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

^{*}

## Abstract

**:**

_{10}supertetrahedron, the Mn

_{7}disk to the Mn

_{6}single molecule magnet.

## 1. Introduction

_{10}Gd

_{10}molecular wheel, precise detection of entanglement between molecular qubits by neutron scattering, and many more [6,7,8,9,10]. In all these cases, the most pertinent features are not sufficiently well covered by a set of numbers such as $S$ and $D$, but require a detailed inspection of the structure of the magnetic ground state and the elementary excitations. These are most cleanly observed at low temperatures (where “low” depends on the excitation spectrum and thus on the detailed system at hand), and in this work we accordingly will be concerned with low temperatures, or in fact zero temperature.

_{10}supertetrahedron [23], a Mn

_{7}disk-like molecule [24], and a Mn

_{6}single molecule magnet [25]. These three molecules were studied previously using inelastic neutron scattering (INS), which permits direct observation of spin wave excitations and which is thus a primary tool for their experimental investigation [3,26]. The availability of these data facilitates a comparison to the results of FCSWT. Furthermore, these molecules are well suited for FCSWT, but each of them emphasizes a different characteristic aspect. FCSWT analyses were reported before for Mn

_{10}and Mn

_{7}[23,24], which are reviewed here and significantly extended. Single molecule magnets were, to the best of our knowledge, not yet analyzed with FCSWT, and indeed some additional new aspects need to be considered.

## 2. Concepts

#### 2.1. Basics

#### 2.2. Dimers

#### 2.3. General Concept

#### 2.4. Analogy to Vibrations and Pictorial Representation

_{10}cluster in the next chapter. The spin Hamiltonian is given by:

## 3. Applications

_{10}, Mn

_{7}, and Mn

_{6}, will be discussed in terms of FCSWT. All three molecules exhibit a fully polarized ground state, and were studied in detail by INS at low temperatures [23,24,25]. For Mn

_{10}and Mn

_{7}the discussion follows Refs. [23,24], for the single molecule magnet Mn

_{6}a FCSWT analysis was not attempted before. For all three molecules the spin wave matrix Equation (9) can be analytically diagonalized using symmetry concepts. This is especially true for the Mn

_{10}molecule, for which the matrix is of dimension $10\times 10$ and yet can be fully diagonalized thanks to the molecular ${T}_{d}$ point group symmetry. Mn

_{10}thus establishes a showcase example for the application of symmetry concepts. The Mn

_{7}disk-like molecule is distinguished by its cyclic structure, which results in spin wave excitations which are similar to “real” waves. Finally, the single molecule magnet Mn

_{6}brings in the additional aspect of a comparatively large Ising anisotropy. In fact, in contrast to most single molecule magnets, which are in the so-called strong exchange limit, Mn

_{6}is distinguished by a strong spin-mixing, which is favorable for experimentally observing the spin wave excitations by INS in such systems.

#### 3.1. The Mn_{10} Supetretrahedron

^{III}

_{6}Mn

^{II}

_{4}(μ

_{4}-O)

_{4}(μ

_{3}-N

_{3})

_{3}(μ

_{3}-Br)(Hmpt)

_{6}(Br)] Br

_{0.7}(N

_{3})

_{0.3}∙2MeOH∙3MeCN (H

_{3}mpt = 3-methylpentan-1,3,5-triol), or Mn

_{10}in short, has received significant interest due to its high-spin $S=22$ ferromagnetic ground state, and the chemical link to other compounds such as the Mn

_{19}cluster with even larger spin-ground states [35]. The magnetic core of the cluster consists of six spin-2 Mn

^{III}ions halfway along the edges of a tetrahedron of four spin-5/2 Mn

^{II}ions to form the supertetrahedron shown in Figure 6.

^{III}ions and between the Mn

^{II}and Mn

^{III}ions, respectively, see Figure 6b. The magnetism of the Mn

_{10}cluster can then be described by a Heisenberg Hamiltonian:

^{III}ions neighboring the $i$-th Mn

^{II}ion. Both couplings are ferromagnetic supporting the experimentally observed $S=22$ ground state, and were determined to ${J}_{a}=18.4$ K, ${J}_{b}=7.4$ K. In the INS spectra, two cold peaks at 3.5 meV (peak I) and 7.0 meV (peak II) were observed, and associated to spin wave excitations [23].

_{10}for ${J}_{a}/{J}_{b}=2.54$, the ratio observed experimentally. A threefold degenerate first excited state ${T}_{2}$ is obtained. In these spin wave modes only the outer Mn

^{II}spins show a spin deviation, while the Mn

^{III}spins are not involved at all in the precession. In the next higher lying ${A}_{1}$ mode, all Mn

^{II}ions precess with identical amplitudes, and so do the Mn

^{III}spins, but with spin deflections opposite to the Mn

^{II}spins. The lowest and highest ${A}_{1}$ modes thus appear to be the symmetric and anti-symmetric pairs with respect to the Mn

^{II}and Mn

^{III}precessions. The ${T}_{2}$ and $E$ modes at higher energies show precession of only the core Mn

^{III}spins: in the threefold degenerate ${T}_{2}$ mode, the precessing spins are further apart than in the twofold degenerate $E$ mode, where neighboring spins precess in counterphase, resulting in the highest-energy magnon of the system.

_{10}is shown. The ferromagnetic ground state is determined by the spin quantum numbers $S=22,\text{}M=22$, and two transitions I and II into states with spin quantum numbers $S=21,\text{}M=21$ are observed. The spin wave energy spectrum and the observed transitions are indicated in the inset to Figure 8a, showing that the first ${T}_{2}$ and second ${A}_{1}$ spin wave modes were detected in this experiment. The intense peak I corresponds to the threefold degenerate ${T}_{2}$ mode, and the weaker peak II to the ${A}_{1}$ magnon.

_{10}with those obtained for the five-spin model previously discussed in Section 2.3, see Figure 5. The five-spin model was introduced in Reference [23] as a low-energy approximation to the full magnetic model of Mn

_{10}, by assuming dominant exchange interactions between the six Mn

^{III}spins (${J}_{a}\gg {J}_{b}$), in which case these behave like a giant spin ${s}_{0}={{\displaystyle \sum}}_{i=5}^{10}{s}_{i}=12$. The five-spin model thus reduces the full problem by ignoring the individual magnetic degrees of freedom within the strongly connected core of $s=2$ spins. However, it preserves a complete description of the weakly bound outer spins, which are responsible for the low-energy spectrum of the system. Comparison of Figure 5b and Figure 7 shows that the low energy spin wave modes ${T}_{2}$ and ${A}_{1}$ in the two models are the same in terms of both energies and wavefunctions. The pictorial representation of the spin waves provides us with an immediate justification for the success of the five-spin model for analyzing the low-energy excitations in Mn

_{10}.

_{10}supertetrahedron. Based on the dimensions of the supertetrahedron, the interference term of this transition is expected at low $Q$ of about 1 Å

^{−1}. On the other hand, the wavefunction corresponding to the second ${A}_{1}$ mode has non-zero components on all spins, so that interference is expected to also occur at larger reciprocal space distances. Figure 8b shows the calculated $Q$-dependence for the ${T}_{2}$ and ${A}_{1}$ modes, which are indeed markedly different for the two cases. The blue and red arrows indicate the reciprocal space distances obtained by the above simple considerations, clearly showing that the dominant interference terms of the scattering modes can be anticipated from the pictorial representation of the wavefunctions.

#### 3.2. The Mn_{7} Disk Molecule

_{3})[Mn

_{7}(N

_{3})

_{6}(teaH)

_{6}], or Mn

_{7}in short, was synthesized from a lower-spin ($S=11$) Mn

_{7}compound by a replacement of a peripheral ligand [24]. The fact that a large change of magnetic properties results from a peripheral chemical modification prompted a detailed investigation of the system [24,37]. The molecular structure is shown in Figure 9. Three spin-2 Mn

^{III}ions and three spin-5/2 Mn

^{II}ions are positioned alternately on the outer hexagon with the remaining spin-5/2 Mn

^{II}ion in the center of the disk.

_{7}are depicted in Figure 10a. As always, for the ${A}_{1}$ spin wave the spin precession is completely in phase. The lowest ${A}_{2}$ spin wave is characterized by a dominating spin precession of the central spin, while the spins on the outer ring are essentially not involved. In the remaining five higher-lying spin wave states, it is vice versa, i.e., here the central spin is essentially not involved (for the two $E$ doublets its deflection is strictly zero), and the excitation is concentrated on the ring.

_{7}molecule is missing in the ring system, since its contribution is dominantly located on the central spin which is not present in the ring system. For the other states, the similarity is however obvious, i.e., all details are reproduced. In both systems, a cosine- or sine-type modulation of the spin deflections on the Mn

^{II}and Mn

^{III}sublattices along the ring can be observed. That is, for the highest ${A}_{2}$ spin wave the wave function (for the spins along the ring) is of the form $a|A,0\rangle -b|A,1\rangle \propto a\left(|1\rangle +|2\rangle +|3\rangle \right)-b\left(|5\rangle +|6\rangle +|4\rangle \right)$, with positive real values $a,b$. For the ‘+’ components of the two $E$ doublets the wave functions are of the form $\alpha |E,0,+\rangle +\beta {e}^{-i\phi}|E,1,+\rangle $ and $-\beta |E,0,+\rangle +\alpha {e}^{-i\phi}|E,1,+\rangle $, respectively, and similarly for the ‘−’ components, with positive real values $\alpha ,\beta $. The energetically degenerate ‘+’ and ‘−’ components of an $E$ doublet can be linearly combined to yield the real-valued, cosine- and sine-type wave functions $\alpha \left(|1\rangle +\mathrm{cos}\phi |2\rangle +\mathrm{cos}\phi |3\rangle \right)+\beta \left(|5\rangle +\mathrm{cos}\phi |6\rangle +\mathrm{cos}\phi |4\rangle \right)$ and $\alpha \left(\mathrm{sin}\phi |2\rangle -\mathrm{sin}\phi |3\rangle \right)+\beta \left(\mathrm{sin}\phi |6\rangle -\mathrm{sin}\phi |4\rangle \right)$, which give rise to the $\left(1,-\frac{1}{2},-\frac{1}{2}\right)$ and $\left(0,-\frac{1}{2}\sqrt{3},\frac{1}{2}\sqrt{3}\right)$ patterns of the spin deflections on the sublattices visible in the plots in Figure 10 for the cosine- and sine-type wave functions. The values for $a,b$ and $\alpha ,\beta $ are of course not exactly identical for Mn

_{7}and the ferromagnetic ring, because of the additional coupling to the central spin in the Mn

_{7}molecule, but they are very close.

_{7}to “real” spin waves on a ferromagnetic chain manifests itself also by comparing energies. A ring consisting of six spins allows for six spin wave modes with only a few distinct $k$ values. Increasing the number of spins would allow for more modes, and more $k$ values. When the number of spins goes to infinity, the $\epsilon \left(k\right)$ dependence becomes continuous and the full dispersion relation $\epsilon \left(k\right)$ of the infinite spin chain is recovered. One can then describe the spectrum in terms of usual running waves for translationally invariant models, rather than the standing waves seen in the finite systems. Figure 11 plots both the spin wave spectrum of Mn

_{7}and the dispersion relation of a ${s}_{1}=2$, ${s}_{2}=5/2$ ferromagnetic chain, which is given as [38]:

_{7}molecule and the theoretical dispersion of the infinite chain gives a surprisingly good agreement. In Mn

_{7}the highest ${A}_{2}$ spin wave is somewhat raised in energy due to the lower ${A}_{2}$ spin wave present in Mn

_{7}. It is finally noted that, as shown before for the case of the Mn

_{10}cluster in Section 3.1, the specific structure of the wave functions is also revealed by a characteristic Q-dependence of the INS intensity, which is discussed in Reference [24].

#### 3.3. The Mn_{6} Single Molecule Magnet

_{6}O

_{2}(Et-sao)

_{6}(O

_{2}CPh)

_{2}(EtOH)

_{4}(H

_{2}O

_{2})

_{2}, or Mn

_{6}in short, consists of six spin-2 Mn

^{III}ions forming two triangles bridged by oxygen atoms, as shown in Figure 12. In contrast to most SMMs, Mn

_{6}exhibits relatively weak exchange couplings in comparison to the single-ion anisotropies, which leads to a breakdown of the giant-spin model usually employed for describing SMMs, with corresponding interesting effects [25]. It also results in relatively low-lying spin wave excitations, which is useful for experimental observation by INS. The ferromagnetic interactions between the Mn

^{III}spins in Mn

_{6}result in a $S=12$ ferromagnetic ground state, with a large Ising-type ZFS due to the Mn

^{III}single-ion anisotropies, as it is characteristic for SMMs [4,5].

_{6}is associated with tunneling terms in the full spin Hamiltonian, which are not included in Equation (38). These terms lead to a mixing of the $|M={M}_{max}\rangle $ and $|M=-{M}_{max}\rangle $ states, with a corresponding non-zero tunneling splitting [5]. Therefore, the fully polarized state $|M={M}_{max}\rangle $ and its counter part $|M=-{M}_{max}\rangle $ are not ground states, and the condition for the applicability of FCSWT is strictly violated. However, the tunneling splitting of the states in the $M=\pm {M}_{max}$ and $M=\pm ({M}_{max}-1)$ sectors is tiny in comparison to the energies of these states, and therefore can be safely neglected in the calculation of these energies. This has justified using the Hamiltonian Equation (38) in the analysis of the INS data, and within the same realm the application of FCSWT to SMMs is justified.

_{6}is ferromagnetic $|M={M}_{max}\rangle $ with ${M}_{max}=12$. The spin wave sector is spanned by six basis states obtained again from Equation (8). The Hamiltonian matrix ${H}_{ij}$ in this basis is calculated as usual from Equation (9), yielding a $6\times 6$ matrix of the form:

_{6}are nondegenerate and reflect the inversion symmetry of the spin model. The energy of the lowest excited ${A}_{g}$ state corresponds to the ZFS due to the non-zero single-ion anisotropy of the Mn

^{III}ions. The precession in the higher-energy modes shifts from being dominant at the exterior 2 and 3 spins towards the strongly coupled spin 1 in the higher-energy modes.

_{6}at low temperatures are shown in Figure 13b. In the INS spectrum, six cold magnetic transitions can be identified (I at 1.14 meV, II at 1.38 meV, III at 2.28 meV, IV at 4.19 meV, V at 4.44 meV, and VI at 4.97 meV). The peaks can be well reproduced with the Hamiltonian Equation (38) using the parameters given before [25]. The six experimentally observed INS transitions are immediately associated to the six spin wave modes in Mn

_{6}, as shown in Figure 13c. Therefore, in the case of the SMM Mn

_{6}, all spin waves modes expected from and predicted by FCSWT have been detected in the experiment.

_{6}O

_{2}(Et-sao)

_{6}(O

_{2}CPh(Me)

_{2})

_{2}(EtOH)

_{6}was also reported. In this compound, five peaks were observed at low temperatures, which could be understood using the same spin wave approach discussed here, and be associated to the five lowest spin wave modes.

## 4. Conclusions

_{10}molecule yields exact results for the energies and wavefunctions, and the visualized spin wave modes of this high-symmetry molecule provide insight into symmetries, energies and interference terms of the spin wave peaks observed in the neutron scattering data; (ii) FCSWT applied to the disk molecule Mn

_{7}results in wave-like spin wave states for the spins along the disk and provides an understanding of how the spin wave character progresses towards the extended systems; and (iii) FCSWT can be applied to single molecule magnets because of small tunneling splittings between the ground-state and the magnon sectors, as shown exemplarily for the Mn

_{6}molecule.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of the energy spectrum of a ferromagnetic spin-1/2 dimer, with the spin wave excitations emerging from the $|M={M}_{max}\rangle $ state indicated by arrows.

**Figure 2.**Sketch of the energy spectrum of a ferromagnetic spin-1 dimer, with the spin wave excitations emerging from the $|M={M}_{max}\rangle $ state indicated by arrows. Left: isotropic dimer; right: dimer with Ising type anisotropy.

**Figure 3.**Sketch of the spin states for a single spin of length ${s}_{i}=3$, with the states classified by the magnetic quantum number ${M}_{i}$ or the boson number ${n}_{i}={s}_{i}-{M}_{i}$. The boson-operator technique for representing spins is valid only for small boson numbers or small spin deflections.

**Figure 4.**Pictorial representation of the spin wave states of a dimer. The two states $|k=0\rangle $ and $|k=1\rangle $ of Equation (7) are plotted both for the three-dimensional “cone” representation (left) and the planar “disk” representation (right). The spin vectors are represented by the arrows. The spins are precessing in phase, which for spins with negative coefficient ${c}_{i,k}$ results in an “opposite” orientation of the arrow.

**Figure 5.**(

**a**) Sketch of the fictive five-spin cluster discussed in the text; (

**b**) Pictorial representation of the five spin wave states in this cluster. The red points correspond to the outer spins $i=1,\dots ,4$ and the blue point to the central spin ${s}_{0}$.

**Figure 6.**(

**a**) Ball-and-stick representation of the Mn

_{10}molecule. (Mn

^{II}: light pink, Mn

^{III}: dark pink, O: red, N: blue, C: black, H atoms are omitted for clarity); (

**b**) sketch of the exchange couplings in the Heisenberg Model Equation (27).

**Figure 7.**Pictorial representation of the ten spin wave states in the Mn

_{10}cluster. The red points correspond the Mn

^{II}spins at the corners of the supertetrahedron and the blue points to the Mn

^{III}spins of the inner octahedron.

**Figure 8.**(

**a**) Experimental and simulated inelastic neutron scattering (INS) data of Mn

_{10}at 1.9 K (from Reference [23]). Two magnetic features I and II are observed, which correspond to excitations from the ferromagnetic ground state to the first ${T}_{2}$ and the second ${A}_{1}$ spin wave mode. The inset shows the spin wave energy spectrum, and arrows indicate the observed INS transitions; (

**b**) calculation of the $Q$-dependence of the observed ${T}_{2}$ and ${A}_{1}$ spin waves. The arrows indicate the expected dominant component of the $Q$-modulation based on the eigenvector plot in Figure 7.

**Figure 9.**(

**a**) Ball-and-stick representation of the Mn

_{7}molecule (Mn

^{II}: green; Mn

^{III}: yellow; O: red; N: blue; C and H atoms are omitted for clarity). The arrows indicate the classical ground state configuration; (

**b**) sketch of the exchange couplings in the Heisenberg model Equation (30).

**Figure 10.**Pictorial representation of the spin wave states in (

**a**) the Mn

_{7}cluster and (

**b**) a hexanuclear ${s}_{1}-{s}_{2}$ ferromagnetic ring with homogenous coupling constants. For the $E$ doublets, the cosine- and sine-type excitations are shown to the left and right, respectively.

**Figure 11.**Spin wave energies of the Mn

_{7}cluster (crosses) and the dispersion relation of a hexanuclear ${s}_{1}-{s}_{2}$ ferromagnetic chain with homogenous coupling constants (lines), plotted as function of the (pseudo) wave vector $k$, discussed in the text.

**Figure 12.**Ball-and-stick representation of the magnetic core of the Mn

_{6}single molecule magnet. (Mn

^{III}: green; O: red; N: blue; C and H atoms omitted).

**Figure 13.**(

**a**) Pictorial representation of the spin wave states in the Mn

_{6}molecule; (

**b)**low-temperature INS data for Mn

_{6}(recreated from Reference [25]). Six magnetic features are indexed, which correspond to transitions from the ferromagnetic ground state to the spin wave modes; (

**c**) calculated spin wave energy spectrum, with arrows indicating the observed INS transitions.

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Prša, K.; Waldmann, O.
Ferromagnetic Cluster Spin Wave Theory: Concepts and Applications to Magnetic Molecules. *Inorganics* **2018**, *6*, 49.
https://doi.org/10.3390/inorganics6020049

**AMA Style**

Prša K, Waldmann O.
Ferromagnetic Cluster Spin Wave Theory: Concepts and Applications to Magnetic Molecules. *Inorganics*. 2018; 6(2):49.
https://doi.org/10.3390/inorganics6020049

**Chicago/Turabian Style**

Prša, Krunoslav, and Oliver Waldmann.
2018. "Ferromagnetic Cluster Spin Wave Theory: Concepts and Applications to Magnetic Molecules" *Inorganics* 6, no. 2: 49.
https://doi.org/10.3390/inorganics6020049