Hund's Rule Research Articles

Theoretical study of the electron structure of superheavy elements with an open 6d shell: Sg, Bh, Hs, and Mt

We use recently developed efficient versions of the configuration interaction method to perform {\em ab initio} calculations of the spectra of superheavy elements seaborgium (Sg, $Z=106$), bohrium (Bh, $Z=107$), hassium (Hs, $Z=108$) and meitnerium (Mt, $Z=109$). We calculate energy levels, ionization potentials, isotope shifts and electric dipole transition amplitudes. Comparison with lighter analogs reveals significant differences caused by strong relativistic effects in superheavy elements. Very large spin-orbit interaction distinguishes subshells containing orbitals with a definite total electron angular momentum $j$. This effect replaces Hund's rule holding for lighter elements.

Spin-triplet f -wave pairing in twisted bilayer graphene near 14 -filling

We investigate the twisted bilayer graphene by a two-orbital Hubbard model on the honeycomb lattice. The model is studied near 1/4 band filling by using the singular-mode functional renormalization group theory. Spin-triplet $f$-wave pairing is found from weak to moderate coupling limit of the local interactions, and is associated with the Hund's rule coupling and incommensurate spin fluctuations at moderate momenta.

Spin waves and spin-state transitions in a ruthenate high-temperature antiferromagnet.

Ruthenium compounds serve as a platform for fundamental concepts such as spin-triplet superconductivity1, Kitaev spin liquids2-5 and solid-state analogues of the Higgs mode in particle physics6,7. However, basic questions about the electronic structure of ruthenates remain unanswered, because several key parameters (including Hund's coupling, spin-orbit coupling and exchange interactions) are comparable in magnitude and their interplay is poorly understood, partly due to difficulties in synthesizing large single crystals for spectroscopic experiments. Here we introduce a resonant inelastic X-ray scattering (RIXS)8,9 technique capable of probing collective modes in microcrystals of 4d electron materials. We observe spin waves and spin-state transitions in the honeycomb antiferromagnet SrRu2O6 (ref. 10) and use the extracted exchange interactions and measured magnon gap to explain its high Néel temperature11-16. We expect that the RIXS method presented here will enable momentum-resolved spectroscopy of a large class of 4d transition-metal compounds.

Ab Initio Probing of the Ground State of Tetraradicals: Breakdown of Hund's Multiplicity Rule.

The electronic structure of organic σ-type polyradical including 2,4,6-tridehydropyridine radical cation (246-TDHP) and three isomers of tetradehydrobenzene (TDHB) have been studied using a computationally robust and cost-effective second-order multireference perturbative model which provides a balanced treatment of nondynamic and dynamic contributions to the electron correlation problem in the ground or excited electronic states which are imperative for predicting structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of polyradicals. Energy gaps are useful to capture insight into the degree of interaction between the radical sites. An important finding of this study is that the tetraradicals considered here possess singlet ground states, contrary to Hund's rule. Present findings are in close agreement with the available high-level ab initio estimates at attainable cost implying that a perturbative description of the systems is adequate. The impact of N+ on the nature of ground state for the 246-TDHP have also analyzed. The singlet-triplet energy gaps for 1245- and 1234-TDHB are smaller than for o-benzyne mainly due to the ring strain. 1235-TDHB is 14.42 and 11.05 kcal/mol lower in energy than 1245- and 1234-isomers, respectively. IVO-SSMRPT predicts 1A1-3B2 and 1A1-5B2 gaps of 25.84 and 105.15 kcal/mol, respectively for the 246-TDHP cation.

Electronic correlations and competing orders in multiorbital dimers: A cluster DMFT study

We investigate the violation of the first Hund's rule in 4$d$ and 5$d$ transition metal oxides that form solids of dimers. Bonding states within these dimers reduce the magnetization of such materials. We parametrize the dimer formation with realistic hopping parameters and find not only regimes, where the system behaves as a Fermi liquid or as a Peierls insulator, but also strongly correlated regions due to Hund's coupling and its competition with the dimer formation. The electronic structure is investigated using the cluster dynamical mean-field theory for a dimer in the two-plane Bethe lattice with two orbitals per site and $3/8$-filling, that is three electrons per dimer. It reveals dimer-antiferromagnetic order of a high-spin (double exchange) state and a low-spin (molecular orbital) state. At the crossover region we observe the suppression of long-range magnetic order, fluctuation enhancement and renormalization of electron masses. At certain interaction strengths the system becomes an incoherent antiferromagnetic metal with well defined local moments.

Giant contribution of the ligand states to the magnetic properties of the Cr2Ge2Te6 monolayer.

The magnetic properties of the Cr2Ge2Te6 monolayer - an important two-dimensional (2D) ferromagnetic (FM) material - are systematically investigated on the basis of ab initio electronic structure calculations within density functional theory (DFT). For these purposes we construct a minimal tight-binding model in the basis of maximally localized Wannier functions, which describes the behavior of the magnetic Cr 3d, Ge 4p, and Te 5p electrons. This model allows us to rationalize the results of conventional DFT calculations at the microscopic level. We explore the abilities of different techniques, including the Green's function perturbation theory and constraint calculations with an external magnetic field, for the analysis of magnetic interactions, that allow us to decompose these interactions in terms of partial contributions coming from different atomic sites. We argue that, although the magnetism of Cr2Ge2Te6 originates from the Hund's rule effects in the partially filled Cr 3d shell, the contributions of the ligand - and particularly Te 5p - states are crucially important and have a significant effect on the behavior of isotropic exchange interactions, magnetocrystalline anisotropy energy (MAE), and antisymmetric Dzyaloshinskii-Moriya (DM) interactions induced by an electric field. In particular, the Te 5p states increase dramatically the FM coupling and DM interactions between the Cr spins. They are also largely responsible for the behavior of MAE, manifesting themselves in non-Heisenberg type effects, which go beyond the scopes of the conventional Mermin-Wagner theorem for the analysis of 2D magnetism of Cr2Ge2Te6.

Role of interstitial hydrogen in SrCoO2.5 antiferromagnetic insulator

Hydrogen exhibits qualitatively different charge states depending on the host material, as nicely explained by the state-of-the-art impurity-state calculation. Motivated by a recent experiment [Nature 546, 124 (2017)], we show that the complex oxide SrCoO$_{2.5}$ represents an interesting example, in which the interstitial H appears as a deep-level center according to the commonly-used transition level calculation, but no bound electron can be found around the impurity. Via a combination of charge difference analysis, density of states projection and constraint magnetization calculation, it turns out that the H-doped electron is spontaneously trapped by a nonunique Co ion and is fully spin-polarized by the onsite Hund's rule coupling. Consequently, the doped system remains insulating, whereas the antiferromagnetic exchange is slightly perturbed locally.

Thermally Persistent High-Spin Ground States in Octahedral Iron Clusters.

Chemical oxidation and reduction of the all-ferrous (HL)2Fe6 in THF affords isostructural, coordinatively unsaturated clusters of the type [(HL)2Fe6] n: [(HL)2Fe6][BArF24] (1, n = +1; where [BArF24]- = tetrakis[(3,5-trifluoromethyl)phenyl]borate), [Bu4N][(HL)2Fe6] (2a, n = -1), [P][(HL)2Fe6] (2b, n = -1; where [P]+ = tributyl(1,3-dioxolan-2-ylmethyl)phosphonium), and [Bu4N]2[(HL)2Fe6] (3, n = -2). Each member of the redox-transfer series was characterized by zero-field 57Fe Mössbauer spectroscopy, near-infrared spectroscopy, single-crystal X-ray crystallography, and magnetometry. Redox-directed trends are observed when comparing the structural metrics within the [Fe6] core. The metal octahedron [Fe6] decreases marginally in volume as the molecular reduction state increases as gauged by the Fe-Feavg distance varying from 2.608(11) Å ( n = +1) to 2.573(3) ( n = -2). In contrast, the mean Fe-N distances and ∠Fe-N-Fe angles correlate linearly with the [Fe6] oxidation level, or alternatively, the changes observed within the local Fe-N4 coordination planes vary linearly with the aggregate spin ground state. In general, as the spin ground state ( S) increases, the Fe-N(H)avg distances also increase. The structural metric perturbations within the [Fe6] core and measured spin ground states were rationalized extending the previously proposed molecular orbital diagram derived for (HL)2Fe6. Chemical reduction of the (HL)2Fe6 cluster results in an abrupt increase in spin ground state from S = 6 for the all-ferrous cluster, to S = 19/2 in the monoanionic 2b and S = 11 for the dianionic 3. The observation of asymmetric intervalence charge transfer bands in 3 provides further evidence of the fully delocalized ground state observed by 57Fe Mössbauer spectroscopy for all species examined (1-3). For each of the clusters examined within the electron-transfer series, the observed spin ground states thermally persist to 300 K. In particular, the S = 11 in dianionic 3 and S = 19/2 in the monoanionic 2b represent the highest spin ground states isolated up to room temperature known to date. The increase in spin ground state results from population of the antibonding orbital band comprised of the Fe-N σ* interactions. As such, the thermally persistent ground states arise from population of the resultant single spin manifolds in accordance with Hund's rules. The large spin ground states, indicative of strong ferromagnetic electronic alignment of the valence electrons, result from strong direct exchange electronic coupling mediated by Fe-Fe orbital overlap within the [Fe6] cores, equivalent to a strong double exchange magnetic coupling B for 3 that was calculated to be 309 cm-1.

Crystal fields and Kondo effect: Magnetic susceptibility of Cerium ions in axial crystal fields

The thermodynamic Bethe ansatz equations of the Coqblin-Schrieffer model have been solved numerically. The full N=6 (J=5/2) degeneracy of the Hund's rule ionic ground state of Ce is taken into account. Results for the temperature dependent magnetic susceptibility parallel and perpendicular to the crystal axis are presented. The deviations, due to the Kondo effect, to the non-interacting ion results are pointed out.

Robust Hund rule without Coulomb repulsion and exclusion principle in quantum antiferromagnetic chains of composite half spins

Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or ) chain was sometimes approximated by a chain with two (or three) spin ’s per site. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why. In this paper, the validity of this approximation is investigated numerically. We diagonalize the Hamiltonians of spin chains with a spin 1 and per site and with two and three spin ’s per site. The low energy excitation spectrum for the spin chain with M spin ’s per site is found to coincide with that of the corresponding conventional chain with one spin per site. In particular, we find that as the system size increases, an increasingly larger block of consecutive lowest energy states with maximal spin per site is observed, robustly supporting the first Hund rule even though the exclusion principle does not apply and the system does not possess Coulomb repulsion. As for why this approximation works, we show that this effective Hund rule emerges as a plausible consequence when applying to composite spin systems the Lieb–Mattis theorem, which is originally for the ground state of ferrimagnetic and antiferromagnetic spin systems.

Hund's rule in the (1s2s)1,3S states of the two-electron Debye atom

We present an investigation of the (1s2s)1,3S excited states of the two-electron atom immersed in a plasma modeled by the Debye or screened Coulomb potential. Three variants of the Debye atom are considered. The validity of Hund's multiplicity rule is confirmed, and the contribution of the interparticle repulsion energy to the singlet-triplet splitting is examined. The feature that this system shares with the unscreened two-electron atom as well as with the confined two-electron atom and the two-electron quantum dot is that the triplet wave function is contracted relative to that of the singlet. This feature affects both the behavior of the 2s-electron ionization energies and the relative magnitudes of the interparticle repulsion energies in the singlet vs. the triplet. Debye screening of the one-body attraction effectively reduces the nuclear charge, enhancing the reversal of the relative magnitudes of the triplet vs. singlet interparticle repulsion energies. Debye screening of the interparticle repulsion acts in an opposite way.

Element-specific magnetic properties of mixed 3d−4f metallacrowns

Single molecule magnets comprising rare earth metals are of high interest due to the unquenched orbital moments of the rare earth ions that result in a large energy barrier for magnetization reversal. We investigate the magnetic properties of polynuclear $3d\text{\ensuremath{-}}4f15$-MC-5 metallacrowns using x-ray magnetic circular dichroism of powder samples at a temperature of 7 K in a magnetic field of 7 T. The sum rule analysis reveals element-specific spin and orbital moments. The magnetic moments of the $3d$ transition metal Ni(II) ions are coupled antiferromagnetically to each other and contribute only little to the total molecular moment. The spin and orbital moments of the rare earth ions are unexpectedly smaller than the ionic values resulting from Hund's rules. We explain the reduction of the orbital magnetic moment by a finite magnetic anisotropy. Considering an energy functional including magnetic anisotropy and Zeeman energy the powder average reveals a magnetic anisotropy of 28 meV (340 K) in the case of Dy(III) and 7 meV (85 K) in the case of Tb(III). The spin moments agree with the ionic value, too, when the expectation values of the dipole operator are considered.

Phases of a phenomenological model of twisted bilayer graphene

We propose a lattice scale two-band generalized Hubbard model as a caricature of the electronic structure of twisted bilayer graphene. Various possible broken symmetry phases can arise, including a nematic phase (which is a form of orbital ferromagnet) and an orbital-triplet spin-singlet superconducting phase. Concerning the mechanism of superconductivity -- we propose an analogy with superconductivity in alkali-doped C$_{60}$ in which a violation of Hund's first rule plays a central role.

Spinon-orbiton repulsion and attraction mediated by Hund's rule

We study the impact of Hund's-rule coupling on orbital excitations, as e.g. measured in inelastic resonant x-ray scattering. We find that the interpretation in terms of spin-orbit separation, which has been derived for one-dimensional systems without Hund's rule, remains robust in its presence. Depending on whether or not the orbital excitation includes a spin excitation, Hund's rule leads to an attractive or repulsive interaction between spinon and orbiton. Attraction and repulsion leave clear signatures through a transfer of spectral weight to the lower resp. upper edge of the spectrum.

First-principles prediction of ferromagnetism in transition-metal doped monolayer AlN

Two-dimensional (2D) magnetic materials with large spin polarization and high Curie temperature (Tc) are essential and indispensable for quantum computation, logic and memory operations, and spintronic devices at nanoscale. In this study, using the first-principles calculations based on the spin polarized density functional theory with the generalized gradient approximation (GGA) plus Hubbard-U corrections (GGA + U), we investigated the magnetic and electronic properties of 2D AlN sheets doped with different atoms of 3d transition metal (TM) elements. Our calculations demonstrate that these TM-doped 2D monolayer AlN sheets are magnetic semiconductors with integer magnetic moment. The magnetic moments of these doped systems originate mainly from the TM atoms, following the Hund's rule and crystal field theory. Furthermore, we find that the magnetic coupling in the Mn- and Ni-doped 2D AlN sheets is ferromagnetic, while the magnetic coupling in the other TM-doped 2D AlN sheets is anti-ferromagnetic. For the ferromagnetic and semiconducting TM-doped monolayer AlN sheets, the calculated Curie temperatures are above 400 K, exceeding that (155 K) of the most-studied dilute magnetic GaMnAs material. Our studies suggest that the Mn- and Ni-doped 2D monolayer AlN sheets would show very promising applications in the spintronic device at nanoscale.

Density functional theory calculations for magnetic properties of Co3W systems

Cheaper permanent magnetic nanostructures with magnetic properties equivalent to those of noble-metal or rare-earth nanomagnets have been experimentally developed for their potential applications in ultrahigh storage densities in magnetic memory. To date, their intrinsic magnetic properties are not well understood under the micro-level of local atomic arrangements and electronic structures. In this work, we performed theoretical investigations on the Co3W bulk, the clean surface, nanoclusters, and the Co|Co3W bilayers and superlattices for their geometrical structures, magnetic moments, and magnetic anisotropy energies (MAEs). We found that the Co3W nanostructures we constructed are stable and have the local minima in the energetic landscape, whose stabilities increase with increasing proportion of W and cluster size. The Co and W atoms in clusters are antiferromagnetically coupled, and their local magnetic moments decrease with increasing proportion of W. The breakdown of the Hund's third rule in W atoms observed in experiment can be interpreted as the competition between the intra-atomic spin-orbit coupling in W atoms and interatomic Co-W hybridizations. The highest MAE of about a few tens of meV is obtained in small cluster sizes, whereas it is an order of magnitude reduction in large cluster sizes. The magnetic systems of Co3W clean surface, Co|Co3W bilayer and superlattice can present large MAEs, and their easy-axes of magnetization are perpendicular to the (001) surface. Our calculated MAEs are of the same order of magnitude as that of the experimental measurements, and the electronic origin is revealed through the second-order perturbation method.

Controlling spin off state by gas molecules adsorption on metal-phthalocyanine molecular junctions and its possibility of gas sensor

Employing Green's function (GF) technique in combination with spin-polarized density functional theory (DFT), we study the electronic structure and magnetic properties of metal phthalocyanine (MPc) (M = Mn, Fe, Co, Ni, Cu, Zn) with or without four different gas molecules (NO, CO, O2 and NO2) adsorbing on the M atom of MPc molecule. The corresponding stable adsorption structural configurations and transport properties of MPc molecular junctions are also investigated. Our results indicate that the magnetic moment of MPc for M = Mn, Fe and Co can be modified by the specific gas molecule adsorption, which is mainly ascribed to competitive relation of HOMO-LUMO Gap and Hund's rules. However, for M = Ni, Cu and Zn, it is difficult to detect gas molecule because the interaction of M atom and these gases is most of weak van der Waals interaction. Remarkably, the spin of MPc molecule can be switched to a magnetic off-state by specific gas absorption, giving rise to a potential application on controllable spintronic devices. In addition, CO, NO, O2 and NO2 gas molecules can be detected selectively by measuring spin filter efficiency of these MPc molecular junctions. On the basis of our results, MPc (M = Mn, Fe, Co) molecular junctions can be considered as a promising nanosensor device to detect individual gas molecules.

Spin-triplet paired phases inside a ferromagnet induced by Hund's rule coupling and electronic correlations: Application to UGe2

We discuss a mechanism of real-space spin-triplet pairing, alternative to that due to quantum paramagnon excitations, and demonstrate its applicability to $\mathrm{UGe_2}$. Both the Hund's rule ferromagnetic exchange and inter-electronic correlations contribute to the same extent to the equal-spin pairing, particularly in the regime in which the weak-coupling solution does not provide any. The theoretical results, obtained within the orbitally-degenerate Anderson lattice model, match excellently the observed phase diagram for $\mathrm{UGe_2}$ with the coexistent ferromagnetic (FM1) and superconducting ($A_1$-type) phase. Additionally, weak $A_2$- and $A$-type paired phases appear in very narrow regions near the metamaganetic (FM2 $\rightarrow$ FM1) and FM1 $\rightarrow$ paramagnetic first-order phase-transition borders, respectively. The values of magnetic moments in the FM2 and FM1 states are also reproduced correctly in a semiquantitative manner. The Hund's metal regime is also singled out as appearing near FM1-FM2 boundary.

Crystal electric field in CeRh2Si2 studied with high-resolution resonant inelastic soft x-ray scattering

The magnetic properties of rare earth compounds are usually well captured by assuming a fully localized f shell and only considering the Hund's rule ground state multiplet split by a crystal electrical field (CEF). Currently, the standard technique for probing CEF excitations in lanthanides is inelastic neutron scattering. Here we show that with the recent leap in energy resolution, resonant inelastic soft X-ray scattering has become a serious alternative for looking at CEF excitations with some distinct advantages compared to INS. As an example we study the CEF scheme in CeRh2Si2, a system that has been intensely studied for more than two decades now but for which no consensus has been reached yet as to its CEF scheme. We used two new features that have only become available very recently in RIXS, high energy resolution of about 30 meV as well as polarization analysis in the scattered beam, to find a unique CEF description for CeRh2Si2. The result agrees well with previous INS and magnetic susceptibility measurements. Due to its strong resonant character, RIXS is applicable to very small samples, presents very high cross sections for all lanthanides, and further benefits from the very weak coupling to phonon excitation. The rapid progress in energy resolution of RIXS spectrometers is making this technique increasingly attractive for the investigation of the CEF scheme in lanthanides.

Modeling of the gate-controlled Kondo effect at carbon point defects in graphene

We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound $\sigma$ orbital and the vacancy induced bound $\pi$ state in an effective two-orbital single impurity model. The local magnetic moments are stabilized by the Coulomb interaction as well as a significant ferromagnetic Hund's rule coupling between the orbitals predicted by a density functional theory calculation. A hybridization between the orbitals and the Dirac fermions is generated by the curvature of the graphene sheet in the vicinity of the vacancy. We present results for the local spectral function calculated using Wilson's numerical renormalization group approach for a realistic graphene band structure and find three different regimes depending on the filling, the controlling chemical potential, and the hybridization strength. These different regions are characterized by different magnetic properties. The calculated spectral functions qualitatively agree with recent scanning tunneling spectra on graphene vacancies.