曾经的曾经For spin pumping in an n-doped sample,where n ?n0and p?p0,Eqs.(9)and(11)give(D’yakonov and Perel’,1971b)
P n?P n?t?0?/?1?n0/G?s?.(13) In contrast to the previous case,the carrier(now hole) lifetime??1/rn0has no effect on P n.However,P n de-pends on the photoexcitation intensity G,as expected for a pumping process.The effective carrier lifetime is ?J?n0/G,where J represents the intensity of the illu-minating light.If it is comparable to or shorter than?s, spin pumping is very effective.Spin pumping works be-cause the photoexcited spin-polarized electrons do not need to recombine with holes.There are plenty of un-polarized electrons in the conduction band available for recombination.The spin is thus pumped in to the elec-tron system.
29After the illumination is switched off,the electron spin den-sity,or equivalently the nonequilibrium magnetization,will de-crease exponentially with the inverse time constant1/T s?1/??1/?s(Parsons,1969).
30The effect of a?nite length for the light absorption on P
n
is discussed by Pierce and Celotta(1984).The absorption length ??1is typically a micron for GaAs.It varies with frequency roughly as?(??)?(???E g)1/2(Pankove,1971).
TABLE I.Angular and spin part of the wave function at?.
Symmetry?J,m j?Wave function
?6?1/2,1/2??S↑?
?1/2,?1/2??S↓?
?7?1/2,1/2???(1/3)1/2?(X?iY)↓?Z↑??
?1/2,?1/2??(1/3)1/2?(X?iY)↑?Z↓??
?8?3/2,3/2??(1/2)1/2(X?iY)↑?
?3/2,1/2??(1/6)1/2?(X?iY)↓?2Z↑??
?3/2,?1/2???(1/6)1/2?(X?iY)↑?2Z↓??
?3/2,?3/2??(1/2)1/2(X?iY)↓?
332Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004
When magnetic?eld B is applied perpendicular to the axis of spin orientation(transverse magnetic?eld),it will induce spin precession with the Larmor frequency ?L??B gB/?,where?B is the Bohr magneton and g is
the electron g factor.31The spin precession,together with the random character of carrier generation or dif-fusion,leads to the spin dephasing(see Sec.III.A.1). Consider spins excited by circularly polarized light(or by any means of spin injection)at a steady rate.In a steady state a balance between nonequilibrium spin gen-eration and spin relaxation is maintained,resulting in a net magnetization.If a transverse magnetic?eld is ap-plied,the decrease of the steady-state magnetization can have two sources:(a)spins which were excited at ran-dom time and(b)random diffusion of spins towards a detection region.Consequently,spins precess along the applied?eld acquiring random phases relative to those which were excited or have arrived at different times.As a result,the projection of the electron spin along the
exciting beam will decrease with the increase of trans-
verse magnetic?eld,leading to depolarization of the lu-
minescence.This is also known as the Hanle effect
(Hanle,1924),in analogy to the depolarization of the
resonance?uorescence of gases.The Hanle effect was
?rst measured in semiconductors by Parsons(1969).The
steady-state spin polarization of the precessing electron
spin can be calculated by solving the Bloch-Torrey equa-
tions(Bloch,1946;Torrey,1956),Eqs.(52)–(54)describ-
ing the spin dynamics of diffusing carriers.
In p-doped semiconductors the Hanle curve shows a
Lorentzian decrease of the polarization(Parsons,1969), P n(B)?P n(B?0)/(1??L T s)2,where P n(B?0)is the polarization at B?0from Eq.(12)and T s?1is the effec-tive spin lifetime given by1/T s?1/??1/?s;see footnote 29.Measurements of the Hanle curve in GaAlAs were
used by Garbuzov et al.(1971)to separately determine both?and?s at various temperatures.The theory of the Hanle effect in n-doped semiconductors was developed by D’yakonov and Perel’(1976),who showed the non-Lorentzian decay of the luminescence for the regimes of both low(?J/?s?1)and high(?J/?s?1)intensity of the exciting light.At high?elds P n(B)?1/B1/2,consistent with the experiments of Vekua et al.(1976)in Ga0.8Al0.2As,showing a Hanle curve different from the usual P n(B)?1/B2Lorentzian behavior(D’yakonov and Perel’,1984a).Recent?ndings on the Hanle effect in nonuniformly doped GaAs and reanalysis of some earlier studies are given by Dzhioev et al.(2003).
C.Theories of spin injection
Reviews on spin injection have covered materials ranging from semiconductors to high-temperature super-conductors and have addressed the implications for de-vice operation as well as for fundamental studies in solid-state systems.32In addition to degenerate conduc-tors,examined in these works,we also give results for nondegenerate semiconductors in which the violation of local charge neutrality,electric?elds,and carrier band bending require solving the Poisson equation.The nota-tion introduced here emphasizes the importance of dif-ferent(and inequivalent)spin polarizations arising in spin injection.
1.F/N junction
A theory of spin injection across a ferromagnet/ normal metal(F/N)interface was?rst offered by Aronov(1976b).Early work also included spin injection into a semiconductor(Sm;Aronov and Pikus,1976; Masterov and Makovskii,1979)and a superconductor (S;Aronov1976a).Spin injection in F/N junctions was subsequently studied in detail by Johnson and Silsbee (1987,1988a),33van Son et al.(1987),Valet and Fert (1993),Hersh?eld and Zhao(1997),and others.Here we follow the approach of Rashba(2000,2002b)and con-sider a steady-state34?ow of electrons along the x direc-tion in a three-dimensional(3D)geometry consisting of a metallic ferromagnet(region x?0)and a paramag-netic metal or a degenerate semiconductor(region x ?0).
The two regions,F and N,form a contact at x?0,as depicted in Fig.7.The relative magnitudes of three char-
31In our convention the g factor of free electrons is positive, g0?2.0023(Kittel,1996).
32See,for example,Osofsky(2000);Goldman et al.(1999, 2001);Johnson(2001,2002a);Maekawa et al.(2001);Jedema, Nijboer,et al.(2002);Schmidt and Molenkamp(2002);Tang et al.(2002);and Wei(2002).
33Johnson and Silsbee base their approach on irreversible thermodynamics and consider also the effects of a temperature gradient on spin-polarized transport,omitted in this section. 34Even some dc spin injection experiments are actually per-formed at low(audio-frequency)bias.Generalization to ac spin injection,with a harmonic time dependence,was studied by Rashba
(2002a).
FIG.7.(Color in online edition)Spatial variation of the elec-trochemical potentials near a spin-selective resistive interface at an F/N junction.At the interface x?0both the spin-resolved electrochemical potentials(??,??↑,↓,denoted with solid lines)and the average electrochemical potential(?F,?N,dashed lines)are discontinuous.The spin diffusion lengths L sF and L sN characterize the decay of?s??↑??↓(or equivalently the decay of spin accumulation and the nonequi-librium magnetization)away from the interface and into the bulk F and N regions,respectively.
333
Zˇutic′,Fabian,and Das Sarma:Spintronics:Fundamentals and applications Rev.Mod.Phys.,Vol.76,No.2,April2004
acteristic resistances per unit area35determine the de-gree of current polarization injected into a nonmagnetic material.These are the contact resistance r c and the two characteristic resistances r N and r F,each given by the ratio of the spin diffusion length and the effective bulk conductivity in the corresponding region.Two limiting cases correspond to the transparent limit,where r c→0, and the low-transmission limit,where r c?r N,r F.
Spin-resolved quantities are labeled by??1or↑for spin up,???1or↓for spin down along the chosen quantization axis.For a free electron,spin angular mo-mentum and magnetic moment are in opposite direc-tions,and what precisely is denoted by‘‘spin up’’varies in the literature(Jonker et al.,2003).Conventionally,in metallic systems(Tedrow and Meservey,1973;Gijs and Bauer,1997),spin up refers to carriers with majority spin.This means that the spin(angular momentum)of such carriers is antiparallel to the magnetization.Spin-resolved charge current(density)in a diffusive regime can be expressed as
j???????,(14) where??is conductivity and the electrochemical poten-tial is
????qD?/????n???,(15) with q proton charge,D?diffusion coef?cient,?n??n??n?0the change of electron density from the equilib-rium value for spin?,and?the electric potential.36
In the steady state the continuity equation is ?j???q??n???????n???????,(16)
and????is the average time for?ipping a?spin to a??spin.For a degenerate conductor37the Einstein relation is
???q2N?D?,(17) where???↑??↓and N?N↑?N↓is the density of d72efe37f111f18583d05a9ding a detailed balance N↑/?↑↓?N↓/?↓↑(Hersh-?eld and Zhao,1997;Kravchenko,2002)together with Eqs.(15)and(17),the continuity equation can be ex-pressed as
?j???q2
N↑N↓
N↑?N↓
??????
?s,(18)
where?s??↑↓?↓↑/(?↑↓??↓↑)is the spin relaxation time. Equation(18)implies the conservation of charge current j?j↑?j↓?const,while the spin counterpart,the differ-ence of the spin-polarized currents j s?j↑?j↓is position dependent.Other‘‘spin quantities,’’X s,unless explicitly de?ned,are analogously expressed with the correspond-ing(spin)polarization given by P X?X s/X.For ex-ample,the current polarization38P j?j s/j,generally dif-ferent from the density polarization P n?(n↑?n↓)/n,is related to the conductivity polarization P?as
P j?2??↑?↓/????s/j?P?,(19) where?s??↑??↓.In terms of the average electro-chemical potential??(?↑??↓)/2,P?further satis?es ????P???s/2?j/?.(20) From Eqs.(15)and(18)it follows that?s satis?es the diffusion equation(van Son et al.,1987;Valet and Fert, 1993;Hersh?eld and Zhao,1997;Schmidt et al.,2000)?2?s??s/L s2,(21) where the spin diffusion length is L s?(Dˉ?s)1/2with the spin-averaged diffusion coef?cient Dˉ?(?↓D↑??↑D↓)/??N(N↓/D↑?N↑/D↓)?d72efe37f111f18583d05a9ding Eq.(15)and the local charge quasineutrality?n↑??n↓?0shows that ?s is proportional to the nonequilibrium spin density ?s??n↑??n↓(s?s0??s?n↑?n↓),
