This review presents the recent progress in computational materials design, experimental realization, and control methods of spinodal nanodecomposition under three- and two-dimensional crystal-growth conditions in spintronic materials, such as magnetically doped semiconductors. The computational description of nanodecomposition, performed by combining first-principles calculations with kinetic Monte Carlo simulations, is discussed together with extensive electron microscopy, synchrotron radiation, scanning probe, and ion beam methods that have been employed to visualize binodal and spinodal nanodecomposition (chemical phase separation) as well as nanoprecipitation (crystallographic phase separation) in a range of semiconductor compounds with a concentration of transition metal (TM) impurities beyond the solubility limit. The role of growth conditions, co-doping by shallow impurities, kinetic barriers, and surface reactions in controlling the aggregation of magnetic cations is highlighted. According to theoretical simulations and experimental results the TM-rich regions appear either in the form of nanodots (the {em dairiseki} phase) or nanocolumns (the {em konbu} phase) buried in the host semiconductor. Particular attention is paid to Mn-doped group III arsenides and antimonides, TM-doped group III nitrides, Mn- and Fe-doped Ge, and Cr-doped group II chalcogenides, in which ferromagnetic features persisting up to above room temperature correlate with the presence of nanodecomposition and account for the application-relevant magneto-optical and magnetotransport properties of these compounds. Finally, it is pointed out that spinodal nanodecomposition can be viewed as a new class of bottom-up approach to nanofabrication.
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