Effect of spatial confinement on spin-wave spectrum: Low temperature deviation from Bloch T3/2 law in Co nanoparticles

Abstract

We present the study on Bloch T3/2 law and its applicability in ferromagnetic cobalt nanoparticles with sizes 25 and 38 nm. Bloch has derived the T3/2 law by assuming long wave-length spin-waves to be excited at low temperatures. But, in nanoparticles the wavelength of the spin wave is confined by the size of the magnetic particle leading to the gap in the spin-wave energy spectrum. The experimental observation leads to the conclusion that Bloch law is valid at temperatures higher than the spin-wave energy gap. However, it is not applicable at low enough temperatures, where the energy gap becomes prominent. We have demonstrated that a theory recently developed by us [Mandal et al., Europhys. Lett. 75, 618 (2006)] explains the variation of magnetization with temperature accurately. In addition, the hysteresis properties of these cobalt nanoparticles are also presented here.