We have performed density-functional-based electronic structure calculations on a single Fe8 molecular nanomagnet. Our calculated total moments and local moments are in excellent agreement with experiment. By including spin–orbit coupling we determine the easy, medium, and hard axes and find the ordering of the principle axes also agrees with experiment. From our calculated anisotropy Hamiltonian, we calculate the oscillations in the tunnel splittings and compare to the experimental results

Khanna, S. N.

Kortus, Jens

Pederson, Mark R.

VCU Scholars Compass

Virginia Commonwealth UniversityVCU Scholars CompassPhysics Publications Dept. of Physics2002Electronic-structure-based investigation ofmagnetism in the Fe8 molecular magnetMark R. PedersonNaval Research LaboratoryJens KortusMax-Planck-Institut für FestkörperforschungS. N. KhannaVirginia Commonwealth UniversityFollow this and additional works at: http://scholarscompass.vcu.edu/phys_pubsPart of the Physics CommonsPederson, M. R., Kortus, J., Khanna, S. N. Electronic-structure-based investigation of magnetism in the Fe8 molecularmagnet. Copyright © 2002 AIP Publishing LLC.This Article is brought to you for free and open access by the Dept. of Physics at VCU Scholars Compass. It has been accepted for inclusion in PhysicsPublications by an authorized administrator of VCU Scholars Compass. For more information, please contact libcompass@vcu.edu.Downloaded fromhttp://scholarscompass.vcu.edu/phys_pubs/131Molecular Magnets I Bruce Harmon, ChairmanElectronic-structure-based investigation of magnetism in the Fe8molecular magnetMark R. Pedersona)Center for Computational Materials Science, Code 6390, Naval Research Laboratory, Washington,DC 20375-5000Jens KortusMax-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstr. 1, D-70569 Stuttgart, GermanyS. N. KhannaDepartment of Physics, Virginia Commonwealth University, Richmond, Virginia 23284-2000We have performed density-functional-based electronic structure calculations on a single Fe8molecular nanomagnet. Our calculated total moments and local moments are in excellent agreementwith experiment. By including spin–orbit coupling we determine the easy, medium, and hard axesand find the ordering of the principle axes also agrees with experiment. From our calculatedanisotropy Hamiltonian, we calculate the oscillations in the tunnel splittings and compare to theexperimental results. © 2002 American Institute of Physics. @DOI: 10.1063/1.1450786#Recent experiments on molecular magnetic clusters1–7have revealed the phenomena of quantum tunneling of mag-netization. For systems with a second-order anisotropyHamiltonian @e.g., W52DSz21E(Sx22Sy2)# there are twotypes of experiments that can be performed. In the first ex-periment, which is primarily applicable to systems that areexactly or approximately uniaxial ~e.g., D@E! with an easyaxis (D.0), a magnetic field is applied parallel to the easyaxis (Bz) and states of different M s on opposite sides of thebarrier are brought into resonance allowing for tunnelingmechanisms. This experiment allows for the direct measure-ment of D since the resonance conditions correspond to mag-netic fields which are an integer multiplied by D and funda-mental constants. For nonuniaxial systems, which can alwaysbe described by D.0 and E non-negligible, Garg has shownthat a second experiment allows for the measurement of@2E(D1E)#1/2.8 In this experiment, the anisotropy Hamil-tonian splits the 2S11 states into S two-fold and one non-degenerate states. Upon application of a magnetic field alongthe hard axis (Bx) the pairs of two-fold states cross with afrequency proportional to @2E(D1E)#1/2 which again al-lows for tunneling mechanisms to occur when the resonancecondition is satisfied. The Mn12–acetate molecule with a mo-lecular formula Mn12O12(OOCCH3)16(H2O)4 is a well stud-ied system which fits into the first category ~e.g., D.0 andE50!.1–3 The octanuclear iron ~III! oxo-hydroxo cluster(Fe8)4–6 is another molecular magnet which allows for theobservation of the second type of tunneling mechanism.As pointed out earlier by van Vleck,9 the magnetic an-isotropy Hamiltonian, which ultimately controls the afore-mentioned tunneling resonances, arises because of spin–orbitcoupling and other relativistic terms. For over a decade, ithas been recognized that the calculation of magneticanisotropies are, in principle, possible within density-functional theory ~DFT!10 and many researchers have per-formed such calculations on solids and films.11 Problems as-sociated with the accurate density-functional-baseddetermination of MAE in the solid state have been identifiedand the role of incomplete orbital polarization has beenshown to be one issue related to inaccuracies in the solid.The electronic and magnetic structure of theMn12–acetate molecule has recently been determined withinthe framework of DFT.12,13 The resulting electronic structureshowed an insulating behavior and all magnetic propertieswere in reasonable agreement with experiment. In particular,the total moment, ferrimagnetic ordering, and second-orderanisotropy parameter matched the experimental data quitewell. The good agreement for the case of Mn12 may be rela-tively unsurprising since the on-site filling of Mn(3d) statesis determined entirely by strong covalent bonding to neigh-boring ligands or to strong Jahn–Teller distortions. In thiswork, we describe recent efforts at deriving the same prop-erties for the Fe8 molecule and compare our results to ex-periment.The DFT calculations14 discussed herein were performedwith the all-electron Gaussian-orbital-based NRLMOLprogram.15 All calculations employed the Perdew–Burke–Ernzerhof generalized-gradient approximation for the densityfunctional.16 NRLMOL combines large Gaussian-orbital-basissets, numerically precise variational integration, and an ana-lytic solution of Poisson’s equation in order to accuratelydetermine the self-consistent potentials, secular matrix, totalenergies, and Hellmann–Feynman–Pulay forces.17 The ex-a!Author to whom all correspondence should be addressed; electronic mail:pederson@dave.nrl.navy.milJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 200271490021-8979/2002/91(10)/7149/3/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:128.172.48.58 On: Mon, 19 Oct 2015 17:53:43ponents for the single Gaussians have been fully optimizedfor DFT calculations.18 The basis set for the fluorinated clus-ter consisted of a total of 1466 contracted orbitals while thebasis set for the brominated cluster consisted of a total of1562 contracted orbitals. Basis sets are available upon re-quest. Using x-ray data deposited at the Cambridge Crystal-lographic Data Center19 we generated the Fe8 cluster. Forthese calculations, we follow the same method used in Ref.12 to start our calculations with overlapping atomic poten-tials. To further allow for the possibility of spin ordering, weadd to this potential an empirical starting potential whichfavors the spin ordering shown in Fig. 1. After the first itera-tion, this potential is removed and all the electronic and spindegrees of freedom are optimized variationally.The structure of the Fe8 cluster is shown in Fig. 1. Theapproximate D2 symmetry observed in the molecule,19 is for-mally broken by the presence of halide atoms and waters ofcrystallization. The central iron atoms are connected by oxo-hydroxo bridges to the four outer iron ions. The large spheresshow the iron Fe~III! ions with a d5 electron configuration.The ferrimagnetic coupling of spins between the eight Featoms results in an S510 spin ground state20 and is illus-trated by arrows inside the spheres in Fig. 1. The organictacn rings are very important for stabilizing the magneticcore of the molecule because the three pairs of nitrogen dan-gling bonds complete a quasi-six-fold environment for the Featoms. Further, the tacn rings separate the Fe8 clusters in thecrystal, resulting in negligible intermolecular dipole fieldswhich are typically on the order of 0.05 T.6 The resultingformal charge states are nominally Fe31, (OH)21 O22, andtacn0 leading to a molecule with an overall formal chargestate of 18 which may then be compensated by the eightnegatively charged halide ionsIn order to make the problem computationally tractable,we have adopted an isolated Fe8 geometry with the D2 sym-metry plus inversion, and replaced the Br anions by F anions.This leads to a 186 atom Fe8 complex with 50 inequivalentatoms. In the actual crystal, the halide ions break this sym-metry. The magnetic core with iron and the oxo-hydroxobridges is unchanged by our chosen symmetry operations.The results of our calculations ~Fig. 1! confirm that thereare two minority spin iron atoms and six majority spin ironatoms, all of which carry a local moments of approximate5 mB . We find a total spin moment of 20 mB . The density ofstates for the Fe8 complex is summarized in Fig. 2. We haveused a Fermi function with an electronic temperature of0.001 a.u. for the occupation of states near the Fermi level.The larger spheres in Fig. 1 are isosurfaces of the spindensity showing that we indeed obtain the same magneticordering observed in experiment by Pontillon et al.21 How-ever some degree of spin polarization is visible in the regionof the nominally closed-shell fluorine atoms. The minor po-larization of the halide anions may be quenched in the solidby a variety of interactions which include additional Made-lung stabilization or polarization due to the inclusion of thewaters of solvation. Alternatively, additional interactionswhich may be improperly accounted for within the existingapproximations to the DFT could be needed. A more detaileddiscussion will appear elsewhere. The recent experiments byPontillon et al.21 confirm not only the ferrimagnetic orderingbut find smaller local moments on the two minority-spin ironatoms than on the six majority-spin iron atoms. In order toascertain changes in the local moments as a function of atomtype, we placed a sphere of 1.32 Å around each iron, andcalculated the net moment inside each sphere. For the twoiron atoms with minority spins, we obtain a moment of23.6 mB , whereas the majority-spin irons have moments3.8 mB and 3.9 mB which is in qualitative accord with experi-ment. The plot of the spin density also confirms the correctspin polarization of the irons by showing a spherical spinFIG. 1. Optimized geometry and spin configuration of the Fe8 cluster. Thelarge balls show isosurfaces of the spin density at 60.08e/a.u.3. Arrowsrepresent the ferrimagnetic spin ordering in the cluster. On top and bottom,a fluorine atom can be seen, which also has small spin polarization.FIG. 2. Total and projected density of states for the fluorinated Fe8 complexnear the Fermi level as calculated within GGA. The halide p states appear atthe Fermi level.7150 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Pederson, Kortus, and Khanna [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:128.172.48.58 On: Mon, 19 Oct 2015 17:53:43density around the Fe as expected for a closed-shell d5 elec-tron configuration.To further address the experiments on this molecule, wehave used a variant of the methodology discussed in Ref. 12to determine the spin–orbit energy as a function of differentmagnetization directions which yields an anisotropy Hamil-tonian with the general form H5(x ,ygxySxSy . Diagonaliz-ing this particular anisotropy Hamiltonian defines the set ofprincipal axes and in the most general case identifies theeasy, medium, and hard axes. We obtain the following eigen-values for the diagonal anisotropy Hamiltonian: gxx526.325 K, gyy526.2165 K, and gzz526.8045 K. Themedium axis is aligned along the two central iron atoms~Fe1–Fe2! in the middle of Fig. 1. All axes are in very goodagreement with experiment.22 From the eigenvalues of thespin Hamiltonian, we obtain a classical barrier of 59 K be-tween the easy and hard axis, and 48 K between the easy andmedium axis. Translating these results into the form pro-posed by Barra et al.4,22 ~e.g., W52DSz21E(Sx22Sy2!would correspond to D520.53 K and E50.054 K, with theanisotropy D parameter too large by a factor of 2, whereas Eagrees very well with experiment.Although our D does not quantitatively agree with ex-perimental findings,4,3,20,22 we point out that our D and Eanisotropy parameters give a better agreement with the mea-sured period of oscillations for the tunnel splitting whichonly depends on D and E. The period of oscillation has beengiven by Garg8 to be DH5(2kB /gmB)A2E(E1D), and wehave numerically verified this analytic result by direct diago-nalization of the anisotropy Hamiltonian. Experiment ob-serves DH;0.4 T,7 whereas we obtain a value of 0.37 T, inbetter agreement than the previous value of 0.26 T deducedfrom the experimentally determined D and E parameters.7However, as discussed in Ref. 7, higher-order terms in theanisotropy Hamiltonian, which have also been experimen-tally determined, also increase the tunnel oscillation to theexperimental value.This work was supported in part by ONR Grant Nos.N0001498 WX20709 and N0001400WX2011. Computationswere performed at the DoD Major Shared Resource Centers.1 R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak, Nature ~London!365, 141 ~1993!.2 J. Friedman, M. P. Sarachik, J. Tejeda, J. Maciejewski, and R. Ziolo, Phys.Rev. Lett. 76, 3820 ~1996!; L. Thomas, F. Lionti, R. Ballou, D. Gatteschi,R. Sessoli, and B. Barbara, Nature ~London! 383, 145 ~1996!.3 C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D. Gatteschi, Phys.Rev. Lett. 78, 4645 ~1997!.4 A. L. Barra, P. Brunner, D. Gatteschi, Ch. E. Schulz, and R. Sessoli,Europhys. Lett. 35, 133 ~1996!.5 W. Wernsdorfer and R. Sessoli, Science 284, 133 ~1999!.6 W. Wernsdorfer, T. Ohm, C. Sangregorio, R. Sessoli, D. Mailly, and C.Paulsen, Phys. Rev. Lett. 82, 3903 ~1999!.7 W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, and D.Mailly, J. Appl. Phys. 87, 5481 ~2000!.8 A. Garg, Europhys. Lett. 22, 205 ~1993!.9 J. Van Vleck, Phys. Rev. 52, 1178 ~1937!.10 Z. Q. Gu and W. Y. Ching, Phys. Rev. B 46, 8545 ~1987!.11 For example see: G. Schneider and H. J. F. Jansen, J. Appl. Phys. 87, 5875~2000! and references therein; H. J. F. Jansen, Phys. Rev. B 38, ~1988!; R.Q. Wu et al., J. Magn. Magn. Mater. 177, 1216 ~1998! and referencestherein; X. F. Zhong, W. Y. Ching, and W. Lai, J. Appl. Phys. 70, 6146~1991!.12 M. R. Pederson and S. N. Khanna, Phys. Rev. B 60, 9566 ~1999!.13 M. R. Pederson and S. N. Khanna, Chem. Phys. Lett. 307, 253 ~1999!; M.R. Pederson and S. N. Khanna, Phys. Rev. B 59, R693 ~1999!.14 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 ~1964!; W. Kohn and L.J. Sham, ibid. 140, A1133 ~1965!.15 M. R. Pederson and K. A. Jackson, Phys. Rev. B 41, 7453 ~1990!; K. A.Jackson and M. R. Pederson, ibid. 42, 3276 ~1990!.16 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865~1996!.17 H. Hellmann, Einfu¨hrung in die Quantentheorie ~Deuticke, Leipzig 1937!;R. P. Feynman, Phys. Rev. 56, 340 ~1939!; P. Pulay, Mol. Phys. 17, 197~1969!.18 D. Porezag and M. R. Pederson, Phys. Rev. A 60, 2840 ~1999!.19 K. Wieghardt, K. Pohl, I. Jibril, G. Huttner, Angew. Chem. Int. Ed. Engl.23, 77 ~1984!; Cambridge Crystallographic Data Centre, Refcode: COC-NAJ; http://www.ccdc.cam.ac.uk/20 R. Caciuffo, G. Amoretti, A. Murani, R. Sessoli, A. Caneschi, and D.Gatteschi, Phys. Rev. Lett. 81, 4744 ~1998!.21 Y. Pontillon, A. Caneschi, D. Gatteschi, R. Sessoli, E. Ressouche, J.Schweizer, and E. Lelievre-Berna, J. Am. Chem. Soc. 121, 5342 ~1999!.22 A. L. Barra, D. Gatteschi, and R. Sessoli, Chem. Eur. J. 6, 1608 ~2000!.7151J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Pederson, Kortus, and Khanna [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:128.172.48.58 On: Mon, 19 Oct 2015 17:53:43

Electronic-structure-based investigation of magnetism in the Fe8 molecular magnet

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Electronic-structure-based investigation of magnetism in the Fe8 molecular magnet

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