Grado de coherencia entre las técnicas

In the discussion on magnetism, it has been assumed that at low enough temperatures spin systems will find a way to minimise their energy and settle into a long-range ordered state. However, for some systems this is not always possible, and thus the termfrustrationis used to describe the inability of a system to satisfy all the competing microscopic interactions in order to establish a unique global ground state. Frustration was first investigated in 1935 by Pauling in the context of water ice [11], but it was not discussed in the context of magnetic systems until the discovery of spin glasses [12, 13] in the 1970’s (although the low temperature phase in these materials is due to both frustration and a high degree of structural disorder). A general consequence of frustration is the establishment of long-range order at temperatures much lower than expected from the strength of the exchange interactions, and this leads systems to exhibit some unusual properties. At very low temperatures some frustrated systems may eventually order, sometimes in very complicated spin arrangements, but others may stay disordered, for example the spin glasses mentioned above and also spin liquids [14, 15] in which the ground state is dynamic.

An empirical measure of frustration proposed by Ramirez [16] is the “frustration parameter”,f, which capitalises on the fact that in the field of frustrated magnetism dis- crepancies betweenθCW and actual TN are very common. The frustration parameter is defined as

f = θCW

TN

(2.40) so thatf >1corresponds to frustration. The temperature regime betweenθCWandTNthus represents a free spin in a strongly interacting environment. However, care must be taken when applying this criteria as this is almost always true for antiferromagnets [3] because in the discussion presented in Section 2.1.5, the relations used to derive the Curie-Weiss law for antiferromagnets it was assumed that the molecular field on one magnetic sublattice depends only on the magnetisation of the other. Nevertheless,f can be a useful guide if the difference betweenθCWandTNis at least an order of magnitude,i.e.wheref >10. Such materials are said to be strongly frustrated, and they cannot be described by using the mean field theory multi-sublattice picture - instead values off >10signify a more complicated

Figure 2.2: In a simple triangular lattice Ising spin system geometric frustration may be observed, since the third spin is unable to satisfy both antiferromagnetic constraints in order to minimise the ground state energy.

state that is not accessible by mean field theory [16].

In this section a few basic concepts of frustrated magnetism are presented, mostly involving a discussion of how the geometry of the underlying lattice coupled with suit- able interactions can frustrate the ordering of the magnetic moments, such that frustration plays a fundamental role in establishing highly degenerate ground states and exotic low- temperature phases. Some different examples of these geometrically frustrated structures will be considered, as well as a few types of novel magnetic phenomena they support. It should be pointed out that not all frustration is geometrical - for example the spin glasses can have site randomness, where the magnetic sites are arranged randomly in a non-magnetic matrix, and bond randomness, where the material possesses more than one type of magnetic atom and the interactions between the moments are unequal - but this is not relevant to the SrLn2O4materials, and will not be considered here.

There are vast amounts of both theoretical and experimental literature available on magnetic frustration, so only a short introduction to the subject intended here, and [16–20] are suggested as formal books and reviews of the topic.

2.2.1 Geometrical frustration

Geometrical frustration arises in systems where the interactions between the magnetic mo- ments are incompatible with their spatial arrangement in a lattice, so that at low tempera- tures not all of the interaction energies can be simultaneously minimised. A simple example would be to consider an Ising spin system. On a square lattice, the energy of all the antifer-

romagnetic interactions between the spins can be minimised since it is possible to anti-align every spin with respect to its nearest neighbours; but on a triangular lattice this is impossi- ble, as the third spin cannot be simultaneously anti-aligned to both of its nearest neighbours, and this situation is illustrated in Fig. 2.2. The spins would have to compromise, such that there will not be a unique microscopic ground state solution; and in the general macroscopic case there will be a multitude of degenerate low temperature states with the same energy (since the spins in all of these different combinations will be equally unable to minimise their energy). Frustration also tends to enhance quantum fluctuations for low-dimensional spin systems.

In the case of the simple triangular antiferromagnet considered above a non-collinear solution can be found where the moments on each triangle are pointing at 120◦with respect to each other. However, unique ground state solutions of some frustrated magnetic materials are not always possible even for Heisenberg spins [21, 22], so these systems will never show long-range order at finite temperatures. The traditional examples are of spin glasses, where the spins eventually get ‘frozen’ in a random metastable state, and spin liquids where the spins are highly correlated, but fluctuate down to the lowest temperatures. The newer “spin- ice” materials [23, 24] are a particularly interesting examples of spin-liquid materials with a strong local Ising anisotropy - here one dimensional spins are arranged on a corner-sharing tetrahedral lattice and there is no unique way to minimise all of the nearest neighbour in- teractions (and satisfy the local two-in two-out spin rule on each tetrahedron) in order to establish a single ground state. This problem is analogous to stacking H2O molecules to form (hexagonal) water ice [25]. Most intriguingly, it can be shown that the excitations in spin ice (the defects that break the two-in two-out rule) can freely propagate throughout the lattice and can be thought of as magnetic monopoles connected by a semi-infinite string of flipped spins [26].

For real frustrated systems thatdomanage to achieve long range order at low tem- peratures, there may be a multitude of extra (small) perturbations to the Hamiltonian, such as further-neighbour exchange, single-ion anisotropy, magnetic dipolar interactions, quan- tum fluctuations, etc., that lift the degeneracy and stabilise the formation of a unique ground state. These materials often result in a variety of interesting magnetic structures, arrived at

Figure 2.3: Examples of lattices based on triangles and tetrahedra include the (left) two dimensional kagom´e, (middle) three dimensional garnet, and (right) pyrochlore structures.

by novel mechanisms such as order-by-disorder (driven by thermal [27] or quantum [28] fluctuations). New magnets which exhibit the effects of geometric frustration are constantly being discovered. The behaviour of these systems is often complex, with a rich variety of low-temperature properties.

2.2.2 Geometrically frustrated lattices

In the presence of antiferromagnetic exchange interactions, many magnets based on corner- or edge-sharing triangles [29] or tetrahedra [30] can exhibit geometric frustration. The corner-sharing materials are more susceptible to the effects of frustration than than edge- sharing networks. There are a large number of such systems in both two and three dimen- sions, and three of these are illustrated in Fig. 2.3. The two dimensional structures include the (simple) edge-sharing triangular lattice, and corner-sharing triangular networks such as the kagom´e [31, 32] and honeycomb [33] structures. In three dimensions, triangular corner- sharing arrangements include the garnet [34, 35] and kagom´e staircase compounds [36]; while corner-sharing tetrahedra are realised in spinels [37] and pyrochlore [30, 38] com- pounds. Other frustrated magnetic networks are found in square lattices with nearest and next-nearest-neighbour interactions [39], spin chains [40] and face-centered cubic lattices in three dimensions [41, 42]. Thus, there are plenty of systems where frustration can play a fundamental role in establishing the low temperature properties, and recently the SrLn2O4

family of compounds (which shall be described below) have also been suggested as lattices that could give rise to frustrated magnetism.