  Due to the significant difference between the experimental measurements and
the theoretical predictions of standard model (SM) for the value of
$\mathcal{R}(D)$ of the semileptonic decay $B\to D\ell\bar{\nu}_{\ell}$, people
speculate that it may be the evidence of new physics beyond the SM. Usually,
the $D$-meson twist-2, 3 distribution amplitudes (DAs) $\phi_{2;D}(x,\mu)$,
$\phi_{3;D}^p(x,\mu)$ and $\phi_{3;D}^\sigma(x,\mu)$ are the main error sources
when using perturbative QCD factorization and light-cone QCD sum rules to study
$B\to D\ell\bar{\nu}_{\ell}$. Therefore, it is important to get more reasonable
and accurate behaviors for those DAs. Motivated by our previous work [Phys.
Rev. D 104, no.1, 016021 (2021)] on pionic leading-twist DA, we revisit
$D$-meson twist-2, 3 DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^p(x,\mu)$ and
$\phi_{3;D}^\sigma(x,\mu)$. New sum rules formulae for the $\xi$-moments of
these three DAs are suggested to obtain more accurate values. The light-cone
harmonic oscillator models for those DAs are improved, and whose model
parameters are determined by fitting the values of $\xi$-moments with the least
squares method.
