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The recently discovered superconductor MgB 2 with T c at 39 K has great potential in superconducting microelectronics. Thermodynamics studies with the calculation of phase diagrams (CALPHAD) modelling technique show that due to the high volatility of Mg, MgB 2 is only thermodynamically stable under fairly high Mg overpressures for likely in situ growth temperatures. This provides a helpful insight into the appropriate processing conditions for MgB 2 thin films, including the identification of the pressure–temperature region for adsorption-controlled growth. The initial MgB 2 thin films were made by pulsed laser deposition followed by in situ annealing. The cross-sectional transmission electron microscopy reveals a nanocrystalline mixture of textured MgO and MgB 2 with very small grain sizes. A zero-resistance transition temperature of 34 K and a zero-field critical current density of 1.3× 10 6 A cm− 2 were …