Magnetization of type-ii superconductors in the range of fields H c 1 ≤ H ≤ H c 2Variational Method

A variational method is proposed to find the magnetic field dependence of the magnetization of type-II superconductors in the mixed state by a self-consistent technique. This model allows for suppression of the order parameter to zero at the centers of Abrikosov vortices and also for the magnetic field dependence of the order parameter. The results can be applied to the entire range of fields H c 1 ≤ H ≤ H c 2 for any values of the Ginzburg-Landau parameter 𝜅>1/2‾√. It is shown that in weak fields where κ ≫ 1 the behavior of the magnetization can be described exactly in the London approximation provided that the correct value of H c 1 is used. Near the second critical field this dependence shows good agreement with the well-known Abrikosov result. It is also shown that using the concept of isolated vortices and applying the principle of superposition of the fields and currents generated by these vortices to calculate the magnetization gives inaccurate quantitative results even in fairly weak fields. By going beyond these concepts, it was possible to allow more accurately for the influence of the vortex cores on the magnetization behavior in the intermediate range of fields H c 1 ≪H ≪ H c 2 and to identify the range of validity of various approximations used widely in the literature.