Ferromagnetic Cluster Spin Wave Theory: Concepts and Applications to Magnetic Molecules

Abstract

Ferromagnetic cluster spin wave theory (FCSWT) provides an exact and concise description of the low-energy excitations from the ferromagnetic ground state in finite magnetic systems, such as bounded magnetic molecules. In particular, this theory is applicable to the description of experimental inelastic neutron scattering (INS) spectra at low temperatures. We provide a detailed conceptual overview of the FCSWT. Additionally, we introduce a pictorial representation of calculated wavefunctions, similar to the usual depiction of vibrational normal modes in molecules. We argue that this representation leads to a better intuitive understanding of the excitations, their symmetry properties, and has links to the energy and wavevector dependence of intensity in the neutron scattering experiments. We apply FCSWT and illustrate the results on a series of examples with available low-temperature INS data, ranging from the Mn 10 supertetrahedron, the Mn 7 disk to the Mn 6 single molecule magnet.Inorganics 2018, 6, 49 2 of 25 assumption of an ordered ground state, which in fact could be regarded as the hallmark feature of the spin wave theories.SWT is widely applied to extended magnets, in one, two, and three dimensions [12][13][14][15]. The elementary excitations correspond to waves, where the spins precess around the polarized orientation in the ground state, with phase differences between the spins described by a wave vector. Applications of SWT to magnetic clusters as small as magnetic molecules, which typically contain a dozen or so magnetic centers or less, is in contrast very scarce. This is so for a number of reasons, one is that the starting assumption of an ordered ground state is strictly never realized, and the validity of the SWT procedure is thus questionable. A notable exception is the class of magnetic molecules in which predominant ferromagnetic exchange interactions between the spins in the molecule stabilize the ferromagnetic or fully polarized ground state. Here, the theoretical methods of SWT become exact at zero temperature, for molecules of any size or composition. Luckily, this comprises quite a number of experimentally relevant magnetic molecules, such as high-spin molecules, Lanthanide containing clusters, or single molecule magnets.Few molecules with antiferromagnetic couplings were analyzed using SWT techniques [16][17][18][19][20][21][22]. The focus in this work is however on the ferromagnetically coupled clusters, for which SWT is guaranteed to work exactly for the ground state and the zero-temperature excitations. When applied to small, bounded ferromagnetic clusters, the SWT techniques and their results are similar in many respects to those in extended systems, but also notable differences exist. Therefore, in order to make the distinction apparent, and to avoid potential confusion, we use the term Ferromagnetic Cluster Spin Wave Theory (FCSWT) for the set of techniques relevant in bounded clusters.It should be mentioned that the use of the word "spin wave" in the context of bounded spin systems...

Inelastic Neutron Scattering Intensities of Ferromagnetic Cluster Spin Waves

“…FCSWT applies to clusters for which the fully polarized states are ground state, or ferromagnetic clusters with Ising‐type magnetic anisotropy in colloquial terms. The FCSWT at zero temperature can be expressed in various ways, as described in ref., with identical results. In this work the “spin lowering operator” technique is followed, as it is most straight forward and amenable for practical implementation.…”

“…The FCSWT calculation is built around the fully polarized ground state |0〉 and yields the spin wave states | k 〉 and their excitation energies ε k . This is perfectly sufficient for discussing the excitation spectrum as observed in INS at zero temperatures, and for understanding the nature of the excitations, as shown in ref …”

“…They however differ by symmetries and geometrical arrangements in space, and magnetism: The Mn 7 and Mn 10 molecules can be described by isotropic exchange‐only spin Hamiltonians whereas Mn 6 is a single‐molecule magnet (SMM) and exhibits an Ising anisotropy in addition. Lastly, the FCSWT for all three systems was worked out before in detail in ref …”

“…The quantity was identified in ref . as the classical amplitude of the deflection of the i ‐th spin from the polarized direction, and was used to establish a pictorial representation of the k ‐th spin wave, allowing an intuitive insight into its physical nature.…”

“…Noteworthy, many molecular systems of interest fall into this category, such as the Mn 19 molecule exhibiting a huge ground state spin of S = 83/2,, the class of Mn 10 molecules with supertetrahedron magnetic cores, many single‐molecule magnets, and other compounds. The spin wave theory for the low‐temperature excitations in bounded ferromagnetic clusters was worked out in detail in a previous work, and called ferromagnetic cluster spin wave theory (FCSWT) in order to avoid confusion and distinguish it from the existing theories. It is mentioned that, in the context of bounded spin systems, the term “spin waves” is not fully satisfying, since the excitations are not dispersive waves as in the extended systems.…”