Exchange Hamiltonian Research Articles

Analysis of the Magnetic Coupling in a Mn(II)-U(V)-Mn(II) Single Molecule Magnet.

[{Mn(TPA)I}{UO2(Mesaldien)}{Mn(TPA)I}]I formula (here TPA=tris(2-pyridylmethyl)amine and Mesaldien=N,N'-(2-aminomethyl)diethylenebis(salicylidene imine)) reported by Mazzanti and coworkers (Chatelain et al. Angew. Chem. Int. Ed. 2014, 53, 13434) is so far the best Single Molecule Magnet (SMM) in the {3d-5f} class of molecules exhibiting barrier height of magnetization reversal as high as 81.0 K. In this work, we have employed a combination of ab initio CAS and DFT methods to fully characterize this compound and to extract the relevant spin Hamiltonian parameters. We show that the signs of the magnetic coupling and of the g-factors of the monomers are interconnected. The central magnetic unit [UV O2 ]+ is described by a Kramers Doublet (KD) with negative g-factors, due to a large orbital contribution. The magnetic coupling for the {Mn(II)-U(V)} pair is modeled by an anisotropic exchange Hamiltonian: all components are ferromagnetic in terms of spin moments, the parallel component JZ twice larger as the perpendicular one J⊥ . The spin density distribution suggests that spin polarization on the U(V) center favors the ferromagnetic coupling. Further, the JZ /J⊥ ratio, which is related to the barrier height, was found to correlate to the corresponding spin contribution of the g-factors of the U(V) center. This correlation established for the first time offers a direct way to estimate this important ratio from the corresponding gS -values, which can be obtained using traditional ab initio packages and hence has a wider application to other {3d-5f} magnets. It is finally shown that the magnetization barrier height is tuned by the splitting of the [UV O2 ]+ 5 f orbitals.

Hot spots along the Fermi contour of high-Tc cuprates analyzed by s-d exchange interaction

We perform a thorough theoretical study of the electron properties of a generic CuO_2 plane in the framework of Shubin–Kondo–Zener s-d exchange interaction that simultaneously describes the correlation between Tc and the Cu4s energy. To this end, we apply the Pokrovsky theory (J Exp Theor Phys 13:447–450, 1961) for anisotropic gap BCS superconductors. It takes into account the thermodynamic fluctuations of the electric field in the dielectric direction perpendicular to the conducting layers. We microscopically derive a multiplicatively separable kernel able to describe the scattering rate in the momentum space, as well as the superconducting gap anisotropy within the BCS theory. These findings may be traced back to the fact that both the Fermi liquid and the BCS reductions lead to one and the same reduced Hamiltonian involving a separable interaction, such that a strong electron scattering corresponds to a strong superconducting gap and vice versa. Moreover, the superconducting gap and the scattering rate vanish simultaneously along the diagonals of the Brillouin zone. We would like to stress that our theoretical study reproduces the phenomenological analysis of other authors aiming at describing Angle Resolved Photoemission Spectroscopy measurements. Within standard approximations one and the same s-d exchange Hamiltonian describes gap anisotropy of the superconducting phase and the anisotropy of scattering rate of charge carriers in the normal phase.

Spin Waves and Magnetic Exchange Hamiltonian in CrSBr.

CrSBr is an air‐stable two‐dimensional (2D) van der Waals semiconducting magnet with great technological promise, but its atomic‐scale magnetic interactions—crucial information for high‐frequency switching—are poorly understood. An experimental study is presented to determine the CrSBr magnetic exchange Hamiltonian and bulk magnon spectrum. The A‐type antiferromagnetic order using single crystal neutron diffraction is confirmed here. The magnon dispersions are also measured using inelastic neutron scattering and rigorously fit the excitation modes to a spin wave model. The magnon spectrum is well described by an intra‐plane ferromagnetic Heisenberg exchange model with seven nearest in‐plane exchanges. This fitted exchange Hamiltonian enables theoretical predictions of CrSBr behavior: as one example, the fitted Hamiltonian is used to predict the presence of chiral magnon edge modes with a spin‐orbit enhanced CrSBr heterostructure.

Moments and multiplets in moiré materials

The observation of strongly correlated states in moiré systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g., to describe Mott insulators where the local moments are coupled spin–valley degrees of freedom. Here, we discuss a numerical renormalization group scheme to explore the formation of spin–valley ordered and unconventional spin–valley liquid states at zero temperature based on a pseudo-fermion representation. Our generalization of the conventional pseudo-fermion functional renormalization group approach for {{mathfrak {s}}}{{mathfrak {u}}}(2) spins is capable of treating diagonal and off-diagonal couplings of generic spin–valley exchange Hamiltonians in the self-conjugate representation of the {{mathfrak {s}}}{{mathfrak {u}}}(4) algebra. To achieve proper numerical efficiency, we derive a number of symmetry constraints on the flow equations that significantly limit the number of ordinary differential equations to be solved. As an example system, we investigate a diagonal SU(2)_{text {spin}}otimes U(1)_{text {valley}} model on the triangular lattice which exhibits a rich phase diagram of spin and valley ordered phases.Graphic

Exchange interactions in d5 Kitaev materials: From Na2IrO3 to α−RuCl3

We present an analytical study of the exchange interactions between pseudospin one-half ${d}^{5}$ ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard $U$, charge-transfer excitations, and cyclic exchange are considered. Hoppings within ${t}_{2g}$ orbitals as well as between ${t}_{2g}$ and ${e}_{g}$ orbitals are included. Special attention is paid to the trigonal crystal field $\mathrm{\ensuremath{\Delta}}$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction $K$ within a wide range of $\mathrm{\ensuremath{\Delta}}$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\mathrm{\ensuremath{\Delta}}$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic $K$ and positive nondiagonal $\mathrm{\ensuremath{\Gamma}}$ values. However, their Heisenberg $J$ terms have opposite signs: AFM $J>0$ in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and FM $J<0$ in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM $J$ is emphasized in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. The differences between the exchange couplings of these two materials originate from the opposite $\mathrm{\ensuremath{\Delta}}$ values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of ${d}^{5}$ spin-orbit Mott insulators.

Combined Dissipative and Hamiltonian Confinement of Cat Qubits

Quantum error correction with biased-noise qubits can drastically reduce the hardware overhead for universal and fault-tolerant quantum computation. Cat qubits are a promising realization of biased-noise qubits as they feature an exponential error bias inherited from their non-local encoding in the phase space of a quantum harmonic oscillator. To confine the state of an oscillator to the cat qubit manifold, two main approaches have been considered so far: a Kerr-based Hamiltonian confinement with high gate performances, and a dissipative confinement with robust protection against a broad range of noise mechanisms. We introduce a new combined dissipative and Hamiltonian confinement scheme based on two-photon dissipation together with a Two-Photon Exchange (TPE) Hamiltonian. The TPE Hamiltonian is similar to Kerr nonlinearity, but unlike the Kerr it only induces a bounded distinction between even- and odd-photon eigenstates, a highly beneficial feature for protecting the cat qubits with dissipative mechanisms. Using this combined confinement scheme, we demonstrate fast and bias-preserving gates with drastically improved performance compared to dissipative or Hamiltonian schemes. In addition, this combined scheme can be implemented experimentally with only minor modifications of existing dissipative cat qubit experiments.

Spin-exchange Hamiltonian and topological degeneracies in elemental gadolinium

We present a comprehensive study of the magnetic exchange Hamiltonian of elemental Gadolinium. We use neutron scattering to measure the magnon spectrum over the entire Brillouin zone, and fit the excitations to a spin wave model to extract the first 26 nearest neighbor magnetic exchange interactions with rigorously defined uncertainty. We find these exchange interactions to follow RKKY behavior, oscillating from ferromagnetic to antiferromagnetic as a function of distance. Finally, we discuss the topological features and degeneracies in Gd, and HCP ferromagnets in general. We show theoretically how, with asymmetric exchange, topological properties could be tuned with a magnetic field.

Critical behavior of the classical spin-1 Ising model for magnetic systems

In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-state spin-1 Ising model, is obtained from an empirical expression of the partition function recently proposed for a spin-1/2 Ising model in an external magnetic field and applied to a 2D magnetic lattice. These results could pave the way to numerical simulations and to measurements able to confirm the analytical predictions.

Theoretical study on various contributions to the magnetization of Pb1-xEuxSe

The results of the calculation on magnetization (M) for diluted magnetic semiconductor Pb1-xEuxSe are reported. Our calculation includes two distinct contributions namely contribution from the localized magnetic impurity (Mi) and contribution from the band effect (Mb). Mi has been calculated using a three-spin cluster model which comprises the clusters consisting of one-spin (M1), two-spins (M2) and three-spins (M3). Moreover, M3 includes both open triplets (Mot) and closed triplets (Mct). The statistical distribution of the different clusters has been considered using random distribution approximation in the dilute limit. The contributions from these clusters to the magnetization have been evaluated using a modified Heisenberg's exchange Hamiltonian in the nearest approximation. The magnetization has been calculated as a function of magnetic field and impurity concentration. The band effect includes the contribution from lattice diamagnetism(χlatt). This mechanism has been considered in the effective mass formalism using a two-band k→.π→model. Our calculated values of M compare very well with experimental data where available.

Towards Kitaev spin liquid in 3d transition metal compounds

This paper reviews the current progress on searching the Kitaev spin liquid state in 3[Formula: see text] electron systems. Honeycomb cobaltates were recently proposed as promising candidates to realize the Kitaev spin liquid state, due to the more localized wavefunctions of [Formula: see text] ions compared with that of [Formula: see text] and [Formula: see text] ions, and also the easy tunability of the exchange Hamiltonian in favor of Kitaev interaction. Several key parameters that have large impacts on the exchange constants, such as the charge-transfer gap and the trigonal crystal field, are identified and discussed. Specifically, tuning crystal field effect by means of strain or pressure is emphasized as an efficient phase control method driving the magnetically ordered cobaltates into the spin liquid state. Experimental results suggesting the existence of strong Kitaev interactions in layered honeycomb cobaltates are discussed. Finally, the future research directions are briefly outlined.

Power maximization of two-stroke quantum thermal machines

We present a detailed study of quantum thermal machines employing quantum systems as working substances. In particular, we study two different types of two-stroke cycles where two collections of identical quantum systems with evenly spaced energy levels are initially prepared at thermal equilibrium by putting them in contact with a cold and a hot thermal bath, respectively. The two cycles differ in the absence or the presence of a mediator system, while, in both cases, non-resonant exchange Hamiltonians are exploited as particle interactions. We show that the efficiencies of these machines depend only on the energy gaps of the systems composing the collections and are equal to the efficiency of "equivalent" Otto cycles. Focusing on the cases of qubits or harmonic oscillators for both models, we maximize the engine power and analyze, in the model without the mediator, the role of the waiting time between subsequent interactions. It turns out that the case with the mediator can bring performance advantages when the interaction times are comparable with the waiting time of the correspondent cycle without the mediator. We find that in both cycles, the power peaks of qubit systems can surpass the Curzon-Ahlborn efficiency. Finally, we compare our cycle without the mediator with previous schemes of the quantum Otto cycle showing that high coupling is not required to achieve the same maximum power.

Thermodynamics of General Heisenberg Spin Tetramers Composed of Coupled Quantum Dimers

Advances in quantum computing technology have been made in recent years due to the evolution of spin clusters. Recent studies have tended towards spin cluster subgeometries to understand more complex structures better. These molecular magnets provide a multitude of phenomena via exchange interactions that allow for advancements in spintronics and other magnetic system applications due to the possibility of increasing speed, data storage, memory, and stability of quantum computing systems. Using the Heisenberg spin–spin exchange Hamiltonian and exact diagonalization, we examine the evolution of quantum energy levels and thermodynamic properties for various spin configurations and exchange interactions. The XXYY quantum spin tetramer considered in this study consists of two coupled dimers with exchange interactions α1J and α1′J and a dimer–dimer exchange interaction α2J. By varying spin values and interaction strengths, we determine the exact energy eigenstates that are used to determine closed-form analytic solutions for the heat capacity and magnetic susceptibility of the system and further analyze the evolution of the properties of the system based on the parameter values chosen. Furthermore, this study shows that the Schottky anomaly shifts towards zero as the ground-state of the system approaches a quantum phase transition between spin states. Additionally, we investigate the development of phase transitions produced by the convergence of the Schottky anomaly with both variable exchange interactions and external magnetic field.

Ferro- vs. antiferromagnetic exchange between two Ni(II) ions in a series of Schiff base heterometallic complexes: what makes the difference?

Three new NiII/ZnII heterometallics, [NiZnL'2(OMe)Cl]2 (1), [NiZnL''(Dea)Cl]2·2DMF (2) and [Ni2(H3L''')2(o-Van)(MeOH)2]Cl·[ZnCl2(H4L''')(MeOH)]·2MeOH (3), containing three-dentate Schiff bases as well as methanol or diethanolamine (H2Dea) or o-vanillin (o-VanH), all deprotonated, as bridging ligands were synthesized and structurally characterized. The Schiff base ligands were produced in situ from o-VanH and CH3NH2 (HL'), or NH2OH (HL"), or 2-amino-2-hydroxymethyl-propane-1,3-diol (H4L'''); a zerovalent metal (Ni and Zn in 1, Zn only in 2 and 3) was employed as a source of metal ions. The first two complexes are dimers with a Ni2Zn2O6 central core, while the third compound is a novel heterometallic cocrystal salt solvate built of a neutral zwitterionic ZnII Schiff base complex and of ionic salt containing dinuclear NiII complex cations. The crystal structures contain either centrosymmetric (1 and 2) or non-symmetric di-nickel fragment (3) with NiNi distances in the range 3.146-3.33 Å. The exchange coupling is antiferromagnetic for 1, J = 7.7 cm-1, and ferromagnetic for 2, J = -6.5 cm-1 (using the exchange Hamiltonian in a form Ĥ = Jŝ1ŝ2). The exchange interactions in 1 and 2 are comparable to the zero-field splitting (ZFS). High-field EPR revealed moderate magnetic anisotropy of opposite signs: D = 2.27 cm-1, E = 0.243 cm-1 (1) and D = -4.491 cm-1, E = -0.684 cm-1 (2). Compound 3 stands alone with very weak ferromagnetism (J = -0.6 cm-1) and much stronger magnetic anisotropy with D = -11.398 cm-1 and E = -1.151 cm-1. Attempts to calculate theoretically the exchange coupling (using the DFT "broken symmetry" method) and ZFS parameters (with the ab initio CASSCF method) were successful in predicting the trends of J and D among the three complexes, while the quantitative results were less good for 1 and 3.

Spin-valley density wave in moiré materials

We introduce and study a minimum two-orbital Hubbard model on a triangular lattice, which captures the key features of both the trilayer ABC-stacked graphene-boron nitride heterostructure and twisted transition metal dichalcogenides in a broad parameter range. Our model comprises first- and second-nearest neighbor hoppings with valley-contrasting flux that accounts for trigonal warping in the band structure. For the strong-coupling regime with one electron per site, we derive a spin-orbital exchange Hamiltonian and find the semiclassical ground state to be a spin-valley density wave. We show that a relatively small second-neighbor exchange interaction is sufficient to stabilize the ordered state against quantum fluctuations. Effects of spin- and valley Zeeman fields as well as thermal fluctuations are also examined.

The s-d exchange model as the underlying mechanism of magnetoresistance in ZnO doped with alkali metals

High field magnetoresistance has been studied in epitaxial n-type ZnO:Na and ZnO:Li thin films in a temperature range between 4 K and 150 K. The resulting negative magnetoresistance can be well fitted using a semiempirical model of Khosla and Fischer based on third order contributions to the s-d exchange Hamiltonian. The parameters obtained from this model were carefully analyzed. One of these parameters is related to a ratio between electron mobilities at zero field (a non-exchange scattering mobility and an exchange or spin dependent one ). From Hall effect measurements was obtained, displaying a weak temperature dependence in accordance with highly n-doped ZnO while the extracted exhibits an anomalous T-dependence. On the other hand, our magnetoresistance data cannot be properly fitted using Kawabata’s expression based on a weak-localization model.

Interfacial spin Seebeck effect in noncollinear magnetic systems

The interplay between spin and heat currents at magnetic insulator|nonmagnetic metal interfaces has been a subject of much scrutiny because of both fundamental physics and the promise for technological applications. While ferrimagnetic and, more recently, antiferromagnetic systems have been extensively investigated, a theory generalizing the heat-to-spin interconversion in noncollinear magnets is still lacking. Here, we establish a general framework for thermally-driven spin transport at the interface between a noncollinear magnet and a normal metal. Modeling the interfacial coupling between localized and itinerant magnetic moments via an exchange Hamiltonian, we derive an expression for the spin current, driven by a temperature difference, for an arbitrary noncollinear magnetic order. Our theory reproduces previously obtained results for ferromagnetic and antiferromagnet systems.

Anisotropic exchange Hamiltonian, magnetic phase diagram, and domain inversion of Nd2Zr2O7

We present thermodynamic and neutron scattering measurements on the quantum spin ice candidate Nd$_2$Zr$_2$O$_7$. The parameterization of the anisotropic exchange Hamiltonian is refined based on high-energy-resolution inelastic neutron scattering data together with thermodynamic data using linear spin wave theory and numerical linked cluster expansion. Magnetic phase diagrams are calculated using classical Monte Carlo simulations with fields along \mbox{[100]}, \mbox{[110]} and \mbox{[111]} crystallographic directions which agree qualitatively with the experiment. Large hysteresis and irreversibility for \mbox{[111]} is reproduced and the microscopic mechanism is revealed by mean field calculations to be the existence of metastable states and domain inversion. Our results shed light on the explanations of the recently observed dynamical kagome ice in Nd$_2$Zr$_2$O$_7$ in \mbox{[111]} fields.

Thermodynamics and Magnetic Excitations in Quantum Spin Trimers: Applications for the Understanding of Molecular Magnets

Molecular magnets provide a playground of interesting phenomena and interactions that have direct applications for quantum computation and magnetic systems. A general understanding of the underlying geometries for molecular magnets therefore generates a consistent foundation for which further analysis and understanding can be established. Using a Heisenberg spin-spin exchange Hamiltonian, we investigate the evolution of magnetic excitations and thermodynamics of quantum spin isosceles trimers (two sides J and one side α J ) with increasing spin. For the thermodynamics, we produce exact general solutions for the energy eigenstates and spin decomposition, which can be used to determine the heat capacity and magnetic susceptibility quickly. We show how the thermodynamic properties change with α coupling parameters and how the underlying ground state governs the Schottky anomaly. Furthermore, we investigate the microscopic excitations by examining the inelastic neutron scattering excitations and structure factors. Here, we illustrate how the individual dimer subgeometry governs the ability for probing underlying excitations. Overall, we feel these calculations can help with the general analysis and characterization of molecular magnet systems.

Single-ion anisotropy and exchange coupling in cobalt(ii)-radical complexes: insights from magnetic and ab initio studies.

The concurrent effects of single-ion anisotropy and exchange interactions on the electronic structure and magnetization dynamics have been analyzed for a cobalt(ii)-semiquinonate complex. Analogs containing diamagnetic catecholate and tropolonate ligands were employed for comparison of the magnetic behavior and zinc congeners assisted with the spectroscopic characterization and assessment of intermolecular interactions in the cobalt(ii) compounds. Low temperature X-band (ν ≈ 9.4 GHz) and W-Band (ν ≈ 94 GHz) electron paramagnetic resonance spectroscopy and static and dynamic magnetic measurements have been used to elucidate the electronic structure of the high spin cobalt(ii) ion in [Co(Me3tpa)(Br4cat)] (1; Me3tpa = tris[(6-methyl-2-pyridyl)methyl]amine, Br4cat2- = tetrabromocatecholate) and [Co(Me3tpa)(trop)](PF6) (2(PF6); trop- = tropolonate), which show slow relaxation of the magnetization in applied field. The cobalt(ii)-semiquinonate exchange interaction in [Co(Me3tpa)(dbsq)](PF6)·tol (3(PF6)·tol; dbsq- = 3,5-di-tert-butylsemiquinonate, tol = toluene) has been determined using an anisotropic exchange Hamiltonian in conjunction with multistate restricted active space self-consistent field ab initio modeling and wavefunction analysis, with comparison to magnetic and inelastic neutron scattering data. Our results demonstrate dominant ferromagnetic exchange for 3+ that is of similar magnitude to the anisotropy parameters of the cobalt(ii) ion and contains a significant contribution from spin-orbit coupling. The nature of the exchange coupling between octahedral high spin cobalt(ii) and semiquinonate ligands is a longstanding question; answering this question for the specific case of 3+ has confirmed the considerable sensitivity of the exchange to the molecular structure. The methodology employed will be generally applicable for elucidating exchange coupling between orbitally-degenerate metal ions and radical ligands and relevant to the development of bistable molecules and their integration into devices.

Spin triplet ground-state in the copper hexamer compoundsA2Cu3O(SO4)3(A=Na,K)

The compounds A2Cu3O(SO4)3 (A=Na,K) are characterized by copper hexamers which are weakly coupled along the b-axis to realize one-dimensional antiferromagnetic chains below TN=3 K, whereas the interchain interactions along the a- and c-axes are negligible. We investigated the energy-level splittings of the copper hexamers by inelastic neutron scattering below and above TN. The eight lowest-lying hexamer states could be unambiguously assigned and parametrized in terms of a Heisenberg exchange Hamiltonian, providing direct experimental evidence for an S=1 triplet ground-state associated with the copper hexamers. Therefore, the compounds A2Cu3O(SO4)3 serve as novel cluster-based spin-1 antiferromagnets to support Haldane's conjecture that a gap appears in the excitation spectrum below TN, which was verified by inelastic neutron scattering.