We model "soft" error rates for writing (WSER) and for reading (RSER) forperpendicular spin-torque memory devices by solving the Fokker-Planck equationfor the probability distribution of the angle that the free layer magnetizationmakes with the normal to the plane of the film. We obtain: (1) an exact, closedform, analytical expression for the zero-temperature switching time as afunction of initial angle; (2) an approximate analytical expression for theexponential decay of the WSER as a function of the time the current is applied;(3) comparison of the approximate analytical expression for the WSER tonumerical solutions of the Fokker-Planck equation; (4) an approximateanalytical expression for the linear increase in RSER with current applied forreading; (5) comparison of the approximate analytical formula for the RSER tothe numerical solution of the Fokker-Planck equation; and (6) confirmation ofthe accuracy of the Fokker-Planck solutions by comparison with results ofdirect simulation using the single-macrospin Landau-Lifshitz-Gilbert (LLG)equations with a random fluctuating field in the short-time regime for whichthe latter is practical.
展开▼
