A CPP MTJ MRAM unit cell utilizes transfer of spin angular momentum as a mechanism for changing the magnetic moment direction of a free layer. The strength of the switching field, Hs of the cell is controlled by the magnetic anisotropy of the cell which, in turn, is controlled by a combination of the shape anisotropy and the stress and magnetostriction of the cell free layer. The coefficient of magnetostriction of the free layer can be adjusted by methods such as adding Nb or Hf to alloys of Ni, Fe, Co and B or by forming the free layer as a lamination of layers having different values of their coefficients of magnetostriction. Thus, by tuning the coefficient of magnetostriction of the cell free layer it is possible to produce a switching field of sufficient magnitude to render the cell thermally stable while maintaining a desirable switching current.