Analytical nonadiabatic coupling and state-specific energy gradient for the crystal field Hamiltonian describing lanthanide single-ion magnets.

Paramagnetic molecules with a metal ion as an electron spin center are promising building blocks for molecular qubits and high-density memory arrays. However, fast spin relaxation and decoherence in these molecules lead to a rapid loss of magnetization and quantum information. Nonadiabatic coupling (NAC), closely related to spin-vibrational coupling, is the main source of spin relaxation and decoherence in paramagnetic molecules at higher temperatures. Predicting these couplings using numerical differentiation requires a large number of computationally intensive ab initio or crystal field electronic structure calculations. To reduce computational cost and improve accuracy, we derive and implement analytical NAC and state-specific energy gradient for the ab initio parametrized crystal field Hamiltonian describing single-ion molecular magnets. Our implementation requires only a single crystal field calculation. In addition, the accurate NACs and state-specific energy gradients can be used to model spin relaxation using sophisticated nonadiabatic molecular dynamics, which avoids the harmonic approximation for molecular vibrations. To test our implementation, we calculate the NAC values for three lanthanide complexes. The predicted values support the relaxation mechanisms reported in previous studies.

Enhancing coherence in molecular spin qubits via atomic clock transitions.

Quantum computing is an emerging area within the information sciences revolving around the concept of quantum bits (qubits). A major obstacle is the extreme fragility of these qubits due to interactions with their environment that destroy their quantumness. This phenomenon, known as decoherence, is of fundamental interest. There are many competing candidates for qubits, including superconducting circuits, quantum optical cavities, ultracold atoms and spin qubits, and each has its strengths and weaknesses. When dealing with spin qubits, the strongest source of decoherence is the magnetic dipolar interaction. To minimize it, spins are typically diluted in a diamagnetic matrix. For example, this dilution can be taken to the extreme of a single phosphorus atom in silicon, whereas in molecular matrices a typical ratio is one magnetic molecule per 10,000 matrix molecules. However, there is a fundamental contradiction between reducing decoherence by dilution and allowing quantum operations via the interaction between spin qubits. To resolve this contradiction, the design and engineering of quantum hardware can benefit from a 'bottom-up' approach whereby the electronic structure of magnetic molecules is chemically tailored to give the desired physical behaviour. Here we present a way of enhancing coherence in solid-state molecular spin qubits without resorting to extreme dilution. It is based on the design of molecular structures with crystal field ground states possessing large tunnelling gaps that give rise to optimal operating points, or atomic clock transitions, at which the quantum spin dynamics become protected against dipolar decoherence. This approach is illustrated with a holmium molecular nanomagnet in which long coherence times (up to 8.4 microseconds at 5 kelvin) are obtained at unusually high concentrations. This finding opens new avenues for quantum computing based on molecular spin qubits.

Binding Sites, Vibrations and Spin-Lattice Relaxation Times in Europium(II)-Based Metallofullerene Spin Qubits.

To design molecular spin qubits with enhanced quantum coherence, a control of the coupling between the local vibrations and the spin states is crucial, which could be realized in principle by engineering molecular structures via coordination chemistry. To this end, understanding the underlying structural factors that govern the spin relaxation is a central topic. Here, we report the investigation of the spin dynamics in a series of chemically designed europium(II)-based endohedral metallofullerenes (EMFs). By introducing a unique structural difference, i. e. metal-cage binding site, while keeping other molecular parameters constant between different complexes, these manifest the key role of the three low-energy metal-displacing vibrations in mediating the spin-lattice relaxation times (T1 ). The temperature dependence of T1 can thus be normalized by the frequencies of these low energy vibrations to show an unprecedentedly universal behavior for EMFs in frozen CS2 solution. Our theoretical analysis indicates that this structural difference determines not only the vibrational rigidity but also spin-vibration coupling in these EMF-based qubit candidates.

Near Isotropic D4d Spin Qubits as Nodes of a Gd(III)-Based Metal-Organic Framework.

Embedding coherent spin motifs in reproducible molecular building blocks is a promising pathway for the realization of quantum technologies. Three-dimensional (3D) MOFs are a versatile platform for the rational design of extended structures employing coordination chemistry. Here, we report the synthesis and characterization of a gadolinium(III)-based MOF, [Gd(bipyNO)4](TfO)3·xMeOH (bipyNO = bipyridine,N,N'-dioxide; TfO = triflate; and MeOH = methanol) (quMOF-1), which presents a unique coordination geometry that leads to a tiny magnetic anisotropy (in terms of D, an equivalent zero-field splitting would be achieved by D = 0.006 cm-1) even compared with regular Gd(III) complexes. Pulsed electron paramagnetic resonance experiments on its magnetically diluted samples confirm the preservation of quantum coherence of single Gd(III) qubit units in this 3D extended molecular architecture (T2 = 612 ns and T1 = 66 µs at 3.5 K), which allows for the detection of Rabi oscillations at 40 K.

Spectroscopic Analysis of Vibronic Relaxation Pathways in Molecular Spin Qubit [Ho(W5O18)2]9-: Sparse Spectra Are Key.

Vibrations play a prominent role in magnetic relaxation processes of molecular spin qubits as they couple to spin states, leading to the loss of quantum information. Direct experimental determination of vibronic coupling is crucial to understand and control the spin dynamics of these nano-objects, which represent the limit of miniaturization for quantum devices. Herein, we measure the magneto-infrared properties of the molecular spin qubit system Na9[Ho(W5O18)2]·35H2O. Our results place significant constraints on the pattern of crystal field levels and the vibrational excitations allowing us to unravel vibronic decoherence pathways in this system. We observe field-induced spectral changes near 63 and 370 cm-1 that are modeled in terms of odd-symmetry vibrations mixed with f-manifold crystal field excitations. The overall extent of vibronic coupling in Na9[Ho(W5O18)2]·35H2O is limited by a modest coupling constant (on the order of 0.25) and a transparency window in the phonon density of states that acts to keep the intramolecular vibrations and MJ levels apart. These findings advance the understanding of vibronic coupling in a molecular magnet with atomic clock transitions and suggest strategies for designing molecular spin qubits with improved coherence lifetimes.

Towards peptide-based tunable multistate memristive materials.

Development of new memristive hardware is a technological requirement towards widespread neuromorphic computing. Molecular spintronics seems to be a fertile field for the design and preparation of this hardware. Within molecular spintronics, recent results on metallopeptides demonstrating the interaction between paramagnetic ions and the chirality induced spin selectivity effect hold particular promise for developing fast (ns-µs) operation times. [R. Torres-Cavanillas et al., J. Am. Chem. Soc., 2020, DOI: 10.1021/jacs.0c07531]. Among the challenges in the field, a major highlight is the difficulty in modelling the spin dynamics in these complex systems, but at the same time the use of inexpensive methods has already allowed progress in that direction. Finally, we discuss the unique potential of biomolecules for the design of multistate memristors with a controlled- and indeed, programmable-nanostructure, allowing going beyond anything that is conceivable by employing conventional coordination chemistry.

Reinforced Room-Temperature Spin Filtering in Chiral Paramagnetic Metallopeptides.

Chirality-induced spin selectivity (CISS), whereby helical molecules polarize the spin of electrical current, is an intriguing effect with potential applications in nanospintronics. In this nascent field, the study of the CISS effect using paramagnetic chiral molecules, which could introduce another degree of freedom in controlling the spin transport, remains so far unexplored. To address this challenge, herein we propose the use of self-assembled monolayers (SAMs) of helical lanthanide-binding peptides. To elucidate the effect of the paramagnetic nuclei, monolayers of the peptide coordinating paramagnetic or diamagnetic ions are prepared. By means of spin-dependent electrochemistry, the CISS effect is demonstrated by cyclic voltammetry and electrochemical impedance measurements for both samples. Additionally, an implementation of the standard liquid-metal drop electron transport setup has been carried out, and this process helped to demonstrate the peptides' suitability for solid-state devices. Remarkably, the inclusion of a paramagnetic center in the peptide increases the spin polarization as was independently proved by different techniques. These findings permit the inclusion of magnetic biomolecules in the CISS field and pave the way to their implementation in a new generation of (bio)spintronic nanodevices.

Exploring the High-Temperature Frontier in Molecular Nanomagnets: From Lanthanides to Actinides.

Molecular nanomagnets based on mononuclear metal complexes, also known as single-ion magnets (SIMs), are crossing challenging boundaries in molecular magnetism. From an experimental point of view, this class of magnetic molecules has expanded from lanthanoid complexes to both d-transition metal and actinoid complexes. From a theoretical point of view, more and more improved models have been developed, and we are now able not only to calculate the electronic structure of these systems on the basis of their molecular structures but also to unveil the role of vibrations in the magnetic relaxation processes, at least for lanthanoid and d-transition metal SIMs. This knowledge has allowed us to optimize the behavior of dysprosocenium-based SIMs until reaching magnetic hysteresis above liquid-nitrogen temperature. In this contribution, we offer a brief perspective of the progress of theoretical modeling in this field. We start by reviewing the developed methodologies to investigate the electronic structures of these systems and then move on focus to the open problem of understanding and optimizing the vibrationally induced spin relaxation, especially in uranium-based molecular nanomagnets. Finally, we discuss the differences in the design strategies for 4f and 5f SIMs, including an analysis of the metallocenium family.

A first peek into sub-picosecond dynamics of spin energy levels in magnetic biomolecules.

We estimate the time- and temperature-evolution of spin energy levels in a metallopeptide by combining molecular dynamics with crystal field analysis. Fluctuations of tens of cm-1 for spin energy levels at fs times gradually average out at longer times. We confirm that local vibrations are key in spin dynamics.

Sublimable Single Ion Magnets Based on Lanthanoid Quinolinate Complexes: The Role of Intermolecular Interactions on Their Thermal Stability.

We report the design, preparation, and characterization of two families of thermally robust coordination complexes based on lanthanoid quinolinate compounds: [Ln(5,7-Br2q)4]- and [Ln(5,7-ClIq)4]-, where q = 8-hydroquinolinate anion and Ln = DyIII, TbIII, ErIII, and HoIII. The sodium salt of [Dy(5,7-Br2q)4]- decomposes upon sublimation, whereas the sodium salt of [Dy(5,7-ClIq)4]-, which displays subtly different crystalline interactions, is sublimable under gentle conditions. The resulting film presents low roughness with high coverage, and the molecular integrity of the coordination complex is verified through AFM, MALDI-TOF, FT-IR, and microanalysis. Crucially, the single-molecule magnet behavior exhibited by [Dy(5,7-ClIq)4]- in bulk remains detectable by ac magnetometry in the sublimated film.

Deciphering the Role of Dipolar Interactions in Magnetic Layered Double Hydroxides.

Layered double hydroxides (LDHs) exhibit unparalleled anion exchange properties and the ability to be exfoliated into 2D nanosheets, which can be used as a building block to fabricate a wide variety of hybrid functional nanostructured materials. Still, if one wants to use LDHs as a magnetic building blocks in the design of complex architectures, the role played by the dipolar magnetic interactions in these layered materials needs to be understood. In this work, we synthesized and characterized a five-membered CoAl-LDH series with basal spacing ranging from 7.5 to 34 Å. A detailed experimental characterization allows us to conclude that the main factor governing the dipolar interactions between magnetic layers cannot be the interlayer spacing. Supporting theoretical modeling suggests instead a relevant role for spin correlation size, which, in the limit, is related to the lateral dimension of the layer. These results highlight the importance of cation ordering in the magnetic behavior of LDHs, and underpin the differences with homometallic-layered hydroxides.

Custom Coordination Environments for Lanthanoids: Tripodal Ligands Achieve Near-Perfect Octahedral Coordination for Two Dysprosium-Based Molecular Nanomagnets.

Controlling the coordination sphere of lanthanoid complexes is a challenging critical step toward controlling their relaxation properties. Here we present the synthesis of hexacoordinated dysprosium single-molecule magnets, where tripodal ligands achieve a near-perfect octahedral coordination. We perform a complete experimental and theoretical investigation of their magnetic properties, including a full single-crystal magnetic anisotropy analysis. The combination of electrostatic and crystal-field computational tools (SIMPRE and CONDON codes) allows us to explain the static behavior of these systems in detail.

Electric field controllable magnetic coupling of localized spins mediated by itinerant electrons: a toy model.

In this paper, we propose a toy model to describe the magnetic coupling between the localized spins mediated by the itinerant electron in partially delocalized mixed-valence (MV) systems. This minimal model takes into account the key interactions that are common for all such systems, namely, electron transfer in the valence-delocalized moiety and magnetic exchange between the localized spins and the delocalized electrons. The proposed descriptive model is exactly solvable which allows us to qualitatively and quantitatively discuss the main features of the whole class of partially delocalized MV systems. In the case of relatively strong exchange coupling, the combined action of these two interactions is shown to give rise to a specific kind of double exchange coupling termed here as "external core" double exchange. In the opposite case of relatively strong electron transfer, the general Hamiltonian is shown to be reduced to the effective Hamiltonian of indirect exchange of the localized spins. We argue a possibility to efficiently control the magnetic coupling of the localized spins using an external electric field acting on the delocalized part of the system. Finally, we discuss the perspectives of the present model for molecular spintronics and spin qubits.

SIMPRE1.2: Considering the hyperfine and quadrupolar couplings and the nuclear spin bath decoherence.

SIMPRE is a fortran77 code which uses an effective electrostatic model of point charges to predict the magnetic behavior of rare-earth-based mononuclear complexes. In this article, we present SIMPRE1.2, which now takes into account two further phenomena. First, SIMPRE now considers the hyperfine and quadrupolar interactions within the rare-earth ion, resulting in a more complete and realistic set of energy levels and wave functions. Second, and to widen SIMPRE's predictive capabilities regarding potential molecular spin qubits, it now includes a routine that calculates an upper-bound estimate of the decoherence time considering only the dipolar coupling between the electron spin and the surrounding nuclear spin bath. Additionally, SIMPRE now allows the user to introduce the crystal field parameters manually. Thus, we are able to demonstrate the new features using as examples (i) a Gd-based mononuclear complex known for its properties both as a single ion magnet and as a coherent qubit and (ii) an Er-based mononuclear complex. © 2016 Wiley Periodicals, Inc.

Rational Design of Lanthanoid Single-Ion Magnets: Predictive Power of the Theoretical Models.

We report two new single-ion magnets (SIMs) of a family of oxydiacetate lanthanide complexes with D3 symmetry to test the predictive capabilities of complete active space ab initio methods (CASSCF and CASPT2) and the semiempirical radial effective charge (REC) model. Comparison of the theoretical predictions of the energy levels, wave functions and magnetic properties with detailed spectroscopic and magnetic characterisation is used to critically discuss the limitations of these theoretical approaches. The need for spectroscopic information for a reliable description of the properties of lanthanide SIMs is emphasised.

Switching of Slow Magnetic Relaxation Dynamics in Mononuclear Dysprosium(III) Compounds with Charge Density.

The symmetry around a Dy ion is recognized to be a crucial parameter dictating magnetization relaxation dynamics. We prepared two similar square-antiprismatic complexes, [Dy(LOMe)2(H2O)2](PF6) (1) and Dy(LOMe)2(NO3) (2), where LOMe = [CpCo{P(O)(O(CH3))2}3], including either two neutral water molecules (1) or an anionic nitrate ligand (2). We demonstrated that in this case relaxation dynamics is dramatically affected by the introduction of a charged ligand, stabilizing the easy axis of magnetization along the nitrate direction. We also showed that the application of either a direct-current field or chemical dilution effectively stops quantum tunneling in the ground state of 2, thereby increasing the relaxation time by over 3 orders of magnitude at 3.5 K.

A Mononuclear Uranium(IV) Single-Molecule Magnet with an Azobenzene Radical Ligand.

A tetravalent uranium compound with a radical azobenzene ligand, namely, [{(SiMe2 NPh)3 -tacn}U(IV) (η(2) -N2 Ph2 (.) )] (2), was obtained by one-electron reduction of azobenzene by the trivalent uranium compound [U(III) {(SiMe2 NPh)3 -tacn}] (1). Compound 2 was characterized by single-crystal X-ray diffraction and (1) H NMR, IR, and UV/Vis/NIR spectroscopy. The magnetic properties of 2 and precursor 1 were studied by static magnetization and ac susceptibility measurements, which for the former revealed single-molecule magnet behaviour for the first time in a mononuclear U(IV) compound, whereas trivalent uranium compound 1 does not exhibit slow relaxation of the magnetization at low temperatures. A first approximation to the magnetic behaviour of these compounds was attempted by combining an effective electrostatic model with a phenomenological approach using the full single-ion Hamiltonian.

Electrically switchable magnetic molecules: inducing a magnetic coupling by means of an external electric field in a mixed-valence polyoxovanadate cluster.

Herein we evaluate the influence of an electric field on the coupling of two delocalized electrons in the mixed-valence polyoxometalate (POM) [GeV14 O40 ](8-) (in short V14 ) by using both a t-J model Hamiltonian and DFT calculations. In absence of an electric field the compound is paramagnetic, because the two electrons are localized on different parts of the POM. When an electric field is applied, an abrupt change of the magnetic coupling between the two delocalized electrons can be induced. Indeed, the field forces the two electrons to localize on nearest-neighbors metal centers, leading to a very strong antiferromagnetic coupling. Both theoretical approaches have led to similar results, emphasizing that the sharp spin transition induced by the electric field in the V14 system is a robust phenomenon, intramolecular in nature, and barely influenced by small changes on the external structure.

Magnetic properties of the layered lanthanide hydroxide series Y(x)Dy(8-x)(OH)20Cl4·6H2O: from single ion magnets to 2D and 3D interaction effects.

The magnetic properties of layered dysprosium hydroxides, both diluted in the diamagnetic yttrium analogous matrix (LYH:0.04Dy), and intercalated with 2,6-naphthalene dicarboxylate anions (LDyH-2,6-NDC), were studied and compared with the recently reported undiluted compound (LDyH = Dy8(OH)20Cl4·6H2O). The Y diluted compound reveals a single-molecule magnet (SMM) behavior of single Dy ions, with two distinct slow relaxation processes of the magnetization at low temperatures associated with the two main types of Dy sites, 8- and 9-fold coordinated. Only one relaxation process is observed in both undiluted LDyH and intercalated compounds as a consequence of dominant ferromagnetic Dy-Dy interactions, both intra- and interlayer. Semiempirical calculations using a radial effect charge (REC) model for the crystal field splitting of the Dy levels are used to explain data in terms of contributions from the different Dy sites. The dominant ferromagnetic interactions are explained in terms of orientations of easy magnetization axes obtained by REC calculations together with the sign of the superexchange expected from the Dy-O-Dy angles.