Academic Staff - Dr Andrew S. Wills - Physical/Inorganic

Magnetic Properties of Solid State Systems Frustrated Magnetism Application of Symmetry to Magnetic Structure Determination Development of Polarised Neutron Scattering Techniques for Chemistry

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Frustrated Magnetism

Introduction

Kagomé antiferromagnets

Pyrochlore antiferromagnets

Introduction

Once a playground for solid state physicists, magnetic frustration has now opened up into one of the underlying themes of science, that links at a fundamental level exotic electronic states (e.g. spin liquids, spin ices, and superconductivity) with the world of biology (e.g. protein folding), artificial intelligence and next generation electronics.

Its simplest realisation is in magnetism, where we are most often concerned with the pair-wise scalar product of the various spin vectors S_i and S_j and correspondingly a Hamiltonian of the form :

where J is the exchange interaction parameter.

Frustration is simply defined as the inability to minimize each of the individual terms in a Hamiltonian simultaneously, meaning that some of the terms in the Hamiltonian prevent others from being minimized, they frustrate them. This is demonstrated by the triangular antiferromagnet where the spins are constrained to lie within the plane of the triangle (they have XY symmetry). If the exchange interactions are antiferromagnetic, it is clear that no spin configuration can minimize each of the terms in the Hamiltian, i.e. it is impossible for all the spins to be antiparallel with both of its neighours:

This situation leads to two main consequences: firstly the destabilization of a simple ordered spin configuration, a Néel ground state, and secondly a degeneracy that results from the compromise that the system must make if it is to minimse the overall Hamiltonian. This degeneracy, not present in an unfrustrated system, arises because there are two compromise ground states for the system: two structures in which the moments make up the 120º compromise configurations. These configurations are distinguished by their chirality, just as we use (R) and (S) to distinguish chiral molecules in chemistry.

In extended solids much research has focused on two particular geometries, the 2-dimensional (2D) kagomé and 3-dimensional (3D) pyrochlore lattices which are made up of vertex sharing triangles and tetrahedra respectively, as this topology of vertex sharing augments the degeneracy of a single triangle to form macroscopically degenerate systems. This massive degeneracy can prevent conventional Néel order all together as there is no reason to select only a single ground state from the complete manifold, allowing a host of unconventional magnetic orderings to occur.

Kagomé antiferromagnets- is hydronium jarosite a topological spin glass?

The jarosites make up the most studied family of kagomé antiferromagnets. Synthesised from conventional and Redox hydrothermal reactions, they crystallise with a highly flexible structure that allows a wide range of compositions to be formed. They have the general formula AB₃(SO₄)₂(OH)₆ (A=Na⁺, K⁺, Rb⁺, NH₄⁺, Ag⁺, H₃O⁺, 1/2Pb²⁺; B=Fe³⁺, Cr³⁺, V³⁺) and provide access to model frustrated magnets in both the classical, S =5/2, and more quantum, S=3/2, 1 limits.

One of the long standing questions in the jarosites hangs over the spin glass-like state of hydronium jarosite, and brings to the fore questions over the nature of spin glasses, and whether frustration and randomness are both required. While disorder is likely to be always present due to the flexibility of the crystallographic structure, the thermodyanmics and kinetics of the spin glass state appear quite different to those in site disordered spin glasses.

Our work first identified hydronium jarosite as a exotic form of spin glass [1, 2, 3] and showed that the ageing at fixed temperature below Tg obeys the same scaling law as in spin glasses, but at the same time is remarkably insensitive to temperature changes [4]. This difference points to a different underlying mechanism in this kagomé antiferromagnet.

Figure - The local coodination of the Fe in the jarrosites corresponds to a distorted octahedron with different apical and equitorial bond lengths.

Recently we showed also that changes to the Fe-coordination are important[5] and that a distortion at the Fe-site maps, characterised by the ratio of apical and equitorial bonds \Delta, correlates well to the spin glass freezing temperature in (H₃O)Fe₃(SO₄)₂(OH)₆ and the Neel ordering of KFe₃(SO₄)₂(OH)₆:

Figure - Relation between the spin glass freezing temperature in hydronium jarosites and the distortion of the Fe geometry

This is agreement follows well a model in which an evolution associated with spin glass dynamics proceeds via a sequence of zero-energy modes that is retarded by spin or exchange anisotropy.[6]

Figure - the zero energy modes in the kagomé antiferromagnet form closed loops or open lines and allow transformation between ground states through a series of excitations that involve only other ground states, and thus cost no energy.

[1] “Structure and magnetism of hydronium jarosite, a model kagomé antiferromagnet”, A.S. Wills and A. Harrison, J. Chem. Soc., Faraday Trans. 92, 2161-2166 (1996)

[2] “Magnetic correlations in deuteronium jarosite, a model S = 5/2 kagomé antiferromagnet”, A.S. Wills, A. Harrison, S. A. M. Mentink, T. E. Mason, and Z. Tun, Europhys. Lett. 42, 325-330 (1998).

[3] “Magnetic properties of pure and diamagnetically doped jarosites, model kagomé antiferromagnets with variable coverage of the magnetic lattice”, A.S. Wills, A. Harrison, C. Ritter, and I. Smith, Phys. Rev. B 61, 6156-6169 (2000).

[4] “Aging in a topological spin glass”, A.S. Wills, V. Depuis, E. Vincent, J. Hamman, and R. Calemczuk, Phys. Rev. B 62, R9264- R9267 (2000). (communication)

[5] W.G. Bisson and A.S. Wills, unpublished work.

[6] I. Ritchey, P. Chandra, and P. Coleman, Phys. Rev. B 47, 15342 (1993).

Pyrochlore antiferromagnets

The 3-dimensional analogue or the kagomé lattice is the pyrochlore lattice, made up of corner sharing tehrahedra:

Figure - The pyrochlore lattice.

Despite the large degree of frustration magnetic long range order is observed in several pyrochlore. In recent studies of Er₂Ti₂O₇, an example of an XY pyrochlore antiferromagnet, where the moments are constrained by anisotropy to lie in the XY planes perpendicular to the local 3x axis, we performed an experiment using spherical neutron polarimetry to determine the low temperature magnetic structure. [1] Of the symmetry types possible:

Figure - The basis function of the different symmetry types of magnetic structure, labelled according to the corepresentation.

order was found to occur with basis function \psi2:

Figure - A fit to the spherical neutron polarimetry data using the basis functions calculated by corepresentational theory and reverse Monte Carlo fitting. The 3 minima correspond to 3 domains with the same structure.

This structure was proposed to occur when quantum fluctuaions break the degeneracy associated with the highly frustrated XY antiferromagnet. [2]

[1] “Magnetic ordering in the XY pyrochlore antiferromagnet Er2Ti2O7:  a Spherical Neutron Polarimetry study”, A. Poole, A.S. Wills, E. Lelievre-Berna, J. Phys. Condens. Matter., 19, 452201 (2007) (Fast Track Communication).

[2] “Er₂Ti₂O₇: Evidence of order by disorder in a frustrated quantum antiferromagnet”, J.D.M. Champion, M.J. Harris, P.C.W. Holdsworth, A.S. Wills, G. Balakrishnan, S.T. Bramwell, E. Cizmár, T. Fennell, J.S. Gardner, J. Lago, D.F. McMorrow, M. Orendác, A. Orendácová, D.McK. Paul, R.I. Smith, M.T.F. Telling and A. Wildes, Phys. Rev. B 68, 020401(R) (2003).

This page last modified 26 January, 2008

University College London
Department of Chemistry
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