Switching Distributions for Perpendicular Spin-Torque Devices Within the Macrospin Approximation

Abstract

We model "soft" error rates for writing (WSER) and for reading (RSER) for perpendicular spintorque memory devices 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 exponential decay of the WSER as a function of the time the current is applied; (3) comparison of the approximate analytical expression for the WSER to numerical solutions of the Fokker-Planck equation; (4) an approximate analytical expression for the linear increase in RSER with current applied for reading; (5) comparison of the approximate analytical formula for the RSER to the numerical solution of the Fokker-Planck equation; and (6) confirmation of the accuracy of the Fokker-Planck solutions by comparison with results of direct simulation using the single-macrospin Landau-Lifshitz-Gilbert (LLG) equations with a random fluctuating field in the short-time regime for which the latter is practical.

Historical Background and Introductory Concepts

“…During the last decade, various analytical and numerical approaches to the calculation of the measurable parameters of STT devices such as the magnetization reversal time, etc. via the magnetic Langevin and/or Fokker-Planck equations including STT have been developed [130,137,[140][141][142][143][144][145][146][147][148][149][150]. These approaches mainly generalize methods originally developed for treating thermal fluctuations in nanoscale ferromagnets at zero STT, such Fig.…”

Estimation of thermal stability factor and intrinsic switching current from switching distributions in spin-transfer-torque devices with out-of-plane magnetic anisotropy

“…For I Ic 0 , the value of the exponent is approximately 1, in agreement with Li and Koch, and increases up to approximately 2.2 with increasing current. For the case of out-of-plane anisotropy, Butler et al 12 solved the Fokker-Planck equation for the switching distribution and reported that the exponent should be n = 2. Pinna et al 13 reported that, in a system with uniaxial anisotropy, n = 2 is a much better fit to the data at long time scales than n = 1, and the crossover regime between deterministic and thermal effects could span very large switching timescales.…”

Physical Principles of Spin Torque

“…The results of this simple model appear in reasonable agreement, with a full Fokker-Planck result generating slightly faster switching times at the low overdrive I=I c $ 1 ð Þlimit. The exponentially decreasing tail of the non-switching probability E r , either in t or in I, remains essentially the same [69,75,77]. This tail, however, is yet to be experimentally observed, probably due to the non-macrospin nature of the devices experimentally studied to date [78,79].…”

“…This tail, however, is yet to be experimentally observed, probably due to the non-macrospin nature of the devices experimentally studied to date [78,79]. Higher-precision statistics at shorter switching time (well below 10ns) and for smaller junctions (probably below 20 nm) would bring experimental situation closer to the macrospin model assumptions, and one would have a better understanding of whether this macrospin model predicted "stagnation" behavior [69,75,77] and is likely to cause practical problems for reliable write operation for a memory element in its small-size limit, for example.…”