Spin dynamics in inhomogeneous and defected low dimensional systems

The systems studied in this dissertation are assumed to be free of thermal fluctuations. The assumption is valid in low temperatures or when the magnetic moments, e.g. of macrospins, are sufficiently large. The equilibrium states of the systems under such an assumption correspond to minimum of energy instead of thermodynamic potentials involving entropy. Another consequence of the absence of fluctuations is that the solutions of the secular equations show real frequencies and, thus, describe infinitely lived, oscillating excitations. The complex frequencies mentioned in semiinfinite systems of Chapter 3 do not contradict this statement because the attenuation there is a result of energy radiation into the semi-infinite medium by a way of ”leaky waves” [14, 15, 34, 35] rather than of dissipation mediated damping. It is known that from among the elementary excitations studied for signal transmission the magnetic ones show the lowest damping and the related large attenuation lengths [36, 37]. The neglect of damping is, therefore, a justified first approximation. Defects of periodic structures are known to support localized states. This is a result of the fact that complex wave vectors are forbidden in the infinite systems to avoid solutions growing to infinity but they are allowed in semi-infinite media provided that they describe spatially attenuated partial waves. In principle the localized states occur outside the frequency ranges of bulk waves, because in the opposite case the energy is radiated into the medium and the secular state transform into resonances also known as leaky waves. Exceptions are known [17, 16, 38] but rare. However the phenomenon has been found here on the interface of two ferromagnetic chains coupled in an antiferromagnetic manner provided that the radiative bands of both chains are different. The design of devices capable of transmitting spin waves at a desired speed in a defined frequency range is routinely effectuated with the use of micromagnetic computations. We have employed the techniques to study configurations and spin waves in magnetic particles of a currently available material (permalloy) and of feasible sizes. As the propagative properties change abruptly when switching between stable-metastable ferromagnetic and antiferromagnetic configurations we have studied conditions allowing one to recover the initial antiferromagnetic configuration which is stable in the absence of external field but does not reappear once the ferromagnetic configuration is set with a field of sufficient intensity in a chain of identical elliptically shaped macrospins. Two ways of achieving this goal have been proposed: i) making every second ellipse narrower that affects the shape anisotropy and ii) introducing anisotropy difference as a material parameter. It is not surprising that a typical classification of modes at k = 0 into acoustic and optical (as is symmetry protected in the phonon case) does not hold when the shapes of macrospins are altered but it is remarkable that this classification works perfectly for the particles of different material anisotropy. The mechanism of FM→AF switching relies on the softening of a mode. One can remark that the soft mode in this transformation is acoustic at k = 0, i.e. analogous to the one driving the FM→FM configuration change. Thus, the paths the system follows in the space of its parameters in both transformations begin in the same way. This result comes from the fact that the shape anisotropy does not allow for any soft mode involving out of plane displacements of the magnetization. An attempt to construct a system mimicking the behavior of a macrospin has turned out successful for as few as two single spins in the macrospin. Not only is this a ”toy” model for macrospin considerations but it is also conceivable in the realm of molecular magnets where the interactions are controlled by appropriate ligands. To behave like macrospins the pairs of spins, called here bi-spins, must show an intraparticle exchange ferromagnetic interactions in addition to the ubiquitous dipolar interactions that are also assumed between different bi-spins. Under such assumptions we have availed ourselves of a liberty to vary all the parameters of chains constructed of the bi-spins. A number of stable/metastable configurations have been found as functions of the distance of the bi-spins in the chain. The most striking is a transformation unifying properties of first and second order phase transition with bi-spin distance as a control parameter. This is in fact an example of ideal switching that transforms two very different configurations without hysteresis. The sparsity of degrees of freedom in the chains of bi-spins allows one to watch the behaviour of all the modes. The rule is that the high frequency modes involve significant out-of plain semiaxes of the precession ellipses whereas the ellipticity (i.e. departure from circularity of the precession ellipse) grows with decreasing frequency so that the semi long axis of the ellipse lies in the plane of the chain. The ellipticity becomes infinite in the soft mode. This is in agreement with the results obtained for massive macrospins. We have demonstrated that the soft mode polarization coincides with the eigenvector of the Hessian matrix associated with the vanishing eigenvalue. A practical realization of a system with variable distance of macrospins would involve macrospins deposited on an elastic substrate deformable by an applied stress. Using an example of a very small macrospin consisting of four voxels arranged in a square we have demonstrated that the voxels behave, in spite of being parts of a continuum, in a very close way to a square particle constituted by single spins with ferromagnetic exchange interactions extended to the nearest neighbours. At this example we have also found that the sequence of configurations under a field applied parallel to the square diagonal is quite universal provided that the exchange interactions prevail over the dipolar ones. The system then undergoes one continuous and one discontinuous transition independently of the exchange interaction parameter. The ratio of the field intensities of both transitions does not depend on the interaction details.