  We make a detailed study on the $D_s$ meson leading-twist LCDA $\phi_{2;D_s}$
by using the QCD sum rules within the framework of the background field theory.
To improve the precision, its moments $\langle \xi^n\rangle _{2;D_s}$ are
calculated up to dimension-six condensates. At the scale $\mu = 2{\rm GeV}$, we
obtain: $\langle \xi^1\rangle _{2;D_s}= -0.261^{+0.020}_{-0.020}$, $\langle
\xi^2\rangle _{2;D_s} = 0.184^{+0.012}_{-0.012}$, $\langle \xi^3\rangle
_{2;D_s} = -0.111 ^{+0.007}_{-0.012}$ and $\langle \xi^4\rangle _{2;D_s} =
0.075^{+0.005}_{-0.005}$. Using those moments, the $\phi_{2;D_s}$ is then
constructed by using the light-cone harmonic oscillator model. As an
application, we calculate the transition form factor $f^{B_s\to D_s}_+(q^2)$
within the light-cone sum rules (LCSR) approach by using a right-handed chiral
current, in which the terms involving $\phi_{2;D_s}$ dominates the LCSR. It is
noted that the extrapolated $f^{B_s\to D_s}_+(q^2)$ agrees with the Lattice QCD
prediction. After extrapolating the transition form factor to the physically
allowable $q^2$-region, we calculate the branching ratio and the CKM matrix
element, which give $\mathcal{B}(\bar B_s^0 \to D_s^+ \ell\nu_\ell) =
(2.03^{+0.35}_{-0.49}) \times 10^{-2}$ and
$|V_{cb}|=(40.00_{-4.08}^{+4.93})\times 10^{-3}$.
