Ab Initio Prediction of Tunneling Relaxation Times and Effective

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Ab initio Prediction of Tunneling Relaxation Times and Effective Demagnetization Barriers in Kramers Lanthanide Single Molecule Magnets Daniel Aravena J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02359 • Publication Date (Web): 28 Aug 2018 Downloaded from http://pubs.acs.org on August 30, 2018

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The Journal of Physical Chemistry Letters

Ab initio Prediction of Tunneling Relaxation Times and Effective Demagnetization Barriers in Kramers Lanthanide Single Molecule Magnets

Daniel Aravena*

Departamento de Química de los Materiales, Facultad de Química y Biología, Universidad de Santiago de Chile, Casilla 40, Correo 33, Santiago, Chile

Email: [email protected]

1 ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT Single molecule magnets (SMMs) are promising candidates for molecule-based quantum information devices. Their main limitation is their cryogenic operative temperature. To achieve devices performing at higher temperatures, demagnetization mechanisms must be suppressed by chemical tuning. Electronic structure calculations can provide useful information to rationalize SMM behavior, but they do not provide a direct prediction for the key experimental parameters describing magnetic relaxation (i.e. tunneling relaxation time ( τ QT ) and effective demagnetization barrier ( U eff )). In this letter, a first principles model is proposed to predict τ QT and U eff for mononuclear, half integer spin SMMs, allowing direct comparison with experiment. Model accuracy was assessed against experimental data for 18 mononuclear LnIII complexes (15 DyIII and 3 ErIII) and applied to three of the current best-performing SMMs, correctly predicting non-trivial relaxation pathways. The model shows that the combination of single-ion anisotropy and spin-spin dipolar coupling can account for the major part of tunneling demagnetization for the studied systems. TOC GRAPHIC

KEYWORDS Single Molecule magnets, ab initio calculations, spin relaxation, lanthanides


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One of the long-term goals in theoretical Molecular Magnetism research is the development of a first-principles method to reliably predict relaxation times for Single-Molecule Magnets (SMMs). Nowadays, magnetic relaxation is recognized as a complex phenomenon, involving many physical mechanisms in a simultaneous way.1–4 Thus, experimental relaxation times represent a superposition of several sources, which are hard to disentangle. Currently, temperature dependence of relaxation time is fitted considering electronic and vibrational effects, having different temperature- and field- dependence (i.e. tunneling, one and two phonon Raman and Orbach mechanisms)5–10. Unfortunately, Raman and Orbach mechanisms can be significantly covariant in the relevant temperature range, hampering the extraction of physically meaningful parameters for each process. Until recently, best-performing SMMs presented measurable relaxation times at temperatures as high as 100K, although their blocking temperatures are significantly smaller (c.a. 1020K)11–15. In 2017, a remarkable improvement in SMM performance was achieved for a dysprosocenium derivative showing a blocking temperature of 60K, reporting relaxation times for temperatures as high as 111K.16,17 Upon increasing temperatures, spin dynamics of SMMs becomes even more complex, as excited states play a progressively important role in demagnetization. In the thermally activated regime (Orbach), all mechanistic complexity is condensed in a single experimentally accessible parameter, the effective demagnetization barrier, U eff . This empirical parameter is convenient to describe temperature dependence of the relaxation time but provides no information about the underlying mechanisms leading to demagnetization. Multireference electronic structure calculations are commonly employed to gain further insight about the level spacing and magnetic anisotropy of SMMs18–25. In favorable cases, U eff can be connected with a specific excited state in a qualitative manner. However, there is no standard protocol to predict effective demagnetization barriers from ab initio data in a univocal way. Regarding tunneling relaxation times, it is common practice to analyze transverse components of relevant spin Hamiltonian parameters (D or g) or matrix elements of the dipole moment operator. These quantities should be proportional to tunneling rates and can be useful to identify quickly relaxing states although they do not provide a prediction for relaxation times. Aiming for the direct prediction of relaxation times, different spin dynamics models have been reported in literature: (i) complementing the description of the record dysprosocenium SMM, Goodwin et al. presented a spin dynamics model of the phonon mediated excitations for the 6 H15/2 DyIII multiplet.16 This model predicts relaxation times for the Orbach and Raman mechanisms and is able to yield an accurate slope for the temperature dependence of the relaxation time, while an exact match can be achieved by adjusting a vibrational broadening parameter. (ii) Lunghi et al. developed a model for the phonon mediated spin relaxation of SMMs, considering both molecular lattice vibrations in an explicit manner.26,27 Their model was applied to understand phonon-mediated under barrier 3 ACS Paragon Plus Environment


relaxation in mononuclear FeII SMMs, highlighting the role of intramolecular vibrations in this context. (iii) Gómez-Coca et al.28 presented a spin dynamics model to unveil the origin of relaxation within Kramers’ ground state for a CoII SMMs, identifying hyperfine interaction as the source of state mixing leading to relaxation and (iv) master equation models based on spin-Hamiltonian parameters, employed to predict relaxation times in polynuclear SMMs as Mn12, Fe4 or Fe8 29–32. Although many aspects of SMM demagnetization have been described by spin dynamics approaches, proposals tend to be case specific and their accuracy is normally not benchmarked against experimental data. Thus, information about transferability of such approaches is lacking. In this article, a new method for the prediction of tunneling demagnetization times (τ QT ) and effective demagnetization barriers ( U eff ) in SMMs is presented. Predictions are directly comparable with experimental parameters derived from ac-magnetic susceptibility measurements. Input data are the molecule orientation in the crystal and the ground and excited state energies and g-factors, as calculated by state of the art multireference calculations. No adjustable parameter is included at any stage of our model, yielding a completely ab initio prediction. The proposed method was assessed considering 18 LnIII mononuclear single molecule magnets, which experimental tunneling times are published in literature. Given the agreement between experimental and calculated relaxation times, the presented model was employed to predict demagnetization pathways for three of the best performing SMMs, including the dysprosocenium molecule. Predictions are consistent with experiments and point to the key factors limiting the ultimate performance of these systems, pointing to critical points in the design of new SMM systems with enhanced properties. Considering that best performing SMMs are lanthanide complexes, it is reasonable to develop a model adapted to this kind of systems. For DyIII molecules, it has been shown that large transverse anisotropy is associated with efficient suppression of SMM behavior while slow magnetic relaxation can be observed in systems with low excitation energies as long as axial anisotropy is preserved.33 Thus, quantum tunneling should be the primary factor determining the appearance of SMM properties. In the thermally activated regime, demagnetization barriers are normally related with first excitation energies due to fast thermally assisted tunneling. Demagnetization though higher excited states is a relatively rare phenomenon leading to improved SMM properties34–37. Thus, a predictive model for demagnetization rates in the tunneling regime should be useful to rationalize SMM properties, even in the thermally activated regime. The ingredients for the proposed model are listed next. At zero magnetic field, resonant tunneling will occur between degenerate states. In the case of half-integer spin systems, the existence of two-fold degeneracy in the absence of magnetic field is warranted by Kramers’ theorem. Then, doublet mixing can be represented 4 ACS Paragon Plus Environment

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by an effective g-tensor in a pseudospin-1/2 framework38. Values for x, y and z components of the g-tensor for each of the eight Kramer’s doublets stemming from the ground 6 H15/2 multiplet were obtained through CASSCF(n,7) (n=9 DyIII and n=11 ErIII) calculations including spin-orbit coupling effects (see SI for further information). For an isolated Kramers’ doublet, quantum tunneling is forbidden by time-reversal symmetry and additional mechanisms should be accounted to explain magnetic relaxation. Sources of demagnetization can be molecular vibrations and spin coupling with neighbor electronic and nuclear magnetic centers. In this model, spin-spin dipolar interaction is enough for a reasonably accurate description of tunneling rates. Thus, environment contributions are restricted to this mechanism. Of course, model extensions including hyperfine and vibration contributions can be designed if needed. Magnetic dilution experiments are an efficient way to suppress dipolar relaxation phenomena. In the tunneling regime, the substitution of magnetic centers by diamagnetic ions (usually YIII) yields a dramatic change in relaxation times. For most cases, the tunneling plateau is suppressed after dilution in the entire frequency and temperature interval of ac-susceptibility experiments. Dipolar interaction between two magnetic centers has the form39:

r

H dip =

r

β 2 µ a ⋅ µb r3

r r r r

−3

β 2 ( µa ⋅ r )( µb ⋅ r ) r5

(1)

r Where β is the Bohr magneton, r is the distance between magnetic centers, µa ,b is the r magnetic moment vector of both interacting sites and r is the vector connecting them. The following replacement is made:

r

3

µ = − ∑ gi Si

(2)

i =1

In the spirit of a single-site relaxation model, center a is considered as the system, perturbed by center b, belonging to the environment. Thus, a spin flip of the first site will be considered as magnetic relaxation while the second site will be regarded as part of the magnetic response of the environment. Both centers are described as pseudospin-1/2 sites, off-diagonal matrix elements for states connected by one spin-flip corresponding to center a are: 3β 2 gbz rz ( − g ax Sbz S ax rx + ig ay Sbz S ay ry ) ↓↑ Hˆ dip ↑↑ = 2r 5 ↓↓ Hˆ dip ↑↓ = − ↓↑ Hˆ dip ↑↑


(3)


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Two-spin flip elements are of the form ↓↓ Hˆ dip ↑↑ =

β 2  r 2 ( g ax gbx S ax Sbx − g ay gby Say Sby ) − 3 g ax gbx Sax Sbx rx2 + 3irx ry ( g ax gby S ax Sby + g ay gbx Say Sbx ) + 3g ay gby Say Sby ry2  4r 5

(4)

Both types of spin-flip terms depend on the transverse components of the site a. In the case of site b, the one spin-flip term is related to µ z while two spin-flip are proportional to µ x, y . The contribution of both mechanisms will be associated to the relative orientation of the magnetic moments. As two spin-flip terms are proportional to the square of transverse gcomponents, this contribution is expected to be small and will be thus, neglected. To derive an expression depending only on the spin of a, replacement in equation 2 is reversed for the environment center b. In this way, spin-flip matrix element for a central magnetic ion interacting with N neighbor spins is: N

3β 2 µiz± riz ( − g ax S ax rix + ig ay S ay riy )

i =1

2ri5

H dip , a = ∑

(5)

Where µiz± can take the values ± µiz for each neighbor center. Thus, each additional neighbor spin will double the number of states in a tree diagram fashion, yielding a 2N+1 manifold. Independent histograms for the distributions of the real and imaginary parts of H dip ,a are constructed and their variances ( σ r2/ i ) are calculated numerically. Fermi Golden Rule allows for the calculation of the relaxation rate k=

2π Edip h

2

(6)

2 Where the distribution of Edip is calculated from the convolution of two squared normal

distributions with zero mean and variance σ r2 and σ i2 . Finally, a large sample of this distribution is generated, the square root of each value is obtained, and the resulting sample is averaged to yield

Edip

Finally, the relaxation time is

τ QT =

1 2k


(7)

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In the Orbach regime, relaxation time dependence with respect to temperature follows an exponential relation (Arrhenius law). To sustain this regime, thermally assisted tunneling must consist in a fast thermal excitation step followed by slower excited state tunneling, which controls the demagnetization rate. Under this regime, each Kramers’ doublet will have a particular demagnetization rate following40 k i (T ) ∝

exp( − E i / k B T ) k QT ,i Z

(10)

Where i represents the index for the Kramers’ doublet starting from number 1 for the ground level,

Ei is the doublet energy obtained through CASSCF calculations, Z is the

partition function,

kB is the Boltzmann constant and kQT,i is the tunneling relaxation rate

for doublet i . The effective demagnetization barrier is defined as M

kQT ,i (T )

i =1

Nk

U eff (T ) = ∑

Ei

Where M is the number of Kramers’ doublets and

(11)

Nk is a normalization factor for kQT,i .

At low T , Ueff will sharply rise due to the progressive depopulation of the ground state, which does not contribute to the effective barrier. When T is larger, the Orbach regime is reached and Ueff is basically constant, representing the effective demagnetization barrier. Demagnetization can be partitioned in contributions from each excited state, allowing for the identification of the dominant relaxation pathways. To test the proposed relaxation model, a set of 18 LnIII (15 DyIII and 3 ErIII) mononuclear complexes was selected from literature. 12,14,41–49 All systems present X-Ray crystal structure and ac-magnetic susceptibility measurements, displaying a clear plateau for the relaxation time in the low temperature range at zero field. In this way, the quantum tunneling relaxation time

τ QT can be extracted for each compound. To include dipolar coupling with

neighbor magnetic ions, crystals with a single crystallographically independent LnIII ion were selected. Thus, the magnitude and orientation of the magnetic moment of all positions can be assigned by considering the symmetry operations of the crystal. For crystals containing more than one symmetry independent magnetic center, calculations must be performed for each inequivalent magnetic ion. For simplicity, individual molecules will be mentioned by their Cambridge Crystallographic Database labels (Refcodes) (see Table S1). Technical details about electronic structure calculations are presented as Supporting Information. 7 ACS Paragon Plus Environment


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As expected for zero field SMMs, all compounds show strong axial magnetic anisotropy, where the transverse components (gx and gy) vary from 0.067 (Refcode: BAJSIQ) to 4*10-6 (NAMFIT). Even if all these numbers appear as small contributions, their differences are important for an accurate prediction of

τ QT . As discussed previously, spin-flip matrix

elements will be roughly proportional on gx and gy. Experimental values for

τ QT are spread

over five orders of magnitude from 9.1*10-5 s (OLUJEM) to 30 s (NAFMIT). Calculated values cover a broader interval from 2.6*10-5 (BAJSIQ) to 8075 s (NAMFIT). Figure 1 presents the relation between experimental and predicted

τ QT values.

In fast tunneling region, there are four cases where the predicted relaxation time is too short in comparison with experiment and fall below the x=y diagonal (DARTOH, DARTUN, GUYRAU, IMOVAJ). Interestingly, these molecules present small cell volumes per unit formula (see Table S2), indicating that LnIII ions are especially close in these crystals. A better description of these close contacts could improve predictions for this group. BAJSIQ is the only example where the ground state is not close to ms = 15/2 and its relaxation time is also underestimated. On the other extreme, slowly relaxing systems appear over the blue diagonal (NAFMIT, ZAVSOH, FUMCUN01). These systems present almost vanishing transverse components of g, and any small contribution from subtle molecular distortions can represent a significant part of the total transverse anisotropy. Despite these deviations, model accuracy is considered remarkable given the complexity of the tunneling phenomenon and the simplicity of the working equations.


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Figure 1. experimental and calculated relaxation times for the 18 LnIII complexes from the validation set. Black squares and red circles correspond to DyIII and ErIII molecules, respectively. The blue diagonal line represents the exact prediction.

Three of the current best performing SMMs were selected to test predictions for relaxation pathways and Ueff . All complexes present large demagnetization barriers and their proposed relaxation pathways are complex and qualitatively different. The first case is the dysprosocenium molecule (MEKDOY), exhibiting an effective demagnetization barrier of 1223 cm-1 and a relaxation pathway dominated by the fifth excited state, as predicted by spin dynamics simulations16. Ueff is predicted to be 1209 cm-1, stemming almost exclusively from the fifth excited state, in accordance with the original proposal. A pictorial representation of this simulation is presented in Figure 2,B. For increasing values of T, Ueff grows rapidly to a constant value when the Orbach regime is reached. Relaxation pathways are also presented: at low T, the only contributing value is the ground doublet, depicted in blue. At the transition from tunneling to thermally activated relaxation, ground state contribution drops, and the fifth excited state emerges as the dominant source of relaxation. 9 ACS Paragon Plus Environment


The second example is a pentagonal bipyramidal SMM reported in 2016 by Ding et al.13 This complex features tertbutoxide ligands at axial positions and five pyridine molecules filling equatorial coordination (Refcode: RAPDUK). Effective demagnetization barrier is also extremely high (1261 cm-1), while its relaxation pathway involves a compact block of states, starting from the third excited state. This assignment was based on inspection of matrix elements of the magnetic transition dipole operator. Again, model predictions are in agreement with the original assignment, as calculated Ueff is 1196 cm-1 and most contributing levels are the third, fifth, sixth and seventh excited doublets (See Figure 2,C and Table 1). The last system to analyze was published by Liu et al.12, it corresponds to a DyIII complex coordinated to a N4O2 polydentate ligand and a bromine anion, adopting a pentagonal bipyramidal coordination geometry (IMOTUB). Demagnetization barrier for this complex is 712 cm-1 and relaxation path is predicted to happen through the third excited state. In this case, second and third excited states are adequate for tunneling given their large matrix elements for the transition dipole moment. Authors choose the third excited state due to its larger transition moment. Predicted Ueff is again close to experiment (736 cm-1) while relaxation has contributions from the second, fourth and sixth excited states (see Figure 2,A and Table 1). Thus, the relaxation pathway is slightly different than the original proposal although the effective demagnetization barrier is consistent with experiment.

Table 1. CASSCF(9,7) state energies (cm-1), effective barriers ( Ueff ) and tunneling relaxation times for three high performance SMMs. The eight Kramers’ doublets composing the ground

6

H15/2 multiplet are considered. States mainly responsible for

demagnetization are highlighted in bold case. REFCODE IMOTUB

MEKDOY

0

log(τQT) (s) -1.094

395.1

-5.409

630.5 724.6

-7.639 -0.585

775.8 785.5

-7.966 -6.492

797.9

-8.020

862.7

-6.084

0

4.249

Ei (cm-1)

464.3

0.929

731.4

-1.938

906.2

-3.778

Ueff,exp (cm-1) 712

Ueff,calc (cm-1) 736

1223

1209


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RAPDUK

1063.2

-4.985

1210.9 1327.4

-7.243 -3.273

1397.6

-4.798

0

3.815

564.6

-0.599

946.3

-4.725

1151.3 1180.8

-6.201 -3.479

1208.6

-5.635

1227.3

-5.866

1243.6

-6.459

1261

1196

IMOTUB



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MEKDOY

RAPDUK

Figure 2. Relaxation contributions of each Kramers’ doublet and predicted effective barrier as a function of T. Ueff is represented as a dashed black line and its value is indicated in the right y-axis. Left y-axis represents the relative contribution of each Kramers’ doublet to relaxation. First and last doublets are represented as blue and red lines, respectively while intermediate levels are weighted combinations of these two colors. Model structures are presented for each compound. Color code: Dy, light green; Br, brown; O, red; N, blue; C, grey; H, white.


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In sum, the proposed theoretical model predicts tunneling relaxation times for Kramers LnIII SMMs in good agreement with available experimental data for a relatively large validation set. Model predictions are unique and cannot be tuned to fit experimental results, since input parameters are the molecular geometry and ab initio parameters derived from standard CASSCF calculations. Under the assumption of fast thermalization, calculated tunneling rates can be employed to derive the effective demagnetization barrier Ueff . Consequently, the two key experimental parameters describing tunneling ( τ QT ) and Orbach ( Ueff ) regimes can be estimated by ab initio methods, providing a direct link to contrast experiment and theory. Complex demagnetization pathways in the thermally assisted tunneling regime can be decomposed in contributions from each Kramers’ doublet. Predicted pathways are in reasonable accordance with literature proposals. From a mechanistic perspective, tunneling demagnetization in Kramers LnIII systems can be associated with spin-flip transitions due to spin-dipolar coupling with neighbor magnetic centers. As expected, this interaction is proportional to the transverse components of the g-tensor. In its current form, the proposed model is aimed to predict

τ QT and Ueff values for

mononuclear, half integer SMMs. To broaden the scope of this approach, it is necessary to extend this model beyond the pseudospin-1/2 approximation to describe integer spin systems. Developments in this direction would be also useful to describe polynuclear systems. Another desirable property would be the inclusion of external magnetic fields, requiring the explicit consideration of phonons due to the importance of the direct relaxation mechanism under this scenario. New advances in this aspect are key for the improvement of current proposals for spin dynamics models.



ASSOCIATED CONTENT

Model parameters for the validation set, Computational Details, Cartesian coordinates for all calculated models and cell volumes per unit formula are included as Supporting Information

AUTHOR INFORMATION *E-mail: [email protected] ORCID-ID: 0000-0003-3140-4852 The author declares no competing financial interest.

ACKNOWLEDGMENTS The author thanks FONDECYT Regular 1170524 and Anillo ACT1404 projects for financial support. Powered@NLHPC: This research was partially supported by the supercomputing infrastructure of the NLHPC (ECM-02).

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Ab Initio Prediction of Tunneling Relaxation Times and Effective

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