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Spin-injection efficiency and magnetoresistance in a hybrid ferromagnetic–semiconductor trilayer with interfacial barriers
Applied Surface Science 252 (2006) 4003–4008 www.elsevier.com/locate/apsusc
Spin-injection efficiency and magnetoresistance in a hybrid ferromagnetic–semiconductor trilayer with interfacial barriers S. Agrawal a,*, M.B.A. Jalil a, S.G. Tan b, K.L. Teo a, T. Liew a,b a
Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore b Data Storage Institute, DSI Building, 5 Engineering Drive 1, Off Kent Ridge Crescent, National University of Singapore, Singapore 117608, Singapore Available online 21 October 2005
Abstract We present a self-consistent model of spin transport in a ferromagnetic (FM)–semiconductor (SC)–FM trilayer structure with interfacial barriers at the FM–SC boundaries. The SC layer consists of a highly doped n2+ AlGaAs–GaAs 2DEG while the interfacial resistance is modeled as delta potential (d) barriers. The self-consistent scheme combines a ballistic model of spin-dependent transmission across the d-barriers, and a driftdiffusion model within the bulk of the trilayer. The interfacial resistance (RI) values of the two junctions were found to be asymmetric despite the symmetry of the trilayer structure. Transport characteristics such as the asymmetry in RI, spin-injection efficiency and magnetoresistance (MR) are calculated as a function of bulk conductivity ss and spin-diffusion length (SDL) within the SC layer. In general a large ss tends to improve all three characteristics, while a long SDL improves the MR ratio but reduces the spin-injection efficiency. These trends may be explained in terms of conductivity mismatch and spin accumulation either at the interfacial zones or within the bulk of the SC layer. # 2005 Elsevier B.V. All rights reserved. PACS: 72.25.Mk; 73.40.Cg; 73.40.Sx Keywords: Spin-injection efficiency; Spin-diffusion length; Magnetoresistance
1. Introduction Semiconductor (SC)-based spintronics which exploits the spin as well as the charge property of carriers, is a fast growing field of research. The long spin coherence of electrons in SC [1], coupled with the ability to control spin orientation by electrical means [2] has opened the possibility of realizing devices such as the Datta–Das spin field effect transistor [3] (spin-FET). A chief prerequisite of such devices is the ability to inject spin-polarized current into semiconductor materials, which are usually non-magnetic. Initial experimental efforts at spin-injection were not very successful with the resultant spin injection efficiency h of the order of 1% only [4,5]. Furthermore, the signals from such a small spin polarization of the current could possibly be ascribed to other effects such as the local Hall effect in the vicinity of the ferromagnetic spin
* Corresponding author. E-mail address: [email protected] (S. Agrawal). 0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.09.033
injector or detector [6,7] although subsequent experiments by Hammar et al. [8] did account for this spurious local Hall effect. The reasons for such low values of spin-injection efficiencies have also been explained in terms of conductivity mismatch [9]. More recently, a much higher spin injection efficiency h exceeding 30% [10,11] has been achieved for injection from a Fe contact into a AlGaAs–GaAs quantum well structure through a Schottky barrier. In addition, an appreciable spinpolarization value of 13% was obtained, albeit at a low temperature of 2 K, for injection from a Heusler alloy into a AlGaAs/GaAs light-emitting diode [12], which also involves a Schottky contact. The crucial element for these recent successes is the incorporation of interfacial barriers, e.g. tunnel or Schottky barriers between the FM and SC layers, to overcome the conductivity mismatch problem, as was theoretically proposed by Rashba [13]. In this paper we consider a FM–SC–FM trilayer structure with interfacial resistive (RI) barriers at the SC–FM boundaries. The role of RI in determining the overall charge and spin transport has been investigated in previous work [14]. However,
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S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
the characteristics of the SC layer also play a prominent role, owing to the relatively large resistivity of the SC layer (about five orders of magnitude larger than that of the FM contacts). Thus, the main focus of this article is to investigate the effects of changing the bulk-SC conductivity and the spin-diffusion length of the SC region on the spin injection efficiency and overall magnetoresistance (MR) of a FM–SC–FM trilayer. In general, an increase in the conductivity and spin-diffusion length of the SC material improves the spin-injection efficiency and MR of the device by maintaining spin coherence, and reducing the conductivity-mismatch [9] problem, respectively. However, the presence of RI complicates the analysis. For instance, a short SC spin-diffusion length enhances the spinaccumulation effect adjacent to the FM–SC interfaces, which can also improve the spin-injection efficiency. It is thus important to analyze the effects of varying conductivity as well as spin-diffusion lengths of the SC region on the spin transport characteristics of the device. In our model, we determine the interfacial resistances via a self-consistent drift-diffusion and ballistic model. The diffusive part of the model is based on the spin-dependent drift-diffusion (DD) equation. The RI values are evaluated by considering ballistic tunneling transmission across d-function potential barriers at the FM–SC interfaces. The incorporation of interfacial d-barriers makes our model similar to that of Heersche et al. [15]. But the latter assumes fully ballistic transport through the structure and is limited to a single junction. However in this paper, we consider a trilayer structure in which the SC layer thickness w is assumed to be larger than the carrier mean free path (MFP), although it is comparable to the spin-diffusion length in the SC material. It is thus necessary to incorporate a diffusive model to model the charge and spin transport in the SC layer away from the interfacial zones. The trilayer material consists of a highly doped n2+ AlGaAs– GaAs 2DEG (SC) layer sandwiched between two Fe (FM) layers. The d-potential barriers at the two FM–SC interfaces are expressed as U½dðxÞ þ dðx wÞ with the barrier height U being spin-dependent, as was assumed by Yu and Flatte [16] and Smith and Silver [17]. The quantitative description of tunneling at the FM–SC interface is rather complicated because the transport properties are strongly dependent on the potential barrier height and thickness, and are highly sensitive to interfacial roughness and impurity states within the barriers. However, as a first approximation, we ignore any type of electron scattering within the barriers and assume purely ballistic transport through them. 2. Model For transport in the bulk FM and SC layers, we consider the drift-diffusion model, which is based on the following equations: @2 ðm " m # Þ m " m # ¼ @x2 l2
(1)
@mc" ð # Þ @x
@ms" ð # Þ @x
¼
¼
e jc" ð # Þ s c" ð # Þ
e js" ð # Þ s s" ð # Þ
(2a)
(2b)
where l is the spin-diffusion length and e is the magnitude of electronic charge. Subscripts ‘c’ and ‘s’ refer to the FM contact c;s c;s and the 2DEG SC layer. mc;s " ð # Þ , j " ð # Þ and s " ð # Þ are, respectively, the electrochemical potentials, current densities and conductivities for majority (minority) spin electrons, each of which are different in the FM and the SC 2DEG region. Eq. (1) is the diffusion equation while Eqs. (2a) and (2b) are Ohm’s law in FM and 2DEG regions, respectively. In the FM contact, jc and sc are expressed in conventional units of A/cm2 and V1 cm1, respectively, while in the 2DEG js and ss are expressed in A/cm and V1, respectively, so that (jc/sc) and (jc/ss) have the same units. We have assumed no spin-flip scattering at the interface so that j" and j# are continuous at the interfaces. Ideally, the diffusion equation and Ohm’s law [Eqs. (1) and (2)] should be supplemented by Poisson’s equation, which relates m to charge accumulation. However, since the SC layer is a highly doped 2DEG, it is a good approximation to assume charge neutrality i.e. zero charge accumulation. Such an approximation has been made by Zhang et al. [18] in their diffusive transport model where they have dropped the terms involving charge accumulation. In addition, as far as transport equations are concerned, there is a freedom in choosing the solution of the Poisson’s equation according to Stiles and Zangwill [19]. For simplicity, we can thus choose a solution corresponding to zero charge accumulation. At the FM–SC interface, there is a discontinuity in both m" and m#, which is given by
Dm0;w ";# ¼
e jc" ; # GL;R " ; # eq:
(3)
L;R where GL;R " ; # eq: ¼ G " ; # =AFM is the equivalent interfacial conductance experienced by electrons as seen from FM side with units of V1 cm2, AFM the cross-sectional area of the ferro1 magnetic layer, and GL;R " ; # (in units of V ) is the reciprocal of the interfacial resistance obtained through Landauer’s formula (see below). a and b are the parameters which describes spin polarization of current and conductivity, respectively, i.e. j" = bj, j# = (1 b)j, s" = as and s" = (1 a)s, where j is the total current density. As the electron density is high in the FM contact regions and is not much affected by spin accumulation, its conductivity ac there can be taken to be constant i.e. the bulk spin polarization value in Fe of 0.7. Likewise for the 2DEG layer, in which the Fermi-level lies in the conduction band, we can also neglect the change in the conductivity due to the small electrochemical potential difference (m" m#). The polarization parameter as in the SC layer
S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
thus becomes spin-independent and takes a constant value of 0.5. The solutions to Eq. (1) for the three regions can be written as ðm " m # Þ ¼ A ex=lc ;
x<0
(4)
ðm " m # Þ ¼ B ex=ls þ C eðwxÞ=ls ; ðm " m # Þ ¼ D eðxwÞ=lc ;
x>w
0
(5)
Dm0" ¼
e j c bL GL" eq:
EB = EF(Fe) EF(2DEG). The Fermi-wave vectors at x = 0 and x ¼ w are, respectively, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe þ dm1;n:l: " Þ 1 k" ¼ ; h2 (11) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1;n:l: Fe 2mFe ðEF þ dm # h0 Þ k1# ¼ h2
(6)
To determine the coefficients A to D and bL,R we can apply Eqs. (2a) and (2b) on both sides of the left and right interfaces and match Eq. (3) at left and right interfaces. bL,R refers to the interfacial value of the spin polarization of current [i.e. bL = b(0), bR ¼ bðwÞ]. With the known interfacial quantities, the position dependence of b(x) and hence m"(x) and m#(x) can also be determined which will be used to calculate the magnetoresistance of the structure. From Eq. (3), we can obtain the interfacial discontinuities in the electrochemical potentials, i.e.
k2" ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFFe þ Dm0" þ dm2;n:l: EB Þ "
k2# ¼
k3" ¼
k3" ¼
(7) k4" ¼
e jc ð1 b Þ GL# eq:
e j c bR Dmw" ¼ R G " eq:
Dmw" ¼
e jc ð1 bR Þ GR# eq:
(8)
(9)
(12)
h2
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFGaAs þ dm3;n:l: # Þ
(13)
h2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe þ Dmw" þ dm4;n:l: " Þ
k4# ¼
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe h0 þ Dmw# þ dm4;n:l: # Þ
(14)
h2
Here dmn:l: " ; # represents the non-linear (non-Ohmic) part of electrochemical potential m in the vicinity of both interfaces, due to spin accumulation. By flux and wavefunction matching at the interfaces, the transmission coefficients across the interfaces are given by
(10)
We now apply the ballistic transport model to evaluate GL;R " ; # , as shown schematically in Fig. 1. The potential is drawn only for the majority-spin and a similar one can be drawn for the minority-spin. The barrier heights have been taken to be same for both left and right interfaces i.e. U L" ; # ¼ U R" ; # . EB is the Fermi-level shift between the 2DEG and FM layers, i.e.
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFFe þ Dm0# þ dm2;n:l: EB Þ #
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFGaAs þ dm3;n:l: Þ "
L
Dm0# ¼
4005
vL;GaAs ";# L;Fe v
2
(15)
2 vR;Fe 2vR;GaAs " ; # ";# ¼ R;GaAs R;GaAs R;Fe R v";# v " ; # þ v " ; # 2U " ; # dt=ih
(16)
T L" ; # ¼
";#
T R" ; #
2vL;Fe ";# L;GaAs vL;Fe 2U L" ; # dt=ih ";# þ v";#
Fig. 1. Schematic diagram of the electrochemical potential profile in the presence of an applied bias voltage across the device. The ballistic model is applied only in the shaded regions in the vicinity of the SC–FM interfaces.
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S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008 LðRÞFe
1ð4Þ
LðRÞGaAs
2ð3Þ
where v " ; # ¼ hk " ; # =mFe and v " ; # ¼ hk " ; # =mGaAs and dt is the thickness of the tunnel barrier which is taken to be thin (1 nm) in order to achieve good spin-tunneling properties L;Fe R;Fe R;GaAs [20,21]. The prefactors vL;GaAs of " ; # =v " ; # and v " ; # =v " ; # Eqs. (15) and (16), respectively, ensure flux continuity across the two FM–SC interfaces. From Landauer’s formula, the interfacial resistance R",# is then given by 1 RL;R ";#
¼ GL;R ";# ¼
e2 L;R T nm h ";#
(17)
where nm is the number of transverse modes. To simplify our calculations, we restrict our analysis to one transverse conductance mode only (nm = 1). This may be achieved for e.g. by constricting the FM–SC interface to a narrow channel, so that it acts as a mode filter which allows only one transverse mode to pass through. L;R A self-consistency loop linking GL;R " ; # on Dm " ; # can be formed by considering the DD results of Eqs. (7)–(10), and the ballistic model results of Eqs. (11)–(17). The self-consistency loop is terminated when the values of GL;R " ; # have converged to within 0.1% accuracy. Table 1 lists all the parameter values assumed in the numerical calculations. 3. Results and discussion We first analyze the asymmetry in interfacial resistance at the left and right interface, the spin-injection efficiency of the device and the magnetoresistance of the device over a range of bulk-SC conductivity values. The percentage asymmetry As in the interfacial resistances is defined as RL As ¼ 1 100 (18) RR RL and RR are the mean values of interfacial resistances experienced by up and down spin electrons, and are expressed as resistance-area products (i.e. in terms of 1/Geq.). The asymmetry As arises even though the trilayer structure is physically symmetric. To account for this, we note that from Eqs. (15) and (16), the transmission probability T depends on the respective kvectors at the FM and SC sides of the interface. These are given by Eqs. (11)–(14), from which we infer that the asymmetry in T is due to (i) the interfacial discontinuity in potential DmL;R " ð # Þ, and (ii) the non-linear part of the electrochemical potentials dmn:l: " ; # which corresponds to spin-accumulation. Both these effects arise when a finite current flows across the structure. Assuming constant current across the device, MR can be expressed as P DR RAP RP DmAP 0 Dm0 ¼ ¼ AP R RAP Dm0
(19)
where DmP;AP ¼ mP;AP ðx ¼ lFM Þ mP;AP ðx ¼ w þ lFM Þ 0 0 0
(20)
Table 1 Parameter values assumed in numerical calculations jc, current density in the FM region sc, FM contact conductivity AFM, cross-sectional area of FM contact js, current density in the 2DEG region m*(GaAs), effective mass of 2DEG GaAs m*(Fe), effective mass of Fe lc, contact spin-diffusion length aLc ¼ aRc , contact polarization parameter dt, barrier thickness U#, barrier height U", barrier height EF(Fe), Fermi-level of Fe EF(GaAs), Fermi-level of GaAs 2DEG h0, molecular field EB, difference in Fermi levels of FM and 2DEG w, 2DEG layer width h, Planck’s constant
1 A/cm2 106 V1 cm1 200 nm 200 nm (in yz plane) jc 200 nm = 2 105 A/cm 0.067me (me = 9.1 1031 kg) 1me 100 nm 0.7 1 nm 500 meV 250 meV 11.10 eV 3.5 meV 0.25 eV EF(Fe) EF(2DEG) 100 nm 6.6 1034 J s
Here, m0 is the linear spin-independent component of the electrochemical potential m",#(x), which is solely due to the applied voltage drop across the device. In Fig. 2, we have plotted asymmetry As, spin-injection efficiency of the structure and magnetoresistance as a function of SC conductivity ss. The 2DEG conductivity ss is varied from 106 to 102 V1, and this may be achieved in practice by increasing the doping density in the 2DEG. The SC spindiffusion length (SDL) is fixed at 1 mm. It can be seen from Fig. 2(a) that there is an appreciable increase in As from 42 to 50% over the range of SC conductivity considered. To explain this, we note that the dominant resistances in the device are that of the SC layer and interfacial resistance RI. As we increase the conductivity of the SC, the contribution of its resistance becomes smaller, and thus the interfacial resistances will be the controlling factor of the overall spin-transport. Thus, the discontinuity in electrochemical potentials DmL;R " ð # Þ as well as the spin accumulation dmn:l: " ; # should be more prominent due to the more significant contribution of the interfacial resistances. Hence, the asymmetry As which arises due to these two factors, will be enhanced. We also observe from Fig. 2(b) and (c) that spin-injection efficiency and MR both initially increase with ss. This is because as ss increases, we reduce the conductivity mismatch between FM and SC, which tends to suppress spin-injection efficiency as well as MR. However, a further increase in ss beyond 103 V1 causes a decrease in the spin-injection efficiency and MR. This is because as ss starts to approach sc of the FM contacts, the magnitude of the interfacial resistances begins to diminish. The decrease in MR is much faster than that of the spin-injection efficiency due to the fact that MR in general varies as the square of spin-injection [see e.g. Ref. [9]]. Thus, in summary we can see that the conductivity of the SC layer plays a crucial role in determining the spin transport
S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
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Fig. 2. (a) Asymmetry, (b) spin-injection efficiency and (c) magnetoresistance plotted as a function of conductivity in the 2DEG region.
Fig. 3. (a) Asymmetry, (b) spin-injection efficiency and (c) magnetoresistance plotted as a function of spin-diffusion length in the 2DEG region.
behavior in terms of spin-injection efficiency and MR of the device. Next we analyze the effect of the spin-diffusion length (SDL) of the SC layer. In Fig. 3, we have plotted the resistance asymmetry, spin injection efficiency and MR, as a function of SDL within the 2DEG. The 2DEG conductivity has been kept fixed at ss = 5.55 106 V1. Both the asymmetry and spininjection efficiency show a decrease with increasing SDL [Fig. 3(a) and (b)]. This may be explained as follows: spininjection is dependent on several factors such as U asymmetry (U" = 0.5U#), interfacial spin accumulation and discontinuity in m. For a short SDL, the spin accumulation must necessarily be concentrated at the interfaces, since it decays rapidly within the SC. Therefore a short SDL would actually increase the spin dependence of transport at the interfacial region. Since spininjection is defined as the spin polarization of current at the FM–SC interface, it is therefore enhanced by short SDL of the 2DEG layer. However, due to the rapid depolarization of spins in the SC layer, the high spin injection efficiency can be utilized for a practical spintronics application only if the spins are acted upon just after injection, e.g. by trapping them into a quantum well in the vicinity of the SC–FM interface. The increased
interfacial spin accumulation also accounts for a large interfacial resistance asymmetry at short SDL. The MR ratio [Fig. 3(c)], on the other hand, shows an opposite dependence on SDL (i.e. increase rather than decrease with longer SDL). This may be explained as follows: in general, the MR ratio reflects the spin dependence of the overall transport through the two interfaces and the SC layer. The SC layer by virtue of its large resistance will have a major impact on the overall resistance. Note that although the individual SC conductivities of the two spin channels are independent of SDL, the overall SC resistance will change since the effect of a finite SDL is equivalent to inserting a spin-flip resistance between the two spin channels. A large SDL will result in a large spinaccumulation being maintained across the entire SC layer. Due to the large spin accumulation, the electrochemical potential gradients will be very different for parallel (P) as compared to anti-parallel (P) magnetization configuration [n.b. that for P alignment, the m profiles for majority/minority carriers must cross within the SC, while for AP alignment, they are essentially parallel]. Thus, a large MR results from a long SDL in the 2DEG layer. Another feature of Fig. 3(c) is that the MR dependence shows a roughly antisymmetrical profile, about
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S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
SDL = 100 nm, which corresponds to the 2DEG width. There are plateaus in the MR ratio for very short and very long SDL values. For very short SDL « 100 nm, spin accumulation is almost negligible in the 2DEG, and so any further decrease in SDL will not cause any change in MR. At the other extreme, i.e. SDL » 100 nm, it is so long that there is negligible spin flip throughout the SC layer, and hence any further lengthening in the SDL will not result in much increase in MR. 4. Conclusion We have presented a self-consistent spin-dependent transport model of a FM–SC–FM trilayer structure, where the SC layer consists of a highly doped n2+ AlGaAs–GaAs 2DEG and with the interfacial resistance at the FM–SC junctions being modeled as delta potential (d) barriers. The model combines the spin drift-diffusion equation applied to the bulk of the trilayer, and the ballistic model applied to charge tunneling across the delta barriers at the FM–SC interfaces. Based on this selfconsistent model, we have analyzed the effects of SC conductivity ss and spin-diffusion length SDL on transport characteristics, such as the asymmetry of interfacial resistances, spin-injection efficiency and MR. We found that in general a large ss tends to improve all three characteristics. A large SDL, however, improves the MR ratio but causes a decrease in the spin-injection efficiency. These trends are explained in terms of conductivity mismatch and spin accumulation either at the interfacial zones or within the bulk of the SC layer. Acknowledgements We would like to thank the National University of Singapore (R-263-000-329-112) and the Agency for Science, Technology and Research (A*STAR) of Singapore (Grant No.: 022 105
0053) for financially supporting the work. One of the authors (SA) would like to thank NUS for his research scholarship. References [1] J.M. Kikkawa, D.D. Awschalom, Nature (London) 397 (1999) 139. [2] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett. 78 (1997) 1335. [3] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. [4] P.R. Hammar, B.R. Bennett, M.J. Yang, M. Johnson, Phys. Rev. Lett. 83 (1999) 203. [5] S. Gardelis, C.G. Smith, C.H.W. Barnes, E.H. Linfield, D.A. Ritchie, Phys. Rev. B 60 (1999) 7764. [6] F.G. Monzon, H.X. Tang, M.L. Roukes, Phys. Rev. Lett. 84 (2000) 5022. [7] B.J. van Wees, Phys. Rev. Lett. 84 (2000) 5023. [8] P.R. Hammar, B.R. Bennett, M.J. Yang, M. Johnson, Phys. Rev. Lett. 84 (2000) 5024. [9] G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip, B.J. van Wees, Phys. Rev. B 62 (2000) R4790. [10] A.T. Hanbicki, O.M.J. van’t Erve, R. Magno, G. Kioseoglou, C.H. Li, B.T. Jonker, G. Itskos, R. Mallory, M. Yasar, A. Petrou, Appl. Phys. Lett. 82 (2003) 4092. [11] C. Adelmann, X. Lou, J. Strand, C.J. Palmstrom, P.A. Crowell, Phys. Rev. B 71 (2005) 121301(R). [12] X.Y. Dong, C. Adelmann, J.Q. Xie, C.J. Palmstrom, X. Lou, J. Strand, P.A. Crowell, J.-P. Barnes, A.K. Petford-Long, Appl. Phys. Lett. 86 (2005) 102107. [13] E.I. Rashba, Phys. Rev. B 62 (2000) R16267(R). [14] S. Agrawal, M.B.A. Jalil, K.L. Teo, Y.F. Liew, J. Appl. Phys. 97 (2005) 103907. [15] H.B. Heersche, Th. Schapers, J. Nitta, H. Takayanagi, Phys. Rev. B 64 (2001) 161307(R). [16] Z.G. Yu, M.E. Flatte, Phys. Rev. B 66 (2002) 35302. [17] D.L. Smith, R.N. Silver, Phys. Rev. B 64 (2001) 45323. [18] S. Zhang, P.M. Levy, A. Fert, Phys. Rev. Lett. 88 (2002) 236601. [19] M.D. Stiles, A. Zangwill, J. Appl. Phys. 91 (2002) 6812. [20] V.F. Motsnyi, J. De Boeck, J. Das, W. Van Roy, G. Borghs, E. Goovaerts, V.I. Safarov, Appl. Phys. Lett. 81 (2002) 265. [21] M. Covington, J. Nowak, D. Song, Appl. Phys. Lett. 76 (2000) 3965.
Spin-injection efficiency and magnetoresistance in a hybrid ferromagnetic–semiconductor trilayer with interfacial barriers S. Agrawal a,*, M.B.A. Jalil a, S.G. Tan b, K.L. Teo a, T. Liew a,b a
Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore b Data Storage Institute, DSI Building, 5 Engineering Drive 1, Off Kent Ridge Crescent, National University of Singapore, Singapore 117608, Singapore Available online 21 October 2005
Abstract We present a self-consistent model of spin transport in a ferromagnetic (FM)–semiconductor (SC)–FM trilayer structure with interfacial barriers at the FM–SC boundaries. The SC layer consists of a highly doped n2+ AlGaAs–GaAs 2DEG while the interfacial resistance is modeled as delta potential (d) barriers. The self-consistent scheme combines a ballistic model of spin-dependent transmission across the d-barriers, and a driftdiffusion model within the bulk of the trilayer. The interfacial resistance (RI) values of the two junctions were found to be asymmetric despite the symmetry of the trilayer structure. Transport characteristics such as the asymmetry in RI, spin-injection efficiency and magnetoresistance (MR) are calculated as a function of bulk conductivity ss and spin-diffusion length (SDL) within the SC layer. In general a large ss tends to improve all three characteristics, while a long SDL improves the MR ratio but reduces the spin-injection efficiency. These trends may be explained in terms of conductivity mismatch and spin accumulation either at the interfacial zones or within the bulk of the SC layer. # 2005 Elsevier B.V. All rights reserved. PACS: 72.25.Mk; 73.40.Cg; 73.40.Sx Keywords: Spin-injection efficiency; Spin-diffusion length; Magnetoresistance
1. Introduction Semiconductor (SC)-based spintronics which exploits the spin as well as the charge property of carriers, is a fast growing field of research. The long spin coherence of electrons in SC [1], coupled with the ability to control spin orientation by electrical means [2] has opened the possibility of realizing devices such as the Datta–Das spin field effect transistor [3] (spin-FET). A chief prerequisite of such devices is the ability to inject spin-polarized current into semiconductor materials, which are usually non-magnetic. Initial experimental efforts at spin-injection were not very successful with the resultant spin injection efficiency h of the order of 1% only [4,5]. Furthermore, the signals from such a small spin polarization of the current could possibly be ascribed to other effects such as the local Hall effect in the vicinity of the ferromagnetic spin
* Corresponding author. E-mail address: [email protected] (S. Agrawal). 0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.09.033
injector or detector [6,7] although subsequent experiments by Hammar et al. [8] did account for this spurious local Hall effect. The reasons for such low values of spin-injection efficiencies have also been explained in terms of conductivity mismatch [9]. More recently, a much higher spin injection efficiency h exceeding 30% [10,11] has been achieved for injection from a Fe contact into a AlGaAs–GaAs quantum well structure through a Schottky barrier. In addition, an appreciable spinpolarization value of 13% was obtained, albeit at a low temperature of 2 K, for injection from a Heusler alloy into a AlGaAs/GaAs light-emitting diode [12], which also involves a Schottky contact. The crucial element for these recent successes is the incorporation of interfacial barriers, e.g. tunnel or Schottky barriers between the FM and SC layers, to overcome the conductivity mismatch problem, as was theoretically proposed by Rashba [13]. In this paper we consider a FM–SC–FM trilayer structure with interfacial resistive (RI) barriers at the SC–FM boundaries. The role of RI in determining the overall charge and spin transport has been investigated in previous work [14]. However,
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the characteristics of the SC layer also play a prominent role, owing to the relatively large resistivity of the SC layer (about five orders of magnitude larger than that of the FM contacts). Thus, the main focus of this article is to investigate the effects of changing the bulk-SC conductivity and the spin-diffusion length of the SC region on the spin injection efficiency and overall magnetoresistance (MR) of a FM–SC–FM trilayer. In general, an increase in the conductivity and spin-diffusion length of the SC material improves the spin-injection efficiency and MR of the device by maintaining spin coherence, and reducing the conductivity-mismatch [9] problem, respectively. However, the presence of RI complicates the analysis. For instance, a short SC spin-diffusion length enhances the spinaccumulation effect adjacent to the FM–SC interfaces, which can also improve the spin-injection efficiency. It is thus important to analyze the effects of varying conductivity as well as spin-diffusion lengths of the SC region on the spin transport characteristics of the device. In our model, we determine the interfacial resistances via a self-consistent drift-diffusion and ballistic model. The diffusive part of the model is based on the spin-dependent drift-diffusion (DD) equation. The RI values are evaluated by considering ballistic tunneling transmission across d-function potential barriers at the FM–SC interfaces. The incorporation of interfacial d-barriers makes our model similar to that of Heersche et al. [15]. But the latter assumes fully ballistic transport through the structure and is limited to a single junction. However in this paper, we consider a trilayer structure in which the SC layer thickness w is assumed to be larger than the carrier mean free path (MFP), although it is comparable to the spin-diffusion length in the SC material. It is thus necessary to incorporate a diffusive model to model the charge and spin transport in the SC layer away from the interfacial zones. The trilayer material consists of a highly doped n2+ AlGaAs– GaAs 2DEG (SC) layer sandwiched between two Fe (FM) layers. The d-potential barriers at the two FM–SC interfaces are expressed as U½dðxÞ þ dðx wÞ with the barrier height U being spin-dependent, as was assumed by Yu and Flatte [16] and Smith and Silver [17]. The quantitative description of tunneling at the FM–SC interface is rather complicated because the transport properties are strongly dependent on the potential barrier height and thickness, and are highly sensitive to interfacial roughness and impurity states within the barriers. However, as a first approximation, we ignore any type of electron scattering within the barriers and assume purely ballistic transport through them. 2. Model For transport in the bulk FM and SC layers, we consider the drift-diffusion model, which is based on the following equations: @2 ðm " m # Þ m " m # ¼ @x2 l2
(1)
@mc" ð # Þ @x
@ms" ð # Þ @x
¼
¼
e jc" ð # Þ s c" ð # Þ
e js" ð # Þ s s" ð # Þ
(2a)
(2b)
where l is the spin-diffusion length and e is the magnitude of electronic charge. Subscripts ‘c’ and ‘s’ refer to the FM contact c;s c;s and the 2DEG SC layer. mc;s " ð # Þ , j " ð # Þ and s " ð # Þ are, respectively, the electrochemical potentials, current densities and conductivities for majority (minority) spin electrons, each of which are different in the FM and the SC 2DEG region. Eq. (1) is the diffusion equation while Eqs. (2a) and (2b) are Ohm’s law in FM and 2DEG regions, respectively. In the FM contact, jc and sc are expressed in conventional units of A/cm2 and V1 cm1, respectively, while in the 2DEG js and ss are expressed in A/cm and V1, respectively, so that (jc/sc) and (jc/ss) have the same units. We have assumed no spin-flip scattering at the interface so that j" and j# are continuous at the interfaces. Ideally, the diffusion equation and Ohm’s law [Eqs. (1) and (2)] should be supplemented by Poisson’s equation, which relates m to charge accumulation. However, since the SC layer is a highly doped 2DEG, it is a good approximation to assume charge neutrality i.e. zero charge accumulation. Such an approximation has been made by Zhang et al. [18] in their diffusive transport model where they have dropped the terms involving charge accumulation. In addition, as far as transport equations are concerned, there is a freedom in choosing the solution of the Poisson’s equation according to Stiles and Zangwill [19]. For simplicity, we can thus choose a solution corresponding to zero charge accumulation. At the FM–SC interface, there is a discontinuity in both m" and m#, which is given by
Dm0;w ";# ¼
e jc" ; # GL;R " ; # eq:
(3)
L;R where GL;R " ; # eq: ¼ G " ; # =AFM is the equivalent interfacial conductance experienced by electrons as seen from FM side with units of V1 cm2, AFM the cross-sectional area of the ferro1 magnetic layer, and GL;R " ; # (in units of V ) is the reciprocal of the interfacial resistance obtained through Landauer’s formula (see below). a and b are the parameters which describes spin polarization of current and conductivity, respectively, i.e. j" = bj, j# = (1 b)j, s" = as and s" = (1 a)s, where j is the total current density. As the electron density is high in the FM contact regions and is not much affected by spin accumulation, its conductivity ac there can be taken to be constant i.e. the bulk spin polarization value in Fe of 0.7. Likewise for the 2DEG layer, in which the Fermi-level lies in the conduction band, we can also neglect the change in the conductivity due to the small electrochemical potential difference (m" m#). The polarization parameter as in the SC layer
S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
thus becomes spin-independent and takes a constant value of 0.5. The solutions to Eq. (1) for the three regions can be written as ðm " m # Þ ¼ A ex=lc ;
x<0
(4)
ðm " m # Þ ¼ B ex=ls þ C eðwxÞ=ls ; ðm " m # Þ ¼ D eðxwÞ=lc ;
x>w
0
(5)
Dm0" ¼
e j c bL GL" eq:
EB = EF(Fe) EF(2DEG). The Fermi-wave vectors at x = 0 and x ¼ w are, respectively, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe þ dm1;n:l: " Þ 1 k" ¼ ; h2 (11) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1;n:l: Fe 2mFe ðEF þ dm # h0 Þ k1# ¼ h2
(6)
To determine the coefficients A to D and bL,R we can apply Eqs. (2a) and (2b) on both sides of the left and right interfaces and match Eq. (3) at left and right interfaces. bL,R refers to the interfacial value of the spin polarization of current [i.e. bL = b(0), bR ¼ bðwÞ]. With the known interfacial quantities, the position dependence of b(x) and hence m"(x) and m#(x) can also be determined which will be used to calculate the magnetoresistance of the structure. From Eq. (3), we can obtain the interfacial discontinuities in the electrochemical potentials, i.e.
k2" ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFFe þ Dm0" þ dm2;n:l: EB Þ "
k2# ¼
k3" ¼
k3" ¼
(7) k4" ¼
e jc ð1 b Þ GL# eq:
e j c bR Dmw" ¼ R G " eq:
Dmw" ¼
e jc ð1 bR Þ GR# eq:
(8)
(9)
(12)
h2
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFGaAs þ dm3;n:l: # Þ
(13)
h2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe þ Dmw" þ dm4;n:l: " Þ
k4# ¼
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mFe ðEFFe h0 þ Dmw# þ dm4;n:l: # Þ
(14)
h2
Here dmn:l: " ; # represents the non-linear (non-Ohmic) part of electrochemical potential m in the vicinity of both interfaces, due to spin accumulation. By flux and wavefunction matching at the interfaces, the transmission coefficients across the interfaces are given by
(10)
We now apply the ballistic transport model to evaluate GL;R " ; # , as shown schematically in Fig. 1. The potential is drawn only for the majority-spin and a similar one can be drawn for the minority-spin. The barrier heights have been taken to be same for both left and right interfaces i.e. U L" ; # ¼ U R" ; # . EB is the Fermi-level shift between the 2DEG and FM layers, i.e.
; h2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFFe þ Dm0# þ dm2;n:l: EB Þ #
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mGaAs ðEFGaAs þ dm3;n:l: Þ "
L
Dm0# ¼
4005
vL;GaAs ";# L;Fe v
2
(15)
2 vR;Fe 2vR;GaAs " ; # ";# ¼ R;GaAs R;GaAs R;Fe R v";# v " ; # þ v " ; # 2U " ; # dt=ih
(16)
T L" ; # ¼
";#
T R" ; #
2vL;Fe ";# L;GaAs vL;Fe 2U L" ; # dt=ih ";# þ v";#
Fig. 1. Schematic diagram of the electrochemical potential profile in the presence of an applied bias voltage across the device. The ballistic model is applied only in the shaded regions in the vicinity of the SC–FM interfaces.
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S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008 LðRÞFe
1ð4Þ
LðRÞGaAs
2ð3Þ
where v " ; # ¼ hk " ; # =mFe and v " ; # ¼ hk " ; # =mGaAs and dt is the thickness of the tunnel barrier which is taken to be thin (1 nm) in order to achieve good spin-tunneling properties L;Fe R;Fe R;GaAs [20,21]. The prefactors vL;GaAs of " ; # =v " ; # and v " ; # =v " ; # Eqs. (15) and (16), respectively, ensure flux continuity across the two FM–SC interfaces. From Landauer’s formula, the interfacial resistance R",# is then given by 1 RL;R ";#
¼ GL;R ";# ¼
e2 L;R T nm h ";#
(17)
where nm is the number of transverse modes. To simplify our calculations, we restrict our analysis to one transverse conductance mode only (nm = 1). This may be achieved for e.g. by constricting the FM–SC interface to a narrow channel, so that it acts as a mode filter which allows only one transverse mode to pass through. L;R A self-consistency loop linking GL;R " ; # on Dm " ; # can be formed by considering the DD results of Eqs. (7)–(10), and the ballistic model results of Eqs. (11)–(17). The self-consistency loop is terminated when the values of GL;R " ; # have converged to within 0.1% accuracy. Table 1 lists all the parameter values assumed in the numerical calculations. 3. Results and discussion We first analyze the asymmetry in interfacial resistance at the left and right interface, the spin-injection efficiency of the device and the magnetoresistance of the device over a range of bulk-SC conductivity values. The percentage asymmetry As in the interfacial resistances is defined as RL As ¼ 1 100 (18) RR RL and RR are the mean values of interfacial resistances experienced by up and down spin electrons, and are expressed as resistance-area products (i.e. in terms of 1/Geq.). The asymmetry As arises even though the trilayer structure is physically symmetric. To account for this, we note that from Eqs. (15) and (16), the transmission probability T depends on the respective kvectors at the FM and SC sides of the interface. These are given by Eqs. (11)–(14), from which we infer that the asymmetry in T is due to (i) the interfacial discontinuity in potential DmL;R " ð # Þ, and (ii) the non-linear part of the electrochemical potentials dmn:l: " ; # which corresponds to spin-accumulation. Both these effects arise when a finite current flows across the structure. Assuming constant current across the device, MR can be expressed as P DR RAP RP DmAP 0 Dm0 ¼ ¼ AP R RAP Dm0
(19)
where DmP;AP ¼ mP;AP ðx ¼ lFM Þ mP;AP ðx ¼ w þ lFM Þ 0 0 0
(20)
Table 1 Parameter values assumed in numerical calculations jc, current density in the FM region sc, FM contact conductivity AFM, cross-sectional area of FM contact js, current density in the 2DEG region m*(GaAs), effective mass of 2DEG GaAs m*(Fe), effective mass of Fe lc, contact spin-diffusion length aLc ¼ aRc , contact polarization parameter dt, barrier thickness U#, barrier height U", barrier height EF(Fe), Fermi-level of Fe EF(GaAs), Fermi-level of GaAs 2DEG h0, molecular field EB, difference in Fermi levels of FM and 2DEG w, 2DEG layer width h, Planck’s constant
1 A/cm2 106 V1 cm1 200 nm 200 nm (in yz plane) jc 200 nm = 2 105 A/cm 0.067me (me = 9.1 1031 kg) 1me 100 nm 0.7 1 nm 500 meV 250 meV 11.10 eV 3.5 meV 0.25 eV EF(Fe) EF(2DEG) 100 nm 6.6 1034 J s
Here, m0 is the linear spin-independent component of the electrochemical potential m",#(x), which is solely due to the applied voltage drop across the device. In Fig. 2, we have plotted asymmetry As, spin-injection efficiency of the structure and magnetoresistance as a function of SC conductivity ss. The 2DEG conductivity ss is varied from 106 to 102 V1, and this may be achieved in practice by increasing the doping density in the 2DEG. The SC spindiffusion length (SDL) is fixed at 1 mm. It can be seen from Fig. 2(a) that there is an appreciable increase in As from 42 to 50% over the range of SC conductivity considered. To explain this, we note that the dominant resistances in the device are that of the SC layer and interfacial resistance RI. As we increase the conductivity of the SC, the contribution of its resistance becomes smaller, and thus the interfacial resistances will be the controlling factor of the overall spin-transport. Thus, the discontinuity in electrochemical potentials DmL;R " ð # Þ as well as the spin accumulation dmn:l: " ; # should be more prominent due to the more significant contribution of the interfacial resistances. Hence, the asymmetry As which arises due to these two factors, will be enhanced. We also observe from Fig. 2(b) and (c) that spin-injection efficiency and MR both initially increase with ss. This is because as ss increases, we reduce the conductivity mismatch between FM and SC, which tends to suppress spin-injection efficiency as well as MR. However, a further increase in ss beyond 103 V1 causes a decrease in the spin-injection efficiency and MR. This is because as ss starts to approach sc of the FM contacts, the magnitude of the interfacial resistances begins to diminish. The decrease in MR is much faster than that of the spin-injection efficiency due to the fact that MR in general varies as the square of spin-injection [see e.g. Ref. [9]]. Thus, in summary we can see that the conductivity of the SC layer plays a crucial role in determining the spin transport
S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
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Fig. 2. (a) Asymmetry, (b) spin-injection efficiency and (c) magnetoresistance plotted as a function of conductivity in the 2DEG region.
Fig. 3. (a) Asymmetry, (b) spin-injection efficiency and (c) magnetoresistance plotted as a function of spin-diffusion length in the 2DEG region.
behavior in terms of spin-injection efficiency and MR of the device. Next we analyze the effect of the spin-diffusion length (SDL) of the SC layer. In Fig. 3, we have plotted the resistance asymmetry, spin injection efficiency and MR, as a function of SDL within the 2DEG. The 2DEG conductivity has been kept fixed at ss = 5.55 106 V1. Both the asymmetry and spininjection efficiency show a decrease with increasing SDL [Fig. 3(a) and (b)]. This may be explained as follows: spininjection is dependent on several factors such as U asymmetry (U" = 0.5U#), interfacial spin accumulation and discontinuity in m. For a short SDL, the spin accumulation must necessarily be concentrated at the interfaces, since it decays rapidly within the SC. Therefore a short SDL would actually increase the spin dependence of transport at the interfacial region. Since spininjection is defined as the spin polarization of current at the FM–SC interface, it is therefore enhanced by short SDL of the 2DEG layer. However, due to the rapid depolarization of spins in the SC layer, the high spin injection efficiency can be utilized for a practical spintronics application only if the spins are acted upon just after injection, e.g. by trapping them into a quantum well in the vicinity of the SC–FM interface. The increased
interfacial spin accumulation also accounts for a large interfacial resistance asymmetry at short SDL. The MR ratio [Fig. 3(c)], on the other hand, shows an opposite dependence on SDL (i.e. increase rather than decrease with longer SDL). This may be explained as follows: in general, the MR ratio reflects the spin dependence of the overall transport through the two interfaces and the SC layer. The SC layer by virtue of its large resistance will have a major impact on the overall resistance. Note that although the individual SC conductivities of the two spin channels are independent of SDL, the overall SC resistance will change since the effect of a finite SDL is equivalent to inserting a spin-flip resistance between the two spin channels. A large SDL will result in a large spinaccumulation being maintained across the entire SC layer. Due to the large spin accumulation, the electrochemical potential gradients will be very different for parallel (P) as compared to anti-parallel (P) magnetization configuration [n.b. that for P alignment, the m profiles for majority/minority carriers must cross within the SC, while for AP alignment, they are essentially parallel]. Thus, a large MR results from a long SDL in the 2DEG layer. Another feature of Fig. 3(c) is that the MR dependence shows a roughly antisymmetrical profile, about
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S. Agrawal et al. / Applied Surface Science 252 (2006) 4003–4008
SDL = 100 nm, which corresponds to the 2DEG width. There are plateaus in the MR ratio for very short and very long SDL values. For very short SDL « 100 nm, spin accumulation is almost negligible in the 2DEG, and so any further decrease in SDL will not cause any change in MR. At the other extreme, i.e. SDL » 100 nm, it is so long that there is negligible spin flip throughout the SC layer, and hence any further lengthening in the SDL will not result in much increase in MR. 4. Conclusion We have presented a self-consistent spin-dependent transport model of a FM–SC–FM trilayer structure, where the SC layer consists of a highly doped n2+ AlGaAs–GaAs 2DEG and with the interfacial resistance at the FM–SC junctions being modeled as delta potential (d) barriers. The model combines the spin drift-diffusion equation applied to the bulk of the trilayer, and the ballistic model applied to charge tunneling across the delta barriers at the FM–SC interfaces. Based on this selfconsistent model, we have analyzed the effects of SC conductivity ss and spin-diffusion length SDL on transport characteristics, such as the asymmetry of interfacial resistances, spin-injection efficiency and MR. We found that in general a large ss tends to improve all three characteristics. A large SDL, however, improves the MR ratio but causes a decrease in the spin-injection efficiency. These trends are explained in terms of conductivity mismatch and spin accumulation either at the interfacial zones or within the bulk of the SC layer. Acknowledgements We would like to thank the National University of Singapore (R-263-000-329-112) and the Agency for Science, Technology and Research (A*STAR) of Singapore (Grant No.: 022 105
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