![]() The superconducting radio-frequency (SRF) resonant cavities in particle accelerators enable high accelerating gradients with low power consumption. For example, an S-I-S structure with a 90-nm-thick Nb 3Sn layer on Nb can boost the superheating field up to ≈500 mT, while protecting the superconducting radio-frequency (SRF) cavity from dendritic thermomagnetic avalanches caused by local penetration of vortices. We show that the impurity concentration profile at the surface and thicknesses of S-I-S multilayers can be optimized to enhance H sh above the bulk superheating fields of both Nb and Nb 3Sn. Simulations were performed for the material parameters of Nb and Nb 3Sn at different values of κ and the mean free paths. The superheating field was calculated taking into account the instability of the Meissner state with a non-zero wavelength along the surface, which is essential for the realistic values of the GL parameter κ. Numerical simulations of the Ginzburg–Landau (GL) equations were performed for a superconductor with an inhomogeneous impurity concentration, a thin superconducting layer on top of another superconductor, and superconductor–insulator–superconductor (S-I-S) multilayers. We report calculations of a DC superheating field H sh in superconductors with nanostructured surfaces. 2Department of Physics and Center for Accelerator Science, Old Dominion University, Norfolk, VA, United States.1Department of Physics and Astronomy, Virginia Military Institute, Lexington, VA, United States.Our discussion could help to provide hints for the development of multiscale modeling scheme in polarization dynamics based on a phase field model. In our opinion, the viable solution is to reconstruct the potential field, making it be compatible with the thermal fluctuation induced by random force. If simply revising the free-energy functional to get rid of such contradiction, the possible phase instability can be eliminated, but it results in the underestimation of thermal fluctuations and the associated polarization dynamical behaviors. In addition, the thermal fluctuations of random force are found to lead to the unexpected phase instability when considering the atomic-scale polarization dynamical behaviors, which is considered to be originated from the incompatibility between the free-energy functional and random force used in the current phase field model. The numerical simulations indicate that the thermal fluctuations induced by random force give rise to a different heat dissipation mechanism during the process of polarization dynamical responses, which is not taken into account in the conventional phase field simulations. It is found that the presence of random force guarantees the thermodynamics of polarization system. Starting from the generalized many-body stochastic dynamics, we discuss the thermodynamics of the polarization dynamics simulated based on the phase field scheme. With the size of polar element reducing to be atomic scale, the underlying physics in the phase field model should be well clarified. The phase field model has been proved as an efficient and indispensable method to capture the polarization evolution behaviors. Polarization dynamical response is a fundamental issue for both physics and functional device applications for ferroelectrics.
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