Size and shape dependence of the magnetic ordering temperature in nanoscale magnetic particles

Natural rocks and synthetic analogues can contain extremely small scaled magnetic minerals varying in shape from approximately equidimensional nanoparticles to lower dimensionally shaped lamellae resembling thin films or whiskers. The magnetic ordering temperatures of such nanomagnetic structures can significantly depend on their size and shape. Here, a general method for detailed numerical or analytical calculations of these ordering temperatures is developed. Based on a modified mean-field approach, the result proves a refined version of a known scaling law that links atomic-layer number to the Curie temperatures of nanostructures. An analytic expression for the dependence of the Curie temperature on the atomic-layer number is obtained for thin films and rectangular nanostructures. It is confirmed by comparison to experimental results.