On the trapped magnetic moment in type-II superconductors

Measurements of the trapped (remanent) magnetic moment, $M_{trap}\left(H\right)$, when a small magnetic field $H$ is turned off after cooling below the superconducting transition temperature, $T_c$, or ramping a magnetic field up and down after cooling in a zero field, have advantages in difficult cases of small samples and large field-dependent backgrounds, which is relevant for hydrogen-based ultra-high-$T_{c}$ superconductors (UHTS). Until recently, there was no need for a separate paper on the trapped magnetic flux for well-known critical state models due to the simplicity of the physics involved. However, recent publications showed the need for such an analysis. This note summarizes the expectations for the Bean model with constant critical current density and the Kim model with field-dependent critical currents. It is shown that if the trapped moment is fitted to the power law, $M_{trap}\propto H^{\alpha}$, the fixed exponent $\alpha=2$ is exact for the Bean model, while Kim models show a wide interval of possible values, $2\leq\alpha\leq4$. Furthermore, accounting for reversible magnetization expands the range of possible exponents to $1\leq\alpha\leq4$. In addition, demagnetizing factors are essential and make the trapped moment orientation dependent even in isotropic materials. As a concrete application, it is shown that flux trapping experiments on H$_{3}$S UHTS compounds can be described well using this generalized approach, lending further support to the type-II superconducting nature of H$_{3}$S under ultra-high pressure.