Spin-orbit Coupling In Systems Research Articles

Yang–Mills field strength and classical trajectories of electrons in Rashba spin–orbit-coupled systems

Yang–Mills field strength and classical trajectories of electrons in Rashba spin–orbit-coupled systems

Current-Induced Magnetization Switching in Light-Metal-Oxide/Ferromagnetic-Metal Bilayers via Orbital Rashba Effect.

The orbital angular momentum (OAM) generation as well as its associated orbital torque is currently a matter of great interest in spin-orbitronics and is receiving increasing attention. In particular, recent theoretical work predicts that the oxidized light metal Cu can serve as an efficient OAM generator through its surface orbital Rashba effect. Here, for the first time, the crucial current-induced magnetic-field-free in-plane magnetization reversal is experimentally demonstrated in CoFeB/CuOx bilayers without any heavy elements. We show that the critical current density can be comparable to that of strong spin-orbit coupling systems with heavy metals (Pt and Ta) and that the magnetization reversal mechanism is governed by coherent rotation in the grains through the second-harmonic and magneto-optical Kerr effect measurements. Our results indicate that light metal oxides can play an equally important role as heavy metals in magnetization reversal, broadening the choice of materials for engineering spintronic devices.

Phase diagram of the square-lattice Hubbard model with Rashba-type antisymmetric spin-orbit coupling

We clarify the ground-state phase diagram of the half-filled square-lattice Hubbard model with Rashba spin-orbit coupling (SOC) characterized by the spin-split energy bands due to broken inversion symmetry. Although the Rashba metals and insulating magnets have been studied well, the intermediate interaction strength of the system remained elusive due to the lack of appropriate theoretical tools to unbiasedly describe the large-scale magnetic structures. We complementarily apply four different methods (sine-square deformed mean-field theory, random phase approximation, Luttinger-Tisza method, and density matrix embedding theory) and succeed in capturing the incommensurate spin-density-wave (SDW) phases with very long spatial periods which were previously overlooked. The transition to the SDW phases from the metallic phase is driven by an unprecedented instability that nests the two parts of the Fermi surface carrying opposite spins. For large SOC, the spiral, stipe, and vortex phases are obtained, when the four Dirac points exist near the Fermi level and their whole linear dispersions nest by a wavelength $\ensuremath{\pi}$, opening a band gap. These two types of transition provide Fermiology that distinguishes the antisymmetric SOC systems, generating a variety of magnetic phases that start from the relatively weak correlation regime and continue to the strongly interacting limit.

Large Spin Hall Conductivity and Excellent Hydrogen Evolution Reaction Activity in Unconventional PtTe1.75 Monolayer.

Two-dimensional (2D) materials have gained lots of attention due to the potential applications. In this work, we propose that based on first-principles calculations, the (2 × 2) patterned PtTe2 monolayer with kagome lattice formed by the well-ordered Te vacancy (PtTe1.75) hosts large and tunable spin Hall conductivity (SHC) and excellent hydrogen evolution reaction (HER) activity. The unconventional nature relies on the A1 @ 1b band representation of the highest valence band without spin-orbit coupling (SOC). The large SHC comes from the Rashba SOC in the noncentrosymmetric structure induced by the Te vacancy. Even though it has a metallic SOC band structure, the ℤ2 invariant is well defined because of the existence of the direct bandgap and is computed to be nontrivial. The calculated SHC is as large as 1.25 × 103 (Ω cm)-1 at the Fermi level (EF ). By tuning the chemical potential from EF - 0.3 to EF + 0.3 eV, it varies rapidly and monotonically from -1.2 × 103 to 3.1 . In addition, we also find that the Te vacancy in the patterned monolayer can induce excellent HER activity. Our results not only offer a new idea to search 2D materials with large SHC, i.e., by introducing inversion-symmetry breaking vacancies in large SOC systems, but also provide a feasible system with tunable SHC (by applying gate voltage) and excellent HER activity.

Vortex gap solitons in spin–orbit-coupled Bose–Einstein condensates with competing nonlinearities

The formation and dynamics of full vortex gap solitons (FVGSs) are investigated in two-component Bose–Einstein condensates with spin–orbit coupling (SOC), Zeeman splitting (ZS), and competing cubic and quintic nonlinear terms, while the usual kinetic energy is neglected, assuming that it is much smaller than the SOC and ZS terms. Unlike previous SOC system with the cubic-only attractive nonlinearity, in which solely semi-vortices may be stable, with the vorticity carried by a single component, the present system supports stable FVGS states, with the vorticity present in both components (such states are called here “full vortex solitons”, to stress the difference from the half-vortices). They populate the bandgap in the system’s linear spectrum. In the case of the cubic self-attraction and quintic repulsion, stable FVGSs with a positive effective mass exist near the top of the bandgap. On the contrary, the system with cubic self-repulsion and quintic attraction produces stable FVGSs with a negative mass near the bottom of the bandgap. Mobility and collisions of FVGSs with different topological charges are investigated too.

Global dynamical analysis of an electronic spin–orbit coupling system

By means of a canonical generalized momentum and a canonical conjugate spin variable, a complete canonical Hamiltonian formalism is designed to describe a coulomb field with electronic spin–orbit coupling in a semi-classical and non-relativistic way. After this operation, unlike the existing Lagrange formulation, the concepts of hidden momentum, hidden angular momentum and spin kinetic energy are not used in the canonical formalism. Besides, it is easy to find that there are four first integrals involving the conserved total energy and the conserved total angular momentum vector in an 8-dimensional phase space of the system. In this sense, the global dynamics is typically integrable, regular and non-chaotic, and each orbit in the phase space is a quasi-periodic 4-dimensional Kolmogorov-Arnold-Moser(KAM) torus.

Particle and spin transports of spin-orbit-coupled Fermi gases through a quantum point contact

The particle and spin transport through a quantum point contact between two Fermi gases with Raman-induced spin-orbit coupling are investigated. We show that the particle and spin conductances both demonstrate the structure of plateau due to the mesoscopic scale of the quantum point contact. Compared with the normal Fermi gases the particle conductance can be significantly enhanced by the spin-orbit coupling effect. Furthermore, the conversion of the particle and spin currents can take place in the spin-orbit coupled system, and we find that it is controlled by the parameter of two-photon detuning. When the parameter of two-photon detuning vanishes the particle and spin currents decouple.

Electron vortices in spin-orbit coupling system

There exists an electron vortex solution with orbital angular momentum quantum in a non-spin-orbit coupling system which has nonconservative orbital angular momentum. We discuss the system with spin-orbit coupling and nonconservative orbital angular momentum, and we can find that the electrons with the total angular momentum numbers also have vortex beam solutions. And the vortex beam is expressed as an entangled wave function of the spin wave function and the vortex wave function. Taking the electrons in the central force field for example, in this paper constructed is a spinor vortex structure which is caused by the propagation of electrons carrying a fixed quantum number of total angular momentum along the <i>z</i>-axis. The spinor vortex structure is under the condition that the orbital angular momentum caused by spin-orbit coupling is non-conserved but the total angular momentum is conserved. The corresponding electron vortex beams in spin-vortex entanglement are solved by perturbation method, and the Foldy-Wouthuysen transformation is utilized to show that the vortex solution of the four-component spinor does exist in the case of relativity, when the electron with a fixed total angular momentum quantum number propagates along the <i>z</i>-axis in the central force field. The spinor provides theoretical support for the existence of the vortex structure for the system where the orbital angular momentum is not conserved but the total angular momentum is conserved due to spin-orbit coupling.

Generalized Rashba Coupling Approximation to a Resonant Spin Hall Effect of the Spin–Orbit Coupling System in a Magnetic FieldSupported by the National Natural Science Foundation of China (Grant Nos. 12075040, 11875231, 11974064, and 12047564), and the Chongqing Research Program of Basic Research and Frontier Technology (Grant No. cstc2020jcyj-msxmX0890).

We introduce a generalized Rashba coupling approximation to analytically solve confined two-dimensional electron systems with both the Rashba and Dresselhaus spin–orbit couplings in an external magnetic field. A solvable Hamiltonian is obtained by performing a simple change of basis, which has the same form as that with only Rashba coupling. Each Landau state becomes a new displaced-Fock state instead of the original Harmonic oscillator Fock state. Analytical energies are consistent with the numerical ones in a wide range of coupling strength even for a strong Zeeman splitting, exhibiting the validity of the analytical approximation. By using the eigenstates, spin polarization correctly displays a jump at the energy-level crossing point, where the corresponding spin conductance exhibits a pronounced resonant peak. As the component of the Dresselhaus coupling increases, the resonant point shifts to a smaller value of the magnetic field. In contrast to pure Rashba couplings, we find that the Dresselhaus coupling and Zeeman splittings tend to suppress the resonant spin Hall effect. Our method provides an easy-to-implement analytical treatment to two-dimensional electron gas systems with both types of spin–orbit couplings by applying a magnetic field.

Edelstein effect in pseudospin Dirac systems

Edelstein effect of spin polarization means an electric field could induce a magnetization in nonmagnetic materials. This effect can be well understood by a Zeeman-like term due to an effective magnetic field in spin-orbit coupling systems. We propose a mechanism of current-induced spin polarization, where the real spin is not directly related to the effective magnetic field induced by an electric field. This effective field only results in the polarization of a general spin. Then the real spin polarization is generated through the coupling between the real spin and the general spin in pseudospin Dirac systems with low symmetry. The values of our spin polarization and the conventional one may have the same order of magnitude though ours involves two processes, which has been further verified by the numerical calculation in the $1{T}^{\ensuremath{'}}\text{\ensuremath{-}}{\mathrm{WTe}}_{2}$ distorted monolayer. It will be helpful for the deeper understanding of the current-induced spin polarization in pseudospin Dirac materials.

Ground energy level transition for two-body interacting Fermionic system with spin-orbit coupling and Zeeman interaction

Experimental realization of artificial gauge field has made it possible to simulate important models with electromagnetic field or spin-orbit interaction in condensed matter physics, which opens a new avenue to engineer novel quantum states and phenomena. The spin-orbit coupled system reveals many significant phenomena in condensed matter physics, such as quantum spin Hall effect, topological insulator and topological superconductor. The combined effect of Zeeman interaction and spin-orbit coupling leads to a nontrivial topological phase. The analytic solution of few-body system provides an in-depth insight into the physical phenomena, which has been studied extensively. Through the analytic study of two-body physics, we show new quantum phenomena for various gauge field parameters. We investigate the two-body interacting fermionic gas with spin-orbit coupling and Zeeman interaction in a ring trap. Through the plane wave expansion method, two-body fermionic system is solved analytically. In the absence of Zeeman interaction, the total momentum of the ground state is zero. With the increase of Zeeman interaction, an energy level crossing occurs between the lowest energy levels for different total momentum spaces and the ground state changes from zero total momentum space to non-zero total momentum space. Considering the Zeeman interaction, the total momentum of the ground state changes from zero to finite value. The single particle analysis shows that the ground energy level transition is induced by Zeeman energy level splitting. The momentum distributions of the ground state are given to provide an intuitive physical picture. This work can be further extended to the exploration of the heteroatom system, lattice system and higher spin system.

Supersolid striped droplets in a Raman spin-orbit-coupled system

We analyze the role played by quantum fluctuations on a Raman Spin-Orbit Coupled system in the stripe phase. We show that beyond mean-field effects stabilize the collapse predicted by mean-field theory and induce the emergence of two phases: a gas and a liquid, which also show spatial periodicity along a privileged direction. We show that the energetically favored phase is determined by the Raman coupling and the spin-dependent scattering lengths. We obtain the ground-state solution of the finite system by solving the extended Gross-Pitaevskii equation and find self-bound, droplet-like solutions that feature internal structure through a striped pattern. We estimate the critical number for binding associated to these droplets and show that their value is experimentally accessible. We report an approximate energy functional in order to ease the evaluation of the Lee-Huang-Yang correction in practical terms.

Complete Multipole Basis Set for Single-Centered Electron Systems

A whole series of expressions for four species of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal) is provided as a complete basis set to describe arbitrary single-centered spinful electron systems. A compact formula to calculate matrix elements of these multipoles is also derived. A visualization method of an electronic state characterized in terms of multipoles is proposed. The complete basis set is useful to narrow down a candidate order parameter of electron systems in phase transition, to describe a property of cross-correlated phenomena, to analyze spectra of x-ray scattering in magnetically ordered state, and so on. We demonstrate a usage of the complete basis set by taking monopole and toroidal dipole orderings, and the mutual relationship among three distinct magnetic dipoles (orbital, spin angular momenta and anisotropic dipole) in a spin-orbit coupled system as prime examples.

Theory of Current-Induced Angular Momentum Transfer Dynamics in Spin-Orbit Coupled Systems.

Motivated by the importance of understanding various competing mechanisms to the current-induced spin-orbit torque on magnetization in complex magnets, we develop a theory of current-induced spin-orbital coupled dynamics in magnetic heterostructures. The theory describes angular momentum transfer between different degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum between different degrees of freedom, achieved in a steady state under an applied external electric field. We then propose a classification scheme for the mechanisms of the current-induced torque in magnetic bilayers. We evaluate the sources of torque using density functional theory, effectively capturing the impact of the electronic structure on these quantities. We apply our formalism to two different magnetic bilayers, Fe/W(110) and Ni/W(110), which are chosen such that the orbital and spin Hall effects in W have opposite sign and the resulting spin- and orbital-mediated torques can compete with each other. We find that while the spin torque arising from the spin Hall effect of W is the dominant mechanism of the current-induced torque in Fe/W(110), the dominant mechanism in Ni/W(110) is the orbital torque originating in the orbital Hall effect of the non-magnetic substrate. Thus the effective spin Hall angles for the total torque are negative and positive in the two systems. Our prediction can be experimentally identified in moderately clean samples, where intrinsic contributions dominate. This clearly demonstrates that our formalism is ideal for studying the angular momentum transfer dynamics in spin-orbit coupled systems as it goes beyond the "spin current picture" by naturally incorporating the spin and orbital degrees of freedom on an equal footing. Our calculations reveal that, in addition to the spin and orbital torque, other contributions such as the interfacial torque and self-induced anomalous torque within the ferromagnet are not negligible in both material systems.

Realization of a Density-Dependent Peierls Phase in a Synthetic, Spin-Orbit Coupled Rydberg System

We experimentally realize a Peierls phase in the hopping amplitude of excitations carried by Rydberg atoms, and observe the resulting characteristic chiral motion in a minimal setup of three sites. Our demonstration relies on the intrinsic spin-orbit coupling of the dipolar exchange interaction combined with time-reversal symmetry breaking by a homogeneous external magnetic field. Remarkably, the phase of the hopping amplitude between two sites strongly depends on the occupancy of the third site, thus leading to a correlated hopping associated to a density-dependent Peierls phase. We experimentally observe this density-dependent hopping and show that the excitations behave as anyonic particles with a non-trivial phase under exchange. Finally, we confirm the dependence of the Peierls phase on the geometrical arrangement of the Rydberg atoms.

Dynamical precession of spin in the two-dimensional spin-orbit coupled systems

We investigate the spin dynamics in the two-dimensional spin-orbit coupled system subject to an in-plane (x-y plane) constant electric field, which is assumed to be turned on at the moment t=0. The equation of spin precession in linear response to the switch-on of the electric field is derived in terms of Heisenberg's equation by the perturbation method up to the first order of the electric field. The dissipative effect, which is responsible for bringing the dynamical response to an asymptotic result, is phenomenologically implemented à la the Landau-Lifshitz-Gilbert equation by introducing damping terms upon the equation of spin dynamics. Mediated by the dissipative effect, the resulting spin dynamics asymptotes to a stationary state, where the spin and the momentum-dependent effective magnetic field are aligned again and have nonzero components in the out-of-plane (z) direction. In the linear response regime, the asymptotic response obtained by the dynamical treatment is in full agreement with the stationary response as calculated in the Kubo formula, which is a time-independent approach treating the applied electric field as completely time-independent. Our method provides a new perspective on the connection between the dynamical and stationary responses.

Newton’s second law in spin–orbit torque

Spin–orbit torque (SOT) refers to the excitation of magnetization dynamics via spin–orbit coupling (SOC) under the application of a charged current. In this work, we introduce a simple and intuitive description of the SOT in terms of spin force. In Rashba SOC system, the damping-like SOT can be expressed as , in analogy to the classical torque–force relation, where Rc is the effective radius characterizing the Rashba splitting in the momentum space. As a consequence, the magnetic energy is transferred to the conduction electrons, which dissipates through Joule heating at a rate of , with je being the applied current. Finally, we propose an experimental verification of our findings via measurement of the anisotropic magnetoresistance effect.

Effect of cubic Dresselhaus interaction on the longitudinal optical conductivity of a spin-orbit coupled system

We investigate the effect of the oft-neglected cubic terms of the Dresselhaus spin-orbit coupling on the longitudinal current response of a two-dimensional electron gas with both Rashba and linear Dresselhaus interactions. For a quantum well grown in the [001] direction, the changes caused by these nonlinear-in-momentum terms on the absorption spectrum become more notable under SU(2) symmetry conditions, when the Rashba and linear Dresselhaus coupling strengths are tuned to be equal. The longitudinal optical response no longer vanishes then and shows a strong dependence on the direction of the externally applied electric field, giving a signature of the relative size of several spin-orbit contributions. This anisotropic response arises from the nonisotropic splitting of the spin states induced by the interplay of Rashba and Dresselhaus couplings. However, the presence of cubic terms introduces characteristic spectral features and can modify the overall shape of the spectra for some values of the relative sizes of the spin-orbit parameters. We compare this behavior to the case of a sample with [110] crystal orientation which, under conditions of spin-preserving symmetry, has a collinear spin-orbit vector field that leads to vanishing conductivity, even in the presence of cubic terms. In addition to the control through the driven frequency or electrical gating, such a directional aspect of the current response suggests new ways of manipulation and supports the use of interband optics as a sensitive probe of spin-orbit mechanisms in semiconductor spintronics.

Effect of Cu2+ substitution in spin-orbit coupled Sr2Ir1−xCuxO4 : Structure, magnetism, and electronic properties

Sr$_2$IrO$_4$ is an extensively studied spin-orbit coupling induced insulator with antiferromagnetic ground state. The delicate balance between competing energy scales plays crucial role for its low temperature phase, and the route of chemical substitution has often been used to tune these different energy scales. Here, we report an evolution of structural, magnetic and electronic properties in doped Sr$_2$Ir$_{1-x}$Cu$_x$O$_4$ ($x$ $\leq$ 0.2). The substitution of Cu$^{2+}$ (3$d^9$) for Ir$^{4+}$ (5$d^5$) acts for electron doping, though it tunes the related parameters such as, spin-orbit coupling, electron correlation and Ir charge state. Moreover, both Ir$^{4+}$ and Cu$^{2+}$ has single unpaired spin though it occupies different $d$-orbitals. With Cu substitution, system retains its original structural symmetry but the structural parameters show systematic changes. X-ray photoemission spectroscopy measurements show Ir$^{4+}$ equivalently converts to Ir$^{5+}$ and a significant enhancement in the density of states has been observed at the Fermi level due to the contribution from the Cu 3$d$ orbitals, which supports the observed decrease in the resistivity with Cu substitution. While the long-range magnetic ordering is much weakened and the highest doped sample shows almost paramagnetic-like behavior the overall system remains insulator. Analysis of resistivity data shows mode of charge conduction in whole series follows 2-dimensional variable-range-hopping model but the range of validity varies with temperature. Whole series of samples exhibit negative magnetoresistance at low temperature which is considered to be a signature of weak localization effect in spin-orbit coupled system, and its evolution with Cu appears to follow the variation of resistivity with $x$.

Optical Selection Rules in Spin–Orbit Coupled Systems on Honeycomb Lattice

We investigate the optical properties in a minimal model of spin-orbit coupled systems on honeycomb lattice at half-filling. The absorption of the circularly polarized light in the charge and antiferromagnetic ordered states is particularly studied, and the spin-valley selective excitation depending on the handedness of the light is found. It is shown that the optical selection rules plainly differ with the type of broken symmetries and topological properties of the electronic states. Consequently, the abrupt change of spin-valley selectivity in the optical absorption occurs at the topological transition caused by changing the magnitude of the order parameters.