Effects of disorder and Coulomb interaction on the spatial dispersion of spin susceptibility in two-dimensional structures

Abstract

The wave-vector- and frequency-dependent spin susceptibility of an interacting disordered two-dimensional electron gas at frequencies near the spin resonance has been considered. It is assumed that the external magnetic field is moderately strong ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\ll}{\ensuremath{\omega}}_{c}\ensuremath{\ll}{\ensuremath{\epsilon}}_{F}$ and the Coulomb interaction is weak $({e}^{2}/\ensuremath{\kappa}\ensuremath{\lambda})/{\ensuremath{\omega}}_{c}\ensuremath{\ll}1$ ($\ensuremath{\lambda}$ stands for the magnetic length). The spin susceptibility has been calculated within the ladder diagram approximation. A way of dealing with the degeneracy of Landau levels has been proposed and a transport-type equation free from the degeneracy indexes has been derived. It has been shown that the resonant part of the spin susceptibility has the form $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\chi}}(\ensuremath{\omega},\mathbf{q})\ensuremath{\sim}{[\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{\mathrm{res}}\ensuremath{-}({d}^{\ensuremath{'}}+i{d}^{\ensuremath{'}\ensuremath{'}}){\mathbf{q}}^{2}]}^{\ensuremath{-}1}$, where in the hydrodynamic limit ${\ensuremath{\omega}}_{s}\ensuremath{\tau}\ensuremath{\ll}1$ (${\ensuremath{\omega}}_{s}$ stands for the Larmour frequency $g{\ensuremath{\mu}}_{B}H$) and to first order in $({e}^{2}/\ensuremath{\kappa}\ensuremath{\lambda})/{\ensuremath{\omega}}_{c}$ the imaginary part, which stems from the diffusion of the spin magnetization due to the stochastic scattering on impurities, has a conventional form, ${d}^{\ensuremath{'}\ensuremath{'}}=\frac{{\ensuremath{\epsilon}}_{F}\ensuremath{\tau}/m}{{({\ensuremath{\omega}}_{c}\ensuremath{\tau})}^{2}}$, while the real part, which characterizes the coherent motion of the spin excitations through the crystal, is $d{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{'}}\ensuremath{\sim}{M}^{\ensuremath{-}1}({\ensuremath{\omega}}_{s}\ensuremath{\tau})$, where $M\ensuremath{\sim}{(\ensuremath{\lambda}{e}^{2}/\ensuremath{\kappa})}^{\ensuremath{-}1}$ is the effective mass of the spin excitation in the clean system. Thus, at ${\ensuremath{\omega}}_{s}\ensuremath{\tau}\ensuremath{\ll}1$, the disorder strongly impedes the coherent motion of the excitations though it does not forbid it completely. The resonant frequency is unaffected by the Coulomb interaction, ${\ensuremath{\omega}}_{\mathrm{res}}={\ensuremath{\omega}}_{s}$, just as in the clean case. The results may be applied to experiments on the spin-flip Raman scattering, where the finite wave-vector susceptibility can be directly measured, as well to the electromagnetic absorption studying by means of a grating superimposed on the structure.