Modification of the Stoner-Wohlfarth astroid by a spin-polarized current: An exact solution

The conventional Stoner-Wohlfarth astroid describes magnetic switching caused by an external magnetic field in a uniaxial, spatially uniform magnet. In spin-transfer devices the shape of the astroid is modified by the current-induced spin torque. The astroid shape modification is calculated here for a setup with the spin polarizer directed along the easy axis of the magnet, in which case the overall axial symmetry of the device is maintained. Our method does not rely on the assumption of small displacement of equilibria by the spin torque and takes into account that the destabilization of equilibrium states can be caused by two mechanisms: equilibrium merging, already active in a conventional astroid, and local equilibrium destabilization produced by the spin torque. The modified astroid has a self-crossing boundary and demonstrates a region with three stable equilibria. It is shown that the region of stable magnetic energy maximum can be reached only through a narrow bottleneck in the applied field space, which sets some stringent requirements for magnetic field alignment in the experiments. Switching diagrams are then calculated for the setups with magnetic field not perfectly aligned with the easy axis. Features not found in the approximate calculations are revealed and suggestions for their experimental observation are provided.