Spintronics(自旋电子学)

Spintronics:Fundamentals and applications

Igor Zˇutic′*

Condensed Matter Theory Center,Department of Physics,University of Maryland

at College Park,College Park,Maryland20742-4111,USA

Jaroslav Fabian?

Institute for Theoretical Physics,Karl-Franzens University,Universita¨tsplatz5,8010Graz,

Austria

S.Das Sarma

Condensed Matter Theory Center,Department of Physics,University of Maryland

at College Park,College Park,Maryland20742-4111,USA

(Published23April2004)

Spintronics,or spin electronics,involves the study of active control and manipulation of spin degrees of freedom in solid-state systems.This article reviews the current status of this subject,including both recent advances and well-established results.The primary focus is on the basic physical principles underlying the generation of carrier spin polarization,spin dynamics,and spin-polarized transport in semiconductors and metals.Spin transport differs from charge transport in that spin is a nonconserved quantity in solids due to spin-orbit and hyper?ne coupling.The authors discuss in detail spin decoherence mechanisms in metals and semiconductors.Various theories of spin injection and spin-polarized transport are applied to hybrid structures relevant to spin-based devices and fundamental studies of materials properties.Experimental work is reviewed with the emphasis on projected applications,in which external electric and magnetic?elds and illumination by light will be used to control spin and charge dynamics to create new functionalities not feasible or ineffective with conventional electronics.

CONTENTS

I.Introduction323

A.Overview323

B.History and background325

1.Spin-polarized transport and

magnetoresistive effects325

2.Spin injection and optical orientation328

II.Generation of Spin Polarization329

A.Introduction329

B.Optical spin orientation331

C.Theories of spin injection333

1.F/N junction333

2.F/N/F junction337

3.Spin injection through the space-charge

region338

D.Experiments on spin injection340

1.Johnson-Silsbee spin injection340

2.Spin injection into metals341

3.All-semiconductor spin injection342

4.Metallic ferromagnet/semiconductor

junctions345 III.Spin Relaxation and Spin Dephasing346

A.Introduction346

1.T1and T2347

2.Experimental probes348

B.Mechanisms of spin relaxation349

1.Elliott-Yafet mechanism349

2.D’yakonov-Perel’mechanism351

a.Bulk III-V semiconductors353

b.Two-dimensional III-V semiconductor

systems354

3.Bir-Aronov-Pikus mechanism356

4.Hyper?ne-interaction mechanism357

C.Spin relaxation in metals358

D.Spin relaxation in semiconductors360

1.Bulk semiconductors360

2.Low-dimensional semiconductor structures361

3.Example:Spin relaxation in GaAs363

a.Bulk n-GaAs363

b.GaAs-based quantum wells365 IV.Spintronic Devices and Applications366

A.Spin-polarized transport366

1.F/I/S tunneling366

2.F/I/F tunneling368

3.Andreev re?ection371

4.Spin-polarized drift and diffusion372

B.Materials considerations373

C.Spin?lters376

D.Spin diodes377

E.Spin transistors380

1.Spin?eld-effect transistors380

2.Magnetic bipolar transistor381

3.Hot-electron spin transistors382

F.Spin qubits in semiconductor nanostructures384

V.Outlook385 Acknowledgments386 References386 I.INTRODUCTION

A.Overview

Spintronics is a multidisciplinary?eld whose central theme is the active manipulation of spin degrees of free-

*Present address:Center for Computational Materials Sci-

ence,Naval Research Laboratory,Washington,D.C.20735,

USA.Electronic address:igorz@d72efe37f111f18583d05a9d

?Electronic address:jaroslav.fabian@uni-graz.at

REVIEWS OF MODERN PHYSICS,VOLUME76,APRIL2004

0034-6861/2004/76(2)/323(88)/$40.00?2004The American Physical Society

323

dom in solid-state systems.1In this article the term spin

stands for either the spin of a single electron s,which

can be detected by its magnetic moment?g?B s(?B is the Bohr magneton and g is the electron g factor,in a

solid generally different from the free-electron value of

g0?2.0023),or the average spin of an ensemble of elec-trons,manifested by magnetization.The control of spin

is then a control of either the population and the phase

of the spin of an ensemble of particles,or a coherent spin manipulation of a single or a few-spin system.The goal of spintronics is to understand the interaction be-tween the particle spin and its solid-state environments and to make useful devices using the acquired knowl-edge.Fundamental studies of spintronics include inves-tigations of spin transport in electronic materials,as well as of spin dynamics and spin relaxation.Typical ques-tions that are posed are(a)what is an effective way to polarize a spin system?(b)how long is the system able to remember its spin orientation?and(c)how can spin be detected?

Generation of spin polarization usually means creat-ing a nonequilibrium spin population.This can be achieved in several ways.While traditionally spin has been oriented using optical techniques in which circu-larly polarized photons transfer their angular momenta to electrons,for device applications electrical spin injec-tion is more desirable.In electrical spin injection a mag-

netic electrode is connected to the sample.When the current drives spin-polarized electrons from the elec-trode to the sample,nonequilibrium spin accumulates there.The rate of spin accumulation depends on spin relaxation,the process of bringing the accumulated spin population back to equilibrium.There are several mechanisms of spin relaxation,most involving spin-orbit coupling to provide the spin-dependent potential,in combination with momentum scattering to provide a randomizing force.Typical time scales for spin relax-ation in electronic systems are measured in nanosec-onds,while the range is from picoseconds to microsec-onds.Spin detection,also part of a generic spintronic scheme,typically relies on sensing the changes in the signals caused by the presence of nonequilibrium spin in the system.The common goal in many spintronic de-vices is to maximize the spin detection sensitivity to the point that it detects not the spin itself,but changes in the spin states.

Let us illustrate the generic spintronic scheme on a prototypical device,the Datta-Das spin?eld-effect tran-sistor(SFET;Datta and Das,1990),depicted in Fig.1. The scheme shows the structure of the usual FET,with a drain,a source,a narrow channel,and a gate for control-ling the current.The gate either allows the current to ?ow(ON)or does not(OFF).The spin transistor is simi-lar in that the result is also a control of the charge cur-rent through the narrow channel.The difference,how-ever,is in the physical realization of the current control. In the Datta-Das SFET the source and the drain are ferromagnets acting as the injector and detector of the electron spin.The drain injects electrons with spins par-allel to the transport direction.The electrons are trans-ported ballistically through the channel.When they ar-rive at the drain,their spin is detected.In a simpli?ed picture,the electron can enter the drain(ON)if its spin points in the same direction as the spin of the drain. Otherwise it is scattered away(OFF).The role of the gate is to generate an effective magnetic?eld(in the direction of?in Fig.1),arising from the spin-orbit cou-pling in the substrate material,from the con?nement ge-ometry of the transport channel,and the electrostatic potential of the gate.This effective magnetic?eld causes the electron spins to precess.By modifying the voltage, one can cause the precession to lead to either parallel or antiparallel(or anything between)electron spin at the drain,effectively controlling the current.

Even though the name spintronics is rather novel,2 contemporary research in spintronics relies closely on a long tradition of results obtained in perse areas of physics(for example,magnetism,semiconductor phys-ics,superconductivity,optics,and mesoscopic physics) and establishes new connections between its different sub?elds(Rashba,2002c;Zˇutic′,2002a).We review here both well-established results and the physical principles

1While there are proposals for spintronic devices based on

deoxyribonucleic acid(DNA)molecules(Zwolak and Di Ven-tra,2002),the whole device,which includes electrodes, voltage/current source,etc.,is still a solid-state system.

2The term was coined by S.A.Wolf in1996,as a name for a DARPA initiative for novel magnetic materials and

devices. FIG.1.(Color in online edition)Scheme of the Datta-Das spin ?eld-effect transistor(SFET).The source(spin injector)and the drain(spin detector)are ferromagnetic metals or semicon-ductors,with parallel magnetic moments.The injected spin-polarized electrons with wave vector k move ballistically along a quasi-one-dimensional channel formed by,for example,an InGaAs/InAlAs heterojunction in a plane normal to n.Elec-tron spins precess about the precession vector?,which arises from spin-orbit coupling and which is de?ned by the structure and the materials properties of the channel.The magnitude of ?is tunable by the gate voltage V G at the top of the channel. The current is large if the electron spin at the drain points in the initial direction(top row)—for example,if the precession period is much larger than the time of?ight—and small if the direction is reversed(bottom).

324Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

relevant to present and future applications.Our strategy is to give a comprehensive view of what has been accom-plished,focusing in detail on a few selected topics that we believe are representative for the broader subject within which they appear.For example,while discussing the generation of spin polarization,we survey many ex-perimental and theoretical studies of both optical orien-tation and electrical spin injection and present a detailed and self-contained formalism of electrical spin injection. Similarly,when we discuss spin relaxation,we give a catalog of important work,while studying spin relax-ation in the cases of Al and GaAs as representative of the whole?eld.Finally,in the section on spin devices we give detailed physical principles of several selected de-vices,such as,for example,the above-mentioned Datta-Das SFET.

There have been many other reviews written on spin-tronics,most focusing on a particular aspect of the?eld. We pide them here,for an easier orientation,into two groups,those that cover the emerging applications3and those covering already well-established schemes and materials.4The latter group,often described as magne-toelectronics typically covers paramagnetic and ferro-magnetic metals and insulators,which utilize magnetore-sistive effects,realized,for example,as magnetic read heads in computer hard drives,nonvolatile magnetic random access memory(MRAM),and circuit isolators (Wang et al.,2002).These more established aspects of spintronics have also been addressed in several books5 and will be discussed in another review6complementary to ours.

Spintronics also bene?ts from a large class of emerg-ing materials,such as ferromagnetic semiconductors (Ohno,1998;Pearton et al.,2003),organic semiconduc-tors(Dediu et al.,2002),organic ferromagnets(Pejak-ovic′et al.,2002;Epstein,2003),high-temperature super-conductors(Goldman et al.,1999),and carbon nanotubes(Tsukagoshi et al.,1999;Zhao et al.,2002), which can bring novel functionalities to the traditional devices.There is a continuing need for fundamental studies before the potential of spintronic applications can be fully realized.

After an overview,Sec.I covers some basic historical and background material,part of which has already been extensively covered in the context of magnetoelec-tronics and will not be discussed further in this review. Techniques for generating spin polarization,focusing on optical spin orientation and electrical spin injection,are described in Sec.II.The underlying mechanisms respon-sible for the loss of spin orientation and coherence, which impose fundamental limits on the length and time scales in spintronic devices,are addressed in Sec.III. Spintronic applications and devices,with the emphasis on those based on semiconductors,are discussed in Sec. IV.The review concludes with a look at future prospects in Sec.V.

B.History and background

1.Spin-polarized transport and magnetoresistive effects

In a pioneering work,Mott(1936a,1936b)provided a basis for our understanding of spin-polarized transport. Mott sought an explanation for an unusual behavior of resistance in ferromagnetic metals.He realized that at suf?ciently low temperatures,where magnon scattering becomes vanishingly small,electrons of majority and mi-nority spin,with magnetic moment parallel and antipar-allel to the magnetization of a ferromagnet,respectively, do not mix in the scattering processes.The conductivity can then be expressed as the sum of two independent and unequal parts for two different spin projections—the current in ferromagnets is spin polarized.This is also known as the two-current model and has been extended by Campbell et al.(1967)and Fert and Campbell(1968). It continues,in its modi?cations,to provide an explana-tion for various magnetoresistive phenomena(Valet and Fert,1993).

Tunneling measurements played a key role in early experimental work on spin-polarized transport.Studying N/F/N junctions,where N was a nonmagnetic7metal and F was an Eu-based ferromagnetic semiconductor(Ka-suya and Yanase,1968;Nagaev,1983),revealed that I-V curves could be modi?ed by an applied magnetic?eld (Esaki et al.,1967)and now show potential for develop-ing a solid-state spin?lter.When unpolarized current is passed across a ferromagnetic semiconductor,the cur-rent becomes spin-polarized(Moodera et al.,1988;Hao et al.,1990).

A series of experiments(Tedrow and Meservey, 1971b,1973,1994)in ferromagnet/insulator/ superconductor(F/I/S)junctions has unambiguously proved that the tunneling current remains spin polarized even outside of the ferromagnetic region.8The Zeeman-

3Reviews on emerging applications include those of Das Sarma et al.(2000a,2000b,2000c,2001);Wolf and Treger (2000);Das Sarma(2001);Wolf et al.(2001);Oestreich et al. (2002);Rashba(2002c);Zˇutic′(2002a,2002b).

4Established schemes and materials are reviewed by Tedrow

and Meservey(1994);Prinz(1995,1998);Gijs and Bauer (1997);Gregg et al.(1997);Ansermet(1998);Bass and Pratt, Jr.(1999);Daughton et al.(1999);Stiles(2004).

5See,for example,the books of Hartman(2000);Ziese and Thornton(2001);Hirota et al.(2002);Levy and Mertig(2002); Maekawa et al.(2002);Parkin(2002);Shinjo(2002);and Chtchelkanova et al.(2003).

6In preparation by S.S.P.Parkin for Reviews of Modern Physics.

7Unless explicitly speci?ed,we shall use the terms‘‘nonmag-netic’’and‘‘paramagnetic’’interchangeably,i.e.,assume that they both refer to a material with no long-range ferromagnetic order and with Zeeman-split carrier spin subbands in an ap-plied magnetic?eld.

8It has been shown that electrons photoemitted from ferro-magnetic gadolinium remain spin polarized(Busch et al., 1969).

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split quasiparticle density of states in a superconductor (Tedrow et al.,1970;Fulde,1973)was used as a detector of spin polarization of conduction electrons in various

magnetic materials.Jullie

`re (1975)measured tunneling conductance of F/I/F junctions,where I was an amor-phous Ge.By adopting the Tedrow and Meservey (1971b,1973)analysis of the tunneling conductance

from F/I/S to the F/I/F junctions,Jullie

`re (1975)formu-lated a model for a change of conductance between the parallel (↑↑)and antiparallel (↑↓)magnetization in the two ferromagnetic regions F1and F2,as depicted in Fig.2.The corresponding tunneling magnetoresistance 9(TMR)in an F/I/F magnetic tunnel junction (MTJ)is de?ned as

TMR ?

?R R ↑↑?R ↑↓?R ↑↑R ↑↑?G ↑↑?G ↑↓

G ↑↓

,(1)

where conductance G and resistance R ?1/G are la-beled by the relative orientations of the magnetizations in F1and F2(it is possible to change the relative orien-tations,between ↑↑and ↑↓,even at small applied mag-netic ?elds ?10G).TMR is a particular manifestation of a magnetoresistance that yields a change of electrical resistance in the presence of an external magnetic ?eld.10Historically,the anisotropic magnetoresistance in bulk ferromagnets such as Fe and Ni was discov-ered ?rst,dating back to the experiments of Lord Kelvin (Thomson,1857).Due to spin-orbit interaction,electri-cal resistivity changes with the relative direction of the

charge current (for example,parallel or perpendicular)

with respect to the direction of magnetization.

Within Jullie

`re’s model,which assumes constant tun-neling matrix elements and that electrons tunnel without spin ?ip,Eq.(1)yields

TMR ?2P 1P 2

1?P 1P 2

,

(2)

where the polarization P i ?(N Mi ?N mi )/(N Mi ?N mi )is expressed in terms of the spin-resolved density of states N Mi and N mi ,for majority and minority spin in F i ,re-spectively.Conductance in Eq.(1)can then be expressed

as (Maekawa and Ga

¨fvert,1982)G ↑↑?N M 1N M 2?N m 1N m 2and G ↑↓?N M 1N m 2?N m 1N M 2to give Eq.

(2).11While the early results of Jullie

`re (1975)were not con?rmed,TMR at 4.2K was observed using NiO as a

tunnel barrier by Maekawa and Ga

¨fvert (1982).The prediction of Jullie

`re’s model illustrates the spin-valve effect :the resistance of a device can be changed by manipulating the relative orientation of the magnetiza-tions M 1and M 2,in F1and F2,respectively.Such ori-entation can be preserved even in the absence of a power supply,and the spin-valve effect,12later discov-ered in multilayer structures displaying the giant magnetoresistance 13(GMR)effect (Baibich et al.,1988;Binasch et al.,1989)can be used for nonvolatile memory applications (Hartman,2000;Hirota et al.,2002;Parkin,

9

Starting with Jullie

`re (1975)an equivalent expression (G ↑↑?G ↑↓)/G ↑↑has also been used by different authors and is often referred to as junction magnetoresistance (Moodera and Mathon,1999).10

The concept of TMR was proposed independently by R.C.Barker in 1975[see Meservey et al.(1983)]and by Slonczewski (1976),who envisioned its use for magnetic bubble memory (Parkin,2002).

11

In Sec.IV we address some limitations of the Jullie

`re model and its potential ambiguities to identify precisely which spin polarization is actually measured.12

The term was coined by Dieny et al.(1991)in the context of GMR,by invoking an analogy with the physics of the TMR.13

The term ‘‘giant’’re?ected the magnitude of the effect (more than ?10%),as compared to the better known aniso-tropic magnetoresistance (?

1%).

FIG. 2.(Color in online edi-tion)Schematic illustration of

electron tunneling in ferro-magnet /insulator /ferromagnet (F/I/F)tunnel junctions:(a)Parallel and (b)antiparallel orientation of magnetizations with the corresponding spin-resolved density of the d states in ferromagnetic metals that have exchange spin splitting ?ex .Arrows in the two ferro-magnetic regions are deter-mined by the majority-spin sub-band.Dashed lines depict spin-conserved tunneling.

326

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ˇutic ′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April 2004

2002).GMR structures are often classi?ed according to whether the current?ows parallel(CIP,current in plane) or perpendicular(CPP,current perpendicular to the plane)to the interfaces between the different layers,as depicted in Fig.3.Most of the GMR applications use the CIP geometry,while the CPP version,?rst realized by Pratt et al.(1991),is easier to analyze theoretically(Gijs and Bauer,1997;Levy and Mertig,2002)and relates to the physics of the tunneling magnetoresistance effect (Mathon and Umerski,1997).The size of magnetoresis-tance in the GMR structures can be expressed analo-gously to Eq.(1),where parallel and antiparallel orien-tations of the magnetizations in the two ferromagnetic regions are often denoted by‘‘P’’and‘‘AP,’’respectively (instead of↑↑and↑↓).Realization of a large room-temperature GMR(Parkin,Bhadra,and Roche,1991; Parkin,Li,and Smith,1991)enabled a quick transition from basic physics to commercial applications in mag-netic recording(Parkin,Jiang,et al.,2003).

One of the keys to the success of magnetoresistance-based applications is their ability to control14the relative orientation of M1and M2.An interesting realization of such control was proposed independently by Berger (1996)and Slonczewski(1996).While in GMR or TMR structures the relative orientation of magnetizations will affect the?ow of spin-polarized current,they predicted a reverse effect.The?ow of spin-polarized current can transfer angular momentum from carriers to ferromag-net and alter the orientation of the corresponding mag-netization,even in the absence of an applied magnetic ?eld.This phenomenon,known as spin-transfer torque, has since been extensively studied both theoretically and experimentally(Bazaliy et al.,1998;Tsoi et al.,1998;My-ers et al.,1999;Sun,2000;Waintal et al.,2000;Stiles and Zangwill,2002),and current-induced magnetization re-versal has been demonstrated at room temperature(Ka-tine et al.,2000).It was also shown that the magnetic ?eld generated by passing the current through a CPP giant magnetoresonance device could produce room-temperature magnetization reversal(Bussmann et al., 1999).In the context of ferromagnetic semiconductors additional control of magnetization was demonstrated optically,by shining light(Koshihara et al.,1997; Boukari et al.,2002;Oiwa et al.,2002)and electrically, by applying gate voltage(Ohno,Chiba,et al.,2000; Boukari et al.,2002;Park et al.,2002)to perform switching between the ferromagnetic and paramagnetic states.

Jullie`re’s model also justi?es the continued quest for highly spin-polarized materials—they would provide large magnetoresistive effects,desirable for device appli-cations.In an extreme case,spins would be completely polarized even in the absence of magnetic?eld.Numeri-cal support for the existence of such materials—the so-called half-metallic ferromagnets15—was provided by de Groot,Janner,and Mueller(1983),and these materials were reviewed by Pickett and Moodera(2001).In addi-tion to ferromagnets,such as CrO2(Soulen et al.,1998; Parker et al.,2002)and manganite perovskites(Park et al.,1998a),there is evidence for high spin polarization in III-V ferromagnetic semiconductors like(Ga,Mn)As (Braden et al.,2003;Panguluri,Nadgorny,et al.,2003). The challenge remains to preserve such spin polarization above room temperature and in junctions with other materials,since the surface(interface)and bulk mag-netic properties can be signi?cantly different(Fisher, 1967;Mills,1971;Falicov et al.,1990).

While many existing spintronic applications(Hart-man,2000;Hirota et al.,2002)are based on the GMR effects,the discovery of large room-temperature TMR (Miyazaki and Tezuka,1995;Moodera et al.,1995)has renewed interest in the study of magnetic tunnel junc-tions,which are now the basis for the several magnetic random-access memory prototypes16(Parkin,Roche, et al.,1999;Tehrani et al.,2000).Future generations of magnetic read heads are expected to use MTJ’s instead of CIP giant magnetoresonance.To improve the switch-ing performance of related devices it is important to re-duce the junction resistance,which determines the RC time constant of the MTJ cell.Consequently,semicon-ductors,which would provide a lower tunneling barrier than the usually employed oxides,are being investigated both as the nonferromagnetic region in MTJ’s and as the basis for an all-semiconductor junction that would dem-

onstrate large TMR at low temperatures(Tanaka and Higo,2001;Tanaka,2002).Another desirable property of semiconductors has been demonstrated by the ex-traordinary large room-temperature magnetoresistance in hybrid structures with metals,reaching750000%at a magnetic?eld of4T(Solin et al.,2000),which could lead to improved magnetic read heads(Solin et al.;2002; Moussa et al.,2003).Magnetoresistance effects of similar magnitude have also been found in hybrid metal/ semiconductor granular?lms(Akinaga,2002).Another approach to obtaining large room-temperature magne-toresistance(?100%at B?100G)is to fabricate ferro-magnetic regions separated by a nanosize contact.For simplicity,such a structure could be thought of as the

14For example,with small magnetic?eld(Parkin,2002)or at high switching speeds(Schumacher et al.,2003a,2003b).

15Near the Fermi level they behave as metals only for one spin,the density of states vanishes completely for the other spin.

16Realization of the early magnetic random-access memory proposals used the effect of anisotropic magnetoresistance (Pohn et al.,1987,1988).

FIG.3.(Color in online edition)Schematic illustration of(a) the current in plane(CIP),(b)the current perpendicular to the plane(CPP)giant magnetoresistance geometry.327

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limiting case of the CPP giant magnetoresonance scheme in Fig.3(b).This behavior,also known as ballis-tic magnetoresistance,has already been studied in a large number of materials and geometries(Bruno,1999;Gar-cia et al.,1999;Tatara et al.,1999;Imamura et al.,2000; Versluijs et al.,2001;Chung et al.,2002).

2.Spin injection and optical orientation

Many materials in their ferromagnetic state can have a substantial degree of equilibrium carrier spin polariza-tion.However,as illustrated in Fig.1,this alone is usu-ally not suf?cient for spintronic applications,which typi-cally require current?ow and/or manipulation of the nonequilibrium spin(polarization).17The importance of generating nonequilibrium spin is not limited to device applications;it can also be used as a sensitive spectro-scopic tool to study a wide variety of fundamental prop-erties ranging from spin-orbit and hyper?ne interactions (Meier and Zakharchenya,1984)to the pairing symme-try of high-temperature superconductors(Vas’ko et al., 1997;Wei et al.,1999;Tsuei and Kirtley,2000;Ngai et al.,

2004)and the creation of spin-polarized beams to mea-sure parity violation in high-energy physics(Pierce and Celotta,1984).

Nonequilibrium spin is the result of some source of pumping arising from transport,optical,or resonance methods.Once the pumping is turned off the spin will return to its equilibrium value.While for most applica-tions it is desirable to have long spin relaxation times,it has been demonstrated that short spin relaxation times are useful in the implementation of fast switching(Nish-ikawa et al.,1995).

Electrical spin injection,an example of a transport method for generating nonequilibrium spin,has already been realized experimentally by Clark and Feher(1963), who drove a direct current through a sample of InSb in the presence of a constant applied magnetic?eld.The principle was based on the Feher effect,18in which the hyper?ne coupling between the electron and nuclear spins,together with different temperatures representing electron velocity and electron spin populations,is re-sponsible for the dynamical nuclear polarization(Slich-ter,1989).19Motivated by the work of Clark and Feher (1963)and Tedrow and Meservey(1971b,1973)and the principle of optical orientation(Meier and Zakharch-enya,1984),Aronov(1976a,1976b),and Aronov and Pikus(1976)established several key concepts in electri-cal spin injection from ferromagnets into metals, semiconductors,20and superconductors.Aronov(1976b) predicted that,when a charge current?owed across the F/N junction(Fig.4),spin-polarized carriers in a ferro-magnet would contribute to the net current of magneti-zation entering the nonmagnetic region and would lead

to nonequilibrium magnetization?M,depicted in Fig. 4(b),with the spatial extent given by the spin diffusion length(Aronov,1976b;Aronov and Pikus,1976).21Such

a?M,which is also equivalent to a nonequilibrium spin accumulation,was?rst measured in metals by Johnson

and Silsbee(1985,1988d).In the steady state?M is re-

17Important exceptions are tunneling devices operating at low bias and near equilibrium spin.Equilibrium polarization and the current?ow can be potentially realized,for example,in spin-triplet superconductors and thin-?lm ferromagnets (Ko¨nig et al.,2001),accompanied by dissipationless spin d72efe37f111f18583d05a9ding an analogy with the quantum Hall effect,it has been suggested that the spin-orbit interaction could lead to dissipationless spin currents in hole-doped semiconductors (Murakami et al.,2003).Rashba(2003b)has pointed out that similar dissipationless spin currents in thermodynamic equilib-rium,due to spin-orbit interaction,are not transport currents which could be employed for transporting spins and spin injec-tion.It is also instructive to compare several earlier proposals that use spin-orbit coupling to generate spin currents,dis-cussed in Sec.II.A.

18The importance and possible applications of the Feher ef-fect(Feher,1959)to polarize electrons was discussed by Das Sarma et al.(2000c)and Suhl(2002).

19Such an effect can be thought of as a generalization of the Overhauser effect(Overhauser,1953b),in which the use of a resonant microwave excitation causes the spin relaxation of the nonequilibrium electron population through hyper?ne cou-pling to lead to the spin polarization of nuclei.Feher(1959) suggested several other methods,instead of microwave excita-tion,that could produce a nonequilibrium electron population and yield a dynamical polarization of nuclei(see also Weger, 1963).

20In an earlier work,spin injection of minority carriers was proposed in a ferromagnet/insulator/p-type semiconductor structure.Measuring polarization of electroluminescence was suggested as a technique for detecting injection of polarized carriers in a semiconductor(Scifres et al.,1973).

21Supporting the?ndings of Clark and Feher(1963),Aronov calculated that the electrical spin injection would polarize nu-clei and lead to a measurable effect in the electron spin reso-nance.Several decades later related experiments on spin injec-tion are also examining other implications of dynamical nuclear polarization(Johnson,2000;Strand et al.,

2003). FIG.4.(Color in online edition)Pedagogical illustration of the concept of electrical spin injection from a ferromagnet(F)into a normal metal(N).Electrons?ow from F to N:(a)schematic device geometry;(b)magnetization M as a function of position—nonequilibrium magnetization?M(spin accumula-tion)is injected into a normal metal;(c)contribution of differ-ent spin-resolved densities of states to both charge and spin transport across the F/N interface.Unequal?lled levels in the density of states depict spin-resolved electrochemical poten-tials different from the equilibrium value?0.

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alized as the balance between spins added by the mag-netization current and spins removed by spin relaxation.22

Generation of nonequilibrium spin polarization and spin accumulation is also possible by optical methods known as optical orientation or optical pumping.In op-tical orientation,the angular momentum of absorbed circularly polarized light is transferred to the medium. Electron orbital momenta are directly oriented by light and through spin-orbit interaction electron spins be-come polarized.In Sec.II.B we focus on optical orien-tation in semiconductors,a well-established technique (Meier and Zakharchenya,1984).In a pioneering work Lampel(1968)demonstrated that spins in silicon can be optically oriented(polarized).This technique is derived from the optical pumping proposed by Kastler(1950)in which optical irradiation changes the relative popula-tions within the Zeeman and hyper?ne levels of the ground states of atoms.While there are similarities with previous studies of free atoms(Cohen-Tannoudji and Kostler,1966;Happer,1972),optical orientation in semi-conductors has important differences related to the strong coupling between the electron and nuclear spin and the macroscopic number of particles(Paget et al., 1977;Meier and Zakharchenya,1984;Hermann et al., 1985).Polarized nuclei can exert large magnetic?elds (?5T)on electrons.In bulk III-V semiconductors,such as GaAs,optical orientation can lead to50%polariza-tion of electron density,which could be further en-hanced in quantum structures of reduced dimensionality or by applying a strain.A simple reversal in the polar-ization of the illuminating light(from positive to nega-tive helicity)also reverses the sign of the electron den-sity d72efe37f111f18583d05a9dbining these properties of optical orientation with semiconductors tailored to have a nega-tive electron af?nity allows photoemission of spin-polarized electrons to be used as a powerful detection technique in high-energy physics and for investigating surface magnetism(Pierce and Celotta,1984).

II.GENERATION OF SPIN POLARIZATION

A.Introduction

Transport,optical,and resonance methods(as well as their combination)have all been used to create nonequi-librium spin.After introducing the concept of spin po-larization in solid-state systems we give a pedagogical picture of electrical spin injection and detection of po-larized carriers.While electrical spin injection and opti-cal orientation will be discussed in more detail later in this section,we also survey here several other tech-niques for polarizing carriers.

Spin polarization not only of electrons,but also of holes,nuclei,and excitations can be de?ned as P X?X s/X,(3) the ratio of the difference X s?X??X??,and the sum X?X??X??of the spin-resolved?components for a particular quantity X.To avoid ambiguity as to what precisely is meant by spin polarization,both the choice of the spin-resolved components and the relevant physi-cal quantity X need to be speci?ed.Conventionally,?is taken to be↑or?(numerical value?1)for spin up, and↓or?(numerical value?1)for spin down,with respect to the chosen axis of quantization.23In ferro-magnetic metals it is customary to refer to↑(↓)as car-riers with magnetic moment parallel(antiparallel)to the magnetization or,equivalently,as carriers with majority or minority spin(Tedrow and Meservey,1973).In semi-conductors the terms majority and minority usually refer to relative populations of the carriers while↑or?and↓or?correspond to the quantum numbers m j with re-spect to the z axis taken along the direction of the light propagation or along the applied magnetic?eld(Meier and Zakharchenya,1984;Jonker et al.,2003).It is im-portant to emphasize that both the magnitude and the sign of the spin polarization in Eq.(3)depend on the choice of X,relevant to the detection technique em-ployed,say optical vs transport and bulk vs surface mea-surements(Mazin,1999;Jonker et al.,2003).Even in the same homogeneous material the measured P X can vary for different X,and it is crucial to identify which physi-cal quantity—charge current,carrier density,conductiv-ity,or the density of states—is being measured experi-mentally.

The spin polarization of electrical current or carrier density,generated in a nonmagnetic region,is typically used to describe the ef?ciency of electrical spin injec-tion.Silsbee(1980)suggested that the nonequilibrium density polarization in the N region,or equivalently the nonequilibrium magnetization,acts as the source of spin electromotive force(emf)and produces a measurable ‘‘spin-coupled’’voltage V s??d72efe37f111f18583d05a9ding this concept,also referred to as spin-charge coupling,Silsbee(1980)pro-posed a detection technique consisting of two ferromag-nets F1and F2(see Fig.5)separated by a nonmagnetic region.24F1serves as the spin injector(spin aligner)and F2as the spin detector.This could be called the polarizer-analyzer method,the optical counterpart of the transmission of light through two optical linear po-larizers.From Fig.5it follows that the reversal of the magnetization direction in one of the ferromagnets

22The spin diffusion length is an important quantity for CPP giant magnetoresonance.The thickness of the N region in Fig. 3should not exceed the spin diffusion length,otherwise the information on the orientation of the magnetization in F1will not be transferred to the F2region.

23For example,along the spin angular momentum,applied magnetic?eld,magnetization,or direction of light propaga-tion.

24A similar geometry was also proposed independently by de Groot,Janner,and Mueller(1983),where F1and F2were two half-metallic ferromagnets,with the goal of implementing spin-based devices to amplify and/or switch current.

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would lead either to V s →?V s ,in an open circuit (in the limit of large impedance Z ),or to the reversal of charge current j →?j ,in a short circuit (at small Z ),a conse-quence of Silsbee-Johnson spin-charge coupling (Sils-bee,1980;Johnson and Silsbee,1987,1988a).Corre-spondingly,as discussed in the following sections,spin injection could be detected through the spin accumula-tion signal as either a voltage or a resistance change when the magnetizations in F1and F2are changed from parallel to antiparallel alignment.

Since the experiments demonstrating the spin accu-mulation of conduction electrons in metals (Johnson and Silsbee,1985),spin injection has been realized in a wide range of materials.While in Sec.II.C we focus on re-lated theoretical work motivated by potential applica-tions,experiments on spin injection have also stimulated proposals for examining the fundamental properties of electronic systems.25

The generation of nonequilibrium spin polarization has a long tradition in magnetic resonance methods (Abragam,1961;Slichter,1989).However,transport

methods to generate carrier spin polarization are not limited to electrical spin injection.For example,they also include scattering of unpolarized electrons in the presence of spin-orbit coupling (Mott and Massey,1965;Kessler,1976)and in materials that lack inversion sym-metry (Levitov et al.,1984),adiabatic (Mucciolo et al.,2002;Sharma and Chamon,2003;Watson et al.,2003)and nonadiabatic quantum spin pumping (Zheng et al.,2003;for an instructive description of parametric pump-ing see Brouwer,1998),and proximity effects (Ciuti et al.,2002a).

It would be interesting to know what the limits are on the magnitude of various spin polarizations.Could we have a completely polarized current ?P j →?;see Eq.(3)],with only a spin current (j ↑?j ↓)and no charge cur-rent (j ↑?j ↓?0)?While it is tempting to recall the Stern-Gerlach experiment and try to set up magnetic drift through inhomogeneous magnets (Kessler,1976),this would most likely work only as a transient effect (Fabian and Das Sarma,2002).It was proposed by D’yakonov and Perel’(1971a,1971c)that a transverse spin current (and transverse spin polarization in a closed sample)would form as a result of spin-orbit coupling-induced skew scattering in the presence of a longitudinal electric ?eld.This interesting effect,also called the spin Hall effect (Hirsch,1999;Zhang,2000),has yet to be demon-strated.An alternative scheme for producing pure spin currents was proposed by Bhat and Sipe (2000),moti-vated by the experimental demonstration of phase-coherent control of charge currents (Atanasov et al.,

1996;Hache

′et al.,1997)and carrier population (Fraser et al.,1999).A quantum-mechanical interference be-tween one-and two-photon absorptions of orthogonal linear polarizations creates an opposite ballistic ?ow of spin-up and spin-down electrons in a semiconductor.Only a spin current can ?ow without a charge current,as

demonstrated by Stevens et al.(2003)and Hu

¨bner et al.(2003),who were able to achieve coherent control of the spin current direction and magnitude by the polarization and relative phase of two exciting laser light ?elds.

Charge current also can be driven by circularly polar-ized light (Ivchenko and Pikus,1997).Using the prin-ciples of optical orientation (see Sec.I.B.2and further discussion in Sec.II.B)in semiconductors of reduced dimensionality or lower symmetry,both the direction and the magnitude of a generated charge current can be controlled by circular polarization of the light.This is called the circular photovoltaic effect (Ganichev and Prettl,2003),which can be viewed as a transfer of the angular momentum of photons to directed motion of electrons.This could also be called a spin corkscrew ef-fect,since a nice mechanical analog is a corkscrew whose rotation generates linear directed motion.A re-lated effect,in which spin photocurrent is driven,is called the spin-galvanic effect (Ganichev and Prettl,2003).The current here is caused by the difference in spin-?ip scattering rates for electrons with different spin states in some systems with broken inversion symmetry.A comprehensive survey of the related effects,from the circular photogalvanic effect (Asnin et al.,1979)to re-cent demonstrations in semiconductor quantum wells

25

For example,studies probing the spin-charge separation in the non-Fermi liquids have been proposed by Kivelson and Rokhsar (1990);Zhao and Hersh?eld (1995);Si (1997,1998);Balents and Egger (2000,2001).Spin and charge are carried by separate excitations and can lead to spatially separated spin and charge currents (Kivelson and Rokhsar,

1990).

FIG.5.(Color in online edition)Spin injection,spin accumu-lation,and spin detection:(a)two idealized completely polar-ized ferromagnets F1and F2(the spin-down density of states

N ↓is zero at the electrochemical potential energy E ??0)with parallel magnetizations are separated by the nonmagnetic re-gion N;(b)density-of-states diagrams for spin injection from F1into N,accompanied by the spin accumulation-generation of nonequilibrium magnetization ?M .At F2in the limit of low impedance (Z ?0)electrical spin is detected by measuring the spin-polarized current across the N/F2interface.In the limit of high impedance (Z ??)spin is detected by measuring the voltage V s ??M developed across the N/F2interface;(c)spin accumulation in a device in which a superconductor (with the superconducting gap ?)is occupying the region between F1and F2.

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(Ganichev et al.,2001;Ganichev,Danilov,et al.,2002; Ganichev,Ivchenko,et al.,2002;Ganichev et al.,2003), is given by Ganichev and Prettl(2003).

There is a wide range of recent theoretical proposals for devices that would give rise to a spin electromotive force(Zˇutic′et al.,2001a,2001b;Brataas et al.,2002; Governale et al.,2003;Long et al.,2003;Mal’shukov et al.,2003;Ting and Cartoixa`,2003),often referred to as spin(-polarized)pumps,cells,or batteries.However, even when it is feasible to generate pure spin current, this does not directly imply that it would be dissipation-less.In the context of superconductors,it has been shown that Joule heating can arise from pure spin cur-rent?owing through a Josephson junction(Takahashi et al.,2001).

B.Optical spin orientation

In a semiconductor the photoexcited spin-polarized electrons and holes exist for a time?before they recom-bine.If a fraction of the carriers’initial orientation sur-vives longer than the recombination time,that is,if???s,26where?s is the spin relaxation time(see Sec.III),

the luminescence(recombination radiation)will be par-tially polarized.By measuring the circular polarization of the luminescence it is possible to study the spin dy-namics of the nonequilibrium carriers in semiconductors (Oestreich et al.,2002)and to extract such useful quan-tities as the spin orientation,the recombination time,or the spin relaxation time of the carriers(Parsons,1969; Ekimov and Safarov,1970;Garbuzov et al.,1971;Meier and Zakharchenya,1984).

We illustrate the basic principles of optical orientation by the example of GaAs,which is representative of a large class of III-V and II-VI zinc-blende semiconduc-tors.The band structure is depicted in Fig.6(a).The

band gap is E g?1.52eV at T?0K,while the spin split-off band is separated from the light and heavy hole

bands by?so?0.34eV.We denote the Bloch states ac-cording to the total angular momentum J and its projec-tion onto the positive z axis m j:?J,m j?.Expressing the wave functions with the symmetry of s,p x,p y,and p z orbitals as?S?,?X?,?Y?,and?Z?,respectively,the band wave functions can be written as listed in Table I(Pierce and Meier,1976,with minor typos removed;see also Kittel,1963).

To obtain the excitation(or recombination)probabili-

ties,consider photons arriving in the z direction.Let??represent the helicity of the exciting light.When we rep-resent the dipole operator corresponding to the??op-tical transitions as27?(X?iY)?Y1?1,where Y l m is the spherical harmonic,it follows from Table I that

??1/2,?1/2?Y11?3/2,?3/2??2

??1/2,1/2?Y11?3/2,?1/2??2?3(4) for the relative intensity of the??transition between the heavy(?m j?3/2?)and the light(?m j?1/2?)hole sub-bands and the conduction band.Other transitions are

analogous.The relative transition rates are indicated in

Fig.6(b).The same selection rules apply to the optical

orientation of shallow impurities(Parsons,1969;Eki-

mov and Safarov,1970).

The spin polarization of the excited electrons28de-pends on the photon energy??.For??between E g and E g??so,only the light and heavy hole subbands contribute.Denoting by n?and n?the density of elec-trons polarized parallel(m j?1/2)and antiparallel(m j ??1/2)to the direction of light propagation,we de?ne the spin polarization as(see Sec.II.A)

P n??n??n??/?n??n??.(5) For our example of the zinc-blende structure,

P n??1?3?/?3?1???1/2(6) is the spin polarization at the moment of photoexcita-tion.The spin is oriented against the direction of light propagation,since there are more transitions from the heavy hole than from the light hole subbands.The cir-cular polarization of the luminescence is de?ned as

26In Si this condition is not ful?lled.Instead of measuring the luminescence polarization,Lampel(1968)has used NMR to detect optical spin orientation.

27For an outgoing light in the?z direction the helicities are reversed.

28Although holes are initially polarized too,they lose spin orientation very fast,on the time scale of the momentum re-laxation time(see Sec.III.D.1).However,it was suggested that manipulating hole spin by short electric?eld pulses,between momentum scattering events,could be useful for ultrafast spin-tronic applications(Dargys,

2002).

FIG.6.Interband transitions in GaAs:(a)schematic band structure of GaAs near the center of the Brillouin zone(?point),where E g is the band gap and?so the spin-orbit split-ting;CB,conduction band;HH,valence heavy hole;LH,light hole;SO,spin-orbit split-off subbands;?6,7,8are the corre-sponding symmetries at the k?0point,or,more precisely,the irreducible representations of the tetrahedron group T d (Ivchenko and Pikus,1997);(b)selection rules for interband transitions between the m j sublevels for circularly polarized light??and??(positive and negative helicity).The circled numbers denote the relative transition intensities that apply for both excitations(depicted by the arrows)and radiative re-combinations.

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P circ??I??I??/?I??I??,(7) where I?is the radiation intensity for the helicity??. The polarization of the??photoluminescence is then

P circ??n??3n????3n??n??

?n??3n????3n??n??

??

P n

2

?

1

4

.(8)

If the excitation involves transitions from the spin split-off band,that is,if???E g??so,the electrons will not be spin polarized(P n?P circ?0),underlining the vi-tal role of spin-orbit coupling for spin orientation.On the other hand,Fig.6suggests that a removal of the heavy/light hole degeneracy can substantially increase P n(D’yakonov and Perel’,1984),up to the limit of com-plete spin polarization.An increase in P n and P circ in GaAs strained due to a lattice mismatch with a sub-strate,or due to con?nement in quantum well hetero-structures,has indeed been demonstrated(Vasilev et al., 1993;Oskotskij et al.,1997),detecting P n greater than 0.9.

While photoexcitation with circularly polarized light creates spin-polarized electrons,the nonequilibrium spin decays due to both carrier recombination and spin relax-ation.The steady-state degree of spin polarization de-pends on the balance between the spin excitation and decay.Sometimes a distinction is made(Pierce and Meier,1976;Meier and Zakharchenya,1984)between the terms optical spin orientation and optical spin pump-ing.The former term is used in relation to the minority carriers(such as electrons in p-doped samples)and rep-resents the orientation of the excited carriers.The latter term is reserved for the majority carriers(electrons in n-doped samples),representing spin polarization of the ‘‘ground’’state.Both spin orientation and spin pumping were demonstrated in the early investigations on p-GaSb(Parsons,1969)and p-and n-Ga0.7Al0.3As(Eki-mov and Safarov,1970,1971;Zakharchenya et al.,1971). Unless speci?ed otherwise,we shall use the term optical orientation to describe both spin orientation and spin pumping.

To derive the steady-state expressions for the spin po-larization due to optical orientation,consider the simple model of carrier recombination and spin relaxation(see Sec.IV.A.4)in a homogeneously doped semiconductor. The balance between direct electron-hole recombination and optical pair creation can be written as

r?np?n0p0??G,(9) where r measures the recombination rate,the electron and hole densities are n and p,with index zero denoting the equilibrium values,and G is the electron-hole pho-toexcitation rate.Similarly,the balance between spin re-laxation and spin generation is expressed by

rsp?s/?s?P n?t?0?G,(10) where s?n??n?is the electron spin density and P n(t

?0)is the spin polarization at the moment of photoex-citation,given by Eq.(5).Holes are assumed to lose their spin orientation very fast,so they are treated as unpolarized.The?rst term in Eq.(10)describes the dis-appearance of the spin density due to carrier recombi-nation,while the second term describes the intrinsic spin relaxation.From Eqs.(9)and(10)we obtain the steady-state electron polarization as(Zˇutic′et al.,2001b)

P n?P n?t?0?

1?n0p0/np

1?1/?s rp

.(11) In a p-doped sample p?p0,n?n0,and Eq.(11)gives P n?P n?t?0?/?1??/?s?,(12) where??1/rp0is the electron lifetime.29The steady-state polarization is independent of the illumination in-tensity,being reduced from the initial spin polarization P n(t?0).30The polarization of the photoluminescence is P circ?P n(t?0)P n(Parsons,1969).Early measure-ments of P n?0.42?0.08in GaSb(Parsons,1969)and P n?0.46?0.06in Ga0.7Al0.3As(Ekimov and Safarov, 1970)showed an effective spin orientation close to the maximum value of P n(t?0)?1/2for a bulk unstrained zinc-blende structure,indicating that?/?s?1.

For spin pumping in an n-doped sample,where n ?n0and p?p0,Eqs.(9)and(11)give(D’yakonov and Perel’,1971b)

P n?P n?t?0?/?1?n0/G?s?.(13) In contrast to the previous case,the carrier(now hole) lifetime??1/rn0has no effect on P n.However,P n de-pends on the photoexcitation intensity G,as expected for a pumping process.The effective carrier lifetime is ?J?n0/G,where J represents the intensity of the illu-minating light.If it is comparable to or shorter than?s, spin pumping is very effective.Spin pumping works be-cause the photoexcited spin-polarized electrons do not need to recombine with holes.There are plenty of un-polarized electrons in the conduction band available for recombination.The spin is thus pumped in to the elec-tron system.

29After the illumination is switched off,the electron spin den-sity,or equivalently the nonequilibrium magnetization,will de-crease exponentially with the inverse time constant1/T s?1/??1/?s(Parsons,1969).

30The effect of a?nite length for the light absorption on P

n

is discussed by Pierce and Celotta(1984).The absorption length ??1is typically a micron for GaAs.It varies with frequency roughly as?(??)?(???E g)1/2(Pankove,1971).

TABLE I.Angular and spin part of the wave function at?.

Symmetry?J,m j?Wave function

?6?1/2,1/2??S↑?

?1/2,?1/2??S↓?

?7?1/2,1/2???(1/3)1/2?(X?iY)↓?Z↑??

?1/2,?1/2??(1/3)1/2?(X?iY)↑?Z↓??

?8?3/2,3/2??(1/2)1/2(X?iY)↑?

?3/2,1/2??(1/6)1/2?(X?iY)↓?2Z↑??

?3/2,?1/2???(1/6)1/2?(X?iY)↑?2Z↓??

?3/2,?3/2??(1/2)1/2(X?iY)↓?

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When magnetic?eld B is applied perpendicular to the axis of spin orientation(transverse magnetic?eld),it will induce spin precession with the Larmor frequency ?L??B gB/?,where?B is the Bohr magneton and g is

the electron g factor.31The spin precession,together with the random character of carrier generation or dif-fusion,leads to the spin dephasing(see Sec.III.A.1). Consider spins excited by circularly polarized light(or by any means of spin injection)at a steady rate.In a steady state a balance between nonequilibrium spin gen-eration and spin relaxation is maintained,resulting in a net magnetization.If a transverse magnetic?eld is ap-plied,the decrease of the steady-state magnetization can have two sources:(a)spins which were excited at ran-dom time and(b)random diffusion of spins towards a detection region.Consequently,spins precess along the applied?eld acquiring random phases relative to those which were excited or have arrived at different times.As a result,the projection of the electron spin along the

exciting beam will decrease with the increase of trans-

verse magnetic?eld,leading to depolarization of the lu-

minescence.This is also known as the Hanle effect

(Hanle,1924),in analogy to the depolarization of the

resonance?uorescence of gases.The Hanle effect was

?rst measured in semiconductors by Parsons(1969).The

steady-state spin polarization of the precessing electron

spin can be calculated by solving the Bloch-Torrey equa-

tions(Bloch,1946;Torrey,1956),Eqs.(52)–(54)describ-

ing the spin dynamics of diffusing carriers.

In p-doped semiconductors the Hanle curve shows a

Lorentzian decrease of the polarization(Parsons,1969), P n(B)?P n(B?0)/(1??L T s)2,where P n(B?0)is the polarization at B?0from Eq.(12)and T s?1is the effec-tive spin lifetime given by1/T s?1/??1/?s;see footnote 29.Measurements of the Hanle curve in GaAlAs were

used by Garbuzov et al.(1971)to separately determine both?and?s at various temperatures.The theory of the Hanle effect in n-doped semiconductors was developed by D’yakonov and Perel’(1976),who showed the non-Lorentzian decay of the luminescence for the regimes of both low(?J/?s?1)and high(?J/?s?1)intensity of the exciting light.At high?elds P n(B)?1/B1/2,consistent with the experiments of Vekua et al.(1976)in Ga0.8Al0.2As,showing a Hanle curve different from the usual P n(B)?1/B2Lorentzian behavior(D’yakonov and Perel’,1984a).Recent?ndings on the Hanle effect in nonuniformly doped GaAs and reanalysis of some earlier studies are given by Dzhioev et al.(2003).

C.Theories of spin injection

Reviews on spin injection have covered materials ranging from semiconductors to high-temperature super-conductors and have addressed the implications for de-vice operation as well as for fundamental studies in solid-state systems.32In addition to degenerate conduc-tors,examined in these works,we also give results for nondegenerate semiconductors in which the violation of local charge neutrality,electric?elds,and carrier band bending require solving the Poisson equation.The nota-tion introduced here emphasizes the importance of dif-ferent(and inequivalent)spin polarizations arising in spin injection.

1.F/N junction

A theory of spin injection across a ferromagnet/ normal metal(F/N)interface was?rst offered by Aronov(1976b).Early work also included spin injection into a semiconductor(Sm;Aronov and Pikus,1976; Masterov and Makovskii,1979)and a superconductor (S;Aronov1976a).Spin injection in F/N junctions was subsequently studied in detail by Johnson and Silsbee (1987,1988a),33van Son et al.(1987),Valet and Fert (1993),Hersh?eld and Zhao(1997),and others.Here we follow the approach of Rashba(2000,2002b)and con-sider a steady-state34?ow of electrons along the x direc-tion in a three-dimensional(3D)geometry consisting of a metallic ferromagnet(region x?0)and a paramag-netic metal or a degenerate semiconductor(region x ?0).

The two regions,F and N,form a contact at x?0,as depicted in Fig.7.The relative magnitudes of three char-

31In our convention the g factor of free electrons is positive, g0?2.0023(Kittel,1996).

32See,for example,Osofsky(2000);Goldman et al.(1999, 2001);Johnson(2001,2002a);Maekawa et al.(2001);Jedema, Nijboer,et al.(2002);Schmidt and Molenkamp(2002);Tang et al.(2002);and Wei(2002).

33Johnson and Silsbee base their approach on irreversible thermodynamics and consider also the effects of a temperature gradient on spin-polarized transport,omitted in this section. 34Even some dc spin injection experiments are actually per-formed at low(audio-frequency)bias.Generalization to ac spin injection,with a harmonic time dependence,was studied by Rashba

(2002a).

FIG.7.(Color in online edition)Spatial variation of the elec-trochemical potentials near a spin-selective resistive interface at an F/N junction.At the interface x?0both the spin-resolved electrochemical potentials(??,??↑,↓,denoted with solid lines)and the average electrochemical potential(?F,?N,dashed lines)are discontinuous.The spin diffusion lengths L sF and L sN characterize the decay of?s??↑??↓(or equivalently the decay of spin accumulation and the nonequi-librium magnetization)away from the interface and into the bulk F and N regions,respectively.

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acteristic resistances per unit area35determine the de-gree of current polarization injected into a nonmagnetic material.These are the contact resistance r c and the two characteristic resistances r N and r F,each given by the ratio of the spin diffusion length and the effective bulk conductivity in the corresponding region.Two limiting cases correspond to the transparent limit,where r c→0, and the low-transmission limit,where r c?r N,r F.

Spin-resolved quantities are labeled by??1or↑for spin up,???1or↓for spin down along the chosen quantization axis.For a free electron,spin angular mo-mentum and magnetic moment are in opposite direc-tions,and what precisely is denoted by‘‘spin up’’varies in the literature(Jonker et al.,2003).Conventionally,in metallic systems(Tedrow and Meservey,1973;Gijs and Bauer,1997),spin up refers to carriers with majority spin.This means that the spin(angular momentum)of such carriers is antiparallel to the magnetization.Spin-resolved charge current(density)in a diffusive regime can be expressed as

j???????,(14) where??is conductivity and the electrochemical poten-tial is

????qD?/????n???,(15) with q proton charge,D?diffusion coef?cient,?n??n??n?0the change of electron density from the equilib-rium value for spin?,and?the electric potential.36

In the steady state the continuity equation is ?j???q??n???????n???????,(16)

and????is the average time for?ipping a?spin to a??spin.For a degenerate conductor37the Einstein relation is

???q2N?D?,(17) where???↑??↓and N?N↑?N↓is the density of d72efe37f111f18583d05a9ding a detailed balance N↑/?↑↓?N↓/?↓↑(Hersh-?eld and Zhao,1997;Kravchenko,2002)together with Eqs.(15)and(17),the continuity equation can be ex-pressed as

?j???q2

N↑N↓

N↑?N↓

??????

?s,(18)

where?s??↑↓?↓↑/(?↑↓??↓↑)is the spin relaxation time. Equation(18)implies the conservation of charge current j?j↑?j↓?const,while the spin counterpart,the differ-ence of the spin-polarized currents j s?j↑?j↓is position dependent.Other‘‘spin quantities,’’X s,unless explicitly de?ned,are analogously expressed with the correspond-ing(spin)polarization given by P X?X s/X.For ex-ample,the current polarization38P j?j s/j,generally dif-ferent from the density polarization P n?(n↑?n↓)/n,is related to the conductivity polarization P?as

P j?2??↑?↓/????s/j?P?,(19) where?s??↑??↓.In terms of the average electro-chemical potential??(?↑??↓)/2,P?further satis?es ????P???s/2?j/?.(20) From Eqs.(15)and(18)it follows that?s satis?es the diffusion equation(van Son et al.,1987;Valet and Fert, 1993;Hersh?eld and Zhao,1997;Schmidt et al.,2000)?2?s??s/L s2,(21) where the spin diffusion length is L s?(Dˉ?s)1/2with the spin-averaged diffusion coef?cient Dˉ?(?↓D↑??↑D↓)/??N(N↓/D↑?N↑/D↓)?d72efe37f111f18583d05a9ding Eq.(15)and the local charge quasineutrality?n↑??n↓?0shows that ?s is proportional to the nonequilibrium spin density ?s??n↑??n↓(s?s0??s?n↑?n↓),

?s?

1

2q

N↑?N↓

N↑N↓

?s.(22)

Correspondingly,?s is often referred to as the(nonequi-librium)spin accumulation39and is used to explain the GMR effect in CPP structures(Johnson,1991;Valet and Fert,1993;Gijs and Bauer,1997;Hartman,2000;Hirota et al.,2002).

The preceding equations are simpli?ed for the N re-gion by noting that????/2,?s?0,and D??Dˉ.Quan-tities pertaining to a particular region are denoted by the index F or N.

Equation(21)has also been used to study the diffu-sive spin-polarized transport and spin accumulation in ferromagnet/superconductor structures(Jedema et al., 1999).Some care is needed to establish the appropriate boundary conditions at the F/N interface.In the absence of spin-?ip scattering40at the F/N interface(which can arise,for example,due to spin-orbit coupling or mag-netic impurities),the spin current is continuous and thus P jF(0?)?P jN(0?)?P j(omitting x?0?for brevity,and superscripts?in other quantities).These boundary con-ditions were used by Aronov(1976b;Aronov and Pikus,

35For this simple geometry various resistances have a com-

mon factor of the cross-sectional area,which can be factored out.This is no longer possible for a more complicated geom-etry(Takahashi and Maekawa,2003).

36More generally,for a noncollinear magnetization,j

?be-comes a second-rank tensor(Johnson and Silsbee,1988a;Mar-gulis and Margulis,1994;Stiles and Zangwill,2002).

37In the nondegenerate case of Boltzmann statistics,the Ein-stein relation implies that the ratio of the diffusion coef?cient and the mobility is k B T/q.

38This is also referred to as a spin injection coef?cient (Rashba,2000,2002b).

39Spin accumulation is also relevant to a number of physical phenomena outside the scope of this article,for example,to the tunneling rates in the quantum Hall regime(Chan et al., 1999;MacDonald,1999).

40The effects of nonconserving interfacial scattering on spin injection were considered by Valet and Fert(1993),Fert and Lee(1996),and Rashba(2002b).

334Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

1976)without relating P j to the effect of the F/N contact or material parameters in the F region.

Unless the F/N contact is highly transparent,??is discontinuous across the interface(Johnson and Silsbee, 1988c;Valet and Fert,1993;Hersh?eld and Zhao,1997; Rashba,2000),and the boundary condition is

j??0???????N?0????F?0??,(23) where

???↑??↓(24) is the contact conductivity.For a free-electron model

?↑ ?↓can be simply inferred from the effect of the exchange energy,which would yield spin-dependent Fermi wave vectors and transmission coef?cients.A mi-croscopic determination of the corresponding contact resistance[see Eq.(27)]is complicated by the in?uence of disorder,surface roughness,and different scattering mechanisms and is usually obtained from model calcula-tions(Schep et al.,1997;Stiles and Penn,2000).Contin-ued work on the?rst-principles calculation of F/N inter-faces(Stiles,1996;Erwin et al.,2002)is needed for a more detailed understanding of spin injection.From Eqs.(23)and(24)it follows that

?sN?0???sF?0??2r c?P j?P??j,(25)?N?0???F?0??r c?1?P?P j?j,(26) where the effective contact resistance is

r c??/4?↑?↓.(27) The decay of?s,away from the interface,is character-ized by the corresponding spin diffusion length ?sF??sF?0?e x/L sF,?sN??sN?0?e?x/L sN.(28) A nonzero value for?sN(0)implies the existence of nonequilibrium magnetization?M in the N region(for noninteracting electrons q?s??B?M/?,where?is the magnetic susceptibility).Such a?M,as a result of elec-trical spin injection,was proposed by Aronov and Pikus (1976)and?rst measured in metals by Johnson and Sils-bee(1985).

By applying Eq.(19),separately,to the F and N re-gions,one can obtain the amplitude of spin accumula-tion in terms of the current and density-of-states spin polarization and the effective resistances r F and r N,?sF?0??2r F?P j?P?F?j,?sN?0???2r N P j j,(29) where

r N?L sN/?N,r F?L sF?F/?4?↑F?↓F?.(30) From Eqs.(29)and(25)the current polarization can be obtained as

P j??r c P??r F P?F?/r FN,(31) where r FN?r F?r c?r N is the effective equilibrium resis-tance of the F/N junction.It is important to emphasize that a measured highly polarized current,representing an ef?cient spin injection,does not itself imply a large spin accumulation or a large density polarization,typi-cally measured by optical techniques.In contrast to the derivation of P j from Eq.(31),determining P n requires

using Poisson’s equation or a condition of the local

charge quasineutrality.41

It is useful to note42that Eq.(31),written as Eq.(18)

in Rashba(2000),can be mapped to Eq.(A11)from

Johnson and Silsbee(1987),where it was?rst derived.43

An equivalent form for P j in Eq.(31)was obtained by

Hersh?eld and Zhao(1997)and for r c?0results from van Son et al.(1987)are recovered.

In contrast to normal metals(Johnson and Silsbee,

1985,1988d)and superconductors,for which injection

has been reported in both conventional(Johnson,1994),

and high-temperature superconductors(Hass et al., 1994;Dong et al.,1997;Vas’ko et al.,1997;Yeh et al., 1999),creating a substantial current polarization P j by direct electrical spin injection from a metallic ferromag-net into a semiconductor proved to be more dif?cult (Hammar et al.,1999;Monzon and Roukes,1999;Filip et al.,2000;Zhu et al.,2001).

By examining Eq.(31)we can both infer some pos-sible limitations and deduce several experimental strat-egies for effective spin injection into semiconductors. For a perfect Ohmic contact r c?0,the typical resistance mismatch r F?r N(where F is a metallic ferromagnet) implies inef?cient spin injection with P j?r F/r N?1,re-ferred to as the conductivity mismatch problem by Schmidt et al.(2000).Even in the absence of the resis-tive contacts,effective spin injection into a semiconduc-tor can be achieved if the resistance mismatch is reduced by using for spin injectors either a magnetic semiconduc-tor or a highly spin-polarized ferromagnet.44

While there was early experimental evidence(Alva-rado and Renaud,1992)that employing resistive(tun-neling)contacts could lead to an ef?cient spin injection,45a systematic understanding was provided by Rashba(2000)and supported by subsequent experimen-tal and theoretical studies(Fert and Jaffres,2001;Smith and Silver,2001;Rashba,2002b;Johnson,2003;Johnson and Byers,2003;Takahashi and Maekawa,2003).As can

41Carrier density will also be in?uenced by the effect of screening,which changes with the dimensionality of the spin injection geometry(Korenblum and Rashba,2002).

42E.I.Rashba(2002d).

43The substitutions are P

j

→?*,P?→p,P?→?,r c→?G(???2)??1,r N→?n/?n?n,r F→?f/?f(?f?p f2),L sN,F→?n,F,and n,f label N and F regions,respectively.?,?n,and?f are of the order of unity.To ensure that resistances and the spin diffusion lengths in Johnson and Silsbee(1987)are positive,one must additionally have(???2)?0and(?i?p i2)?0,i?n,f(for nor-mal and ferromagnetic regions,respectively).In particular,as-suming???n??f?1,a detailed correspondence between Eq.

(31)and Eq.(A11)in Johnson and Silsbee(1987)is recovered. For example,r c→?G(???2)??1yields Eq.(27),where?→G.

44From Eq.(30)a half-metallic ferromagnet implies a large r F.

45The in?uence of the resistive contacts on spin injection can also be inferred by explicitly considering resistive contacts (Johnson and Silsbee,1987;Hersh?eld and Zhao,1997).

335

Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

be seen from Eq.(31),a spin-selective resistive contact r c?r F,r N(such as a tunnel or Schottky contact)would contribute to effective spin injection with P j?P?being dominated by the effect r c and not the ratio r F/r N.46 This limit is also instructive to illustrate the principle of spin?ltering(Esaki et al.,1967;Moodera et al.,1988; Hao et al.,1990;Filip et al.,2002).In a spin-discriminating transport process the resulting degree of spin polarization is changed.Consequently the effect of spin?ltering,similar to spin injection,leads to the gen-eration of(nonequilibrium)spin polarization.47For ex-ample,at low temperature EuS and EuSe,discussed in Sec.IV.C,can act as spin-selective barriers.In the ex-treme case,initially spin-unpolarized carriers(say,in-jected from a nonmagnetic material)via spin?ltering could attain a complete polarization.For a strong spin-?ltering contact P??P?F,the sign of the spin accumu-lation(nonequilibrium magnetization)is reversed in the F and N regions,near the interface[recall Eq.(25)],in contrast to the behavior sketched in Fig.7,where ?sF,N?0.

The spin injection process alters the potential drop across the F/N interface because differences of spin-dependent electrochemical potentials on either side of the interface generate an effective resistance?R.By in-tegrating Eq.(20)for N and F regions separately,it fol-lows that Rj??N(0)??F(0)?P?F?sF(0)/2,where R is the junction d72efe37f111f18583d05a9ding Eqs.(26),(30),and(31) allows us to express R?R0??R,where R0?1/?(R0?r c if?↑??↓)is the equilibrium resistance,in the ab-sence of spin injection,and

?R??r N?r F P?F2?r c P?2??r F r c?P?F?P??2?/r FN,

(32) where?R?0is the nonequilibrium resistance.Petukhov has shown(Jonker et al.,2003a)that Eqs.(31)and(32) could be obtained by considering an equivalent circuit scheme with two resistors R?↑,R?↓connected in parallel, where R???L sF/??F?1/???2L sN/?N and R?↑?R?↓?4r FN.For such a resistor scheme,by noting that j↑R?↑?j↓R?↓,Eq.(31)is obtained as P j??P R???(R?↑?R?↓)/(R?↑?R?↓).?R in Eq.(32)is then obtained as the difference between the total resistance of the nonequi-librium spin-accumulation region of the length L sF ?L sN[given by the equivalent resistance R?↑R?↓/(R?↑?R?↓)]and the equilibrium resistance for the same re-gion,L sF/?F?L sN/?N.

The concept of the excess resistance?R can also be explained as a consequence of the Silsbee-Johnson spin-charge coupling(Silsbee,1980;Johnson and Silsbee,1985,1987)and illustrated by considering the simpli?ed schemes in Figs.5and7.Accumulated spin near the F/N interface,together with a?nite spin relaxation and a ?nite spin diffusion,impedes the?ow of spins and acts as a‘‘spin bottleneck’’(Johnson,1991).A rise of?sN must be accompanied by the rise of?sF[their precise alignment at the interface is given in Eq.(25)]or there

will be a back?ow of the nonequilibrium spin back into

the F region.Because both spin and charge are carried

by electrons in spin-charge coupling,the back?ow of

spin driven by diffusion creates an additional resistance

for the charge?ow across the F/N interface.Based on an

analogy with the charge transport across a clean

N/superconductor(S)interface(see Sec.IV.A.3),van Son et al.(1987)explained?R by invoking the conse-quences of current conversion from spin-polarized,far

to the left of the F/N interface,to completely unpolar-

ized,at far right in the N region.

The increase in the total resistance with spin injection

can be most dramatic if the N region is taken to be a

superconductor(S);see Fig.5(c).Spin injection depletes

the superconducting condensate and can result in

switching to a normal state of much higher resistance

(Dong et al.,1997;Vas’ko et al.,1997;Takahashi et al.,

1999;Wei et al.,1999;Yeh et al.,1999).A critical review

of possible spurious effects in reported experiments

(Gim et al.,2001)has also stimulated the development

of a novel detection technique which uses scanning tun-

neling spectroscopy combined with pulsed quasiparticle

spin injection to minimize Joule heating(Ngai et al.,

2004;see Sec.IV.A.1).In the S region the quasiparticle energy is E k?(?k2??2)1/2,where?k is the single-particle excitation energy corresponding to the wave vector k and?is the superconducting gap[see Fig.5(c)].Such a dispersion relation results in a smaller diffusion coef?-cient and a longer spin-?ip time than in the N region, while their product,the spin diffusion length,remains the same(Yamashita et al.,2002).Consequently,Eq.

(21)also applies to the diffusive spin-polarized transport

and spin accumulation in ferromagnet/superconductor

structures(Jedema et al.,1999;Yamashita et al.,2002).

Opening of a superconducting gap implies that a super-

conductor is a low carrier system for spin,which is car-

ried by quasiparticles(Takahashi and Maekawa,2003).

In the preceding analysis,appropriate for bulk,homo-

geneous,three-dimensional N and F regions and degen-

erate(semi)conductors,Poisson’s equation was not in-voked and the local charge neutrality?n↑??n↓was used only to derive Eq.(22).48Focusing on bulk samples in which both the size of the F and N regions and the cor-responding spin diffusion lengths are much larger than the Debye screening length,one can?nd that the quasineutrality condition,combined with Eqs.(15)and (17),yields

46A similar result was stated previously by Johnson and Sils-

bee(1988a).

47While most of the schemes resemble a CPP geometry[Fig. 3(b)],there are also proposals for generating highly polarized currents in a CIP-like geometry[Fig.3(a)](Gurzhi et al.,2001, 2003).

48For spin injection in nondegenerate semiconductors(with the carriers obeying the Boltzmann statistics)there can be large effects due to built-in?elds and deviation from local charge neutrality,as discussed in Sec.II.C.3.

336Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

?????P N?s/2,(33) where the density-of-states spin polarization of P N van-ishes in the N region.At the contact x?0there is a potential drop,even when r c?0,which can be evaluated from Eqs.(26)and(33)as

?N?0???F?0???r c?1?P?P j?j?P N F?0??sF?0?/2.

(34) The creation of nonequilibrium spin in the N region re-sults in the spin emf in the F/N structure which can be used to detect electrical spin injection,as depicted in Fig.5.Within a simpli?ed semi-in?nite geometry for the F and N regions,we consider an effect of spin pumping in the N region,realized either by electrical spin injec-tion from another F region[as shown in Fig.5(b)]or by optical pumping(see Sec.II.B).The resulting potential drop can be calculated by modifying?sN in Eq.(28),?sN??sN??????sN?0???sN????e?x/L sN,(35) where?sN(?)represents the effect of homogeneous spin pumping in the N region.To calculate the open circuit voltage(j?0)the continuity of spin current at x?0should be combined with the fact that P j j?j s. From Eq.(19)it follows that

j s?0??2?↑?↓

?F

?sF?0?

L sF

??

1

2

?N

?sN?0???sN???

L sN

,

(36)

while the discontinuity of?s in Eq.(25)yields49?sF?0???r F/r FN??sN???,j s?0???sN???/2r FN,

?sN?0????r c?r F?/r FN??sN???.(37) By substituting this solution into Eq.(34),we can evalu-ate the contact potential drop as

?N?0???F?0???r F P N F?r c P???sN???/2r FN.(38) The total potential drop(recall j?0)at the F/N junction50is(Rashba,2002b)

??FN??N?????F?????P j?sN???/2.(39) where P j is given in Eq.(31).In the context of the spin-detection scheme from Fig.5and high impedance mea-surements at the N/F2junction,the spin-coupled voltage V s(Silsbee,1980;Johnson and Silsbee,1985)was also found to be proportional to current polarization and the spin accumulation(?s??s??M;Johnson and Silsbee, 1988b).

2.F/N/F junction

The above analysis of the F/N bilayer can be readily extended to the geometry in which two in?nite F regions are separated by an N region of thickness d.The quan-tities pertaining to the two ferromagnets are de?ned as in the case of an F/N junction and labeled by the super-scripts L and R(left and right regions,respectively).It follows from Eq.(19),by assuming the continuity of the spin current at L,R,that the difference of the spin-resolved electrochemical potential,responsible for the spin accumulation,is

?sF L?2r F L?P j L?P?F L?je x/L sF L,x?0,(40)

?sN?2r N?P j R cosh?x/L sN??P j L cosh??d?x?/L sN???j/sinh?d/L sN?,0?x?d,(41)

?sF R??2r F R?P j R?P?F R?je(d?x/L sF R),x?d,(42) where the current spin polarization P j L,R at the two con-tacts in the F/N/F geometry can be expressed(Rashba, 2002b)in terms of the P j calculated for F/N junction with the in?nite F and N regions in Eq.(19)and the appropriate effective resistances.By P j?L,R we denote the P j calculated in Eq.(31)for the left and right contact (with the appropriate parameters for the F/N/F junction) as if it were surrounded by the in?nite F and N regions. Analogously to the F/N junction,the consequence of the spin injection is an increase in the resistance R?R0??R,over the equilibrium value R0?(?L)?1?(?R)?1.The nonequilibrium resistance?R is also al-ways positive for spin-conserving contacts(Rashba,

2000,2002b),in agreement with experiments on all-semicondcutor trilayer structures(Schmidt et al.,2001; see Sec.II.D.3).

Many applications based on magnetic multilayers rely on the spin-valve effect,in which the resistance changes due to the relative orientations of the magnetization in the two F regions.The geometry considered here is rel-evant for CPP giant magnetoresistance(Gijs and Bauer, 1997;Bass and Pratt,1999;Parkin,2002)and the all-metallic spin injection of Johnson and Silsbee(1985).In particular,the resistance change between antiparallel and parallel magnetization orientations in the two ferro-magnets can be expressed using current polarization of an in?nite F/N junction P j?L,R(Rashba,2002b):

?R?R↑↓?R↑↑?4P j?L P j?R

r FN L r FN R r N

D sinh d/L sN,(43) where r F L,R,r c L,R,and r N are de?ned as in the case of an F/N junction and

D??r F L?r c L??r c R?r F R??r N2?r N?r F L?r c L?r c R

?r F R?coth?d/L sN?.(44) Up to a factor of2,Eq.(43)has also been obtained by Hersh?eld and Zhao(1997)using Onsager relations.In the limit of a thin N region,d/L sN→0,?R remains?-nite.In the opposite limit,for d?L sN,

49A missprint in?

sF (0)from Rashba(2002b)has been cor-

rected.

50A similar potential drop was also calculated across a ferro-magnetic domain wall(Dzero et al.,2003).337

Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

?R?P j?L P j?R exp??d/L sN?.(45) For a symmetric F/N/F junction,where r c,F L?r c,F R,it fol-lows that

?R?4r N?r c P??r F P?F?

D sinh?d/L sN?.(46)

Considering the spin injection from F into a ballistic N region in the presence of diffusive interfacial scattering, where the phase coherence is lost and the Boltzmann equation can be applied,it is instructive to reconsider the effect of contact resistance(Kravchenko and Rashba,2003).We introduce the Sharvin resistance R Sharvin(Sharvin,1965),arising in ballistic transport be-tween two in?nite regions connected by a contact(an ori?ce or a narrow and short constriction)of radius much smaller than the mean free path,a?l.In a3D geometry the resistance is

R Sharvin?

4?l

3?a2

??e2h k2A2???1,(47)

where h/e2?25.81k?is the quantum of resistance per spin,A is the contact area,and k is the Fermi wave vector.The opposite limit,of diffusive transport through the contact with a?l,corresponds to the Maxwell or Drude resistance R Maxwell??/2a.The studies of interme-diate cases provide an interpolation scheme between the R Maxwell and R Sharvin for various ratios of a/l(Wexler, 1966;Jansen et al.,1980;de Jong,1994;Nikolic′and Allen,1999).Following Kravchenko and Rashba(2003) the effective contact resistance r c?r c↑?r c↓(recall that it is de?ned per unit area)is obtained as

r c???4R Sharvin/A??1?t?L?t?R?/t?L,(48) where t?L,R represent the transmission coef?cients for electrons reaching the contact from the left and from the right and satisfy t L?t Rр1.For r c which would exceed the resistance of the N and F bulk regions,the spin injection ef?ciency can attain P j?(r c↑?r c↓)/r c (Kravchenko and Rashba,2003),showing,similarly to the diffusive regime,the importance of the resistive con-tacts to ef?cient spin injection.Connection with the re-sults in the diffusive regime can be obtained (Kravchenko and Rashba,2003)by identifying r c??1/4??,where the contact conductivity??was intro-duced in Eq.(24).

While most of the experimental results on spin injec-tion are feasible in the diffusive regime,there are many theoretical studies treating the ballistic case and phase-coherent transport in both F/N and F/N/F junctions(Hu and Matsuyama,2001;Hu,Nitta,et al.,2001;Mireles and Kirczenow,2001;Matsuyama et al.,2002).Simple models in which the N region is a degenerate semicon-ductor often adopt an approach developed?rst for charge transport in junctions involving superconductors, discussed in Sec.IV.A.3.Considering spin-orbit coupling and the potential scattering at the F/N interface mod-eled by the?function,Hu and Matsuyama(2001)have examined ballistic spin injection in the F/N junction. They show that even a spin-independent barrier can be used to enhance the spin injection and lead to an in-crease in the conductance polarization.First-principles calculations were also used for ballistic spin injection from a ferromagnetic metal into a semiconductor (Mavropoulos et al.,2002;Wunnicke et al.,2002;Zwi-erzycki et al.,2003).In the limit of coherent(specular) scattering51and high interfacial quality it was shown that different band structure in the F and the N regions would contribute to a signi?cant contact resistance and an ef?cient spin injection(Zwierzycki et al.,2003).

3.Spin injection through the space-charge region Interfaces making up a semiconductor often develop a space-charge region—a region of local macroscopic charges.Typical examples are the Schottky contact and the depletion layer in p-n junctions.While phenomeno-logical models,such as the one introduced in Sec.II.C.1, capture a remarkable wealth of spin injection physics, they carry little information about spin-dependent pro-cesses right at the interfaces.Microscopic studies of spin-polarized transport and spin-resolved tunneling through space-charge regions are still limited in scope. The dif?culty lies in the need to consider self-consistently simultaneous charge accumulation and electric-?eld generation(through Poisson’s equation), both affecting transport.Non-self-consistent analyses of a Schottky-barrier spin injection were performed by Al-brecht and Smith(2002,2003)and Prins et al.(1995), while Osipov and Bratkovsky(2003)proposed an ef?-cient spin injection method using a?-doped Schottky contact.

Let us now consider spin injection through the deple-tion layer in magnetic p-n junctions(Fabian et al., 2002a;Zˇutic′et al.,2002,2003).The physics is based on drift and diffusion52limited by carrier recombination and spin relaxation,as described in more detail in Sec. IV.A.4.The transport equations are solved self-consistently with Poisson’s equation,taking full account of electric?eld due to accumulated charges.Additional examples of magnetic p-n junctions are discussed in Sec. IV.D.

The system is depicted in Fig.8.The p-n junction has a magnetic n region53with a net equilibrium electron spin P n0R,where R stands for the right(here n)region. Holes are assumed to be unpolarized.An important is-sue to be resolved is whether there will be spin accumu-

51The wave-vector component along the interface is con-served during scattering.

52Tunneling or?eld emission becomes important,for ex-ample,in thin Schottky barriers or in p-n junctions and het-erostructures at large reverse biases(Kohda et al.,2001; Johnston-Halperin et al.,2002;Van Dorpe,Liu,et al.,2003). 53Equilibrium magnetization can be a consequence of doping with magnetic impurities,yielding large carrier g factors,and applying magnetic?eld,or of using a ferromagnetic semicon-ductor(Ohno,1998;Pearton et al.,2003).

338Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004

lation in the p region if a forward bias is applied to the

junction.In other words,will spin be injected across the depletion layer?Naively the answer is yes,since spin is carried by electrons,but the result shown in Fig.8sug-gests a more complicated answer.At small biases there is no spin injection.This is the normal limit of diode operation,in which the injected carrier density through the depletion region is still smaller than the equilibrium carrier density.Only with bias increasing to the high-injection limit (typically above 1V)is spin injected.The explanation for the absence of spin injection at small biases and for nondegenerate doping levels (Boltzmann statistics is applicable)is as follows.On the n side there are more spin-up than spin-down electrons,n ↑?n ↓.If 2q ?is the spin splitting of the conduction band,n ↑(?)/n ↑(??0)?exp(q ?/k B T ).Under a forward bias,electrons ?ow to the p region.The ?ow is limited by thermal activation over the barrier (given by the built-in electrostatic potential minus bias),which is,for the spin-up electrons,greater by q ?.For Boltzmann sta-tistics,the rate of transmission of spin-up electrons over the barrier is ?exp(?q ?/k B T ).Since current is propor-tional to both the carrier density and the transmission rate,the two exponential factors cancel out.Similarly for spin down.As a result,the spin-resolved current is unaffected by 2q ?and there is no spin current ?owing through the depletion layer.There is no spin accumula-tion.Spin injection appears only at large biases,where it is driven by electric drift leading to nonequilibrium spin population already in the n region (Fabian et al.,2002a;Z

ˇutic ′et al.,2002).In addition to spin injection,spin ex-traction has also been predicted in magnetic p -n junc-tions with a magnetic p region (Z

ˇutic ′et al.,2002).Under a large bias,spin is extracted (depleted)from the non-magnetic n region.

Electric ?eld in the bulk regions next to the space charge is important only at large biases.It affects not only spin density,but spin diffusion as well.That spin injection ef?ciency can increase in the presence of large electric ?elds due to an increase in the spin diffusion length (spin drag)was ?rst shown by Aronov and Pikus (1976)and was later revisited by other authors.54To be important,the electric ?eld needs to be very large,55more than 100V/cm at room temperature.While such large ?elds are usually present inside the space-charge regions,they exist in the adjacent bulk regions only at the high injection limit and affect transport and spin in-jection.In addition to electric drift,magnetic drift,in magnetically inhomogeneous semiconductors,can also enhance spin injection (Fabian et al.,2002a).

The following formula was obtained for spin injection at small biases (Fabian et al.,2002a):

P n L ?

P n 0L ?1??P n 0R ?2???P n R ?1?P n 0L P n 0R

?

1??P n 0R ?2??P n R ?P n 0L ?P n 0R

?

,(49)

where L (left)and R (right)label the edges of the space-charge (depletion)region of a p -n junction.Cor-respondingly,?P n

R

represents the nonequilibrium elec-tron polarization,evaluated at R ,arising from a spin

source.The case discussed in Fig.8is for P n 0L ??P n

R

?0.Then P n L

?0,in accord with the result of no spin injection.For a homogeneous equilibrium magnetiza-tion (P n 0L ?P n 0R ),?P n L ??P n R

;the nonequilibrium spin polarization is the same across the depletion layer.Equation (49)demonstrates that only nonequilibrium spin,already present in the bulk region,can be trans-ferred through the depletion layer at small biases (Z

ˇutic ′et al.,2001b;Fabian et al.,2002a).Spin injection of non-equilibrium spin is also very effective if it proceeds from

the p region (Z

ˇutic ′et al.,2001b),which is the case for a spin-polarized solar cell (Z

ˇutic ′et al.,2001a).The result-ing spin accumulation in the n region extends the spin diffusion range,leading to spin ampli?cation—increase of the spin population away from the spin source.These results were also con?rmed in the junctions with two differently doped n regions (Pershin and Privman,2003a,2003b).Note,however,that the term ‘‘spin polar-ization density’’used in Pershin and Privman (2003a,2003b)is actually the spin density s ?n ↑?n ↓,not the spin polarization P n .

54

See,for example,Margulis and Margulis (1994);Flensberg

et al.(2001);Z

ˇutic ′et al.(2001b);Fabian et al.(2002a);Yu and Flatte

′(2002a);Bratkovsky and Osipov (2003);Martin (2003);and Vignale and D’Amico (2003).55

The critical magnitude is obtained by piding a typical en-ergy,such as the thermal or Fermi energy,by q and by the spin diffusion length.At room temperature the thermal energy is 25meV ,while the spin diffusion length can be several

microns.

FIG.8.(Color in online edition)Spin injection through the

space-charge region of a magnetic p -n junction.The geometry is depicted in the inset,which shows a junction with a spin-split conduction band in the n region with spin-polarized electrons (solid circles)and unpolarized holes (empty circles).Under applied forward bias V the charge current ?ows to the right.The curves,labeled by V ,show the electron density polariza-tion pro?les P n (x )for the depicted geometry and GaAs ma-terials parameters.The equilibrium density polarization in the n region is about 0.5.At low bias (0.8V)there is no spin injection.Spin injection,manifested by the increase of P n in the p region,appears only at large biases (1.2and 1.5V),

where it is driven by electric drift (Z

ˇutic ′et al.,2002).Spin polarization of the current is discussed by Z ˇutic ′et al.(2001a)and Fabian et al.(2002a).Adapted from Z

ˇutic ′et al.,2002.339

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Theoretical understanding of spin injection has fo-cused largely on spin density while neglecting spin phase,which is important for some proposed spintronic applications.The problem of spin evolution in various transport modes (diffusion,tunneling,thermionic emis-sion)remains to be investigated.Particularly relevant is the question of whether spin phase is conserved during spin injection.Malajovich et al.(2001)showed,by studying spin evolution in transport through a n -GaAs/n -ZnSe heterostructure,that the phase can in-deed be preserved.

D.Experiments on spin injection 1.Johnson-Silsbee spin injection

The ?rst spin polarization of electrons by electrical spin injection (Johnson and Silsbee,1985)was demon-strated in a ‘‘bulk wire’’of aluminum on which an array of thin ?lm permalloy (Py)pads (with 70%nickel and 30%iron)was deposited spaced in multiples of 50?m,center to center (Johnson and Silsbee,1988d)to serve as spin injectors and detectors.In one detection scheme a single ferromagnetic pad was used as a spin injector while the distance to the spin detector was altered by selecting different Py pads to detect V s and through the spatial decay of this spin-coupled voltage infer L sN .56This procedure is illustrated in Fig.9,where the separa-tion between the spin injector and detector L x is vari-able.

Johnson and Silsbee (1985)point out that in the de-picted geometry there is no ?ow of the charge current for x ?0and that in the absence of nonequilibrium spins a voltage measurement between x ?L x and x ?b gives zero.Injected spin-polarized electrons will diffuse sym-metrically (at low current density the effect of electric

?elds can be neglected),and the measurement of voltage will give a spin-coupled signal V s related to the relative orientation of magnetizations in F1and F2.57The re-sults,corresponding to the polarizer-analyzer detection and the geometry of Fig.9,are given in Fig.10.An in-plane ?eld (B ?z ?),of a magnitude several times larger than a typical ?eld for magnetization reversal,B 0?100G,is applied to de?ne the direction of magnetiza-tion in the injector and detector.As the ?eld sweep is performed,from negative to positive values,at B 01there is a reversal of magnetization in one of the ferromag-netic ?lms accompanied by a sign change in the spin-coupled signal.As B z is further increased,at approxi-mately B 02,there is another reversal of magnetization,resulting in parallel orientation of F1and F2and a V s of magnitude similar to that for the previous parallel orien-tation when B z ?B 01.

A more effective detection of the spin injection is re-alized through measurements of the Hanle effect,also discussed in Secs.II.

B and III.A.2,and described by Bloch-Torrey equations [Bloch,1946;Torrey,1956;see Eqs.(52)–(54)].The inset of Fig.10summarizes results from a series of Hanle experiments on a single sample.For the Hanle effect B must have a component perpen-dicular to the orientation axes of the injected spins.Only projection of B perpendicular to the spin axis applies a torque and dephases spins.The magnitude of B ,applied at an angle ?to the z axis in the y -z plane,is small enough that the magnetizations in ferromagnetic thin ?lms remain in the x -z plane (see Fig.9).If,at B ?0,

injected nonequilibrium magnetization is ?M (0)z

?then at ?nite ?eld ?M precesses about B with a cone of angle 2?.After averaging over several cycles,only ?M (0)cos ?,the component ?B ,will survive.The volt-age detector 58senses the remaining part of the magneti-zation projected on the axis of the detector ?M (0)cos ??cos ?(Johnson and Silsbee,1988a).The

56

The spin relaxation time in a ferromagnet is often assumed to be very short.Correspondingly,in the analysis of the experi-mental data,both the spin diffusion length and ?M are taken to vanish in the F region (Silsbee,1980;Johnson and Silsbee,1985,1988a,1988d).

57

This method for detecting the effects of spin injection is also referred to as a potentiometric method .58

Recall from the discussion leading to Eq.(39)that the spin-coupled signal is proportional to ?M

.

FIG.9.Schematic top view of nonlocal,quasi-one-dimensional

geometry used by Johnson and Silsbee (1985):F1and F2,the two metallic ferromagnets having magnetizations in the x -z plane;dotted lines,equipotentials characterizing electrical cur-rent ?ow;gray shading,diffusing population of nonequilibrium spin-polarized electrons injected at x ?0,with darker shades corresponding to higher density of polarized electrons.From Johnson,

2002a.

FIG.10.Spin injection data from bulk Al wire sample.Nega-tive magnetic ?eld is applied parallel to the magnetization (?z axis)in the two ferromagnetic regions.As the ?eld is increased,at B 0,1magnetization in one of the ferromagnetic regions is reversed,and at B 0,2the magnetization in the other region is also reversed (both are along ?z axis).Inset:ampli-tude of the observed Hanle signal as a function of orientation angle ?of magnetic ?eld.From Johnson and Silsbee,1985.

340

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predicted angular dependence for the amplitude of the

Hanle signal(proportional to the depolarization of?M in a?nite?eld)??M(0)??M(0)cos2??is plotted in the inset together with the measured data.59Results con?rm

the?rst application of the Hanle effect to dc spin injec-

tion.

The Hanle effect was also studied theoretically by

solving the Bloch-Torrey equations for an arbitrary ori-

entation,characterized by the angle?,between the mag-netization in F1and F2(Johnson and Silsbee,1988a). From the Hanle curve?V s(B?)?measured at T?4.3 (36.6)K,the parameters L s?450(180)?m and P??0.06(0.08)were extracted.60This spin injection tech-nique using the few-pV resolution of a superconducting quantum interference device(SQUID)and with an esti-mated P??0.07provided an accuracy able to detect P n?5?10?12,causing speculation that a single-spin sen-sitivity might be possible in smaller samples(Johnson and Silsbee,1985,1988d).While in a good conductor, such as Al,the observed resistance change?R was small (?n?),the relative change at low temperatures and for L x?L s was?R/R?5%,where?R is de?ned as in Eq.

(1),determined by the relative orientation of the mag-

netization in F1and F2,and R is the Ohmic resistance

(Johnson,2002a).Analysis from Sec.II.C.2shows that

measurement of?R could be used to determine the

product of injected current polarizations in the two F/N

junctions.

The studies of spin injection were extended to the

thin-?lm geometry,also known as the‘‘bipolar spin

switch’’or‘‘Johnson spin transistor’’(Johnson,1993a,

1993b)similar to the one depicted in Fig.5(a).The mea-

sured spin-coupled signals61in Au?lms were larger than

the values obtained in bulk Al wires(Johnson and Sils-

bee,1985,1988d).A similar trend,V s?1/d,potentially important for applications,was already anticipated by Silsbee(1980).The saturation of this increase can be inferred from Eqs.(43)and(44)for d?L sN and been discussed by Fert and Lee(1996)and Hersh?eld and Zhao(1997).

When polarizer-analyzer detection was used,one of

the?tting parameters from the measured data P?some-

times exceeded1—which corresponds to complete inter-

facial polarization.The origin of this discrepancy re-

mains to be fully resolved(Johnson,1993b,2002a;Fert

and Lee,1996;Hersh?eld and Zhao,1997;Geux et al.,

2000).Results obtained from the Hanle effect,on simi-lar samples,gave the expected P??1values(Johnson, 2002a).62

A modi?cation of the bipolar spin-switch structure was used to demonstrate spin injection into a niobium ?lm(Johnson,1994),realizing the theoretical assertion of Aronov(1976a)that nonequilibrium spin could be injected into a superconductor.Two insulating Al2O3?lms were inserted between F1and F2(both made of Py)and a Nb?lm[see Fig.5(b)].The measurements were performed near the superconducting transition temperature T c with the data qualitatively similar,above and below T c,to the spin-coupled voltage,as obtained in the magnetic-?eld sweep from Fig.10.The results were interpreted as support for enhanced depletion of the superconducting condensate(and correspondingly the reduction of the critical current I c)by spin-polarized quasiparticles,as compared to the usual spin-unpolarized quasiparticle injection.Related measure-ments were recently performed in a CPP geometry(Gu et al.,2002),and the penetration depth of the quasipar-ticle in the Nb?lms was measured to be?16nm,as compared to2nm in Johnson(1994).The corresponding temperature dependence of CPP giant magnetoresis-tance is well explained by the theory of Yamashita,Ima-mura,et al.(2003)and the modi?cation of Andreev re-?ection(see Sec.IV.A.3)by spin polarization.

The spin injection technique of Johnson and Silsbee was also applied to semiconductors.Initial experiments on using a metallic ferromagnet to inject spin into a two-dimensional electron gas(2DEG)showed only a very low(?1%)ef?ciency(Hammar et al.,1999)for which various explanations were offered(Hammar et al.,2000; Monzon et al.,2000;van Wees,2000).However,stimu-lated by the proposal of Rashba(2000)to employ spin-selective diffusive contacts(Sec.II.C.1),the subsequent measurements have showed substantially more ef?cient spin injection into a2DEG after an insulating layer was inserted(Hammar and Johnson,2001,2002).The geom-etry employed is depicted in Fig.9.In interpreting the results,the spin-orbit coupling and the energy-independent density of states at the Fermi level were taken into account(Silsbee,2001).This topic is reviewed by Tang et al.(2002).

2.Spin injection into metals

An important part of the operation of CPP giant mag-netoresistance structures is the presence of nonequilib-

59The range of the angle?,in the inset,is corrected from the one originally given in Fig.3of Johnson and Silsbee(1985).

60The?tting parameters are?

s ,P?,and?(Johnson and Sils-

bee,1988d),and since the diffusion coef?cient is obtained from Einstein’s relation L s is known.

61d?100nm was much smaller than the separation between F1and F2in bulk Al wires(Johnson and Silsbee,1985),and the amplitude of the Hanle effect was about104larger (Johnson,2002a).

62Theoretical estimates for V

s

from which P??1was inferred are modi?ed when one considers the Coulomb interaction and proximity effects—near the N/F interface the spin splitting of the carrier bands in the N region will be?nite even at equilib-rium.Model calculations(Chui,1995;Chui and Cullen,1995), which treat the F/N/F junction as a whole,show that the mag-netic susceptibility?in N can be much smaller than the free-electron value and can increase the predicted V s?1/?.These corrections to the free-electron picture of an F/N/F junction are smaller for larger d,as in the bulk-wire geometry of Johnson and Silsbee(1985),where theoretical estimates of V s did not lead to P??1.

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rium spin polarization in nonmagnetic metallic regions.Studies of spin-injection parameters in such systems have been reviewed by Bass and Pratt (1999)and Gijs and Bauer (1997).However,until recently,except for the work of Johnson and Silsbee,there were few other experimental studies directly concerned with spin injec-tion into metals.A series of experiments (Jedema et al.,2001;Jedema,Costache,et al.,2002;Jedema,Heersche,et al.,2002a;Jedema,Nijboer,et al.,2002)at both low (4.2K)and room temperature,were performed using the van der Pauw geometry depicted in Fig.11.In vari-ous structures (Jedema,2002)the two ferromagnetic re-gions (made of Py,Co,or Ni)were chosen to be of dif-ferent sizes to provide different coercive ?elds,allowing an independent reversal of magnetization in F1and F2.The cross-shaped nonmagnetic region was made of Al or Cu (Jedema,2002).Nonlocal measurements,similar to the approach shown in Figs.5and 9(discussed by Johnson,1993b;Johnson and Silsbee,1988d),were shown to simplify the extraction of spurious effects (for example,anisotropic magnetoresistance and the Hall signal)from effects intrinsic to spin injection,as com-pared to the local or conventional spin-valve geometry.In the ?rst type of experiment the cross-shaped region was deposited directly over the F region (Fig.11),and the spin-coupled resistance ?R ,de?ned analogously to Eq.(1),was measured as a function of an in-plane mag-netic ?eld.A theoretical analysis (Jedema et al.,2001;Jedema,Nijboer,et al.,2002)was performed assuming no interfacial resistance (r c ?0)and the continuity of the electrochemical potentials at the F/N interface (see Sec.II.C.1).For a spin injection from Py into Cu,the maximum current polarization obtained was P j ?0.02at 4.2K.The results for ?R (Jedema et al.,2001)scaled to the size of the samples used by Johnson (1993a,1993b)were interpreted to be three to four orders of magnitude smaller.As discussed in Secs.II.C.1and II.C.2,the pres-ence of interfacial spin-selective resistance can substan-tially change the spin injection ef?ciency and in?uence the resistance mismatch between the F and N regions [see Eq.(19)].Estimates of how these considerations would affect the results of Jedema et al.(2001)were given by Jedema,Heersche,et al.(2002b)as well as by others (Johnson and Byers,2003;Takashi and Maekawa,2003),who analyzed the importance of multidimensional geometry.In addition to comparing characteristic values

of the contact resistance obtained on different samples,63for a conclusive understanding it will be crucial to have in situ measurements.

In analyzing data for the van der Pauw cross,a two-dimensional geometry has an important effect—while the electric current is following the paths depicted in Fig.11,the spin current,through the diffusion of non-equilibrium spin,would have similar ?ow in all four arms (Johnson,2002a).This is different from the usual (quasi-)one-dimensional analysis in which spin and charge currents ?ow along the same paths.For a full understanding of the van der Pauw cross geometry,two-dimensional modeling might be necessary (Johnson and Byers,2003;Takahashi and Maekawa,2003).

In the second type of experiment,tunneling contacts were fabricated by inserting Al 2O 3as an insulator into the regions where F1and F2overlapped with the cross.By applying a transverse ?eld B z (see Fig.11)the pre-cession of the injected nonequilibrium spin was con-trolled and the amplitude of the Hanle effect was mea-sured (Jedema,Costache,et al.,2002;Jedema,Heersche,et al.,2002a)as outlined in Sec.II.D.1.From Co/Al 2O 3/Al/Al 2O 3/Co structures L s ?0.5?m was ex-tracted at room temperature.The analysis of the Hanle signal was performed by averaging contributions of dif-ferent lifetimes (D’yakonov and Perel’,1984,p.40).This proved to be equivalent to Johnson and Silsbee’s (1988d)solution to the Bloch-Torrey equations.

3.All-semiconductor spin injection

If a magnetic semiconductor could be used as a robust spin injector (spin aligner)into a nonmagnetic semicon-ductor it would facilitate the integration of spintronics and semiconductor-based d72efe37f111f18583d05a9dparable resis-tivities of magnetic and nonmagnetic semiconductors could provide ef?cient spin injection [see Eq.(31),with r F ?r N ]even without using resistive contacts.Ulti-mately,for a wide range of applications and for compat-ibility with complementary metal-oxide semiconductors (CMOS;Wong et al.,1999),it would be desirable to be able to inject spin into silicon at room temperature.Early studies (Osipov et al.,1990,1998;Viglin et al.,1991,1997),which have since largely been ignored,used a Cr-and Eu-based chalcogenide ferromagnetic semi-conductor (FSm)(Nagaev,1983)as the spin injector.64The experiments were motivated by the theoretical

63

For example,the measured resistance of clean F/N contacts in CPP giant magnetoresistance (Bussmann et al.,1998)was used to infer that there is also a large contact resistance in all-metal spin injection experiments (Johnson,2002b).64

These materials,while more dif?cult to fabricate than the subsequent class of III-V ferromagnetic semiconductors,have the desirable properties of providing injection of spin-polarized electrons (with spin lifetimes typically much longer than for holes)and large spin splitting ??0.5eV at 4.2K for n -doped HgCr 2Se 4(Nagaev,1983)]with nearly complete spin polarization and a Curie temperature T C of up to 130K (HgCr 2Se 4)(Osipov et al.,

1998).

FIG.11.(Color in online edition)Schematic representation of

(a)local and (b)nonlocal geometry used to measure the ef-fects of spin injection and spin accumulation.

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work of Aronov(1976a;Aronov and Pikus,1976)pre-dicting that the electron spin resonance signal,propor-tional to the steady-state magnetization,would be changed by spin injection.The measurements of Osipov et al.(1990)and Viglin et al.(1991)prompted a related prediction(Margulis and Margulis,1994)that spin injec-tion could be detected through changes in electric dipole spin resonance(EDSR).EDSR is the spin-?ip reso-nance absorption for conduction electrons at Zeeman frequency,which is excited by the electric-?eld vector of an incident electromagnetic wave.The theory of EDSR, developed by Rashba and Sheka(1961),is extensively reviewed by Rashba and Sheka(1991). Ferromagnetic semiconductor spin injectors formed p-n and n-n heterostructures with a nonmagnetic semi-conductor InSb.The choice of InSb is very suitable,due to its large negative(??50)g factor(McCombe and Wagner,1971),for detecting the effects of spin injection through electron spin resonance.The observed absorp-tion and emission of microwave power(Osipov et al., 1998)was tuned by an applied magnetic?eld(from35 GHz at?400G up to1.4THz at20kG)and only seen when electrons?owed from FSm into an Sm region.The injection to the lower Zeeman level increased the ab-sorption of electron spin resonance,while injection to the higher Zeeman level,leading to population inver-sion,generated microwave emission.

The most recent experiments using semiconductor

spin injectors can be grouped into two different classes. In one approach(II,Mn)VI paramagnetic semiconduc-tors were employed as the spin aligners.These included CdMnTe(Oestreich et al.,1999),BeMnZnSe(Fiederling et al.,1999),and ZnMnSe(Jonker et al.,2000).In the second approach ferromagnetic semiconductors like (Ga,Mn)As(Ohno,Young,et al.,1999;Chun et al.,2002; Mattana et al.,2003)were used.Both approaches were also employed to inject spins into CdSe/ZnSe(Seufert et al.,2004)and InAs(Chye et al.,2002)quantum dots, respectively.

In(II,Mn)VI materials,at low Mn concentration and at low temperatures,there is a giant Zeeman splitting ?E?g*?B H(Furdyna,1988;Gaj,1988)of the conduc-

tion band,in which g*is the effective electron g factor. Such splitting arises due to sp-d exchange between the spins of conduction electrons and the S?5/2spins of the localized Mn2?ions.The g*factor for H 0can exceed65100and is given by(Brandt and Moshchalkov, 1984;Furdyna,1988)

g*?g??M/?g Mn?B2H?,(50) where g is the H?0II-VI‘‘band’’value g,generally different from the free-electron value,magnetization M??S z??B s?(g Mn?B SH)/(k B T)?,B s is the Brillouin function(Ashcroft and Mermin,1976),and?is the ex-change integral for s-like?6electrons(see Table I in Sec.II.B),given by(Furdyna,1988)

???S?J sp?d?S?/V0,(51) where J sp?d is the electron-ion exchange coupling,and V0is the volume of an elementary cell.From Eqs.(50) and(51)it follows that g*?g*(H)can even change its sign.Similar analysis also applies to g factors of holes, with the Zeeman splitting of a valence band being typi-cally several times larger than that of a conduction band (Brandt and Moshchalkov,1984).

(II,Mn)VI materials can be incorporated in high-quality heterostructures with different optically active III-V nonmagnetic semiconductors which,by providing circularly polarized luminescence,can also serve as spin detectors.In this case carriers are excited by electrical means and we speak of electroluminescence rather than photoluminescence.The selection rules for the recombi-nation light are the same as discussed in Sec.II.B. Figure12depicts a scheme for realization of all-semiconductor electrical spin injection and optical de-tection(Fiederling et al.,1999;Jonker et al.,2000).Dis-played is a spin light-emitting diode(LED;Jonker et al., 1999)in a Faraday geometry where both the applied B ?eld and the direction of propagation of the emitted light lie along the growth direction.Similar to an ordi-nary LED(Sze,1981),electrons and holes recombine in a quantum well or a p-n junction and produce electrolu-minescence.However,in a spin LED,as a consequence of radiative recombination of spin-polarized carriers,the emitted light is circularly polarized.In experiments of

65At low temperatures(?1K)Cd

0.95Mn0.05Se has?g*??500

(Dietl,1994),while in n-doped(In,Mn)As?g*??100at30K (Zudov et al.,2002).Such large g factors,in the presence of a highly inhomogeneous magnetic?eld could lead to the charge carrier localization(Berciu and Janko′,

2003).FIG.12.(Color in online edition)Schematic device geometry and band diagram of a spin LED:(a)Recombination of spin-polarized electrons injected from the(II,Mn)VI spin aligner and unpolarized holes injected from the p-doped GaAs,in the intrinsic GaAs quantum well,producing circularly polarized light;(b)conduction and valence bands of a spin aligner in an external magnetic?eld;(c)sketch of the corresponding band edges and band offsets in the device geometry.In the quantum well,spin-down electrons and unpolarized holes are depicted by solid and empty circles,respectively.Adapted from Fieder-ling et al.,1999.

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