Significant charge transfer between a single-molecule magnet Mn12 and a Bi substrate

Polyhedron 66 (2013) 157–161

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Significant charge transfer between a single-molecule magnet Mn12 and a Bi substrate Kyungwha Park a,⇑, Jun-Zhong Wang b a b

Department of Physics, Virginia Tech, Blacksburg, VA 24061, United States School of Physical Science and Technology, Southwest University, Chongqing 400715, China

a r t i c l e

i n f o

Article history: Available online 21 March 2013 Keywords: Single-molecule magnet Density-functional theory Bismuth Spin–orbit coupling

a b s t r a c t We investigate electronic and magnetic properties of a single-molecule magnet Mn12 adsorbed on Bi(1 1 1) without any linker molecules, using a first-principles method. This study is motivated by a scanning tunneling microscopy experiment on individual Mn12 molecules on a Bi substrate. We apply density-functional theory including spin–orbit coupling, on-site Coulomb repulsion U, and dipole corrections. With geometry relaxation, the Mn12 molecule remains slightly tilted relative to the surface such that its magnetic easy axis is 6° away from the axis normal to the surface. Upon adsorption, a gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the Mn12, is reduced to 0.43 eV, compared to the corresponding gap of 1.07 eV for an isolated Mn12. The total magnetic moment of the adsorbed Mn12 increases to 21 lB . This is due to charge transfer from the Bi slab to the Mn12. The tilted geometry of the Mn12 allows to favor one of the outer Mn sites for charge transfer and magnetic moment change. Although bulk Bi is semimetal, there are surface states near the Fermi level, which facilitates significant charge transfer and a change in the magnetic moment. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Bulk forms of single-molecule magnets (SMMs) have shown quantum effects such as quantum tunneling of magnetization [1,2] and quantum phase interference [3], despite their large magnetic moments. To investigate such quantum effects at lower dimensions or even at the single-molecule level, in the past decade, SMMs have been deposited onto various substrates [4–10] or bridged between electrodes [11–13]. In this case, two important questions can be addressed: (i) Do magnetic cores of SMMs remain intact upon adsorption? (ii) How do interactions between a substrate and SMMs affect electronic and magnetic properties of SMMs? The previous experimental effort suggested that prototype SMMs Mn12 may not be stable on Au, in that SMMs may be broken into smaller clusters or that the oxidation states of some Mn ions change [7]. To stabilize SMMs on a metallic substrate, long alkane chains were used to first functionalize a Au surface [5] or an insulating layer was placed between SMMs Mn12 and a metallic substrate [10]. A recent scanning tunneling microscopy (STM) experiment shows a possibility of deposition of individual SMMs Mn12, [Mn12O12(COOCH3)16(H2O)4], (not aggregates) onto a Bi(1 1 1) substrate without any linker molecules [14]. Recently, Bi-based alloys were shown to belong to a new class of matter referred to as topological insulators [15,16]. Although bulk Bi is ⇑ Corresponding author. Tel.: +1 5402315533. E-mail addresses: [email protected], [email protected] (K. Park). 0277-5387/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.poly.2013.03.021

semimetal, a Bi slab has surface states with large spin–orbit coupling (SOC) near the Fermi level EF . Interactions between SMMs Mn12 and a Bi substrate via strongly spin–orbit-coupled surface states, are interesting but they have not been explored yet. We examine properties of a SMM Mn12, [Mn12O12(COOH)16(H2O)4], adsorbed on Bi(1 1 1) without any linker molecules, using density-functional theory (DFT) including on-site Coulomb repulsion U and SOC. Motivated by the STM experiment [14], we consider a Mn12 molecule slightly tilted relative to the surface such that its magnetic easy axis is several degrees tilted from the axis normal to the surface (z axis). The tilting of the magnetic easy axis remains with geometry relaxation. We find that a gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) levels of the Mn12 is reduced to 40% of the HOMO–LUMO gap of an isolated Mn12, and that the total magnetic moment increases by 1 lB . This change is due to significant charge transfer from the Bi slab to the Mn12. Our results suggest mechanisms of changes of oxidation states of Mn sites in an asymmetric fashion. We discuss our computational method and model in Section 2 and present our results and discussion in Section 3. A brief conclusion is followed in Section 4.

2. Computational method and model We use Perdew–Burke–Ernzerhof (PBE) generalized-gradient approximation (GGA) [17] for exchange–correlation potential and

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projector-augmented-wave (PAW) pseudopotentials [18], implemented in VASP [19]. For Bi, 6s and 6p orbitals are treated as valence states, while for Mn, 3d and 4s orbitals as valence states. We include U ¼ 4 eV only for the Mn d orbitals following the scheme in Ref.[20] to take into account additional electron correlation in localized d electrons. The value of U is chosen to coincide with the experimental HOMO–LUMO gap for bulk forms of Mn12 molecules [21]. We consider SOC self-consistently within DFT and include dipole corrections along the z axis. In the current study, van der Waals interactions are not included. Bulk Bi forms a rhombohedral unit cell with two inequivalent pffiffiffi atoms and the p following lattice vectors: ~ a1ffiffiffi¼ ða=2;  3a=6; ffiffiffi p c=3Þ; ~ a2 ¼ ða=2;  3a=6; c=3Þ, and ~ a3 ¼ ð0; 3a=3; c=3Þ, where experimental lattice constants a ¼ 4:5332 Å and c ¼ 11:7969 Å [22]. For bulk band structure, the kinetic energy cutoff of 105 eV is used, and k points of 11  11  11 are sampled. Along the (1 1 1) direction, Bi atoms form covalently bonded bilayers with intra-bilayer distance of 1.590 Å, and inter-bilayer distance of 2.343 Å [22]. We construct a Bi(1 1 1) slab of 4  4  6 atoms to fully cover one Mn12 and to separate it from neighboring Mn12 molecules in the unit cell. For slab band structure, the kinetic energy cut off of 400 eV is used with k points sampling of 3  2  1. Regarding the Mn12, to save computational cost, we consider a simplified form of Mn12 (of 100 atoms) where CH3 molecules in terminating ligands are replaced by H atoms [23]. It was confirmed that this simplification did not alter electronic and magnetic properties of an isolated Mn12 molecule [23]. We build a unit cell of the whole structure consisting of the simplified form of Mn12 physisorbed on the Bi(1 1 1) slab with a vacuum layer of 11 Å [Fig. 1(a)]. Topographic images of the STM experiment [14] indicate that 50% of identified individual Mn12 molecules have a flat-lying orientation with slight tilting relative to the surface (tilting angle of about 6°), as shown in Fig. 1(a). Thus, we start with this orientation and small tilting for geometry relaxation. All atoms in the whole structure is then relaxed except for the bottommost Bi atomic layer such that the magnitude of the forces exerted on all the atoms is less than 0.05 eV/Å. In the geometry relaxation, the U term and self-consistent SOC are included. We find a significant effect of surface reconstruction upon Mn12 adsorption. The intra-bilayer distance increases in the range of 1.628-1.646 Å and the inter-bilayer distance increases to 2.487-2.515 Å. To examine properties of the whole structure, self-consistent calculations are performed, using the optimized geometry, until the total energy converges up to 9  106 eV. The kinetic energy cutoff of 400 eV is used, and k points of 3  2  1 are sampled. Corrections caused by the induced dipole moment along the z axis are included. We set an initial vertical position of the Mn12 molecule such that there is no chemical bonding between the Mn12 and the Bi. We consider three initial vertical positions of the Mn12. (In the three cases, the initial shortest vertical distances between the Mn12 and Bi are 1.57, 2.07, and 2.57 Å, and after geometry optimization, the distances become 1.80, 2.03, 2.33 Å.) The energy difference among them is less than 50 meV, and all the three cases give the same physical properties including the same tilting angle of 6° and the same total magnetic moment of the Mn12. Here we show the result with the initial vertical distance of 1.57 Å.

3. Results and discussion 3.1. Bulk Bi and bare Bi slab We calculate band structure of bulk Bi without SOC (not shown) and with SOC at several symmetry points, using the experimental lattice constants [22] [Fig. 2(a)]. Due to time-reversal symmetry and inversion symmetry, each band has double degeneracy:

Fig. 1. (a) Side view (in the yz plane) of SMM Mn12 on a Bi slab of six atomic layers. (b) Geometry of the twelve Mn atoms in the Mn12 in the xy plane. For a S4 symmetric Mn12, the inner Mn ions, Mn(I), are Mn4+ (S ¼ 3=2) and the outer Mn ions, Mn(II) and Mn(III), are Mn3+ (S ¼ 2).

Eðk; "Þ ¼ Eðk; #Þ ¼ Eðk; #Þ, where E is an energy as a function of momentum k and electron spin " (#). The band structure without SOC qualitatively differs from that with SOC. Only with SOC, a shallow electron pocket is shown at the L point, and a small hole pocket appears at the T point. This result agrees with experiments [22] and other DFT calculations [24]. Now we compute band structure of bare Bi slabs of 1  1  6 (not shown) and 4  4  6 with SOC using the experimental lattice constants [22] in the two-dimensional (2D) Brillouin zone (BZ) [Fig. 2(b)]. The Bi slabs are not passivated with H. Time-reversal symmetry and inversion symmetry also allow double degeneracy in each band of the slabs. States near EF shown in Fig. 2(b) are surface states localized at the surface atoms.

3.2. Isolated Mn12 molecule We discuss electronic and magnetic properties of an isolated Mn12 molecule with S4 symmetry. Note that the geometry of Mn12 considered in this subsection, is not exactly the same as that of the Mn12 in the whole structure due to geometry relaxation upon adsorption. Based on S4 symmetry, the twelve Mn ions can be classified into three inequivalent types referred to as Mn(I), Mn(II), and Mn(III) [Fig. 1(b)]. In the ground state, each inner Mn site, Mn(I), has Mn4+ (S ¼ 3=2), while each outer Mn site [Mn(II) and Mn(III)] has Mn3+ (S ¼ 2). Antiferromagnetic superexchange coupling between the Mn sites results in the total ground-state spin of S ¼ 10. The previous DFT calculation showed that without

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2

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Fig. 3. Projected density of states (PDOS) onto the inner and outer Mn ions and the O anions for an isolated S4 symmetric Mn12 (a) with U ¼ 4:0 eV for the Mn d orbitals only, and (b) without U. The bold arrows in (a) and (b) represent the HOMO and LUMO levels. The zero of the energy coincides with the HOMO level. Due to SOC, the majority and minority-spin contributions cannot be distinguished.

0.5

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Fig. 2. Calculated band structures of (a) bulk Bi and (b) a bare Bi slab of 4  4  6 at symmetry points with SOC.

U, the gap between the HOMO and the LUMO was 0.38 eV [23] [Fig. 3(b)]. Without U, it was found that the HOMO arises from the outer Mn ions, and that the LUMO originates from both the inner and outer Mn ions [Fig. 3(b)] [23]. In addition, the magnetic anisotropy barrier (MAB) was calculated to be 65 K,1 which agrees with experiments [25,26]. Now with U ¼ 4 eV for the Mn d orbitals, some changes in the electronic structure and magnetic anisotropy are expected. As shown before [27], the HOMO–LUMO gap opens up to 1.07 eV. The LUMO comes from the outer Mn ions only, while the HOMO still arises from the outer Mn ions [Fig. 3(a)]. This change in the origin of the LUMO could be qualitatively rationalized as follows: with U, the inner Mn sites are not favored to add extra electrons due to a larger energy cost compared to the outer Mn sites. Interestingly, with U, the MAB decreases to 60 K due to the increased HOMO–LUMO gap. 3.3. Whole structure: Mn12 on Bi(1 1 1) After relaxation of the whole geometry, we find that the adsorbed Mn12 molecule remains to be slightly tilted relative to the surface, such that its magnetic easy axis is 6° away from the z axis [Fig. 1(a)]. Due to the tilting of the Mn12, among the twelve Mn sites, Mn site 5 is closest to the surface and Mn site 9 is next closest to the surface [Fig. 1(b)]. The adsorption energy of the Mn12 is 1.42 eV. In the relaxed geometry, the shortest distance between 1 This value differs from the reported value in Ref. [23] by 1.7 K because of different parameter values used in this study to lower computational cost, such as a lower energy cutoff (400 eV) and neglect of semicore states (3p6 ) in Mn orbitals.

the bottommost H (O) atom of Mn12 and the Bi surface is 2.88 (3.54) Å, and the vertical distance between them is 1.80 (2.62) Å. Upon adsorption, the double degeneracy in the bands is now lifted due to the presence of the Mn12. We calculate density of states (DOS) of the whole structure projected onto the Mn12 molecule and the Bi slab (Fig. 4). From the DOS projected onto the Mn12, the orbital level right below (above) EF is assigned to be the HOMO (LUMO) of the adsorbed Mn12. The HOMO–LUMO energy gap is now 0.43 eV, which is only 40% of the corresponding gap for an isolated Mn12 molecule. The HOMO arises mainly from Mn site 9, while the LUMO arises from Mn sites 6, 7, 10, 12 (Figs. 1(b) and 4). For the HOMO and LUMO, contributions of the Bi slab are negligible. However, the DOS peak at 0.66 eV (HOMO1) originates from both the Mn12 and the Bi slab. In addition, significant DOS is found between the HOMO and LUMO of the adsorbed Mn12 in the total DOS [Fig. 4(a)]. Projection of these states onto the individual Bi atoms shows that they are surface states of the Bi slab (localized onto the surface atoms).

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from a dyz orbital of Mn site 9 and Bi surface states, and the HOMO originates from dx2 of Mn site 9, while the LUMO is from dxy orbitals of Mn sites 6 and 7, and dx ; dxz , and dyz orbitals of Mn sites 10 and 12. For the isolated Mn12, the HOMO is from dz2 of Mn sites 5-8, dyz of Mn sites 9, 12, and dxz of Mn sites 10, 11, while the LUMO and LUMO+1 arise from dxy of Mn site 5-8 and dx2 of Mn sites 9-12. Thus, we find correspondence between the HOMO1 (HOMO) of the adsorbed Mn12 and the HOMO (LUMO) of an isolated Mn12. We also calculate the magnetic anisotropy barrier of the Mn12 molecule adsorbed on Bi including SOC self-consistently within DFT+U. We find that the barrier decreases to 46 K, which is consistent to the large change of the magnetic moment of the Mn site 9. The change of the Mn oxidation state from 3þ to 2þ lowers a local magnetic anisotropy barrier of the Mn site due to the absence of Jahn–Teller distortion. A similar reduction of the barrier was discussed for an isolated Mn12 molecule with one extra electron localized at one of the outer Mn sites [28]. Our analysis leads to significant charge transfer from the Bi slab to the Mn12 via mainly the Mn site 9, although there are no linker molecules to chemically bound the Mn12 to the Bi substrate. We emphasize that the charge transfer is not due to the fact that the Bi slab considered is thin because the transfer is shown even for a slab of five Bi bilayers (10 atomic layers). First of all, the following two scenarios can be speculated as possible causes of the significant charge transfer: (i) The simplification made for the Mn12 (by replacing the CH3 terminating ligands by H) reduces the distance between the Mn12 and the Bi surface [Fig. 1(a)], which can increase charge transfer and allow mixing between the Bi surface states and the molecular orbitals near EF . (ii) The interactions between the Bi slab and the Mn12 are overestimated due to the neglect of van der Waals interactions. However, the STM experiment of the Mn12 on Bi without the simplification, also indicates a greatly reduced HOMO–LUMO gap such as 0.38 eV [14], which is consistent with our calculated result. In addition, recently, it was shown that SOC in Bi-based topological insulators can enhance an adsorption

To examine changes in charge and magnetic moment of the Mn12 due to the adsorption, we compute layer-projected charge and magnetic moment density for the whole structure, the bare Bi slab, and the isolated Mn12, using the same unit cell and parameter values (Fig. 5). In this case, we use the geometries of the bare Bi slab and the isolated Mn12 taken from the whole structure. Note that the Mn12 geometry taken from the whole structure does not have S4 symmetry any more. A change in charge density Dqz is defined to be [q(whole)  q(bare Bi)  q(Mn12)], and a change in magnetic moment density Dmz is similarly defined. At the midpoint of the surface and the bottommost H atom of the Mn12 (10.648 Å), we find that 0.8 electrons are transferred from the Bi slab to the Mn12. The magnetic moment increases by 1 lB (Fig. 5). To understand the large changes of the HOMO–LUMO gap and of the magnetic moment, we calculate magnetic moments of individual Mn sites (by placing a sphere around each atom), and compare to those of an isolated Mn12 with S4 symmetry. The moments of Mn sites 1–4 are 3:11; 3:17; 3:17, and 3:11 lB , respectively. The moments of Mn sites 5-12 are 3,98, 3.93, 3.93, 3.93, 4.60, 3.94, 3.94, 3.93 lB , respectively (Fig. 1). Compared to the moments of Mn sites in an isolated Mn12, the moments of Mn sites 2, 3, 5 decrease by 0.06 lB , while the moment of Mn site 9 increases by 0.68 lB . The moments of Mn sites 10, 11, 12 increases by 0.05, 0.05, 0.04, respectively. For the rest of the Mn sites, the changes are 0.01 lB or less than that. Overall, there are small increases in the magnitudes of magnetic moments of the Mn sites upon adsorption, but the largest change occurs at Mn site 9, which gives rise to the total magnetic moment of 21 lB for the adsorbed Mn12. Note that only the Mn site 9 contributes to the HOMO of the adsorbed Mn12, and that it is the next closest to the surface among the twelve Mn sites. Although the Mn site 5 is closest to the surface, it is more likely that the charge transfers to the Mn site 9, because of its larger contributions to the LUMO of an isolated Mn12. Now we discuss orbital contributions of the HOMO and LUMO levels compared to those of an isolated Mn12 with S4 symmetry. For the adsorbed Mn12, the HOMOl (at 0.66 eV) arises mainly

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Fig. 5. Layer-projected (a) charge density qz and (b) magnetic moment density mz as a function of z for the isolated Bi slab, the isolated Mn12, and the whole structure. The bold arrows in (a) and (b) indicate the region where the Mn atoms are located. Change in (c) charge density Dqz and (d) magnetic moment density Dmz vs. z.

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energy of small molecules such as CO and O2 by about 0.2 eV, and modify the bond lengths [29]. 4. Conclusions We examined electronic and magnetic properties of Mn12 adsorbed onto a Bi(1 1 1) surface without linker molecules, using DFT, on-site Coulomb repulsion, and SOC, which was motivated by a recent STM experiment. We found significant charge transfer from the Bi substrate (metallic surface states) to the Mn12, which was shown in the reduced HOMO–LUMO gap and the increase in the total magnetic moment. The current work is interesting in that the whole structure explicitly shows potential mechanisms that the oxidation states of the Mn ions change and that transferred charges are asymmetrically distributed, although the simplified Mn12 may lead to overestimated charge transfer. A further calculation of electronic and magnetic properties of the Mn12 on Bi including the van der Waals interaction for the Mn12 without the simplification is for a future study. Acknowledgments K.P. was supported by US NSF DMR-0804665 and DMR1206354, and J.-Z. W. was supported by National Natural Science Foundation of China (Nos. 10974156, 21173170, 91121013). Computational support was provided by the San Diego Supercomputer Center (SDSC) trestles under DMR060009N and by Virginia Tech Advanced Research Computing. References [1] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [2] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature 383 (1996) 145. [3] W. Wernsdorfer, R. Sessoli, Science 284 (1999) 133. [4] A. Cornia, A.C. Fabretti, M. Pacchioni, L. Zobbi, D. Bonacchi, A. Caneschi, D. Gatteschi, R. Biagi, U. Del Pennino, V. De Renzi, L. Gurevich, H.S.J. Van der Zant, Angew. Chem., Int. Ed. 42 (2003) 1645.

161

[5] S. Voss, M. Fonin, U. Rudiger, M. Burgert, U. Groth, Y.S. Dedkov, Phys. Rev. B 75 (2007) 045102. [6] Z. Salman, K.H. Chow, R.I. Miller, A. Morello, T.J. Parolin, M.D. Hossain, T.A. Keeler, C.D.P. Levy, W.A. MacFarlane, G.D. Morris, H. Saadaoui, D. Wang, R. Sessoli, G.G. Condorelli, R.F. Kiefl, Nano. Lett. 7 (2007) 1551. [7] M. Mannini, P. Sainctavit, R. Sessoli, C.C. dit Moulin, F. Pineider, M.-A. Arrio, A. Cornia, D. Gatteschi, Chem. Eur. J. 14 (2008) 7530. [8] M. Mannini, F. Pineider, P. Sainctavit, C. Danielei, E. Otero, C. Sciancalepore, A.M. Talarico, M.-A. Arrio, A. Cornia, D. Gatteschi, D. Sessoli, Nat. Mat. 8 (2009) 194. [9] V. Corradini, F. Moro, R. Biagi, V. De Renzi, U. del Pennino, V. Bellini, S. Carretta, P. Santini, V.A. Milway, G. Timco, R.E.P. Winpenny, M. Affronte, Phys. Rev. B 79 (2009) 144419. [10] S. Kahle, Z. Deng, N. Malinowski, C. Tonnoir, A. Forment-Aliaga, N. Thontasen, G. Rinke, D. Le, B. Turkowski, T.S. Rahman, S. Rauschenbach, M. Ternes, K. Kern, Nano Lett. 12 (2012) 518. [11] H.B. Heersche, Z. de Groot, J.A. Folk, H.S. van der Zant, C. Romeike, M.R. Wegewijs, L. Zobbi, D. Barreca, E. Tondello, A. Cornia, Phys. Rev. Lett. 96 (2006) 206801. [12] M.-H. Jo, J.E. Grose, K. Baheti, M.M. Deshmukh, J.J. Sokol, E.M. Rumberger, D.N. Hendrickson, J.R. Long, H. Park, D.C. Ralph, Nano Lett. 6 (2006) 2014. [13] A.S. Zyazin, J.W.G. van den Berg, E.A. Osorio, H.S.J. van der Zant, N.P. Konstantinidis, M. Leijnse, M.R. Wegewijs, F. May, W. Hofstetter, C. Danieli, A. Cornia, Nano Lett. 10 (2010) 3307. [14] K. Sun, K. Park, J.-L. Xie, J.-Y. Luo, H.-K. Yuan, Z.-H. Xiong, J.-Z. Wang, Q.-K. Xue, unpublished. [15] M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82 (2010) 3045. [16] X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83 (2011) 1057. [17] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [18] P.E. Blöchl, Phys. Rev. B 50 (1994) (1994) 17953; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [19] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169; G. Kresse, J. Furthmüller, Comp. Mat. Sci. 6 (1996) 15. [20] V.I. Anisimov, F. Aryastiawan, A.I. Lichtenstein, J. Phys.: Condens. Matter 9 (1997) 767. [21] D.W. Boukhvalov, M. Al-Saqer, E.Z. Kurmaev, A. Moewes, V.R. Galakhov, L.D. Finkelstein, S. Chuizbaian, M. Neumann, V.V. Dobrovitski, M.I. Katsnelson, A.I. Lichtenstein, B.N. Harmon, K. Endo, J.M. North, N.S. Dalal, Phys. Rev. B 75 (2007) 014419. [22] Ph. Hoffmann, Prog. Surf. Sci. 81 (2006) 191. [23] S. Barraza-Lopez, M.C. Avery, K. Park, Phys. Rev. B 76 (2007) 224413. [24] X. Gonze, J.-P. Michenaud, J.-P. Vigneron, Phys. Rev. B 41 (1990) 11827. [25] A.L. Barra, D. Gatteschi, R. Sessoli, Phys. Rev. B 56 (1997) 8192. [26] S. Hill, J.A.A.J. Perenboom, N.S. Dalal, T. Hathaway, T. Stalcup, J.S. Brooks, Phys. Rev. Lett. 80 (1998) 2453. [27] S. Barraza-Lopez, M.C. Avery, K. Park, J. Appl. Phys. 103 (2007) 07B907. [28] K. Park, M.R. Pederson, Phys. Rev. B 70 (2004) 054414. [29] H. Chen, W. Zhu, D. Xiao, Z. Zhang, Phys. Rev. Lett. 107 (2011) 056804.