In condensed matter physics, geometrical frustration (or in short, frustration) is a phenomenon where the combination of conflicting inter-atomic forces leads to complex structures. Frustration can imply a plenitude of distinct ground states at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much-studied examples include amorphous materials, glasses, and dilute magnets.
The term frustration, in the context of magnetic systems, was introduced by Gerard Toulouse in 1977. Frustrated magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950. Related features occur in magnets with competing interactions, where both ferromagnetic as well as antiferromagnetic couplings between pairs of spins or magnetic moments are present, with the type of interaction depending on the separation distance of the spins. In that case commensurability, such as helical spin arrangements may result, as had been discussed originally, especially, by A. Yoshimori, T. A. Kaplan, R. J. Elliott, and others, starting in 1959, to describe experimental findings on rare-earth metals. A renewed interest in such spin systems with frustrated or competing interactions arose about two decades later, beginning in the 1970s, in the context of spin glasses and spatially modulated magnetic superstructures. In spin glasses, frustration is augmented by stochastic disorder in the interactions, as may occur experimentally in non-stoichiometric magnetic alloys. Carefully analyzed spin models with frustration include the Sherrington–Kirkpatrick model, describing spin glasses, and the ANNNI model, describing commensurable magnetic superstructures. Recently, the concept of frustration has been used in brain network analysis to identify the non-trivial assemblage of neural connections and highlight the adjustable elements of the brain.
Magnetic ordering
Geometrical frustration is an important feature in magnetism, where it stems from the relative arrangement of spins. A simple 2D example is shown in Figure 1. Three magnetic ions reside on the corners of an equilateral triangle with antiferromagnetic interactions between them; the energy is minimized when each spin is aligned opposite to neighbors. Once the first two spins align antiparallel, the third one is frustrated because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy.
Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration. If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel. There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and between 3 and 4) are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated.
Geometrical frustration is also possible if the spins are arranged in a non-collinear way. If we consider a tetrahedron with a spin on each vertex pointing along the easy axis (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away. The net magnetic moment points upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out (i.e. are antiferromagnetically aligned), as do forwards and backwards. There are three different equivalent arrangements with two spins out and two in, so the ground state is three-fold degenerate.
Water ice
Although most previous and current research on frustration focuses on spin systems, the phenomenon was first studied in ordinary ice. In 1936 Giauque and Stout published The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15 K to 273 K, reporting 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. The resulting entropy, S1 = 44.28 cal/(K·mol) = 185.3 J/(mol·K) was compared to the theoretical result from statistical mechanics of an ideal gas, S2 = 45.10 cal/(K·mol) = 188.7 J/(mol·K). The two values differ by S0 = 0.82 ± 0.05 cal/(K·mol) = 3.4 J/(mol·K). This result was then explained by Linus Pauling to an excellent approximation, who showed that ice possesses a finite entropy (estimated as 0.81 cal/(K·mol) or 3.4 J/(mol·K)) 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 (276 pm), while the O–H bond length measures only 0.96 Å (96 pm). Every oxygen (white) ion is surrounded by four hydrogen ions (black) and each hydrogen ion is surrounded by 2 oxygen ions, as shown in Figure 5. Maintaining the internal H2O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions. There are two equivalent positions a hydrogen may occupy 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 O2− and 2N protons. Each O–O bond has two positions for a proton, leading to 22N 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 < 22N()N. Correspondingly the configurational entropy S0 = kBln(Ω) = NkBln() = 0.81 cal/(K·mol) = 3.4 J/(mol·K) 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.
Spin ice
A mathematically analogous situation to the degeneracy in water ice is found in the spin ices. A common spin ice structure is shown in Figure 6 in the cubic pyrochlore structure with one magnetic atom or ion residing on each of the four corners. Due to the strong crystal field in the material, each of the magnetic ions can 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> cubic axes, which coincide with the lines connecting each tetrahedral vertex to the center. Every tetrahedral cell must have two spins pointing in and two pointing out in order to minimize the energy. 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 low temperature.
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 (see also the Villain model) or by lattice structure such as in the triangular, face-centered cubic (fcc), hexagonal-close-packed, tetrahedron, pyrochlore and kagome lattices with antiferromagnetic interaction. 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 ferromagnets With the help of lithography techniques, it is possible to fabricate sub-micrometer size magnetic islands whose geometric arrangement reproduces the frustration found in naturally occurring spin ice materials. Recently R. F. Wang et al. reported 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.
These artificially frustrated ferromagnets can exhibit unique magnetic properties when studying their global response to an external field using Magneto-Optical Kerr Effect. In particular, a non-monotonic angular dependence of the square lattice coercivity is found to be related to disorder in the artificial spin ice system.
Geometric frustration without lattice Another type of geometrical frustration arises from the propagation of a local order, resulting in questions regarding the stability of a solid.
It is sometimes possible to establish some local rules, of chemical nature, which lead to low-energy configurations and therefore govern structural and chemical order. This is not generally the case and often the local order defined by local interactions cannot propagate freely, leading to geometric frustration. A common feature of all these systems is that, even with simple local rules, they present a large set of, often complex, structural realizations. Geometric frustration plays a role in fields of condensed matter, ranging from clusters and amorphous solids to complex fluids.