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We derive the AC Josephson frequency using the gauge invariant equations ofmotion with including the battery-contact effect. The frequency is given as asum of a contribution from an Aharonov-Bohm phase that arises whencharged-partices pass through an electric field in the insulator and that froma chemical potential difference arising form the battery contact; each of themcontributes $|q|V/h$ to the frequency where $q$ is the charge, thus, the sum is$2|q|V/h$. The observed Josepshon frequency, $2|e|V/h$, hence, means that thecharge on the tunneling particles is $e$. A variety of derivations so far missone of the above two contributions; the original derivation misses the firstcontribution due to the lack of inclusion of the electric field in theinsulator. The present result indicates that the phase of the order-parameterdoes not arise from the number fluctuations of Cooper pairs. We present analternative origin for it; we argue that a Berry phase that arises fromspin-vortices is the origin.

A gauge invariant derivation of the AC Josephson frequency and a reconsideration of the origin of the phase of the order-parameter