Magnetization of type-II superconductors in the Kim-Anderson model.

Abstract

Within the construct of the complete Kim-Anderson model for the critical-current density, we have calculated the initial magnetization curves and full hysteresis loops of type-II superconductors immersed in an external field H=${\mathit{H}}_{\mathrm{dc}}$+${\mathit{H}}_{\mathrm{ac}}$cos(\ensuremath{\omega}t), where ${\mathit{H}}_{\mathrm{dc}}$ (\ensuremath{\ge}0) is a dc bias field and ${\mathit{H}}_{\mathrm{ac}}$ (>0) is an ac field amplitude. We denote the maximum and minimum values of H by ${\mathit{H}}_{\mathit{A}}$ (=${\mathit{H}}_{\mathrm{dc}}$+${\mathit{H}}_{\mathrm{ac}}$) and ${\mathit{H}}_{\mathit{B}}$ (=${\mathit{H}}_{\mathrm{dc}}$-${\mathit{H}}_{\mathrm{ac}}$). According to the Kim-Anderson model, the critical-current density ${\mathit{J}}_{\mathit{c}}$ is assumed to be a function of the local internal magnetic-flux density ${\mathit{B}}_{\mathit{i}}$, ${\mathit{J}}_{\mathit{c}}$(${\mathit{B}}_{\mathit{i}}$)=k/(${\mathit{B}}_{0}$+\ensuremath{\Vert}${\mathit{B}}_{\mathit{i}}$\ensuremath{\Vert}), where k and ${\mathit{B}}_{0}$ are constants. We consider an infinitely long cylinder with radius a, and the applied field along the cylinder axis. The field for full penetration is ${\mathit{H}}_{\mathit{p}}$=[(${\mathit{B}}_{0}^{2}$+2${\mathrm{\ensuremath{\mu}}}_{0}$ka${)}^{1/2}$-${\mathit{B}}_{0}$]/${\mathrm{\ensuremath{\mu}}}_{0}$. A related parameter is ${\mathit{H}}^{\mathrm{*}}$=[(${\mathit{B}}_{0}^{2}$-4${\mathrm{\ensuremath{\mu}}}_{0}$ka${)}^{1/2}$-${\mathit{B}}_{0}$]/${\mathrm{\ensuremath{\mu}}}_{0}$. Magnetization equations for full hysteresis loops are derived for three different ranges of ${\mathit{H}}_{\mathit{A}}$: 0${\mathit{H}}_{\mathit{A}}$\ensuremath{\le}${\mathit{H}}_{\mathit{p}}$, ${\mathit{H}}_{\mathit{p}}$\ensuremath{\le}${\mathit{H}}_{\mathit{A}}$\ensuremath{\le}${\mathit{H}}^{\mathrm{*}}$, and ${\mathit{H}}^{\mathrm{*}}$\ensuremath{\le}${\mathit{H}}_{\mathit{A}}$. Each of these three cases is further classified for several ranges of ${\mathit{H}}_{\mathit{B}}$. To describe completely the descending and ascending branches of the full hysteresis loops for all cases, 58 stages of H are considered and the appropriate magnetization equations are derived. In addition to these equations for a cylinder, the corresponding equations for a slab are presented. Comparison with previous work by Ji et al. and by Chen and Goldfarb in the appropriate limits supports the validity of the present derivation.