Non-linear magnetization dynamics probed with X-rays: 1. Broken cylindrical symmetry of uniform modes
Résumé
We discuss how X-ray magnetic circular dichroism (XMCD) and X-ray magnetic linear dichroism (XMLD)
may complement each other to probe the nonlinear nature of the resonant precession of either spin or
orbital magnetization components in aligned ferro-, ferri- or even antiferro-magnets. The Landau–
Lifshitz–Gilbert (LLG) equation is solved in a rotating frame locked to the microwave pump field, while
treating as time-dependent perturbations the terms which, in the formulation of the free energy density,
break down the cylindrical symmetry of precession. Concretely, we analyze the time-oscillating
deviations of the magnetization from the steady-state solutions of the LLG equation hereafter called
SS-modes. At any perturbation order, one may derive magnetic dipole components which oscillate at
harmonic frequencies of the pump frequency and could be probed with XMCD. Under bichromatic
pumping, frequency mixing arises from a time-dependent Zeeman coupling between two rotating
frames locked to each individual pump field. Similarly, we expect magnetic quadrupole components to
oscillate at the same frequencies. For consistency, their derivation requires a perturbation calculation up
to second order. The latter time-reversal even, rank-2 magnetic tensor components can be probed only
with XMLD. Beyond the (reciprocal) linear dichroism classically measured in ferri- or antiferromagnetic
samples, a non-reciprocal XMLD signal is to be expected when space parity is lost. Nonlinear effects
strongly depend upon the relative orientations of the external bias field and of the pump field with
respect to the symmetry axes of the magnetic system. This holds true for the foldover lineshape
distortions, harmonic generation, frequency mixing or multiquanta excitations.