, driving emergent square ice ordering and complex antiferromagnetic patterns.
• Monte Carlo simulations and magnetic force microscopy experiments confirm a rich phase diagram with tunable magnetic states, promising reconfigurable magnonic and memory applications.

Pyramid artificial spin ice (pyramid ASI) is a three-dimensional nanomagnetic system in which single-domain nanomagnets are patterned onto square-based pyramidal substrates, combining geometric frustration, distinct dipolar interaction topologies, and tunable magnetic control. This architecture extends the established principles of artificial spin ices (frustrated planar arrays of dipolar-coupled nanomagnets) into a three-dimensional regime, enabling emergent phenomena—such as coarse-grained square ice ordering and vertex-level effective spin manifolds—not present in conventional two-dimensional systems. The central physical mechanisms include the interplay between multiple vertex types (flat, mixed, and pyramid apex) and selective field-driven switching of nanomagnet subsets, leading to a rich phase diagram and reconfigurable magnetic orders (Berchialla et al., 1 Sep 2025).

1. System Architecture and Fundamental Geometry

Pyramid ASI consists of arrays of square-based pyramids etched or patterned into a substrate, with two sets of disconnected nanomagnets: "tilted" nanomagnets are deposited on the four lateral pyramid faces, oriented at a nontrivial angle with respect to the substrate, while "in-plane" nanomagnets reside on the planar regions between pyramids.

From a top-down perspective, the pattern retains the periodicity of a square lattice, but vertically, the three-dimensionality introduces local environments (vertex types) not seen in planar ASI. Vertices are categorized as:

• Flat vertices: Four in-plane nanomagnets meet.
• Pyramid/apex vertices: Four tilted nanomagnets converge at the pyramid apex.
• Mixed vertices: Two tilted and two in-plane nanomagnets meet at the pyramid’s mid-edge.

All elements are physically separated and interact via long-range dipolar coupling, thus preserving the Ising macrospin nature of each nanomagnet and eliminating exchange-coupled domain walls at the vertices (Berchialla et al., 1 Sep 2025).

2. Control of Magnetic State via External Fields

Pyramid ASI enables selective control over its microstates by exploiting the three-dimensional orientation of nanomagnets relative to the applied magnetic field:

• An out-of-plane magnetic field ($H_z$) preferentially couples to the tilted nanomagnets on the pyramid faces, efficiently reversing their magnetization owing to the alignment of their long axis with the field direction.
• The in-plane nanomagnets are largely unaffected by $H_z$, permitting the decoupled manipulation of sublattices.
• Combining $H_z$ with rotating, oscillating in-plane fields ($H_A$), one can drive the system between various low-energy vertex configurations, traversing distinct sectors of the energy landscape.

This selective driving is exploited in tailored demagnetization protocols. For instance, a fixed-polarity (unipolar) demagnetization sequence—where $H_A$ oscillates between zero and a positive maximum in the presence of a finite $H_z$—forces the tilted sublattice into a uniform (Type IV) state. As a result, all mixed vertices adopt Type III configurations, setting the stage for emergent vertex-level ordering (Berchialla et al., 1 Sep 2025).

3. Emergent Square Ice and Coarse-Grained Ordering

In the regime where the tilted nanomagnets are saturated (Type IV state), the mixed vertices’ net moments define effective macrospins that reside on a secondary square lattice with double the period of the underlying nanomagnet lattice. Within this coarse-grained manifold:

• These effective spins arrange into head-to-tail closed loops, directly mirroring the ground-state configuration of conventional square ice.
• The emergent square ice state, observed and confirmed in both experiment and point-dipole Monte Carlo simulations, originates from the interplay of local constraints (vertex types) and nontrivial three-dimensional geometry.
• The presence of multiple, physically distinguishable vertex types allows for complex global organization unattainable in purely two-dimensional ASI.

This emergent ordering mechanism substantially broadens the accessible range of frustrated phases and magnetic charge propagation phenomena in engineered nanomagnetic materials (Berchialla et al., 1 Sep 2025).

4. Phase Diagram, Monte Carlo Simulations, and Geometric Dependence

Monte Carlo simulations using a point-dipole Hamiltonian were performed as a function of two key parameters:

• Pyramid face angle ($\theta$ or Op), which sets the out-of-plane orientation of the tilted nanomagnets and directly modulates the interaction strength and sign at mixed and apex vertices.
• Applied out-of-plane field ($H_z$), which controls the subset of macrospins that switch.

The simulation results resolve five prominent regimes:

• Low $H_z$, Op < 50°: The system forms a 2D-like square ice phase of Type I vertices (alternating moments).
• Low $H_z$, Op > 50°: Mixed vertices favor Type II, forming collinear antiferromagnetic chains.
• Intermediate $H_z$: Type III local configurations emerge at apexes and propagate anisotropically according to Op, organizing into additional AFM patterns.
• High $H_z$: All tilted nanomagnets are locked into Type IV; mixed vertices then show a pure Type III vertex lattice, yielding the coarse-grained emergent square ice.

These theoretical findings are paralleled in experiment, where magnetic force microscopy (MFM) allows direct identification of vertex-type populations. Under tuned demagnetization protocols, the experimental vertex distributions closely track the simulated phase boundaries and support the emergence of long-range order in the coarse-grained square ice (Berchialla et al., 1 Sep 2025).

5. Experimental Techniques: Fabrication, Field Protocols, and Imaging

Realization of pyramid ASI involves:

• Substrate engineering: Etching or patterning of square-pyramidal features into silicon, followed by nanomagnet deposition (typically via electron-beam lithography and thermal evaporation) on both pyramid faces and the interstitial plane.
• Demagnetization protocols: Employed to access (quasi-)ground or low-energy states. Alternating- and fixed-polarity field rotation sequences at various angles are applied; the field is carefully ramped to minimize net magnetization and nucleate domain wall-like excitations.
• Magnetic force microscopy (MFM): The three-dimensional topography allows unambiguous discrimination between in-plane and tilted nanomagnet signals. MFM quantitatively resolves the microstate of each macrospin and reconstructs local and global spin configurations, vertex types, and the extent of emergent ordering (Berchialla et al., 1 Sep 2025).

6. Applications, Functional Manipulation, and Future Prospects

Pyramid ASI’s ability to realize and switch among highly tunable, structurally protected frustrated states in three dimensions has key implications:

• Reconfigurable magnonics, logic, and memory: Selective field-driven control over nanomagnet subpopulations enables tailored information writing and readout.
• Frustrated charge dynamics: Manipulation of local constraints through geometry and field allows for the paper of magnetic charge propagation, monopole-like defects, and domain wall kinetics in controlled environments.
• Custom frustration engineering: Adjustment of the pyramid base shape or face angle can locally tune interactions, opening the path to designs based on other base lattices (kagome, triangular), potentially yielding unprecedented collective states.
• Integration with spintronic or quantum platforms: Three-dimensional frustration and hierarchical coarse-graining suggest compatibility with emerging computing paradigms, including artificial neural networks and quantum-mimetic devices.

Further research aims to clarify the thermalization, defect, and relaxation dynamics in such 3D architectures; to explore the interplay of disorder, frustration, and slow dynamics; and to assess the role of long-range order and effective temperature in hierarchical, multi-layered nanomagnetic systems (Berchialla et al., 1 Sep 2025).

7. Relation to Broader Artificial Spin Ice Context

Pyramid ASI can be interpreted as an extension of both planar artificial spin ice and multi-layered or stacked ASI models (Chern et al., 2013, Dubitskiy et al., 2015). The introduction of three-dimensionally variable vertex topologies and selective control mechanisms distinguishes it from earlier bisymmetric or vertex-frustrated 2D lattices, and provides a versatile platform for both fundamental studies of geometrical frustration and the development of device-scale reconfigurable magnetic arrays. The presence of emergent vertex-level effective spin manifolds, macroscopic degeneracy, and tunable connectivity cements pyramid ASI’s role as a cornerstone system for the exploration of collective behavior in frustrated nanomagnetic materials (Berchialla et al., 1 Sep 2025).