  Nonlinear spin polarization occurring in the second order of driving electric
current is the dominant source of nonequilibrium magnetization in
centrosymmetric or weakly noncentrosymmetric nonmagnetic materials, and induces
nonlinear spin-orbit torque in magnets. Up to now, only the intrinsic mechanism
based on anomalous spin polarizability dipole, which is the spin counterpart of
Berry curvature dipole, has been studied, while disorder induced mechanisms are
still missing. Here, we derive these contributions, which include not only the
anomalous distribution function due to skew scattering and coordinate shift,
but also interband coherence effects given by disorder induced spin shift and
electric field induced anomalous scattering amplitude. We demonstrate these
terms and show their importance in a minimal model. A scaling law for nonlinear
current-induced spin polarization is constructed, which may help analyze
experimental data in the future.
