Phenomenological Spin Hamiltonian Research Articles

Zero-Field Splitting Tensor of the Triplet Excited States of Aromatic Molecules: A Valence Full-π Complete Active Space Self-Consistent Field Study.

A method to predict the D tensor in the molecular frame with multiconfigurational wave functions in large active space was proposed, and the spin properties of the lowest triplets of aromatic molecules were examined with full-π active space; such calculations were challenging because the size of active space grows exponentially with the number of π electrons. In this method, the exponential growth of complexity is resolved by the density matrix renormalization group (DMRG) algorithm. From the D tensor, we can directly determine the direction of the magnetic axes and the ZFS parameters, D- and E-values, of the phenomenological spin Hamiltonian with their signs, which are not usually obtained in ESR experiments. The method using the DMRG-CASSCF wave function can give correct results even when the sign of D- and E-values is sensitive to the accuracy of the prediction of the D tensor and existing methods fail to predict the correct magnetic axes.

First-principles calculation of the parameters used by atomistic magnetic simulations

While the ground state of magnetic materials is in general well described on the basis of spin density functional theory (SDFT), the theoretical description of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. Time-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are computationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin Hamiltonian and SDFT-based electronic structure calculations, giving access to the dynamical and finite-temperature properties for example via spin-dynamics simulations using the Landau–Lifshitz–Gilbert (LLG) equation or Monte Carlo simulations. We present an overview on the various methods to calculate the parameters of the various phenomenological Hamiltonians with an emphasis on the KKR Green function method as one of the most flexible band structure methods giving access to practically all relevant parameters. Concerning these, it is crucial to account for the spin–orbit coupling (SOC) by performing relativistic SDFT-based calculations as it plays a key role for magnetic anisotropy and chiral exchange interactions represented by the DMI parameters in the spin Hamiltonian. This concerns also the Gilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral multispin interaction parameters of the extended Heisenberg Hamiltonian, as well as spin–lattice interaction parameters describing the interplay of spin and lattice dynamics processes, for which an efficient computational scheme has been developed recently by the present authors.

Block orbital-selective Mott insulators: A spin excitation analysis

We present a comprehensive study of the spin excitations - as measured by the dynamical spin structure factor $S(q,\omega)$ - of the so-called block-magnetic state of low-dimensional orbital-selective Mott insulators. We realize this state via both a multi-orbital Hubbard model and a generalized Kondo-Heisenberg Hamiltonian. Due to various competing energy scales present in the models, the system develops periodic ferromagnetic islands of various shapes and sizes, which are antiferromagnetically coupled. The 2$\times$2 particular case was already found experimentally in the ladder material BaFe$_2$Se$_3$ that becomes superconducting under pressure. Here we discuss the electronic density as well as Hubbard and Hund coupling dependence of $S(q,\omega)$ using density matrix renormalization group method. Several interesting features were identified: (1) An acoustic (dispersive spin-wave) mode develops. (2) The spin-wave bandwidth establishes a new energy scale that is strongly dependent on the size of the magnetic island and becomes abnormally small for large clusters. (3) Optical (dispersionless spin excitation) modes are present for all block states studied here. In addition, a variety of phenomenological spin Hamiltonians have been investigated but none matches entirely our results that were obtained primarily at intermediate Hubbard $U$ strengths. Our comprehensive analysis provides theoretical guidance and motivation to crystal growers to search for appropriate candidate materials to realize the block states, and to neutron scattering experimentalists to confirm the exotic dynamical magnetic properties unveiled here, with a rich mixture of acoustic and optical features.

Observation of a Large Magnetic Anisotropy and a Field-Induced Magnetic State in SrCo(VO4)(OH): A Structure with a Quasi One-Dimensional Magnetic Chain.

A new member of the descloizite family, a cobalt vanadate, SrCo(VO4)(OH), has been synthesized as large single crystals using high-temperature and high-pressure hydrothermal methods. SrCo(VO4)(OH) crystallizes in the orthorhombic crystal system in space group P212121 with the following unit cell parameters: a = 6.0157(2) Å, b = 7.645(2) Å, c = 9.291(3) Å, V = 427.29(2) Å3, and Z = 4. It contains one-dimensional Co-O-Co chains of edge-sharing CoO6 octahedra along the a-axis connected to each other via VO4 tetrahedra along the b-axis forming a three-dimensional structure. The magnetic susceptibility of SrCo(VO4)(OH) indicates an antiferromagnetic transition at 10 K as well as unusually large spin orbit coupling. Single-crystal magnetic measurements in all three main crystallographic directions displayed a significant anisotropy in both temperature- and field-dependent data. Single-crystal neutron diffraction at 4 K was used to characterize the magnetically ordered state. The Co2+ magnetic spins are arranged in a staggered configuration along the chain direction, with a canting angle that follows the tipping of the CoO6 octahedra. The net magnetization along the chain direction, resulting in ferromagnetic coupling of the a-axis spin components in each chain, is compensated by an antiferromagnetic interaction between nearest neighbor chains. A metamagnetic transition appears in the isothermal magnetization data at 2 K along the chain direction, which seems to correspond to a co-alignment of the spin directions of the nearest neighbor chain. We propose a phenomenological spin Hamiltonian that describes the canted spin configuration of the ground state and the metamagnetic transition in SrCo(VO4)(OH).

Steplike metamagnetic transitions in a honeycomb lattice antiferromagnet Tb2Ir3Ga9

Single crystals of a honeycomb lattice antiferromagnet, Tb$_2$Ir$_3$Ga$_9$ were synthesized, and the physical properties have been studied. From magnetometry, a long-range antiferromagnetic ordering at $\approx$12.5 K with highly anisotropic magnetic behavior was found. Neutron powder diffraction confirms that the Tb spins lie along the $\va $-axis, parallel to the shortest Tb-Tb contact. Two field-induced spin-flip transitions are observed when the field is applied parallel to this axis, separated by a plateau corresponding roughly to M$\approx$M$_{\rm{s}}$/2. Transport measurements show the resistivity to be metallic with a discontinuity at the onset of N\'eel order. Heat capacity shows a $\lambda$-like transition confirming the bulk nature of the magnetism. We propose a phenomenological spin-Hamiltonian that describes the magnetization plateau as a result of strong Ising character arising from a quasidoublet ground state of the Tb$^{3+}$ ion in a site of \textit{C$_s$} symmetry and expressing a significant bond dependent anisotropy.

Longitudinal integration measure in classical spin space and its application to first-principle based simulations of ferromagnetic metals

The classical Heisenberg type spin Hamiltonian is widely used for simulations of finite temperature properties of magnetic metals often using parameters derived from first principles calculations. In itinerant electron systems, however, the atomic magnetic moments vary their magnitude with temperature and the spin Hamiltonian should thus be extended to incorporate the effects of longitudinal spin fluctuations (LSF). Although the simple phenomenological spin Hamiltonians describing LSF can be efficiently parameterized in the framework of the constrained Local Spin Density Approximation (LSDA) and its extensions, the fundamental problem concerning the integration in classical spin space remains. It is generally unknown how to integrate over the spin magnitude. Two intuitive choices of integration measure have been used up to date – the Murata-Doniach scalar measure and the simple three dimensional vector measure. Here we derive the integration measure by considering a classical limit of the quantum Heisenberg spin Hamiltonian under conditions leading to the proper classical limit of the commutation relations for all values of the classical spin magnitude and calculate the corresponding ratio of the number of quantum states. We show that the number of quantum states corresponding to the considered classical spin magnitude is proportional to this magnitude and thus a non-trivial integration measure must be used. We apply our results to the first-principles simulation of the Curie temperatures of the two canonical ferromagnets bcc Fe and fcc Ni using a single-site LSF Hamiltonian with parameters calculated in the LSDA framework in the Disordered Local Moment approximation and a fixed spin moment constraint. In the same framework we compare our results with those obtained from the scalar and vector measures.

Determination of exchange coupling constants in linear polyradicals by means of local spins

This work extends the previously reported studies (Oliva et al. in Theor Chem Acc 132:1329, 2013, Theor Chem Acc 134:9, 2015) on electronic structures of simple polyhedral polyradicals constructed from s = ½ closo-carborane CB11H12 • structural units. Linear polyradical structures obtained from these units connected by means of –CH2– bridges are described in terms of their energies and local spins. The resulting spin states of these chains have been mapped onto a phenomenological Heisenberg spin Hamiltonian, providing the evaluation of spin exchange coupling constants and performing an analysis of their transferability. The eigenvalues of this Hamiltonian allow one to determine the ground spin state and the suitable combinations of spin orientations of the magnetic sites. We prove that the minimal energy in all these systems corresponds to the highest-spin state.

The Role of Anisotropic Exchange in Single Molecule Magnets: A CASSCF/NEVPT2 Study of the Fe4 SMM Building Block [Fe2(OCH3)2(dbm)4] Dimer

The rationalisation of single molecule magnets’ (SMMs) magnetic properties by quantum mechanical approaches represents a major task in the field of the Molecular Magnetism. The fundamental interpretative key of molecular magnetism is the phenomenological Spin Hamiltonian and the understanding of the role of its different terms by electronic structure calculations is expected to steer the rational design of new and more performing SMMs. This paper deals with the ab initio calculation of isotropic and anisotropic exchange contributions in the Fe(III) dimer [Fe 2 (OCH 3 ) 2 (dbm) 4 ]. This system represents the building block of one of the most studied Single Molecule Magnets ([Fe 4 RC(CH 2 O) 3 ) 2 (dpm) 6 ] where R can be an aliphatic chain or a phenyl group just to name the most common functionalization groups) and its relatively reduced size allows the use of a high computational level of theory. Calculations were performed using CASSCF and NEVPT2 approaches on the X-ray geometry as assessment of the computational protocol, which has then be used to evinced the importance of the outer coordination shell nature through organic ligand modelization. Magneto-structural correlations as function of internal degrees of freedom for isotropic and anisotropic exchange contributions are also presented, outlining, for the first time, the extremely rapidly changing nature of the anisotropic exchange coupling.

Phenomenological simulation and density functional theory prediction of 57 Fe Mössbauer parameters: application to magnetically coupled diiron proteins

The use of phenomenological spin Hamiltonians and of spin density functional theory for the analysis and interpretation of Mossbauer spectra of antiferromagnetic or ferromagnetic diiron centers is briefly discussed. The spectroscopic parameters of the hydroxylase component of methane monooxygenase (MMOH), an enzyme that catalyzes the conversion of methane to methanol, have been studied. In its reduced diferrous state (MMOHRed) the enzyme displays 57Fe Mossbauer and EPR parameters characteristic of two ferromagnetically coupled high spin ferrous ions. However, Mossbauer spectra recorded for MMOHRed from two different bacteria, Methylococcus capsulatus (Bath) and Methylosinus trichosporium OB3b, display slightly different electric quadrupole splittings (ΔEQ) in apparent contradiction to their essentially identical active site crystallographic structures and biochemical functions. Herein, the Mossbauer spectral parameters of MMOHRed have been predicted and studied via spin density functional theory. The somewhat different ΔEQ recorded for the two bacteria have been traced to the relative position of an essentially unbound water molecule within their diiron active sites. It is shown that the presence or absence of the unbound water molecule mainly affects the electric field gradient at only one iron ion of the binuclear active sites.

Theoretical determination of spin Hamiltonians with isotropic and anisotropic magnetic interactions in transition metal and lanthanide complexes

The determination of anisotropic magnetic parameters is a task of both experimental and theoretical interest. The added value of theoretical calculations can be crucial for analyzing experimental data by (i) allowing assessment of the validity of the phenomenological spin Hamiltonians, (ii) allowing discussion of the values of parameters extracted from experiments, and (iii) proposing rationalizations and magneto-structural correlations to better understand the relations between geometry, electronic structure, and properties. In this review, we discuss the model Hamiltonians that are used to describe magnetic properties, the computational approaches that can be used to compute magnetic parameters, and review their applications to transition metal and (to a lesser extent) lanthanide based complexes. Perspectives concerning current methodological challenges will then be presented, and finally the need for further joint experimental/theoretical efforts will be underlined.

Broken Symmetry Approach to Density Functional Calculation of Magnetic Anisotropy or Zero Field Splittings for Multinuclear Complexes with Antiferromagnetic Coupling

Antiferromagnetic coupling in multinuclear transition metal complexes usually leads to electronic ground states that cannot be described by a single Slater determinant and that are therefore difficult to describe by Kohn-Sham density functional methods. Density functional calculations in such cases are usually converged to broken symmetry solutions which break spin and, in many cases, also spatial symmetry. While a procedure exists to extract isotropic Heisenberg (exchange) coupling constants from such calculations, no such approach is yet established for the calculation of magnetic anisotropy energies or zero field splitting parameters. This work proposes such a procedure. The broken symmetry solutions are not only used to extract the exchange couplings but also single-ion D tensors which are then used to construct a (phenomenological) spin Hamiltonian, from which the magnetic anisotropy and the zero-field energy levels can be computed. The procedure is demonstrated for a bi- and a trinuclear Mn(III) model compound.

Magnetic anisotropy from density functional calculations. Comparison of different approaches: Mn12O12 acetate as a test case

Magnetic anisotropy is the capability of a system in a triplet or higher spin state to store magnetic information. Although the source of the magnetic anisotropy is the zero-field splitting of the ground state of the system, there is a difference between these two quantities that has to be fully rationalized before one makes comparisons. This is especially important for small spins such as triplets, where the magnetic anisotropy energy is only half of the zero-field splitting. Density functional calculations of magnetic anisotropy energies correspond to a high-field limit where the spins are aligned by the external magnetic field. Data are presented for the well-studied molecular magnet Mn(12)O(12) acetate. Both perturbative and self-consistent treatments, different quasirelativistic Hamiltonians (zeroth order regular approximation, Douglas-Kroll, effective core potentials) and exchange-correlation functionals are compared. It is shown that some effects usually considered minor, such as the inclusion of the exchange-correlation potential in the effective one-particle spin-orbit operator, lead to sizable differences when computing magnetic anisotropy energies. Higher-order contributions, that is, the difference between self-consistent and perturbative results, increase the magnetic anisotropy energy somewhat but do not introduce sizeable quartic terms or an in-plane anisotropy. In numerical experiments, on can switch off and on spin-orbit coupling at individual atomic sites. This procedure yields single-site contributions to the overall magnetic anisotropy energy that could be used as parameters in phenomenological spin Hamiltonians. If ferrimagnetic systems are treated with broken symmetry density functional methods where the Kohn-Sham reference function is not a spin eigenfunction, corrections are needed which depend on the size of the exchange couplings in the system and must therefore be evaluated case by case.

Zero-field splitting and g-factor studies for 3d [formula omitted] ions in tetragonal and trigonal symmetry

In this paper, the rigorous analytical expressions for the zero field splittings D, a, F and g-factors of a 3d 5 ion in tetragonal and trigonal crystal fields are derived. In the derivations, the energy levels equivalence between microscopic Zeeman interaction and phenomenological spin-Hamiltonian was used. The deduced expressions do not depend on any interaction model and are universal expressions. As some examples, we studied the specific examples of MnCO 3 , Rb 2 CdF 4 : Mn 2 + and KZnF 3 : Mn 2 + . Good agreement between calculation and experiments shows that the formulas are reasonable.

Exchange anisotropy, disorder, and frustration in diluted, predominantly ferromagnetic, Heisenberg spin systems

Motivated by the recent suggestion of anisotropic effective exchange interactions between Mn spins in Ga$_{1-x}$Mn$_x$As (arising as a result of spin-orbit coupling), we study their effects in diluted Heisenberg spin systems. We perform Monte Carlo simulations on several phenomenological model spin Hamiltonians, and investigate the extent to which frustration induced by anisotropic exchanges can reduce the low temperature magnetization in these models and the interplay of this effect with disorder in the exchange. In a model with low coordination number and purely ferromagnetic (FM) exchanges, we find that the low temperature magnetization is gradually reduced as exchange anisotropy is turned on. However, as the connectivity of the model is increased, the effect of small-to-moderate anisotropy is suppressed, and the magnetization regains its maximum saturation value at low temperatures unless the distribution of exchanges is very wide. To obtain significant suppression of the low temperature magnetization in a model with high connectivity, as is found for long-range interactions, we find it necessary to have both ferromagnetic and antiferromagnetic (AFM) exchanges (e.g. as in the RKKY interaction). This implies that disorder in the sign of the exchange interaction is much more effective in suppressing magnetization at low temperatures than exchange anisotropy.

SPIN-HAMILTONIAN FORMALISMS IN ELECTRON MAGNETIC RESONANCE (EMR) AND RELATED SPECTROSCOPIES

The historical development leading to, and the current status of, the spin-Hamiltonian (SH) formalisms, characterizing electron magnetic resonance (EMR), also referred to as electron paramagnetic (or spin) resonance (EPR or ESR), is reviewed. The spin-Hamiltonian concept is briefly outlined to set a framework for definitions of relevant terms. Meanings of the terms which are often confused with each other, e.g. physical versus effective Hamiltonian, real versus effective versus fictitious spin, microscopic SH (MSH), zero-field-splitting (ZFS) Hamiltonian, generalized SH (GSH), and phenomenological SH (PSH), are elucidated. Various general approaches to ‘derive’ MSH as well as to ‘construct’ GSH for transition ions are discussed. This enables clarification of relationship between the ZFS Hamiltonian and (a) the crystal-field (CF), (b) electronic spin-spin, and (c) nuclear-quadrupole Hamiltonians. The inadmissibility of the odd-order ZFS terms (l = 3, 5) is also discussed. A brief general classification of the major operator and parameter notations used in the literature to describe ZFS Hamiltonian is provided. Other important areas relevant to EMR, where the SH concept plays a central role, are outlined.

ChemInform Abstract: Evolution of Magnetic Resonance Spectroscopy

A review of magnetic resonance spectroscopy from its beginning to the current developments is presented. The phenomenological spin-Hamiltonian of nuclear magnetic resonance, the methods and constrains of high resolution NMR, evolution of Fourier transform NMR spectroscopy are presented. A brief introduction to 2D-FT NMR methods like COSY, NOESY and EXSY is given. Magnetic resonance imaging by 2D and 3D-FT NMR spectroscopy is described EPR spin-Hamiltonian the advantage of loop-gap resonators over the conventional cavity resonators are presented. The modern trends in EPR spectroscopy, its applications to biological systems, recent developments in EPR spectrocopy with the availability of fast computers and sophisticated EPR components including FT EPR spectrocopy, elelctron spin echoes, electron spin echo envelop modulation, and 2D-FT EPR spectrocopy are discussed. Electron-Zeeman resolved EPR spectrocopy, EPR imaging technique and in vivo EPR spectrocopy, electron-nuclear double resonance, Mossbauer and nuclear quadrupole resonance spectrocopies also find an expression in this report.

Optically detected magnetic resonance studies of CdS nanoparticles grown in phosphate glass

CdS nanoparticles, with an average diameter between 2.3 and 5.5 nm, were grown in phosphate glasses. The photoluminescence of the aforementioned materials showed a broad band, Stokes shifted from the corresponding absorption edge by about 1.2 eV and blue shifted upon reduction of the particle size. The ODMR spectra of the discussed materials are composed of three resonance signals. Theoretical simulation of the latter spectra, utilizing phenomenological spin Hamiltonian, revealed that the luminescence band is associated with a recombination between trapped electrons and holes, at asymmetric sites, more likely located at the surface of the nanoparticles. Moreover, this simulation showed a distribution of electron–hole distances around the circumference of the nanoparticles.

The Analysis of EPR Spectra Using Tesseral Tensor Angular Momentum Operators

The advantages of using a tesseral rather than one of the conventional tensor angular momentum operator formulations for the zero field splitting terms in a phenomenological spin Hamiltonian are discussed in detail. The EPR spectral data for Gd 3+ impurity ions in the NH 4 Ln(SO 4 ) 2 . 4H 2 O lattice where Ln = La, Ce, Pr, Nd, Sm, Eu are used to demonstrate these advantages. The tesseral zero field splitting parameters have been calculated from the conventional zero field splitting parameters reported in the literature. It is demonstrated that the values of these parameters are more consistent with C 2h rather than D 2h symmetry and may not be consistent with a C2 monoclinic symmetry crystalline electric field. It is also shown that there exist definitions of both the Hermitian adjoint and time reversal operations for these tensor angular momentum operators, which are independent of their representation. These definitions are consistent with the well known conclusion that only even rank zero field splitting terms can be used in the parameterization of the EPR spectra for systems with an odd number of electrons.

G-factor formulas of3d5ions in trigonal-symmetry fields and their applications

The g-factor formulas of ${3\mathrm{d}}^{5}$ ions in ${\mathrm{C}}_{3\mathrm{v}}$ symmetry are developed based on the equivalence between the phenomenological spin Hamiltonian and the microscopic Zeeman interaction. The formulas are universal ones and do not depend on the specific interaction model. Diagonalization of the complete Hamiltonian within the whole ${3\mathrm{d}}^{5}$ configuration yields the theoretical zero-field splitting D, a, a-F, and g factors. The results for ${\mathrm{Mn}}^{2+}$ at the ${\mathrm{Li}}^{+}$ site in ${\mathrm{LiNbO}}_{3}$ agree well with the experimental data. The ${\mathrm{Mn}}^{2+}$(Li)-${\mathrm{O}}^{2\mathrm{\ensuremath{-}}}$ average bond length deduced (R=2.0217 \AA{}) is comparable with those obtained from extended x-ray-absorption fine structure measurements, namely (2.06\ifmmode\pm\else\textpm\fi{}0.05) \AA{}. The d-d transition bands and EPR a value predicted can be verified by further experiments.

A relativistic effective Hamiltonian for S-state ions

This paper uses the 'effective operator technique' to develop a relativistic formulation to describe the crystalline electric field splitting of an S-state ion. This formulation leads to an effective Hamiltonian which is capable of describing the EPR and ENDOR spectra observed from S-state ions such as Gd3+. It replaces the traditional phenomenological spin Hamiltonian and reveals that all those total angular momentum tensor operators permitted by symmetry considerations are included. It is shown that Kramer's theorem which is valid only for spin states must be modified to apply to only the lowest degenerate level of the ground state of any ion since all tensor operators Tqk(J) where k is odd and k not=1 are allowed because J, and not S, is a good relativistic quantum number.