Relativistic Spin-orbit Interaction Research Articles

Spin‐Controlled Helical Quantum Sieve Chiral Spectrometer (Adv. Mater. 24/2023)

Identification of Chiral Enantiomers In article number 2209125, Hossam Haick and co-workers report a molecular-spin-sensitive-antenna (MSSA) made up of organically functionalized graphene layers on a fibrous helical cellulose network. The MSSA structure combines relativistic spin–orbit interaction coupled with organic spin-coherence and offers spatiotemporal chiral molecule separation, recognition, and selective identification of enantiomers. Coupled with neuromorphic artificial intelligence, it enables rapid classification of chiral molecules, resulting in portable spin-controlled spectrometry.

Noncollinear antiferromagnetic textures driven high-harmonic generation from magnetic dynamics in the absence of spin-orbit coupling

We demonstrate the generation of high order harmonics in carrier pumping from precessing ferromagnetic or antiferromagnetic orders, excited via magnetic resonance, in the presence of topological antiferromagnetic textures. This results in an enhancement of the carrier dynamics by orders of magnitude, enabling for an emission deep in the THz frequency range. Interestingly, the generation process occurs in an intrinsic manner, and is solely governed by the interplay between the s-d exchange coupling underlying the noncollinear antiferromagnetic order and the dynamical s-d exchange constant of the magnetic drive. Therefore, the relativistic spin–orbit interaction is not required for the emergence of high harmonics in the pumped currents. Accordingly, the noncollinear topological antiferromagnetic order is presented as an alternative to spin–orbit interaction for the purpose of harnessing high harmonic emission in carrier pumping. Furthermore, we demonstrate the emergence of high harmonics from random magnetic impurities. This suggests the universality of the magnetically induced high harmonic emission in the presence of real and/or momentum space noncollinear textures. Our proposal initiates a tantalizing prospect for the utilization of topological textures in the context of the highly active domains of ultrafast spintronics and THz emission.

On-demand electrical control of spin qubits

Once called a 'classically non-describable two-valuedness' by Pauli, the electron spin forms a qubit that is naturally robust to electric fluctuations. Paradoxically, a common control strategy is the integration of micromagnets to enhance the coupling between spins and electric fields, which, in turn, hampers noise immunity and adds architectural complexity. Here we exploit a switchable interaction between spins and orbital motion of electrons in silicon quantum dots, without a micromagnet. The weak effects of relativistic spin-orbit interaction in silicon are enhanced, leading to a speed up in Rabi frequency by a factor of up to 650 by controlling the energy quantization of electrons in the nanostructure. Fast electrical control is demonstrated in multiple devices and electronic configurations. Using the electrical drive, we achieve a coherence time T2,Hahn ≈ 50 μs, fast single-qubit gates with Tπ/2 = 3 ns and gate fidelities of 99.93%, probed by randomized benchmarking. High-performance all-electrical control improves the prospects for scalable silicon quantum computing.

Spin splitting of the conduction band by exchange interaction in the valence band through a k·p interband process in ferromagnetic semiconductors

The momentum-dependent spin splitting in the conduction band couples orbital motion to spin and enables electrical control of spin. Currently, this control relies on the relativistic spin-orbit interaction (SOI), which limits useful materials to those containing heavy elements. Recently, Naka et al. [Nat. Commun. 10, 4305 (2019)] have found a momentum-dependent spin splitting originating from the exchange interaction, which is expected to extend spintronic materials to those without heavy elements. In this paper, we propose a mechanism of the exchange-induced orbital-spin coupling by extending the $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ theory. As an example, we consider an $n$-type ferromagnetic semiconductor (nFMS) of ${T}_{d}$ point group symmetry with the $p\text{\ensuremath{-}}d$ exchange interaction between an electron in the valence band and the spin of a magnetic ion and evaluate the spin splitting in the conduction band of ${\mathrm{\ensuremath{\Gamma}}}_{6}$ irreducible representation from the eight-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian. We find that the lowest-order spin splitting in bulk is of the second order of momentum, which results in a nonzero splitting at ${k}_{x}={k}_{y}=0$ in a quantum well with a nonzero quantized momentum ${k}_{z}$. An estimation shows that the $p\text{\ensuremath{-}}d$ exchange interaction is the dominant origin of the conduction-band spin splitting in InFeAs nFMS. We also calculate the intrinsic anomalous Hall conductivity of bulk InFeAs generated by the $p\text{\ensuremath{-}}d$ exchange, which provides both the coupling of orbital motion to spin and that of spin to nFMS magnetization. We find that the $p\text{\ensuremath{-}}d$ exchange-induced Hall conductivity exhibits an accelerated increase with Fe density, in contrast to that produced by the $s\text{\ensuremath{-}}d$ exchange and the Dresselhaus SOI. This finding suggests that the extended $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ mechanism of orbital-spin coupling is expected to help find remarkable phenomena and useful applications in a wide variety of materials and structures.

Transition energies and oscillator strengths for the intrashell and intershell transitions of the C-like ions in a thermodynamic equilibrium plasma environment.

We present a theoretical study of the transition energies ω and the oscillator strengths gf for the C-like ions (with Z from 14-36) subject to plasma environment for atomic transitions, which meet the spatial and temporal criteria of the Debye-Hückel (DH) approximation. Two strong dipole-allowed transitions, viz., the intrashell transition 2s2p^{3}^{3}D_{1}→2s^{2}2p^{2}^{3}P_{0}, and the intershell transition 2s^{2}2p3d^{3}D_{1}→2s^{2}2p^{2}^{3}P_{0} are investigated in detail. We found that both ω and gf increase for the intrashell transition under the Debye-Hückel screening potential V_{DH} in terms of the Debye length D, which is linked to the ratio between the plasma density N_{e} and its temperature kT. In contrast, both ω and gf decrease for the intershell transition. Our theoretically estimated data have led to a general scaling feature for the change in ω of both intershell and intrashell transitions for ions with different nuclear charge Z. A similar general feature for the change in gf is also found for the intrashell transition. However, due to the change of the electron correlations between electrons in different shells with respect to the relativistic spin-orbit interaction as Z varies, the variation of gf subject to the surrounding plasma is more complicated for the intershell transition. The results presented in this work may facilitate the plasma diagnostic to determine the plasma temperature and density for the astrophysical objects and the controlled fusion facilities.

High- n limit of quantum defects for high- ℓ strontium Rydberg states

Studies of high-$n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160$, strontium $5sng, 5snh$, and $5sni$ Rydberg states are reported. Measurements using microwave spectroscopy provide the quantum defects for these states and furnish a valuable test of quantum defects reported for lower-$\ensuremath{\ell}$ states in the high-$n$ limit. With increasing $\ensuremath{\ell}$ the relativistic spin-orbit interaction becomes comparable to the electron-electron interaction resulting in the breakdown of the standard $LS$-coupling scheme. Heavier alkaline-earth quasi-two-electron systems differ significantly from a true two-electron system such as helium because indirect spin-orbit effects are dramatically enhanced. We explore the high-$n$ limit of this transition between $LS$ and $jj$ coupling. For high-$\ensuremath{\ell}$ states the quantum defects are determined predominantly by the polarizability of the inner electrons.

Astatine Facing Janus: Halogen Bonding vs. Charge-Shift Bonding.

The nature of halogen-bond interactions was scrutinized from the perspective of astatine, potentially the strongest halogen-bond donor atom. In addition to its remarkable electronic properties (e.g., its higher aromaticity compared to benzene), C6At6 can be involved as a halogen-bond donor and acceptor. Two-component relativistic calculations and quantum chemical topology analyses were performed on C6At6 and its complexes as well as on their iodinated analogues for comparative purposes. The relativistic spin–orbit interaction was used as a tool to disclose the bonding patterns and the mechanisms that contribute to halogen-bond interactions. Despite the stronger polarizability of astatine, halogen bonds formed by C6At6 can be comparable or weaker than those of C6I6. This unexpected finding comes from the charge-shift bonding character of the C–At bonds. Because charge-shift bonding is connected to the Pauli repulsion between the bonding σ electrons and the σ lone-pair of astatine, it weakens the astatine electrophilicity at its σ-hole (reducing the charge transfer contribution to halogen bonding). These two antinomic characters, charge-shift bonding and halogen bonding, can result in weaker At-mediated interactions than their iodinated counterparts.

Orbital Effects in Solids: Basics, Recent Progress, and Opportunities.

The properties of transition metal compounds are largely determined by nontrivial interplay of different degrees of freedom: charge, spin, lattice, and also orbital ones. Especially rich and interesting effects occur in systems with orbital degeneracy. For example, they result in the famous Jahn-Teller effect, leading to a plethora of consequences for static and dynamic properties, including nontrivial quantum effects. In the present review, we discuss the main phenomena in the physics of such systems, paying central attention to the novel manifestations of those. After shortly summarizing the basic phenomena and their descriptions, we concentrate on several specific directions in this field. One of them is the reduction of effective dimensionality in many systems with orbital degrees of freedom due to the directional character of orbitals, with the concomitant appearance of some instabilities that lead in particular to the formation of dimers, trimers, and similar clusters in a material. The properties of such cluster systems, which are largely determined by their orbital structure, are discussed in detail, and many specific examples of those in different materials are presented. Another big field that has acquired special significance relatively recently is the role of the relativistic spin-orbit interaction. The mutual influence of this interaction and the more traditional Jahn-Teller physics is treated in detail in the second part of the review. In discussing all of these questions, special attention is paid to novel quantum effects.

Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory

Thermodynamics of plutonium monocarbide is studied from first-principles theory that includes relativistic electronic structure and anharmonic lattice vibrations. Density-functional theory (DFT) is expanded to include orbital-orbital coupling in addition to the relativistic spin-orbit interaction for the electronic structure and it is combined with anharmonic, temperature dependent, lattice dynamics derived from the self-consistent ab initio lattice dynamics (SCAILD) method. The obtained thermodynamics are compared to results from simpler quasi-harmonic theory and experimental data. Formation enthalpy, specific heat, and Gibbs energy calculated from the anharmonic model are validated by direct comparison with a calculation of phase diagram (CALPHAD) assessment of PuC and sub-stochiometric PuC0.896. Overall, the theory reproduces CALPHAD results and measured data for PuC rather well, but the comparison is hampered by the sub-stoichiometric nature of plutonium monocarbide. It was found that a bare theoretical approach that ignores spin-orbit and orbital-orbital coupling (orbital polarization) of the plutonium 5f electrons promotes too soft phonons and Gibbs energies that are incompatible with that of the CALPHAD assessment of the experimental data. The investigation of PuC suggests that the electronic structure is accurately described by plutonium 5f electrons as “band like” and delocalized, but correlate through spin polarization, orbital polarization, and spin-orbit coupling, in analogy to previous findings for plutonium metal.

Anomalous Lattice Softening Near a Quantum Critical Point in a Transverse Ising Magnet.

We have investigated the elastic response of a transverse Ising magnet CoNb_{2}O_{6} by means of ultrasound velocity measurement. A huge elastic anomaly in the C_{66} mode is observed near a quantum critical point when sweeping a magnetic field perpendicular to the Ising axis. This anomaly appears to become critical only for the Faraday configuration (field parallel to the sound propagation direction) but is much less pronounced for the Voigt geometry (field perpendicular to the sound propagation direction). We propose that the relativistic spin-orbit interaction plays a crucial role in the quantum critical regime resulting in the elastic anomaly, which is enhanced by quantum fluctuations.

Assessing Relativistic Effects and Electron Correlation in the Actinide Metals Th to Pu

Density functional theory (DFT) calculations are employed to explore and assess the effects of the relativistic spin–orbit interaction and electron correlations in the actinide elements. Specifically, we address electron correlations in terms of an intra-atomic Coulomb interaction with a Hubbard U parameter (DFT + U). Contrary to recent beliefs, we show that for the ground-state properties of the light actinide elements Th to Pu, the DFT + U makes its best predictions for U = 0. Actually, our modeling suggests that the most popular DFT + U formulation leads to the wrong ground-state phase for plutonium. Instead, extending DFT and the generalized gradient approximation (GGA) with orbital–orbital interaction (orbital polarization; OP) is the most accurate approach. We believe the confusion in the literature on the subject mostly originates from incorrectly accounting for the spin–orbit (SO) interaction for the p1/2 state, which is not treated in any of the widely used pseudopotential plane-wave codes. Here, we show that for the actinides it suffices to simply discard the SO coupling for the p states for excellent accuracy. We thus describe a formalism within the projector-augmented-wave (PAW) scheme that allows for spin–orbit coupling, orbital polarization, and non-collinear magnetism, while retaining an efficient calculation of Hellmann–Feynman forces. We present results of the ground-state phases of all the light actinide metals (Th to Pu). Furthermore, we conclude that the contribution from OP is generally small, but substantial in plutonium.

Dynamical magnetoelectric phenomena of skyrmions in multiferroics

Abstract Magnetic skyrmions, nanoscopic spin vortices carrying a quantized topological number in chiral-lattice magnets, are recently attracting great research interest. Although magnetic skyrmions had been observed only in metallic chiral-lattice magnets such as B20 alloys in the early stage of the research, their realization was discovered in 2012 also in an insulating chiral-lattice magnet Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . A characteristic of the insulating skyrmions is that they can host multiferroicity, that is, the noncollinear magnetization alignment of skyrmion induces electric polarizations in insulators with a help of the relativistic spin-orbit interaction. It was experimentally confirmed that the skyrmion phase in Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ is indeed accompanied by the spin-induced ferroelectricity. The resulting strong magnetoelectric coupling between magnetizations and electric polarizations can provide us with a means to manipulate and activate magnetic skyrmions by application of electric fields. This is in sharp contrast to skyrmions in metallic systems, which are driven through injection of electric currents. The magnetoelectric phenomena specific to the skyrmion-based multiferroics are attracting intensive research interest, and, in particular, those in dynamical regime are widely recognized as an issue of vital importance because their understanding is crucial both for fundamental science and for technical applications. In this article, we review recent studies on multiferroic properties and dynamical magnetoelectric phenomena of magnetic skyrmions in insulating chiral-lattice magnet Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . It is argued that the multiferroic skyrmions show unique resonant excitation modes of coupled magnetizations and polarizations, so-called electromagnon excitations, which can be activated both magnetically with a microwave magnetic field and electrically with a microwave electric field. The interference between these two activation processes gives rise to peculiar phenomena in the gigahertz regime. As its representative example, we discuss a recent theoretical prediction of unprecedentedly large nonreciprocal directional dichroism of microwaves in the skyrmion phase of Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . This phenomenon can be regarded as a one-way window effect on microwaves, that is, the extent of microwave absorption changes significantly when its incident direction is reversed. This dramatic effect was indeed observed by subsequent experiments. These studies demonstrated that the multiferroic skyrmions can be a promising building block for microwave devices.

Helicoidal excitonic phase in an electron-hole double-layer system

We propose helicoidal excitonic phase in a Coulomb-coupled two-dimensional electron-hole double layer (EHDL) system with relativistic spin-orbit interaction. Previously, it was demonstrated that layered InAs/AlSb/GaInSb heterostructure is an ideal experimental platform for searching excitonic condensate phases, while its electron layer has non-negligible Rashba interaction. We clarify that due to the Rashba term, the spin-triplet (spin-1) exciton field in the EHDL system forms a helicoidal structure and the helicoid plane can be controlled by an in-plane Zeeman field. We show that due to small but finite Dirac term in the heavy hole layer the helicoidal structure of the excitonic field under the in-plane field results in a helicoidal {\it magnetic} order in the electron layer. Based on linearization analyses, we further calculate momentum-energy dispersions of low-energy Goldstone modes in the helicoidal excitonic phase. We discuss possible experimental probes of the excitonic phase in the EHDL system.

Multireference Ab Initio Studies of Magnetic Properties of Terbium-Based Single-Molecule Magnets.

We investigate how different chemical environments influence magnetic properties of terbium(III) (Tb)-based single-molecule magnets (SMMs), using first-principles relativistic multireference methods. Recent experiments showed that Tb-based SMMs can have exceptionally large magnetic anisotropy and that they can be used for experimental realization of quantum information applications, with a judicious choice of chemical environment. Here, we perform complete active space self-consistent field calculations including relativistic spin-orbit interaction for representative Tb-based SMMs such as TbPc2 and TbPcNc in three charge states. We calculate the low-energy electronic structure from which we compute the Tb crystal-field (CF) parameters and construct an effective pseudospin Hamiltonian. Our calculations show that the ligand type and fine points of molecular geometry do not affect the gap between the ground-state and first-excited doublets, whereas the latter varies weakly with oxidation number. On the other hand, higher-energy levels have a strong dependence on all these characteristics. For neutral TbPc2 and TbPcNc molecules, the Tb magnetic moment and ligand spin are parallel to each other and the coupling strength between them does not depend much on the ligand type and details of the atomic structure. However, ligand distortion and molecular symmetry play a crucial role in transverse CF parameters which lead to tunnel splitting. The tunnel splitting induces quantum tunneling of magnetization by itself or by combining with other processes. Our results provide insights into the mechanisms of magnetization relaxation in the representative Tb-based SMMs.

All-electron fully relativistic Kohn-Sham theory for solids based on the Dirac-Coulomb Hamiltonian and Gaussian-type functions

We present the first full-potential method that solves the fully relativistic 4-component Dirac-Kohn-Sham equation for materials in the solid state within the framework of atom-centered Gaussian-type orbitals (GTOs). Our GTO-based method treats one-, two-, and three-dimensional periodic systems on an equal footing, and allows for a seamless transition to the methodology commonly used in studies of molecules with heavy elements. The scalar relativistic effects and spin-orbit interaction are handled variationally. The full description of the electron-nuclear potential in the core region of heavy nuclei is straightforward due to the local nature of the GTOs and does not pose any computational difficulties. We show how the time-reversal symmetry and a quaternion algebra-based formalism can be exploited to significantly reduce the increased methodological complexity and computational cost associated with multiple wave-function components coupled by the spin-orbit interaction. We provide a description of how to employ the matrix form of the multipole expansion and an iterative renormalization procedure to evaluate the conditionally convergent lattice sums arising for periodic systems. We investigate the problem of inverse variational collapse that arises if the Dirac operator containing a repulsive periodic potential is expressed in a basis containing diffuse functions, and suggest a possible solution. We demonstrate the validity of the method on silver halide crystals with large relativistic effects, and two-dimensional honeycomb structures (silicene and germanene) exhibiting the spin-orbit-driven quantum spin Hall effect. Our results are well-converged with respect to the basis limit using standard bases developed for molecular calculations, and indicate that the rule of removing basis functions with small exponents should not be applied when transferring the molecular basis to solids.

Direct detection of spin Nernst effect in platinum

Generation of spin current lies at the heart of spintronic research. The spin Hall effect and the spin Seebeck effect have drawn considerable attention in the last few years to create pure spin current by heavy metals and ferromagnets, respectively. In this work, we show the direct evidence of heat current to spin current conversion in non-magnetic Platinum by the spin Nernst effect (SNE) at room temperature. This is the thermal analogue of the spin Hall effect in non-magnets. We have shown that the 8 K/μm thermal gradient in Pt can lead to the generation of pure spin current density of the order of 108 A/m2 by virtue of SNE. This opens up an additional possibility to couple the relativistic spin-orbit interaction with the thermal gradient for spintronic applications.

Terahertz radiation by subpicosecond spin-polarized photocurrent originating from Dirac electrons in a Rashba-type polar semiconductor

The spin-splitting energy bands induced by the relativistic spin-orbit interaction in solids provide a new opportunity to manipulate the spin-polarized electrons on the sub-picosecond time scale. Here, we report one such example in a bulk Rashba-type polar semiconductor BiTeBr. Strong terahertz electromagnetic waves are emitted after the resonant excitation of the interband transition between the Rashba-type spin-splitting energy bands with a femtosecond laser pulse circularly polarized. The phase of the emitted terahertz waves is reversed by switching the circular polarization. This suggests that the observed terahertz radiation originates from the subpicosecond spin-polarized photocurrents, which are generated by the asymmetric depopulation of the Dirac state. Our result provides a new way for the current-induced terahertz radiation and its phase control by the circular polarization of incident light without external electric fields.

MS-CASPT2 study of the ground and low lying states of CsH.

Correlated ab initio methods [CASPT2 and R-CCSD(T)] in conjunction with the ANO-RCC basis sets in large contraction were used to calculate potential energy curves (PECs) of the ground and excited electronic states of CsH+ (doublets and quartets) with the inclusion of the scalar relativistic effects and spin-orbit interaction. The ground X2Σ+ state is a rather fragile van der Waals molecular ion. The binding energy of this X2Σ+ state provided by both computational methods is estimated to be 0.02-0.04eV, and is compared with the reported experimental binding energy (0.51-0.77eV). This large binding energy can be attributed to the A2Σ+ state, and can thus explain the apparent disagreement between theory and experiment. The spectroscopic constants of all bound states were calculated from the PECs and compared with previous published data for X2Σ+ and A2Σ+ states. Graphical abstract Low-lying Ω states of cesium hydride cation.

Superexchange theory of electronic polarization driven by relativistic spin-orbit interaction at half filling

By applying Berry-phase theory for the effective half-filled Hubbard model, we derive an analytical expression for the electronic polarization driven by the relativistic spin-orbit (SO) coupling. The model itself is constructed in the Wannier basis, using the input from the first-principles electronic structure calculations in the local-density approximation, and then treated in the spirit of the superexchange theory. The obtained polarization has the following form: ${\bf P}_{ij} = \boldsymbol{\epsilon}_{ji} \boldsymbol{\cal P}_{ij} \cdot [\boldsymbol{e}_i \times \boldsymbol{e}_j]$, where $\boldsymbol{\epsilon}_{ji}$ is the direction of the bond $\langle ij \rangle$, $\boldsymbol{e}_i$ and $\boldsymbol{e}_j$ are the directions of spins in this bond, and $\boldsymbol{\cal P}_{ij}$ is the pseudovector containing all the information about the crystallographic symmetry of the considered system. The expression describes the ferroelectric activity in various magnets with noncollinear but otherwise nonpolar magnetic structures, which would yield no polarization without SO interaction, including the magnetoelectric (ME) effect, caused by the ferromagnetic canting of spins in the external magnetic field, and spin-spiral multiferroics. The abilities of this theory are demonstrated for the the analysis of linear ME effect in Cr$_2$O$_3$ and BiFeO$_3$ and properties multiferroic MnWO$_4$ and $\beta$-MnO$_2$. In all considered examples, the theory perfectly describes the symmetry properties of the induced polarization. However, in some cases, the values of this polarization are underestimated, suggesting that other effects, besides the spin and electronic ones, can also play an important role.

Magnetochiral nonreciprocity of volume spin wave propagation in chiral-lattice ferromagnets

Spin current, i.e. the flow of spin angular momentum or magnetic moment, has recently attracted much attention as the promising alternative for charge current with better energy efficiency. Genuine spin current is generally carried by the spin wave (propagating spin precession) in insulating ferromagnets, and should hold the chiral symmetry when it propagates along the spin direction. Here, we experimentally demonstrate that such a spin wave spin current (SWSC) shows nonreciprocal propagation characters in a chiral-lattice ferromagnet. This phenomenon originates from the interference of chirality between the SWSC and crystal-lattice, which is mediated by the relativistic spin-orbit interaction. The present finding enables the design of perfect spin current diode, and highlights the importance of the chiral aspect in SWSC.