  Carbon-enhanced metal-poor (CEMP) stars in the Galactic halo have a wide
range of neutron-capture element abundance patterns. To identify their origin,
we investigated three modes of $s$-process nucleosynthesis that have been
proposed to operate in extremely metal-poor (EMP) Asymptotic Giant Branch (AGB)
stars: the convective 13C burning, which occurs when hydrogen is engulfed by
the helium flash convection in low-mass AGB stars, the convective 22Ne burning,
which occurs in the helium flash convection of intermediate-mass AGB stars, and
the radiative 13C burning, which occurs in the $^{13}$C pocket that is formed
during the inter-pulse periods. We show that the production of $s$-process
elements per iron seed ($s$-process efficiency) does not depend on metallicity
below $[{\rm Fe}/{\rm H}]=-2$, because 16O in the helium zone dominates the
neutron poison. The convective 13C mode can produce a variety of $s$-process
efficiencies for Sr, Ba and Pb, including the maxima observed among CEMP stars.
The 22Ne mode only produce the lowest end of $s$-process efficiencies among
CEMP models. We show that the combination of these two modes can explain the
full range of observed enrichment of $s$-process elements in CEMP stars. In
contrast, the 13C pocket mode can hardly explain the high level of enrichment
observed in some CEMP stars, even if considering star-to-star variations of the
mass of the 13C pocket. These results provide a basis for discussing the binary
mass transfer origin of CEMP stars and their subgroups.
