Solid State Physics: Advances in Research and Applications, Vol. 58

SOLID STATE PHYSICS VOLUME 5 8

Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors

HENRY EHRENREICH F U N S SPAEPEN Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts

VOLUME 5 8

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Contents

CONTRIBUTORS ............................................................... PREFACE....................................................................

vii

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Vlll

An Introduction to Semiconductor Spintronics NITINSAMARTH I. Introduction

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II. Ferromagnetic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Coherent Spintronics with Conventional Semiconductor Heterostructures . . . . . . . . . . . . Iv. Semiconductor Spintronic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. ConcludingRemarks ......................................................

1 6 34 60 71

Electron Spin Dynamics in Semiconductors EX. BRONOLD, A. SAXENA, AND D.L. SM!mr

...................... I. Introduction ... . ... . . . . ... ... ... ... . II. Spin Dependence of Semiconductor Electronic Structure . . . III. Kinetic Theory of Spin Dynamics. . . . . . . . . . . . . . . . . . . . . . IV. Application of the Kinetic Theory. . . . ......................... V. Spin Injection from Polarized Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Summary and Future Directions . . . . . . . . . . . . . . . . . . . ............

74 80 104 125 138 164

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167 175

AUTHOR INDEX . . . . . . .

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Contributors to Volume 58 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

EX. BRONOLD (. ..), InstitutfirrPhysik, Emst-moritz-AmdtUniversitat Greifswald, D-17487 Greifwald, Germany N m SAMARTH (1), Department of Physics and Materials Research Institute, The Pennsylvania State University, University Park, PA 16802 A. SAXENA(. . .), Theoretical Division, Los Alamos National Lab, Los Alamos, New Mexico 87545

D.L. SMITH (. . .), Theoretical Division, Los Alamos National Lab, Los Alamos, New Mexico 87545

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Preface

This volume is concerned entirely with the subject that has become popularly known as “spintronics” or “magnetoelectronics”.The utilization of the electronic spin degree of freedom in devices has already had a revolutionary impact. The impetus for development of the relevant science and technology was provided by the discoveryof the metallic giant magnetoresistance(GMR) effect, independently, by Fert and Gruenberg in 1988. This discovery led to a highly sensitive magnetic sensor which was utilized in read-heads of magnetic hard disk drives. Commercial devices were announced by IBM in November 1997,just nine years after discovery of the GMR effect. By now they constitute a multi-billion dollar business. Other applications such as magnetic random access memory (MRAM), the spin field effect transistor and spin controlled laser emitting diodes have been demonstrated or are under development. These applications rely on the ability to control the spin degrees of freedom in solids and aim to reduce power consumption, to overcome speed limitations associated with ordinary electronics, or in the more distant future to implement schemes for quantum information processing and computing. The enabling research effort is directed in part towards obtaining a deeper understanding of the physics of spin lifetimes and transport and tunneling of spinpolarized carriers across heterojunctions in new combinations of materials. In addition to their extensive utilization in conventional electronics, semiconductor systems, the subject of this volume, are particularly attractive for studies of spinbased phenomena in solids because circularly polarized light can be used to inject and detect spin orientation. This volume contains two articles by Samarth and by Bronold, Saxena and D. L. Smith dealing respectively with an introductionto semiconductorspintronics and a theoretical description of electron spin dynamics relevant to experimental measurements.While both articles are clear and self-contained,it might be helpful for the uninitiated to begin with some more qualitative recent overviews in readily accessible journals (Science and Scientific American). The articles by G. Prinz, by S. A. Wolf et al, and by D. D. Awschalom, M. E. Flatte and N. Sam& are to be recommended in this respect. They are referenced on p. 74 as numbers, 4, 1, and 10 respectively. In the first article Samarth presents a selective overview of semiconductorspintronics which emphasizes some of the basic ingredients of this rapidly burgeoning ix

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field. It considers ferromagnetic semiconductors and their heterostructures, focusing on the III-Vferromagnetic materials, and, in particular Gal-,Mn,As. The authorterms research on the latter system as a cautionary tale because of its increasing complexity. Neverthelessthe basic ideas are understandable within a relatively simple model which utilizes mean field theory. Because of the importance of III-V semiconductordevices in optoelectronics, and because the Curie temperatures are quite large even at the present stage of early materials development, these semiconductor alloys offer the possibility of ready incorporation into existing technology. One of the principal materials problems emphasized in this article is that the crystal growth still incorporates many defects in addition to the substitutional incorporation of Mn at concentrations far larger than the 10’’ cmP3characteristic of conventional doping. Samarth’sdiscussion of exchange interactions and carrier mediated ferromagnetics in these systems within the mean field approximation, the quite detailed discussion of low temperature MBE crystal growth and the elucidation of the experimentally determined magnetic properties and magnetotransport of Gal-,Mn,As should be particularly useful. The next major topic of this article concerns optical studies of spin coherence in ordinary bulk semiconductors and quantum dots. The relatively brief but extensively referenced presentation is devoted to both experimental techniques and measurements in GaAs and ZnSe/GaAs heterostructures. The coherent manipulation of spins of importance in quantum information processing schemes is also touched upon. The review concludes with a survey of existing and proposed semiconductor spin devices, for example the spin field effect transistor, light emitting diodes and the ferro- magnetic tunnel junction. While some prototypes have been realized in the laboratory, others remain to be demonstrated. The problems to be solved are challenging to both experimentalists and theorists. However, the rewards of commercial applications may be very substantial. As already suggested, spin based electronics requires a quantitativeunderstanding of non-equilibrium electron-spin-band phenomena better than that currently available. Research must be focused on the generation of non-equilibrium spin polarized distributions, the transport of electron- spin distributions through bulk material and across interfaces between two materials possibly having different spin ground states, the relaxation dynamics of non-equilibrium spin polarized electron distributions and the interaction of spin distributions with optical and magnetic probes. The associated experimental phenomena in semiconductors were emphasized in the first article of this volume. The second article by Bronold, Saxena and D. L. Smith concerning electron spin dynamics in semiconductorssupplies the theoretical framework for describing these phenomena. It also provides a phenomenological discussion of experimental results. While requiring some theoretical background on the graduate student level, the article presents a superbly detailed, didactically presented exposition

PREFACE

xi

of the underlying theory which distinguishes clearly between the relatively well understood and open areas which still require intensive research. Like Samarth’s article, this review focuses on the physical processes in group IV, 111-V and II-VI semiconductors having the diamond or zinc-blende crystal structures. It begins with a discussion of the interaction governing spin-dependent processes. This is followed by a kinetic theory for electron spin dynamics, and application to several semiconductor systems. In particular it offers a theoretical description of the important topic of spin injection at interfaces between a spinpolarized contact and a non-magnetic semiconductor. Section I1 is devoted to a review of the relevant semiconductorelectronic structures, the development of the spin-dependent Hamiltonian, the various scattering mechanisms, hyperfine interactions, spin-dependent optical properties that can lead to non-equilibrium distributions, and the g factor. Section III develops a semi-classical kinetic equation for the density matrix describing non-equilibrium spin distributions including explicit collision integrals. Keldysh Green’s function techniques are used to derive a Fokker-Planck equation for the non-equilibrium spin polarization. This equation permits the calculation of spin relaxation rates which are applied to bulk semiconductors and quantum wells in Section IV. The ability to inject spin polarized currents into a semiconductoris basic to most semiconductor devices. Section V utilizes spin-dependent transport equations to describe spin-injection. It is suggested that efficient spin injection into a semiconductor is difficult. For example, properly designed interface doping profiles turn out to be crucial. The physical origin of the tunneling spin dependence essential to the MRAM is explored and the current experimental situation is summarized. Specific situations under discussion concern an n-type doped large g-factor semimagnetic semiconductor polarized by a magnetic field and a ferromagnetic metal injecting spins into a nonmagnetic semiconductor Spatial inhomogenieties associated with interfaces, superlattices and heterostructures present both theoretical and experimental problems, and point to the need for further investigations. The overall progress of spintronics will be determined by the solutions of problems such as these. In addition the materials issues necessary for reproducible scaled-up fabrication techniques require further development.

HENRY EHRENREICH FRANS SPAEPEN

SOLID STATE PHYSICS VOLUME 5 8

SOLID STATE PHYSICS, VOL.58

An Introduction to Semiconductor Spintronics NITIN SAMARTH Department of Physics and Materials Research Institute, The Pennsylvania State University. Universiiy Park, PA 16802

I.

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Introductio

......................... mediated Ferromagnetism in Di Magnetic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 2. Ferromagnetism in the 111-V Diluted Magnetic Semiconductors . . . . . . . . . . . . . . 3. Crystal Growth and Defects in Gal-,Mn,As . . . . 4. Magnetic Properties of Gal-,Mn,As . . . . . . . . . . . . . . . . . .................... 5. Magneto-transport in Gal-,Mn,As . . . . . . 6. Measurements of Spin Polarization in Gal-x ....................... 7. Criteria for Identifying a Ferromagnetic Semiconductor . . . . . . . . . . . . . . . . . . . . III. Coherent Spintronics with Conventional Semiconductor Heterostructures . . . . . . . . . . . 8. Optical Measurements of Spin Coherence: Experimental Techniques . . . . . . . . . . 9. Measurement of Spin Coherence in Bulk Semiconductors. . . . . . . . . . . . . . . . . . . 10. Spin Coherence Measurements in Semiconductor Quantum Dots . . . . . . . . . . . . . 1 1. Electrical Manipulation of Spin Coherence in Semiconductor Heterostructures . . 12. All-optical Coherent Manipulation of Spin in Semiconductors . . . . . . . . . . . . . . . 13. Spin Coherence in Hybrid Ferromagnet/Semiconductor Heterostructures . . . . . . . IV. SemiconductorSpintronic Devices. . . . . ................... 14. The Datta-Das Spin-Field Effect Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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V.

Concluding Remarks . . . . . . . . .

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1 6

8 13 14 21 27 29 33 34 36 41 43 45 55 57 60 60 63 65 71

1. Introduction When Richard Feynman boldly speculated in 1959 about the technological possibilities latent in the manipulation of “quantized energy levels, or the interactions of quantized spins, etc.,”’ even he might not have imagined the current explosion of scientific and technological activity directed at the quantum functionality embodied in his visionary speech. Contemporary materials engineering R. P. Feynman, The Pleasure of Finding Things Out, Perseus Publishing, Cambridge, Massachussets (1999). 136. 1 ISBN 0-12-6077584 ISSN 0081-1947/04

8 2004 Elxvier Science (USA) All rights reserved.

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techniques now enable the fabrication of a wide variety of compositionally modulated materials (“heterostructures”) with excellent control of explicitly quantum mechanical phenomena such as quantum confinement and quantum tunneling. These heterostructures have had a lasting impact on quantum device technologies, as well as on fundamental discoveries in condensed matter physics. For instance, semiconductorheterostructures play a vital role in contemporary opto-electronics and microwave frequency devices, while simultaneously providing a rich arena for studying correlated electron physics in low dimensions.’ On a parallel but quite separate track, advances in the fabrication of metallic magnetic heterostructures now allow us to read memory of unprecedented density with the “giant magnetoresistance”(GMR) effect3v4and also form the basis for an emerging nonvolatile memory known as “magnetic random access memory” (MFWM).’ It is noteworthy that this push toward metallic “spintronics” (an acronym for “spin transport-based electronics”) emerged from an interest in answering fundamental questions about spin transport and tunneling in metallic ferromagnetic heterostructures. The integration of semiconductorheterostructureswith magnetic materials may be viewed as a natural outcome of these remarkable advances and has led to a burgeoning new field-“semiconductor spintronics”-that lies squarely at the nexus between these usually disparate areas of science and technology.”* Semiconductor spintronics is broadly aimed at the manipulation of spin-dependent phenomena in conventional semiconductor heterostructures? as well as in “hybrid” systems that combine magnetic and semiconducting In this chapter, we offer an introductoryguide to this rapidly developingfield, with a focus on three major areas of research: spintronics with ferromagnetic semiconductors and their heterostructures; coherent spintronics with conventionalsemiconductorheterostructures; and, finally, semiconductor spintronic devices. There are two complementary but not mutually exclusive approaches toward the implementation of semiconductor spintronics.

R. Dingle, Ed.,Applications of Multiquantum Wells, Selective Doping, and Superlattices: Semiconductor and Semimetals, vol. 24, Academic, New York (1988). M. N. Baibich et al., Phys. Rev. Lett. 61,2472 (1988). G . A. Prinz,Science 282, 1660 (1998). S. Tehrani er al., Proceedings of rhe IEEE 91,703 (2003). D. D. Awschalom, D. Loss, and N. Samarth, Eds., Semiconductor Spintronics and Quantum Computation, Springer-Verlag, Berlin (2002). S. Wolf er al., Science 294, 1488 (2001). H. Akinaga and H. Ohno, IEEE Trans. Nanorechnology 1, 1 (2002). D. D. Awschalom and J. M. Kikkawa, Phys. Today 52.33 (1999). lo H. Ohno, Science 281,951 (1998). I’ N. Samarth er al., Solid Srar. Comm. 127, 173 (2003).

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. Spin polarized electrons (or holes) can be introduced into a conventional semiconductor through optical pumping,” via ferromagnetic contact^,'^ or by the proximity effect of a vicinal ferromagnetic s u ~ f a c e ;this ~ ~spin-polarized *~~ population of charge carriers can subsequently be manipulated via electric and/or magnetic fieldsthat either may be externally applied or may arise due to intrinsic effects within a semiconductor crystal (such as strain-induced piezoelectricity, spin-orbit coupling, and nuclear hyperfine fields).16 2. Alternatively, in a “magnetic semiconductor,” charge carriers can be exchange coupled with magnetic ions incorporated into the semiconductor lattice i t ~ e l f ; ’ ~ such - ’ ~ interactions often enhance “bare” Zeeman effects, resulting in large magneto-optical and magneto-resistiveeffects, and can also lead to a collective ordering of the magnetic ions.” Both of the preceding approaches raise many interesting issues from the vantage point of condensed matter physics. These include questions such as the fundamental understanding of the transport and injection of spin in conventional semiconductors and their heterostructures, the understanding and control of quantum coherent spin phenomena in semiconductors, and the nature and origin of ferromagnetism in magnetic semiconductors. Progress toward the resolution of many of these issues is of course inextricably linked with the development and control of new materials, which consequently forms a substantial and critical componentof semiconductor spintronics. This is perhaps best exemplified by the current state of understanding about ferromagnetic semiconductors,where basic materials science issues (for instance, defects and phase separation) can often muddy attempts to grasp the essential physics through idealized models. As implied by the introductory paragraph, semiconductor spintronics is motivated by the prospects for new technological possibilities that utilize the spin degree of freedom. The idea of using spin-dependentphenomena for opto-electronic devices was perhaps first recognized many decades ago, with the invention of the InSb-based Raman spin-flip laser.” Other device applications have focused on l2 F. Meier and B. P. Zachachrenya,Eds., Optical Orientation, Modem Problems in Condensed Maner Science, vol. 8, North-Holland, Amsterdam (1984). l3 S. Datta and B. Das, Appl. Phys. k t t . 56,665 (1900). l4 R. K. Kawakami etal., Science 294, 131 (2001). Is R. J. Epstein et al., Phys. Rev. B 65, 121202(R) (2002). l6 D. K. Young et al., Semi. Sci. Tech. 17,275 (2002). ” J. K. Furdyna, J. Appl. Phys. 64, R29 (1988). T. Dietl, “(Diluted) Magnetic Semiconductors:’ in Handbook of Semiconductors, ed. S. Mahajan, V O ~ .3B, North-Holland, Amsterdam (1994). 125 1. D. D. Awschalom and N. Samarth, J. Mag. Magn. Mate,: 200, 130 (1999). T. Dietl, A. Haury, and Y. Merle D’Aubigne, Phys. Rev. B 55, R3347 (1997). 21 C. K. N. Patel and E. D. Shaw, Phys. Rev. B3.1279 (1971).

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the use of the Faraday effect in magnetic semiconductors for magneto-optical devices such as magnetic field sensors and optical isolators?2 culminating in commercially available optical isolators derived from narrow gap 11-VI diluted magnetic semiconductor^?^ The new device schemes emerging in contemporary discussions vary from immediately obvious extensions of existing electronic and opto-electronic device concepts to visionary functionality that cannot be achieved using existing technology. Examples of the former arise from the incorporation of a magnetic semiconductor within standard devices. While qualitative proposals for such spin-tunable devices hark back several decades,” it is only recently that serious attention has turned to devices such as spin-dependent resonant tunneling diode^?^^^^ spin-polarized light emitter^,^'-^^ and ferromagnetic heterojunction bipolar transistor^.^^-^' The functional characteristics of such devices are magnetic field tunable, hence providing a new degree of freedom that could be exploited for the rapid modulation of optical or electronic signals. As a further extension, qualitatively new device concepts such as reconfigurable logic may emerge from the manipulation of ferromagnetic states within such spin devices via optical or electrical perturbation^.^^-^^ An example of this new functionality is shown in Fig. 1, wherein a gate voltage applied to a field effect transistor controls the isothermal magnetization of a ferromagnetic semiconductor by varying the Curie temperature. Although the effect observed in this experiment is small and only demonstrated at low temperatures, it embodies the unique functionality possible using a ferromagnetic semiconductoras opposed to a ferromagneticmetal. The optical and electrical manipulation of carrier-mediatedferromagnetismis possible in semiconductors because of the low carrier density. Yet more revolutionary ideas center around the alluring possibility of establishing, storing, and manipulating coherent quantum states in semiconductors for quantum information p r o ~ e s s i n g .~n ~ ~particular, a recent series of experiments

P. I. Nikitin and A. A. Beloglazov, Sensors and Actuators A 41-42,547 (1994). K. Onodera, H. Ohba, and T. Kawamura, Oyo Buturi 70,300 (2001) (Japanese version only). 24 J. Kossut and J. K. Furdyna, in Diluted Magnetic (Semimagnetic) Semiconductors, eds. R. L. Agganval, J. K. Furdyna, and S. Von Molnar, Materials Research Society, Pittsburgh (1987). 97. 25 H. Ohno et al., Appl. Phys. Lett. 73,363 (1998). 26 A. Slobodoskyy et al., Phys. Rev. Lett, 90,246601 (2003). R. Fiederling et al., Narure (London),402,787 (1999). Y.Ohno er al., Nature (London) 402,790 (1999). 29 B. T. Jonker et al., Phys. Rev. B 62,8180 (2000). 30 M. E. Flatt6, Z. G . Yu,E. Johnston-Halperin, and D. D. Awschalom, Appl. Phys. Lett. 82,4740 (2003). 31 J. Fabian, I. iutiC, and S. Das Sarma, Appl. Phys. Lett. 84.85 (2004). 32 S . Koshihara et al., Phys. Rev. Lett. 78,4617 (1997). 33 H. Ohno et al., Nature (London) 408,944 (2000). 34 A. M. Nazmul, S. Kobayashi, S. Sugahara, and M. Tanaka, cond-mat/0309532. 35 D. P.Divincenzo, Science 270,255 (1995). 22 23

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B (mT) FIG. 1. Anomalous Hall resistance RH in an electrically gated ferromagnetic semiconductor(In,Mn)As-as a function of magnetic field for different electrical gate voltages. The magnetic field is applied along the easy axis of the ferromagnet (normal to the sample plane), and RH is proportional to the sample magnetization. (From Ref. 33.)

in doped semiconductors has revealed electron spin relaxation and decoherence times that can be as long as several nanoseconds in certain materials (even at room temperature), hence allowing the coherent transfer of spin across both homogeneous and inhomogeneous semiconductor^.^^^ Theoretical proposals for quantum information processing rely on these long coherence times to exploit coherent electron spin states in mesoscopically patterned quantum and have prompted ongoing experimental efforts that probe coherent spin-dependent transport in such quantum system^!^^^ We begin this chapter with an overview of ferromagnetic semiconductors and their heterostructures, with principal focus on the 111-V ferromagnetic 36

J. M. Kikkawa, I. P. Smorchkova,N. Samarth,and D. D.Awschalom, Science 277, 1284 (1997).

37 J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80,4313 (1998). 38 J. M. Kikkawa and D. D. Awschalom, Nature (London) 397,139 (1999). 39 1. Malajovich, J. M. Kikkawa, D. D. Awschalom, J. J. Berry, and N. Samarth,Phys.

Rev. Lett. 84, 1015 (2000). I. Malajovich, J. J. Berry, N. Samarth,and D. D. Awschalom, Nature (London) 411,770 (2001). 41 B. Beschoten et al., Phys. Rev. B 63, R121202 (2001). 42 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 43 L. M. K. Vandersypen et al., in Quantum Computing and Quantum Bits in Mesoscopic Systems, Eds. A. Leggett, B. Ruggiero, and P. Silvestrini, (Kluwer AcademicPlenum Publishers, 2003). 44 J. A. Folk, R. M. Potok, C. M. Marcus, and V. Umansky, Science 299,679 (2003).

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semiconductors. Here, we attempt to cover an emerging story about an important ferromagnetic semiconductor (Gal-,Mn,As). This is a cautionary tale with surprising twists and turns in recent years, resulting in a picture that is complex in its details, yet understandable within a relatively simple model. Next, we turn our attention to optical studies of spin coherence in semiconductors.Because an extensive review of these latter experiments has been provided very recently? we will only give a condensed introduction to the principal concepts and results. Finally, we finish with a tour through a gallery of semiconductor spintronic "devices," where the quotation marks purposely signify the very elementary level of sophistication of current technological exploration in this field. II. FerromagneticSemiconductors The occurrence of ferromagnetism in magnetic semiconductors is neither a new nor a rare phenomenon, and was extensively studied several decades ago in materials that include the manganites?' semiconductor spinels (such as CdCrzSed), the europium chalcogenides (e.g., EUO),%*~~ and the lead chalcogenides (e.g., Pbl-,-ySnyMn,Te)."8 The recent resurgence of interest in ferromagnetic semiconductors has largely been ignited by the discovery of ferromagnetism in the Mn-doped 111-V semiconductors Gal-,Mn,As and 1 n l - , M n , A ~ . " ~ ~ ~ These new ferromagnetic semiconductors are derived from materials of great technological relevance (e.g., 111-V semiconductor devices form the backbone of much of contemporary opto- and high-frequency electronics). The Curie temperature (Tc) in some of these materials was found to be encouragingly high even at an early stage of materials development (1 10 K in Gal-,Mn,As with x 0.05).52 Finally, the ferromagnetism in these materials is demonstrably carrier mediated and can be modulated by external electrical or optical signals. Hence, the GaAsand InAs-based ferromagnetic semiconductorsat least have the potential for ready incorporation into existing technologies. As we shall see in this section, this technological potential is yet to be realized, principally because the thermodynamicsof crystal growth extracts a heavy price (defects) for the substitutional incorporation of Mn into a 111-V semiconductor lattice. Nonetheless, the ferromagnetic semiconductors Gal-,Mn,As and Inl-,Mn,As continue to serve admirably as "canonical" E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance: The Physics of Manganires and Related Compounds, Springer-Verlag. (Berlin). T. Kasuya and A. Yanase,Rev. Mod. Phys. 40,684 (1968). 47 A. Mauger and C. Godart, Physics Reports 141,51 (1986) 48 T. Story, R. R. Galazka,R. B. Frankel. and P. A. Wolff, fhys. Rev. Lett. 56,777 (1986). 49 H. Munekata et al., Phys. Rev. Lett. 63, 1849 (1989). 50 H. Ohno et al., fhys. Rev. Len. 68,2664 (1992). 51 H. Ohno et 01.. Appl. fhys. Lett. 69 363 (1996). 52 F. Matsukura, H. Ohno, A. Shen, and Y.Sugawara, Phys. Rev. E 57, R2037 (1998). 45

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systems wherein one can test fundamental and applied concepts relevant to semiconductor spintronics. It is also noteworthy that mean field theories developedto understand the origins of ferromagnetismin the 111-Mn-V semiconductors20~53-56 have lead to predictions of ferromagnetismin a much wider class of magnetically doped semicond~ctors.~~ Complementary to this mean field approach that relies on carrier-mediated ferromagnetism, first principles calculations have also predicted the existence of ferromagnetic ground states in many different These predictions have prompted intense interest in the crystal growth of magnetic semiconductors that include new 11-VI magnetic semiconductors such as (Zn,Cr)Te,@magnetic 111-V p h o ~ p h i d e s ? ~and , ~ ~a n t i m ~ n i d e s magnetic ;~~ group N semiconductors;’’ as well as various other complex phosphides and o ~ i d e s ? ~ - ~ ~ In some cases, such as (Zn,Mn)Te and (Cd,Mn)Te, the theoretical predictions have been vindicated by experiment;7c78 in others, such as the nitrides and oxides, the current experimental data-while indicating tantalizingly high values of 7’‘-still need to be complemented by additional measurements to rule out extrinsic origins for the observed f e r r o m a g n e t i ~ m . ~ ~ ~ ~ ~ T. Jungwirrh et al., Phys. Rev. B 59.9818 (1999). T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B, 63,195205 (2001). ” J. Konig, I. Schliemann,T. Jungwirth, and A. H. MacDonald, in Electronic Structure andMagnetism ofComplexMaterials, eds. D. J. Singh and D. A. Papaconstantopoulos,Springer Verlag, Berlin (2002); see also condmat/Ol11314. 56 T.Jungwirth et al., Phys. Rev. B 66,012402(2002). 57 T.Dietl et al., Science 287,1019 (2000). ” J. Blinowski, P. Kacman, J.A. Majewski, Phys. Rev. B 53,9524(1996). 59 M. Jain, L. Kronik, J. R. Chelikowsky, and V. V. Godlevsky, Phys. Rev. B 64,25205(2001). M. van Schilfgaarde and 0. N. Mryasov, Phys. Rev. B 63,233205(2001). K.Sat0 and H. Katayama-Yoshida. Semicond. Sci. Technol. 17.67 (2002). S.Sanvito, G. Theurich, N. A. Hill, J. ofSupercond. 15.85 (2002). P. Mahadevan and A. Zunger, cond-mat/O309509. H. Saito, V. Zayets, S. Yamagata, and K. Ando, Phys. Rev. Len. 90,207202(2003). 65 M. E. Overberg et al., Appl. Phys. Len. 79, 1312 (2001). M. L.Reed et al., Appl. Phys. Lett. 79.3473 (2001). 67 S.Sonoda et al., J. Cryst. Growth 237-239, 1358 (2002). N. Theodoropoulou et al., Appl. Phys. Lett. 78,3475(2001). 69 N. Theodoropoulou et al., Phys. Rev. Lett. 89, 107203 (2002). 70 T.Wojtowicz et al., Appl. Phys. Len. 82,4310(2003). ” Y. D.Park et al., Science 295,651 (2002). 72 G. A. Medvedkin et aL, Jpn. J. Appl. Phys. 39,L949 (2000). 73 Y.Matsumoto et al., Science 291,854(2001). 74 S.A. Chambers et al., Appl. Phys. Lett. 79,3467(2001). 7’ S.B. Ogale e f al., Phys. Rev. Len. 91,077205(2003). 76 A. Haury et nl., Phys. Rev. Len. 79,51 1 (1 997). 77 D. Ferrand et al., Phys. Rev. B 63,085201(2001). 78 H. Boukari et al., Phys. Rev. Len. 88,207204(2002). 79 S.A.Chambers et al.,Appl. Phys. Lett. 82,1257 (2003). S. Dhar et al., Appl. Phys. Lett. 82,2077(2003). 53

@

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NITIN SAMARTH

We begin this section by discussing the underlying basis for carrier-mediated ferromagnetismin magnetic semiconductors.We then follow this with a discussion of experimental studies (crystal growth and physical properties) of a "canonical" case: Gal-,Mn,As. Finally, we conclude with the formulation of a set of experimental criteria that we believe are useful for identifying whether a given material is an intrinsic ferromagnetic semiconductor with carrier-mediated ferromagnetism. 1. EXCHANGE INTERACTIONSAND CARRIER-MEDIATED FERROMAGNETISM INDILUTED MAGNETIC SEMICONDUCTORS

The understanding of carrier-mediated ferromagnetism in an electrically doped, randomly diluted magnetic semiconductor lattice can be approached via a number of possible routes. Thus far, approaches to this problem have included mean field theories within a continuum a p p r o ~ i m a t i o n , ~dynamical ~ * ~ ~ - ~ ~mean field theory that incorporates effects of disorder and impurity band formation?' percolation model^,^"^^ and defect-specific model^.^^^^^ A critical discussion of all such approaches is beyond the scope of this introductory review, and we limit ourselves to a very simple-and certainly incomplete-picture based on the mean field approach. Imagine that the cations in a direct gap zinc-blende semiconductor are randomly replaced with the transition metal atom Mn, which has an atomic shell configuration 3d54s2. We assume that the d-electrons remain well-localized, forming a local moment with S = 5/2 in the absence of any crystal field effects. In addition to the local moments, we now assume that the semiconductor itself is doped either n- or p-type; the dopant impurities may either be the Mn ions themselves (as in the case of Gal-,Mn,As, where Mn acts as an acceptor) or they could be some other nonmagnetic donors or acceptors (such as in the case of p-Znl-,Mn,Te, where the Mn ions are isoelectronic). For carrier densities above the metal-insulator transition, the idealized system then consists of delocalized, itinerant carriers that interact with local moments over extended length scales. At low carrier densities below the metal-insulator transition, we expect the carriers to be localized, interacting with local moments that lie within the localization radius. There are two classes of exchange interactionsthat arise within such an idealized magnetic semiconductor lattice.

S. Das Sarma, E. H. Hwang, and A. Kaminski, Solid Srar. Commun. 127.99 (2003) and references therein. M. Berciu and R. N. Bhatt, Phys. Rev. Letr. 87, 107203 (2001), and references therein. 83 G. Alvarez, M. Mayr, and E. Dagotto, Phys. Rev. Len. 89,277202 (2002). 84 A. Kaminski and S. Das Sarma, Phys. Rev. Len. 88,247202 (2002). 85 C. Timm, F. Schiifer, and F. von Oppen, Phys. Rev. Lerr. 89, 137201 (2002). 86 P. Mahadevan and A. Zunger, Phys. Rev. B 68,075202 (2003).

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INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

9

The d-d superexchange between the d-electrons of the magnetic ions: This is a short-ranged antiferromagnetic interaction and is mediated by the intervening anion.87 In the 11-VI diluted magnetic semiconductors, this interaction is characterized by an exchange integral J d 10 K, while in Gal-,Mn,As, J d 5 1 K. The s-d(p-d) exchange between the d-electrons and the band electrons (holes): This interaction is purely ferromagnetic (potential exchange) for conduction band states and is predominantly antiferromagnetic (kinetic exchange) for valence band states. qpically, the p-d exchange integral is much larger than the s-d exchange.88

-

One might anticipate that the two systems of spins (free carriers and local moments) in a diluted magnetic semiconductor will influence each other through the sp-d exchange interaction. This interplay is indeed observed in experiments. Localized charge carriers (e.g., acceptor-boundholes) can align the Mn ions within their vicinity, forming a “bound magnetic p ~ l a r o n ; ”conversely, ~~ the alignment of the Mn ions by an external magnetic field can enhance the spin splitting of conduction and valenceband states via an exchange field,88resulting in a significant spin polarization of the electron or hole gas.88qw As we will soon see, the sp-d exchange between free electronsholes and Mn d-electrons is also responsible for mediating a collective, ferromagnetic alignment of the Mn spins under the right circumstances. We now discuss the magnetic behavior of a diluted magnetic semiconductor as a function of the free carrier density. At low carrier densities, the global magnetic properties of a diluted magnetic semiconductorare relatively insensitiveto the presence of free carriers. The system can be modeled quite simply as a collection of S = 5 / 2 spins randomly distributed on an fcc lattice, and coupled by antiferromagneticd-d superexchange. Depending on the magnetic ion concentration and temperature, the system exhibits paramagnetic, spin glass, or short-range antiferromagnetic order. (The spin glass phase originates in the frustration that is inherent in the randomly diluted fcc lattice.) In the paramagnetic phase, for an external magnetic field H.2, the magnetization M, is described by a Brillouin function that is empirically modified to account for the antiferromagneticMn-Mn interactions. Mz = NO(&) = NOS,,B5/2(5CLBH/kBT,ff),

(2.1)

where No is the number density of magnetic ions and B 5 / 2 ( ~ is ) the Brillouin function for S = 5 / 2 . Because antiferromagnetic spin-spin correlations between ” B. E. Larson, K.C. Hass, H. Ehrenreich, and A. E. Carlsson, Solid State Commun. 56,347 (1985). ” J. Kossut, in Diluted Magnetic Semiconductors: Semiconductor and Semimetals, vol. 25, eds. J. K.

Furdyna and J. Kossut, Academic Press, New York (1988), 183. 89 P. A. Wolff, in Ref. 88,413. I. P. Smorchkova, N. Samarth, J. M. Kikkawa, and D. D. Awschalom, Phys. Rev. Lett. 78 3571 (1997).

10

NITIN SAMARTH

Mn2+ ions reduce the magnetization from that of non-interacting Mn2+ ions, the standard Brillouin function is adjusted by using the parameters S,,, (which is the saturation value for the spin of an individual Mn2+ ion and is smaller than 5/2) and the rescaled temperature Teff = T To. We note that for a given distribution of magnetic spins on the lattice, there is a well-defined statistical distribution of magnetic ions, with isolated spins, pairs of spins, triplets, etc. Due to the antiferromagnetic coupling between nearest neighbor Mn2+ spins, the magnetization at modest fields is dominated by the paramagnetic response of isolated single spins. At very high magnetic fields, the presence of the antiferromagneticcoupling between Mn spins reveals itself directly in abrupt changes in magnetization?’ In addition, as we briefly mentioned, the localization of the free carriers at low temperatures (on the insulating side of the metal-insulator transition) can in fact significantly perturb the magnetic moments within the localization radius, forming “bubbles” of magnetization known as bound magnetic polarons. In fact, some of the first predictions for ferromagnetism in the diluted magnetic semiconductors were based on a scenario of the percolation of bound magnetic polarons.82Models along these lines continue to attract attention, particularly within the context of ferromagnetic ordering in Gal-,Mn,As at low concentrations where the states are strongly l ~ c a l i z e d . ~ ~ * ~ ~ As the carrier density increases, the effects of the sp-d exchange on the global magnetization can eventually overcome the antiferromagnetic coupling between local moments, and-under the right circumstances-this can lead to spontaneous, long-range ferromagnetic ordering of the local moments mediated by the delocalized carriers. This has been treated in mean field approaches that use a continuum a p p r o ~ i m a t i o n ? ~ ~wherein ~ ~ - ~ ’ the random arrangement of magnetic ions on a semiconductor lattice is replaced with a uniform background. Such a simplification isjustified because the typical distance between magnetic ions is much smaller than the Fermi wavelength h~ of the electrons9r holes in the extended band states. In such a model, a Mn ion located at position R experiences a local exchange field proportional to the hole spin polarization.

+

where ( ~ ( 2 represents )) the hole spin density at the Mn sites, and J is the s-d(p-d) exchange integral. The Mn ions then align according to Eq. 2.1 in the net magnetic field given by the sum of the local exchange field and any external magnetic field. When the local exchange field is strong enough to overcome the effects of the antiferromagnetic Mn-Mn exchange, a spontaneous ferromagnetic alignment of the system becomes possible. 9’ S. A. Crooker, N. Samatth, and D.D.Awschalom, Phys. Rev. B 61, 1736 (2000), and references therein.

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

11

At least two complementary approaches have been used to determine the conditions for such carrier-mediated ferromagnetism. The carrier spin density can be calculated by solving the free particle Schrdinger equation in the presence of a kinetic-exchange field determined by the polarization of the Mn ions.53 Additional effects such as Coulomb potentials in heterostructures can also be included in this calculation, as well as exchange-correlation effects within the local spin density approximation. An alternative approach uses a methodology (inspired by early work by Zener)92wherein the Ginzburg-Landau free energy functional is minimized with respect to the Mn magnetization for a given temperature, external magnetic field, and carrier density.57 Both these approaches essentially produce the same physical result, which correlates the Curie temperature with a variety of system parameters according to the following equation:

Here, g* is the g-factor of the carriers mediating the sp-d exchange, and xf is the interactingcarrier spin susceptibility.The carrier spin susceptibilityis transparently related to the Fermi wave vector kF through the total energy of the system. In particular, when contributions from the kinetic exchange dominate those from the potential exchange, Tc m*kF, where kF is the Fermi wave vector and m* is the where x is the fraction effective mass of the carriers. It follows that TC of lattice sites occupied by Mn ions and n is the carrier density. These insights provide a “confidence test” for the mean field approach. We anticipate that TC will be higher in materials with larger effective mass and larger carrier-ion exchange. Both these characteristics favor ferromagnetism in p-type diluted magnetic semiconductors over n-type materials; the absence of evidence for ferromagnetism in the n-doped 11-Mn-VI materials supports this prediction. In addition, by evaluating kinetic and exchange contributions to the carrier spin susceptibility using detailed band structure, the mean field approach provides the means to quantitativelycalculate various physical parameters such as Tc, the magnetic anisotropy coefficients,spin stiffness,and carrier spin polarization. The use of a realistic Kohn-Luttingerdescription of the valence band structure is particularly crucial in the context of hole-mediated ferromagnetism. How representative of reality is the model system that we have just described when compared with real materials? It turns out that that it is in fact well represented by the 11-Mn-VI diluted magnetic semiconductors,where the substitutional Mn ions are isoelectronic and do not perturb the host semiconductorband structure qualitatively, except to alter the band parameters such as the energy gap and the effective mass in a well-understood fashion.I7 Here, the predictions of the mean field theory correspond very well to observationsof ferromagnetismin the p-doped

-

92 C. Zener, Phys.

Rev. A 81,440 (1950).

-

12

NITIN SAMARTH

11-Mn-VI diluted magnetic semiconductors such as (Zn,Mn)Te and (modulation doped) (Cd,Mn)Te.93Quantitative comparisons are possible in these materials because relevant parameters such as the exchange integrals are experimentally well k n 0 ~ n . The l ~ mean field approach also appears to provide a very credible explanation of many qualitative aspects of ferromagnetism in the 111-Mn-V materials, although quantitative comparisons are limited by uncertainties in the experimental knowledge of parameters such as the exchange integral and the effective mass.54 The understanding of magnetic anisotropy in Gal-,Mn,As provides a good example of the qualitative predictions made possible by the mean field model. The source of magnetic anisotropy in Gal-,Mn,As can be explicitly traced to the effects of strain and spin-orbit interaction on the valence bands. At first glance, it is non-intuitive that Gal -,Mn,As should exhibit any magnetic anisotropy. Because the Mn ions are in an S = 512 state, one would not expect any influence of spinorbit coupling on the d-shell electrons. However, the valence band holes mediating the ferromagnetic interactions between the Mn ions experience large spin-orbit coupling effects. As a result, there is significant anisotropy associated with the p-d exchange interaction itself, and this manifests itself in observable magnetic anisotropy effects. Initial studies of strained Gal-,Mn,As samples suggested two broad regimes of strain-induced magnetic anisotropy: Under compressive strain, the easy axis is in-plane, while under tensile strain, the easy axis is perpendicular to the ~ l a n e . 9It~is now recognized that the magnetic anisotropy in the III-Mn-V ferromagnetic semiconductors can be considerably more complex because the specifics depend on an interplay between strain and hole spin p~larization.’~ We will describe other comparisons between predictions of the mean field theory and experiments later in this section. Despite the successful application of the mean field model to various aspects of Gal-,Mn,As, it is important to bear in mind some lingering and unresolved questions that continue to rankle in some quarters. Some of these misgivings have already been addressed in our preceding discussion. For instance, the observation of ferromagnetism in Gal -,Mn,As at dilute magnetic composition where the carriers are certainly strongly localized suggests that a percolation picture is likely to be more appropriate in this regime rather than in one wherein delocalized carriers mediate ferromagnetic interactions between the local moments. Questions also still persist about the influence of Mn on the valence band structure of the In-V semiconductor. In particular, measurements of magnetic circular dichroism?’ photoemission?6 and infrared as well as theoretical T. Died, cond-maU0307503. al., J. Ctysr. Gmwrh 175/176, 1069 (1997). 95 B. Beschoten er al., Phys. Rev. Len. 83,3073 (1999). 96 J. Okabayashi er al., Phys. Rev. B 64,125304 (2001). 97 Y.Nagai et 41.. Jpn. J. Appl. Phys. 40,623 1 (2001). 98 E. J. Singley er al., Phys. Rev. Len. 89,097203 (2002). 93

94 A. Shen et

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

13

calculationsw suggest that the presence of Mn in Gal-,Mn,As leads to a more complicated picture than one in which delocalized holes occupy the unperturbed bands of the host semiconductor and interact with well-localized Mn d-shell electrons. A recent calculation of the local density of states shows that even the presence of a single Mn atom leads to spin-polarized resonances with the valence band that significantly enhance the local density of states at the valence band edge, while also yielding an inherently anisotropic Mn-Mn interaction.lWEven more suprisingly, as we shall see shortly, the crystal growth of the 111-Mn-V materials is invariably accompanied by the formation of defects whose character and distribution undoubtedly affects the magnetic, electrical, and structural properties of these materials.86 Despite all these issues, the mean field model is the only one that presently provides relatively straightforwardpredictions that can be compared with experiments and- overall, which, are borne out by the experimental studies. Hence, the mean field approach is likely to continue to play a critical role in our understanding of the 111-Mn-V materials until a better methodology is developed. We now provide an overview of experiments focusing on these materials.

2. FERROMAGNETISM IN THE 111-V DILUTED MAGNETIC SEMICONDUCTORS The 111-Mn-V semiconductors present a somewhat complex situation that originates in both intrinsic as well as extrinsic causes. First, Mn acts as an acceptor in the 111-V lattice, so that the electrical and magnetic properties are inextricably coupled. More problematically,Mn has a very low solubility in a 111-V semiconductor: Under typical crystal growth conditions used for the fabrication of GaAs crystals, Mn can only be introduced at dopant-levelconcentrations (-10l8 ~ m - ~Attempts ). to exceed this low solubility using either conventionalbulk crystal growth or standard epitaxial growth conditions leads to the formation of thermodynamically stable clusters of the ferromagnetic semimetal MnAs rather than a homogeneous Gal-,Mn,As semiconductor crystal. This difficulty is resolved by using lowtemperature molecular-beam epitaxy (LT-MBE) at substrate temperatures in the range 200 to 250°C. This allows the incorporationof several percent Mn into a 111-V semiconductor lattice, and led to the demonstration of hole-mediated ferromagnetism first in Inl-,Mn,AsS0 and then in G ~ ~ - , M ~ , ASince S . ~ ~these pioneering efforts, carrier-induced ferromagnetism has also been conclusively demonstrated in other 111-Mn-V-based systems such as Inl-xMnxSb?OGa~-,-yInxMnyAs,101~102

99 H.

Akai, Phys. Rev. Len. 81,3002 (1998).

loo J. M Tang and M. E. Flatte, cond-mat/03051 18.

T. Slupinski, H. Munekata, and A. Oiwa, Appl. Phys. Left. 80, 1592 (2002). H. Kobayashi, and M. Tanaka, Appl. Phys. Lerr. 83,2175 (2003).

'02 S. Ohya,

14

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SAMARTH

and Ga~-,-yA1,Mn,As.’03 We focus here exclusively on Gal-,Mn,As, which is the most thoroughly studied of the 111-V semiconductors, and hence serves as a good model system.

3. CRYSTAL GROWTHAND DEFECTSIN Gal-,Mn,As Gal-,Mn,As samples are grown by LT-MBE on (100) epiready GaAs substrates using high-purity elemental sources evaporated from standard effusion cells for Ga and Mn, and either a standard effusion cell source for As4 or a cracker for ASZ. Although recent studies suggest that an As cracker source leads to the formation of fewer As antisite defects,’04there is no clear evidence to date that the ferromagnetic properties of Gal-,Mn,As grown using As2 are in any way superior to those of Gal-,Mn,As grown using As4.1°5 The substrates are mounted on molybdenum blocks with indium (or in some systems on indium-free holders) and the growth temperature is monitored and controlled using a radiatively coupled thermocouple situated behind the substrate mounting block. Sample growth is also monitored periodically using in sifu reflection high-energy electron diffraction (RHEED) at 1&15 k e y this allows the measurement of the surface crystallography as well as growth rates. We now provide some details about the typical growth conditions and protocol for fabricating single Gal -,Mn,As epilayers. It is worthwhile mentioning a nontrivial technical issue that often arises in this seemingly straightforward growth scheme: the radiatively coupled thermocouple scheme used in most MBE systems is notoriously poor at reading the correct substrate temperature because it varies with the emissivity of the substrate holder. Hence, some care needs to be taken to maintain reproducibility of the substrate temperature from one sample growth to another. As a further complication, the substrate temperature can change appreciably (tens of degrees Celsius) during sample growth as the emissivity of the substrate and the mounting block varies with material deposition. Additional transients in substrate temperature can also occur during sample growth that involves frequent opening and closing of the effusion cell shutters (for instance, during the growth of heterostructures). Because optical pyrometers do not work in this low temperature range, the best solution is to resort to other optical schemes such as in-siru band-gap measurements. Optical band-edge thermometry coupled with a temperature controller has been successfully used for achieving good consistency in the sample growth of complex systems such as “digital ferromagnetic heterostructures,” lo3

K.Takamura, F. Matsukura, D. Chiba, and H.Ohno, Appl. fhys. Lett. 81,2590 (2002). R. P.Campion et al., J. Cyst. Growth 251,311 (2003). K. C. Ku et al., Appl. Phys. Left 82,2302 (2003).

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

15

wherein fractional monolayers of MnAs are repeatedly incorporated into a GaAs lattice.'("j As with any MBE growth, the principal parameters that need to be optimized are the beam fluxes and the substrate temperature. Explorations of the parameter space presented by these three physical variables have identified a fairly broad regime for viable Gal-,Mn,As growth. Although a substrate temperature around 250°C is often used for the growth of G ~ ~ - , M ~ , Asuccessful S,~~ growth has been demonstrated at substrate temperatures as low as 150°C using "migration enhanced epitaxy" techniques wherein the As shutter is sequentially opened and closed to enhance the surface diffusion of the Ga adatoms.Io7 Substrate temperatures higher than 300°C appear to result in the formation of a NiAs phase of the metallic ferromagnet MnAs, while temperatures lower than 150°C result in surface segregation of elemental deposits of the source materials. A wide range of Ga:As beam flux ratios have also been reported from different laboratories. For instance, when As4 is generated from an uncracked source, the beam equivalent pressure ratio of As:Ga can range from 15: 1 to 100:1 (as directly measured on an ion gauge), without any obvious differences in the magnetic properties of the resulting samples. Although ferromagnetic Gal-,Mn,As can certainly be obtained over this broad range of parameter space, we will show subsequently that the specific growth conditions directly determine the magnetic, electronic, and structural properties of Gal-,Mn,As by altering the nature of the defects in the material. The first steps in the growth of Gal-,Mn,As epilayers are identical to those required for any standard GaAs-based heterostructure:after desorbingthe oxide from the surface of an epiready GaAs substrate, a buffer layer of GaAs is first deposited under standard (high-temperature)conditions (i.e., substrate temperature= 700"C, As4:Ga beam flux ratio of -15). The use of buffer layers (often in combination with smoothing GaAs/(Ga,Al)As superlattices) is by now an established procedure for establishing a high-quality, defect-free surface for subsequent epitaxial growth. During the growth of high-temperature GaAs, the RHEED pattern shows a (2 x 4) reconstruction. Once a buffer layer of the required thickness has been established, the substrate temperature is lowered to around 250°C for LT-MBE and the surface reconstruction changes to c(4 x 4). A thin (50 nm) epilayer of LT-GaAs is then deposited; RHEED shows an unreconstructed surface during this growth procedure. Finally, the Gal-,Mn,As epilayer is deposited, with the Mn:Ga flux ratio determining the percentage of Mn incorporated into the crystal. When the growth is carried out with As4, at a substrate temperature of around 250"C, and with an As:Ga beam equivalentpressure ratio around 15, the RHEED shows a clear R. K. Kawakami et al., Appl. Phys. Lett. 77,2379 (2000). Sadowski et al.,Appl. Phys. Lett. 78,3271 (2001).

'07 J.

16

NITIN SAMARTH

FIG.2. W E D diffractionpattern during LT-MBEof Gal-xMnxAs.The electron beam is incident along (a) [110] and (b) [liO].

(1 x 2) reconstruction (Fig. 2). Oscillations in the intensity of the specular RHEED spot can be used for determining both the growth rate (typically in the range of a monolayer/second) and the estimated Mn concentration; the latter is simply estimated from the fractional increase in growth rate for Gal-,Mn,As compared to GaAs. The resulting epitaxial layers have surface roughness characterized by an rms value of around 1 nm over areas of a square micron. X-ray diffraction measurements of thick epilayers show that the structural quality is comparable to that of high-quality GaAs (as characterized by the width of the x-ray diffraction peaks). Even though LT-MBE presents the best means of fabricating Gal-,Mn,As, the technique has a well-known drawback: The low substrate temperatures required for epitaxial growth of Gal-,Mn,As result in a large concentration of point defects, often with densities comparable to the intended Mn concentration itself. It has long been recognized, for instance, that GaAs grown by LT-MBE (often referred to as LT-GaAs) has a large density of arsenic antisites.lo8It is hence not surprising that LT-MBE-grown Gal-,Mn,As also has a large concentration of As-antisite defects, as well as other defects such as vacancies and interstitials. It was recognized even in early studies of Gal-,Mn,As that some of these defects may be donors that compensate the holes provided by Mn acceptors, and hence set constraints on TC if the ferromagnetism is hole-mediated." Indeed, the initial studies of Gal-,Mn,As showed that TC increases with increasing Mn content to its maximum value of 110 K at x x 0.05, and then drops for larger values of x . As expected, this reduction in TC was accompanied by a drop in the free hole density, indicating strong compensation of the holes by defects. The understanding of the role of defects in the physical properties of Gal-,Mn,As changed with the realization that the structural, electrical, and magnetic properties of as-grown Gal -,Mn,As epilayers could be substantially altered by post-growth annealing at temperatures comparable to the growth lo'

D.C. Look, J. Appl. Phys., 70,3148 (1991).

INTRODUCTION To SEMICONDUCTOR SPINTROMCS

17

9

8

7 h

E 6

G -E 5 @ Q

4

3 2

0

50

100

150

200

250

300

T (K) FIG.3. Temperature dependence of the resistivity (pm) in Gal-,Mn,As samples for different annealing times. Note the peak in pm near TC and the nonmonotonic changes with annealing time. (From Ref. 110.)

temperature. Specifically, these experiments showed that the annealing of Gal-,Mn,As epilayers at around 260°C for short time intervals (15 minutes to 2 hours) could enhance the hole density, the conductivity, the Curie temperature, and the saturation magnetization, while simultaneously reducing the lattice constant (see Figs. 3,4,5). More importantly, the annealing had a dramatic effect on the magnetization of the samples, and the elusive Curie temperature of TC= 110 K could be reproducibly attained for a range of Mn composition (0.05 2 x 2 0.085)."' Subsequentannealing experimentshave confirmed these findings,' 12-' l4 and additionally identified different sample growth and processing parameters that result in Curie temperatureswell in excess of the 110 K "ceiling," and with consistentlyreproducible values in the range of 140-160 K (for instance, see Fig. 6).'05*1'2,'13.115 One study has even reported a Curie temperature as high as 172 Kin a modulationY.Hashimoto, S. Katsumoto, and Y. Iye, Appl. Phys. Lett. 78, 1691 (2001). S. J. Potashnik et al.,Appl. Phys. Lett. 79, 1495 (2001). ' I ' S. J. Potashnik et al., Phys. Rev. E 66,012408 (2002). 'I2 K. W. Edmonds et al., Appl. Phys. Lett. 81,4991 (2002). 'I3 K. W. Eklmonds et al.,Appl. Phys. Lett. 81,3010 (2002). 'I4 B. S. Sorensen e f al., Appl. Phys. Lert. 82,2287 (2003). 'I5 D. Chiba, K. Takamura, F. Matsukura, and H. Ohno, Appl. Phys. Let?. 82, 3020 (2003). '09 T. Hayashi, 'lo

18

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SAMARTH

FIG.4. Temperature dependence of the magnetization of a Gal-,Mn,As sample for different annealing times. Note that while the unannealed sample has a kink in M ( T ) near 60 K, samples annealed for 30 min or more show smooth temperature dependence.The onset of ferromagnetism can be seen at 110 K. (From Ref. 110.)

doped heterostructure built from a digital ferromagnetic heterostructure. We note that this behavior is markedly different from that found in very early hightemperature annealing studies of Gal -,Mn,As epilayers wherein MnAs clusters were formed and the properties inherent to Gal-,Mn,As degraded.'17 Studies carried out by different groups have thus far identified two different regimes of time-dependentbehavior that result from the low-temperatureannealing of LT-MBE grown Gal-,Mn,As, as follows: At annealing temperatures of around 260°C,110the time dependence of the annealing is nonmonotonic, with an "optimal" window of 1-2 hours within which sample properties such as Tc, conductivity, hole density, and saturation magnetization are enhanced. Annealing for longer periods typically degrades these properties. Interestingly, the lattice constant continuously decreases, regardless of the annealing time. The microscopic origins of this nonmonotonic behavior of the magnetic and electronic properties are still not understood, but the data suggest the presence of two competing effects. 'I6 A. M. Nazmul, S. Sugahara, and M. Tanaka, Phys. Rev. B 67,241308(R) (2003). 'I7 A. VanEscheral., Phys. Rev. B56, 13103 (1997).

INTRODUCTIONTO SEMICONDUCTORSPINTRONICS

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1.8

6

g

1.5 1.2

Y 7

0.9

-

5 m

5.685 5.675 5.665 100

g

80

Y

60 40 0

10

5

22

24

tAnneal (hr)

FIG.5. Various properties of Gal-,Mn,As as a functionof annealingtime: (a) Resistivityand carrier concentration estimated from the Hall resistance at 300 K;(b) lattice constant at 300 K from x-ray diffraction; (c) ferromagnetic transition temperature taken from the maximum in the slope of M(T). In each quantity, there are two regimes of behavior, with fernmagnetism and conductivity optimized at the crossover between the two at annealing times of 12 h. (From Ref. 110.)

At much lower annealing temperatures (around 180"C), the time dependence of annealing is monotonic, with sample properties logarithmically improving with time;Il3 as we will show shortly, a systematic study of the time-dependent changes in sample conductivityhave now yielded a credible picture that explains these observations. Experiments have also shown that the proximity of a free surface influences the outcome of low-temperature annealing. For a fixed set of conditions (e.g., annealing temperature and annealing duration) and sample composition (Mn content), annealing produces higher Curie temperaturesin thinner samples (see Fig. 6).105*114 In addition, the beneficial aspects of annealing appear to be suppressed in Gal-,Mn,As samples that are capped with an epitaxial layer of G ~ A S . ' ~ ~ ~ " ~

20

NITIN SAMARTH

2 h

:

h

Y

v 5.

t-"

c

C

a

5

1

z

0 0

50

100

Temperature

150

0

(K)

40 80 Thickness (nm)

120

FIG.6. (a) Magnetization (in units of Bohr m a g n e t o n a n ) vs. temperature for Gal-xMn,As ( x = 0.085) epilayers with t = 15 nm (squares) and 50 nm (circles), as grown and after annealing (field in-plane). (b) Curie temperature for Gal-,Mn,As ( x = 0.085) epilayers of varying thickness. (From Ref. 105.)

Systematic studies show that a GaAs layer as thin as a few monolayers can significantly suppress the enhancement of the Curie temperature by annealing."' All these annealing studies suggest the presence of metastable defects in Gal-,Mn,As that can readily diffuse at modest temperatures. This picture is substantiated by Rutherford backscattering and particle-induced x-ray emission studies that indicate the Mn ions in Gal-,Mn,As occupy two types of sites: substitutional (MnGa) and interstitial (Mnl)."9~'20These experiments also show that low-temperature annealing reduces the number of Mnl, although not specifically identifying where these defects migrate during the annealing process. Because Mn interstitials act as double donors, the deleterious compensating effects of these defects is reduced during the annealing process, consistent with the observed increase in hole density. A theoretical calculation also suggests that Mn, may be antiferromagnetically coupled with Mnca. Although this may be qualitatively consistent with the observed increase in the saturation magnetization after annealing, there is no quantitative or direct confirmation of this conjecture. The scenario of metastable Mn interstitials that diffuse during annealing has been made even more substantial by in situ conductivity measurements carried out during the annealing of single Gal-,Mn,As epilayers at 190°C.'22 These

'*'

M. B. Stone er af., Appl. Phys. Lett. 83,4568 (2003). 'I9 K. M. Yu et af., Phys. Rev. E 65,201303(R) (2002). IZ0 X-ray absorption studies of as-grown and annealedsamples also indicated the presence of two types 'I8

of Mn sites; however, the non-substitutional site was not interpreted as an interstitial. See Y.Ishiwata et al., Phys. Rev. E 65,233201 (2002). 12'

J. Blinowski and P.Kacman, Phys. Rev. E 67, 121204 (2003). K. W. Edmonds et af., Phys. Rev. Lett. 92,037201 (2004).

INTRODUCTION TO SEMICONDUCTORSPINTRONICS

21

experiments show that the time dependence of the conductivity may be modeled by a straightforward 1 D diffusion process wherein interstitials migrate toward the free surface of a sample. In this model, the increase in conductivity with annealing time is entirely attributed to the decrease in hole compensation as interstitials diffuse out of the “bulk” of a sample. Although the experimental data are fit to a simple model that assumes symmetric out-diffusion toward both the free surface and the interface with the GaAs buffer layer, it is speculated that the actual diffusion process must be asymmetric, with preferential diffusion toward the free surface. This is because the accumulation of Mn interstitials (donors) into the lower GaAs buffer layer should build up a p-n junction that sets up a Coulomb barrier. In contrast, one might reasonably speculate that Mn interstitials accumulating at the free surface of the sample are efficiently passivated by processes such as oxidation. This model is consistent with the observation that the free surface of a Gal-,Mn,As sample influences the outcome of annealing: for a fixed set of annealing conditions, interstitials are removed more efficiently from thinner samples, resulting in a higher Curie temperature; in addition, the enhancement of Curie temperature is suppressed in samples that are capped with GaAs because the diffusion of Mn interstitials into the cap layer creates a Coulomb barrier that limits the out-diffusion of these compensating defects. 4. MAGNETIC PROPERTIES OF Gal-,Mn,As

The previous discussion of defects in Gal-,Mn,As perhaps leaves us with a sense of dread. It appears that it would be very difficultto understand the magnetic behavior of Gal-,Mn,As epilayers given the conclusion that an epilayer of the material may have an inhomogeneoushole density along the direction of epitaxy. Nonetheless, under certain circumstances, Gal-,Mn,As epilayers can show macroscopic magnetic properties that are reasonably well-behaved in a conventional sense. We illustrate this by examining how the magnetic properties of Gal-,Mn,As epilayers vary with Mn composition, focusing on one particular study of a series of ferromagnetic Gal-,Mn,As samples with 0.012 2: x 2 0.083,all of which were grown using a consistent set of parameters.”’ All the Gal-,Mn,As epilayers in this study are 123 f 2 nm thick and grown on a buffer structure consisting of a standard (high-temperature-grown) 100-nmGaAs epilayer followed by a 25-nm low-temperature-grown GaAs epilayer. As we will see later, depending on the hole spin polarization and the amount of strain, such compressive strain leads to an in-plane easy axis of the magnetization, oriented either along the [ 1001 or the [110]crystal axis. In all the samples used in this study, the in-plane anisotropy is quite weak, and magnetization is measured with the field along the [ 1101 direction using a commercial superconducting quantum interference device magnetometer in a field of 0.005 T after cooling in a 1 T field. Magnetization data taken to T 2 320 K show no evidence of MnAs precipitates. Samples in the continuous

22

NITIN SAMARTH 35

I

I

I

I

I

0

50

100

150

200

30 25

10

5 0

T (K) FIG. 7. The magnetization and resistivity as a function of temperature for Gal-,Mn,As with x = 0.06 (circles) as-grown (triangles) annealed. The thick solid line is a Heisenberg model fit to the magnetizationdata for T < 40 K as described in the text.

series were grown in the order of increasing Mn source temperature which yielded increasing Mn concentrations. The Mn concentrations of the annealed samples were determined by electron probe microanalysis (EPMA) wherein the La lines of Mn, Ga, and As were detected and compared with those of the calibration standards (GaAs and the mineral rhodonite). Note that care needs to be taken while carrying out the EPMA measurements because the buffer layer and substrate underneath the Gal-,Mn,As epilayer is GaAs. This requires using a low-energy electron beam (3 keV) to ensure that the penetration depth (100 nm) is smaller than the Gal-,Mn,As layer thickness. The EPMA results show a monotonically increasing value of x with the Mn source temperature as expected. Figure 7 shows the typical temperature dependence of the resistivity p ( T ) and the magnetization M ( T ) for a Gal-,Mn,As sample with x = 0.06, before and after annealing.The resistivity follows a variable range-hoppingform above Tc,but reaches a maximum at Tc, and then decreases as the temperature decreases further. This nonmonotonic temperature dependence of p ( T ) is still not well understood, although it has been attributed to critical scattering by magnetic fluctuations5*and magneto-impurity scattering.lZ3 The magnetization, M ( T ) , displays a sharp rise at Tc. as expected for longrange ferromagnetic order. In the as-grown samples, M ( T ) shows a linear increase and a kink below Ti-; such unconventional temperature dependence of the spontaneous magnetization is often observed in as-grown samples, and has been attributed to a variety of causes: multiple exchange interaction^,"^ noncollinear lZ3

Sh. Yu. Yuldashev et aL, Appl. Phys. Lett. 82 1206 (2003).

INTRODUCTION TO SEMICONDUCTORSPINTRONICS

3

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0.02 0.03 0.04 0.05 0.06 0.07 0.08 X

FIG. 8. (a) and (b) The transition temperature (Tc) and the relaxed lattice constant as a function of Mn concentration (x) for annealed Gal-,Mn,As samples (solid circles). Data from unannealed samples (open circles) are shown for comparison. Previous measurements of the lattice constant of unannealed samples are shown by the solid5' and dashed"' lines. (c) The exchange energy ( J ) as a function of x. The solid circles represent values obtained from fits to M ( T ) ;the open triangles represent values determined from Tc. (From Ref. 111.)

ferr~magnetism,'~~ and disorder.125 Recent studies have also found that-in some cases-the unusual form of M(T) may have a more straightforwardexplanation: the temperature dependence of the hole spin polarization can result in the easy axis of magnetization switching between different crystalline axes. Under such circumstances, the measurement of M ( T ) for a fixed direction of magnetic field can even yield a nonmonotonic temperature dependence.lo3 By contrast, the annealed samples display a much more conventional ferromagnetic M ( T ) at low temperatures as we will discuss. We now discuss how the physical properties of Gal-,Mn,As evolve with Mn content ( x ) . Figure 8 displays the concentration dependence of TC (as determined by the onset of a ferromagnetic moment in M ( T ) ) for as-grown and annealed samples. Schliemann and A. H. MacDonald, Phys. Rev. Lett. 88, 137201 (2002). Sarma, E. H. Hwang, and A. Kaminski, Phys. Rev. B 67, 155201 (2003). 126 M. Sawicki et al., cond-ma1/0212511. '24 J.

125 S. Das

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NITIN SAMARTH

The results for the as-grown samples qualitatively resemble the behavior obtained in early studies:52 TC increases with x for x I0.05, and then decreases. In the annealed samples, however, for x 2 0.05, TC reaches 110 K and then becomes independent of Mn concentration up to the highest value of x. While it may be tempting to plot a compositional phase diagram for Gal-,Mn,As based on such data, such an exercise is not particularly meaningful, given the number of interdependent parameters that determine Tc(Mn concentration, defect-dependent hole density, and thickness). For instance, as we have already seen in Fig. 6, the plateau in TC shown in Fig. 8 has an extrinsic origin due to incomplete annealing of the Mn interstitials in these 120-nm-thick epilayers. Once samples with a conventional, homogeneous magnetization are identified, we can use measurements of M ( T ) to yield the exchange energy J associated with the ferromagnetic state. As shown by the solid line in Fig. 7, in the low-temperature limit, M ( T ) for annealed samples follows the behavior expected within a standard three-dimensional Heisenberg model M ( T ) = Mo - 0.117pB-

kBT 2sJd ’

(2.4)

where M O is the zero temperature magnetization, d is the spacing between Mn ions, and J is the exchange interaction. The extracted values of J agree reasonably well with the mean field prediction^.^^ This agreement suggests that-despite the reservations expressed earlier-ferromagnetism in Gal -,Mn,As can be understood within a conventional model. From the fits to M ( T ) , we extract the zero temperature magnetization (Mo) that corresponds to the number of spins in the ferromagnetic state. As shown in Fig. 9, M O increases with Mn concentration but becomes almost constant for x I 0.05. As one might anticipate from the observation that some fraction of Mn occupies interstitial (and presumably magnetically inactive) sites, the moment per Mn ion is well below the full saturation value for spin 5/2 moments. The decrease in this value with increasing x throughout the entire range of doping to about half of the expected full moment for all the Mn spins implies that, as the concentration of spins in the material increases, a decreasing fraction are actually participating in the ferromagnetism. The most likely explanation is that the local electronic structure associated with certain defects precludes individual Mn moments from participating in the ferromagnetism, and that these defects are enhanced with increasing Mn content. A more exotic explanation for the magnetization “deficit” suggests that some fraction of the spins form a spin-glass-like state at low temperatures in parallel with the ferromagneti~m.’~~ while initial studies have not found any irreversibility between field-cooled and zero-field-cooled data G. Schott, W. Faschinger, and L.W. Molenkamp, Appl. Phys. Left. 79, 1807 (2001).

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

:a 3 Y

25

4M 5 (c) , 0 0.02

0.06

0.04

0.01

X RG.9. (a) and (b) The zero-temperature magnetization (Mo) and the magnetization per Mn atom as a function of Mn concentration ( x ) for the annealed Gal-,Mn,As samples. Note that the latter decreases monotonically with x for the entire range of samples studied. (c) The estimated domain wall thickness for the range of samples studied. (From Ref. 1 1 1.)

at 0.01 tesla, and applied fields of up to 70 kOe (which should quench a spin glass state) do not reveal significant additional moment, the search for this possibility continues. Fits to the magnetization data also yield estimates of the domain wall thickness ( t ) using a simplified model. We consider a two-dimensional Nel wall with no in-plane anisotropy (verified by coercive field measurements on our samples) with a thickness of the form t = [nJS2/2Mi2d]1/2. The values obtained ford, J, and MO then imply an average domain wall thickness of 17.1 f .6 nm, which is remarkably constant over the entire range of Mn concentrations studied (Fig. 9). Although this thickness has not been measured experimentally,it is not inconsistent with direct measurements of magnetic domains in both compressively and tensile strained Gal-,Mn,As, where domain sizes range from a few p m to several hundred micron^.'^*,^^^ An example of the temperature dependence of the magnetic domain structure in (tensile strained) Gal-,Mn,As is shown in Fig. 10. A detailed discussion of magnetic domains in Gal-,Mn,As is beyond the scope of this review, and we refer the reader to an analysis that treats magnetic domain

129

T. Shono et al., Appl. Phys. Lett., 77, 1363 (2000). U. Welp et al., Phys. Rev. Lett. 90, 167206 (2003).

26

NITIN SAMARTH

Rc.10. Magnetic images of Gal-,Mn,As obtained using a scanning Hall probe microscope.Measurement temperatures are (a) 9 K, (b) 20 K, (c) 30 K,(d) 65 K,(e) 70 K,and (f) 77 K.The image areas are 4.8 x 4 . 8 ~m2 for 9-30 K and 7.3 x 7.3 bm2 for 63-77 K. The horizontal axes are [loo]. The grey and black regions denote positive and negative EL,respectively. (From Ref. 128.)

structure in Gal-,Mn,As from the viewpoint of mean field theory, yielding good agreement with the principal experimental observations.130 Finally, we address the connection between the Curie temperature and the hole density in Gal -,Mn,As. As we have seen earlier, the mean field approach predicts a simple connection between carrier density and Tc.Unfortunately, the measurement of the hole density in Gal-,Mn,As is non-trivial. Measurements of the Hall effect contain an overwhelming contribution from the anomalous Hall term, and the reliable extraction of the standard Hall contribution either necessitates measurements at extremes of temperature (below 1 K) and magnetic field (greater than 15 T), or a careful and detailed modeling of the Hall data.'I2 Raman measurements provide an alternative means of determining the hole density through the observation of the coupled plasmodlongitudinal optical phonon mode. l3I* 13' Using a combination

T. Dietl, J. Konig, and A. H. MacDonald, Phys. Rev. B 64,241201 (2001). M. J. Seong et al., Phys. Rev. B 66 033202(R) (2002). '32 W. Limmer et al., Phys. Rev. B 66,205209 (2002). I3I

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

27

p(x1d’ cm-3)

Rc.11. Curie temperatureTC (obtained from SQUID magnetization measurements) vs hole density p (obtained from Raman scattering) for a wide range of post-growth annealed Gal-,Mn,As samples of dtferenr Mn content. The samples with TC < 110 K have thickness t 123 nm, while the two samples shown with TC 140 K have t 550 nm. The solid line is a fit to TC p’I3.

-

--

-

of SQUID and Raman measurements, we obtain an empirical correlation between the hole density and TC for well-behaved annealed samples (see Fig. 11). Suprisingly, the data appear to indicate that-within the error bars set by the Raman measurements-Tc p1l3,but do not show the expected dependence on the Mn content. The reasons for this observation are presently unclear. It is possible that the observed behavior is an accidental consequence of an inhomogeneous hole density along the growth axis in annealed samples.

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5. MAGNETO-TRANSPORT IN Gal-,Mn,As Beginning with the earliest studies of the 111-Mn-V ferromagnetic semiconductors, it was recognized that magneto-transport measurements in these materials provide a convenient means of probing the ferromagnetic properties.” As we briefly mentioned earlier, the Hall effect in these ferromagnetic semiconductors has contributions from both the standard Hall voltage as well as the “anomalous Hall effect” (AHE) and is described by pq = RH B

+ RAPOM,

(2.5)

where RH is the ordinary Hall coefficient, R A is the AHE coefficient,and M is the sample magnetization.An example of the AHE in Gal -,Mn,As is shown in Fig. 12.

28

NITIN SAMARTH 401

I

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I T=120K

FIG. 12. Measurement of the anomalousHall effect in a tensile strained layer of Gal-,Mn,As with a TC close to 135 K. The magnetic field is applied along the easy axis, normal to the sample plane. The Hall resistance follows a hysteresis loop identical to that measured in direct magnetometry.

Experimental studies of ferromagnets in general indicate that the anomalous Hall effect in many systems is well described by the following empirical ansatz:

wherein the exponent y is determined by the dominant scattering mechanism, with values ranging from 1 to 2 for the “skew-scattering” and “side-jump” mechanisms, respectively. A detailed understanding of the AHE in Gal-,Mn,As is relatively recent, and appears to involve a very different picture wherein the AHE originates not in the interplay between spin-orbit coupling and elastic scattering, but rather from an intrinsic Berry’s phase effect related to the presence of the spin-orbit coupling i t ~ e 1 f . This I ~ ~ theory also addresses all aspects of the anisotropic magnetoconductivitytensor in these ferromagnetic semiconductors,such as the anisotropic magnetoresistance (AMR)measured when a magnetic field is applied in the plane of a sample, parallel to the direction of a propagating current. Here, the longitudinal electric field has a contribution that arises from differences in the sample resistance when a current propagates parallel and perpendicular to the sample magnetization. An interesting consequence of this anisotropic magnetoconductivity tensor is the “planar Hall effect,” wherein a transverse electric field arises in a measurement geometry identical to that used for A M R measurements (i.e., the magnetic field is in plane).’34Although the planar effect is well-known in metallic ferromagnets,it is T. Jungwirth et al., Appl. Phys. Len. 83,320 (2003). H. X.Tang, R. K. Kawakami, D. D. Awschalom, and M. L. Roukes, Phys. Rev. Left. 90, 107201 (2003). ‘33 lM

INTRODUCTIONTO SEMICONDUCTORSPINTRONICS

29

FIG. 13. (a)-(c) Planar Hall resistance for Hall bars (1 mm, 100 pm, 6 p m wide) at 4.2 K as a function of an in-plane magnetic field oriented at an angle of 20” from the current direction. (d) Fielddependent sheet resistance of a 100-pm-wide Hall bar, showing the anisotropic magneto-resistance effect. (e) Sketch of the relative orientationsof sensing current I, external field H,and magnetization M.A SEM micrograph of a 6-pm-wide device is also shown. (f) Barkhausen jumps that are evident solely in 6-pm-wide devices near the resistance transitions. (From Ref. 134.)

typically characterized by very small Hall resistances in metallic systems (-jA2). However, experiments carried out in Gal-,Mn,As samples reveal an extremely large planar Hall effect with Hall resistance changes of the order of 100 S2. This “giant planar Hall effect” is an elegant and sensitive probe of the angle-dependent magnetization and also provides a means of studying magnetic domain reversals (Fig. 13). Quite remarkably, the large domain sizes in Gal-,Mn,As enable the observation of Barkhausen jumps even in devices as large as a few microns (Fig. 13(f)). Ongoing measurementsof this planar Hall effect show that the signals tend to be even larger in more insulating digital Gal-,Mn,As a l l 0 ~ s . l ~ ~

6. MEASUREMENTS OF SPINPOLARJZATION IN Gal-,Mn,As The hole spin polarization (P) in Gal-,Mn,As is of direct relevance for applications in spintronic devices, and is also of fundamental importance because it is not immediately obvious how the presence of a spin-orbit interaction might affect the spin polarization. In the latter context, mean field theory suggests that the destructive effect of the spin-orbit interaction is suppressed with increasing 135

E. Johnston-Halperin et al., Phys. Rev. B 68, 165328(2003).

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NITIN SAMARTH

band ~ p l i t t i n g The . ~ ~ large spontaneous spin splitting of the valence band observed in Gal-,Mn,As (-44 meV at low temperatures for x = 0.035)25is hence expected to restore a highly spin-polarized hole gas even for high carrier densities. Furthermore, the observation of a large tunneling magneto-resistance in magnetic tunnel junctions derived from this material reinforces the anticipation that P may be large even for small Mn concentration^.'^"'^^ This is consistent with band structure calculations that predict full spin polarization for x > 0. 125139and even at much lower Mn concentration^.^^ Measurements of the hole spin polarization have recently been obtained using Andreev reflection spectroscopy in both Gal -,Mn,As/Ga planar junctions" and using point contact spectro~copy.~~' Such measurements rely on the Andreev reflection process, which converts quasiparticle currents in a normal metal to supercurrent in a superconductorat the interface between them. The essential physics underlying the measurement of spin polarization using Andreev reflection may be understood using a simple picture. Consider a single electron in the normal metal with energy smaller than the superconducting gap A. In order for this electron to enter the superconductor, it must form a Cooper pair with an electron of opposite momentum and opposite spin, resulting in a N/S junction conductance that is larger than its normal state value below the gap. When the normal metal is replaced by a ferromagnet, Andreev reflection is suppressed due to the spin imbalance, leading to a suppression of the subgap conductance. Measurements of this suppressed conductance directly probe the spin polarization in the ferromagnet. A detailed analysis of the conductance spectra of superconductor/normal metal junctions with arbitrary interfacial scattering strength can be carried out using a theory developedby Blonder, Tinkham and Klapwijk (BTK).142 The interfacial scattering strength is measured with a dimensionless parameter Z, with Z = 0 for a metallic contact and Z >> 1 for a tunnel junction. In this model, Z is a phenomenological parameter that accounts for the effects of the physical barrier (potential scattering) as well as that of the Fermi velocity mismatch. A modified version of the BTK model includes the effects of spin polarization in superconductor/ferromagnet junctions.143In this modified BTK approach, P can beeasily estimatedforhightransparency junctions: when Z = 0, P = 1 - (G(0)/2). We now discuss in some detail the Andreev reflection measurements carried out using Gal-,Mn,As/Ga planar junctions. M. Tanaka and Y. Higo, Phys. Rev. Len. 87,026602 (2001). S. H. Chun er al., Phys. Rev. E 66, 100408(R) (2002). 13* R. Mattan er aL, Phys. Rev. Len. 90,166601 (2003). '39 T. Ogawa, M. Shirai, N. Suzuki, and I. Kitagawa, J. Mag. Maen. Marer. 197,428 (1999). J. G . Braden et aZ., Phys. Rev. Len. 91,056602 (2003). 141 R. J. Panguluri er nL, unpublished. 14* G . E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. E 25,4515 (1982). 143 I. h t i C and S. Das Sarma, Phys. Rev. E 60,R16322 (1999).

13'

13'

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INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

The Gal-,Mn,As/Ga planar junctions are fabricated as follows: First, a 20-nmthick, p-doped GaAs:Mn buffer layer is grown on a heavily p-doped (001) GaAs:Zn substrate using MBE under standard conditions for high-quality GaAs growth. A Gal-,Mn,As epilayer (typically around 100 nm thick and with x 0.05) is then grown by LT-MBE as described earlier. The as-grown Gal-,Mn,As epilayer has 65 K. Immediately after this layer is grown, the substrate temperature is a Tc lowered to 100°Cand a thick layer (thickness larger than 500 nm) of Ga is deposited under UHV conditions in the same MBE chamber. The conductance spectra of the samples are measured in 1 mm2 planar junctions, using phase-sensitive detection in a 3 He cryostat. l b o of the contacts are made on the conducting substrate, while the other two contacts are on top of the Ga electrode. Qpical normal state junction resistances are 10-100 52, while the serial resistance from the Gal-,Mn,As layer is at least seven orders of magnitude smaller. Figure 14 shows the normalized conductance as a function of bias voltage 0.05. This conductaken at 370 mK for a Gal-,Mn,As/Ga junction with x tance spectrum is typical of that for a high transparency metallic contact between a superconductor and a ferromagnet with high P; the conductance peaks at &A corresponding to quasiparticle tunneling are completely absent and the subgap conductance is suppressed, instead of enhanced, from G N due to the large imbalance of spin populations in the ferromagnet. The application of the modified BTK estimate to the zero-bias conductance yields an estimated spin polarization close to 90% for this Gal -,Mn,As sample. Additional analysis, however, indicates that the details are somewhat more complicated. While the conductance spectrum has all the qualitative features of a superconductor/ferromagnet contact, the modified BTK theory fit to the entire spectrum is not completely satisfactory. Further, the approximate energy gap for Ga inferred from the shoulders of the spectrum is roughly 1.4 meV, which corresponds to a superconducting transition temperature much higher than that for bulk crystalline Ga (1.1 K). These discrepancies have been attributed to a distribution of the energy gap and the transition temperature in the Ga film. It is known that several phases of Ga have transition temperatures substantially higher than 1.1 K, and that the transition can occur at temperatures as high as 8.4 K in amorphous thin films of Ga. The Ga film in these devices is grown at a low temperature (-lOOC) and has a granular morphology, suggesting that differences in grain size and crystallinity may result in local variations of transition temperature and energy gap in the film. As shown in Fig. 14, the conductance spectra can indeed be fit to the modified BTK theory by including a distribution of energy gaps in the superconductor. It is noteworthy that an identical distribution is used in both fits in this figure. This distribution is created as an ad hoc weighting and reflects that large portions of the Ga film have transition temperatures around 1.1 K and 8.4 K. The fits yield values Z 0, and P = 0.90 and P = 0.85 respectively, consistent with values calculated from G(0). Furthermore, the suppression of G(0) persists far above the

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32

NlTIN SAMARTH

FIG. 14. Measurement of conductance-voltage characteristics for two planar junctions of Gal-,Mn,As/Ga made from the same wafer. The fits to the data are based on the modified BTK theory, using a distribution of superconducting transition temperatures and gaps. We note that an identical distribution was used for both fits. (From Ref. 140.)

superconductingtransition temperaturefor bulk Ga (1.1 K), vanishing only when T approaches 8.4 K. This corroborates the interpretation that the broad conductance dip does not have its origins in simple thermal broadening or inelastic effects. The large Fermi velocity mismatch between the Gal-,Mn,As semiconductor layer and the metallic Ga layer needs to be considered a bit more carefully. Gal -,Mn,As samples grown under identical conditions as the junctions used in these measurements typically have a hole density of 3 x 1020 ~ m - ~ . In the absence of additional information, we assume that the (heavy) holes in Gal-,Mn,As have the same effective mass as in GaAs (0.45 me). This yields a roughly order of magnitude Fermi velocity mismatch between the two components of the heterostructure (4.6 x lo5 m/s for Gal-,Mn,As compared to 2.0 x lo6 d s for Ga). The effect of this mismatch is in some sense already incorporated within the BTK analysis by the parameter 2,which measures the overall interfacial

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INTRODUCTION TO SEMICONDUCTORSPINTRONICS

33

scattering strength. This large mismatch should result in a substantial 2 even in the absence of any physical barrier at the interface, in contradiction with the extracted values of 2 0. The resolution of this apparent problem lies in a generalization of the BTK analysis for superconductor/semiconductorjunctions. '43 This modified analysis leads to a highly decreased junction transparency for a superconductor/conventionalsemiconductorcontact, signified by a substantial decrease N pronounced peaks at &A in the conductance spectrum of G ( 0) from ~ G and even when the interfacial potential scattering is completely absent (2 = 0). However, in a ferromagnetic semiconductor, the spin polarization actually enhances junction transparency: the conductance peaks at &A from the Fermi velocity mismatch can be completely suppressed by a moderate spin polarization in the ferromagnetic semiconductor, while those due to potential scattering are nor affected by the spin polarization. Hence, the complete absence of any peaks at &A in Fig. 14 is quite consistent with a high transparency of the Gal-,Mn,As/Ga interface (Z = 0) and a high spin polarization for the Gal-,Mn,As. The modified BTK analysis also shows that the zero bias conductance G(0) decreases with increasing spin polarization of the ferromagnet, implying that G(0) is a good measure of the spin polarization in high transparency superconductor/semiconductor junctions. While these Andreev reflection experiments yield a very high intrinsic spin polarization for Gal-,Mn,As, it appears that is extremely difficult to maintain this high spin polarization at many types of metal/Gal-,Mn,As interfaces. A number of different superconductor/Gal-,Mn,As heterostructures have been fabricated and measured, and-in many cases-the conductance spectra do not show any evidence for Andreev reflection, suggesting that the spin polarization at the surface of Gal-,Mn,As is extremely sensitive to the nature and quality of the interface formed with other materials. The successful observation of Andreev reflection in Ga/Gal-,Mn,As heterostructures is all the more intriguing given the very low melting temperature of bulk Ga (-30" C); as expected, annealing of such samples at even very low temperatures results in a significant degradation of the Andreev reflection signal.

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7. CRITERIA FOR IDENTIFYING A FERROMAGNETIC SEMICONDUCTOR Given the large number of recent reports on the observation of above-room temperature in ferromagnetism in a variety of material systems, it is important to sound a note of extreme caution: it is well-recognized from the earliest studies of the 111-Mn-V materials that the formation of metallic ferromagnetic clusters within the semiconductor host can often result in misleading results. When the metallic clusters are large, their unambiguous presence is readily revealed in structural studies (such as x-ray diffraction or transmission electron microscopy)

34

NITIN SAMARTH

as well as in magnetic force microscopy. In fact, several studies have explicitly engineered metallic ferromagnetic clusters within crystalline semiconductor hosts with the aim of achieving interesting effects such as enhanced magneto-resistance or magneto-optical properties.lM Because one of the primary goals of semiconductor spintronics is to obtain optical or electrical control of ferromagnetism, it is important to identify ferromagnetic semiconductors wherein there exists a clear coupling between the carrier density and the ferromagnetic order. The best way to ascertain this requirement is to cany out multiple complementary measurements that probe the ferromagnetic transition: the anomalous Hall effect, magnetic circular dichroism, temperature-dependent conductivity, magneto-optical Kerr effect, and SQUID magnetometry are some of the common tools that can be employed for such tests (see, for instance, Fig. 15). As an example, we direct the reader to a recent study of (Zn,Cr)Te, where both magnetic circular dichroism and direct magnetometry yield consistent Curie temperatures close to room temperature.@

111. Coherent Spintronics with Conventional Semiconductor Heterostructures The drive toward semiconductor spintronic devices using conventional semiconductors focuses on two broad classes of functionality, as follows: Devices that employ the generation, transport, and detection of spin-polarized states but whose functionality-while explicitly dependent on quantum mechanics4oes not exploit the full potential of the wave function (i.e. phase). Examples include the Datta-Das proposal for a “spin transistor” wherein ferromagnetic contacts are used to inject a spin-polarized current into a semiconductorI3 and ultrafast spin switches that rely on spin-selective optical transitions in semiconductor^.'^^ Phase coherent devices that exploit both the amplitude and phase of a wave function. Such spintronic devices might use, for instance, the interference between two coherently occupied spin states whose time variation occurs at a frequency A E l h, where A E is their energy separation. Because typical spin splittings in semiconductors are in the range of meV, the rapidly varying oscillations of a classical observable such as the spin orientation (magnetization) can occur at GHz-THz frequencies, providing the basis for ultrafast devices. Another possibility is that this quantum interference may be manipulated by external electric,

‘44 H. Shimizu, T. Hayashi, T. Nishinaga, and M. Tanaka, Appl. Phys. Lett. 74,398 (1999). 145

R.Takahashi, H. Itoh, and H. Iwamura, Appl. Phys. Lett. 77,2958 (2000).

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

35

3 2 h

=

E

1

' 8 O v c -1 C 7

Q

g

-2

5

-3 -600 -400 -200

0

200

400

600

H (Oe)

-

6

-

4

-

2

0

2

4

6

H (T)

l00

40

80

120

160 l

Temperature (K) hG. 15. Complementary measurementsof magnetization in a Gal-,Mn,As epilayer, showing how different probes such as direct magnetometry,electrical transport (anomalous Hall effect), and magnetooptical spectroscopy(magnetic circulardichroism)can be used to make surethat fernmagnetismindeed arises from a coupling between local moments and carriers. (From Ref. 105.)

36

NITIN SAMARTH

magnetic, or optical fields for quantum information p r o ~ e s s i n g . ’ ~We , ’ ~also ~ refer the reader to proposals for quantum computation based on nuclear spins in phosphorus-doped silicon14*and electron spins in quantum dot^."^,^^ Both these classes of devices raise a number of fundamental questions associated with the injection,’49propagation,150 relaxation,15’ and dephasing of spin states in semiconductors. It is important to emphasize the distinction between “spin relaxation” and “spin dephasing or decoherence” in this context:’52the former is associated with a change in population of spin polarized states and is characterized by the longitudinal spin relaxation time T I ,while the latter is associated with loss of phase relationship between a coherent superposition of quantum states and is characterized by the transverse spin relaxation time (often referred to as the spin decoherence time) T2. In this section, we limit our discussion to an overview of experiments that address spin decoherence (and hence the time T2) in situations relevant to the development of coherent spintronic devices.

8. OPTICAL MEASUREMENTS OF SPIN COHERENCE: EXPERIMENTAL WCHNIQUES The general approach toward measurements of coherent spin states in semiconductors begins with the optical preparation of a superposition of basis states. This is easily accomplished by using the electric dipole selection rules that govern optical transitions across the energy gap in direct gap semiconductors such as GaAs and ZnSe (Fig. 16). For instance, circularly polarized photon pulses from an ultrafast laser can excite a population of spin polarized electrons from the lowest energy heavy hole state to the lowest energy conduction band ~ t a t e . ~If~an? ’external ~~ magnetic field is applied perpendicular to the direction of the propagation of the circularly polarized light pulse (the “Voigt geometry”), then each promoted electron is in a coherent superposition of the basis states defined by the magnetic field. An alternative scheme uses linearly polarized photons to accomplish a similar coherent superposition with the magnetic field applied parallel to the propagating light (the “Faraday” g e ~ m e t r y ) . ’ ~In~either , ’ ~ ~ case, the coherent evolution of such Y. Kato et al., Science 299, 1201-1204(2003). J. A. Gupta, R. Knobel, N. Samarth,and D. D. Awschalom, Science 292,2458 (2001). 14’ B. E. Kane, Nahcre (London) 393, 133 (1998). 149 G. Schmidt et al., Phys. Rev. B 62, R4790 (2000). Is’ M. E. Flank and J. M. Byers, Phys. Rev. Lett. 84,4220 (2000). W. H. Lau, J. T.Olesberg, and M. E. Flattk, Phys. Rev. B 64, 161301(R) (2001). Is* M. E. Flank, J. M. Byers and W. H. Lau, in Semiconductor Spintronics and Quantum Computation, eds. D. D. Awschalom, D. Loss, and N. Samarth,Springer-Verlag, Berlin (2002). 107. Is3 A. P. Heberle, J. J. Baumberg, and K. Kohler, Phys. Rev. Lett. 75, 2598 (1995). J. J. Baumberg ef al., Phys. Rev. Lett. 72,717 (1994). Is5 S.Bar-Ad and I. Bar-Joseph, Phys. Rev. Lett. 68,349 (1992). 14’

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS H=O

H>O

31

mi +I12

cB(r6)

-112

I#

-312

+

VB u-8)

-112 +I12 +3/2

0-

FIG. 16. Selection rules for optical excitations in a direct bandgap zinc-blende semiconductor such as GaAs or ZnSe.

a superposition results in a phase difference between the two energy eigenstates that varies linearly in time as A E t / h . In a semi-classical sense, we may view all the optically excited spins as adding constructively to yield a net magnetization that precesses at the Larmor frequency. The coherent superposition can decay due to a number of possible interactions, including spin-spin scattering with holes, local magnetic impurities, or other electrons, as well as spin-orbit scattering due to phonons or impurities. In addition, the duration of the coherent superposition is of course limited by finite lifetime effects caused by optical recombination with a hole. If all the optically excited conduction electrons are identical and non-interacting, then the decay of the net magnetization reflects the intrinsic spin decoherence of individualelectrons. For an exponential decay, this homogeneous decoherence is characterized by a transverse spin relaxation time T2. Additionally, in real systems, inhomogeneouseffects such as variations in g-factors or local magnetic field result in an extra dephasing of the spin polarization. Hence, it is customary to refer to the experimentally measured spin relaxation time as T;, indicatingthat the measured decay time only establishes a lower bound on the intrinsic decoherence time T2 unless inhomogeneouseffects can be eliminated. We will simply refer to T; as the “spin lifetime,” keeping in mind these considerations. How can we detect the time evolution of the coherently prepared spin population? There are a variety of time-resolved optical probes that can accomplish such a measurement. Measurements of time-resolved circular polarization of photoluminescence (PL) emitted during electron-hole recombination provides a direct measure of the electron and hole spin population; the electronic magnetization may then be inferred through photon spin c~nservation.’~~ These experiments detect the time-resolved intensity of the co- and counter-circularlypolarized components of the PL along the axis of optical excitation from a GaAs/AlGaAs quantum well after optical excitation with circularly polarized light. The experiment--carried out in the Voigt geometry-monitors the projection of the injected spins-parallel

38

NI" SAMARTH

or anti-parallel to their initial orientation, respectively. Periodic oscillations are seen in the luminescence intensity as a function of time, corresponding to the Larmor precession of excited spins and starting in phase for parallel (co-circularly polarized) detection and out of phase for antiparallel (counter-circularly polarized) detection. The oscillation periodicity determines the g-factor; in this case, the measured value of the g-factor identifies the spin precession with an electron rather than a bound electron-hole pair (an exciton). We note, however, that such a technique does not convey any information about processes that may be evolving after all the excited electrons and holes have recombined. This limitation of time-resolved PL has led the the developmentof an alternate, more powerful method that is based on the time-resolved Faraday and Kerr effects to reveal information about coherent spin evolution. Full details of this technique have been provided in an extensive review.6 Here, the optical probe measures the evolution of the electron spin population on time scales much longer than the radiative recombination time. Time-resolved FaradayKerr spectroscopy uses pump-probe techniques to preparehnterrogate a spin population using a stream of short optical pulses at a wavelength tuned just above the absorption edge of a semiconductor. These optical pulses are divided by a beam splitter into pump and probe pulses whose polarization can be appropriately set or modulated (Fig. 17). As described earlier, the pump pulse can be either circularly or linearly polarized, depending on the specific geometry of the experiment, while the probe pulse is always linearly polarized, and the angle of rotation of this probe polarization interrogates the spin polarization produced by the pump at some time delay. Pumpprobe time delays are produced either by a mechanical delay line ( r = 0 --+ 1 ns) or via a dual laser system (Ar > 1 ns). An extension of the dual laser technique allows the energies of the pump and probe pulses to be tuned independently,which-as we shall see later-is critical for experiments that probe coherent spin transfer across heterostructures. In a transmission geometry, we refer to this measurement as the time-resolved Faraday rotation (TRFR),while the equivalent measurement in a reflection geometry is called the time-resolved Kerr rotation (TRKR).The choice

(a)

(b)

FIG. 17. Schematic view of pump-probe Faraday rotation measurementcarried out in (a) the Faraday geometry and (b) the Voigt geometry.

39

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

-0.21 0

1

2

3

4

5

6

7

8

Time (ns)

RG.18. Example of TRFX measurement in a modulation-doped ZnSe/(Zn,Cd)Se quantum well. (Adapted from Ref. 36.)

of transmission versus reflection geometry is usually determined by experimental convenience, because both measurements yield identical information. TRFR measurements of coherent electron spin dynamics in semiconductors were first applied to probe quantum beating of excitons in diluted magnetic semiconductor quantum wells in the Faraday geometry.'54 Coherent exciton beats are observed at a frequency given by the exciton spin splitting (i.e., the g-factors of both the electrons and holes are involved) and the decoherence is very rapid (a few picoseconds) due to the rapid spin scattering of holes. Although these Faraday geometry measurements yield important insights into many body effects such as exciton-exciton correlation^,'^^ the rapid decoherence of the hole states obscures valuable information about the electron spin coherence. This can be revealed by carrying out the TRFR measurementin the Voigt geometry (see Fig. 17(b)) wherein circularly polarized pump pulses are employed to produce a coherent superposition of spin states in the basis defined by a magnetic field orthogonal to the propagation of the light. Whereas holes spin-relax rapidly due to valence band mixing or are pinned in certain quantum geometries, the electron spins continue to precess around the magnetic field at a characteristic Larmor frequency. The resulting time-dependent Faraday rotation has the form

The frequency of the oscillatory dependence of the Faraday rotation directly measures the electronic g-factor, and the time dependence of the decaying amplitude yields the transverse spin lifetime, T;. An example of such a measurement is shown in Fig. 18, wherein the TRFR is measured in a modulation-doped 156 T. Ostreich, K.

Schonhammer, and L. J. Sham, Phys. Rev. Len. 75,2554 (1995).

40

NITINSAMARTH

ZnSe/Znl -,Cd,Se single quantum well that contains a two-dimensional electron gas (2DEG). The frequency of the measured oscillations corresponds to an electronic g-factor for g = 1.1; the decay of these oscillations yields a spin dephasing time T; of several nanoseconds. When the spin dephasing time becomes longer than several nanoseconds, it becomes difficult to generate the correspondingly longer pump-probe time delays using mechanical delay lines. Instead, a complementarytechnique called resonant spin amplification (RSA) has been developed to extract spin lifetimes that exceed the pulse repetition interval (rrep 13 ns). Under such conditions, the spin signals from successive excitations can constructively or destructively interfere. The total spin polarization is then resonantly enhanced whenever g p B Br,,/h is a multipleof 2n,a condition that is met periodically in the applied field. Hence, by measuring the Faraday rotation at a fixed pump-probe delay and sweeping the magnetic field, one obtains a series of field-dependent resonances that are periodic in B at a frequency proportional to both Ar and g (Fig. 19). The RSA measurements are described by

-

, (b)

A t = 1 ns

I I I I I I I

-0.2

-0.1

0

0.1

0.2

Field (T)

FIG. 19. Examples of resonant spin amplification measured at T = 5 K in small magnetic fields for an n-doped GaAs epilayer (n = 10l6 for two different pump-probe time delays ((a)Ar = 10 ps and (b) Ar = 1 ns). The data show the central resonance and the gwBBtrep/R = f 2 n peaks. The spacing between the resonant peaks measures the electronic g-factor, while the width of the peaks can be related to T;. Fits are derived from Eq.3.2 and are offset for clarity. Zeroes are indicated by dotted lines. (Adapted from Ref. 37.)

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

41

modeling the observed spin polarization M, as the sum of exponentially decaying oscillations, @(At

M,(At, B) =

+ nt,p)Ae-(A'+"r"p)'T~

n

where n represents a sum over all laser pulses and the step function 0 ensures that only preceding pump pulses contribute to M, at any particular delay At. Data are fit by adjusting T;, A, and g separately for each resonance (except at zero field, where g is interpolated). This technique is capable of resolving spin lifetimes as long as 5 p s and 0.1.

9-

9. MEASUREMENT OF SPINCOHERENCE IN BULKSEMICONDUCTORS Measurementsof T; have been carried out in single crystals of ZnSe?6 G ~ A sand , ~ ~ GaN4I using both the TRFR/TRKR and RSA techniques. Despite large variations in defect densities (-10'' cmP2 in GaAs to -10" cm-2 in GaN), the general variation of the spin decoherence time with carrier density is quite similar in all these direct band gap semiconductors: T; is a nonmonotonic function of carrier density, abruptly increasing at very low doping levels, reaching a maximum in the vicinity of the metal-insulator transition, and steadily decreasing thereafter. This is exemplified in Fig. 20, which shows the oscillatory temporal evolution of the TRFR in n-GaAs for the different doping concentrations at T 5 K and B = 4 T. The figure shows that T; also depends on magnetic field (see inset). Note that for n = 1 x 10l6, the zero-field polarization exhibits virtually no decay in the 1 ns measurement interval, making it difficult to quantify the spin lifetime using time domain measurements;it is precisely in such circumstancethat RSA measurements can be used to accurately measure long spin lifetimes. In addition to the variation of T; with doping, the data also show a change in the spin precession frequency. This effect is related to an energy dispersion in the GaAs conduction band g factor that manifests itself because of shifts in the absorption edge with doping. As we remarked at the beginning of this section, a number of mechanisms can contribute to spin decoherence. Prominent among these are spin-orbit scattering during collisions with phonons or impurities (the Elliot-Yafet (EY) m e ~ h a n i s m ) ' ~ ~ or spin dephasing from precession about anisotropic internal magnetic fields (the In the former case, the spin decoherence rate Dyakanov-Perel (DP) rnechani~m.'~~ r o< r p . where r p is the momentum scatteringrate. In the latter case, the broadening accumulatesbetween collisions, so r (rP)-l. Electron-hole spin scattering

-

'51 R.J. Elliot, Phys. Rev. %, 266 (1954). lS8 M. I. D'yakonov and V.I. Perel, Sov. Phys. JETP 33,1053 (1971);[Sov.Phys. Solid Srare 13,3023

(1972)l.

42

NITIN SAMARTH

FIG. 20. TRFR for undoped and n-type GaAs at B = 4 T. Data are normalizedjust after zero pumpprobe delay. Plots are offset for clarity, with zeroes marked by dotted lines. The inset shows T* vs field. Data are taken at T 5 K with Nex 2 x lOI4 2 x loL4,1.4 x loi5,3 x IOI5 cm for 5 x 10l8 n = 0, 10l6, respectively. (Adapted from Ref. 37.)

-

-

5

(sometimesknown as the Bir-Aronov-Pikus m e c h a n i~ m ) ,while ' ~ ~ important in ptype GaAs, is not a major factor in n-type samples because the number of holes injected yields a spin relaxation that is too slow by several orders of magnitude and is independent of n. An additional contribution to the spin relaxation rate may arise from a spread in electronic g factors, Ag, which results in an inhomogeneous dephasing of 4 given by A 4 X AgpBBt/h.'60 However, such a process would lead to an inverse relationship between T; and B, which is not observed. The different spin scattering processes can be identified by measuring the temperature dependence of T..12p161 In n-GaAs, this identification is feasible ) a distinct temperafor very lightly doped samples (n = 1 x 10l6 ~ m - ~where ture dependence is observed. Figure 21 shows such experimental data, compared

159 G. Bir, A. Aronov, and G. Pikus, Zh. Ekrp. Teo,: Fir. 69, 1382 (1975) [Sov. Phys. JETP 42, 705 (1976)l. A. Abragam, The Principles of Nuclear Magnetism, Clarendon, Oxford (1961). 16' G. Fishman and G. Lampel, Phys. Rev. B 16,820 (1977); K. Zerrouati et al., Phys. Rev. B 37,1334 (1988).

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

lo2

43

u 10

100

Temperature (K) FIG.21. Temperature dependence of T* in n-GaAsfor n = 10l6 at E = 0 and B = 4T. The excitation density is Nex = 2 x 1014 cm- 3 . Dashed lines indicate DP and EY predictions. (Adapted from Ref. 37.)

to predictions for the EY and DP contributions for isotropic charged impurity scattering.16* In these calculations, r p is estimated from the measured mobility as e / m * p , and contributions to the electron kinetic energy from doping are included. The low-field behavior shows good agreement with DP for T 3 30 K, below which a weaker temperature dependence T; x (kT)-'/* is suggestive of electron-electron scattering. Estimates taking N = n N,,, however, show that this mechanism is actually too strong and may require a more explicit consideration of doping effects.163These data support a transition to the EY mechanism below 30 K, accompanied by a strong field dependence that suppresses the high-field spin lifetimes.

+

10. SPINCOHERENCE MEASUREMENTS IN SEMICONDUCTOR QUANTUM DOTS The current interest in employing semiconductor nanostructures for coherent spintronics-particularly quantum d ~ t s ~ ~ - h amotivated s systematic measurements of spin decoherence in OD systems.It is anticipatedthat spin scatteringmechanisms may be curtailed by the quantum confinement of carriers to nanometersized clusters where the density of states and the energy spectrum are strongly

"*

A. G . Aronov, G. E. Pikus, and A. N. Titkov, 2%. Eksp. Teor. Fiz.84,1170 (1983) [Sov.Phys. JETP 57,680 (1983)l. 163 P. Boguslawski, Solid State Commun. 33,389 (1980).

44

NITIN SAMARTH

Time (ps) FIG. 22. Spin precession in 4 nm QDs at T = 6 K and T = 280 K,showing only a modest decrease in spin lifetime with temperature. (From Ref. 168.)

modified compared to bulk systems, resulting in longer spin lifetimes. Although epitaxial growth techniques and post-growth patterning can yield OD confinement in “self-assembled” quantum dots,’@ time-resolved Faraday rotation is difficult to measure in such samples because the absorption coefficient is quite small.’65 In contrast, these optical experiments are easily performed on chemically synthesized nanocrystals that can be controllably fabricated with nearly spherical shapes in sizes ranging from 1.5 to 10 nm in diameter, with relatively narrow size distributions of -5-10%.166,167 CdSe quantum dots are of particular interest here because they are small enough to provide access to the “strong-confinement” regime where the size of the QD is smaller than the bulk exciton Bohr diameter (-12 nm). Time-resolved Faraday rotation measurements of the spin dynamics in these quantum dots have revealed nanosecond-scale long spin lifetimes that persist up to near room temperature,168as shown in Fig. 22. Because these quantum dots are not n-doped, the observation of extended spin precession lifetimes at room temperature is somewhat unusual in light of measurements in nominally undoped bulk semiconductors where spin lifetimes are very short. Although there is no definitive reason for this enhanced spin lifetime in OD systems, it might be E. D.Jones, A. Mascarenhas, andP. Petroff, Eds.,OproelecrronicMareriaIs:Ordering, Composition, Modulation, and Self-assembled Structures, Materials Research Society, Pittsburgh (196). We note that the Faraday (and Ken) effect rely on differences in dispersion for opposite circular polarizations of light; hence, the strength of these effects is also related through the Kramers-Kronig relations to the corresponding absorption. 166 A. P. Alivisatos. Science 271,933 (1996). C. B. Murray,D.J. Noms, and M. G. Bawendi, J. Am. Chem. Soc. 115,8706 (1993). J. A. Gupta, X.Peng, A. P. Alivisatos, and D.D.Awschalom, Phys. Rev. B 59, R10421 (1999).

’”

“’

INTRODUCTION TO SEMICONDUCTORSPINTRONICS

45

attributable to the increased stability of quantum confined energy levels. Interestingly, studies have found that coherent spin dynamics in semiconductor quantum dots can be generally observed even in samples with large distributions in size and c o m p ~ s i t i o n . This ’ ~ ~ has been elegantly demonstrated with measurements of time-resolved Faraday rotation in CdS,Sel-, quantum dots grown in glass matrices. Despite the larger size distribution and possible composition fluctuations, comparable nanosecond-scale spin lifetimes (attributed solely to electrons) were also measured in these OD systems. More recently, these optical studies of spin coherence in OD systems have been extended to artificial solids of semiconductor quantum dots wherein alternate layers of CdSe quantum dots of two different sizes are bridged by dithiol-conjugated molecule^.'^^ By using a two-color pump-probe TRFR measurement, spin coherence is optically excited resonantly at the absorption peak of the larger quantum dots, and then probed as a function of energy, spanning the range from the exciton ground state of the larger dots to that of the smaller dots. In the latter case, the experiments probe the transfer of spin coherence from the larger dots to the smaller dots via a molecular channel. This is supported by the observationof TRFR oscillations corresponding to two distinct g-factors as the probe energy is scanned from the exciton ground state of the large dots to that of the small dots. Quite surprisingly, the spin transfer efficiency increases with temperature, and is about 20% at room temperature. These experiments clearly demonstrate that the “molecular wiring” of nanostructures may provide a useful route for the coherent transfer of spin information. 11. ELECTRICAL MANIPULATION OF SPIN COHERENCE IN SEMICONDUCTORHETEROSTRUCTURES

We now discuss experiments wherein the optically pumped spin coherence is manipulated by an external electric field. In the first two examples, the electric field produces a drift of the optically excited spin coherence across macroscopic distance in both homogeneous and inhomogeneous semiconductors, while in the third example, the electric field displaces the wave function of spin-polarized carriers in a heterostructure. The first example of such an experiment focuses on n-doped bulk GaAs crystals wherein an external electric field provided the means to laterally drag optically excited spin polarization across macroscopic distances.38 By focusing the pump and probe beams onto spatiallydistinct regions of the sample, and then adjusting the lateral pumpprobe separation, Ax, along the direction of the in-plane electric field, measurementsof the Faraday rotation of the probe beam map both the time and spatial variation of the spin coherence. Figure 23(a) shows A. Gupta and D. D. Awschalom, Phys. Rev. B 63,085303(2001). M. Ouyang and D. D. Awschalom, Science 301,1074(2003).

169 J.

I7O

46

NITIN SAMARTH

0.0’ -0.05

I 0.05

0 B (T)

.s5 .Id

9

> m

U

s

(b)

A

f10 ps difference

resonant

.C-

Ax

(Pm)

FIG. 23. (a) RSA measurements for a 30-pm-thick, 1 x 10l6cmP3 Si-doped GaAs sample. Data are taken with a 100-pm pump diameter and a tightly focused probe. The inset shows TRFR versus pumpprobe delay taken at B = 0.0030 T. (b) Spatial scans of the pumpprobe interaction with 12-pm

FWHM beams, normalized and offset in amplitude for clarity. The dotted line is the resonant response at A t = 0 ps and B = 0. The thick solid line shows the unbroadened spin injection profile, obtained by subtracting the corresponding data at Ar = -10 ps. The lower data are taken at B = 0 with an applied electric field E = 16 V cm-I. (FromRef. 38.)

resonant spin amplification data near zero field in an n-doped sample, obtained by scanning the magnetic field at Af = 50 ps. The inset shows the associated temporal behavior of the spin polarization taken slightly off the B = 0 T resonance, with the offset at zero time delay arising from the injection of new spins on the arrival of the pump pulse. The data shows that pumpprobe pairs reappear at 76 MHz, and past spin injections leave an imprint of negative polarization at t -= 0. Spatial scans of the pumpprobe overlap are taken at a fixed delay with the pump maximally focused,as shown in Fig. 23(b) for a field of B = 0 T. Because there is no precession at this field, spin accumulates from consecutivepump pulses, and the spatial profile at any given delay is broadened because of spin diffusion. The spin injection profile is extracted by taking the difference signal between scans obtained immediately

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

47

before and after spin injection (At = f 1 0 ps). The upper part of Fig. 23(b) compares the spatial profile obtained by this method (solid line) to the wider profile taken at A f = - 10 ps (dotted line). The spatial resolution of these measurements is limited to 18 p m by the full width at half-maximum of the former. A macroscopic displacement of the electron spin polarization is obtained by the application of an in-plane electric field. The lower portion of Fig. 23(b) shows that an electric field of 16 V cm-’ produces a lateral displacement and an asymmetry in the spin distribution. This asymmetry stems from a separation of the zero-field spin resonance into constituent “spin packets” created by distinct pump events. Under the influence of an electric field, packets created at different times drift variable distances that are proportional to their ages. Hence, these spins no longer constructively reinforce each other at the resonance magnetic fields. The data in Fig. 23(b) show that the measured spin polarization is that of free electrons with a drift distance that is linear in the electric field, and that the spins are carried by negative charges. Because there are no indications of any spin polarization traveling opposite to the electron spins, the assumption that hole spins scatter rapidly in these systems appears to be valid. Finally, we note that control measurementson insulating (undoped) samples show no changes in spin profile at fields up to 400 V cm-*. A quantitative measure of spin drift and diffusion can be obtained by fitting the oscillatory data to obtain the amplitude of each spin packet at every position. By fitting the broadening of the spin packet as it evolves in both time and spatial position, a spin diffusion constant D,is obtained that exceeds the electron diffusion constant, D, = p d k T / e , by more than one order of magnitude. This surprising discrepancy has been explained by the observation that transport of spin packets is dominated by conduction electron properties, while the transport of charge packets is dominated by the valence band properties. The significant difference in the effective mass of electrons and holes hence results in a large effective spin diffusion constant. Because designs for coherent semiconductordevices will probably involve more complex, inhomogeneous semiconductor systems (such as p-n junctions and heterojunction bipolar transistors), it is important to examine the coherent transmission of spin information across interfaces in heterostructures. Such measurements have been carried out in ZnSe/GaAs h e t e r o ~ t r u c t u r e s .The ~ ~ *particular ~ choice of heterostructure is determined by several considerations: It is convenient to have a heterostructure with small lattice mismatch (in this case, % 0.25%), but with semiconductors that have very different bandgaps (E,) and electronic g-factors. In particular, at 4.2 K, E , = 1.5 eV, g = -0.44 for GaAs, and E , = 2.82 eV and g = 1.1 for ZnSe. Optical pulses resonantly tuned to the very different energy gaps of the two semiconductors then allow one to selectively excite spin coherence in the semiconductor with a lower energy gap, and unambiguously measure its time evolution in either of the two different heterojunction constituents. Furthermore, because the band offset is largely taken up by the valence band, the band alignment

-

5

48

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SAMARTH

between GaAs and ZnSe favors the propagation of a coherent spin polarization from the source layer. Consequently,electrons created in GaAs easily diffuse into ZnSe, while holes created in GaAs are blocked by the sizeable band offset. The use of doped GaAs substrates and transparent indium tin oxide contacts on the upper ZnSe surface allows the application of an electric field that enables the study of bias-driven spin transport across a heterojunction. The ZnSe/GaAs heterostructures used in these studies are grown by molecular beam epitaxy on a variety of GaAs substrates: semi-insulating, n-doped (n 3 x 10l6 ~ m - and ~ ) also p-doped (p ~ m - ~ZnSe ) . epilayers are n-doped with C1 and transparently contacted by 40 nm of indium-tin-oxide. The epilayers are 100, 150, 200, and 300 nm thick. We note that the critical thickness for misfit dislocation formation in ZnSe on GaAs is -150 nm; hence, the structures include heterostructuresthat have a wide range of defect density. The n-doping in the ZnSe epilayers is in the range 5-15 x 10” cmP3 and is provided by C1 donors. As with the study of coherent spin transfer in molecular-bridgedquantum dots, the two-color pumpprobe optical experiments use pump and probe pulses tuned to different energies. Electron spins oriented along the sample normal are created using a circularly polarized pump pulse tuned to either the ZnSe or GaAs absorption threshold (1.52 and 2.80 eV, respectively, at 5 K); we will denote these pump pulses as Pznseor P G ~accordingly. ~ , We note that while the primary absorption of Pzns occurs in ZnSe, P G passes ~ ~through the ZnSe epilayer unimpeded because it is far below the bandgap, exciting spins in the GaAs substrate. The Kerr rotation of a linearly polarized probe pulse at the ZnSe absorption threshold then records the dynamics of the normal component of the total ZnSe spin, S,(At), where At is the pumpprobe temporal interval. When spin polarization is both pumped and probed in the ZnSe layer, the data are identical to the measurementsdiscussed earlier for bulk crystals, with the TRFR a signal appearing suddenly at At = 0 and then decaying exponentially at a rate characteristic of T.znse.In contrast, pumping in the GaAs layer and probing in the ZnSe layer produces a ZnSe magnetization that grows over the first few hundred picoseconds and then decays with a spin lifetime appropriate to the ZnSe host. Figure 24(a) shows a low-temperature TRKR measurement of spin accumulation in ZnSe originating from spin polarization that was excited in the GaAs layer. Data are shown at B = 0 T as the pump energy E P is tuned through the GaAs absorption threshold. This suggests spin migration from GaAs to ZnSe, a suspicion confirmed by the sudden growth of S, as E p is tuned through the GaAs absorption threshold (Fig. 24(b)). Conversely, the selectivity of the probe in measuring only ZnSe spins is seen in Fig. 24(c), where t!?K (and hence S,) is maximized at the ZnSe absorption threshold. An estimate of the fraction of GaAs spins entering ZnSe is 2.5-10% at B = 0 T, based on the Kerr response obtained by introducing a known number of spins directly into the ZnSe. This means that most of the optically excited electron spins remain trapped in the GaAs.

-

-

49

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

?

?

m

Y

Y

J

.............................................................................................

a

............................................................................................. 1.503 eV

0

At I

fn

6000

4000

2000

1

1

8000

(PS)

I

80 -

fn (I

0

8

% a 1.50

1.52

1.54

2.78

2.80

FIG. 24. (a) TRKR data for spin excitation in the SI GaAs substrate at T = 5 K and B = OT. The dashed lines mark OK = 0.(b) Pump energy dependence of OK at a fixed delay Ar = 800 ps using a probe energy of 2.80 eV. (c) Probe energy dependence of O K at Ar = 800 ps and E p = 1.52 ev. (From Ref. 39.)

The spin arrival distribution can be estimated by extrapolating the exponential decay of the transferred spins back into the time of their arrival. This is shown in Fig. 25.(a), where the dashed line fits the exponential decay from 2 ns to 9 ns and the shaded region marks the discrepancy between this fit and the measured spin profile. Figure 25(b) shows this difference on a log scale for both substrates, where the data are normalized by the extrapolated spin amplitude and equal the fraction of spins yet to cross the interface. The data are well described by e-A'/r, where t is the accumulation time of spin transfer and equals 210 and 440 ps for SI and n-GaAs substrates, respectively. Generally, t-' = to-' T;CGaAS-l, where to is the spin accumulation time for non-decaying GaAs spins and T;CGaAs is the substrate transverse spin lifetime. The application of a transverse magnetic field induces coherent spin precession at the Larmor frequency. Figure 24(c) shows the evolution of S, at 50 mT, where

+

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NITIN SAMARTH

FIG. 25. (a) TRKR at B = 0 T (solid line) for spins excited in an n-GaAs substrate.The dashed line shows the fit described within the text. The decay of the shaded amplitude (fit - data) gives a measure of the spin accumulation time in ZnSe. (b) A plot of the fraction of spins yet to cross into ZnSe. Both SI (solid) and n-doped (open) substrates are shown, with r = 210 ps and 440 ps, respectively.(c) Data are similar to (a) but with B = 50 mT. The temporal phase shift, 8 , is indicated. (From Ref. 39.)

spin precession results in an oscillatory profile, S, (At) = Ae-At/T;cos(oAt +#). For At >> t, the data due to pumping in GaAs resembles that obtained from pumping in ZnSe. We note that the values of o and T; determinedby fitting this data yield the g-factor and spin lifetime of the ZnSe layer, providing a clear indication of spin transfer. In addition to the coherent spin precession, the time-resolvedKen rotation measurementsat B > 0 also provide insights into the dephasing of the spin system during interlayer spin transport. This dephasing arises from conventional inhomogeneous effects as well as from a field-dependent suppression of T;C.Spin evolution is particularly inhomogeneous during spin transfer because GaAs and ZnSe have different g-factors (gG& and gznse,respectively). Hence, we expect significant spin dephasing phenomena to occur only during spin accumulation. This is confirmed by a phenomenological model of the coherent spin transfer that considersinhomogeneouseffects arising from differencesbetween and gase. The temperature dependence of spin accumulation provides further insight into the incident carrier kinetics. Figure 26(a) shows that at B = OT, t is roughly constant up to 100 K and decreases sharply thereafter. These changes may reflect

51

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

v)

Q

v

t-

100

0

10

- ‘ O O : I : eT(K)

100

0

200

400

600

800

At (PS)

FIG.26. (a) Temperature dependence of T at B = 0 T. (b) TRKR measured (crosses)and fit (dashed line) at T = 300 K and B = 0.5 T. All data taken on a semi-insulating substrate. (From Ref. 39.)

a temperature dependence of the interface potential, the substrate spin lifetime, or perhaps an increase in the mean carrier velocity, but more extensive study is necessary to confirm these trends as T appears to vary with probe energy. Of significant importance to technology, Fig. 26(b) shows that the coherent transfer of spins from GaAs to ZnSe occurs even at 300 K, where no significant decrease is observed in the measured signal. The measurements described thus far are carried out with a zero bias. We now examine the effect of an external electric field on the coherent spin transfer, beginning with measurements at B = OT, where bias-dependent changes in net spin transfer are more evident in the absence of spin precession. In Fig. 27, time scans show an increase in the amplitude of the spin signal in ZnSe as the applied bias is increased, while a reverse bias reduces the number of spins that cross the interface.The data also show a striking offset to the spin polarization:The long spin lifetimes in GaAs result in the layer acting as a “reservoir” whose spin population decays only slightly before receiving a boost from the next pump pulse. Hence, the GaAs layer sources a spin current that never reaches zero for E > 0. The inset in Fig. 27 shows the change in the total spin polarizationtransferred from the reservoir to the epilayer, where modest electric fields increase spin transfer nearly 500%. It can be shown that the total spin contribution in the ZnSe layer is given by

, and T&, are the GaAs and ZnSe where 4 = tan-’(te&o~aAs- ( o z o ~ ~ ) )TGaAs spin lifetimes, wa~,and wnsetheir Larmor frequencies (which contain the g’ the effecfactor explicitly), trepis the laser repetition time, reil = t - l + T G ~ , - is tive spin accumulation time, and t is the spin accumulation time. This equation is

52

NITIN SAMART€I

FIG.27. Changes in the efficiency of the spin transfer with bias. Time evolution of the spin transfer at different biases with B =OT. The time interval is long enough to include two successive pump pulses. The 300-nm-thick n = 1.5 x 10l8 cm-3 ZnSe epilayer was used with E = -16.5.0, 4.5,7.5,9.8, 12.22 V/cm (at higher voltages heating occurred). No offset added. (Inset) Percentile change in the spin transfer from the reservoir to the epilayer, calculated using the sum of the three mechanisms amplitudes (A +B +C) as a measure of the spin transfer at different biases, and comparhg them to the unbiased case. (From Ref. 40.)

obtained by summing over spins crossing the interface at different times ti with an exponentially decreasing probability o<e-ri/5.During spin accumulation, the dephasing process phase shifts the spin precession and suppresses the spin amplitude, A = S (1 (tanc$)*)-'/', as the applied magnetic field increases. In contrast to spin dephasing due to local field fluctuations or g-factor dispersion, this inhomogeneousprocess ceases once spin accumulation in the ZnSe is complete (At >> r). The experimental spin precession data is modeled by using the functional form for S, (At) to extrapolate spin behavior from B = 0 T to non-zero fields. A fit to the B = OT data determines the zero-field values of S, t,and T;C.Using nominal values of gGA = -.44and g B = 1.1, determined from spin precession measurements on these systems, Eq. (3.3) then predicts how & ( A t ) might evolve as the magnetic field increases (assuming that A, 5 , and T;C are field independent). Theoretical plots based on this model exhibit many qualitative similarities with the actual data, including a decrease in the spin amplitude and a negative phase shift of spin precession. More detailed fits to such data actually reveal the presence of parallel fast and slow transfer mechanisms, A and B, each described by functions S, and SB of the form of Eq. (3.3) but with amplitudes A and B, respectively, and corresponding accumulation times teff~% t~ % 20 ps and t e f f%~t g 500 ps (where we used T;CGaAs>> T A ,B ) . Long spin lifetimes require summation over prior pump pulses as well.

+

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

53

FIG.28. Time progression of the spin transfer from the reservoir to the epilayer with and without a bias. An offset is added for clarity; the dotted lines mark the zeros. (a) 300-nm-thick n = 1.5 x 10" cm-3 ZnSe epilayer. (b) 100-nm-thick n = 5 x loi7cm-3 ZnSe epilayer. (c) Decomposition of the fit to the biased scan in (b) into its three different conduction channels. (From Ref. 40.)

In the presence of an electric field, modeling of the TRKR measurementsreveals the onset of a fundamentally new mode of spin transfer. Without an external bias (Figs. 28(a) and (b) for E = 0), spin accumulation in the epilayer lasts only a few hundred ps and can leave a large spin polarization trapped in the reservoir. Once transferred, spins precess about B at a rate determined by the ZnSe g-factor, and decay with a time reflecting the ZnSe spin lifetime. In epilayers with lifetimes longer than 4 ns, the appearance of a second g-factor is revealed by the presence of beats (Fig. 28(a)); in contrast, in epilayers with shorter spin lifetimes (Fig. 28(b)), after a few ns, the spin signal is present only under a biased condition and exhibits a precession rate different from that of ZnSe. A Fourier transform of the biased time scans shows that the second g-factor is within 2% of the nominal gG& FZ 0.44 value for GaAs. The biased spin transfer data may be understood by adding a persistent spin flow (mechanism C), described by Fq.(3.3) with an amplitude C and tc = 00. In this case, spin transfer never turns off, and tcefi = tc x T2;jaAsrepresents not the spin accumulation time but a decay in the persistent spin current that mirrors spin polarization decay in the reservoir. The total spin signal in the epilayer is therefore described by SZnse(Ar)= SA S E Sc, shown in Fig. 28. Figure 28(c) shows the dynamics of the three parallel channels contributing to spin injection: the spins that

+ +

54

NITIN SAMARTH

cross over tens of ps (A), over a few hundred ps (B), and the persistent spin current that appears with a bias (C). Fitting variables are the amplitudes A, B, and C (the latter is taken as non-zero only for positive biases), and the risetimes T A and tg. The other values are fixed and obtained from complementary measurements, such as resonance spectra or time scans from degenerate pumpprobe arrangements. Note that the persistent spin current contribution is only sensitive to spin that has arrived within a time T;znse. Hence, fort > T;, the epilayer polarization reflects the precession and the lifetime of the spin current itself, which follows the reservoir spin dynamics. This explains why the average spin polarization in the epilayer is characterized by wasand T;"Cas although spins reside in ZnSe (and each spin has Wnse and T.&,se). Similar physics also arises from the built-in interfacial electric fields in a p-n junction, and is demonstrated by coherent spin transfer across a p-GaAsl n-ZnSe heterojunction. As a control, an undoped GaAs substrate is placed together with the p-type substrate in the growth chamber, and 300 nm of n-ZnSe (n = 1.5 x 10" ~ m - is~ deposited ) simultaneously on both. An -4000% increase in spin transfer due to the heterojunction voltage is observed, compared to the control (degenerate pumpprobe measurements of the ZnSe epilayers are nearly identical, with a slight change in spin lifetime). We note that unlike the n-doped GaAs "spin reservoirs," the spin lifetimes in the p-doped GaAs substrate are extremely short (a few ps) so that persistent spin currents do not explain the observed increase. Instead, the data suggest an enhancement of spontaneous transfer mechanisms similar to A and B, but with a non-exponential spin accumulation profile. The electrical control of coherent electron spin precession has also been shown possible in single parabolic Al,Gal-,As quantum well ~ a m p 1 e s . Here, l ~ ~ the electric field serves to displace the wave function of the coherent spin population within a parabolic confinement potential, hence exploiting changes in the sign and magnitude of the electron g-factor with alloy composition (in bulk A1,Gal-,As, g = -0.44 for x = 0 and g = 0.40 for x = 0.3).17*Because an applied bias does not change the shape of the parabolic potential, but is able to displace the minimum of the effectivepotential to a region with higher aluminum content, this allows for a continuous displacement of the complete electron wave function from one material to another without distortion. (This is in contrast to a square quantum well where only the tail of the wave function would be pushed into the barrier.) TRKR measurementsin a series of spin-engineeredstructures show that gate-voltagemediated quasi-staticcontrol of coherent spin precession is obtained over a 13GHz frequency range in a fixed magnetic field of 6 T,including complete suppressionof precession and a reversal of the sign of g (see Fig. 29). Such manipulations are possible up to room temperature, suggesting potential technologically relevant functionality. G.Salk et al., Naiure (London)414,619 (2001). C. Weisbuch and C. Hermann, fhys. Rev. B 15,816 (1977).

INTRODUCTIONTO SEMICONDUCTORSPINTRONICS

55

FIG.29. Voltage-controlledspin coherence. a TRKR measurements of the electron spin precession in the QW with minimum A1 concentration of 7% at 5 K and B = 6T. As a positive voltage U8 is applied between back and front gate, the electron wave function is displaced toward the back gate into regions with more A1 concentration (schematic in b), leading to an increase of g. At U, = 2V, no precession is observed, corresponding to g = 0. Circles are data points, and the solid lines are fits to the data. (From Ref. 171).

A remarkable extension of these quasi-static electrical control experiments has recently used gighertz frequency voltages to coherently control the electron spin precession in parabolic quantum wells by resonantly modulating the anisotropic g-tensor.'46 This result has direct implications for the development of quantum computation schemes in semiconductors because it provides a route toward highspeed, all-electrical local manipulation of spin-based qubits.

12. ALL-OPTICAL COHERENT MANIPULATION OF SPIN IN SEMICONDUCTORS

In addition to preparing, probing, and transporting coherent electron spin populations in semiconductors, it is also important to construct schemes for the coherent manipulation of spins for operations in quantum information processing schemes. The general method for achieving this can be sought in analogy with spin echo techniques employed in pulsed NMR and ESR.lm A recently developed method achieves similar results using optical methods for producing coherent spin rotations on lOOfs time ~ c a 1 e s .The l ~ ~mechanism for this process relies on the generation of an effective magnetic field by a below-bandgap laser pulse through the optical '73 J. Preskill, quant-pW9712048at http://xxx.lanl.gov (1997).

56

NITIN SAh4ARTH

RG.30. (a) Schematic depiction of coherent spin control experiment. (b)The Stark field (Hs& rotates away from the x - y plane when t& # 0. (c) Data from an undoped quantum well sample showing the reduction in amplitude of the TRKR caused by the tipping pump. The tipping pulse in these data is tuned 52 meV below the resonance, and has an intensity of 0.5 GW cm-2. (d) Comparison of the difference signal with and without the tipping pulse for different relative orientations of the Stark field and the total electron spin, showing that the tipping effect is a maximum when these are orthogonal. (e) Similar data to (d), except that the energy of the tipping pulse is now slightly above the absorption threshold, so as to excite an additional spin population. The magnitude of the tipping effect is then a maximum when this additional spin population has a magnetization parallel to that of the original one. This test allows the separation of state filling effects from those due to the optical Stark effect. (From Ref. 147.)

s

Stark e f f e ~ t . ’ ~When ~ ’ ’ ~the ~ pulse is circularly polarized, initially degenerate states in the conduction band experience different Stark shifts due to optical selection rules, resulting in meV-scale spin splittings that correspond to effective field strengths of up to 20 T. Any net torque between an existing electron spin population and the effective field then leads to an impulsive “tip of the electron” spin by angles up to --n/2. Because the Stark shift only lasts for the duration of the laser pulse, the effective magnetic field is turned “on” for only 150fs, thus enabling ultrafast coherent control over electron spins in semiconductors. The experiment is carried out by extending the time-resolved Faraday rotation setup described earlier to include a “tipping pulse” (TP),which creates an effective magnetic field (HStark) that coherently rotates the electron spins as they precess about the static field. This effect arises because of a nonzero torque given by

-

J. Dupont-Roc, Phys. Rev. A 5,968 (1972). M. Rosatzin, D. Suter, and J. Mlynek, Phys. Rev. A 42, 1839 (1990).

174 C. Cohen-Tannoudji and

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INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

57

? cx x &stark. The torque occurs only yhen the tipping pulse is incident on the sample at a time delay AT^ such that S ( A t p ) 11 & j , a situation that occurs at zero crossings in the detected Faraday rotation signal. Figure 30(a) shows a scan comparing the spin precession taken with and without the TF’in a magnetic QW ~ a m p 1 e . The l ~ ~ tipping pulse rotates the spin magnetization out of the x - y plane; precession following the tipping event traces a cone about HO(Fig. 30(b)), thus giving a reduced value of the measured component S,. A priori, the angle of rotation can be calculated from the relation: @tip = arccos(A/Ao), where A and A0 are the amplitudes of precession with and without the TF’,re~pective1y.l~~ Such a calculation for the data in Fig. 30 yields a value for @tip that exceeds n/2 radians, evidenced by the reversal in sign of the oscillations. IN HY~RID FERROMAGNET/ 13. SPINCOHERENCE SEMICONDUCTOR HETEROSTRUCTURE~

Measurements of time-resolved FaradayKerr rotation in hybrid heterostructures that integrate ferromagnetic materials such as M ~ A s into ’ ~ semiconductor ~ heterostructures have yielded some remarkable new discoveries. We briefly mention a few of the essential results here. Ultrafast optical pump-probe measurements on ferromagnet/GaAs heterostructures reveal that the dynamics of coherent electron spins in the GaAs layer are strongly affected by the ferromagnet. Naively, one might expect these modifications to result from fringe fields or from direct exchange interactions. Contrary to these expectations, the principal effect is the hyperpolarization of nuclear spins in the GaAs layer through an “imprinting” effect of the ferr~magnet.’~ These polarized nuclei, in turn, generate large effective magnetic fields on the coherent electron spins through the hyperfine interaction, leading to electron spin precession in the GaAs layer that is dominated by interactions with nuclei. This surprising effect has been observed in a variety of ferromagnedn-GaAs(100 nm) samples, with the ferromagnetic layer ranging from a single submonolayer of MnAs, to digital and random (Ga,Mn)As alloys, as well as MnAs epilayers. Figure 3 l(a) shows the time-resolved Kerr effect for a sample wherein a single fractional layer of MnAs is placed at the surface of an n-doped GaAs epilayer. Happ is swept from 10oOOe to lo00 Oe in steps of 10Oe. The time interval between field steps is -40 s, the time needed to obtain each time-resolved scan. Near 250 G, a sharp change in the Larmor frequency is observed, corresponding to a magnetization reversal as revealed in the magnetic hysteresis loop obtained by magneto-optic Kerr effect (MOKE) measurements (Fig. 31(c) upper panel). Figure 31(b) shows that the sharp change appears near 250 G as the field sweep is reversed. M. Tanaka et al., Appl. Phys. Lett. 65, 1964 (1994).

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NITIN SAMARTH

FIG. 31. This set of figures shows how different types of proximal ferromagnetic layers affect the spin precession of electrons in an n-doped semiconductor. Panels (a) and (b) show sequential timeresolved Faraday rotation scans for a sample where the only magnetic layer is a single fractional monolayer of MnAs. The scans are taken at 5 K as the applied magnetic field is (a) ramped up from -lo00 G to lo00 G in 10 G steps, and (b) ramped down from lo00 G to -lo00 G. The amplitude of rotation .9F is represented in grayscale. (c) Top: Magnetic hysteresis loop of a single-layer sample at 5 K, measured by MOKE. Bottom: Larmor frequency as a function of field, obtained by fitting the time-resolved Faraday rotation scans in (a) and (b) at each applied field. The arrows indicate whether the field is increasing or decreasing. (d) through (f) MOKE hysteresis loop and Larmor frequency as a function of field at 5 K for the (d) digital (Ga,Mn)As sample, (e) (Ga,Mn)As sample, and (f) MnAs sample. The field step in ( d H f ) is 20 G. The total field Btot is defined by the relation V L = (gpLg Btm/ h). (From Ref. 14.)

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

59

This measurement procedure is repeated for each of the samples, and the results are summarized in Figures 31(c)-(f). The Larmor frequency U L = gpLg B / h is determined by fitting the data at each applied field with Eq.(3.1). For all samples, the sharp change in U L occurs when B,, matches the coercive field observed in the magnetic hysteresis loop. This is a clear indication that the ferromagnetic layer strongly influences the electron spin dynamics. However, the observations cannot be explained by either fringe fields or direct exchange interactions with the magnetic moments, because they would produce local fields that are proportional to the magnetization and therefore do not account for the disappearance of U L at BqP = 0 G, where there is significant remanent magnetization. An alternative explanation is that the nuclear spins in the n-GaAs layer become polarized along the magnetization of the ferromagnetic layer. These polarized nuclei, in turn, generate effective magnetic fields Bn (I)(via hyperfine interaction), where (I)is the average nuclear spin. Because the precession frequency is given by

-

V L = (gpB/h)(Bapp

+

(3.4)

the disappearance of UL at Bapp= 0 G can be attributed to the depolarization of nuclear spins at zero field due to dipole-dipole interactions.177 An intuitive understanding of the behavior of the nuclear spins is obtained by measuring their response to a magnetization reversal. It is found that the spin precession frequency of the electrons continuously evolves over several minutesfollowing a sudden magnetization reversal. This time dependence is characteristic of nuclear polarization and-again4annot be explained by fringe fields or direct exchange interactions. The precession frequency may be used as a “Larmor magnetometer” of the nuclear polarization; Its time dependence shows that the nuclear polarization tracks the direction of the magnetizationand requires -20 min to reach steady-state. The fact that (I)tracks the magnetization explains why U L changes sign when Bappcrosses the coercive field of the ferromagnet (Fig. 31(c)-(f)). Direct evidence for the nuclear polarization is provided by all-optical NMR,17* wherein the helicity of the pump beam is modulated at a given fundamental frequency (f = 40,50, or 55 M z) , resulting in an effective ac-magnetic field through the hyperfine interaction. When the resonance condition y Bapp= nf is satisfied, the nuclei depolarize; here y is the isotope-specific gyromagnetic ratio, and n is an integer labeling the harmonics of the frequency modulation. On the singlelayer sample, the Faraday rotation is measured as a function of applied field for each modulating frequency at a fixed pumpprobe time delay Af = 1500 ps. The resulting data show resonances associated with the isotopes 69Ga,71Ga,and 75As, confirming that the nuclei are indeed polarized. 177 D. Paget, G . Lampel, B.

17*

Sapoval, and V. I. Safarov, Phys. Rev. B 15.5780 (1977). J. M. Kikkawa and D. D. Awschalom, Science 287,473 (2000).

60

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SAMARTH

A surprising offshoot of these nuclear “imprinting” experiments is the observation of electron spin precession initiated in ferromagnet/semiconductor heterostructures by a linearly polarized optical excitation in a semiconductor.l5 Because linearly polarized light has zero angular momentum, the only reasonable explanation for these observations is that the ferromagnetic layer imparts angular momentum to the electron spin system. This effect is observed with timeresolved Faraday rotation, using linearly polarized pump and probe beams in a Voigt geometry with the sample magnetizationroughly perpendicularto the applied field. Detailed measurements indicate that the spin coherence is established very rapidly (within 50 ps) and possibly even during the optical excitation. This “ferromagnetic proximity effect” has been observed in different combinations of ferromagnets and semiconductors and is attributed to the process of dynamic nuclear polarization. While the physical origin of the spontaneous electron spin coherence is currently unclear, the opposite sign of polarization generated by these two materials provides flexibility in orienting regions of electron and nuclear spin in a semiconductor.

IV. Semiconductor Spintronic Devices Because much of the motivation for semiconductor spintronics originates in the desire to explore new technological opportunities, it is worthwhile to have a look at the different device schemes that have been theoretically proposed and examined in experiments. Some examples of such devices are shown in Fig. 32: three of the examples shown in this figure (the spin FET, the spin LED, and the ferromagnetic semiconductortunnel junction) have already been explored in experiments that we will describe shortly. The fourth example-the ferromagnetic heterojunctionbipolar transistor-is still only a theoretical proposal?0 although preliminary attempts to fabricate such devices have already begun.” We note that all these examples are extensions of existing device concepts, but incorporate some spin-dependent functionality. The discussion here is not meant to be comprehensive, but rather reflects the principal focus of activity in recent years.

14. THEDATTA-DAS SPIN-FIELD EFFECTTRANSISTOR

In 1989,Datta and Das introduced a concept for a semiconductor spintronic device that has since become the motivating factor for many experiment^.'^ The device is now commonly referred to as a spin field effect transistor (“spinFET’) and is often used as an important motivation for semiconductor spintronics research. We note that the authors themselves modestly characterized the original idea as

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

61

FIG. 32. Schematic examples of several possible semiconductor spintronic devices.

an “electronic analog of the electro-optic modulator,” without making claims to any technological applications. While one might imagine that the reliance on spin properties to switch such a device between the “on” and “off” states may possibly require less power and perhaps lead to faster device operation compared to standard MOSFET technology, there is no rigorous argument-as far this author knowsto expect such enhanced functionality. Nonetheless, experimental and theoretical research toward the spinFET has raised some important fundamental questions about spin injection at metalhemiconductor interfaces. A very detailed account of the theoretical and experimental efforts carried out in this context has been provided by Tang ef al. in Ref. [6]. Here, we addressjust some of the basic issues. The spinFET essentially has the generic design shown in Fig. 32(a): a spinpolarized current is injected from a ferromagnetic contact into an n-doped semiconductor channel where it is subject to an externally applied electrical field (via a Schottky gate) and then finally the spin-resolved current is detected by another ferromagnetic contact. In narrow gap semiconductors such as InAs, the electrons

62

NITIN SAMARTH

experience a strong spin-orbit coupling that-in the presence of an asymmetric confining potential-results in a Rashba interaction of the form'79

where the carriers are confined to the x - y plane by an asymmetric confinement potential along the z-axis. The parameter ct is proportional to the ;-component of the interfacial electric field, ~3represents the Pauli matrices, and k is the crystal momentum of the carriers. As a direct consequence of this Hamiltonim, the eigenstates for the electron spin along the z-axis undergo a momentum-dependent spin splitting even in the absence of an external magnetic field. (This spin splitting may be viewed in terms of the relativistic transformation of the electric field into a magnetic field in the rest frame of the electron.) The application of an external electric field via the Schottky gate modulates the interfacial field (and confinement asymmetry), hence resulting in a precession of the electron spin as the current propagates under the gate. If a modulation of the gate voltage results in a complete reversal of the spin polarization of the current, the device can be effectively switched off. A number of experimental efforts have been undertaken over the years to measure the characteristic spin-dependent switching behavior expected in a spinFET. 180-185 However, careful measurements show that the observed signatures of spin-dependent switching in many of the studied devices originate in a spurious effect: The fringe fields from the patterned ferromagnetic contacts create a local Hall effect that easily leads to misleading results.Ig0 Measurements using a nonlocal geometry to eliminate contributions from the local Hall effect demonstrate quite conclusively that-in carefully designed structures-there is no evidence for an observable spin switching effect, leading inevitably to the conclusion that spin injection from metallic ferromagnetic contacts into a semiconductor must be very inefficient.lu This has lead to a picture wherein the spin injection efficiency is found to be limited, at least in the diffusive regime, by the large mismatch in conductivity between a semiconductor and a metal.149 Before we move on to discuss other devices, it is worth mentioning a variant of the spinFET that does not depend on ferromagnetic source and drain contacts.lg6 In this device, the gate in a standard MOSFET is replaced by two ferromagnetic

'I

''*

E. I Rashba, Fit. Tverd. Tela (Leningrad)2, 1224 (1960). [Sov. Phys. Solid Stare 2, 1109 (1960).] F. G. Monzon and M. L. Roukes, J. Magn. Magn. Muter. 195, 19 (1999). S. Gardelis et aL, Phys. Rev. B 60,7764 (1999). W.Y.Lee et al., J. Appl. Phys. 85,6682 (1999). P. R. Hammar, B. R. Bennett, M. J. Yang,and M. Johnson, Phys. Rev. Lett. 83,203 (1999). A. T. Filip et al., Phys. Rev. B 62,9996 (2000). C. M. Hu et al., Phys. Rev. B 63 125333 (2001). C. Ciuti, J. P. McGuire, and L. J. Sham, Appl. Phys. Lett. 81,4781 (2002).

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Schottky mesas. As a current flows under each mesa, it becomes spin polarized by the proximity e f f e ~ t , ’ ~ , ’hence ~ . ’ ~resulting ~ in a magnetoresistancethat can be modulated by the relative orientations of the magnetization of the two ferromagnetic “imprinting” mesas. Because the electrical contacts here are all conventional and the spin polarization occurs only within the device, such a device is not limited by spin injection efficiency.

15. SPIN-DEPENDENT LIGHT-EMITTING DIODES The difficultiesin the electrical detection of spin injection from metallic ferromagnets into semiconductors led to the development of experiments that use optical methods to detect spin injection. The generic scheme for such experiments is shown in Fig. 32(c): the spin injector can be a metallic or semiconducting ferromagnet to allow for spin-polarized injection even in the absence of a magnetic field, or one may use a paramagnetic semiconductor in a non-zero magnetic field. Spin-polarized electrons (holes) injected by the magnetic contact then recombine with unpolarized holes (electrons) in the active quantum well region of the device, resulting in light emission whose degree of circular polarization is determined by the spin polarization of the spin-injected carrier species. The initial demonstrations of such “spinLED’ devices employed paramagnetic n-(Zn,Mn)Se and n-(Zn,Be,Mn)Se27.29as well as Gal -,Mn,As for spin injection.’* In both cases, the conductivity mismatch is not an issue because the spin injection takes place from a semiconductor. In the (Zn,Mn)Se-based devices, the experiment involves the injection of spin-polarized electrons and is carried out in a vertical Faraday geometry, wherein the external magnetic field, injected spins, and emitted light all lie normal to the plane of the quantum well emitter. Under such conditions, the selection rules (see Fig. 16) allow a straightforward correspondence between the polarization of the emitted light and the spin polarization of the recombining carriers (in this case, electrons). By contrast, the case of Gal-,Mn,As-based spinLEDs presents a somewhat more complicated situation. Because the easy axis of the magnetization in Gal-,Mn,As lies in plane for growth on standard GaAs-based heterostructures, the emitted polarization is detected parallel to the plane of the quantum well. A naive approach to the selection rules would lead us to believe that there should be no circular polarization detected in such a geometry. Furthermore, because such a device involves the spin injection of holes with shorter spin diffusion lengths than electrons, the observed effect is expected to be small. Indeed, as one might have anticipated, the degree of circular polarization observed in the vertical geometry (Zn,Mn)Se electron injection spin LEDs (as high as -45%) is substantially larger than that measured 18’

C. Ciuti, J. P. McGuire, and L. J. Sham, Phys. Rev. Len. 89, 156601 (2002).

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in the side geometry Gal-,Mn,As-based hole spin injection devices (less than a percent). However, a series of very careful control tests unambiguously prove that the observed circular polarization signal in the latter is not spurious. Furthermore, another vertical geometry version of the Gai-,Mn,As-based spinLED fabricated using a tensile strained layer shows quite conclusively that hole spin injection and transport does occur in the Gai-,Mn,As-based devices, and that-as expected-the emitted light has a stronger circular polarization (up to -7% in the standard vertical geometry)."' Finally, a recent calculation shows that accounting for the orbital coherence of injected holes leads to a much higher degree of circular polarization for side geometry spinLEDs that involve spin-polarized h01es.I~~ The success of the initial op!ical spin detection experiments has sparked significant interest in spinLEDs, resulting in a large number of experiments probing the fundamental aspects of these devices. The suggestion that the conductivity mismatch problem can be overcome using a tunnel or Schottky contact between a ferromagnetic metal and a s e m i c o n d u ~ t o rhas ' ~ ~also ~ ~motivated ~~ several studies that involve spinLEDs with high-resistance contacts between a ferromagnetic metal and the semiconductor One of these experiments reports a very high spin injection efficiency (-30%) that persists up to room t e m p e r a t ~ r e . ' ~ ~ Measurementshave also probed the role of defects and interfaces in determining the spin injection effi~iency.'~~ Systematic spin-polarized photocurrent spectroscopy (essentially, the reverse of a spinLED) has pointed out that careful studies of optical pumping in spinLEDs provide important complementary insights into the injection and transport of spin-polarized e1 e ~ t r o ns. l~ ~ While much of the attention on spinLEDshas been directed toward the measurement and demonstration of efficient spin injection into semiconductors, a recent set of studies has revealed that these devices also provide a vehicle for fundamental measurements of electrically detected nuclear polarization signal^.'^^.'^' In these experiments, electrical spin injection is measured at low fields from an Fe film into a GaAs-based quantum well. By measuring only the component of the spin that precesses after injection into the semiconductor,the effective magnetic field inducing the precession is probed as a function of electrical bias conditions and shows a striking increase at very high injection current densities. The magnetic field D. K. Young et al., Appl. Phys. Left. 80, 1598 (2002). 2.G. Yu, W. H. Lau, and M. E. Flatte, cond-mat/O308220. E. I. Rashba, Phys. Rev. E 62, R16267 (2OOO). D. L. Smith and R. N. Silver, Phys. Rev. E 64,045323 (2001). H. J. Zhu et al., Phys. Rev. Lea 87,016601 (2001). 193 A. T. Hanbicki et al.,Appl. Phys. Lett. 80, 1240 (2002). 194 V. F.Motsnyi et al., Appl. Phys. Lett. 81,265 (2002). 195 R. M. Stroud et nl., Phys. Rev. Lett. 89, 166602 (2002). I% A. F.Isakovic ef al., Phys. Rev. E 64, 161304 (R) (2001). Ig7 J. Strand et nl., Phys. Rev. Lea 91,036602 (2003) 19' J. Strand et al., Appl. Phys. Len. 83,3335 (2003). 19' 19'

INTRODUCTION TO SEMICONDUCTOR SPINTRONICS

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dependence of the spin polarization of the emitted light shows a striking resemblance to the hysteresis loops observed in the magnetic-field-dependent timeresolved Faraday rotation measurements of nuclear polarization by the proximity effect (recall Fig. 3 1). Detailed modeling of the electroluminescence polarization loops indeed demonstrates that spin-polarized injection of electrons from Fe into GaAs leads to the dynamic nuclear polarization of nuclei with GaAs via the Overhauser effect. SEMICONDUCTOR TUNNEL JUNCTIONS 16. FERROMAGNETIC

As we stated in the introduction to this chapter, fundamental studies of spindependent transport and tunneling in metallic ferromagnetic heterostructureshave been of critical importance to the development of metallic “spintronic” devices for high-density information storage. Several diffeTent classes of ferromagnetic semiconductor tunnel junctions have been studied in recent years, generically represented by the device in Fig. 32(b).We note that the driving motivation behind these semiconductor-basedtunnel junctions is not to develop a technology that can compete against the mature tunnel junction technology of metallic ferromagnets, but rather to gain new fundamental insights into the physics of spin-dependent tunneling in semiconductors.In such magnetic tunneljunction devices, the vertical resistance is small when the magnetization of the two ferromagnets is parallel, and large when the the magnetization is antiparallel. A model developed for metallic ferromagnet tunnel junctions simply relates the difference in resistance between the parallel (R P )and antiparallel (R A ) states to the spin polarization ( P I and P2) in each fex~0magnet.l~~

A large tunnel magnetoresistance (TMR) (-80%) has been observed in allsemiconductor Gal -,Mn,As/AlAsGal -,Mn,As tunnel junctions at low temperat u r e ~In. ~these ~ ~ experiments, a sliding shutter is used to create a wedge-shaped tunnel barrier between the ferromagnetic layers, so as to allow a reliable exploration of the variation of TMR with barrier thickness. Although the usual approach employed for understanding TMR in metallic magnetic tunnel junctions does not have any explicit dependence on the barrier thickness,199the experimental data show that a different physical picture is at work in these all semiconductor systems. In particular, the TMR has a non-monotonic dependence on the thickness of the AlAs barrier: A maximum TMR is observed at a barrier thickness of around 1.6 nm, and it decreases for both thinner and thicker barriers. The small TMR in very M. Julliere, Phys. Left.A 54,225 (1975).

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thin barrier samples is difficult to understand in detail. It may possibly be caused by extrinsic reasons (for instance, surface roughness), or perhaps may even have an intrinsic origin (such as the onset of interlayer coupling). The rapid decrease in TMR for larger barrier thickness is qualitatively consistent with a model that considers conservation of the in-plane component of the wave vector of the tunneling carriers and also treats in detail the tunneling between the Fermi surfaces of the majority (minority) spins in the two ferromagnetic layers. Another variation of the all-semiconductormagnetic tunnel junction is the recent study of spin accumulation in a nominally undoped, low-temperature-grown GaAs spacer layer that is sandwiched between two Gal -,Mn,As/AlAs tunnel junction^.'^^ Here, one of the Gal-,Mn,As layers acts as a spin injector and the other as a spin detector. The observation of a TMR signal suggests the spin-polarized injection of holes into the GaAs spacer, followed by sequential tunneling into the detector layer before the holes can undergo spin relaxation in the spacer. We now focus on a different type of hybrid magnetic tunnel junction that shows sizeable tunnel magneto-resistance when a ferromagnetic metal (MnAs) is interfaced with a ferromagnetic semiconductor (Gal -,Mn,As). This latter experiment was initiated to cany out an electrical measurement of spin injection via a tunnel barrier from a metal into a semiconductor. As we mentioned earlier, the inefficient injection of spin-polarized currents from metallic ferromagnets into semiconducIt is now recognized tors arises because of the large mismatch in condu~tivities.'~~ that this problem can be overcome by using either ferromagnetic semiconductors or highly spin-polarized paramagnetic semiconductors for spin inje~tion.*~.~' Alternatively, spins can be injected from a ferromagnetic metal via a tunnel barrier,193and the conductivity mismatch problem is essentially circumvented by the large contact resi~tance."~ A direct scheme for detecting spin injection in this case is to measure the TMR between metallic ferromagnetic tunnel contacts that sandwich a semicondu~tor;'~~ one tunnel barrier serves as a spin injector and the other a spin detector, and the physics is completely analogous to that of a traditional magnetic tunnel junction (MTJ).200 In principle, an ideal device for such an experiment would consist of a heterostructure such as MnAs/AlAs sandwichingan n-doped layer of GaAs. However, the fabrication of high-quality epitaxial metdsemiconductodmetal heterostructures needed for such a scheme presents a difficult materials challenge, because it is almost impossible to grow a good-quality epitaxial semiconductor layer on top of a metal. For instance, even in the most successful attempts to make such epitaxial MTJs (MnAslAIAslMnAs),the magnetoresistanceeffects are small (% 1%), presumably due to the poor quality of the interfaces and the crystalline structure.2o1 2oo J. 20'

S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74,3273(1995).

S.Sugahara and M. Tanaka, Appl. Phys. Lett 80, 1969 (2002).

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An alternative is to replace the bottom ferromagnet by Gal-,Mn,As, because this

allows the epitaxial overgrowth of AlAs and GaAs of high structural quality. We have implemented the first steps toward such a device by measuring the TMR of a single Gal-,Mn,As/AlAs/MnAs tunneljunction. Although the current experiment is limited to detecting spin injection at temperatures below the relatively low Curie temperature of Gal-,Mn,As (Tc = 70 K), the high Curie temperature of MnAs (Tc = 320 K) allows for future room-temperature experiments in different configurations. We note the low Curie temperature of the Gal-,Mn,As layer provides a built-in control experiment because we can measure the TMR in both the paramagnetic and ferromagnetic states of Gal-,Mn,As. Unlike traditional MTJs based on metallic ferromagnets?OOor the recently developed all-ferromagneticsemiconductor MTJs wherein the ferromagnetic layers have comparable condu~tivities,’~~ the ferromagnetic components in these hybrid heterostructures have conductivities differing by four orders of magnitude (-1pn-cm for MnAs and -10 ma-cm for Gal -,Mn,As).202~I lo Samples are fabricated by MBE on p+-GaAs (001) substrates after the growth of a 40-nm-thick p-GaAs buffer layer. We have studied a wide variety of sample configurations involving GaAs, (Ga,Mn)As, (Ga,Al)As, and MnAs, but focus here on a systematic set of 4 samples wherein Gal-,Mn,As ( x = 0.03, thickness 120 nm), GaAs (thickness 1 nm), AlAs (thickness AM^ = 1 nm, 2 nm, 5 nm, and 10 nm), GaAs (thickness 1 run), and MnAs (thickness 45 nm) are grown sequentially at 250” C. The thin GaAs spacer layers placed between the ferromagnetic layers and the tunnel barrier seem crucial to the observationof distinct TMR characteristic^.'^^ Reflection high-energy electron diffraction measurements during the growth confirm the epitaxy of MnAs in the “type-B” orientation.203Photolithographyand wetetching techniques define 300-pm-diameter mesas etched down into the p-GaAs region for vertical transport measurements. The DC current-voltage characteristics of a mesa between the top MnAs layer and the back of the p-GaAs substrate are measured using a 4-probe method in a continuous flow He cryostat over the range 4.2-300 K with an in-plane magnetic field ranging up to 2 kG provided by an electromagnet; additional transport measurements down to 330 mK use a 3 He cryostat with a superconducting magnet. Finally, magnetization is measured on 10 mm2 pieces of the unpatterned wafer using a Quantum Design superconducting quantum interference device (SQUID) magnetometer. Magnetization hysteresis loops measured with the magnetic field applied along the easy axis of “type-B” MnAs (parallel to [ 1701 GaAs [203]) reveal two distinct transitions at 20 Oe and 500 Oe, indicatingthe switchingof magnetizationdirection of Gal-,Mn,As and MnAs, respectively. Figure 33 shows the TMR for 4 samples mz J. J. Beny et al., Phys. Rev. B 64,052408 (2001). ’03

S. H. Chun et al., Appl. Phys. Let?. 78,2530 (2001).

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GaMnAs 1.2

1.0 -

J

I

-1

>

-

0 1 Magnetic Field (kOe)

RG. 33. Magnetoresistances ofhybridjunctionswith adifferentAlAs barrierthickness ( T = 4.2 K). The curves are shifted for clarity. (Adapted from Ref. 137)

of different tunnel barrier thicknesses, normalized at the high field value. A sudden resistance drop accompanies the switching of the MnAs magnetization from antiparallel to parallel with respect to the Gal-,Mn,As magnetization. We note that the TMR shows a nonmonotonic dependence on the AlAs barrier thickness d ~ lwith ~ ~ a striking , effect of around 30% for the sample with d ~= 5l nm. ~ ~ These results are interpreted in straightforwardanalogy with metallic MTJs: The tunneling probability is larger when the two ferromagnetic layers are magnetically aligned than when they are anti-aligned. Quantitatively, the change in the tunnel resistance is given by:199 (4.3) where RA and Rp are the junction resistances with antiparallel and parallel moments, Pc and PM are the spin polarizations of Gal-,Mn,As and MnAs respectively. The spin polarizations PC 0.9 and P.M 0.45 have been recently measured using Andreev reflection.204Hence, one can estimate the spin injection efficiency through the tunnel barrier as a ratio of the observed TMR to the ideal TMR predicted by the preceding equation. Using these measurements, the observed TMR is close to 25% of the .ideal TMR. Spurious effects that might artificially enhance TMR, such as magnetoresistance due to fringe fields, can be ruled out in our measurements because the magnetoresistancesof the individual MnAs and Gal-,Mn,As layers are smaller than 0.5% for the field range shown in Fig. 33.

-

*04 R. J.

-

Panguluri eraZ., Phys. Rev. B 68,201301(R) (2003).

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FIG.34. Comparison between the temperature dependences of the Gal-,Mn,As magnetization ~ (Adapted ~ from at 50 Oe and the TMR measured with 100 mA for a junction with d ~ = 55l nm. Ref. 137.)

Further, the systematic variation of the TMR with barrier thickness rules out possible effects of fringe fields due to the nearby MnAs layer on Gal-,Mn,As. We now focus on the sample with d ~ = l5 nm ~ in~order to examine the physics of these MTJs in some depth. In contrast with all-metal MTJs where there is a negligible change in magnetization with temperature, the magnetization and hence the spin-polarization of Gal-,Mn,As depend strongly on temperature. Figure 34 shows the temperature dependence of the TMR along with that of the bulk magnetization. Both disappear at the Curie temperature of Gal-,Mn,As (around 70 K). The figure also shows that the temperature dependences of the TMR and the magnetization are quite different. This discrepancy may be related to the faster decay of surface magnetism, as is found in other half-metallic systems?05 or it may indicate that the spin polarization in Gal-,Mn,As is not directly proportional to the magnetization. The correlation of the TMR with TC is a unique feature of these junctions where one can probe both ferromagnetic and paramagnetic states by changing the temperature. The voltage dependence of the TMR at 4.2 K shows a rapid decay of the TMR at voltages as low as 100 mV. This qualitatively resembles similar behavior in metallic MTJs, but the relative scale of the voltage is much smaller in these hybrid M T J s . The ~ ~ rapid decrease in the TMR with voltage is likely related to the smaller spin splitting in the Gal-,Mn,As layer ( ~ 1 0 meV) 0 compared to typical splittings in metallic ferromagnets (-1 eV). Figure 35 shows the conductance-voltage '05

J.-H. Park eral., fhys. Rev. L.et?. 81, 1953 (1998).

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Voltage (mV)

FIG. 35. Zero field conductance curves of the junction used in Fig. 34 at selected temperatures.

T = 20,40,60,80, and 100 K for the curves between 4.2 and 120 K. The dashed line superimposedon the 120 K data is a fit to the Brinkman-Dynes-Rowel1model over the range 640 mV. The inset shows the schematic diagram of the model. (From Ref. 137.)

(G-V) characteristics measured at zero magnetic field for several temperatures between 4.2K and 240 K. A distinct zero bias anomaly develops below the TC of Gal-,Mn,As, suggesting a small energy gap around the Fenni energy. Another noticeable feature is the asymmetry in the G-V curves. The conductance under positive bias (wherein the MnAs is at a higher potential) is larger than that under negative bias, and-at temperatures above the Curie temperature of Gal-,Mn,As-the minimum conductance occurs away from zero bias. A full analysis of the G-V characteristics below the TC of Gal-,Mn,As is not possible because the detailed valence band structure of this material is presently not known from experiment. Instead we focus on the G-V characteristics above Tc, where the conductancevoltage curves are parabolic within the voltage range f40 mV (see for instance the data for T 1 120K in Fig. 35). The asymmetric shape and the occurrence of minimum conductance at a finite voltage lead us to apply the Brinkman-Rowell-Dynes(BDR) tunneling model that was originally developed to calculate the tunneling across metal-insulator-metal junctions with different barrier heights at the interfaces.206Although Gal-,Mn,As is not a metal, the BDR model is still applicable for voltages less than the Fermi *06 W.

F. Brinkman, R. C. Dynes, and J. M. Rowell, J. Appl. fhys. 41, 1915 (1970).

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energy of Gal-,Mn,As (0.16 eV if we assume a hole density of 1 x 10” ~ m - ~A) . best fit to the BDR model-shown for the data at 120 K in Fig. 35-allows us to extract the barrier heights at the MnAs/AlAsinterface (& ) and the Gal -,Mn,As/AlAs interface (#z), as well as the barrier thickness ( ~ A I A ~(see ) the inset of Fig. 35). The best value for 41 is 0.15 f 0.01 eV, which is much smaller than that obtained (0.8 eV) in studies of MnAs/AlAs/MnAs MTJs grown on (1 11) GaAs.’O1 This discrepancy can be attributed to the different orientation of the MnAs growth or to a subtle change at the interfa~e.’~’We note that measurements of Fe/GaN/Fe MTJs grown on (001) GaAs yield a small barrier height (0.11 eV), comparable to our results.208A simple estimate of 4 2 is given by the difference between the known AlAs-GaAs valence band offset (0.55 eV) and the Fenni energy of Gal-,Mn,As (0.16 eV). Our result from fitting the data (0.40 f0.OleV) is close to this estimate (0.39 eV). Equally good agreement is found for the barrier thickness: We determine d m s = 6.7 f0.1 nm assuming light hole states participate in ~ ~f=0.1 nm assuming heavy holes. the tunneling through AlAs, while d ~ l 4.4 A mixture of light and heavy holes may possibly provide a better description of reality, consistent with the designed AlAs thickness (5 nm). The successful application of the BDR model implies that the conduction is indeed due to tunneling processes.

V. Concluding Remarks In this review, we have attempted to cut a somewhat broad and sweeping swath through a field that has shown enormous growth in recent years. A number of fundamentally interesting concepts have emerged from the concerted theoretical and experimental efforts in this area, ranging from advances in our basic understanding of ferromagnetism in magnetically doped semiconductors to the coherent spin dynamics of electrons, magnetic ions, and nuclei in semiconductors. Whether or not studies of semiconductor spintronics will lead toward a practical technology of some impact remains an open question, but the historical interplay between fundamental discovery in condensed matter physics and the development of technology leaves this author optimistic that important technological applications of spin-dependent phenomena in semiconductors will undoubtedly surface in the years ahead. In the meantime, there is a wealth of new physics waiting to be unravelled at the intersection of semiconductor physics and magnetism.

207

208

W. Van Roy et al., J. Cryst. Growth 227-228, 852 (2001). S . Nemeth et al., J. Cryst. Growth 227-228.888 (2001).

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Acknowledgments The author has enjoyed many years of enjoyable and rewarding interactions with a number of talented colleagues and students. In particular, he wishes to acknowledge David Awschalom, Peter Schiffer, Michael Flatt6, Allan MacDonald, and H. Ohno for many insightful discussions and collaboration. The author is also deeply grateful to the Office of Naval Research, the Defense Advanced Research Projects Agency, and the National Science Foundation for providing the wherewithal for the work described in this chapter. Finally, the author thanks the Nature Publication Group, Science Magazine, American Institute of Physics and the American Physical Society for permission to reproduce published figures.

SOLID STATE PHYSICS. VOL.58

Electron Spin Dynamics in Semiconductors EX. BRONOLD',A. SAXENA~, AND D.L. S M ~ *

'Institutfir Physik, Emst-Moritz-Amdt Universitat Greijiwald, D-17487 Greijswald, Germany 'Theoretical Division, Los Alamos National Lab. LQS Alamos, New Mexico 87545 USA

I.

Introduction 1. 2. 3. 4. 5. 6. 7.

...............................................

nductor Electronic Structure ........... Spin-dependent Hamiltonian . . ................... Band States Away from the Zone Center. ................................ Spin-flipscattering .................................. Inversion Asymm ..................... Electron-Hole E ..................... Hypertine Interaction ............................. Spin-dependent Optical Properties ......................

9. Effective Mass Hamiltonian for n-Qpe Semiconductors ....................

................

74 80 83 86 91 92 96 98

106 112

12. Fokker-Planck Equation for the Spin Density . . . . . . . . . . . . . . . . . . 13. Calculation of the Spin Relaxation Rates ..................... N. Application of the Kinetic Theory 14. Spin Relaxation in Bulk Sem

..........................

VI.

17. Constant Conductivity Model . 18. Spin Injection at Schottky Contacts. ....................... 19. Spin-dependent Tunneling ............................... 20. Experiments on Spin Injection .................. SummaryandFutureDirections ............................................

138

162 164

13 ISBN 0-12-6077584 ISSN 0081-1947/04

0 2004 Elsevier Science (USA)

All rights reserved.

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EX.BRONOLD, A. SAXENA, AND D.L. SMITH

1. Introduction The ability to control and measure electron spin degrees of freedom in solids has been proposed'-'* as the operating principle for a new generation of novel electrical devices with the potential to overcome power consumption and speed limitations associated with conventional electronic circuits, and also as a means to physically implement schemes for quantum information processing and computing. Recent fundamental discoveries related to non-equilibrium electron spin dynamics and transport in solids make the study of these phenomena extremely rich from both basic science and technology points of vie^.'^-^^ Devices based on spin-dependent transport in metallic systems, such as giant magneto-resistive structures and magnetic tunnel junctions utilizing spin-dependent tunneling, are

I S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molniu; M.L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001) and references therein. G. E. Pikus, V. A. Marushchak, and A. N. Titkov, Sov. Phys. Semicond. 22, 115 (1988). L. J. Sham. J. Phys.: Condens. Muter 5, A5 1 (1993). G. Prinz, Phys. Today 48:4,24 (1995). L. Vina, J. Phys.: Condens. Muter 11,5929 (1999). J. Fabian and S. Das Sarma, J. Vac. Sci. Tech. 17, 1708 (1999). R. Fitzgerald, Phys. Today 53:4,21 (2OOO). S. Das Sarma, Amer: Sci. 89,516 (2001). D. Grundler, Phys. World 15,39 (2002). l o D. D. Awschalom, M. E. Flatti, and N. Samarth, Sci. Am. 286.66 (2002). ' I Semiconductor Spintmnics and Quantum Computation, eds. D. D. Awschalom, D. Loss, and N. Samarth, Springer, Berlin (2002). I' H. Ohno., Eds., Semiconductor Science and Technology (Special issue) 17:4,275404 (April 2002). l 3 S. A. Crooker, J. J. Baumberg, F. Flack, N. Samarth, and D. D. Awschalom, Phys. Rev. Lett. 77,

*

2814 (1996). l4 S . A. Crooker, D. A. Tulchinsky, J. Levy, D. D. Awschalom, R. Garcia, and N. Samarth,Phys. Rev. Lett. 75,505 ( 1995). I5 J. M. Kikkawa, D. D. Awschalom, I. P. Smorchkova, and N. Samarth, Science 277, 1284 (1997). l6 J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80,4313 (1998). I7 J. M. Kikkawa and D. D. Awschalom, Nature 397, 139 (1999). H. Ohno, Science 281,951 (1998). j 9 W. H. Lau, J. T. Olesberg, and M. E. Flatti, Phys. Rev. B 64, 161301(R) (2001). 2o C. Ciuti, J. P. McGuire, and L. J. Sham, Phys. Rev. Lett. 89, 156601 (2002). 2 1 J. A. Gupta, D. D. Awschalom, X.Peng, and A. P. Alivisatos, Phys. Rev. B 59, R10421 (1999). 22 W. H. Rippard, and R. A. Buhrman, Phys. Rev. Len. 84,971 (2000). 23 B. T. Jonker, Y. D. Park, B. R. Bennett, H.-D. Cheong, G. Kioseoglou, and A. Petrou, Phys. Rev. B 62,8180 (2000). 24 R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, and L. W. Molenkamp, Nature (London) 402,787 (1999). 25 R. M. Stroud, A. T. Hanbicki, Y. D. Park, G. Kioseoglou, A. G. Petukhov, B. T. Jonker, G. Itskos, and A. Petrou, Phys. Rev. Lett. 89, 166602 (2002). 26 A. F. Isakovic, D. M. Carr, J. Strand, B. D. Schultz, C. J. Palmstrom, and P. A. Crowell, Phys. Rev. B 64,016304 (2001).

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75

having a major technological Promising new electron-spin-based device concepts utilizing semiconductors have been Realizing the full potential of spin-based electronics requires a more complete understanding of non-equilibrium electron-spin-based phenomena than is currently available, and there is extensive on-going research in this area. Essential points of focus for this research area include: generation of non-equilibrium spin-polarized electron distributions; transport of electron spin distributions through a given material and across interfaces between two materials perhaps with different spin-dependent ground states; the relaxation dynamics of a non-equilibrium spin-polarized electron distribution; and the interaction of spin distributionswith optical and magnetic probes.

27 H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Schonherr, and K. H. Ploog, Phys. Rev. Len. 87,016601 (2001). 28 A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou, Appl. Phys. Leff. 80, 1240 (2002). 29 V. F. Motsnyi, J. De Boeck, J. Das, W. Van Roy, G. Borghs, E. Goovaerts, and V. I. Safarov, Appl. Phys. Lett. 81,265 (2002). 30 G. Vignale and M. E. Hatt6, Phys. Rev. Leff. 89,098302 (2002). 31 M. E. Flank and J. M. Byers, Phys. Rev. Left. 84,4220 (2000). 32 G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. E 62,4790 (2OW. 33 E. I. Rashba, Phys. Rev. E 62, 16267 (2000). 34 D. L. Smith and R. N. Silver, Phys. Rev. B 64,045323 (2001). 35 I. Zutic, J. Fabian, and S. Das Sarma, Phys. Rev. Leff. 88,066603 (2002). 36 I. Zutic, J. Fabian, and S. Das Sarma, Phys. Rev. E 64, 121201 (2001). 37 J. Fransson, 0. Eriksson, and I. Sandalov, Phys. Rev. Len.89, 179903 (2002). 38 M. N. Baibich, J. M. Broto, A. Fert, F. N. Vandau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Len. 61,2472 (1988). 39 J. Bamas, A. Fuss, R. E. Camley, P.Grunberg, and W. Zinn, Phys. Rev. E 42.81 10 (1990). B. Dieny, V. S Speriosu, S. Metin, S. S. P.Parkin, B. A. Gurney, P. Baumgart, and D. R. Wilhoit, J. Appl. Phys. 69,4792 (1991). 41 B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. E 43, 1297 (1991). 42 S. Datta and B. Das, Appl. Phys. Lett. 56,665 (1 990). 43 D. Hagele, M. Oestreich, W. W. Rhle, N. Nestle and K. Eberl, Appl. Phys. Len. 73, 1580 (1998). 44 M. Oestreich, J. Hubner, D. Hagele, P. J. mar, W. Heimbrodt, W. W. M e , D. E. Ashenford, and B. Lunn, Appl. Phys. Len. 74, 1251 (1999). 45 M. Oestreich, Nafure 402,735 (1999). 46 J. Rudolph, D. Hagele, H. M. Gibbs, G. Khitrova, M. Oestreich, Appl. Phys. Lett. 82,4516 (2003). 47 E. A. de Andrada e Silva and G. C. La Rocca, Phys. Rev. E 59, 15583 (1999). 48 D. 2. Y.Ting and X . Cartoixa, Appl. Phys. Left. 81,4198 (2002). 49 J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Len. 82,2658 (2003). 50 D. Schmeltzer, A. R. Bishop, A. Saxena, and D. L. Smith, Phys. Rev. Len. 90, 116802 (2003). 5 1 C. Ciuti, J. P. McGuire, and L. J. Sham, Appl. Phys. Left. 81,4781 (2002). ” I. Zutic, J. Fabian, and S. Das S m a , Appl. Phys. Left 79, 1558 (2001). 53 S. K.Upadhyay, R. N. Louie, and R. A. Buhrman, Appl. Phys. Left. 74,3881 (1999).

76

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Semiconductorsare particularly attractive for studies of spin-based phenomena in solids because circularly polarized light can be used to inject and detect specific spin orientation of electrons and because the use of these materials for integrated electronics is well established. Non-equilibrium spin distributions in semiconductors can be generated by: (1) polarized optical techniques utilizing spin-dependent optical selection rules; (2) spin-polarized electrical injection across interfaces between materials with different magnetic properties; and (3) current transport in spatially inhomogeneous magnetic fields. Spin distributions and currents can be probed and characterized by optical, magnetic, and electrical transport techniques. Non-equilibrium spin distributions can be manipulated by electric and magnetic fields and by semiconductorheterostructured e ~ i g n ? ~The , ~spin ~,~ coherence ~ lifetimes of electrons in these materials can be quite long and may be optimized by tuning electron density and by selecting specifically designed heterostructures. Many physical probes, including optical and electrical probes, couple directly to the spatial component of the electron wave function but do not couple directly to the spin component. The spin-orbit interaction mixes the spatial and spin components of the electron wave function and allows, for example, an optical probe to couple to the electron spin. Thus the spin-orbit interaction is useful in allowing the electron spin degrees of freedom to be manipulated and measured using convenient probes. However, the spin-orbit interaction also provides mechanisms for spin decoherence. For example, most scattering processes involve the spatial part of the electron wave function and in the absence of the spin-orbit interaction would not lead to spin decoherence. The spin-orbit interaction allows a loss of spin coherence in scattering events in which the direct coupling is to the spatial part of the wave function. Thus the spin-orbit interaction provides a means to conveniently manipulate the electron spin degree of freedom but also introduces mechanisms for loss of spin cohejence. The role of the spin-orbit interaction is a major theme in spin-based device concepts. Controlling the strength of the spinorbit interaction spatially by heterostructure design is an important approach to spin-based electronic device d e ~ i g n . ~ ~ . ~ ~ The coupling between electron spin and orbital motion that results from the spin-orbit interaction produces optical selection rules that allow a non-equilibrium electron spin population to be generated by the absorption of circularly polarized light. Optical transitions across the energy gap of a semiconductorare governed by electric dipole selection rules. Circularly polarized light with a wavelength correspondingto the semiconductorabsorption edge induces transitions from the highest S. Gardelis, C. G. Smith, C. H. W. Barnes, E. H. Linfield, and D. A. Ritchie, Phys. Rev. B 60,7764 (1999). ” G. A. Prim, Science 282, 1660 (1998). 56 Y. Kato, R. C. Myers, D. C. Driscoll, A. C. Gossard, J. Levy, and D. D. Awschalom, Science 299, 1201 (2003). ” H. Kosaka, A. A. Kiselev, F. A. Baron, K. W. Kim, and E. Yablonovitch, Elecrmnics Letters 37,464 (2001). 54

77

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

energy valence band states to the lowest energy conduction band states, which produce a spin-polarized electron distribution in direct bandgap semiconductors. The same optical selection rules allow the dynamics of a non-equilibriumelectron spin population to be probed optically using polarization-sensitive transmission and reflection, Kerr- and Faraday-rotation spectroscopies, respectively. A nonequilibrium spin distribution produces a difference in the index of refraction for left- and right-hand circularly polarized light. As a result, the polarization axis of linearly polarized light, which is a coherent superposition of left- and right-hand circularly polarized light, rotates upon reflection from or transmission through the semiconductor with the spin-polarized electron distribution. These spectroscopies are readily adaptable to time- and spatially-resolvedmeasurements.' For example, circularly polarized optical pulses can create spin-polarized states in a given spatial region of a bulk semiconductoror a quantum well. A magnetic field can be applied, in which case the spin population precesses in the applied field. The amplitude, phase, and spatial position of the resultant spin population can be tracked in timeresolved experiments using Kerr- or Faraday-rotation spectroscopy. Electron spin polarization can also be detected by polarization-sensitiveluminescence measurements because, due to the optical selection rules, a non-equilibrium distribution of spins gives rise to circularly polarized luminescence. In this approach, the spin-polarized electron distribution is allowed to recombine radiatively with a hole distribution-for example, in a p-n junction structure-and the polarization properties of the resulting luminescence are analyzed. Using these optical techniques, electron spin lifetimes have been investigated in a variety of semiconductors under various conditions and long spin lifetimes have been ob~erved.~*'~.'~-'~.~&~'

'

58

W. A. J. A. van der Poel, A. L. G.J. Severens, H. W. van Kesteren, and C. T. Foxon, Phys. Rev. B

39,8552 (1989).

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, Phys. Rev. Len.67,3432 (1991). V. Srinivas, Y.J. Chen, and C. E. C. Wood, Phys. Rev. B 47, 10907 (1993). " J. Wagner, H. Schneider, D. Richards, A. Fischer, and K. Ploog, Phys. Rev. B 47,4786 (1993). R. Terauchi, Y.Ohno, T. Adachi, A. Sato, F. Matsukura, A. Tackeuchi, and H. Ohno, Jpn. J. Appl. Phys. (Purr I ) 38,2549 (1999). 63 A. Tackeuchi, T. Kuroda, S. Muto, Y.Nishikawa, and 0. Wada, J. Appf. Phys. (Put? 1 ) 38,4680 59

'*

(1999).

Y.Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno, Phys. Rev. Leu. 83,4196 (1999). A. L. C. Triques, J. Urdanivia, F. Iikawa, M. Z. Maidle, J. A. Brum, and G.Borhgs, Phys. Rev. B 59, R7813 (1999).

A. Malinowski, R. S. Britton, T. Grevatt, R. T. Harley, D. A. Ritchie, and M. Y. Simmons, Phys. Rev. B 62,13034 (2000). 67 T. F. Boggess, J. T. Olesberg, C. Yu,M. E. Flattk, and W. H. Lau, Appf. Phys. Len. 77, 1333 (2000). J. S. Sandhu, A. P. Heberle, J. J. Baumberg, and J. R. A. Cleaver, Phys. Rev. Len. 86,2150 (2001). 69 J. T. Olesberg, W. H. Lau, M. E. Flattk, C. Yu,E. Altunkaya, E. M. Shaw, T. C. Hasenberg, and T. F. Bogess, Phys. Rev. B 64,201301 (2001). 70 R. I. Dzhioev, K. V. Kavolin, V. L. Korenev, M. V. Lazarev, B. Y.Meltser, M. N. Stepanova, B. P. Zakharchenya, D. Gammon, and D. S. Katzer, Phys. Rev. B 66,245204 (2002). 71 R. I. Dzhioev, V. L. Korenev, I. A. Merkulov, B. P. Zakharchenya, D. Gammon, A. L. Efros, and D. S. Katzer, Phys. Rev. Lett. 88,256801 (2002). 66

78

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

The lifetimes depend strongly on doping levels and temperature. Experiments have identified ranges of doping concentrations in semiconductors where lifetimes are strongly enhanced, in some cases exceeding 100 ns." Long electron spin lifetimes can be achieved up to room temperatures in certain nanostructures and quantum dots.' Because of the long spin coherence times a non-equilibrium electron spin distributioncan be driven across macroscopic distances in homogeneous semiconductor crystals and also across heterointe~faces.'~.~~ A magnetic field couples directly to spin and can be used to create spin polarization. Most semiconductors have effective g-factors of order unity and the magnetic spin splittings are small so that electron spin populations can be significantly polarized only at very low temperature by high magnetic fields. However, in certain Mn-doped 11-VI semiconductors, called semimagnetic semiconductors, very large effective g-factors (of order 100)occur and strong electron spin polarization can be achieved with more moderate magnetic fields and temperature^.^^ These semimagnetic semiconductors can be incorporated into heterostructures with nonmagnetic semiconductors and used as a source of spin-polarized electrons. Ferromagnetic semiconductorsconsisting of Mn-doped 111-V semiconductors have recently been developed and show many interesting proper tie^.^^^^^-^^ They can also be incorporated into heterostructures to make novel device structures. To date, these ferromagnetic semiconductorshave all been p-type doped and have Curie temperatures below room temperature. Current flow in an inhomogeneous magnetic field can be used to create a spin current and a spin current can be manipulated by electrical means. There is also a direct coupling between the electron spin and the nuclear spin through the hyperfine interaction. This interaction enables a non-equilibrium electron spin population to transfer spin polarization to the nuclear spin population. It allows the electron spin polarization to be detected by nuclear spin probes such as nuclear magnetic resonance (NMR). In addition, the optical techniques can be combined with NMR to give an all-optical form of NMR.The latter approach enables an all-optical coherent manipulation of electron spins in a way analogous to nuclear spin echoes.

I. Malajovich, J. J. Berry, N. Samarth,and D. D. Awschalom, Nature (London)411,770 (2001). I. Malajovich, J. M. Kikkawa, D. D. Awschalom,J. J. Berry, and N. Samarth,J. Appl. Phys. 87,5073 (2@3-9. 74 J. K. Furdyna, J. Appl. Phys. 53,7637 (1982). 75 S. A. Cmker, D. D. Awschalom, J. J. Baumberg, F. Hack, and N. Samarth,Phys. Rev. B 56,7574 (1997). 76 Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura, H. Ohno, and D. D. Awschalom, Nature 402, 790 (1999). 77 C. Camilleri, F. Teppe, D. Scalbert, Y. G. Semenov, M. Nawrocki, M. Dyakonov, J. Cibert, S. Tatarenko, and T. Wojtowicz, Phys. Rev. B 64,085331 (2001). 78 M. Berciu and R. N. Bhatt, Phys. Rev. B 66,085207 (2002). 79 T. Dietl, Semiconductor Science and Technology (Special issue), 17:4,371-392 (April 2002).

72

73

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

79

Semiconductor device concepts that utilize spin-dependent phenomena rely on electrical injection and transport of spin currents. Structures designed to achieve electrical spin injection utilize spin-polarized contacts as the source of polarized electrons. ’ b o main types of spin-polarized contacts have been investigated; ferromagnetic metal contacts in which the electron spin is polarized because of the ferromagnetic ground state of the c o n t a ~ t , ~ and ” ~ semimagnetic ~ . ~ ~ ~ ~ semiconductor contacts with large g-factors in which the electron spin is polarized by an external magnetic fie1d.23v24*84 Ferromagnetic metals with Curie temperatures well above room temperature are an ideal source of spin-polarized electrons for electrical device purposes because there is no requirement for low temperatures or externally applied magnetic fields. However, the problem of achieving spin injection from ferromagnetic contacts has proven much more difficult than originally expected.85A fundamental physical problem was shown to result because of the conductivity mismatch between the metallic contact and the semicond~ctor.~~ Approaches to overcome this problem and to design efficient spin injection structures have been Spin injection has now been reported from semimagnetic semiconductor contact^,^^-^^*^ ferromagnetic metal ~ o n t a c t s , 2 ” ~ferromagnetic ~,~~~ metal scanning tunneling microscopy (STM) tip^?^,^' and hot electron injection structure^.^^ In this article we discuss physical processes that determine the dynamics of electron spins in semiconductors. We concentrate on group IV,111-V, and 11-VI semiconductors with the diamond or zinc-blende crystal structure. We first discuss the basic interactions that govern spin-dependent processes, then present a kinetic theory to describe electron spin dynamics, use the kinetic theory to discuss electron spin dynamics for several semiconductor systems, consider spin injection at A. T. Hanbicki, 0.M. J. van ’t Erve, R. Magno, G . Kioseoglou, C. H. Li, B. T. Jonker, G. Itskos, R. Mallory, M. Yasar, and A. Petrou, Appl. Phys. Len. 82,4092 (2003). B. T. Jonker, Proc. IEEE 91,727 (2003). J. Strand, B. D. Schultz, A. F. Isakovic, C. J. Palmstr@m,and P. A. Crowell, Phys. Rev. Lett. 91, 036602 (2003). 83 A. F. Isakovic, D. M. Cam, J. Strand, B. D. Schultz, C. J. Palmstrom, and P. A. Crowell, J. Appl. Phys. 91,7261 (2002). 84 R. Fiederling, P. Grabs, W. Ossau, G . Schmidt, and L. W. Molenkamp, Appl. Phys. Lea 82,2160 (2003). 85 A. G. Aronov and G . E. Pikus, Fir. Tekh. Polupmvodn. 10, 1177 (1976) [Sov. Phys. Semicod. 10, 698 (1976)]. 86 J. D. Albrecht and D. L. Smith, Phys. Rev. B 66, I13303 (2002). 87 J. D. Albrecht and D. L. Smith, Phys. Rev. B 68,035340 (2003). 88 Z. G . Yu and M. E. Flattb, Phys. Rev. B 66,201202 (2002). 89 Z. G . Yu and M. E. Flattb, Phys. Ra! B 66,235302 (2002). 9o S. F. Alvardo and P. Renaud, Phys. Rev. Lett. 68, 1387 (1992). 91 S. F. Alvardo, Phys. Rev. Lea 75,513 (1995). 92 X. Jiang, R. Wang, S. van Dijken, R. Shelby, R. Macfarlane, G . S. Solomon, J. Harris, and S. S. P. Parkin, Phys. Rev. Lett. 90,256603 (2002).

80

EX.BRONOLD, A. S A X E " , AND D.L. SMITH

interfacesbetween a spin-polarizedcontact and a nonmagnetic semiconductor,and conclude with a summary and discussion of possible future research directions.

II. Spin Dependence of Semiconductor Electronic Structure The most studied and technologicallyimportant semiconductorshave the diamond or zinc-blende crystal structures. Figure 1 shows the zinc-blende crystal structure and Brillouin zone. The structure is based on a face-centered cubic lattice with a two-atom basis. The diamond crystal structure is the same as the zinc-blende except that the two atoms in the unit cell are the same. The Brillouin zones for the two structures are the same, but for the diamond structure there is an additional inversion symmetry operator in the point The electronic structure for diamond and zinc-blende semiconductors is dominated by bonding and antibonding states made up from s- and p-type atomic orbitals.93Figure 2 shows the band structure of Ge on a large energy scale. There are four main groups of states at the zone center (labeled by symmetry point r). The lowest energy state, at about -12 eV, is formed by a bonding configuration of s-type atomic orbitals (for Ge they are 4s atomic orbitals). The bonding combination of orbitals is phased so that the amplitudes add in the spatial region between the two atoms making up the unit cell. For s-type orbitals, the two orbitals have the same sign for the bonding state because the s-functions are even. For a symmetric case like Ge or Si in which the two atoms in the unit cell are the same, the amplitudes of the two atomic orbitals making up the bonding state are equal. For an asymmetric case like GaAs, the amplitude of the anion orbital is larger than that of the cation orbital for the bonding state. The second group of states, at about 0 eV, is formed by a bonding configuration of p-type atomic orbitals (for Ge they are 4p atomic orbitals). The bonding combination of orbitals is phased so that the amplitudes add in the spatial region between the two atoms making up the unit cell, but for p-type orbitals the two orbitals have opposite sign in the bonding state because the p-type functions are odd. For a symmetric case like Ge the magnitude of the two atomic orbitals is the same, but for an asymmetric case like GaAs, the magnitude of the anion orbital is larger than that of the cation orbital. There are three bonding p-bands because of the 3-fold degeneracy of p-orbitals. The bonding s-band and three bonding p-bands are occupied by electrons in the electronic ground state of pure Ge and make up the valence bands of the material. The third group of states, at about 0.8 eV, is formed by an antibonding configuration of s-type atomic orbitals. The antibonding combination of orbitals is phased so that the amplitudes subtract in the spatial region between the two atoms making up the unit cell. For s-type orbitals, the two orbitals have the opposite sign in the 93 K.Seeger, Semiconductor

Physics: An Inrmducrion, Springer, Berlin (2002).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

---

81

.

--*' I

.

FIG.1. The zinc-blende crystal structure (left) and the correspondingBrillouin zone (right). 6

>

p

c w

-4

-6 -8

- 10 -1 2

RG.2. The energy band structure of Ge. Bonding and antibonding S and Porbitals are schematically illustrated.

antibonding state. For a symmetric case, the amplitude of the two atomic orbitals is the same, and for an asymmetric case, the amplitude of the cation orbital is larger than that of the anion orbital. The fourth group of states, at about 4 eV, is formed by an antibonding configuration of p-type atomic orbitals. The antibonding combination is phased so that the amplitudes subtract in the spatial region between the two atoms making up the unit cell, and for p-type orbitals, the two orbitals have the same sign for the antibonding state. For a symmetric case the amplitude of the two atomic orbitals is the same, but for an asymmetric case, the amplitude of the cation orbital is larger than that of the anion orbital. There are three bonding p-bands because of the 3-fold degeneracy of p-orbitals. The antibonding s-band and three antibonding p-bands are empty of electrons in the ground state of pure Ge and make up the conduction bands of the material.

82

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

The band structure of other diamond or zinc-blende structure semiconductors are qualitatively similar to that of Ge on a large energy scale like that of Fig. 2. The same four groups of states-that is, bonding and antibonding s- and p-statesoccur at the zone center for these other materials. The energy ordering is sometimes different from that of Ge; for example, in Si the antibonding p-levels are lower in energy than the antibonding s-levels at the zone center. For semiconductors the lowest energy electron and hole states at the bottom of the conduction band and the top of the valence band, respectively, are especially important. For most diamond or zinc-blende structure semiconductors, the top of the valence band occurs at the zone center and consists of the bonding p-levels as for Ge. For Ge, the bottom of the conduction band occurs at the edge of the Brillouin zone in the [ 1111 directions, the L symmetry points. (There are four inequivalent minima corresponding to the four distinct edges of the Brillouin zone in the [ 1111 directions.) The conduction band states at the L points of Ge are slightly lower in energy than the conduction band state at the r point. For many of the zinc-blende structure 111-V and 11-VI semiconductorsthis ordering is reversed and the conduction band minimum occurs at the zone center and consists of the antibonding s-levels. For Si, the conduction band minima occur along the [ 1001axes, the A symmetry directions, approximately 0.85 of the way to the zone edge (also a local minimum in Ge). (There are six inequivalent minima corresponding to the six distinct [ 1001 directions.) Because electrons typically occupy states near the conduction band minimum, it is rather important at which point in the zone this energy minimum occurs even if the energy splitting between the various local minima is small on the energy scale of Fig. 2. The spin-orbit interaction results from a relativistic effect that couples the electron orbital motion with its spin.94 For atomic systems with continuous rotational symmetry, so that angular momentum is a conserved quantity, the spin-orbit interaction couples the orbital and spin angular momenta. For s-atomic states the orbital angular momentum is L = 0, there is no splitting of the state when the orbital angular momentum is coupled with the electron spin, and the total angular momentum is J = 1/2. For a p-atomic state with orbital angular momentum L = 1, coupling with the electron spin produces a four-fold degenerate set of J = 3/2 states and a two-fold degenerate set of J = 1/2 states. These two sets of states are split apart by the spin-orbit interaction, with the J = 1/2 states lower in energy than the J = 3/2 states. In solids such as diamond or zinc-blende structure semiconductors, there is no continuous rotational symmetry and angular momentum is not a conserved quantity. However, there are discrete rotational, and for the diamond, but not the zinc-blende structure, inversion symmetries under which the Hamiltonian is invariant. At symmetry points in the zone, for which a set of rotation or inversion symmetry operators does not change the wave vector, these symmetry operators constitute a point group that characterizes the transformation 94

J. D. Bjorken and S. D. Drell, RelufivisficQuantum Mechanics, McGraw-Hill, New York (1964).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

83

properties of the Hamiltonian eigenfunctions. At the zone center of diamond (point group Oh) or zinc-blende (point group T d ) structure semiconductors, coupling of the orbital motion with the electron spin is analogous to that for atomic systems with continuous rotational symmetry for the four sets of states formed from the bonding and antibondings-and p-orbitals. The states formed from the bonding and antibonding s-orbitals both form 2-fold degenerate states analogous to the J = 1/2 states for the coupling of an s-level with electron spin in atomic systems. The states formed from the bonding and antibonding p-orbitals both form a 4-fold degenerate set of states analogous to the J = 3/2 states and a 2-fold degenerate set of states analogous to the J = 1/ 2 states for the coupling of a p-level with electron spin in atomic systems. For electrons at each of the conduction band minima of Ge and Si, coupling of the orbital motion with electron spin forms a 2-fold degenerate set of states (not including the degeneracy from the various conduction band valleys). The strength of the spin-orbit interaction scales with the atomic number; it is small for light elements and increases for the heavier elements.93The splitting of the bonding p-states at the zone center is a convenient measure of the spin-orbit interaction strength. For the group IV semiconductors this splitting is: diamond, negligible; Si, 0.044 eV; Ge; 0.29 eV; and Sn (diamond structure) 0.77 eV?5 Except for the heaviest elements this splitting is fairly small on the scale of energy bands. However, this interaction is very important in determining the spin dynamics of electrons and holes. The hyperfine interaction couples electron spin to nuclear spin for nuclear isotopes that do not have zero spin. This coupling leads to electron spin decoherence and also allows a means of manipulating electron spin through nuclear magnetic resonance t e c h n i q ~ e s For . ~ ~ many semiconductors spin-orbit interaction effects are more important than hyperfine interaction effects in determining the electron spin dynamics. However, for the lighter elements such as diamond and other carbon-based materials or Si, hyperfine interaction effects dominate. Hyperfine interaction effects can also dominate in reduced dimensional structures such as quantum dots because of phase space consideration^.^^ 1. SPIN-DEPENDENT HAMILTONIAN The one-electron Hamiltonian used to calculate the semiconductor band structure has the form

’* P. Lawaetz, Phys. Rev. B 4,3460 (1971). C. P. Slichter, Principles ofMagneric Resonance, Springer-Verlag, Berlin ( 1990). ’’A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Len. 88, 186802 (2002). %

84

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

where 5 is the electron momentum operator, e is the magnjtude of the electron charge, m is the bare electron mass, c is the speed of light, A is the vector potential describing an external electromagnetic field, V ( ; ) is a self-consistent crystal potential calculated using a mean-field description of the_electron-electroninteraction, c? are the Pauli matrix electron spin operators, B describes an external magnetic field (i= V x and

A),

The term 2 c? is the spin-orbit interaction. It results from the interaction of the magnetic moment of the electron due to its spin with the magnetic moment set up by its orbital motion. The form for the interaction is derived from taking a nonrelativisticlimit of the Dirac equation in which positron solutions are separated Some additional small terms that appear in this treatment, but which do not change the symmetry of the Hamiltonian, have been neglected in Eq. (2.1). The eigenfunctions of the Hamiltonian, Eq.(2.1), have the form of two component spinors

(2.3) where the arrows indicate spin up or down along a chosen quantization axis. In general the Hamiltonian eigenstates are not spin eigenstates. If either of the functions f ( r ) or g ( r ) vanish then 111.) is a spin eigenstate for the chosen quantization axis. If f ( r ) and g ( r ) are proportional, that is f ( r ) = Cg(r) for some constant C, then the spatial and spin terms factor and 111.) is a spin eigenstate along an axis different than the chosen quantization axis. We first consider a situation without external probes so $ a t and vanish. Eigenstates of the spin-independentpart of the Hamiltonian [5 V @ ) ]at the zone center give the four sets of states formed from the bonding and antibonding s- and p-orbitals. We are particularly interested in the states from the bonding p-orbitals that form the top of the valence band and the state from the antibonding s-orbitals that form the bottom of the conduction band for the direct bandgap 111-V and 11-VI semiconductor^?^ It is convenient to label the three degenerate bonding p-states as Ix), ly), and lz) and the antibonding s-state as Is). When the spin-orbit interaction is included, these spatial states are coupled with spin. The ClebschGordan coefficients for coupling in the zone center point group are the same as those for coupling L = 1 and S = 1/2 angular momentum states and it is convenient to label the states in analogy with angular momentum states. The four

A +

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

85

states at the top of the valence band are9*

(2.4)

and the two states split to lower energy are

where f and 3- refer to the [Ool] quantization axis. The two states formed from coupling the antibonding s-states with spin are

1 1

1.'

2'

-A)2

= Is

t)

= 1s 4).

It is important to notice that the Hamiltonian eigenstates corresponding to the antibonding s-orbitals, which are at the bottom of the conduction band in direct energy gap semiconductors, are also spin eigenstates, whereas the Hamiltonian eigenstatescorrespondingto the bonding p-orbitals, at the top of the valence band, are not spin eigenstates. (When higher lying d-orbitals are included in the basis set, the zone center conduction band states are no longer exact spin eigenstates, but the admixture of the spin-orbit split d-states is small and these zone center states closely approximate spin eigenstates.) This difference leads to qualitatively different spin dynamics for electronsand holes. For Si and Ge the spatial part of the electron wave function is also nondegenerate (not counting the valley degeneracy, the spin-orbit interaction does not mix these states with different wave vectors) and the Hamiltonian eigenstates at the conduction band minima are also spin eigenstates. 98 G. F.Koster, J. 0.Dimmock, R. G. Wheeler,and H.Statz, Properties of the Thirty-huoPoint Groups, MIT Press, Cambridge, MA (1963).

86

EX.BRONOLD, A. S A X E ” , AND D.L. SMITH

2. BANDSTATES AWAYFROM THE ZONE CENTER If the Hamiltonian eigenstates are known at a particular point in the zone, usyally the zone center, it is convenient to describe states at nearby k-points using k . perturbation the01-y.~~ Specifically, suppose that the eigenvalues and eigenstates of the one-electron Hamiltonian, including the spin-orbit interaction, are known at the zone center (2.7) where ub(i) and &b(O)are the eigenstates and eigenvaluesand b is a band index. For the states at the conduction band minimum and near the valence band maximum, the states ub(T) have the form of Eqs. (2.4-2.5) and (2.6), respectively. The one electron wave functions away from the zone center have the Bloch form

1 ~ / - - eik.; b,k

-

‘b,i?

(2.8)

where Ub,jis the periodic part of the Bloch function. These periodic functions can be expanded in the set of zone center states ub,j(r) =

Cb.b’(i)Ub’(r)r

(2.9)

b’

where cb,b’(z) are expansion coefficients. Substituting into the one-electron Schrodinger equation and commuting the exponential across the one-electron Hamiltonian gives an equation for the expansion coefficients (ubl

bJ

(g+

V(r)

+

Iub’)Cb,b’(z)= 0.

. (T

hi.? + h2k2 -+ -- Eb(k) 2m m (2.10)

(We have neglected the commutator of the exponential with the momentum operator in 6 ;this term is usually very small.) Because the states Ub(r) satisfy Eq. (2.7) we have

It is often convenient to divide the zone center states into two groups: (1) those close in energy to, and (2) those far away in energy from, the states that are of 99 Handbook on Semiconductors, Vol. 1. ed. P.T.Landsberg, North-Holland, Amsterdam (1992). see L. M. Roth, p. 489.

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

87

primary interest. The states close in energy are treated explicitly and those far away in energy are treated perturbatively where ( h i . $/m)is the perturbation. If the states treated perturbatively are included through second order, this approach gives an eigenvalue equation of the form

(2.12) where

(2.13) and the matrix elements are $b& = ( u b l $ l U b ’ ) , the zone center states that are treated explicitly are labeled by the set (b), and those treated perturbatively are labeled by the set {d),and ?i = (&b(O)+ & b l ( 0 ) ) / 2 . This approach gives an effective Hamiltonian to describe states close in energy to a band extremum. Sometimes it is necessary to go to higher than second order to include processes of interest. This can be achieved by employing the usual methods of degenerate perturbation theory. The four-fold degenerate zone center states in Eq. (2.4) describe the top of the valence band for most diamond and zinc-blende structure semiconductors. The two-fold degenerate zone center states in Eq. (2.5) are split to lower energy by the spin-orbit interaction. The two-fold degenerate zone center states in Eq. (2.6) describe the bottom of the conduction band in most direct bandgap zinc-blende structure semiconductors. The zone center conduction band minimum staJes are spin eigenstates whereas the zone center valence band states are not.+For k away from the zone center, the zone center states mix as described by the k . $ perturbation theory.99 Various schemes, corresponding to the choice of which states to be explicitly included in the set (b} and which to be included in the set ( d ) and treated perturbatively, can be used. One frequently used approach is to explicitly include the eight states in Eqs. (2.4), (2.5), and (2.6) for the conduction and valence bands and to treat higher- and lower-lying states perturbatively. A second common scheme is to treat the two states of Eq. (2.6) explicitly for the conduction band and the six states of Eqs. (2.4) and (2.5) for the valence bands. In either case the eight states in Eqs. (2.4), (2.5). and (2.6) become mixed as is moved away from the zone center. An important consequence of this mixing is that the conduction band states are no longer spin eigenstates as is moved away from the zone center, but rather become an admixture of both spin types in the Hamiltonian eigenstates. The group IV semiconductors have an inversion symmetry operator whereas the group 111-V and II-VI semiconductors do not. This difference in symmetry has

88

EX.BRONOLD, A. SAXENA,AND D.L. SMITH

important consequences in the electron spin dynamics of these materials. The time reversal symmetry operator changes the sign of the wave vector and also reverses spin. For a system with time reversal symmetry, that is nonmagnetic materials in the absence of an applied magnetic field, application of the time reversal operator to an eigenstate of the Hamiltonian generates a second eigenstate of the Hamiltonian with the same energy eigenvalue, all spins reversed, and the sign of the wave vector where b is a band label that reversed. If we label the original eigenstate by (b, includes the spin labels of the components of the wave function ($e state may -k), where -b not be a spin eigenstate), and label the transformed state by (4, implies that the signs of all spin components have been reversed, the Hamiltonian eigenvalues satisfy

z),

+

-I

& b ( k )= & 4 ( - k ) .

(2.14)

The inversion symmetry operator changes the sign of the wave vector but does not influencespin. For a system with inversion symmetry, the Hamiltonianeigenvalues satisfy

-

-

&b(k) = &b(-k).

(2.15)

For systems with both time reversal and inversion symmetry, sequential application of the two operators implies

-

-0

&b(k) = &-b(k).

z)

z)

(2.16)

In this case because (b, and (4, label two distinct states with the same wave vector, all bands are at least two-fold degenerate. Such a degeneracy occurs in the group IV but not in the group In-V or 11-VI semiconductors. Along special symmetry directions the bands may be two-fold degenerate in the 111-V or 11-VI semiconductors, for-example for along the [ 1001 or [ 1113 directions, but for a general direction in k space they are nondegenerate.Figure 3 shows a schematic of the conduction band near the zone center for a direct gap semiconductor with and without inversion symmetry. For the case with inversion symmetry, the bands are two-fold degenerate and the minimum energy is at the zone center. For the case without inversion symmetry, the bands are nondegenerateand the energy minimum is shifted slightly away from the zone center. The magnitude of the energy splitting for the case without inversion symmetry is typically very small. For a Hamiltonian with inversion symmetry, the conduction band states remain two-fold degenerate as is moved away fro? the zone center. At the zone center, the eigenstates are given by Eq.(2.6) and as k is moved away from the zone center some amplitude for the bonding p-levels (Eqs. (2.4) and (2.5)) is mixed to form the periodic piece of the Hamiltonian eigenstates. Because the bands are two-fold degenerate, any linear combination of the two states is also a Hamiltonian eigenstate. In particular,it is possible to take the linear combinations so that the amplitude

z

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS E

89

E

RG.3. Schematic of the zone center conduction band minimum for a diamond structure material with inversion symmetry (left) and a zinc-blende structurematerial without inversion symmetry (right).

for 1s 4)vanishes for one of the eigenstates and the amplitude for 1s t) vanishes in the other. Near the zone center, the amplitude for the antibonding s-levels is much larger than for the bonding p-levels in the eigenstates corresponding to the conduction band. Therefore these Hamiltonian eigenstates are approximately, but not exactly, spin eigenstates. For a Hamiltonian without inversion symmetry, the conduction band states are not degenerate. The lack of inversion symmetry that results in this splitting manifests itself in the crystal potential V ( 3 ) .For the asymmetric case, it is convenientto split the crystal potential into an inversion symmetric and an antisymmetric piece (2.17)

The symmetric Hamiltonian, in which V,(3) is used for the potential, has twofold degenerate bands and this degeneracy is split when V,(;) is included. For most cases of interest the energy splitting of the bands by V,(3) is very small. However, in general the two split states contain large amplitudes of both 1s t) and 1s 4) together with much smaller amplitudes for the va$ous bonding p-levels in Eqs. (2.4) and (2.5). Thus for a general direction of k away from the zone center, the two band states are not even approximately spin eigenstates along the quantization axis. Because the spatial part of the two large amplitude components of the wave function, i.e. 1s f ) and 1s $), is the same, it is possible to choose a new spin quantization axis for which the two band states will be approximately spin eigenstates (i.e., neglecting the small amplitude bo;ding p-levels). However, the symmetry of the splitting depends on direction in k-space and therefore the quantization axis along which the t y o band states are approximate spin eigenstates is different for every direction in k-space. It is sometimes convenient to use the symmetric Hamiltonian,in which V, (3) is used for the crystal potential, to provide a basis in which the bands are two-fold degenerate and the bands states are chosen as

90

EX.BRONOLD, A. SAXENA, AND D.L. SMITH &

FIG.4. Schematic of the zone center valence band maximum showing heavy-hole, light-hole, and split-off hole bands.

approximate spin eigenstates along a convenient quantization axis, and to include the inversion asymmetric potential V, )(; as a perturbation. For most diamond and zinc-blende structure semiconductors the four-fold degenerate zone center states in Eq.(2.4) describe the top of the valence band and the two-fold degenerate zone center states in Eq. (2.5) are split to lower energy by the spin-orbit interaction and form the split-off valence band. For a Hamiltonian with inversion symmetry, the four-fold de_generatezone center states split into two sets of two-fold degenerateband states ask is moved away from the zone center. The way in which the four states split into the two two-fold degenerate bands depends on the direction of i .For along the [Ool] quantization axis, the 13/2; f 3 / 2 ) states split away from the 13/2; f 1 / 2 ) states. The 13/2; f 3 / 2 ) states are at higher energy and form the heavy-hole bands while the (3/2; f 1 / 2 ) states form the lower energy light-hole bands. Figure 4 shows a schematic representation of the valence band near the zone center. The heavy-hole band is at higher energy and the light-hole band at lower energy. The split-off hole band is below the light-hole band. For other i directions, the splitting combination of the four zone center states to form heavy- and light-hole bands can be found by applying a spin 3/2 rotation operator to the set of 13/2; f 3 / 2 ) and 13/2; f 1 / 2 ) states fork along the [Ool] quantization axis. The new heavy- and light-hole states will have the combination (3/2; f 3 / 2 ) and 13/2; f 1 / 2 ) , respeztively, but with all labels (including the spin quantization) referenced to the k direction. The splitting combination ,can also be found by constructing and diagonalizing the i 3 Hamiltonian Hb,b’(k) in+Eq. (2.13). It is important to notice that the splitting occurs for afbitrarily small k and that the splitting combination depends on the direction of k. The conduction band levels I1/2; f1/2) also become mixed into the valence band states as i moves away from the zone center. For a Hamiltonian without inversion symmetry, the heavy- and light-hole bands are each further split and there is no band degeneracy for a general direction in i-space. These effects are not as important for the spin dynamics of holes as they are for electrons because the hole states, even for inversion symmetric structures,are not even approximatelyspin eigenstates near the zone center whereas +

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

91

electron states for inversion symmetric structures are approximately spin eigenstates near the zone center. Hole spin dynamics are governed by large zeroth-order effects for which small higher-order effects such as the inversion asymmetry band splitting can usually be neglected. But for electron spin dynamics the analogous zeroth-order effects vanish and the small higher-order effects dominate. In bulk 111-V and 11-VI semiconductors, inversion symmetry is broken because the two atoms in the unit cell are different. In heterostructures formed using diamond or zinc-blende structure semiconductors, inversion symmetry can also be broken by the nature of the heterostructure. This breaking of inversion symmetry by an asymmetric heterostructure also leads to a band splitting. The first contribution is called bulk inversion asymmetry (BIA) and the second contribution structure inversion asymmetry (SIA).lm,lolSIA can be as important as BIA in III-Vand 11-VI heterostructures.

SCATTERING 3. SPIN-FLIP Most scattering processes in nonmagnetic semiconductors do not directly couple to the spin degree of freedom of an electron or hole; that is, the interaction Hamiltonian that describes the scattering process does not contain a spin operator. If the Hamiltonian eigenstates of the system were also spin eigenstates, such scattering events would not affect the spin state. However, neither the electron nor the hole states are generally true spin eigenstates. Holes are not even approximately spin eigenstates. It is possible to assign quasi-angular momentum labels (i.e., 13/2; f 3 / 2 ) for the heavy-holes and 13/2; f 1 / 2 ) for the light-holes) to the hole states, but the diFction of the quasi-angular momentum q3antization axis is along the direction of k and thus different for every direction in k-space. Therefore every scattering event that changes the wave vector also changes the quasi-angular momentum for holes. Because momentum scattering events occur at a rapid time scale, hole quasi-angular momentum dephasing also occurs at a rapid time scale. The four-fold degeneracy of the zone center valence band states is responsible for the fact that a quasi-spin variable that can survive a wave vector changing scattering event cannot be defined for holes. In reduced dimensional structures, such as quantum wells, a combination of quantum confinement and strain effects often breaks the four-fold degeneracy of the zone center valence band states."* In these cases it is sometimes possible to define a quasi-spin variable for holes, which can survive a wave vector changing scattering event. Near the zone center electron states are approximate but not exact spin eigenstates. Because they are not exact spin eigenstates, there is a small but non-zero probability of the approximate spin quantum number being changed in a wave loo lo' lo*

E. I. Rashba, Sov. Phys.-Solid State 2,1109 (1960). Yu. A. Bychov and E. I. Rashba, J. Phys. C 17,6093 (1984). D. L. Smith and C. Mailhiot, Rev. Mod. Phys. 62, 173 (1990).

92

EX. BRONOLD, A. SAXENA,AND D.L. SMITH

vector changing scattering event. As a specific example consider electron scattering from a charged impurity. The interaction matrix element used to calculate transition rates is (2.18)

where the impurity is taken to be singly charged and located at the origin of spatial coordinates and EO is the static dielectric constant of the semiconductor. Because the periodic part of the Bloch function Ub,i,both before and after scattering, contains bonding-p orbitals of both spin types, the overlap matrix (Ub,i(Uw,p) does not vanish if the dominant antibonding s-orbital components of Ub,i have different spin before and after scattering. However, the transition matrix element is much larger for scattering processes in which the dominant antibonding s-orbital components of Ub,i have the same spin before and after scattering. In this case the main contribution to the overlap integral is from the overlap of the dominant antibonding s-orbital components. For the spin-flip scattering process, this term does not contribute because of the orthogonality of the corresponding spinors, but the overlap integral is nonzero because of the bonding-p orbitals in the two states. The overlap integral is second order in the amplitude of bonding-p orbitals. Spinconserving scattering is much more probable than spin-flip scattering, but spin-flip scattering does occur. For other scattering mechanisms in which the interaction Hamiltonian does not contain spin operators, such as electron-phonon scattering, the same overlap (Ub,jlUb,,i,)appears in the transition matrix element and essentially the same considerations as for charged impurity scattering apply. This spin decoherence mechanism, resulting from spin-flip scattering when the interaction Hamiltonian does not contain spin operators, is known as the Elliott-Yafet (EY) spin relaxation me~hanism.'~~*~@' It can be the dominant spin relaxation mechanism in certain cases. The spin relaxation rate increases as the wave vector scattering rate increases if spin relaxation is dominated by the EY mechanism. The EY mechanism takes place both in materials with and without inversion symmetry. 4. INVERSION ASYMMETRY For semiconductorswithout inversion_symmetry,the conduction band states are not degenerate for a general direction in k-space. Usually the two split states contain large amplitudes of both spin states although the energy splitting of the bands is generally very small. Often it is convenient to use a symmetric Hamiltonian '03

R. J. Elliott, Phys. Rev. %, 266 (1954).

'04 Y.

Yafet, Solid State Physics, Vol. 14, 4 s . F. Seitz and D. Turnbull, Academic Press, New York

(1963), p.1.

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

93

FIG. 5. Schematic of spin precession due to inversion asymmetry. The axis about which spin precesses depends on the eleceon wave vector.

to provide a basis in which the bands are two-fold degenerate and the band states chosen as approximate spin eigenstates along a fixed quantization axis, and to include the inversion asymmetry as a perturbation. In this approach theF is a piece of the Hamiltonian that mixes the two degenerate band states at a given k point. Thus, if an electron occupies one of the two-fold degenerateband states at a given time, its probability amplitude will precess in time because of the term in the Hamiltonian that mixes the two degenerate band states. This precession corresponds to a spin rotation because the band states are approximate spin eigenstates. The situation is schematically illus$ated in Fig. 5 . The direction of the precession axis depends on the direction in k space of the state. For most cases the rate of spin precession is slow compared to wave vector changing scattering times because the inversion asymmetric contribution to the crystal potential is very small. Therefore, the electron scatters after only a small angular spin precession. Most probably spin is conserved in the wave vector changing scattering event because the overlap integral for spin-flip scattering is much smaller than that for spin-conserving scattering. In the new band state after the scattering event, the electron spin will contjnue to precess. However, the axis about which the spin precesses depends on the k point of the band state and will be different before and after the wave vector changing scattering event. Thus an electron initially in a band state labeled by (approximate) spin and wave vector would spin precess about an axis determined by its wave vector, but after a small angular precession it would scatter,changing its wave vector but probably not its spin, and then precess about a new axis determined by the new wave vector. This process leads to spin relaxation. Because the axis of spin precession changes after scattering, wave vector changing scattering reduces the efficiency of spin relaxation. This process is called the Dyakonov-Perel (DP) mechanism for spin relaxation and it is often the dominant spin relaxation mechanism in zinc-blende structure semiconductors that lack inversion ~ymmetry."~It does not occur in diamond structure semiconductors with inversion symmetry. *05 M.

I. D'yakonov and V. I. Perel, Sov. Phys. Solid Stare 13,3023 (1972).

94

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

In Sec. I11 we will discuss the DP mechanism in detail and consider the possibility of interference between DP and EY processes. Here we present a simple model discussion to illustrate the basic physical features of the DP spin relaxation mechanism for electrons in direct gap zinc-blende semiconductors. We describe the electron spin state using a two-dimensional density matrix p , which we divide into a slowly varying piece and a rapidly fluctuating piece p = Po

+ Sp.

(2.19)

We are interested in the long-term behavior of the slowly varying piece po and wish to integrate out the rapidly fluctuating behavior described by Sp. In this simple model the electron spatial degrees of freedom have been integrated out and only the spin degree of freedom is maintained. A formal procedure for doing this is described in Sec. III. The spin-dependent Hamiltonian is described by a two-dimensional matrix with the form. -

4

V=B.fJ,

(2.20)

6

where 6 are the Pauli matrices and the three components of fluctuate in time independently with the same average amplitude. The rapid fluctuations of the model Hamiltonian are meant to describe the fact that the spin dependence of the Hamiltonian changes as the wave vector is changed by scattering events. The time scale for the fluctuations is the wave vector scattering time, t. Fluctuations in the density matrix are produced by the fluctuations in the Hamiltonian. The equation of motion for the density matrix is dp i (2.21) - = --[V, p]. A dt Because the density matrix is unitary, there are independent equations for (p11p22) and p21. We separate equations for the slowly varying components of p (2.22)

and the rapidly fluctuating components of p (2.24)

Here, the angular brackets indicate a time average large compared to the fluctuation time scale but small compared to the rate at which the slowly varying components of p change.

ELECTRON SPIN DYNAMICS IN SEMICONDU(JT0RS

95

To solve these equations for the slowly varying components of p. we evaluate time averages such as (V12 S p 2 1 ) . We integrate the differential equation for 6p21 from zero to a time that is large compared to the fluctuation time scale but small compared to the rate at which the slowly varying components of p change. We then multiply by Vl2 and time average over this time scale. This procedure gives integrals of the t’

f

J dt‘V12 (t ’) J dt’’ V2 1 (t”) I =

0

0

j dt‘

(2.26)

0

Changing variables in this integral gives

(2.27)

Aty=O

(2.28)

For y > 0, the integral decreases because V&’) and V21(t’ - y ) lose phase coherence. We take the statistical properties of V12(t’)such that

Using this approach to evaluate the time averages then gives (2.30) (2.31) (2.32)

96

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

where (p2)is the square of the rms average fluctuation amplitude of the potential. Substituting into the equations for the slowly varying components gives (2.33)

and (2.34)

Thus both the differencein diagonal components of the density matrix, i.e., the spin density, and the off-diagonal component of the density matrix, which describes coherent spin rotation, decay exponentially with decay constant 5 _1 -- -8(p2).

T

h2

(2.35)

In this simple model the longitudinal ( T I )and transverse (T2) decay times, which describe decay of the difference in diagonal components and the off-diagonal component of the spin density matrix, respectively, are the same. But this is not a general result and in general TI and T2 are not the same. (In the model, letting the three components of j?j fluctuate with different average amplitudes would give different results for TI and T2). The model results show that the spin relaxation time increases linearly as the wave vector scattering time decreases and quadratically with the ms average fluctuation amplitude of the potential. These conclusions remain in the more complete description of the DP scattering mechanism. Similar effects appear in nuclear magnetic resonance and are called motional narrowing.

5. ELECTRON-HOLEEXCHANGE SCATTERING In some cases, such as following photoexcitation, both electrons and holes are present in the same spatial region of a semiconductor. In this case, electronhole scattering can lead to spin relaxation. In addition to the direct electron-hole scattering term, with transition matrix element (2.36)

(2.37)

there is also an exchange scattering term, with transition matrix element (2.38)

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

97

which contains the overlap (2.39) The direct scattering matrix element behaves in a manner that is generally similar to the charged impurity scattering matrix element. The important overlap integral ( Ue,icI.Ye,,,,) is the same in both cases. The hole overlap integral is of order unity. For spin-conserving scattering the electron overlap integral is of order unity and for spin-flip scattering it is second order in the amplitude of bonding-p orbitals, a small quantity. This direct scattering matrix element describes an EY process. The exchange scattering term has a different structure. The two overlap integrals (U e , tIUhr,ih,) and ( uh,ih IUe,,it,) both depend on the amplitude of bonding-p orbitals in the conduction band states or antibonding s-orbitals in the valence band states. Each overlap integral is first order in these small quantities so that the overall matrix element is second order. The spin-conserving and spin-flip exchange matrix elements are of the same order. Spin-flip electron hole scattering is called the Bir-Aronov-Pikus (BAP)spin relaxation rnechanism.lM It is present for materials both with and without inversion symmetry and can be important for cases where electrons and holes are both present.

6. HYPERFINE INTERACTION The EY, DP, and BAP spin relaxation processes all fundamentally result from the spin-orbit interaction, which couples the electron spin to its orbital motion and whose strength scales with atomic weight. Electron spin can also be relaxed by the hyperfine interaction between electron and nuclear spins. This is a direct coupling to electron spin, which does not rely on the spin-orbit interaction. In the heavier 111-V and 11-VIsemiconductors,spin relaxation effects resulting from the spin-orbit interaction are usually dominant, but direct electron-nuclear spin coupling can be important in some cases such as reduced dimensional structures, e.g. quantum wells.97 For solids made from lighter elements, hyperfine interaction effects are often dominant. The electron has spin angular momentum and a corresponding magnetic dipole moment. Nuclei can also have a spin angular momentum and corresponding magnetic dipole moment. The spin angular momentum of a nucleus depends on the isotope; for example in Si-28 it is spin zero whereas in Si-29 it is spin 1/2. If a nucleus is not spin zero and therefore has a spin angular momentum and corresponding magnetic dipole moment, there is an interaction between the magnetic dipole moments of the electron and the nucleus. For electron states with a nonzero amplitude at the nuclear site (that is, an s-wave component at the nuclear site), the Bir, A. G. Aronov, and G. E.F'ikus, Sov. Phys.-JETP 42,705(1976)[a. Eksp. Teor: Fiz.69, 1382 (1975)l. '06 G. L.

98

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

interaction Hamiltonian is96 (2.40)

where 7 is the nuclear spin operator, ye is the electronic gyromagnetic ratio, y, is the nuclear gyromagnetic ratio, and R, is the nuclear position. There is a weaker interaction for electronic states that have zero amplitude at the nuclear site. 7. SPIN-DEPENDENT OPTICAL PROPERTIES We are often interested in the lowest energy optical transitions between the valence and conduction band states in semiconductors. We specifically consider direct bandgap zinc-blende structure materials. These optical transitions involve photons in the visible or near infrared frequency region. An electromagnetic optical field described by a vector potential 2 (F) couples to electrons in a solid as described by the Hamiltonian of Eq. (2.1). For optical fields in the visible or near infrared frequency region, the optical wavelength is long compared to the length scale over which the electron wave function varies and the position dependence of the vector potential can usually be neglected. After this dipole approximation the vector potential commutes with the electron momentum operator. The important term describing the interaction between an optical field at visible or near infrared frequency and electrons in a solid comes from the cross term after the first term in the Hamiltonian is squared e -

HI = -A mc

. I;.

(2.41)

This term couples to the spatial part of the electron wave function through the electron momentum operator I;. Interband matrix elements of I; can be quite large. For example, coupling between the zone center valence band bonding b-levels and the conduction band antibonding s-levels through I; is alloyed in zeroth order and the matrix elements are large. The term proportional to A .A , after the first term in the Hamiltonian is squared, describes two photon processes such as photon scattering. It is important for higher energy photons, but does not couple to electrons if the photon wavelength is long enough that the position dependence of 2 (?) can be neglected. The term proportional to . ~3couples directly to the electron spin. It is small at visible and near infrared frequencies because = V x 2 and therefore the term is inversely proportional to the photon wavelength. In addition the electron spin operator does not couple the zone center valence band bonding p-levels and the conduction band antibonding s-levels at the zone center. The term proportional to a,which describes the spin-orbit interaction, is small because the spin-orbit interaction is small and because the electron spin operator does not couple the valence and conduction band levels at the zone center.

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

99

Thus for the lowest energy optical transitions between the valence and conduction band states in zinc-blende structure direct energy gap semiconductors, the dominant contribution to the interaction Hamiltonian is given by H I in Eq.(2.41). Optical selection rules can be used both to generate and to probe a nonequilibrium spin distribution in direct energy gap zinc-blende structure semicon{uctors. The polarization of the optical field deterrnirfesthe vector components of A . In particular the vector potential can be written as A = Ah where the unit vector h describes the optical polarization. The case of circular polarization is of particular interest for optical selection rules. For light propagating in the ? direction (i.e., the optical wave vector is in the ? direction) the polarization vectors for right and left circularly polarized light are h+ = (2 - i Q) and 2- = (2 if), respectively. Because of spin-orbit splitting in the valence band, left and right circularly polarized light couples differently to electron spin states quantized along the z-axis (see Fig.6.)(Note, (P,lp,lS) = (PylpylS) = (P,lp,lS)= p,wherepisamomentum matrix element between the zone center Bloch states, but (Pxlpy(S) = 0, etc.)

5

5 +

I(O+) = 1

RG.6. Schematic of spin-dependent optical selection rules for optical transitions involving circularly polarized light near the fundamental absorption edge of a direct bandgap zinc-blende structure semiconductor. For the given polarization, I(a) denotes the relative intensity.

100

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

The relative strength and polarization of the allowed optical transitions at the absorption threshold, for light propagating in the z direction, is labeled and indicated by arrows in Fig. 6. Summing the contributions from the allowed transitions, the degree of electron polarization is equal to one-half of the optical polarization, i.e., for complete optical polarization the spin polarization is one-half. The same optical selection rules apply for the time-reversed process of luminescence, and circularly polarized optical luminescence can be used to detect a non-equilibrium spin distribution.lo7These optical selection rules require that transitions from the split-off hole band are not energetically allowed. If transitions from the split-off hole band are energetically allowed and have the same transition strength as transitions from top of the valence band states so that they are included with equal weight in the sums shown if Fig. 6, there is no electron spin selectivity. In practice, optical transitions must be tuned near the absorption edge to have spin selectivity. The spin-dependent optical selection rules can be used to generate a nonequilibrium electron spin distribution. For example, an n-type doped semiconductor might be photoexcited with a short circularly polarized optical pulse tuned to an optical frequency just above the energy gap. Electrons and holes will be generated by the optical pulse. Because of the optical selection rules, the photogenerated electrons will be preferentially spin polarized (the preferential orientation determined by the chirality of circular polarization) with quantization axis along the optical propagation direction. Depending on the energy of the optical pulse the local density of photoexcited electrons may be larger or smaller than the initial density. The holes will rapidly scatter and lose any information concerning their quasi-spin variable. Electrons and holes recombine on a time scale that is usually short compared to electron spin relaxation time. Thus after electron-hole recombination has been completed, there will be an electron density equal to the original electron density, prior to photoexcitation,but which is preferentially spin polarized with quantization axis along the optical propagation direction. The degree of spin polarization is not easy to directly quantify because some degree of spin relaxation will likely take place during the period of electron-hole recombination. The spin-dependent optical selection rules can also be used to probe a nonequilibrium electron spin distribution. One approach consists of detecting circularly polarized light from the radiative recombination of a spin-polarized electron In this approach the photoludistribution and an unpolarized hole di~tribution."~ minescence must be collected so that the propagation direction of the light is along the suspected spin quantization axis. Because of the optical selectionrules of Fig. 6, in time-reversed form, a spin-polarized electron distribution will produce circularly polarized photoluminescence. Polarization-sensitiveoptical transmission, reflection, and emission, such as Kerr- and Faraday-rotation spectroscopies and lo' See, e.g., Optical Orientation, Modem Pmblem in Condensed Matter Science, eds. F.Meier and B. P.Zachachrenya, North Holland, Amsterdam (1984), Vol. 8, and references therein.

101

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

spin-polarizedphotoluminescence,can also provide a means to probe the dynamics of a non-equilibrium spin population in direct gap semiconductor^.^*'^*^^-^^^^^^^ These measurement techniques are sensitive to magneto-optical anisotropy, arising from non-equilibrium spin distributions, which change the polarization state of incident light on reflection from or transmission through a material (Kerr and Faraday geometries, respectively). Because of the spin-dependent optical selection rules, a non-equilibrium electron spin distribution will lead to a change in refractive index for left- and right-hand circularly polarized light propagating in the direction of the spin quantization axis. As a result the material is birefringent and will thus rotate the polarization axis of linearly polarized light. The sensitivity of these measurements is limited by quantum-mechanicalcounting statistics of the photons in the probe beam, enabling exceedingly sensitive measurements (< 10 microdegrees). A great advantage of Kerr- and Faraday-rotation spectroscopies is that they are readily adapted to ultrafast methods, such as a time-resolved Faraday rotation experiment shown in Fig. 7. Here, one measuresthe pump-induced changes in the polarization state of a linearly polarized, time-delayed probe pulse, revealing the dynamics of electron spins on a femtosecond time scale. Spatial resolution is also possible because the probe beam can be focused.

7

40

m c

Y

-e

20

2

0

m

'0

..............

P

F

2 '0 8 3

-20

= -40 '0

.':

-pump RCP

..

............pump LCP

i'f ,.1 : 1.' 0 Tesla I.

I

I

I

I

0

10

20

30

40

Time (picoseconds) FIG. 7. Induced Faraday rotation of a linearly polarized probe beam following optical excitation with a circularly polarized pump beam. With no magnetic field or with a magnetic field oriented along the propagation direction of the pump beam, the decay of the Faraday rotation signal determines the longitudinal relaxation time TI.With a magnetic field oriented normal to the propagation direction of the pump beam, the Faraday rotation signal oscillates in time because the spin population precesses about the magnetic field. The decay of the oscillating Faraday rotation signal determines the transverse relaxation time T2. Changing the chirality of the circularly polarized light changes the sign of the spin polarization and of the Faraday rotation signal. The inset shows a schematic of the experimental configuration. (Figure courtesy of S. A. Crooker.)

102

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

8. g-FACTORS Often an external vector ;i(3) and/or scalar potential 4(;) that is slowly varying on the interatomic scale is applied to a semiconductor. vpically, these external potentialsare not strong enough to completely disrupt the underlyingband structure of the material and it is convenientto describe states near the conduction or valence band extrema as (2.42)

where u b ( ; ) are the zone center solutions to Eq. (2.7) and fb(;) are slowly varying functions of position that must be determined. A similar approach was discussed previously in the description of Bloch states with no external potential, but away from the zone center. In that case the wave vector was a known quantum number and the functions fb(;) were all proportional to the same plane wave, fb(3) = Cb(k)ei’“. Only the coefficients Cb(k)needed to be determined. But here it is necessary to determine the set of functions fb(;). Substituting the expansion, Eq. (2.42), into the one-electron SchrMinger equation, adding the contribution from the external scalar potential -e#(;) to the Hamiltonian, and taking a projection on ( u b ’ ( ; ) I gives

(2.43)

+ $.

where 6 = The vector potential term in the spin-orbit interaction and the momentum operator acting on f b ( i ) in the spin-orbit interaction have been neglected because these terms are usually very small. We are interested in states that are close to a band edge. As in the description of Bloch states away from the zone center, states are divided into a set close in energy to the band extrema being considered, labeled by { b ]and treated explicitly, and a set further in energy from the band extrema being considered, labeled by (4 and treated perturbatively. If the states treated perturbatively are included through second order,

103

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

Considering specifically the effect of a magnetic field on the zone center conduction band states and taking the set (b} to include the two antibonding s-orbitals coupled to spin, where &,b’ = 0,gives

(2.45) where &b = &b’ = &, is the conduction band minimum and i , j label the Cartesian coordinates. The effective mass Hamiltonian is

where

(2.47) The components of $ do not commute with each other because 2 (F) is a function of position. It is conynient,to split DLi, into symmetric and antisymmetric parts 0::) where D:;? = (Dtb, D&) and in the components of P ,Orb,= 0;:’ = (Dyb, - DLk). The effective mass Hamiltonian can be written as

+

+

(2.48) where the brackets [, ] and (, ] designate commutators and anticommutators, reaA spectively. The _corrqutator [Pi, Pj] = - $) is proportional to a component of V x A = B . Then the effective mass Harmltonian becomes

(2

(2.49) where i , j , k in the last term are a cyclic permutation of the Cartesian coordinates (i.e., i and j take on different values for each of the terms in the sum on k). This term describes an indirect coupling of the magnetic field to the spin degree of freedom through the orbital motion of the electron because of the spin-orbit interactio?. It can often dominate the direct coupling term, which contains the factor &,’ B . It is possible to produce fairly large g-factors if the spin-orbit interaction is strong,

-

104

EX.BRONOLD, A. SAXENA,AND D.L.SMITH

particularly if the energy gap is ~ m al l . 9Terms ~ through second order have been included in Eq. (2.49).Sometimes it is necessary to include higher-order terms to describe specific physical effects. As a simple example it is instructive to consider a case in which the magnetic field is along the z-axis, the set of states ( b )corresponds to the two states at the conduction band minimum, and the set of states (d) that are summed over consists of the four states at the valence band maximum and the two zone center split-off states. In this case it is necessary to calculate the matrix D;?. Performing the sum over the valence band states gives

(2.50)

where E , and A are the energy gap and the spin-orbit splitting of the valence band, respectively. The indirect coupling of the magnetic field to the spin degree of freedom because of the spin-orbit interaction modifies the electron g-factor to become

(2.51)

The second term has opposite sign to the first. It is often the dominant term, for example in GaAs g* M -0.44, and thus has the opposite sign to that of a free electron.95In general, additional bands should be included in the sum over states for quantitatively reliable results. Equation (2.51) is quantitative only for very small gap semiconductors.

111. Kinetic Theory of Spin Dynamics Electron or hole spin polarization in nonmagnetic semiconductors is a nonequilibrium state resulting from a response to an external probe, such as absorption of circularly polarized light or electrical injection from a spin polarized contact, which relaxes to an unpolarized equilibrium state after the external probe is removed. The main relaxation processes in nonmagnetic semiconductors have been identified in the seminal papers by Elliott,'03 Yafet,lMDyakonov and Perel,"' and Bir-Aronov-Pikus,'" and valuable insight into the spin dynamics has been obtained in the 1970s and 80s by analyzing various aspects of magneto-transport,

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

105

magneto-optics, and optical orientation experiments.'08-116Starting in the 1990s, improved experimental techniques and proposals to use the electron spin as an operating principle in electronic devices inspired theoretical studies concerned with a quantitative description of the non-equilibrium spin dynamics in b ~ l k " ~ - ' ~ ' and in reduced-dimensionalsemiconductor ~ t r u c t u r e s . ~ * ' ~ *Because ' * ~ - ' ~ ~spin dephasing and relaxation are detrimental to spin-based devices, a common theme of the recent theoretical investigationsis the manipulationof decay channels, by either external fields or heterostructure design, and the identification of the conditions under which the electron spin is particularly robust. To describe relaxation of a non-equilibrium spin distribution to its unpolarized equilibrium state requires a kinetic theory of spin relaxation dynamics. In this section we describe a systematic kinetic theory of spin relaxation, based on non-equilibrium Green function techniques, and apply it in the following section to specific physical situations. Because electron spin lifetimes are usually much E. L. Ivchenko, Sov. Phys. Solid State 15, 1048 (1973). D. Margulis and V1. A. Margulis, Fiz. Tverd. Tela (Lenningrad) 25, 1590 (1983) [Sov. Phys. Solid Stare 25,918 (1983)l. 'lo J.-N. Chazalviel, Phys. Rev. B 11, 1555 (1975). ''I B. P. Zakharchenya, E. L. Ivchenko, A. Ya. Ryskin, and A. V. Varfolomeev, Fiz. Tvenl. Tela (Leningrad)18,230 (1976) [Sov. Phys. Solid State 18, 132 (1976)l. ' I 2 G. Fishman and G. Lampel, Phys. Rev. B 16,820 (1977). ' I 3 P. Boguslawski, Solid Srate Commun. 33,389 (1980). 'I4 A. G. Aronov, G. E. Pikus,and A. U. Titkov, Zh.kksp. Teo,: Fiz. 84,1170 (1983) [Sov. Phys. JETP 57,680 (1983)]. 'I5 A. D. Margulis and V1. A. Margulis, Fiz. Tverd. Tela (Leningrad)28, 1452(1986) [Sov. Phys. Solid Srate 28,817 (1986)l. 'I6 G. E. Pikus and A. N. Titkov in Optical Orientation, Modem Problems in Condensed Marter Science, eds. F. Meier and B.P. Zakharchenya, Elsevier, Amsterdam (1984). 'I7 M. W. Wu and C. 2.Ning, Eu,: Phys. J. B 18,373 (2000); phys. stat. sol. ( b ) 222,523 (2000). 'I8 M. W. Wu, J. Phys. SOC.Jpn. 70,2195 (2001). 'I9 P. H. Song and K.W. Kim, Phys. Rev. B 66,035207 (2002). I2O F. X. Bronold, I. Martin, A. Saxena, and D. L. Smith, Phys. Rev. B 66,233206 (2002). 12' Yu.G. Semenov, Phys. Rev. B 67, 115319 (2003). '22 M. I. D'yakonov and V. Yu. Kachorovskii,Sov. Phys. Semicond. 20, 110 (1986). 123 G. Bastard and R. Ferreira, Surj: Sci. 267,335 (1992). '24 M. W. Wu and H. Metiu, Phys. Rev. E 61,2945 (2000). N. S. Averkiev and L. E. Golub, Phys. Rev. B 60,15582 (1999); N. S. Averkiev, L. E. Golub, and M. Willander, Semiconductors (Fiz. Tekh. Polup. 36.97 (2002)) 36,91 (2002). 126 M. M. Glazov and E. L. Ivchenko, JETP k r r . 75,403405 (2002). 12' J. Kainz, U. Rossler, and R. Winkler, Phys. Rev. B 68,075322 (2003). I*' V. I. Puller, L. G. Mourokh, N. J. M. Horing, and A. Yu. Smimov, Phys. Rev. B 67, 155309 (2003). 129 M. Q. Weng and M. W. Wu, Phys. Star. Sol. (b) 239, 121 (2003). I3O M. Q. Weng and M. W. Wu, J. Phys.: Condens. Matter 15,5563 (2003). 13' M. Q. Weng and M. W. Wu, Phys. Rev. B 66,235109 (2002). 132 F. X. Bronold, A. Saxena, and D. L. Smith, Phys. Rev. E (2004) submitted. lo'

'09 A.

106

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

larger than hole spin lifetimes in semiconductors,we concentrate on spin relaxation of electrons. Important spin relaxation processes in n-type semiconductorsinclude the Elliott-Yafet process,'03,'04 which leads to spin-flip scattering and, in semiconductors without inversion symmetry, the Dyakonov-Perel process,' 0 5 in which spin states precess because of spin off-diagonal Hamiltonian matrix elements resulting from a combination of the spin-orbit interaction and inversion asymmetry. n p i cally, the Dyakonov-Perel mechanism dominates the spin dynamics in n-type III-V semiconductors. An external magnetic field, in many cases required for control and manipulation of electron spin, can also influence electron spin dynamics.108~1w We treat the various spin relaxation processes on an equal footing and calculate longitudinal ( T I )and transverse (T2) spin relaxation times as a function of temperature, electron density, and magnetic field.'20*'32First, we introduce an effective mass Hamiltonian for conduction band electrons. We specifically consider bulk semiconductors and briefly describe necessary modifications for quantum wells when quantum well results are presented in the following section. Next, we derive a semiclassical kinetic equation, in which the spin degree of freedom is treated quantum mechanically and the spatial degrees of freedom are treated classically, starting from a quantum kinetic approach. Collision integrals that appear in the semiclassical kinetic equation are then evaluated. We utilize a diffusion approximation to reduce the semiclassical kinetic equation to a Fokker-Planck equation for the spin polarization. Finally, we calculate the spin relaxation rates using the Fokker-Planck equation. 9. EFFECTIVE MASSHAMILTONIAN FOR n - m E SEMICONDUCTORS

We consider the conduction band electrons in III-V semiconductors in the presence of an external magnetic field. We use the effective mass approach discussed previously, take the set of explicitly treated states, { b } ,to be the two states at the conduction band minimum, and include a large set of states, { d } ,perturbatively. In order to include the effect of bulk inversion asymmetry and wave vector dependence of the electron g-factor, the perturbation calculation is carried through fourth order. This approach gives an effective mass Hamiltonian of the form'333'34 (3.1)

z

where j? = + ( e / h c ) z(F) (1(3) is the vector potential). €(I?) describes the dispersion of a Kramers degenerate conduction band. The second order contributions contained in the first term on the right-hand side of Eq. (2.49) give parabolic N. R. Ogg, Pmc. Phys. SOC.London 89,431 (1966). V.G. Golubev, V. I. Ivanov-Omskii,I. G. Minervin, A. V. Osutin, and D. Polyakov, a. Eksp. Teo,: Fiz.88,2052 (1985) [Sov.Phys. JETP61, 1214 (198591. '33

'34

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

107

dispersion and fourth order terms not inciuded in Eq. (2.49) contain nonparabolHere A ~ z = , psg*B is the Larmor frequency for electrons icity effects in ~(2). at the zone center. The term that contains this factor comes from the third and fourth terms on the right-hand side of Eq. (2.4_9)which appear in the second order perturbation calculation. A 6 , (t) ~ = 260&4(K)describes the splitting of the conduction band degeneracy due to the inversion asymmetric piece of the crystal potential. It comes from third order terms that were not included in Eq. (2.49). This term would vanish for a material with inversion symmetry because matrix elements that involve the momentum operator (which is inversion odd) appearing an odd number of times vanish if the states have inversion symmetry. The term

hG2,(k)= 2 a 4 K 2 i

i ~2) + 2a6?(2,2)

-I-hs(2,

(3.2)

describes the wave vector dependence of the conduction electron g-factor away from the zone center. It comes from fourth order terms not included in Eq.(2.49). This term can lead to spin relaxation because electrons with different wave vectors precess in a magnetic field at different rates and therefore lose spin coherence.'0g-'20 For brevity we refer to this magnetic-field-dependen: sein relaxation mechanism as a variable g-factor (VG) mechanism. Z u ( K ) , ?(K,B) and the parameters SO, a 4 , u5, a6 are explicitly given in terms of momentum matrix elements between the zone center states.133,134 Collision terms arising from electron-impurity,electron-electron,and electronphonon scattering H, = Hei He, Hep are added to the effective mass Hamiltonian. The electron-impurity scattering term has the form

+

+

where the scattering matrix element h'fbb! (Z, element contains the overlap factor Ibb'(Z9

2) is given in Q. (2.18).

i')= (ub,ilubf,G)9

his matrix

(3.4)

which is of order unity if b=b' and is small, but not zero otherwise. The electron-phonon scattering matrix element has a similar structure to Eq.(3.3) but with p_hofon creation and annihilation operators appearing in the matrix element h'fbb'(k, k ' ) . The electron-phonon scattering matrix element has the same overlap factor ( Ub,iI Uw,i,),which determines the ratio of spin conserving to spin-flip scattering contributions,as in the electron impurity scattering matrix. The electronelectron scattering term has the form

108

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

6,

The matrix element h ! f b l b 2 b 3 b 4 ( i 1 , i 3 , &) contains two overlap factors of the form (Ub,;lUb,,p). Bothadirectandanexchange termappearinthe suminEq. (3.5). The direct term is usually dominant. 10. SEMICLASSICAL KINETIC EQUATION FOR THE DENSITY MATRIX

Complete information about spin relaxation is contained in the time dependence of the electron density matrix +

+

N b i b z ( k l 9 k29

t

= (CGb, ( f ) C k ; b 2 ( f ) ) ’

(3.6)

where the brackets imply a statistical average. The diagonal part of N b l b 2(k; , k ; , t ) gives the distribution functions, and the off-diagonal elements describe quantum mechanical coherence between states. Because scattering events occur on a rapid time scale, phase coherence between states with different wave vect2rs @ rapidly lost. Thus we are primarily interested in the components of N b l h ( k l r k 2 , t ) that are diagonal in but not necessarily diagonal in b. This is the physical basis of the semiclassical approximation. To derive a kinetic equation for the density matrix it is convenient to start with Keldysh Green function^.'^^^^^^ In the first step we reduce the kinetic equation for the Green function to a kinetic equation for the density matrix. This equation is not closed, however, and to obtain a closed equation for the density matrix we adopt a semiclassical approximation which requires treating momentum scattering processes as instantaneous on the time scale of spin relaxation. Calculating the self-energiesthat appear in the semiclassical kinetic equation in the Born approximation, linearizing with respect to spin polarization (assuming small polarization), and expandingthe self-energiesup to second order in the momentum transfer yields a Fokker-Planck equation for the spin density. Introducing a numerical index 1 that stands for bltl and 2 for i 2 b 2 r 2 , we write the Keldysh Green functions in the notation of Ref. [ 1371

z

where T and i’ are time ordering and anti-time ordering operators, respectively. Each component of the Keldysh Green function is a 2 x 2 matrix in spin space. The four components of the Keldysh Green function are not independent, but rather satisfy G++ G-- = G+- G-+. The density matrix can be obtained from any

+

135 13’

+

L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965). L. D. Landau and E. M. Lifshitz, Physicul Kinetics, Pergamon Press, New York (1981). A. V. Kutznetsov, Phys. Rev. B 44,8721 (1991).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

109

one of the components of the Keldysh Green function at coincident times t + t'. Although the Keldysh Green function is a more complex object than the density matrix, it is often convenient to use this approach to obtain kinetic equations for the density matrix because one can write a formally exact Dyson equation for the Keldysh Green function which involves a self-energy that can be calculated from well-defined diagrammatic rules.136 Introducing a self-energy

gives Dyson equations in differential form, which, introducing relative and center times t = tl - q and T = (tl t2)/2, have the form137

+

Here, the brackets [, ] and {, } denote commutator and anticommutator, respec) 8b1bZSE1E2,a, is a Pauli matrix, tively, GI2 = G(ltl,2t2) = G(12; ~ t )~,( 1 2 = and the energy 2(12) = S ( 1 2 )(~1). We adopt a convention that numerical indices written as a subscript include the time variable, whereas numerical indices written as an argument do not. Matrix multiplication with respect to the Keldysh indices is implied and the internal momentum, spin, and time variables are summed (integrated) over. The first equation describes the time evolution on the macroscopic, kinetic timescale. In particular, relaxation processes are contained in this equation. The second equation, by contrast, describes the time evolution on the microscopic, spectral timescale. Because we are primarily interested in relaxation processes that occur on the kinetic timescale, we focus on the first equation to derive a kinetic equation for the density matrix. To obtain an equation for the time evolution of the density matrix from the Keldysh Green functions, we use the relation ha~N(12; T) =

lim G"(12; Tt).

r+O+

(3.10)

Separating the self-energy into a singular and a regular part'37 212

= A(12; t l ) S ( t l - t2)

+ 512,

(3.1 1)

gives e (hat - fic(V~ ~ ( xi 2) ) . V;

N ( i t ) = i [ N ( i t ) ,A++(;t)]

+ lim C ( i ,t , t'), t+t'

(3.12)

110

EX.BRONOLD,A. SAXENA,AND D.L. SMITH

where we set the center time T + t. The term on the left-hand side that contains the magnetic field comes from the term [Z, &;I12 in Eq. (3.9) and describes the cyclotron motion of electrons in the magnetic field. The first term on the righthand side comes from the singular part of the self-energy and describes the coherent motion of the density matrix. The terms in the Hamiltonian that contribute to this singular part of the self-energy include the inversion asymmetric term described by &A(K), terms describing the interactio? of an electron spin with an external magnetic field described by 6~ and fi2,(K),and mean field terms such as the Hartree-Fock contribution due to electron4ectron scattering. Dissipation and relaxation are contained in the second term, which contains scattering processes. Explicitly it reads d t ” [ G + + ( i ,t , t”)e++(i, t“, t’)

f+f‘

+ G+$,

t , t”)e-+(i,t ” , t’)

(3.13)

- Z++(i, t , f”)G++(i,t ” , t’) - e+-
(3.14)

Writing two-time functions in terms of relative and center time variables, e.g., A ( t , t’) = A(r - t’, ( t t ’ ) / 2 ) = A ( r , T ) and defining a Fourier transform with respect to the relative time,

+

(3.15) we find for the equal time limit O0

f+f‘

dw

t ) A ( w , t ) B ( w ,t ) ,

(3.16)

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

111

where we introduced the gradient operator,’38 (3.17) and relabeled the center time T + t. The second term on the right-hand side of Eq. (3.13) becomes

+ G + - ( i , o,t)e-+(i,o,t ) ) - rEG(o, t)(e++(i, w , t) G + + ( z,o,t ) -

e+-(i,0 ,t ) G - + ( i , w , r ) ) ] .

(3.18)

This is an exact result but the right-hand side still contains the Green functions and the equation for time evolution of the density matrix is not yet closed. The semiclassical approximation uses the fact that the Green functions and selfenergies vary slowly on the macroscopic center timescale. As a consequence, the gradient operator r + 1. Assuming weak interactions, the full Green functions in Fq. (3.18) can be replaced by the non-interacting Green functions

w k w , t ) + G ; G , 41No(;)-*N(;,,)’

(3.19)

where the non-interacting density matrix is replaced by the full density matrix N ( i , t). This procedure yields a closed form semiclassical kinetic equation for the time evolution of the electron density matrix

For a magnetic field in the z direction &(Vi ~ ( xi i) ) . Vi = -in=& where J!., is the z-component of the angular momentum operator in k-space and Qc = eAB/m*c is the cyclotron frequency. The first term on the right-hand side of Q. (3.20) describes the coherent motion in a self-consistentfield. The second and third term, respectively, correspond to the scattering-out and the scattering-in terms in a matrix-Boltzmann equation. The matrix structure is aconsequence of the quantum H. Haug and A. P.Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, Berlin (1996).

112

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

mechanical description of the spin. Only momentum scattering has been treated classically. 11. CALCULATION OF THE COLLISION INTEGRALS The general semiclassical approach to determine the self-energies in Eq. (3.20) is to represent interaction processes in terms of diagrams,_calculatethe diagrams using standard to obtain e p q ( Z , f, t') and APq(k, t ) , and then replace all internal Green functions by the non-injeracting Green function with the noninteracting density matrix replaced by N ( k t ) . This heuristic strategy is equivalent to the Kadanoff-Baym Ansatz for the spectral function and can be put on a rigorous footing.137 Figure 8 shows the self-energy in the Born approximation for electron-electron (a-d), electron-impurity (e), and electron-phonon (f) interaction. The HartreeFock diagrams (a and b) contribute to the instantaneous self-energy A++. They are second order in the spin polarization and, therefore, for sufficiently small spin polarizations, negligible. There are two second order diagrams due to electronelectron scattering, the direct (c) and the exchange (d) Born diagrams. Because soft scattering processes dominate, we neglect the exchange diagram (d). The direct Born diagram contributes to and %-+. However, is second order in the spin polarization and therefore negligible in the limit of small spin polarizations. The and components contributing to the collision integral are in contrast linear in the spin polarization and cannot be neglected. Diagrams (e) and (f), corresponding to the Born approximation for electronimpurity and electron-phonon scattering, contribute to % P q . As in the case of electron-electron scattering, the term can be again neglected, if the spin poand components contribute in linear order larization is small, whereas the in the spin polarization to the collision integral. In all scattering self-energies we neglect the effect of the magnetic field. Writing in the Born approximation IS"] = ZF[N] Zip[N] Zf[N]for the second and third terms on the righthand side of Eq. (3.20) and calculating the torque force contributions to A++ in leading order in the magnetic field, the semiclassical kinetic equation for N ( h ) reduces to

e++,e+-, e+-

e-+

e++

++ +- -+

+

+

(Aa, - i s 2 , i Z > N ( i t )

where IS"] is the Born form for the collision integral. guation (3.21) is a Boltzmann-type equation for the 2 x 2 density matrix N ( k , t). Equations of this type can be solved n u m e r i ~ a l l y . " ~A ~ ' ~ ~ ~ ~ strictly numerical solution does not, however, provide insight into the interplay of the various scattering processes. We focus therefore on small spin polarizations,

113

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

P

L

2

....

. ..,: . ... .... ....... ;.'. ... ,:;.. ..... ...

;

:....:

......

FIG. 8.

:...':

......

Diagrams used to calculate the self-energy in the scattering integral.

where further insight can be gained by linearizing the Born collision intzgral with respect to the spin polarization. The equilibrium density matrix N,,(k) = N ( i , t + 00) is not diagonal in the spin basis because of the inversion asymmetric crystal potential. The basis states are not zero field Hamiltonian eigenstates when $ , ~ (.i ) is included in the Hamiltonian. Expanding the equilibrium density matrix in terms of Pauli matrices yields &I,

(3.22)

fd)

where = TrNeq(i) = f+(i)+f-(i) is thesumoftheequilibriumdistribution functions of the spin-up and spin-down electrons and &(i)is the equilibrium spin

114

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

polarization. Accordingly, we also write for the density matrix at arbitrary times (3.23) with SS(z, t) denoting the non-equilibrium spin polarization which, by d_efi!ition, vanishes for r + 00. Assuming the total spin polarization SS(k, r ) to and be small, we linearize the Born collision integral with respect to both S,,(z)

$,,(z) +

SS(Z,

t)

ZB[Nl = I B [ N e q l

+IB[ft

(3.24)

We apply TrG[. . .] on both sides of Eq. (3.21), use Z B [ N , ~= ] 0 and d~,(z) x S,,(z) = 0, because the equilibrium density matrix Ne4(z) commutes with HIA, and arrive at a kinetic equation for the non-equilibrium spin polarization [ha, - i Q c i z ]SS
+ d,

+ d,(z)) x S&z,

t)

+ J B [ ~AS], ,

(3.25)

where JB[f,SS] = TrG ZB [f,SS]. The collision integral can be simplified by splitting the scattering matrix into a leading spin-conserving diagonal part and a small off-diagonal part that describes spin-flip scattering. The dhgonal part of the overlap m $ i x (3.4) is nearly equal to the unit matrix and Z(z, k') 2: 1 S Z ( i A c ) , with S l ( k , 2') << 1. Expanding the , =JB[~, collision integrals to second order in SZ(k, k') gives J B [ ~SSI JB[f,6?](') JB[f,6S](2).For semiconductors with zinc-blende symmetry, the first order term J B [ ~SS](') , does not contribute to spin relaxation. The kinetic equation for the non-equilibrium spin polarization becomes

+

+

+

+ dlA + dg(Z)> x s S < ~t ,) + J F ' [ f , SS] + Jf"f, 83.1.

[ha, - i n c i z IS?(;, t ) = A&

(3.26)

The Elliott-Yafet process is encoded in Jf'[f, 851, whereas the Dyakonov-Perel process and the variable g-factor process result from the _combinedaction of the spin-conservingscattering processes comprising J t ' [ f , SS] and the torque forces due to and 6, respectively. Independent of the scattering process, the structure of the collision integrals in Eq. (3.26) is ( u = ei, ee, and ep)

(z)

(z),

+

JLO'[f, SS] = C [ W " ( i ; ij)SS(Z

+ ;,

t ) - W " ( i ;Cj)SS(Z, t ) ] ,

(3.27)

4

J 4 " f ; SS] =

(3.28)

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

115

where, for concise notation, we introduced a spin-flip vector

(3.29) with I+-$, it)the off-diagonal element of the overlap matrix (3.4). This is a result of the Born approximation and the linearization with respect to the spin polarization. In general, the structure of the collision integrals depends on the scattering process. Here, however, the scattering process enters only through W"(&G), the probabilities for a transition between momentum state i 4 and Z. For electronimpurity scattering, for instance,

+

W " ( i ; i j ) = 2nlvilu(q)12s(&(i- i j ) - E ( i ) ) ,

(3.30)

while for electron-electron scattering,

w y i ;i j ) = 4rrlV(q)12 C { [ 1 - f ( i - i j ) - f(2+ij)]f(2) It

+ f ( i- ij)f(i'+ i j ) ) 8 ( & ( i ) +&(it) - &(i - i j ) - &(it +ij)),

(3.31)

with U (q) and V (q) statically screened Coulomb potentials. Similar expressions hold for electron-phonon scattering. For electron-impurity scattering, which is elastic, W e i ( i 4; 4) = W e i ( i ;G); moreover Wei(k;i ) is independent of the equilibrium distribution of the spin-up and the spin-down electrons. In general, however, the transition probabilities depend on the equilibrium distribution of the electrons, and, in the case of electron-phonon scattering, also on the equilibrium distribution of the phonons.

+

EQUATION FOR THE SPIN DENSITY 12. FOKKER-RANCK The simple form of the collision integrals (Eqs. (3.27) and (3.28)) suggests we conceptualize the dynamics of the non-equilibrium spin polarization in terms of a test spin polarization, scattering off a virtual bath of impurities, field electrons, and phonons. Usually this picture can only be applied to electron-impurity and electron-phonon scattering, where the scattering partners belong to different species, and not to electron4ectron scattering, where the scattering partners belong to the same species. It is only within the linearized spin dynamics, which essentially treats the non-equilibrium spin polarization as a separate species, that the test-bath analogy can be used. We now take full advantage of the simplicity of the collision integrals and expand the collision integrals with respect to the momentum transfer 4. As a result the kinetic equation for the non-equilibrium spin polarization (Eq. 3.26) becomes a differential equation.

116

(e

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

Spin-conserving processes due to elastic scattering yield terms of the form = ei)

whereas the inelastic spin-conserving processes give rise to terms of the form (i = ep, ee)

where

(3.34)

4)

are the moments-of the trFsition probability W”(i; for a transition between and k induced by the scattering process u. Nofe, for inelastic momentum state k scattering the differential operators act on the moments C:,,,,,in( k ) whereas for elastic scattering the moments are in front of the differential operators. Equations (3.32) and (3.33) are exact representations of the collision integrals. In many cases small momentum transfer dominates, however, and it is sufficient to keep terms in the expansion only up to second order (diffusion appro~imation),’~~

+4

= D ( z ) S S ( i ,t ) ,

(3.35)

where the first two terms on the right-hand side come from elastic scattering events and the last two terms describe inelastic scattering processes. In Eq. (3.35) we introduced for the first and second moment

(3.36) 1 =-

C qiqj wyi; ij),

2 a 139 Ref.

[136], Chap. II.

(3.37)

117

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

which have the meaning of i-dependent dynamical friction and diffusion coefficients, respectively.Within the diffusion approximationthe spin-consewing (Born) collision integrals are therefore represented by a Fokker-Planck differential operator and, in analogy with the theory of stochastic processes, an intuitive physical picture for the spin-conserving Faftering processes emerges: They cause the nonequilibrium spin polarization SS(k, t) to perform a random walk in momentum space, Yhich, for the various scattering processes u = ei, ep, and ee,is charact2rized by k-dependentdynamical friction and diffusioncoefficientsA; (k) and B&(k). Expanding the spin-flip collision integral .If) up to second order in the mo+ - + mentum transfer 4, and using g ( k , k) = 0, gives

(3.38) with a spin-flip tensor

r(i)= 4 C ~ ~ ~ ( i ) ~ ~ ~ ( (3.39) i ) ij

+

given in t e y s of the total diffusion coefficient Bij (i)= BG (i) (i).,qd a tensor Gi,(k),which describes the rate of change of the spin-flip vector g ( k , k’):

(3.40)

with

(3.41) which makes the quadratic dependence of the spin-flip tensor on the off-diagonal elements of the overlap matrix, which describe the admixture of the bonding pstates into the conduction electron states (cf. discussion in Section 1.3), particularly clear. Using symmetry-adapted coordinates, a radial variable E = k2 and a generalized angle variable w, which, for bulk semiconductors, comprises two angles, the polar angle 8 and the azimuth angle 4, and for quantum wells w is simply the polar angle 4, the differential operator describing the spin-conserving scattering processes becomes

(3.42)

118

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

where the operator i 2 ( w )denotes the square of the angular momentum operator in wave vector space. The off-shell term b(6)is characterized by the relaxation rates l / t f ( ~and ) l / t d ( ~ )which , denote the rate with which the non-equilibrium spin polarization loses energy and the rate with which it diffuses in momentum space, respectively. Only inelastic scattering processes contribute to the rates l / t f ( 6 ) and l / t d ( ~ ) For . elastic scattering processes = 0 and only the on-shell term, describing randomization of the direction of k, remains. It is given by the second term on the right-hand side of Eq. (3.42) and is proportional to the on-shell relaxationrate ~ / T L ( E ) = l/t~(~)+l/t~(6),duetobothelasticandinelasticscattering processes. The rates characterizing the differential operator b(i)are obtained, for the various scattering processes, from a direct calculation of the coefficients A:$) and BG(i) and the resulting differential operator b(i)is cast in the specific form given in Eq. (3.42). An explicit calculation of the symmetry-adapted form of the relax-ation tensor r, which is defined in terms of the total diffusion coefficient Bi, (k), due to both elastic and inelastic scattering processes, shows moreover that it can be expressed in terms of the same scattering rates. Thus, the three scattering , l / t l ( ~completely ) specify the two collision integrals rates l / t f ( ~ )l ,/ r d ( ~ )and in the kinetic equation for the non-equilibrium spin polarization (Eq.3.26). The symmetry-adapted Fokker-Planck equation for the non-equilibrium spin polarization ~ S ( Ew,, t) becomes, in atomic units (with a magnetic field applied along the z-axis),

e(~)

This equation is the basis for the calculation of the spin relaxation rates presented in the next section. It contains Larmor precession and orbital motion of the electrons, accounts for inelasticity within the diffusion approximation, and treats Elliott-Yafet, Dyakonov-Perel, and variable g-factor spin relaxation processes on an equal footing. The Dyakonov-Perel and variable g-factor processes are motional narrowing-type spin relaxation processes originating from the interplay of spin-conserving scattering processes, encoded in the differential operator b - ( 1 / 4 t l ) L 2 ,and the torque forces induced by h, and h,, respectively. The Elliott-Yafet process, originating from spin-flip scattering, is included through the relaxation tensor I?. 13. CALCULATION OF THE SPIN RELAXATION RATES

The Fokker-Planck equation for the non-equilibrium spin polarization contains the various momentum scattering processes through the three scattering rates 1 / r f ( E ) , l / t d ( ~ )and , l / t ~ ( e )which , are total scattering rates, due to whatever scattering

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

119

processes arejncluded. In this section we derive from Eq. (3.43) a Bloch equation for the total, k-averaged spin polarization. As a result, we obtain general expressions for the spin relaxation rates. The calculation is independent of the explicit form of the scattering rates. In the spirit of Ref. [ 1051we employ a perturbative approach that treats all t e y s due to inversion asymmetry as small. We consider the torque forces due to C~,A and d,, respectively, as 0 (1) terms, and the term involving the spin-flip tensor I' as an O(2) term (cf. discussion in sections II.3 and 111.12). Results are valid for 161~ !&It* < 1 and II'lr* < 1, where t* denotes a representative value for one of the scattering times of the previous section and I.. I is a vector or tensor norm. We are interested in the final stages of spin relaxation, where the energy relaxation of the non-equilibrium spin polarization is basically completed, but the spin subsystems are not yet balanced. Each spin subsystem is then almost equilibrated and the non-equilibrium spin polarization can be written as

+

(3.44) The total, ;-averaged non-equilibrium spin polarization, divided by the number of electrons initially contributing to the non-equilibrium spin polarization, is then = (G)z, where we used -af/as = pff with f = 1 - f and given by defined a ;-average ( . . . ) i= ( s d z f f . . . ) / ( J d if f ) . The spin polarization of the saf"e1e consists of an equilibrium and a non-equilibrium contribution, and SS(k, r ) , respectively. In expenmen? without k resolution, howeyer, only the \otal+non-equilibrium spin polarization So(t ) can be measured. The k-average of S,,(k) vanishes because of the angle integration. We apply a relaxation time approximation on the scale of the spin relaxation time rs.Toward that end, we write

$.,(z)

and require that, on the timescale t,, ( ~ $ ( l ) )= ~ 0, with (. . . ) w the average over all angles, and a , q ( ' ) ( E , w , t) 2: 0. The first condition imposes that on the timescale rs the energy dependence of the angle averaged non-equilibriu-m spin polarization is approximately given by -af/as. From the definition of the k-average it follows then that $(O) is equal to the total non-equilibrium spin polarization SO.The second constraint imposes, on the timescale ts,quasi-stationarity for the fluctuations around the total non-equilibrium spin polarization. Inserting Eqs. (3.44) and (3.45) into the Fokker-Planck equation for S?, imposing the two constraints, and collecting terms through second order, we obtain a set of two coupled equations that govern the time evolution of the total nonequilibrium spin polarization on the timescale rs (we suppress the arguments of

120

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

the various functions):

The first term on the right-hand side of Eq. (3.46) is the Larmor precession, the ~ process, while the second term describes spin relaxation due to t h Elliott-Yafet third term, which depends on the fluctuations W('),eventually gives rise to spin relaxation due to the Dyakonov-Perel and the variable g-factor process, respectively. As in the simple model discussed in Section 11.4, the motional narrowing type relaxation processes are driven by the fluctuations around an average spin polarization. The system of three difJerential equations (Eq.3.47) for the three components of the fluctuation vector W(') are coupled by the various torque forces. They can be decoupled by expanding + ( I ) in terms of spherical harmonics (bulk) or trigonometric functions (quantum well). We describe here the procedure for bulk in some detail. The calculation for a quantum well is analogous. In bulk, we can introduce for each momentum channel (1, m) a diagonal matrix Rlm( E ) , +which+relat:s the (1, m) component of$(') to the (1, m)component ofthe vector (nIA +a,) x ~ o ( t > . The elements of Rl,,,( E ) satisfy the decoupled differential equation_s. Separating the energy and angle dependencies of the vectors ~ I and A ng,we first write

(3.48) (3.49) where we defined energy-dependent precession rates

(3.50) (3.5 1) with CIA = 260/Roai, C,(B) = 2 p BIRoa;, ~ Ro and a0 the Rydberg energy and the Bohr radius, respectively, and p~ the Bohr magneton. We then expand the angle-dependent parts in terms of spherical harmonics, K;A(~,0) = El,,,C;,,, Ylm(4,0) and K : ( $ , 6 ) = El,,,D;,,,Y,,,,($,e), with i = x , y , z , and rewrite the cross products in Eqs. (3.46) and (3.47), which involve the vectors 61~ and Gg in

121

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

terms of matrices. For example,

(3.52) and

(3.53) where we defined the matrices CI, and DI,, containing the expansion coefficients C!, and Dim,respectively. Similar expressionshold for the cross products involving $(I). An explicit calculation of the matrices CI, and DI, shows that CI, is nonzero only for I = 3, whereas DI, contributes only for 1 = 0,2. For a magnetic field along the z-axis the matrix-form of the Larmor term, for instance in Eq.(3.47), becomes 6~x $(l) = Q L * ( ' ) . With the previous definitions, the solution of Eq. (3.47) can be expressed as '@)(&,

4 , 8 , t ) = GrnUtRlm(E)U[C/rn(4,0)

+ Dlrn(4, e)lSo(t)Yrm(4,e), (3.54)

where U is the unitary matrix that diagonalizes QL, and RI,(E) is a diagonal matrix whose elements, ~ Y , ( E )are , the solution of the differential equation ( u = 1,2, and 3)

+

+

with I/r&) = l(1 1 ) / 4 r ~ ( & QL ) , = QL(6,1 - &2), CI(&)= ( ~ /T I A( & ) ) & ~ (l/rg(~))(&2 610). The physical meaning of the functions rY,(&)is that of a relaxation time. The relaxation times are different for the Dyakonov-Perel process (1 = 3) and the variable g-factor process (I = 0,2). Due to the orbital motion, the relaxation times depend moreover on m . Using Eq. (3.54) in Eq.(3.46) and performing the 2-average in the third term on the right-hand side, we finally obtain a Bloch equation for the total non-equilibrium

+

122

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

spin polarization,

where the total spin relaxation tensor is given by

rt = F E Y + r D p + r v G with

the contributions due to the Elliott-Yafet, the Dyakonov-Perel, and the variable g-factor processes, respectively. In Eqs. (3.58) and (3.59) the brackets denote d s & f ( ~ ) f ( E ) ( . . .)/4n de,Ef(&)f(~). an energy average ((. . .))& = Because of the orthogonalityof the angular dependencesof the three spin relaxation processes, the total spin relaxation tensor obeys a Matthiessen-type rule, i.e., it is the sum of three individualcontributions.This result arises from the high symmetry of the zinc-blende structure materials; in lower symmetry structures cross terms involving the three processes can occur. The total spin relaxation tensor I?, contains off-diagonal terms (r:' and r,Y"), which modify the Larmor frequency (magnetic field along the z-axis), and do not affect the spin lifetimes. The diagonal elements give the longitudinal (1/ TI = r?) and the transverse (1/T2 = r? = r,") spin relaxation rates. From Eqs. (3.57)(3.59) we explicitly obtain (i = 1,2)

Jr

Jr

with the EY contributions 1

2

T,EY- TEY 2 -

32n

(3.60)

the DP contributions

(3.61) (3.62) '2

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

123

and the variable g-factor contributions

(3.63)

where we defined

Sometimes it is sufficient to treat all scattering processes elastically. Within the elastic approximation, the differential operator & E ) is neglected and the generalized momentum relaxation times tk (k) satisfy an algebraic equation that is readily solved to yield

fk =

TI(&)

+

+

[CLE3&3 C,2(B)E2(6r2 &o)]

1

+ [ W c + S2;)tAs)l2

(3.66)

The constants CIAand C, (B), characterizing the Dyakonov-Perel and the variable g-factor process, have been introduced previously. The corresponding constant for the Elliott-Yafet process is C,, = S2(A 2~,)Rorn/2Ac,m*, where Sz = 2A2/(A cg)(2A 3 ~ , ) ,A is the spin-orbit splitting, E , is the bandgap, Ro and a0 are the Rydberg energy and the Bohr radius, respectively, m* and m are the conduction electron mass and the mass of a bare electron, respectively, and /.LB is the Bohr magneton. Note, the Elliott-Yafet spin relaxation rate is given by an integral over a scattering rate and therefore increases with increasing scattering. The Dyakonov-Perel and variable g-factor spin relaxation rates, on the other hand, are proportional to integrals over generalized scattering times and therefore decrease with increasing scattering. For small magnetic fields, the variable g-factor process is negligible and only the Elliott-Yafet process, which is independent of the magnetic field, and the Dyakonov-Perel process contribute. Due to the field dependence of the generalized relaxation times tY, ,the Dyakonov-Perel process is quenched by an external magnetic field.lo8This effect will be discussed in the next section. If the field is ~< 1 and S2cq << 1, and the generalized relaxation times small enough, Q z , < become field independent. As a result, Eqs. (3.61) and (3.62) yield identical, magnetic-field-indpendentrelaxation rates,

+

+

+

(3.67)

124

EX.BRONOLD, A. SAXENA,AND D.L. SMITH

where the generalized momentum relaxation time t ( ~satisfies )

For the moderate fields usually employed in optical orientation and Faraday rotation experiments, the spin lifetimes due to the Dyakonov-Perel process are deter) ~~(E)/~IA(E), minedby Eqs. (3.67)and(3.68). Intheelasticapproximation, t ( ~ = and we recover the well-known resultIo5

(3.69) which strongly resembles the result obtained from the simple model discussed in Section 11.4.The (appropriately averaged) Dyakonov-Perel rate increases with increasing scattering time t 3 , i.e., with decreasing scattering, and is proportional to the square of the fluctuation amplitude (here given by 1/r1.4). Starting with basic kinetic equations for the Keldysh Green function we derived a Fokker-Planck equation (Eq. (3.43)) to describe the time dependence of the non-equilibriumspin polarization. We developed an approach to approximately extract from the Fokker-Planck equationJor the non-equilibrium spin eolarization S s ( i , t) the time evolution of the total, k-averaged spin polarization So(?) on the timescale t 2: ts of spin relaxation. The spin relaxation rates are given in terms of generalized relaxation times, which satisfy the differential equation (3.55). The scattering rates appearing in Eq. (3.55)are obtained from the first and second moments of the transition probabilities for the various momentum scattering processes. Within the elastic approximation the differential equation becomes an algebraic equation and the calculation of the spin relaxation rates is reduced to quadratures, which can be done either numerically or with saddle point techniques exploiting the peaked structure of the integrands. In cases where inelasticity plays an important role, the generalized relaxation times have to be obtained numerically. We only addressed the linear regime, where the spin polarization is small, and the spin decay is exponential. Nonlinear effects due to large spin polarizations can be either investigated on the level of the matrix-Boltzmann equation for the electronic density m a t r i ~ ’ * or ~ ~in’ terms ~ ~ of a “Fokker-Planck-Landau equation” where the friction and diffusion coefficients A: and BG (Eqs. (3.36)and (3.37))as well as the_spin flip tensor I’(Eq.(3.39))d y e n d on the non-equilibrium spin polarization SS. The differential operator b ( k ) in Eq. (3.352 w y l d then be an (implicitly time-dependent) integro-differential operator b ( k , SS). In the following section we apply the semiclassical kinetic theory to specific physical situations.

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

125

IV. Application of the Kinetic Theory

When an n-type doped semiconductor is photoexcited with a short circularly polarized optical pulse tuned to an optical frequency just above the energy gap, electrons and holes are generated. Because of the spin-dependent optical selection rules, the photogenerated electrons will be preferentially spin polarized with quantization axis along the optical propagation direction. The holes will rapidly scatter and lose any information concerning their initial quasi-spin variable. Electrons and holes recombine on a timescale that is short compared to electron spin relaxation. After electron-hole recombination has been completed, there will be an electron density equal to the original electron density, prior to photoexcitation, that is preferentially spin polarized with quantization axis along the optical propagation direction. The exact degree of spin polarization depends on the energy of the optical pulse, and is difficult to calibrate precisely. The electron momentum distribution will thermalize on a fast timescale compared to the electron spin relaxation rate. Polarization-sensitive optical transmission, reflection, and emission can be used to probe the dynamics of the non-equilibrium spin p ~ p u l a t i o n ~ , ' ~ . ' ~ - ' ~ ~ ~ ~ ~ ~ An external magnetic field can be imposed, causing the spin distribution to precess about the magnetic field in order to distinguish longitudinal ( T I )and transverse (T2) relaxation rates. Figure 7 shows the decay of a Faraday rotation signal with and without a magnetic field normal to the spin polarization axis. The longitudinal and transverse spin relaxation rates are determined by the rate of decay of the Faraday rotation signal for these two cases. We apply the kinetic theory developed in the previous section to describe the dynamics of a photogeneratedelectron spin polarization, first in bulk GaAs,'*O and then in GaAs quantum wells.'32 We assume that the initial spin polarization is small so that spin decay is exponential and characterized by longitudinal or transverse relaxation times.

14. SPINRELAXATION IN BULKSEMICONDUCTORS At heavy doping and low temperature, electrons are degenerate. In this case electron-electron scattering is reduced because of the Pauli Principle, electronphonon scattering is reduced because of a combination of phase space considerations and small thermal phonon populations, and electron-impurity scattering dominates the scattering rates. At higher temperatures and lower doping levels, the electrons are nondegenerate and inelastic scattering rates become important. Electron-phonon scattering is often dominant in these cases. For the calculation of the spin relaxation rates presented in this section we included scattering on ionized impurities, acoustic phonons, and longitudinal optical (LO) phonons. Because we are primarily interested in the magnetic field dependence of the spin relaxation rates which, at least qualitatively, does not depend on the approximation adopted to describe the scattering processes, we treat all scattering processes within the

126

EX.BRONOLD, A. S A X E " , AND D.L. SMITH 106 1

1

0.10

1 .oo

10.00 '

Density [l Ole ~ r n - ~ ]

FIG.9. Calculated transverse spin relaxation time for electrons in GaAs at low temperature as a function of electrondensity. The Elliott-Yafet, Dyakonov-Perel,and total times are shown. Data labeled with dots are from Ref. [16] and data labeled, triangles are from Ref. [141].

elastic approximation.I2' The parameters needed to specify h, ( i )have been previously obtained, partly experimentally by measuring combined cyclotron resonances (a4,ab) and partly theoretically within a five-level Kane model (a$: (u4, as, a6) = (97, -8,49) x 10-"eVcm20e-'.'" Theparameterdefining hl~(k) is given by SO = 0 . 0 6 f i 3 / , / w . 1 ' 4 The remaining parameters, such as the effective CB electron mass or the deformation potential, are available from standard data bases." Figure 9 shows the results of a calculation of the T2 lifetime at low temperature as a function of density for electrons in bulk GaAs. The scattering is dominated by charged impurity (donor) scattering and the density of charged donors is taken to be the same as the electron density (i.e., the materials are uncompensated). The contribution to spin lifetime from the Elliott-Yafet (EY) process, Dyakonov-Perel (DP) process, and the total lifetimes are shown. The DP process dominates the spin relaxation at all densities. The spin relaxation time is a strongly decreasing function of density. The spin polarization arises from electrons on the Fermi surface. As the density increases the Fermi surface rises higher in the conduction band. Both the band splitting, due to the inversion asymmetric component of the crystal potential-which is responsible for the DP spin relaxation process-and the '40 Semiconductors, Vol. 17, eds. 0.Madelung, Landoldt-Bornstein, New Series, Group 111, SpringerVerlag, Berlin (1986).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

127

admixture of bonding p-states into the conduction band wave functions-which is responsible for the EY spin relaxation process-increase for states higher in the conduction band. Therefore, both the DP and EY spin relaxation rates increase as the Fermi energy increases. Also shown on Fig. 9 are measured T2 lifetimes for GaAs from Refs. [16,141] at low temperature and degenerate electron densities. The calculated results are in reasonably good agreement with the measurements. There are no adjustable parameters in the calculation. Our theory, which is based on a band description of the electronic states, is not applicable to the low density data also shown in Ref. [16] because, under the conditions of the experiment, electrons are most probably bound to the donors. A small magnetic field (51Tesla) was applied to the samples in the experimental results reproduced in Fig. 9 in order to distinguish longitudinal and transverse relaxation times. This field is not large enough to significantly influence the spin dynamics, except, of course, for the precession of the spin polarization about the magnetic field. That is, in this range of magnetic fields the longitudinal and transverse relaxation times do not change with magnetic field. For larger magnetic fields the relaxation rates themselves become functions of the field. The FokkerPlanck equation (3.43) for the spin polarization c_ont+ns terms describing the EY (term containing ( i ) and ) DP (term containing alA(k)) spin relaxation processes and a term that describes spin precession about an applied magnetic field (term containing GL.) There are two additional terms that depend on magnetic field; one that describes the orbital motion of the conduction band electrons in the magnetic field (termcontaining 5 2 ~and ) one that describes the_chFgein electron g-factor for states away form the zone center (term containing 52, (k)). The EY spin relaxation process arises due to spin-flip scattering events and is not significantly influenced by a magnetic field. By contrast, the DP spin relaxation process originates from an interplay of spin-conserving wave vector scattering events described by the . cyclotron orbital operator b - ( 1 / 4 t l ) i 2 and the torque forces due to 5 2 1 ~ The motion described by -i SZci, leads to a quenching of the DP process similar to that from wave vector scattering events because the axis of the torque force depends on wave vector and that axis is changed by the cyclotron orbital motion.lo8The wave vector dependence of the conduction band electron g-factor opens an additional spin relaxationprocess, which tends to reduce the spin lifetimesin applied magnetic fields.Iw As a result of the variations in the g-factor, electrons with different wave vectors precess about a transverse magnetic field at different rates and thus lose spin coherence. We refer to this process as a variable g-factor (VG) mechanism. The VG mechanism primarily effects the T2 lifetime.'20 In Fig. 10, we show calculated longitudinal ( T I )and transverse (Tz) spin relaxation times for GaAs as a function of magnetic field at low temperature and an electron density of n = 10l8 cm-3.120We show separately the contributions to the spin relaxation times from the EY, DP, and VG processes and the total spin 141

Data courtesy of S. A. Crooker.

128

EX.BRONOLD, A. SAXENA, AND D.L. SMITH 10'0

,

1

5

10

15

20

Magnetic field

! '

-B

m

---DP ---VG

'* ld

\.

r

30

-- EY

106 . I.

105

25

'. '.

10' \,

\.',

'O0O

'-.

5

10

15 20

25 30

102

Magnetic field

m

FIG.10. The top and bottom panels show, respectively, calculated TI and T2 spin relaxation times The contributions from the in GaAs as a function of magnetic field for T = 0 and n = 10l8 EY (long dash), DP (short dash), and VG (dot-dash) processes and the total relaxation time (solid) are shown in the main panel. The insets (same axes as the main panel) show the total relaxation times for 1 x 10l8~ m - 5~x, 10l8cmP3, and 1 x 1019cm-3 n = 5 x 10l6 1 x 1017 ~ m - 5~x, loi7 (top to bottom). The solid circles are experimental data from Ref. [16].

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

129

relaxation time including all three processes. In the insets of Fig. 10 we give the total spin relaxation time for various electron densities at low temperature. For the temperature and density conditions in Fig. 10, the electrons are degenerate and electron-ionized impurity scattering dominates. The VG process makes a small contribution to T I ,which is dominated by the DP process at zero magnetic field. As the magnetic field is increased, the DP process is quenched. Thus, TI increases monotonically with increasing magnetic field, saturating at high field at a value determined by the EY process, which is not affected by the magnetic field. If the material parameters had been such that the EY process dominated the DP process for TI relaxation at zero magnetic field, TI relaxation would not be significantly affected by the applied field. By contrast the VG process makes a significant contribution to T2 relaxation. At small applied magnetic fields the T2 lifetime increases with increasing magnetic field, but as the field continues to increase the VG process begins to dominate the relaxation so that T2 has a maximum around -2 1 Tesla and begins to decrease for larger magnetic fields. If the material parameters had been such that the EY process dominated the DP process for T2 relaxation at zero magnetic field, the T2 relaxation would monotonically decrease with increasing magnetic field. The solid circles in the lower panel of Fig. 10 are measured T2 spin lifetimes in GaAs at 5 K from Ref. [16] at electron densities of 10l8cm-3 and 5 x 10l8~ m - There ~ . is reasonably good agreement between the calculation and measured results. There were no adjustable parameters in the calculation. The magnetic fields in Ref. [16] are not high enough to completely capture the effects due to the VG process. The calculations in Fig. 10 assume uncompensated samples so that the ionized donor concentration is the same as the electron density. The scattering rate in these calculations is dominated by electron-ionized impurity scattering and this scattering rate depends on the impurity concentration. If compensated samples are considered, the density of ionized scattering centers could be increased for a fixed electron density. Within the Born approximation, the cross section for electron scattering from positively charged donors and negatively charged acceptors is the same. In Fig. 10, the DP process dominates the EY process at zero magnetic field. The spin relaxation rate due to the DP process decreases with increased scattering rate, whereas the spin relaxation rate due to the EY process increases with increasing scatteringrate. For compensated samples, the scatteringrate should increase and the spin relaxationrate due to the DP process should decrease, whereas the spin relaxation rate due to the EY process should increase with increasing compensation. At some level of compensation the DP and EY rates should cross. In Fig. 12, we show calculated transverse (T2) spin relaxation times for GaAs as a function of magnetic field at low temperature and an electron density of n = IO'* cm-3 for various compensation levels. The concentration of impurity scattering centers (donors plus acceptors) divided by the electron density is labeled by x . For the uncompensated case (x = I), the DP process dominates at zero field

130

EX.BRONOLD, A. SAXENA,AND D.L. SMITH

-

_ _ _ _ _ _------. ---

-

400 -

2000""""'""'""'""'""' 5 10 15

20

25

30

Magnetic field [TI FIG. 1 1. Calculated transverse spin lifetime for GaAs as a function of magnetic field for T = 0, n = 10l8~ m - and ~ , for various degrees of compensation. The compensation is characterized by x, the ratio of donor plus acceptor impurity concentration divided by electron density.

and the spin relaxation time first increases with increasing magnetic field, peaks, and then decreases as the VG process takes over. For more heavily compensated samples ( x > l), the EY process dominates at zero field and the spin relaxation time is insensitive to field. In Fig. 11, we show the various contributions to the TI and T2 spin relaxation for GaAs as a function of magnetic field at T = 100 K and an electron density of n = lOI7 cm-3.120In the insets of Fig. 11, we show the total spin relaxation time as ~ . the a function of magnetic field for various temperatures at n = 1017 ~ m - For temperature and density conditions in Fig. 11, the electrons are non-degenerate and electron-LO-phonon scattering is the dominant scattering process. As for the degenerate electron case, the VG process makes a small contribution to T I , which is again dominated by the DP process at zero magnetic field. The DP process is quenched by the field so that TI increases with field at small fields and saturates at a value determined by the EY process at large fields. Similar to the degenerate electron case, the VG process makes a substantialcontribution to T2 relaxation. At small fields the T2 lifetime increases with increasing field and at large fields the VG process begins to dominate the relaxation so that T2 again has a maximum at some finite magnetic field. The sign of the slope in T2 at small magnetic fields is again a clear signature of whether the EY process (T2 decreases with increasing field) or DP process (T2 increases with increasing field) dominates T2 relaxation at zero magnetic

131

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

5

10

”‘0

5

10

15

Magnetic field

109

-- EY

15

20 25

20

25

30

30

m

. . . ,

. .

.

10

20 25

I

--- DP 108 i.i --total --VQ

B

r

!

107

!

‘.

106 -

‘8,.

Io2O

‘.

., I . . . .

in3

I

0

5

.

5

. , . . . .

10

I

15

. . . .

15

Magnetic field

.

20

. .

30

. . . .

25

30

m

FIG.12. The top and bottom panels show, respectively, calculated TI and T2 spin relaxation times in GaAs as a function of magnetic field for T = 100 K and n = 10’’ The contributions from the EY (long dash), DP (short dash), and VG (dot-dash) processes and the total relaxation time (solid) are shown in the main panel. The insets (same axes as the main panel) show the total relaxation times for T = 150,200,250, and 300 K (top to bottom).

132

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

field. The qualitative behavior of longitudinal and transverse spin relaxation times with increasing magnetic field is similar for degenerate and non-degenerate electrons, but the magnitude of the change is larger for non-degenerate electrons. We described calculations of the longitudinal ( T i ) and transverse ( 7 3 spin relaxation times of conduction band electrons in n-type GaAs focusing on the interplay between the EY, DP, and VG spin relaxation processes as a function of the magnetic field. Scattering on ionized impurities, acoustic phonons, and longitudinal optical (LO) phonons was taken into account within the elastic diffusion approximation. For all field strengths we neglected quantization effects due to the external magnetic field.' 1'*115 That is, we implicitly assumed the strong scattering limit, where Landau levels are blurred and electrons effectively reside in (restored) band states. In p-type materials, the Bir-Aronov-Pikus (BAP) process provides an additional, efficient spin relaxation ~ h a n n e l . ' ~ .The ~ ' ~BAP ~ ' ~process ~ can be incorporated into the kinetic approach outlined in the previous sections by adding an additional collision integral J j h x to the kinetic equation for the non-equilibrium spin polarization, which describes exchange scattering of electrons on equilibrium holes. The diffusion approximation cannot be applied to Jihx, however, and the integral form of the kinetic equation (3.26) has to be used to calculate the spin relaxation rates due to the BAP process. WELLS 15. SPIN &LAXATION IN QUANTUM In this section, we discuss spin relaxation of conduction band electrons in an n-type GaAs quantum well grown in the [Ool] direction. As for the bulk case, there is spin relaxation due to spin-flip scattering (Elliott-Yafet process) and, in quantum wells without inversion symmetry, due to spin-off diagonal Hamiltonian matrix elements. In addition to inversion asymmetry from the bulk semiconductor, there can also be structural asymmetryloo*'oldue to either an external electric field or an internal electric field, caused, for example, by the particular choice of the barrier materials. Structural inversion asymmetry also induces a momentum-dependent torque force, giving rise to a motional narrowing type spin relaxation process, similar to the Dyakonov-Perel process due to bulk inversion asymmetry. Usually the inversion asymmetry induced spin relaxation processes dominate the ElliottYafet process. In a magnetic field, motional narrowing processes due to bulk and structural inversion asymmetry are quenched, as in the bulk materials, due to the orbital motion of the conduction band electrons, and the variable g-factor process takes over at large magnetic fields. We consider spin relaxation of conduction band electrons in a symmetric n-type GaAs quantum well at zero magnetic field. Spin relaxation is dominated by bulk inversion asymmetry (Dyakonov-Perel process). '22 The electronic structure of the quantum well is calculated for an infinite confinement potential. As in the bulk case, we treat the two states at the conduction band minimum explicitly and include a large set of states perturbatively, up to third order, to include the effect

133

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

of bulk inversion asymmetry. For energies close to the ban! minimum, the secular equation for the conduction band envelope functions @ b ( k , z ) reads (b = -)

+,

with

(4.2) where i is the in-plane momentum vector, and Rz = - i d / d z , E(Z) = A2k2/2m*, - E ~ eg ; and m* are the bulk energy gap and effective mass, respectively. The boundary conditions for infinite confinement are @ b ( z r z = z t ~ / 2 )= 0, where L is the width of the quantum well. The second term on the right-hand side of Eq. (4.1) gives the spin off-diagonal part to the non-interacting effective mass Hamiltonian for a quantum well. Because the term is small, it is treated in first order perturbation theory. Specializing for the lowest conduction subband and measuring energies from the minimum of the subband, the free part of the effective mass Hamiltonian becomes

E =E

with (neglecting cubic terms in k)

(4.4) where SO is the constant characterizing the bulk inversion asymmetry. As in bulk, the leading order term in the effectivemass Hamiltonianfor conduction electrons is Kramers degenerate, with the degeneracy being lifted by a much smaller term that arises from inversion asymmetry. More refined models for the electronic structure of a quantum well, including not only the bulk but also structural inversion asymmetry and taking finite band-off sets into account, are sometimes necessary for a quantitative description of specific physical effects. 142-'4s For a complete description, a collision term H, has to be added. It has the same form as in bulk, except that the wave vectors are two-dimensional and the matrix elements for the various '41

F.Malcher, G . Lommer, and U. Rossler, Superlattices and Microstructures 2,267 (1986).

e Silva, G. C. La Rocca, and F. Bassini, Phys. Rev. B 50,8523 (1994); ibid. 55, 16293 (1997). 144 P. Pfeffer and W. Zawadzki, Phys. Rev. B 59, R5312 (1999). 145 C.-M. Hu, J. Nina, T. Akazaki, H. Takayanagi, J. Osaka, P. Pfeffer, and Zawadzki, Phys. Rev. B 60, 7736 (1999). 143 E. A. de Andrada

134

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

scattering processes have to be calculated using the quFtum well eigenfunctions, which can be obtained from the envelope functions @ b ( k , z ) in the usual way. They are again not spin eigenfunctions because of the admixture of the bonding p-states. Any momentum scattering process has therefore a small probability to flip the spin. We neglect the Elliott-Yafet process here and only keep the spin-conserving parts of the scattering matrix elements. The spin relaxation rates are obtained employing the semiclassical theory given in Section III. The formal structure remains the same as for the bulk case, except that the angular dependence is now des_cribedby a single variable, the polar angle 4, with obvious implications for the k-space angular momentum operators, I!2 and I!,, and the k-average (. . .)i.For weak magnetic fields the Dyakonov-Perel process dominates. Moreover, neglecting the effect of the weak magnetic field, the Dyakonov-Perel spin relaxation tensor is diagonal, but not, as in the bulk case, proportional to the unit tensor. In a quantum well grown along [Ool], which is = r$ and l/Tl = r&,= 2 r s . also the spin quantization axis, 1/T2 = The longitudinal spin relaxation rate 1/ TI is therefore twice the transverse spin relaxation rate,

rs

with the generalized relaxation time t (E)given by

, / ZCfA the (bulk) constant where we introduced l/tBfA(&) = C ~ ( ~ / L ) ~with defined in Section 111.13. The energy average in two dimensions is defined by ((. . . ) ) E = d ~ f ( ~ (E)(. ) f . .)/2n dEf(E)P(E).Therelaxationrates I / T ~ ( E ) and the and l/t&), characterizing the off-shell differential operator on-shell rate l / t l ( ~ )are calculated for a two-dimensional electron gas along the lines given in section III.12. The different prefactor in Eq. (4.5) comes from the reduced dimension. We calculate the transverse Dyakonov-Perel spin relaxation time T2 for a quantum well at low enough temperatures, where scattering from phonons is negligible, and electron-ionized-impurityand electron-electron scattering dominate. The Coulomb potentials are screened statically. Electron-electron scattering is treated inelastically and the differential equation (4.6) for the generalized relaxation time t (E) is solved numerically. The spin relaxation time T2 is obtained from Eq.(4.5) by numerical integration. In Fig. 13 we show the transverse spin relaxation time (T2) as a function of the electron density for a 25 nm modulation doped GaAs quantum well at T = 10 K, 20 K, and 30 K. In modulation doped quantum wells the donors are spatially

[F

[F

a(&),

135

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS I

.. 1

I

Y\ \' I

I

1

10

\ 100

Density (1010ern-*) FIG. 13. Calculatedtransverse spin relaxationdue to the DP spin relaxation mechanism for a 25-nm

GaAs quantum well as a function of electron density at T = 10.20 and 30 K. The quantum well is assumed to be modulation doped so that it contains no donors and electrowelectron scattering is the dominant scattering process. The solid dots are data at T = 10 K from Ref. [68] .

separated from the electron gas and usually sufficiently screened so that electronionized-impurity scattering is negligible and electron-electron scattering is the dominant scattering process. We therefore took only electron-electron scattering into account. For fixed temperature the spin relaxation time first increases with doping, reaches a maximum, and then decreases with doping. The non-monotonic doping dependence of the spin relaxation time reflects the doping dependence of the electron-electron scattering rate: At low densities, where the non-degenerate electron gas is very dilute, electron-electron scattering is suppressed because of the lack of scattering partners, whereas at high densities, where the electron gas is degenerate, the scattering rate is blocked due to the Pauli principle. For intermediate electron densities, where the cross-over from non-degenerate to degenerate occurs, the scattering rates and therefore the spin lifetimes are maximal. A similar effect has been discussed before in the context of current relaxation in optically pumped semiconductors.'46 For a fixed density, the spin relaxation time increases with increasing temperature in the degenerate regime and decreases in the non-degenerate regime. The former is because the Pauli blocking decreases with temperature, giving rise to an increasing scattering rate and, therefore, to an '46

M. Combescot and R. Combescot, Phys. Rev. B 35,7986 (1987).

136

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

increasing spin lifetime. At low densities, on the other hand, increasing temperature increases the average thermal energy and spin relaxation preferentially occurs from higher lying states in the conductionband, where the inversion-asymmetry-induced torque force on the spin polarization is large. As a result, the spin relaxation time decreases with temperature at densities where the electron gas is non-degenerate. In Fig. 13 we also show experimental data for T = 10 K.68For electron densities above n = 5 * 10'' cm-* we get reasonably good agreement between theory and experiment. At the lower densities, conduction band electrons are most probably localized by binding to charged donors or other potential fluctuations and our theory, which is based on free electron band states, does not apply. In quantum wells that are not modulationdoped, impurities due to donors and acceptors (in compensated samples) act as very efficient additional scattering centers. This leads, if the motional narrowing spin relaxation processes prevail, to a substantial increase in the spin relaxation time. In Fig. 14 we show for various degrees of compensation the transverse spin relaxation time for a 25-nm GaAs quantum well: x = 0 (modulation doped), x = 1 (uncompensated sample with impurity density equal to electron density), and x =4 (compensated sample with impurity, donor plus acceptor,density four times the electron density).The temperature is T = 20 K. For all densities the spin relaxation time increases with compensation x . The increase is, however, not uniform, with the largest increase occumng at intermediate 2500 h

rn

4 2000

!i

F

.1500 c

8

c?

8

1000

9 rn

C

2

500

I-

.x.:p. -.*. \ ,..---. -.---

0/

*--.

0

I

,

lo Density (1010 c m 2 ) 1

\

-. '*--.____

10

FIG. 14. Calculated transverse spin relaxation time due to the DP spin relaxation mechanism for a 25-nm GaAs quantum well as a function of electron density at T = 20 K for three values of ionized

(donor plus acceptor) impurity concentration: no ionized impurities, ionized impurity concentration equal to the electron density, and ionized impurity concentration equal to four times the electron density.

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

137

IC

.O 600 iij

-3 Q

400

2 0

2 200 II

10 Temperature (K)

100

FIG.15. Calculatedtransverse spin relaxationdue to the DP spin relaxation mechanism for a 25-nm GaAs quantum well as a function of temperature at an electron density of 3 x 10'O cm2. The solid curve is for a modulation doped quantum well in which electron-electron scattering dominatesand the dashed curve is for a doped quantumwell with a donor concentrationequal to the electron concentration in which both electron-electron and electron-impurity scattering are important.

densities, where the cross-over between a non-degenerate and a degenerate electron gas takes place. In this density regime the inelasticity of the electron-electron scattering is particularly strong and substantially enhances the spin lifetime. For a fixed electron density, increasing temperature pushes the electron gas from the degenerate into the non-degenerate regime. At temperatures where the electron gas is in the cross-overregime, electron-electron scattering is again particularly strong, and we expect a maximum of the spin lifetime. Figure 15 shows the temperature dependence of the transverse spin relaxation time for a 25 nm GaAs quantum well with n = 3 x 10'' cm-*. We contrast a modulation doped quantum well (x = 0 ) with a doped quantum well, where the donor density is equal to the electron density (x = 1). In the quantum well with x = 1, spin lifetimes are for all temperatures longer than in the modulation doped case, because of the additional electron-ionized-impurity scattering processes. The enhancement is again not homogeneous, with the largest increase occurring at temperatures where the electron gas crosses from the degenerate to the non-degenerateregime. The maximum is more pronounced in the modulation doped quantum well, however, because at low temperatures Pauli blocking effectively reduces the scattering rate, and therefore the spin lifetime. In the x = 1 doped case, in contrast, the decrease at low temperatures saturates at a value set by the rate of the electron-ionized-impurity scattering rate. In both cases, the electrons are non-degenerate at high enough

138

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

temperatures, with an average thermal energy increasing with temperature, which leads to a sampling of the inversion-asymmetry-induced torque force at higher energies and, as a result, in both cases to a decreasing spin lifetime. We presented results for the transverse spin relaxation time (T2)in a symmetric quantum well at zero magnetic field and low temperatures, where the main scattering processes are due to electron-ionized-impurity and electron-electron scattering. For temperatures above 100 K, electron-phonon scattering determines the temperature dependence of the spin relaxation time.19*128 In the framework of our approach, electron-phonon scattering simply adds an additional term to the relaxation rates l/tf(e), l / t d ( & ) , and l / t l ( e ) , which, for elevated temperatures, would be the dominant term and therefore decisive for the temperature dependence of the spin relaxation time. In an asymmetric quantum well, the bulk inversion asymmetry term is replaced with a total asymyetry-term A j. that is a-sum of the _contrib_utions_ of bulk and structural asymmetry ~ D B I(k) AhtOlal(_k) =h!&,~ (k) h Q s ~ ~ ( kIf) the . angular dependences of h & ~ ( k ) and h & ~ ( k ) are the same, the spin relaxation time is given by Eq. (4.5) with + l/ttota[(e)= ~ / Q I A ( E ) l / t ~ , ~ ( eand ) t ( e ) obtained from Eq. (4.6) ~/QIA(E) with the same modification. If the angular dependences are different 1/T2 = 1/ T:” 1/ T:lA, with each contribution calculated individually. The different angle dependencies then give rise to different prefactors in Eq. (4.5). The possibility of designing quantum wells in which the structure inversion-asymmetryinduced spin relaxation compensates the bulk inversion-asymmetry-inducedspin relaxation has been d i s c ~ s s e d . The ’ ~ ~spin ~ ~lifetimes ~~ would then be limited by either the Elliott-Yafet process or by the spin decay due to hyperfine interaction. An external magnetic field can be included semiclassically as in the bulk case, giving rise to quenching of the Dyakonov-Perel spin relaxation process and, at high magnetic fields, to the variable g-factor mechanism.

+

+

+

V. Spin Injection from Polarized Contacts

Most semiconductor device concepts that exploit the electron spin degree of freedom require an electrical means of injecting spin-polarizedcurrents into a semiconductor. Optical techniques for spin injection are usually not practical for device purposes. Structuresdesigned to achieve electrical spin injection utilize spin-polarized contacts as the source of polarized electrons. Two main types of spin-polarized contacts have been investigated;ferromagneticmetal contacts in which the electron spin is polarized because of the ferromagnetic ground state of the c o n t a ~ t , ~ ” ~ ~ * ~ ~ and semimagnetic semiconductor contacts with large g-factors in which the electron spin is polarized at low temperature by a large external magnetic field.23*24-84 Ferromagnetic metals with Curie temperatures well above room temperature are an ideal source of spin-polarized electrons for electrical device purposes because

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

139

there is no requirement for low temperatures or externally applied magnetic fields. Semimagneticsemiconductorsare 11-VIsemiconductorsheavily doped with a transition metal, typically Mn. The Mn substitutes for the group I1 element. Because Mn has two electrons in its valence s-p orbitals, it acts like a group I1 element and does not electrically dope a 11-VI semiconductor. However, Mn also has a halffilled d-level with spin 5/2, which is quite stable. The semiconductor can be doped n-type by the addition of electrically active impurities in the usual way. In these n-type materials an exchange interaction between the electrons in the semimagnetic semiconductorconduction band and the half-filled Mn d-levels leads to very large, often of order 100, effective g-factors for the conduction band electrons. Because of their large g-factors the conduction band electrons in a semimagnetic semiconductor can be strongly spin polarized by a readily achievable magnetic field (a few Tesla) at low temperatures. The conduction band electrons in a conventional semiconductor,with a g-factor of order unity, will not be significantly spin polarized under the same experimental conditions. Thus the n-type doped semimagnetic semiconductor can be used as a spin-polarized contact to a conventional semiconductor in which spins are not significantly polarized at thermal equilibrium.Device concepts based on spin polarization of semimagnetic semiconductorsare probably not technologically practical because low temperatures and high magnetic fields are required, but they can provide important test structures to explore ideas about spin injection. Measurements of spin-polarized electron injection are often made using a spinIn these experiments, electrons are injected into an n-type LED configuration.' semiconductor from a spin-polarized contact and are transported to a region in space, typically a quantum well, where they recombine radiatively with unpolarized holes transported from an adjacent p-type doped region. The relative intensity of right- and left-circularly polarized light emitted from the quantum well gives a measure of the spin polarization of the electron density in the recombinationregion. A schematic illustration of this experimental configuration is shown in the upper panel of Fig. 16. Because of the nature of the spin-dependent optical selection rules, the luminescence must be collected so that the optical propagation direction is along the axis of spin polarization. Experimentally it is usually most convenient to collect luminescence propagating normal to the sample surface, either from the top of the sample by using a thin (about 10 nm) semitransparentc ~ n t a c t ' ~or, ~ ~ , ~ ~ from the bottom of the sample by using a transparent ~ubstrate.'~ In this detection geometry, the direction of spin polarization in the spin-polarized contact must also be normal to the sample surface. If the spin-polarized contact is a ferromagnetic metal, the direction of spin polarization will most likely be in the plane of the metal film, not normal to the sample surface, because of shape anisotropy. The direction of polarization of the ferromagnetic contact can be brought normal to the sample surface by application of an external magnetic field of a few Tesla. This field is not necessary, in principle, for spin injection, but it is required for the spin-LED

',"

140

EX.BRONOLD, A. SA)(ENA, AND D.L. SMITH Optical detection

Ferromagnetic contact

n-doped Polarized semi. luminescence

semi. Tunnel barrier

Quantum well

Electrical detection

Ferromagnetic contact

Ferromagnetic contact barrier

barrier

FIG.16. Schematicof electron spin injectioddetection structuresusing ferromagneticcontacts.Top panel shows a ferromagnetic injecting contact and a spin-LED detection approach in which circularly polarized luminescence is used to detect spin polarization. Bottom panel shows ferromagnetic injecting and detecting contacts in which a voltage difference for parallel and antiparallel configurations of the contact magnetic orientations is used to detect spin-polarized currents.

detection scheme if luminescence is collected propagating normal to the sample surface. It is possible to use a spin-LED detection scheme in which luminescence propagating parallel to the sample surface is collected. In this case a large external magnetic field is not required. However, this geometry presents experimentaldifficulties for collection of luminescenceand is not the most commonly used approach. A spin current can also be detected electrically using spin-polarized ~ 0 n t a c t s . l ~ ~ A schematic illustration of this approach is shown in the lower panel of Fig. 16. l k o spin-polarized contacts are used, one to inject spin-polarized electrons and the second to collect the spin-polarized current. The spin polarization direction of the injecting contact will determine the polarization direction of the spin current in the semiconductor.A difference in resistance at the collecting contact for spin current polarized parallel or antiparallel to the collecting contact polarization is used to detect spin current. In this approach it is not necessary to align the spin polarization of the contacts normal to the sample surface as it is for the spin-LED detection approach using the common experimental geometry. '41

P. Hammer, B. R. Bennett, M.J. Yang, and M.Johnson, Phys. Rev. Lett.83,203 ( 1999).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

141

Initially one might expect that a spin-polarized contact would allow spinpolarized electrical injection into a semiconductor in a straightforward fashion. The spin-polarized electrons in the contact result in a spin-polarized current in the contact when a voltage bias is applied and one might expect that the current would remain spin polarized as it flows into the semiconductor.However, the problem of achieving spin injection has proven much more difficult than originally expected.148The fundamental physical problem was first discussed by Schmidt et al. in terms of a conductivity mismatch between the metallic contact and the semicond~ctor.~~ They showed that efficient spin injection could be achieved if the contact and semiconductor had similar conductivities, but not if their conductivities differed by orders of magnitude as is the case for a metal and a semiconductor. Efficient spin injection was first observed using spin-polarized contacts fabricated from semimagnetic semiconductors that had a similar conductivity to the nonmagnetic semiconductor into which the spin current was i n j e ~ t e d . ~ ~It- "was .~ later shown that the impediment to efficient spin injection discovered by Schmidt et al.32could be circumvented by the use of spin-dependent tunnel barrier^.^^,^^ This approach allows ferromagnetic metals to be used as spin injecting contacts and avoids the need for the low temperatures and high magnetic fields that are required for semimagnetic semiconductor spin injecting contacts. The discussions of Refs. [32-341 are based on a simple model in which both the contact and the semiconductor have a spatially constant electron density and therefore a spatially constant conductivity. Important insight was provided by this model, but it is not completely realistic because it does not account for the depletion region that occurs at real semiconductor/metalinterfaces, because of the Schottky energy barrier that forms at the interfaces, or at real semiconductor heterostructures,because of the energy band off-sets that occur at semiconductor hetero~tructures.'~~ The effects of interfacial depletion layers were subsequently investigated and they were shown to be very detrimental to efficient spin i n j e ~ t i o n .However, ~ ~ . ~ ~ an approach based on selective interface doping profiles was proposed to eliminate these problems. Thus two important requirements to achieve efficient spin injection from a ferromagnetic metal contact are the formation of a spin-dependent tunnel barrier to circumvent the conductivity mismatch problem and the use of selective interface doping profiles to overcome problems that arise from interfacial depletion regions. In the remainder of this section we first discuss the spin-dependent transport equations that will be used to describe spin injection, then discuss the constant conductivity m ~ d e l ~to * -show ~ ~ why efficient spin injection into a semiconductor 14* F. G . Monzon, H. X. Tang, and M. L. Roukes Phys. Rev. Len. 84,5022 (2000); B. J. van Wees, Phys. Rev. Lett. 84,5023 (2000); P. Hammer, B. R. Bennett, M. J. Yang, and M. Johnson, Phys. Rev. Lett. 84,5024 (2000). '41 S. M. Sze, Physics of Semiconductor Devices, 2nd ed.,John Wiley & Sons, New York, (1981). p. 294.

142

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

is difficult and how spin-dependent tunnel barriers can be used to circumvent the difficulty. Next, we describe the effect of depletion regions on spin injection and discuss how properly designed interface doping profiles can be used to eliminate the problems they present. We then discuss the physical origin of spin dependence in tunneling and conclude the section with a brief discussion of the current experimental situation of electrical spin injection.

TRANSPORTEQUATIONS 16. SPIN-DEPENDENT We use spin-dependent drift-diffusion equations J, = p(en,E

+

ax

(5.1)

and the continuity equation (5.2) where J , is the current due to electrons of spin type r,~, p is the spin-independent electron mobility in the semiconductor,n, is the density of electrons of spin r , ~ , E is the electric field, and G, and R, are generation and recombination rates for electrons of spin r , ~ ,coupled to Poisson’s equation to describe spin-dependent electrical transport in semiconductors.149 This basic approach assumes rapid wave vector scattering so that the electron wave vector distribution is in local quasi-equilibrium. However, electrons of different spin may be driven out of quasi-equilibrium with respect to each other because spin relaxation is slow compared to wave vector scattering. The drift-diffusion transport model is a strong scattering approximation appropriate for relatively high temperatures, such as room temperature. It is the approach used to describe most semiconductor device operation. The spin-dependent electron density can be written in terms of the electrostatic potential 4 and a spin-dependent electro-chemical potential h,, (5.3) where ni is the intrinsic carrier density of the semiconductor. Using this form for the electron density the spin-dependent drift-diffusion equation can be written as

where the spin-dependentconductivity is u, = epn,.150Sometimes it is convenient to write the spin-dependent d@ft-diffusionequation in terms of an exponential of P. C. van Son, H. van Kempen,and P. Wyder, Phys. Rev. Lett. 58,2271 (1987).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

143

the electro-chemical potential,

For spin-polarized electrons there is a contribution to the recombination rate from both spin-flip scattering and electron-hole recombination, (5.6)

where tr and ts are the recombination and spin-flip times, respectively, and an analogous expression applies forr4. It is often convenientto go into a representation describing the electron charge and spin degrees of freedom

so that (5.8)

and

The corresponding generation and recombination rates are g, = gtfg4, r+ = n + / t , , and r- = n - ( 2 / t s + l / t r ) . The steady state continuity equations become (5.10) Substituting the drift-diffusion form into the continuity equation gives a transport equation for Q,, -!&

= -eg&e

-e@ f kT

(5.11)

where gf = g r t / [ p L n ( k T / e ) ( n i / 2 ) ] A: , = ( k T / e ) p n t r , and A? = ( k T / e ) p , , ( 2 / t s 1/rr)-'.We often consider cases in which there are no generation terms. If the electric field is small, the electron density slowly varying in space and the difference in electro-chemical potentials for the two spin types small compared with k T , we can expand Eq. (5.1 1 ) and get a diffusion equation for the difference in electro-chemical potentials,

+

(5.12) The diffusion equation is not valid in the depletion region of a semiconductor but it can be applied to the conductive region outside of the depletion region.

144

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

We first consider a constant conductivity model in which both the contact and the semiconductor are characterized by a spatially uniform electron density and conductivity, and the electric field in both materials is uniform and small. Driftdiffusion equations of the form of Eq.(5.4) and the spin diffusion equation of the form of Eq.(5.12) characterize the spin-dependent transport in each material. We then consider a Schottky diode structure with a large and rapidly varying electric field in the depletionregion. The spin diffusion equation is not valid in the depletion region and Eq. (5.1 1) is used to describe the spin-dependent transport. 17. CONSTANT C o m u c m MODEL

We consider electron injection from a spin-polarizedcontact, which may be a ferromagnetic metal or a spin-polarized doped semiconductor,into a lightly doped nonmagnetic semiconductor. The electron states in the nonmagnetic semiconductor satisfy nondegenerate statistics. We include the possibility of spin-dependentinterface resistance due, for example, to a tunnel barrier at the contact/semiconductor interfa~e.'~'A nonmagnetic insulating tunnel barrier with a spin-polarized contact can have spin-dependent resistance because the wave functions for electrons of the two spin types are different in the spin-polarized contact. A ferromagnetic insulator tunnel barrier can also have spin-dependent transmission properties. A spin-polarizedcontact/semiconductor structure is described by using the spin dependent drift-diffusionEq.(5.4) to describe current flow. The conductivities are, of course, different in the contact and the semiconductor. If electrons with different spins are driven out of local quasi-thermalequilibrium, so that pt is not equal to p ~ , at some point in space, the difference in the two electro-chemicalpotentials relaxes as described by the diffusion Eq. (5.12). At the contact/semiconductor interface, electrons of different spin can be driven out of quasi-thermalequilibrium by current flow. Far from the interface, located at x = 0, ( p t - p l ) returns to zero in both the contact and the semiconductor. The total steady state current density is a constant function of position. We assume no strong spin-flip scattering at the interface so that the individual current components for the two spin types are continuous at the interface. It is not difficult to include interfacial spin-flip scattering in the model, but it just adds additional unknown parameters and further reduces the degree of spin injection. Current flow at the interface is described using an interface conductance (5.13)

15'

See, for example, Y.Qi, Y. Xing, and J. Dong, Phys. Rev. B 58,2783 (1998).

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

145

where j," is the current density at the interface, G, is the interface conductance (1/G, is the interface resistance), and Ap, is an interfacial discontinuity in the electro-chemicalpotential for electrons of spin type q . If the interface conductance is infinite, the electro-chemical potential is continuous at the interface, whereas for finite values of G, a discontinuity in p,,develops. Because the steady state current is constant, it is convenient to define a variable B by j , = Bj where j is the total electron current density ( j , = (1 - B ) j ) . /? is continuous at the interface. The electron density as a function of position is fixed, independent of the current density j , by electrostatic constraints. This is the constant conductivity approximation, which is relaxed in the next section. The conductivities for the two spin types in the semiconductor are proportional to the corresponding electron densities with the same proportionality constant. The total conductivity is then fixed independent of current density. It is convenient to define a variable a! by at = a!a where is the total conductivity (a, = (1 - a!)a).a! is not continuous at the interface. Because the electron density is much greater in the contact than in the semiconductor, the fractional spin density can be more easily changed in the semiconductorthan in the contact material. Therefore, in the semiconductor a!, is taken to be a function of current density and position, but in the contact a!, is taken independent of current density and position. (Subscripts s and c are used to refer to the semiconductor and the contact, respectively.) The semiconductorsatisfiesnondegeneratestatistics with the conductivityproportional to the electron density, (5.14)

where n,, is the density of electrons with spin r,~ in the semiconductor, k is Boltzmann's constant, and T is temperature. The contact is on the left and the semiconductor on the right-hand side of the interface so that the current density, j , is negative for electron injection into the semiconductor. Solving Eq. (5.12) with the stated boundary conditions gives (p+- pk) = A e R (pt - p ~ = ) BeK

--I

x < 0, x > 0.

(5.15)

Equation (5.13) for the interfacial discontinuity in electro-chemicalpotential gives a relation between the expansion coefficients A and B (5.16)

146

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

where B is evaluated at the interface. The drift-diffusion equation evaluated at the two sides of the interface combine to give

(5. 17)

where a, and B are evaluated at the interface. Equations (5.16) and (5.17) can be solved to give the injected current spin polarization

-jf - J1 j , +j,

- (28 - 1) =

+ (2% - + (&) - (&) , R, + Rs + (&) + (&)

(2% - 1)Rc

1)RS

(5.18)

where position-dependentquantities are evaluated at the interface and A (5.19) *a (1 -a)' R, and R, for the contact and the semiconductor, respectively, are important parameters in the model. They correspond to the sum of the bulk resistivities for the two spin types (i.e., (& &)) times the spin diffusion length. They have the units of interface resistance, ohm-cm2. At very low current density there is no spin-densitypolarization in the semiconductor and as= 1/2. But at larger current density, spin polarization of the electron density in the semiconductor can occur and a, can deviate from 1 /2. In this high current density regime a, is found by solving Eq. (5.14) and

R=

+

Equation (5.20) follows from Eqs. (5.16)and (5.17).Here B is (/A+ - /A$) evaluated at the semiconductor side of the interface. It is linear in j at small j . Position dependences are determined once the interface quantities are found by

The position dependence of a,(x) and R,(x) follows from Eqs. (5.14), (5.15) and (5.19). The position dependence of the chemical potentials can be found by

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

147

integrating Eq. (5.21) in space, using Eq. (5.1) ej

/ - ~= t -X

(T,

+A(l -

-

C Y ~ ) ( . ~1)

x < 0,

where pt in the contact at the interface is set to zero as an arbitrary zero of energy and p: is the discontinuity in pt at the interface found from Eq. (5.13). The position dependence of p ~is. found from Eqs. (5.15) and (5.22). At thermal equilibrium electron spins are polarized in the contact but not in the semiconductor. To achieve spin injection, the system must be driven out of equilibrium by an electric current in such a way that the electrons injected into the semiconductor are spin polarized. There is a fundamental difference in how spin polarization is maintained in the contact as compared with the semiconductor. The contact maintains a spin polarization due to different densities of states for spin-up and -down electrons. It does not require a splitting in the electro-chemicalpotential for the different spin types. By contrast, the semiconductorhas the same density of states for spin-up and spin-down electrons, so a splitting of the electro-chemical potentials for the spin types is required for spin polarization. Because the electron density and therefore the electrical conductivity is high in the contact material and also because the spin diffusion length is comparatively short in the contact, it is difficult to drive the electron population in the contact far from local quasi-thermal equilibrium with a physically attainable current density. If the bulk contact is truly metallic, this is essentially impossible. At a spin-polarized contacthemiconductor interface without significant interface resistance, the electrons in the contact and in the semiconductor are in good thermal contact and therefore they stay in local equilibrium with each other. Because the electrons in the contact stay near local quasi-thermal equilibrium, so do those in the semiconductor, with the result that strong spin injection is difficult to achieve. This is the essential physical problem in achieving strong electrical spin injection. Figure 17 gives a pictorial illustration of this point. The left panel schematically shows energy band structures for the spin-polarized contact and the nonmagnetic semiconductor.In both cases the Fenni energy is the same for the two spin types at equilibrium. However, for the spin-polarized contact the energy bands are shifted for the two spin types and at equilibrium there are more spin-up than spin-down electrons and the density of states at the F e d surface is different for the two spin types so that the conductances are different. By contrast, in the semiconductor the energy bands for the two spin types are the same; at equilibrium, the number of electrons, the density of states at the Fenni surface, and the conductivities are the same for the two spin types. The right panel schematically illustrates the electrochemical potential for spin-up and spin-down electrons near the interface in which

148

EX.BRONOLD, A. S A X E " , AND D.L. SMITH Polarized Contact

Semiconductor

Polarized Contact

Semiconductor

RG.17. Left panel is a schematic of energy bands for spin-up and spin-down electrons for a spinpolarized contact and a nonmagnetic semiconductor.Right panel schematically shows electro-chemical potentialsfor spin-upand -down electrons near a (spin-polarizedcontact)/(nonmagneticsemiconductor) interface when the structure is under bias.

the structure is under electrical bias. Far from the interface (i.e., a is significantly larger distance than a spin diffusion length) the electro-chemical potentials for spin-up and -down electrons are the same on both sides of the interface and the slope in electro-chemicalpotential corresponds to the electric field in the material. It is orders of magnitude larger in the semiconductor than in a metal contact (the figure is not to scale). In the spin-polarized contact the current is polarized far from the contact because of the different conductivities for the two spin types, whereas in the semiconductorfar from the interface the conductivitiesfor the two spin types are the same and the current is not spin polarized. Moving toward the interface from the contact side, the electro-chemical potentials for the two spin types can split apart within a spin diffusion length of the interface with the electro-chemical potential for the high conductivity spin type above that for the low conductivity spin type. This splitting will be necessary if there is to be spin injection. Because the slope in electro-chemical potential is smaller for the larger conductivity spin type, spin polarization of the current is reduced. The ratio of slopes in electrochemical potential times conductivity gives the spin polarization of the current. If there is no tunnel barrier so that electrons on the two sides of the interface are in good thermal contact (G, very large), the electro-chemical potentials are continuous at the interface. On the semiconductor side, the conductivities for the two spin types are the same, unless the splitting in electro-chemical potentials is larger than k T , and a polarization of spin current must come from a different ratio in electro-chemical potential slopes. However, because the conductances of the metal contact and semiconductor differ by orders of magnitude, it is clear from Fig. 17 that the ratio of electro-chemical potential slopes in the semiconductor cannot differ significantly from unity. Thus efficient spin injection will not occur

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

149

if the conductivities of the contact and semiconductor differ widely and electrons on the two sides of the interface are in good thermal contact. However, if electrons on the two sides of the interface are prevented from being in good thermal contact by a tunnel barrier, a discontinuity in electro-chemical potential can develop at the interface. If the tunnel barrier has a spin-dependent conductance, the discontinuities in electro-chemical potential will be different for the two spin types. The development of interfacial discontinuities in electro-chemical potential that are different for the two spin types allows efficient spin injection. Mathematically this behavior is described by Eq. (5.18), which gives the injection current spin polarization. At low current density as = 1/2 and a, > 1/2. (We take the dominant spin type as spin up.) The conductivity of the contact is typically much larger than that of the semiconductor and the spin diffusion length of the semiconductor is typically larger than that of the contact, so that typically R, >> R,. If the interface resistance is small so that the terms involving+-$ can be neglected, the injected current spin polarization is small at low current density. From Eqs. (5.14) and (5.20), we see that a, at the interface increases to a=, for small interface resistance, at sufficiently large current density. At such large current densities, the injection current spin polarization becomes (2a, - 1); that is, the spin polarization of the contact, which can be a large value. However, the values of current density required to achieve this condition are unphysically large for truly metallic contacts. If the interface resistance is large, electrons on the two sides of the interface are not in good thermal contact with one another. Thus it is possible for the spin populations on the semiconductor side of the interface to be out of local quasi-thermal equilibrium even though those on the contact side are in local quasi-thermal equilibrium. This situation is described by Eq. (5.18) when the interface conductance terms dominate the bulk terms containing the R parameters. Strong spin injection can be achieved in this case if the interface resistances for the two spin types differ significantly. Figure 18, shows a calculation of the current spin polarization, defined as as a function of position. The calculation is for zero interface resistance, a current density of 1 A/cm2, 80% spin polarization in the contact (i.e., a, = 0.9), so that deep in the bulk of the contact J+ is nine times larger than Jk, and a contact spin diffusion length of 100 nm (these values for the spin polarization and spin diffusion length of the contact are used throughout this section) at room temperature. The semiconductor spin diffusion length is 1 p m and its resistivity is 1 Q-cm. (These values for the semiconductor parameters are also fixed throughout this section.) The bulk resistivity of the contact is varied in Fig. 18. For a low bulk resistivity contact, a larger electro-chemical gradient develops within a spin diffusion length of the interface for spin-down electrons, and the currents due to the two spin types approach each other near the interface. As a result the spin injection is weak. As the resistivity of the contact material is increased it becomes possible to drive the higher resistivity contact material further out of quasi-thermal equilibrium

fi,

150

EX. BRONOLD, A. SAXENA, AND D.L. SMITH 1.o

I

I

I

Polarized Contact

I

I

Semiconductor

n

-300 -200 -100

0

100

200

300

Position (nm)

FIG.18. Calculated current spin polarization at room temperature as a function of position near a spin-polarized contact/semiconductorinterface with no interface resistance for various values of the bulk resistivity of the contact material.

0.8

0.2

1

\

I 10” ohm-cm2

0.0 -300 -200 -100 I

I

0 100 Position (nm)

200

300

RG. 19. Calculated current spin polarization at room temperature as a function of position near a spin-polarized contactlsemiconductorinterface with a spin-selective interface resistance and metallic bulk resistivity of the contact material for various values of the spin-down interface resistance.

and more significant spin injection can occur. The contact resistivities used in the calculations for Fig. 18 are all significantly larger than typical metallic values. For true metallic resistivities the spin polarization current is very small at the interface. If a tunnel barrier with different resistances for the two spin types is introduced at the contact/semiconductor interface, strong spin injection can result. Figure 19

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

151

shows a calculation of the current spin polarization as a function of position for a metallic bulk contact resistivity of lop5S2-cm and a current density of 1 Akm2 at room temperature. The interface resistance for spin-up electrons is a tenth that of spin-down electrons and the interface resistance of spin-down electrons is varied. When an interface with spin selective resistance is included, a discontinuity in the electro-chemical potential difference that enhances spin injection develops at the interface, because the spin-up electrons have a smaller interface resistance. The discontinuity that occurs at a spin selectiveinterface is a promising approach to get a significantdifference in electro-chemicalpotentials at the semiconductorside of the interface, and therefore significant spin injection. However, if the interface resistance is spin independent, there is a discontinuity in the electro-chemical potential difference at that interface with a sign that reduces spin injection because the current for spin-up electrons is larger than for spin-down electrons and therefore the interfacial drop in electro-chemical potential is larger for spin-up electrons, leading to a reduction in spin injection. Figure 20 shows the calculated position dependence of the spin-up (solid lines) and spin-down (dashed lines) electron electro-chemicalpotentialsfor a bulk contact resistivity of lo-' n-cm with zero interface resistance for both spin type electrons (upper panel) and with a spin-down electron interface resistance of ohm-cm2 and a spin-up electron interface resistance a tenth of the spin-down value (lower panel) at a current density of 1 Akm2 at room temperature. Figure 21 shows the calculated electro-chemical potentials at room temperature as a function of position for the same material parameters at a current density of lo3 A/cm2. For the cases without contact resistance, the electro-chemicalpotentials are continuous at the interface. A slightly greater gradient develops in the spin-down electron electro-chemical potential than in the spin-up one and the currents for the two spin types become nearly equal at the interface (compare with Fig. 18). Even for the comparatively large current density of lo3 A/cm2, the spin injection is quite small. For true metallic values for the contact resistivity, the separation between the spin-up and spin-down electron electro-chemical potentials cannot be seen on the scale of Fig. 20. For the case with a spin selective interface resistance, there is a discontinuity in the electro-chemical potentials at the interface. A separation in these potentials occurs because of the different interface resistivities. For a 1 Akm2 current density the injection current is strongly spin polarized (compare with Fig. 19) but the magnitude of the injected current is not large enough to effectively polarize the electrons already in the semiconductor. This is indicated by the fact that the separation between the spin-up and spin-down electro-chemical potentials is small compared to kT.For a lo3 A/cm2 current density, the injection current is also strongly spin polarized and the magnitude of the injected current is large enough to effectively polarize the electrons in the semiconductoras indicated by the fact that the separation between the spin-up and spin-downelectro-chemical potentials is larger than kT.

152

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

-600 -400 -200

0

200

400

600

Position (nm) FIG. 20. Calculated position dependence of spin-up (solid lines) and spin-down (dashed lines) electron electro-chemical potentials for a bulk contact resistivity of lo-' $2-cm with zero interface $2 - cm2 and a resistance (upper panel) and with a spin-down electron interface resistance of spin-up electron interface resistance a tenth of the spin-down value (lower panel) at a current density of 1 A/cm2 at room temperature.

The upper panel of Fig. 22 shows the room temperature injection current spin polarizationas a function of current density for zero interface resistance and various bulk contact resistivities. For high conductivity contacts, a larger electro-chemical gradient develops within a spin diffusion length of the interface for spin-down electrons and the currents due to the two spin types of electrons become nearly equal at the interface. As a result the spin injection is weak. For lower conductivity contacts it is easier to drive the contact out of quasi-thermal equilibrium and the spin injection is stronger. The lower panel in Fig. 22 shows the calculated spin density, defined as (2a, - 1) = in the semiconductor at the interface as a function of current density. Compare with the upper panels of Fig. 20 and Fig. 21, which show the corresponding electro-chemical potentials. In this structure the injected electrons are not confined close to the interface in the semiconductor by,

s,

153

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

. 5i

j .

PolarizedContact

Semiconductor

0 -

-2

i

-

-600 -400

-200

0

200

400

600

Position (nm)

RG.21. Calculated position dependence of spin-up (solid lines) and spin-down (dashed lines) electron electro-chemicalpotentials for the same material parameters as in Fig. 19 at a current density of lo3 A/cm2 at room temperature.

for example a quantum well, and fairly high current densities are required to achieve a strong density polarization even when the injection current is strongly polarized. For semiconductor structures in which the injected electrons are confined near the interface, strong density polarization can be achieved at lower current densities. The upper panel of Fig. 23 shows the calculated room temperature injection current polarization as a function of current density for varying interfaceresistance for the spin-down electrons. The calculation is for a contact with a metallic bulk resistivity ( Q-cm) at room temperature. The spin-up electrons have a tenth of the spin-down electron interface resistance. Electrons on the two sides of the interface do not remain in local equilibrium with each other because of the interface resistance. Spin injection is reduced as the interface resistance is reduced. The lower panel in Fig. 23 shows the calculated spin density in the semiconductor near the interface as a function of current density. Compare with the lower panels of Fig. 20 and Fig. 21, which show the corresponding electro-chemical potentials. The injected electrons are not confined in the semiconductor in this structure, and

154

EX.BRONOLD, A. SAXENA,AND D.L.SMITH 1 00 1 ohm-crn

C

0 .c

w

10-1

'C

m 0

a

c

C

g

a

10-2

Current Density (A/cm2) FIG.22. Calculated injection current spin polarization (upper panel) and electron density spin polarization (lower panel) at room temperature as a function of current density at a spin-polarized contactlsemiconductor interface with no interface resistance for various values of the bulk resistivity of the contact material.

fairly high current densities are required to achieve a strong density polarization even when the injection current is strongly polarized. Without a tunnel barrier, spin injection is very weak for metallic ferromagnetic contacts. At thermal equilibrium, electron spins are polarized in the contact but not in the semiconductor.To achieve spin injection, the system must be driven out of equilibrium by an electric current in such a way that the electrons injected into the semiconductorare spin polarized. It is difficult to drive the electron population in a metallic contact far from local quasi-thermal equilibrium with a physically attainable current density because of its high electrical conductivity and comparatively short spin diffusion length. For a spin-polarized contact/semiconductor interface with no interface resistance, the electrons in the contact and in the semiconductor are in good thermal contact and therefore the electrons in the semiconductor also stay in local quasi-thermal equilibrium. As a result, spin injection is weak. This is the essential physical problem in achieving strong electrical spin injection. To achieve strong spin injection it is necessary to provide a mechanism that allows the

ELECIRON SPIN DYNAMICS IN SEMICONDUCTORS

155

10-3 ohmzm2

l@ ohm-cm2

-10-1

id lo3 Current Density (Ncm2)

100

101

lo4

FIG.23. Calculated injection current spin polarization (upper panel) and electron density spin polarization (lower panel) at room temperature as a function of current density for a spin-polarized contact/semiconductorinterface with a spin selective interface resistance and metallic bulk resistivity of the contact material for various values of the spin-down interface resistance.

applied current density to drive electrons out of quasi-thermal equilibrium either in the bulk contact or at the contact/semiconductor interface. A tunnel barrier with spin-dependent resistance provides such a mechanism and can significantly enhance spin injection. An insulating tunnel barrier with a spin polarized contact has spin-dependent interface resistance because of the difference in Fermi wave vectors for the two spin types in the contact material. A ferromagnetic insulator tunnel barrier can also have spin-dependent interface resistance.

18. SPIN INJECTION AT SCHOTTKY CONTACTS The discussion of the previous section centered on a conductivity mismatch between the contact and the semiconductor that can limit polarization of the injected carriers, and the possibility of using spin-dependent tunnel barriers to circumvent this difficulty. The contact and semiconductor were described simply

156

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

I I I

Tunneling region e$+

‘I

0

x=w

FIG.24. Energy diagram of the Schottky barrier, which includes the possibility of a narrow tunneling region near the interface. The highly doped region near the interface, through which electrons tunnel, is indicated by the dashed portion of the conduction band profile and corresponding doping profile. cm-3 (smallerreverse Two calculated diode characteristics are shown for vbj = 0.2 V and Nd = saturation current) and lo” ~ m - ~ .

as uniform conductive media. But real metal/semiconductor structures consist of a Schottky contact with band bending in a depletion region.’49We now consider spin-polarized electron injection from a reverse-biased Schottky contact with a depletion region in the semiconductor. The use of interfacial doping profiles to control the depletion region is investigated. We analytically solve spin-dependent continuity and drift-diffusion equations in the depletion region and examine the influence of the interface and the depletion region on the spin-polarized current and carrier densities in the semiconductor. An energy diagram for a Schottky barrier structure is shown in Fig. 24. The energy diagram includes the possibility of a narrow tunnelling region near the interface produced by a heavily doped region near the interface. Such a doping profile is illustrated in the upper panel of Fig. 24. The doping can be designed to form a sharp potential profile through which electrons tunnel. The heavily doped region reduces the effective Schottky energy barrier that determines the properties of the depletion region. The total barrier e& is divided into two parts, a tunnelling region with barrier height e$t and an effective Schottky barrier height eVbi. The potential drop in the depletion region consists of the effective Schottky barrier height plus the applied reverse bias eVR. TWO parameters of the tunneling region, its tunneling resistance and the magnitude of the reduction of the effective Schottky barrier,

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

157

can be separately controlled by the parameters of the doping profile, for example, the height and width of the heavily doped region. The inset of Fig. 24 shows calculated current-voltage characteristics for two Schottky contacts with different bulk doping levels. Spin-injection experiments are typically performed in reverse bias in which electrons are transported from the contact to the semiconductor. The calculation decouples into a part for charge currents and densities and a part for spin currents and densities. The calculation for charge currents and densities is standard.'49We use a depletion approximation for the electrostatics and the diffusiodthermionic emission model for the electron current and density. We treat the spin components in the contact and in the charge-neutral part of the semiconductor outside of the depletion region using spin-dependent drift-diffusion equations, a spin diffusion equation, and a spin-dependent interface resistance to describe tunneling as in the previous section. In the depletion region the conductivity varies with the local electron concentration a,,= epn,. Because of the large electric fields in the depletion region a spin diffusion equation is not valid and we use spin-dependent continuity equations Eq. (5.1 1). Because the electrostatic potential in the depletion region is quadratic, Eq. (5.11) has the form

a2a+ (-ax + b)-a R ax2 ax

-

- rS2 = 0,

(5.23)

where a , b and r are known constants that follow from the electrostatic solution in the depletion region. &.(5.23) can be transformed to a confluent hypergeometric equation by a change of variables and thus solved analytically in terms of two matching ~0efficients.l~~ These coefficients are determined by matching to the solutions for (pt - p ~ =) p- in the contact and in the charge-neutral region outside of the depletion region. Once the matching coefficients are known, the spin-polarized currents and electron densities can be calculated. The model can be applied both to metaysemiconductor contacts and to heterojunction contacts with injection from a heavily doped, spin-polarized semimagnetic semiconductor into a less heavily doped unpolarized semiconductor with a higher energy conduction band.72We first consider parameters appropriate to the heterostructure case. In Fig. 25 we show the calculated spin-current polarization, ( j , - j ~ ) / ( j +j J ) ,as a function of position for a series of structures with different barrier heights (negligibly small, 0.05,0.1, and 0.15 eV), an injection current density of 10 A cm-2, and a bulk doping of 5 x 10'6cm-3. The symbol x on the curves indicates the edge of the depletion region. Results from the constant conductivity model of the previous section for the same parameters are also shown. The contact is taken to be (95%) spin polarized and with a conductivity twice that of the

+

152Let 52 = (-ax + b)g(z)where z = (-ax + b)2/2u.Then g ( z ) satisfies a confluent hypergeometic equation (CHE) and has a solution of the form aM + ,9U where (r and ,9 are matching coefficients and M and U are independent solutions to the CHE.

158

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

0.8

-g c

0.4

VW=O.l5V

a --f,

0

~

0.2

t

0.0

‘

T = W K N,= 5.10’6crnJ J = 10 NcmZ

-1 00

0

20

40

^.

601

0.10 Y

V, = 0.15 V

100 Position (nrn)

200

300

FIG. 25. Calculatedcurrentpolarizationas a functionof position for various Schottkybarrier heights. The inset shows the difference in electro-chemical potentials near the interface.

collecting semiconductor. It is assumed that the contact has a lower mobility but is more heavily doped than the collecting semiconductor so that depletion occurs in the collecting semiconductor.The interface resistance is zero. A mobility of 5000 cm2/Vs, a spin diffusion length of 1 p m for the collecting semiconductor, and a spin diffusion length of 100 nm in the contact at T = 300 K is used throughout. The top two curves, which are indistinguishable, show that in the limit of small energy barrier we recover the results of the constant conductivity model. There is a strong decrease in spin injection with increasing barrier height for fixed doping. The inset of Fig. 25 shows the difference in electro-chemicalpotentials for spin-up and spin-down electrons, p - , as a function of position for the same conditions as in the main panel of Fig. 25. As the barrier height increases there is a rapid drop in the difference in electro-chemical potentials for spin-up and spin-down electrons across the depletion region. This rapid drop in p- across the depletion region is the cause of the decreased spin injection with increasing barrier height seen in the main panel of Fig. 25. The drop results because the depletion region has a low and rapidly varying electron density. The heterostructure situation depicted in Fig. 25 is idealized in the sense that spin-polarized n-type semiconductor injectors that do not require high magnetic fields and low temperatures are still being sought. However, it is feasible to grow ferromagnetic metals on semiconductors, for example, films of Fe on G ~ A sIn. ~ ~ Fig. 26 we show the calculated current spin polarization as a function of position

159

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

.? "

-

"

T=300K 0.30 N, = 5.1016cms RJ= 2 . Rr = 1cr3R~crn2 V, = 0.4 V

(a)

*

,

v,

= 0.2 v R~=2.R~=103Rcm~

,

T=300K

-1 00

.. a

N, = 5.10'6cmJ

~

-\

0

100

Position (nm) FIG.26. Calculated current polarization as a function of position for (a) various barrier height with fixed bulk doping, (b) various bulk doping with fixed barrier height, and (c) various interface resistance values with fixed doping and barrier height.

from a metallic contact (contact resistivity equal to a-cm). We have, for comparison purposes, computed all curves for 90% of the reverse saturation current density (which, of course, varies with barrier energy and bulk doping). In Fig. 26a ), polarization curves corwe show, for fixed bulk doping (5 x 1OI6 ~ m - ~current responding to different effective barrier heights (0.1, 0.2.0.3, and 0.4 eV) and a spin-selectiveresistance at the interface of a-cm2 for spin-down current and half this value for spin-up current. A typical energy barrier for FdGaAs is -0.7 eV and we have assumed a barrier lowering due to a heavily doped region near the interface. Figure 26b shows an analogous series of curves for a fixed energy barrier (0.2 eV) and different bulk doping densities (5 x 10l6,1 x 1017,5 x 1017, and 1 x lo1*cmP3)with the same interface resistance. Figure 26c shows a series of curves in which the barrier height (0.2 eV) and bulk doping (5 x 10l6~ m - ~ ) lop4, lod5 S2-cm2). are held fixed and the interface resistance is varied One can see that a combination of small effective barrier and high doping levels leads to stronger injected current polarization. Minimizing the barrier height and

160

EX. BRONOLD, A. SAXENA, AND D.L. SMITH

maximizing the bulk doping reduces the adverse effects of the depletion region on spin injection. It is also important to have a significant spin-dependent interface resistance. A significant depletion region at a Schottky contact is undesirable for spin injection. Design of the doping profile is very important to maximize spin injection. A heavily doped region near the interface can be used to form a sharp potential profile through which electrons tunnel and reduce the effective Schottky energy barrier that determines the properties of the depletion region. The doping profile should be chosen so that the potential drop in the depletion region is as small as possible, but the tunneling region must also have a significant interface resistance (of order !2-cm2). Spin injection measurements using a spin-LED configuration are sensitive to the electron density polarization in the optical recombination region. The electron density in this recombination region should be as low as possible, consistent with a small depletion region, so that it can be more easily spin polarized. 19. SPIN-DEPENDENT TUNNELING Tunnel barriers with spin-dependent interface resistance are important in order to achieve efficient spin injection at ferromagnetic metalhemiconductor interfaces. An insulatingtunnel barrier with a spin-polarizedcontact has spin-dependent resistance because of the difference in wave functions for electrons of the two spin types in the contact material. Spin-dependent tunneling resistance out of a ferromagnetic contact is the essential principle behind magnetic random access memory (MRAM) that employs metal/insulator/metal structures. A ferromagnetic insulator tunnel barrier can also have spin-dependent transmission properties. These are two examples of interface structures with spin-dependent resistance. We consider a very simple free electron model, shown in Fig. 27, to illustrate how spin-dependent tunneling can arise and lead to a spin-dependent interface conductance.The contact on the left in Fig. 27 is spin polarized and the conduction band minimum for spin-up electrons, Ect, is at lower energy than the conduction band minimum for spin-down electrons, E,J. The Fermi energy is the same for spin-up and spin-down electrons in the spin-polarizedcontact but because the bands are shifted by the ferromagnetic exchange interactions the Fermi wave vectors are different for the two spin types. As a result the spatial part of the wave functions for electrons with different spin at the Fermi surface are different. We also allow for the possibility that the insulating barrier region may be magnetic so that the energy barrier E g may be spin dependent. A straightforward calculation of the transmission coefficient for this simple free electron model gives

*

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

161

RG.27. Schematic diagram of a spin-dependent tunneling structure. Spin-dependent tunneling results either because the left contact is magnetic and the wave functions for an incident electron is different for the two spin types or because the insulating barrier is magnetic and the tunnel barrier is different for the two spin types.

where

(5.25)

Here E is the electron energy, kll is the interface parallel component of the electron wave vector, and C is the thickness of the tunnel barrier. The transmission coefficient can be spin dependent either because of the spin dependence of the magnetic contact, from k,, ,or because of the spin dependence of a magnetic tunnel barrier, from q,. Note that spin dependence arising from magnetization in the contact (appearing in the spin dependence of k,,) would not show up in a WKE3 approximation for the transmission coefficient. The tunneling current density is calculated in the usual way,

where f ( ~ is) the Fermi function and V is the voltage bias. For small voltage bias, the difference in the two Fermi functions can be replaced by an energy derivative,

162

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

and the current becomes

where kfccl,kfs, and qfo are evaluated at the Fermi energy and the spin-dependent interface conductance is

where kfs is assumed small compared to qfcl and kfco. The interface conductance can be spin dependent either because of the contact magnetization or because of a magnetic tunnel barrier. In the common case of a nonmagnetic tunnel barrier with a large barrier height, so that 4 f . l > 1 and qfo > kf,, ,the interface conductance is proportional to kfc. and (5.29) Although it should be possible to get a reasonably strong spin dependence from the contact magnetization,use of magnetic tunnel barriers in which the spin-dependent parameter appears in an exponent rather than as a linear prefactor would be a more powerful approach. The model for spin-dependenttunneling,discussed gives insight into the process but is highly idealized. The development of more realistic descriptions of spindependent tunneling, which include real materials properties, is an area of ongoing research. Realistic models that give guidance in selecting materials systems and interface structures that would yield strong spin-dependent tunneling would be very useful.

ON SPIN INIECTION 20. EXPERIMENTS

Experimentally, spin injection into a nonmagnetic semiconductor has been investigated using spin-polarized contacts consisting of an n-type doped large g-factor semimagnetic s e m i c o n d ~ c t o ? ~polarized *~~ by an external magnetic field or a ferromagnetic metal?G28 In most spin injection experiments, detection of spinpolarized injection is made using a spin-LED (light-emitting diode) configuration (see the upper panel of Fig. 16). Spin injection from a ferromagnetic contact with use of a change in voltage at a second ferromagnetic contact as a signature of spin injection (see the lower panel of Fig. 16) has also been attempted.'47Using this second approach, which operationally is a magnetoresistancemeasurement, effects of the order of 1% or less were reported. But it is difficult to unambiguously establish

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

163

spin injection by observations of such small effects.'48 In addition, it was pointed out that there are other possible explanations for the observed magnetoresistance effects. Spin-polarized electron injection has also been reported using ferromagnetic metal scanning tunneling microscopy (STM) tips and an optical detection s ~ h e m e .The ~ * optical ~ ~ signals are relatively weak in these experiments because the STM tunneling currents are rather small. Semimagnetic semiconductors provide a convenient source of spin-polarized electrons at low temperature and high applied magnetic field. The electrons can be nearly 100% spin polarized and the conductance of the semimagnetic semiconductor contact can be comparable to that of nonmagnetic semiconductor into which the spins are to be injected. These conditions remove the conductivity mismatch impediment to efficient spin injection from metal contacts that was first recognized by Schmidt et al.32The conduction band offsets can be made comparatively small and thus also minimize problems associated with a depletionregion at the heterostructure interface. In addition, the two materials forming the interface can both be zinc-blende structure semiconductors and high-quality interfaces can be grown for lattice-matched structures using molecular beam epitaxy (MBE). Particularly well-suited materials systems for this purpose are based on GaAlAsEnMnSe heterostructures that are well lattice matched and in which the conduction band offsets can be varied with alloy composition. Efficient spin injection was first reported by Fiederling et aLZ4using a Beo.o~Mno.o3Z~.~Se/Alo.o3Gao.9.1As injection structure with a GaAs quantum well for the optical emission region used for the spin detection. Spin injection current polarization as large as 90% was reported. Very similar results using similar structures were also reported by Jonker et al.23These experimental results clearly demonstrate that efficient spin injection is possible. The use of spin-polarized semimagnetic semiconductor contacts has demonstrated the possibility of efficient electrical spin i n j e c t i ~ n . ~However, ~ * ~ ~ -this ~ approach only works at low temperatures with large externally applied magnetic fields and is not practical from a technological point of view. The use of ferromagnetic contacts with a Curie temperature well above room temperature is a more practical approach. There has been exciting recent work on ferromagnetic semiconductors based on Mn doping of III-V semiconductors, for example Gal-,Mn,As?6 However, to date these materials all have a Curie temperature below room temperature and they are p-type materials, because Mn dopes III-V semiconductors p-type, and thus are not suitable for electron injection. The use of ferromagnetic metal contacts for electron spin injection is desirable. According to the model results discussed previously, two problems must be overcome for efficient spin injection from ferromagnetic contacts: (1) there must be a spindependent tunnel barrier, and (2) the effective Schottky barrier must be reduced by a large interface doping profile. The materials system that has been the most heavily investigated experimentally for spin injection from a metallic ferromagnet is Fe/GaAlAs using polarized optical luminescence as the detection scheme.

164

EX.BRONOLD, A. SAXENA,AND D.L.SMITH

For this interface the intrinsic Schottky barrier height is about 0.7eV, which must be substantially reduced by interface doping. The spin-dependent tunnel barrier is formed by the heavily doped interface region. Spin current polarization of 2%,27 a few %26, and as high as 32%28-80have been reported. In all cases the GaAs or GaAlAs layer near the interface was heavily doped so that there was most likely a tunneling region that reduced the effective Schottky barrier compared to its intrinsic value. The doping profiles were different in these three cases, which may account for the difference in the magnitude of the spin polarization of the current that was reported. In one case" current-voltage characteristics were analyzed to argue that the tunneling transport indeed took place at the Schottky contact. Experimentally efficient spin injection has clearly been observed using spinpolarized semimagnetic semiconductor contacts. Spin injection has also been observed using metallic ferromagnetic contacts, but there is a discrepancy in the magnitude of the spin current polarization reported by different groups. This difference in experimental results could result from different interface doping profiles in the samples. The theoretical model predicts that spin injection at a Schottky contact is sensitive to the interface doping profile. A direct quantitative comparison between the calculated results and experimental data is difficult at this point because the ferromagnetic metalhemiconductor interface is difficult to completely characterize and the degree of spin-dependent tunneling of a particular interface structure is yet to be understood. However, there is a qualitative correspondence between the theoretical and experimental results.

VI. Summary and Future Directions The ability to control the electron spin degree of freedom in semiconductorsis being explored as the operating principle for a new generation of novel electrical devices with the potential to overcomepower consumption and speed limitations associated with conventionalelectronic circuits, and also as a means to physically implement schemes for quantum information processing and computing.'~''.'2 Realizing the full potential of spin-based electronics requires a more complete understanding of non-equilibrium electron spin-based phenomena than is currently available. This article discussed electron spin dynamics in semiconductors.Emphasis was placed on a semiclassical kinetic description of spin relaxation and on a discussion of spin injection from spin-polarized contacts with transport described at the driftdiffusion equation level of approximation. Electron states at the bottom of the conduction band in direct bandgap zincblende structure semiconductors are approximate spin eigenstates whereas hole states at the top of the valence band are not approximately spin eigenstates. Hole states can be labeled by a quasi-angular momentum, but the quantization axis

ELECTRON SPIN DYNAMICS IN SEMICONDUCTORS

165

of the quasi-angular momentum depends on the hole wave vector and therefore changes after a scattering event. Therefore, electron spin relaxation occurs on a significantly longer timescale than hole quasi-angular momentum relaxation. Relaxation of a non-equilibriumelectron spin distribution in 111-V semiconductors is usually dominated by effects that result from the spin-orbit interaction. The Dyakonov-Perel spin relaxation process,105which results from a combination of the spin-orbit interaction and inversion asymmetry and leads to a precession of electron spin about an axis that depends on the electron wave vector, typically dominates; but the Elliott-Yafet spin relaxation p r o c e ~ s , ' ~which ~ . ' ~ results from the spin-orbit interaction mixing small-amplitude minority spin components into the electron wave function and leads to spin-flip scattering, can be important in systems with strong wave vector scattering. Both the DP and EY processes are manifestations of the spin-orbit interaction and occur because the bonding pstates that make up the top of the valence band are mixed into the conduction band states away from the zone center. Effects that increase the energy of the states electrons occupy, such as increased temperature for nondegenerate electrons or increased density for degenerate electrons, tend to decrease spin coherence times because of the larger amplitude of the bonding p-states in these electron states. In an applied magnetic field spin relaxation can also take place due to variable g-factors (the VG process). Detailed calculation of the spin lifetimes as a function of temperature, density, and external fields requires a kinetic theory, which takes not only the spin-orbit interaction-inducedband structure effects into account, but also the kinematics and dynamics of the various scattering processes. Most semiconductor device concepts that exploit the electron spin degree of freedom require an electrical means of injecting spin-polarizedcurrents in a semiconductor. Ferromagneticmetals with Curie temperatureswell above room temperature are an ideal source of spin-polarizedelectrons for electrical device purposes. Semimagnetic semiconductors with large g-factors can be strongly spin polarized by an external magnetic field at low temperature. They are probably not practical technologically as spin injection contacts, but they can provide important test structures to explore ideas about spin injection and transport. For a semiconductor and contact in which electrons on both sides of the interface are in good thermal contact, efficient spin injection can be achieved if the contact and semiconductor have similar conductivities, but not if their conductivities differed by orders of magnitude as is the case for a metal and a semiconductor. This impediment to efficient spin injection can be circumvented by the use of spin-dependent tunnel barriers. This approach allows ferromagnetic metals to be used as spin-injecting contacts and avoids the need for the low temperatures and high magnetic fields that are required for semimagnetic semiconductor spin-injecting contacts. The depletion region that occurs at real semiconductor/metal interfaces, because of the Schottky energy barrier that forms at the interfaces, or at real semiconductor heterostructures, because of the energy band offsets that occur at semiconductor

166

EX.BRONOLD, A. SAXENA, AND D.L. SMITH

heterostructures,is very detrimental to efficient spin injection. The use of selective interface doping profiles can eliminate these problems. Important requirements to achieve efficient spin injection from a ferromagnetic metal contact are the formation of a spin-dependent tunnel barrier to circumvent the conductivity mismatch problem and the use of selective interface doping profiles to overcome problems that arise from interfacial depletion regions. At present, spin selective tunneling is not adequately understood and the development of more realistic descriptions of this process, which include real materials properties and give guidance in selecting materials systems and interface structures that would yield strong spin-dependent tunneling, would be very useful. Spin relaxation and decoherence processes that occur at interfaces are also not well understood. In addition to the bulk inversion asymmetry of the constituent bulk materials and the structure inversion asymmetry due to a particular heterostructure design, interfaces can have an additional source of asymmetry if the bulk constituents at the interface share no common atoms. This native interface asymmetry can have a large effect on spin lifetime^.^^ Most of the kinetic theories used to describe spin relaxation assume translational invariance and are not suitable to investigate spin relaxation at or due to an interface. It is possible that spatial inhomogeneity opens new spin decay channels, which do not To develop an adequate description appear in spatially homogeneous sit~ations.'~' of spin dynamics in spatially inhomogeneous systems, such as interfaces, superlattices, and heterostructures under bias, constitutes an important future research direction.

Acknowledgments We are pleased to acknowledge many collaborators who have made major contributions to this work, including John Albrecht, Scott Crooker, Ivar Martin, Paul Ruden, and Richard Silver. This research was supported by the Los Alamos Directed Research and Development Program and the Defense Advanced Research Projects Agency.

Author Index Numbers in parentheses are reference numbers and indicate that an author’s name is not cited in the text.

A Abragam, A., 42(160), 55(160) Adachi, T., 77(62,64), 101(62,64),125(62,64) Akai, H., 13(99) Akazaki,T., 133(145) Akinaga, H., 2(8) Albrecht, J.D., 79(86,87), 141(86, 87) Alivisatos, A.P., 44(166, 168). 74(21) Altunkaya, E., 77(69), lOl(69). 125(69),166(69) Alvardo, S.F., 79(90,91), 163(90,91) Alvarez, G., 8(83), lO(83) Ando, K., 7(64) Andrada e Silva, E.A. de, 75(47), 133(143) Aronov, A., 42(159) Aronov, A.G., 42(159), 43(162), 79(85), 97(106), 104, 105(114), 132(106,114) Ashenford, D.E., 75(44) Averkiev, N.S.,105(125), 138(125) Awschalom, D.D., 2(6,9), 3(19), 5(6,36,37,38, 39,40), 6(6), 9(90), 10(91), 28(134), 29(134), 36(36, 147). 38(6), 39(36), 40(37), 41(36,37), 42(37), 43(37), 44(168),45(38, 169, 170). 46(38), 47(39,40), 49(39), 50(39), 51(39), 52(39), 53(39),56(147), 57(169),59(178), 60(30),74(1, 10. 11, 13, 14, 15, 16, 17,21), 76(56),77(11, 13, 15, 16, 17),78(1, 11.72, 73,75,76), 101(13, 15, 16, 17). 125(13, 15, 16, 17). 127(16), 128(16),129(16), 139(11), 157(72),160(1), 163(76), 164(1, 11)

B Baibich, M.N., 2(3), 75(38) Bar-Ad, S., 36(155) Bar-Joseph, I., 36(155) Barnas, J., 75(39) Barnes, C.H.W., 76(54) Baron, F.A., 76(57) Bassini, F., 133(143) Bastard, G., 105(123)

Baumberg, J.J., 36(153, 154). 37(153), 39(154), 74(13), 77(13,68), 78(75), 101(13,68), 125(13,68), 135(68), 136(68) Baurngart, P., 75(40) Bawendi, M.G., 44(167) Beloglazov, A.A., 4(22) Bennett, B.R., 62(183), 74(23), 79(23), 138(23), 139(23), 140(147), 141(23,148). 162(23, 147). 163(23. 148) Berciu, M.,8(82), 10(82), 78(78) Berry, J.J., 5(39,40), 47(39,40), 49(39), 50(39), 51(39), 52(40), 53(40), 67(202),78(72,73), 157(72) Beschoten, B., 5(41), 12(95),41(41), 78(76), 163(76) Bhatt, R.N., 8(82), 10(82),78(78) Bir, G., 42(159) Bir, G.L.,97(106), 104, 132(106) Bishop, A.R., 75(50) Bjorken, J.D., 82(94) Blinowski, J., 7(58), ZO(121) Blonder, G.E.,30(142) Boeck, J. De, 74(29),79(29), 138(29) Bogess, T.F., 77(69), 101(69), 125(69), 166(69) Boggess, T.F., 77(67), 101(67), 125(67) Boguslawski, P., 43(163), 105(113) Borghs, G., 74(29), 75(29), 138(29) Borhgs, G., 77(65), 101(65), 125(65) Boukari, H., 7(78) Braden, J.G., 30(140). 32(140) Brinkman, W.F., 70(206) Britton, R.S.,77(66), 101(66), 125(66) Bronold, EX., 105(120, 132). 106(120, 132). 107(120), 125(120, 132), 126(120), 127(120) Broto, J.M., 75(38) Brum. J.A., 77(65), 101(65), 125(65) Buhrman, R.A., 74(1,22), 75(53), 78(1), 160(1), Burkard, G., 75(49) Bychov, Yu. A., 91(101), 132(101) Byers, J.M., 36(150, 152). 74(31)

167

168

AUTHOR INDEX C

Camilleri, C., 78(77) Camley, R.E., 75(39) Campion, R.P., 14(104) Carlsson, A.E., 9(87) Cam, D.M., 74(26), 79(26,83), 138(26, 83), 139(26), 162(26) Cartoixa, X., 75(48) Chambers, S.A., 7(74,79) Chazalviel, J.-N., 105(110) Chazelas, J., 75(38) Chelikowsky, J.R., 7(59), 30(59) Chen, Y.J., 77(60), 101(60),125(60) Cheong, H.-D., 74(23), 79(23), 138(23), 139(23),141(23), 162(23),163(23) Chiba, D., 14(103), 17(115), 19(115), 23(103) Chtchelkanova, A.Y., 74(1), 78(1), 160(1) Chun, S.H., 30(137), 67(203),68(137),69(137), 70(137) Cibert, J., 78(77) Ciuti, C., 62(186),63(187),74(20), 75(51) Cleaver, J.R.A., 77(68), 101(68), 125(68), 135(68), 136(68) Cohen-Tannoudji, C., 56(174) Combescot, M., 135(146) Combescot, R., 135(146) Creuzet, G., 75(38) Crooker, S.A., 10(91),74(13, 14). 77(13), 78(75), 101(13), 125(13), 126(141), 127(141) Crowell, P.A., 74(26), 79(26,82,83), 138(26, 82.83). 139(26),162(26) Cunningham, J.E., 77(59), 101(59),125(59)

Dijken, S. van, 79(92) Dimmock, J.O., 85(98) Dingle, R., 2(2) Divincenzo, D.P., 4(35), 5(42), 36(42) Dong, J., 144(151) Drell, S.D., 82(94) Driscoll, D.C., 76(56) Dupont-Roc, J., 56(174) D’yakonov, M.I., 41(158),78(77), 93(105), 104(105), 105(122), 106(105), 119(105), 124(105),132(122),164(105) Dynes, R.C., 70(206) Dzhioev, R.I., 77(70,71), 101(70,71), 125(70,71)

E Eberl, K.,75(43) Edmonds, K.W., 17(112, 113), 19(113), 20(122),26(112) Efros, A.L., 77(71), 101(71),125(71) Egues, J.C., 75(49) Ehrenreich, H., 9(87) Eitenne, P., 75(38) Elliott, R.J., 41(157), 92(103), 104(103), 106(103), 165(103) Epstein, R.J., 3(15), 60(15), 63(15) Eriksson, O., 74(37) Erve, O.M.J. van ’t, 79(80), 138(80), 164(80)

F D Dagotto, E., 6(45), 8(83), lO(83) Damen, T.C., 77(59), 101(59),125(59) Das, B.,3(13), 34(13),60(13), 75(42), 76(42), 138(29) Das, J., 74(29), 79(29) Dana, S., 3(13), 34(13), 60(13), 75(42), 76(42) D’Aubigne, Y. Merle, 3(20), 8(20), lO(20) Daughton, J.M., 74(1), 78(1), 160(1), i64(1) Dhar, S., 7(80) Dieny, B., 75(40,41) Dietl, T., 3(18,20), 7(20,54,57), 8(20,54,57), 10(20,54,57), 11(57), 12(54.93),26(130), 30(54), 78(79)

Fabian, J., 4(31), 74(6,35,36), 75(52) Ferrand, D., 7(77), 74(32), 79(32), 141(32), 163(32) Ferreira, R., 105(123) Fert, A., 75(38) Feynman, R.P., 1 Fiederling, R., 4(27), 63(27), 66(27), 74(24), 79(24,84), 138(24,84), 141(24),162(24), 163(24,84) Filip, A.T., 62(184),74(32), 79(32), 141(32), 163(32) Fischer, A., 77(61), 101(61),125(61) Fishman, G., 42(161), 105(112) Fitzgerald, R., 74(7) Hack, F., 74(13), 77(13), 78(75), 101(13), 125(13)

169

AUTHOR INDEX Flatt6, M.E.,4(30), 36(150, 151, 152). 60(30), 64(189),74(10, 19.30.31). 77(67,69), 79(88, 89), 101(67,69), 105(19), 125(67,69), 138(19),166(69) Folk, J.A., 5(44) Foxon, C.T., 77(58), 101(58), 125(58) Frankel, R.B., 6(48) Fransson, J., 74(37) Friederich, A,, 75(38) Furdyna, J.K., 3(17),4(24), 11(17), 12(17), 78(74) Fuss, A., 75(39)

G Galazka, R.R., 6(48) Gammon, D., 77(70,71), 101(70,71), 125(70,71) Garcia, R., 74(14), 139(11) Gardelis, S., 62(181),76(54) Gibbs, H.M., 75(46) Glazman, L., 83(97),97(97) Glazov, M.M., 105(126),138(126) Godart, C., 6(47) Godlevsky, V.V., 7(59), 30(59) Golubev, V.G., 106(134), 107(134) Golub, L.E., 105(125), 138(125) Goovaerts, E., 74(29), 79(29), 138(29) Gossard, A.C., 76(56) Grabs, P., 79(84), 138(84), 163(84) Grevatt, T., 77(66), 101(66), 125(66) Grunberg, P., 75(39) Grundler, D., 74(9) Gupta, J.A., 36(147),44(168),45(169), 56(147), 57(169),74(21) Gurney, B.A., 75(40,41)

H Hagele, D., 75(43,44,46) Hammar, P.R., 62(183), 140(147), 141(148), 162(147), 163(148) Hanbicki, A.T., 64(193), 66(193),74(25,28), 79(28,80), 138(28,80), 139(28),162(28), 164(28,80) Harley, R.T., 77(66), 101(66), 125(66) Harris, J., 79(92) Hasenberg, T.C., 77(69), 101(69), 125(69), 166(69)

Hashimoto, Y., 17(109) Hass, K.C., 9(87) Haug,H., 111(138) Haury, A., 3(20), 7(76), 8(20), lO(20) Hayashi, T., 17(109), 34(144) Heberle, A.P., 36(153),37(153), 77(68), 101(68), 125(68), 135(68), 136(68) Heimbrodt, W., 75(44) Hermann, C., 54(172) Higo, Y., 30(136), 65(136),67(136) Hill, N.A., 7(62) Horing, N.J.M., 105(128), 138(128) Hosaka, H., 76(57) Hubner, J., 75(44) Hu, C.-M., 62(185), 133(145) Hwang, E.H., 8(81), 23(125), 24(125)

I Iikawa, F., 77(65), 101(65), 125(65) Isakovic, A.F., 64(196), 74(26), 79(26,82,83), 138(26,82,83), 139(26), 162(26) Ishiwata, Y.,20(120) Itoh, H.,34( 145) Itskos, G., 74(25,28), 79(28, 80). 138(28,80), 139(28), 162(28),164(28,80) Ivanov-Omskii, V.I., 106(134), 107(134) Ivchenko, E.L., 105(108,111, 126). 127(108), 132(111), 138(126) Iwamura, H., 34(145) Iye, Y., 17(109)

J Jain, M., 7(59), 30(59) Jauho, A.P., 11 l(138) Jiang, X., 79(92) Johnson, M., 62(183), 140(147), 141(148), 162(147),163(148) Johnston-Halperin, E., 29( 135), W30) Jones, E.D., 44(164) Jonker, B.T., 4(29), 63(29), 74(23,25,28), 79(23,28,80,81), 138(23,28,80,81), 139(23,28), 141(23), 162(23,28), 163(23), 164(28,80) Julliere, M., 65(199),68(204) Jungwirth, T., 7(53), 7(55,56), 8(53,55,56), 10(53,55,56), 11(53),24(56), 28(133)

170

AUTHOR INDEX

K Kachorovskii, V. Yu., 105(122), 132(122) Kacman, P., 7(58), 20(121) Kainz, J., 105(127), 138(127) Kaminski, A., 8(81,84), 10(84),23(125), 24( 125) Kane, B.E., 36(148) Kasuya, T., 6(46) Katayama-Yoshida, H., 7(61) Kato, Y., 36(146), 55(146), 76(56) Katsumoto, S., 17(109) Katzer, D.S., 77(70,71), 101(70,71), 125(70,71) Kavolin, K.V., 77(70), 101(70), 125(70) Kawakami, R.K., 3(14), 15(106),28(134), 29(134), 57(14), 58(14), 63(14) Kawamura, T.,4(23) Keim, M., 74(24), 79(24), 138(24), 141(24), 162(24), 163(24) Keldysh, L.V., 108(135) Kesteren, H.W. van, 77(58), 101(58), 125(58) Khaetskii, A.V., 83(97), 97(97) Khitrova, G., 75(46) Kikkawa, J.M., 2(9), 5(36,37,38,39), 9(90), 36(36), 39(36), 40(37), 41(36,37), 42(37), 43(37), 45(38), 46(38), 47(39), 49(39), 50(39), 51(39), 59(178), 74(15, 16, 17). 77(15, 16, 17). 78(73), lOl(15, 16, 17). 125(15,16, 17). 127(16), 128(16), 129(16) Kim, K.W., 76(57), 105(119) Kinder, L.R., 66(200), 67(200), 69(200) Kioseoglou, G., 74(23,25,28), 79(23,28, SO), 138(23,28,80), 139(23,28), 141(23), 162(23. 28), 163(23), 164(28,80) Kiselev, A.A., 76(57) Kitagawa, I., 30(139) Klapwijk, T.M., 30( 142) Klar, P.J., 75(44) Knobel, R., 36(147), 56(147) Kobayashi, H., 13(102) Kobayashi, S., 4(34) Kohler, K., 36(153), 37(153) Konig, J., 7(55), 8(55), 10(55),26(130) Korenev, V.L., 77(70,71), 101(70,71), 125(70,71) Kosaka, H.,76(57) Koshihara, S.,4(32) Kossut, J., 4(24), 9(88) Koster, G.F., 85(98)

Kostial, H., 74(27), 79(27), 138(27), 139(27), 162(27) Kronik, L., 7(59), 30(59) Ku, K.C., 14(105), 17(105), 19(105),20(105), 35( 105) Kuroda, T., 77(63), 101(63), 125(63) Kutznetsov, A.V., 108(137), 109(137), 112(137)

L Lampel, G., 42(161), 59(177), 105(112) Landau, L.D., 108(136), 109(136), 112(136), 116(136, 139) Landoldt-Bomstein, 126(140) Landsberg, P.T.,86(99), 87(99) Larson, B.E., 9(87) Lau, W.H., 36(151, 152),64(189),74(19), 77(67,69), lOl(67.69). 105(19), 125(67,69), 138(19), 166(69) Lawaetz, P., 83(95) Lazarev, M.V., 77(70), 101(70), 125(70) Lee, W.Y., 62(182) Levy, J., 74(14), 76(56) Li, C.H., 79(80), 138(80), 164(80) Lifshitz, E.M., 108(136), 109(136), 112(136), 116(136, 139) Limmer, W., 26(132) Linfield, E.H., 76(54) Lommer, G., 133(142) Look, D.C., 16(108) Loss, D., 2(6), 5(6,42), 6(6), 36(42), 38(6), 43(42), 74(11). 75(49), 77( 11). 78( 1 1). 83(97), 97(97), 139(11), 164(11) Louie, R.N., 75(53) Lunn, B., 75(44)

M MacDonald, A.H., 7(55), 8(55), 10(55),23(124), 26(130) Macfarlane,R., 79(92) Madelung, O., 126(140) Magno, R., 79(80), 138(80), 164(80) Mahadevan, P.,7(63), 8(86), 13(86) Maialle, M.Z., 77(65), 101(65), 125(65) Mailhiot, C., 91(102) Majewski, J.A., 7(58) Malajovich, I., 5(39,40), 47(39,40), 49(39), 50(39), 51(39), 52(40), 53(40), 78(72,73), 157(72)

171

AUTHOR INDEX Malcher, F., 133(142) Malinowski, A., 77(66), 101(66), 125(66) Mallory, R., 79(80), 138(80),164(80) Marcus, C.M., 5(44) Margulis,A.D., 105(109, 115). 107(109), 132(115) Margulis, VLA., 105(109,115),107(109), 132(1 15) Martin, I., 105(120), 106(120), 107(120), 125(120). 126(120). 127(120) Marushchak, V.A., 74(2) Mascarenhas, A., 44(164) Matsukura, F., 6(52), 7(54), 8(54), 10(54), 12(54), 14(103),17(115),19(115), 22(52), 23(103), 24(52), 30(54), 77(62,64), 78(76), 101(62,64), 125(62,64), 163(76) Matsumoto, Y., 7(73) Mattan, R., 30(138),66(138) Mauger, A., 6(47) Mauri, D., 75(41) Mayr, M., 8(83), lO(83) McGuire, J.P., 62(186),63(187),74(20), 75(51) Medvedkin, G.A., 7(72) Meier, F., 3(12), 42(12), 100(107) Meitu, H., 105(124) Meltser, B.Y., 77(70), 101(70), 125(70) Merkulov, LA., 77(71), 101(70), 125(70) Meservey, R., 66(200),67(200),69(200) Metin, S.,75(40) Metiu, H., 105(124), 112(124) Minervin.1.G.. 106(134). 107(134) Mlynek, J., 56(175) 74(24,32), 79(24,32,84), Molenkamp, L.W., 138(24,84),141(24,32),162(24),163(24,32, 84) Molnhr, S. von, 74(1), 78(1), 160(1),164(1) Monzon, F.G., 62(180), 141(148), 163(148) Moodera, J.S.,66(200),67(200), 69(200) Motsnyi, V.F., 64(194). 74(29), 79(29), 138(29) Mourokh, L.G., 105(128),138(128) Mryasov, O.N., 7(60) Munekata, H., 6(49), 13(101) Murray, C.B., 44(167) Muto, S., 77(63), 101(63),125(63) Myers, R.C., 76(56)

Nazmul, A.M., 4(34), 18(116) Nemeth, S., 71(208) Nestle, N., 75(43) Nikitin, P.I., 4(22) Ning,C.Z., 105(117), 112(117) Ning, M., 105(129),112(129) Nishikawa, Y.,77(63), 101(63), 125(63) Nishinaga, T., 34(144) Nitta, T., 133(145) Norris, D.J., 44(167)

0

Oestreich, M., 75(43,44,45,46) Ogale, S.B.,7(75) Ogawa, T., 30(139) Ogg, N.R., 106(133),107(133) Ohba, H., 4(23) Ohno, H., 2(8, lo), 4(25,33), 5(33). 6(50,51, 52), 7(54), 8(54), 10(54), 12(54), 13(50,51), 14(103),15(51),16(10), 17(115),19(115). 22(52), 23(103), 24(52), 27(10), 30(25,54), 63(28), 66(28), 74(12, 18), 77(62,64), 78(18, 76). 101(62,64), 125(62,64),139(11), 163(76), 164(12) Ohno, Y., 4(28), 77(62,64), 78(76), 101(62,64), 125(62,64),163(76) Ohya, S.,13(102) Oiwa, A.. 13(101) Okabayashi, J., 12(96) Olesberg, J.T., 36(151),74(19), 77(67,69). lOl(67, 69), 105(19), 125(67, 69). 138(19), 166(69) Onodera, K.,4(23) Oppen, F.von. 8(85) Osaka, J., 133(145) 74(24), 79(24,84), 138(24,84), Ossau, W., 141(24),162(24), 163(24.84) Ostreich, T., 39(156) Osutin, A.V., 106(134), 107(134) Ouyang, M., 45(170) Overberg, M.E., 7(65)

P

N Nagai, Y., 12(97) Nawrocki, M., 78(77)

Paget, D., 59(177) Palanguri, R.J., 30(141) Palmstrom, C.J., 74(26), 79(26,82,83), 138(26, 82,83), 139(26),162(26)

172

AUTHOR INDEX

Panguluri, R.J., 68(204) Parkin, S.S.P., 75(40,41), 79(92) Park, J.-H., 69(205) Park, Y.D., 7(71), 74(23,25), 79(23), 138(23), 139(23), 141(23), 162(23), 163(23) Patel, C.K.N., 3(21) Peng, X.,44(168), 74(21) Perel, V.I., 41(158), 93(105), 106(105), 1 19(105). 124(105). 165(105) Petroff, F., 75(38) Petroff, P., 44(164) Petrou, A,, 74(23,25,28), 79(23,28, 80), 138(23,28,80), 139(23,28), 141(23), 162(23, 28), 163(23), 164(28,80) Petukhov, A.G., 74(25) Pfeffer, P., 133(144, 145) Pikus, G.E., 42(159), 43(162), 74(2), 79(85), 97(106), 104, 105(114, 116). 132(106, 114, 116) Ploog, K.,77(61), 101(61), 125(61) Ploog, K.H., 74(27), 79(27), 138(27), 139(27), 162(27) Poel, W.A.J.A. van der, 77(58), 101(58), 125(58) Polyakov, D., 106(134), 107(134) Potashnik, S.J., 17(1 10, 111). 18(1 lo), 19(1lo), 21(110, 111). 2301 1). 25(111), 67(110) Potok, R.M., 5(44) Preskill, J., 55(173) Prim,G.A., 2(4), 74(4), 76(55), 158(55) Puller, V.I., 105(128), 138(128)

Q Qi, Y., 144(151)

R Ramsteiner, M., 74(27), 79(27), 138(27), 139(27), 162(27) Rao, T.,105(129), 112(129) Rashba, E.I., 62(179), 64(190),66(190), 74(33), 79(33), 91(100, 101). 132(100, 101). 141(33) Reed, M.L., 7(66) Renaud, P., 79(90), 163(90) Reuscher, G., 74(24), 79(24), 138(24), 141(24), 162(24), 163(24) Rhle, W.W., 75(43,44) Richards, D., 77(61), 101(61), 125(61)

Rippard, W.H., 74(22) Ritchie, D.A., 76(54), 77(66), 101(66), 125(66) Rocca, G.C. La, 75(47), 133(143) Rosatzin, M., 56(175) Rossler, U., 105(127), 133(142), 138(127) Roukes, M.L., 28( 134). 29( 134). 62(180). 74( 1). 78(1), 141(148), 160(1), 163(148), 164(1) Rowell, J.M., 70(206) Rudolph, J., 75(46) Ryskin,A. Ya.. 105(111), 112(111)

S Sadowski, J., 15(107) Safarov, V.I., 59(177), 74(29), 75(29), 138(29) Saito, H., 7(64) Salis, G., 54(171), 55(171) Samarth, N., 2(6, 1I), 3(19), 5(6,39,40), 6(6), 9(90), 10(91),36(36, 147). 38(6), 39(36), 41(36), 47(40), 49(39), 50(39), 51(39), 52(40), 53(40), 56(147), 60(1 I), 74(10, 11, 13, 14, 15). 77(11, 13, 15), 78(11,72,73,75), 101(13, 15). 125(13, 15), 139(11), 157(72), 164(11) Sandalov, I., 74(37) Sandhu, J.S., 77(68), 101(68), 125(68), 135(68), 136(68) Sanvito, S., 7(62) Sapoval, B., 59(177) Sanna, S. Das, 4(31), g(81.84). 10(84),23(125), 24(125), 30(143), 33(143), 74(6,8,35,36), 75(52) Sato, A., 77(62), 101(62), 125(62) Sato, K.,7(61) Sawicki, M., 23(126) Saxena, A., 75(50), 105(120, 132). 106(120, 132), 107(120), 125(120, 132). 126(120), 127(120) Scalbert, D., 78(77) Schser, F., 8(85) Schilfgaarde,M. van, 7(60) Schliemann, J., 7(55), 8(55), 10(55),23(124) Schmeltzer, D., 75(50) Schmidt, G., 36(149), 62(149), 66(149), 74(24, 32). 79(24,32,84), 138(24,84), 141(24,32), 162(24), 163(24,32,84) Schneider, H., 77(61), 101(61), 125(61) Schonhammer, K.,39(156) Schonherr, H.-P., 74(27), 79(27), 138(27), 139(27), 162(27)

173

AUTHOR INDEX

Schott, G., 24(127) Schultz, B.D., 74(26), 79(26,82,83), 138(26, 82,83), 139(26), 162(26) Seeger, K., 80(93), 83(93), 84(93) Semenov, Y.G., 78(77) Semenov, Yu.G., 105(121) Seong, M.J., 26(131) Severens, A.L.G.J.,77(58), 101(58), 125(58) Shah, J., 77(59), 101(59), 125(59) Sham, L.J.,39(156), 62(186), 63(187), 74(3, 20). 75(51), 77(59), 101(59), 105(3), 125(59) Shaw, E.D., 3(21), 77(69), 101(69), 125(69), 166(69) Shelby, R.,79(92) Shen, A., 6(52), 12(94), 22(52), 24(52) Shimizu, H., 34(144) Shirai, M., 30(139) Shono, T., 25(128), 26(128) Silver, R.N., 64(191), 74(34), 79(34), 141(34) Simmons, M.Y., 77(66), 101(66), 125(66) Singley, E.J., 12(98) Slichter, C.P., 83(96), 95(96) Slobodosky, A., 4(26) Slupinski, T., 13(101) Smimov, A. Yu., 105(128), 138(128) Smith, C.G., 76(54) Smith, D.L., 64(191), 74(34), 75(50), 79(34,86, 87), 91(102), 105(120, 132). 106(120, 132), 107(120), 125(120, 132). 126(120), 127(120), 141(34,86,87) Smorchkova, I.P., 5(36), 9(90), 36(36), 39(36), 41(36), 74(15), 77(15), 101(15), 125(15) Solomon, G.S., 79(92) Song, P.H., 105(119) Sonoda, S., 7(67) Son, P.C. van, 142(150) Sorensen, B.S., 17(114), 19(114) Speriosu, V.S., 75(40,41) Srinivas, V., 77(60), 101(60), 125(60) Statz, H., 85(98) Stepanova, M.N., 77(70), 101(70), 125(70) Stone, M.B., 20( 1 18) Story, T., 6(48) Strand, J., 64(197, 198). 74(26), 79(26,82,83), 138(82, 83, 126). 139(26), 162(26) Stroud, R.M., 64(195), 74(25) Sugahara, S., 4(34), 18(116), 66(201), 71(201) Sugawara, Y., 6(52), 22(52), 24(52) Suter, D., 56(175) Suzuki, N., 30(139)

Sze, S.M., 141(149), 142(149), 156(149), 157(149)

T Tackeuchi, A., 77(62,63), lOl(62.63). 125(62, 63) Takahashi, R.,34( 145) Takamura, K.,14(103), 17(115), 19(115), 23(103) Takayanagi, H., 133(145) Tanaka, M., 4(34), 13(102), 18(116), 30(136), 34(144), 57(176), 65(136), 66(201), 67(136), 7 1(201) Tang, H.X., 28(134), 29(134), 141(148), 163( 148) Tang, J.M., 13(100), 21(100) Tatarenko, S., 78(77) Tehrani, S., 2(5) Teppe, F., 78(77) Terauchi, R.,77(62,64), 101(62,64), 125(62, 64) Theodoropoulou, N., 7(68,69) Theurich, G., 7(62) Timm, C., 8(85) Ting, D.Z.Y., 75(48) Tinkham, M., 30( 142) Titkov, A.N.,43(162),74(2), 105(114, 116). 132(114) Treger, D.M., 74(1), 78(1), 160(1) Triques, A.L.C., 77(65), 101(65), 125(65) Tulchinsky, D.A., 74(14)

U Umansky, V., 5(44) Upadhyay, S.K., 75(53) Urdanivia, J., 77(65), 101(65), 125(65)

V Vandau, EN., 75(38) Vandersypen, L.M.K., 5(43), 36(43) VanEsch,A., 18(117),22(117) VanKempen, H.,142(150) Van Roy, W., 71(207), 74(29), 79(29), 138(29) Varfolomeev, A.V., 105(111), 132(111) Vignale, G., 74(30) Vina, L.,74(5), 77(5,59), lOl(5.59). 125(5,59)

174

AUTHOR INDEX W

Waag, A., 74(24), 79(24), 138(24), 141(24), 162(24) Wada, O., 77(63), 101(63), 125(63) Wagner, J., 77(61), 101(61), 125(61) Wang, R., 79(92) Wassermeier, M., 74(27), 79(27), 138(27), 139(27), 162(27) Wees, B.J. van, 74(32), 79(32), 141(32, 148), 163(32, 148) Weisbuch, C., 54(172) Welp, U., 25(129) Weng, M.Q., 105(129,130, 131). 112(129, 130, 131). 166(131) Wheeler, R.G., 85(98) Wilhoit, D.R., 75(40,41) Willander, M., 105(125), 138(125) Winkler, R., 105(127), 138(127) Wojtowicz, T., 7(70), 13(70), 78(77) Wolff, P.A., 6(48), 9(89) Wolf, S.A., 2(7), 74(1), 78(1), 160(1), 164(1) Wong, T.M., 66(200), 67(200), 69(200) Wood, C.E.C., 77(60), 101(60), 125(60) Wu, M.W.,, 105(117, 118, 124, 129, 130, 131), 112(117, 118, 124, 129, 130, 131). 166(131) Wyder, P., 142(150)

X Xing, Y., 144(151)

Y Yablonovitch, E., 76(57) Yafet. Y., 92(104). 104(104), 106(104), 165(104) Yamagata, S., 7(64) Yanase, A., 6(46) Yang, M.J., 62(183), 140(147), 141(148), 162(147), 163(148) Yasar, M., 79(80), 138(80), 164(80) Young, D.K., 3(16), 64(188), 78(76), 163(76) Yu, C., 77(67,69), 101(67,69), 125(67,69), 166(69) Yu, K.M., 20(119), 60(30) Yuldashev, Sh. Yu., 22(123) Yu, Z.G., 64(189), 79(88,89)

Z Zachachrenya, B.P., 3(12), 77(70,71), 100(107), 101(70,71), 105(111), 125(70,71), 132(111) Zawadski, W., 133(144, 145) Zayets, V., 7(64) Zener, C., 11 Zhu, H.J., 64(192), 74(27), 79(27), 138(27), 139(27), 162(27) Zinn, W., 75(39) Zunger, A., 7(63), 8(86), 13(86) Zutic, I., 4(31), 30(143), 33(143), 74(35,36), 75(52)

Subject Index

A

“Digital ferromagneticheterostructures,” 14-15 Dyakanov-Perel (DP) mechanism spin coherence, 4 1 ,4 2 4 3 spin dynamics, 93, 118, 120-124, 165 spin relaxatiodquantum well, 132-133, 134, 135, 136-137, 138 zinc-blende semiconductors, 94-96

AHE (anomalous Hall effect), 26.27-28,34,35 Andreev reflectionfspectroscopy,30.33.68 Anisotropic magnetoresistance (AMR), 28 Annealing experiments, 17-21 Anomalous Hall effect (AKE),26,27-28,34,35

B Band states (away from the zone cluster), 86-9 1 “Bare” k m a n effects, 3 Barkhausen jumps, 29 BDR (Brinkman-Rowell-Dynes)model, 70-71 Berry’s phase effect, 28 BIA (bulk inversion asymmetry), 91 Bir-Aronov-Phs mechanism, 4142,97 Blonder, Tinkham,Klapwijk (BTK) theory, 30, 31.32-33 Brillouin function of magnetization, 9-10 Brinkman-Rowell-Dynes (BDR) model, 70-71 BTK (Blonder, Tinkham, Klapwijk) theory, 30, 31.32-33 Bulk inversion asymmetry (BIA), 91

E Effective mass Hamiltonian (n-type semiconductors), 106-108 Electric dipole selection rules, 36-37, 63 Electron-hole exchange scattering, 96-97 Electron spin degree of freedom new technologies, 3,4, 164, 165 spin dynamics, 76,91,94, 103, 104,106,138 Elliott-Yafet (EY) mechanism spin coherence, 41,4243 spin dynamics, 92,114,118, 120-124.165 spin relaxatiodquantum wells, 132, 138 Excitons, 38,39 EY mechanism. See Elliot-Yafet (EY) mechanism

C

CdSe quantum dots, 44-45 Collision integrals, 112-1 15 Condensed matter physics discoveries, 2 Conductivity mismatch, 63,64,66, 79, 141 Conductivity model, 144155 Constant conductivity model, 144-155 Coulomb barrier, 2 1

D Datta-Das spin field effect transistor (“spinFET), 3 4 , 6 0 6 3 Defect-specific models, 8 Density matrix equation, 108-1 12 Diamond crystal structure, 80.82.83

F Faraday effect, 3-4 Faraday geometry, 36.39 Faraday-rotation spectroscopies, 77, 100-101. See also Time-resolved Faraday rotation Ferromagnetic heterojunction bipolar transistor, 60 Ferromagnetic metal scanning tunneling microscopy (STM), 79 Ferromagnetic semiconductors approaches to understanding, 8 bound magnetic polarons, 9.10 Brillouin function of magnetization, 9-10 carrier-mediated fernmagnetism basis, 8-13

175

176

SUBJECT INDEX

criteria for identifying, 33-34, 35 crystal growth, 6.7, 13 d-d superexchange, 9 dopant impurities, 8 example functionality, 4.5 exchange interactions with, 8-9 free carriers, 9, 10 Kohn-Luttinger valence band structure, 11 local moments, 9, 12 magnetic anisotropy, 12 Mn-doped HI-V, 6-8.8-14 overview, 6-8 p-as-d exchanges, 9, 10, 11 Ferromagnetic semiconductordGa1-,Mn,As bound magnetic polarons, 10 energy gaps, 3 1 Fermi velocity mismatch, 32-33 Ga planar junctions, 3CL31, 32 growth of annealing experiments, 17-21 conditions for, 1 4 1 6 Curie temperature, 17-18, 19,21 defects, 14, 15, 16-17.20-21 MBE systems and, 14, 15, 16, 18 substrate temperature, 14-15 magnetic properties, 21-27 m a g n e t i s d n concentration, 25-27 magnetisdtemperature relationship, 2 1-24, 25-26 magneto-transport, 27-29 overview, 6-7, 8 spin polarization measurements, 29-33 Ferromagnetic semiconductorsnnl-,Mn,As, 6-7,8 Ferromagnetic semiconductor tunnel junctions, 60,61,65-71 Fokker-Planck equation (spin density), 108, 115-1 18 Free-particle Schrijdinger equation, 11

G Ge energy band structure, 80,8 1, 82.83 G-factors (of semiconductors) description, 11,40,47, 50, 78 effects of, 37.52.53 effects on, 38.54

electronic structure, 102-104 measuring, 39.45 “Giant magnetoresistance” (GMR) effect, 2 Ginzburg-Landau free energy functional, 11

H Hall effect anomalous Hall effect (AHE), 26.27-28, 34,35 “planar Hall effect,” 28-29 spinFET, 62 Hamiltonian n-type semiconductors, 106-108 spin-dependent Hamiltonian, 83-85 Heterostructures overview, 1-2 Hybrid ferromagnetlsemiconductor heterostructures, 57-60 Hyperfine interaction, 97-98

I “Imprinting” effect (ferromagnet), 57-60 Information processing. See Quantum information processing InSb-based spin-flip laser, 3 Interstitial (Mnl) sites, 20-21 Inversion asymmetry, 92-96

K Keldysh Green functions, 108-1 1 I Kerr-rotation spectroscopies, 77, 100-101. See also Time-resolved Kerr rotation Kinetic theory application overview, 125 spin relaxation bulk semiconductors, 125-132 quantum wells, 132-138 Kinetic theorylspin dynamics collision integrals, 112-1 15 density matrix equation, 108-1 12 effective mass Hamiltonidn-type semiconductors, 106- 108 Fokker-Planck equation (spin density), 108, 115-118 overview, 104-106 spin relaxation rates, 118-124 Kohn-Luttinger valence band structure, 11

177

SUBJECT INDEX

L Larmor frequency/precession, 37,38,39,49, 57,58 “Larmor magnetometer,” 59

M Magnetic circular dichroism, 12-13, 34, 35 Magnetic field tunable devices, 4 “Magnetic random access memory” (MRAM), 2, 160 Magnetic semiconductors. See also Ferromagnetic semiconductors crystal growth, 7 examples, 7 as semiconductor spintronics approach, 2-3 Magnetic tunnel junction (MTJ), 66-67 Magneto-optical devices, 3 4 Magneto-optic-Kerr effect (MOKE), 57-58 Matrix-Boltzmann equation, 11 1-1 12 MBE systems, 14, 15, 16, 18 Mean field theories continuum approximation, 8, 10 ferromagnetism predictiondexperiments, 11-13 ferromagnetism understanding, 8 spin-orbit interaction, 29-30 Microwave frequency devices, 2 “Migration enhanced epitaxy” techniques, 15 MOKE (magneto-optic-Kerreffect), 57-58 MOSFET technology, 61.62-63 Motional narrowing, 96, 132 MRAM (“magnetic random access memory”), 2, 160 MTJ (magnetic tunnel junction), 66-67

N NMR (nuclear magnetic resonance), 78 Nuclear magnetic resonance (NMR), 78

0 Opto-electronics, 2.3-4 Overhauser effect, 65

P Particle-induced x-ray emission studies, 20 Pauli principle, 135-136, 137 Percolation models, 8 Persistent spin flow, 5 1,53 Phase coherent devices, 34.36 “Planar Hall effect,” 28-29 Polarized optical luminescence, 100, 163 Proximity effect, 3,60,63,65

Q Quantum confinement,2 Quantum dots CdSe quantum dots, 4 4 4 5 phase coherent devices, 36 quantum information processing, 5 spin coherence, 4 3 4 5 Quantum information processing experiments,4-5 phase coherent devices, 36 spin coherence, 55 spin dynamics, 74 Quantum tunneling, 2 Quasi-angular momentum, 91, 164-165 Quasi-thermal equilibrium, 149, 152, 153, 154-155

R Rashba interaction, 62 Reflection high-energy electron diffraction (RHEED), 14, 15-16 Resonant spin amplification (RSA), 40-41, 45-46 RHEED (reflection high-energy electron diffraction), 14, 15-16 RSA (resonant spin amplification),40-41, 45-46 Rutherford backscattering studies, 20

S Scanning tunneling microscopy (STM), 79, 163 Schottky contacts conductivity mismatches, 64 depletion region, 160 spin injection, 155-160, 163-164, 165

178

SUBJECTINDEX

Schrijdingerequation, I 1 Sciencdtechnology perspectives, 2, 74 Second contribution structure inversion asymmetry (SIA), 91 Selection rules (electric dipole), 36-37.63 Semiconductor electronic structure band states (away from the zone cluster), 86-91 electron-hole exchange scattering, 96-97 g-factors, 102-104 hypefine interaction, 97-98 inversion asymmetry, 92-96 overview, 80-84 spin-dependent Hamiltonian, 83-85 spin-dependent optical properties, 98-101 spin-flip scattering, 91-92, 132 Semiconductor spintronic devices Datta-Das spin field effect transistor, 34, 60-63 ferromagnetic heterojunction bipolar transistor, 60 ferromagnetic semiconductor tunnel junctions, 60.61.65-71 overview, 60,61,74-75 spin-dependent light-emitting diodes, 60.61, 63-65 Semiconductor spintronics overview, 2 4 SIA (second contribution structure inversion asymmetry), 91 Spin coherence all-optical coherent manipulation, 55-57 bulk semiconductors,4 1 4 3 control of, 55-57 decay, 37 electrical manipulation of, 45-55 hybrid ferromagnetkemiconductor heterostructures, 57-60 magnetization reversal, 59 optical measurements, 3 6 4 1 overview, 34,36 quantum information processing, 55 semiconductor quantum dots, 43-45 spin transfer with bias, 51-52.53 spin transfer without bias, 51,53 temperature dependence, 50-5 1 voltage-controlled spin coherence, 54-55 Spin decoherence time, 36.37 Spin density (Fokker-Planckequation), 108, 115-118 Spin-dependent Hamiltonian, 83-85

Spin-dependent light-emitting diodes (“spinLED) semiconductorspintronic devices, 60.61, 63-65 spin injection experiments, 162 spin-polarized electron injection, 139-140 Spin-dependent optical properties, 98-101 Spin-dependent transport equations, 142-144 “Spin dephasingldecoherence,”36 Spin drift, 47 Spin dynamics (in semiconductors) future directions, 166 kinetic theory, 104-124 kinetic theory applications, 125-138 overview, 74-80, 164-166 semiconductorelectronic structure, 80-104 “SpinFET” (spin field effect transistor), 34, 60-63 Spin field effect transistor (“spinFET’), 34, 60-63 Spin-flip scattering, 91-92, 132 Spin injection (from polarized contacts) conductivities, 147-149 constant conductivity model, 144-155 contact resistivity, 149-150, 152, 15&155 experiments in, 162-164 overview, 138-142 quasi-thermal equilibrium, 149, 152, 153, 154-155 Schottky contacts, 155-160, 163-164, 165 spin-dependent transport equations, 142-144 spin-dependent tunneling, 141-142, 144, 149, 150, 154, 160-162, 163-164 spin-up/spin-down electro-chemical potentials, 151-153,158 SpidED. See Spin-dependent light-emitting diodes Spin lifetime factors affecting, 4345.77-78, 124, 126, 127, 135-138.166 measuring, 37-41,45,48,50-54.122 RSA, 40 Spin-orbit coupling, 28 Spin packets, 47 Spin-polarized contact types, 79 Spin polarized electrons (holes) diluted magnetic semiconductors, 9 as semiconductor spintronics approach, 2-3 spinLEDs, 64 Spin-polarized injection inefficiency,62.66

179

SUBJECT INDEX Spin relaxation bulk semiconductors, 125-132 quantum wells, 132-138 rate calculation, 118-124 “spin de-phasing/decoherence” vs., 36 Spin relaxation time, 37 “Spin transistor,” 34.60-63 Spintronics (spin transport-based electronics) origins, 2 Spin-up/spin-downelectrons electro-chemical potentials, 151-153, 158 Fermi energy, 160 SQUID magnetometry. 34 Stark effecthhifts, 55-56 STM (scanning tunneling microscopy), 79 Substitutional(Mm,) sites, 20

Time-resolved Kerr rotation (spin coherence). See also Kerr-rotation spectroscopies bulk semiconductors, 41 optical measurements, 38-39 semiconductor heterostructures, 4849.50, 51,53,57-58 semiconductor quantum dots, 44,45 ‘Tipping pulse” (TP), 56-57 TMR (tunnel magnetoresistance), 65-66, 67-69 Traditional magnetic tunnel junction (MTJ), 66-67 Transverse spin relaxation time, 36.37 TRFR. See Time-resolved Faraday rotation TRKR.See Time-resolved Kerr rotation Tunnel magnetoresistance (TMR), 65-66, 67-69

T Technology/scienceperspectives, 2.74 Temperature dependence spin coherence, 50-5 1 TMR,69-70 Time-resolved Faraday rotation (spin coherence). See also Faraday-rotation spectroscopies bulk semiconductors, 41.42 optical measurement, 38-40 semiconductor heterostructures, 4 5 4 6 , 4 8 , 57.60

V Voigt geometry, 36.37-38,39,60 Voltage and TMR, 69-7 1

2 Zeeman effects (bare), 3 OD systems, 4 3 4 5 Zinc-blende crystal structure, 80.81.82.83 ZnSdGaAs heterostructures, 4748.49-50, 51-52

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