ScienceWeek

CONDENSED MATTER: NANOMAGNETS AND SPIN ICE

The following points are made by Steven T. Bramwell (Nature 2006 439:273):

1) How can we understand disordered states of matter such as liquids, glasses and disordered magnets? First, we must know how they are organized; second, we must know how that organization responds to changes in external constraints such as temperature, pressure, or magnetic field. This knowledge is beyond our reach in most cases: disordered states are intrinsically complicated and do not reveal themselves clearly to experiment. But thankfully, there are exceptions. One of them is a magnetic state known as "spin ice" in which the magnetic moments of ions --their "spins", in analogy to the property of electron spin --remain disordered even at low temperatures, revealing much about the basic physics of disorder.

2) New work[1] report the creation of spin-ice behavior in an array of nanoscale magnets. Such "artificial" spin ice, which is stable at room temperature and possesses magnetic moments large enough to be observed directly, offers a new approach to understanding and exploiting the properties of disordered systems.

3) In conventional spin ice[2,3], magnetic ions form a network of linked tetrahedra (for example, ions of the lanthanide element holmium in the compound holmium titanate). The spins of these ions point either in or out of the tetrahedra. The dipolar interaction with their neighbors' magnetic fields favors an in out arrangement of neighboring moments, but not all pairs of neighbors can be satisfied simultaneously: the system is "frustrated". The best compromise -- two moments pointing in, two pointing out, in any one tetrahedron -- constitutes the local organizing principle. The same rule controls the hydrogen structure of ice[4], and is fully satisfied by a huge number of equivalent arrangements of spins (or hydrogen atoms). This means that the chance of achieving an ordered structure is effectively nil. So spin ice, just like normal ice, remains disordered, with a non-zero entropy (a key measure of disorder) as its temperature approaches absolute zero.

4) The Schiffer et al[1] artificial spin ice consists of a two-dimensional array of 80,000 elongated magnetic islands, each a few hundred nanometers long. The magnetic moment of every island is aligned parallel to its long axis, as in a bar magnet, and is coupled to its neighbors by the ubiquitous dipolar interaction. For a square geometry, the two in, two out rule is approximately satisfied ("square ice"[5[). But because the magnetic moments involved are about three million times bigger than those of holmium ions, they interact more strongly and have less tendency to flip. The artificial spin-ice state is therefore stable at room temperature; for conventional spin ice, this is only the case at temperatures below 1 kelvin. Schiffer et al[1] encouraged their system to settle into a minimum energy state by cycling the applied magnetic field, and then determined the directions of individual magnetic moments using a technique known as magnetic force microscopy. Statistical analysis of these directions confirmed that a spin-ice state, characterized by a preponderance of the two in, two out configuration, had indeed been created.

References (abridged):

1. Wang, R. F. et al. Nature 439, 303 306 (2006)

2. Harris, M. J. , Bramwell, S. T. , McMorrow, D. F. , Zeiske, T. & Godfrey, K. W. Phys. Rev. Lett. 79, 2554 2557 (1997)

3. Ramirez, A. P. , Hayashi, A. , Cava, R. J. , Siddharthan, R. & Shastry, B. S. Nature 399, 333 355 (1999)

4. Pauling, L. J. Am. Chem. Soc. 57, 2680 2684 (1935)

5. Lieb, E. H. Phys. Rev. Lett. 18, 692 694 (1967)

Nature http://www.nature.com/nature

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Related Material:

ON THE ISING MODEL

Notes by ScienceWeek:

In theoretical physics, one approach that has proved to be of great general utility is to begin with an attempt to identify and understand the simplest model exhibiting the same essential features as the physical problem in question. In condensed-matter physics, such a model is the so-called "Ising model", an approach that has been applied to ferromagnetism, and also to a number of other systems. In general, the Ising model consists of an array of entities in one, two, or three dimensions, with each entity capable of being in one of two possible states, with each entity interacting only with its nearest neighbors, with a condition that when two neighboring entities are in the same state the total energy of the pair is reduced compared to when the same two neighboring entities are in opposite states. These are the elements of the model, with other conditions imposed depending on how the model is used. Various versions of the model have been of great utility in studies of cooperative phenomena in condensed-matter systems, and the model itself has an interesting human story attached to it.

The following points are made by Brian Hayes (American Scientist 2000 88:384):

1) The Ising model was invented in 1920 by Wilhelm Lenz, who proposed it as a simplified version of a ferromagnet (Physik. Z. 1920 21:613). In 1925, a student of Lenz, Ernst Ising, chose the model as the subject of his doctoral dissertation at the University of Hamburg (DE), and the model has subsequently borne Ising's name.

2) Lenz and Ising formulated the original model in terms of "spins", although the concept of rotation is never used. In the original model, a spin is merely one of two states, characterized by an arrow pointing either up or down but in no other direction. The spins are arranged in a grid or lattice pattern. Spins at neighboring sites prefer to point the same way: the energy is lower when adjacent spins are parallel, and the energy is higher when adjacent spins are antiparallel. Except for these nearest neighbor preferences, the spins do not interact at all. Thermal fluctuations tend to randomize the spins. Finally, an external magnetic field may impose a bias on the spin directions.

3) Hayes points out that the Ising model is indeed a crude picture of a ferromagnet: a) the Ising spins correspond to spinning electrons in iron atoms; b) the lattice represents the crystal structure; c) the nearest neighbor interaction mimics the overlap of quantum mechanical wave functions in adjacent iron atoms. The one element in the model that has no obvious counterpart in real systems is the requirement that spins take on only two possible orientations.

4) Ising's doctoral dissertation examined whether the 1-dimensional version of the model exhibited a Curie point. The results were negative: the 1-dimensional Ising model exhibits no phase transition at any temperature above absolute zero. Ising apparently believed this negative result would hold in higher dimensions as well, but in this conjecture he was wrong.

5) Ising's published results (Z. fur Physik 1925 31:253) were essentially ignored until 1936, when Rudolf Peierls (1907-1995) showed that a 2-dimensional Ising model might exhibit a temperature-dependent phase transition . An exact calculation of such a system, a mathematical tour de force, was made by Lars Onsager (1903-1976) in 1944. Exact calculations for 3-dimensional Ising models have remained intractable, but approximations and computer simulations involving the model have proved extremely useful, and the value of the model has grown rather than diminished through the years. An important approximation method is known as "the renormalization group": the simplest version of this algorithm gathers sets of spins into blocks, replaces each block with a single new spin, and finally adjusts the couplings between spins to compensate for the coarsening of the lattice.

6) Concerning Ernst Ising, there is no record of Ising ever publishing anything else in physics. After receiving his doctorate, Ising taught physics in German public high schools, but as a Jew he was dismissed from his teaching post when Hitler came to power in 1933. Ising then taught at a Jewish boarding school in Potsdam (DE), until that school was destroyed in the Kristallnacht pogrom of 1938. Ising and his wife fled Germany, but they escaped only as far as Luxembourg before the war overtook them. They managed to survive the occupation, and they finally reached the US in 1947. Ising taught physics and mathematics in Minot, North Dakota (US), and then taught for almost 30 years more at Bradley University in Peoria, Illinois (US). In 1998, Ernst Ising died at the age of 98.

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