  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.
