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

Li, X. G.; Liu, Z. J.; Xie, X. Y.; Kang, A. G.; Fu, W. N.

doi:https://doi.org/10.1155/2016/4259846

Advances in Materials Science and Engineering

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Research Article | Open Access

Volume 2016 | Article ID 4259846 | https://doi.org/10.1155/2016/4259846

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

X. G. Li,¹Z. J. Liu,¹X. Y. Xie,¹A. G. Kang,¹and W. N. Fu²

Academic Editor: Santiago Garcia-Granda

Received03 Jun 2016

Revised13 Oct 2016

Accepted01 Nov 2016

Published08 Dec 2016

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.

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 (-W [19, 20], -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. Section 2 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

Figure 1 is a sketch of the in-plane SHE-RAM element. The short axis lies in direction with length . The long axis of the free layer lies in direction with length , which is also the width of the heavy metal layer, as shown in Figure 1(b).

(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:where and are the normalized magnetization of the free layer and spin polarization vector, respectively, = 1.76 × 10¹¹ T⁻¹ s⁻¹ is the gyromagnetic ratio, is the Gilbert damping constant, and is the spin torque strength which can be expressed as [28]where is the reduced Planck constant, is the charge current density applied to the heavy metal layer in the direction, is the electron charge, is the permeability of vacuum, is the material dependent effective spin Hall angle, and is the thickness of the free layer. In our latter discussion, we assume = 0.01, = 1 × 10⁶ A/m, and = 0.32 to simulate the CoFeB-based MTJ adjacent to an 8 nm -W thin film [20]. is the stochastic field related to thermal fluctuation. Based on the framework of Brown [29], it can be expressed as follows:where , and 3 and refers to the Cartesian component of the stochastic field. is the Boltzmann constant, is the temperature in kelvin, and is the volume of the free layer. represents the effective field. Since bulk anisotropy is ruled out, the energy density of the free layer with a zero applied field iswhere , , and are the three components of the normalized magnetization and , , and are the demagnetizing factors along the corresponding directions. Since , we haveand can be derived from the magnetic energy density of the free layer aswith the easy axis anisotropy field in the long axis direction with strength and the demagnetizing field normal to the film plane with strength .

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, [30, 31], with a “squat elliptic cylinder” limit (): wherewith aspect ratio , , and , refer to the complete elliptic integral of the first and second kind, respectively. In previous research works, simplified shape anisotropy was commonly used as [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.

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:And the critical current density for magnetization switching is given by [37, 38] as where 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 . With results from (6), the analytical critical current densities for an in-plane SHE-RAM can be written as

3. Zero Temperature Simulation Results

3.1. Analytic Results

In this section, we exclude the influence from temperature and use a free layer with = 1.5 nm and = 2. Figure 3 shows and dependence on the long axis length of the free layer. It can be seen that drops more rapidly and approaches 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. leads to small cone angle precession illustrated by the cyan circle, while leads to a clamshell-shaped trajectory illustrated in red. Current density higher than leads to magnetization switching. The results from the analytical expressions agree with that obtained from macrospin modeling.

3.2. Macrospin and Micromagnetic Results

The critical current densities based on macrospin and micromagnetic simulations are compared in Figure 5. 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 , indicating uniform magnetization for small cone angle precession. The discrepancy in indicates that noncoherent oscillating or switching behavior takes place for larger nanomagnet size. For samples with exchange stiffness constant = 1 × 10⁻¹¹ J/m, rises due to increased exchange energy when > 40 nm and keeps around 1.2 × 10¹¹ A/m² when > 70 nm, as shown by the blue triangle. For samples with = 2 × 10⁻¹¹ J/m, the discrepancy is alleviated and can be observed when > 60 nm, as shown by the green square.

Figure 6 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.

(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 by . It is found that thermal stability 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 and free layer thickness . It is found that can be effectively improved by tapering the free layer; this gain however saturates when , as illustrated by the solid lines without symbol in Figure 8. The improvement of 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 9 shows the critical current density necessary for a free layer with = 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 does not lead to an obvious increase in switching current density, which is due to a smaller demagnetizing field. We used = 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 is more susceptible to nonuniform behavior, and this is due to a larger area compared with free layers with .

(a)

(b)

The spin torque efficiency is defined as . For SHE-RAM, , where is the thickness of the tungsten layer. In order to analyze the geometric dependence of for devices satisfying the retention requirement, the thermal stability is fixed at 40 by adjusting the thickness of the free layer, while the length of long axis, , and the aspect ratio, , are varied. It is found that the spin torque efficiency decreases with , as shown in Figure 10. Moreover, improvement of can be observed in the free layer with higher aspect ratio. This improvement also saturates when , similar to that of thermal stability. The inset of Figure 10 shows the thickness adjustment of free layer with .

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.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61274089) and the International Scientific and Technological Cooperation Project of Shanxi Province, China (2014081029-2).

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Copyright © 2016 X. G. Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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