曾经的曾经will be a back?ow of the nonequilibrium spin back into
the F region.Because both spin and charge are carried
by electrons in spin-charge coupling,the back?ow of
spin driven by diffusion creates an additional resistance
for the charge?ow across the F/N interface.Based on an
analogy with the charge transport across a clean
N/superconductor(S)interface(see Sec.IV.A.3),van Son et al.(1987)explained?R by invoking the conse-quences of current conversion from spin-polarized,far
to the left of the F/N interface,to completely unpolar-
ized,at far right in the N region.
The increase in the total resistance with spin injection
can be most dramatic if the N region is taken to be a
superconductor(S);see Fig.5(c).Spin injection depletes
the superconducting condensate and can result in
switching to a normal state of much higher resistance
(Dong et al.,1997;Vas’ko et al.,1997;Takahashi et al.,
1999;Wei et al.,1999;Yeh et al.,1999).A critical review
of possible spurious effects in reported experiments
(Gim et al.,2001)has also stimulated the development
of a novel detection technique which uses scanning tun-
neling spectroscopy combined with pulsed quasiparticle
spin injection to minimize Joule heating(Ngai et al.,
2004;see Sec.IV.A.1).In the S region the quasiparticle energy is E k?(?k2??2)1/2,where?k is the single-particle excitation energy corresponding to the wave vector k and?is the superconducting gap[see Fig.5(c)].Such a dispersion relation results in a smaller diffusion coef?-cient and a longer spin-?ip time than in the N region, while their product,the spin diffusion length,remains the same(Yamashita et al.,2002).Consequently,Eq.
(21)also applies to the diffusive spin-polarized transport
and spin accumulation in ferromagnet/superconductor
structures(Jedema et al.,1999;Yamashita et al.,2002).
Opening of a superconducting gap implies that a super-
conductor is a low carrier system for spin,which is car-
ried by quasiparticles(Takahashi and Maekawa,2003).
In the preceding analysis,appropriate for bulk,homo-
geneous,three-dimensional N and F regions and degen-
erate(semi)conductors,Poisson’s equation was not in-voked and the local charge neutrality?n↑??n↓was used only to derive Eq.(22).48Focusing on bulk samples in which both the size of the F and N regions and the cor-responding spin diffusion lengths are much larger than the Debye screening length,one can?nd that the quasineutrality condition,combined with Eqs.(15)and (17),yields
46A similar result was stated previously by Johnson and Sils-
bee(1988a).
47While most of the schemes resemble a CPP geometry[Fig. 3(b)],there are also proposals for generating highly polarized currents in a CIP-like geometry[Fig.3(a)](Gurzhi et al.,2001, 2003).
48For spin injection in nondegenerate semiconductors(with the carriers obeying the Boltzmann statistics)there can be large effects due to built-in?elds and deviation from local charge neutrality,as discussed in Sec.II.C.3.
336Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004
?????P N?s/2,(33) where the density-of-states spin polarization of P N van-ishes in the N region.At the contact x?0there is a potential drop,even when r c?0,which can be evaluated from Eqs.(26)and(33)as
?N?0???F?0???r c?1?P?P j?j?P N F?0??sF?0?/2.
(34) The creation of nonequilibrium spin in the N region re-sults in the spin emf in the F/N structure which can be used to detect electrical spin injection,as depicted in Fig.5.Within a simpli?ed semi-in?nite geometry for the F and N regions,we consider an effect of spin pumping in the N region,realized either by electrical spin injec-tion from another F region[as shown in Fig.5(b)]or by optical pumping(see Sec.II.B).The resulting potential drop can be calculated by modifying?sN in Eq.(28),?sN??sN??????sN?0???sN????e?x/L sN,(35) where?sN(?)represents the effect of homogeneous spin pumping in the N region.To calculate the open circuit voltage(j?0)the continuity of spin current at x?0should be combined with the fact that P j j?j s. From Eq.(19)it follows that
j s?0??2?↑?↓
?F
?sF?0?
L sF
??
1
2
?N
?sN?0???sN???
L sN
,
(36)
while the discontinuity of?s in Eq.(25)yields49?sF?0???r F/r FN??sN???,j s?0???sN???/2r FN,
?sN?0????r c?r F?/r FN??sN???.(37) By substituting this solution into Eq.(34),we can evalu-ate the contact potential drop as
?N?0???F?0???r F P N F?r c P???sN???/2r FN.(38) The total potential drop(recall j?0)at the F/N junction50is(Rashba,2002b)
??FN??N?????F?????P j?sN???/2.(39) where P j is given in Eq.(31).In the context of the spin-detection scheme from Fig.5and high impedance mea-surements at the N/F2junction,the spin-coupled voltage V s(Silsbee,1980;Johnson and Silsbee,1985)was also found to be proportional to current polarization and the spin accumulation(?s??s??M;Johnson and Silsbee, 1988b).
2.F/N/F junction
The above analysis of the F/N bilayer can be readily extended to the geometry in which two in?nite F regions are separated by an N region of thickness d.The quan-tities pertaining to the two ferromagnets are de?ned as in the case of an F/N junction and labeled by the super-scripts L and R(left and right regions,respectively).It follows from Eq.(19),by assuming the continuity of the spin current at L,R,that the difference of the spin-resolved electrochemical potential,responsible for the spin accumulation,is
?sF L?2r F L?P j L?P?F L?je x/L sF L,x?0,(40)
?sN?2r N?P j R cosh?x/L sN??P j L cosh??d?x?/L sN???j/sinh?d/L sN?,0?x?d,(41)
?sF R??2r F R?P j R?P?F R?je(d?x/L sF R),x?d,(42) where the current spin polarization P j L,R at the two con-tacts in the F/N/F geometry can be expressed(Rashba, 2002b)in terms of the P j calculated for F/N junction with the in?nite F and N regions in Eq.(19)and the appropriate effective resistances.By P j?L,R we denote the P j calculated in Eq.(31)for the left and right contact (with the appropriate parameters for the F/N/F junction) as if it were surrounded by the in?nite F and N regions. Analogously to the F/N junction,the consequence of the spin injection is an increase in the resistance R?R0??R,over the equilibrium value R0?(?L)?1?(?R)?1.The nonequilibrium resistance?R is also al-ways positive for spin-conserving contacts(Rashba,
2000,2002b),in agreement with experiments on all-semicondcutor trilayer structures(Schmidt et al.,2001; see Sec.II.D.3).
Many applications based on magnetic multilayers rely on the spin-valve effect,in which the resistance changes due to the relative orientations of the magnetization in the two F regions.The geometry considered here is rel-evant for CPP giant magnetoresistance(Gijs and Bauer, 1997;Bass and Pratt,1999;Parkin,2002)and the all-metallic spin injection of Johnson and Silsbee(1985).In particular,the resistance change between antiparallel and parallel magnetization orientations in the two ferro-magnets can be expressed using current polarization of an in?nite F/N junction P j?L,R(Rashba,2002b):
?R?R↑↓?R↑↑?4P j?L P j?R
r FN L r FN R r N
D sinh d/L sN,(43) where r F L,R,r c L,R,and r N are de?ned as in the case of an F/N junction and
D??r F L?r c L??r c R?r F R??r N2?r N?r F L?r c L?r c R
?r F R?coth?d/L sN?.(44) Up to a factor of2,Eq.(43)has also been obtained by Hersh?eld and Zhao(1997)using Onsager relations.In the limit of a thin N region,d/L sN→0,?R remains?-nite.In the opposite limit,for d?L sN,
49A missprint in?
sF (0)from Rashba(2002b)has been cor-
rected.
50A similar potential drop was also calculated across a ferro-magnetic domain wall(Dzero et al.,2003).337
Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004
?R?P j?L P j?R exp??d/L sN?.(45) For a symmetric F/N/F junction,where r c,F L?r c,F R,it fol-lows that
?R?4r N?r c P??r F P?F?
D sinh?d/L sN?.(46)
Considering the spin injection from F into a ballistic N region in the presence of diffusive interfacial scattering, where the phase coherence is lost and the Boltzmann equation can be applied,it is instructive to reconsider the effect of contact resistance(Kravchenko and Rashba,2003).We introduce the Sharvin resistance R Sharvin(Sharvin,1965),arising in ballistic transport be-tween two in?nite regions connected by a contact(an ori?ce or a narrow and short constriction)of radius much smaller than the mean free path,a?l.In a3D geometry the resistance is
R Sharvin?
4?l
3?a2
??e2h k2A2???1,(47)
where h/e2?25.81k?is the quantum of resistance per spin,A is the contact area,and k is the Fermi wave vector.The opposite limit,of diffusive transport through the contact with a?l,corresponds to the Maxwell or Drude resistance R Maxwell??/2a.The studies of interme-diate cases provide an interpolation scheme between the R Maxwell and R Sharvin for various ratios of a/l(Wexler, 1966;Jansen et al.,1980;de Jong,1994;Nikolic′and Allen,1999).Following Kravchenko and Rashba(2003) the effective contact resistance r c?r c↑?r c↓(recall that it is de?ned per unit area)is obtained as
r c???4R Sharvin/A??1?t?L?t?R?/t?L,(48) where t?L,R represent the transmission coef?cients for electrons reaching the contact from the left and from the right and satisfy t L?t Rр1.For r c which would exceed the resistance of the N and F bulk regions,the spin injection ef?ciency can attain P j?(r c↑?r c↓)/r c (Kravchenko and Rashba,2003),showing,similarly to the diffusive regime,the importance of the resistive con-tacts to ef?cient spin injection.Connection with the re-sults in the diffusive regime can be obtained (Kravchenko and Rashba,2003)by identifying r c??1/4??,where the contact conductivity??was intro-duced in Eq.(24).
While most of the experimental results on spin injec-tion are feasible in the diffusive regime,there are many theoretical studies treating the ballistic case and phase-coherent transport in both F/N and F/N/F junctions(Hu and Matsuyama,2001;Hu,Nitta,et al.,2001;Mireles and Kirczenow,2001;Matsuyama et al.,2002).Simple models in which the N region is a degenerate semicon-ductor often adopt an approach developed?rst for charge transport in junctions involving superconductors, discussed in Sec.IV.A.3.Considering spin-orbit coupling and the potential scattering at the F/N interface mod-eled by the?function,Hu and Matsuyama(2001)have examined ballistic spin injection in the F/N junction. They show that even a spin-independent barrier can be used to enhance the spin injection and lead to an in-crease in the conductance polarization.First-principles calculations were also used for ballistic spin injection from a ferromagnetic metal into a semiconductor (Mavropoulos et al.,2002;Wunnicke et al.,2002;Zwi-erzycki et al.,2003).In the limit of coherent(specular) scattering51and high interfacial quality it was shown that different band structure in the F and the N regions would contribute to a signi?cant contact resistance and an ef?cient spin injection(Zwierzycki et al.,2003).
