  Unexpected fluctuating charge field near a semiconductor quantum dot has
severely limited the coherence time of the localized spin qubit. It is the
interplay between the spin-orbit coupling and the asymmetrical confining
potential in a quantum dot, that mediates the longitudinal interaction between
the spin qubit and the fluctuating charge field. Here, we study the $1/f$
charge noise induced spin dephasing in a nanowire double quantum dot via
exactly solving its eigen-energies and eigenfunctions. Our calculations
demonstrate that the spin dephasing has a nonmonotonic dependence on the
asymmetry of the double quantum dot confining potential. With the increase of
the potential asymmetry, the dephasing rate first becomes stronger very sharply
before reaching to a maximum, after that it becomes weaker softly. Also, we
find that the applied external magnetic field contributes to the spin
dephasing, the dephasing rate is strongest at the anti-crossing point $B_{0}$
in the double quantum dot.
