(Geometrical) frustration is a phenomenon in
condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the...
in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.
This may lead to highly
degenerateIn physics two or more different physical states are said to be degenerate if they are all at the same energy level. Physical states differ if and only if they are linearly independent. An energy level is said to be degenerate if it contains two or more different states...
ground states with a nonzero
entropyEntropy is a concept of information maintaining great importance in physics, chemistry, and information theory...
at zero temperature. Or in simple terms, the substance can't be completely and totally
frozenFrozen may refer to:* the result of freezingIn film:* Frozen , a film by Wang Xiaoshuai* Frozen , a film by Juliet McKoen* Frozen , a film by Shivajee ChandrabhushanIn theatre:...
, ever, because the structure it forms prevents collapse to a single minimal-energy state - something in there can always move, even at
absolute zeroAbsolute zero is a temperature marked by a 0 entropy configuration. It is the coldest temperature theoretically possible and cannot be reached by artificial or natural means, because it is impossible to decouple a system fully from the rest of the universe...
, even without input of
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
.
The term
frustration, in the context of
magneticIn physics, the term magnetism is used to describe how materials respond on the microscopic level to an applied magnetic field; to categorize the magnetic phase of a material. For example, the most well known form of magnetism is ferromagnetism such that some ferromagnetic materials produce their...
systems, is due to Gerard Toulouse (1977). Frustrated
magneticIn physics, the term magnetism is used to describe how materials respond on the microscopic level to an applied magnetic field; to categorize the magnetic phase of a material. For example, the most well known form of magnetism is ferromagnetism such that some ferromagnetic materials produce their...
systems have been studied for many years. Early work includes a study of the
Ising modelThe Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It has since been used to model diverse phenomena in which bits of information, interacting in pairs, produce collectiveeffects.-Definition :...
on a triangular lattice with nearest-neighbor
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
s coupled
antiferromagneticallyIn materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...
by
G. H. WannierGregory Hugh Wannier was a Swiss physicist.He attended Princeton as a graduate student and later taught at several American universities before a stint in industry....
, published in 1950. Related research on magnets with
competing interactions, where
different couplings, each favoring simple (e.g. ferro- and antiferromagnetic), but different structures,
are present. In that case
incommensurateGenerally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable...
, such as
helicalHelimagnetism is an incommensurate form of magnetic ordering that results from the competition between Ferromagnetic and Antiferromagnetic exchange interactions, and is typically only observed at liquid helium temperatures. Spins of neighbouring magnetic moments arrange themselves in a spiral or...
spin arrangements may result, as had been discussed originally by A. Yoshimori, T. A. Kaplan, R. J. Elliott, and others, starting in 1959. A renewed interest in such spin systems with competing or frustrated interactions arose about two decades later in the context of
spin glassA spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
es and spatially modulated magnetic superstructures. In spin glasses, frustration is
augmented by
stochasticStochastic means random.A stochastic process is one whose behavior is non-deterministic in that a system's subsequent state is determined both by the process's predictable actions and by a random element....
disorder in the interactions. Well-known spin models with competing or frustrated interactions include the
Sherrington-Kirkpatrick modelIn statistical physics, a system is said to present quenched disorder when some parameters defining its behaviour are random variables which do not evolve with time, i.e.: they are quenched or frozen. As a typical example, we may cite spin glasses...
, describing spin glasses, and the
ANNNI modelThe abbreviation ANNNI model stands for 'Axial Next-Nearest Neighbor Ising model'. It is a highly cited variant of one of the best known models in statistical physics, the Ising model. In that variant, competing ferromagnetic and...
, describing
commensurate and incommensurateGenerally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable...
magnetic superstructures.
Magnetic ordering
Geometrical frustration is an important feature in
magnetismIn physics, the term magnetism is used to describe how materials respond on the microscopic level to an applied magnetic field; to categorize the magnetic phase of a material. For example, the most well known form of magnetism is ferromagnetism such that some ferromagnetic materials produce their...
, where it stems from the topological arrangement of
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
s. A simple 2D example is shown in Figure 1. Three magnetic ions reside on the corners of a triangle with
antiferromagneticIn materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...
interactions between them—the energy is minimized when each spin is aligned opposite to its neighbors. Once the first two spins align anti-parallel, 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. Thus the ground state is twofold
degenerateIn physics two or more different physical states are said to be degenerate if they are all at the same energy level. Physical states differ if and only if they are linearly independent. An energy level is said to be degenerate if it contains two or more different states...
.
Similarly in three dimensions, four spins arranged in a
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
(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 momentThe magnetic moment of a system is a measure of the strength and the direction of its magnetism. More technically , the term magnetic moment of a system usually refers to its magnetic dipole moment, and quantifies the contribution...
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
iceIce is a solid phase, usually crystalline, of a non-metallic substance that is liquid or gas at room temperature, such as carbon dioxide ice , ammonia ice, or methane ice. However, the predominant use of the term ice is for water ice, technically restricted to one of the 15 known crystalline phases...
. In 1936 Giauque and Stout published
The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15 to 273°K, reporting
calorimeterA calorimeter is a device used for calorimetry, the science of measuring the heat of chemical reactions or physical changes as well as heat capacity. The word calorimeter is derived from the Latin word calor, meaning heat. Differential scanning calorimeters, isothermal microcalorimeters, titration...
measurements on water through the freezing and vaporization transitions up to the high temperature gas phase. The
entropyEntropy is a concept of information maintaining great importance in physics, chemistry, and information theory...
was calculated by integrating the heat capacity and adding the
latent heatThe expression latent heat refers to the amount of energy released or absorbed by a chemical substance during a change of state that occurs without changing its temperature, meaning a phase transition such as the melting of ice or the boiling of water...
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 PaulingLinus Carl Pauling was an American chemist, peace activist, author, and educator. He was one of the most influential chemists in history and ranks among the most important scientists in any field of the 20th century. Pauling was among the first scientists to work in the fields of quantum...
, 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
cubicThe cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals....
ice phase the
oxygenOxygen Oxygen Oxygen (acid, literally "sharp", from the taste of acids) and -γενής (-genēs) (producer, literally begetter) is the element with atomic number 8 and represented by the symbol O...
ions form a tetrahedral structure with an O-O bond length 2.76
ÅThe ångström or angstrom is an internationally recognized unit of length equal to 0.1 nanometre or 1 metres. It is named after Anders Jonas Ångström...
(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 H
2O 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 of O
2- and 2
N 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 H
2O 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 =
kBln(
Ω) =
NkBln(3/2) = 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 iceSpin ice refers to a property of pure ice crystals formed from water or a material with similar structure, wherein the atomic structure permits non-zero residual entropy. Originally postulated by Linus Pauling in 1935, these structures were shown to exhibit the properties of the long sought-after...
s. 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 fieldCrystal field theory is a model that describes the electronic structure of transition metal compounds, all of which can be considered coordination complexes. CFT successfully accounts for some magnetic properties, colours, hydration enthalpies, and spinel structures of transition metal complexes,...
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 axesMiller indices are a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers , , and , the Miller indices. They are written and denote planes orthogonal to a direction in the basis of the...
, 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 Ho
2Ti
2O
7, Dy
2Ti
2O
7, and Ho
2Sn
2O
7. These materials all show nonzero residual entropy at zero kelvins.
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
triangularIn geometry, the triangular tiling is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees...
,
face-centered cubicThe cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals....
(fcc), hexagonal-close-packed,
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
,
pyrochlorePyrochlore 2Nb2O6 is a solid solution between the niobium end member , and the tantalum end member . The mineral is associated with the metasomatic end stages of magmatic intrusions...
and
kagome latticeA kagome lattice is an arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points...
s with antiferromagnetic interaction. So frustration is divided into two categories: the first corresponds to the
spin glassA spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
, 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
RKKYRKKY stands for Ruderman-Kittel-Kasuya-Yosida and refers to a coupling mechanism of nuclear magnetic moments or localized inner d or f shell electron spins in a metal by means of an interaction through the conduction electrons....
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
Although many properties of spin ice materials have been studied experimentally, little has been revealed about the local accommodation of spin to frustration within the system, since that individual spins cannot be probed without altering the state of the system. Fortunately, 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, and they can be probed without altering the moment configuration.
In 2006 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. As shown in Figure 7a, to mimic the frustration of spin ice, a two-dimensional analog is created by frustrated arrays consisting of square lattices, in which a single lattice is represented by four ferromagnetic islands meeting at a vertex. For a pair of moments at one vertex, it is favorable to have one pointing in and the other pointing out, while unfavorable to have both pointing out or pointing in, due to energy minimization (Figure 7b). For the four moments at one vertex, there are 16 kinds of configurations, as in Figure 7c. The lowest energy vertex configurations is Type I and II, which have two moments pointing in toward the centre of the vertex, and two pointing out. The percentage of Type I and II are 12.5% and 25% respectively.
Using lithographically fabricated arrays, it is possible to engineer frustrated systems to alter the strength of interactions, the geometry of the lattice, the type and number of defects, and other properties which impact the nature of frustration. The lattice parameters range from 320 nm to 880 nm, with a fixed island size of 80 nm × 220 nm laterally and 25 nm thick, which is small enough for magnetic moments to point lengthwise along the islands and big enough to be stable at 300 K. Figure 8 is AFM (Atomic force microscopy) and MFM (Magnetic force microscopy) images of the frustrated lattice. The black and white halves in Figure 8b indicate the north and south poles of the ferromagnetic island. From the MFM images, the moment configuration of array can be easily determined. The vertex types can be directly observed as described in Figure 7c: the pink vertex is Type I, the green vertex is Type III and the blue vertex is Type II. Thus the artificial spin ice is demonstrated.
In this work on a square lattice of frustrated magnets, Wang et al. 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.
Geometric Frustration without Lattice
There is another way to have ``geometrical frustration” which results from the propagation of a local order. A main question that a condensed matter physicist faces is to explain the stability of a solid.
It is some time possible to establish some local rules, of chemical nature, which leads
to low energy configurations and therefore govern structural and chemical order. But this is not a general case and often the local order define by local interactions cannot propagate freely.
It is what is concerned, in this case, by the 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. Even if this recalls another field of physics, that of frustrated spin systems
spin glassA spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
, as introduced above, the method of theoretical investigation largely differ . ”geometric frustration” plays a crucial role in very different fields of condensed matter, ranging from clusters and amorphous solids to complex fluids. It uses geometrical tools.
The general method of approach follows two steps. First, the constraint of perfect space-filling is relaxed by allowing for space curvature. An ideal, un-frustrated, structure is defined in this curved space. Then, specific distortions are applied to this ideal template in order to embed it in the three dimensional Euclidean space. Then the final structure is a mixture of ordered regions, where the local order is similar to that of the template, and defects arising from the embedding. Among the possible defects, disclinations will play an important role.