Dynamical Properties of Unconventional Magnetic Systems

It is surprising that the Programme Committee of the Institute has asked me to give an introduction to the programme of this Institute because I am not a member of the Committee. This is the 14^th in this series of NATO Advanced Study Institutes, and they are well known for being on areas of research which are both topical and interesting and associated with condensed matter physics and statistical mechanics. They have played an important role both in disseminating the latest results, and in bringing together the different approaches of theorists and experimentalists from differing parts of the world. The title of this Institute was chosen by the Programme Committee to be Dynamical Properties of Unconventional Magnetic Systems. I suggest that we examine why the Committee choose this title and what they hoped we would learn from the Institute.

Nearly one-dimensional magnetic systems can be obtained by growing materials in which the magnetic ions have strong exchange interactions along atomic chains but much weaker interactions between the chains. By suitable choice of the magnetic ions and their environment the interactions can be varied to give realisations of Ising chains, XY chains, and Heisenberg chains, and their excitations can be studied in detail using neutron scattering techniques. In ordered three-dimensional antiferromagnets, the excitations are long lived spin waves. Spin 1/2 chains do not have long range order even at low temperature, and furthermore the quantum fluctuations are important and the excitations are very different from those of three-dimensional antiferromagnets. In these lectures I review the excitations observed for the Ising system, CsCoCl₃, and show that the results can be described in terms of domain walls. CsGeO₃ is an example of a spin-Peierls system in which the coupling to the lattice causes a alternation of the exchange constants and the ground state becomes a non-magnetic dimer state. The excitations are then long lived but different from spin waves and best described as the triplet excitations of the ground state dimers. The last example is KCuF₃ which is close to a Heisenberg spin 1/2 chain for which the excitations are spinons and the neutron scattering can be described as arising from pairs of spinons. The experimental results are described and it is argued that spinons are also best understood in terms of the excitations of dimers.

A review of quantum coherence effects in mesoscopic spin systems is presented. We begin with a general introduction to the topic of mesoscopic effects in magnetism and give some specific examples of current interest. We review then theoretical results in single domain magnetism of superparamagnetic type and mention recent measurements on antiferro-magnetic grains (ferritin) and their interpretation in terms of macroscopic quantum coherence. Introducing the effects originating from spin parity in the context of ferromagnetic grains, we discuss antiferromagnetic particles with excess spins and molecular magnets such as the ferric wheel. It is shown that tunneling in such magnets can be tuned by external magnetic fields and is directly observable via the magnetization and the Schottky anomaly in the specific heat. The main part of the review will be devoted to non-uniform magnets and specifically to the quantum dynamics of domain walls or magnetic solitons. In a semiclassical analysis based on coherent spin-state path-integrals an effective model for the domain wall dynamics is derived which includes the effects of spin-wave dissipation and of quantum spin phases (Berry phases). In the presence of a Peierls potential (e.g. due to the discrete lattice) the soliton center can tunnel coherently between the lattice sites and form a Bloch band. Integer and half-odd integer spins have different energy dispersions resulting from interference between soli-ton states of opposite chirality-the internal rotation sense of the soliton. These effects occur in ferro- and antiferromagnets due to the presence of a topological spin phase. For antiferromagnetic chains, this spin phase occurs in addition to the Pontryagin-index phase. We will discuss experimental consequences of this Bloch band structure and show that —in analogy to Bloch oscillations of crystal electrons— static magnetic fields induce large oscillations in the sample magnetization. We will also discuss the extreme quantum limit of spin-1/2 chains in the Ising regime, and show that, quite remarkably, the semiclassical analysis is valid even in this regime. In particular, for antiferromagnetic Ising chains the low-energy excitations are solitons (Villain modes) which have been observed in neutron scattering experiments on CsCoCl₃. We show that the prediction of chirality effects could be tested via the measurement of the off-diagonal components of the dynamical structure factor. The concept of chirality is shown to be of universal character in a variety of magnetic systems, a notable example being the motion of a hole in a 2D antiferromagnetic background. To illustrate the observability of the predicted effects we will give explicit estimates for a number of magnetic systems, in particular for Yttrium iron garnet but also for one-dimensional ferromagnetic chains such as the CoCl₂ salt compound. A common thread in the discussion of quantum dynamics in magnets is provided by the Berry phases and their associated interference effects which can lead to surprising spin parity effects.

Low energy properties of non-metallic magnets can be described by a spin Hamiltonian of the form where the double summation visits each spin pair once.

We review experiments on Li(Ho,Tb)_pY_1-pF₄ which test ideas about quantum critical phenomena in the clean (ferromagnetic) and disordered (spin glass) limits. The outstanding problem of the ‘anti-glass’ discovered for p=0.045 is also described. Finally, we point out the ubiquity of quantum critical points in condensed matter physics.

The excitation spectrum of spin-Peierls antiferromagnets is discussed taking into acount phonon dynamics but treating inter-chain elastic couplings in mean field theory. This gives a ladder of soliton-anti-soliton boundstates, with no soliton continuum, until soliton deconfinement takes place at a transition into a non-dimerized phase.

The physics of non-zero temperature dynamics and transport near quantum-critical points is discussed by a detailed study of the O(N)-symmetric, relativistic, quantum field theory of a N-component scalar field in d spatial dimensions. A great deal of insight is gained from a simple, exact solution of the long-time dynamics for the N = 1 d = 1 case: this model describes the critical point of the Ising chain in a transverse field, and the dynamics in all the distinct, limiting, physical regions of its finite temperature phase diagram is obtained. The N = 3, d = 1 model describes insulating, gapped, spin chain compounds: the exact, low temperature value of the spin diffusivity is computed, and compared with NMR experiments. The N = 3, d = 2, 3 models describe Heisenberg antiferromagnets with collinear Néel correlations, and experimental realizations of quantum-critical behavior in these systems are discussed. Finally, the N = 2, d = 2 model describes the superfluid-insulator transition in lattice boson systems: the frequency and temperature dependence of thé the conductivity at the quantum-critical coupling is described and implications for experiments in two-dimensional thin films and inversion layers are noted.

The study of electronic spin dynamics and coherence is central to gaining insights into a variety of contemporary problems in condensed matter physics. In particular, there is substantial current interest in understanding spin-dependent phenomena in materials systems that involve interfacial and confined geometries such as semiconductor quantum wells [1], metallic magnetic multilayers [2], granular magnetic materials [3], and magnetic semiconductor (MS) quantum structures [4]. For instance, a detailed understanding of electronic spin scattering in low dimensional systems can provide important clues about the fundamental origins of the giant magneto-resistance observed in multilayered and granular magnets. Furthermore, investigations of spin coherence in quantum semiconductor structures are important for exploring coherent spin behavior in solid state systems as a basis for quantum computation and for new generations of magneto-electronic devices [5,6]. In this chapter, we focus on experiments that probe the fundamental spin dynamical properties of a low dimensional (i.e. d < 3) electron or exciton gas that is strongly exchange coupled to a controlled distribution of local moments. We have recently carried out a systematic and detailed study of dynamical spin interactions in such systems by exploiting the concurrent development of sophisticated magnetic nanostructures (e.g. digital magnetic heterostructures and magnetic two-dimensional electron gases) and spatio-temporally resolved spin dynamical probes (e.g. ultrafast Faraday rotation and near-field scanning optical microscopy).

Rare-earth superlattices have been grown by a number of groups using molecular beam epitaxy, and the chemical and magnetic structures determined with x-ray and neutron-diffraction techniques. Superlattices of Dy/Y, Ho/Y, Ho/Lu, and Ho/Dy show helical structures in which the magnetic moments are aligned in ferromagnetic sheets within each basal plane, but the orientation of the moments changes from one plane to the next. The magnetic order then propagates coherently from one magnetic layer to the next with a coherence in both the turn angle and chirality of the helix. In contrast there is no coherence in the ordering of the Ho moments from one Ho layer to the next for Ho/Pr superlattices for which Pr has the dhcp structure and Ho the hep structure. There is also no coherence in the magnetic structures of Ho/Sc superlattices for which the lattice mis-match is so large that the Ho and Sc layers have different basal-plane lattice constants. The behaviour is more complex for Ho/Er, Ho/Tm, Er/Y and Er/Lu superlattices which show coherent structures at temperatures where the moments order either in the basal plane or along the c-axis but when both components order the coherence length decreases as the second component of the moment increases. It is argued that these results are consistent with a model in which if the conduction electrons responsible for the magnetism can propagate through the superlattice a coherent structure is obtained but if they are confined to particular layers when the structure is coherent only over single layers.

The spinwave modes occurring in layered magnetic structures are discussed. Experimental examples obtained by means of light scattering are presented. Emphasis is put on coupled modes occurring in magnetic double layers, i.e. two ferromagnetic films, separated by a nonferromagnetic interlayer. The coupling can be via oscillating dipolar fields caused by the preccessing spins, or via the exchange interaction. It is shown, how the coupled modes can be used, to explore such interactions. These investigations have led to the discovery of the interlayer exchange interaction.

We have carried out extensive neutron and synchrotron scattering studies to unravel the spin density and charge density waves of the incommensurate antiferromagnetic Cr spin structure in epitaxially grown thin Cr(001) films, including surface and interface effects. These studies show that, unlike bulk Cr, in thin Cr(001) films of less than 300 nm the spin density waves are almost entirely longitudinal with a single wave vector, \( \vec{Q} \), propagating in the out-of-plane direction. A thin ferromagnetic Fe cap layer of only 2–3 nm thickness causes a complete re-orientation of \( \vec{Q} \) from longitudinal out-of-plane to transverse in-plane. This re-orientational transition can be understood in terms of frustration effects at the Fe/Cr interface, introduced by monoatomic high steps and kinks. In very thin Cr(001) layers sandwiched between Fe(001) layers a second re-orientation takes place to a single \( \vec{Q} \), out-of-plane transverse spin density wave with the Cr spins now lying in the plane. When the Cr layer thickness, t _Cr , is reduced to smaller than the extension of the spin density modulation wavelength, Λ _SDW , the incommensurate spin density wave collapses and becomes commensurate antiferromagnetic. Lateral Cr thickness fluctuations cause the commensurate antiferromagnetic spin structure to break up into domains. These domains, in turn, mediate a strong non-collinear exchange coupling of adjacent Fe layers.

Neutron scattering experiments with full polarization analysis have been performed with a single crystal of chromium to study the low-energy spin fluctuations in the transverse spin density wave (TSDW) state. A number of remarkable results have been found. Inelastic scattering observed close to the TSDW satellite positions at (1±δ,0,0) does not behave as expected for magnon scattering. In particular, the scattering corresponds to almost equally strong magnetization fluctuations both parallel and perpendicular to the ordered moments of the TSDW phase. As the Neel temperature is approached from below, scattering at the commensurate wave vector (1,0,0) increases in intensity as a result of critical scattering at “silent” satellites (1,0,±δ) being included within the spectrometer resolution function. This effect, first observed by Sternlieb et al, does not account for all of the inelastic scattering around the (1,0,0) position, however. Rather, there are further collective excitations, apparently emanating from the TSDW satellites, which correspond to magnetic fluctuations parallel to the ordered TSDW moments. These branches have a group velocity that is close to that of (1,0,0) longitudinal acoustic (LA) phonons, but assigning their origin to magneto-elastic scattering raises other unanswered questions.

The magnetism of lamellar copper oxides, which are the parent materials of high temperature superconductors, is dominated by the spin 1/2 Cu⁺² ions on the CuO₂ planes. The magnetic behavior of these planes at high temperature is described well by the planar quantum Heisenberg antiferromagnetic (AFM) model, which has long range order only at T = 0. In fact they have three dimensional AFM order due to weak spin anisotropics and interplane couplings. Starting from a Hubbard model with spin orbit and Coulomb exchange couplings, we first derive an effective single-bond magnetic Hamiltonian which contains these anisotropics and couplings. Using a simple Coulomb interaction, as used by Moriya, reveals an interesting hidden symmetry which implies a rotationally symmetric interaction. For orthorhombic LCO, this symmetry is removed only when one adds the different bonds on the lattice, yielding a net Dzyaloshinskii-Moriya antisymmetric anisotropy. In tetragonal symmetry, the zero point quantum spin wave energy (QZPE) generates additional four-fold symmetry terms and delicate higher order interplane interactions, which help select a ground state among states which would otherwise be degenerate due to frustration. Adding several more interplanar interactions yields the full effective magnetic Hamiltonian, which is then used to identify the magnetic structures and competitions among them, leading to phase diagrams in parameter space. These are also used to discuss the critical phenomena which occur near various possible transitions. Specific attention will be devoted to the structures of tetragonal Sr₂CuO₂Cl₂, Nd₂CuO₄ and Pr₂CuO₄ In the former, frustration among layers is lifted by pseudodipolar interactions and by QZPE. In the latter two, the rare earth also participates in the magnetism. Finally, we give full analysis of Sr₂Cu₃O₄Cl₂, which contains Cu₃O₄ planes with an extra Cu ion in the center of every second Cu plaquette. Each of the two types of Cu ions reaches AFM order separately, and the system also exhibits an interesting ferromagnetic moment. A theory in which the intersublattice coupling is described by pseudodipolar interactions allows us to deduce various coupling constants. Similarities in the relative Cu-O-Cu geometries enable us to relate these coupling constants to the nearest and next nearest neighbor interactions in many chain, ladder and lamellar cuprates. These have consequences concerning the magnetic behavior of the latter systems: competing nearest and next nearest neighbor couplings may explain the spin gap observed in the chains, and the new pseudodipolar interactions remove frustration in the interladder coupling.

A nucleation picture of magnetization switching in single-domain ferromagnetic nanoparticles with high local anisotropy is discussed. Relevant aspects of nucleation theory are presented, stressing the effects of the particle size on the switching dynamics. The theory is illustrated by Monte Carlo simulations and compared with experiments on single particles.

The aim of this review is to describe some aspects of the dynamic properties of magnetic multiparticle systems in the micrometer range. These particles include monodisperse magnetic microspheres, and magnetic holes, i.e. nonmagnetic particles dispersed in ferrofluids.

The complementary use of analogue simulations and computer simulations to explore the dynamic properties is demonstrated.

The ability to describe the complex dynamics of magnetic holes in quantitative symbolic dynamic terms using the notion of knot- and braid theory will be discussed. In particular, statistical analysis of braid words, diffusion, memory effects and correlations for the complex dynamics is illustrated.

An ageing phenomenon profoundly affects the dynamic physical properties of spin glasses and other frustrated magnetic systems. The dynamics of spin glasses including the ageing phenomenon is first recapitulated followed by a brief description of the dynamics and an analogous ageing behaviour that occurs in: the ferromagnetic phase of re-entrant spin glasses, concentrated magnetic particle systems and some diluted antiferromagnets at the percolation limit.