Session A12: Focus Session: Spin Hall Effect

Generating Spin Currents in Semiconductors with the Spin Hall Effect

There is a growing interest in exploiting electron spins in semiconductor nanostructures for the manipulation and storage of information in emergent technologies based upon spintronics and quantum logic. Recently we have explored two mechanisms for electrically generating spin polarization in non-magnetic materials: current-induced spin polarization and the spin Hall effect. Current-induced spin polarization results in spins being polarized by the internal magnetic field arising from spin-orbit coupling, and the spin Hall effect refers to the generation of a spin current transverse to a charge current in the absence of an applied magnetic field. Recent measurements in ZnSe reveal that both of these effects are robust to room temperature\footnote{N. P. Stern, S. Ghosh, G. Xiang, M. Zhu, N. Samarth, and D. D. Awschalom, \textit{Phys. Rev. Lett.} \textbf{97}, 126603 (2006)}. Although spin current is difficult to measure directly, the spin Hall effect creates spin accumulation at the edges of a channel which has been measured in bulk epilayers of n-doped semiconductors and in two- dimensional hole and electron systems. More recently, we investigate spin currents generated by the spin Hall effect in GaAs structures that distinguish edge effects from spin transport\footnote{V. Sih, W. H. Lau, R. C. Myers, V. R. Horowitz, A. C. Gossard and D. D. Awschalom, \textit{Phys. Rev. Lett.} \textbf{97}, 096605 (2006).}. We fabricate mesas with transverse channels to allow spins to drift into regions in which there is minimal electric current. Using optical techniques, we observe the electrical generation of a transverse spin current, which can drive spin polarization nearly 40 microns into a transverse channel. Using a model that incorporates the effects of spin drift, we determine the transverse spin drift velocity from the magnetic field dependence of the spin polarization. These results reveal opportunities for an electrical spin source in non-magnetic materials.   

Persistent Spin Helix

Spin-orbit coupled systems generally break the spin rotation symmetry. However, for a model with equal Rashba and Dresselhauss coupling constant (the ReD model), and for the {\$}[110]{\$} Dresselhauss model, a new type of SU(2) spin rotation symmetry is discovered. This symmetry is robust against spin-independent disorder and interactions, and is generated by operators whose wavevector depends on the coupling strength. It renders the spin lifetime infinite at this wavevector, giving rise to a Persistent Spin Helix (PSH). We obtain the spin fluctuation dynamics at, and away, from the symmetry point, and suggest experiments to observe the PSH.   

Topological Insulators in Three Dimensions

We study three dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where the QSH effect is distinguished by a single $Z_2$ topological invariant, in three dimensions there are 4 invariants distinguishing 16 ``topological insulator'' phases. There are two general classes: weak (WTI) and strong (STI) topological insulators. The WTI states are equivalent to layered 2D QSH states, but are fragile because disorder continuously connects them to band insulators. The STI states are robust and have surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, giving rise to a novel ``topological metal'' surface phase. We show that the $Z_2$ invariants can be easily determined for systems with inversion symmetry. This allows us to predict specific materials are STI's, including semiconducting alloy Bi$_{1-x}$ Sb$_x$ as well as $\alpha-$Sn and HgTe under uniaxial strain.\newline \newline 1. Liang Fu, C.L. Kane, E.J. Mele, cond-mat/0607699. \newline 2. Liang Fu, C.L. Kane, cond- mat/0611341.   

Skew-scattering contribution in Rashba-type 2D systems with short-range scalar impurity potential.

There is a renewed interest in the anomalous Hall effect (AHE) motivated by the fabrication of materials that are both ferromagnetic and semiconducting, diluted magnetic semiconductors (DMS). Experimental and theoretical studies have shown that the skew-scattering contribution can have a dominant role in magnetotransport, especially in the low-impurity-concentration limit. The Hamiltonian describing Rashba-type systems is sufficiently simple to allow analytical solutions, yet it has the features of a typical DMS: (1) spin-orbit coupling, (2) more than one band with momentum-dependent Berry's curvature, and (3) it allows for inter- and intra-band scattering on impurities. We estimate skew-scattering contribution to two-dimensional Rashba-coupled systems in the leading order of expansion in disorder strength. We consider short-range disorder potentials and derive the general formula for arbitrary spin-orbit coupling through a high-order Born approximation.   

Low field phase diagram of spin-Hall effect in the mesoscopic regime

When a mesoscopic two dimensional four-terminal Hall cross-bar with Rashba and/or Dresselhaus spin-orbit interaction (SOI) is subjected to a perpendicular uniform magnetic field $B$, both integer quantum Hall effect (IQHE) and mesoscopic spin-Hall effect (MSHE) may exist when disorder strength $W$ in the sample is weak. We have calculated the low field `phase diagram' of MSHE in the ($B, W)$ plane for disordered samples in the IQHE regime. For weak disorder, MSHE conductance $G_{sH}$ and its fluctuations \textit{rmsG}$_{sH}$ vanish identically on even numbered IQHE plateaus, they have finite values on those odd numbered plateaus induced by SOI, and they have values $G_{sH}=1/2$ and \textit{rmsG}$_{sH}=0$ on those odd numbered plateaus induced by Zeeman energy. For moderate disorder, the system crosses over into a regime where both $G_{sH}$ and \textit{rmsG}$_{sH}$ are finite. A larger disorder drives the system into a chaotic regime where $G_{sH}=0$ while \textit{rmsG}$_{sH}=0$ is finite. Finally at large disorder both $G_{sH}$ and \textit{rmsG}$_{sH}$ vanish. We present the physics behind this `phase diagram'.   

Quantum spin Hall phase and surface spin current in Bi and Sb

In the quantum spin Hall (QSH) phase, the bulk is gapped while edge states are gapless and carry spin currents. Experimental studies for the QSH phase are called for. To search for candidates of the 2D QSH phase, we relate the spin Hall conductivity in insulators with magnetic response of the orbital magnetization to the Zeeman field. In this respect, bismuth is promising since it is a strong diamagnet enhanced by spin-orbit coupling. For a 2D (111)-bilayer bismuth, we calculate the $Z_2$ topological number, the band structure for the strip geometry, the spin Chern number, and the parity at the time-reversal symmetric wavenumbers. We predict that the (111)-bilayer bismuth will be a QSH phase [1]. On the other hand, it was proposed recently that 3D bismuth is a simple insulator, and not the QSH phase, by parity consideration [2]. Transition from the 2D QSH topological phase to the 3D simple insulator phase is described by gradually increasing inter-bilayer hopping, thereby band-touching occurs at high- symmetry points and parities of the wavefunctions are exchanged. Similar discussion applies for Sb, where 2D bilayer is a simple insulator and 3D bulk is the QSH phase. Finally, we compare the theory with the ARPES data showing surface spin-splitting (spin current) for various surfaces of Bi and Sb. [1] S. Murakami, cond-mat/0607001 (to appear in Phys. Rev. Lett.). [2] L. Fu, C. L. Kane, cond-mat/0611341.   

Mesoscopic Spin Hall Effect

We investigate the spin Hall effect in ballistic chaotic quantum dots with spin-orbit coupling. We show that a longitudinal charge current can generate a pure transverse spin current. While this transverse spin current is generically nonzero for a fixed sample, we show that when the spin-orbit coupling time is short compared to the mean dwell time inside the dot, it fluctuates universally from sample to sample or upon variation of the chemical potential with a vanishing average. For a fixed sample configuration, the transverse spin current has a finite typical value $\simeq e^2 V/h$, proportional to the longitudinal bias $V$ on the sample, and corresponding to about one excess open channel for one of the two spin species. We discuss spin current correlations and noise.   

Spin torque contribution to the frequency dependent spin Hall conductivity in spin-orbit coupled systems

The spin Hall effect in spin-orbit coupled systems has lately attracted great attention. Since the electron spin is not a conserved quantity in spin-orbit coupled systems, the conventional form for the spin current operator turn out to be ill-defined. A fundamental issue is then a proper definition of spin current in such systems. Recently J. Shi et. al. [1] introduced an unambiguous and proper definition of spin current which adds to the conventional part, a spin source term (spin torque) associated to the spin processional motion. In this work, using the linear response Kubo formalism, and employing the new definition for the spin current operator, we study the frequency dependent spin Hall conductivity for a two dimensional electron gas in the presence of Rashba and Dresselhaus spin-orbit coupling. We show that the optical spectrum of the charge and spin conductivity changes dramatically when the proper definition is used, as new and strong resonances appear. It is shown that the spin torque contribution to the spin Hall conductivity clearly dominates over the conventional part. These results may encourage experimentalists to measure the spin Hall current and/or spin accumulation in the frequency domain in such systems, as to establish the vality of the new definition of the spin current operator. [1] J. Shi, P Zhang, D. Xiao and Q Niu, Phys Rev. Lett \textbf{96}, 076604 (2006)   

Search for the Persistent Spin Helix in a 2-Dimensional Electron Gas

The persistent spin helix is an infinitely long-lived helical spin density wave that is predicted to occur in 2-dimensional electron systems with equal-strength Rashba and Dresselhaus spin-orbit coupling [Bernevig \textit{et al}., cond-mat/0606196]. The infinite lifetime of the helix would result from the combined effects of diffusion and precession in the spin-orbit effective field. These effects would also greatly enhance the lifetime of spin excitations at the helix wave vector in systems where Rashba $\ne $ Dresselhaus. We use the transient spin grating technique to search for this effect in GaAs quantum wells. In these experiments, two non-collinear, orthogonally polarized pump pulses from a Ti:Sapphire oscillator generate a holographic spin grating in the interference region on the sample. The subsequent decay of the spin grating is monitored by diffraction of a time-delayed probe pulse. The wave vector of the spin grating can be tuned by varying the angle between the interfering pump beams, making this technique ideally suited for observing the persistent spin helix.   

Berry Curvature and the $Z_2$ Topological Invariants of Spin-Orbit-Coupled Bloch bands.

The (``anomalous'') integer quantum Hall effect can occur in non-interacting models of band insulators with broken time-reversal- ($T$-)symmetry where the sum of Chern invariants of occupied bands of Bloch states is non-zero. These topological invariants can be computed from the zeroes of certain functions in the Brillouin zone (BZ), but have a simpler formulation as BZ-integrals of Berry curvature. Recently, Kane and Mele found that $T$-invariant 2D systems with strong spin-orbit coupling possess a ``$Z_2$'' ($+$ or $-$) analog of the Chern invariant, which they formulated in terms of zero-counting arguments (3D generalizations have also been found). I give an alternate formulation in terms of Berry-curvature integrals, in the case that spatial-inversion- ($I$-)symmetry is broken, but $T$-symmetry is not. In 2D, such bands generically form a genus-5 2-manifold, with antipodal points paired by Kramers degeneracy: the $Z_2$ invariant is obtained by integration over a Kramers-distinct half-manifold; the 3D case is similar. I also discuss the case of doubly-degenerate bands with unbroken $I$-symmetry: despite recent suggestions, it does not appear that the $Z_2$ invariant of such systems can be obtained purely from knowledge of the parity quantum numbers at $T$-invariant points in the BZ.   

Quantum Spin Hall Effect in HgTe in a Magnetic Field

Recently, the quantum spin Hall effect has been proposed in HgTe quantum wells. It has been shown that this system exhibits the quantum spin Hall effect and the Hamiltonian is analogous to two copies of the quantum anomalous Hall effect. Here we examine the features of this system in a strong magnetic field. We use an analytic transfer matrix formalism to study the system on a lattice in a strip geometry in the presence of a strong perpendicular magnetic field. We characterize the bulk band structure and edge states for various applied field strengths and discuss possible experimental signatures of the quantum spin Hall effect. We also discuss possible discrepancies between the continuum and lattice picture.   

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

We show that the Quantum Spin Hall Effect, a state of matter with topological properties distinct from conventional insulators, can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the electronic state changes from a normal to an ``inverted'' type at a critical thickness d$_c$. We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss the methods for experimental detection of the QSH effect.