The Multiple Faces, and Phases, of Magnetic Anisotropy - Inorganic

May 8, 2019 - A rough classification of the magnetic properties is widely done in terms of the qualitative descriptors of magnetic anisotropy: “easy...
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Cite This: Inorg. Chem. XXXX, XXX, XXX−XXX

The Multiple Faces, and Phases, of Magnetic Anisotropy Mauro Perfetti* and Jesper Bendix* Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

Inorg. Chem. Downloaded from pubs.acs.org by UNIV OF LOUISIANA AT LAFAYETTE on 05/08/19. For personal use only.

S Supporting Information *

ABSTRACT: The notion of magnetic anisotropy is very central to the field of molecule-based magnetism, where it is considered to be a key quantity that must be rationalized and controlled in order to improve the performances of, e.g., single-molecule magnets. A rough classification of the magnetic properties is widely done in terms of the qualitative descriptors of magnetic anisotropy: “easy-axis” and “easyplane”. They can be based on different physical properties, in casu: free energy, magnetization, or magnetic susceptibility. However, this degree of freedom leads in some cases, including very simple ones like [V(H2O)6]3+, to incommensurate descriptions of a system being simultaneously easyaxis and easy-plane, depending only on the choice of the physical quantity on which the descriptor is based. Moreover, it has recently been pointed out that the magnetic anisotropy of a chemical system can be addressed and switched using external stimuli like temperature and magnetic field. These external parameters are, though, not the only ones capable of triggering anisotropy switching for actual chemical systems under experimentally relevant conditions. Indeed, this applies also to pressure, as discussed here. In this paper, we try to illustrate the multifaceted nature of magnetic anisotropy and assist the overview using anisotropy phase diagrams.

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base complexes, e.g., trensal complexes, show this behavior.17 Likewise, a change in magnetic anisotropy due to variation of either the temperature or of the magnetic field was observed for transition-metal systems in an iron(II) dimer18 and a manganese(II) grid,19 respectively. The relatively common occurrence and obvious potential applications of magnetic anisotropy switching make an analysis of the phenomenon interesting and pertinent to the field of molecule-based magnetism. Historically, the most common quantities that were used to quantify magnetic anisotropy are magnetization (M) and magnetic susceptibility (χ).20 This choice is indeed very convenient because these quantities can be easily measured in a standard magnetometer. However, there is a third quantity, conceptually more fundamental, that is useful to describe magnetic anisotropy, namely, the free energy in an applied field. The free energy is defined as the difference between the internal energy and the product of temperature and entropy [see section 1 of the Supporting Information (SI) for the explicit derivation]. Thus, when a magnetic field is applied to a magnetic compound at constant pressure, the free energy takes the form21

INTRODUCTION Most magnetic materials exhibit a spatial dependence of their magnetic properties. This characteristic, called magnetic anisotropy, is intimately connected to many phenomena and properties of practical interest. For example, in bulk materials and thin layers, magnetic anisotropy is intensively studied for its relationship with the blocking temperature,1 magnetoresistance,2 and magnetoelasticity.3 In molecular materials, a qualitative and quantitative description of magnetic anisotropy is essential to characterize and understand several classes of compounds like single-molecule magnets (SMMs)4−7 and magnetic refrigerators.8−10 However, the experimental detection of magnetic anisotropy is far from trivial, and the results can be counterintuitive.11−13 An intriguing aspect of magnetic anisotropy in magnetic materials is the possibility of observing the phenomenon of “spin reorientation” (magnetic anisotropy switching) as a function of external stimuli like temperature, magnetic field, and pressure, as reported for extended structures like ferrites and intermetallic compounds.14,15 However, the presence of multiple competing interactions in these materials complicates the analysis and understanding of this phenomenon. Recently, we demonstrated that the switching of magnetic anisotropy can also be observed, and importantly modeled, in molecular systems, such as lanthanide complexes.16 Shortly after, magnetic anisotropy switching was then predicted, and experimentally verified, to be a quite common phenomenon in molecular compounds. For example, lanthanide phthalocyaninate complexes, polyoxometallates, and polydentate Schiff© XXXX American Chemical Society

Special Issue: Paradigm Shifts in Magnetism: From Molecules to Materials Received: March 4, 2019

A

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 1. (a) Free energy, (b) magnetization, and (c) susceptibility surfaces for the vanadium(III) hexaaqua cation at T = 2 K and B = 1 T. The color scale emphasizes the magnitude of the plotted quantity. The isotropic contribution has been subtracted from each plot.

ij n yz = −kBT lnjjjj∑ e−Ei[B(θ , φ)] / kBT zzzz j i=1 z k {

this compound can be described as a S = 1 system with a positive axial zero-field-splitting (ZFS) parameter, D = 4.8581 cm−1, and anisotropic g⊥ = 1.8690 and g∥ = 1.9549 (see the SI for details about the formalism used22). The free energy, magnetization, and susceptibility were simulated using a program that encompasses the Easy Spin package for MATLAB.25 A plot of these three quantities26 for [V(H2O)6]3+ at T = 2 K and B = 1 T is reported in Figure S1. It is immediately evident that the shape of the free energy is rather spherical. This spherical (isotropic) part often dominates over the anisotropic part of the free energy, magnetization, and susceptibility (see after) and, especially at high temperatures, provides a challenge in the experimental detection of magnetic anisotropy. Visualization of magnetic anisotropy is facilitated when the isotropic part is subtracted from the plots. Indeed, subtracting the spherical contribution by setting the lowest value of the plot to zero allows an immediate recognition of (at least) one principal direction. We have done so, for V(H2O)63+, in Figure 1, where it is easy to locate the extrema of each quantity being either along z or in the xy plane. If we then define magnetic anisotropy as the difference between the two extreme values of the free energy reported in Figure 1a (for convention, we will evaluate the z − x difference), we get ΔF = Fz − Fx = F∥ − F⊥ > 0. A positive value implies easy-plane magnetic anisotropy (the energy is minimum in the xy plane). If instead we use magnetization as the quantity defining magnetic anisotropy of the system, we get ΔM = Mz − Mx = M∥ − M⊥ < 0. The difference is now negative because the sample is more difficult to magnetize along z than in the xy plane. This leads again to a description of the system as possessing easy-plane magnetic anisotropy. Finally, the same reasoning can be applied, with the difference between the parallel and perpendicular components of the susceptibility tensor Δχ = χz − χx = χ∥ − χ⊥ < 0. Because Δχ is a negative number, the anisotropy would be termed easy-plane. Thus, the three plots in Figure 1 describe the same physical situation and are internally consistent in labeling this “easyplane”. After our reasoning, it might thus seem rather immaterial if one measures or computes one or the other quantity to describe magnetic anisotropy of the system. In the following, we demonstrate that this is, interestingly, not the case: on the contrary, the use of one or the other quantity to define the anisotropy can lead to an opposite description of magnetic anisotropy. Moreover, we will also emphasize that in addition to choosing a physical property as the basis for the descriptors of magnetic anisotropy, it is also necessary to recognize that magnetic anisotropy of the system is not unambiguously determined by the chemical and ensuing electronic structure.

F[T , B(θ , φ)] = −kBT ln Z

(1)

where kB is the Boltzmann constant, T is the absolute temperature, B(θ,φ) is the magnetic field vector, and Ei is the energy of the ith state. The argument of the logarithm is the partition function (Z), and it accounts for population of the states as a function of their energy. As explicitly emphasized in eq 1, F[T, B(θ,φ)] depends on both the temperature and direction in which the field is applied. If we take the derivative of the free energy with respect to the magnetic field, we directly obtain the magnetization (see section 1 of the SI for the mathematical derivation). M[T , B(θ , φ)] = −

∂F[T , B(θ , φ)] ∂B(θ , φ) n

=

{

∑i = 1 −

∂Ei[B(θ , φ)] −Ei[B(θ , φ)] / kBT e ∂B

}

Z (2)

Finally, the magnetic susceptibility is the derivative of magnetization with respect to the magnetic field. χ [T , B ( θ , φ )] =

∂ 2F[T , B(θ , φ)] ∂M[T , B(θ , φ)] =− ∂B(θ , φ) ∂[B(θ , φ)]2 (3)

The key for providing a model description of the aforementioned three quantities is knowledge of the eigenenergies and eigenstate compositions. These quantities can be obtained from the modeling, and their correctness is contingent on the applicability of the effective Hamiltonian employed. The form of the Hamiltonian used to extract eigenvectors and eigenvalues differs depending on the analyzed system. In section 2 of the SI, we give a condensed description of the Hamiltonians used here, while an encompassing and accurate review of the different conventions and common mistakes related to this matter can be found in the literature.22 The literature contains a plethora of examples where magnetic anisotropy was quantified starting either from the difference between the maximum and minimum magnetizations or from susceptibility values in different directions.20 However, the physical property on which the magnetic anisotropy descriptor is based is not immaterial. In order to emphasize this point, we will use the vanadium(III) hexaaqua cation [V(H2O)6]3+ as a simple chemical example.23 According to electron paramagnetic resonance (EPR) measurements,24 B

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 2. (a) Free energy, (b) magnetization, and (c) susceptibility surfaces for the vanadium(III) hexaaqua cation simulated for B = 12 T. The color scale emphasizes to the magnitude of the quantity. The isotropic contribution has been subtracted from each plot.

Hence, we proceed to consider the effect of the external parameters on the descriptors of magnetic anisotropy, specifically illustrated by the profound influence of magnetic field, temperature, and pressure. A key point of our analysis is that experimentally accessible ranges of T, B, and P can be used to switch magnetic anisotropy in many real systems.

To have more concrete statistics on the mismatch between the magnetic anisotropy descriptors, we picked several examples of transition-metal-based single-ion compounds. The parameters describing these systems have been accurately determined using EPR spectroscopy.24 Our analysis considered all of the possible spin values for an isolated transition-metal ion (except the trivial S = 1/2 case), that is, S = 1, 3/2, 2, and 5 /2. The range of D values is −16 to +14 cm−1. Rhombicity (E ≠ 0), fourth-order parameters, and anisotropic g values are also included (if determined). While at low fields the description of magnetic anisotropy obtained starting from the three quantities is the same (and it coincides with the direction obtained from F in Table 1), near saturation conditions (e.g., T = 2 K and B = 12 T), differences are the rule (Table 1).

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RESULTS AND DISCUSSION 1. Description of Magnetic Anisotropy from F, M, and χ. If we plot the quantities depicted in Figure 1 (F, M, and χ) for the exact same system but at different experimental conditions (B = 12 T instead of 1 T), the situation is altered fundamentally, and we get the plots depicted in Figure 2. While the free-energy shape (Figure 2a) does not modify qualitatively (easy-plane anisotropy compared with Figure 1a), both the magnetization and susceptibility plots are distinctly different from their counterparts in Figure 1. This change is related to a combination of level crossing and magnetic saturation. Indeed, the high mixing of the states produced by the field and selective population of the lowest state produce a new maximum in the magnetization plot, along z (Figure 2b). This anisotropy shape calculated from magnetization is called easy-axis, which is opposite to the one extracted from the free energy. The anisotropy obtained from the susceptibility (Figure 2c) is instead easy-cone because the magnetic susceptibility has a maximum along a cone tilted by θ = 26° (or, equivalently, 154°) from the z axis. A plot of the same quantities at the same conditions, but without the subtraction of the spherical part, is reported in Figure S2, where the dominant role of the isotropic part is evident. At this point, the reader could infer that the mismatch in the magnetic anisotropy description obtained from the three physical quantities can only be encountered if a level crossing occurs. This is, however, not the case. An easy way to prove this is to simulate another system: a spin S = 1 with isotropic g = 2 and a negative D parameter (D = −20 cm−1). The large negative ZFS parameter assures no ground-level crossings. Figure S3 shows that, in the saturation regime (T = 2 K and B = 12 T), the anisotropy obtained starting from the free energy and magnetization can be defined as easy-axis, while the anisotropy obtained from the susceptibility is easy-plane. In order to understand this discrepancy, we can start from eq 3. At saturation, the molecule cannot significantly vary its magnetization, and especially so along the easy direction (the z axis, where the susceptibility has indeed a minimum). However, the direction where the sample can still vary its magnetization significantly is where magnetization is the farthest from saturation, that is, in the xy plane.

Table 1. Principal Direction of Magnetic Anisotropy for Several Transition-Metal-Based Mononuclear Compounds24 at T = 2 K and B = 12Ta compound V(H2O)63+ in CsGa(SO4)2· 12H2O vanadium(III) doped into Al2O3 V(acac)3 VBr3(THF)3 chromium(III) doped into SrF2 Mn(acac)3 Mn(cyclam)I2 manganese protoporphyrin IX FeIII-EDTA Mohr’s salt Co(PPh3)2Cl2 Ni(PPh3)2Cl2

lowest F direction

highest M direction

highest χ direction

xy plane

z

c

xy plane

z

z

y z y

z z y

z x z

z y y

z y y

x c c

y y z y

y y z y

c x x z

a The symbol c refers to either a cone (for axial systems) or to a direction that is not a principal magnetic axis (for rhombic systems).

In our original example [the vanadium(III) hexaaqua cation], the free-energy shapes are qualitatively similar (Figures 1a and 2a); thus, magnetic anisotropy described starting from the free energy can be consistently described at both fields as easy-plane. This is, however, not a general rule, as we will show in the next sections, because magnetic anisotropy is an elastic quantity that can change as a function of external stimuli like field, temperature, and pressure. Anyway, from now on in this paper, we will define magnetic anisotropy based on the free energy. This choice is related to the fact that the free energy is C

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 3. Free energy plots for a total angular momentum J = 15/2, with gJ = 4/3 (e.g., Dy3+), at T = 2 K and B = 5 T: (a) B02 = −1 × 10−2 cm−1 and B14 = 1 × 10−3 cm−1; (b) B02 = −1 × 10−2 cm−1 and B24 = 1 × 10−3 cm−1; (c) B02 = −1 × 10−2 cm−1 and B34 = 1 × 10−3 cm−1; (d) B02 = −1 × 10−2 cm−1 and B44 = 1 × 10−3 cm−1. The isotropic contribution has been subtracted.

Figure 4. Anisotropic free-energy plots for [PPh4][Ln{Pt(SAc)4}2].16 (a) Ln = Ho, T = 18 K, and B = 3 (red), 2.5 (green), and 1.5 (blue) T; (b) Ln = Er, T = 45 K, and B = 12 (red), 10 (green), and 8 (blue) T. Data in the 180−360° range were obtained by symmetry. The black lines are a guide to the eye.

high-symmetry lanthanide compounds. Conveniently, the large spin−orbit coupling in lanthanide ions produces a large separation of the crystal-field levels in zero magnetic field. This leads to an energy-level splitting where the contribution of each level to magnetic anisotropy is relatively easily pictured and understood. For the sake of simplicity, we will limit our analysis here to mononuclear compounds, while some comments on polynuclear (transition-metal) systems and the consequences of magnetic coupling will be deferred to section 3. The description of magnetic anisotropy in lanthanides is commonly done in terms of crystal-field operators. The number and nature of these operators depend on the symmetry of the complex, and it can be as high as 27 for C1 symmetry. In Figure 3, we have plotted the free energy of a Dy3+ ion with different sets of crystal-field parameters (see also Figure S4 for a different perspective). The overall shape of the plots reflects the presence of different off-diagonal crystal-field parameters, which, in turn, implies different molecular geometries. 2.1. Magnetic Field. The effect of an external magnetic field is the removal of degeneracies of the levels and concomitant mixing. The different slopes (more generally, the curvature) of the levels determine the direction where the energy is lowest (see eq 1) and thus relate with the magnetic anisotropy. As an experimental example to describe the effect of a field on magnetic anisotropy, we will use magnetic torque data recorded on two lanthanide complexes.17 Torque magnetometry has proven to be a leading technique in determining the

the basic quantity from which magnetization and susceptibility can be derived (see eqs 1−3). 2. Effect of External Stimuli. The previous example pointed out how the chosen quantity for defining magnetic anisotropy can influence the resulting qualitative description. However, within a given definition, magnetic anisotropy can be modified using external stimuli like magnetic field, temperature, and pressure. It is important to remark that the presence of a nonzero magnetic field is necessary to observe magnetic anisotropy in paramagnetic complexes. Therefore, the anisotropy descriptors, e.g., easy-axis or easy-plane, become meaningless in zero field. Interestingly, the magnetic anisotropy cannot also be unequivocally defined for a chemical system using the zerofield limit (e.g., considering very minute fields) because, as we demonstrate in section 2.2, the temperature also plays a crucial role in defining magnetic anisotropy. By a combination of experimental data and simulations, the following sections demonstrate how field, temperature, and pressure can affect magnetic anisotropy. To isolate the influence of a single external parameter on magnetic anisotropy, we initially keep the others fixed. However, because magnetic anisotropy is directly connected to the free energy, it behaves like a state function, for which phase diagrams in two external parameters can be illustrative. The phenomenon of magnetic anisotropy switching was observed in both transition-metal18,19 and lanthanide complexes;16,17 however, in this section, we will mainly focus on D

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 5. Free-energy plots for Er (pink diamonds) and Ho (orange dots): (a) T = 2 K and B = 1 T; (b) T = 18 K and B = 3 T (Ho) and T = 45 K and B = 12 T (Er); (c) T = 150 K and B = 8 T (Ho) and T = 150 K and B = 10 T (Er). An isotropic contribution has been subtracted. Data in the 180−360° range were obtained by symmetry. The black lines are a guide to the eye.

magnetic anisotropy of molecules27 and molecular thin films.28 This is due to its particular sensitivity to the anisotropic part of the free energy. Indeed, the magnetic torque (τ) acting on the anisotropic single crystal placed inside a homogeneous magnetic field can be equivalently expressed as the vector product between magnetization and the magnetic field or as the derivative of the free energy with respect to the rotation angle.27 The second definition can be rearranged to emphasize the direct connection between torque and free energy as follows: Fani(ε) = F(ε) − Fiso = −

∫0

plane. A more detailed explanation of the origin of this phenomenon can be found in the literature.16,17 A final question that must be addressed in this section is the role of level crossings in switching the anisotropy. In a simplistic picture of, e.g., a linearly coordinated lanthanide system, the distinctly different slopes of the different |±mJ⟩ doublets (or pseudodoublets, for non-Kramers’ ions) would immediately lead one to expect switching anisotropy to relate to field-induced crossing of the levels. Level crossings can, indeed, promote a change in the magnetic anisotropy, as experimentally verified;19 however, modeling demonstrates that they are neither a necessary nor a sufficient requirement to observe a reshaping of the magnetic anisotropy with the magnetic field.16,17 2.2. Temperature. The temperature modifies the population of the levels, affecting the partition function and thus the free energy, according to eq 1. In Figure 5, we show a polar plot of the free energy of Ho and Er together in each panel at low, intermediate, and high temperatures for the respective systems. Again, the data are experimental, derived from integration of the determined torque response16 using eq 4. The data are superimposed to the molecular structure. A thermal evolution of the anisotropy of the free energy is quite evident: the Ho derivative has an easy-axis anisotropy at low temperature (F is minimum along the z axis), while the Er complex is easy-plane (F is minimum in the xy plane). See Figure 5a. At high temperature (Figure 5c), the anisotropy is reversed: Ho is now easy-plane, and Er is easy-axis. The transition temperature is different in the two compounds: around 18 K for Ho and around 45 K for Er (Figure 5b). The transition in the temperature can be very abrupt; thus, it is very challenging to follow the complete reshaping experimentally. However, the simulation reported in Figure S7 shows that, considering, e.g., B = 0.5 T on the Ho compound, a complete easy-axis to easy-plane switch can be obtained in an interval as small as 0.25 K (16.55−16.80 K). We can conveniently visualize at the same time the effect of the field and temperature by plotting the magnetic anisotropy at selected T and B values in a “phase diagram” plot. We have done so for the Ho compound in Figure 6. The chosen temperature range (1−50 K, 0.025 K step) allows one to fully illustrate the effect of the temperature on this compound because above 50 K the anisotropy at fields up to 12 T is

ε

τ ( ε) d ε

(4)

where Fani(ε) and Fiso are the anisotropic and isotropic parts of the free energy [F(ε)], respectively, in the plane perpendicular to the rotation axis and ε is the rotation angle in the plane perpendicular to the rotation axis.27 In Figure 4, we report a polar plot representing the anisotropic part of the free energy of [PPh4][Ln{Pt(SAc)4}2] (SAc− = thioacetate; Ln = Ho, Er; hereafter Ho and Er, for simplicity) obtained from integration of the experimentally measured torque,16 using eq 4. The experimental data were superimposed to the molecular structure. At the selected temperatures (T = 18 K for Ho and 45 K for Er), the increasing magnitude of the magnetic field produces a reshaping of the magnetic anisotropy. As expected, at high fields (red symbols), the magnetic anisotropy increases compared to that at low fields (blue symbols). In the case of Ho (Figure 4a), the magnetic anisotropy changes from easy-plane at B = 1.5 T to easy cone at B = 2.5 and 3 T.29 This can be immediately visualized by looking at the value of the free energy in the xy plane, which gradually increases, thus disfavoring the application of the field in that direction. A zoom on the central part of the plot is reported in Figure S5. The effect of the field on the magnetic anisotropy of the Er compound is less abrupt because the anisotropy always remains easy-plane at all of the fields reported in Figure 4b (lowest-energy direction along the xy plane). However, a clear change of the free energy along the z axis can be observed. A simulation of the same quantity at B = 50 T (Figure S6) shows that the effect of increasing the field at this temperature is to equally lower the energy along the z axis and along the xy E

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 6. Magnetic anisotropy phase diagram for Ho. The color scale refers to the angle between the z axis and the lowest-energy direction [θ = 0° (blue) corresponds to easy-axis anisotropy, while θ = 90° (red) corresponds to easy-plane anisotropy; the other colors are easycone anisotropy]. The 3D plots are free-energy surfaces simulated at selected experimental conditions.

Figure 7. Magnetic anisotropy phase diagram for an ideal Er3+-based system with only diagonal crystal-field parameters (see the text). The y axis simulates the effect of pressure. P = 1 stands for ambient pressure, while P = 1.5 is approximately 12 GPa (see the text). The magnetic field was kept constant at B = 1 T.

always easy-plane. The field range (0.1−12 T, 0.05 T step) is commensurate with most of the experimental setups commonly available in laboratories. As a final remark on this section, we point out that the temperature also has an influence on the interatomic distances and thus on the values of the crystal-field parameters, which become temperature-dependent. However, this effect was advocated to have little influence on the magnetic anisotropy30 because of the relatively small thermal contraction/expansion of inorganic complexes, which in most cases is much below 1% in volume.31 2.3. Pressure. The effect of pressure on the magnetic anisotropy is significantly less explored in comparison with the temperature and magnetic field because of the practical limitations that restrict the number of available techniques and suitable systems.32 However, pressure has a significant impact on the crystal structure (bond lengths and angles); thus, it changes the electronic structure of the molecule. The shift of the energy levels (sometimes accompanied by level crossings) can be followed spectroscopically to determine the trend of the crystal-field parameters under pressure.33 Unfortunately, this trend is, in general, not trivial to predict, and it is not the same for all of the parameters and compounds. An exception is constituted by the Pr3+ and Nd3+ ions in monocapped square-antiprismatic coordination geometry imposed in the LaOCl matrices, for which B02 and B06 decrease, while B04 remains almost constant under pressure.34 We thus simulated a fictious Er3+ system (J = 15/2, gJ = 6/5) having B02 = 0.18 cm−1, B04 = −2 10−2 cm−1, and B06 = 3.6 × 10−5 cm−1 at ambient pressure, and we progressively increased B02 and B06, while keeping B04 constant to simulate the effect of increasing pressure. The magnetic field was kept constant at B = 1 T. In Figure 7, we show a T versus P diagram, where P = 1 simulates ambient pressure, while P = 1.5 corresponds to a 50% increase of B02 and B06. Following the literature example, P = 1.5 would correspond to approximately 12 GPa. At ambient pressure, this fictious compound is expected to be easy-axis at all temperatures (horizontal cut of Figure 7, corresponding to P = 1). However, when the parameters are slightly modified, a

change in anisotropy is expected. In particular, an easy-plane region appears at low temperature as the pressure increases. In the region P = 1.25−1.35, even two switches of magnetic anisotropy are expected in the temperature interval 1−300 K. If P > 1.35, the compound is instead easy-plane at all of the (simulated) temperatures. 3. Coupling. The coupling between magnetic centers can influence the magnetic anisotropy and produce a material exhibiting anisotropy switching in experimentally relevant regions of the parameter space. As amply demonstrated in the literature, it is much easier to achieve large couplings between transition metals than between lanthanides because of the high radial extension of the d orbitals compared to the 4f orbitals. Indeed, to the best of our knowledge, an inversion of the magnetic anisotropy was observed in clusters19 and dimers18 of transition metals but so far never in molecular structures based on lanthanide- or actinide-coupled systems. The example that we will use in this section is a previously reported dimeric compound consisting of two anisotropic FeII (S = 2) ions antiferromagnetically coupled (jAFM = 1.74 cm−1).18 The magnetic reference frames of the two FeII ions are collinear and can be modeled with a negative ZFS parameter D = −8.5 cm−1 and an isotropic g = 2.23. If we consider the two ions to be uncoupled, the magnetic anisotropy of the system (defined from the free energy) is expected to be easy-axis at any given temperature and field. However, when an antiferromagnetic isotropic coupling term (necessary to reproduce the experimental data18) is added to the description of the system, the magnetic anisotropy at low temperature and low field becomes easy-plane (red region in Figure 8). When the temperature or field is raised, the anisotropy is instead easy-axis (blue region in Figure 8).

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CONCLUSIONS A rational design of magnetic molecules often implies targeting of a very specific shape and magnitude of magnetic anisotropy. This requires strict control over the energy and composition of F

DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX

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in general, not possible with the sole use of powder-averaged techniques. Instead, the combination of spectroscopy and single-crystal magnetometric measurements seems to be very effective. (3) Stating that a system has a certain anisotropy descriptor (for example, it is “easy-axis”) has no meaning without defining all of the external parameters like the temperature, field, and pressure. This is because magnetic anisotropy is an elastic quantity whose many facets can not only be engineered chemically but also be tuned using external stimuli.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.9b00636. Derivation of the formulas, notes about the employed Hamiltonians, and supplementary plots (PDF)

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Figure 8. Magnetic anisotropy phase diagram for the iron(II) dimer (see the text for parameters). The 3D plots are the free energy simulated at selected experimental conditions.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Mauro Perfetti: 0000-0001-5649-0449 Jesper Bendix: 0000-0003-1255-2868

the crystal-field levels. For example, the ideal SMMs are characterized by a systematic ordering and low mixing of the | mJ⟩ states, leading to a strong easy-axis anisotropy at all of the accessible temperatures and fields.35,36 For these molecules, no anisotropy switching is expected to take place. However, these systems are very special cases. On the contrary, most metal complexes, including SMMs, are characterized by nontrivial ordering and high mixing of the states and can potentially exhibit anisotropy switching. Such systems are not simply suboptimal SMMs although these properties are to some extent complementary. Rather, the quite narrow switching intervals in the temperature and field make them interesting bi- or multimodal systems. Notably, the magnetic anisotropy switche broadens the concept of magnetic switching in the d block, where high-spin/low-spin switching (spin-crossover) is commonplace, and it brings a new degree of freedom to the f block, where switching of the magnetic properties has received little attention. The possibility of synthesizing a molecule exhibiting ad hoc magnetic anisotropy is certainly appealing, but it is also challenging, and it requires both rational design and synthetic control. Targeting anisotropy switching is in some ways simpler than targeting optimal SMMs because many different orderings of the electronic states can produce the desired effect. For the very same reason, it is less easy to establish simple design criteria for anisotropy switches. That said, we argue that the following three bullet points are central in the modeling, discussion, and potential exploitation of magnetic anisotropy in molecular systems: (1) To describe and compare the magnetic anisotropy of molecular systems, it is important to agree on the physical quantity that defines it. Indeed, starting from the free energy, magnetization or susceptibility can, in some cases, lead to completely different results. (2) The key to understanding and modeling magnetic anisotropy is an accurate determination of all of the relevant parameters (crystal-field parameters, ZFS parameters, g tensor, couplings, etc.) that describe the system under study. This is,

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge useful discussions with Prof. H. Weihe and Prof. L. Sorace and access to the torquemagnetometry setup in Florence, provided by Prof. R. Sessoli, which has furnished data for several of the examples discussed. J.B. acknowledges support from Independent Research Fund Denmark (Grant 8021-00410B) and the Velux Foundations.

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REFERENCES

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DOI: 10.1021/acs.inorgchem.9b00636 Inorg. Chem. XXXX, XXX, XXX−XXX