Spin Injection and Detection in Magnetic Nanostructures
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a r X i v :c o n d -m a t /0212317v 1 [c o n d -m a t .m e s -h a l l ] 13 D e c 2002Spin Injection and Detection in Magnetic NanostructuresS.Takahashi and S.MaekawaInstitute for Materials Research,Tohoku University,Sendai 980-8577,Japan(August 21,2002)We study theoretically the spin transport in a nonmagnetic metal connected to ferromagnetic injector and detector electrodes.We derive a general expression for the spin accumulation signal which covers from the metallic to the tunneling regime.This enables us to discuss recent controversy on spin injection and detection experiments.Extending the result to a superconducting device,we find that the spin accumulation signal is strongly enhanced by opening of the superconducting gap since a gapped superconductor is a low carrier system for spin transport but not for charge.The enhancement is also expected in semiconductor devices.PACS numbers:72.25Ba,72.25.Hg,72.25.Mk,73.40.GkThere has been considerable interest recently in spin transport in magnetic nanostructures.1The spin polar-ized electrons injected from ferromagnets (F)into non-magnetic materials (N)such as a normal metal,semicon-ductor,and superconductor create nonequilibrium spin accumulation in N.2,3,4,5,6,7,8,9,10,11,12The efficient spin injection,accumulation,and transport are central is-sues to be explored in manipulating the spin degree of freedom of the electron.Johnson and Silsbee 3have demonstrated that the injected spins penetrate into N over the spin-diffusion length of µm scale using the spin injection and detection techniques in F1/N/F2trilayer structures.Very recently,Jedema et al.have made a permalloy/copper/permalloy (Py/Cu/Py)structure 4and observed spin accumulation at room temperature.Subsequently,they have shown that the efficiency of spin injection and accumulation is greatly improved in a cobalt/aluminum/cobalt (Co/I/Al/I/Co)structure with tunnel barriers (I).5In this paper,we study the spin injection and detec-tion in a device of F1/N/F2structure by taking into account the geometry of nonlocal measurement.4,5By proper modeling of the system in the diffusive transport regime,we derive an analytical expression for the spin accumulation signal which covers from the metallic to the tunnel regime.A controversial issue on the analysis of spin accumulation has been raised in the structures of metallic contacts.13We discuss the issue based on the present analytical expression.Extending the result to the device containing a superconductor,we find that the spin signal is greatly enhanced by opening of superconducting rge spin signals are also expected in semiconduc-tor devices.We consider a spin injection and detection device con-sisting of a nonmagnetic metal N connected to ferromag-nets of injector F1and detector F2as shown in Fig.1.F1and F2are the same ferromagnetic films of width w F and thickness d F ,and are separated by distance L .N is a normal metal film of width w N and thickness d N .The magnetizations of F1and F2are aligned either parallel or antiparallel.Since the spin-diffusion length λN of N(λCu ∼1µm,4λAl >∼1µm3,5,6)is much larger than the length λF of F (λPy ∼5nm 14),we consider the devicehaving dimensions of λF ≪(d N ,d F )≪(w N ,w F )≪λN .This situation,which corresponds to the experimental geometry,4,5facilitates a description for the spin and charge transport in the device.The electrical current j σfor spin channel σis driven by the electric field E and the gradient of the carrier density deviation δn σfrom equilibrium:j σ=σσE −eD σ∇δn σ,where σσand D σare the electrical conductivity and the diffusion constant.Making use of δn σ=N σδǫσand σσ=e 2N σD σ(N σis the density of states in the spin sub-band and δǫσis the shift in the chemical potential of car-riers from its equilibrium value)gives j σ=−(σσ/e )∇µσ,where µσ=ǫσ+eφis the electrochemical potential (ECP)and φthe electric potential.The continuity equations for charge and spin in the steady state are ∇·(j ↑+j ↓)=0and ∇·(j ↑−j ↓)=−eδn ↑/τ↑↓+eδn ↓/τ↓↑,where τσσ′is the scattering time of an electron from spin state σto σ′.Using these equations and detailed balanc-ing N ↑/τ↑↓=N ↓/τ↓↑,one obtains 3,15,16,17,18,19∇2(σ↑µ↑+σ↓µ↓)=0,(1)∇2(µ↑−µ↓)=λ−2(µ↑−µ↓),(2)with the spin-diffusion length λ=√2(τ−1↑↓+τ−1↓↑)and D−1=(N ↑D −1↓+N ↓D −1↑)/(N ↑+N ↓).The material parameters in N are spin-independent :FIG.1.Basic structure of a spin injection and detection device.The bias current I flows from F1to the left side of N.The spin accumulation at distance L is detected by measuring the spin-dependent voltage V 2between F2and N.σ↑N=σ↓N=1λN2i=1 P J R i1−P2J+p F R F1−p2F× 2 i=1 1+2R i1−P2J+2R F1−p2F −e−2L(1−p2F)2R NR F1−e−2L/λN.(4)When one of the junctions is a transparent contact andthe other is a tunnel junction,i.e.,(R1≪R F≪R N≪R2)or(R2≪R F≪R N≪R1),we haveR s=2p F P JR Ne−L/λN.(5)When bothjunctions are tunneling junctions(R1,R2≫R N),we have3,5R s=P2J R N e−L/λN.(6) Note that R s in the above limiting cases is indepen-dent of R i.Equations(4)-(6)indicate that the resis-tance mismatch factor(R F/R N)is removed systemati-cally when a transparent contact is replaced with a tun-nel junction.19,20,21,22Thus,the maximum spin signal is achieved when all the junctions are tunnel junctions. Figure2show the spin accumulation signal R s in Eqs.(4)-(6)for p F=0.73,14P J=0.4,23and R F/R N= 10−2.4We see that R s increases by one order of mag-nitude by replacing a transparent contact with a tunnel barrier.The value R N=3Ω25taken from the Py/Cu/Py structure yields R s=1mΩat L=λN.If one takes into account the cross-shaped Cu,4one expects one-third of the above value,which is in reasonable agreement with the experimental value0.1mΩ.4In the Co/I/Al/I/Co structure,R N=3Ωis estimated26and R s=100mΩis obtained at L=λN,which is10times larger than the experimental value10mΩ.5This discrepancy may be attributed to the reduction in P J due to the spin-flip scattering at the barriers.17,19A question arises on whether the contacts of metal-lic F1/N/F2structures is transparent(R i≪R F)4or tunneling-like(R i≫R N).13,24The experimental values123L/λN10– 410– 310– 210– 1R s /RNp F =0.73P J = 0.4R F /R N = 0.01 F/N/FF/I/N/F, F/N/I/F F/I/N/I/F FIG.2.Spin accumulation signal R s vs distance L betweenF1and F2.Solid line:F/I/N/I/F.Long-dashed line:F/I/N/F and F/N/I/F.Short-dashed line:F/N/F.of Py/Cu (R i A J ∼5×10−12Ωcm 2,14ρF ∼10−5Ωcm,4and λF ∼5nm 14)yields R i ∼R F ,which is strictly speak-ing neither transparent or tunneling-like.However,the values of R s for R i =R F calculated from Eq.(3)are about 2times larger than those for the transparent case in Fig.2,indicating that the Py/Cu/Py structure lies on the verge of transparent regime.However,depending on sample fabrication processes,there will be cases that belong to the intermediate regime (R F ≪R i ≪R N ),for which one should useR s =4P 2J R 2Ne −L/λN2eP J R N I ,is smaller than ∆,i.e.,I <2∆/(eP J R N ),for which the suppression of ∆due to spin accumulation can be neglected.27We also neglect charge imbalance created by injection of QP charge into S,which originates from the conversion of injected QPs into condensate,and produces the excess voltage due to charge accumulation at F2.28However,the effect is spin-independent and does not contribute to R s .In the superconducting state,the equation for the spin splitting (µ↑−µ↓)is the same as Eq.(2)with λN in the normal-state,29which is intuitively understood as fol-lows.Since the dispersion curve of the QP excitation en-ergy is given by E k =3˜v2k ˜τimp and ˜τ−1imp =σ′˜τ−1σσ′results in λS =2(1+ǫk /E k ).Using thegolden rule formula,31we can calculate (∂S i /∂t )sf and obtain I s i =[2f 0(∆)/(e R N )]a i ,where 2f 0(∆)represents the QP populations and f 0(∆)=1/[exp(∆/k B T )+1].From the matching condition of the spin currents across the barriers,we obtain the spin signal R s in the superconducting stateR s =V s /I =12f 0(∆)P 2JR N2f −10(∆)below T c .Note that the T -dependent factor in Eq.(9)is0.51T/T c1S p i n s i g n a lV s /V54326V s /IFIG. 3.Temperature dependence of the spin signals R s =V s /I and V s /V in a F/I/S/I/F structure.The values of the spin signal are normalized to those at the superconducting critical temperature T c .the same as that in the spin-relaxation time τs =[χs (T )/2f 0(∆)]τsf ,34which is derived from (∂S /∂t )sf =−S /τs .Figure 3shows the temperature dependence of R s =V s /I and V s /V .The values are normalized to those at T c .The strong increase of V s /I reflects the T -dependence of the resistivity of the spin current below T c .The signal V s /V increases with the same T -dependence as τs (T ),indicating that the spin-relaxation time in S is directly obtained by measuring V s vs T at constant V .To test these predictions,it is highly desirable to measure V s of Co/I/Al/I/Co structures 5by lowering T below T c .A large enhancement of spin signals is also expected in degenerate semiconductors,because the resistivity is much larger compared with normal metals and the spin-diffusion length is relatively long.In degenerate semicon-ductors,the spin current is given by j s =−µm n c ∇x (µ↑−µ↓),where µm is the mobility and n c the carrier con-centration.For Si-doped GaAs with n c =1018cm −3and µm =2×103cm 2/Vs at room temperature,35ρN =1/(eµm n c )=0.1Ωcm.For (Mn,Ga)As,ρF =0.01∼0.1Ωcm.36It follows from Eqs.(5)and (6)that R s ∝ρF for a (Ga,Mn)As/I/GaAs(n -type)/(Ga,Mn)As device and R s ∝ρN for a F1/I/GaAs(n -type)/I/F2de-vice.Therefore,we expect that R s is larger by several orders of magnitude than that of metal case.This result is promising for applications for spintronic devices.In summary,we have studied the spin injection and de-tection in the F1/N/F2structure,and derived an expres-sion for the spin accumulation signal which covers from the metallic to the tunneling regime.This enables us to resolve the recent controversy of spin injection and de-tection experiments.Extending the result to a supercon-ducting device,we have found that the signal is strongly enhanced below T c ,because superconductors become a low carrier system for spin transport by opening of the gap and a larger spin splitting is required for carrying the same spin current.Our finding can be tested in su-perconducting devices such as Co/I/Al/I/Co by loweringtemperature below T c .A large spin accumulation signal is also expected in semiconductor devices.This work is supported by a Grant-in-Aid for Scientific Research from MEXT and CREST.√。
