Quantum Theory of Domain-Wall Motion

Abstract

Several workers have examined the enhancement of nuclear magnetic resonance within a Bloch wall, and have demonstrated the existence of both bound and free excitations on the Bloch wall structure. The free states correspond to precessional excitations akin to ordinary spin-wave excitations, while the bound states form a covenient basis for the representation of domain-wall motion. We derive the spectrum of both types of excitations, including exchange, anisotropy, and dipole field contributions for an infinite uniaxial ferromagnet. In contrast to earlier treatments, we treat the dipole field exactly (in the magnetostatic approximation), and show that this leads to a translational spectrum in which many states are degenerate with the translation, which is the translational mode excited by a uniform external magnetic field. The existence of such degeneracy is required for damping by imperfections to occur. The precessional spectrum is greatly different from the usual spin-wave spectrum, and, in particular, is not a symmetric function of k. The dipole fields lead to strong interactions, not conserving momentum, between the precessional modes; such interactions may explain the increase in ferromagnetic-resonance linewidth which is observed experimentally in the presence of a domain wall (in low dc magnetic fields). The motion of the domain wall, when it is bound to a certain position in the crystal by linear restoring forces, is studied by a Green's function technique. The domain-wall effective mass so obtained is identical to the expression given by D\oring, and the domain-wall damping parameter proves to be simply related to the energy dispersion of the uniform translational mode. We calculate this energy dispersion due to scattering by the dipole fields, and due to fluctuations, as used by Clogston et al. to explain the linewidth in disordered systems, such as the ferrites. The damping due to intrinsic scattering processes is proportional to ${T}^{2}$, while the damping due to fluctuations is essentially temperature-independent. In disordered systems, such as ferrite, the resonance line-width and domain-wall damping due to fluctuations should agree to within a factor of order unity. The motion is not describable by the Landau-Lifshitz equation. This communication is intended to demonstrate that a formulation for the quantum-mechanical study of domain-wall motion exists, and has the properties necessary to explain the losses which occur during such motion; it is not intended to lead to any quantitative results which can be directly compared with experiment. We also consider the specific heat contribution due to the domain wall, and we find that this is proportional to $T$ above about ${10}^{\ensuremath{-}2}$ \ifmmode^\circ\else\textdegree\fi{}K. It should be possible to observe such a specific heat contribution in YIG below 1 \ifmmode^\circ\else\textdegree\fi{}K.