Spintronics(自旋电子学)(7)

?s?

1

2q

N↑?N↓

N↑N↓

?s.(22)

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

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

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

35For this simple geometry various resistances have a com-

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

36More generally,for a noncollinear magnetization,j

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

using Poisson’s equation or a condition of the local

charge quasineutrality.41

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

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

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

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

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

In contrast to normal metals(Johnson and Silsbee,

1985,1988d)and superconductors,for which injection

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

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

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

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

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

42E.I.Rashba(2002d).

43The substitutions are P

j

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

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

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

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

335

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

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

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

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

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

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

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