Spin-wave modes and low-temperature specific heat in the spin-glassEuxSr1−xS:x=0.54 and 0.40

Abstract

We study the linear spin-wave modes and their contribution to the specific heat in the insulating spin-glass ${\mathrm{Eu}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}\mathrm{S}$ for $x=0.54 \mathrm{and} 0.40$. The spin waves are identified with the collective excitations obtained by linearizing the equations of motion of the spins about their equilibrium orientations. The latter were determined by minimizing the energy by successive rotations of the spins into the directions of their local fields. The energies and localization indices of the modes were calculated by diagonalizing the dynamical matrices of ensembles of 270 ($x=0.45$) and 345 ($x=0.40$) spins with periodic boundary conditions. The densities of states have appreciable weight down to energies on the order of 0.1 K. The localization indices indicate that except for the high end of the spectrum nearly all the modes are delocalized. Comparison with experimental data indicates that the spin waves are responsible for seventy to eighty percent of the specific heat at $T\ensuremath{\approx}0.3$ K with an even greater percentage expected at lower temperatures. The discrepancy between the experimental and theoretical values of the specific heat increases with increasing temperature, an effect which we attribute to the breakdown of the linear theory.