Modern trends in semiconductor spintronics

Dedication: Dedicated to Leonid V. Keldysh on the occasion of his 75th birthday Fundamentals of semiconductor spintronics and some recent developments in this ﬁeld are reviewed. Special attention is paid to electrical manipulation of electron spins, spin injection, and producing nonequilibrium spin populations via spin orbit interaction.


Introduction
Spin is the only internal degree of freedom of electron.Traditionally, solid state physics related to electron charge and electron spin were only loosely related.With very few exceptions, semiconductor electronics and optoelectronics are based on employing electron charge only.Currently, miniaturization of semiconductor devices approaches the quantum limits established by mesoscopic physics.At this spatial scale, in strongly confined systems, the basic laws of nature involve electron spin actively in most of electron phenomena.In physical processes in semiconductor micro-and nanostructures investigated during the last decade, electron charge and electron spin contribute at equal footing.It is widely believed that the new physics based on the involvement of electron spin into the phenomena that were traditionally regarded as "charge related," will transform semiconductor electronics into spintronics.Combining electrical, magnetic, and optical phenomena for designing devices with new functionalities is the ambitious goal of semiconductor spintronics.
Many of the spectacular phenomena that are the subject of current research on low-dimensional semiconductor microstructures like quantum wells base on physical mechanisms that bear similarity to the mechanisms discovered and investigated at the epoch when our world was still perceived as three-dimensional (3D), and artificial low-dimensional structures were considered as exotic.It was a curiosity-driven research without any direct connection to applications.A considerable part of it was performed in the former Soviet Union.The underlying physics was widely discussed at public events like conferences and seminars and during private discussions inside the community of which Leonid Keldysh has always been a distinguished member since publishing his very first paper in which the Franz-Keldysh effect was discovered.Therefore, I feel it is appropriate to review in this Volume the essential basics and some of more recent developments in spintrinics, especially because its theoretical apparatus heavily relies on Keldysh technique for nonequilibrium responses (Keldysh, 1965).It is true for analytical theories, calculational procedures, and also for simulations for finite systems of different geometries.

Electrical spin operation
The final goal of spintronics is operating electron spins at a nanometer scale.Time dependent magnetic fields B (t) usually have large spatial scale while the spatial scale for time dependent electric fields E (t) applied through gates may be much less.Such fields couple to electron spins through spin-orbit interaction.For nonrelativistic electrons in vacuum it is described by the Thomas term where v is the electron velocity and V ( r ) is the scalar potential.Because of the Dirac gap 2m 0 c 2 in the denominator, in vacuum this interaction is very weak for a moderate and smooth external potential.As a result, electrical operation of electron spins in vacuum is less effective than the magnetic operation through the Zeeman interaction by a factor v/c 1 .The situation changes drastically for an electron in a crystal because of a strong potential V ( r ) and a large electron velocity v near nuclei.In this case, the Dirac gap is substituted by the forbidden gap E G ∼ 1 eV, and the enhancement factor is typically as large as m 0 c 2 /E G ∼ 10 6 .Of course, there is an additional factor ∆ SO /E G , ∆ SO being the spin orbit splitting of the bands.It is small for light elements, but of the order of unity for the compounds from the middle and lower parts of the periodic table.It is this enhancement factor that can make the electrical operation of electron spins in crystals highly efficient (Rashba, 1960).Electric dipole spin resonance (EDSR) driven by a field E (t) of the frequency ω , coinciding with spin flip frequency ω s , has been observed in InSb by McCombe et al. (1967), and in a number of following studies.
An additional factor influencing spin-orbit coupling is the symmetry of the system.The effective Hamiltonian of spin orbit coupling depends on the momentum k , H SO = H SO ( k ) .This Hamiltonian is an invariant of the appropriate symmetry group (Bir andPikus 1974, Winkler 2003).The lower is the symmetry, the lower powers of k appear in H SO = H SO ( k ) , and the stronger spin orbit coupling is.For electrons in the zinc blende modification of A 3 B 5 crystals, the expansion of H SO = H SO ( k ) begins with k 3 terms (Dressel-haus 1955).However, for the wurtzite modification of the same crystals or for electrons confined in an asymmetric quantum well, the Hamiltonian is where ẑ is a unit vector along the hexagonal axis or the confinement direction.H α is linear in k , usually dominates in narrow gap systems, and is frequently used as a model Hamiltonian for 2D systems.Electrically driven spin dynamics in quantum wells strongly depends on the polarization of the field E (t) .EDSR is especially strong with in-plane field E (t) while is suppressed with E (t) ẑ because orbital dynamics in z direction is restricted by the confining potential (Rashba and Efros, 2004).Schulte et al. (2005) reported dominance of EDSR over the traditional electron paramagnetic reseonance (EPR) by a factor of about 10 4 for AlAs quantum wells that are typical of weak spin-orbit coupling.EDSR has been seen in in-plane polarization in agreement with above arguments.
Besides spin-orbit coupling coming from orbital motion through the modified Thomas term, spatial dependence of the Zeeman term provides an additional mechanism of coupling.Kato al. (2003) took advantage of the small gfactor of Al x Ga 1−x As alloys that crosses zero at appropriate content x and used the dependence of ĝ -tensor on the spatial coordinate z in the grows direction.Hence, z dependence of the Hamiltonian H Z = ( σ ĝ(z) B ) was strong, and efficient spin operation through time dependent gate voltage was achieved.
These examples illustrate the large variety of options that spin-orbit coupling provides for electric-field-driven spin dynamics in semiconductors with different mechanisms and strength of this coupling.

Spin interference phenomena
The spin interference device proposed by Datta and Das (1990), usually termed as spin transistor, became one of the most influential concepts in the field of semiconductor spintronics.The basic idea of it was formulated in terms of the Hamiltonian H α .The energy spectrum consists of two chiral branches with energies (3) In eigenstates, spins of electrons propagating in k direction are polarized ⊥ k ; in mixed states, spins precess about an effective magnetic field B α = 2α( k ×ẑ)/gµ B .This precession, described by a characteristic momentum k α = mα/h 2 and characteristic length α = h2 /mα , was observed by Kalevich and Korenev (1990).If two ferromagnets F 1 and F 2 with the magnetization parallel to the propagation direction k serve as a source and drain, the conductivity of a straight wire of a length L reaches maxima when Lk α = πn and minima when Lk α = π(2n + 1)/2 , with integer n .Actually, the same result follows when considering interference of electrons propagating with the same energy but with different momenta k λ related by the equation λ (k) = .
The basic ideas underlying the spin transistor are (i) injecting spin polarized electrons into a specimen with spin split bands, (ii) using and detecting spin dependent electron phases, and (iii) electrical control of spin orbit coupling constant α by gate voltage.Gate control of α has been achieved (Nitta et al. 1997, Engels et al. 1997), but a working device has never been reported.Nevertheless, the ideas underlying the spin transistor concept strongly influenced the following research.
The quantum phases acquired by electrons due to spin orbit coupling were related to Berry phases (Aronov and Lyanda-Geller 1993) and the role of singular points at electron trajectories have been clarified (Bercioux at al. 2005); extensive literature on the subject is currently available.At the experimental side, there are several reports on observing these phases and gate control of them, of which the most recent are the papers by Koga et al. (2005) on loop arrays in InGaAs quantum wells and by Koenig et al. (2005) on HgTe quantum rings.
From the conceptual point of view, these recent experimental achievements support the spin transistor concept because they demonstrate convincingly the modulation of electrical resistance based on gate controlled spin interference.In these experiments quantum loops have been used instead of straight wires and, what is critically important, such setups do not require injection of spin polarized electrons which constitutes a problem by itself as will be explained in the next section.

Spin injection: Optical and electrical
Spin injection into semiconductors has been achieved long ago through excitation by circularly polarized light (Meier and Zakharchenya 1984).
Discovery of the Dyakonov-Perel (1972) spin relaxation mechanism and prediction of the photogalvanic effect (Ivchenko andPikus 1978, Belinicher 1978) strongly influenced the following research.The former originates from spin precession in the field B α ( k ) interrupted by fast collisions and results in spin relaxation time τ −1 s ≈ τ p (2αk F /h) 2 and spin diffusion length scale of α ; τ p being the momentum relaxation time and k F the Fermi momentum.The latter is a charge current induced by optically produced nonequilibrium spin polarization.It critically depends on the light polarization and allows detecting nonequilibrium spin populations by electrical measurements.
More recent research was focused on electrical spin injection that is of higher importance for devices.
Discovery of spin injection from metallic ferromagnets into superconductors (Tedrow and Meservey 1973) and paramagnetic metals (Johnson and Silsbee 1985) suggested using similar sources for spin injection into spin transistor.However, early attempts of achieving this goal were unsuccessful, and the origin of the failure has been understood in terms of conductivity mismatch (Schmidt et al. 2000).Namely, spin flow across a contact of two conductors that differ in electron spin polarization degree is controlled by the conductor with higher resistance.Hence, the spin injection coefficient γ into a semiconductor from a ferromagnetic metal can be evaluated through the ratio of their conductivities, γ ∼ σ S /σ F ∼ 10 −5 .This conductivity mismatch can be remedied by using resistive contacts (tunnel or Schottky) between the metallic ferromagnet and a semiconductor, or by employing semimagnetic semiconductors or half-metals as spin sources.The last five years have witnessed an impressing increase in the spin injection coefficient, but the ultimate solution has not been found yet; for review see Žutić et al. (2004).Another important problem regarding ferromagnetic spin emitters, irrespective of their nature, are the stray magnetic fields that perturb electron spin dynamics.Therefore, various concepts of spin emitters that do not include ferromagnetic elements and are based completely on spin-orbit coupling have been proposed.
One of the ideas was initiated by Voskoboynikov et al. (1999) and de Andrada e Silva and La Rocca (1999) and is based on the fact that electrons described by the Hamiltonian H α are spin polarized by the effective field B α ( k ) either along this field or opposite to it.However, because spin-orbit coupling respects time inversion symmetry, the total spin magnetization averaged over the equilibrium distribution vanishes.Violating time reversal symmetry by inducing in-plane electron drift and filtering the nonequilibrium part of spin populations by electron tunneling across a double or a triple barrier should produce spin polarized tunnel current.A prototype resonant-tunneling device fabricated with an AlSb/InAs/GaSb/AlSb heterostructure has been reported by Moon et al. (2004).
Spin injection cannot be achieved without magnetic elements in a two-terminal device.Thus, multi-terminal quantum rings were proposed as efficient injectors of pure spin currents, i.e., spin currents that are not accompanied with charge currents (Kiselev andKim 2003, Souma andNikolić 2005).
Furthermore, spin-orbit coupling allows creating beams of spin-polarized electrons either by means of magnetic focusing in a weak perpendicular field or even in the absence of it.Folk et al. ( 2003) achieved emission of spin-polarized electrons from an open quantum dot by applying in-plane magnetic field.Rokhinson et al. (2004) employed spin-orbit coupling in the bulk to spatially separate, in a magnetic focusing experiment, two spin components of a spin-unpolarized hole beam injected from a quantum point contact.Media with spin-split energy spectrum are similar to double-refractive crystals, hence, they allow spatial separation of two spin-polarized components of an originally unpolarized electron beam.Khodas et al. (2004) have established conditions under which such separation can be efficient.Spin polarization of electrons scattered from a lithographic barrier has been already achieved by Chen et al. (2005).
Generally, electron dynamics in media with spin-orbit coupling includes a number in exciting aspects.Spatially dependent α provides a solid-state analog of the Stern-Gerlach experiment that is free of complications related to the effect of Lorentz force (Ohe et al. 2005), and various types of spin orbit coupling result in different versions of Zitterbewegung (Schliemann et al. 2005, Zawadzki 2005).

Transport in media with spin orbit coupling
With the spin-orbit coupling playing the central role in semiconductor spintronics, developing a consistent theory of spin transport in systems with spin orbit coupling is an important but also a demanding goal.The eventful history of a related phenomenon, the anomalous Hall effect (AHE), indicates existence of essential theoretical problems.As distinct from the regular Hall effect, the AHE is a transverse charge current due to electron magnetization M rather than due to an external magnetic field B ; therefore, it is fundamentally based on spinorbit coupling.The theory of AHE has been initiated by Karplus and Luttinger (1954), and 20 year long efforts resulted in the conclusion that the AHE is an extrinsic effect controlled by competing terms related to electron scattering.Extensive cancellations make calculations tricky (Nozières and Lewiner 1973).Some of the extrinsic contributions to the anomalous Hall current do not depend on the scattering time, and recent research indicates existence of an intrinsic contribution to the anomalous Hall current that can be expressed in terms of the Berry curvature in k -space (Jungwirth et al. 2002, Onoda and Nagaosa 2002, Haldane 2004).Experiment still does not provide any convincing evidence regarding the mechanisms involved in the AHE, and this challenging problem remains a subject of current research.
Theory of spin transport meets even more fundamental difficulties.In absence of spin-orbit coupling, spin transport in ferromagnets can be comfortably described in the framework of the Mott two-channel model whenever τ p τ s .Indeed, under these conditions spin is conserved, and spin transport theory is similar to the charge transport theory, and such an approach is routinely used in the theory of spin injection from ferromagnets into semiconductors, see Sec. 4.However, spin nonconservation stemming from spin-orbit coupling changes the situation drastically.Absence of the continuity equation for spin makes the very existence of a consistent definition of spin current problematic.Absence of magnetization current in Maxwellian equations implies a similar conclusion.Only spin magnetization M s ( r ) is an observable quantity, hence, it should play in the spin transport theory the cental role.
Under these considerations, two approaches have been applied.First, in an attempt to develop a theory of spin transport similarly to the theory of charge transport, the following definition of spin currents was applied (e.g., Murakami et al. 2003, Sinova et al. 2004): Here ... stands for the integration over the electron distribution function, and the anticommutator of v j and σ i is used because in media with spin orbit coupling the velocity v = h−1 ∂H( k )/∂ k depends on Pauli matrices σ .This definition is the simplest one and is widely used.However, because spin nonconservarion makes the physical meaning of j i j obscure, different definitions have been also proposed; some of them result in the opposite sign of j i j .Second, because only spin magnetization is measured, Boltzmann and diffusive equations have been derived for it (Mishchenko et al. 2004, Burkov et al. 2004, Tang et al. 2005, Liu and Lei 2005, Khaetskii 2005, Shytov et al. 2005); such an approach does not rely on the spin current concept.
Because the energy spectrum of Eq. ( 3) comprises two branches, both interbranch and intrabranch transitions contribute to j i j .Remarkably, when chemical potential µ > 0 , the interbrach contribution to spin conductivity is universal in a perfect crystal, j z x /E y = e/4πh (Sinova et al. 2004).To find the intrabranch contribution one has to violate momentum conservation by impurities or by a magnetic field.Then, the inter-and intraband contributions cancel (Inoue et al. 2004), and this is an intrinsic property of the free electron Hamiltonian of Eq. (3) (Dimitrova 2005).For spin-orbit Hamiltonians including higher powers of k , spin currents do not vanish, e.g., for heavy holes because their spectrum includes k 3 terms.

Macro-and mesoscopic spin Hall effect
An in-plane electric field E induces homogeneous spin polarization of H α electrons in the ( E × ẑ) direction (Edelstein 1990).This was observed by several groups (Silov et al. 2004, Kato et al. 2004a, Ganichev et al. 2004).A different related effect is the spin Hall effect (Dyakonov and Perel 1971), that usually manifests itself as spin polarization S ẑ near sample edges.It has been observed in 3D n -GaAs by Kato et al. (2004) with distinct signatures of the extrinsic mechanism.The magnitude of the effect is in agreement with a theory including no free parameters (Engel et al. 2005); see also Tse and Das Sarma (2005).On the contrary, a stronger effect observed in p -GaAs by Wunderlich et al. (2005) was ascribed to the intrinsic spin Hall effect.
It is tempting to attribute spin accumulation near sample flanks to the bulk current j z x and estimate it as S z ≈ (h/2)(j z x α /D) , D being the diffusion coefficient.Such an estimate is based on the assumption that bulk currents j z x have a meaning of transport currents.However, several arguments indicate strict limitations on such an assumption.First, for H α -electrons the bulk current vanishes, j z x = 0 .However, spins can accumulate at the flanks under the condition that they can either cross them and move into α = 0 regions (Adagideli and Bauer 2005) or can relax on them; this seeming paradox is related to spin nonconservation.Second, spin currents j i j are even with respect to time inversion, hence, this tensor not necessarily vanishes in equilibrium while spin transport and spin accumulation vanish.
Spin accumilation can be found only through solving the transport problem.However, evaluation of the spatial dispersion of j i j (q) suggests the following Conjecture (Rashba 2005).Fourier components of spin currents of Eq. (4) at the wave vector k so are scaled by the "universal" value of eE/h (Sinova et al. 2004) and can be employed for estimating spin transport rate.Hence where k so is the spin precession momentum that reduces to k α for H α -electrons.Numerical coefficients in these equations depend on the specific form of H SO and on the boundary conditions.The first of Eqs. ( 5) indicates e/h as the "universal" scale for spin conductivities.The above discussion is applicable only to macroscopic samples of size L > ∼ α .For spintronic applications, mesoscopic three-or four-terminal devices with L < ∼ α seem to be of most importance.Applying the Keldysh technique in conjunction with the Landauer-Büttiker formalism to the mesoscopic spin Hall effect proved to be successful (Nikolić et al. 2005).As far as numerical data and analytical estimates can be compared, numerical data seem to support the estimates of Eq. (5).

Related systems
A theory of quantized spin Hall conductivity through the edge states of insulating graphene was developed by Kane and Mele (2001) in the framework of the Haldane (1988) model, and the phase diagram for the dissipationless regime was found.These spin currents show remarkable robustness to a random potential (Sheng et al. 2005).
For typical semiconductors, spin-orbit splitting is not strong and k α k F , hence, most theories have been developed for this limit.Recently, large spin-orbit splitting of surface states have been observed for metals and semimetals, of which the largest one is h2 k 2 α /2m ≈ 0.2 eV (Ast et al. 2005).If such states can be used for interference devices, their sizes can be dramatically reduced.There is current interest in a noncentrosymmetric superconductor Ce 2 Pt 3 Si where spin orbit splittings at the Fermi level of 0.1 eV have been estimated (Mineev and Samokhin 2005).Finally, spin-orbit effects in hopping conductors (Entin-Wohlman et al. 2005) and optical lattices (Dudarev et al. 2004) were also discussed.
The review does not cover a number of different avenues of semiconductor spintronics such as coherent op-eration of quantum dots in the context of quantum computing (Loss andDiVincenzo 1998, Petta et al. 2005).