Spin-current-induced magnetoresistance in trilayer structure with nonmagnetic metallic interlayer

We have theoretically investigated the spin Hall magnetoresistance (SMR) and Rashba–Edelstein magnetoresistance (REMR), mediated by spin currents, in a ferrimagnetic insulator/nonmagnetic metal/heavy metal system in the diffusive regime. The magnitude of both SMR and REMR decreases with increasing thickness of the interlayer because of the current shunting effect and the reduction in spin accumulation across the interlayer. The latter contribution is due to driving a spin current and persists even in the absence of spin relaxation, which is essential for understanding the magnetoresistance ratio in trilayer structures.

M agnetoresistance effects are important for applications in sensor and memory devices and for investigation of spin-dependent transport. [1][2][3] Recently, a new type of magnetoresistance has been demonstrated, in which a spin current generated via spin-orbit coupling from an applied charge current plays a central role. [4][5][6][7][8][9][10][11][12] The generated spin current interacts with magnetization and magnetic field and changes its magnitude. 6,8,11,13) When the spin current is converted back to a charge current by spin-orbit coupling, 14,15) the spin-current transport modulates the conductivity of the system, and then gives rise to magnetoresistance. This spin-current-induced magnetoresistance is now recognized as an indispensable tool for investigating the spin transport in ferromagnetic material=metal heterostructures. [16][17][18][19] The spin Hall magnetoresistance (SMR) refers to the resistance change due to the spin current generated by the spin Hall effect (SHE), which interacts with a ferromagnetic material. 4,6) Typically in materials consisting of heavy metals such as Pt and W, 4,9) a spin current is induced by an applied charge current, j c , due to the SHE. This spin current propagates in the material and forms spin accumulation, s , around the system edges. 20) When s acts on a ferromagnetic material, it exerts spin transfer torque on the magnetization m, which is given by 13) Here, the spin current is defined to be positive when it flows out of the F layer, and e > 0, G r , m, and s j interface respectively denote the elementary charge, the real part of the mixing conductance per unit area, and the magnetization unit vector, the spin accumulation at the interface. Since j STT s is absorbed by the magnetization, the resultant spin accumulation s decreases, which results in the reduction in backflow spin current and thus the additional charge current due to the SHE (Fig. 1). As j STT s depends on the magnetization direction, the spin-current absorption finally appears as a magnetoresistance effect, under which the total conductivity σ shows a different dependence on m from the conventional anisotropic magnetoresistance, 6) i.e., where n, σ 0 , and Δσ SMR respectively denote the unit vector normal to the junction interface and the conductivities insensitive and sensitive to the magnetization. In the context of SMR research, an interlayer with a large spin-diffusion-length is often inserted between the ferromagnetic material and heavy metal layers to distinguish the SMR from the static magnetic proximity effect. 4,5,21) For example, the effects of Cu and Au insertion were examined in ferrimagnetic insulator yttrium iron garnet (Y 3 Fe 5 O 12 : YIG)= heavy metal Pt bilayer systems, and the existence of the SMR has been confirmed. However, the SMR magnitude decreases induced by the reciprocal process of the spin-charge conversion in the system, which is observable as magnetoresistance. This magnetoresistance effect relies on the spin transport between the conversion layer and the interface of the F layer. more rapidly than expected even when a current shunting effect is considered. Although such trilayer SMR provides a versatile method to extract spin diffusion length, 22) an analytic expression for quantitative interpretation has not yet been obtained.
In this letter, we derive the interlayer thickness dependence of the SMR in the diffusive regime based on the spin diffusion equation and magneto-circuit theory, 1,13) and show that the intrinsic magnitude of the SMR decreases as a result of spin-current transport over the trilayer structure. We consider a system composed of a ferrimagnetic insulator (F), a nonmagnetic metal (N), and a heavy metal (H) used for converting a charge current to a spin current. We find that the intrinsic magnitude of the SMR decreases even when the spin relaxation in the N layer is negligibly small. This is due to the reduction in the spin accumulation for driving spin transfer torque. In addition to the SMR, we also calculated the recently observed Rashba-Edelstein magnetoresistance (REMR), caused by spin-charge conversion at the N=H interface, 12,15,23,24) because it is also affected by the spin accumulation reduction. The derived expression for the trilayer structure will be useful for understanding the spin-currentinduced magnetoresistance effects.
The system coordinate is shown in Fig. 1. The layers are stacked in the z-direction, and we assume translational symmetry in the xy plane. The thicknesses of the F, N, and H layers are d F , d N , and d H , respectively. The N=H interface is at z = 0 and the F=N interface z = −d N . Hereafter, we will use the superscript and subscript ¼ ðF; N; HÞ to represent the corresponding quantity in the α layer. In this F=N=H system, we calculated the distribution of spin accumulation s ðzÞ and spin current flowing in the z-direction j s ðzÞ with its vector direction representing its spin polarization using the diffusion equation @ 2 z s ðzÞ ¼ s ðzÞ= 2 , where λ α is the spin diffusion length. 1) We imposed the boundary conditions with Eq. (1) and j STT s ¼ j N s ðÀd N Þ at the F=N interface (z = −d N ), zero spin current at the H surface (z = d H ), and N=H s N s ð0Þ ¼ H s ð0Þ at the N=H interface. For the SMR, continuity of j s ð0Þ is assumed at the N=H interface and, for the REMR, the derived condition in Eq. (9) is assumed. The spin current in the α layer is given by 6) where the vector represents the spin polarization, and σ α denotes the conductivity. j SHE; s ¼ SHE z Â E represents the SHE-induced spin current, where SHE denotes the spin Hall angle, z the unit vector of the z-axis, and E the applied electric field. We assumed j SHE,N s ¼ 0 because 2 SHE , which is proportional to Δσ SMR , 6) is smaller in the N layer in experiments than in the H layer. 4,5,21) The SMR magnitude is determined from the difference in the magnitude of the spin current flowing through the N=H interface with j  1)]. In the trilayer structure, this spin current in the N layer is given by where À ¼ ð =2 Þ tanhðd = Þ. G 0 r denotes the effective mixing conductance between the F and H layers in the trilayer configuration F=N=H. G 0 r is smaller than G r , so the effective spin transfer torque on the F layer is reduced, which in turn reduces the SMR magnitude. This reduction originates from (i) the decay of spin current and (ii) the reduction in s for driving j s in the N layer. Although the former is partially considered in the earlier analysis, 4,5) the latter is not. The contribution from (ii) can be clearly recognized in the no-spinrelaxation limit (λ N ≫ d N ), where we obtain G 0 r ¼ ðG À1 r þ 2d N = N Þ À1 , which still shows the apparent dependence of G 0 r on the N layer. In this limit, Eq. (3)  ). As one can see, when the absorption is introduced, μ s for driving j s decreases with j s propagation, and the dependence of j H s on m becomes weaker than that in the bilayer system [see Fig. 2 The magnetoresistance ratio MR ≡ Δσ SMR =σ 0 is given by where R shunt ¼ H d H =ð P d Þ is the current shunting factor for the SMR. For local magnetoresistance effects, such as the anisotropic magnetoresistance (AMR), its intrinsic magnetoresistance ratio MR int is related to MR in the multilayers via MR = R shunt MR int by considering parallel circuiting. 9,12) Obviously, Eq. (8) shows the dependence of MR int on G 0 r and Γ N , which is a distinct contribution from the shunting effect. Since À H % G r ≧ G 0 r holds in typical experiments (see the inset in Fig. 3), MR is approximately proportional to G 0 r and thus is reduced by the spin transport in the N layer. Figure 3 shows that MR int for the SMR depends on the thickness d N . As expected from the discussion in the last paragraph, MR int decreases even when λ N ≫ d N . In this regime, Γ N becomes zero and only the effective mixing conductance G 0 r ¼ ðG À1 r þ 2d N = N Þ À1 contributes to the SMR, which again shows the importance of the reduction in the spin accumulation across the N layer.
The experimentally reported MR int values of YIG=Cu=Pt trilayers in Ref. 4 are smaller than that expected from Eq. (8). This difference can be attributed to the thickness dependence of the spin diffusion length due to that of the conductance and the spin memory loss effect at the interfaces. [25][26][27] The spin diffusion length decreases at a smaller thickness because of the interfacial scattering, 5,12,28) such that the decreased spin-current transmission reduces the SMR magnitude. Similarly, the spin memory loss effect reduces the spin current transmission through the interface. Besides these factors, the spin accumulation reduction considered in this report is important, especially for the larger thickness limit, where most of the spin transport is determined by the bulk properties.
Finally, we apply our calculations to the REMR. Since the REMR appears in F=N=H trilayer structures, it is relevant to the trilayer SMR, which has not been clarified yet.
Under the Rashba-Edelstein effect (REE), an applied electric field induces spin accumulation at the N=H interface, which is given by where λ REE is the REE coefficient, and N and τ s denote the density of states and spin relaxation time for the N=H interface, respectively. 23) Note that the second term on the righthand side of Eq. (9), given by Eqs. (5) and (6) with j SHE s ¼ 0, is the contribution of outflow spin currents driven by the REE and is crucial to describing the effect of j STT s on s , i.e., the resultant magnetoresistance. We use Eq. (9) for the boundary condition instead of the continuity of j s . Considering the spinto-charge conversion relation, j add c ¼ À REE ðeN= s Þ N=H s Â zðzÞ, 15,23) and Eqs. (4), (5), and (9), the magnetoresistance ratio of the REMR is given by whereR shunt ¼ int =ð int þ P d Þ denotes the shunting factor for the REMR and σ int expresses the sheet conductivity of the N=H interface. As in Eq. (8), MR REMR in Eq. (10) also decreases with increasing d N . Figure 4 shows the calculated thickness dependence of MR int REMR ¼ MR REMR =R shunt . In Ref. 12, the d N dependence of MR REMR is analyzed by assuming constant MR int REMR , and the experimental data shows a sharper peak at a smaller d N than the calculated data. This difference can be explained by the MR int REMR reduction revealed here. Note that our calculations are for systems with magnetic insulators, but the experiment was performed with a metallic ferromagnet. 12) This may affect the MR value,  Fig. 4. (Color online) d N thickness dependence of the magnetoresistance ratio MR int REMR of the REMR at N ¼ 10; 100; 1 nm, where the contribution from the current shunting effect is factored out by MR int REMR ¼ MR REMR =R shunt . We assumed N = Ag and H = Bi, so that σ N = 3.6 × 10 6 m −1 ·Ω −1 for the N layer, 12) and σ H = 7.5 × 10 4 m −1 ·Ω −1 and λ H = 50 nm for the H layer, 29) τ s = 3.2 × 10 −15 s, N ¼ 9:1 Â 10 36 m −2 ·J −1 , and σ int = 2.3 × 10 −4 Ω −1 for the N=H interface are used. 12) The REE coefficient is set as λ REE = 0.3[1 − exp(−d N =t int )] nm with t int = 0.5 nm to include the suppression of the interface formation. 12) For the F=N interface, G r = 5 × 10 14 m −2 ·Ω −1 is assumed. The expected maximum MR value is about 0.001 in this calculation, which is in good agreement with the observed value. 12) The inset shows the calculated values of G 0 r , Γ α , and e 2 N= s .
but only a small difference may be expected for the thickness dependence, as in the SMR case. 9) In summary, we formulated the magnitude of the spincurrent-induced magnetoresistance effects in a ferrimagnetic insulator=nonmagnetic metal=heavy metal trilayer structure with the spin Hall and Rashba-Edelstein effects. We showed that the spin-current-induced magnetoresistances are sensitive to the spin-current transport in the interlayer, which gives rise to the reduction in the magnetoresistance ratio in addition to the shunting of the applied charge current. Our derived thickness dependence will be useful for quantitatively understanding the spin transport of the inserted layer using the spin-current-induced magnetoresistances.