What is meant by the concept of critical current when talking about superconducting phase diagrams and transitions? How does this relate to critical field and temperature?
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Critical current, usually critical current density, is the current that is strong enough to force superconductor into normal state.
If the current is constant, the only thing you have to worry is the magnetic field, if the current is alternating, it will heat up the superconductor, and then you may also need to worry about temperature.
Assuming the current is constant, the main influence of the current is through the magnetic field it forces through the superconductor. Once the magnetic field will cross the critical (thermodynamic) value the superconductor, the superconductor, in that region of space, will go into normal state. The critical field is temperature dependent. The lower the temperature the higher the critical field. Type-I superconductors have one critical field, type-II will have two critical fields. For large bits of superconductor you can end up with mixed state, where part of the material is supercondcting and the other is not, irrespective of superconductor type. For type-II you can also have the Meissner state.
The relationship between the current and magnetic field is not trivial, and is geometry-dependent. See, for example:
R. P. Huebener and R. T. Kampwirth, “Meissner shielding currents and magnetic flux penetration in thin-film superconductors,” J. Low Temp. Phys., vol. 6, pp. 275–285, 1972
Also there will be specifics relating to types of superconductors. Type-I superconductors will be easier to treat analytically, but type-II will be able to sustain higher critical currents due to flux penetration and pinning (but this is a whole new can of worms). Practically, the best thing is to look for papers that report experimental measurements of critical current and current density.