Discovery of superparamagnetism in sub-millimeter-sized magnetite porous single crystals

• This is the first finding of SPM in sub-millimeter-sized magnetite single crystals.
• Such SPM is due to the stress-induced variation of exchange constants α, β, and ν.
• Three general criterions, (i) ν1=ν2=βν1=ν2=β, (ii) α1=α2=αα1=α2=α, and (iii) αβ=1αβ=1, was proposed.

In this work, sub-millimeter-sized magnetite porous single crystals were found to exhibit unique superparamagnetism rather than the known ferrimagnetism. This superparamagnetism was intimately related to the hydrothermal formation process, during which high lattice stress of ca. 6 GPa and large lattice strain of ca. −1.21×10−2−1.21×10−2 would change the exchange constants of α, β, and ν   to concurrently meet criterions of (i) ν1=ν2=βν1=ν2=β, (ii) α1=α2=αα1=α2=α, and (iii) αβ=1αβ=1. These criterions, deduced from the molecular-field theory, were proposed to be the general transition conditions for any ferrimagnetic material exhibiting superparamagnetism when their size was beyond their superparamagnetic size limit.