  We investigate the origin of the abundance ratios and scatter of the
neutron-capture elements Sr, Ba and Eu in the stellar halo of a Milky Way-mass
galaxy formed in a hydrodynamical cosmological simulation, and compare them
with those of $\alpha$-elements. For this, we implement a novel treatment for
chemical enrichment of Type II supernovae which considers the effects of the
rotation of massive stars on the chemical yields and differential enrichment
according to the life-times of progenitor stars. We find that differential
enrichment has a significant impact on the early enrichment of the interstellar
medium which is translated into broader element ratio distributions,
particularly in the case of the oldest, most metal-poor stars. We find that the
[element/Fe] ratios of the $\alpha-$elements O, Mg and Si have systematically
lower scatter compared to the neutron-capture elements ratios Sr, Ba and Eu at
[Fe/H]$<-2$, which is $\sim 0.1-0.4$ dex for the former and between $\sim 0.5$
and $1$ dex for the latter. The different scatter levels found for the
neutron-capture and $\alpha$-elements is consistent with observations of old
stars in the Milky Way. Our model also predicts a high scatter for the [Sr/Ba]
ratio, which results from the treatment of the fast-rotating stars and the
dependence of the chemical yields on the metallicity, mass and rotational
velocities. Such chemical patterns appear naturally if the different ejection
times associated to stars of different mass are properly described, without the
need to invoke for additional mixing mechanisms or a distinct treatment of the
$\alpha-$ and neutron-capture elements.
