Artificial kagome arrays of nanomagnets : A frozen dipolar spin ice

Frustration is a general concept in physics and can be found in many condensed matter systems. Frustration arises when all pairwise interactions can not be satisfied at once, for example due to the system geometry. In some cases, frustration effects lead to an extensively degenerate ground state, i.e a low temperature manifold built with a large number of configurations with identical energy. Pauling’s description of the low-temperature proton disorder in water ice was perhaps the first example of frustration in condensed matter physics, and remains its paradigm. At the end of the nineties, new magnetic compounds have been synthesized, in which the disorder of the magnetic moments at low temperature is analogous to the proton disorder in water ice. Because of this analogy, these compounds have been named spin ice. Recently, this correspondence between the physics of water ice and its magnetic equivalents has been pushed further with the realization of artificial, two dimensional arrays of ferromagnetic islands. One important advantage of these mesoscopic lattices compared to the spin ice compounds found in condensed matter physics is to allow magnetic imaging of individual spins, i.e. the visualization in real space of how spins accommodate frustration.
Using lithography techniques, we have fabricated geometrically frustrated arrays of nanomagnets on a kagome lattice, i.e. a lattice of triangles sharing their corners on which the nanomagnets are located (see figure). The magnetic configuration of each nanomagnet is then probed using X-ray PhotoEmission Electron Microscopy (X-PEEM). Due to their elongated shape, magnetization within the nanomagnets can only point along the long axis of the elements. A nanomagnet can thus be considered as an Ising spin. Each center of the triangles, called a vertex, is then defined by the spin state of a set of three Ising spins (see figure). Eight states are possible : two have a high energy and correspond to the 3-in or 3-out spin configurations, and six have a lower energy and correspond to the 1-in/2-out or 2-in/1-out spin configurations. Each spin of the array being a magnetic dipole, the X-in/Y-out configuration of a vertex can also be characterized by the magnetic charge it carries : the 3-in or 3-out configuration are equivalent to a magnetic charge state ±3, while the other spin configurations are equivalent to a charge state of ±1. In other words, these spin models on a kagome lattice can also be seen as a magnetic charge model on a hexagonal network.

Combining Monte Carlo simulations and X-PEEM magnetic imaging on artificial arrays of cobalt nanomagnets, we found an important result. Contrary to what was thought until now, the long range (i.e. beyond nearest neighbors) dipolar interactions between the nanomagnets can not be neglected. This result has profound consequences : while the main interest for frustrated compounds arises from the massive degeneracy of their ground state, this degeneracy is fully lifted when long range, dipolar interactions are included in the model. Understanding whether or not these long range interactions influence the local spin configurations in artificial arrays of nanomagnets is thus essential, especially because these arrays are often considered as a playground to study magnetic frustration effects on a mesoscopic scale.

Figure:X-PEEM magnetic image of an artificial array of 470x70x10 nm3 nanomagnets. The partly superimposed dots highlight the nodes of the kagome lattice. Field of view is 5 µm. The direction of the incoming X rays is indicated by the hn arrow. Black and white contrast is observed according to the sign of the magnetization component along this X ray direction, as sketched in the two panels below the image. The spin state of each nanomagnet, and thus the magnetic charge carried by each vertex, are measured.

One argument we used to demonstrate this result was to compare predictions from dipolar spin ice models and our experimental observations. In particular, as the system reaches low-energy spin configurations when we demagnetize the arrays, it goes through a (predicted) phase transition where spins fluctuate while the magnetic charges at the vertices crystallize to form a perfectly ordered arrangement of alternating +1 and –1 magnetic charges. In this phase, all vertices are in one of the six low-energy configurations (we say that the system satisfies the ice rule), but these configurations are such that a vertex carrying a magnetic charge +1 can only be surrounded by three vertices carrying a charge -1 (and vice versa). The novelty of our work was to observe the emergence of this phase, in which magnetic charges crystallize due to the presence of long range interactions between the nanomagnets.

Contributors
  • Institut Néel (France) : N. Rougemaille, B. Canals, O. Fruchart
  • Institut Jean Lamour (France) : F. Montaigne, A. Duluard, D. Lacour, M. Hehn
  • Synchrotrons SOLEIL (France) and ELETTRA (Italy) : R. Belkhou, S. El Moussaoui, A. Bendounan, F. Maccherozzi

Our related publication :

  • Artificial Kagome Arrays of Nanomagnets : A Frozen Dipolar Spin Ice, N. Rougemaille, F. Montaigne, B. Canals, A. Duluard, D. Lacour, M. Hehn, R. Belkhou, O. Fruchart, S. El Moussaoui, A. Bendounan, and F. Maccherozzi, Phys. Rev. Lett. 106, 057209 (2011). http://arxiv.org/abs/1102.2455

Corresponding author  :

N. Rougemaille

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