Superconductors in a Strong Low-Frequency ac Electric Field

The electric-field induced ball formation has been observed for high temperature superconducting particles, MgB$_2$ powder, and low temperature superconducting particles in a low frequency ac electric field. Different from the situation with a static electric field, the superconducting particles in an ac field first form chains along the field direction if the electric field is below a critical value $E_{c1} $. As soon as the field exceeds $E_{c1}$, the chains are broken and the particles aggregate into balls. The experiment has found that $E_{c1}$ is a function of frequency $\omega$. To understand the experimental results, we consider a bulk superconductor in an ac field. The electric field is along the x direction and the bulk superconductor has its surface at $x=0$, perpendicular to the field and is located at $x\ge 0$. The electric field penetrates into the superconductor: for $x>0$, ${\vec E}(x)={\vec e_x}E \exp(-x/l_s)\cos(\omega t)$, where $l_s$ is the electric-field's penetration depth and $E$ is the electric field at the surface of $x=0$. With this model, we have found that if the electric field is strong enough, Cooper pairs near the surface are depleted and a positive surface energy is produced. This induced surface energy is responsible for the formation of superconducting balls. The critical electric field to produce the positive surface energy $E_{c1}$ is found to be related to the binding energy of a Cooper pair$\Delta(T)=2\epsilon_f-\epsilon$ and the frequency $\omega$. As $\omega$ increases, $E_{c1}$ goes up, too. A comparison between the theory and experimental results will also be made.