Session B33: Focus Session: Spin Dependent Phenomena in Semiconductors: II

Imaging Drift and Diffusion of Accumulation from the Spin Hall Effect

The spontaneous generation of spin polarization near sample edges by the spin Hall effect when electron currents flow in a metal or semiconductor with spin-orbit coupling has attracted recent attention due to the elegant and complex spin-orbit physics as well as the potential for all-electrical spin generation in spintronics devices. Optical techniques in semiconductors allow for spatial resolution of the electrically generated spin accumulation, a feature not present in all-electrical measurements. We use Kerr rotation microscopy to image the spatial and temporal evolution of spin accumulation produced by the extrinsic spin Hall effect in n-GaAs devices. Measurements in a variety of device geometries, including arms transverse to a channel, reveal the unambiguous contribution of longitudinal spin drift in accumulation profiles\footnote{N. P. Stern, D. W. Steuerman, S. Mack, A.C. Gossard, and D. D. Awschalom, \textit{Appl. Phys. Rev. Lett.} \textbf{91}, 062109 (2007)}. We develop one- and two- dimensional drift-diffusion modeling to explain the observed features, providing a more complete understanding of observations of spin accumulation and the spin Hall effect.   

Fractional Charge and Topological Pumping in the Quantum Spin Hall Insulators

We study the physics at the edge of a 2d topological (quantum spin Hall) insulator. This system is known to be topologically non-trivial and a profound manifestation of topologically non-trivial states of matter is the occurrence of fractional charge. In this work, we show that a magnetic domain wall at the edge of the quantum spin Hall insulator carries one half of the unit of electron charge, and we propose an experiment to directly measure this fractional charge on an individual basis. As an additional consequence, a rotating magnetic field can induce a topologically pumped dc electric current, and vice versa. Finally, we discuss an interacting version of this model in which the fractional charge is carried by the fundamental excitations. These physical phenomena can be derived from a generic topological effective action for topological insulators and are directly related to the physics of the second Chern number as will be described in another talk.   

Topological Field Theory of Time-Reversal Invariant Insulators

We show that the fundamental time reversal invariant (TRI) insulator exists in the $4+1$ dimension, where the effective field theory is described by the $4+1$ dimensional Chern-Simons theory, and the topological properties of the electronic structure is described by the second Chern number. These topological properties are the natural generalizations of the time reversal symmetry breaking (TRSB) quantum Hall insulator in $2+1$ dimension. The TRI quantum spin Hall insulator in $2+1 $ dimension and the topological insulator in $3+1$ dimension can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through dimensional reduction. The effective topological field theory, and the $Z_2$ topological number for the TRI insulators $2+1$ and $3+1$ dimensions are naturally obtained from this procedure of dimensional reduction. All physically measurable topological response functions of the TRI insulators are completely described by our effective topological field theory. As a consequence of the effective theory, we predict the existence of fractional charge induced by a magnetic domain wall on the edge of quantum spin Hall insulator, which will be presented as a separate talk.   

A many-body generalization of the $Z_2$ topological invariant for the quantum spin Hall effect

We propose a many-body generalization of the $Z_2$ topological invariant for the quantum spin Hall insulator, which does not rely on single-particle band structures. The invariant is derived as a topological obstruction that distinguishes topologically distinct many-body ground states on a torus. It is also expressed as a Wilson-loop of the SU(2) Berry gauge field, which is quantized due to the time-reversal symmetry.   

Shot noise in the mesoscopic spin Hall effect

The spin Hall effect has recently attracted a lot of attention as a promising all-electrical scheme to generate and manipulate pure spin currents by utilizing spin-orbit (SO) coupling in semiconductor nanostructures. Injection of unpolarized charge current through the longitudinal leads of a four-terminal two-dimensional electron gas (2DEG) with the Rashba SO coupling and/or SO-dependent scattering off extrinsic impurities is responsible not only for the pure spin Hall current in the transverse electrodes, maximized when the ballistic sample size is comparable to the {\it mesoscale} defined by the spin precession length, but also for random time-dependent current fluctuations. We extend the Landauer-B\"uttiker scattering formalism to calculate spin-resolved shot noise in multiterminal nanostructures for arbitrary polarization of the injected current and analyze the shot noise of transverse pure spin Hall current and zero charge current or transverse spin current and non-zero charge Hall current, driven by unpolarized or spin-polarized longitudinal charge current respectively. Since any spin-flip event (instantaneous or due to precession) within the 2DEG acts as an additional source of noise, we demonstrate that these spin and charge shot noises offer a unique tool to differentiate between intrinsic and extrinsic microscopic SO mechanisms behind the spin Hall and charge Hall effects in paramagnetic systems.   

Beyond the spin-Hall effect

Considerable progress has been made in recent years in the electrical manipulation of spins in semiconductors. An extraordinary amount of experimental and theoretical work have culminated in the prediction and discovery of the spin-Hall effect and of spin generation by an electric field. In the past year alone the spin-Hall effect was observed at room temperature and several groups successfully measured spin currents directly. However, we will demonstrate that the symmetry of crystal lattices allows spin currents other than the spin-Hall current and the spin current response to an electric field can be very complex. We will discuss samples in which such currents are expected to arise. The presence of non- spin-Hall currents has important and interesting consequences with regard to the manipulation of spins by electric fields. We will moreover show that spin currents and bulk spin densities in an electric field arise from linearly independent contributions to the density matrix, and that the presence of a nonequilibrium spin density has a profound effect on the spin current [1, 2]. [1] D. Culcer and R. Winkler, arxiv:0708.4009, to appear in Phys. Rev. Lett. [2] D. Culcer and R. Winkler, arxiv:0710.5260, submitted to Phys. Rev. B.   

Universal quantized spin-Hall conductance fluctuation in graphene

We report a theoretical investigation of quantized spin-Hall conductance fluctuation of graphene devices in the diffusive regime. Two graphene models that exhibit quantized spin-Hall effect (QSHE) are analyzed. Model-I is with unitary symmetry under an external magnetic field $B\ne 0$ but with zero spin- orbit interaction, $t_{SO}=0$. Model-II is with symplectic symmetry where $B=0$ but $t_{SO} \ne 0$. Extensive numerical calculations indicate that the two models have exactly the same universal QSHE conductance fluctuation value $0.285 e/4\pi$ regardless of the symmetry. Qualitatively different from the conventional charge and spin universal conductance distributions, in the presence of edge states the spin-Hall conductance shows an one-sided log-normal distribution rather than a Gaussian distribution. Our results strongly suggest that the quantized spin-Hall conductance fluctuation belongs to a new universality class.   

Spin-Hall edge spin polarization in a ballistic 2D electron system

Universal properties of spin-Hall effect in ballistic 2D electron systems are addressed. The net spin polarization across the edge of the conductor is second order, $\sim\lambda^2$, in spin-orbit coupling constant independent of the form of the boundary potential, with the contributions of normal and evanescent modes each being $\sim\sqrt{\lambda}$ but of opposite signs. This general result is confirmed by the analytical solution for a hard-wall boundary, which also yields the detailed distribution of the local spin polarization. The latter shows fast (Friedel) oscillations with the spin-orbit coupling entering via the period of slow beatings only. Long-wavelength contributions of evanescent and normal modes exactly cancel each other in the spectral distribution of the local spin density.   

Network Model for $Z_2$ Quantum Spin-Hall Effects with Disorder

We study the effects of static disorder on the $Z_2$ quantum spin-Hall effect for noninteracting electrons propagating in two dimensions. To this end, a network model realizing the $Z_2$ quantum spin-Hall effect is constructed to account for the effects of static disorder on the propagation of noninteracting electrons subjected to spin-orbit couplings with the time-reversal symmetry. This network model is different from past network models belonging to the symplectic symmetry class in that the propagating modes along the links of the network has a single Kramers doublet. By investigating this network model numerically, it is found that a two-dimensional metallic phase of finite extent is embedded in insulating phases. We also find that the Anderson localization-delocalization transition between the metallic and $Z_2$ insulating phases belong to the conventional symplectic universality class in two dimensions.   

Intrinsic spin-Hall effect in the presence of an in-plane magnetic field

The intrinsic spin-Hall effect (SHE) induced by a driving electric field E$_{x}$ in the presence of an in-plane magnetic field $\vec {B}$ in a 2D semiconductor strip is studied. In the diffusive regime, the spatial distribution of spin densities S$_{i}$ (\textit{i=x, y ,z}) is calculated from a spin diffusion equation derived from nonequilibrium Green's function. For the case of Rashba spin-orbit interaction (SOI), we find that the spin polarization S$_{z}$ normal to the 2D strip remains zero with or without the in-plane magnetic field. For the case of Dresselhaus SOI, where cubic term is included, the symmetry of S$_{z}$ with respect to the in-plane magnetic field depends on the orientation of the $\vec {B}$ field. With $\vec {B}$ along $\hat {x}$, S$_{z}$ exhibits symmetric dependence on B$\hat {x}$. However, with a transverse in-plane magnetic field, along $\hat {y}$, at the edge of the strip exhibits asymmetric dependence on B$\hat {y}$. These results lead to a possible diagnostic tool for the identification of the SOI in the system.   

Phase transition between quantum spin Hall and ordinary insulating phases

We theoretically study the phase transition between the quantum spin Hall (QSH) and insulator phases, which involves a change of the $Z_2$ topological number. We deal with 2D and 3D systems without impurity and interaction. We introduce a parameter $m$ controlling the phase transition, and we study whether the gap closes or not by one-parameter tuning. In general, level repulsion prevents the gap from closing. In fact, the physics of the $Z_2$ topological number is encoded in the problem whether the gap closes by tuning a single parameter. In 2D [1], as well as in the 3D inversion-symmetric systems [2], the gap closes at one point, $m=m_0$, whereas in 3D inversion-asymmetric systems [2], there appears a finite regime for $m$ ($m_1 [Preview Abstract]

A voltage probe of the spin Hall effect

The spin Hall effect does not generally result in a transverse voltage. We predict that in systems with inhomogeneous electron density in the direction perpendicular to main current flow, the spin Hall effect is instead accompanied by a transverse voltage. We find that, unlike the ordinary Hall effect, this voltage is quadratic in the longitudinal electric field for a wide range of parameters accessible experimentally. We also predict spin accumulation in the bulk and sharp peaks of spin Hall induced charge accumulation near the edges. Our results can be readily tested experimentally, and would allow the electrical measurement of the spin Hall effect in non-magnetic systems and without injection of spin-polarized electrons. \newline [1] Yu. V. Pershin and M. Di Ventra , J. Phys.: Cond. Matt. (in press), arXiv:cond-mat/0703310.   

Spin-charge Separated Excitations in a Topological Insulator

We construct a simple, controllable, two dimensional model based on a topological band insulator. It has many attractive properties. The main conclusions are: (1) The quasiparticles exhibit spin-charge separation. (2) It suggests an alternative way to classify $Z_2$ topological insulator without resorting to the sample boundary. (3) The quasiparticle condensation triggers a phase transition from a spin liquid to an insulating easy-plane ferromagnet.