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

By W. H. Butler, T. Mewes, C. Mewes, Pieter B. Visscher, William H. Rippard, S. Russek, R. Heindl. IEEE transactions on magnetics, 2012

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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)τ] where the reduced time τ is related to the switching time, Gilbert damping and precession frequency through τ = αω0t, and the reduced current i is the ratio of the applied current to the critical current density for switching i=I̅/I0 . This exponentially decaying tail in WSER is not easily reduced by tilting the pinned layer magnetization.

Subfields: Atomic and Molecular Physics, and Optics, Biomedical Engineering, Electronic, Optical and Magnetic Materials

View Paper: https://www.semanticscholar.org/paper/ab48c855386812a73534bce6f0fd892b74acd58c

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Citations in the corpus

Basic principles of STT-MRAM cell operation in memory arrays (2013, methodology): “For many simple cases it can also be evaluated using the Fokker–Planck equation on the evolution of the probability density function [72, 73, 79].”
Stochastic p-bits for Invertible Logic (2016, methodology): “This can be analytically shown from a 1D Fokker-Planck equation [42], and we have reproduced this behavior directly from sLLG simulations.”