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Model of spin transport in noncollinear magnetic systems: Effect of diffuse interfaces
Journal of Magnetism and Magnetic Materials 484 (2019) 238–244
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Research articles
Model of spin transport in noncollinear magnetic systems: Effect of diffuse interfaces N. Saenphuma, J. Chureemarta, R.W. Chantrellb, P. Chureemarta, a b
T
⁎
Computational and Experimental Magnetism Group, Department of Physics, Mahasarakham University, Mahasarakham 44150, Thailand Department of Physics, University of York, York YO10 5DD, United Kingdom
A R T I C LE I N FO
A B S T R A C T
Keywords: Spin transport Diffuse interface Magnetoresistance Spin torque
We propose a theoretical model to study the spin transport, particularly the interfacial resistance in a noncollinear system with a diffuse interface. The realistic interface region arising from the sputtering process is first modeled by using Fick’s law. Subsequently, the spin transport behavior in magnetic system is determined via a generalized spin accumulation model taking into account spin dephasing term. Our model is able to deal with magnetic systems with arbitrary orientation of magnetization by employing the transfer matrix approach. The model is first used to investigate spin accumulation in a bilayer system consisting of a ferromagnet (FM) and a non-magnet (NM). The influence of important transport parameters: spin dephasing, interdiffusion, injected current density and spin polarization on magnetoresistance (MR) are studied. Next we investigate the transport properties of a FM/NM/FM system where the interface disorder giving rise to enhanced spin dephasing is modeled. We found that the interdiffusion of the FM/NM/FM interface gives rise to large spin scattering, which enhances the GMR. The results also show that including the disorder arising from interface diffusion is an important contribution to the transport properties.
1. Introductions Spin transport in magnetic nanostructures has attracted much attention due to possible applications in spintronic devices such as magnetic tunnel junction (MTJ) sensors [1], spin-torque oscillators [2,3] and magnetoresistive random access memory (MRAM) [4]. The functionality of spintronic devices relies on spin transport properties, in particular the behavior of a spin polarized current, which can be used to switch the direction of the magnetization using the spin torque phenomenon or to detect the direction of magnetization using the phenomenon of giant magnetoresistance (GMR) or tunneling magnetoresistance (TMR). Understanding of the fundamental mechanisms of spin transport in magnetic structures, including the study of magnetoresistance (MR) effects is necessary for the development of next-generation spintronic devices [5–8]. There are various physical parameters strongly affecting both the spin transport and spin-transfer torque (STT) phenomena. In addition, the interface structure and its properties will also play a crucial role in determining resistance arising from spin-dependent scattering at the interface [9,10]. In spintronic devices such as spin-valves, the spin transport behavior across interfaces is central to device functionality. Generally, sputtered
⁎
devices exhibit diffuse interfaces. It becomes increasingly important factor as device sizes reduce, with consequent reduction in magnetic layer thicknesses to 2 nm and below. The concept of spin accumulation is often used to quantitatively describe the spin transport phenomenon. The spin accumulation arises at the interface region due to the difference of the spin transport properties between layers [11–13]. In addition, the STT arising from the s-d exchange interaction between the transverse spin accumulation and local magnetization is strongly dependent on the relative orientation of magnetization between layers. In terms of the spin torque, this arises from the transverse component of the accumulation which is oscillatory and decays rapidly from the interface. It implies the importance of detailed treatment of interfaces, in particular their structure and properties. There are a number of experimental investigations of the effects of diffuse interface, a practical outcome of the sputtering process of realistic devices [14–17], on spin transport properties. The results indicate that the interface structure is an important issue in device design. However, the associated theoretical problem has received little attention since most models focused on ideal atomically smooth interfaces [13,18–20]. Recent work [21] proposed a simple model of diffuse interfaces neglecting the effect of spin dephasing and applied the model to
Corresponding author. E-mail address: [email protected] (P. Chureemart).
https://doi.org/10.1016/j.jmmm.2019.04.010 Received 11 February 2019; Received in revised form 20 March 2019; Accepted 3 April 2019 Available online 04 April 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 484 (2019) 238–244
N. Saenphum, et al.
dm ℓ m − m∞ + (J /ℏ) m × M + (J /ℏ) L M × (m × M) = − . dt ℓ⊥ τsf
investigation of a system with a collinear magnetization configuration. In this work, we develop a more realistic spin transport model taking the effect of diffuse interfaces into account. This model is able to study the spin transport behavior of the system with noncollinear magnetization, consequently allowing calculation of the magnetoresistance. The aim of this paper is to understand the spin transport mechanism in systems with diffuse interfaces. The paper is organized as follows. Section 2 describes the theoretical method used to calculate the spin accumulation in noncollinear structures. We first present the formalism of generalized spin accumulation which will be used to calculate the interfacial resistance. Secondly, the interface region will be modeled as a simple interdiffusion process. Subsequently, we relate the spatial dependence of the transport properties to the local concentration within the interface. In order to represent the enhanced scattering due to disorder within the interface we propose a local enhancement of the spin dephasing in the spin accumulation model. In Section 3 our approach will be applied to a ferromagnetic/non-magnetic (FM/NM) bilayer system to investigate the importance of the spin dephasing term and to a FM/NM/FM structure to investigate the spin transport behavior and evaluate the GMR ratio. The effect of spin dephasing, diffuse interface, spin polarization and injected current density on GMR are also studied. The proposed model is expected to be applicable, within the atomistic model approach, for the investigation of complex phenomena such as interface diffusion and finite size effects.
(2)
Our definition of spin accumulation requires the damping term driving the system to an equilibrium value, m∞ as shown in RHS of Eq. (2). We note that neither the spin current (dependent on the gradient m ) nor the spin torque (dependent on the component of m transverse to the magnetization) are affected by this definition. In addition, the spin accumulation is assumed to reach the equilibrium in the direction of the magnetization. The magnetization current or spin current (jm ) can be written in terms of the modulus of the electric current (je ) and the spin accumulation according to a spinor form as follows:
∂m ∂m ⎞ ⎤ − ββ′M ⎛M· jm = βje M − 2D0 ⎡ , ⎢ ∂ x ∂x ⎠ ⎥ ⎝ ⎦ ⎣
(3)
where β is the spin polarization for the conductivity, β′ is the spin polarization for the diffusion constant, D0 is the diffusion constant and x is the direction of the injected current. The stationary solution of the spin accumulation (m ) decomposed into longitudinal and transverse components with respect to magnetization direction can be determined under the assumption that the relaxation time (τsf ) is much shorter than the timescale of magnetization changes. Similarly to the generalized spin accumulation model in Ref. [24], to determine the components of the spin accumulation, we re∂j ∂m place dm in Eq. (2) by ∂t + ∂xm leading to dt
∂j ∂m =− ∂xm ∂t
2. Theoretical model
− (J /ℏ) m × M ℓ
− (J /ℏ) ℓL M × (m × M) − ⊥
Spin accumulation is the physical quantity which is extensively used to describe the spin transport in magnetic structures. Most spin accumulation models based on drift-diffusion equations have captured the precessional motion around the local magnetization based on an equation of motion [13,18,21] expressed as follows,
(4)
From Eqs. (3) and (4), we finally arrive at βj ∂M 1 ∂m =− 2De ∂x 2D0 ∂t 0
− dm m + (J /ℏ) m × M = − , dt τsf
m − m∞ . τsf
(1)
m×M λJ2
+ −
∂2m ∂x 2
(
− ββ′M M·
M × (m × M) λ ϕ2
−
∂2m ∂x 2
)
m − m∞ , λsf2
(5)
where λJ = 2ℏD0 / J , the spin dephasing length λ ϕ = 2ℏD0 ℓ⊥/(J ℓL) and λsf = 2D0 τsf . In this work, we discretize the magnetic structure into a series of thin layers and assume that the magnetization is uniform throughout a layer. Therefore, the first RHS term of above equation, a so-called source term, is set to zero. In this case, the source term can be taken into account by imposing the boundary condition at the interface between layers which is the continuity of spin current [12]. Next the longitudinal (m‖) and transverse (m⊥ ) components of spin accumulation can be derived from Eq. (5) by using the transfer matrix approach [24] based on a rotated coordinate system b1̂ , b2̂ and b3̂ which is parallel and perpendicular to the local magnetization. The longitudinal component will be parallel to b1̂ , and the two components of the transverse spin accumulation are along the directions b2̂ and b3̂ . The use of a rotated coordinate system allows retention of the form of solution adopted by Zhang, Levy and Fert [18], leading to a complete solution of spin accumulation in the following form
where m is the spin accumulation, M is the unit vector for the local magnetization of the free layer, J is the exchange energy between the electron spin and the local magnetization, ℏ is the reduced Planck constant and τsf is the spin-flip relaxation time of the conduction electrons. Recently, an additional damping term was introduced to represent the dephasing effect [22–24], arising from local variation of properties, ℓ of the form (J /ℏ) ℓL M × (m × M) where ℓL and ℓ⊥ are the (Lamor) spin ⊥ precession length and spin coherence length [22], respectively. Typically, the spin coherence length is relatively small compared with the spin diffusion length. Accordingly, the spin dephasing is negligible in most drift-diffusive approaches under the assumption that the transverse spin current is fully absorbed at the interface. However, the spin dephasing term representing the interplay between spin precession and impurity scattering becomes very important to describe the role of transverse spin accumulation in the spin transport behavior and its absence may lead to missing some relevant physics. In particular, the spin scattering within a diffuse interface might be expected to have a significant bearing on the transport properties. Here we suggest to represent this phenomenology by an enhanced spin dephasing local to the interface as will be described later. Here we use a definition of the spin accumulation based on the difference of spin-up and spin-down density of states (DOS) [21], which simplifies the treatment of multiple layers with different equilibrium values of spin accumulation (m∞) . Specifically, we modify the formalism of spin accumulation by taking the additional transverse damping ↑ ↓ − neq ) M, into acterm and the nonzero equilibrium value, m∞ = (neq count as follows:
m‖(x )=[m‖ (∞) + [m‖ (0) − m‖ (∞)] e−x / λsdl ] b1̂ m⊥,2 (x )=2e−k1 x [ucos(k2 x ) − v sin(k2 x )] b2̂ m⊥,3 (x )=2e−k1 x [usin(k2 x ) + v cos(k2 x )] b3̂ ,
(6)
with
(k1 ± ik2) =
−2 λ trans ± iλJ−2 .
(7)
−2 The transverse length scale (λ trans ) is defined as λ trans = λ ϕ−2 + λsf−2 incorporating the damping contributions from the Bloch-like term in the ZLF model and the transverse damping introduced by Petitjean et. al. [22]. The coefficients m‖ (0), u and v are constants which can be determined from the interface condition of the continuity of spin current.
239
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N. Saenphum, et al.
where RAi is the resistance-area product at position i, Δm is the difference of spin accumulation across the layers, jm is the spin current, a is the lattice constant of the material, kB T is 10 meV or 1.6 × 10−21 J, t is the thickness of discretized layer and e is the electron charge.
at x = 0 . Here we present an initial study of a FM/NM bilayer to determine the effects of the disorder as represented by the spin dephasing. The magnetization of the FM layer is aligned in the y direction. We treat the magnetization in the interface region as a continuously rotating vector. Across the FM/NM interface, it is allowed to gradually change in yz plane and its magnitude is considered from the spatial concentration of magnetic ions calculated by using Eq. (8). The spatial variation of the magnetization is introduced in order to investigate the importance of the localized spin dephasing on the transverse spin accumulation. The solution of spin accumulation using Eq. (6), allowing treatment of systems with spatially varying magnetization, is then applied to the FM/NM system. It is expected that the disorder introduced by the interface interdiffusion will introduce effects beyond the spatial variation of the transport properties. Here we represent disorder effects as an increased local spin dephasing in the interface resulting from the spatial variation of the local environment. Thus the degree of disorder and presence of impurities in the interface are represented by the value of λ ϕ . We assume that the variation of λ ϕ across the interface reflects the, approximately Gaussian, concentration profile. Consequently, λ ϕ within the interface is given by
3. Results
λ ϕ = λ ϕ,A CA (x ) + λ ϕ,MAX e− 2 ⎡⎣ σ ⎤⎦ + λ ϕ,B CB (x ),
We apply the model to the calculation of the transport properties of simple systems with diffuse interfaces. We first consider FM/NM system to describe the model of the diffuse interface and to investigate the effect of the locally enhanced spin dephasing at the interface. We then proceed to study a FM/NM/FM system which is a simple model of the reference layer and free layer in a spin valve. Finally, we present a parametric study of the effects of enhanced scattering at the interface, predicting an increase of the MR ratio with increased spin dephasing. This is a finding consistent with experiment.
where x is the position, A, B label the phases separated by the interface and CA, CB are the local concentrations of each phase within the interface. λ ϕ,MAX is a phenomenological parameter which determines the maximum value of the dephasing (occurring at the center of the interface), and which therefore represents the degree of disorder. σ is the standard deviation related to the interface thickness. It is straightforward to show that σ = x 0 /( 2ln2 ) . For the interface with x1/2 = 1 nm, we use the calculated value of σ = 0.165. We first focus on the influence of λ ϕ on the spin transport behavior. The spin transport parameters of the FM used here are those of NiFe with the following values: β = 0.7, β′ = 0.95, λsf = 8 nm and λJ = 3 nm . The value of spin transport parameters of NM are taken as β = β′ = 0, λsf = 600 nm and λ ϕ,FM = λ ϕ,NM = 2 nm . We set value of s-d exchange integral (J) closed to zero giving rise to very high value of λJ . The effect of interface disorder, represented by the degree of dephasing, is investigated by varying the coefficient λ ϕ,MAX from 0 to 100 nm as demonstrated in Fig. 1. The effect of the spin dephasing length on the spin accumulation is then investigated. Fig. 2(a)–(c) displays the spatial variation of transverse and longitudinal spin accumulations of the FM/NM bilayer which are perpendicular and parallel to the local magnetization, respectively. The appearance of transverse spin accumulation arises from the spatially varying magnetization
It is noted that m‖ (∞) = |m∞|. The total spin torque (τST ) , naturally including the adiabatic (AST, τAST ) and non-adiabatic torques (NAST, τNAST ), can be obtained from the divergence of the spin current by considering the conservation of the total spin angular momentum [12,22,25] as follows, ∂j
τST= ∂xm = −(J /ℏ)(m × M) ℓ
− (J /ℏ) ℓL [M × (m × M)] ⊥
=−
2D0 (m λJ2
× M) −
2D0 [M λ ϕ2
× (m × M)].
The spin accumulation allows the calculation of the spin torque and also the calculation of a local resistance which can be related to the gradient of spin accumulation and the spin current [26,27] as follows,
RAi =
|Δm|a2tkB T , jm e 2
(8)
1
3.1. Model of diffuse interface The proposed model is implemented to first investigate the effect of spin dephasing [18,24,28] localized to the interface. Before applying the general solution of spin accumulation to the FM/NM structure to investigate the spin transport behavior, we outline the interface model for the calculation of spin accumulation since the interdiffusion between atoms at the interface region has a significant effect on spin transport parameters, such as the spin polarization of conductivity and diffusion, which become spatially dependent within the interface. The interfacial region can be simply modeled by using Fick’s law [21]. The magnetic ion concentration at any given position of the system is calculated from Fick’s law and used to determine the spatial variation of the spin transport parameters across the interface. Consider an interface between two species of ion, (A,B). The spatial concentration of a given ion (A) at any given position x of the system can be written as
CA (x ) =
NA (x ) , [NA (x ) + NB (x )]
x 2
(11)
(9)
where Nα (x ) (α = (A, B ) ) is the number of local ions α at any position x, determined by the solution of Fick’s law. The calculated concentration from Eq. (9) is used to model the spatial variation of the diffusive transport parameters, P (x ) . For simplicity this is taken as a linear combination of the bulk parameters weighted by the local concentrations and is given by,
P (x ) = PA CA (x ) + PB [1 − CA (x )],
(10)
where Pα is the diffusive transport parameter of species α . The width of the interface is characterized by the parameter x1/2 , essentially the full width at half height such that x1/2 = 2x 0 and C (x 0)/ C (0) = 0.5, where C (x ) is the concentration at a distance x from the interface, positioned
Fig. 1. Spatial profile of spin dephasing length of FM(10 nm)/NM(5 nm) bilayer system with different λ ϕ, MAX : the position x = 0 is the center of the interface. The positions x < 0 and x > 0 show the layer of FM and NM respectively. 240
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Fig. 3. Schematic illustration of spin-valve stack with diffuse interfaces: the structure is divided into many thin layers with 0.0035 nm thick and the magnetization of interface region slowly changes as the domain wall.
resistance of the structure strongly depends on the relative angle (θ ) between the magnetization of the two ferromagnets. The magnetization of the region far from the interface is allowed to be homogeneous. Across the first interface region (FM/NM), the direction of magnetization is set to be the same as that of the first FM regarded as the pinned layer where its magnitude is gradually decreased due to the interdiffusion between FM and NM ions. For the second interface region (NM/FM), the magnetization rotates continuously between the layers which is analogous to a domain wall and its magnitude is considered from the spatial concentration of magnetic ions. A charge current of density je = 50 MA/cm2 , typical magnitude of current density for GMR investigation [29,30], is injected into a structure comprising NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) to observe the spin accumulation and spin current. Then the MR value can be calculated. The angular dependence of MR is determined by varying the relative angle between the magnetization of two ferromagnets in the range 0 < θ < 2π . The transport parameters of the ferromagnet are chosen for NiFe with β = 0.7, β′ = 0.95, λJ = 3 nm, λ ϕ = 2 nm and the magnitude of equilibrium spin accumulation |m∞| = 1.118 × 108 C/m3 whereas the spin polarization parameters and equilibrium value of spin accumulation are zero for Cu [21]. The spatial variation of spin current and spin accumulation are calculated using Eqs. (3) and (6) respectively. The resistance at any given position is determined from Eq. (8) and the total MR of the structure is subsequently obtained by summing n the resistance-area product along the x-direction, MR = ∑i = 1 RAi . Fig. 4 shows the calculated MR for different relative angles between the magnetization of the ferromagnets. The resistance of the anti-parallel configuration is higher than that of the parallel orientation due to the fact that the spin accumulation tends to follow the direction of
Fig. 2. Spatial profile of spin accumulation for the FM(10 nm)/NM(5 nm) bilayer system with parameters given in the text.
across the interface between FM/NM. We obviously observe that the spin dephasing length only significantly affects the transverse spin accumulation because the longitudinal component of spin accumulation strongly depends on the spin diffusion length. This is very important since the transverse component is related to the enhancement of spin torque in the system. Neglect of the spin dephasing effect in spin transport model is equivalent to setting λ ϕ = ∞, which is applicable for materials with λ ϕ > λJ . However, for the case of soft materials such as NiFe, λ ϕ > λJ , it seems important to include the spin dephasing effect into the spin transport model. In addition, we also notice that the peak in m⊥ moves further into the NM with increasing λ ϕ . These peaks observed at interface region correspond to the value of k1 where −2 −2 (k1 ± ik2) = λ trans ± iλJ−2 and λ trans = λ ϕ−2 + λsf−2 . In NM, the value of k1 is mainly dominated by λ trans since λJ,NM is very high. 3.2. Spin transport in a magnetic trilayer system In this section, to demonstrate the advantage of the proposed model as a powerful tool for GMR calculation, the generalized formalism of spin accumulation described above is applied to a structured film to calculate the magnetoresistance (MR) for a trilayer FM/NM/FM stack with arbitrary noncollinear magnetization configuration. This is the simplest model of the reference and sensing layer of a spin-valve stack. We focus on the effects of interface mixing and its effects on the spin accumulation and spin current. As illustrated in Fig. 3, the trilayer system consisting of two ferromagnets separated by a nonmagnet is discretized into many thin layers in order to model the spatial concentration of magnetic ions used to calculate the spatial variation of transport parameters and magnetization. The intermixing between ions of two different materials produces a diffuse interface region which is modeled as described earlier. The
Fig. 4. Angular dependence of magnetoresistance of the spin-valve structure with injected current density of je = 50 MA/cm2 : The relative angle between magnetization in two ferromagnets is varied. 241
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Fig. 6. ΔRA and MR ratio of NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) with different interfacial thicknesses.
Fig. 5. MR ratio and ΔRA as a function of spin dephasing length (λ ϕ, MAX ) in NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) structure.
magnetization. For the anti-parallel case, spin accumulation changes rapidly at the interface leading to a high gradient of spin accumulation. We make a comparison with fitting function, a + bsin2 (θ /2) , which is used to describe the GMR effect [31–35]. The result shows a good agreement and the parameters a and b in fitting function give the values of RP and ▵R , respectively. The GMR ratio is found to be 21%. This shows that the proposed model is an effective tool for GMR calculation, giving a realistic value of the GMR ratio and an angular variation of resistance consistent with experiment [32]. Next, we perform a study of the effect of spin dephasing on the enhancement of resistance change (ΔRA) and MR ratio for the structure NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) with interface thickness of 1 nm. As illustrated in Fig. 5, it is observed that increasing spin dephasing length (λ ϕ, MAX ) results in a significant decrease of the resistance change and MR ratio. The spin dephasing length varies from 0.001 to 1000 giving rise to nearly 20% change in MR ratio. The effect becomes especially strong for λ ϕ comparable with λJ . To study the important factors influencing the spin transport properties, the effect of spin dephasing, interdiffusion, spin polarization and current density on resistance change and GMR will be next investigated.
3.3. Parametric studies We have carried out a parametric investigation of the interfacial scattering effect on GMR using the spin dephasing to characterize the disorder arising from the interdiffusion. The proposed model allows us to study this effect in a structure of NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) by varying the interfacial thickness controlled by the degree of intermixing between atoms of two adjacent materials. A charge current of density je = 50 MA/cm2 is injected to the structure with interfacial thickness varied from 0.3 to 1.5 nm. We have investigated the magnetoresistance and resistance change of the trilayer structure with different degrees of intermixing. Fig. 6 demonstrates that the increasing interface thickness results in a larger spin dependent scattering which eventually leads to higher ΔRA and MR ratio. This confirms that the nature of the interface becomes a significant factor in the spin transport behavior. Our results give the same trend with previous experimental [36–38] and numerical [39] studies. Refs. [36–38] experimentally investigated the effect of annealing temperature on GMR of the currentperpendicular-to-plane (CPP) spin-valves using heusler alloys. It is well known that the interface thickness is associated with the annealing temperature, therefore we could make an indirect comparison between the trend of the experimental results and ours. In addition, it is found experimentally that high spin scattering at the interface provides the improvement of ΔRA and the enhancement of the MR ratio, which is confirmed by the model calculations. We also study the influence of the spin polarization parameter (β ) and injected current density ( je ) on MR and resistance change in the
Fig. 7. (a) MR ratio and ΔRA as a function of spin polarization parameter, and (b) RAAP, RAP and normalized ΔRA as a function of injected current density in NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) structure.
trilayer structure. As shown in Fig. 7(a), high MR ratio can be achieved by using material with a high spin polarization parameter. This is due to the fact that the CPP-GMR is associated with bulk and interface spin scattering which strongly depends on the spin polarization of the material [37,40–42]. Furthermore, we demonstrate the dependence of ΔRA on the injected current density. An electric current density ranging from 10 to 16 MA/cm2 is injected into the structure. Increasing injected current density gives rise to a reduction of MR and ΔRA as shown in Fig. 7(b). To make the comparison with experimental study in Ref. [43], the value of ΔRA is normalized by that of je = 10 MA/cm2 . Our results give the same trend with previous experiments by Kimura et al. [43] which measured the magnetoresistance of Py/Cu spin-valve structure as a function of injected current density. In summary, spin dependent interfacial scattering, in addition to the spin transport parameters and current density, plays an important role in achieving high output MR for the potential read sensors in hard drives with high areal density. Our proposed model provides a good description of spin transport in magnetic structures and provides a practically useful tool for appropriated 242
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read sensor design with high performance.
barriers, Nat. Mater. 3 (2004) 862–867. [8] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, Giant room-temperature magnetoresistance in single-crystal fe/mgo/fe magnetic tunnel junctions, Nat. Mater. 3 (2004) 868–871. [9] S. Zhang, P.M. Levy, Enhanced spin-dependent scattering at interfaces, Phys. Rev. Lett. 77 (1996) 916–919. [10] M. Ishikawa, H. Sugiyama, T. Inokuchi, K. Hamaya, Y. Saito, Effect of the interface resistance of cofe/mgo contacts on spin accumulation in silicon, Appl. Phys. Lett. 100 (7) (2012). [11] S. Wang, Y. Xu, K. Xia, First-principles study of spin-transfer torques in layered systems with noncollinear magnetization, Phys. Rev. B 77 (2008) 184430 . [12] A. Shpiro, P.M. Levy, S. Zhang, Self-consistent treatment of nonequilibrium spin torques in magnetic multilayers, Phys. Rev. B 67 (2003) 104430 . [13] M. Ruggeri, C. Abert, G. Hrkac, D. Suess, D. Praetorius, Coupling of dynamical micromagnetism and a stationary spin drift-diffusion equation: a step towards a fully self-consistent spintronics framework, Physica B (2015). [14] P. Ho, G.C. Han, R.F.L. Evans, R.W. Chantrell, G.M. Chow, J.S. Chen, Perpendicular anisotropy l10-fept based pseudo spin valve with ag spacer layer, Appl. Phys. Lett. 98(13). [15] J. Jung, Z. Jin, Y. Shiokawa, M. Sahashi, Effect of spin-dependent interface resistance at (001)-oriented fe/ag interface on giant magnetoresistance effect, Appl. Phys. Express 8 (7) (2015) 073005 . [16] Q. Zhang, P. Li, Y. Wen, C. Zhao, J.W. Zhang, A. Manchon, W.B. Mi, Y. Peng, X.X. Zhang, Anomalous hall effect in fe/au multilayers, Phys. Rev. B 94 (2016) 024428 . [17] L. O’Brien, D. Spivak, J.S. Jeong, K.A. Mkhoyan, P.A. Crowell, C. Leighton, Interdiffusion-controlled kondo suppression of injection efficiency in metallic nonlocal spin valves, Phys. Rev. B 93 (2016) 014413 . [18] S. Zhang, P.M. Levy, A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett. 88 (2002) 236601 . [19] Z. Nedelkoski, P.J. Hasnip, A.M. Sanchez, B. Kuerbanjiang, E. Higgins, M. Oogane, A. Hirohata, G.R. Bell, V.K. Lazarov, The effect of atomic structure on interface spinpolarization of half-metallic spin valves: Co2mnsi/ag epitaxial interfaces, Appl. Phys. Lett. 107(21). [20] P. Bal, J. Barna, J.-P. Ansermet, Transverse spin penetration length in metallic spin valves, J. Appl. Phys. 113(19). [21] P. Chureemart, R. Cuadrado, I. D’Amico, R.W. Chantrell, Modeling spin injection across diffuse interfaces, Phys. Rev. B 87 (2013) 195310 . [22] C. Petitjean, D. Luc, X. Waintal, Unified drift-diffusion theory for transverse spin currents in spin valves, domain walls, and other textured magnets, Phys. Rev. Lett. 109 (2012) 117204 . [23] C.A. Akosa, W.-S. Kim, A. Bisig, M. Kläui, K.-J. Lee, A. Manchon, Role of spin diffusion in current-induced domain wall motion for disordered ferromagnets, Phys. Rev. B 91 (2015) 094411 . [24] P. Chureemart, I. D’Amico, R.W. Chantrell, Model of spin accumulation and spin torque in spatially varying magnetisation structures: limitations of the micromagnetic approach, J. Phys.: Condens. Matter 27 (14) (2015) 146004 . [25] S. Lepadatu, Unified treatment of spin torques using a coupled magnetisation dynamics and three-dimensional spin current solver, Scientific Rep. 7 (1) (2017) 12937 . [26] A. Fert, H. Jaffrès, Conditions for efficient spin injection from a ferromagnetic metal into a semiconductor, Phys. Rev. B 64 (2001) 184420 . [27] N.L. Chung, M.B.A. Jalil, S.G. Tan, S. Bala Kumar, Interfacial resistance and spin flip effects on the magnetoresistance of a current perpendicular to plane spin valve, J. Appl. Phys. 103(7). [28] C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, G. Hrkac, D. Praetorius, D. Suess, A three-dimensional spin-diffusion model for micromagnetics, Sci. Rep. 5 (2015) 14855 26442796[pmid]. [29] K. Nikolaev, P. Kolbo, T. Pokhil, X. Peng, Y. Chen, T. Ambrose, O. Mryasov, allheusler alloy current-perpendicular-to-plane giant magnetoresistance, Appl. Phys. Lett. 94 (22) (2009) 222501 . [30] T. Nakatani, N. Hase, H. Goripati, Y. Takahashi, T. Furubayashi, K. Hono, Co-based heusler alloys for cpp-gmr spin-valves with large magnetoresistive outputs, IEEE Trans. Magn. 48 (5) (2012) 1751–1757. [31] C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, A. Manchon, D. Praetorius, D. Suess, Simultaneous resolution of the micromagnetic and spin transport equations applied to current-induced domain wall dynamics, arXiv:1512.05519v4. [32] S. Urazhdin, R. Loloee, W. Pratt Jr, Noncollinear spin transport in magnetic multilayers, Phys. Rev. B 71 (10) (2005) 100401 . [33] S. Urazhdin, R. Loloee, W. Pratt Jr, Spin transport at interfaces in magnetic multilayers, J. Appl. Phys. 99 (8) (2006) 08G504 . [34] F. Montaigne, C. Tiusan, M. Hehn, Angular dependence of tunnel magnetoresistance in magnetic tunnel junctions and specific aspects in spin-filtering devices, J. Appl. Phys. 108 (6) (2010) 063912 . [35] T. Qu, R. Victora, Angular dependence of current perpendicular to plane giant magnetoresistance in multilayer nanowire, J. Appl. Phys. 111 (7) (2012) 07C516 . [36] T. Kubota, Y. Ina, Z. Wen, H. Narisawa, K. Takanashi, Current perpendicular-toplane giant magnetoresistance using an l 1 2 ag 3 mg spacer and co 2 fe 0.4 mn 0.6 si heusler alloy electrodes: spacer thickness and annealing temperature dependence, Phys. Rev. Mater. 1 (4) (2017) 044402 . [37] Y. Du, T. Furubayashi, T.T. Sasaki, Y. Sakuraba, Y.K. Takahashi, K. Hono, Large magnetoresistance in current-perpendicular-to-plane pseudo spin-valves using co2fe(ga0.5ge0.5) heusler alloy and agzn spacer, Appl. Phys. Lett. 107 (11) (2015) 112405 . [38] S. Li, Y.K. Takahashi, T. Furubayashi, K. Hono, Enhancement of giant magnetoresistance by l21 ordering in co2fe(ge0.5ga0.5) heusler alloy current-perpendicular-
4. Conclusions The spin transport behavior in magnetic structure is significantly affected by the presence of diffuse interfaces due to the intermixing of ions which is expected for the sputtering growth process. In this work, the spin transport in bilayer systems with noncollinear configuration of magnetization is studied by using the generalized spin accumulation model including the effect of diffuse interfaces. The expected scattering due to disorder in the interface is modeled by introducing a strong local spin dephasing at the interface. The effect of spin dephasing length is first studied. The results suggest that inclusion of locally enhanced spin dephasing into the model is a reasonable representation of the spin scattering associated with diffuse interfaces. The modified spin accumulation model is then applied to a FM/NM/FM structure with variable direction of magnetization to calculate the magnetoresistance, which is related to the gradient of spin accumulation and the spin current. The total resistance of parallel and anti-parallel configurations is calculated giving the GMR value. Finally, we parametrically study the effect of the interdiffusion, spin polarization and injected current density on device performance. We found that the interdiffusion significantly affects the resistance change. In addition, high MR output can be achieved by using ferromagnetic materials with high spin polarization. The results show that the formalism of spin accumulation, modified to include local variation of properties is a realistic approach to the calculation of spin transport in magnetic nanostructures with diffuse interfaces. As device size and magnetic film thicknesses decrease, it is expected that interface diffusion and finite size effects will become increasingly important. The model described here is shown to be a reasonable representation of the enhanced spin scattering at interfaces and can form an important basis both for physical investigation and device design. Acknowledgements N.S. would like to acknowledge PhD funding from Seagate technology (Thailand). P.C. gratefully acknowledges the funding from Mahasarakham University and Thailand Research Fund under Grant No. MRG6080048. J. C. would like to acknowledge the funding from Mahasarakham University and IAPP 1R2/100151. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.jmmm.2019.04.010. References [1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, H. Ohno, A perpendicular-anisotropy cofeb–mgo magnetic tunnel junction, Nat. Mater. 9 (9) (2010) 721–724. [2] H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru, M. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yuasa, T. Taniguchi, H. Arai, H. Imamura, Spin-torque oscillator based on magnetic tunnel junction with a perpendicularly magnetized free layer and in-plane magnetized polarizer, Appl. Phys. Express 6 (10) (2013) 103003 . [3] Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner, V.E. Demidov, S.O. Demokritov, I.N. Krivorotov, A perpendicular-anisotropy cofeb–mgo magnetic tunnel junction, Nat. Commun. 5. [4] A.V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R.S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W.H. Butler, P.B. Visscher, D. Lottis, E. Chen, V. Nikitin, M. Krounbi, Basic principles of stt-mram cell operation in memory arrays, J. Phys. D: Appl. Phys. 46 (7) (2013) 074001 . [5] M.N. Baibich, J.M. Broto, A. Fert, F.N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. Chazelas, Giant magnetoresistance of (001)fe/(001)cr magnetic superlattices, Phys. Rev. Lett. 61 (1988) 2472–2475. [6] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39 (1989) 4828–4830. [7] S.S.P. Parkin, C. Kaiser, A. Panchula, P.M. Rice, B. Hughes, M. Samant, S.-H. Yang, Giant tunnelling magnetoresistance at room temperature with mgo (100) tunnel
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N. Saenphum, et al.
suppressing comn anti-sites in co2mn (ge0. 75ga0. 25) heusler alloy thin film, Appl. Phys. Lett. 108 (12) (2016) 122404 . [42] Ikhtiar, S. Kasai, Y. Takahashi, T. Furubayashi, S. Mitani, K. Hono, Temperature dependence of magneto-transport properties in co2fe (ga0. 5ge0. 5)/cu lateral spin valves, Appl. Phys. Lett. 108 (6) (2016) 062401 . [43] T. Kimura, N. Hashimoto, S. Yamada, M. Miyao, K. Hamaya, Room-temperature generation of giant pure spin currents using epitaxial co2fesi spin injectors, NPG Asia Mater. 4 (2012) e9 .
to-plane pseudo spin valves, Appl. Phys. Lett. 103 (4) (2013) 042405 . [39] B. Elsafi, F. Trigui, Z. Fakhfakh, Influence of interfacial scattering on giant magnetoresistance in co/cu ultrathin multilayers, J. Appl. Phys. 109 (3) (2011) 033910 . [40] H. Goripati, T. Furubayashi, Y. Takahashi, K. Hono, Current-perpendicular-to-plane giant magnetoresistance using co2fe (ga1- x ge x) heusler alloy, J. Appl. Phys. 113 (4) (2013) 043901 . [41] S. Li, Y. Takahashi, Y. Sakuraba, N. Tsuji, H. Tajiri, Y. Miura, J. Chen, T. Furubayashi, K. Hono, Large enhancement of bulk spin polarization by
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Research articles
Model of spin transport in noncollinear magnetic systems: Effect of diffuse interfaces N. Saenphuma, J. Chureemarta, R.W. Chantrellb, P. Chureemarta, a b
T
⁎
Computational and Experimental Magnetism Group, Department of Physics, Mahasarakham University, Mahasarakham 44150, Thailand Department of Physics, University of York, York YO10 5DD, United Kingdom
A R T I C LE I N FO
A B S T R A C T
Keywords: Spin transport Diffuse interface Magnetoresistance Spin torque
We propose a theoretical model to study the spin transport, particularly the interfacial resistance in a noncollinear system with a diffuse interface. The realistic interface region arising from the sputtering process is first modeled by using Fick’s law. Subsequently, the spin transport behavior in magnetic system is determined via a generalized spin accumulation model taking into account spin dephasing term. Our model is able to deal with magnetic systems with arbitrary orientation of magnetization by employing the transfer matrix approach. The model is first used to investigate spin accumulation in a bilayer system consisting of a ferromagnet (FM) and a non-magnet (NM). The influence of important transport parameters: spin dephasing, interdiffusion, injected current density and spin polarization on magnetoresistance (MR) are studied. Next we investigate the transport properties of a FM/NM/FM system where the interface disorder giving rise to enhanced spin dephasing is modeled. We found that the interdiffusion of the FM/NM/FM interface gives rise to large spin scattering, which enhances the GMR. The results also show that including the disorder arising from interface diffusion is an important contribution to the transport properties.
1. Introductions Spin transport in magnetic nanostructures has attracted much attention due to possible applications in spintronic devices such as magnetic tunnel junction (MTJ) sensors [1], spin-torque oscillators [2,3] and magnetoresistive random access memory (MRAM) [4]. The functionality of spintronic devices relies on spin transport properties, in particular the behavior of a spin polarized current, which can be used to switch the direction of the magnetization using the spin torque phenomenon or to detect the direction of magnetization using the phenomenon of giant magnetoresistance (GMR) or tunneling magnetoresistance (TMR). Understanding of the fundamental mechanisms of spin transport in magnetic structures, including the study of magnetoresistance (MR) effects is necessary for the development of next-generation spintronic devices [5–8]. There are various physical parameters strongly affecting both the spin transport and spin-transfer torque (STT) phenomena. In addition, the interface structure and its properties will also play a crucial role in determining resistance arising from spin-dependent scattering at the interface [9,10]. In spintronic devices such as spin-valves, the spin transport behavior across interfaces is central to device functionality. Generally, sputtered
⁎
devices exhibit diffuse interfaces. It becomes increasingly important factor as device sizes reduce, with consequent reduction in magnetic layer thicknesses to 2 nm and below. The concept of spin accumulation is often used to quantitatively describe the spin transport phenomenon. The spin accumulation arises at the interface region due to the difference of the spin transport properties between layers [11–13]. In addition, the STT arising from the s-d exchange interaction between the transverse spin accumulation and local magnetization is strongly dependent on the relative orientation of magnetization between layers. In terms of the spin torque, this arises from the transverse component of the accumulation which is oscillatory and decays rapidly from the interface. It implies the importance of detailed treatment of interfaces, in particular their structure and properties. There are a number of experimental investigations of the effects of diffuse interface, a practical outcome of the sputtering process of realistic devices [14–17], on spin transport properties. The results indicate that the interface structure is an important issue in device design. However, the associated theoretical problem has received little attention since most models focused on ideal atomically smooth interfaces [13,18–20]. Recent work [21] proposed a simple model of diffuse interfaces neglecting the effect of spin dephasing and applied the model to
Corresponding author. E-mail address: [email protected] (P. Chureemart).
https://doi.org/10.1016/j.jmmm.2019.04.010 Received 11 February 2019; Received in revised form 20 March 2019; Accepted 3 April 2019 Available online 04 April 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 484 (2019) 238–244
N. Saenphum, et al.
dm ℓ m − m∞ + (J /ℏ) m × M + (J /ℏ) L M × (m × M) = − . dt ℓ⊥ τsf
investigation of a system with a collinear magnetization configuration. In this work, we develop a more realistic spin transport model taking the effect of diffuse interfaces into account. This model is able to study the spin transport behavior of the system with noncollinear magnetization, consequently allowing calculation of the magnetoresistance. The aim of this paper is to understand the spin transport mechanism in systems with diffuse interfaces. The paper is organized as follows. Section 2 describes the theoretical method used to calculate the spin accumulation in noncollinear structures. We first present the formalism of generalized spin accumulation which will be used to calculate the interfacial resistance. Secondly, the interface region will be modeled as a simple interdiffusion process. Subsequently, we relate the spatial dependence of the transport properties to the local concentration within the interface. In order to represent the enhanced scattering due to disorder within the interface we propose a local enhancement of the spin dephasing in the spin accumulation model. In Section 3 our approach will be applied to a ferromagnetic/non-magnetic (FM/NM) bilayer system to investigate the importance of the spin dephasing term and to a FM/NM/FM structure to investigate the spin transport behavior and evaluate the GMR ratio. The effect of spin dephasing, diffuse interface, spin polarization and injected current density on GMR are also studied. The proposed model is expected to be applicable, within the atomistic model approach, for the investigation of complex phenomena such as interface diffusion and finite size effects.
(2)
Our definition of spin accumulation requires the damping term driving the system to an equilibrium value, m∞ as shown in RHS of Eq. (2). We note that neither the spin current (dependent on the gradient m ) nor the spin torque (dependent on the component of m transverse to the magnetization) are affected by this definition. In addition, the spin accumulation is assumed to reach the equilibrium in the direction of the magnetization. The magnetization current or spin current (jm ) can be written in terms of the modulus of the electric current (je ) and the spin accumulation according to a spinor form as follows:
∂m ∂m ⎞ ⎤ − ββ′M ⎛M· jm = βje M − 2D0 ⎡ , ⎢ ∂ x ∂x ⎠ ⎥ ⎝ ⎦ ⎣
(3)
where β is the spin polarization for the conductivity, β′ is the spin polarization for the diffusion constant, D0 is the diffusion constant and x is the direction of the injected current. The stationary solution of the spin accumulation (m ) decomposed into longitudinal and transverse components with respect to magnetization direction can be determined under the assumption that the relaxation time (τsf ) is much shorter than the timescale of magnetization changes. Similarly to the generalized spin accumulation model in Ref. [24], to determine the components of the spin accumulation, we re∂j ∂m place dm in Eq. (2) by ∂t + ∂xm leading to dt
∂j ∂m =− ∂xm ∂t
2. Theoretical model
− (J /ℏ) m × M ℓ
− (J /ℏ) ℓL M × (m × M) − ⊥
Spin accumulation is the physical quantity which is extensively used to describe the spin transport in magnetic structures. Most spin accumulation models based on drift-diffusion equations have captured the precessional motion around the local magnetization based on an equation of motion [13,18,21] expressed as follows,
(4)
From Eqs. (3) and (4), we finally arrive at βj ∂M 1 ∂m =− 2De ∂x 2D0 ∂t 0
− dm m + (J /ℏ) m × M = − , dt τsf
m − m∞ . τsf
(1)
m×M λJ2
+ −
∂2m ∂x 2
(
− ββ′M M·
M × (m × M) λ ϕ2
−
∂2m ∂x 2
)
m − m∞ , λsf2
(5)
where λJ = 2ℏD0 / J , the spin dephasing length λ ϕ = 2ℏD0 ℓ⊥/(J ℓL) and λsf = 2D0 τsf . In this work, we discretize the magnetic structure into a series of thin layers and assume that the magnetization is uniform throughout a layer. Therefore, the first RHS term of above equation, a so-called source term, is set to zero. In this case, the source term can be taken into account by imposing the boundary condition at the interface between layers which is the continuity of spin current [12]. Next the longitudinal (m‖) and transverse (m⊥ ) components of spin accumulation can be derived from Eq. (5) by using the transfer matrix approach [24] based on a rotated coordinate system b1̂ , b2̂ and b3̂ which is parallel and perpendicular to the local magnetization. The longitudinal component will be parallel to b1̂ , and the two components of the transverse spin accumulation are along the directions b2̂ and b3̂ . The use of a rotated coordinate system allows retention of the form of solution adopted by Zhang, Levy and Fert [18], leading to a complete solution of spin accumulation in the following form
where m is the spin accumulation, M is the unit vector for the local magnetization of the free layer, J is the exchange energy between the electron spin and the local magnetization, ℏ is the reduced Planck constant and τsf is the spin-flip relaxation time of the conduction electrons. Recently, an additional damping term was introduced to represent the dephasing effect [22–24], arising from local variation of properties, ℓ of the form (J /ℏ) ℓL M × (m × M) where ℓL and ℓ⊥ are the (Lamor) spin ⊥ precession length and spin coherence length [22], respectively. Typically, the spin coherence length is relatively small compared with the spin diffusion length. Accordingly, the spin dephasing is negligible in most drift-diffusive approaches under the assumption that the transverse spin current is fully absorbed at the interface. However, the spin dephasing term representing the interplay between spin precession and impurity scattering becomes very important to describe the role of transverse spin accumulation in the spin transport behavior and its absence may lead to missing some relevant physics. In particular, the spin scattering within a diffuse interface might be expected to have a significant bearing on the transport properties. Here we suggest to represent this phenomenology by an enhanced spin dephasing local to the interface as will be described later. Here we use a definition of the spin accumulation based on the difference of spin-up and spin-down density of states (DOS) [21], which simplifies the treatment of multiple layers with different equilibrium values of spin accumulation (m∞) . Specifically, we modify the formalism of spin accumulation by taking the additional transverse damping ↑ ↓ − neq ) M, into acterm and the nonzero equilibrium value, m∞ = (neq count as follows:
m‖(x )=[m‖ (∞) + [m‖ (0) − m‖ (∞)] e−x / λsdl ] b1̂ m⊥,2 (x )=2e−k1 x [ucos(k2 x ) − v sin(k2 x )] b2̂ m⊥,3 (x )=2e−k1 x [usin(k2 x ) + v cos(k2 x )] b3̂ ,
(6)
with
(k1 ± ik2) =
−2 λ trans ± iλJ−2 .
(7)
−2 The transverse length scale (λ trans ) is defined as λ trans = λ ϕ−2 + λsf−2 incorporating the damping contributions from the Bloch-like term in the ZLF model and the transverse damping introduced by Petitjean et. al. [22]. The coefficients m‖ (0), u and v are constants which can be determined from the interface condition of the continuity of spin current.
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Journal of Magnetism and Magnetic Materials 484 (2019) 238–244
N. Saenphum, et al.
where RAi is the resistance-area product at position i, Δm is the difference of spin accumulation across the layers, jm is the spin current, a is the lattice constant of the material, kB T is 10 meV or 1.6 × 10−21 J, t is the thickness of discretized layer and e is the electron charge.
at x = 0 . Here we present an initial study of a FM/NM bilayer to determine the effects of the disorder as represented by the spin dephasing. The magnetization of the FM layer is aligned in the y direction. We treat the magnetization in the interface region as a continuously rotating vector. Across the FM/NM interface, it is allowed to gradually change in yz plane and its magnitude is considered from the spatial concentration of magnetic ions calculated by using Eq. (8). The spatial variation of the magnetization is introduced in order to investigate the importance of the localized spin dephasing on the transverse spin accumulation. The solution of spin accumulation using Eq. (6), allowing treatment of systems with spatially varying magnetization, is then applied to the FM/NM system. It is expected that the disorder introduced by the interface interdiffusion will introduce effects beyond the spatial variation of the transport properties. Here we represent disorder effects as an increased local spin dephasing in the interface resulting from the spatial variation of the local environment. Thus the degree of disorder and presence of impurities in the interface are represented by the value of λ ϕ . We assume that the variation of λ ϕ across the interface reflects the, approximately Gaussian, concentration profile. Consequently, λ ϕ within the interface is given by
3. Results
λ ϕ = λ ϕ,A CA (x ) + λ ϕ,MAX e− 2 ⎡⎣ σ ⎤⎦ + λ ϕ,B CB (x ),
We apply the model to the calculation of the transport properties of simple systems with diffuse interfaces. We first consider FM/NM system to describe the model of the diffuse interface and to investigate the effect of the locally enhanced spin dephasing at the interface. We then proceed to study a FM/NM/FM system which is a simple model of the reference layer and free layer in a spin valve. Finally, we present a parametric study of the effects of enhanced scattering at the interface, predicting an increase of the MR ratio with increased spin dephasing. This is a finding consistent with experiment.
where x is the position, A, B label the phases separated by the interface and CA, CB are the local concentrations of each phase within the interface. λ ϕ,MAX is a phenomenological parameter which determines the maximum value of the dephasing (occurring at the center of the interface), and which therefore represents the degree of disorder. σ is the standard deviation related to the interface thickness. It is straightforward to show that σ = x 0 /( 2ln2 ) . For the interface with x1/2 = 1 nm, we use the calculated value of σ = 0.165. We first focus on the influence of λ ϕ on the spin transport behavior. The spin transport parameters of the FM used here are those of NiFe with the following values: β = 0.7, β′ = 0.95, λsf = 8 nm and λJ = 3 nm . The value of spin transport parameters of NM are taken as β = β′ = 0, λsf = 600 nm and λ ϕ,FM = λ ϕ,NM = 2 nm . We set value of s-d exchange integral (J) closed to zero giving rise to very high value of λJ . The effect of interface disorder, represented by the degree of dephasing, is investigated by varying the coefficient λ ϕ,MAX from 0 to 100 nm as demonstrated in Fig. 1. The effect of the spin dephasing length on the spin accumulation is then investigated. Fig. 2(a)–(c) displays the spatial variation of transverse and longitudinal spin accumulations of the FM/NM bilayer which are perpendicular and parallel to the local magnetization, respectively. The appearance of transverse spin accumulation arises from the spatially varying magnetization
It is noted that m‖ (∞) = |m∞|. The total spin torque (τST ) , naturally including the adiabatic (AST, τAST ) and non-adiabatic torques (NAST, τNAST ), can be obtained from the divergence of the spin current by considering the conservation of the total spin angular momentum [12,22,25] as follows, ∂j
τST= ∂xm = −(J /ℏ)(m × M) ℓ
− (J /ℏ) ℓL [M × (m × M)] ⊥
=−
2D0 (m λJ2
× M) −
2D0 [M λ ϕ2
× (m × M)].
The spin accumulation allows the calculation of the spin torque and also the calculation of a local resistance which can be related to the gradient of spin accumulation and the spin current [26,27] as follows,
RAi =
|Δm|a2tkB T , jm e 2
(8)
1
3.1. Model of diffuse interface The proposed model is implemented to first investigate the effect of spin dephasing [18,24,28] localized to the interface. Before applying the general solution of spin accumulation to the FM/NM structure to investigate the spin transport behavior, we outline the interface model for the calculation of spin accumulation since the interdiffusion between atoms at the interface region has a significant effect on spin transport parameters, such as the spin polarization of conductivity and diffusion, which become spatially dependent within the interface. The interfacial region can be simply modeled by using Fick’s law [21]. The magnetic ion concentration at any given position of the system is calculated from Fick’s law and used to determine the spatial variation of the spin transport parameters across the interface. Consider an interface between two species of ion, (A,B). The spatial concentration of a given ion (A) at any given position x of the system can be written as
CA (x ) =
NA (x ) , [NA (x ) + NB (x )]
x 2
(11)
(9)
where Nα (x ) (α = (A, B ) ) is the number of local ions α at any position x, determined by the solution of Fick’s law. The calculated concentration from Eq. (9) is used to model the spatial variation of the diffusive transport parameters, P (x ) . For simplicity this is taken as a linear combination of the bulk parameters weighted by the local concentrations and is given by,
P (x ) = PA CA (x ) + PB [1 − CA (x )],
(10)
where Pα is the diffusive transport parameter of species α . The width of the interface is characterized by the parameter x1/2 , essentially the full width at half height such that x1/2 = 2x 0 and C (x 0)/ C (0) = 0.5, where C (x ) is the concentration at a distance x from the interface, positioned
Fig. 1. Spatial profile of spin dephasing length of FM(10 nm)/NM(5 nm) bilayer system with different λ ϕ, MAX : the position x = 0 is the center of the interface. The positions x < 0 and x > 0 show the layer of FM and NM respectively. 240
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Fig. 3. Schematic illustration of spin-valve stack with diffuse interfaces: the structure is divided into many thin layers with 0.0035 nm thick and the magnetization of interface region slowly changes as the domain wall.
resistance of the structure strongly depends on the relative angle (θ ) between the magnetization of the two ferromagnets. The magnetization of the region far from the interface is allowed to be homogeneous. Across the first interface region (FM/NM), the direction of magnetization is set to be the same as that of the first FM regarded as the pinned layer where its magnitude is gradually decreased due to the interdiffusion between FM and NM ions. For the second interface region (NM/FM), the magnetization rotates continuously between the layers which is analogous to a domain wall and its magnitude is considered from the spatial concentration of magnetic ions. A charge current of density je = 50 MA/cm2 , typical magnitude of current density for GMR investigation [29,30], is injected into a structure comprising NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) to observe the spin accumulation and spin current. Then the MR value can be calculated. The angular dependence of MR is determined by varying the relative angle between the magnetization of two ferromagnets in the range 0 < θ < 2π . The transport parameters of the ferromagnet are chosen for NiFe with β = 0.7, β′ = 0.95, λJ = 3 nm, λ ϕ = 2 nm and the magnitude of equilibrium spin accumulation |m∞| = 1.118 × 108 C/m3 whereas the spin polarization parameters and equilibrium value of spin accumulation are zero for Cu [21]. The spatial variation of spin current and spin accumulation are calculated using Eqs. (3) and (6) respectively. The resistance at any given position is determined from Eq. (8) and the total MR of the structure is subsequently obtained by summing n the resistance-area product along the x-direction, MR = ∑i = 1 RAi . Fig. 4 shows the calculated MR for different relative angles between the magnetization of the ferromagnets. The resistance of the anti-parallel configuration is higher than that of the parallel orientation due to the fact that the spin accumulation tends to follow the direction of
Fig. 2. Spatial profile of spin accumulation for the FM(10 nm)/NM(5 nm) bilayer system with parameters given in the text.
across the interface between FM/NM. We obviously observe that the spin dephasing length only significantly affects the transverse spin accumulation because the longitudinal component of spin accumulation strongly depends on the spin diffusion length. This is very important since the transverse component is related to the enhancement of spin torque in the system. Neglect of the spin dephasing effect in spin transport model is equivalent to setting λ ϕ = ∞, which is applicable for materials with λ ϕ > λJ . However, for the case of soft materials such as NiFe, λ ϕ > λJ , it seems important to include the spin dephasing effect into the spin transport model. In addition, we also notice that the peak in m⊥ moves further into the NM with increasing λ ϕ . These peaks observed at interface region correspond to the value of k1 where −2 −2 (k1 ± ik2) = λ trans ± iλJ−2 and λ trans = λ ϕ−2 + λsf−2 . In NM, the value of k1 is mainly dominated by λ trans since λJ,NM is very high. 3.2. Spin transport in a magnetic trilayer system In this section, to demonstrate the advantage of the proposed model as a powerful tool for GMR calculation, the generalized formalism of spin accumulation described above is applied to a structured film to calculate the magnetoresistance (MR) for a trilayer FM/NM/FM stack with arbitrary noncollinear magnetization configuration. This is the simplest model of the reference and sensing layer of a spin-valve stack. We focus on the effects of interface mixing and its effects on the spin accumulation and spin current. As illustrated in Fig. 3, the trilayer system consisting of two ferromagnets separated by a nonmagnet is discretized into many thin layers in order to model the spatial concentration of magnetic ions used to calculate the spatial variation of transport parameters and magnetization. The intermixing between ions of two different materials produces a diffuse interface region which is modeled as described earlier. The
Fig. 4. Angular dependence of magnetoresistance of the spin-valve structure with injected current density of je = 50 MA/cm2 : The relative angle between magnetization in two ferromagnets is varied. 241
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Fig. 6. ΔRA and MR ratio of NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) with different interfacial thicknesses.
Fig. 5. MR ratio and ΔRA as a function of spin dephasing length (λ ϕ, MAX ) in NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) structure.
magnetization. For the anti-parallel case, spin accumulation changes rapidly at the interface leading to a high gradient of spin accumulation. We make a comparison with fitting function, a + bsin2 (θ /2) , which is used to describe the GMR effect [31–35]. The result shows a good agreement and the parameters a and b in fitting function give the values of RP and ▵R , respectively. The GMR ratio is found to be 21%. This shows that the proposed model is an effective tool for GMR calculation, giving a realistic value of the GMR ratio and an angular variation of resistance consistent with experiment [32]. Next, we perform a study of the effect of spin dephasing on the enhancement of resistance change (ΔRA) and MR ratio for the structure NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) with interface thickness of 1 nm. As illustrated in Fig. 5, it is observed that increasing spin dephasing length (λ ϕ, MAX ) results in a significant decrease of the resistance change and MR ratio. The spin dephasing length varies from 0.001 to 1000 giving rise to nearly 20% change in MR ratio. The effect becomes especially strong for λ ϕ comparable with λJ . To study the important factors influencing the spin transport properties, the effect of spin dephasing, interdiffusion, spin polarization and current density on resistance change and GMR will be next investigated.
3.3. Parametric studies We have carried out a parametric investigation of the interfacial scattering effect on GMR using the spin dephasing to characterize the disorder arising from the interdiffusion. The proposed model allows us to study this effect in a structure of NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) by varying the interfacial thickness controlled by the degree of intermixing between atoms of two adjacent materials. A charge current of density je = 50 MA/cm2 is injected to the structure with interfacial thickness varied from 0.3 to 1.5 nm. We have investigated the magnetoresistance and resistance change of the trilayer structure with different degrees of intermixing. Fig. 6 demonstrates that the increasing interface thickness results in a larger spin dependent scattering which eventually leads to higher ΔRA and MR ratio. This confirms that the nature of the interface becomes a significant factor in the spin transport behavior. Our results give the same trend with previous experimental [36–38] and numerical [39] studies. Refs. [36–38] experimentally investigated the effect of annealing temperature on GMR of the currentperpendicular-to-plane (CPP) spin-valves using heusler alloys. It is well known that the interface thickness is associated with the annealing temperature, therefore we could make an indirect comparison between the trend of the experimental results and ours. In addition, it is found experimentally that high spin scattering at the interface provides the improvement of ΔRA and the enhancement of the MR ratio, which is confirmed by the model calculations. We also study the influence of the spin polarization parameter (β ) and injected current density ( je ) on MR and resistance change in the
Fig. 7. (a) MR ratio and ΔRA as a function of spin polarization parameter, and (b) RAAP, RAP and normalized ΔRA as a function of injected current density in NiFe(5 nm)/Cu(2 nm)/NiFe(5 nm) structure.
trilayer structure. As shown in Fig. 7(a), high MR ratio can be achieved by using material with a high spin polarization parameter. This is due to the fact that the CPP-GMR is associated with bulk and interface spin scattering which strongly depends on the spin polarization of the material [37,40–42]. Furthermore, we demonstrate the dependence of ΔRA on the injected current density. An electric current density ranging from 10 to 16 MA/cm2 is injected into the structure. Increasing injected current density gives rise to a reduction of MR and ΔRA as shown in Fig. 7(b). To make the comparison with experimental study in Ref. [43], the value of ΔRA is normalized by that of je = 10 MA/cm2 . Our results give the same trend with previous experiments by Kimura et al. [43] which measured the magnetoresistance of Py/Cu spin-valve structure as a function of injected current density. In summary, spin dependent interfacial scattering, in addition to the spin transport parameters and current density, plays an important role in achieving high output MR for the potential read sensors in hard drives with high areal density. Our proposed model provides a good description of spin transport in magnetic structures and provides a practically useful tool for appropriated 242
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read sensor design with high performance.
barriers, Nat. Mater. 3 (2004) 862–867. [8] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, Giant room-temperature magnetoresistance in single-crystal fe/mgo/fe magnetic tunnel junctions, Nat. Mater. 3 (2004) 868–871. [9] S. Zhang, P.M. Levy, Enhanced spin-dependent scattering at interfaces, Phys. Rev. Lett. 77 (1996) 916–919. [10] M. Ishikawa, H. Sugiyama, T. Inokuchi, K. Hamaya, Y. Saito, Effect of the interface resistance of cofe/mgo contacts on spin accumulation in silicon, Appl. Phys. Lett. 100 (7) (2012). [11] S. Wang, Y. Xu, K. Xia, First-principles study of spin-transfer torques in layered systems with noncollinear magnetization, Phys. Rev. B 77 (2008) 184430 . [12] A. Shpiro, P.M. Levy, S. Zhang, Self-consistent treatment of nonequilibrium spin torques in magnetic multilayers, Phys. Rev. B 67 (2003) 104430 . [13] M. Ruggeri, C. Abert, G. Hrkac, D. Suess, D. Praetorius, Coupling of dynamical micromagnetism and a stationary spin drift-diffusion equation: a step towards a fully self-consistent spintronics framework, Physica B (2015). [14] P. Ho, G.C. Han, R.F.L. Evans, R.W. Chantrell, G.M. Chow, J.S. Chen, Perpendicular anisotropy l10-fept based pseudo spin valve with ag spacer layer, Appl. Phys. Lett. 98(13). [15] J. Jung, Z. Jin, Y. Shiokawa, M. Sahashi, Effect of spin-dependent interface resistance at (001)-oriented fe/ag interface on giant magnetoresistance effect, Appl. Phys. Express 8 (7) (2015) 073005 . [16] Q. Zhang, P. Li, Y. Wen, C. Zhao, J.W. Zhang, A. Manchon, W.B. Mi, Y. Peng, X.X. Zhang, Anomalous hall effect in fe/au multilayers, Phys. Rev. B 94 (2016) 024428 . [17] L. O’Brien, D. Spivak, J.S. Jeong, K.A. Mkhoyan, P.A. Crowell, C. Leighton, Interdiffusion-controlled kondo suppression of injection efficiency in metallic nonlocal spin valves, Phys. Rev. B 93 (2016) 014413 . [18] S. Zhang, P.M. Levy, A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett. 88 (2002) 236601 . [19] Z. Nedelkoski, P.J. Hasnip, A.M. Sanchez, B. Kuerbanjiang, E. Higgins, M. Oogane, A. Hirohata, G.R. Bell, V.K. Lazarov, The effect of atomic structure on interface spinpolarization of half-metallic spin valves: Co2mnsi/ag epitaxial interfaces, Appl. Phys. Lett. 107(21). [20] P. Bal, J. Barna, J.-P. Ansermet, Transverse spin penetration length in metallic spin valves, J. Appl. Phys. 113(19). [21] P. Chureemart, R. Cuadrado, I. D’Amico, R.W. Chantrell, Modeling spin injection across diffuse interfaces, Phys. Rev. B 87 (2013) 195310 . [22] C. Petitjean, D. Luc, X. Waintal, Unified drift-diffusion theory for transverse spin currents in spin valves, domain walls, and other textured magnets, Phys. Rev. Lett. 109 (2012) 117204 . [23] C.A. Akosa, W.-S. Kim, A. Bisig, M. Kläui, K.-J. Lee, A. Manchon, Role of spin diffusion in current-induced domain wall motion for disordered ferromagnets, Phys. Rev. B 91 (2015) 094411 . [24] P. Chureemart, I. D’Amico, R.W. Chantrell, Model of spin accumulation and spin torque in spatially varying magnetisation structures: limitations of the micromagnetic approach, J. Phys.: Condens. Matter 27 (14) (2015) 146004 . [25] S. Lepadatu, Unified treatment of spin torques using a coupled magnetisation dynamics and three-dimensional spin current solver, Scientific Rep. 7 (1) (2017) 12937 . [26] A. Fert, H. Jaffrès, Conditions for efficient spin injection from a ferromagnetic metal into a semiconductor, Phys. Rev. B 64 (2001) 184420 . [27] N.L. Chung, M.B.A. Jalil, S.G. Tan, S. Bala Kumar, Interfacial resistance and spin flip effects on the magnetoresistance of a current perpendicular to plane spin valve, J. Appl. Phys. 103(7). [28] C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, G. Hrkac, D. Praetorius, D. Suess, A three-dimensional spin-diffusion model for micromagnetics, Sci. Rep. 5 (2015) 14855 26442796[pmid]. [29] K. Nikolaev, P. Kolbo, T. Pokhil, X. Peng, Y. Chen, T. Ambrose, O. Mryasov, allheusler alloy current-perpendicular-to-plane giant magnetoresistance, Appl. Phys. Lett. 94 (22) (2009) 222501 . [30] T. Nakatani, N. Hase, H. Goripati, Y. Takahashi, T. Furubayashi, K. Hono, Co-based heusler alloys for cpp-gmr spin-valves with large magnetoresistive outputs, IEEE Trans. Magn. 48 (5) (2012) 1751–1757. [31] C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, A. Manchon, D. Praetorius, D. Suess, Simultaneous resolution of the micromagnetic and spin transport equations applied to current-induced domain wall dynamics, arXiv:1512.05519v4. [32] S. Urazhdin, R. Loloee, W. Pratt Jr, Noncollinear spin transport in magnetic multilayers, Phys. Rev. B 71 (10) (2005) 100401 . [33] S. Urazhdin, R. Loloee, W. Pratt Jr, Spin transport at interfaces in magnetic multilayers, J. Appl. Phys. 99 (8) (2006) 08G504 . [34] F. Montaigne, C. Tiusan, M. Hehn, Angular dependence of tunnel magnetoresistance in magnetic tunnel junctions and specific aspects in spin-filtering devices, J. Appl. Phys. 108 (6) (2010) 063912 . [35] T. Qu, R. Victora, Angular dependence of current perpendicular to plane giant magnetoresistance in multilayer nanowire, J. Appl. Phys. 111 (7) (2012) 07C516 . [36] T. Kubota, Y. Ina, Z. Wen, H. Narisawa, K. Takanashi, Current perpendicular-toplane giant magnetoresistance using an l 1 2 ag 3 mg spacer and co 2 fe 0.4 mn 0.6 si heusler alloy electrodes: spacer thickness and annealing temperature dependence, Phys. Rev. Mater. 1 (4) (2017) 044402 . [37] Y. Du, T. Furubayashi, T.T. Sasaki, Y. Sakuraba, Y.K. Takahashi, K. Hono, Large magnetoresistance in current-perpendicular-to-plane pseudo spin-valves using co2fe(ga0.5ge0.5) heusler alloy and agzn spacer, Appl. Phys. Lett. 107 (11) (2015) 112405 . [38] S. Li, Y.K. Takahashi, T. Furubayashi, K. Hono, Enhancement of giant magnetoresistance by l21 ordering in co2fe(ge0.5ga0.5) heusler alloy current-perpendicular-
4. Conclusions The spin transport behavior in magnetic structure is significantly affected by the presence of diffuse interfaces due to the intermixing of ions which is expected for the sputtering growth process. In this work, the spin transport in bilayer systems with noncollinear configuration of magnetization is studied by using the generalized spin accumulation model including the effect of diffuse interfaces. The expected scattering due to disorder in the interface is modeled by introducing a strong local spin dephasing at the interface. The effect of spin dephasing length is first studied. The results suggest that inclusion of locally enhanced spin dephasing into the model is a reasonable representation of the spin scattering associated with diffuse interfaces. The modified spin accumulation model is then applied to a FM/NM/FM structure with variable direction of magnetization to calculate the magnetoresistance, which is related to the gradient of spin accumulation and the spin current. The total resistance of parallel and anti-parallel configurations is calculated giving the GMR value. Finally, we parametrically study the effect of the interdiffusion, spin polarization and injected current density on device performance. We found that the interdiffusion significantly affects the resistance change. In addition, high MR output can be achieved by using ferromagnetic materials with high spin polarization. The results show that the formalism of spin accumulation, modified to include local variation of properties is a realistic approach to the calculation of spin transport in magnetic nanostructures with diffuse interfaces. As device size and magnetic film thicknesses decrease, it is expected that interface diffusion and finite size effects will become increasingly important. The model described here is shown to be a reasonable representation of the enhanced spin scattering at interfaces and can form an important basis both for physical investigation and device design. Acknowledgements N.S. would like to acknowledge PhD funding from Seagate technology (Thailand). P.C. gratefully acknowledges the funding from Mahasarakham University and Thailand Research Fund under Grant No. MRG6080048. J. C. would like to acknowledge the funding from Mahasarakham University and IAPP 1R2/100151. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.jmmm.2019.04.010. References [1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, H. Ohno, A perpendicular-anisotropy cofeb–mgo magnetic tunnel junction, Nat. Mater. 9 (9) (2010) 721–724. [2] H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru, M. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yuasa, T. Taniguchi, H. Arai, H. Imamura, Spin-torque oscillator based on magnetic tunnel junction with a perpendicularly magnetized free layer and in-plane magnetized polarizer, Appl. Phys. Express 6 (10) (2013) 103003 . [3] Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner, V.E. Demidov, S.O. Demokritov, I.N. Krivorotov, A perpendicular-anisotropy cofeb–mgo magnetic tunnel junction, Nat. Commun. 5. [4] A.V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R.S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W.H. Butler, P.B. Visscher, D. Lottis, E. Chen, V. Nikitin, M. Krounbi, Basic principles of stt-mram cell operation in memory arrays, J. Phys. D: Appl. Phys. 46 (7) (2013) 074001 . [5] M.N. Baibich, J.M. Broto, A. Fert, F.N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. Chazelas, Giant magnetoresistance of (001)fe/(001)cr magnetic superlattices, Phys. Rev. Lett. 61 (1988) 2472–2475. [6] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39 (1989) 4828–4830. [7] S.S.P. Parkin, C. Kaiser, A. Panchula, P.M. Rice, B. Hughes, M. Samant, S.-H. Yang, Giant tunnelling magnetoresistance at room temperature with mgo (100) tunnel
243
Journal of Magnetism and Magnetic Materials 484 (2019) 238–244
N. Saenphum, et al.
suppressing comn anti-sites in co2mn (ge0. 75ga0. 25) heusler alloy thin film, Appl. Phys. Lett. 108 (12) (2016) 122404 . [42] Ikhtiar, S. Kasai, Y. Takahashi, T. Furubayashi, S. Mitani, K. Hono, Temperature dependence of magneto-transport properties in co2fe (ga0. 5ge0. 5)/cu lateral spin valves, Appl. Phys. Lett. 108 (6) (2016) 062401 . [43] T. Kimura, N. Hashimoto, S. Yamada, M. Miyao, K. Hamaya, Room-temperature generation of giant pure spin currents using epitaxial co2fesi spin injectors, NPG Asia Mater. 4 (2012) e9 .
to-plane pseudo spin valves, Appl. Phys. Lett. 103 (4) (2013) 042405 . [39] B. Elsafi, F. Trigui, Z. Fakhfakh, Influence of interfacial scattering on giant magnetoresistance in co/cu ultrathin multilayers, J. Appl. Phys. 109 (3) (2011) 033910 . [40] H. Goripati, T. Furubayashi, Y. Takahashi, K. Hono, Current-perpendicular-to-plane giant magnetoresistance using co2fe (ga1- x ge x) heusler alloy, J. Appl. Phys. 113 (4) (2013) 043901 . [41] S. Li, Y. Takahashi, Y. Sakuraba, N. Tsuji, H. Tajiri, Y. Miura, J. Chen, T. Furubayashi, K. Hono, Large enhancement of bulk spin polarization by
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