# Influence of Shape Anisotropy on Magnetization Dynamics Driven by Spin Hall Effect

Authors: X. G. Li; Z. J. Liu; X. Y. Xie; A. G. Kang; W. N. Fu
Journal: Advances in Materials Science and Engineering
 (2016)

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## Abstract

As the lateral dimension of spin Hall effect based magnetic random-access memory (SHE-RAM) devices is scaled down, shape anisotropy has varied influence on both the magnetic field and the current-driven switching characteristics. In this paper, we study such influences on elliptic film nanomagnets and theoretically investigate the switching characteristics for SHE-RAM element with in-plane magnetization. The analytical expressions for critical current density are presented and the results are compared with those obtained from macrospin and micromagnetic simulation. It is found that the key performance indicators for in-plane SHE-RAM, including thermal stability and spin torque efficiency, are highly geometry dependent and can be effectively improved by geometric design.


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## Body

## 1. Introduction

The effect of spin-transfer torque [1–3] has been investigated extensively in the last decade due to its potential applications in spin-transfer torque magnetic random-access memory (STT-RAM) [4–7], nanoscale microwave generators [7–10], and spin based logic devices [11]. The functional element utilizing spin-transfer torque is commonly a magnetic tunnel junction (MTJ) with a magnetic free layer and reference layer separated by a tunneling barrier. For STT-RAM, the bit is stored by the relative orientation of magnetization of free layer and reference layer, and the polarized current is used to change its state between parallel (P) and antiparallel (AP). In order to balance the demands for low threshold current, high thermal stability, and small element size, MTJ stacks with different materials and structures have been proposed. In the last few years, MTJ with perpendicular magnetization has attracted much attention since the strong interfacial magnetic anisotropy in CoFeB/MgO bilayer reduces the switching current density by canceling out the demagnetization field [12–14]. However, as the element dimension keeps scaling down, time dependent dielectric breakdown (TDDB) becomes a critical issue restraining the performance of perpendicular MTJ [15]. More recently, a three-terminal MTJ utilizing spin Hall effect (SHE) [16] has been demonstrated to be an effective solution [4, 17, 18]. Pure spin current, due to spin orbit interaction, can be generated by applying a current to a heavy metal (β-W [19, 20], β-Ta [21], and Pt [22]) adjacent to the free layer of MTJ.Apparently, the SHE-RAM allows for further scaling down of the nonvolatile memory element. However, there is a critical issue from the scaling perspective, that is, the influence of shape anisotropy on switching characteristics, which needs to be carefully examined. In previous macrospin based researches, simplified expressions of the shape anisotropy were often used with the assumption that the area of free layer is large compared with its thickness. However, as the element dimension is scaled down, this simplification frequently becomes a nontrivial source of error when studying the switching current and thermal stability of STT-RAM elements [23–26]. The problem can be addressed by either micromagnetic modeling or accurate demagnetizing factors for macrospin model.In this paper, we introduce the analytic demagnetizing factors for elliptic film nanomagnets and theoretically investigate the geometric dependent critical current density for SHE-RAM with in-plane magnetization. We present the analytic expressions for both current densities leading to magnetization instability and switching. The analytic results are compared with macrospin and micromagnetic numerical simulation, and good consistency is found below a critical size. We further analyze the geometric dependent thermal stability and spin torque efficiency in room temperature, and it is found that these key performance indicators can be effectively improved by geometric design.The remainder of this paper is organized as follows. Section2 gives the theoretical model and analytic equations for demagnetizing factors and critical current density. Section 3 presents the dependence of zero temperature critical current density on free layer geometry and the comparisons between analytic expression, macrospin, and micromagnetic simulation. Section 4 introduces thermal fluctuation by stochastic LLG equation, and the element characteristics concerning thermal fluctuation are investigated. Section 5 summarizes the paper and draws the conclusion. SI units are used for equations and results in this paper.


## 2. Model Detail

### 2.1. LLGS Equation

Figure1 is a sketch of the in-plane SHE-RAM element. The short axis lies in y direction with length b. The long axis of the free layer lies in x direction with length a, which is also the width of the heavy metal layer, as shown in Figure 1(b).Figure 1
The sketch of in-plane SHE-RAM element.
(a) (b)The magnetization dynamics of the in-plane MTJ driven by SHE polarized current are similar to that of a conventional STT-RAM and can be described by the Landau-Lifshitz-Gilbert equation with an SHE spin torque term [27] as follows:(1)dmdt=-γm×Heff+Hthe+αm×dmdt+aJm×m×σ,where m and σ are the normalized magnetization of the free layer and spin polarization vector, respectively, γ = 1.76 × 1011 T−1 s−1 is the gyromagnetic ratio, α is the Gilbert damping constant, and aJ is the spin torque strength which can be expressed as [28](2)aJ=γħJeθSHeff2μ0etfMs,where ħ is the reduced Planck constant, Je is the charge current density applied to the heavy metal layer in the y direction, e is the electron charge, μ0 is the permeability of vacuum, θSHeff is the material dependent effective spin Hall angle, and tf is the thickness of the free layer. In our latter discussion, we assume α = 0.01, Ms = 1 × 106 A/m, and θSHeff = 0.32 to simulate the CoFeB-based MTJ adjacent to an 8 nm β-W thin film [20]. Hthe is the stochastic field related to thermal fluctuation. Based on the framework of Brown [29], it can be expressed as follows:(3)Hthe=0,Hthe,it1Hthe,jt2=2kbTαμ0γMsVδijδt1-t2,where i,j=1,2, and 3 and Hthe,i refers to the Cartesian component of the stochastic field. kb is the Boltzmann constant, T is the temperature in kelvin, and V is the volume of the free layer. Heff represents the effective field. Since bulk anisotropy is ruled out, the energy density of the free layer with a zero applied field is(4)Etotal=Edemag=12μ0Ms2Nxmx2+Nymy2+Nzmz2,where mx, my, and mz are the three components of the normalized magnetization m and Nx, Ny, and Nz are the demagnetizing factors along the corresponding directions. Since mx2+my2+mz2=1, we have(5)Etotal=12μ0Ms2Nx-Nymx2+12μ0Ms2Nz-Nymz2+12μ0Ms2Ny,and Heff can be derived from the magnetic energy density of the free layer as(6)Heff=Ny-NxMsmx-Nz-NyMsmz,with the easy axis anisotropy field Hk in the long axis direction with strength Ny-NxMs and the demagnetizing field Hd normal to the film plane with strength Nz-NyMs.


### 2.2. Demagnetizing Factors

The analytic demagnetizing factors for elliptical disk can be expressed as a power series expansion of the ratio between the disk thickness and long axis length,τ=tf/a [30, 31], with a “squat elliptic cylinder” limit (b>2tf): (7)Nx=18π2λJ1-J2,Ny=λ8π2J1+J2,Nz=1-Nx-Ny,where(8)J1=16τL0-12+ln⁡τ4I0+τ28L1+14-ln⁡τ4I1+Oτ4,J2=16τγ1+εL-1-L0+32-ln⁡τ41+εI-1-I0-τ2241+εL0-L1-512+ln⁡τ41+εI0-I1+Oτ4,I-1=E1-1λ2,I0=K1-1λ2,I1=β2K1-1λ2,L-1=1+λ22λ2K1-1λ2-ln⁡λ2+1E1-1λ2,L0=-ln⁡λ2K1-1λ2,L1=λ21-ln⁡λ2E1-1λ2-1+λ22λ2K1-1λ2,with aspect ratio λ=a/b, ε=(λ2-1)/(λ2+1), and K, E refer to the complete elliptic integral of the first and second kind, respectively. In previous research works, simplified shape anisotropy was commonly used as Hks=21/b-1/atfMs [32–35]. It however leads to overestimation as Figure 2 depicts. For an MTJ sample with a size of 45 nm × 15 nm × 3 nm, the shape anisotropy field is overestimated by 70%, leading to poor predictions of device characteristics.Figure 2
The deviation percentage caused by simplified shape anisotropy with variedτ and λ.


### 2.3. Current Density for Instability and Switching

The analytical expression of the critical current density necessary for the in-plane MTJ free layer magnetization to become unstable has been introduced by Sun [36] as follows:(9)Jins=2eαtfμ0MsħηHk+Hd2.And the critical current density for magnetization switching is given by [37, 38] as (10)Jsw=4eαtfμ0MsπħηHdHd+Hk,where η is the spin polarization, and for SHE device it should be the effective spin Hall angle. An intermediate current density will drive the magnetization towards oscillation state, which is not sensitive to the initial angle θ0. With results from (6), the analytical critical current densities for an in-plane SHE-RAM can be written as(11)Jins=eαtfμ0Ms2ħθSHeff1-3Nx,Jsw=4eαtμ0Ms2πħθSHeffNz-NyNz-Nx.


## 2.1. LLGS Equation

Figure1 is a sketch of the in-plane SHE-RAM element. The short axis lies in y direction with length b. The long axis of the free layer lies in x direction with length a, which is also the width of the heavy metal layer, as shown in Figure 1(b).Figure 1
The sketch of in-plane SHE-RAM element.
(a) (b)The magnetization dynamics of the in-plane MTJ driven by SHE polarized current are similar to that of a conventional STT-RAM and can be described by the Landau-Lifshitz-Gilbert equation with an SHE spin torque term [27] as follows:(1)dmdt=-γm×Heff+Hthe+αm×dmdt+aJm×m×σ,where m and σ are the normalized magnetization of the free layer and spin polarization vector, respectively, γ = 1.76 × 1011 T−1 s−1 is the gyromagnetic ratio, α is the Gilbert damping constant, and aJ is the spin torque strength which can be expressed as [28](2)aJ=γħJeθSHeff2μ0etfMs,where ħ is the reduced Planck constant, Je is the charge current density applied to the heavy metal layer in the y direction, e is the electron charge, μ0 is the permeability of vacuum, θSHeff is the material dependent effective spin Hall angle, and tf is the thickness of the free layer. In our latter discussion, we assume α = 0.01, Ms = 1 × 106 A/m, and θSHeff = 0.32 to simulate the CoFeB-based MTJ adjacent to an 8 nm β-W thin film [20]. Hthe is the stochastic field related to thermal fluctuation. Based on the framework of Brown [29], it can be expressed as follows:(3)Hthe=0,Hthe,it1Hthe,jt2=2kbTαμ0γMsVδijδt1-t2,where i,j=1,2, and 3 and Hthe,i refers to the Cartesian component of the stochastic field. kb is the Boltzmann constant, T is the temperature in kelvin, and V is the volume of the free layer. Heff represents the effective field. Since bulk anisotropy is ruled out, the energy density of the free layer with a zero applied field is(4)Etotal=Edemag=12μ0Ms2Nxmx2+Nymy2+Nzmz2,where mx, my, and mz are the three components of the normalized magnetization m and Nx, Ny, and Nz are the demagnetizing factors along the corresponding directions. Since mx2+my2+mz2=1, we have(5)Etotal=12μ0Ms2Nx-Nymx2+12μ0Ms2Nz-Nymz2+12μ0Ms2Ny,and Heff can be derived from the magnetic energy density of the free layer as(6)Heff=Ny-NxMsmx-Nz-NyMsmz,with the easy axis anisotropy field Hk in the long axis direction with strength Ny-NxMs and the demagnetizing field Hd normal to the film plane with strength Nz-NyMs.


## 2.2. Demagnetizing Factors

The analytic demagnetizing factors for elliptical disk can be expressed as a power series expansion of the ratio between the disk thickness and long axis length,τ=tf/a [30, 31], with a “squat elliptic cylinder” limit (b>2tf): (7)Nx=18π2λJ1-J2,Ny=λ8π2J1+J2,Nz=1-Nx-Ny,where(8)J1=16τL0-12+ln⁡τ4I0+τ28L1+14-ln⁡τ4I1+Oτ4,J2=16τγ1+εL-1-L0+32-ln⁡τ41+εI-1-I0-τ2241+εL0-L1-512+ln⁡τ41+εI0-I1+Oτ4,I-1=E1-1λ2,I0=K1-1λ2,I1=β2K1-1λ2,L-1=1+λ22λ2K1-1λ2-ln⁡λ2+1E1-1λ2,L0=-ln⁡λ2K1-1λ2,L1=λ21-ln⁡λ2E1-1λ2-1+λ22λ2K1-1λ2,with aspect ratio λ=a/b, ε=(λ2-1)/(λ2+1), and K, E refer to the complete elliptic integral of the first and second kind, respectively. In previous research works, simplified shape anisotropy was commonly used as Hks=21/b-1/atfMs [32–35]. It however leads to overestimation as Figure 2 depicts. For an MTJ sample with a size of 45 nm × 15 nm × 3 nm, the shape anisotropy field is overestimated by 70%, leading to poor predictions of device characteristics.Figure 2
The deviation percentage caused by simplified shape anisotropy with variedτ and λ.


## 2.3. Current Density for Instability and Switching

The analytical expression of the critical current density necessary for the in-plane MTJ free layer magnetization to become unstable has been introduced by Sun [36] as follows:(9)Jins=2eαtfμ0MsħηHk+Hd2.And the critical current density for magnetization switching is given by [37, 38] as (10)Jsw=4eαtfμ0MsπħηHdHd+Hk,where η is the spin polarization, and for SHE device it should be the effective spin Hall angle. An intermediate current density will drive the magnetization towards oscillation state, which is not sensitive to the initial angle θ0. With results from (6), the analytical critical current densities for an in-plane SHE-RAM can be written as(11)Jins=eαtfμ0Ms2ħθSHeff1-3Nx,Jsw=4eαtμ0Ms2πħθSHeffNz-NyNz-Nx.


## 3. Zero Temperature Simulation Results

### 3.1. Analytic Results

In this section, we exclude the influence from temperature and use a free layer withtf = 1.5 nm and λ = 2. Figure 3 shows Jins and Jsw dependence on the long axis length of the free layer. It can be seen that Jsw drops more rapidly and approaches Jins as the lateral dimension scales down, which also indicates that a smaller current density span allows for oscillation to occur. The magnetization oscillating trajectories with different current densities are shown in Figure 4. Jins leads to small cone angle precession illustrated by the cyan circle, while Jsw leads to a clamshell-shaped trajectory illustrated in red. Current density higher than Jsw leads to magnetization switching. The results from the analytical expressions agree with that obtained from macrospin modeling.Figure 3
J i n s and Jsw dependence on long axis length of the free layer with tf = 1.5 nm and λ = 2. Intermediate current density leads to stable oscillation of m.Figure 4
Magnetization oscillating trajectory with different current densities for in-plane free layer with initial statemx≈1. Higher current density leads to oscillation with larger cone angle and finally switching. The free layer size is 40 nm × 20 nm × 1.5 nm, with Jins = 7.93 × 1010 A/m2 and Jsw = 9.05 × 1010 A/m2.


### 3.2. Macrospin and Micromagnetic Results

The critical current densities based on macrospin and micromagnetic simulations are compared in Figure5. An open-source code, Mumax [39], was used for micromagnetic simulation with a mesh of 64 × 32 × 1 set for all geometries for better GPU performance. The initial magnetization tilt angle is 5 degrees. Good consistency can be observed in Jins, indicating uniform magnetization for small cone angle precession. The discrepancy in Jsw indicates that noncoherent oscillating or switching behavior takes place for larger nanomagnet size. For samples with exchange stiffness constant Aex = 1 × 10−11 J/m, Jsw rises due to increased exchange energy when a > 40 nm and keeps around 1.2 × 1011 A/m2 when a > 70 nm, as shown by the blue triangle. For samples with Aex = 2 × 10−11 J/m, the discrepancy is alleviated and can be observed when a > 60 nm, as shown by the green square.Figure 5
The critical current densities obtained from macrospin model and micromagnetic simulation.Figure6 shows the magnetization components during the switching process of an 80 nm × 40 nm × 1.5 nm free layer. The applied current density is 1.2 × 1011 A/m2. It can be seen that the magnetization experiences a period of oscillation before switching occurs. Figure 7 shows the corresponding magnetization spatial distribution. The noncoherent magnetization behavior can be observed at the two ends of the free layer, which is more obvious in Figure 7(d). Magnetization switching occurs when the magnetization in the central part of the free layer oscillates across the short axis.Figure 6
Free layer magnetization components during a switching process. The applied current density is 1.2 × 1011 A/m. The magnetization oscillating time before switching is 22 ns.Figure 7
Magnetization spatial distribution corresponding to Figure6. The snapshots (a)~(f) are picked at 3 ns, 7 ns, 11 ns, 15 ns, 19 ns, and 23 ns during the switching process.
(a) (b) (c) (d) (e) (f)


## 3.1. Analytic Results

In this section, we exclude the influence from temperature and use a free layer withtf = 1.5 nm and λ = 2. Figure 3 shows Jins and Jsw dependence on the long axis length of the free layer. It can be seen that Jsw drops more rapidly and approaches Jins as the lateral dimension scales down, which also indicates that a smaller current density span allows for oscillation to occur. The magnetization oscillating trajectories with different current densities are shown in Figure 4. Jins leads to small cone angle precession illustrated by the cyan circle, while Jsw leads to a clamshell-shaped trajectory illustrated in red. Current density higher than Jsw leads to magnetization switching. The results from the analytical expressions agree with that obtained from macrospin modeling.Figure 3
J i n s and Jsw dependence on long axis length of the free layer with tf = 1.5 nm and λ = 2. Intermediate current density leads to stable oscillation of m.Figure 4
Magnetization oscillating trajectory with different current densities for in-plane free layer with initial statemx≈1. Higher current density leads to oscillation with larger cone angle and finally switching. The free layer size is 40 nm × 20 nm × 1.5 nm, with Jins = 7.93 × 1010 A/m2 and Jsw = 9.05 × 1010 A/m2.


## 3.2. Macrospin and Micromagnetic Results

The critical current densities based on macrospin and micromagnetic simulations are compared in Figure5. An open-source code, Mumax [39], was used for micromagnetic simulation with a mesh of 64 × 32 × 1 set for all geometries for better GPU performance. The initial magnetization tilt angle is 5 degrees. Good consistency can be observed in Jins, indicating uniform magnetization for small cone angle precession. The discrepancy in Jsw indicates that noncoherent oscillating or switching behavior takes place for larger nanomagnet size. For samples with exchange stiffness constant Aex = 1 × 10−11 J/m, Jsw rises due to increased exchange energy when a > 40 nm and keeps around 1.2 × 1011 A/m2 when a > 70 nm, as shown by the blue triangle. For samples with Aex = 2 × 10−11 J/m, the discrepancy is alleviated and can be observed when a > 60 nm, as shown by the green square.Figure 5
The critical current densities obtained from macrospin model and micromagnetic simulation.Figure6 shows the magnetization components during the switching process of an 80 nm × 40 nm × 1.5 nm free layer. The applied current density is 1.2 × 1011 A/m2. It can be seen that the magnetization experiences a period of oscillation before switching occurs. Figure 7 shows the corresponding magnetization spatial distribution. The noncoherent magnetization behavior can be observed at the two ends of the free layer, which is more obvious in Figure 7(d). Magnetization switching occurs when the magnetization in the central part of the free layer oscillates across the short axis.Figure 6
Free layer magnetization components during a switching process. The applied current density is 1.2 × 1011 A/m. The magnetization oscillating time before switching is 22 ns.Figure 7
Magnetization spatial distribution corresponding to Figure6. The snapshots (a)~(f) are picked at 3 ns, 7 ns, 11 ns, 15 ns, 19 ns, and 23 ns during the switching process.
(a) (b) (c) (d) (e) (f)


## 4. Thermal Effect and Stability Analysis

As the free layer size is scaled down below 100 nm, the thermal fluctuation becomes significant and will have an impact on the SHE-RAM characteristics. In this section, we use (3) to account for the influence of thermal fluctuation, and the geometry dependent thermal stability and the spin torque efficiency in room temperature (300 K) are investigated.The energy barrier of the free layer can be described byEb=μ0Ms2V(Ny-Nx)/2. It is found that thermal stability Δ=Eb/kbT scales almost linearly with the long axis length rather than the area of free layer, as shown in Figure 8. The reason is that the shape anisotropy which determines the energy barrier decreases with the dimension of the free layer. This tendency is similar to that of perpendicular MTJs used for conventional STT-RAM [23]. We further investigate the influences from aspect ratio λ and free layer thickness tf. It is found that Δ can be effectively improved by tapering the free layer; this gain however saturates when λ>3, as illustrated by the solid lines without symbol in Figure 8. The improvement of Δ with free layer thickness can be attributed to the enhanced easy axis anisotropy and a larger volume, as shown by lines with solid symbol.Figure 8
Geometric dependence of thermal stability.Figure9 shows the critical current density necessary for a free layer with tf = 3.5 nm to switch in 5 ns in room temperature. The results from both macrospin and micromagnetic modeling are shown. It can be noted that, despite the improvement in thermal stability, higher λ does not lead to an obvious increase in switching current density, which is due to a smaller demagnetizing field. We used Aex = 1 × 10−11J/m for micromagnetic modeling. The discrepancy between results from macrospin and micromagnetic modeling for larger geometries indicates the nonuniform magnetization during switching process that, for smaller geometries, as shown in Figure 9(a), might be caused by coarse mesh. It can be noted that the free layer with λ=1.5 is more susceptible to nonuniform behavior, and this is due to a larger area compared with free layers with λ=3.Figure 9
Current densities lead to magnetization switch in 5 ns in room temperature for free layers with (a)λ = 3 and (b) λ = 1.5.
(a) (b)The spin torque efficiency is defined asκ=Δ/Ic. For SHE-RAM, Ic=Jcatw, where tw is the thickness of the tungsten layer. In order to analyze the geometric dependence of κ for devices satisfying the retention requirement, the thermal stability Δ is fixed at 40 by adjusting the thickness of the free layer, while the length of long axis, a, and the aspect ratio, λ, are varied. It is found that the spin torque efficiency decreases with a, as shown in Figure 10. Moreover, improvement of κ can be observed in the free layer with higher aspect ratio. This improvement also saturates when λ>3, similar to that of thermal stability. The inset of Figure 10 shows the thickness adjustment of free layer with λ=3.Figure 10
Geometric dependence of spin torque switching efficiency with fixed thermal stabilityΔ = 40. The inset shows the corresponding adjustment of thickness of free layer with λ = 3.


## 5. Conclusion

In this paper, we have studied the influence of shape anisotropy on switching dynamics of elliptic nanomagnets by introducing accurate demagnetizing factors and theoretically investigated the characteristics of SHE-RAM with in-plane magnetization. Analytic expressions of the key performance indicators for in-plane SHE-RAM are obtained including the critical switching current density, thermal stability, and spin torque efficiency, which are found to be highly geometry dependent. In particular, the thermal stability and spin torque efficiency can be effectively improved by selecting the appropriate aspect ratio of the free layer. Moreover, the spin torque efficiency tends to increase as the SHE-RAM element is scaled down. The demagnetizing factors can also be applied to the macrospin approach to give more reliable characteristic predictions for sub-100 nm spin torque based device.



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*Source: 4259846-2016-12-08.xml*