Enhancing current-induced torques by abutting additional spin polarizer layer to nonmagnetic metal layer

Recently, the switching of a perpendicularly magnetized ferromagnet (FM) by injecting an in-plane current into an attached non-magnet (NM) has become of emerging technological interest. This magnetization switching is attributed to the spin-orbit torque (SOT) originating from the strong spin-orbit coupling of the NM layer. However, the switching efficiency of the NM/FM structure itself may be insufficient for practical use, as for example, in spin transfer torque (STT)-based magnetic random access memory (MRAM) devices. Here we investigate spin torque in an NM/FM structure with an additional spin polarizer (SP) layer abutted to the NM layer. In addition to the SOT contribution, a spin-polarized current from the SP layer creates an extra spin chemical potential difference at the NM/FM interface and gives rise to a STT on the FM layer. We show that, using typical parameters including device width, thickness, spin diffusion length, and the spin Hall angle, the spin torque from the SP layer can be much larger than that from the spin Hall effect (SHE) of the NM.

to the y-direction. In our theoretical model, as an approximation, we divide the device geometry into a combined sub-structure of the SP/NM (x-direction) and NM/FL (z-direction) layers and treat them separately.

Model Calculation Methods
First, we calculate the spin accumulation for the SP/NM sub-structure. For simplicity, here we ignore the upper FM layer (FL) (the interface scattering at the SP/NM interface is also neglected). By solving the one-dimensional drift-diffusion equation 16 , one obtains the y-component of the spin accumulation in the NM layer as follows.
where ρ F and ρ N are the resistivities of SP and NM layers, L is the length (or width) of the fixed FM layer (SP), l sf F and l sf N are the spin-flip diffusion lengths of the FM and NM layers, respectively, E x is the applied electric field in the x-direction, and β is the spin polarization of the SP layer (− 1 ≤ β ≤ 1). Here x = 0 is chosen as the SP/NM interface.
Next, we focus on the NM/FL sub-structure. When we apply the external electric field in the x-direction, in the NM layer, the charge and spin current densities flowing in the z-direction are given by ref. 17 (2) where σ is the conductivity of NM layer, σ SH is the spin Hall conductivity of NM layer, and ε ijz is the Levi-Civita symbol. Note that the usual definition of the spin current is 2 , , where the subscripts i and z stand for the spin orientation and the flow directions of the spin current, respectively. For the FL, the equations for the charge and spin currents are as follows.
where M is the unit vector along the magnetization and β 0 is the spin polarization of the FL. The SP layer provides additional spin accumulation to the NM layer, which decays exponentially in the x-direction. Because it is difficult to obtain an analytical solution for the two-dimensional spin diffusion equation, as an approximation, we take an ansatz for the spin accumulation in the NM layer by simply adding an average spin accumulation from the SP layer (Equation (1)) to the solution of the diffusion equation in the z-direction. This gives where we use an approximation , and w is the width of the structure.
From Eqs (1) and (2), we obtain the averaged current flowing in z-direction µ = ∂ + σ σ j z e z e K l SH sf N . As in ref. 18, we assume that the sample is infinite in the y-direction and sufficiently thin in the z-direction. In the steady state where C is a constant and F(x) represents the external electric field contribution, F(x) ∝ eE x x. Note that the current flowing in the z-direction is zero, j z = 0 (this condition practically holds for an usual three terminal device including a magnetic tunnelling junction which has large resistance in z-direction), thus giving Using Eqs (2), (4), and (6), we obtain At the NM/FL interface, the charge and spin currents are given by refs 17,19-21 where the subscripts T and L represent the transverse and longitudinal components, G ↑ (G ↓ ) is the interface conductivity of majority (minority) spin, G ↑↓ is the mixing conductance, and ∆µ ∆µ ( ) s is the charge (spin) chemical potential drop over the interface. As in ref. 17, we assume that the spin dephasing length in the FL is very short, so that j ( ) z s T is fully absorbed at the NM/FL interface. Then, the spin torque is written as where γ is the gyromagnetic ratio and M s is the magnetization per unit volume. The spin torque can then be rewritten in terms of damping-like and field-like torques as follows.
1 tanh( / ) In Eqs (11) and (12), the spin torques, which are localized at the interface of the NM/FL, are averaged over the FL layer's thickness t F . The first term of each torque corresponds to the spin Hall contribution without the SP layer, which was computed by Haney et al. 17 . The spin polarizer effects are reflected in the second term. Because the spins in the NM layer are partially polarized by the SP layer, even without the SHE, a nonzero spin chemical potential drop at the NM/FL interface is induced that produces a spin torque. We note that the spin accumulation caused by the SP layer exerts a torque on the FL even without a direct current-flow perpendicular to the interface. This kind of spin torque has been experimentally observed in non-local geometry 22 and also has been shown as a lateral spin torque caused by inhomogeneous magnetization 21,23,24 . 1 17 . The applied electric field in the NM is chosen as E x = 2 × 10 4 V/m so that the charge current density in the x-direction is approximately σ N E x ≈ 10 11 A/m 2 . Other parameters are chosen as follows: magnetization of the FL is M S = 1.0 MA/m, the length (or width) of SP layer is L = 20 nm, the spin Hall angle is θ SH = 0-0.9, and the spin polarization is β = 0-1.0. Figure 2 shows the y-component spin accumulation µ y s in the NM of the NM (5 nm)/FL (1.5 nm) structure for θ SH = 0.3. Note that the net amount of spin accumulation at the upper edge (z = 0) significantly increases as β increases, which in turn enhances the spin torques in Eqs. (11) and (12). The dependence of the four parameters (θ SH , t N , l sf N , and w) on the damping-like spin torques T D is summarized in Fig. 3. As shown in Fig. 3(a), the resultant spin torques increase with increasing θ SH . The dependence of T D on t N and l sf N in Eq. (11) is complicated. In our parameterization, the resulting spin torques increase with increasing t N (Fig. 3(b)) and have maximum values with varying l sf N (Fig. 3(c)). Because the additional spin torque from the spin polarization decays along the x-direction, for nonzero β, the spin torque decreases with increasing NM width w (blue and red lines of Fig. 3(d)). For all plots, the spin polarization dramatically enhances the spin torques. We note the field-like torques T F depicted in Fig. 4 are an order of magnitude smaller than the damping-like spin torques T D because, in our parameterization,

Results
Re[ ], and the dependence of the four parameters on both torques is similar. We propose that several other device geometries may also improve the switching performance. For example, another SP layer may be abutted on the other side (right side) of the NM. Because of spin diffusion, the magnitudes of spin torques from the SP layer decay along the x-direction; however, the decrease of the SP-layer effect would be compensated by another SP layer (oppositely magnetized) on the other side of the NM. In addition, by tilting the magnetization to z-direction, the proposed device structure may allow field-free switching. We note that conventional SOT switching requires an additional in-plane magnetic field for deterministic switching, which is not suitable for device engineering. Recently several reports have resolved this problem by breaking structural inversion symmetry 27,28 or exchange bias from an antiferromagnetic layer [29][30][31][32] .

Discussion
We have investigated the effects of both the SP layer and the spin Hall effect on the spin torque acting on the magnetization of the FM layer in an NM/FM heterostructure. Because of SP layer, the conduction electrons in the NM are partially polarized and provide an additional spin chemical potential change at the NM/FL interface. In our device structure, therefore, the SP layer was an additional source of spin torque. We also investigated the dependence of spin torque on parameters including θ SH , t N , l sf N , and w. Note that the resultant spin torques are maximized by minimizing the width of the sample, which can be beneficial for ultra-dense memory applications. Additionally, we have shown that, in typical parameterization, the spin torque from the SP layer is more effective than that from the spin Hall effect. Thus, the presence of the SP layer can significantly enhance the spin torque. With respect to the switching efficiency, we believe that our result provides insights into possible practical applications of STT-MRAM devices, which employ the current in-plane geometry.
We would like to note that the proposed structure can be applied to an MTJ utilizing standardized microelectronic fabrication technologies. For example, when a bottom-pinned MTJ (an FL on the top in the MTJ) is considered, an NM conductor can be patterned after film deposition, and then followed by SP layer deposition, patterning and field setting. Nevertheless, process development effort is necessary to achieve precise dimensional control.