Magnetic Exchange Interactions in the Molecular Nanomagnet Mn_{12}
A. Chiesa, T. Guidi, S. Carretta, S. Ansbro, G. A. Timco, I. Vitorica-Yrezabal, E. Garlatti, G. Amoretti, R. E. P. Winpenny, P. Santini
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Magnetic exchange interactions in the molecular nanomagnet Mn A. Chiesa,
1, 2
T. Guidi, S. Carretta, S. Ansbro,
4, 5
G. A. Timco, I.Vitorica-Yrezabal, E.Garlatti, G. Amoretti, R. E. P. Winpenny, and P. Santini Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a di Parma, I-43124 Parma, Italy Institute for Advanced Simulation, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany ISIS facility, Rutherford Appleton Laboratory, OX11 0QX Didcot, UK School of Chemistry and Photon Science Institute,The University of Manchester, M13 9PL Manchester, UK Institut Laue-Langevin, 71 Avenue des Martyrs CS 20156, Grenoble Cedex 9 F-38042, France (Dated: January 18, 2021)The discovery of magnetic bistability in Mn more than 20 years ago marked the birth of molec-ular magnetism, an extremely fertile interdisciplinary field and a powerful route to create tailoredmagnetic nanostructures. However, the difficulty to determine the interactions within the core ofcomplex polycentric molecules often prevents their understanding and can hamper addressing im-portant fundamental and applicative issues. Mn is an outstanding example: although it is theforefather and most studied of all molecular nanomagnets, an unambiguous determination even ofthe leading magnetic exchange interactions is still lacking. Here we exploit four-dimensional inelas-tic neutron scattering to portray how individual spins fluctuate around the magnetic ground state,thus fixing the exchange couplings of Mn for the first time. Our results demonstrate the power offour-dimensional inelastic neutron scattering as an unrivalled tool to characterize magnetic clusters. The ability to store magnetic information in a singlemolecule was reported for the first time in the Mn poly-metallic complex [1]. Many further breakthroughs fol-lowed from studies of this molecule, including the obser-vation of macroscopic quantum tunneling of magnetiza-tion [2, 3], and the discovery that it can be used to builddevices based on the Grover algorithm [4]. The phrase”single molecule magnet” was invented to describe thephysics of Mn , and this molecule inspired the entirefield of molecular magnetism, which continues to produceremarkable science [5–18]. However, the understandingof complex polycentric molecules is often limited due tothe difficulty to determine the interactions within thecore, thus hampering the addressing of important fun-damental and applicative issues. Mn is a particularlystriking example: in spite of hundreds of papers there isnot even an unambiguous description for the leading in-teractions of this archetypal molecule, twenty-five yearsafter it fathered a new field of science. Thus, the debateabout Mn is still completely open, as witnessed by re-cent studies [19–21].The phenomenology of molecular nanomagnets resultsfrom a number of interactions in the magnetic core, whereisotropic exchange couplings are usually leading and var-ious types of anisotropic single- and two-ion terms actperturbatively. The interplay of these interactions re-sults in a multitude of physical behaviors, usually de-scribed in terms of simplified effective models. Theseare parametrized to capture distinctive low-temperatureand low-frequency properties, but in many cases withcomplex cores the determination of the fundamental un-derlying spin Hamiltonian is still a challenge. Moleculesdisplaying magnetic remanence like Mn are usually de-scribed in terms of phenomenological ”giant spin” mod- els, where a single quantum spin S ( S = 10 in Mn ) rep-resents the magnetic core as a whole [22, 23]. Althoughthis approach is cost-effective in terms of model complex-ity, it leaves in the shadows the nature of the giant spin atthe atomic level, hindering the tailoring of the magneticcore for improved performance in fundamental or applica-tive issues [24]. Moreover, the many-spin character of thecore emerges already in the low-energy physics (see, e.g.,[11, 25]). Here we close this long-standing unresolvedcase: we exploit four-dimensional inelastic neutron scat-tering [8] to portray the spin precession patterns, whichare unambiguous fingerprints of the magnetic Hamilto-nian, and we thus pinpoint the exchange couplings ofMn for the first time. Our results open unprecedentedprospects in understanding magnetic spin clusters andmotivate the synthesis of new polycentric nanostructures,where the set of interactions is optimal for specific fun-damental issues or applications.Most of the proposed models for Mn are based on aset of four isotropic exchange parameters, reported inthe schematic representation in Fig. 1a (with J = J ′ ).The spin Hamiltonian also includes anisotropic terms ac-counting for the uniaxial behavior of the system and itreads: H = X m 10 and M = ± S = 10 multiplet. The slight asymmetry of the peak is dueto the instrumental resolution function of the TOF spectrom-eter. Peaks II - VI are inter-multiplet transitions to differentexcited S = 9 multiplets. The broad peak at 3 meV is aphonon, as shown by the monotonic increase as Q of theassociated form factor. that the spins are not locked in a maximally-alignedstate due to quantum fluctuations (see Table S1 in [26]).The local distribution of moments, µ n , is stable over arange of exchange constants and hence is not sufficientto fix the magnitude of the exchange interactions J mn uniquely. It is intuitively clear that exchange is probedmore effectively through excitations that break theinternal alignment of Mn spins in their ground state.Just like spin waves in bulk magnetic compounds, theirenergies and structure directly reflect the values ofexchange constants. These excitations correspond topeaks II - VI in Fig. 1b and represent inter-multiplet FIG. 2. (a) Form factor for the intra-multiplet transition | S = 10 , M = ± i → | S = 10 , M = ± i , i.e., S ( E, Q ) for E = 1.25 meV (giant-spin excitation I ). The inset shows theequivalent real-space information, that is the distribution ofthe static magnetization of the giant spin over the three in-equivalent Mn sites, µ n = h S = 10 , M = 10 | s zn | S = 10 , M =10 i . The values (in µ B ) µ =-1.2 (2), µ = 1.7 (0.15) and µ =2 (0.15) are extracted directly from the form-factor andcompare well with polarized neutron diffraction (-1.17, 1.84,1.90) [32] and NMR (-1.3, 1.8, 1.8) [33] data on a slightly dif-ferent variant of Mn . (b) S ( E, Q ) as a function of Q x , Q y and E , and integrated over the full Q z range [30]. The energywindow contains the intermultiplet peaks II and III of Fig.1b. transitions between the ground | S = 10 , M = ± i doublet and a set of excited | S = 9 , M = ± i doublets.Although these Q -integrated energy spectra, togetherwith susceptibility (Fig. S2 in [26]), provide constraintson the set of exchange constants, they are not selec-tive. Conversely, a clear identification of the | S = 9 i wavefunctions is achieved thanks to the measured Q -dependencies, which contain detailed information onthe composition and symmetry of the states involved inthe transition. For example, Fig. 2b shows S ( E, Q ) as afunction of Q x , Q y and E , and integrated over the full Q z range. The energy interval spans peaks II and III of Fig. 1b, whereas constant-energy cuts of S ( E, Q ) for E corresponding to all the peaks in Fig. 1b are shownin Fig.3. The great amount of information available inthese experimental data is immediately evident. The S ( E, Q ) data fully characterize the low-lying multiplets,and make it possible to identify the five exchangeparameters. The simulation of these data (Figure 3c)unequivocally establishes the five exchange parameters(in meV): J = -1.2(1), J = 3.2(2), J = 6.6(3), J =0.55(5), J ′ = 0.30(5). The agreement between calcula-tion and experiment is very good and the model also fitsthe magnetic susceptibility (Fig. S2 in [26]) and peakpositions (Figure 1b). There is just a slight discrepancyfor the position of peak VI , whose fine-tuning requiresadditional small parameters in Eq. 1 [26]. As expected inbroad terms from the internal structure of the giant-spin(Fig. 2a), antiferromagnetic couplings between Mn and Mn ions are leading. The coupling between thefour Mn ions is ferromagnetic, whereas that betweenthe eight Mn ions is weakly antiferromagnetic.The information on eigenstates is so rich that even subtlevariations of exchange parameters alter these maps. Forinstance, a single parameter is usually assumed for theexternal Mn ring, i.e., J = J ′ in Fig. 1a. Althoughthese constants are an order of magnitude smaller thanthe leading ones, by enabling J = J ′ we can quantifythem separately [26]. The effect of the difference J − J ′ stands out in Fig. 3, showing also (panel d) simulationsobtained with J = J ′ = 0 . 42 meV. The intensitydistribution in the intermultiplet maps is noticeablydifferent, reflecting a change in composition of theexcited S = 9 multiplets. It is worth noting that smallmodel variations of this type have significant impact onthese maps, but negligible effects on the energy spectrumand susceptibility.The information on eigenstates collected in reciprocal( E, Q ) space can be made intuitive by using an equiva-lent description in terms of time and position variables,i.e., by portraying the precession pattern of the twelveMn spins associated with each excitation. Indeed, the Q dependence of a peak at energy E p reflects the spatialpattern of the spins preceding around z with frequency E p /h , after a resonant perturbation has brought amolecule from its M = 10 ground state into a superposi-tion state with a small component on the correspondingexcited M = 9 state [26]. These precession motions arein a one to one correspondence with the form factor S ( E p , Q ), as both are set by the same reduced matrixelements [26]. For a generic weak perturbation (e.g., a δ -pulse), the resulting motion will then be a weightedsuperposition of these single-frequency contributions.Precession patterns, directly extracted from experi-mental data, are shown in Fig. 3e and represent themolecular counterpart of spin-wave excitations in bulkferromagnets. The difference in the spin dynamics FIG. 3. (a) INS energy spectrum (same as Fig.1b). (b) Constant-energy cuts for S ( E, Q ), integrated over the full Q z range,obtained from measurements at T =1.5 K for incident neutron energies of 4.2 meV (first column, peak I ) and 15.4 meV (peaks II , III , IV , V , VI ) [30]. Each map is normalized to its maximum. (c) Corresponding simulated maps, obtained with parameters(in meV) J = -1.2 (1), J = 3.2 (2), J = 6.6 (3), J = 0.55 (5), J ′ = 0.30 (5) , d =-0.315 (2). Eigenstates are listed in Table S1[26]. Row (d) highlights the effect of a slight variation of exchange parameters ( J = J ′ =0.42 meV is assumed). Peaks IV and V are too close in energy to extract individual maps, and only their sum is addressed. (e) Precession pattern of the individualMn spins for excitations I , II and III . For each excitation, arrows represent the twelve vectors ( h s xn ( t ) , s yn ( t ) i ) describing thespatial pattern of the spins preceding around z , after a resonant perturbation has brought a molecule from its M = 10 groundstate into a superposition state with a small component on the corresponding excited M = 9 state. All the spins precede withthe same frequency E/h and dashed circular arrows indicate the direction of the spin precessions for two representative sites.Preparing the system in an initial state with opposite M would induce an opposite precession of the spins. The two panels forexcitation II correspond to a pair of degenerate states (Table S1 [26]). For peaks IV , V and VI experimental form factors aremore noisy or unresolved. Their precession pattern is not directly deduced from data, and is obtained by simulations of thebest-fit Hamiltonian (Fig. S5 in [26]). associated to the various transitions is evident: intransition I all the spins rigidly precess conserving thesame total-spin modulus of the ground state, as expectedfor a giant spin excitation. Conversely, the precession pattern of all the other peaks is characterized by a zerototal spin, demonstrating the inter-multiplet nature ofthe transitions. In addition, the different symmetriesof the excited states (Table S1, [26]) produce clearsignatures in the precession patterns.The present results finally characterize the exchangeinteractions in the archetypal single-molecule magnetMn , enabling us to draw for the first time a soundpicture of the eigenstates beyond the giant spin model.This will be the starting point to address importantissues in the understanding of this molecule, which arestill not really solved after more than twenty years ofresearch. For instance, the relaxation dynamics of Mn should be influenced by the low-lying excited multiplets,partially overlapping with the ground one (e.g., theselead to additional relaxation and tunneling pathwayswith respect to the giant spin model). In general, theseresults open remarkable perspectives in understandingnanomagnets with complex polycentric core. These arestill relatively little explored and understood but are offundamental importance, with potential applications inthe longer term. We mention among others moleculeswhere the role of anisotropy is not perturbative, like inpresence of Co [34] or f -electron ions [18]. These canconvey their large anisotropy through exchange to thewhole core, thus producing large anisotropy barriers orexotic magnetic states (e.g., toroidal or chiral). On theopposite side, we mention small-anisotropy moleculeswhere the pattern of exchange couplings results in frus-tration, which is important both for fundamental andapplicative issues [16]. 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