Size-dependent Curie temperature of Ni nanoparticles from spin-lattice dynamics simulations

The magnetic properties of Ni nanoparticles (NPs) with diameter D are investigated using spin-lattice dynamics (SLD) simulations. Using exchange interactions fitted to ab-initio results we obtain a Curie temperature, Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}, similar, but lower, than experiments. In order to reproduce quantitatively the bulk Curie temperature and the experimental results, the exchange energy has to be increased by 25% compared to the ab-initio value. During the simulated time, Ni NPs remain ferromagnetic down to the smallest sizes investigated here, containing around 500 atoms. The average magnetic moment of the NPs is slightly smaller than that determined experimentally. By considering a core-shell model for NPs, in which the shell atoms are assigned a larger magnetic moment, this discrepancy can be removed. Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document} is lower for a moving lattice than for a frozen lattice, as expected, but this difference decreases with NP size because smaller NPs include higher surface disorder which dominates the transition. For NPs, Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document} decreases with the NP diameter D by at most 10% at D=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=2$$\end{document} nm, in agreement with several experiments, and unlike some modeling or theoretical scaling results which predict a considerably larger decrease. The decrease of Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document} is well described by finite-size scaling models, with a critical exponent that depends on the SLD settings for a frozen or moving lattice, and also depends on the procedure for determining Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}. Extrapolating the inverse of the magnetization as function of temperature near Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document} gives a lower Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document} than the maximum of the susceptibility.