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

TL;DR

This paper models error rates in perpendicular spin-torque memory devices using the Fokker-Planck equation, providing analytical expressions and numerical validation for switching times and error rates.

Contribution

It introduces exact and approximate analytical formulas for error rates and switching times, validated through numerical solutions and simulations within the macrospin approximation.

Findings

Analytical expression for zero-temperature switching time.

Approximate formulas for writing and reading error rates.

Validation of formulas with numerical solutions and LLG simulations.

Abstract

We model "soft" error rates for writing (WSER) and for reading (RSER) for perpendicular spin-torque 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.