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arXiv:2108.06465 · Physical review. B./Physical review. B

Dissipation-relaxation dynamics of a spin- 12 particle with a Rashba-type spin-orbit coupling in an ohmic heat bath

Tomohiro Hata, Eiji Nakano, Kei Iida, Hiroyuki Tajima, Junichi Takahashi

Published Oct 27, 2021• 4 citations
arXivHTMLPDF

Abstract

Spin-orbit coupling (SOC), which is inherent in a Dirac particle that moves under the influence of electromagnetic fields, manifests itself in a variety of physical systems, including nonrelativistic ones. For instance, it plays an essential role in spintronics developed in the past few decades, particularly by controlling spin current generation and relaxation. In the present work, by using an extended Caldeira-Leggett model, we elucidate how the interplay between spin relaxation and momentum dissipation of an open system of a single spin-$\frac{1}{2}$ particle with Rashba-type SOC is induced by the interactions with a spinless, three-dimensional environment. Staring from the path-integral formulation for the reduced density matrix of the system, we have derived a set of coupled nonlinear equations that consists of a quasiclassical Langevin equation for the momentum with a frictional term and a spin precession equation. The spin precesses around the effective magnetic field generated by both the SOC and the frictional term. It is found from analytical and numerical solutions to these equations that a spin torque effect included in the effective magnetic field causes a spin relaxation and that the spin and momentum orientations after a long time evolution are largely controlled by the Rashba coupling strength. Such a spin relaxation mechanism is qualitatively different from, e.g., the one encountered in semiconductors in which essentially no momentum dissipation occurs due to the Pauli blocking.

DOI
10.1103/physrevb.104.144424

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This dissertation contains a collection of topics on the spin-orbit coupling of cold atoms and semiconductors. We first consider the effects of an optically induced spin-orbit interaction on a system of Bosonic atoms. The spin-orbit term couples the emergent pseudo-spin- 1/2 degree of freedom to momentum. The single particle energy spectrum has a low energy band with two minima at non-zero momentum. At low temperatures, a many-body system will condense into these minima. In the presence of vanishing interactions, the ground state wavefunction is found to be N00N state of "left" and "right"-moving state. An experimentally observable signature of the condensate is predicted that can be observed using time-of-flight imaging. We then predict a bulk manifestation of the spin-Hall effect in an inhomoge- neous spin-orbit-coupled system. The phenomena is predicted in the framework of the spin diffusion equations generalized to include arbitrary Rashba and linear and cubic Dresselhaus terms. This framework shows that a bulk spin-density wave with a wavevector oriented perpendicular to an applied electric field will induce a charge- density wave characterized by a π/2-phase shift and a non-monotonic time-varying amplitude. The optimal values of spin-orbit coupling for observation of the effect are determined. We propose a scheme for using atom interferometry to measure weak time- dependent accelerations. This proposal uses an ensemble of dilute trapped bosons with two pseudo-spin states coupled to a synthetic magnetic field, but with opposite effective charges. The synthetic field acts to couple spin to momentum continuously, which continuously imparts the acceleration on the phase of the internal states. We use time reversal pulses to reduce noise. The sensitivity of such a system is estimated to be S ∼ 10−7 m/s^2/√Hz Finally, we consider the semiclassical dynamics of a trapped spin-orbit coupled system. We find non-linear dynamics parameterized by the value of the initial displacement of the trap, and the anisotropy of the spin-orbit energy spectrum. We show that the dynamics can give a Berry's phase, and propose an experiment to measure this phase. We then propose a generalization of the 4-level scheme that allows for spin-orbit coupling described by a vector potential that is proportional to the angular momentum operator.

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Spin-orbit coupling (SOC), which is inherent in a Dirac particle that moves under the influence of electromagnetic fields, manifests itself in a variety of physical systems, including nonrelativistic ones. For instance, it plays an essential role in spintronics developed in the past few decades, particularly by controlling spin current generation and relaxation. In the present work, by using an extended Caldeira-Leggett model, we elucidate how the interplay between spin relaxation and momentum dissipation of an open system of a single spin-12\frac{1}{2}21​ particle with Rashba-type SOC is induced by the interactions with a spinless, three-dimensional environment. Staring from the path-integral formulation for the reduced density matrix of the system, we have derived a set of coupled nonlinear equations that consists of a quasiclassical Langevin equation for the momentum with a frictional term and a spin precession equation. The spin precesses around the effective magnetic field generated by both the SOC and the frictional term. It is found from analytical and numerical solutions to these equations that a spin torque effect included in the effective magnetic field causes a spin relaxation and that the spin and momentum orientations after a long time evolution are largely controlled by the Rashba coupling strength. Such a spin relaxation mechanism is qualitatively different from, e.g., the one encountered in semiconductors in which essentially no momentum dissipation occurs due to the Pauli blocking.
Brownian motion of a mobile impurity in a bath is affected by spin-orbit coupling (SOC). Here, we discuss a Caldeira-Leggett-type model that can be used to propose and interpret quantum simulators of this problem in cold Bose gases. First, we derive a master equation that describes the model and explore it in a one-dimensional (1D) setting. To validate the standard assumptions needed for our derivation, we analyze available experimental data without SOC; as a byproduct, this analysis suggests that the quench dynamics of the impurity is beyond the 1D Bose-polaron approach at temperatures currently accessible in a cold-atom laboratory---motion of the impurity is mainly driven by dissipation. For systems with SOC, we demonstrate that 1D spin-orbit coupling can be gauged out even in the presence of dissipation---the information about SOC is incorporated in the initial conditions. Observables sensitive to this information (such as spin densities) can be used to study formation of steady spin polarization domains during quench dynamics.
A higher-order relativistic k.p model is developed to describe plasmon excitations in two-dimensional (2D) electronic systems with spin-orbit coupling (SOC) and magnetic-exchange interactions. Derived entirely from ab initio band structure, the model allows for a non-Rashba spin-momentum locking and enables a direct coupling of the exchange field to the real spin of electrons. Using the BiTeI trilayer (hexagonal C3v symmetry) and the Si-terminated surface state of TbRh2Si2 (cubic C4v symmetry) as prototypes, we show that the exchange field induces strong, symmetry-dependent modifications of the band structure and plasmon dispersion. In BiTeI, it breaks the sixfold symmetry and leads to anisotropic, nonreciprocal plasmon modes, while in TbRh2Si2 it suppresses the characteristic triple spin winding and alters the plasmon damping. The results reveal that the interplay between SOC and exchange magnetism enables magnetic control of collective charge excitations in 2D spin-orbit systems beyond the Rashba paradigm.
We exploit the variational and Hamiltonian structures of quantum hydrodynamics with spin to unfold the correlation and torque mechanisms accompanying spin-orbit coupling (SOC) in electronic motion. Using Hamilton's action principle for the Pauli equation, we isolate SOC-induced quantum forces that act on the orbital Madelung--Bohm trajectories and complement the usual force terms known to appear in quantum hydrodynamics with spin. While the latter spin-hydrodynamic forces relate to the quantum geometric tensor (QGT), SOC-induced orbital forces originate from a particular current operator that contributes prominently to the spin current. This distinction between force terms reveals two fundamentally different mechanisms generating quantum spin-orbit correlations. Leveraging the Hamiltonian structure of the hydrodynamic system, we also elucidate spin transport features such as the correlation-induced quantum torques and the current shift in the spin Hall effect. This Hall shift leads to a new definition of the transport spin current thereby addressing an open question in spintronics. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation.
We studied the influence of spin-orbit interaction (SOI) in interacting low dimensional quantum systems at zero temperature within the framework of the functional renormalization group (fRG). Among the several types of spin-orbit interaction the so-called Rashba spin-orbit interaction is especially intriguing for future spintronic applications as it may be tuned via external electric fields. We investigated its effect on the low energy physics of an interacting quantum wire in an applied Zeeman field which is modeled as a generalization of the extended Hubbard model. To this end we performed a renormalization group study of the two particle interaction, including the SOI and the Zeeman field exactly on the single particle level. Considering the resulting two band model, we formulated the RG equations for the two particle vertex keeping the full band structure as well as the non trivial momentum dependence of the low energy two particle scattering processes. In order to solve these equations numerically we defined criteria that allowed us to classify whether a given set of initial conditions flows towards the strongly coupled regime. We found regions in the models parameter space where a weak coupling method as the fRG is applicable and it is possible to calculate additional quantities of interest. Furthermore we analyzed the effect of the Rashba SOI on the properties of an interacting multi level quantum dot coupled to two semi innite leads. Of special interest was the interplay with a Zeeman field and its orientation with respect to the SOI term. We found a renormalization of the spin-orbit energy which is an experimental quantity used to asses SOI effects in transport measurements, as well as renormalized effective g factors used to describe the Zeeman field dependence. In particular in asymmetrically coupled systems the large parameter space allows for rich physics which we studied by means of the linear conductance obtained via the generalized Landauer-Büttiker formalism from the renormalized self energy. The multi level quantum dot exhibits Kondo physics at finite Zeeman fields which can be suppressed if the field and the SOI term are not aligned in parallel. Due to broken particle hole symmetry we observed a gate voltage dependence for finite field conductance plateaus. The computed dependencies have been observed in experiments and thus highlight the flexibility of the considered model. Finally we computed the linear conductance for a multi level quantum dot model that has been proposed to describe SOI and impurity scattering effects in carbon nanotubes (CNT). The SOI was seen to affect the already complicated level structure of the CNT and thus profoundly changes the linear conductance, especially the location of conductance resonances with respect to an applied magnetic field and gate voltage. Again asymmetries influence the conductance and the effects are explained by means of the effective tunnel couplings to the eigenstates of the CNT. We finished with an improved analysis of a reduced version of this model that includes two particle vertex renormalization. The computed linear conductance is in excellent agreement with experimental data.
We present an extension of the dynamical cluster approximation (DCA) that incorporates Rashba spin-orbit coupling (SOC) to investigate the interplay between disorder, spin-orbit interaction, and nonlocal spatial correlations in disordered two-dimensional systems. By analyzing the average density of states, momentum-resolved self-energy, and return probability, we demonstrate how Rashba SOC and nonlocal correlations jointly modify single-particle properties and spin-dependent interference. The method captures key features of the symplectic universality class, including SOC-induced delocalization signatures at finite times. We benchmark the DCA results against those obtained from the numerically exact kernel polynomial method, finding good agreement. This validates the computationally efficient, mean-field-based DCA framework as a robust tool for exploring disorder, spin-orbit coupling, and nonlocal correlation effects in low-dimensional systems, and paves the way for simulating multiorbital and strongly correlated systems that were previously inaccessible due to computational limitations.
This dissertation contains a collection of topics on the spin-orbit coupling of cold atoms and semiconductors. We first consider the effects of an optically induced spin-orbit interaction on a system of Bosonic atoms. The spin-orbit term couples the emergent pseudo-spin- 1/2 degree of freedom to momentum. The single particle energy spectrum has a low energy band with two minima at non-zero momentum. At low temperatures, a many-body system will condense into these minima. In the presence of vanishing interactions, the ground state wavefunction is found to be N00N state of "left" and "right"-moving state. An experimentally observable signature of the condensate is predicted that can be observed using time-of-flight imaging. We then predict a bulk manifestation of the spin-Hall effect in an inhomoge- neous spin-orbit-coupled system. The phenomena is predicted in the framework of the spin diffusion equations generalized to include arbitrary Rashba and linear and cubic Dresselhaus terms. This framework shows that a bulk spin-density wave with a wavevector oriented perpendicular to an applied electric field will induce a charge- density wave characterized by a π/2-phase shift and a non-monotonic time-varying amplitude. The optimal values of spin-orbit coupling for observation of the effect are determined. We propose a scheme for using atom interferometry to measure weak time- dependent accelerations. This proposal uses an ensemble of dilute trapped bosons with two pseudo-spin states coupled to a synthetic magnetic field, but with opposite effective charges. The synthetic field acts to couple spin to momentum continuously, which continuously imparts the acceleration on the phase of the internal states. We use time reversal pulses to reduce noise. The sensitivity of such a system is estimated to be S ∼ 10−7 m/s^2/√Hz Finally, we consider the semiclassical dynamics of a trapped spin-orbit coupled system. We find non-linear dynamics parameterized by the value of the initial displacement of the trap, and the anisotropy of the spin-orbit energy spectrum. We show that the dynamics can give a Berry's phase, and propose an experiment to measure this phase. We then propose a generalization of the 4-level scheme that allows for spin-orbit coupling described by a vector potential that is proportional to the angular momentum operator.