Spin-orbit Energy Research Articles

Topological superconductivity in the extended Kitaev-Heisenberg model

We study superconducting pairing in the doped Kitaev-Heisenberg model by taking into account the recently proposed symmetric off-diagonal exchange $\Gamma$. By performing a mean-field analysis, we classify all possible superconducting phases in terms of symmetry, explicitly taking into account effects of spin-orbit coupling. Solving the resulting gap equations self-consistently, we map out a phase diagram that involves several topologically nontrivial states. For $\Gamma<0$, we find a competition between a time-reversal symmetry breaking chiral phase with Chern number $\pm1$ and a time-reversal symmetric nematic phase that breaks the rotational symmetry of the lattice. On the other hand, for $\Gamma \geq 0$ we find a time-reversal symmetric phase that preserves all the lattice symmetries, thus yielding clearly distinguishable experimental signatures for all superconducting phases. Both of the time-reversal symmetric phases display a transition to a $\mathbb{Z}_2$ non-trivial phase at high doping levels. Finally, we also include a symmetry-allowed spin-orbit coupling kinetic energy and show that it destroys a tentative symmetry protected topological order at lower doping levels. However, it can be used to tune the time-reversal symmetric phases into a $\mathbb{Z}_2$ non-trivial phase even at lower doping.

Excited interatomic potential energy surfaces of Rb + He that correlate with Rb terms 52S through 72S.

The excited state interatomic potential energy surfaces for Rb + He are computed at the spin-orbit multi-reference configuration interaction level of theory using all-electron basis sets of triple and quadruple-zeta quality that have been contracted for Douglas-Kroll-Hess (DKH) Hamiltonian and includes core-valence correlation. Davidson-Silver corrections (MRCI+Q) are employed to ameliorate size consistency error. An extrapolation of CASSCF energies is performed using the procedure of Karton and Martin whereas extrapolation of correlation energy is performed using an expression involving the inverse powers of (lmax + 1/2), the highest angular momentum value present in the basis set. The spin-orbit energies in the limit of complete basis set are obtained by replacing the energy eigenvalues in the spin-orbit matrix by the relativistic-corrected MRCI+Q energies extrapolated to the complete basis set limit. MRCI diabatic potential energy surfaces for a few selected 2Σ states are calculated to study the general topology and avoided crossings and repulsive form of the 6s 2Σ+ state. Important features of the potential energy surfaces are discussed with implications for alkali laser spectroscopy.

The Role of Self-Interaction Corrections, Vibrations, and Spin-Orbit in Determining the Ground Spin State in a Simple Heme

Without self-interaction corrections or the use of hybrid functionals, approximations to the density-functional theory (DFT) often favor intermediate spin systems over high-spin systems. In this paper, we apply the recently proposed Fermi–Löwdin-orbital self-interaction corrected density functional formalism to a simple tetra-coordinated Fe(II)-porphyrin molecule and show that the energetic orderings of the S = 1 and S = 2 spin states are changed qualitatively relative to the results of Generalized Gradient Approximation (developed by Perdew, Burke, and Ernzerhof, PBE-GGA) and Local Density Approximation (developed by Perdew and Wang, PW92-LDA). Because the energetics, associated with changes in total spin, are small, we have also calculated the second-order spin–orbit energies and the zero-point vibrational energies to determine whether such corrections could be important in metal-substituted porphins. Our results find that the size of the spin–orbit and vibrational corrections to the energy orderings are small compared to the changes due to the self-interaction correction. Spin dependencies in the Infrared (IR)/Raman spectra and the zero-field splittings are provided as a possible means for identifying the spin in porphyrins containing Fe(II).

Conductance through a helical state in an Indium antimonide nanowire

The motion of an electron and its spin are generally not coupled. However in a one-dimensional material with strong spin-orbit interaction (SOI) a helical state may emerge at finite magnetic fields, where electrons of opposite spin will have opposite momentum. The existence of this helical state has applications for spin filtering and cooper pair splitter devices and is an essential ingredient for realizing topologically protected quantum computing using Majorana zero modes. Here, we report measurements of a quantum point contact in an indium antimonide nanowire. At magnetic fields exceeding 3 T, the 2 e2/h conductance plateau shows a re-entrant feature toward 1 e2/h which increases linearly in width with magnetic field. Rotating the magnetic field clearly attributes this experimental signature to SOI and by comparing our observations with a numerical model we extract a spin-orbit energy of approximately 6.5 meV, which is stronger than the spin-orbit energy obtained by other methods.

Critical Magnetic Field of Ultra-Thin Superconducting Films and Interfaces

We derive an analytic expression for the temperature dependent critical magnetic field parallel to ultrathin superconducting films with Rashba spin-orbit interaction. Thereby we cover the range from small to large spin-orbit interactions $\lambda$ compared with the gap parameter $\Delta_0$. We find that at a critical spin-orbit energy $\lambda_c$ a first-order phase transition takes place at which the pairing momentum of the Cooper pairs changes discontinuously. We speculate that this might give raise to new phenomena. With increasing $\lambda/\Delta_0$, the pair formation changes from interband to intraband pairing. For $\lambda>\lambda_c$, a dimensional cross-over of the critical field from two to one dimension is taking place.

Superconducting instability of a non-centrosymmetric system

The Fermi gas approach to the weak-coupling superconductivity in the non-centrosymmetric systems lead to a conclusion of an approximately spin-orbit coupling independent critical temperature of the singlet states as well as the triplet states defined by the order parameter aligned with the antisymmetric spin-orbit coupling vector. We indicate that the above results follow from a simplified approximation of a density of states by a constant Fermi surface value. Such a scenario does not properly account for the spin-split quasiparticle energy spectrum and reduces the spin-orbit coupling influence on superconductivity to the bare pair-breaking effect of a lifted spin degeneracy. Applying the tight-binding model, which captures the primary features of the spin-split energy band, i.e., its enhanced width and the spin-orbit coupling induced redistribution of the spectral weights in the density of states, we calculate the critical temperature of a non-centrosymmetric superconductor. We report a general tendency of the critical temperature to be suppressed by the antisymmetric spin-orbit coupling. We indicate that, the monotonic decrease of the critical temperature may be altered by the spin-orbit coupling induced van Hove singularities which, when driven to the Fermi level, generate maxima in the phase diagram. Extending our considerations to the intermediate-coupling superconductivity we point out that the spin-orbit coupling induced change of the critical temperature depends on the structure of the electronic energy band and both – the strength and symmetry of the pair potential. Finally, we discuss the mixed singlet-triplet state superconducting instability and establish conditions concerning the symmetry of the singlet and triplet counterparts as well as the range of the spin-orbit coupling energy which make such a phase transition possible.

Controlling plasmon modes and damping in buckled two-dimensional material open systems

Full ranges of both hybrid plasmon-mode dispersions and their damping are studied systematically by our recently developed mean-field theory in open systems involving a conducting substrate and a two-dimensional (2D) material with a buckled honeycomb lattice, such as silicene, germanene, and a group IV dichalcogenide as well. In this hybrid system, the single plasmon mode for a free-standing 2D layer is split into one acoustic-like and one optical-like mode, leading to a dramatic change in the damping of plasmon modes. In comparison with gapped graphene, critical features associated with plasmon modes and damping in silicene and molybdenum disulfide are found with various spin-orbit and lattice asymmetry energy bandgaps, doping types and levels, and coupling strengths between 2D materials and the conducting substrate. The obtained damping dependence on both spin and valley degrees of freedom is expected to facilitate measuring the open-system dielectric property and the spin-orbit coupling strength of individual 2D materials. The unique linear dispersion of the acoustic-like plasmon mode introduces additional damping from the intraband particle-hole modes, which is absent for a free-standing 2D material layer, and the use of molybdenum disulfide with a large bandgap simultaneously suppresses the strong damping from the interband particle-hole modes.

Reactions of exotic nuclei with the quark-meson coupling model

The nucleon-nucleon interaction is an important requirement for investigations of nuclear structure and reactions, as well as for astrophysical models such as r-process nucleosynthesis and neutron stars. The traditional approach to low-energy nuclear physics is to treat nucleons as immutable objects interacting via phenomenological forces. The use of phenomenological interactions, rather than one derived from a microscopic theory, raises questions as to the reliability of predictions for exotic regions of the nuclear chart. The quark-meson coupling (QMC) model uses a relativistic mean-field approach to provide a microscopically derived nucleon-nucleon interaction, which takes into account the quark structure of the nucleon.The Skyrme energy density functional is a popular phenomenological tool in studies of nuclear structure and reactions. In this work, the QMC density functional was used to produce a set of Skyrme parameterisations, in the hopes that they will give more reliable predictions for exotic nuclei. In conjunction with Hartree-Fock-Bogoliubov (HFB) calculations, the Skyrme-QMC (SQMC) parameterisations have been used to model the ground-state properties of individual nuclei and nucleus-nucleus potentials for Ca + Sn reactions. The SQMC parameterisation performs with an accuracy comparable to modern phenomenological functionals. From this, one can investigate the importance of the isovector terms of the nucleon-nucleon interaction, which are particularly significant for exotic, neutron-rich regions of the nuclear chart.One of the notable successes of the QMC model is its derivation of nuclear spin-orbit coupling. The isovector dependence of the spin-orbit equation of state is remarkably similar to that of the modern UNEDF1 phenomenological density functional. HFB calculations along the Sn isotopic chain reveal that the isovector properties of the spin-orbit term impact binding energies to a level that will be significant for astrophysical r-process modelling.

Role of the electromagnetic momentum in the spin-orbit interaction

The role played by the linear and angular momentum of the electromagnetic fields in the understanding of several aspects of quantum mechanics is discussed. A non-relativistic semi-classical model of the spin-orbit interaction, where the electromagnetic interaction energy U is calculated in the frame of the nucleus, is presented. Taking into account the electron hidden momentum P h = c -1 μ × E, the spin-orbit energy splitting turns out to be Δℰ so = (1 / 2)U, the factor 1 / 2 emerging directly by requiring that the energy variation be a minimum. After quantization, the radius of the orbit is found to be spin-dependent, anticipating a feature of the Dirac equation. Finally, a test of the hidden momentum P h , which may corroborate the approaches based on the hidden momentum and related interpretations of electrodynamics, is proposed and shown to be viable with present technology.

Tuning ferromagneticBaFe2(PO4)2through a high Chern number topological phase

There is strong interest in discovering or designing wide gap Chern insulators. Here we follow a Chern insulator to trivial Mott insulator transition versus interaction strength U in a honeycomb lattice Fe-based transition metal oxide, discovering that a spin-orbit coupling energy scale $\xi$=40 meV can produce and maintain a topologically entangled Chern insulating state against large band structure changes arising from an interaction strength U up to 60 times as large. Within the Chern phase the minimum gap switches from the zone corner K to the zone center $\Gamma$ while maintaining the topological structure. At a critical strength $U_c$, the continuous evolution of the electronic structure encounters a gap closing then reopening, upon which the system reverts to a trivial Mott insulating phase. This Chern insulator phase of honeycomb lattice Fe$^{2+}$ BaFe$_2$(PO$_4$)$_2$ corresponds to a large Chern number C=-3 that will provide enhanced anomalous Hall conductivity due to the associated three edge states threading through the bulk gap of 80 meV.

Stability and accuracy control of k · p parameters

The k · p method is a successful approach to obtain band structure, optical and transport properties of semiconductors and it depends on external parameters that are obtained either from experiments, tight binding or ab initio calculations. Despite the widespread use of the k · p method, a systematic analysis of the stability and the accuracy of its parameters is not usual in the literature. In this work, we report a theoretical framework to determine the k · p parameters from state-of-the-art hybrid density functional theory including spin–orbit coupling, providing a calculation where the gap and spin–orbit energy splitting are in agreement with the experimental values. The accuracy of the set of parameters is enhanced by fitting over several directions at once, minimizing the overall deviation from the original data. This strategy allows us to systematically evaluate the stability, preserving the accuracy of the parameters, providing a tool to determine optimal parameters for specific ranges around the Γ-point. To prove our concept, we investigate the zinc blende GaAs that shows results in excellent agreement with the most reliable data in the literature.

The Kondo temperature of a two-dimensional electron gas with Rashba spin–orbit coupling

We use the Hirsch–Fye quantum Monte Carlo method to study the single magnetic impurity problem in a two-dimensional electron gas with Rashba spin–orbit coupling. We calculate the spin susceptibility for various values of spin–orbit coupling, Hubbard interaction, and chemical potential. The Kondo temperatures for different parameters are estimated by fitting the universal curves of spin susceptibility. We find that the Kondo temperature is almost a linear function of Rashba spin–orbit energy when the chemical potential is close to the edge of the conduction band. When the chemical potential is far away from the band edge, the Kondo temperature is independent of the spin–orbit coupling. These results demonstrate that, for single impurity problems in this system, the most important reason to change the Kondo temperature is the divergence of density of states near the band edge, and the divergence is induced by the Rashba spin–orbit coupling.

High-order moments of spin-orbit energy in a multielectron configuration.

In order to analyze the energy-level distribution in complex ions such as those found in warm dense plasmas, this paper provides values for high-order moments of the spin-orbit energy in a multielectron configuration. Using second-quantization results and standard angular algebra or fully analytical expressions, explicit values are given for moments up to 10th order for the spin-orbit energy. Two analytical methods are proposed, using the uncoupled or coupled orbital and spin angular momenta. The case of multiple open subshells is considered with the help of cumulants. The proposed expressions for spin-orbit energy moments are compared to numerical computations from Cowan's code and agree with them. The convergence of the Gram-Charlier expansion involving these spin-orbit moments is analyzed. While a spectrum with infinitely thin components cannot be adequately represented by such an expansion, a suitable convolution procedure ensures the convergence of the Gram-Charlier series provided high-order terms are accounted for. A corrected analytical formula for the third-order moment involving both spin-orbit and electron-electron interactions turns out to be in fair agreement with Cowan's numerical computations.

Laplace-transformed atomic orbital-based Møller-Plesset perturbation theory for relativistic two-component Hamiltonians.

We present a formulation of Laplace-transformed atomic orbital-based second-order Møller-Plesset perturbation theory (MP2) energies for two-component Hamiltonians in the Kramers-restricted formalism. This low-order scaling technique can be used to enable correlated relativistic calculations for large molecular systems. We show that the working equations to compute the relativistic MP2 energy differ by merely a change of algebra (quaternion instead of real) from their non-relativistic counterparts. With a proof-of-principle implementation we study the effect of the nuclear charge on the magnitude of half-transformed integrals and show that for light elements spin-free and spin-orbit MP2 energies are almost identical. Furthermore, we investigate the effect of separation of charge distributions on the Coulomb and exchange energy contributions, which show the same long-range decay with the inter-electronic/atomic distance as for non-relativistic MP2. A linearly scaling implementation is possible if the proper distance behavior is introduced to the quaternion Schwarz-type estimates as for non-relativistic MP2.

Roles of NN-interaction components in shell-structure evolution

Since the importance of the monopole interaction was first emphasized in 1960s, roles of monopole strengths of two-body nucleon–nucleon interaction in shell structure have been discussed. Through the monopole strengths, we study the roles in shell-structure evolution, starting from explicit forms of the interaction. For the tensor component of the interaction, we show the derivation of the relation, (2j>+1)Vjj>+(2j<+1)Vjj<=0, with a detailed manipulation. We show that one-body spin–orbit term appears in the multipole expansion of two-body spin–orbit interaction. Only the spin–orbit components can affect the spin–orbit energy splitting between spin–orbit partners, when the spin–orbit partner orbits are fully occupied.

Spin–orbit coupling rule in bound fermion systems

Spin–orbit coupling characterizes quantum systems such as atoms, nuclei, hypernuclei, quarkonia, etc, and is essential for understanding their spectroscopic properties. Depending on the system, the effect of spin–orbit coupling on shell structure is large in nuclei, small in quarkonia and perturbative in atoms. In the standard non-relativistic reduction of the single-particle Dirac equation, we derive a universal rule for the relative magnitude of the spin–orbit effect that applies to very different quantum systems, regardless of whether the spin–orbit coupling originates from the strong or electromagnetic interaction. It is shown that in nuclei the near equality of the mass of the nucleon and the difference between the large repulsive and attractive potentials explain the fact that spin–orbit splittings are comparable to the energy spacing between major shells. For a specific ratio between the particle mass and the effective potential whose gradient determines the spin–orbit force, we predict the occurrence of giant spin–orbit energy splittings that dominate the single-particle excitation spectrum.

Bond energies of ThO+ and ThC+: A guided ion beam and quantum chemical investigation of the reactions of thorium cation with O2 and CO

Kinetic energy dependent reactions of Th(+) with O2 and CO are studied using a guided ion beam tandem mass spectrometer. The formation of ThO(+) in the reaction of Th(+) with O2 is observed to be exothermic and barrierless with a reaction efficiency at low energies of k/kLGS = 1.21 ± 0.24 similar to the efficiency observed in ion cyclotron resonance experiments. Formation of ThO(+) and ThC(+) in the reaction of Th(+) with CO is endothermic in both cases. The kinetic energy dependent cross sections for formation of these product ions were evaluated to determine 0 K bond dissociation energies (BDEs) of D0(Th(+)-O) = 8.57 ± 0.14 eV and D0(Th(+)-C) = 4.82 ± 0.29 eV. The present value of D0 (Th(+)-O) is within experimental uncertainty of previously reported experimental values, whereas this is the first report of D0 (Th(+)-C). Both BDEs are observed to be larger than those of their transition metal congeners, TiL(+), ZrL(+), and HfL(+) (L = O and C), believed to be a result of lanthanide contraction. Additionally, the reactions were explored by quantum chemical calculations, including a full Feller-Peterson-Dixon composite approach with correlation contributions up to coupled-cluster singles and doubles with iterative triples and quadruples (CCSDTQ) for ThC, ThC(+), ThO, and ThO(+), as well as more approximate CCSD with perturbative (triples) [CCSD(T)] calculations where a semi-empirical model was used to estimate spin-orbit energy contributions. Finally, the ThO(+) BDE is compared to other actinide (An) oxide cation BDEs and a simple model utilizing An(+) promotion energies to the reactive state is used to estimate AnO(+) and AnC(+) BDEs. For AnO(+), this model yields predictions that are typically within experimental uncertainty and performs better than density functional theory calculations presented previously.

Spin–orbit interaction in a multilayered spherical quantum dot

The conventional spin–orbit interaction due to the presence of a central or off-central impurity located in a multilayered spherical quantum dot (MSQD) has been investigated. The different effective masses of dots and barriers have been taken into consideration. The spin–orbit interaction has been calculated in the excited state (2p). The variational method has been applied by using a new form of the trial wave function in addition to the conventional form that has been used in previous work. The new form has the advantage of satisfying the boundary conditions at the interfaces between dots and barriers in the case of different masses. The case of central impurity has been further explored by using the exact analytical solution in the state (2p) of the radial Schrodinger equation in the presence of the impurity. Moreover, the calculations have been performed in the case of a MSQD of thin inner barrier, for which the electron is most probably localized in the outer dot. Thus for an off-central impurity placed in the inner dot the spin–orbit energy increases continuously as the impurity location moves towards the surface of the inner dot. The study has yielded new analytical expressions for the electron energy, binding energy and spin–orbit expectation energy for both central and off-central impurities.

Double reflection of spin-orbit-coupled cold atoms

Artificial spin-orbit coupling in neutral cold atom have been experimentally implemented in alkali-metal atoms. Nowadays people begin to explore its possible applications. One of the most interesting applications is the atomic mirror, which is a key element in atom optics. And spin-orbit coupling provides the atomic beam with the possibility that the atomic spin can flip during its propagation, thus can be used to prepare the quantum-state-selective atomic mirror. In 2008, Juzeliūnas, et al. [Juzeliūnas G, et al. 2008 Phys. Rev. Lett. 100 200405] studied a spin-orbit-coupled matter wave packet of cold atom gas impinging on an infinite step potential created by the optical light field. Results showed that there is not only ordinary specular reflection, but also non-specular one. The reflected atoms split into two beams and double reflection takes place. Based on the previous study, here we consider a matter wave packet of spin-orbit-coupled cold atom gas impinging on a finite step potential created by the optical light field. Due to the effect of the spin-orbit coupling, in addition to the propagating state, the eigenstates of cold atoms include evanescent state and oscillating evanescent state. Under suitable conditions double reflection will take place. If there are just evanescent waves in the step potential, total internal reflection will take place. In other words, when there is propagating wave in the step potential, partial reflection will take place. By taking into account both the total internal reflection and partial reflection, we study not only the polarization rate but also the reflectivity each as a function of incident energy, incident angle and spin-orbit coupling strength. The properties different from those of previous studies are found. In the case of total internal reflection, we find that the polarization rate of the reflected atoms is sensitive to incident angle instead of the spin-orbit coupling strength and incident energy. While in the case of partial reflection, all these factors strongly affect the polarization rate and reflectivity. We carefully study these properties and find that, on one hand, high efficiency atomic mirror can be acquired in the case of total internal reflection, and on the other hand, we can acquire the different polarization rates by adjusting the incident angle, the spin-orbit coupling strength and incident energy in the case of partial reflection.

Towards a new class of heavy ion doped magnetic semiconductors for room temperature applications.

The article presents, using Bi doped ZnO, an example of a heavy ion doped oxide semiconductor, highlighting a novel p-symmetry interaction of the electronic states to stabilize ferromagnetism. The study includes both ab initio theory and experiments, which yield clear evidence for above room temperature ferromagnetism. ZnBixO1−x thin films are grown using the pulsed laser deposition technique. The room temperature ferromagnetism finds its origin in the holes introduced by the Bi doping and the p-p coupling between Bi and the host atoms. A sizeable magnetic moment is measured by means of x-ray magnetic circular dichroism at the O K-edge, probing directly the spin polarization of the O(2p) states. This result is in agreement with the theoretical predictions and inductive magnetometry measurements. Ab initio calculations of the electronic and magnetic structure of ZnBixO1−x at various doping levels allow to trace the origin of the ferromagnetic character of this material. It appears, that the spin-orbit energy of the heavy ion Bi stabilizes the ferromagnetic phase. Thus, ZnBixO1−x doped with a heavy non-ferromagnetic element, such as Bi, is a credible example of a candidate material for a new class of compounds for spintronics applications, based on the spin polarization of the p states.