Switching Distributions for Perpendicular Spin-Torque Devices Within the Macrospin Approximation
Switching Distributions for Perpendicular Spin-Torque Devices Within the Macrospin Approximation
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DOI：
10.1109/tmag.2012.2209122
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发表时间：
2012-12-01
影响因子：
2.1
通讯作者：
Heindl, Ranko
中科院分区：
文献类型：
--
作者：
Butler, W. H.;Mewes, Tim;Heindl, Ranko
我们的“软”的错误率为写入（WSER）和阅读（RSER）的自旋扭矩存储器设备，具有一个自由层的易磁化轴垂直于膜平面的概率分布的角度，自由层的磁化与膜的平面的法线求解福克-普朗克方程。我们得到：1）作为初始角的函数的零温度开关时间的精确的、封闭形式的解析表达式; 2）作为施加电流的时间的函数的磁化方向的分布函数和WSER的指数衰减的近似解析表达式; 3）分布函数和WSER的近似解析表达式与Fokker-Planck方程的数值解的比较; 4）对于钉扎层与垂直自由层不共线的情况，分布函数和WSER的近似解析表达式; 5）对于施加用于阅读的电流，RSER的线性增加的近似解析表达式; 6）比较了RSER的近似解析公式与Fokker-Planck方程的数值解; 7）通过与单巨自旋Landau-Lifshitz模型直接模拟结果的比较，证实了Fokker-Planck解的准确性。吉尔伯特方程与随机波动场在短期制度，后者是实用的。我们发现，WSER在很长时间内衰减为exp [-2（i - 1）τ]，其中减少的时间τ与开关时间、吉尔伯特阻尼和进动频率通过τ = alpha omega（0）t相关，并且减少的电流是所施加的电流与用于开关的临界电流密度i =（I）/bar /I-0的比率。通过倾斜钉扎层磁化，不容易减少WSER中的这种指数衰减尾部。
We model "soft" error rates for writing (WSER) and for reading (RSER) for spin-torque memory devices that have a free layer with easy axis perpendicular to the film plane by solving the Fokker-Planck equation for the probability distribution of the angle that the free layer magnetization makes with the normal to the plane of the film. We obtain: 1) an exact, closed form, analytical expression for the zero-temperature switching time as a function of initial angle; 2) an approximate analytical expression for the distribution function of the direction of the magnetization and the exponential decay of the WSER as a function of the time the current is applied; 3) comparison of the approximate analytical expressions for the distribution function and WSER to numerical solutions of the Fokker-Planck equation; 4) an approximate analytical expression for the distribution function and WSER for the case in which the pinned layer is not collinear with the perpendicular free layer; 5) an approximate analytical expression for the linear increase in RSER with current applied for reading; 6) comparison of the approximate analytical formula for the RSER to the numerical solution of the Fokker-Planck equation; and 7) confirmation of the accuracy of the Fokker-Planck solutions by comparison with results of direct simulation using the single-macrospin Landau-Lifshitz-Gilbert equations with a random fluctuating field in the short-time regime for which the latter is practical. We find that the WSER decays at long times as exp [-2(i - 1)tau] where the reduced time tau is related to the switching time, Gilbert damping and precession frequency through tau = alpha omega(0)t, and the reduced current is the ratio of the applied current to the critical current density for switching i = (I) over bar /I-0. This exponentially decaying tail in WSER is not easily reduced by tilting the pinned layer magnetization.