Semiconductor Spintronics: Progress and Challenges

Emmanuel I. Rashba

Dept. of Physics, Harvard University, Cambridge, MA 02138, U.S.A.

1.   Introduction

Spin is the only internal degree of freedom of the electron, and utilizing it in the new generations of semiconductor devices is the main goal of semiconductor spintronics. Contemporary semiconductor electronics is based on electron charge only. It is expected that spin-based phenomena will provide electronic devices with new functionality, and achieving quantum computing with electron spins is among the most ambitious goals of spintronics.^1,2 During the last five years, there has been impressive progress in this field, both in experiment and in developing theoretical concepts. Since the goals are highly challenging and require overcoming a number of difficult problems, research is developing along several avenues. For example, the prospects of spin-based computing with quantum dots require an increase in the spin coherence time of gate-controlled double quantum dots by several orders of magnitude, and a great advance in this direction has been achieved recently by applying spin echo techniques.³ In what follows, we concentrate on fundamentals and on recent developments related to a different branch of spintronics: the use of spin-orbit coupling to achieve direct electrical control of electron spins in semiconductor nanostructures. Compared with magnetic control, electrical control holds promise of much higher efficiency, as well as access to electron spins on the nanoscale.

Strong enhancement of spin-orbit coupling in crystals as compared to vacuum originates from the large gradients of crystal field V(r) and high electron velocities v near nuclei. The enhanced spin-orbit coupling affects the wave-functions and energy spectra of Bloch states. In vacuum, the dimensionless parameter of spin-orbit coupling is about E(k)/m₀c² ~ 10^-6, with the electron energy E(k) ~ 1 eV and the Dirac gap m₀c² ~ 1 MeV. In semiconductors, the similar parameter is Δ_SO/E_G ~ 1, where Δ_SO ~ 1 eV is the spin-orbit splitting of valence bands and the E_G ~ 1 eV is the bandgap. This enhancement makes semiconductors promising for electrical manipulation of electron spins.⁴

2.   Basic concepts of semiconductor spintronics

Apparently, the first practical application of electrically-driven spin transitions belongs to laser physics and dates back as far as 1971.⁵ More recent research was initiated by the 1990 paper of Datta and Das, who advanced the idea of a spin-transport device based on spin interference in media with spin-orbit coupling.⁶ Afterwards, this device became known as a spin transistor. Despite the fact that the attempts to fabricate such a device have not been successful to date and its feasibility has been questioned,⁷ the basic principles underlying it strongly influenced subsequent research.

A toy-model spin-orbit Hamiltonian describing electrons in asymmetric two-dimensional (2D) systems (Rashba term^8,9) is

where α is the spin-orbit coupling constant, σ is the Pauli matrix vector, k is electron wavevector, and z₀ is a unit vector perpendicular to the confinement plane. When rewritten as

where μ_B is the Bohr magneton and B_α(k)is an effective momentum-dependent spin-orbit field, the Hamiltonian H_α describes spin precession in the field B_α(k). The same phenomenon can be also understood in terms of two eigenstates of the Hamiltonian H_α with the same propagation direction k₀ and energy E, but with different momenta k_± depending on the spin-orbit coupling constant α (spin birefringence). Therefore, if spin-polarized electrons are injected at x = 0 along the direction k₀ in a spin state that is not an eigenstate for the field B_α(k), the resistance of the device is controlled by the α-dependent phase of the electron wave function near the spin-polarized drain at x = L.

The proposed Datta and Das device is based on the following principles:

• spin injection from a ferromagnetic source and spin detection by a ferromagnetic drain;
• electrical control of spin-orbit coupling α by a Schottky gate;
• spin precession in the spin-orbit field B_α; and
• spin interference.

Lately, there has been impressive progress in developing ferromagnetic injectors, including better understanding of the role of contacts between spin injectors and semiconductor microstructures. At the same time, much research effort has been focused on generating and injecting spin populations electrically, by means of spin-orbit coupling, as avoiding ferromagnetic elements would allow the elimination of stray magnetic fields. Electrical control of spin-orbit coupling^10,11 and spin precession¹² in the field B_α have been reported long ago, while spin interference has been observed only recently, as discussed in Section 4 below.

All-semiconductor electrically controlled spintronics needs a better understanding spin transport in media with spin-orbit coupling that is rather nontrivial. In what follows, we review some of the recent progress in this field.

3.   The origin of spin coupling to external electric field

Electrically induced quantum transitions are usually described in terms of oscillator strengths that are subject to the oscillator sum rule (Thomas-Kuhn-Reiche theorem). This sum rule follows from the standard commutation relation

When this commutation relation is written as a sum over the intermediate states, it becomes a sum of the terms

which are oscillator strengths of n ← transitions. In the absence of spin-orbit coupling, calculating the commutator of the coordinate x and the Hamiltonian, we find

and after substitution into Eq. (4) we arrive at

where m₀ is the electron mass in vacuum. For local states, the oscillator sum rule

includes nondiagonal n ≠ terms only, as the diagonal n = terms vanish because matrix elements of the coordinate x in Eq. (5) are finite.

However, because Bloch states are extended, diagonal matrix elements of x diverge. Hence, diagonal matrix elements of k may survive. Observing that the oscillator strengths f_n← of Eq. (6) coincide, with accuracy up to the factor m₀, with the summands in the standard expression of k·p theory for the inverse effective mass m in the th Bloch band, one arrives at the equation

Therefore, m₀/m is the oscillator strength, f_← for the transition from the state “into itself”.¹³ This is precisely the oscillator strength that manifests itself in the Drude and cyclotron absorption.

The problem we need to solve is what happens to this oscillator strength in a noncentrosymmetric system when spin-orbit coupling enters the picture, α ≠ 0, and a spin-degenerate band splits into two subbands. This situation is illustrated in Fig. 1. For each state, the total oscillator strength m₀/m is divided between the transition “into itself” and the transition between branches. For the transitions from the bottom of the band, the inter-branch transition energy equals 2E_SO, with E_SO = mα²/², and the oscillator strength is divided equally between both transitions. For a given wavevector k, electron spins have opposite directions on two branches of the energy spectrum and hence inter-branch transitions are spin-flip transitions. These transitions have high oscillator strengths, comparable to the oscillator strength of the cyclotron resonance. With increasing electron energy, intensities of inter-branch transitions decrease, but only as k_α/k_F, where k_α = mα/² is the spin precession momentum in the field B_α(k), and k_F is Fermi momentum. Therefore, their intensities remain high for reasonable α values. Outside the spectral region of interbranch transitions, their Kramers-Kronig transform describes spin coupling to electric fields;¹⁴ spectral dependence of the corresponding sponse can be only found from detailed transport equations. In a strong magnetic-field B, inter-branch transitions transform into the electric dipole spin resonance (EDSR), whose intensity is usually much higher than the intensity of electron paramagnetic resonance (EPR).^4,8

The important role that intrabranch transitions play in spin transport will be discussed in Section 6 below.

4.   Experimental achievements: Spin populations and spin interference

Because of the spin coupling to electric field, propagation of electric current across a sample is accompanied by spin accumulation in the bulk of three-dimensional (3D) and 2D systems.^15,16 For thin 3D layers and 2D systems, this was recently observed by Kato et a1.,¹⁷ Silov et al., and Ganichev et al.¹⁹ Another related phenomenon is spin Hall effect^20-22 that manifests itself in spin accumulation near the edges of a sample, for review see Refs. 23 and 24. Spin Hall effect observed in n-GaAs²⁵ was attributed to an extrinsic mechanism, whereas the effect observed in p-GaAs²⁶ was taken to be intrinsic – that is, the former effect originates from impurity scattering, whereas the latter is due to spin-orbit coupling in the bulk.

Figure 1. Spin-degenerate spectrum in the absence of spin orbit coupling (a) and spin-split spectrum of 2D electrons with spin-orbit coupling of Eq. (1) (b). In (a), the oscillator strength for transitions “into itself” (a circle with an arrow) equals m_o/m. In (b), this oscillator strength is divided between transitions “into itself” and transitions between two branches (vertical arrow). For the transitions from the bottom of the band (shown in the figure), the oscillator strength m_o/m, is equally divided between both types of transitions.

Spin interference phenomena, besides their promise for applications, are important from the fundamental point of view because they are related to quantum phases that essentially depend on the shape of the electron paths. In particular, these phases differ for rings, where electron motion is close to adiabatic, and polygons where motion near vertices is strongly nonadiabatic. Spin interference on a large array of InGaAs square loops was reported by Koga et a1.,²⁷ and on a single HgTe/HgCdTe ring by Koenig et al.²⁸

5.   Enhancing spin responses to electric fields

Long delay in the experimental observation of electrically driven spin populations in 2D systems after their prediction was caused by the small magnitude of these effects. The weakness of the effect also hinders its utilization in semiconductor devices. In this Section, we discuss some options for enhancing spin response.

• Spin response to an inhomogeneous dc electric field F(r) = F₀exp(iq·r) diverges when q → 2k_α.²⁹ This behavior can be easily understood if one takes into account that adding wavevector q results in a mutual displacement of two Fermi surfaces with the same energy, and for q = 2k_α these surfaces touch. This results in “spin breakdown” because spin can be flipped at zero cost in energy. Therefore, one should choose inhomogeneous fields with large q ≈ 2k_α spectral components. It also suggests that optimal sizes of elements with α ≠ 0 (rings or diamonds) designed for injecting spins into α = 0 wires should be about the spin precession length _α = l/k_α.
• Frequencies of time-dependent fields should be nearly resonant, i.e. for B = 0 the frequency should be about 2αk_F, whereas for EDSR conditions it should be close to the EDSR frequency gμ_BB. It is also important that strong scattering, / >> αk_F where is the momentum relaxation time, results in the narrowing of the EDSR line to the inverse spin relaxation time 1/_S.³⁰ The explicit shape of the line, with _S equal to the Dyakonov-Perel’ relaxation time,³¹ has been recently derived by Duckheim and Loss.³²
• In quantum wells, the in-plane polarization of the electric field F is more efficient than the out-of-plane polarization by a factor (ω₀²/ω_Cω_S)² where ω₀, ω_C, and ω_S are, respectively, the confinement frequency and the frequencies of the cyclotron and spin resonance.³³
• Spin response is stronger in p-type materials.³⁴
• Surface states on the (111) face of Bi³⁵ and on Bi/Ag(lll) monolayer alloys³⁶ show giant spin-orbit splittings with E_SO up to 0.4 eV. Thin layers of such materials can provide ultra-short spin precession lengths _α.

New options arise if two different spin-orbit coupling mechanisms are combined. The symmetry group C_2V of (001) quantum wells in A_IIIB_V materials, in addition to an invariant of Eq. (1), has a different invariant (Dresselhaus term) that is linear in k:

Pikus noticed that the 3D Dresselhaus k³-spin splitting³⁷ reduces to Eq. (9) in the limit of narrow quantum wells.³⁸ Combining H_α and H_β provides new options for spintronic devices^39,40 especially in the vicinity of the magic points α = ±β. At these points, stable spin superstructures with a k_α-dependent period have been predicted recently.⁴¹

6.   Conceptual theoretical problems

Generation of spin populations by a driving electric field is possible only due to spin nonconservation. As a result, theory of spin transport essentially differs from theory of charge transport. This difference is already obvious from Maxwell’s equations that include four electric variables F, D, charge density p and current J, but only two magnetic variables, B and H (or magnetization M = (B – H)/4π). Therefore, absence of magnetic monopoles results not only in the absence of a magnetic analog of ρ, but also in the absence of magnetization current. Introduction of such a current is justified only under some special conditions, particularly, in the framework of the Mott two-fluid theory of electron transport in ferromagnets without spin-orbit coupling.⁴² Spin-orbit coupling results in spin nonconservation. As a result, time derivative of spin magnetization, ∂S/∂t, cannot be represented as a divergence of any vector. Therefore, there is no unambiguous definition of spin current, and the form of the extra term depends on the spin current definition; this term is known as torque.⁴³ Usually, ith component of the spin current j_i is defined as

where an anticommutator is taken because in the media with spin orbit coupling the velocity v depends on Pauli matrices σ, and <...> stands for averaging over the electron distribution; however, different definitions for j_i have also been proposed.⁴⁴ The concept of spin currents has been used in literature for many years, but it attracted more attention after Murakami et al.⁴⁵ and Sinova et al.⁴⁶ reported some unexpected properties of these currents for 3D holes and 2D electrons, respectively. Since then, these quantities have become a popular playground for comparing spin responses of particles described by various spin-orbit Hamiltonians to dc and ac electric fields.

In particular, dc spin-Hall conductivity defined as σ_SH = j_x^z/F_y, when calculated using the kubo formula for a perfect system with spin-orbit Hamiltonian H_α, comes out as σ_SH = e/4π for arbitrary chemical potential μ > 0. This “universal conductivity” raised hopes that there might be possible to find some simple results for spin accumulation near the sample edges; spin accumulations are the only quantities currently accessible for experimental detection. However, calculations of σ_SH that properly account for electron scattering have shown that σ_SH vanishes, for a review see Refs. 23 and 24. The simplest formal argument, explaining this spin current cancellation, was provided by Dimitrova⁴⁷, who noticed that j_x^z, is proportional to the mean value of the derivate dσ_y/dt that should vanish in a stationary state. A different argument demonstrating that this cancellation follows from the form of the free Hamiltonian only, irrespective of the potentials of non-magnetic scatterers, is based on vanishing the spin current j_x^z in a perfect sample placed in an external magnetic field perpendicular to the confinement plane.⁴⁸ Physically, the vanishing of σ_SH comes from the fact that there exists an intrabranch contribution to σ_SH, similar to the intrabranch oscillator strength of Section 3, that cancels the universal contribution e/4π. From this standpoint, impurities and the magnetic field play a similar role: by violating momentum conservation, they give rise to the intrabranch contribution.

It is currently well understood that the above cancellation is an exceptional property of the terms H_α and H_β in conjunction with a quadratic nonrelativistic Hamiltonian H₀ = ²k²/2m, and it underscores the fact that while spin response to electric fields per se originates from the spin-orbit coupling term in the free electron Hamiltonian (Section 3), the specific form of the spin response can be found only by rigorous solution of the proper transport equations.

Boltzmann equations for systems with a spin-split energy spectrum were derived in a number of papers.^49-51 In principle, they can describe transport problems for arbitrary values of the parameter αk_F/, but they are usually solved in the diffusive limit αk_F/ << 1.^43,52,53 In this limit, the problem of boundary conditions becomes nontrivial because of spin nonconservation. Indeed, spin is not conserved even on a perfect boundary between α ≠ 0 and α = 0 regions because the currents defined by Eq. (10) persist in thermodynamic equilibrium in the α ≠ 0 region but vanish in the adjacent α = 0 region.⁵⁴ Numerical work shows that these “equilibrium currents”, which are not related to any real spin transport, are especially strong near boundaries.⁵⁵ Therefore, boundary conditions for diffusive equations cannot be derived from spin conservation, but only from a consistent solution of the transport equations near boundaries on a scale small compared with the spin diffusion length L_SD. For the Dyakonov-Perel’ spin relaxation mechanism^3l L_SD is roughly equal to _α. This problem is still awaiting a solution.⁵⁶ For an H_α semiconductor, it is expected that a dc current flowing along a perfect hard-wall boundary would produce only tiny spin accumulation near the edge.^52,57

A different problem concerns the relative influence of extrinsic and intrinsic mechanisms and their interplay. Extrinsic mechanisms are related to impurity scattering and are traditionally discussed in terms of skew scattering and side jump contributions. Intrinsic mechanisms are usually attributed to the spin-orbit coupling terms in the Hamiltonian H(k). A similar problem has existed in the theory of anomalous Hall effect (AHE) for over 50 yeas:⁵⁸ but still remains somewhat controversial. The early period has been summarized in the paper by Nozieres and Lewiner,⁵⁹ where a set of competing (and partly canceling) terms had been derived and compared for a centrosymmetric semiconductor. They attributed AHE to extrinsic mechanisms. Remarkably, mean free time drops out from the side jump that therefore depends only on the parameters of a perfect crystal; this conclusion agrees with the previous result by Luttinger.⁶⁰ More recently, AHE has been related to a Berry phase in k-space that is essentially intrinic,^61-63 and it seems probable now that Berry curvature is an elegant mathematical language for describing the mechanism that in simplified models was understood as a side jump contribution. Within the framework of the Boltzmann equation, side jump appears as the next order correction in the small parameter /E_F to the skew scattering term in the Hall conductivity σ_xy (E_F is the Fermi energy). In the meantime, some experimental data suggest that this correction term dominates in the dirty regime.^60,64 A topological interpretation of the side jump contribution to σ_xy seems to be the most natural explanation of its remarkable ubiquity. However, the fact that the side jump contribution has, in the framework of Ref. 59, the same magnitude but opposite sign in the clean and dirty limits, indicates that the problem still persists.

The problems that make theory of AHE so tricky are also inherent in the theory of the spin Hall effect. Moreover, while the definition of the anomalous Hall current is straightforward, the ambiguity of the spin current concept makes calculating the spin Hall effect much trickier. It has been shown that the data of Ref. 25 can be reasonably described by the extrinsic mechanism,^65,66 while the data of Ref. 26 seem to point to the role of intrinsic mechanisms.⁶⁷ Remarkably, the side jump term of Ref. 65 coincides with Berry curvature (_k × r_SO), where r_SO is the spin-orbit contribution to the coordinate operator in the crystal-momentum representation. From this standpoint, side jump can be understood as an intrinsic effect that originates from the operator r rather than the Hamiltonian H(k). Meanwhile, there is no doubt that in noncentrosymmetric crystals H(k) contributes to spin transport, and this contribution cannot be expressed in terms of Berry curvature. Indeed, r_SO = (μ_k|i_k|μ_k) is exactly the same for the Hamiltonians H_α with α = constant and α = α(k²), while spin currents do vanish in the first case and not in the second⁶⁸ (μk are eigen-spinors). The same holds for Hamiltonians with parabolic and nonparabolic H₀ parts.⁶⁹ Also, _kμk is not defined at k = 0 for the Hamiltonians of H_α type.

It has been shown recently,⁷⁰ that the joint effect of intrinsic and extrinsic terms in H(k) on spin currents is singular. In H_α semiconductors, spin current j_x^z defined according Eq. (10) vanishes for arbitrary α ≠ 0, i.e. spin precession in the field β_α nullifies the extrinsic spin current after the integration over the whole sample. This can be understood as the result of averaging the spins, polarized by skew scattering, over the electron trajectories, and seems to underscore the fact that spin accumulation near boundaries cannot be derived from spin currents of Eq. (10). The same conclusion comes from the observation that spin relaxation on the boundary leads to the spin Hall effect even when bulk spin currents vanish.^71,72

Analysis of the existing data on spin currents and spin Hall effect suggest that, at a qualitative level, they can be better related if instead of q = 0 components of spin currents corresponding to averaging over the entire infinite homogeneous space, one considers their Fourier components at q ≈ k_α. Such an approach corresponds to the idea that when it comes to spin accumulation S at the edge, only the adjacent layer of width _α matters. Fourier components j_i^z(k_α) do not vanish for the H_α Hamiltonian and have the same magnitude of about eE/ as for the Hamiltonians that are nonlinear in k (e.g. the k³ heavy hole spin-orbit Hamiltonian),⁷³ as long as the definition of k_α is generalized by expressing it in terms of the spin-orbit splitting δ_SO at the Fermi level, k_α → k_SO = mδ_SO/2²k_F. Moreover, spin Hall currents j_SH defined in this way can be related to spin accumulations as follows:

After such redefinition of spin currents, they acquire some universality in establishing the basic scales and connection to spin accumulations near the edges.⁷² Equation (11) shows that j_SH(k_SO) coincides in order of magnitude with the spin current calculated by Sinova et al.,⁴⁶ but has a somewhat different physical meaning. Numerical constants in Eq. (11) essentially depend on the specific form of the spin-orbit coupling and on the boundary conditions and can be only found from detailed transport equations. There is no doubt that physical quantities like S are continuous functions of all parameters, including α. Also, near the sample edge, spin magnetization S(x) is an oscillating function of x (with a period about k_SO^-1) that usually changes sign, x being separation from the edge. Hence, it is difficult to expect any universal relation even between the signs of the bulk spin current and the spin accumulation at the edge.

7.   Conclusions

Spin-orbit coupling is currently considered a key to creating and manipulating spin populations electrically, on a nanometer scale. Recent years have witnessed impressive progress in this field, both in experiment and in theory, as described in this chapter. The very possibility of creating nonequilibrium spins by electric fields is based on spin nonconservation. This fact, in turn, results in an essential difference between the spin-transport theory in media with spin-orbit coupling and the traditional theory of charge transport.

Acknowledgments

This work was supported by the Harvard Center for Nanoscale Systems. Inspiring discussions with H.-A. Engel, B. I. Halperin, A. H. MacDonald, C. M. Marcus, D. Loss, and Q. Niu are gratefully acknowledged.

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