Simulation and design of artificial model magnets
Why do some materials stick to the front of your refrigerator and some do not? How can helium remain liquid down to the lowest achievable temperatures while all other materials freeze? Why do only a handful of superconducting materials allow electrons to flow with zero resistance above liquid nitrogen temperature? From the simplest to the most exotic, the properties of materials always originate from the organization of matter at the atomic scale. In quantum materials, quantum coherence survives up to the human scale and offers opportunities to solve the energy, environmental and health challenges of the 21st century. However, in the absence of a general method to solve the equations of quantum mechanics ruling the microscopic world, predicting the macroscopic behavior of materials is a daunting task.
In its report “Directing Matter and Energy: Five Challenges for Science and the Imagination” , the U.S. Department of Energy identified that understanding and controlling the emergent properties of quantum materials is one of five grand challenges for basic energy science. Here, we call for the creativity and curiosity of young scientists to expose how rich collective phenomena emerge in matter by simulating and constructing artificial model magnets.
Collective properties of model magnets: emergence and frustration
Emergence, or in the words of Nobel laureate P.W. Anderson the concept that “More is Different” , is central to understanding the electronic properties of metals, magnets and superconductors. As an analogy, consider a city populated by millions of people. It hosts complex cultural phenomena that are different from that of singular persons, but nonetheless a culture emerges from the social interaction and concerted collective behavior of many individuals. When researchers try to predict how the interaction between billions of atoms leads to the properties of a given material, they are confronted with a problem so complex that no computer can solve nor will be able to solve for the foreseeable future. Alternative and creative routes to understand the emergent properties of quantum materials are therefore necessary.
One such approach is to study model magnets, for which the individual behavior is simple but the emergent collective properties are complex. In model magnets, atomic-scale magnetic moments or “spins” are located at the nodes of a periodic lattice and interact with their neighbors [Figure 1a]. While “spins” are formally quantum objects, they can be represented within a good approximation by three-component unit vectors S=(Sx,Sy,Sz) with |S|=1. The energy of a pair i,j of spins is given by the Heisenberg interaction energy Eij =J Si · Sj where · is the familiar dot product and J is a constant that can be positive or negative. Minimizing the interaction energy for an isolated pair of spins is simple and leads to a ferromagnetic state for J<0 (spins are aligned ) or to an antiferromagnetic state for J>0 (spins are anti-aligned ¯). Minimization of the total energy for a system of N spins interacting on an extended lattice can be infinitely more complicated. Simple ordered patterns spanning long-distances are easy to stabilize in the case of square or honeycomb lattices in two-dimensions (2D) or simple cubic or diamond lattices in three-dimensions (3D) [Figure 1b]. However, for certain lattice geometries comprising triangular patterns such as the triangular and kagome lattices in 2D or the pyrochlore lattice in 3D [Figure 1c], the energies of each pair of interacting spins cannot be simultaneously minimized. These systems are called “frustrated magnets” . Due to frustration, spins can adopt complex ordering patterns  or even remain perpetually fluctuating because many different states have the exact same energy . Clearly, frustrated magnets are ideal to search for new emergent behaviors.
From neutron scattering to artificial magnets
At Oak Ridge National Laboratory, we use neutron scattering to study quantum materials and frustrated magnets made of atomic-scale spins. Because neutrons carry a magnetic moment, they are sensitive to the magnetic properties of a given material and produce interference patterns that reveal how spins are arranged and fluctuate over time. Our experiments are performed by sending intense beams of neutrons on a sample of interest and use a “spectrometer” made of large pixelated area detectors to detect the scattered neutrons. While neutron scattering is one of the best experimental techniques to probe the quantum world, it is limited by the need for large and clean samples that can be extremely difficult to synthesize. Another complication is that the accurate interpretation of neutron scattering data relies on high-fidelity simulations of the materials properties and understanding of the resolution of the neutron spectrometer.
In an effort to scale the emergent behaviors observed in atomic-scale magnets up to the macroscopic world, researchers recently demonstrated the concept of “macroscopic magnetic frustration” [6,7]. Using inch-sized ferromagnetic bar magnets and additive manufacturing techniques such as 3D printing, they constructed mechanical rotors mimicking atomic-scale spins [Figure 2a]. By arranging these rotors on a plane, they created a synthetic version of the kagome lattice [Figure 2b]. Because the magnetic rotors are dipoles, they interact with their neighbors through the dipole-dipole interaction force described by classical electromagnetism and can display rich collective behavior, for instance when relaxing from a state where they were all aligned by an external magnetic field [6,7]. These artificial model magnets may present key advantages over their atomic-scale counterparts because the spin patterns can be seen with the naked-eye and evolve slowly enough to observe out-of-equilibrium phenomena.
Understanding the collective properties of frustrated magnets is currently a central focus at Oak Ridge National Laboratory. In order to analyze neutron scattering experiments on real materials, we are seeking original approaches to simulate the behavior of models magnets and design macroscopic artificial magnets that expose their emergent behavior. How would you solve such challenge? Propose a research plan that is designed to answer the following question(s).
Simulations. How would you simulate the behavior of classical dipole moments interacting with their neighbors on the 2D and 3D lattices of Figure 1? What is the difference between J>0 and J<0? What patterns do the spins adopt? How do spins reach their equilibrium directions when they relax from an “all-aligned” state? What are the effects of constraining the rotation of individual spins into a plane?
Design. Is it possible to design a human-scale artificial magnet that realizes the three-dimensional pyrochlore lattice? Can you propose a practical method to construct this artificial magnet? How do the length scales of the force between two magnetic dipoles influence the scale of the design?
 U.S. Department of Energy, Office of Science, Energy Frontier Research Centers Website.
 Ramirez, A. P. (1994). “Strongly Geometrically Frustrated Magnets”. Annual Review of Materials Science 24, 453. http://dx.doi.org/10.1146/annurev.ms.24.080194.002321
 Chalker, J. T. , Holdsworth, P. C. W., and Shender, E. F. (1992) “Hidden order in a frustrated system: Properties of the Heisenberg Kagomé antiferromagnet”. Physical Review Letters 68, 855.
 Moessner, R., and Chalker, J. T. (1998). “Properties of a Classical Spin Liquid: The Heisenberg Pyrochlore Antiferromagnet”. Physical Review Letters 80, 2929.
 Mellado, P., Concha, A., and Mahadevan, L. (2012). “Macroscopic Magnetic Frustration”.
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 Nisoli, C., Moessner, R., Schiffer P. , (2013). “Colloquium: Artificial spin ice: Designing and imaging magnetic frustration”, Review of Modern Physics 85, 1473.
Figure 1. (a) In model magnets, spins reside at the nodes of a lattice. The nodes are represented by blue spheres. The interaction between spins is pictured by a blue line. The spins are three-dimensional vectors: when isolated, a spin can rotate to point in any direction. When two spins interact, they become co-aligned and form a ferromagnetic state if the sign of the interaction J is negative. If the sign of the interaction J is positive, the spins are anti-align and form an antiferromagnetic state. (b) Examples of non-frustrated lattices. (c) Examples of frustrated lattices.
(a) Three interacting artificial spins
(b) Kagome lattice of artificial spins
Figure 2 (a) Taken from Reference . Picture of three magnetic rotors forming a spin triangle. (b) Taken from Reference . Rotors organized to form a kagome lattice.