Iron doped gold cluster nanomagnets: ab initio determination of barriers for demagnetization

Magnetic properties of small- and nano-sized iron doped gold clusters are calculated at the level of second order multireference perturbation theory. We first assess the methodology for small Au6Fe and Au7Fe clusters, which are representative of even and odd electron count systems. We find that larger active spaces are needed for the odd electron count system, Au7Fe, which exhibits isotropic magnetization behaviour. On the other hand, the even electron count system, Au6Fe, exhibits strong axial magnetic anisotropy. We then apply this methodology to the tetrahedral and truncated pyramidal nano-sized Au19Fe (with S = 3/2) and Au18Fe (with S = 2) clusters. We find that face substitutions result in the most stable structures, followed by edge and corner substitutions. However, for Au18Fe, corner substitution results in strong magnetic anisotropy and a large barrier for demagnetization while face substitution does not. Thus, although corner and face substituted Au18Fe have the same spin, only corner substituted Au18Fe can act as a single nanoparticle magnet.


Introduction
Small-to nano-sized gold clusters show considerably different properties compared to bulk gold. 1 Remarkable structural variations, such as two dimensional akes, and three dimensional compact, cage and tube structures have been found. [2][3][4][5][6][7] Gold clusters usually exhibit small (S ¼ 0, 1/2) spin quantum numbers, leading to closed shell or doublet electronic ground states but doping them with a transition metal atom can lead to open-shell clusters with signicantly higher spin-quantum numbers. [8][9][10][11][12][13][14] For example, Tam et al. 12 investigated the structure and stability of transition metal doped golden pyramids Au 19 M (M ¼ Cr, Mn, Fe) at the level of generalized gradient density functional theory (DFT). All systems show higher ground state spin quantum numbers, while the structural modications are minor. Similar ndings have also been reported by Wang et al. 14 and Yang et al. 10 for golden cages containing a centrally trapped transition metal atom (M@Au 16 and M@Au 24 ). Doping a gold cluster with a transition metal atom therefore provides an opportunity to combine the structural diversity of gold clusters with the unique properties of high-spin systems. We note that experimental evidence of Au m Fe n (m ¼ 1-35, n ¼ 1-3) structures has been found by Mawale et al. using mass spectrometry aer laser desorption ionization of a goldiron nanoower. 15 Single molecule magnets (SMMs) can be seen as a special class of open-shell systems with distinct properties. For example, the rst investigated SMM, the dodecanuclear manganese acetate cluster [Mn 12 O 12 (CH 3 COO) 16 (H 2 O) 4 ]$2CH 3 -COOH$4H 2 O 16-18 exhibits a ground state spin quantum number of S ¼ 10 and can be magnetized by an external magnetic eld. Once the eld is switched off, the system relaxes via various channels back to the initial non-magnetized state. Characteristic for SMMs is a high relaxation barrier, which signicantly slows the demagnetization process at low temperatures. The higher the demagnetization barrier, the longer the magnetization can be retained, which increases the potential for information storage and other applications. The demagnetization barrier is proportional to |D|S 2 , 19 where S is the spin quantum number and D is the axial zero-eld splitting (ZFS) parameter. Together with the rhombicity parameter, E/D, these enter the eld-free part of the spin-Hamiltonian. If complemented with the eld-dependent part, the spin-Hamiltonian can be written as: 20,21 whereŜ k are spin operators while b B , B, and g are the Bohr magneton, the external magnetic eld, and the g-tensor, respectively. The latter quantity, which describes the interaction with the external magnetic eld, can be written as g ¼ 1g e + Dg where g e is the g-value for the free electron and Dg are the shis. The spin-Hamiltonian describes the splitting of the (2S + 1) manifold of the electronic ground state, which results from spin-orbit and spin-spin interactions. 22 As outlined by Atanasov et al., 20 besides large S, SMM candidates should have large D (with negative sign) ensuring that the barrier for demagnetization is large. Also, they should have a small rhombicity parameter ensuring that "tunneling" through the demagnetization barrier is slow. As such, to assess the capability of an open-shell system to be used as a potential SMM, it is necessary to calculate the ZFS parameters. Finally, we note that the two requirements for SMM characteristics result in a large axial magnetic anisotropy of the system (which can be checked by inspecting eqn (1), for D ( 0, E/D ¼ 0). The central aim of this paper is the ab initio determination of the ZFS parameters (and, as such, magnetic anisotropies) for iron doped gold clusters. Several studies have investigated transition metal doped gold clusters using DFT. [8][9][10][11][12][13][14] These studies focused on predicting stable structures and their spin quantum numbers. On the other hand, several studies examined magnetic properties of transition metal complexes, i.e., a transition metal ion surrounded by a primarily organic framework, with wave function based, multireference perturbation theory methods. [23][24][25] In addition, Aravena et al. studied transition metal ions in an inorganic polyoxometalate environment. 26 To our present knowledge, our study is the rst to report calculations of magnetic properties of transition metal doped gold clusters at the level of multireference perturbation theory. In particular, we use an approach based on SA-CASSCF 27 /NEVPT2 (ref. [28][29][30][31] (state averaged complete active space self-consistent eld/second order n-electron valence perturbation theory) as implemented in the ORCA program 32 (for a detailed description, see the ESI †). The outcomes of these ab initio calculations are connected to the spin-Hamiltonian by an effective Hamiltonian method. 20,33 Further, we use the ab initio results to calculate direction-dependent magnetizations in order to assess the magnetic anisotropy. The initial structures are obtained using density functional theory with the revTPSS 34 and B3LYP 35 functionals for gold and iron-doped gold clusters, respectively (see ESI † for further details on the theory, the resulting optimized geometries and spin densities). In all calculations, scalar relativistic effects have been taken into account by using the respective effective core potentials.
In the following section, the results are divided into two parts: (I) small Au 6 Fe and Au 7 Fe clusters, which represent even and odd electron count systems, are used to examine the methodology and critical calculation parameters. (II) Nanosized truncated pyramidal Au 18 Fe and tetrahedral Au 19 Fe clusters, which represent realistic and thermodynamically stable models. Finally, we summarize the results and provide an outlook for future investigations. As mentioned in the Introduction, we begin by examining the central parameters of the CASSCF method, the size and composition of the active space, for two small test systems: (1) Au 7 Fe, which represents clusters with an odd number of electrons, is derived from a three dimensional Au 8 cluster ( Fig. 1(a)). One gold atom was substituted and the resulting geometry was optimized for several spin states. The most stable (S ¼ 3/2) geometry is shown in Fig. 1 (2) Au 6 Fe, which represents clusters with an even number of electrons, is derived from a three dimensional Au 7 cluster ( Fig. 1(c)). Again, one gold atom was substituted and the resulting geometry was optimized for several spin states. The most stable (S ¼ 2) geometry is shown in Fig. 1(d).
Using these optimized geometries, we calculated electronic ground and excited states for both systems with the SA-CASSCF/ NEVPT2 method for several active spaces, which are denoted as CAS(M,N) where M is the number of electrons that are distributed over N spatial orbitals. For the even electron count system, Au 6 Fe, state averaging was done over 5, 45 states having a spin quantum number of S ¼ 2, 1, respectively. These numbers represent the complete manifold of conguration state functions arising from 6 electrons in the ve iron 3d orbitals. States with higher spins (S ¼ 3) would require charge transfer type excitations, which are expected to have signicantly higher energies. This was conrmed by a test calculation for CAS (8,7) (see below).
The situation is different for the odd electron count system, Au 7 Fe. Here an extra electron, originating from the additional gold atom, gives rise to three scenarios: (I) the electron stays in a delocalized 6s-type orbital interacting with the six 3d electrons. (II) The 6s electron is transferred into one of the localized 3d orbitals, resulting in seven 3d electrons. (III) A 3d electron is transferred from the iron into a delocalized 6s-type orbital, leaving ve 3d electrons. It is expected that larger active spaces are needed to describe these scenarios appropriately. Further, from a computational and methodological point of view, not all possible states should be included in the calculation. Therefore, we only include 5, 20, and 20 states for S ¼ 5/2, 3/2, and 1/2, respectively. Our decision to include only a subset of all possible excited states is justied by the following two considerations: (1) The total number of possible states for an active space with 6 orbitals and 7 electrons (CAS(7,6)) is already 300. 36 Due to the state averaging procedure, the ground state wavefunction is described less accurately. In addition, the multireference perturbative treatment of each state becomes rather cumbersome. Of course, this problem increases exponentially for larger active spaces.
(2) As shown by Atanasov et al. 20 it can be expected that higher lying excited states contribute less to the zero-eld splitting parameters, because of prefactors that include inverse excitation energies.
In addition, a second approach with seven electrons in only ve d-orbitals has been tested. One assumes that the spin polarization arises only from the magnetic dopant. All possible states can be included in this case, however, the drawback of this approach is that only S ¼ 3/2 and S ¼ 1/2 states are possible. This not only excludes the possibility of a S ¼ 5/2 ground state, but also neglects couplings between S ¼ 3/2 and S ¼ 5/2 states which might be important.
In Table 1, the rst nonrelativistic excitation energy, E ex , the axial ZFS parameter, D, the rhombicity parameter, E/D, and the shis of the g-tensor are listed for different calculation set-ups. Starting with Au 7 Fe (upper part of Table 1), ve different active space compositions have been examined. For the smallest possible active space, (a)-CAS(7,6), which consists of seven electrons in six orbitals (ve 3d orbitals and one 6s orbital), it was not possible to converge the calculation to a reasonable result. This minimum active space was therefore augmented with 2 and 4 Au 6s-type orbitals, leading to (b)-CAS (9,8) and (c)-CAS(11,10), respectively. E ex is found to be similar for both calculations at around 2900 cm À1 . Including more excited states ((d)-CAS(9,8)) or a second d-shell ((e)-CAS(7,11)) leads to a slightly reduced energy gap. Finally, also the results of the (f)-CAS(7,5) do not deviate signicantly from all the other calculation setups. For all calculations, a ground state spin quantum number of S ¼ 3/2 and a rather small absolute axial ZFS parameter is found. This, and the fact that all shis of the gtensor are quite close together, indicate a dominant isotropic magnetization behaviour. The values for the rhombicity parameter vary but, for such small |D| values, this is not surprising. We conclude that larger active spaces are necessary to appropriately describe the odd electron count system. The active space (c)-CAS (11,10) is sufficiently large while still computationally feasible and will therefore be used for further investigations. In addition, we will compare the results to calculations of the second "d-only" (f)-CAS (7,5) approach.
Similar calculations can be done for the even electron count system, Au 6 Fe. The results are shown in the lower part of Table  1. Set-up (a)-CAS (6,5), represents the minimum active space size, consisting of ve iron 3d orbitals with 6 electrons. For the set-ups (b) to (d), the active spaces are systematically expanded, by including one occupied and one unoccupied, delocalized valence orbital which both have dominant Au 6s character. For these set-ups, the ground state spin quantum number is S ¼ 2, in accordance with the B3LYP calculations. The rst excitation energies vary, but no trend can be found with respect to increasing active space size. Similarly, the axial ZFS parameter and the rhombicity parameter vary slightly as a function of the active space size. A higher multiplicity state was included in the h set-up, (e)-CAS (10,9), which, due to its high energy, has a negligible inuence on the ZFS parameter. Finally, a second dshell was included in the active space, resulting in set-up (f)-CAS (6,10). Again, only minor changes of a few wave numbers are observed. The qualitative result is similar for all tested setups: a large negative axial ZFS parameter in combination with a small rhombicity parameter, E/D, indicates a large axial magnetic anisotropy. This is supported by the shis for the gtensor, where one component (Dg zz ) is much larger than the other two. We conclude that, for the system with an even electron count, the smallest active space, (a)-CAS (6,5), captures all essential effects and is therefore used for further investigations.
The values shown in Table 1 demonstrate that the two systems exhibit very different properties. For example, for Au 6 Fe a high absolute axial ZFS parameter of 45 cm À1 with a negative Table 1 The first nonrelativistic excitation energy (E ex ), the axial ZFS parameter (D), the rhombicity parameter (E/D), and the shifts for the g-tensor are calculated with different the active space sizes in the SA-CASSCF/NEVPT2 approach. In the first column the active space CAS(M,N) is given, where M and N are the numbers of electrons and spatial orbitals, respectively. The second column labels the character of orbitals in the active space. The number of states included in the calculation is given in the third column. We use the following terminology: for a calculation with N(S 1 , Paper sign and a very small rhombicity parameter of 0.03 is observed (for set-up (a)-CAS (6,5)). The resulting axial magnetic anisotropy is also reected in the reported shis for the g-values, where one shi is signicantly larger than the other two (which are close to zero). The system is easily magnetized along one preferred orientation and, furthermore, it exhibits a high demagnetization barrier of |D|S 2 ¼ 180 cm À1 . On the other hand, Au 7 Fe has a small axial ZFS parameter, which indicates more isotropic magnetization behaviour. This is supported by the shis for the g-values, which are quite close together. Therefore, demagnetization is expected to occur signicantly faster.

Au 6 Fe & Au 7 Fezero-eld splitting & magnetizations
In order to examine the properties of Au 6 Fe and Au 7 Fe in more detail, we explicitly simulate the inuence of an external magnetic eld. This is shown in the upper and lower parts of Fig. 2, which correspond to Au 7 Fe and Au 6 Fe, respectively. Each cluster, as well as three coloured arrows (each indicating one direction of the magnetic eld) are shown in the insets (b) and (d). The dependence of the state energies and calculated magnetizations as a function of the external magnetic ux density is shown in insets (a) and (c). The colour of each magnetization graph corresponds to the colour of one arrow on the right to illustrate the direction of the external eld. Each energy data point is coloured by its Boltzmann population, from 0 (black) to 1 (red), at a temperature of 10 K. For Au 7 Fe, shown in the upper part of Fig. 2 (insets (a) and (b)), the S ¼ 3/2 electronic ground state splits into two degenerate M S ¼ AE3/2 and M S ¼ AE1/2 magnetic states, independent of the value of E/D (Kramers theorem 20 ). As expected from the small |D| value, the energetic separation is small. Further splitting is observed upon interaction with the external magnetic eld. The three directions correspond to the principal axes of the g-tensor (i.e., where the g-tensor is diagonal). The axes are then chosen such that their corresponding principal values full g zz > g yy > g xx . The relative state energies vary with the magnetic eld, and a magnetization is observed. As expected from the calculated shis of the g-values and the small |D| value, an almost perfect isotropic magnetization behaviour is found. For Au 6 Fe, shown in the lower part of Fig. 2 (insets (c) and (d)), the behaviour is quite different. As expected from the negative value of D, the magnetic ground states exhibit M S ¼ AE2 followed by M S ¼ AE1 and M S ¼ 0. The magnetic ground states (M S ¼ AE2) are almost perfectly degenerate, a small energetic gap can be observed for M S ¼ AE1, which is due to the small but nonnegligible value of the rhombicity parameter, E/D. Switching on an external magnetic eld leads to two different observations, which are dependent on the orientation of the eld: (1) For a magnetic eld along a direction parallel to the main anisotropic axis, there is a strong splitting of the M S ¼ AE2 and M S ¼ AE1 states as shown on the right panel of inset (c) in Fig. 2. As a function of the external magnetic ux density, the formerly equally populated magnetic ground states exhibit increasingly different Boltzmann populations and the magnetization increases dramatically.
(2) For a magnetic eld along a direction perpendicular to the main anisotropic axis, very different behaviour is expected and observed, as shown in the middle and le panels of inset (c) in Fig. 2. The relative energies depend much less on the external magnetic ux density. As such, the Boltzmann populations are essentially constant, and no signicant magnetization is observed.

Au 18 Fe & Au 19 Festabilities
We now examine two nano-sized iron doped gold clusters. The rst class of systems is derived from the tetragonal Au 20 cluster 37,38 shown at the top of Fig. 3. The three optimized iron doped Au 19 Fe clusters are shown underneath in energetic order. The systems have odd electron counts and all ground states exhibit a spin quantum number of S ¼ 3/2. The DFT derived relative energies are given in parenthesis in Fig. 3 and in Table  2. The face substituted system is most stable, followed by the edge (0.25 eV higher) and corner (0.64 eV higher) substitutions. The same result is found in the study of Tam et al. 12 In Table 2, we further list the relative energies of the NEVPT2 calculations, using (c)-CAS (11,10), which support the energetic order of the B3LYP calculations. For most systems however, NEVPT2 predicts slightly higher relative energies.
The second class of systems is derived from the truncated pyramid Au 19 cluster 38 shown at the lower part of Fig. 3. The six optimized iron doped Au 18 Fe clusters are shown underneath in Fig. 2 Relative state energies (solid lines with bullets) and magnetizations (solid lines) for the Au 7 Fe (top) and Au 6 Fe (bottom) are shown as a function of an external magnetic field. The principal axes of the gtensor (calculated with the spin-Hamiltonian formalism) have been used as directions, which are indicated by coloured arrows on the right and correspond to one magnetization. Each energy data point is coloured by its Boltzmann population, from 0 (black) to 1 (red), at a temperature of 10 K. energetic order. The systems have even electron counts and all ground states exhibit a spin quantum number of S ¼ 2. The face substituted systems Au 18 Fe-A and Au 18 Fe-B are most stable, followed by the edge (Au 18 Fe-C and Au 18 Fe-D, 0.3 eV higher in energy) and corner (Au 18 Fe-E and Au 18 Fe-F, 0.1 eV higher in energy) substitutions. Quite surprising is the large energetic separation of 0.8 eV between the two corner substituted clusters, which is not observed for the two other substitution schemes (i.e. face and edge substitutions). Similar to the Au 19 Fe systems, the NEVPT2 calculations predict the same energetic order with slightly higher relative energies.

Au 18 Fe & Au 19 Fezero-eld splittings and magnetizations
We now examine the magnetic properties of Au 18 Fe and Au 19 Fe. The rst nonrelativistic excited state energy, the axial and rhombicity parameters, and the shis of the main g-values for each system are listed in Table 2. The calculated values for the Au 19 Fe systems are all qualitatively different from each other. The face substituted system, Au 19 Fe-A, has a very small E ex of only 24.7 cm À1 and the reported values are therefore given in parenthesis. Further, the ground state spin quantum number calculated with the NEVPT2 formalism for this cluster is S ¼ 5/2, in contradiction to the B3LYP calculations. However, the gap between the ground S ¼ 5/2 state and the lowest S ¼ 3/2 state is quite small (only 63.3 cm À1 ). Calculations with basis set extrapolation and without state averaging might be needed in order to give a reliable prediction of the ground state spin quantum number but this is beyond the scope of the present paper. For the other two systems this problem does not occur. For the edge (Au 19 Fe-B) and corner (Au 19 Fe-C) substituted systems, we nd rst nonrelativistic excitation energies of 848 cm À1 and 3849 cm À1 , respectively. Both systems exhibit a ground state spin quantum number of S ¼ 3/2, in agreement with B3LYP. The axial ZFS and rhombicity parameters indicate a small rhombic anisotropy (D ¼ À9. The ground states of the even electron count systems, Au 18 Fe, all exhibit a spin quantum number of S ¼ 2. Starting with the most stable face substituted systems, Au 18 Fe-A and Au 18 Fe-B, we nd an axial ZFS parameter of À29.5 cm À1 and 19.3 cm À1 , respectively. Both have a pronounced rhombicity parameter which is reected in the tabulated shis for the gvalues. For both systems, all three values differ signicantly ( Table 2, last column on the right). The two edge substituted systems, Au 18 Fe-C and Au 18 Fe-D, which are approximately 0.3 eV higher in energy, exhibit magnetic properties similar to the face substituted systems although the values for E ex are higher in energy. Again, a rhombic anisotropy is observed for both systems (the shis for the g-values differ for each direction and a high rhombicity parameter E/D is found).
Very different magnetic properties are observed for the systems highest in energy, i.e., the corner substituted iron doped gold clusters Au 18 Fe-E and Au 18 Fe-F. For both systems, a high absolute axial ZFS with negative sign is observed. Further, relatively small rhombicity parameters (0.011 & 0.015 for Au 18 Fe-E and Au 18 Fe-F, respectively) are found, which indicate strong axial magnetic anisotropies. The shis for the g-values also indicate strong axial anisotropy: for both systems two g-value shis are close to zero, while one is signicantly larger.
In order to analyze the magnetic anisotropy in more detail, we calculated magnetizations and relative state energies from the ab initio calculations (as previously for the small clusters). This is shown in Fig. 4 for the most promising candidates Au 18 Fe-E (top) Fig. 3 The revTPSS/def2-TZVP optimized, pyramidal Au 20 structure is shown at the top. The derived and optimized iron doped clusters are shown below with increasing relative energies, given in brackets in units of eV. For all structures, the S ¼ 3/2 spin quantum results in the most stable geometries. Below the truncated pyramidal Au 19 and the six iron doped clusters are shown analogously. Here, the most stable clusters are found for the S ¼ 2 spin quantum numbers. and Au 18 Fe-F (bottom). The geometries and orientations (indicated by coloured arrows) of the external magnetic eld are shown in (b) and (d). Insets (a) and (c) show, for each system, the relative state energies and magnetizations as a function of the external magnetic ux density. Again, note that the colour of one magnetization corresponds to the colour of one direction of the magnetic eld. For Au 18 Fe-E, a ZFS in two M S ¼ AE2, two M S ¼ AE1 and one M S ¼ 0 is observed. For two directions along the principal axes of the g-tensor the state energies remain almost constant with respect to the external magnetic ux density. However, for one direction, the state energies vary drastically with increasing external magnetic ux density, as it can be seen in the right inset of panel (c). The lowest two states (M S ¼ AE2) vary most with the external ux density, followed by the third and fourth states (M S ¼ AE1). The h state (M S ¼ 0) shows no dependence. As a consequence, the ground state is dominantly populated and the magnetization increases dramatically. For the second candidate, Au 18 Fe-F, a similar behaviour is observed. However, the ZFS is much more pronounced compared to Au 18 Fe-E and the direction of main magnetization axis is oriented differentlyalmost perpendicular to the base of the truncated pyramid. For both systems, an impressive and almost perfect axial magnetic anisotropy is observed and we can estimate the demagnetization barrier to be |D|S 2 ¼ 141.2 cm À1 and 207.6 cm À1 for Au 18 Fe-E and Au 18 Fe-F, respectively. Both systems are therefore candidates for nanomagnets.

Summary
In conclusion, we nd that the corner substituted systems Au 18 Fe-E and Au 18 Fe-F exhibit a very pronounced axial magnetic anisotropy. For both systems, a large absolute |D| with negative sign and an almost negligible rhombicity parameter E/D is found. The high axial anisotropy is further reected in the shis for the g-values and in the explicitly calculated relative state energies and magnetization (Fig. 4), where the axial anisotropy is demonstrated impressively. Thus our ab initio calculations indicate that both systems possess single nanoparticle magnetic properties and are therefore candidates for further (experimental) investigation. In Table 2 Relative energies, given in eV and calculated with two different electronic structure methods are listed for the three investigated Au 19 Fe and six Au 18 Fe clusters. The systems are labeled from A-C (Au 19 Fe) and A-F (Au 18 Fe) and correspond to the geometries shown in Fig. 3. For Au 18 Fe and Au 19 Fe, calculations have been done with (a)-CAS (6,5), (c)-CAS (11,10) and (f)-CAS (7,5), respectively. Further, the first nonrelativistic excitation energy (E ex ), the axial and rhombicity parameter, and the shifts for the main values of the g-tensor are given  18 Fe-E (top) and Au 18 Fe-F (down) clusters are shown as a function of an external magnetic field. The principal axes of the g-tensor (calculated with the spin-Hamiltonian formalism) have been used as directions, which are indicated by coloured arrows on the right and correspond to one magnetization. Each energy data point is coloured by its Boltzmann population, from 0 (black) to 1 (red), at a temperature of 10 K.
future investigations, other (transition or f-shell) metal substitutions could be examined. Future studies could focus on the explicit time-dependent demagnetization dynamics. This would include the evaluation of transition rates between the different magnetic levels and a temperature-dependent propagation of the state populations until thermal equilibrium is achieved. Other interesting questions arise further from couplings of multiple spin centres (for example, Au 17 Fe 2 ) or iron doped gold clusters bonded to ligands that could be used to tailor the electronic and magnetic properties of the system. A challenge is to identify thermodynamically stable clusters out of the huge geometrical space. Here, global optimization techniques like genetic algorithms 39 could be very helpful.

Conflicts of interest
There are no conicts to declare.