Christopher
Ehlert
* and
Ian P.
Hamilton
*
Department of Chemistry and Biochemistry, Wilfrid Laurier University, 75 University Ave W, Waterloo, ON N2L3C5, Canada,. E-mail: cehlert@wlu.ca; ihamilton@wlu.ca
First published on 12th February 2019
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 Au_{6}Fe and Au_{7}Fe 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, Au_{7}Fe, which exhibits isotropic magnetization behaviour. On the other hand, the even electron count system, Au_{6}Fe, exhibits strong axial magnetic anisotropy. We then apply this methodology to the tetrahedral and truncated pyramidal nano-sized Au_{19}Fe (with S = 3/2) and Au_{18}Fe (with S = 2) clusters. We find that face substitutions result in the most stable structures, followed by edge and corner substitutions. However, for Au_{18}Fe, corner substitution results in strong magnetic anisotropy and a large barrier for demagnetization while face substitution does not. Thus, although corner and face substituted Au_{18}Fe have the same spin, only corner substituted Au_{18}Fe can act as a single nanoparticle magnet.
Single molecule magnets (SMMs) can be seen as a special class of open-shell systems with distinct properties. For example, the first 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 = 10 and can be magnetized by an external magnetic field. Once the field 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 significantly 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-field splitting (ZFS) parameter. Together with the rhombicity parameter, E/D, these enter the field-free part of the spin-Hamiltonian. If complemented with the field-dependent part, the spin-Hamiltonian can be written as:^{20,21}
(1) |
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.^{8–14} 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.^{23–25} In addition, Aravena et al. studied transition metal ions in an inorganic polyoxometalate environment.^{26} To our present knowledge, our study is the first 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. 28–31) (state averaged complete active space self-consistent field/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) Nano-sized 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.
(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 = 3/2) geometry is shown in Fig. 1(b).
(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 = 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 = 2, 1, respectively. These numbers represent the complete manifold of configuration state functions arising from 6 electrons in the five iron 3d orbitals. States with higher spins (S = 3) would require charge transfer type excitations, which are expected to have significantly higher energies. This was confirmed 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 five 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 = 5/2, 3/2, and 1/2, respectively. Our decision to include only a subset of all possible excited states is justified 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-field splitting parameters, because of prefactors that include inverse excitation energies.
In addition, a second approach with seven electrons in only five 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 = 3/2 and S = 1/2 states are possible. This not only excludes the possibility of a S = 5/2 ground state, but also neglects couplings between S = 3/2 and S = 5/2 states which might be important.
In Table 1, the first nonrelativistic excitation energy, E_{ex}, the axial ZFS parameter, D, the rhombicity parameter, E/D, and the shifts of the g-tensor are listed for different calculation set-ups. Starting with Au_{7}Fe (upper part of Table 1), five 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 (five 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^{−1}. 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 significantly from all the other calculation setups. For all calculations, a ground state spin quantum number of S = 3/2 and a rather small absolute axial ZFS parameter is found. This, and the fact that all shifts of the g-tensor 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.
orbitals | # of states N({S_{i}}) | E _{ex} [cm^{−1}] | D [cm^{−1}] | E/D | g-Shifts [Δg_{xx}, Δg_{yy}, Δg_{zz}] | |
---|---|---|---|---|---|---|
Au _{ 7 } Fe | ||||||
(a)-CAS(7,6) | 5x 3d, Au_{s} | N(5/2, 3/2, 1/2) = [5, 20, 20] | — | — | — | — |
(b)-CAS(9,8) | Au_{s}, 5x 3d, 2x Au_{s} | N(5/2, 3/2, 1/2) = [5, 20, 20] | 2801.1 | −2.39 | 0.089 | [0.12, 0.12, 0.13] |
(c)-CAS(11,10) | 2x Au_{s}, 5x 3d, 3x Au_{s} | N(5/2, 3/2, 1/2) = [5, 20, 20] | 2947.0 | 1.36 | 0.147 | [0.10, 0.12, 0.12] |
(d)-CAS(9,8) | Au_{s}, 5x 3d, 2x Au_{s} | N(5/2, 3/2, 1/2) = [5, 40, 40] | 2611.5 | −1.47 | 0.212 | [0.12, 0.12, 0.14] |
(e)-CAS(7,11) | 5x 3d, 5x 4d, Au_{s} | N(5/2, 3/2, 1/2) = [5, 20, 20] | 1656.7 | −3.86 | 0.003 | [0.35, 0.35, 0.45] |
(f)-CAS(7,5) | 3d | N(3/2, 1/2) = [10, 40] | 1853.4 | −1.66 | 0.005 | [0.36, 0.36, 0.41] |
Au _{ 6 } Fe | ||||||
(a)-CAS(6,5) | 5x 3d | N(2, 1) = [5, 45] | 447.6 | −45.11 | 0.030 | [−0.02, 0.13, 0.93] |
(b)-CAS(8,7) | Au_{s}, 5x 3d, Au_{s} | N(2, 1) = [5, 45] | 183.0 | −60.71 | 0.003 | [−0.10, −0.01, 1.10] |
(c)-CAS(10,9) | 2x Au_{s}, 5x 3d, 2x Au_{s} | N(2, 1) = [5, 45] | 413.7 | −50.93 | 0.008 | [−0.03, 0.04, 0.94] |
(d)-CAS(12,11) | 3x Au_{s}, 5x 3d, 3x Au_{s} | N(2, 1) = [5, 45] | 391.0 | −48.23 | 0.010 | [−0.02, 0.04, 0.90] |
(e)-CAS(10,9) | 2x Au_{s}, 5x 3d, 2x Au_{s} | N(3, 2, 1) = [1, 5, 45] | 483.2 | −44.99 | 0.039 | [−0.02, 0.12, 0.89] |
(f)-CAS(6,10) | 5x 3d, 5x 4d | N(2, 1) = [5, 45] | 395.5 | −42.81 | 0.020 | [−0.02, 0.11, 0.92] |
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 five 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 = 2, in accordance with the B3LYP calculations. The first 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 fifth set-up, (e)-CAS(10,9), which, due to its high energy, has a negligible influence on the ZFS parameter. Finally, a second d-shell 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 set-ups: 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 shifts for the g-tensor, where one component (Δg_{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^{−1} with a negative 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 reflected in the reported shifts for the g-values, where one shift is significantly 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} = 180 cm^{−1}. On the other hand, Au_{7}Fe has a small axial ZFS parameter, which indicates more isotropic magnetization behaviour. This is supported by the shifts for the g-values, which are quite close together. Therefore, demagnetization is expected to occur significantly faster.
For Au_{7}Fe, shown in the upper part of Fig. 2 (insets (a) and (b)), the S = 3/2 electronic ground state splits into two degenerate M_{S} = ±3/2 and M_{S} = ±1/2 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 field. 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 fulfil g_{zz} > g_{yy} > g_{xx}. The relative state energies vary with the magnetic field, and a magnetization is observed. As expected from the calculated shifts 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} = ±2 followed by M_{S} = ±1 and M_{S} = 0. The magnetic ground states (M_{S} = ±2) are almost perfectly degenerate, a small energetic gap can be observed for M_{S} = ±1, which is due to the small but non-negligible value of the rhombicity parameter, E/D. Switching on an external magnetic field leads to two different observations, which are dependent on the orientation of the field:
(1) For a magnetic field along a direction parallel to the main anisotropic axis, there is a strong splitting of the M_{S} = ±2 and M_{S} = ±1 states as shown on the right panel of inset (c) in Fig. 2. As a function of the external magnetic flux density, the formerly equally populated magnetic ground states exhibit increasingly different Boltzmann populations and the magnetization increases dramatically.
(2) For a magnetic field along a direction perpendicular to the main anisotropic axis, very different behaviour is expected and observed, as shown in the middle and left panels of inset (c) in Fig. 2. The relative energies depend much less on the external magnetic flux density. As such, the Boltzmann populations are essentially constant, and no significant magnetization is observed.
System | E(B3LYP) [eV] | E(SA-NEVPT2) [eV] | E _{ex} [cm^{−1}] | D [cm^{−1}] | E/D | g-Shift [Δg_{xx}, Δg_{yy}, Δg_{zz}] |
---|---|---|---|---|---|---|
Au _{ 19 } Fe – CAS(11,10) | ||||||
Au_{19}Fe-A | 0.00 | 0.00 | 24.7 | (9.69) | (0.050) | ([−1.43, 0.02, 0.32]) |
Au_{19}Fe-B | 0.25 | 0.34 | 848.4 | −9.52 | 0.197 | [0.16, 0.18, 0.34] |
Au_{19}Fe-C | 0.88 | 0.90 | 3848.9 | 3.37 | 0.138 | [0.11, 0.17, 0.19] |
Au _{ 19 } Fe – CAS(7,5) | ||||||
Au_{19}Fe-A | 0.00 | 0.00 | 903.3 | 38.80 | 0.03 | [0.12, 0.91, 0.94] |
Au_{19}Fe-B | 0.25 | 0.25 | 1154.1 | −20.47 | 0.31 | [0.27, 0.47, 0.72] |
Au_{19}Fe-C | 0.88 | 0.42 | 1839.1 | 11.31 | 0.04 | [0.29, 0.47, 0.49] |
Au _{ 18 } Fe – CAS(6,5) | ||||||
Au_{18}Fe-A | 0.00 | 0.00 | 85.3 | −29.52 | 0.200 | [−0.25, 0.59, 0.96] |
Au_{18}Fe-B | 0.14 | 0.06 | 111.1 | 19.34 | 0.107 | [−0.28, 0.70, 0.85] |
Au_{18}Fe-C | 0.47 | 0.76 | 458.2 | −20.96 | 0.199 | [0.04, 0.31, 0.66] |
Au_{18}Fe-D | 0.52 | 0.76 | 508.6 | 19.49 | 0.179 | [0.00, 0.42, 0.61] |
Au_{18}Fe-E | 0.60 | 0.93 | 714.4 | −35.25 | 0.011 | [0.07, 0.08, 0.73] |
Au_{18}Fe-F | 1.41 | 1.79 | 183.5 | −51.91 | 0.015 | [−0.00, 0.03, 1.08] |
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 energetic order. The systems have even electron counts and all ground states exhibit a spin quantum number of S = 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.
The ground states of the even electron count systems, Au_{18}Fe, all exhibit a spin quantum number of S = 2. Starting with the most stable face substituted systems, Au_{18}Fe-A and Au_{18}Fe-B, we find an axial ZFS parameter of −29.5 cm^{−1} and 19.3 cm^{−1}, respectively. Both have a pronounced rhombicity parameter which is reflected in the tabulated shifts for the g-values. For both systems, all three values differ significantly (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 shifts 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 shifts for the g-values also indicate strong axial anisotropy: for both systems two g-value shifts are close to zero, while one is significantly 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) and Au_{18}Fe-F (bottom). The geometries and orientations (indicated by coloured arrows) of the external magnetic field 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 flux density. Again, note that the colour of one magnetization corresponds to the colour of one direction of the magnetic field. For Au_{18}Fe-E, a ZFS in two M_{S} = ±2, two M_{S} = ±1 and one M_{S} = 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 flux density. However, for one direction, the state energies vary drastically with increasing external magnetic flux density, as it can be seen in the right inset of panel (c). The lowest two states (M_{S} = ±2) vary most with the external flux density, followed by the third and fourth states (M_{S} = ±1). The fifth state (M_{S} = 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 differently – almost 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} = 141.2 cm^{−1} and 207.6 cm^{−1} for Au_{18}Fe-E and Au_{18}Fe-F, respectively. Both systems are therefore candidates for nanomagnets.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8na00359a |
This journal is © The Royal Society of Chemistry 2019 |