Debye temperature of iron nanoparticles: Finite-size effects in the scaling regime

Abstract

The Debye temperature ${\mathrm{\ensuremath{\Theta}}}_{\mathrm{D}}$ characterizes the vibrations of a solid and marks the transition between quantized and classical behaviors of nuclear motion. While thermodynamical theories suggest that for nanoparticles ${\mathrm{\ensuremath{\Theta}}}_{\mathrm{D}}$ should be lower than the bulk limit and increase with increasing nanoparticle size, various experimental measurements have reported intriguing variations with size, including values above the bulk limit and possibly decreasing with size. In this paper we have theoretically determined the Debye temperature of iron nanoparticles of up to about 7 nm diameter at the atomistic level of detail, using complementary approaches based on the equilibrium heat capacity or the mean-square atomic displacement. Both methods consistently indicate steady finite-size effects in the scaling regime, with no evidence for values higher than the bulk or varying in opposite ways, but they also produce marked quantitative differences. Further comparison with the melting temperature ${T}_{m}$ indicates that the two quantities correlate with each other through ${T}_{m}\ensuremath{\propto}{\mathrm{\ensuremath{\Theta}}}_{\mathrm{D}}^{\ensuremath{\alpha}}$, but with an exponent $\ensuremath{\alpha}$ close to 3 that deviates from the expected value of 2 that classical thermodynamical theories predict based on pure bulk ingredients.