Theoretical Study on Gilbert Damping of Nonuniform Magnetization Precession in Ferromagnetic Metals

We study Gilbert damping in bulk metallic ferromagnets containing magnetic and nonmagnetic impurities in the presence of nonuniform magnetization precession. In this model, a microscopic expression for the Gilbert damping tensor is obtained using the linear response theory with respect to the interaction between magnetization and conduction electrons. We especially focus on a diagonal element of the tensor, which is a conventional Gilbert damping constant, and evaluate it numerically as a function of the wave vector q of magnetization precession. We show that the impurity scattering dominates Gilbert damping for 0 < vertical bar q vertical bar < k(F up arrow) - k(F down arrow), while the Stoner excitation dominates for k(F up arrow) - k(F down arrow) < vertical bar q vertical bar < k(F up arrow) - k(F down arrow) (k(F sigma) is the Fermi wave number for sigma spin electrons).