Magnetic Dipole Term Research Articles

Metamaterial-induced-transparency engineering through quasi-bound states in the continuum by using dielectric cross-shaped trimers

This study presents a novel approach to activate a narrowband transparency line within a reflecting broadband window in all-dielectric metasurfaces, in analogy to the electromagnetically-induced transparency effect, by means of a quasi-bound state in the continuum (qBIC). We demonstrate that the resonance overlapping of a bright mode and a qBIC-based nearly-dark mode with distinct Q-factor can be fully governed by a silicon trimer-based unit cell with broken-inversion-symmetry cross shape, thus providing the required response under normal incidence of a linearly-polarized light. Our analysis that is derived from the far-field multipolar decomposition and near-field electromagnetic distributions uncovers the main contributions of different multipoles on the qBIC resonance, with governing magnetic dipole and electric quadrupole terms supplied by distinct parts of the dielectric “molecule”. The findings extracted from this research open up new avenues for the development of polarization-dependent technologies, with particular interest in its capabilities for sensing and biosensing.

Computed Gyration Tensors of Knotted Chiral and Achiral Topological Stereoisomers of C60 Cyclocarbons.

The electronic origins of the computed optical rotations of the simplest chiral and achiral chemical knots with comparatively simple compositions and large, anticipated magnetoelectric polarizabilities are provided. Linear response theory (LRT) is used to calculate the gyration at 1064 nm of two knotted polyyne chains, topological stereoisomers of cyclo[60]carbon. One isomer is analogous to the trefoil knot with approximate D3 symmetry and the other to the figure eight knot with approximate S4 symmetry. The response in each case can be attributed largely to the magnetic dipole term that arises in a near degenerate E-like excited state. An oriented achiral figure eight knot is as optically active in some directions as the chiral knot in any direction, and its absolute eigenvalues are larger.

Constraining new physics in entangled two-qubit systems: top-quark, tau-lepton and photon pairs

The measurement of quantum entanglement can provide a new and most sensitive probe to physics beyond the Standard Model. We use the concurrence of the top-quark pair spin states produced at colliders to constrain the magnetic dipole term in the coupling between top quark and gluons, that of tau -lepton pair spin states to bound contact interactions and that of tau -lepton pairs or two-photons spin states from the decay of the Higgs boson in trying to distinguish between CP-even and odd couplings. These four examples show the power of the new approach as well as its limitations. We show that differences in the entanglement in the top-quark and tau -lepton pair production cross sections can provide constraints better than those previously estimated from total cross sections or classical correlations. Instead, the final states in the decays of the Higgs boson remain maximally entangled even in the presence of CP-odd couplings and cannot be used to set bounds on new physics. We discuss the violation of Bell inequalities featured in all four processes.

Optical activity of solids from first principles

Within the framework of independent particle approximation, the optical activity tensor of solids is formulated as from different contributions: the magnetic dipole, electric quadrupole, and band dispersion terms. The first two terms have similar counterparts in the theory of finite systems, while the last term is unique for crystals. The magnetic dipole and electric quadrupole transition moments are calculated with a sum-over-states formulation. We apply the formulation to calculate and analyze the optical rotation of elemental tellurium and the circular dichroism of $(6,4)$ carbon nanotube. Decomposed optical activity into different contributions are discussed. The calculated spectra agree well with experiments. As a showcase of achiral crystals, we calculate the optical activity of wurtzite GaN.

Polarization analysis by means of individual soft x-ray absorption spectra of rare earths

X-ray magnetic circular dichroism (XMCD), which by virtue of the sum rules provides element-specific spin and orbital moments, is obtained from the difference between two polarized spectra by reversing the direction of either the light helicity or the applied magnetic field. Usually, it is tacitly assumed that these two spectra are obtained using the same absolute degree of light and magnetic polarization. This is, however, not always possible and depends on circumstances that can be beyond control. First, we recapitulate the conventional XMCD sum rule method to obtain the values of the moments and emphasize some of the complications in the case of the rare-earth ${M}_{4,5}$ edges, such as the presence of strong core-hole $jj$ overlap, linear dichroism, and magnetic dipole term $\ensuremath{\langle}{T}_{z}\ensuremath{\rangle}$. Instead, we propose an alternative method. Using the individual polarized x-ray absorption spectra obtained at the Ho and Dy ${M}_{5}$ edges, where each of the $\mathrm{\ensuremath{\Delta}}J=\ensuremath{-}1,0$, and $+1$ transitions are separated by $\ensuremath{\sim}2$ eV in photon energy, we are able to determine independently the degree of circular dichroism in a single spectrum. Since light is a transverse wave, we need to include, apart from the circular dichroism, also a linear dichroism contribution in order to fit the circularly polarized spectra. In the measurements on paramagnetic rare-earth dopants it was found that reversing the field produces the same degree of circular dichroism, while reversing the helicity yields a $\ensuremath{\sim}$20% difference in the degree of circular dichroism.

Theoretical study of the electronic structure of hafnium (Hf,Z=72) and rutherfordium (Rf,Z=104) atoms and their ions: Energy levels and hyperfine-structure constants

Energy levels, magnetic dipole, and electric quadrupole hyperfine structure of the superheavy element rutherfordium (Rf, $Z$=104) and its first three ions are calculated using a combination of the configuration interaction, coupled-cluster single-doubles, and many-body perturbation theory techniques. The results are to be used in future interpretations of the measurements in terms of nuclear magnetic dipole and electric quadrupole momenta. To have a guide on the accuracy of the study, we perform similar calculations for hafnium (Hf, $Z$=72) and its ions. Hf is a lighter analog of Rf with a similar electronic structure. Good agreement with the experiment for Hf and with available previous calculations of the energy levels of Rf is demonstrated.

Non-Loudon-Fleury Raman scattering in spin-orbit coupled Mott insulators

We revisit the theory of magnetic Raman scattering in Mott insulators with strong spin-orbit coupling, with a major focus on Kitaev materials. We show that Kitaev materials with bond-anisotropic interactions are generally expected to show both one- and two-magnon responses. It is further shown that, in order to obtain the correct leading contributions to the Raman vertex operator $\mc{R}$, one must take into account the precise, photon-assisted microscopic hopping processes of the electrons and that, in systems with multiple hopping paths, $\mc{R}$ contains terms beyond those appearing in the traditional Loudon-Fleury theory. Most saliently, a numerical implementation of the revised formalism to the case of the three-dimensional hyperhoneycomb Kitaev material $\beta$-Li$_2$IrO$_3$ reveals that the non-Loudon-Fleury scattering terms actually dominate the Raman intensity. In addition, they induce a qualitative modification of the polarization dependence, including, e.g., the emergence of a sharp one-magnon peak at low energies which is not expected in the traditional Loudon-Fleury theory. This peak is shown to arise from microscopic photon-assisted tunneling processes that are of similar type with the ones leading to the symmetric off-diagonal interaction $\Gamma$ (known to be present in many Kitaev materials), but take the form of a bond-directional magnetic dipole term in the Raman vertex. These results are expected to apply across all Kitaev materials and mark a drastic change of paradigm for the understanding of Raman scattering in materials with strong spin-orbit coupling and multiple exchange paths.

Sum rules of L -edge x-ray magnetic circularly polarized emission for 3d transition metals

We propose sum rules of x-ray magnetic circularly polarized emission (XMCPE) at $L$ edges for $3d$ transition metals. By making use of combinations of incident and emitted photon helicities, $z$-component expectation values of spin, orbital, magnetic dipole, and quadrupole terms can be obtained separately. The fundamental difference in the sum rules between x-ray magnetic circular dichroism and XMCPE arises from the variety of electron transitions involving core states split by the spin-orbit interaction. The additional electron transition in XMCPE causes complicated angular dependence of the sum rule relation for the spin moment. Our findings promote future $L$-edge XMCPE measurements, which have not been observed at present.

Determination of spin chirality using x-ray magnetic circular dichroism

A 3-fold symmetric kagome lattice that has negative spin chirality can give a non-zero x-ray magnetic circular dichroism (XMCD) signal, despite that the total spin moment amounts to zero. This is explained by a hitherto unnoticed rule for the rotational symmetry invariance of the XMCD signal. A necessary condition is the existence of an anisotropic XMCD signal for the local magnetic atom, which can arise from a spin anisotropy either in the ground state or the final state. The angular dependence of the XMCD as a function of the beam direction has an unusual behavior. The maximum dichroism is not aligned along the spin direction, but depends on the relative orientation of the spin with respect to the atomic orientation. Therefore, different geometries can result in the same angular dependence, and the spin direction can only be determined if the atomic orientation is known. The consequences for the x-ray magneto-optical sum rules are given. The integrated XMCD signals are proportional to the anisotropy in the orbital moment and the magnetic dipole term, where the isotropic spin moment drops out.

Resonance energy transfer mediated by a chiral molecule.

The problem of resonant energy transfer (RET) between an electric dipole donor, D, and an electric dipole acceptor, A, mediated by a passive, chiral third-body, T, is considered within the framework of molecular quantum electrodynamics theory. To account for the optical activity of the mediator, magnetic dipole and electric quadrupole coupling terms are included in addition to the leading electric dipole interaction term. Fourth-order diagrammatic time-dependent perturbation theory is used to obtain the matrix element. It is found that the Fermi golden rule rate depends on pure multipole moment polarizabilities and susceptibilities of T, as well as on various mixed electric and magnetic multipole moment response functions. The handedness of T manifests through mixed electric-magnetic dipole and mixed electric dipole-quadrupole polarizabilities, which affect the rate and, respectively, require the use of fourth-rank and sixth-rank Cartesian tensor averages over T, yielding non-vanishing isotropic rate formulae applicable to a chiral fluid medium. Terms of a similar order of magnitude proportional to the product of electric dipole polarizability and either magnetic dipole susceptibility or electric quadrupole polarizability of T are also computed for oriented and freely tumbling molecules. Migration rates dependent upon the product of the pure electric dipole or magnetic dipole polarizability with the mixed electric-magnetic or electric dipole-quadrupole analogs, which require fourth- and fifth-rank Cartesian tensor averaging, vanish for randomly oriented systems. Asymptotically limiting rate expressions are also evaluated. Insight is gained into RET occurring in complex media.

Self-induction and magnetic effects in electron transport through a photon cavity

We explore higher order dynamical effects in the transport through a two-dimensional nanoscale electron system embedded in a three-dimensional far-infrared photon cavity. The nanoscale system is considered to be a short quantum wire with a single circular quantum dot defined in a GaAs heterostructure. The whole system, the external leads and the central system are placed in a constant perpendicular magnetic field. The Coulomb interaction of the electrons, the para- and diamagnetic electron-photon interactions are all treated by a numerically exact diagonalization using step-wise truncations of the appropriate many-body Fock spaces. We focus on the difference in transport properties between a description within an electric dipole approximation and a description including all higher order terms in a single photon mode model. We find small effects mostly caused by an electrical quadrupole and a magnetic dipole terms that depend strongly on the polarization of the cavity field with respect to the transport direction and the photon energy. When the polarization is aligned along the transport direction we find indications of a weak self-induction that we analyze and compare to the classical counterpart, and the self-energy contribution of high-order interaction terms to the states the electrons cascade through on their way through the system. Like expected the electron-photon interaction is well described in the dipole approximation when it is augmented by the lowest order diamagnetic part for a nanoscale system in a cavity in an external magnetic field.

Modifying the Magnetic Anisotropy of an Iron Porphyrin Molecule by an on-Surface Ring-Closure Reaction

The magnetic properties of adsorbed metalloporphyrin molecules can be altered or tuned by the substrate, additional axial ligands, or changes to the molecules’ macrocycle. These modifications influence the electronic configuration of the fourfold-coordinated central metal ion that is responsible for the metalloporphyrins’ magnetic properties. We report a substantial increase in the effective spin moment obtained from sum-rule analysis of X-ray magnetic circular dichroism for an iron metalloporphyrin molecule on Au(111) through its conversion from iron(II)-octaethylporphyrin to iron(II)-tetrabenzoporphyrin in a surface-assisted ring-closure ligand reaction. Density functional theory calculations with additional strong Coulomb correlation (DFT+U) show that the on-surface reaction alters the conformation of the molecule, increasing its planarity and the ion–surface distance. A spin-Hamiltonian fit of the magnetization as a function of field reveals a substantial increase in the intra-atomic magnetic dipole term (Tz) and a decrease in the magnitude of the easy-plane anisotropy upon ring closure. This consequence of the ring closure demonstrates how new magnetic properties can be obtained from on-surface reactions, resulting here in significant modifications to the magnetic anisotropy of the Fe ion, and sheds light onto the molecule–substrate interaction in these systems.

Photon-momentum transfer in diatomic molecules: An ab initio study

For a molecule, the two-center interference and the molecular scattering phase of the electron are important for almost all the processes that may occur in a laser field. In this study, we investigate their effects in the transfer of linear photon momentum to the ionized electron by absorbing a single photon. The time-dependent Schr\"odinger equation of H$_2^+$ is numerically solved in {the multipolar} gauge in which the electric quadrupole term and the magnetic dipole term are explicitly expressed. This allows us to separate the contributions of the two terms in the momentum transfer. For different configurations of the molecular and the laser orientation, the transferred momentum to the electron is evaluated at different internuclear distances with various photon energies { and two-center interferences are identified in the whole region. At small electron energies and small internuclear distances, we find significant deviations from the prediction of the classical double-slit model due to the strong mediation of the Coulomb potential.} Finally, even for a large internuclear distance, our results show that a varying molecular scattering phase is important at all electron energies, which is beyond the simple prediction of the linear combination of the atomical orbitals.

Electronic ground state of Ni2+

The Φ9/24 ground state of the Ni2+ diatomic molecular cation is determined experimentally from temperature and magnetic-field-dependent x-ray magnetic circular dichroism spectroscopy in a cryogenic ion trap, where an electronic and rotational temperature of 7.4±0.2 K was reached by buffer gas cooling of the molecular ion. The contribution of the spin dipole operator to the x-ray magnetic circular dichroism spin sum rule amounts to 7Tz=0.17±0.06μB per atom, approximately 11% of the spin magnetic moment. We find that, in general, homonuclear diatomic molecular cations of 3d transition metals seem to adopt maximum spin magnetic moments in their electronic ground states.

Influence of spin-orbit coupling on the magnetic dipole termTα

The influence of the spin-orbit coupling (SOC) on the magnetic dipole term $T_{\alpha}$ is studied across a range of systems in order to check whether the $T_{\alpha}$ term can be eliminated from analysis of x-ray magnetic circular dichroism spectra done via the spin moment sum rule. Fully relativistic Korringa-Kohn-Rostoker (KKR) Green function calculations for Co monolayers and adatoms on Cu, Pd, Ag, Pt, and Au (111) surfaces were performed to verify whether the sum over magnetic dipole terms $T_{x}+T_{y}+T_{z}$ is zero and whether the angular dependence of the \ta\ term goes as $3\cos^{2}\theta-1$. It follows that there are circumstances when the influence of the SOC on $T_{\alpha}$ cannot be neglected even for 3$d$ atoms, where the SOC is nominally small. The crucial factor appears to be the dimensionality of the system: for 3$d$ adatoms, the influence of SOC on $T_{\alpha}$ can be significant while for monolayers it is always practically negligible. Apart from the dimensionality, hybridization between adatom and substrate states is also important: small hybridization enhances the importance of the SOC and vice versa.

Coherent Excitation-Selective Spectroscopy of Multipole Resonances

Thin films are central to modern technologies ranging from semiconductors to metamaterials. The authors observe that by placing a subwavelength thin film at the node of an electromagnetic standing wave, it is possible to separate electric from magnetic dipole terms, or dipole from quadrupole terms, in the absorption spectrum. The technique is twice as sensitive as conventional measurements, functions at very low laser power, and reveals resonances that are invisible to existing spectroscopies. This approach could see application in analytical chemistry, condensed matter physics, nanotechnology, and forensic science.

Mixed-valence state of Ce and its individual atomic moments in Ce2Fe14B studied by soft X-ray magnetic circular dichroism

We performed soft X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) measurements at Fe L2,3-edges and Ce M4,5-edges on Ce2Fe14B at 300 K. The valence and total atomic moment of Ce in Ce2Fe14B were determined to be ∼+3.12 and ∼0.100 μB, respectively, which mainly stem from a large portion of the well-localized 4f1 configurations. Besides, the individual orbital and spin moments of Ce in Ce2Fe14B as well as those of Fe were determined for the first time with consideration of the magnetic dipole term.

Co monolayers and adatoms on Pd(100), Pd(111), and Pd(110): Anisotropy of magnetic properties

We investigate to what extent the magnetic properties of deposited nanostructures can be influenced by selecting as a support different surfaces of the same substrate material. Fully relativistic ab initio calculations were performed for Co monolayers and adatoms on Pd(100), Pd(111), and Pd(110) surfaces. Changing the crystallographic orientation of the surface has a moderate effect on the spin magnetic moment and on the number of holes in the d band, a larger effect on the orbital magnetic moment but sometimes a dramatic effect on the magnetocrystalline anisotropy energy (MAE) and on the magnetic dipole term T_alpha. The dependence of T_alpha on the magnetization direction alpha can lead to a strong apparent anisotropy of the spin magnetic moment as deduced from the X-ray magnetic circular dichroism (XMCD) sum rules. For systems in which the spin-orbit coupling is not very strong, the T_alpha term can be understood as arising from the differences between components of the spin magnetic moment associated with different magnetic quantum numbers m.

Effect of Medium Chirality on the Rate of Resonance Energy Transfer

Resonance energy transfer (RET) between two chromophores in an absorptive and dispersive chiral medium is investigated using a quantum electrodynamical formulation. To accurately describe such an environment involves the introduction of electric displacement and auxiliary magnetic field operators that are solutions of the Drude-Born-Fedorov equations and the time-harmonic Maxwell equations. Perturbation theory within the electric and magnetic dipole approximation is used in the derivation of the probability amplitude for energy transfer. Expressions for the contributions to the RET rate arising from the pure electric dipole term, replacing one of the chiral chromophores by its corresponding enantiomer in the mixed electric-magnetic dipole term, and the pure magnetic dipole contribution are obtained. In the near-zone limit in a nonabsorptive medium, the medium chirality amplifies the pure electric dipole contribution to the rate relative to that in a racemic mixture and also increases the discriminatory contribution, but to a lesser extent relative to the pure electric dipole term. On the other hand, under the same conditions, the medium chirality does not affect the pure magnetic dipole contribution to the rate. Measurements of the rate could be used to obtain information on the magnitude of the chirality admittance or concentration of chiral species, by comparing with the rate in an environment comprised of a racemic mixture of the enantiomers. This method could allow for the analysis of macroscopically heterogeneous systems that are comprised of enantiomers and where the chromophores experiencing RET are located in regions of interest.

Extensions to the HELIUM code to handle intense X-ray light

The X-ray free electron lasers currently being built promise light of sufficient intensity to cause direct double photoionization of the inner K-shells of Ne and Ar atoms. The Ne and Ar K-shell electrons experience a binding energy that is almost the same in the neutral atom as in the corresponding two-electron ion, Ne8+ or Ar16+. We report on new numerical methods we have implemented for calculating double ionization rates for these two-electron positive ions interacting with an intense X-ray free electron laser pulse. The numerical method builds upon an existing code (HELIUM) for solving the full-dimensional time-dependent Schrödinger equation that, operating within the electric dipole approximation, has found successful application in the optical to XUV wavelength range. The primary extensions to HELIUM are the inclusion of magnetic dipole and electric quadrupole interaction terms appropriate to extreme laser intensities and wavelengths approaching atomic dimensions.