  A theoretical formula is derived for the threshold current to switch a
perpendicular magnetization in a ferromagnet by the spin Hall effect. The
numerical simulation of the Landau-Lifshitz-Gilbert equation indicates that
magnetization switching is achieved when the steady-state solution of the
magnetization in the presence of the current is outside an energetically
unstable region. Based on the numerical result, an analytical theory deriving
the threshold current is developed by focusing on the first-order perturbation
to the unstable state. The analytical formula clarifies that the magnitude of
the magnetic field applied to the current direction should be larger than 15\%
of the perpendicular magnetic anisotropy field, and the current is less than
the derived threshold value.
