Linear Spin-orbit Research Articles

Harmonic spin–orbit angular momentum cascade in nonlinear optical crystals

Optical angular momentum-based photonic technologies demonstrate the key role of the optical spin–orbit interaction that usually refers to linear optical processes in spatially engineered optical materials1. Re-examining the basics of nonlinear optics of homogeneous crystals under circularly polarized light2,3, we report experiments on the enrichment of the spin–orbit angular momentum spectrum of paraxial light. The demonstration is made within the framework of second-harmonic generation using a crystal with three-fold rotational symmetry. Four spin–orbit optical states for the second harmonic field are predicted from a single fundamental state owing to the interplay between linear spin–orbit coupling and nonlinear wave mixing; three of these states are experimentally verified. Besides representing a spin-controlled nonlinear route to orbital angular multiplexing4, modal vortex light sources5,6, high-dimensional parametric processes7 and multi-state optical magnetization8, our findings suggest that the fundamentals of nonlinear optics are worth revisiting through the prism of the spin–orbit interaction of light.

Inelastic light scattering by intrasubband spin-density excitations in GaAs-AlGaAs quantum wells with balanced Bychkov-Rashba and Dresselhaus spin-orbit interaction: Quantitative determination of the spin-orbit field

Inelastic light scattering experiments on low-energy intrasubband spin-density excitations (SDEs) are performed in (001)-grown modulation-doped GaAs-AlGaAs single quantum wells in in-plane external magnetic fields. The investigated samples possess balanced linear Bychkov-Rashba (α) and Dresselhaus (β) spin-orbit strengths in two different configurations, α=β and α=−β. Both configurations lead to an extreme anisotropy of the spin splitting of the conduction band, where the in-plane directions of maximum spin splitting for both configurations are perpendicular to each other. The spin splitting asymmetry can be directly detected via the SDE by breaking of the time-reversal symmetry due to transfer of a momentum q in the quantum-well plane. In addition, the application of an in-plane magnetic field Bext⊥q allows us to modulate the effective magnetic field. Via a numerical line-shape analysis of the experimental SDE spectra, we determine the relevant parameters of the samples. We find that the linear spin-orbit strength |α|=β is comparable for both samples, while the electron g factors are markedly different. Furthermore, we experimentally quantify the values of the maximum internal spin-orbit fields, which are as high as Bso∼18T for both samples.

Spin relaxation in wurtzite nanowires

We theoretically investigate the D'yakonov-Perel' spin-relaxation properties in diffusive wurtzite semiconductor nanowires and their impact on the quantum correction to the conductivity. Although the lifetime of the long-lived spin states is limited by the dominant $k$-linear spin-orbit contributions in the bulk, these terms show almost no effect in the finite-size nanowires. Here, the spin lifetime is essentially determined by the small $k$-cubic spin-orbit terms and nearly independent of the wire radius. At the same time, these states possess in general a complex helical structure in real space that is modulated by the spin-precession length induced by the $k$-linear terms. For this reason, the experimentally detected spin relaxation largely depends on the ratio between the nanowire radius and the spin-precession length as well as the type of measurement. In particular, it is shown that while a variation of the radius hardly affects the magnetoconductance correction, which is governed by the long-lived spin states, the change in the spin lifetime observed in optical experiments can be dramatic. We compare our results with recent experimental studies on wurtzite InAs nanowires.

Spin-polarized transport in helical membranes due to spin–orbit coupling

Spin-dependent electron transmission through a helical membrane, taking account of linear spin–orbit interaction, has been investigated by numerically solving the Schrödinger equation in cylindrical coordinates. It is shown that the spin precession is affected by the magnitude of geometric parameters and chirality of the membrane. This effect is also explained analytically using perturbation theory in the weak coupling regime. In the strong coupling regime, the current spin polarization is evident when the number of the open modes in leads is larger than that of the open channels in the membrane. Moreover, we find that the chirality of the helical membrane can determine the orientation of current spin polarization. Therefore, one may get totally opposite spin currents from helical membranes rolled in different directions.

Antiferromagnetic order in a semiconductor quantum well with spin–orbit coupling

An argument is made on the existence of a low-temperature itinerant antiferromagnetic (AF) spin alignment, rather than persistent helical (PH), in the ground state of a two dimensional electron gas in a semiconductor quantum well with linear spin–orbit Rashba–Dresselhaus interaction at equal coupling strengths, α. This result is obtained on account of the opposite-spin single-particle state degeneracy at k=0 that makes the spin instability possible. A theory of the resulting magnetic phase is formulated within the Hartree–Fock approximation of the Coulomb interaction. In the AF state the direction of the fractional polarization is obtained to be aligned along the displacement vector of the single-particle states.

Spin hot spots in single‐electron GaAs‐based quantum dots

AbstractSpin relaxation of a single electron in a weakly coupled double quantum dot is calculated numerically. The phonon‐assisted spin flip is allowed by the presence of the linear and cubic spin–orbit couplings and nuclear spins. The rate is calculated as a function of the interdot coupling, the magnetic field strength and orientation, and the dot bias. In an in‐plane magnetic field, the rate is strongly anisotropic with respect to the magnetic field orientation, due to the anisotropy of the spin–orbit interactions. The nuclear spin influence is negligible. In an out‐of‐plane field, the nuclear spins play a more important role due selection rules imposed on the spin–orbit couplings. Our theory shows a very good agreement with data measured by Srinivasa et al. [Phys. Rev. Lett. 110, 196803 (2013)], allowing us to extract information on the linear spin–orbit interactions strengths in that experiment. We estimate that they correspond to spin–orbit lengths of about 5–15 m.

Nonlinear modes in binary bosonic condensates with pseudo–spin-orbital coupling

We consider a binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudo-spin, i.e., anti-ferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov-de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo) magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.

Spin-orbit coupling in hydrogenated graphene.

First-principles calculations of the spin-orbit coupling in graphene with hydrogen adatoms in dense and dilute limits are presented. The chemisorbed hydrogen induces a giant local enhancement of spin-orbit coupling due to sp(3) hybridization which depends strongly on the local lattice distortion. Guided by the reduced symmetry and the local structure of the induced dipole moments, we use group theory to propose realistic minimal Hamiltonians that reproduce the relevant spin-orbit effects for both single-side semihydrogenated graphene (graphone) and for a single hydrogen adatom in a large supercell. The principal linear spin-orbit band splittings are driven by the breaking of the local pseudospin inversion symmetry and the emergence of spin flips on the same sublattice.

Jahn-Teller, pseudo-Jahn-Teller, and spin-orbit coupling Hamiltonian of a d electron in an octahedral environment

Starting from the model of a single d-electron in an octahedral crystal environment, the Hamiltonian for linear and quadratic Jahn-Teller (JT) coupling and zeroth order as well as linear spin-orbit (SO) coupling in the (2)T(2g) + (2)E(g) electronic multiplet is derived. The SO coupling is described by the microscopic Breit-Pauli operator. The 10 × 10 Hamiltonian matrices are explicitly given for all linear and quadratic electrostatic couplings and all linear SO-induced couplings. It is shown that the (2)T(2g) manifold exhibits, in addition to the well-known electrostatic JT effects, linear JT couplings which are of relativistic origin, that is, they arise from the SO operator. While only the e(g) mode is JT-active in the (2)E(g) state in the nonrelativistic approximation, the t(2g) mode becomes JT-active through the SO operator. Both electrostatic as well as relativistic forces contribute to the (2)T(2g) - (2)E(g) pseudo-JT coupling via the t(2g) mode. The relevance of these analytic results for the static and dynamic JT effects in octahedral complexes containing heavy elements is discussed.

Relativistic Jahn–Teller and pseudo-Jahn–Teller couplings in [formula omitted] systems

The Hamiltonian for linear and quadratic Jahn–Teller (JT) and pseudo-JT (PJT) coupling as well as zeroth-order and linear spin–orbit (SO) coupling in the 2E+2B2 electronic multiplet in D2d systems is derived. The SO coupling is described by the microscopic Breit–Pauli operator. It is shown that the 2E state exhibits a 2E×e JT effect which is of relativistic origin, that is, it arises from the SO operator. The relativistic PJT coupling of the 2E and 2B2 states involves the normal modes of b1,b2 and e symmetry. The 2E×(b1+b2+e) JT Hamiltonian is analytically transformed to a SO-adapted electronic basis in which the zeroth-order SO operator is diagonal. In the special case of 2E-2B degeneracy, the (2E+2B2)×(b1+b2+e) JT+PJT Hamiltonian also is transformed to a SO-adapted basis.

Phase Diagram of the Spin Hall Effect

We obtain analytic formulas for the frequency-dependent spin-Hall conductivity of a two-dimensional electron gas (2DEG) in the presence of impurities, linear spin-orbit Rashba interaction, and external magnetic field perpendicular to the 2DEG. We show how different mechanisms (skew scattering, side jump, and spin precession) can be brought in or out of focus by changing controllable parameters such as frequency, magnetic field, and temperature. We find, in particular, that the dc spin-Hall conductivity vanishes in the absence of a magnetic field, while a magnetic field restores the skew-scattering and side jump contributions proportionally to the ratio of magnetic and Rashba fields.

A study of spin–orbit vibronic-coupling effects in the [formula omitted] state of CCX (X = O, S, Se) and CNY (Y = N, P, As)

The vibronic structure of the A ∼ 3 Π excited electronic state of CCX (X = O, S, Se) and CNY (Y = N, P, As) has been calculated by considering Renner–Teller coupling together with spin–orbit coupling. The vibronic and spin–orbit coupling parameters have been determined by accurate ab initio electronic-structure calculations. The effect of the linear (in the bending mode) spin–orbit coupling mechanism [S. Mishra et al., J. Chem. Phys. 126 (2007) 134312] on the vibronic structure of the 3Π electronic state has been discussed. The vibronic structures of the A ∼ 3 Π electronic state of CCSe and CNAs are found to exhibit pronounced perturbations due to the linear spin–orbit coupling term.

Frequency dependence of induced spin polarization and spin current in quantum wells

Dynamic response of two-dimensional electron systems with spin-orbit interaction is studied theoretically on the basis of quantum kinetic equation, taking into account elastic scattering of electrons. The spin polarization and spin current induced by the applied electric field are calculated for the whole class of electron systems described by $\mathbf{p}$-linear spin-orbit Hamiltonians. The absence of nonequilibrium intrinsic static spin currents is confirmed for these systems with arbitrary (nonparabolic) electron energy spectrum. Relations between the spin polarization, spin current, and electric current are established. The general results are applied to the quantum wells grown in [001] and [110] crystallographic directions, with both Rashba and Dresselhaus types of spin-orbit coupling. It is shown that the existence of the fixed (momentum-independent) precession axes in [001]-grown wells with equal Rashba and Dresselhaus spin velocities or in symmetric [110]-grown wells leads to vanishing spin polarizability at arbitrary frequency $\ensuremath{\omega}$ of the applied electric field. This property is explained by the absence of Dyakonov-Perel-Kachorovskii spin relaxation for the spins polarized along these precession axes. As a result, a considerable frequency dispersion of spin polarization at very low $\ensuremath{\omega}$ in the vicinity of the fixed precession axes is predicted. Possible effects of extrinsic spin-orbit coupling on the obtained results are discussed.

Jahn‐Teller Effect in the 4T1(1) and 4T2(1) States of Tetrahedrally Coordinated Mn2+

AbstractAn interpretation of the 4T1(1) → 6A1 luminescence and 6A1 → 4T2(1) absorption spectrum of Mn2+ in ZnS is presented. These transitions show only two closely spaced zero‐phonon lines which is in contrast to static crystal field theory. It is shown that strong Jahn‐Teller coupling to E‐vibrations, which fully quenches the linear spin–orbit interaction, can account for the observed zero‐phonon line doublets and the size of the splittings. The results apply also to Mn2+ in CdS, GaP, and ZnSe, where similar zero‐phonon doublets have been observed.