Perspectives of Giant Magnetoresistance

Transcription

1 University of Nersk - Lincoln DigitlCommons@University of Nersk - Lincoln Evgeny Tsyml Pulictions Reserch Ppers in Physics nd Astronomy 1 Perspectives of Gint Mgnetoresistnce Evgeny Y. Tsyml University of Oxford, tsyml@unl.edu Dvid G. Pettifor University of Oxford, dvid.pettifor@mterils.ox.c.uk Follow this nd dditionl works t: Prt of the Condensed Mtter Physics Commons Tsyml, Evgeny Y. nd Pettifor, Dvid G., "Perspectives of Gint Mgnetoresistnce" (1). Evgeny Tsyml Pulictions This Article is rought to you for free nd open ccess y the Reserch Ppers in Physics nd Astronomy t DigitlCommons@University of Nersk - Lincoln. It hs een ccepted for inclusion in Evgeny Tsyml Pulictions y n uthorized dministrtor of DigitlCommons@University of Nersk - Lincoln.

2 1 pulished in Solid Stte Physics, ed. y H. Ehrenreich nd F. Spepen, Vol. 56 (Acdemic Press, 1) pp Perspectives of Gint Mgnetoresistnce E.Y.Tsyml nd D.G.Pettifor Deprtment of Mterils, University of Oxford, Prks Rod, Oxford OX1 3PH, UK I. Introduction II. Origin of GMR 1. Spin-dependent conductivity. Role of nd structure 3. Resistor model III. Experimentl survey 4. Composition dependence 5. Nonmgnetic lyer thickness dependence 6. Mgnetic lyer thickness dependence 7. Roughness dependence 8. Impurity dependence 9. Outer-oundry dependence 1. Temperture dependence 11. Angulr dependence IV. Free-electron nd simple tight-inding models 1. Semiclssicl theory 13. Quntum-mechnicl theory 14. Tight-inding models V. Multind models 15. Bllistic limit 16. Semiclssicl theory 17. Tight-inding models 18. First-principle models VI. CPP GMR VII. Conclusions

3 I. INTRODUCTION Gint mgnetoresistnce (GMR) is one of the most fscinting discoveries in thin-film mgnetism, which comines oth tremendous technologicl potentil nd deep fundmentl physics. Within decde of GMR eing discovered in 1988 commercil devices sed on this phenomenon, such s hrd-disk red-heds, mgnetic field sensors nd mgnetic memory chips, hd ecome ville in the mrket. These chievements would not hve een possile without detiled understnding of the physics of GMR, which requires quntum-mechnicl insight into the electronic spin-dependent trnsport in mgnetic structures. The discovery of GMR ws to gret extent due to the significnt progress in thin-film deposition techniques, which mde it possile to fricte lyers of vrious mterils with nerly monolyer precision. Thin films of nnometer thickness cn nowdys e routinely fricted using moleculr em epitxy (MBE), sputtering nd electrodeposition. By stcking such thin films in multilyers one cn crete lyered systems with properties, which re totlly distinct from those of the constitutive ulk mterils. Metllic mgnetic multilyers, which consist of severl ferromgnetic lyers seprted y non-mgnetic lyers, re very ttrctive s they exhiit rod vriety of unique electronic, mgnetic, nd trnsport properties. Like other mgnetoresistive effects, GMR is the chnge in electricl resistnce in response to n pplied mgnetic field. It ws discovered tht the ppliction of mgnetic field to Fe/Cr multilyer resulted in significnt reduction of the electricl resistnce of the multilyer. 1 This effect ws found to e much lrger thn either ordinry or nisotropic mgnetoresistnce nd ws, therefore, clled gint mgnetoresistnce or GMR. A similr, though diminished effect ws simultneously discovered in Fe/Cr/Fe trilyers. As ws shown lter, high mgnetoresistnce vlues cn lso e otined in other mgnetic multilyers, such s Co/Cu. The chnge in the resistnce of the multilyer rises when the pplied field ligns the mgnetic moments of the successive ferromgnetic lyers, s is illustrted schemticlly in Fig.1. In the sence of the mgnetic field the mgnetiztions of the ferromgnetic lyers re ntiprllel. Applying the mgnetic field, which ligns the mgnetic moments nd sturtes the mgnetiztion of the multilyer, leds to drop in the electricl resistnce of the multilyer. In order to oserve GMR one hs to provide n opportunity to reorient the mgnetic moments of the ferromgnetic lyers reltive to one nother. In mgnetic multilyers this cn e chieved due to the effect of ntiferromgnetic interlyer coupling, 3 which is prticulr cse of interlyer exchnge coupling. The interlyer exchnge coupling is medited y the itinernt electrons in the metllic spcer lyer nd is n nlogue of the Rudermn-Kittel-Ksuy-Yosid (RKKY) interction etween loclized mgnetic moments in non-mgnetic host metl. The interlyer exchnge coupling oscilltes etween ferromgnetic nd ntiferromgnetic s function of the thickness of the nonmgnetic lyer. 4 By choosing n pproprite thickness of the non-mgnetic lyer it is, therefore, possile to crete n ntiprllel configurtion of the ferromgnetic lyers nd then reorient (lign) the moments y n pplied mgnetic field. The presence of n ntiferromgnetic interlyer coupling is not, however, necessry condition for GMR to occur. Antiprllel lignment cn lso e otined y introducing different coercivities of the successive ferromgnetic lyers. 5-7 In this cse the mgnetic moments of the soft nd hrd mgnetic lyers switch t different vlues of the pplied mgnetic field providing field rnge in which they re ntiprllel nd the resistnce is higher. Another wy to chnge the lignment of the mgnetiztions is to use spin vlve. 8 In the spin vlve the mgnetiztion of one ferromgnetic lyer is pinned y the exchnge coupling with n djcent ntiferromgnetic lyer, wheres the mgnetiztion of the other ferromgnetic lyer is free to rotte with the pplied mgnetic field. Although the mesured vlues of GMR re higher in mgnetic multilyers, spin vlves re more ttrctive from the point of view of pplictions, ecuse only smll mgnetic fields need to e pplied to chnge the resistnce. Mgnetic grnulr solids represent nother system, which displys the GMR effect. 9 In these mterils ferromgnetic precipittes re emedded in non-mgnetic host metl film. The rndomly-oriented mgnetic moments of the precipittes cn e ligned y the pplied mgnetic field which results in resistnce drop. The vrious types of systems in which GMR is oserved re shown in Fig.. Fig.1 c R M Schemtic representtion of the GMR effect. (): Chnge in the resistnce of the mgnetic multilyer s function of pplied mgnetic field. (): The mgnetiztion configurtions (indicted y the rrows) of the multilyer (trilyer) t vrious mgnetic fields: the mgnetiztions re ligned ntiprllel t zero field; the mgnetiztions re ligned prllel when the externl mgnetic field H is lrger thn the sturtion field H S. (c): The mgnetiztion curve for the multilyer. GMR is distinguished from oth ordinry mgnetoresistnce nd nisotropic mgnetoresistnce (AMR) which re lso present in lyered nd grnulr mgnetic systems. Ordinry mgnetoresistnce rises from the effect of the Lorentz force on the electron trjectories due to the pplied mgnetic field. In contrst to GMR, it does not sturte t the sturtion mgnetic field nd is usully smll in metls (less thn 1% in fields of the order of 1 Tesl). AMR origintes from the spin-orit interction nd cuses the resistnce to depend on the reltive orienttions of the mgnetiztion nd the electric current. The mgnetic field rnge in which the AMR effect occurs is governed y the field needed to chnge the direction of the mgnetic moment. For exmple, permlloy (Ni 8 Fe ) films, which re presently employed in sensor pplictions, exhiit the AMR effect of 1-%, the resistnce chnge tking plce in field rnge of few Guss. 1 Contrry to nisotropic mgnetoresistnce, GMR rises R AP R P H S H S H H 3

4 due to the dependence of the resistivity in lyered nd grnulr mgnetic structures on the locl mgnetic configurtion rther thn on the orienttion of the pplied mgnetic field with respect to the electric current. Fig. sustrte FM NM FM NM FM NM FM NM FM NM FM sustrte Vrious structures in which GMR cn e oserved: mgnetic multilyer (), pseudo spin vlve (), spin vlve (c) nd grnulr thin film (d). Note tht the lyer thickness is of the order of few nnometers, wheres the lterl dimensions cn vry from micrometers to centimetres. In the mgnetic multilyer () the ferromgnetic lyers (FM) re seprted y nonmgnetic (NM) spcer lyers. Due to ntiferromgnetic interlyer exchnge coupling they re ligned ntiprllel t zero mgnetic field s is indicted y the dshed nd solid rrows. At the sturtion field the mgnetic moments re ligned prllel (the solid rrows). In the pseudo spin vlve () the GMR structure comines hrd nd soft mgnetic lyers. Due to different coercivities, the switching of the ferromgnetic lyers occurs t different mgnetic fields providing chnge in the reltive orienttion of the mgnetiztions. In the spin vlve (c) the top ferromgnetic lyer is pinned y the ttched ntiferromgnetic (AF) lyer. The ottom ferromgnetic lyer is free to rotte y the pplied mgnetic field. In the grnulr mteril (d) mgnetic precipittes re emedded in the non-mgnetic metllic mteril. In the sence of the field the mgnetic moments of the grnules re rndomly oriented. The mgnetic field ligns the moments in certin direction. In ddition to ordinry, nisotropic nd gint mgnetoresistnce, there lso exists colossl mgnetoresistnce (CMR) which ws found in doped mngnite perovskites such s L 3-x C x MnO 3 (for recent review see ref.11). The CMR effect cn e extremely lrge resulting in resistnce chnge of few orders in mgnitude. CMR origintes from metl-insultor trnsition in the vicinity of the Curie temperture nd requires mgnetic fields of the order of severl Tesl. The ltter property mkes the pplicility of CMR mterils firly limited. On the other hnd, tunneling mgnetoresistnce (sometimes referred to s junction mgnetoresistnce) roused considerle interest recently due to possile pplictions in the mgnetic sensor nd storge industry. Tunneling mgnetoresistnce (TMR) is oserved in mgnetic tunnel junctions, in which ferromgnetic metllic lyers re seprted y thin insulting spcer lyer (for recent reviews see ref.1). Similr to GMR, TMR is determined y the reltive orienttion of the mgnetic moments of the ferromgnetic lyers. Although oth these phenomen my hve similr pplictions, they re very distinct from the point of view of the physics involved. GMR is oserved in mgnetic metllic multilyer structures nd 4 FM - hrd NM FM - soft d sustrte c sustrte AF FM - pinned NM FM - free therefore the physics of GMR is relted to spin-dependent electronic trnsport in complex metl systems. On the other hnd, TMR is oserved in lyered systems where mgnetic metllic lyers re seprted y n insulting spcer lyer nd is consequence of spin-polrized tunneling. Most experiments on GMR re performed y mesuring the electric current in the plne of the multilyer, i.e. within the current-in-the-plne (CIP) geometry. This geometry is currently used for the industril pplictions of GMR. Mesuring the current perpendiculr to the multilyer plne, i.e. within the current-perpendiculr-to-the-plne (CPP) geometry, is much more difficult. This is due to the very smll thickness of the multilyer nd consequently the very low CPP resistnce, which is not esy to detect. There re severl wys to solve this prolem, one of which is to perform the experiments using superconducting contcts. 13 Although this technique hs the dvntge of reltively simple smple preprtion, mesurements cn e performed only t low tempertures. Other techniques, which void this prolem, re sed on lithogrphiclly defined pillr structures, 14 on growing the mgnetic multilyers on prestructured (grooved) sustrtes 15,16 (Fig.3) or on electrodeposition of the multilyer nnowires into the pores of n insulting polymer mtrix 17,18 (Fig.3). CPP GMR ppered to e very ttrctive, ecuse its mgnitude is higher thn the corresponding mgnitude of CIP GMR. In ddition we will see tht CPP experiments provide importnt informtion out the mechnisms of gint mgnetoresistnce. Fig.3 multilyer deposition.µm 71 o InP InP (1) (111) Techniques for current-perpendiculr-to-the-plne (CPP) GMR, which cn e used t room temperture: grooved sustrtes nd nnowires. (): Schemtic representtion of the grooved InP sustrte nd multilyer evported t n ngle to the surfce. Current flow is indicted y the rrows. After Gijs et l. 16 (): Schemtic representtion of the rry of multilyer nnowires in n insulting polymer mtrix. After Pirux et l. 18 Since the discovery of GMR the theoreticl tretment of this effect ecme the suject of much ttention. First theories of GMR were sed minly on free-electron models. Although these theories re useful for the qulittive understnding of electronic trnsport in mgnetic multilyers nd cn provide vlule insight into the phenomenon, they cnnot e pplied to the quntittive tretment of GMR, due to the complex spin-polrized electronic structure of the mgnetic multilyers. It is wellknown tht the nd structures of trnsition metl ferromgnets which re mostly used in the GMR structures re chrcterized y unfilled d nds which cn not e descried y single prolic nd t the Fermi energy. Recent dvnces in electronic trnsport nd nd structure theory hve mde it possile to develop relistic multind models nd perform first-principle clcultions of GMR. 5 Polycronte 4nm Cu lyer Co lyer Cu film

5 These models provided new conceptul insights into the phenomenon nd hve extended our fundmentl understnding of GMR. Reviews on GMR hve een pulished y Fert nd Bruno, 19 Levy nd Dieny 1 covering the field upto Other reviews y Gijs nd Buer, Ansermet, 3 Bss nd Prtt, 4 Fert nd Pirux 5 nd Gijs 6 re devoted specificlly to the CPP GMR. Very recently two review ppers y Coehoorn 7 nd y Brthelemy et l. 8 hve ppered. The first one highlights the theoreticl nd experimentl results, which re of prticulr interest for pplictions of spin vlves in red heds. The second one discusses the nture of GMR y ccenting the importnce of CPP geometry nd gives full list of experimentl ppers. The present review is devoted to the physics of gint mgnetoresistnce. We emphsize the role of the spin-polrized electronic nd structure, which is crucil for understnding GMR. In section II, the origin of GMR is explined nd simple resistor model is introduced. In section III, we overview the experimentl dt on CIP GMR in mgnetic multilyers nd spin vlves nd discuss the dependence of GMR on composition, lyer thickness, roughness, impurities, outer oundries nd temperture. The theoreticl formultions of GMR within free-electron nd simple tight-inding models re reviewed in section IV oth from the semiclssicl nd quntum mechnicl viewpoints. Multind models for GMR re reviewed in section V. Strting from the llistic regime of conduction, we discuss oth the semiclssicl nd quntum mechnicl pproches to GMR within the diffusive limit. The mechnisms, which re responsile for GMR, re discussed nd the interprettion of selected experimentl results is presented. A seprte section VI is devoted to CPP GMR, which hs recently ttrcted much ttention due to new experimentl nd theoreticl results. In conclusion, we indicte directions for future work on GMR. ligned mgnetic lyers (the top pnel in Fig.4), the up-spin electrons pss through the structure lmost without scttering, ecuse their spin is prllel to the mgnetiztion of the lyers. On the contrry, the down-spin electrons re scttered strongly within oth ferromgnetic lyers, ecuse their spin is ntiprllel to the mgnetiztion of the lyers. Since conduction occurs in prllel for the two spin chnnels, the totl resistivity of the multilyer is determined minly y the highly-conductive up-spin electrons nd ppers to e low. For the ntiprllel-ligned multilyer (the top pnel in Fig.4), oth the up-spin nd down-spin electrons re scttered strongly within one of the ferromgnetic lyers, ecuse within the one of the lyers the spin is ntiprllel to the mgnetiztion direction. Therefore, in this cse the totl resistivity of the multilyer is high. R R up spin down spin up spin down spin R R II. ORIGIN OF GMR R R R R GMR cn e qulittively understood using the Mott model, which ws introduced s erly s 1936 to explin the sudden increse in resistivity of ferromgnetic metls s they re heted ove the Curie temperture. 9 There re two min points proposed y Mott. First, the electricl conductivity in metls cn e descried in terms of two lrgely independent conducting chnnels, corresponding to the up-spin nd down-spin electrons, which re distinguished ccording to the projection of their spins long the quntiztion xis. The proility of spin-flip scttering processes in metls is normlly smll s compred to the proility of the scttering processes in which the spin is conserved. This mens tht the up-spin nd down-spin electrons do not mix over long distnces nd, therefore, the electricl conduction occurs in prllel for the two spin chnnels. Second, in ferromgnetic metls the scttering rtes of the up-spin nd down-spin electrons re quite different, whtever the nture of the scttering centers is. According to Mott, the electric current is primrily crried y electrons from the vlence sp nds due to their low effective mss nd high moility. The d nds ply n importnt role in providing finl sttes for the scttering of the sp electrons. In ferromgnets the d nds re exchnge-split, so tht the density of sttes is not the sme for the upspin nd down-spin electrons t the Fermi energy. The proility of scttering into these sttes is proportionl to their density, so tht the scttering rtes re spin-dependent, i.e. re different for the two conduction chnnels. Although, s we will see elow, this picture is too simplified in view of the strong hyridiztion etween the sp nd d sttes, it forms useful sis for qulittive understnding of the spin-dependent conduction in trnsition metls. Using Mott s rguments it is strightforwrd to explin GMR in mgnetic multilyers. We consider colliner mgnetic configurtions, s is shown in Fig.4, nd ssume tht the scttering is strong for electrons with spin ntiprllel to the mgnetiztion direction, nd is wek for electrons with spin prllel to the mgnetiztion direction. This is supposed to reflect the symmetry in the density of sttes t the Fermi level, in ccordnce with Mott s second rgument. For the prllel- Fig.4 Schemtic illustrtion of electron trnsport in multilyer for prllel () nd ntiprllel () mgnetiztions of the successive ferromgnetic lyers. The mgnetiztion directions re indicted y the rrows. The solid lines re individul electron trjectories within the two spin chnnels. It is ssumed tht the men free pth is much longer thn the lyer thicknesses nd the net electric current flows in the plne of the lyers. Bottom pnels show the resistor network within the two-current series resistor model. For the prllel-ligned multilyer (), the up-spin electrons pss through the structure lmost without scttering, wheres the down-spin electrons re scttered strongly within oth ferromgnetic lyers. Since conduction occurs in prllel for the two spin chnnels, the totl resistivity of the multilyer is low. For the ntiprllel-ligned multilyer (), oth the up-spin nd downspin electrons re scttered strongly within one of the ferromgnetic lyers, nd the totl resistivity of the multilyer is high. The sme rguments cn e used for understnding GMR in grnulr mterils. In the sence of mgnetic field, the mgnetic moments of the ferromgnetic grnules re rndomly-oriented. This implies tht oth up- nd down-spin electrons re scttered strongly y the grnules, the mgnetic moments of which re close to ntiprllel. The resistnce in this cse is lrge. When sturting mgnetic field is pplied, the mgnetic moments re ligned nd the resistnce is low, like in the cse of the prllel-ligned multilyer. Therefore, s ws originlly suggested y Biich et l., 1 spin-dependent scttering is the primry origin of GMR. An understnding of the microscopic mechnisms, which cuse spin-dependent scttering in mgnetic systems, is one of the most importnt questions, which this review ttempts to nswer. In ddition, we will see tht there re other mechnisms distinct from spin-dependent 6 7

6 scttering which re importnt for understnding GMR nd which will lso e ddressed in this review. We strt now from qulittive picture for the spin-dependent conduction in ferromgnets. 1. Spin-dependent conduction According to Mott s first rgument, the conductivity of metl is the sum of the independent conductivities for the up-spin nd down-spin electrons: σ = σ σ. (1.1) + Within ech conduction chnnel the conductivity is determined y vrious fctors. In order to illustrte their role we use the Drude formul (e.g., ref.3) which cn e expressed s follows: e kf σ Drude = λ. π! 6π (1.) Here σ Drude is the Drude conductivity per spin, e /π! Ω -1 is the spin conductnce quntum, k F is the Fermi momentum, nd λ is the men free pth, which is the product of the relxtion time τ nd the Fermi velocity v F, i.e. λ=v F τ. (1.3) We do not disply explicitly the spin indices here it is ssumed tht ll the ove quntities re in generl spin-dependent. Although the Drude formul is vlid only for free electrons, it is useful to understnd qulittively the fctors ffecting the spin-dependent conductivity. The conductivity is determined y the electrons which hve the Fermi energy. Due to the Puli exclusion principle the electrons which lie elow the Fermi level cn not gin energy responding to the smll pplied electric field, ecuse ll the sttes t higher energies re occupied. As consequence, only electrons t the Fermi level cn contriute to the electric current. As cn e seen from Eq.(1.), the conductivity is proportionl to the cross sectionl re of the Fermi surfce ~ k F, which chrcterizes the numer of electrons contriuting to the conduction. The men free pth depends of the Fermi velocity nd the relxtion time, the ltter cn e estimted from the Fermi golden rule 1 τ = π Vsct n( E F! ). (1.4) Here V sct is n verge vlue of the scttering potentil nd n(e F ) is the density of electronic sttes t the Fermi energy E F for the pproprite spin. Although ll the quntities which enter expressions (1.)-(1.4) re in generl spin-dependent, the origin of the spin dependence is different. The Fermi momentum k F nd the Fermi velocity v F re intrinsic properties of the metl nd entirely determined y the electronic nd structure of the metl. In ferromgnetic metls these quntities re different for the up- nd down-spin electrons. The density of sttes t the Fermi energy n(e F ) is lso determined y the spin-polrized nd structure. It is the density of sttes, which ws referred to y Mott, rguing tht the scttering rtes in ferromgnetic metls re spin-dependent. On the contrry, the scttering potentil which enters Eq.(1.4) is not n intrinsic property of the metl. It is generted y the sctterers such s defects, impurities, or lttice virtions. The scttering potentil cn e either spin-dependent or spin-independent, which is determined y the prticulr mechnism of scttering. For exmple, spin-dependent scttering potentils produced y impurities in dilute mgnetic lloys leds to the spin symmetry of the conductivity in these lloys This hs to e tken into ccount when treting GMR in mgnetic lyered systems in which ferromgnetic lyers re often lloys such s permlloy Ni 8 Fe. Spin-dependent scttering potentils might lso 8 contriute to GMR t the interfces etween ferromgnetic nd non-mgnetic lyers. In rel mgnetic multilyers these interfces re not idel. Interfcil roughness nd/or sustitutionl disorder (i.e. mixing of the djcent metl toms t the interfce) re lwys present in experiments. Rndomness of the tomic potentils t the interfce results in enhnced interfcil scttering. If for one spin orienttion the tomic potentils of the mgnetic nd nonmgnetic toms re similr ut for the other spin orienttion they re dissimilr, then one would expect strong spin-dependent scttering due to the spin dependence in the scttering potentil. The reltive importnce of spin-dependent scttering potentils cn, however, e diminished in rel GMR structures which re fr from eing perfect. Vrious types of defects such s grin oundries, stcking fults nd misfit disloctions re lwys present in the multilyers. Becuse the relxtion time in Eq.(1.4) is determined y the configurtionlly-verged vlue of the scttering potentil squred, vrious types of scttering centers cn mke this verge vlue spin-independent. In these circumstnces the spin-polrized nd structure of the multilyer ecomes decisive nd usully gives the dominnt contriution to the spin dependence of the men free pth nd the conductivity.. Role of nd structure The electronic nd structure of the multilyer is proly the most importnt property which determines the spin-dependent conduction nd consequently is responsile for the GMR. In most experiments on GMR the ferromgnetic 3d trnsition metls Co, Fe nd Ni, nd their lloys, such s permlloy Ni 8 Fe, re used in comintion with non-mgnetic spcer lyers, such s Cr or the nole metls Cu, Ag nd Au. The electronic nd structure of these metls is chrcterized y numer of similr fetures which we discuss elow. Due to the spin-orit coupling of the 3d trnsition metls eing very wek the electronic structure for the up-spin nd down-spin electrons cn e considered independently. The 3d elements re chrcterized y the presence of the 4s, 4p nd 3d vlence sttes, which re distinguished y their oritl momentum. The 4s nd 4p sttes crete dispersive sp nd which is similr to free-electron nd. The sp electrons hve high velocity, low density of sttes nd consequently long men free pth, i.e. they my e thought to e minly responsile for the conductivity in 3d metls. On the contrry, the d nd is loclized in reltively nrrow energy window nd is chrcterized y high density of sttes nd low velocity of electrons. In the intervl of energy where the sp nd d nds cross, they cn not e considered s independent ecuse of the strong sp-d hyridiztion, which modifies sustntilly the nd structure. It chnges drmticlly the properties of the sp electrons, which is reflected in the nd ending nd results in reduced velocity ssocited with the sp nd. These fetures re evident from Fig.5, in which the electronic nd structure of Cu is shown. In ferromgnetic 3d metls the d nd is exchnge-split. Due to the loclized nture of the d electrons, two d electrons experience strong Coulom repulsion provided tht they hve ntiprllel spins nd occupy the sme oritl. To reduce the energy it is dvntgeous for the d electrons to hve prllel-oriented spins, ecuse the Puli exclusion principle does not permit two electrons with the sme spin to pproch ech other closely (i.e. occupy the sme oritl) nd hence the Coulom interction is reduced. Therefore, the Coulom repulsion in conjunction with the Puli principle leds to the ferromgnetic exchnge interction nd fvors the formtion of spontneous mgnetic moment. However, putting ll the electrons into sttes with the sme spin direction increses the totl kinetic energy, the increse eing lrger the wider the d nd or lower the d-nd density of sttes. There re, therefore, two competing tendencies, which hve to e lnced in order to find whether ferromgnetic ordering is fvored. The condition which hs to e stisfied for the ppernce of ferromgnetism is the fmous Stoner criterion Jn(E F )>1, where J is the exchnge constnt (which tkes the vlue of out 1eV for 3d trnsition metls) nd n(e F ) is the density of sttes for given spin t the Fermi energy. 34 The Stoner criterion is stisfied for cc Fe, fcc Co nd fcc Ni. Due to the exchnge splitting of the d nds, the numer of occupied sttes is different for the up-spin nd down- 9

7 spin electrons, giving rise to the non-zero mgnetic moments of.µ B, 1.7µ B nd.6µ B for Fe, Co nd Ni respectively. In order to distinguish etween the high nd low-occupied spin sttes, the terms mjority-spin electrons nd minority-spin electrons re usully used. The nd structure of Co s representtive of the ferromgnetic 3d metls is shown in Figs.5,c. Fig.5 c Energy (ev) Energy (ev) Energy (ev) Cu Co-mjority Co-minority L Γ X Electronic nd structures (left pnels) nd the density of sttes (right pnels) of Cu () nd fcc Co for the mjority-spin () nd minority-spin (c) electrons. The nd structure of non-mgnetic Cu is sme for the up-spin nd down-spin electrons. It is chrcterized y the fully occupied d nds nd the presence of dispersive sp nd t the Fermi energy, which result in high conductivity of Cu. The electronic structure of ferromgnetic Co is different for the two spin orienttions nd is chrcterized y the exchnge-split d nds. The Fermi level lies within the sp nd for the mjority-spin electrons, which leds to high conductivity of mjority-spin chnnel. The Fermi level lies, however, within the d nd for the minority-spin electrons resulting in low conductivity of the minority-spin chnnel. In the ltter cse the sp electrons re strongly hyridized with the d electrons, which diminishes their contriution to conduction. The conductivity is determined y the position of the Fermi energy with respect to the d nds. In the cse of Cu, the d nds re fully occupied nd the Fermi level lies within the sp nd (Fig.5). Due to the high velocity of the electrons within the sp nd nd the low density of sttes with W K DOS E F E F E F resultnt low proility of scttering, the men free pth is long nd Cu is very good conductor. This is lso the cse for the other nole metls Ag nd Au. On the other hnd, in the cse of ferromgnetic metl like Co, s result of the exchnge splitting, the mjority d nd is fully occupied, wheres the d minority nd is only prtly occupied (Fig.5,c). The Fermi level lies, therefore, within the sp nd for the mjority spins ut within the d nd for the minority spins. The exchnge splitting of the spin nds leds to crucil difference in the conductivity etween the mjority- nd minority-spin electrons. For the mjority spins the sitution is similr to tht in Cu: the conductivity is governed y the sp electrons nd is high. On the contrry, the conductivity of the minority-spin electrons is not entirely determined y the sp electrons. Due to the strong sp-d hyridiztion which mixes the sp nd d sttes the contriutions of oth the sp nd d electrons ecome importnt. The minority nds represent hyridized spd nds which re not dispersive nd hve high density of sttes. This mkes the men free pth ssocited with these nds reltively short nd the minority-spin conductivity low, despite sizele fctor proportionl to the re of the multind Fermi surfce. These rguments, which re sed on the spin-polrized nd structure, explin the strong spin symmetry in the conductivity of ulk Co. The presence of the interfces in mgnetic multilyer dds new importnt feture to our discussion ove of spin-dependent trnsport in ulk elementl ferromgnets. The two djcent metls creting the interfce hve different nd structures, which led to potentil step t the interfce nd results in the trnsmission proility cross the interfce eing less thn 1. If the interfce seprtes ferromgnetic nd non-mgnetic metls the trnsmission will e spin-dependent due to the spin dependence of the nd structure of the ferromgnetic lyer. This cn e illustrted y considering the nd structures of Co nd Cu, which re shown in Fig.5. As is seen y compring Fig.5 nd 5, the nd structure of Cu is similr to the nd structure of the mjority spins in Co. This good nd mtching implies high trnsmission for the mjority-spin electrons cross the Co/Cu interfce. On the contrry, s is seen from Fig.5 nd 5c, there is reltively lrge nd mismtch etween Cu nd the minority spins in Co nd consequently the trnsmission of the minority-spin electrons cross the Co/Cu interfce is expected to e poor. Therefore, the interfces of the Co/Cu multilyer ct s spin-filters. When the filters re ligned, the mjority spin-electrons cn pss through reltively esily. When the filters re ntiligned, the electrons in oth spin chnnels re reflected t one of the interfces. This spin-dependent trnsmission is n importnt ingredient of the electronic trnsport in GMR structures. Bnd mtching lso plys n importnt role in the spin-dependent interfce scttering due to the intermixing of toms ner the interfces. If we ignore the chnge in the chemicl stte of the toms, i.e. ssume tht their tomic energy levels nd mgnetic moments re identicl to those in the ulk of the djcent lyers, then the intermixing t the interfce produces rndom potentil which is strongly spin-dependent. This spin dependence is direct consequence of the good nd mtching for the mjority spins in Co/Cu, which implies smll scttering potentil, nd the poor nd mtching for the minority spins in Co/Cu, which implies lrge scttering potentil. A similr ehvior tkes plce in Fe/Cr multilyers, where very smll scttering potentil (good nd mtching) is expected for the minority-spin electrons, ut lrge scttering potentil (d nd mtching) is expected for the mjority-spin electrons. Thus, the mtching or mismtching of the nds etween the ferromgnetic nd nonmgnetic metls results in spin-dependent scttering potentils t disordered interfces, which cn contriute to GMR. 3. Resistor model Physicl insight into the origin of the current-in-the-plne (CIP) GMR cn e otined using the very simple resistor model. 35,36 Although this model is not le to provide quntittive description of the CIP GMR, it is useful s strting point for understnding this phenomenon. 1 11

8 According to the resistor model ech metllic lyer (nd ech interfce) is treted s n independent resistor. Within ech spin conduction chnnel the resistors re dded in prllel or in series depending on the reltionship etween the men free pth nd the lyer thickness. If the men free pth is short compred to the lyer thickness, then ech lyer conducts the electric current independently nd the resistors should e dded in prllel. It is ovious tht under this circumstnce the resistnce of the prllel nd ntiprllel configurtions re the sme nd consequently the GMR is zero. The ove oservtion indictes tht for otining non-zero GMR the men free pth should e sufficiently long. This is consistent with the qulittive picture of GMR (the top pnels of Fig.4), which is sed on the possiility for the electrons to propgte cross the spcer lyer freely, sensing the mgnetiztions of the two consecutive ferromgnetic lyers. In the limit the long men free pth eing long compred to the lyer thickness, the proility of scttering within the multilyer is the sum of scttering proilities within ech lyer nd ech interfce. 37 Therefore, within given spin chnnel the totl resistnce is the sum of resistnces of ech lyer nd ech interfce, i.e. the resistors re connected in series. This limiting cse is more relevnt to the mgnetic multilyers exhiiting gint mgnetoresistnce. In order to uild up the resistor network for the multilyer, we consider unit cell which consisting of the four lyers, two ferromgnetic nd two non-mgnetic, s is shown in the top pnels of Fig.4 nd 4. We choose the glol spin-quntiztion xis colliner to the mgnetiztion directions. Within ech ferromgnetic lyer the electron spin cn e either prllel or ntiprllel to the mgnetiztion direction. In the former cse the electron is loclly mjority-spin electron nd in the ltter cse minority-spin electron. The mjority- nd minority-spin resistivities of the ferromgnetic lyer re different nd re equl to ρ nd ρ respectively. The resistnce of the ilyer, which consists of the ferromgnetic lyer nd the spcer lyer, for either of the two spin chnnels is equl to R ρ d, (3.1) = NM d NM + ρ,, FM where ρ NM nd d NM denote the resistivity nd the thickness of the non-mgnetic spcer lyer nd t FM is the thickness of the ferromgnetic lyer. For simplicity the interfce resistnce etween the ferromgnetic nd spcer lyers hs een omitted. Using the resistnces which re defined y Eq.(3.1) the equivlent network of resistors for the prllel nd ntiprllel mgnetiztions re shown in the ottom pnels of Figs.4 nd 4. The totl resistnce of the prllel-ligned multilyer is then given y R R R P = N, (3.) R + R where N is the numer of the four-lyer unit cells within the multilyer. The totl resistnce of the ntiprllel-ligned multilyer equls to R + R R AP = N. (3.3) Thus, the mgnetoresistnce rtio is determined y the simple expression R R = R R R AP P P = ( R R ) 4R R. (3.4) Note tht we use definition in which GMR is normlized to the low resistnce R P. Although within this definition the GMR cn e lrger thn 1%, this definition is used in most ppers devoted to GMR, nd therefore we dopt it in this review. 1 Using Eqs. (3.1) nd (3.4), it is esy to pinpoint the min fctors which determine GMR. Let us first ssume tht the resistnce of the spcer lyer is smll s compred to the resistnce of the ferromgnetic lyers. In this limit the expression for GMR is ( ρ ρ ) R ( α 1) = =, (3.5) R 4ρ ρ 4α where the spin symmetry prmeter is defined y α = ρ / ρ. As is ovious from Eq. (3.5), the mgnitude of GMR is strongly dependent on the symmetry in the resistivity etween the two spin conduction chnnels in ferromgnetic lyers. Lrge symmetry, i.e. α>>1 or α<<1, is n importnt requirement for otining high vlues of GMR. If there is no spin symmetry in the resistivity of the two spin chnnels, i.e. α=1, then the GMR will e zero. Now we use Eq.(3.5) to mke n estimte of GMR in Co/Cu nd Fe/Cr multilyers. If we ssume tht α is entirely determined y the spin symmetry in the scttering rtes in Eq.(1.4), due to the spindependent density of sttes t the Fermi energy, we find α 7 for Co nd α 3 for Fe. For the Co/Cu multilyer this leds to GMR vlue of 13%, which is very close to the est pulished experimentl result of 1%. 38 However, for Fe/Cr Eq.(3.5) gives GMR of 3%, which is fr elow the highest ever oserved vlue of %. 39 This is not surprising ecuse the ove model is too simplified s GMR depends on mny other fctors such s the properties of the FM/NM interfce which were ignored in this estimte. The finite resistnce of the spcer lyer my lso e tken into ccount, which leds to R ( α 1) =, (3.6) R 4( α + pd NM / d FM )( 1+ pd NM / d FM ) where p = ρ NM / ρ. Hence, for given vlue of α, the GMR will increse with decresing pd d NM / FM. Therefore, in order to otin higher GMR, it is importnt to hve low resistivity of the non-mgnetic spcer lyer. As function of the spcer thickness d NM, the GMR decreses monotoniclly nd t lrge spcer thickness it flls off s 1/ d NM. Although the drop in GMR with incresing spcer thickness is lso found in experiments, the ctul dependence on d NM is different compred to this simplified model. As discussed lter, the CIP GMR is found to decrese exponentilly with d NM for lrge spcer thicknesses. The reson for this disgreement is tht the series-resistor model is not pplicle for d NM lrge compred to the men free pth. In the ltter cse, more sophisticted models hve to e pplied. The series-resistor model is le to ccount for the inverse GMR effect. 4 Eq.(3.5) suggests tht the resistnce of the prllel configurtion is lwys smller thn the resistnce of the ntiprllel configurtion. In most cses this sttement is correct. However, there re exceptions. This cn e seen, y considering multilyer, which comprises different ferromgnetic lyers. As is esy to show, in this cse the GMR equls to R ( α1 1)( α 1) =, (3.7) 1 R α1(1 + q) + α (1 + q ) where α 1 nd α re the symmetry prmeters for the two different ferromgnetic lyers, i.e. (1) (1) () () α 1 = ρ / ρ nd α = ρ / ρ, nd q is the rtio of the up-spin resistivities in the two (1) () ferromgnets, i.e. q = ρ / ρ. It is cler from Eq.(3.7) tht in the cse when the two ferromgnetic lyers hve different symmetries in resistivity, i.e. α 1 >1 nd α <1 or vice vers, then one cn expect n inverse GMR. The series-resistor model cn e redily generlized to include spin-dependent interfce resistnces, y dding dditionl resistors in the network. As will e discussed in section VI, this 13

9 model is etter pproximtion for the description of GMR within the CPP geometry, rther thn the CIP geometry discussed here. For this reson it hs often een used to otin vlues of the spindependent ulk nd interfce resistnces from CPP experimentl dt. III. EXPERIMENTAL SURVEY In this section we overview the min experimentl results on GMR. Where it is not specified, we discuss the current-in-the-plne (CIP) geometry. A seprte section VI will e devoted to GMR within the current-perpendiculr-to-the-plne (CPP) geometry. Gint mgnetoresistnce ws discovered in 1988 y the group of Alert Fert on Fe/Cr mgnetic multilyers 1 nd the group of Peter Grünerg on Fe/Cr/Fe trilyers. In oth cses the smples were grown using MBE nd hd [1] orienttion of the lyers. The Cr spcer lyers were out 1 nm thick, so tht the Fe lyers were coupled ntiferromgneticlly providing n ntiprllel lignment of their mgnetiztions t zero pplied mgnetic field. As the pplied field is incresed, the mgnetic moments of the ferromgnetic lyers progressively rotte towrds the field, leding to decrese in the resistnce of the multilyer (trilyer). At sturtion the mgnetiztions end up in configurtion of prllel lignment with the lowest vlue of the resistnce. Fig.6 shows the vrition in the resistnce of the Fe/Cr multilyer mesured y Biich et l. 1 The highest mgnitude of GMR in these experiments ws found of 79% (using the definition of GMR given y formul (.7)) t T=4.K. The GMR effect ws scried to the spin-dependent trnsmission of the conduction electrons etween the Fe nd Cr lyers. 1. R/R(H=) (Fe 3nm/Cr 1.8nm) 3 polycrystlline Fe/Cr multilyers 4 nd lter found sizele GMR of 1% on Co/Cu multilyers. 44 In the mgnetic multilyers the successive ferromgnetic lyers re exchnge-coupled through nonmgnetic spcer lyer. Prkin et l. found tht the sign of the coupling oscilltes etween ferromgnetic nd ntiferromgnetic with incresing thickness of the spcer lyer. The mgnitude of GMR is lso oscillting from finite vlue to zero s the spcer thickness increses, s shown in Fig.7 for the Fe/Cr multilyer. These oscilltions in GMR reflect the oscilltions in the interlyer coupling. Sizele vlues of GMR re oserved when the coupling is ntiferromgnetic, since this provides n ntiprllel lignment of the mgnetiztions in the successive ferromgnetic lyers t zero mgnetic field, s for d Cr =1nm nd d Cr =.5nm in Fig.7. No GMR ( much diminished GMR in Fig.7) is oserved when the coupling is ferromgnetic which prevents the chnge in the reltive lignment of the mgnetiztions s the pplied field is vried, s for d Cr =1.8nm in Fig.7. As we sw erlier, ntiferromgnetic coupling is not necessry condition for GMR to occur. All tht is necessry is tht the mgnetic moments of the lyers re not locked y the ferromgnetic coupling, ut cn e reoriented y n pplied field. R/R(%) Cr thickness (nm) (Fe 3nm/Cr 1.nm) 35 Fig.7 Sturtion mgnetoresistnce t 4.K versus Cr thickness for Si(111)/Cr(1nm)/ [Fe(nm)/Cr(t)] N Cr(5nm) multilyers deposited t vrious tempertures: tringles nd squres - 4 C (N=3); circles - 15 C (N=). After Prkin et l. 4.6 H S (Fe 3nm/Cr.9nm) 6 Fig Mgnetic field (kg) Normlized resistnce versus pplied mgnetic field for severl ntiferromgneticlly coupled Fe/Cr multilyers t 4.K. Arrows indicte the sturtion field H S, which is required to overcome the ntiferromgnetic interlyer coupling etween the Fe lyers nd lign their mgnetiztions prllel. After Biich et l. 1 In 199 significnt step towrds the industril ppliction of GMR ws mde y Prkin et l. 4 who demonstrted tht GMR cn e oserved in multilyers deposited y sputtering rther thn the much slower MBE growth process. They succeeded in otining similr GMR vlues on sputtered 14 H S Although the highest vlues of GMR were mesured in ntiferromgneticlly-coupled mgnetic multilyers, these multilyers re not the est mterils for technologicl pplictions. This is due to the lrge mgnetic fields, which re required to sturte the mgnetiztion of the multilyers nd to otin sizele chnge in the resistnce. For exmple, s is evident from Fig.6, the sturtion fields in the Fe/Cr multilyers re of the order of 1- kg which is three orders of mgnitude higher thn the fields required for pplictions. The sensitivity, which is defined s R/R per unit mgnetic field, is much too smll. It is of the order of.1%/g, s compred, e.g., to 1%/G AMR in permlloy. A serch of low field GMR structures in which n ntiprllel configurtion of the mgnetiztions could e chieved y different mens, s compred to the niferromgnetic interlyer coupling, resulted in the invention of pseudo spin vlves nd spin vlves. The pseudo spin vlves, shown in Fig., comine hrd nd soft mgnetic lyers, which hve different coercivities. The mgnetic moments of the soft nd hrd mgnetic lyers switch t different vlues of the pplied mgnetic field providing field rnge in which they re ntiprllel nd the resistnce is higher. 5-7 In the experiments of Brns et l., 5 Co/Au/Co trilyers were used with n Au spcer lyers thick enough so tht there ws no exchnge coupling etween the Co films. The first Co 15

10 lyer ws evported on [11]-oriented GAs, wheres the second one ws grown on the Au spcer lyer, which resulted in different coercive fields of the two Co films. Fig.8 shows the hysteresis loop for the Co/Au/Co trilyer otined t room temperture y the mgneto optic Kerr effect (MOKE). It is seen tht there is rnge of mgnetic fields where the mgnetiztions of the Co lyers re ligned ntiprllel, s is indicted y the rrows in Fig.8. Fig.8 shows the resistnce trce otined y scnning through the hysteresis loop. At sufficiently high mgnetic fields the mgnetiztions of oth ferromgnetic films re prllel nd the resistnce is low. The resistnce increses, however, ech time the ntiprllel lignment is chieved during scn through the hysteresis loop. MOKE signl R (Ω) H (kg) with the following structure: Si/Ni 8 Fe (15nm)/Cu(.6nm)/Ni 8 Fe (l5nm)/femn(1nm)/ag(nm). 8 The mgnetiztion curve shows two seprte hysteresis loops. The loop with the smller coercivity corresponds to the reversl of the free NiFe lyer, while the loop shifted y exchnge nisotropy to round H B =9G corresponds to the reversl of the mgnetiztion of the pinned NiFe lyer. Thus, s the field H is swept, the mgnetiztions of the two NiFe lyers chnge from prllel lignment for H lower thn G or higher thn 135G to ntiprllel lignment etween these two vlues. It is thus pprent tht the chnge in resistnce of Fig. 9 is relted to the chnge in reltive orienttion etween the mgnetiztions of the two ferromgnetic lyers. The steep resistnce chnge in the smll field rnge close to H= is now used for mny low field pplictions, such s sensors, red heds, nd mgnetic rndom ccess memories. M (1-3 emu) R/R(%) - 1 H B H (G) Fig.8 MOKE () nd resistnce () hysteresis loops of Co/Au/Co trilyer with Co thickness of 1nm nd Au interlyer thickness of 6nm t room temperture. The two Co films hve different coercive fields. A rnge of mgnetic fields, in which the mgnetiztions of oth Co lyers re ligned ntiprllel nd the resistnce is high, is indicted y rrows. After Brns et l. 5 Fig.9 Mgnetiztion curve () nd reltive chnge in resistnce () for Si/Ni 8 Fe (15nm)/ Cu(.6nm)/Ni 8 Fe (l5nm)/fe 5 Mn 5 (1nm)/Ag(nm) spin vlve. The field is pplied prllel to the exchnge nisotropy field creted y FeMn. H B denotes the exchnge-is field. After Dieny et l.8 A structure, which gives much etter performnce from the point of view of pplictions, is the spin vlve introduced y Dieny et l. 8 The simplest form of the spin vlve structure is shown in Fig.c. It consists of mgneticlly soft ferromgnetic lyer (free lyer), non-mgnetic metl spcer lyer nd second ferromgnetic lyer (pinned lyer), which is exchnge-coupled to n ntiferromgnetic lyer. The exchnge coupling etween the ntiferromgnetic lyers nd the djcent ferromgnetic lyer cretes unidirectionl exchnge nisotropy, i.e. pins the mgnetiztion of this ferromgnetic lyer in certin direction (for review see, e.g., ref.45). The mgnetic hysteresis loop of the pinned ferromgnetic lyer is therefore centred out non-zero is field, H B. On the contrry, the mgnetic hysteresis loop of the free lyer is centred close to zero field, provided the mgnetic coupling etween the ferromgnetic lyers cross the spcer lyer is wek enough. The mgnetic moments of the two ferromgnetic lyers re thus ligned ntiprllel in the field rnge etween zero nd H B. This ehvior is illustrted in Figs.9(,) which show, respectively, the mgnetiztion curve nd the chnge in resistnce reltive to prllel lignment, mesured t room temperture, for smple Mgnetic grnulr mterils represent nother structure, which displys the GMR effect. 9 They consist of non-mgnetic metl lloyed with ferromgnetic metl, which precipittes into grnules, s is schemticlly shown in Fig.d. The size of the grnules depends on the soluility of the ferromgnetic mteril in the nonmgnetic mtrix nd on growth nd nneling conditions nd cn e s smll s nm. Although the grnules cn e mgneticlly coupled, in the sence of the pplied field their mgnetic moments re rndomly-oriented. Applying the mgnetic field ligns the moments of the grnules, which results in the resistnce drop. This ehviour is illustrted in Fig.1, which displys the field dependence of the reltive chnge in the resistnce for heterogeneous Co x Cu 1-x lloys for two concentrtions of Co: x=.19 nd x=.8. 9 Although the mjor prt of the smples were found to e disordered fcc lloys, the presence of Co-rich clusters results in sizele mgnetoresistnce t low tempertures. The sturtion fields, which re required to lign the moments, re s high s in the ntiferromgneticlly-coupled multilyers, i.e. of the order of 1kG. This fct mkes the pplicility of grnulr mterils firly limited. In ddition, the mgnitude of GMR in grnulr mterils t room temperture is strongly reduced due to superprmgnetic relxtion, which origintes from therml fluctutions of the mgnetic moments of the grnules

11 R/R(%) Fig.1 Mgnetic field dependence of R/R=[R(H)-R(H=kG)]/R(H=kG) in grnulr Co x Cu 1-x films. Curves nd mesured t T=1K, curve c mesured t T=1K. After Berkowitz et l. 9 In the following sections we review experimentl results on composition, thickness, roughness, impurity, outer oundry, temperture nd ngulr dependence of GMR in mgnetic multilyers nd spin vlves. 4. Composition dependence c H (kg) Since the discovery of GMR lrge numer of mgnetic multilyer structures, which disply the GMR effect, hve een discovered. It ws found tht the mgnitude of GMR vries considerly depending on the chemicl constituents of the multilyer. The highest pulished vlues of GMR to dte re % in Fe/Cr multilyers 39 nd 1% in Co/Cu multilyers. 44 Sizele vlues of GMR were lso otined in the multilyers: Co/Ag % t room temperture (RT), 46 Ni/Ag 8% t 4.K, 47 Ni/Cu 9% t 4.K, 48 Ni 8 Fe /Cu 18% t RT, 49 Ni 8 Fe /Ag 17% t RT, 5 nd Ni 8 Fe /Au 1% t RT. 51 On the other hnd, low GMR vlues of the order of 1% or less were mesured in Fe/Mo, 5 Fe/Au, 53 Co/Cr, 4 Co/Al, 54 nd Co/Ir 55 multilyers. No GMR ws found in Ni 8 Fe /NM/ Ni 8 Fe /Fe 5 Mn 5 spin vlve structures with T, Al, Cr nd Pd s the nonmgnetic (NM) spcer lyers. 56 Why re some of the multilyers highly mgnetoresistive, wheres the others re not? All the ove multilyers contin ferromgnetic 3d metls, which should hve pronounced spin symmetry in their conductivity due to the presence of exchnge split d nds. It ppers tht the spin symmetry in the nd structure is necessry ut not sufficient condition for high GMR vlues. As ws noted in section, GMR to gret extent is determined y the ferromgnetic metl/nonmgnetic metl pir, rther thn y n individul mteril considered seprtely. For exmple, GMR ws found to e much lower in Co/Cr nd Fe/Cu multilyers (3% in Co/Cr 4 nd 5.5% in Fe/Cu 57 ), s compred to the Fe/Cr nd Co/Cu multilyers. There re two fctors, which re crucil for otining high vlues of GMR. These re the nd mtching nd the lttice mtching etween the ferromgnetic nd nonmgnetic metls. As hs een lredy explined in section, good nd mtching for one spin orienttion etween ferromgnetic metl (FM) nd non-mgnetic metl (NM) implies high trnsmission for this spin cross the FM/NM interfce. On the other hnd, lrge nd mismtch for the other spin orienttion implies tht the trnsmission of this spin is poor. In ddition, roughness nd intermixing ner the 18 x=.19, 1min t 484C x=.19, s deposited x=.8, s-deposited interfces results in spin-dependent scttering s consequence of the lterl rndomness in the tomic potentils. Lrge spin dependence in scttering rises if the tomic potentils of the two types of toms re similr (mtched) for one spin orienttion ut strongly dissimilr (mismtched) for the other spin orienttion. Lttice mtching of the susequent lyers is lso very importnt fctor for GMR. Lttice mismtch leds to the formtion of misfit disloctions nd other structurl defects t the interfces. Scttering y these defects in the nonmgnetic spcer lyer is spin-independent resulting in reduction of GMR. Although the scttering y defects in ferromgnetic lyer could e spindependent, the spin symmetry in the scttering potentils will vry depending on structurl detils. The presence of vrious types of defects will mke the verge of the scttering potentil only wekly-dependent on the spin, which cn led to reduced vlues of GMR. These two conditions, i.e. nd nd lttice mtching, re lmost perfectly stisfied in Co/Cu nd Fe/Cr multilyers. There is n excellent nd mtching etween the mjority-spin electrons of Co nd Cu nd the minority-spin electrons of Fe nd Cr. On the other hnd, there is strong nd mismtch etween the minority spins in Co nd Cu nd the mjority spins of Fe nd Cr. The lttice mtching is lso lmost perfect in these systems. Thin films of Co grow in the fcc structure with the lttice prmeter of 3.56 Å, which is only % less thn the lttice prmeter of 3.61 Å in fcc Cu. Both Fe nd Cr hve the cc structure nd their lttice prmeters re lmost identicl, i.e..87 Å in Fe nd.88 Å in Cr. Thus, it is not surprising tht the highest vlues of GMR re otined in Co/Cu nd Fe/Cr multilyers. Ni nd permlloy (Ni 8 Fe ) hve the fcc structure with lttice prmeter close to tht in Co nd Cu. Like Co, these mterils re strong ferromgnets with entirely filled mjority-spin d nds, so tht there is good nd mtching etween the mjority-spin electrons in Ni nd Ni 8 Fe on the one hnd nd in Cu on the other hnd. This fct explins the reltively high vlues of GMR in Ni/Cu nd Ni 8 Fe /Cu multilyers. Nevertheless, the mgnitude of GMR in these multilyers is normlly less tht in Co/Cu multilyers (e.g., refs.48,49). This difference cn e explined y the stronger disorder in mgnetic moments t the Ni/Cu nd Ni 8 Fe /Cu interfces s compred to the Co/Cu interfce, s will e discussed in section 8. The nole metls Ag nd Au cn serve s good spcer mterils in Co-, Ni- nd Ni 8 Fe sed multilyers nd spin vlves. These metls hve electronic nd tomic structure similr to Cu, lthough not s good nd nd lttice mtching with the 3d ferromgnets. For exmple, Ni 8 Fe /Ag, nd Ni 8 Fe /Au permlloy-sed multilyers show GMR vlues of out % t room temperture nd revel high sensitivity of the resistnce to the pplied field,.%/g, nd low interlyer coupling. 5,51 This comintion mkes them ttrctive for pplictions. Unfortuntely, the growth of these multilyers represents rel prolem. For exmple the Ni 8 Fe /Ag multilyer hs to e deposited t liquid-nitrogen tempertures in order to ttin the required integrity of the lyers. Other nonmgnetic mterils re poor for using s spcer lyers in 3d-ferromgnet-sed multilyers. For exmple, Al, though good conductor, displys n unimpressive performnce in GMR structures (e.g., ref.54). It produces strong spin-independent scttering t the interfces due to the electronic structure mismtch for oth spin orienttions. This is similr to wht one would expect in Co/Cr multilyers. T is d conductor due to high density of sttes t the Fermi energy. It is not surprising tht GMR is negligile in systems where T is used s spcer lyer. It is interesting tht the systems with highest GMR, such s Fe/Cr, Co/Cu, Co/Ag, Ni 8 Fe /Au nd Ni 8 Fe /Ag, re ll immiscile. This fct indictes tht intermixing t the interfces is not fvourle to GMR, nd contrdicts the expecttion tht intermixing produces strong spin-dependent scttering potentils. One of the resons for this might e reduction in the mgnetic moments in the intermixed regions which negtively effects GMR (e.g., ref.58). In ddition, the intermixing my result in misligned spins, which re wekly coupled with the ferromgnetic lyer, or mgneticlly ded lyers. We come ck to this issue in section 7 nd 8. 19

12 A numer of ttempts hve een mde to use hlf-metllic mterils in GMR structures Hlfmetllic compounds re chrcterized y the coexistence of metllic ehviour for one electron spin nd insulting ehviour for the other spin. The electronic density of sttes is, therefore, 1% spinpolrized t the Fermi level, nd the conductivity is dominted y single-spin chrge crriers. Clcultions predict, for exmple, tht ulk NiMnS 6 nd CrO 63 re hlf-metllic ferromgnets. Idelly incorportion of 1% spin-polrized ferromgnet into GMR multilyer should led to switching etween finite nd infinite resistnce within the CPP geometry s the mgneteztion of lterntive ferromgnetic lyers switches from prllel to ntiprllel, i.e. to n infinitely lrge GMR. Unfortuntely, experiments so fr show n unimpressive ehvior of GMR systems sed on hlf metls. The highest vlue of GMR of out 7% is found in CPP mesurements on NiMnS/Cu/NiMnS trilyers t liquid helium temperture. 6 This result is fr short of the infinite vlue of CPP GMR expected from hlf-metllic-sed structure. A possile reson for this is the poor qulity of the NiMnS films, in prticulr t the NiMnS/spcer interfces, resulting in reduced spin polriztion nd/or significnt spin-flip scttering due to misligned mgnetic moments. 5. Nonmgnetic lyer thickness dependence When considering the dependence of GMR on the non-mgnetic lyer thickness in mgnetic multilyers nd spin vlves one should compre the resistnces of the perfectly prllel nd ntiprllel mgnetic configurtions. The presence of the interlyer exchnge coupling leds to oscilltions in GMR, similr to those displyed in Fig.7. This oscilltory contriution to GMR reflects the extent of ntiprllel lignment, which is chieved t zero mgnetic field, rther thn n intrinsic vrition in GMR. Spin vlves re in this sense etter for studying the spcer thickness dependence of GMR thn mgnetic multilyers. This is due to the pinned ferromgnetic lyer, which keeps the direction of its mgnetiztion nd helps to mintin n ntiprllel lignment of the mgnetiztions in certin field intervl, provided tht the ferromgnetic interlyer coupling is not stronger thn the exchnge-is field. However, t smll spcer thicknesses the mgnetic lyers my ecome strongly coupled ferromgneticlly due to the presence of pinholes in the nonmgnetic film, leding to decresed GMR rtio. The dependence of GMR on the non-mgnetic lyer thickness in spin vlves ws studied y Dieny et l. 64 Fig.11 shows the vrition of GMR s function of the thickness of the non-mgnetic lyer (NM) in spin vlve structures with composition: Si/Co(7nm)/NM(d NM )/Ni 8 Fe (5nm)/ Fe 5 Mn 5 Mn(8nm) with NM=Cu nd Au. As is seen from the figure, the vlue of GMR decreses monotoniclly with incresing non-mgnetic lyer thickness. This decrese cn e qulittively scried to two fctors. (i) With incresing spcer thickness the proility of scttering increses s the conduction electrons trverse the spcer lyer, which reduces the flow of electrons etween the ferromgnetic lyers nd consequently reduces GMR. (ii) The incresing thickness of the nonmgnetic lyer enhnces the shunting current within the spcer, which lso reduces GMR. These two contriutions to GMR cn e phenomenologiclly descried y the following expression: 1 R R = R R ( d NM / lnm ) ( 1+ d d ) exp NM /. (5.1) The exponentil fctor represents the proility tht n electron is not scttered within the NM lyer. The fctor in the denomintor descries the shunting effect due to the NM lyer. The prmeter l NM is relted to the men free pth of the conduction electrons in the spcer lyer. One expects tht l NM will e less thn the men free pth in the spcer lyer λ NM, due to the fct tht electrons which most effectively contriute to GMR hve out-of-plne velocities. Dieny et l. 1 proposed tht for systems of prcticl interest l NM is pproximtely equl to hlf of the men free pth λ NM. The prmeter d is n effective thickness, which depends on the conductnce of the system in the sence of the NM lyer. ( R/R) is normliztion coefficient. Although formul (5.1) is purely phenomenologicl expression, it contins significnt prt of the physics involved. As we will see in section 1, within Boltzmnn pproch to free electrons the simple exponentil in expression (5.1) is replced y more complicted exponentil integrls over vrious incidence ngles of the conduction electrons with respect to the plne of the lyers. Nevertheless, the typicl vrition of GMR versus non-mgnetic lyer thickness remins qulittively the sme. It ws found tht the Cu nd Au thickness dependence of GMR, illustrted in Fig.11, cn e fitted well y using decy lengths of l Cu =6nm nd l Au =5nm respectively. 1 These decy lengths re determined y scttering in the spcer, due to phonons, grin oundries, nd other defects, nd re correlted with the men free pth λ NM. The smller vlue found for Au is consistent with the higher resistivity of Au, deduced from mesurements on sputtered smples: λ Au =8.5nm (ρ=7µωcm) for Au versus λ Cu =11.5nm (ρ=5µωcm) for Cu. Fig.11 R/R(%) Thickness (nm) Mgnetoresistnce t room temperture versus thickness of the nole-metl lyer in spin vlves Si/Co(7nm)/NM(d NM )/Ni 8 Fe (5nm)/Fe 5 Mn 5 (8nm)/NM(1.5nm) with NM = Cu nd Au. The solid lines represent fits ccording to Eq.(3.1). After Dieny et l. 64 As is evident from Fig.11, the vlues of GMR re higher for the Cu spcer lyer thn for the Au spcer lyer. Indeed, extrpoltion to zero interlyer thickness gives R/R of 6.4% for Cu versus 4.1% for Au. This fct ws scried to lower trnsmission through the ferromgnetic/nole-metl interfces for Au thn for Cu, which reduces the intensity of the flow of electrons tht continuously escpe from ech ferromgnetic lyer cross the interfces. 64 The low GMR vlues for Au my lso reflect the higher spin-orit scttering expected of the hevier element, which leds to spin-flip scttering in the spcer lyer. The effect of the microstructure my lso e importnt: the lrge lttice mismtch etween ferromgnetic lyers nd Au my result in misfit disloctions nd e n dditionl cuse of the lower GMR. GMR in mgnetic multilyers versus thickness of the non-mgnetic spcer lyer ehves in similr fshion s in spin vlves. Figs.1, disply vlues of GMR in Co/Cu multilyers mesured t reltively lrge Cu thicknesses, so tht the interlyer exchnge coupling is smll. 65 Note tht the interlyer exchnge coupling decreses with incresing Cu thickness much fster thn GMR, such tht the exchnge coupling fields ecome much weker thn the sturtion fields. Therefore, GMR in Cu Au 1

13 Figs.1 results from the rndom rrngement of mgnetic domins in successive mgnetic lyers. Prkin et l. 65 found tht t T=4.K GMR decys pproximtely s 1/d Cu (Fig.1), which is consistent with Eq. (5.1) provided tht the decy length l NM is lrge. As ws explined ove, this ehvior is the direct consequence of the shunting of the electric current due to incresing thickness of the spcer lyers. At room temperture the scttering within the spcer lyers diminishes the flow of electrons from one mgnetic lyer to neighoring mgnetic lyers nd therefore reduces the mgnitude of GMR. Such scttering is relted to volume scttering within the interior of the spcer lyers due to electron-phonon interctions. The vlue of GMR then decys s in Eq. (5.1) with l NM =3nm nd is shown in Fig.1 y the solid line. nd 1nm. As ws rgued y Dieny, 1 the position of the mximum depends on the loction of the spin-dependent scttering centers. In the cse of interfcil spin-dependent scttering, the mximum is locted t smller thicknesses thn for ulk spin-dependent scttering. The ppernce of the mximum is explined y the following rguments. 64 The decrese in GMR t lrge mgnetic lyer thickness is due to the incresing shunting of the current in the inner prt of the ferromgnetic lyers. The decrese in GMR t low thickness is due to the scttering t the outer oundries (sustrte, uffer lyer or cpping lyer). This scttering significntly ffects GMR when the thickness of the ferromgnetic lyer ecomes smller thn the longer of the two men-free pths ssocited with the up- nd down-spin electrons (see lso section 9). R/R(%) R/R(%) K 95K 1 d Cu exp( d Cu /3) d Cu R/R(%) Thickness (nm) Co NiFe Ni Cu thickness (nm) Fig.13 Mgnetoresistnce in FM(d FM )/Cu(.nm)/Ni 8 Fe (5nm)/Fe 5 Mn 5 (8nm)/Cu(1.5nm) spin vlve versus thickness of the ferromgnetic free lyer FM = Co, Ni 8 Fe nd Ni t room temperture. The solid lines represent fits ccording to Eq.(3.). After Dieny et l. 66 Fig.1 Sturtion mgnetoresistnce versus Cu spcer lyer thickness for severl series of multilyers of the form, Si(l1l)/Ru(5nm)/[Co(l.1nm)/Cu(d Cu )] 6 /Ru(1.5nm). Dt re shown for tempertures of 4. K () nd 95 K (). The ctul curves shown in the figure hve the form of R/R = /(1.3+d Cu ) nd R/R =8.9/(.43+d Cu )exp(-d Cu /31.8) t 4. nd 95 K respectively. d Cu is given in nm. After Prkin et l. 65 As we see from the experimentl results nd will see from the theoreticl nlysis within the semiclssicl free-electron models (section 1), the men free pth ppers to e the scling length for the thickness dependence of GMR within the CIP geometry. GMR decys monotoniclly s function of the spcer lyer thickness. The highest vlues of GMR cn e chieved when the spcer lyer is s thin s possile nd therefore hs only smll mount of ulk scttering. The reduction of the spcer lyer thickness is however limited y pinholes through the nonmgnetic mteril, which prevent the ntiprllel lignment of the mgnetiztions nd therefore suppress the mgnetoresistnce. 6. Mgnetic lyer thickness dependence A typicl vrition of the mgnitude of GMR versus the thickness of the free ferromgnetic lyer in the FM(d FM )/Cu(.nm)/Ni 8 Fe (5nm)/Fe 5 Mn 5 (8nm)/Cu(1.5nm) spin vlve versus the thickness of the ferromgnetic free lyer FM = Co, Ni 8 Fe nd Ni is plotted in Fig As is evident from the figure, the three curves hve very similr shpes chrcterized y rod mximum etween 6 Phenomenologiclly, the vrition of spin-vlve MR with the thickness of the ferromgnetic lyers cn e firly well represented y the following expression 66 (see Fig.13): R R = R R 1 exp( d FM / lfm ) ( 1+ d d ) FM /. (6.1) The numertor descries the vrition of the scttering rtes of the electrons with thickness d FM. It chrcterizes the ngle-verged proility for n electron with the longest men free pth to e scttered within the ferromgnetic lyer efore eing scttered diffusively t the outer oundry of the spin vlve. This fctor is responsile for the decrese of GMR t low thicknesses d FM : if the ferromgnetic lyers re too thin the contrst etween the spin-dependent men free pths decreses due to the stronger diffuse scttering of the electrons with the longer men free pth t the outer oundries. l FM is therefore relted to the longest men free pth in the ferromgnetic lyer λ FM. As rgued y Dieny, 8 it is expected tht l FM ½λ FM. The denomintor descries the shunting of the current within the ferromgnetic lyers, so tht d is n effective thickness which represents the shunting of the current in the rest of the structure, i.e. in ll lyers except the ferromgnetic lyer whose thickness is vried. ( R/R) is normliztion coefficient. Although more ccurte expression descries GMR within the Boltzmnn pproch (see section 1), formul (6.1) contins significnt prt of the physics involved. 3

14 In mgnetic multilyers with lrge numer of repetitions, mximum in the vlue of GMR is normlly oserved when the thickness of the mgnetic lyers is vried from monolyer to few nm, i.e. less thn in spin vlves. For exmple, Sto et l. 67 hve found tht the optiml thickness of the permlloy lyers in (Cu/Ni 8 Fe ) 6 multilyers is typiclly 1-3 nm. The min difference etween the spin vlves nd multilyers is the reduced effect of the outer oundry scttering for the former. Although the decrese of GMR t high FM lyer thickness cn still e explined y the shunting current within the FM lyers, the decrese of mgnetoresistnce t smll thicknesses hs different origin nd cn e explined s follows. 1 In the cse of ulk spin-dependent scttering, the decrese of GMR t low d FM is due to insufficient scttering of the electrons with short men free pths, which reduces the spin-symmetry in the conductivity. The criticl thickness elow which the electrons with short men free pths re insufficiently scttered is the men free pth of the electrons in the ferromgnetic lyers, i.e. of the order of 1-nm in permlloy. 68 In the cse of interfcil spin-dependent scttering, this criticl thickness of the ferromgnetic mteril is the minimum thickness required to estlish the electronic properties of the FM/NM interfce. 7. Roughness dependence As ws known from erliest experiments on Fe/Cr multilyers, GMR is very sensitive to the growth conditions nd the structure of the interfces. It is expected tht interfce roughness will enhnce the mgnetoresistnce due to n increse in spin-dependent scttering. A numer of experiments iming to correlte the interfce roughness nd GMR in mgnetic multilyers hve een performed. Indeed, few experiments hve demonstrted tht the vlue of GMR in Fe/Cr multilyers hs tendency to increse with roughness. For exmple, Fullerton et l. 69 fricted sputtered Fe/Cr multilyers with vrile roughness y chnging the sputtering gs pressure nd vrying the sputtering power. The structure of the smples ws thoroughly chrcterized y high nd low-ngle X-ry diffrction. They found tht GMR is higher when the intensity of the low-ngle diffrction pek is smller, which implies rougher interfces. From the mgnetiztion mesurements they showed tht the enhncement in GMR is not due to the improvement in ntiferromgntic lignment nd concluded tht spin-dependent scttering t the interfces is enhnced y roughness. More recently Schd et l. 7 fricted series of high-qulity epitxil Fe/Cr(l) multilyers chrcterized y negligile numer of ulk defects, so tht the dominnt contriution to scttering resulted from interfce roughness. The interfce roughness ws vried through nneling t different tempertures nd ws quntittively nlyzed y speculr nd diffuse synchrotron X-ry diffrction technique. Schd et l. found tht the mgnitude of GMR increses with decresing the lterl correltion length of roughness, ξ x. This cn e seen from Fig.14, in which the tringles show the sturtion resistivity, ρ S, the chnge in the resistivity, ρ, nd the GMR rtio, ρ/ρ S, for the smples with constnt roughness mplitude η, nd vrile lterl correltion length ξ x. A further increse in the nneling temperture leds to incresing roughness height, η, resulting in further enhncement of GMR, s cn e seen from the circles in Fig.14. This enhncement of GMR with interfcil roughness oserved in monocrystlline Fe/Cr multilyers is in contrst to wht ws oserved erlier on polycrystlline Fe/Cr superlttices in experiments y the sme group. 71 A reduction of GMR ws found with incresing the mplitude of the interfce roughness hving constnt correltion length, s is shown in Fig.14. This fct demonstrtes tht spin-dependent scttering is very sensitive to the detils of the microstructure of the interfces. For exmple, polycrystlline smples could provide efficient diffusion chnnels long the grin oundries, so tht nneling cn fcilitte creting interdiffused interfces, which ccording to ref.39 reduces GMR. In ddition, steps t the interfces of the polycrystlline smples often pper t the grin oundries, resulting in vrile spin symmetry in the scttering potentil due to different structurl nd compositionl environment t the steps. On verge, the scttering potentil ssocited with the roughness cn ecome spin-independent resulting in reduced GMR. Fig.14 ρ s (µωcm) ρ (µωcm) ρ/ρ s (%) η/ξ x (r.un.) Trnsport properties of (1)-oriented monocrystlline () nd polycrystlline () Fe/Cr multilyers. Vritions in the ntiferromgnetic coupling re tken into ccount y dividing ρ nd ρ/ρ S y (1-M R /M S ), where M R nd M S re the remnent nd sturtion mgnetiztions respectively. () The tringulr dt points correspond to the smples with the constnt roughness mplitude η, so tht only the lterl correltion length ξ x ws vrying. () The lterl correltion length of the roughness is constnt ξ x 9nm. After Schd et l. 7,71 From the experiments on Fe/Cr multilyers one cn conclude tht incresing the density of steps nd the roughness height t compositionlly-shrp monocrystlline regions of the interfces enhnces GMR in these multilyers. On the other hnd, interfce roughness ssocited with interdiffused regions nd high density of defects, such s grin oundries, is likely to reduce GMR. Opposite to these Fe/Cr multilyers, no enhncement of GMR hs een oserved in Co/Cu multilyers with incresing interfce roughness. For exmple, Kno et l. 7 vried the shrpness of the interfces in sputtered Co/Cu multilyers y chnging the sustrte temperture. The degree of roughness ws determined y the X-ry stellite pek intensity. They found tht the GMR rtio decreses s the sustrte temperture increses nd concluded tht roughness reduces GMR. Suzuki nd Tg 73 succeeded in prepring Co/Cu superlttices with well-controlled interfcil roughness y mgnetron sputtering. The interfces etween Co nd Cu were modified y codeposition, so tht the thickness of the intermixed CoCu lyer ws vried from to.5 nm. In these smples, only the interfcil region ws modified, while the morphology nd the crystllinity of the multilyer remin unchnged. They found tht interfcil roughness minly contriutes to the residul resistivity nd the spin dependence of the scttering t the interfces is wek. They lso concluded tht the GMR rtio decreses with incresing interfcil roughness. The suppression of GMR in Co/Cu multilyers with incresing interfcil roughness is proly due to significnt chnge in the mgnetic stte of the Co toms in the intermixed regions. The mgnetic moments of these toms might e reduced nd misligned with the mgnetiztion of the Co lyer. The spin symmetry of scttering y these toms is oviously strongly reduced, resulting in decrese of GMR with incresing roughness. Fe/Cr multilyers re electroniclly more stle with respect to roughness, so tht the electronic stte of the toms t the stepped interfces is similr to tht in the ulk of the lyers. This fct is supported y first-principle clcultions which show tht the ρ s (µωcm) ρ (µωcm) η (nm) 4 5

15 tomic moments t the rough Fe/Cr interfces do not depend significntly on the nerest-neighour environment Impurity dependence Since the mgnitude of GMR is relted to the symmetry in the scttering rtes within the two conduction chnnels, it ws expected tht modifying the spin-dependent scttering y introducing pproprite impurities either t the interfces or in the ulk of the ferromgnetic lyers would enhnce GMR. Scttering symmetries hve een indirectly determined from mesurements of the resistivity of mgnetic ternry lloys. 31,3 A numer of ttempts hve, therefore, een mde to find correltion etween the mgnitude of the scttering symmetries in ulk mgnetic lloys nd the mgnitude of GMR in mgnetic multilyers. Gurney et l. 74 inserted thin lyers (.4nm) of vrious impurities, such s Au, Ag, V, Mn, Al, Ge, nd Ir t the interfces etween Fe nd Cr lyers in Fe/Cr multilyers. They found tht inserting Au, Ag, Al, Ge, or Ir strongly reduces GMR, wheres, inserting V nd Mn does not effect GMR much, s compred to n inserted Cr lyer of sme thickness. Johnson nd Cmley interpreted the results of these experiments in terms of different spin-dependent scttering symmetries of these impurities in ulk iron. 75 They found tht the impurities, such s Mn nd V, which hve spindependent scttering symmetry similr to tht of Cr in Fe, do not chnge GMR sustntilly. On the other hnd the impurities, such s Al nd Ir, with spin-dependent scttering symmetry opposite to tht of Cr in Fe, led to rpid degrdtion of the GMR. We note, however, tht ccording to Mrrows nd Hickey, 76 inserting the impurities t the interfce of ntiferromgneticlly-coupled multilyers could esily destroy the interlyer coupling, resulting in reduction of GMR due to imperfect ntiferromgnetic lignment. Prkin 77 demonstrted tht inserting very thin lyer of Co t the Ni 8 Fe /Cu interfces results in drmtic increse in GMR. Fig.15 shows the room temperture resistnce response to the pplied mgnetic field in Si/Ni 8 Fe (5.3nm)/Cu(3.nm)/Ni 8 Fe (.nm)/fe 5 Mn 5 (9nm)/Cu(1nm) spin vlve nd in the sme spin vlve with.5nm thick Co lyers dded t ech Ni 8 Fe /Cu interfce. As is seen from this figure, the vlue of GMR increses y fctor of two, demonstrting the strong effect of the inserted Co lyer. A creful nlysis of GMR s function of the Co lyer thickness shows tht the mgnetoresistnce cn e enhnced from.9% up to 6.4%, the Co lyer thickness scle of the enhncement eing just.3nm (see Fig.15). The positive effect of the Co lyers on GMR in the permlloy-sed spin vlves ws found to e strongly loclized t the interfces. By vrying the distnce of the.5nm Co lyer from the interfce, d, no significnt increse in GMR ws found for d>.5nm (see Fig.15). Contrry to inserting Co lyers t the interfces of permlloy-sed spin vlves, dding permlloy lyer t the Co/Cu interfces in Si/Co(5.7nm)/Cu(.4nm)/Co(.9nm)/ Fe 5 Mn 5 (1nm)/Cu(1nm) spin vlves reduces the vlue of GMR from 6.8% to 3.9% with the Ni 8 Fe thickness scle of just.8nm (see Fig.15d). The results of these experiments re explined y the dominnt contriution to GMR from spindependent scttering t the interfces. Theoreticlly Inoue et l. 78 rgued tht the minority-spin scttering t Co/Cu interfces is lrger thn t Ni 8 Fe /Cu interfces, due to the lrger mismtch in the minority-spin d tomic energy levels for the former. Experimentlly, on the other hnd, the effect ppers to rise from the stilising role of Co on the mgnetic moments t the interfces. In generl, the mgnetic moments t interfces (especilly rough interfces) cn e very different from those in the ulk. Speriosu et l. 79 found tht t room temperture there is sustntil reduction in the mgnetiztion of permlloy ner the interfces with Cu, which is equivlent to mgneticlly-ded lyer of.nm thickness. On the contrry, much thinner ded-lyer of only.1nm ws found for Co/Cu multilyer. The non-mgnetic lyers t the interfces re detrimentl to GMR. These lyers re source of strong spin-independent scttering. Misoriented spins lso reduce GMR due to spin mixing nd spin-flip scttering. The presence of reduced Ni moments nd non-colliner Fe moments on interdiffused Ni 8 Fe /Cu interfces is supported y first-principle clcultions. 8 Plcing smll mount of Co t the interfce drmticlly increse the collinerity of mgnetic moments nd stilise the mgnetic moments of Ni t the Ni 8 Fe /Cu interfce, therey enhncing GMR. R/R (%) Fig.15 6 d 4 d FeMn NiFe Co Cu Co NiFe -4 - Field (G).5nm Co t interfces no Co t interfces R/R (%) Effect of thin lyer inserted t the interfces in spin vlves. () Resistnce versus mgnetic field for Si/NiFe(5.3)/Cu(3.)/NiFe(.)/FeMn(9)/Cu(1) spin vlve without (open circles) nd with (filled circles).5nm thick Co lyers dded t ech NiFe/Cu interfce. Dependence of the sturtion mgnetoresistnce on () Co interfce lyer thickness, d Co, in Si/NiFe(5.3-d Co )/Co(d Co )/Cu(3.)/Co(d Co )/NiFe(.-d Co )/FeMn(9)/Cu(1) spin vlves, (c) distnce d of.5nm thick Co lyer from the NiFe/Cu interfces in Si/NiFe(4.9- d)/co(5)/nife(d)/cu(3)/nife(d)/co(5)/nife(1.8-d)/femn(9)/cu(1) spin vlves, nd (d) NiFe interfce lyer thickness, d NiFe, in Si/Co(5.7-d NiFe )/NiFe(d NiFe )/Cu(.4)/FeNi(d NiFe )/ Co(.9-d NiFe )/FeMn(1)/Cu(1) spin vlves. Note tht NiFe stnds for permlloy nd the lyer thicknesses re given in nm. Experiments re performed t room temperture. After Prkin. 77 A numer of interesting exmples of impurity effects hve een otined within the CPP geometry. As ws explined in Sec.3 using simple series resistor model, the GMR cn e inverted (i.e. cquires n opposite sign) if the spin symmetries in scttering re opposite in consecutive ferromgnetic lyers. Vouille et l. 81 hve found tht Ni 95 Cr 5 /Cu/Co/Cu multilyers, in which Ni lyers re doped y 5% of Cr impurities, disply the inverse GMR when Ni 95 Cr 5 thickness is more thn nm. According to Cmpell nd Fert 3 Cr impurities in Ni sctter more strongly the mjorityspin electrons, which mkes the ulk scttering spin symmetry of the NiCr lyers less thn unity, i.e. α NiCr <1. This is opposite to the Co lyers, which re chrcterized y α Co >1, resulting in the inverse GMR. The inverse GMR ws lso found in other multilyers of the type FM/Cu/Co/Cu, where FM=Ni 1-x Cr x, Co 1-x Cr x, Co 1-x Fe x nd Fe 1-x V x. 81,8 The comprison of the ulk scttering spin symmetries α FM extrcted from these experiments with the previous results otined for ulk lloys 31,3 shows tht the sign of α FM is the sme ut the mgnitude is generlly much smller. The inversion of GMR in the experiments of Hsu et l. 8 nd Vouille et l. 81 ppers to occur t thicknesses of the FM lyer ove certin criticl thickness, so tht there ws crossover in the sign of GMR t this thickness. This ws scried to the competition etween ulk scttering in the FM, [1 exp(-d/.3)] c exp(-d/.3) d [1 exp(-d/.8)] d d d d FeMn NiFe Co Cu Co NiFe Lyer thickness (nm) d d FeMn NiFe Co Cu Co NiFe FeMn Co NiFe Cu NiFe Co 6 7

16 which hs spin symmetry α ulk <1, nd interfce scttering t the FM/Cu interfce, which ws ssumed to hve spin symmetry α int >1. We note, however, tht t low FM thickness the interfce resistnce might ecome dependent on the lyer thickness, which would mke the interprettion of these experiments more complicted (see section VI). 9. Outer oundry dependence As we sw ove, the vlue of GMR is higher in mgnetic multilyers thn in spin vlves. The ltter consist of just two ferromgnetic lyers seprted y nonmgnetic spcer lyer. With incresing numer of FM/NM ilyers within multilyer the vlue of GMR monotoniclly increses until it reches sturtion. Fig.17 shows n exmple of such vrition of GMR, otined y Plskett nd McGuire 83 in experiments on Cu(1nm)/[Co(1nm)/Cu(1nm)] N multilyers. They found tht t T=4.K GMR grows up to 35% with incresing N up to 18, clerly displying tendency to sturte (see Fig.17). The lower vlues of GMR in these experiments s compred to those in ref.38 re explined y wek ntiferromgnetic coupling. One of the fctors, which my ply role in incresing GMR with the numer of ilyers, is n improvement in the structurl qulity for the thicker multilyers. However, the mjor fctor, which is responsile for the ehvior of GMR versus N shown in Fig.17, is the presence of diffuse scttering t the outer oundries of the multilyer. Indeed, if the longest men free pth is much lrger tht the totl thickness of the mgnetic multilyer, diffuse outer-oundry scttering reduces the conductivity of the good conduction chnnel nd hence effects negtively GMR. Plskett nd McGuire used the Fuchs-Sondheimer expression for thin film resistivity 84,85 in order to estimte the men free pths, s is shown in Fig.17. They found men free pth of 47 nm for the sturted stte of the Co/Cu multilyer, which lthough possily n overestimte gives the right order of mgnitude. We see, therefore, tht outer oundry scttering is very importnt chrcteristic of spin vlves ecuse their thickness is comprle to (or even less thn) the men free pth. Fig.16 Position dependence of the gint mgnetoresistnce for vrious trnsition metl impurities in the Co lyer in Si/T(5nm)/Co(.5nm-x)/δ/Co(x)/Cu(3nm)/Co(x)/δ/Co(.5nm-x)/FeMn(8nm)/ T(.5nm) spin vlves. Elements for which α vlues re ville from refs.31,3 hve these vlues noted t the top of ech pnel. The grph width represents the Co lyer thickness. As x increses in ech grph the δ-lyer moves from the Co/Cu interfce to the outermost surfce of the Co lyers. The grph for Co cn e regrded s the control experiment. After Mrrows nd Hickey. 76 ρ/ρ (%) Numer of ilyers ρd ( 1 µω cm nm) 6 4 H= H=18kG 1 3 Totl thickness (nm) Very interesting experiments hve een performed recently y Mrrows nd Hickey 76 who inserted very thin (su-monolyer) δ-lyer of vrious elements in Co/Cu/Co spin vlve t vrious distnces from the interfces. Some of their results re shown in Fig.16, in which the GMR rtio is plotted ginst the position of the dopnt δ-lyer for vriety of elements from the centrl prt of the trnsition metl series. As is evident from the figure, the vlue of GMR cn e reduced or enhnced depending on the nture of the impurity nd its loction within the spin vlve. For exmple, ferromgnetic 3d impurities Fe nd Ni enhnce GMR when they re close to ut just ehind the Co/Cu interfce. For mterils with α<1, i.e. Cr, Mo, T, nd Ru, the mgnetoresistnce is totlly suppressed when they re plced t the interfce, ut it is recovered s they move ck into the ulk of the Co lyer. Mterils to the right of Co, tht might e expected to hve α>1, nmely Pd, Pt, nd the nole metls, cn suppress the GMR somewht t the interfce, ut it is very rpidly recovered s they move into the lyer. It ws lso found tht 4f mgnetic impurities re ltogether dmging to the GMR. A quntittive interprettion of these experiments is chllenge for first-principle modeling. 8 Fig.17 GMR s function of the numer of ilyers N () nd resistivity s function of the multilyer thickness d () in Cu(1nm)/[Co(1nm/Cu(1nm)] N multilyers, mesured t T=4.K. GMR is determined from the resistivity t H= nd H=18kG except for N=18, for which H=5kG. Solid lines in () represent the Fuchs-Sondheimer pproximtion for thick film resistivity, i.e. ρd=ρ ulk (d+3λ/8), where ρ ulk is the ulk resistivity nd λ is the men free pth. After Plskett nd McGuire. 83 The outer oundry scttering in spin vlves occurs t the interfces etween the pinned lyer nd the pinning lyer nd etween the sense lyer nd the cp lyer, which is used for protecting the structure from oxidtion or corrosion. Antiferromgnetic Fe 5 Mn 5 8 nd ferrimgnetic T 3 Co re frequently used s pinning lyers. The scttering t FM/FeMn nd FM/TCo interfces is commonly regrded s entirely diffuse due to the highly disordered tomic nd mgnetic structures of the ntiferromgnetic lyers nd the interfces. In ddition, lthough these mterils re highly resistive (resistivity is out 1µΩcm t room temperture), they still contriute to the shunting current 9

17 within the spin vlve structure, therey reducing GMR. The sme effect occurs t the FM/T interfce when T is used s the protective cp lyer. GMR in spin vlves cn e improved if ntiferromgnetic NiO is used s pinning lyer. Anthony et l. 87 fricted ottom spin vlves of the type NiO/NiFe/Co/Cu/Co/NiFe nd otined GMR rtio of 13% t room temperture (thin Co lyers t the interfces were used to enhnce GMR 77 ). An even higher vlue of GMR, nmely 15%, ws otined y Swgten et l. 88 in spin vlve of the type NiO/Co/Cu/Co/Cu/NiO, in which the top NiO lyer served s protective lyer nd the top Cu lyer mgneticlly insulted the free lyer from exchnge ising. These reltively high vlues of GMR re the result of the fct tht NiO is n insultor nd consequently does not contriute to the shunting current. Moreover, due to the reltively lrge nd gp of NiO (out 4eV) it impedes penetrtion of electrons into the ulk nd therefore provides mechnism for speculr scttering t the interfce etween the pinned nd pinning lyers. Contrry to diffuse scttering, speculr scttering reflects electrons ck through the spin vlve llowing them to propgte cross the spcer lyer mny times, therey incresing spin filtering effects. The degree of speculr scttering t FM/NiO interfces is relted to the interfce qulity ecuse disorder cn esily mke the scttering diffuse. A certin degree of speculr scttering ws rgued to e the origin of high vlues of GMR in the experiments performed y Egelhoff et l. 89,9 on NiO-sed symmetric spin vlves. A symmetric spin vlve 87 contins one free ferromgnetic lyer sndwiched etween two pinned mgnetic lyers, seprted y nonmgnetic spcer lyers, which llows higher mgnitudes of GMR to e otined thn in ordinry spin vlves. The symmetric spin vlve, which ws studied y Egelhoff et l., hd the structure of NiO/Co/Cu/Co/Cu/Co/NiO nd displyed GMR vlues exceeding % t room temperture. Speculr scttering of electrons t Co/CoO interfces ws lso suggested to e the reson for the enhncement of GMR y up to 17% t room temperture, otined in ottom spin vlves of the type NiO/Co/Cu/Co, in which the top Co lyer ws slightly oxidized. 91,9 Sugit et l. 9 regrded the high mgnitude of GMR of 8% otined in n epitxilly-grown symmetric spin vlve with n α- Fe O 3 pinning lyer, i.e. α-fe O 3 /Co/Cu/Co/Cu/Co/α-Fe O 3, s n indiction of speculr electron reflection t the Co/α-Fe O 3 interfces. The importnce of overlyers deposited t the top outer oundry of the NiO/Co/Cu/Co spin vlve ws demonstrted y Egelhoff et l. 93 They found tht the deposition of out monolyers (ML) of Au, Ag, or Cu increses GMR, wheres the deposition of ML of T, Si, C, or Ni 8 Fe decreses GMR. These results were interpreted y Egelhoff et l. s evidence of enhnced speculr scttering for the cse of Au, Ag, nd Cu, ut suppressed speculr scttering for the cse of T, Si, C, nd Ni 8 Fe t the interfce with Co. However, recent studies hve demonstrted tht the enhncement of GMR occurs for ny top Co lyer thickness. 94 This fct contrdicts the plusile explntion in terms of speculr scttering, ecuse the speculr scttering should reduce GMR if the Co thickness is much lrger thn the optimum thickness (see section 1). For such Co thicknesses the speculrly reflected electrons re not le to cross the spcer lyer nd contriute insted to the shunting current. Other effects seem to e importnt nd require further investigtions. 1. Temperture dependence Mny experiments hve found tht GMR decreses with incresing temperture. Typiclly, the mgnitude of GMR is fctor of two or three smller t room temperture thn t liquid helium temperture. For exmple, GMR drops y fctor of 3.1 in Fe/Cr multilyers 95 nd y fctor of 1.8 in Co/Cu multilyers 38 in this temperture intervl. The mjor fctor, which contriutes to the temperture vrition of GMR, is inelstic scttering y phonons. Although electron-phonon scttering conserves spin, it enhnces the sturtion resistivity of the multilyer, effecting negtively the GMR rtio. In ddition, it shortens the men free pth in the spcer lyer, which prevents the flow of electrons etween successive ferromgnetic lyers nd hence reduces GMR. Scttering y phonons in ferromgnetic metls is spin-dependent due to the spin- 3 dependent density of sttes t the Fermi energy (see sec.3). If other spin-dependent scttering processes re importnt (e.g., spin-dependent scttering t the interfces due to interfce roughness), which re chrcterized y different spin symmetry, the contriution from electron-phonon scttering necessrily chnges GMR. Another non-trivil mechnism reducing GMR is relted to internd trnsitions tht re driven y the pplied electric field in the presence of spin-independent scttering potentils. This will e discussed in section 17. Fig.18 AR Co/Cu (fωm ) γ interfce spin symmetry Co/Cu interfce resistnce AR Co/Cu AR Co/Cu. 1 3 Temperture (K) The temperture dependence of the spin-dependent prmeters for Co/Cu multilyers, s determined from the experiments on grooved sustrtes using the two-current series resistor model: () the spin symmetry prmeter, γ, of the Co/Cu interfce, () the resistnces of the Co/Cu interfce, (c) the spin symmetry prmeter of the ulk Co, β, (d) the resistivity of Co nd Cu lyers. γ is defined s γ=(α Co/Cu -1)/(α Co/Cu +1), where α Co/Cu is the rtio of the interfce resistnces AR Co/Cu nd AR Co/Cu. β is defined s β=(α Co -1)/(α Co +1), where α Co is the rtio of the ulk Co resistivities ρ Co nd ρ Co. After Oepts et l. 96 Another fctor, which might influence the temperture vrition of GMR, is electron-mgnon scttering. Contrry to phonons, scttering y mgnons is ssocited with spin-flip processes which intermix the mjority- nd minority-spin current chnnels. At high tempertures when spin fluctutions ecome sizele the electron-mgnon scttering would inevitly suppress GMR. However, whether this effect is importnt t room temperture nd elow for GMR in mgnetic multilyers sed on 3d ferromgnets, which re chrcterized y very high Curie tempertures, still remins uncler. Dieny et l. 66 hve pointed out tht the presence of roughness nd interdiffusion t the interfces wekens the mgnetic interctions due to the decresed mgnetic moments nd the reduced numer of mgnetic nerest neighours. These loose spins my e more strongly ffected y temperture-dependent spin-flip processes thn the spins in the ulk of the lyers. Dieny et l. hve estlished correltion etween the Curie temperture of the ferromgnetic metl nd the slope of the decrese in GMR in spin vlves. They found tht the therml vrition of GMR in spin vlves is weker for those ferromgnets, which hve higher Curie temperture. They rgued tht this 31 β ρ (µωcm) ulk Co spin symmetry ulk resistivities.5ρ Co.5ρ Co ρ Cu 1 3 Temperture (K) d c

18 correltion is consequence of spin-flip scttering minly t the interfces due to the reduced Curie temperture of the interdiffused interfcil regions. Very interesting experiments on the temperture dependence of GMR with quntittive interprettion of the results were reported y Oepts et l. 96 They mesured the temperture vrition of GMR in Co/Cu multilyers deposited on grooved sustrtes, s is shown in Fig.3, so tht the geometry in these experiments ws similr to CPP GMR. The results were nlysed in terms of spindependent ulk nd interfce resistnces within the two-current series resistor model nd re shown in Fig.18. They found tht the spin-dependent interfce resistnces re wekly temperture-dependent (Figs.18,), which implies tht up to room temperture the mjor contriution to the interfce resistnces comes from elstic scttering. On the other hnd, s cn e seen from Fig.18d, the ulk resistivities of Co nd Cu increse y more thn fctor of two etween 4.K nd 3K, reflecting the sizele contriution from inelstic scttering. However, it ws found tht the spin symmetry of the resistivity in the Co lyers, α Co, is lmost independent of temperture (Fig.18c). The results of these experiments demonstrte, firstly, tht phonons, rther thn mgnons, influence the spindependent resistnce of the Co/Cu multilyer. Scttering y mgnons would inevitly reduce the spin symmetry of the ulk nd interfce resistnces with incresing temperture due to spin-flip processes. Secondly, the spin symmetry is out the sme for elstic nd inelstic scttering in the Co lyers. This is evidence tht the scttering spin symmetry is minly determined y intrinsic properties of ulk Co, nmely y its nd structure. 11. Angulr dependence We hve so fr considered GMR tht rises from prllel nd ntiprllel mgnetiztions of the successive ferromgnetic lyers. In this section we discuss the vrition of the mgnetoresistnce s function of the ngle etween the mgnetiztions θ=θ 1 -θ. In spin vlves which comprise free nd pinned mgnetic lyer continuous chnge of the ngle θ cn e otined y pplying rotting field, which rottes the mgnetiztion of the free lyer, ut keeps the direction of the mgnetiztion of the pinned lyer fixed. Such n experiment ws performed y Dieny et l. 8 It ws found tht there re two components contriuting to the mgnetoresistnce: the nisotropic mgnetoresistnce (AMR), which vries s the cosine squred of the ngle etween the rotting mgnetiztion nd the sensing current, nd gint mgnetoresistnce. By sutrcting the contriution from AMR, Dieny et l. found tht GMR vries linerly with cosθ, nd cn e phenomenologiclly descried y the formul R ( θ ) = RP + ( RAP RP )(1 cosθ ) /, (11.1) where R P nd R AP re the resistnces of the spin vlve for the prllel nd ntiprllel mgnetiztions respectively. Such liner vrition of the resistnce with cosθ ws lso oserved in Fe/Cr multilyers. 97 A theoreticl considertion of the ngulr dependence of GMR within quntum-mechnicl pproch nd free-electron model predicts tht for constnt potentil within the multilyer the conductnce, rther thn the resistnce, should vry linerly with cosθ. 98,99 The sme ehviour is lso predicted within the semiclssicl free-electron model. 7 Such vrition ws found in the experimentl study of GMR in Co/Ag/NiFe/Ag multilyers [1] nd is shown in Fig.19. Although the difference etween the two descriptions is second order in the GMR rtio, it my e sizele if the vlue of GMR is reltively high, which is the cse for the Co/Ag/NiFe/Ag multilyers studied in ref.1. This is evident from Fig.19, which shows lmost perfect liner vrition of the conductnce with cosθ, ut displys slight non-linerity in the resistnce dependence. Although the free-electron theory predicts significnt deprtures from linerity when potentil steps t the interfces re present, 98 no experiments to dte hve provided ny evidence for such deprtures. First-principle clcultions 11 of the ngulr vrition of GMR suggest tht the functionl dependence is essentilly of the form (1-cosθ), which grees with experiments. 3 Fig.19 Resistnce Normlized resistnce nd conductnce versus the cosine of the reltive ngle etween the mgnetiztions of the soft permlloy lyers nd hrd lyers composed of Co clusters for [Co(.4nm)/Ag(4nm)/NiFe(4nm)/Ag(4nm)] 15 mutliyer. After Steren et l. 1 The ngulr vrition of GMR hs een lso studied within the CPP geometry. 1 Although the devitions from liner dependence of the conductnce versus cosθ were found to e more pronounced thn in the CIP geometry, they still remin reltively smll. IV. FREE-ELECTRON AND SIMPLE TIGHT-BINDING MODELS A lrge numer of vrious theoreticl models hve een developed to descrie GMR. These models differ minly in the wy tht they tret the electronic structure nd the electronic trnsport. The electronic structure cn e descried either within simple free-electron pproximtion or within n ccurte multind pproch. The min dvntge of the free-electron theories is tht they re physiclly more trnsprent nd, though simple, cn still cpture some importnt physics of GMR. This is lso the cse for simple tight-inding models, which pproximte the electronic structure y single tight-inding nd. Multind models re, however, essentil for quntittive description of GMR. Within these models the electronic structure is descried either using prmeterized tightinding nds or first-principle clcultions within the locl density pproximtion. The electronic trnsport cn e considered either within semiclssicl Boltzmnn theory or within quntummechnicl theory. The Boltzmnn theory of trnsport is verstile formlism, which hs een widely used for treting GMR. It reks down, however, in mgnetic multilyers of prcticl interest ecuse the sund energy splitting is comprle with the life-time rodening due to scttering. In these cses quntum-mechnicl theory within multind tretment of the electronic structure is the est wy to descrie GMR. We egin our review of theoreticl models for GMR y discussing the semiclssicl Boltzmnn theory within the free-electron model. 1. Semiclssicl theory (1-cosθ)/ The resistor model, which hs een introduced in section 3, is too simple to descrie correctly CIP GMR in mgnetic multilyers nd spin vlves. This model is sed on the ssumption tht the men free pth is long for oth spin chnnels s compred to lyer thicknesses. This pproximtion is not justified for rel lyered systems ecuse the men free pth within one of the spin chnnels is comprle to or even less thn the lyer thickness. In ddition, the resistor model is unle to predict the symptotic ehvior of GMR for lrge lyer thicknesses Conductnce

19 A more sophisticted quntittive insight into spin-dependent trnsport cn e otined using the semiclssicl Boltzmnn theory of trnsport (e.g., refs.3,13). This theory considers electron trnsport using clssicl dynmics, which mkes it different from the quntum-mechnicl liner response theory of trnsport tht will e considered in section 13. Nevertheless, the semiclssicl theory includes mny spects of quntum mechnics. For exmple, within this pproch quntummechnicl sttistics is used nd scttering cn e clculted quntum-mechniclly ssuming relistic nd structure. Boltzmnn theory is sed on semiclssicl description of the electrons in metls in the presence of externl fields using sttisticl distriution function. The distriution function f(r,k,t) is defined s the numer of electrons with given position r nd wve-vector k t time t. We ssume tht the two spin sttes of the electrons re uncoupled nd, therefore, the distriution function cn e considered independently for the up- nd down-spin chnnels. The Boltzmnn trnsport eqution is otined y lncing the chnge in the distriution function cused y the pplied electric field nd the scttering processes tht ct to ring it ck towrds equilirium, i.e. df ( r, k, t) f = r r f k k f +. (1.1) dt t sctt The first term in this eqution descries the electron drift due to their velocity, the second term reflects the ccelertion of the electrons due to the pplied field nd the scttering term descries scttering of the electrons y imperfections in the lttice, such s defects or impurities. It cn e written in terms of the proility P for n electron to sctter etween momentum k nd k : f t sctt = k k { P [ ] [ kk 1 f ( r, k, t) f ( r, k, t) P ] } k k 1 f ( r, k, t) f ( r, k, t) k, (1.) where the right-hnd term descries scttering-out processes, in which n electron from n occupied stte of momentum k sctters into unoccupied sttes k, nd the left-hnd term descries "sctteringin processes, in which electrons from occupied sttes of momentum k sctter into n unoccupied stte k. We re interested in stedy stte solution, when the distriution function is no longer chnging so tht df/dt= in eqution (1.1). In this cse, tking into ccount the principle of microscopic reversiility, i.e. P = P, nd ssuming uniform pplied electric field,, we otin k k k k e v ( r f ( r,, k f ( r, = Pkk [ f ( r, k ) f ( r, ], (1.3)! k where e is the solute vlue of the electron chrge nd v is the electron velocity. Aiming t liner response theory, it is convenient to represent the distriution function s f ( r, = f ( + g( r,, where g(r, is the devition of the distriution function f(r, from the equilirium Fermi-Dirc F ] k due to the pplied electric field. Sustituting this form into Eq. (1.3) nd retining only the lowest order contriution with respect to, we otin distriution f ( ) = { 1+ exp[( E( E ) / kt } 1 f ( v ( r g ( r, e, v( = Pkk [ g( r, k ) g( r, ]. (1.4) E( k This is generl representtion of the linerized Boltzmnn kinetic eqution for the description of the electric current, the density of which is given y e j ( r) = Ω v( g( r,, (1.5) k where Ω is the volume of the system. However, the evlution of Eq.(1.4) is not esy to perform ecuse of the scttering-in term P k kk g( r, k ), which links the vlues of the distriution function t vrious moment. The Boltzmnn eqution cn e considerly simplified using the relxtion time pproximtion. Within the relxtion time pproximtion the scttering-in term is neglected which results in f ( g( r, v( r g( r, e, v( =, (1.6) E( τ ( where τ ( is the relxtion time for n electron to sctter out of momentum stte k, which is defined y 1 τ ( = P kk. (1.7) k In generl, neglecting the scttering-in term is not trivil pproximtion nd hs to e justified (e.g., ref.3). For ulk homogeneous system it is strightforwrd within the relxtion time pproximtion to µν derive the expression for the conductivity tensor σ which is defined y µ µν ν = σ, ν j, (1.8) where the indices µ nd ν denote the Crtesin components. In this cse r g ( r, = nd it follows from Eq. (1.6) tht f ( g( = eτ ( v(,. (1.9) E( Tking the zero-temperture limit, i.e. f ( ) / E ( = [ E ( E ] n k δ, nd sustituting Eq.(1.9) into Eq.(1.5), we otin the well-known expression for the conductivity per single spin chnnel within the relxtion time pproximtion: 3 σ µν e = Ω k n µ ν v ( k )v ( τ ( δ[ E( ]. (1.1) E F In the cse of films nd multilyers which re ssumed to e homogeneous in the xy plne of the lyers ut inhomogeneous in the z direction perpendiculr to the plnes (due to the interfces nd oundries), the distriution function g(z,v) is dependent on z, ut independent of x nd y. In this cse the solution of the Boltzmnn eqution (1.6) tkes the form ± f ( ± # z g ( z, = eτ (, v( 1 + A ( exp. (1.11) E( τ ( v z ( Here signs ± refer to whether the z-component of the electron velocity is positive or negtive. The coefficients A ± re determined from mtching the oundry conditions t the interfces nd outer oundries in terms of reflection nd trnsmission proilities nd will e considered elow. The current density cn e otined from Eq. (1.5). We note tht the solution of the Boltzmnn eqution tkes the form of eqution (1.11) only for the CIP geometry which we consider in this section. In this cse the current nd pplied field cn e ssumed to e uniform within the plne of the multilyer. For the CPP geometry the electric field is position- nd spin-dependent ecuse mgnetic multilyers re inhomogeneous in the direction of the n F 34 35

20 electric current nd, therefore, eqution (1.11) does not hold. This point will e further discussed in the next section. Up to this point we hve not specified wht is the nd structure of the system under considertion. The ove derivtions re vlid for the multind electronic structure (ssuming tht the nd index is included in nd cn e pplied for the clcultions of conductivity nd GMR within the semiclssicl pproximtion in this generl cse. This will e discussed in section 16. Below we consider free-electron model. Within free-electron model the nd structure of mgnetic multilyer or thin film is descried using single prolic nd which is independent of the spin direction. The complicted electronic structure of the trnsition metls is therefore significntly simplified y neglecting contriution from the d nds nd their strong hyridiztion with the sp nds. Within the freeelectron pproximtion the expression for the conductivity per spin, which cn e found from Eq.(1.5) y integrting over the film thickness, is simplified (e.g., ref.14): N 1 d i di 1 d = i 3 λ i + ( ) iµ + ( ) λiµ σ dµ (1 µ ) µ A i µ 1 e λ Ai µ 1 e. (1.1) d i ρi 4 ρi Here we hve ssumed for simplicity tht the relxtion time is independent of k nd introduced the lyer-dependent men free pths λ i =τ i v F. In Eq. (1.1) µ refers to the cosine of the momentum perpendiculr to the interfces, d is the totl thickness of the multilyer, d i nd ρ i re the thickness nd the resistivity of the metl lyer i, nd, s efore, we omit spin indices. The first term in this expression gives the conductivity if the vrious lyers were crrying the electric current in prllel. The second term is responsile for finite size effects. The coefficients A ± cn e found using the oundry conditions. Fuchs nd Sondheimer pplied the semiclssicl free-electron model to the conductivity of homogeneous nonmgnetic thin film. 84,85 The oundry conditions for the film were determined using the following rguments. If the film is plced t <z<d, the distriution function t z= must hve no electrons with v z > other thn those speculrly reflected from the surfce ecuse there re no electrons outside the film. Therefore, defining frction p of the electrons which re speculrly reflected, the oundry condition t z= is g + (, v z z ) = pg (, v ). (1.13) Similrly t the opposite side of the film, i.e. t z=d, the electrons with v z < could only e those which re speculrly reflected from the oundry: + g d, v ) = pg ( d, v ). (1.14) ( z z In the cse p=1, which corresponds to perfect reflection from the oundries, the conductivity of the film is identicl to the conductivity of n infinite homogeneous metl. 15 If p<1, frction of electrons, (1-p), is scttered diffusively. The cse of p= corresponds to perfectly diffuse scttering, when ll the reflected electrons lose memory of their velocity. In the presence of diffuse scttering the conductivity of thin film decreses with the film thickness. The effect is significnt if the film thickness is reduced down to ecome comprle to the men free pth, i.e. d~λ=v F τ. For thicker films the current density is close to the ulk vlue in the center of the film, ut lowered in the vicinity of the oundries t distnces of the order of λ. In the cse of lyered structures, prt from the proility p for speculr reflection t the outer oundries, dditionl oundry conditions t the interfces re required. 16 These oundry conditions cn e imposed y ssuming tht the electrons re coherently trnsmitted with proility T, coherently reflected with proility R, or diffusely scttered with proility D t the interfce. The electrons which re diffusely scttered re ssumed to e simply lost so tht the trnsmission nd 36 reflection proilities re relted y the expression D=1-T-R. Thus, t the interfce etween lyers i nd i+1, g, (1.15) + + i+ 1 = Tgi + Rgi+ 1 + i = Tgi+ + Rgi g 1. (1.16) Crci nd Sun 16 used these oundry conditions to tret the conductnce in nonmgnetic multilyers, ssuming for simplicity tht the electrons cn e trnsmitted or scttered, ut cnnot e speculrly reflected, i.e. R ws set equl to zero. Cmley nd Brns 17 generlized this semiclssicl free-electron model to tret GMR in mgnetic multilyers nd spin vlves y ssuming tht the electric current is spin-dependent nd is crried in prllel y the up-spin nd down-spin electrons. Within their model scttering in the ulk ferromgnetic lyers is treted y introducing spin-dependent relxtion times (or equivlently spindependent men free pths which enter the expression for the conductivity (1.1)). Scttering t the interfces is tken into ccount y ssuming spin-dependent trnsmission coefficients in the oundry conditions (1.15) nd (1.16), i.e. T T, speculr reflection t the interfces eing neglected. Using this semiclssicl free-electron model the conductivity nd GMR in mgnetic multilyers nd spin vlves cn, therefore, e nlysed in terms of numer of phenomenologicl prmeters such s spin-dependent men free pths, spin-dependent trnsmission proilities t the interfces nd speculr reflection coefficients t the outer oundries. This model hs een extensively used for clcultions of GMR nd interpreting experimentl dt. 5,75,14,17-11 Below, we discuss the min predictions for GMR, which follow from this semiclssicl free-electron model. The mgnitude of GMR decreses monotoniclly with incresing rtio of the thickness of the nonmgnetic spcer lyer d NM to the men free pth in the spcer lyer λ NM, similr to tht displyed in Figs.11,1. There re two contriutions to the decrese of the GMR rtio R/R ρ/ρ P = σ/σ AP : the first one contriutes to the drop of σ=σ P -σ AP, wheres the second one results in the increse of σ AP. The drop of σ reflects the reduction in the numer of electrons, which cn rech n opposite FM/NM interfce efore eing scttered within the spcer lyer. Asymptoticlly, the decrese of σ is exponentil with chrcteristic decy length equl to the men free pth in the spcer lyer, i.e. exp(-d NM /λ NM ). However, this symptotic regime is reched only for d NM >>λ NM, ecuse ccording to formul (1.1) the conductnce my e expressed s n integrl over n exponentil with respect to vrious incidence ngles of the electrons. This symptotic regime does chrcterize those multilyers in which Cu (or other nole metls) is used s the spcer lyer. This is due to the reltively lrge men free pth in Cu, typiclly of the order of nm t room temperture s is estimted from the Drude formul. The regime which is relevnt to these experiments is chrcterized y more complicted vrition of R/R thn simple exp(-d NM /λ NM ). However, s ws demonstrted in section 5, resonle description cn e otined y introducing n effective scttering length l NM ccording to Eq. (3.1). Another contriution to the decrese of GMR s function of d NM comes from the shunting current within the spcer lyer, which leds to n increse of σ AP pproximtely in liner fshion with d NM. If ulk scttering is negligile compred to the scttering t the interfces, then the chnge in σ with d NM will e smll. In this cse the shunting current ecomes the dominnt contriution nd σ/σ AP drops s 1/d NM. According to the semiclssicl free-electron model the vrition of GMR s function of the thickness of the mgnetic spcer lyer is different depending on whether the ulk or interfce spindependent scttering is dominnt. In the cse of ulk scttering the GMR rtio exhiits mximum t certin thickness similr to tht in Fig.13. This cn e seen from the clculted results presented y the solid lines in Fig., which disply the mgnitude of GMR s function of the ferromgnetic lyer thickness d FM for multilyers with vrious numer of FM/NM ilyers. 19 In this clcultion ulk spin-dependent scttering is introduced through the spin-dependent men free pths in the ferromgnetic lyers, nmely λ FM =1nm nd λ FM =.6nm, nd diffuse scttering is ssumed t the 37

21 outer oundries. As is evident from this figure, the position of the GMR mximum shifts towrds lower FM lyer thicknesses with incresing the numer of ilyers N. The presence of the GMR mximum nd its shift with N is direct consequence of the ulk spin-dependent scttering nd the diffuse scttering t the outer oundries. In the cse of smll numer of ilyers, e.g., for the trilyer structure with N=1, the position of the GMR mximum is relted to the long men free pth in the FM lyer, so tht d FM ~ λfm. At this FM lyer thickness the up-spin electrons re le to contriute to the conduction efore eing diffusely scttered t the outer oundries. On the other hnd, in the cse of lrge numer of the FM/NM ilyers, the position of the GMR mximum is determined y the shorter of the two men free pths, so tht d FM ~ λfm. This is ecuse for lrge N the scttering t the outer oundries ecomes unimportnt so tht the mximum vlue of GMR will e otined when the ferromgnetic lyer thickness is sufficient to provide scttering of the downspin electrons. interfce. The rest of the FM lyer is inctive nd serves only s shunt, which reduces GMR inversely proportionl to d FM. The semiclssicl model predicts tht with incresing numer of FM/NM ilyers within multilyer the vlue of GMR increses until it reches sturtion, which is similr to tht found experimentlly nd presented in Fig.17. This tendency cn lso e seen in Fig., ccording to which t fixed vlue of the FM lyer thickness the increment of the GMR growth decreses for lrger N. This ehviour of GMR versus N is due to the diffuse scttering t the outer oundries of the multilyer. As hs lredy een explined in section 9, if the longest men free pth is lrger thn the totl thickness of the multilyer, then the diffuse outer-oundry scttering reduces the conductivity of the good spin chnnel nd hence effects GMR negtively. The mgnetoresistnce rtio ecomes independent of the numer of ilyers when the totl thickness of the multilyer is much lrger thn the longest men free pth. 8 N=5 3 p= p=.5 p=1 1 R/R (%) R/R (%) 1 R/R (%) d FM (nm) 1 FM lyer thickness (nm) 1 3 d FM (nm) Fig. Mgnetoresistnce versus ferromgnetic lyer thickness in (FM/NM) N FM multilyers for ulk (the solid lines) nd interfce (the dshed line) scttering s clculted using the semiclssicl free-electron model. λ NM =nm, d NM =nm, nd diffuse outer oundry scttering re ssumed in the clcultion. The other prmeters re set s follows: N is vried, λ =1nm, λ =.6nm, T = T =1 for ulk spin-dependent scttering nd FM N=, λ FM Dieny. 14,19 FM = λ FM =.6nm, T =1, T =.1 for interfce spin-dependent scttering. After Fig.1 The effect of speculr scttering t the top outer oundry of FM/NM/FM trilyer s clculted from the semiclssicl free-electron model. The mgnitude of GMR is plotted s function of the top FM lyer thickness for vrious proilities of speculr reflection p t the top outer oundry. Bulk spin dependent scttering, T = T =1 nd λ FM / λ FM =1, nd speculr reflection t the ottom outer oundry, p ottom =1, re ssumed in the clcultion. The other prmeters of the model re set s follows: d FM-ottom =nm, d NM =nm, ρ FM =15µΩcm, nd ρ NM =6µΩcm. After Biley. 111 In the cse of interfce spin-dependent scttering GMR decreses monotoniclly s function of the FM lyer thickness. This cn e seen from the dshed line in Fig., which is clculted for the multilyer with n infinite numer of repetitions N y introducing spin-dependent trnsmission coefficients t the FM/NM interfces, i.e. T =1 nd T = The decrese of GMR reflects the fct tht the ulk scttering in the ferromgnetic lyers is ssumed to e spin-independent nd, therefore, incresing the FM lyer thickness enhnces the reltive contriution of this type of scttering. When d FM ecomes much longer tht the men free pth in the FM lyer, λ FM, GMR is inversely proportionl to d FM, i.e. λ / d FM FM.18 This dependence cn e explined y the rgument tht only those electrons which leve the FM region of thickness λ FM djcent to the interfce hve sufficiently high proility not to e scttered within this FM lyer nd, therefore, rech the opposite Incresing the speculr scttering t the outer oundries strongly enhnces GMR in FM/NM/FM trilyers, provided tht the FM lyers re not too thick. This effect is evident from Fig.1, which shows the clculted mgnetoresistnce s function of the top FM lyer thickness in the spin vlve with vried proility of speculr scttering p t the top outer oundry of the trilyer. 111 Bulk spindependent scttering in the ferromgnets nd speculr reflection t the ottom outer oundry simulting NiO pinning lyer re ssumed in this clcultion. The enhncement of GMR with the incresing mount of speculrity p is due to the stronger speculr scttering from the top surfce, which unlike diffuse scttering reflects electrons ck llowing them to cross the spin vlve mny times, therey incresing spin-filtering effects. Note tht the optimum thickness of the FM lyer t which the mximum GMR is oserved lso depends on p nd decreses with top surfce speculrity p, which is similr to wht ws found for the multilyers with n incresing numer of FM/NM 38 39

22 ilyers (Fig.). This is not surprising ecuse, s hs een recognized y Brns et l., 5 multilyer with n infinite numer of periods cn e simulted y considering ilyer with speculr scttering t the outer oundries in which the FM lyer thickness is tken s hlf the ctul one. At lrger FM lyer thicknesses the GMR rtio for the speculr top surfce ecomes lower thn the GMR rtio for the diffuse-scttering top surfce. As is seen from the insert in Fig.1, the crossover occurs t d FM 1nm. This is consequence of the current shunting which ecomes dominnt over GMR enhncement. Indeed, when the FM lyer thickness is lrger thn the longest men free pth within this lyer the speculr-reflected electrons re not le to rech the spcer lyer nd insted contriute to the shunting current. The semiclssicl free-electron model predicts liner vrition in the conductnce s function of cosθ, where θ is the ngle etween the mgnetiztions of the two ferromgnetic lyers in spin vlve. When the mgnetic moments of the FM lyers re not ligned, the momentum trnsfer etween the up- nd down-spin conduction chnnels is determined y the trnsmission coefficients which re given y T = cos = T ( θ / ) nd T = sin = T ( θ / ). 17 Using these expressions nd ssuming diffuse scttering t the outer oundries nd no speculr reflection t the interfces, it cn e shown tht the conductnce of the trilyer, Γ, vries in liner fshion with cosθ, 7 i.e. Γ( θ ) = ΓAP + ( ΓP ΓAP )(1 cosθ ) /. (1.17) This result is to first order in the GMR rtio equivlent to expression (11.3), ccording to which the resistnce vries linerly with cosθ. However, the difference etween these two descriptions cn ecome sizele for those systems with reltively lrge vlues of GMR (see lso section 11). We see tht the semiclssicl free-electron model predicts correctly numer of importnt fetures of GMR which re oserved experimentlly (section III). For exmple, it qulittively explins the vrition of GMR versus ferromgnetic nd nonmgnetic lyer thickness, the effect of speculr/diffuse scttering t the outer oundries, the enhncement of GMR with the incresing numer of repetitions within multilyer, nd the ngulr vrition of the conductnce in spin vlves. The gret dvntge of this model is the ese of ppliction to prticulr lyered system, which llows understnding qulittive trends in the trnsport properties. However, s ws mentioned ove the semiclssicl free-electron model ignores the relistic nd structure of the multilyer nd, therefore, cn not e pplied for quntittive description of GMR. Although much experimentl dt cn e fitted well using the semiclssicl free-electron model, the prmeters, which re extrcted from the fitting, should e treted with cution. For exmple, Cmley nd Brns 17 hve found tht in order to ccount for the increse in GMR from room temperture to liquid helium temperture they hd to use men free pth of 6nm t 4.K, which is unrelisticlly long even for the MBE-grown thin films. A qulittive filure of the semiclssicl free-electron model to descrie consistently in-situ conductnce experiments in NiO/Co/Cu/Co spin vlves ws demonstrted recently y Biley et l. 11,113 They found striking fetures in the experimentl thickness-dependent conductnce which is displyed in Fig.. As is evident from the figure, ddition of out 1 monolyer of Co to NiO/Co/Cu surfce cuses the net film conductnce to decrese. The reverse cse of Cu on NiO/Co shows strong positive curvture of the conductnce, indicting reduction of the conductivity in Cu ner the interfce with Co. Detiled microstructurl chrcteriztion using in-situ Auger electron spectroscopy nd ex-situ X-ry diffrction mesurements indicted tht the defect concentrtion does not vry noticely s function of thickness. These microstructurl mesurements suggest tht the ulk scttering prmeters ρ nd λ should e considered to e constnt within ech lyer, nd tht the surfce scttering prmeter p does not chnge etween the lyers. Under these constrints, it ppers to e impossile to fit even qulittively the highly symmetric scttering ehvior mesured during the formtion of Co/Cu versus Cu/Co interfces. As cn e seen from Fig., depending on the choice of the interfcil trnsmissivity prmeter T the thickness-dependent conductnce either do not disply ny conductnce step t the interfces or disply step t oth Co/Cu nd Cu/Co interfces, 4 neither eing oserved experimentlly. As we will see in section 17, clcultions incorporting relistic nd structure resolve the oserved inconsistency etween the free-electron model nd the experiments. Sheet conductnce (Ω -1 ) Fig Co lyer Totl thickness (nm) Experimentl () nd clculted () thickness-dependent conductnce of NiO/Co(nm)/ Cu(t Cu )/Co(4nm) spin vlves. (): Conductnce is mesured in-situ during the deposition of the spin vlves with vrious thickness of the Cu lyers: d Cu =5.8nm (A), 1.1nm (B), 1.6nm (C),.3nm (D). The position of the interfce with the ottom Co lyer is the sme for ll the smples nd is mrked y the verticl line. Note the strong devitions from linerity in the vicinity of the interfces: drop in film conductnce for Co on Cu nd positive curvture for Cu on Co. After Biley et l. 11 (): Clcultions re performed using free-electron semiclssicl model using prmeters which provide est fit of the dt in Fig.1. Interfce scttering (T<1) must e introduced to produce the conductnce drop oserved during deposition of Co on Cu, producing complementry drop for Cu on Co which is not oserved. After Biley et l. 113 Some of the nd structure effects cn e cptured within n extended free-electron model, which hs een proposed y Hood nd Flicov. 114 They introduced lyer- nd spin-dependent effective msses nd the relevnt nd fillings. They then studied the effect of speculr scttering from the resulting potentil steps t the interfces. Unfortuntely, the numer of free prmeters in such phenomenologicl model is so lrge tht the nlysis of the experimentl dt in terms of this model ecomes uncontrolled. 13. Quntum-mechnicl theory A B C D Sheet conductnce (Ω -1 ) In ddition to the lck of n ccurte description of the electronic structure, the semiclssicl freeelectron model suffers from the inility to descrie quntum effects, which ecome importnt t smll film thicknesses. The confinement of electrons in thin film leds to discretiztion of the energy levels. The corresponding quntum effects ecome oservle when the typicl spcing δe etween the energy levels ner the Fermi energy ecomes lrger thn the level rodening / τ rising from vrious scttering mechnisms. 115 Since δ E ~!v / F d, where d is the film thickness, the condition for oserving quntum size effects is d < λ, where λ is the men free pth. The filure of the semiclssicl theory ecomes pprent if one considers the conductivity of thin film with diffusively-reflecting surfces. 116 In the limit when the men free pth ecomes much longer thn the film thickness the conductivity tends to infinity, implying tht in the sence of ulk scttering the T=1. T=.5 Co lyer. 4 6 Totl thickness (nm)

23 scttering y the rough surfces induces no dissiption of electricl current. This unphysicl result is direct consequence of ignoring quntum size effects within the semiclssicl theory. 117 In order to resolve this deficiency of the semiclssicl model quntum-mechnicl pproch to electronic trnsport is required. There re severl different quntum-mechnicl formultions of trnsport theory which include those of Kuo, 118 Lnduer 119 nd Keldysh. 1 The Kuo (liner response) formlism considers the electronic trnsport in disordered metllic system s liner response to n pplied electric field. 11 The Lnduer formlism descries the conductnce from the point of view of the trnsmission of electrons through conductor nd is pplicle to mesoscopic trnsport. 1 The Keldysh (non-equilirium Green s function) formlism is conceptully more complicted thn the Kuo nd Lnduer formlisms, ut is more generl ecuse it provides description of the quntum trnsport in the presence of dissiptive interctions. 1,13 All these theories cn e used for clcultions of GMR within quntum-mechnicl pproch. In the present review we outline sic principles of the Kuo theory, which is the most widely used for the tretment of GMR (for detiled formultion of the liner response theory see, e.g., ref.11). The strting point of the Kuo formlism is the density mtrix. The density mtrix is the quntum-mechnicl opertor, which descries the sttisticl properties of quntum-mechnicl system. It is the nlogue of the distriution function within the semiclssicl theory, which we hve discussed in section 1. The density mtrix ρ t stisfies the quntum-mechnicl eqution of motion dρ i dt [ H, ρ ] t! = t t, (13.1) where H t is the Hmiltonin of the system nd [,] denotes commuttor. This eqution descries the evolution of the system ffected y time-dependent perturtion U(t) due to the pplied electric field. We ssume tht the electric field tkes the form,exp(εt), so tht it is uniform in spce, is pplied t t = nd grows diticlly to its vlue, t t =. The ltter is tken into ccount y n infinitesiml positive ε, so tht the limit ε should e tken in the finl result. The single-electron Hmiltonin of the system cn then e represented y H εt t = H + U ( t) = H + e, re, (13.) where H is the time-independent Hmiltonin of the unpertured system. Eqution (13.1) is the quntum-mechnicl nlogue of the semiclssicl eqution (1.1). It descries the time evolution of the system. Initilly, i.e. t t =, the system is t equilirium nd is chrcterized y the unpertured density mtrix ρ, ccording to the Fermi-Dirc distriution ( H EF ) / kt 1 ρ = [ e + 1]. Due to the pplied electric field (13.) the system diticlly follows the perturtion. Within the Kuo formlism we re looking for the solution of the eqution (13.1) to first order with respect to the pplied electric field. We cn, therefore, represent the density mtrix s ρ t =ρ+δρ(t), where δρ(t) is smll time-dependent devition from the equilirium Fermi-Dirc distriution ρ due to the pplied electric field. The linerized eqution for the density mtrix (13.1) then tkes the form dδρ( t) i! = [ H, δρ( t)] + [ U ( t), ρ]. (13.3) dt Now we rewrite this opertor eqution in terms of the mtrix elements y introducing sis of eigensttes α of the unpertured Hmiltonin H. The equilirium density mtrix ρ hs the sme eigensttes s H so tht ρ αβ 1 = δ αβ = δ ( ) ( ) / αβ f Eα, (13.4) Eα EF kt e + 1 ( Eα EF ) / kt 1 where f ( Eα ) = [ e + 1] is the Fermi-Dirc distriution function nd E α is n eigenvlue of H. The opertor r which enter the term [U(t),ρ] through eqution (13.) is non-digonl in the α representtion. It is convenient to represent the mtrix elements of r in terms of the mtrix elements of the velocity opertor v using the reltion i! v = [ r, H ], where we cn use H insted of H t y neglecting high order terms. Tking this into ccount we find: f ( E ) f ( E )! αβ. (13.5) E E α β εt [ U ( t), ρ] = i e, v e αβ α β The time dependence of δρ(t) is determined y the perturtion (13.5), so tht δρ(t)=δρexp(εt). Using equtions ( ) we find the solution for the density mtrix t t= i! e, vαβ f ( Eα ) f ( Eβ ) δραβ =. (13.6) Eβ Eα + iε Eα Eβ This solution cn now e used for clculting the electricl current. For sptilly homogeneous system the current-density opertor is defined y j=-e v/ω, where Ω is the totl volume of the system. We need to clculte the expecttion vlue of j which is determined y j = Tr( jδρ), where we hve used the fct tht Tr ( jρ) =, s in equilirium there is no net current in the system. Using the definition of the conductivity (1.8) nd tking the limit ε we otin σ µν e π µ ν =! vαβ v βαδ ( Eα Eβ )[ f ( Eα )] Ω, (13.7) αβ i where we hve tken into ccount tht Re lim = πδ ( x) nd used the fct tht s E α E β, which ε x + iε is required y the δ-function, f ( E ) f ( E )]/( E E ) f ( E ). Using the identity ( α Eβ ) = deδ ( E Eα ) δ ( E Eβ [ α β α β α δ E ) (13.8) nd tking the zero-temperture limit t which [ f ( E)] = δ ( E EF ), formul (13.7) for the conductivity cn e netly represented y σ e π µ ν = Tr{v δ ( EF H )v δ ( E Ω µν! F H )}. (13.9) The ove formul for the conductivity would generlly depend on the prticulr type of the disorder responsile for the scttering. In order to otin result tht is independent of the prticulr disorder configurtion ut depends only on verge chrcteristics (e.g., defect density or impurity concentrtion), one hs to perform configurtionl verge of this expression. In this form the ove expression is known s the Kuo-Greenwood formul. 14 In mny cses it is convenient to crry out the configurtionl verging using techniques which hve een developed for Green s functions (e.g., refs.11,15,16). Therefore it is useful to rewrite the Kuo-Greenwood formul in the following form σ µν e! µ = Tr{v ImG( E Ωπ ν )v ImG( F E F )}. (13.1) Here the ngulr rckets stnd for the configurtionl verge nd we hve introduced the Green s function, which is defined y 4 43

24 G( E) = lim 1, (13.11) ε E H + iε nd hve tken into ccount tht δ ( E H ) = ImG( E) / π. The generl techniques for configurtionl verging in disordered homogeneous systems hve een descried in detil elsewhere, e.g., in refs.11,15,16. Here we riefly summrize the min results. We ssume tht the totl Hmiltonin of the system cn e represented y H=H +V, where H descries the undistured periodic system nd V is scttering potentil due to the defects or impurities. By configurtionl verging we replce this system which is chrcterized y the rndom non-periodic potentil y n effective medium which possess trnsltionl invrince. The configurtionl verging leds to the renormliztion of the Green s function so tht 1 G( E) = E H Σ( E), (13.1) where Σ is the self energy. The ove eqution cn e considered s the definition of Σ, which is n energy-dependent non-hermitin opertor. Its rel prt shifts the energy levels of the undistured system, wheres the imginry prt chrcterizes the rodening of the levels due to the finite scttering lifetime. ImΣ(E F ) determines, therefore, the relxtion time τ, which hs een introduced within the semiclssicl theory (section 1). It follows from eqution (13.1) tht the conductivity tensor requires n verge over the product of two Green s functions, i.e. σ GG. In generl, performing this verging explicitly is complicted prolem. This is the reson why very often the conductivity is pproximted y the product of the verge of the Green s functions, i.e. σ G G. By mking this pproximtion one ignores the contriution from the vertex corrections in the liner response formlism. 11 This pproximtion is equivlent to neglecting the scttering-in term in the semiclssicl theory, which llows the introduction of the relxtion time pproximtion (section 1). Similr to the relxtion time pproximtion, the neglect of the vertex corrections in quntum-mechnicl liner-response theory is non-trivil pproximtion nd hs to e justified (see ref.11 for discussion). We note tht the ove derivtion of the formul for conductivity is vlid for homogeneous system. In this cse the current nd pplied field cn e ssumed to e uniform so tht one cn define the conductivity ccording to eqution (1.8). This is the conductivity, which is given y the Kuo- Greenwood formul (13.1). In generl cse of n inhomogeneous system the current density is determined y the non-locl conductivity ccording to µ µν ν j ( r) = d r σ ( r, r ), ( r ). (13.13) ν The electric field,(r) in this eqution is the locl electrosttic field, which rises from the ppliction of the potentil difference cross the smple. As ws explined y Levy, this is n internl field, which is not the sme s the field pplied externlly. For inhomogeneous mgnetic systems the locl internl field is position- nd spin-dependent. If the rte t which the electrons re scttered vries from one region to nother, then electricl conduction will led to sptil redistriution of chrge. This chrge redistriution in inhomogeneous medi results in nonuniform internl electric field. In mgnetic systems electron conduction is spin-dependent. Spin-polrized electric currents in inhomogeneous medi led to sptil redistriution of spin s well s chrge. 17 This phenomenon is known s spin ccumultion or current-driven mgnetiztion. The internl electric field in eqution (13.13) my, therefore, e different for different spins. In mgnetic multilyers the effect of position- nd spin-dependent internl field is very importnt for perpendiculr trnsport ecuse these systems re inhomogeneous in the direction of electric current. For prllel trnsport, however, the internl electric field is constnt, ecuse these lyered systems re homogeneous in the plne of the lyers. The Kuo-Greenwood formul (13.1) cn, therefore, e used for the tretment of CIP GMR. The quntum-mechnicl model for GMR ws first introduced y Levy et l. 18 They used the Kuo formlism to clculte the conductivity of free electrons scttered y spin-dependent potentil V ( r ) = ( v + j m 1)( r r ), (13.14) which is produced y rndom-point sctterers. In the ove formul r is rndom position of the sctterer, m is unit vector in the direction of mgnetiztion, σ is the Puli mtrix, v nd j chrcterize the strength of the spin-independent nd spin-dependent contriution to the scttering potentil respectively, which my e different for different lyers nd interfces. Performing the configurtionl verging in momentum spce, Levy et l. derived simple formul for the locl conductivity, which cn e expressed in terms of the locl spin-dependent scttering rte (z) s ne σ ( z) =, (13.15) m ( z) where z is coordinte perpendiculr to the lyers. The locl scttering rte (z) is determined y the pproprite verge of the z-dependent scttering rtes within the different lyers. The totl CIP conductivity of the multilyer cn e found y integrting over the multilyer thickness L, i.e. 1 σ = σ ( z) dz. (13.16) L L In the limit of the men free pth eing long compred to the lyer thickness, Ojd, this model reduces to the series-resistor model, which ws introduced in section 3. In prticulr, in the cse of interfce scttering the result for GMR is given y formul (3.5), in which the symmetry prmeter D is determined y the interfcil scttering prmeters in eqution (13.14), nmely j int / vint = ( α 1) /( α + 1). In the cse of ulk scttering the vlue of GMR cn e found from formul (3.6), in which D is determined y ulk scttering prmeters in eqution (13.14), nmely j ulk / v ulk = ( α 1) /( α + 1). The sme model of free electrons scttered y spin-dependent potentil (13.14) ws used y Cmlong et l. 19,13 to descrie GMR within rel-spce pproch. According to Cmlong 13 the rel-spce pproch works etter ecuse it voids the locl pproximtion for the conductivity which ws used in the momentum spce pproch of ref.18. Solving eqution (13.1) within wek scttering pproximtion nd in the dilute limit of impurity concentrtion, Cmlong et l. derived expressions for the non-locl conductivity for oth the CIP nd CPP geometries. They found tht the semiclssicl pproch for multilyers nd the rel-spce quntum theory produce the sme mgnetotrnsport properties, provided the effect of quntum interference nd quntum-size effects re neglected. A similr conclusion ws rrived t y Vedyyev et l. 131 who used the rel-spce quntum-mechnicl pproch for the conductivity of free electrons ffected y ulk spin-dependent scttering in spin vlves. 13 The quntum models of Levy et l. 18 nd Cmlong nd Levy 19,13 hve een compred with the exct solution for the conductivity y Zhng nd Butler. 133 Similr to refs they used the model of free electrons with rndom point sctterers nd clculted the position-dependent conductivity without the vertex corrections. However, insted of deriving the expression for the selfenergy from Hmiltonin y configurtionl verging, they ssumed tht the imginry prt of the self-energy is phenomenologicl prmeter nd compred pproximtions used y the previous uthors within the quntum nd semiclssicl models. Zhng nd Butler found tht the Cmlong nd Levy s pproch is identicl to the Fuchs-Sondheimer theory within the relxtion time pproximtion nd speculr oundry conditions t the interfces. As cn e seen from Fig.3, this pproch predicts the totl CIP conductivity of multilyer in good greement with the exct solution. This fct demonstrtes tht in the sence of potentil discontinuities t the interfces 44 45

25 quntum effects, such s oscilltions in the conductivity, re effectively verged out nd the semiclssicl solution provides good pproximtion for GMR within the free-electron model. It is lso cler from Fig.3 tht lthough the model of Levy et l. 18 predicts the correct thin nd thick limits, it does not ccurtely reproduce the exct results for intermedite film thicknesses. Fig.3 compres the clculted results for GMR within the different models. As is seen from the figure, the model of Levy et l. strongly overestimtes GMR, wheres the Cmlong nd Levy s or semiclssicl results re much closer to the exct curve. 15 σ (1 s -1 ) Fig Totl thickness (.u.) σ/σ P Comprison of exct clcultions (solid lines) with different free-electron models for GMR, the model of Levy et l. 18 (dotted lines) nd the model of Cmlong nd Levy 19,13 (dshed lines). Note tht Cmlong nd Levy s theory is identicl to the Fuchs-Sondheimer theory with p=1. (): The clculted conductivity (in units 1 15 s -1 =1113Ω -1 cm -1 ) s function of the totl thickness of period for nonmgnetic multilyer with period of two lyers nd with men free pths in ech lyer of 1=36.u. nd =36.u. (1.u.=.59nm). The thickness of the first lyer is twice tht of the second. The horizontl lines show the limits of thin nd thick films. (): The clculted GMR of mgnetic multilyer s function of the ferromgnetic lyer thickness. Thin interfcil lyers of 4.u. thickness nd men free pths of =5.u., =1.u. re introduced to model strong spin-dependent scttering t the interfces. Other prmeters re s follows: d NM =5.u., NM= NM=47.u., FM=1.u. nd FM=.u. After Zhng nd Butler. 133 The ove quntum models for GMR ssume tht the origin of GMR lies in spin-dependent scttering potentils t the interfces or in the ulk metls. By using the free-electron pproximtion these models ignore the vrition of the intrinsic spin-dependent electronic potentil of the lyered structures. An extension of the free-electron theory for mgnetic multilyers to include the intrinsic potentil within the Kronig-Penney model hs een performed y Zhng nd Levy 134 in the momentum-spce pproch nd Bulk nd Brns 135 in the rel-spce pproch. They found tht the spin-dependent steps t the interfces could enhnce or reduce GMR depending on the prmeters chrcterizing the potentil. It ws lso shown tht the GMR phenomenon could occur in structures with no spin symmetry in the relxtion times, ut with spin-dependent electronic structure (which origintes in the ove models from introducing spin-dependent potentil within the multilyer). However, quntittive predictions re difficult ecuse of the unknown prmeters chrcterizing the spin-dependent potentil. Only full nd structure clcultion cn predict why potentil steps t the interfces my e importnt. The effect of the intrinsic step-like potentil on GMR in spin-vlve structures ws lso considered y Vedyyev et l. 136 nd Brns nd Bruynserede. 137 They found tht their results re FM lyer thickness (.u.) strongly influenced y quntum-size effects t smll lyer thicknesses nd predicted oscilltions in GMR s function of the lyer thicknesses. Unfortuntely, these quntum-size effects in GMR which mke the quntum free-electron models conceptully different from the semiclssicl free-electron models hve not yet een oserved. Aprt from the quntum-size effects, the vrition of CIP GMR s function of mgnetic nd non-mgnetic lyer thickness within the quntum free-electron models qulittively reproduces semiclssicl results. 19,133 Although quntittively the predictions of the semiclssicl nd quntum models might differ, the experimentl dt cn e fitted well using oth pproches. 131 However, neglecting the relistic nd structure, which is the grossest pproximtion inherent in the free-electron models, mkes quntittive comprison of these models with experiments unrelile. Including n ccurte spin-polrized nd structure to the models for GMR will e considered in section V. 14. Tight-inding models A single-nd tight-inding model is nother simple wy to descrie the electronic nd structure of metl. Unlike the free-electron theory, the tight-inding theory descries the electronic structure in terms of loclized tomic oritls, which overlp due to the onding etween neighoring toms. 138 The propgting (Bloch) sttes, which re responsile for the electronic trnsport in metls, cn e uilt up from the tomic oritls y solving the respective Schrödinger eqution. The tight-inding pproch is especilly suited to numericl clcultions of the conductnce ecuse it discretizes the sptil continuum in terms of tomic cites. A single-nd pproximtion to the tight-inding model is, of course, strong simplifiction to the nd structure. However, s we will see in section 17, the tight-inding model cn e generlized to multind description of the electronic structure which mkes it very powerful tool for modeling GMR. A single-nd tight-inding Hmiltonin tkes the form H = i Ei i + i hij j, (14.1) i ij where E i re i re the on-site tomic energy nd the tomic oritl t site i respectively. The tightinding hopping mtrix elements h ij re usully ssumed to e non-zero only etween nerestneighor lttice sites. This simple model is quite flexile nd llows elortion. In prticulr, the Stoner exchnge splitting of the spin nds cn e included in this model y tking different on-site tomic energies for the up- nd down-spin electrons. Disorder cn e introduced y ssuming rndomness in the on-site energies or in the hopping mtrix elements. Impurities cn e included y putting toms with their on-site energies, which re different from those of the host toms. Using this tight-inding model for studying GMR helps to elucidte the microscopic origin of spin-dependent scttering, n importnt issue which hs not een ddressed within the phenomenologicl freeelectron models descried in sections 1 nd 13. As ws discussed in the previous section, in order to clculte the trnsport properties of disordered metl n pproprite configurtionl verge hs to e performed. This configurtionl verging cn e crried out nlyticlly using the Green s function s methods (e.g., ref.15) within certin pproximtions, e.g., low density of impurities/defects, wek scttering potentils, nd neglecting vertex corrections. The drwck of this pproch is tht frequently the vlidity of the pproximte solution is difficult to monitor. An lterntive pproch is to perform the verging numericlly y generting numer of rndom configurtions of disorder/impurities within sufficiently lrge cell. An efficient method for clculting conductnce using this pproch hs een developed in the field of mesoscopic physics. 139 This method utilizes the discrete nture of the tightinding model nd is sed on the Kuo formul nd the recursive Green s function technique. The disdvntge of the recursive method is tht it is computtionlly demnding. One is limited to reltively smll cell size, which my led to rtifcts due to the effect of oundry conditions. 47

26 Nevertheless, this method hs ecome powerful tool for studying GMR, especilly within the CPP geometry (see section VI). Below we outline the sic ides of this pproch. The geometry of the system, which is normlly considered for clculting the conductnce within the recursive technique, is shown schemticlly in Fig.4. The smple under considertion, e.g., mgnetic multilyer, is plced etween two semiinfinite leds. The smple cn e, in generl, disordered ut the leds re ssumed to e perfect. We note tht lthough the system, which is shown in Fig.4, represents n infinite wire, periodic oundry conditions cn, if required, e imposed in the trnsverse direction to uilt n infinite multilyer. It is ssumed tht t infinity the leds re connected to reservoirs, which re t thermodynmic equilirium. The electric current in the system is driven y smll electrochemicl potentil difference etween the reservoirs. Such formultion of the prolem is typicl for the Lnduer pproch to the electronic trnsport in mesoscopic systems. 1 It hs een proved tht the Lnduer formlism cn e derived directly from the Kuo formlism, 14 the ltter eing n efficient method for clculting the conductnce within the recursive technique. 139 Fig.4 Within the ove formultion the conductnce rther thn the conductivity (13.1) is the suject of interest. The Kuo formul (13.1) for the zero-temperture conductnce * per spin cn e rewritten for the cse of simple cuic lttice with lttice constnt s e! Γ = Tr{v ImG( E )v Im ( )} F G E F. (14.) π Here v is the projection of the velocity opertor to the direction of current (the z direction in Fig.4) nd G ( EF ) is the Green s function of the totl coupled system, which includes the right nd the left electrodes nd the smple, t the Fermi energy E F. The Kuo formul (14.) cn e evluted y cutting the system in the trnsverse direction (i.e. in the xy plne in Fig.4) nd clculting the mtrix elements of the Green s function nd the velocity opertor etween the tomic plnes l nd l+1. Due to the current conservtion condition the result for the conductnce is independent of l nd the ltter cn e chosen ritrrily. The velocity opertor cn e represented s i v = hil, jl+ 1{ j, l + 1 i, l i, l j, l + 1}, (14.3)! ij () CIP geometry Leds Current () CPP geometry where i, l is the tomic oritl of tom i in plne l nd h il,jl+1 re hopping mtrix elements etween plnes l nd l+1. Normlly it is convenient to tke l eing the lst tomic plne of the smple L so tht L+1 is the first lyer of the right led. The mtrix elements of the Green s function of the totl system 48 Smple (multilyer) Geometry for the CIP () nd CPP () GMR clcultion within the tight-inding recursion method. A smple ( mgnetic multilyer) is plced etween semiinfinite perfect leds. The electric current flows in the z direction. x y z t plnes L, L+1 nd etween them cn e evluted using the recursion technique s descried elow. First, one finds the mtrix elements of the surfce Green s function for the detched semi-infinite leds. A simple lgeric expression cn e derived within the single-nd tight-inding model nd simple-cuic geometry. 141 In generl cse, the surfce Green s function cn e expressed in terms of the Green s function for the ulk metl, 14 the ltter eing clculted in momentum spce using stndrd techniques. Then, the smple is grown y dding tomic lyers with impurity toms distriuted rndomly, lyer y lyer, onto the left led. At every step the Green s function mtrix elements etween tomic sites in the lst dded lyer re clculted y solving numericlly the Dyson eqution: 1 l+ 1 = ( H l+ 1 hl + 1, l glhl, l+ 1) g. (14.4) Here g l is the Green s function of the left semiinfinite led with dded l lyers ( l L ) of the smple (efore dding the lyer l+1), g l+1 is the Green s function of the left semiinfinite led with dded l+1 lyers of the smple (fter dding the lyer l+1), H l+1 is the Hmiltonin of the dded lyer l+1, h l,l+1 is the hopping (onding) etween toms in the new lyer l+1 nd toms in the previous lyer l. Once the smple hs een fully-grown, the lst lyer is onded to the right led in order to otin the Green s function G( EF ) of the full system, which enters the expression for the conductnce (14.). We note tht this method gives n exct solution for the given model nd geometry nd descries the conductnce for prticulr disorder/impurity configurtion. The configurtionl verging should e performed numericlly y generting numer of rndom disorder/impurity configurtions. Asno et l. 143 implemented this pproch for studying GMR in Fe/Cr mgnetic multilyer within the CIP nd CPP geometries. They used single-nd s-vlent tight-inding model within simple cuic geometry with (1) orienttion of tomic plnes nd constnt hopping h etween nerest-neighors (so tht h il,jl+1 =hgij in eqution (14.3)). They introduced Stoner exchnge splitting J of the spin nds in Fe, so the on-site tomic energies of the mjority- nd minority-spin electrons were equl E Fe =E Fe J nd E Fe=E Fe +J respectively. They ssumed tht Cr ws non-mgnetic with on-site tomic energy chosen to e equl to the minority on-site tomic energies of Fe, i.e. E Cr =E Fe. Due to this the minority-spin electrons do not experience potentil step (or roughness potentil) for the prllel lignment of the mgnetiztions. Using this model Asno et l. studied the effect of interfce roughness nd ulk disorder on CIP nd CPP GMR. The interfce roughness ws introduced through sustitutionl rndomness t the interfcil lyers. This implied tht in the Fe lyer, ech tom djcent to the Cr lyer ws replced y the Cr tom with proility c. Similrly, in the Cr lyer ech tom djcent to Fe lyer ws replced y the Fe tom with the sme proility c. The ulk disorder ws modeled y rndom vrition of the on-site tomic energies of the Fe nd Cr toms with uniform distriution of width J, which ws ssumed to e spin-independent nd ws llowed to vry in the clcultions. The resulting conductnce ws verged over 1 rndom configurtions of roughness or disorder. As is evident from Fig.5, the mgnitude of CPP GMR is much lrger thn the mgnitude of CIP GMR nd they ehve differently s function of the interfce roughness, the mesure of which is the intermixing concentrtion c of the two monolyers forming the interfce. Not unexpectedly, within the model considered the interfce roughness hs eneficil effect on CIP GMR, ecuse it is the only mechnism of spin-dependent scttering nd, therefore, CIP GMR increses with c (the full squres in Fig.5). The presence of steps in the electronic potentil t the interfces hs little influence on CIP GMR, which is found to e close to zero in the sence of roughness. On the contrry, sizele CPP GMR is found in the sence of ny roughness, the ltter only wekly reducing GMR (the open circles in Fig.5). This is the result of the spin-dependent potentil of the multilyer, which effects differently the numer of electrons contriuting to the conduction for the prllel nd ntiprllel configurtions (see lso section 15). Figure 5 shows the mgnitude of 49

27 GMR s function of ulk disorder within the multilyer, which is mesured y the width of the distriution in the on-site tomic energies J. As is seen from the figure, the vlue of GMR decreses with incresing J for oth the CIP nd CPP geometries. This is direct consequence of the incresing scttering in the good minority conduction chnnel, which reduces the conductnce within the prllel configurtion of the multilyer. R/R AP.4.. CPP CIP c R/R AP γ CPP CIP result might e the consequence of the smll multilyer period, which ws used in computtions, so tht the men free pth is much longer thn the lyer thickness. The recursive technique ws used y Todorov et l. 146 for clculting CIP GMR of the FM/NM/FM trilyer to study the effect of nd structure nd interfce roughness within single-nd tight-inding model. There re three mjor differences etween the models used y Todorov et l. nd Asno et l. 143 First, Todorov et l. introduced different hopping integrls for the up-spin nd downspin electrons in the ferromgnetic metl. This ws supposed to reflect the difference in the mjoritynd minority-electron dispersions nd the density of sttes t the Fermi energy, which re typicl for rel ferromgnetic metls. The on-site tomic energies were ssumed for simplicity to e the sme for ll the toms. Second, the interfce roughness ws introduced s steps of rndom length l step long the trilyer with correltion length l cor defined y l cor = l step. This llowed studying the effect of the roughness correltion length (i.e. the step density) on GMR. Third, the clcultions were performed for rnge of trilyer lengths L, nd the liner region of the R versus L reltion ws used to otin the Ohmic resistivity ρ=a(dr/dl), where A is the trilyer cross section. This eliminted the effect of contct resistnces, which ffected the vlue of GMR tht re otined directly from R for fixed L. 143 The rnge of L, from which ρ ws clculted, exceeded the effective men free pth in the trilyer y fctor etween 5 nd 1, so tht the diffusive regime of conduction is eing modeled. Fig.5 Clculted GMR of (Fe/Cr) multilyer within single nd tight-inding model s function of interfce roughness c () nd ulk disorder (). Ech metl within the multilyer lyer consists of three tomic monolyers of cross section 1 1 toms. Roughness is mesured in terms of the intermixing concentrtion c of the two monolyers forming the interfce. Disorder is introduced s rndom vrition in the on-site tomic energy levels of width. The tight-iding prmeters in units of the hopping integrl h re s follows: E Fe =.5, E Cr =.5, J=1, E F =. The results re verged over 1 roughness/disorder configurtions. In figure () c is fixed t.. Note the definition of GMR s R/R AP. After Asno et l. 143 A similr tight-inding model ws used y Itoh et l. 144 who studied the comined effect of roughness nd potentil rriers t the interfces on CIP nd CPP GMR in mgnetic multilyers. Unlike Asno et l., 143 they used the coherent potentil pproximtion (CPA) 16,145 to perform their configurtionl verging. Within the CPA the disordered lloy (t the interfce) is replced n effective medium which is chrcterized y the Hmiltonin H eff =H +6, where the non-hermitin selfenergy opertor 6 is defined y eqution (13.1). The pproximte solution for 6 cn e found y the self-consistent condition tht the scttering of electrons in the effective medium y ny site ( singlesite CPA) or ny cell ( single-cell CPA) vnished on verge. Itoh et l. 144 used the single-cell CPA, which is etter pproximtion thn the single-site CPA due to lrger numer of toms, over which the configurtionl verging is performed exctly. Within the single-cell CPA they derived selfconsistent nlytic expressions for the conductivity including the vertex corrections. They found tht in the CIP geometry the vertex corrections vnish, wheres in the CPP geometry the vertex corrections do not necessrily vnish. Since the single-cell CPA is computtionlly expensive, Itoh et l. crried out numericl clcultions in the wek-scttering limit for thin metl lyers consisting of three tomic monolyers. Their results for GMR re consistent with those predicted y Asno et l. 143 within the recursive technique. They find tht the CPP GMR is lrger thn the CIP GMR, which is consequence of the spin-dependent potentil steps t the interfces introduced in their model. The difference etween the CPP nd CIP geometries decreses strongly with decresing step size. The interfcil roughness is fvorle to CIP GMR ut hs smll negtive effect on the CPP GMR. They lso find tht the contriution from the vertex correction to the CPP conductnce is negligile. This GMR (%) Fig l () GMR (%) d FM () Clculted GMR of FM/NM/FM trilyer within single-nd tight-inding model s function of the correltion length of roughness l cor () nd s function of ferromgnetic lyer thickness d FM (). The width of the trilyer is 8 nd the NM lyer thickeness is 4. The nd structure of the FM metl is pproximted y the two spin nds with different hopping integrls, nmely h =1 nd h =1/3. Other prmeters re s follows: E F =.5, h NM =1, ulk disorder prmeter γ=.5, E FM =E NM =, in figure () d FM =4, in figure () l cor =. The solid lines re eye guides. After Todorov et l. 146 Todorov et l. find sizle CIP GMR in the sence of the interfce roughness. The effect rises from the difference in the men free pth nd ulk resistivity etween the up-spin nd the down-spin chnnels in the mgnetic mteril, which in turn comes from the respective difference in the density of sttes nd the Fermi velocity. GMR decreses with incresing ulk disorder, pproching zero in the limit of the men free pth for oth spins ecoming smller thn the trilyer thickness. The ove results indicte tht the spin-dependent nd structure, in conjunction with the intrinsic ulk disorder in rel systems, is sufficient condition for GMR. The chemiclly shrp interfcil roughness generlly enhnces the effect. The contriution of the roughness, however, ecomes quntittively significnt only in the limit of sufficiently dense interfcil steps (smll 3 1 with roughness without roughness 5 51

28 correltion length of the roughness), sufficiently wek ulk scttering, nd sufficiently thin mgnetic lyers. This is evident from Figs.6 nd 6. Fig.6 shows GMR s function of the roughness correltion length. As is seen from the figure, with incresing correltion length GMR decreses, pproching its vlue without roughness. These results re consistent with the experimentl dt y Schd et l. 7 on GMR in epitxil Fe/Cr multilyers discussed in section 7 (see lso Fig.14). Fig.6 shows the dependence of GMR on the mgnetic lyer thickness. As cn e seen from the figure, GMR decreses with incresing d FM, ut in the cse with the roughness, the decrese is fster thn in the cse without the roughness. This result is consistent with the different ehvior of ulk nd interfce spin-dependent scttering predicted y the semiclssicl free-electron model (see section 1). Although some nd structure effects hve een included in the models of Asno et l. 143 nd Itoh et l. 144 y introducing the spin-dependent multilyer potentil nd in the model of Todorov et l. 146 y using the spin-dependent nd widths, in generl single-nd tight-inding models re not le to descrie ccurtely the spin-polrized nd structure of rel mgnetic lyered systems. A more ccurte nd structure ws considered y Itoh et l. 147 for studying the mteril dependence of mgnetoresistnce nd thermoelectric power in Fe/TM nd Co/TM multilyers, where TM = Sc, Ti, V, Cr, Mn, Ru, Rh, or Pd. They used the tight-inding d-nd model nd clculted self-consistently the electronic structure nd the mgnetic moments of rough Fe/TM nd Co/TM interfces. They ignored, however, the contriution from the sp nds nd their hyridiztion with the d nds which is, s we will see elow, crucil for GMR. V. MULTIBAND MODELS As hs lredy een noted ove, free-electron nd single-nd s-vlent tight-inding models oversimplify the electronic structure of the mgnetic multilyers which re used in GMR experiments. These models cnnot ccount relisticlly for the ngulr chrcter of the d oritls, which is prtly responsile for the mgnetism in trnsition metl ferromgnets. They neglect, moreover, the contriution from the d nds to the electronic trnsport, which is importnt due to the presence of the d nds t the Fermi energy, s for the minority-spin electrons in Co (see Fig.3c). These models lso ignore the hyridiztion etween the sp nd d electrons which is, s we will see, very importnt for GMR. Only ccurte multind models cn dequtely tke into ccount the ove effects. Powerful methods of electronic nd structure clcultions hve een developed during the lst few decdes, which re sed on density-functionl theory (for review see, e.g., ref.148). These methods do not involve ny empiricl prmeters nd re therefore referred to s -initio or first principles. Density-functionl theory is generl pproch for clculting the ground-stte properties of n intercting electron gs in the presence of n externl potentil. It is sed on the Hohenerg nd Kohn theorem 149 which sttes tht the ground-stte energy of the mny ody system is unique functionl of the electronic chrge density n(r). Kohn nd Shm 15 showed tht minimizing the totl energy under ssumption of the electronic chrge density eing sum over ll occupied single-prticle sttes ψ i (r), i.e. n( r) = ψ i ( r), (V.1) i occ leds to the set of self-consistent equtions! n( r ) δexc[ n( r)] + Vext ( r) + e dr ψ i ( r) = Eiψ i ( r) m +. (V.) r r δn( r) Here the wve functions ψ i (r) nd eigen-energies E i re leled ccording to stte index i. Eqution (V.) is similr to single-prticle Schrödinger eqution with n effective Hmiltonin, in which the first term is the kinetic energy, the second term is the externl potentil tht includes the electrosttic potentil from the nuclei, the third term is the Hrtree potentil of the totl electronic chrge distriution, nd the fourth term is the exchnge-correltion potentil. The exchnge-correltion potentil includes ll mny-ody effects nd, therefore, cnnot e clculted exctly. The common pproximtion is the locl density pproximtion (LDA). Within the LDA the exchnge-correltion energy E xc [n(r)] for n inhomogeneous system is ssumed to e given y the exchnge-correltion energy density Hxc[n(r)] for homogeneous electron gs of chrge density n(r) so tht Exc [ n( r)] = H xc[ n( r)] n( r) dr. The LDA is formlly justified for slowly vrying densities, ut it hs een found in prctice to provide good description of the ground stte properties of non-mgnetic trnsition metls, in ddition to the more free-electron-like sp-vlent metls. In prticulr, density functionl formlism cn e extended to mgnetic systems 151 y introducing the spin densities n (r) nd n (r), so tht the chrge density is n(r)= n (r)+ n (r). In this cse the single-prticle wve functions ecome spin-dependent, the exchnge-correltion energy ecomes functionl of the spin densities, E xc [n (r),n (r)], nd the LDA is generlized y the LSDA, which is the locl spin-density pproximtion. 15 The Kohn-Shm equtions (V.) hve to e solved self-consistently, ecuse oth the Hrtree nd exchnge-correltion potentils depend on the output density. The resulting singleprticle eigenfunctions ψ i (r) nd eigenvlues E i cn e used for clculting trnsport properties. 153 A vriety of techniques hve een developed to solve the Kohn-Shm equtions, mong them the ugmented plne wve method (APW), 154 the Green s function Korring-Kohn-Rostoker (KKR) method 155 nd the liner-muffin-tin oritl method (LMTO). 156 All these methods cn e implemented within the full-potentil pproch, which provides numericlly exct solution. Most of clcultions re, however, performed using shpe pproximtions for the potentil, such s the muffin-tin pproximtion or the tomic sphere pproximtion (ASA), which work well for closepcked crystl structures. These pproximtions mke the computtions much fster without significnt loss in ccurcy. If pplied to the electronic nd structure of the ulk trnsition 3d metls ll these methods give similr results. An exmple of the nd structure clculted using the LMTO- ASA method is given in Fig.5 for ulk Cu nd Co. The multind tight-inding pproximtion is more empiricl pproch for clculting the electronic nd structure, which is generliztion of the single-nd pproximtion considered in section 14. Although not s ccurte s the -initio models, the multind tight-inding models provide vlule insight into electronic trnsport y simplifying the physics involved. They lso llow considerle sving in computer time compred to full-scle first-principles clcultions. The multind models for trnsition metls include the vlence s, p nd d oritls, which re chrcterized y different on-site tomic energy levels. The effects of onding nd hyridiztion enter through twocenter hopping integrls etween tomic oritls with chrcteristic ngulr moment. The ngulr vrition in the hopping mtrix elements with neighoring sites hs een tulted y Slter nd Koster. 157 Spin polriztion is treted y introducing two different sets of tight-inding prmeters for the mjority nd minority spins. For the 3d ferromgnets sizle spin dependence ppers, however, only for the on-site energies of the d oritls, which reflects the Stoner exchnge spitting of the spin nds. 34 The vlues of the tight-inding prmeters, i.e. the hopping integrls nd the on-site tomic energies, cn e otined y fitting the tight-inding nds to the nds clculted y the firstprinciple methods descried ove. 158 This mkes it possile to represent ccurtely the electronic structure of the trnsition metls. The electronic structure for multilyer cn e uilt either y fitting the -initio nd structure for this multilyer or from the tight-inding prmeters of the respective ulk metls. In the ltter cse the hopping integrls etween non-equivlent species cn e set s the geometric men of the respective hopping integrls for ech species nd the on-site energies cn e set to equte the Fermi energies of the respective ulk metls. In this pproximtion the tight-inding model for the multilyer does not descrie self-consistently the rerrngement of the electronic chrge t the interfces. Some self-consistency cn e introduced using the condition of locl chrge 5 53

29 neutrlity, 159 in which the reltive position of the on-site tomic energies is djusted to gurntee zero net chrge on ech tom. This is resonle pproximtion for metllic multilyers. The multind tight-inding model cn e derived from first-principles using the tight-inding liner muffin-tin oritl (TB-LMTO) method, 16 in which the sis muffin-tin oritls re screened y the presence of their surrounding neighors into more loclized or tight-inding form. Using the tulted LMTO structure constnts set of nerest-neighor tight-inding prmeters cn e generted for mgnetic lyered system y solving the Dyson eqution for the screening selfconsistently. These prmeters cn then e used s input for clculting conductnce. The ove first-principle nd tight-inding methods cn e used for clculting the electronic structure of mgnetic thin-film systems. However, in order to nlyze trnsport properties, these models hve to e extended to include disorder, which genertes rndom potentil nd sctters the electrons t the Fermi energy. Including disorder is importnt ecuse ll the experiments on GMR performed to dte re crried out in the diffusive regime, in which the conductnce is limited y scttering. Within the diffusive regime the smple dimensions re much lrger thn the men free pth. A relistic description of disorder is one of the key prolems for predicting GMR ecuse depending on the model for disorder the result for GMR cn e very different. We will discuss vrious pproched to include disorder nd to tret GMR in the diffusive regime of conduction in sections Opposite to the diffusive trnsport regime, the llistic regime ssumes tht the smple dimensions re much smller thn the men free pth. 161 In this regime the conductnce is not ffected y scttering ut is determined entirely y the nd structure nd the device geometry. All the complictions resulting from the disorder in the diffusive regime do not exist in the llistic regime, which llows prmeter-free first-principle clcultions of GMR to e performed. 16,163, Although the llistic regime hs not yet een proed experimentlly in GMR mterils, considering this limit for conduction is useful to demonstrte the importnce of the spin-polrized nd structure for GMR. This point will e illustrted in the next section. 15. Bllistic limit The conductnce of disorder-free system is determined y the kinemtic motion of the electrons. Even though the electrons pssing through the smple re not scttered, the conductnce of the smple is finite due to its finite cross-section. The mgnitude of the conductnce is determined y the electronic nd structure of the mteril from which the smple is fricted nd cn e clculted from first principles. In order to derive the expression for the llistic conductnce Γ we write down, first, the expression for the net current which flows cross the smple of cross section A due to pplied voltge V. The net current per spin, I, is given y the difference in the numer of electrons propgting in opposite directions, i.e. dk I = ds evη ( { f [ Eη ( ev ] f [ Eη ( ]}, (15.1) 3 η (π A ) where f(e) is the Fermi distriution function, ds is cross sectionl element of the smple, nd the velocity of nd η cn e determined from the nd dispersion E η ( y vη ( = de η ( /! dk. Here s efore we hve omitted the spin indices. Assuming tht the voltge is smll nd tking the zero-temperture limit we rrive t the conductnce Γ = I V V = Ae η dk (π ) 3 1 n v ( δ[ E ( E ], (15.) η η where n is unit vector in the trnsport direction nd the fctor ½ ppers ecuse only electrons moving in one direction contriute to the current. Mthemticlly this formul represents weighted 54 F density of sttes t the Fermi energy. This formul is suitle for prcticl clcultions of the llistic conductnce nd cn e pplied to mgnetic multilyers. The formul (15.) cn e rewritten in different wy. Performing explicitly the integrtion in eqution (15.) with respect to the momentum k z long the direction of trnsport z we otin e A e Γ = d Qη ( ) = N π! 4π k k, (15.3) η π! where k is the trnsverse momentum, i.e. the momentum perpendiculr to the current, nd Q η ( k ) is the numer of roots in the eqution E η ( k, k z ) = EF for given k nd spin. This representtion is equivlent to the Lnduer expression for the llistic conductnce (see, e.g., ref. for discussion). According to eqution (15.3) the conductnce is determined y the numer of conducting chnnels N opened for the current-crrying electrons, which is determined y the density of the trnsverse modes t the Fermi energy. We note tht the llistic conductnce of n idel periodic structure does not depend on the length L of the smple long the direction of the current (which mkes the conductivity infinite). As long s L is much smller thn the men free pth, the conductnce per unit re is determined only y the numer of open conducting chnnels, i.e. entirely y the nd structure. We lso note tht unlike conductivity, the llistic conductnce in systems with cuic symmetry is, in generl, not isotropic. Fig.7 Conductnce (1 15 Ω -1 m - ) Energy (ev) E F Conductnce (1 15 Ω -1 m - ) Using the LMTO-ASA method nd formul (15.) Schep et l. 163 clculted the llistic conductnce of ulk Cu nd Co metls, which is shown in Figs.7, respectively. Although only the vlue t the Fermi energy is relevnt, the conductnce is plotted s function of the nd filling to indicte the contriutions from different nds. Schep et l. found tht t the Fermi energy the llistic conductnce of Cu is well descried y the free-electron estimte (see the dshed line in Fig.7). For the energy rnge etween 1.5 nd 4eV elow the Fermi energy the llistic conductnce is significntly enhnced due to the high density of electrons of minly d chrcter. This is opposite to the diffusive regime in which the hevy d electrons contriute little to the conductnce (see the Energy (ev) Bllistic conductnce in the (1) direction for ulk fcc Cu () nd ulk fcc Co () s function of electron energy within rigid nd structure. The verticl line displys the Fermi energy. The dshed line in figure () shows the free-electron result. The solid nd dshed lines in figure () show the conductnce of the mjority nd minority spins respectively. After Schep et l. 163 E F

30 discussion in section ). The enhncement is, however, considerly less pronounced thn tht in the corresponding density of sttes (compre to Fig.5). This is due to the weighting in eqution (15.) with the velocity, the ltter eing much smller for the reltively flt nds in this energy rnge. For the energies more thn 6eV elow the Fermi energy there is only one nd which hs minly s chrcter. In mgnetic Co the degenercy etween the up- nd down-spin electrons is lifted. As is seen from Fig.7, the llistic conductnce depends on the spin direction due to the spin dependence of the electronic structure. For the mjority-spin electrons the llistic conductnce resemles tht in Cu. The minority-spin d nds re shifted to higher energies resulting in the higher minority-spin conductnce t the Fermi energy. In contrst to the results for Cu nd for the mjority spins of Co, Schep et l. found no resonle free-electron estimte for the minority-spin conductnce of Co, which reflects the complicted nd structure of Co for this spin direction t the Fermi energy. The difference in the electronic structure for the mjority nd minority spins in ulk Co t the Fermi energy leds to difference in the spin conductnce, which inevitly should ffect the conductnce of Co/Cu multilyer nd hence result in GMR. This hs, indeed, een found y Schep et l., 16 who predicted tht in the llistic regime the CPP conductnce increses y more thn fctor of two when the reltive orienttion of the mgnetiztions of djcent mgnetic lyers in Co/Cu multilyer is chnged from ntiprllel (AP) to prllel (P). Such sizle difference in the conductnce reflects the contriution from electrons of different oritl moment to the current. According to eqution (15.3) the llistic conductnce is determined y the geometric properties of the Fermi surfce, nmely the projection of the Fermi surfce in the direction of the current Q η ( k ). Schep et l. investigted this projection in k spce for the Co/Cu multilyers nd found significnt difference etween the mjority nd minority spins. Although for the mjority spins within the P configurtion Q η ( k ) resemles the free-electron picture, the minority spins within the P configurtion nd oth spins within the AP configurtion re not free-electron-like. The mjority-spin s electrons shunt the current within the P mgnetiztions mking the conductnce for this configurtion much higher thn tht for the AP configurtion, which is minly determined y the d electrons. Fig.8 shows the vrition of the llistic conductnce nd GMR for Co n /Cu n multilyers s function of the lyer thickness n for the CPP nd CIP geometries. 163 We see from Fig.8 tht for smll lyer thicknesses the CPP conductnce decreses rpidly with n, which rises from the evnescent sttes in the Co nd Cu lyers nd from the devitions in the potentil ner the interfces from the ulk vlue. 163 For lrger lyer thicknesses the conductnces pproch constnt vlues corresponding to CPP GMR of out 1%. We see from Fig.8 tht the GMR in the CIP geometry is much lower thn in the CPP geometry nd decreses with the lyer thickness. In this cse the minority-spin electrons disply higher conductnce within the P configurtion reflecting tht in ulk Co. We note tht the clculted vlues of CIP GMR re much lower thn the experimentl vlues otined for the Co/Cu multilyers (e.g., ref.44). This indictes tht scttering must e included in order to otin greement with experiments. As is seen from Fig.8, oth the CIP nd CPP conductnces disply smll dmped oscilltions s function of the numer of monolyers. These oscilltions originte from quntum size effects nd were first predicted y Mthon et l. 164 for the CPP GMR within the llistic trnsport regime. Using single-nd tight-inding model nd the recursive pproch Mthon et l. predicted the presence of two types of oscilltion. The first type of oscilltion results from sttionry points t the Fermi surfce nd hs period, which is determined y the wve vector spnning the Fermi surfce of the spcer lyer in the direction perpendiculr to the plnes. This type of oscilltion hs the sme period s tht predicted erlier for the oscilltory exchnge coupling. 165 The second type of oscilltion results from potentil steps t the interfces nd is determined y the trnsmission cutoffs of the trnsmitting sttes close to speculr reflection. Using similr pproch to clculte the llistic conductnce nd CPP GMR in Co/Cu multilyers within multind tight-inding model Mthon et l. 166 found the presence of oth types of oscilltion periods. They showed tht the conductnce oscilltions for the mjority-spin electrons within the P configurtion hve periods, which re determined y the sttionry points t the Fermi surfce of Cu. On the other hnd, the conductnce oscilltions of the minority-spin electrons in the P configurtion nd either-spin electrons in the AP configurtions re dominted y the periods which re determined y the conductnce cutoff due to mismtch etween the Co nd Cu nds cross the Co/Cu interfces. We note tht neither type of oscilltion hs yet een oserved in experiments on CPP trnsport, which is most proly due to the interfce nd ulk disorder in rel multilyers. The ltter fct is confirmed y clcultions in the diffusive regime of conduction, which will e presented in section VI. Fig.8 Conductnce (1 15 Ω -1 m - ) Thickness (ML) Schep et l. 16 hve demonstrted the crucil role of the hyridiztion etween the dispersive sp nds nd the loclized d nds for GMR within the llistic regime. They switched off the sp-d hyridiztion y setting the mtrix elements etween the sp nd d oritls in the LMTO structure constnts equl to zero. This resulted in significnt drop of CPP GMR, e.g., from 1% to 3% for the (1)-oriented Co 5 /Cu 5 multilyer. The origin of this effect cn e explined s follows. The unhyridized sp nd d nds crry the electric current independently nd cn thus e considered s two prllel conduction chnnels. In the sence of hyridiztion the sp electrons re not spinpolrized nd, therefore, do not contriute to GMR. On the other hnd, lthough the d electrons in Co re spin-polrized, they do not contriute to the conductnce ecuse in the sence of the sp-d hyridiztion they re not coupled to the sp nds of Cu (which re the only nds t the Fermi level) nd, therefore, cn not e trnsmitted cross the spcer lyer. The importnce of the sp-d hyridiztion for GMR suggests tht theories which ignore this hyridiztion re leving out crucil ingredient. We will see in section 17 tht the sp-d hyridiztion lso plys decisive role for GMR within the diffusive regime Dependence of GMR (solid circles) nd conductnce of the (1)-oriented Co n /Cu n multilyers for the mjority-spin (dimonds) nd the minority-spin (squres) electrons in the prllel configurtion nd for either-spin electrons in the ntiprllel configurtion (open circles) within CPP () nd CIP () geometries. After Schep et l CPP GMR (%) CIP GMR (%) 56 57

31 16. Semiclssicl theory Accurte models for the nd structure cn e incorported within semiclssicl trnsport theory to clculte GMR in mgnetic multilyers in the diffusive regime of conduction. The simplest wy to proceed is to use the relxtion time pproximtion, s ws descried in section 1. In this cse the conductivity of ulk homogeneous system cn e clculted using expression (1.1). Although mgnetic multilyers re not homogeneous in the direction perpendiculr to the plnes, one cn still use this expression within the CIP geometry, ecuse s ws discussed in section 13 the internl electric field is constnt in this cse due to the homogeneity of the multilyers in the plne of the lyers. The vlidity of the relxtion time pproximtion is often not known, ut even within this pproximtion n ccurte quntum-mechnicl clcultion of the relxtion time is not trivil nd, s we will see elow, requires sophisticted techniques. In ddition, the relxtion time depends on the model for disorder which determines the mechnism of scttering. The simplest pproch is to ssume constnt relxtion time, s ws done y Oguchi. 167 In this cse formul (1.1) for the conductivity in the ν direction per spin cn e rewritten s d νν k ν σ = e τ vη ( δ[ Eη ( E ] 3 F, (16.1) η (π ) where we hve introduced the integrtion over the Brillouin zone nd displyed explicitly the summtion over nd index η. As is seen, the conductivity is fctorised into scttering term τ nd the electronic structure term. The electronic structure term is similr to tht in the llistic regime of conduction (compre to eqution (15.)) nd cn e evluted without ny free prmeters. Within the pproximtion of stte- nd spin-independent relxtion time τ the GMR rtio does not depend on τ nd is entirely determined y the nd structure. Oguchi clculted self-consistently the electronic nd structure of Co 3 /Cu 3 multilyer using the LMTO method nd found vlues of GMR of 47% for CIP nd 17% for CPP. These vlues re higher thn those otined within the llistic regime (lthough of the sme order of mgnitude), which is due to the dditionl velocity-weighting fctor in formul (16.1) which enhnces the spin symmetry. Oguchi ws the first to point out the importnce of the nd structure (the Fermi velocities) for GMR. However, his ssumption of stte- nd spinindependent relxtion time is difficult to justify. In prticulr, neglecting the spin dependence of the τ is unrelistic, t lest due to the density of sttes fctor in eqution (1.4) which is spin-dependent. Zhn et l. 168 used expression (16.1) for the spin conductivity to clculte GMR in Fe/Cr multilyers, ssuming tht the relxtion time is spin-dependent (ut still stte-independent). Scttering ws introduced into their model y ssuming the presence of Cr impurities in the Fe lyers within the low-concentrtion limit. They used n explicit expression for the spin-dependent scttering proility π! ηk, η k = Tηk, η k δ ( Eηk Eη k P ), (16.) in which the T mtrix descries the scttering of n electron y n impurity emedded in the multilyer. This expression cn e considered s generliztion of the Fermi golden rule (1.4) to include ll orders of perturtion theory with respect to the scttering potentil. The mtrix elements T η k,η k were clculted using n ppliction of multiple-scttering theory within the first-principle KKR Green s function formlism, which ws developed erlier in works y Dederichs nd couthors. 169 Using these mtrix elements otined for Cr impurity in ulk Fe, for either spin direction the stte-independent relxtion time ws clculted from expression (16.) y summtion over stte indices ccording to 1 τ η = ηk, η k η k ( P (16.3) 58 nd then y verging over the Fermi surfce. It ws found tht the rtio of the spin-relxtion times α = τ / τ =.11, which implies tht the mjority spins re scttered strongly t Cr defect, wheres the minority spins pss the defect wekly scttered. The relxtion time for the ntiprllel configurtion ws estimted y series-resistor rule, i.e. 1/ τ (1/ AP = τ + 1/ τ ) /. Zhn et l. clculted CIP nd CPP mgnetoresistnce within n optimised LCAO (liner comintion of tomic oritls) method using lrger unit cells (Fe 3 /Cr n nd Fe m /Cr 4 with n,m 1) thn those which were used y Oguchi. 167 They predicted relly gint effects of out 6% for CIP nd of out 5% for CPP, which re much higher thn the experimentl vlues for Fe/Cr multilyers. The pproximtion of stte-independent relxtion time ws lifted in the work y Binder et l. 17 of the sme group, who clculted the stte-dependent relxtion times nd the spin-dependent conductivity of the Co/Cu multilyer within screened KKR method using semiclssicl trnsport theory. They introduced different 3d-metl impurities t the Co interfce lyer nd compred the resulting resistivities nd GMR for the stte-dependent nd stte-independent relxtion times. Their results show tht the pproximtion of stte-independent relxtion time overestimtes the resistivity of the mjority-spin electrons y fctor of two nd of the minority-spin electrons y out 1%, which leds to overestimting GMR. Nevertheless, the predicted vlues of GMR for the Fe, Ni nd Cu impurities re still huge (out 4% for Fe nd 6% for Ni). For Ti, V nd Cr impurities n inverse GMR effect ws otined. Binder et l. lso investigted the vlidity of the relxtion-time pproximtion. They extended their semiclssicl theory to clculte the scttering-in term (see section 1) y solving the Boltzmnn eqution itertively. They found tht the relxtion time pproximtion gives very stisfctory results for the systems considered. Butler et l. 171 used different model for disorder to clculte CPP GMR in the (111)-oriented Co n /Cu m multilyers (up to n=m=6) within the semiclssicl pproximtion. They ssumed tht the Co/Cu interfces re interdiffused with 1% intermixing of the Co nd Cu toms. The nd structure of the disordered multilyers ws clculted using the single-site coherent potentil pproximtion within the KKR Green s function method. Assuming sufficiently wek impurity scttering they computed the relxtion time for ech stte t the Fermi energy, which is inversely proportionl to the imginry prt of the self-energy in eqution (13.1). Then, using the stte-dependent relxtion time they solved the Boltzmnn eqution disregrding the scttering-in term nd clculted the conductivity of the multilyers for the prllel nd ntiprllel mgnetiztions. Similr to the works y Zhn et l. 168 nd Binder et l., 17 the vlues of CPP GMR otined for the Co/Cu multilyers were huge vrying from out 1% to % for different lyer thicknesses. Similr clcultions performed y Butler et l. 171 for the (111)-oriented Fe Ni 8 /Cu multilyers found even lrger GMR vlues. Neset 17 reched similr conclusion for GMR in Co/Cu nd Fe/Cr multilyers. He clculted the stte-dependent relxtion time ccording to equtions (16.) nd (16.3) ssuming tht scttering is cused y interdiffused toms t the interfces. The spin-polrized nd structure nd the scttering T-mtrix were computed y expnding the Bloch wve-functions into lrge numer of tomic oritls, using full-potentil multiple-scttering pproch within the locl density pproximtion. The semiclssicl expression for conductivity within the relxtion time pproximtion (1.1) ws used to compute GMR. Consistent with previous clcultions mgnetoresistnce due to this scttering mechnism ws found to e very lrge. The highly overestimted vlues of GMR otined y Zhn et l., 168 Binder et l., 17 Butler et l., 171 nd Neset 17 is direct consequence of the fct tht the electronic structure of the impurities is nerly identicl to the electronic structure of the host toms in one of the two spin chnnels which leds to no scttering nd shunting within this spin chnnel. This cn e illustrted y clcultions of the numer of vlence electrons per tom in the Co/Cu nd Fe/Cr systems shown in Figs.9,. 173 As is seen from Fig.9, in the Co/Cu system the numer of Co mjority-spin electrons mtches very closely the numer of Cu electrons, wheres there is strong mismtch in the numer of minority- 59

32 spin electrons. This mtching in the mjority-spin chnnel implies tht the tomic colt nd copper potentils pper very similr to the mjority-spin electrons leding to no scttering within this spin chnnel when Cu impurities re imedded in Co or vice vers. The sme conclusion ws otined for Ni 8 Fe /Cu multilyer ecuse the Fermi energy scttering mplitudes for mjority-spin colt, mjority-spin nickel, nd mjority-spin iron (s n impurity in nickel) re ll very similr. 173 For the Fe/Cr system the mtching in the numer of vlence electrons per tom occurs for the minority spins leding to wek scttering of electrons in the minority-spin chnnel (see Fig.9). As ws suggested y Butler et l. 173 the spin-orit interction, which ws not considered in the ove clcultions, might prtly explin the discrepncy etween theory nd experiment. Indeed, in ulk lloys the spin-orit interction, lthough smll, is sufficient to void the ner short circuits due to lck of scttering in either spin chnnel (e.g., ref.174). Another reson could e the presence of misligned mgnetic moments t rough interfces due to the reduced exchnge coupling, which s we hve seen in section 8 might ply role. However, we elieve tht the principle disgreement etween the theory nd experiment origintes from the neglect of scttering y intrinsic defects, which re lwys present in the multilyer. In the ove clcultions it is ssumed tht the introduced impurities re the only source of scttering, which is not the cse in rel experiments on GMR. We will see in section 17 tht relistic model for disorder ccounts for the experimentlly-oserved vlues of GMR in the Co/Cu nd Fe/Cr multilyers. where ψ η ( k, ri ) is the Bloch wve function of the multilyer without the impurities nd n( r i, EF ) is the locl density of sttes. In order to void short circuit effects due to sttes with low scttering 1 mplitude t the impurity position they dded constnt scttering rte τ in eqution (16.4). Since the scttering rte in eqution (16.4) is proportionl to the spin-dependent density of sttes t the impurity site n( r i, EF ), strong position dependence of GMR is expected. Zhn et l. found tht n( r i, EF ) is enhnced nd hve strongest spin symmetry t the Co interfce in Co 9 /Co 7 mutlilyer, which is explined y the formtion of quntum-well nd interfce sttes. Due to this the mgnitude of GMR is enhnced y the impurities with no spin symmetry (α=1) when they re plced t the Co interfce. This is evident from the tringles in Fig.3, which shows the clculted vlues of GMR for vrious positions nd scttering spin symmetries of the impurities. For impurities with stronger scttering of the minority thn the mjority electrons, α>1, the existing spin symmetry in the locl density of sttes is mplified nd leds to n even stronger enhncement of GMR (the circles in Fig.3). For the opposite spin symmetry, α<1, the spin symmetry of the locl DOS nd the scttering potentil compenste ech other nd GMR is reduced. As is seen from Fig.3, impurities plced within the Cu lyer re ineffective for GMR, which is scried to the smll locl DOS in Cu. Electrons/tom Co Cu Co Electrons/tom Fe Cr Fe GMR (%) 3 1 Co Cu Lyer numer Lyer numer Impurity position (ML) Fig.9 Clculted numer of vlence electrons for the (111)-oriented Co/Cu/Co () nd (1)- oriented Fe/Cr/Fe () trilyers for the mjority-spin ( ) nd minority-spin ( ) electrons. Note the mtching for the mjority spins in the Co/Cu/Co nd mtching for the minority spins in the Fe/Cr/Fe. After Butler et l. 173 Fig.3 Clculted CIP (opened symols) nd CPP GMR (closed symols) of the (1)-oriented Co 9 /Cu 7 multilyer s function of the impurity position within the multilyer for the impurities with vrious symmetries in the scttering potentil α = V / V : α=1 (tringles), α=4 (circles), nd α=.5 (squres). The solid nd dshed lines show the vlues 1 of CPP nd CIP GMR respectively clculted for the scttering rte τ (V= in eqution (16.3)). The verticl dotted line seprtes the Co nd Cu lyers. After Zhn et l. 175 An interesting clcultion ws performed y Zhn et l., 175 who investigted the effect of the impurity position on GMR in Co/Cu multilyers. In order to study impurities with different scttering spin symmetries they ssumed tht the spin-dependent scttering potentil V is prmeter, therey neglecting detils of the impurity potentil nd its dependence on the site position r i within the supercell. The scttering spin symmetry ws, therefore, defined s α = V / V. They considered the wek scttering limit in which the scttering rte (the inverse relxtion time) is determined y the Born pproximtion τ ( k, r ) 1 η π! 1 i = V ψ η ( k, ri ) n( ri, EF ) + τ, (16.4) It is interesting to compre the predictions of Zhn et l., 175 with the results of experiments y Mrrows nd Hickey, 76 (see section 8). Experimentlly it ws found tht impurities with α<1 suppress GMR, usully to gret extent when they re plced t the interfce, nd still hve considerle effect when they lie severl lttice constnts wy from the interfce; impurities with α>1 sometimes do provide n enhncement of GMR, ut it is only to e found when they re few ML ehind the Co/Cu interfce; nd impurities in the spcer lyer hve drmtic effect y lowering GMR. These findings re t odds with the theoreticl predictions of Zhn et l. Mrrows nd Hickey 76 suggested tht to reconcile the dt with the theory one must dmit in ddition to the scttering y impurities considerle spin-dependent scttering in the ulk of the Co lyers, or tht the impurities cn ffect 6 61

33 the locl density of sttes up to severl tomic sites wy. Very recent clcultions performed y the sme group 176 demonstrte, however, etter greement etween the theory nd the experiment. A necessry condition for the vlidity of the semiclssicl Boltzmnn theory is sufficiently low impurity/defect density. In lyered systems the Boltzmnn formlism reks down when the sund energy splitting ecomes comprle to the life-time rodening / τ due to scttering y impurities nd/or defects. As we will see in the next section, this is the cse for rel multilyer structures, which re used in experiments on GMR. The quntum-mechnicl tretment of trnsport is necessry in order to mke quntittive comprison etween theory nd experiment. 17. Tight-inding models Experimentl dt show tht mgnetic lyered systems contin lot of intrinsic structurl defects such s vcncies, stcking fults, lttice distortions, misfit disloctions nd grin oundries tht re produced during the process of deposition (e.g., ref.177,178). These defects influence the residul resistivity mking it much lrger in thin films thn in corresponding ulk metls. For exmple, Rijks et l. 178 found significnt scttering t the grin oundries nd other defects in sputter-deposited copper nd permlloy thin films which ws dependent on the lyer thickness. Scttering y these defects must, therefore, e included in relistic model for GMR. However, simultneous modeling of ll these defects including their structurl nd electronic properties is very complicted nd proly pointless ecuse we do not hve sufficient experimentl informtion out distriution, concentrtion, strength nd other prmeters chrcterizing these defects. A simplified pproch to include defect scttering in model for GMR ws developed y Tsyml nd Pettifor 179 within multind tight-inding theory nd quntum-mechnicl pproch to electronic trnsport. In this section we descrie their model nd min results which follow from this model. The multind description of the electronic structure within the tight-inding representtion cn e otined y generlizing the single-nd Hmiltonin (14.1) to include the vlence s, p nd d oritls. We consider perfect crystl which is divided into equivlent unit cells m consisting of tomic sites j with ssocited tomic wve functions mjα, where α denotes the oritl chrcter. Within the two-center, orthogonl tight-inding pproximtion the single prticle Hmiltonin of the perfect system H is determined y the on-site tomic energies E jα nd y the hopping integrls jα H mj α (index denotes the origin of the system). In order to solve the Schrödinger eqution H ηk = E ( ηk, (17.1) η the eigenstte η k with nd index η nd wve vector k cn e expnded in terms of the tomic functions: ηk = 1 ik( R m + r j ) e cη, jα ( mjα, N mjα (17.) where N is the totl numer of cells in the crystl, R m is the m-th lttice vector nd r j is the sis vector ( site position within the unit cell). The expnsion coefficients cη, jα ( re otined y solving the tight-inding seculr eqution H jα, j α ( cη, j α ( = Eη ( cη, jα (, (17.3) j α where the mtrix elements of the Hmiltonin re defined y ik ( R m + rj rj ) H α α = e jα H mj α j, j ( k ). (17.4) m The tight-inding prmeters, i.e. hopping integrls nd the on-site tomic energies, cn e otined y fitting the tight-inding nds of the elementl metls to the nds computed from firstprinciples. 158 This mkes it possile to descrie relisticlly the nd structure of mgnetic multilyers tking into ccount s, p nd d oritls, their full hyridiztion nd spin-polriztion. Mtrix elements of the velocity opertor, which re necessry for clculting the conductivity, cn e µ µ found from the reltion! v ( = H ( / k. The next step is including defect scttering in the model. We ssume tht the totl Hmiltonin of the disordered system cn e represented s the sum of the perfect Hmiltonin H nd the scttering potentil V, H = H + V. The scttering potentil which reflects the presence of intrinsic defects within the multilyer is ssumed to effect the on-site tomic energy levels rndomly so tht α V m j α = V mj αδ mm δ jj δ αα mj. (17.5) Here for simplicity the scttering potentil is tken to e digonl with respect to the cell, site nd oritl indices. The digonl elements V re tken to e rndomly distriuted such tht mjα V =, (17.6) mjα V mj αv m j α = γ δ mm δ jj δ αα, (17.7) where... denotes the configurtion verge nd γ is the men squre displcement of the on-site tomic energies. The prmeter γ is prmeter in the theory. It chrcterizes the degree of disorder within the system reflecting the scttering y defects nd cn e chosen to provide relistic sturtion resistivity of the multilyer. We stress tht in generl the scttering potentil, which is produced y prticulr type of defect or impurity in mgnetic system, is expected to e spin-dependent. However, GMR structures re chrcterized y vrious types of defects. Physiclly this mens tht the spin symmetry nd the strength of the scttering potentil cn vry from defect to defect. This fct is supported y firstprinciple clcultions, 18 which show different spin dependence of scttering produced y stcking fults, twin oundries nd Cu impurities in ulk Co. On verge, it is resonle to ssume no spin dependence in the scttering potentil, which in our model implies spin-independent prmeter γ. Note tht s we will see elow the scttering rte remins spin-dependent s the result of the spin symmetry in the density of sttes t the Fermi level. The zero-temperture conductivity per spin is expressed y the Kuo-Greenwood formul (13.1), which requires configurtionl verging s ws descried in section 13. This verging cn e performed in the wek scttering limit, which is justified for GMR structures. Indeed, it is wellknown tht within the single-nd model the wek scttering limit is vlid provided the rndom potentil is such tht γ <<, where is the ndwidth (e.g., ref.16). This expression my e generlized to the cse of the multind system s γ n( E F ) << 1, where n( EF ) is the verge density of sttes per oritl t the Fermi energy. We will find tht in order to otin typicl experimentl vlues of the resistivity of the 3d-metl multilyer structures, i.e. 1-1 µωcm, one needs γ to e in the rnge from.1 to 1eV. Since n ( EF ) is of the order of.1ev -1, the wek scttering pproximtion is justified. In the wek scttering limit the self-energy, which enters eqution (13.1) for the configurtion-verged Green s function G, is determined y Σ = VG V. 16 Using the properties of the scttering potentil defined y equtions ( ) nd performing the verging explicitly, we otin Σ jα, j α ( E ) = Ωdk γ η, jα δ δ 3 jj αα η (π ) E η ε F E ( i F + c (, (17.8) 6 63

34 where Ω is the volume of the unit cell. It follows tht the imginry prt of the self-energy will e proportionl to the prtil density of sttes t the Fermi energy n j α ( EF ) relted to site j nd oritl α. In prticulr, we cn define the prtil scttering rtes y τ 1 jα π = ImΣ ( ) jα, jα EF = γ n!! jα ( E F ), (17.9) which re spin-dependent due to the spin-dependence in the densities of sttes. The finl expression for the conductivity tkes the form µν σ = µν σ j α = α j jαj α e π! dk (π ) 3 Λ µ ν jα, j α ( Λ j α, jα where the men free pth opertor Λ µ ( is defined y [ E H ( Σ( E )] 1 (, (17.1) µ µ Λ ( =! v ( Im (17.11) F F µν nd the self-energy is given y formul (17.8). The prtil conductivities σ j α corresponding to site j nd oritl α cn e used to determine the locl (lyer-dependent) contriutions to the conductivity from non-equivlent toms (lyers) j µν µν σ j = σ jα (17.1) α nd the prtil contriutions from the s, p nd d oritls µν µν σ = σ, (17.13) l α l j jα where l denotes s, p or d oritls. Fig.31 illustrtes the model y pplying it to clculting the conductivity of ulk fcc Cu s function of energy, or equivlently s function of the Fermi energy within the rigid nd pproximtion. Compring the solid line in the ottom pnel of Fig.31 to the DOS in the top pnel in Fig.31, we see tht the conductivity of Cu is strongly suppressed within the d nds (-5eV < E < 1.5eV). The decomposition of the conductivity into its prtil contriutions shows tht in this intervl of energies the conductivity is minly determined y the d electrons, the weight of the sp chrcter vrying from out 1% to 3%. The very low contriution of the sp electrons to the conductivity within the d nds is connected with the strong hyridiztion etween the sp electrons nd the d electrons, which fltten the sp nds reducing their velocity. Outside the d nd the conductivity rpidly increses nd the contriution of the sp electrons ecomes dominnt. The Fermi level in ulk copper lies high ove the d nds, so tht most of the current is crried y the sp electrons, the d contriution eing just 3%. As is evident from Fig.31, the d nds of Co re exchnge-split due to ferromgnetism, which mkes the contriutions of the two spins to the conductivity different. Similr to Cu, for energies within the d nds the conductivity is low nd minly determined y the d electrons. Aove nd elow the d nds the conductivity rpidly increses with crossover from minly d to minly sp contriutions (not shown). The Fermi level in Co lies ove the top of the d nd for the mjorityspin electrons ut flls inside the d nd for the minority-spin electrons. This results in the sizele difference etween the conductivities for the two spins t the Fermi energy. As cn e seen from the ottom pnel in Fig.31, σ exceeds σ y fctor of five. Note tht the spin symmetry in the conductivity α=σ /σ depends on the degree of disorder γ. We lso note tht the spin symmetry is opposite to the prediction in the llistic regime (see Fig.7), ccording to which the minority-spin conductnce in Co is higher thn the mjority-spin conductnce. The results of these model clcultions for the ulk metls show tht the nd structure plys very importnt role in the conductivity. In prticulr, (i) the contriution from the d electrons is sizele nd in generl cn not e neglected; (ii) the sp-d hyridiztion is importnt due to reducing the conductivity of the sp electrons within the d nds; nd (iii) the exchnge splitting of the d nds results in the strong spin symmetry of the conductivity t the Fermi energy. DOS (ev -1 ) σ (µω -1 cm -1 ) Fig totl sp d E F Energy(eV) DOS (ev -1 ) σ (µω -1 cm -1 ) All these effects mnifest themselves in the GMR multilyers. Fig.3 shows the density of sttes (the top pnel), conductivity (the middle pnel) nd GMR (the ottom pnel) of Co 4 /Cu 4 (1) multilyer clculted s function of energy. Similr to ulk Co, the DOS of the prllel-ligned Co/Cu multilyer is symmetric etween the mjority nd minority spins. This is reflected in the conductivity σ. For the energies tht lie within the d nd the sp electrons re slowed down y the hyridiztion with the d electrons nd the conductivity is low. Aove the top of the d nds σ increses rpidly. This increse is due to the ccelertion of the sp electrons which re less ffected y the sp-d hyridiztion ove the d nds. The top of the d nds for the mjority spins lies t pproximtely.5ev elow the Fermi energy nd for the minority spins t out 1eV ove the Fermi energy. Therefore, in the intervl of the energies from -.5eV to 1eV the difference etween σ nd σ is most pronounced. The presence of the d levels up to 1eV ove E F for the AP configurtion mkes the conductivity per spin in this cse similr to the conductivity of the minority spin electrons for the P configurtion. This results in the crucil difference etween the σ P nd σ AP nd, consequently, in GMR. As seen from the ottom pnel in Fig.3 the lrge vlues of GMR re predicted only in the intervl of energies etween the top of the mjority-spin d nd nd the top of the minority-spin d nd. As the Fermi level in Co/Cu multilyers lies within this intervl sizele vlue of GMR cn e oserved in this system mjority minority mjority minority E F Energy (ev) Density of sttes (top pnels) nd conductivity (ottom pnels) s function of electron energy for ulk fcc Cu () nd for ulk fcc Co (). Conductivity is clculted for γ=.5ev nd decomposed into the contriutions from the sp nd d electrons for Cu nd from the mjority nd minority spins for Co. The verticl line shows the position of the Fermi level. After Tsyml nd Pettifor

35 An interesting effect, which is evident from the ottom pnel in Fig.3, is tht GMR in the Co/Cu system increses with incresing energy ove the Fermi level. A pronounced pek in R/R ppers for hot electrons t out 1eV ove the Fermi level, tking the lrge vlue of nerly 4% for the CPP geometry. The enhncement of GMR in Co/Cu multilyers hs een oserved y Monsm et l. 181 who hve mesured more thn 39% CPP mgnetocurrent chnge in spin-vlve trnsistor. The energy of the hot electrons in this device ws defined y the Schottky rrier heights of out.7ev for the collector nd out.6ev for the emitter. The results of those clcultions give cler interprettion of this enhnced GMR effect, which origintes from the incresing spin symmetry in the electron velocities for hot electrons up to 1eV ove the Fermi energy. We note, however, tht n ccurte quntittive description of the spin-vle trnsistor requires including nonelstic scttering which is not tken into ccount in the ove model. Recently huge chnge in the mgnetocurrent of 56% t 1 K hs een mesured y Jnsen et l. 18 in spin vlve trnsistor with Ni 8 Fe /Au/Co se. DOS (ev -1 ) E mjority (P) F Co/Cu minority (P) DOS (ev -1 ) Fe/Cr E F mjority (P) minority (P) DOS exhiits pronounced vlley for the minority spins, with the Fermi level lying lmost t the ottom of this vlley, which follows from the similrity of the minority DOS of Fe nd the DOS of Cr. We note tht the presence of this vlley in the minority DOS of ulk Fe mkes the minority-spin conductivity higher thn the mjority-spin conductivity, i.e. α=σ /σ <1, which is opposite to ulk Co. 179 For the prllel-ligned multilyer the σ displys n enhncement for electron energies lying in the region of this vlley (see the dotted line in the middle pnel of Fig.3). On the contrry, the σ for the P orienttion nd the conductivity for the AP orienttion do not chnge essentilly in this intervl of energies (the dshed nd the solid lines in the middle pnel of Fig.3). This results in the difference etween σ P nd σ AP nd, consequently, in GMR for the energies close to the Fermi level (the ottom pnel of Fig.3). The vrition of GMR s function of energy shows tht contrry to the Co/Cu multilyers GMR decreses ove the Fermi level. The mgnitude of GMR decreses with incresing disorder. As is seen from Fig.33 the vlue of GMR in Co 4 /Cu 4 multilyer drops y n order of mgnitude, i.e. from out 15% to 15% s the disorder prmeter γ chnges from.4ev to 1.eV. The corresponding sturtion resistivity vries from 8µΩcm to 55µΩcm (Fig.33), spnning the rnge of experimentl vlues. This drop in GMR with incresing disorder hs n importnt underlying physics. 179 It is relted to the internd trnsitions driven y the pplied electric field, which mkes the quntum mechnicl description different from the semiclssicl pproch. In metls, in the sence of disorder these trnsitions my e importnt only in n exceedingly smll region of k-spce ner points where two nds ecome degenerte. 3 In the presence of disorder, however, the electronic levels re rodened y the vlue of Im Σ. If this rodening ecomes comprle with the distnce etween nds even n infinitesiml smll electric field cn led to the internd trnsitions. In this cse, the semiclssicl pproximtion reks down. σ (µω -1 cm -1 ) mjority (P) minority (P) AP σ (µω -1 cm -1 ) mjority (P) minority (P) AP σ (µω -1 cm -1 ) mjority (P) minority (P) AP R/R (%) 4 CIP CPP Energy (ev) R/R (%) 4 CIP CPP Energy (ev) R/R (%) γ (ev) Fig.3 Density of sttes (top pnels), conductivity (middle pnels) nd GMR (ottom pnels) of the Co 4 /Cu 4 () nd Fe 4 /Cr 4 () multilyers s function of electron energy clculted within the tight-inding pproximtion. The verticl line shows the position of the Fermi level. After Tsyml nd Pettifor. 179 Fig.33 Conductivity of the Co/Cu multilyer s function of the root-men-squre vrition of the on-site tomic energies γ, which chrcterizes the degree of disorder within the multilyer. After Tsyml nd Pettifor. 179 The Fe/Cr system ehves differently from the Co/Cu system. As is evident from Fig.3 the Fermi level in the Fe 4 /Cr 4 multilyer lies within the d nds for oth spin orienttions. However, the The drop of GMR in the Co/Cu multilyer within incresing disorder, which is shown in Fig.33, cn, therefore, e explined s follows. The conductivity of the prllel-ligned multilyer is minly determined y the mjority-spin electrons, which hve predominntly free-electron-like sp chrcter 66 67

36 due to the position of E F ove the d nds. Therefore, with incresing γ σ P decreses pproximtely s 1/γ. On the other hnd, the conductivity of the ntiprllel-ligned multilyer is primrily due to the d electrons ecuse E F flls inside the d nd for this lignment. In this cse, due to the internd trnsitions etween the s, p nd d levels, the σ AP decreses more slowly thn 1/γ so tht GMR drops with incresing γ. This dependence is not predicted y the semiclssicl tretment of conductivity, ccording to which the conductivity for oth the P nd AP lignments is inversely proportionl to γ so tht GMR is independent of γ. Within the semiclssicl pproch the GMR vlues found for Co 4 /Cu 4 multilyer re out 4% for the CIP geometry nd out 1% for the CPP geometry. These vlues re much higher thn those predicted within the quntum description (Fig.33) nd those oserved experimentlly (section III). We note tht the internd trnsitions cn lso contriute to the reduction of GMR with temperture (section 1). The predicted vlues of GMR for the Co/Cu nd Fe/Cr multilyers re in good greement with experiment. For exmple, GMR mesured in Fe/Cr multilyers t T=4.K ws found to e 79% for reltively thin Cr lyers of.9nm. 1 For the Fe 4 /Cr 4 multilyer which hs similr Cr thickness nd for the disorder prmeter γ=.6ev which provides similr sturtion resistivity of 3µΩcm (compred to the experimentl resistivity of 31µΩcm), the clculted result of 67% is consistent with tht mesured experimentlly. Another exmple is GMR in Co/Cu multilyers which ws found to e 1% t T=4.K. 38 The thicknesses of the Co nd Cr lyers in these experiments were 1nm nd.9nm respectively nd the sturtion resistivity ws out 1µΩcm. The model predicts tht the Co 4 /Cr 4 multilyer with γ=.4ev (the sturtion resistivity of 8µΩcm) hs GMR of 15% which is lso very close to the experimentl vlue. There is, however, n exception. The model is not le to explin the high GMR vlue of % mesured Schd et l. 39,7 on high-qulity epitxil Fe/Cr multilyers (see lso section 7). As ws emphsized y Schd et l., the high mgnitudes of GMR were only oserved for ultrthin mgnetic lyers, which disply n enhnced density of steps t the interfces nd t very low tempertures. The spin-dependent scttering potentils produced y the interfcil roughness re oviously importnt in this cse, ut hve not een included in the ove theoreticl model. Lyer-dependent conductivity σ j defined y eqution (17.1) provides importnt informtion out contriutions to the spin current nd GMR from different lyers. 183 Fig.34 shows results for Co 16 /Cu 1 /Co 16 trilyer in which the thicknesses of the Co nd Cu lyers, nmely 3nm nd nm respectively, re representtive of the experiments on spin vlves. The clcultions were performed ssuming ulk scttering with γ =.7eV tht gives sturtion resistivity of 3µΩcm nd GMR of 16% in greement with experimentl dt y Egelhoff et l. 91,93 As is seen from Fig.34, for the prllel (P) configurtion the locl conductivity of the Co lyers is much higher for the mjority spins thn for the minority spins, which reflects the spin-dependent conductivity in the ulk Co (see Fig.31 t E=E F ). The locl conductivities σ Cu in the Cu lyers re higher thn those in the Co lyers nd re different for different spins. This is ecuse the trnsport properties of electrons propgting in the Cu lyer re ffected strongly y the spin-dependent scttering in the djcent Co lyers. Due to the much higher minority-spin DOS t the Co sites, electrons contriuting to the conductivity in the Cu lyer sctter more strongly in the minority chnnel nd, therefore, σ Cu > σ Cu. As is seen from the open symols in Fig.34, for the ntiprllel (AP) mgnetiztions there is n symmetry in the locl conductivities of Cu nd Co. The σ Co in the locl minority-spin chnnel is lmost the sme for the AP configurtion s for the P configurtion. On the other hnd, the σ Co in the locl mjority-spin chnnel re reduced ecuse the mjority crriers cn propgte through the Cu lyer nd sctter in the opposite Co lyer which hs high locl minority-spin DOS. As cn e seen from Fig.34, this mechnism leds to GMR, which comes from electrons contriuting to the conductivity in the Co lyers. Surprisingly, significnt contriution to GMR origintes from the Cu spcer lyer, the locl mgnetoconductnce eing highest ner the interfces. As we will see in the next section, these enhnced vlues of GMR cn prtly e explined y electron chnneling within the Cu spcer lyer. The enhncement of mgnetoconductnce in the spcer lyer nd t the interfces ws lso found for Fe/Cr/Fe spin vlves. 183 σ loc (µω -1 cm -1 ) GMR (%).6 mjority (P) minority (P) spin 1 (AP) spin (AP) Co Cu Co Lyer numer Fig.34 Lyer-dependent conductivity () nd mgnetoconductnce () in the Co 16 /Cu 1 /Co 16 (1) spin vlve in the presence of ulk scttering with γ=.7ev (symols). The mgnetoconductnce is defined s (σ j P -σ j AP )/σ AP, where σ j P nd σ j AP re the locl conductivities for the P nd AP configurtions t the lyer j. The dotted nd dshed lines in figure show respectively the mgnetoconductnce for enhnced outer-oundry scttering with γ oundry =.8eV nd enhnced interfce scttering with γ interfce =.8eV. After Tsyml nd Pettifor. 183 The locl contriutions to GMR re very sensitive to the properties of the interfces etween the ferromgnetic nd the spcer lyers. As cn e seen from the dshed line in Fig.34, incresing disorder t the interfces results in the strong reduction of the contriution to GMR from the spcer lyer nd t the interfces. A strong reduction of GMR ws lso found if mgneticlly-ded (prmgnetic) Co lyer is put t the interfce. 183 Contrry to interfce disorder, disorder t the outeroundries of the spin vlve reduces the contriution to GMR only from the Co lyers, especilly from those lyers which re close to the outer oundries (the dotted line in Fig.34). This is lso the cse for prmgnetic Co lyers t the outer-oundries (not shown). The ltter fct suggests tht using T s uffer lyer or FeMn s pinning lyer is not fvorle for otining lrge vlues of GMR in spin vlves, ecuse these metls hve high density of sttes t the Fermi level due to the d- nd contriution. An ccurte description of the nd structure is crucil to elucidte the striking fetures in the experimentl thickness-dependent conductivity mesured in-situ in NiO/Co/Cu/Co spin vlves y Biley et l., 11 which hve een descried in section 1 nd shown in Fig.. The experimentl dt re the consistent with strong scttering of the conduction electrons in Cu t the interfces with Co due to high density of empty Co d sttes t the Cu oundries. This scttering explins oth the positive curvture of the conductnce s function of the Cu lyer thickness when Cu is plced on the NiO/Co ilyer nd the conductnce drop when the Co lyer is plced on top of the NiO/Co/Cu trilyer. The results of the comprison of the experimentl dt with the clcultions re shown in 68 69

37 Figs.35,. The positive curvture in the conductnce versus Cu thickness d Cu displyed in Fig.35 is the result of the decresing scttering of electrons in the Cu lyers locted fr from the interfce with the ottom Co lyer. The conductnce drop shown in Fig.35 is consequence of the reduction of the locl conductnce in the Cu lyer when the Co lyer is plced on top. The conductnce drop increses within the Cu lyer thickness ecuse for the lrger d Cu the locl conductnce in the Cu lyer ecomes less effected y the scttering t the ottom Co lyer. As is seen from Fig.35, the greement etween the experimentl dt nd the clcultion is improved if dditionl disorder is introduced to the top Cu/Co interfce, which is ttriuted to the segregtion of low surfce energy Cu through Co in the experiment. 11 Conductnce (mω -1 ) 3 1 experiment theory 1 d Cu (nm) The decisive role of the spin-polrized nd structure for spin-dependent trnsport in mgnetic lyered systems is supported y studying the thermoelectric power (TEP) in Co/Cu nd Fe/Cr multilyers Tsyml et l. 184 Experimentl studies show tht TEP is positive in Fe/Cr mgnetic multilyers. 185 The mgneto-thermoelectric power (MTEP), i.e. the chnge in TEP ssocited with n pplied mgnetic field, is lso positive t room temperture. On the contrry, s ws confirmed y numerous experimentl studies, 186 oth TEP nd MTEP re negtive in Co/Cu multilyers. According to the Mott formul the TEP is relted to the chnge in the conductivity t the Fermi energy 187 π k T lnσ S =. (17.14) 3e E E F Conductnce drop (mω -1 ) 4 d Cu (nm) As is seen from the middle pnels in Figs.3,, for oth prllel nd ntiprllel mgnetiztions t the Fermi energy the conductivity of the Co/Cu multilyer increses with incresing energy wheres the conductivity of the Fe/Cr multilyer decreses. According to the Mott formul (17.14) this implies tht the thermoelectric power is negtive in the Cu/Cu multilyers wheres it is positive in the Fe/Cr multilyers. This theoreticl prediction is in greement with experimentl oservtions. 185,186 Tsyml et l. 184 hve lso shown tht their model ccounts for the experimentlly oserved sign of 6 4 experiment theory (ulk disorder) theory (ulk nd interfce disorder) Fig.35 Comprison of experimentl nd theoreticl results for the conductnce in NiO/Co/Cu(d Cu ) films () nd for the conductnce drop in NiO/Co/Cu(d Cu )/Co spin vlves () s function of the Cu lyer thickness d Cu. The experimentl curve in figure hs een offset y.3mω -1. In figure the greement is improved if dditionl disorder is introduced to the top Cu/Co interfce tht is ttriuted to segregtion of low surfce energy Cu through Co in the experiment. After Biley et l. 11 the mgneto-thermoelectric power t room temperture, when the mgnetiztions of the consecutive ferromgnetic lyers chnge their lignment. We see, therefore, tht the model for GMR, which incorportes spin-independent disorder in the on-site tomic energy levels nd relistic spin-dependent electronic structure within tight-inding representtion of electronic trnsport, provides relile pproch for studying GMR in mgnetic multilyers nd spin vlves. It ccounts for the experimentlly oserved vlues of GMR in Co/Cu nd Fe/Cr mgnetic multilyers nd shows tht the effects of the electronic nd structure, such s its spin-polriztion, sp-d hyridiztion nd the contriution of the d electrons to the conductivity, re crucil for the relistic description of conductivity nd GMR in mgnetic lyered systems. 18. First-principle models First-principle models for the electronic structure in conjunction with the quntum-mechnicl formultion of the electronic trnsport ecome powerful tool to study the conductivity nd GMR in mgnetic lyered systems. On the one hnd, the mjor dvntge of the first-principle pproch is the possiility to tret the electronic structure in self-consistent wy tking into ccount the chrge nd spin trnsfer t the interfces. On the other hnd, s we sw in the previous section, the quntummechnicl formultion of the trnsport theory is necessry for the relistic modeling of GMR. The principl chllenge to the -initio simultions lies in relistic modeling of the scttering. As ws discussed in section 16, ssuming tht interdiffusion t the interfces is the only mechnism of scttering in mgnetic multilyers results in highly overestimted vlues of GMR. A relile pproch must include other contriutions to the resistivity, such s from intrinsic structurl defects which were considered in the previous section. Unfortuntely, proper first-principle tretment of ll the possile mechnisms of scttering in lyered structures is very complicted. Relizing this, Butler et l. 188 use simplified pproch. Arguing tht in relistic GMR systems scttering comes from severl sources such s impurities, grin oundries, vcncies, voids, nd phonons, they pproximte these scttering processes y phenomenologicl locl scttering rte, which could e in generl spin-dependent. They use the Kuo-Greenwood formul for the conductivity in which they verge the two Green s functions independently, therey neglecting the vertex corrections. They ssume tht the effect of this verging is tht ech tomic potentil cquires n imginry term tht descries the scttering rte in its vicinity. Butler et l. introduce non-locl site-dependent conductivity σ µν ij which determines the electricl current t site (lyer) i, j i, relted to the locl electric field t site (lyer) j,,j, µ j i = σ ij jν µν ν,, (18.1) where it is ssumed tht j i is the verge of the current density over the tomic cell t tht site nd the locl field,j is constnt over ech tomic cell. If the electric field is pplied prllel to the lyers (CIP geometry) the locl fields re uniform y symmetry nd equl to the verge pplied field. This llows otining the overll conductivity y summting over ll lyers ccording to µν 1 µν σ = d i σ ij, (18.) d ij where d i is the thickness of lyer i nd d is the totl film thickness. The non-locl conductivity cn e found ccording to the Kuo-Greenwood formul µν! e µ ν σ ij = dr dr v Im G( r, r ) v Im G( r, r) πω, (18.3) i Ωi Ω j 7 71

38 where the integrtion is performed over the tomic cells t sites i nd j, Ω i nd Ω j eing the volumes of the corresponding cells. The ngulr rckets denote the configurtionl verge, which is introduced s phenomenologicl imginry prt of the self-energy in the Green s function G ( r, r ). In order to clculte G ( r, r ) Butler et l. use the KKR formlism, which explicitly constructs the Green s function in terms of the locl solutions of the Schrödinger eqution nd the scttering pth opertor of multiple scttering theory (see ref.188 nd references therein). These re determined y the tomic potentils which re otined self-consistently y using the locl spin density pproximtion to density functionl theory. The non-locl lyer-dependent-conductivity is useful for etter understnding of the mechnism of mgnetoresistnce. An exmple of the clcultion of the non-locl conductivity for Co 7 /Cu 1 /Co 7 trilyer 188 is shown in Fig.36. In this clcultion the phenomenologicl life-times re constrined to provide typicl experimentl resistivities for sputtered Cu nd Co films. It is ssumed tht the scttering rte for the minority crriers in the ulk Co lyers is 7 times tht for the mjority crries, reflecting the difference in the density of sttes in ulk Co. Due to intermixing t the interfces the scttering rte of the minority spins t the interfcil lyer is set 1 times tht for the minority spins, the fctor of 1 eing sed on coherent potentil pproximtion clcultions of the resistivity due to Cu impurities in Co nd Co impurities in Cu. In ddition, strong spin-independent scttering t the outer oundries of the trilyer is included. Fig.36 σ ij (1 15 s -1 ) σ ij (1 15 s -1 ) Co j i σ ij (1 15 s -1 ) Cu c ij (1 15 s -1 ) Co i Cu Co Co j Non-locl lyer-dependent conductivities (in units 1 15 s -1 =1113Ω -1 cm -1 ) for the Co 7 /Cu 1 /Co 7 (111) trilyer: () mjority chnnel for the prllel (P) lignment of mgnetiztions, () minority chnnel for the P lignment, (c) spin chnnel, which is loclly mjority in the left side Co lyer, for the ntiprllel (AP) lignment, nd (d) gint mgnetoconductnce, which is defined s the difference etween the non-locl conductivities for the P nd AP lignments. The tomic lyers re leled y indices i nd j. After Butler et l. 188 As is ovious from Fig.36 the conductivity is strongly non-locl, which is reflected in lrge vlues of σ ij for i j especilly within the Cu lyers due to long men free pth in Cu. A comprison of Fig.36 with Fig.36 shows tht the conductivities re more locl for the minority crriers thn for the mjority crriers in Co ecuse of higher DOS nd consequently shorter men free pth for the former. As is seen from Figs.36,, nd c, for oth prllel nd ntiprllel lignments the conductivities hve tendency to e confined within the Cu or Co lyers displying 7 Co i i Cu Co j j d decrese t the interfces. This difficulty in pssing etween lyers results from the significnt proility for reflection t the interfces s consequence of the potentil steps creted y the potentil mismtch etween Co nd Cu. As is evident from Fig.36d, lrge contriution to GMR comes from the Cu spcer lyer, which is consequence of electron chnneling nd will e discussed elow. Another contriution to GMR comes from the Co lyers. This contriution rises from electrons tht re ccelerted in one Co lyer, trvel through the Cu lyer to the other Co lyer, nd contriute to the current there. These show up in Fig.36d s the contriutions in the ner nd fr corners, which correspond to i nd j eing in the Co lyers on opposite sides of the Cu spcer lyer. The lyer-dependent conductivities (not shown), which re clculted from the non-locl conductivities y σ i = σ j ij, re qulittively similr to those clculted within the tight-inding model 183 (see Fig.34). The presence of potentil steps t the interfces results in contriution to GMR, which is distinct from spin-dependent scttering. This contriution origintes from electron chnneling s hs een demonstrted within free-electron models. 114,135,136 Chnneling cn occur for the current prllel to the plnes if lrge portion of the electrons in lyer is speculrly reflected from oth its interfces. If the scttering rte in tht lyer is lower thn it is in neighoring lyers, the reflected electrons in tht lyer see lower effective scttering rte thn they would in the sence of reflection. In mgnetic multilyers, chnneling contriutes to GMR due to the electrons with prllel moment which re strongly reflected for one spin, ut not the other. In this cse, chnneling does not occur for electrons of either spin in the ntiferromgnetic lignment ecuse oth trnsmit through one or the other interfce. On the other hnd, for the ferromgnetic lignment, electrons of one spin re confined to the lyer, nd if tht lyer hs lower scttering rte, these electrons cuse short circuit effect, therey giving contriution to GMR. 189 Using first-principle clcultions Butler et l. 19 hve shown tht chnneling cn contriute to GMR in Co/Cu/Co spin vlves. The mjority-spin electrons in Co hve lower Fermi momentum thn electrons in Cu. This leds to the totl internl reflection of those electrons whose momentum prllel to the interfce exceeds the Fermi momentum of Co. These reflected electrons re, therefore, confined to the Cu lyers, nd, if the scttering rtes re significntly lower in the Cu tht in the Co, they give lrge contriution to the mjority current nd lrge contriution to GMR. This contriution to GMR shows up in Fig.36d s contriution to σ ij tht is confined to the region ner the center of the ij-plne where oth i nd j lel Cu tomic lyers. The importnce of chnnelling is nicely illustrted y Stiles 189 who clculted the trnsmission proilities cross vrious interfces. In this clcultion the time-independent scttering sttes re determined y reking spce up into lyers. 191 The potentil is computed for ech lyer from ulk electronic structure clcultion using the linerized-ugmented-plne-wve method. Generlized Bloch sttes for lyer re computed from the potentil in the lyer. Generlized Bloch sttes re relted to Bloch sttes y llowing the component of the wve vector norml to the interfce to e complex. They form complete set of sttes, which includes the usul Bloch sttes nd ll evnescent sttes, nd consequently descrie ny time-independent solution of the Schrödinger eqution for ritrry oundry conditions. The generlized Bloch sttes for the two mterils re mtched together cross the interfce to construct the electron scttering sttes, giving the reflection nd trnsmission mplitudes directly. Fig.37 shows the results of the clcultions for trnsmission cross Cu/Co(1) interfce. 189 The mjority Fermi surfce in Co is similr to the Fermi surfce in Cu, ut is smller. This similrity leds to the lmost complete trnsmission of the mjority-spin electrons from Co into Cu (the left ottom pnel in Fig.37). The smller size of the mjority Co Fermi surfce leds to the complete reflection for the sttes in Cu with the lrgest moment prllel to the interfce (the right top pnel in Fig.37). These electrons contriute to chnneling within the Cu lyer in the Co/Cu/Co trilyer when mgnetiztions of the Co lyers re prllel. 19 Although the trnsmission of the minority-spin electrons is not simply chrcterized (due to the complicted nture of the minority Fermi surfces in 73

39 Co), it is seen from the right top pnel in Fig.37 tht the sme Cu sttes trnsmit much etter into the minority Co sttes. Chnneling y these electrons cn give, therefore, contriution to GMR if the scttering rte in Cu is much smller thn it is in Co. Fig.37 From Co into Cu From Cu into Co k y (nm -1 ) k y (nm -1 ) Mjority Minority k x (nm -1 ) k x (nm -1 ) Trnsmission proility The degree of chnneling which is oserved in rel experiments depends strongly on the detiled nture of the interfces etween the Co nd Cu lyers. Interfcil roughness cn reduce or eliminte the contriution of chnneling to GMR. 19 As we hve seen in the previous section, strong disorder (or mgneticlly-ded Co lyer) t the interfces lso results in sizle reduction of the contriution to GMR from the spcer lyer (the dshed line in Fig.34). Evidence of chnneling in Fe/Au(1) multilyer is found in experiments y Dekdjevi et l., 19 which is consistent with the theoreticl prediction of strong reflection of electrons propgting in Au from Fe/Au interfces. 189 Using the model developed in ref.188, Butler et l. 193 hve performed clcultions of the conductnce nd mgnetoconductnce in T/Co/Cu/Co/FeMn/T spins vlves nd compred results with experimentlly mesured vlues s function of the Co lyer thickness. In the clcultions, the conductnce of the T nd FeMn lyers is neglected ecuse their resistivities re very high, greter tht 1µΩcm. The interfce etween the Co nd the T nd etween the Co nd the FeMn is modeled y n tomic lyer of Co with very high scttering rte. The scttering rtes within the Co nd Cu lyers re djusted phenomenologiclly to e consistent with experimentlly-mesured resistivities of thick films. Vrious models for scttering re considered, such s spin-independent scttering, ulk scttering with different spin symmetry in the scttering rtes, interfcil spindependent scttering nd oth ulk nd interfce scttering. Butler et l. find tht the mgnetoconductnce nd its dependence on the Co lyer thickness re consistent with clcultions tht include ulk spin-dependent scttering within the Co lyers nd some chnneling of electrons in 74 k y (nm -1 ) k y (nm -1 ) k x (nm -1 ) k x (nm -1 ) Spin-dependent trnsmission proilities for the Co/Cu(1) interfce. The trnsmission proilities re shown for vrious points on the Fermi surfce projected onto the interfce Brillouin zone. For minority Co, ech sheet of the Fermi surfce is shown in frction of the Brillouin zone. After Stiles. 189 the Cu lyer. They conclude tht it is not necessry to invoke dditionl diffuse spin-dependent scttering t the interfces to explin the experimentl dt. This is due to the strong spin-dependence of the trnsmission nd reflection coefficients t the interfces, which follow from the electronic nd structures of Co nd Cu. This conclusion is consistent with the prediction of the tight-inding model y Tsyml nd Pettifor. 179,183 All the models considered ove ssume electricl trnsport in prllel within the two spinconduction chnnels, the totl conductivity eing the sum of the spin conductivities. Strictly speking this is not the cse due to the spin-orit interction. The spin-orit interction is reltivistic effect, which couples the electronic nd the spin degrees of freedom nd therefore mkes the two-current model inpplicle. Although the spin-orit interction is reltively smll in the trnsition 3d metls nd does not effect strongly the electronic structure of these metls, it cn ffect trnsport properties of ulk lloys, for exmple, voiding the short circuits tht cn rise in either spin chnnel (e.g., ref.174). It is questionle, however, whether this effect is importnt for the description of GMR in 3d-metl multilyers, ecuse the short circuit effects, which pper if the intermixing t the interfces is the only mechnism of scttering (see section 16), re voided due to structurl disorder nd phonon scttering in rel systems. The spin-orit interction might e importnt when hevy elements like gold or pltinum re used s spcer lyer or introduced s impurities. Fig.38 ρ (µωcm) GMR (%) P AP Cu thickness (ML) The spin-orit coupling is explicitly tken into ccount in fully-reltivistic clcultion of GMR in Co/Cu multilyers y Bls et l. 194 They use screened KKR method dpted to lyered systems nd single-site coherent-potentil pproximtion (CPA) to incorporte the effect of intermixing t the interfces on electricl trnsport. Within their pproch mgnetic multilyer is emedded etween two semiinfinite sustrtes, which mkes the system non-periodic in the direction 75 GMR (%) c Concentrtion Resistivity () nd GMR () of disorder-free Co 3 /Cu(d Cu )/Co 3 (1) trilyer emedded in semiinfinite Cu(1) sustrtes for prllel (P) nd ntiprllel (AP) mgnetiztions s function of the Cu lyer thickness d Cu nd GMR (c) of Co 5 /Co 1-x Cu x /Co x Cu 1-x /Cu 7 / Co x Cu 1-x /Co 1-x Cu x /Co 6 (1) trilyer versus concentrtion x of interdiffused Co nd Cu t the Co/Cu interfces. After Bls et l. 194