Geometrically frustrated magnet

Geometrical frustration and ice-rules

The word frustration was introduced to describe the situation where a system cannot simultaneously minimize the interaction energies between its components [cite journal
last = Schiffer
first = P.
authorlink =
coauthors =
title = Comments
journal = Con. Mat. Phys.
volume = 18
issue = 21
pages =
publisher =
location =
date = 1996
url =
doi =
id =
accessdate = ] Frustration systems were discovered and have been studied for more than 50 years. Early work includes the famous example of the Ising spins on an antiferromagnetic triangular network that was studied by G. H. Wannier in 1950. Related early work on magnets with competing interactions leading to helical and incommensurate spin structures was done by A. Yoshimori, J. Villain, T. A. Kaplan, and R. Elliott starting in 1959. A renewed and great interest in such systems arose almost two decades later in the context of spin glasses and spatially modulated magnetic superstructures. The concept of frustration has been put forward by G. Toulouse and J. Villain within the framework of spin glasses. The frustration is generated by the competition of different kinds of interaction and/or by the lattice geometry. As a result, in the ground state all bonds are not fully satisfied, and the state is highly degenerate. Well-known spin models with competing interactions are, for instance, the Sherrington-Kirkpatrick model describing spin glasses and the ANNNI model, describing commensurate and incommensurate superstructures.

From water ice to spin ice

Although most current research on frustration is based on spin systems, the very first discovered frustration lies in the common water ice. In 1936 Giauque and Stout published the famous paper, "The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15 to 273K", thus starting the epoch of frustration. In their paper, Giauque and Stout reported calorimeter measurements on water through the freezing and vaporization transitions up to the high temperature gas phase. The entropy was calculated by integrating the heat capacity and adding the latent heat contributions; the low temperature measurements were extrapolated to zero, using Debye’s then recently derived formula. [cite journal
last = Debye
first = Peter
authorlink =
coauthors =
title =
journal = Ann. Der Physik
volume = 39
issue =
pages = 789
publisher =
location =
date = 1912
url =
doi =
id =
accessdate = ] The resulting entropy, S₁=44.28 cal/K·mol was compared to the theoretical result from statistical mechanics of an ideal gas, S₂=45.10 cal/K·mol. The two values differ by S₀=0.82±0.05 cal/K·mol. This result was then explainedcite journal
last = Pauling
first = Linus
authorlink =
coauthors =
title =
journal = J. Am. Chem. Soc.
volume = 57
issue =
pages = 2680
publisher =
location =
date = 1935
url =
doi =
id =
accessdate = ] by Linus Pauling, to an excellent approximation, who showed that ice possesses a finite entropy (estimated as 0.81 cal/K·mol) at zero temperature due to the configurational disorder intrinsic to the protons in ice.

In the hexagonal or cubic ice phase the oxygen ions form a tetrahedral structure with an O-O bond length 2.76 Å, while the O-H bond length measures only 0.96 Å. Every oxygen (white) ion is surrounded by four hydrogen ions (black) and each hydrogen ion is surrounded by 2 oxygen ions, as shown in Fig 1a. Maintaining the internal H₂O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions. On the contrary, there are two equivalent positions on the line of the O-O bond, a far and a near position. Thus a rule leads to the frustration of positions of the proton for a ground state configuration: for each oxygen two of the neighboring protons must reside in the far position and two of them in the near position, so-called ‘Ice Rules’. Pauling proposed that the open tetrahedral structure of ice affords many equivalent states satisfying the ice rules.

Pauling went on to compute the configurational entropy in the following way: consider one mole of ice, consisting of N of O^2- and 2N of protons. Each O-O bond has two positions for a proton, leading to 2^2N possible configurations. However, among the 16 possible configurations associated with each oxygen, only 6 are energetically favorable, maintaining the H₂O molecule constraint. Then an upper bound of the numbers that the ground state can take is estimated as Ω<2^2N(6/16)^N. Correspondingly the configurational entropy S₀=k_Bln(Ω)=Nk_Bln(3/2)=0.81 cal/K·mol is in amazing agreement with the missing entropy measured by Giauque and Stout.

Although Pauling’s calculation neglected both the global constraint on the number of protons and the local constraint arising from closed loops on the Wurtzite lattice, the estimate was subsequently shown to be of excellent accuracy.

Upon understanding protons’ frustration of 2 equivalent positions on the adjacent oxygen contact lines, it was quite a natural thought to ‘mimic’ the positional frustration in a 2-state spin system, Spin Ice.

Spin ice

The name “spin ice” and the first concept of frustrated spin alignment were initially proposed in Ref. 4. A common spin ice structure is shown in Fig 1b. Spin ice materials normally adopt the cubic pyrochlore structure with four magnetic atoms/ions residing on the four corners. Due to the strong crystal field in the material, each of the magnetic ions could be represented by an Ising ground state doublet with a large moment. This suggests a picture of Ising spins residing on the corner-sharing tetrahedral lattice with spins fixed along the local quantization axis, the <111> axes: in this case the trigonal <111> axes connect the vertices of each tetrahedron to its center. [cite journal
last = Bitko
first = D.
authorlink =
coauthors = T. F. Rosenbaum and G. Aeppli
title =
journal = Phys. Rev. Lett.
volume = 77
issue =
pages = 940
publisher =
location =
date = 1996
url =
doi =
id =
accessdate = ] [cite journal
last = Friedman
first = J. R.
authorlink =
coauthors = M. P. Sarachik, J. Tejada and R. Ziolo
title =
journal = Phys. Rev. Lett.
volume = 76
issue =
pages = 3830
publisher =
location =
date = 1996
url =
doi =
id =
accessdate = ] Every tetrahedral cell must have two spins pointing in and two pointing out in order to minimize the energy. Using inelastic neutron scattering, Rosenkranz et al. determined the energy level scheme forHo₂Ti₂O₇. By rescaling the crystal field parametersdetermined for Ho₂Ti₂O₇, they were able to estimatethose suitable to describe Dy₂Ti₂O₇. They found that in Ho₂Ti₂O₇ and Dy₂Ti₂O₇, the first excited crystal field levels are of the order of 300 K above the ground state. Consequently, these two materials are indeed well described by a classical Ising model characterizing the crystal field ground state with the excited crystal field levels playing essentially no role in the spin ice phenomenology.

Currently the spin ice model has been approximately realized by real materials, most notably the rare earth pyrochlores Ho₂Ti₂O₇, Dy₂Ti₂O₇, and Ho₂Sn₂O₇. These materials all show nonzero residual entropy at T=0 as that of the water ice.

Extension of Pauling’s model: general frustration

The spin ice model is only one subdivision of frustrated systems. The word frustration was initially introduced to describe a system’s inability to simultaneously minimize the competing interaction energy between its components. In general frustration is caused either by competing interactions due to site disorder (the Villain model [cite journal
last = Villain
first = Jacques
authorlink =
coauthors =
title =
journal = J. Phys. C
volume = 10
issue =
pages = 1717
publisher =
location =
date = 1977
url =
doi =
id =
accessdate = ] ) or by lattice structure such as in the triangular, face-centered cubic (fcc), hexagonal-close-packed, tetrahedron, pyrochlore and kagome lattices with antiferromagnetic interaction. [cite journal
last = Vannimenus
first = J.
authorlink =
coauthors = G. Toulouse
title =
journal = J. Phys. C
volume = 10
issue =
pages = L537
publisher =
location =
date = 1977
url =
doi =
id =
accessdate = ] So frustration is divided into two categories: the first corresponds to the spin glass, which has both disorder in structure and frustration in spin; the second is the geometrical frustration with an ordered lattice structure and frustration of spin. The frustration of a spin glass is understood within the framework of the RKKY model, in which the interaction property, either ferromagnetic or anti-ferromagnetic, is dependent on the distance of the two magnetic ions. Due to the lattice disorder in the spin glass, one spin of interest and its nearest neighbors could be at different distances and have a different interaction property, which thus leads to different preferred alignment of the spin.

Artificial geometrically frustrated ferro-magnets

With the help of new nanometer techniques, it is possible to fabricate nanometer size magnetic islands analogous to those of the naturally occurring spin ice materials. Recently R.F.Wang et al reported [cite journal
last = Wang
first = R. F.
authorlink =
coauthors = C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, P. Schiffer
title =
journal = Nature
volume = 439
issue =
pages = 303
publisher =
location =
date = 2006
url =
doi =
id =
accessdate = ] the discovery of an artificial geometrically frustrated magnet composed of arrays of lithographically fabricated single-domain ferromagnetic islands. These islands are manually arranged to create a two-dimensional analog to Spin Ice. The magnetic moments of the ordered ‘spin’ islands were imaged with magnetic force microscopy (MFM) and then the local accommodation of frustration was thoroughly studied. In their previous work on a square lattice of frustrated magnets, they observed both ice-like short-range correlations and the absence of long-range correlations, just like in the spin ice at low temperature. These results solidify the uncharted ground on which the real physics of frustration can be visualized and modeled by these artificial geometrically frustrated magnets, and inspires further research activity.

References

*P. Schiffer, Comments "Con. Mat. Phys.", 18, 21 (1996)

*P. Debye, "Ann. Der Physik", 39,789 (1912)

*L. Pauling, "J. Am. Chem. Soc.", 57, 2680 (1935)

*M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske and K. W. Godfrey, "Phys. Rev. Lett.", 79, 2554 (1997)

*D. Bitko, T. F. Rosenbaum and G. Aeppli, "Phys. Rev. Lett.", 77, 940 (1996)

*J. R. Friedman, M. P. Sarachik, J. Tejada and R. Ziolo, "Phys. Rev. Lett.", 76, 3830 (1996)

*S. Rosenkranz, A.P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan and B. S. Shastry, "J. Appl. Phys.", 87, 5914 (2000)

*J. Villain, "J. Phys. C", 10, 1717 (1977)

*J. Vannimenus, G. Toulouse, "J. Phys. C", 10, L537 (1977)

*cite journal
last = Fisher
first = Michael E.
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coauthors = Walter Selke
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journal = Phys. Rev. Lett.
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pages = 1502
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*cite journal
last = Stewart
first = R.(ed)
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journal = J. Phys. Condens. Matter
volume = 16
issue =
pages = S553-S922
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*R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, P. Schiffer, "Nature", 439, 303 (2006)

*C. Nisoli, R.F. Wang, J. Li, W. F. McConville, P.l E. Lammert, P. Schiffer, and V. H. Crespi "Phys. Rev. Lett." 98, 217203 (2007)

Footnotes