  The concept of negativity is adapted in order to explore the quantum and
thermal entanglement of the mixed spin-(1/2,$S$) Heisenberg dimers in presence
of an external magnetic field. The mutual interplay between the spin size $S$,
XXZ exchange and uniaxial single-ion anisotropy is thoroughly examined with a
goal to tune the degree and thermal stability of the pairwise entanglement. It
turns out that the antiferromagnetic spin-(1/2,$S$) Heisenberg dimers exhibit
higher degree of entanglement and higher threshold temperature in comparison
with their ferromagnetic counterparts when assuming the same set of model
parameters. The increasing spin magnitude $S$ accompanied with an easy-plane
uniaxial single-ion anisotropy can enhance not only the thermal stability but
simultaneously the degree of entanglement. It is additionally shown that the
further enhancement of a bipartite entanglement can be achieved in the mixed
spin-(1/2,$S$) Heisenberg dimers, involving half-odd-integer spins $S$. Under
this condition the thermal negativity saturates at low-enough temperatures in
its maximal value regardless of the magnitude of half-odd-integer spin $S$. The
magnetic field induces consecutive discontinuous phase transitions in the mixed
spin-(1/2,$S$) Heisenberg dimers with $S\!>\!1$, which are manifested in a
surprising oscillating magnetic-field dependence of the negativity observed at
low enough temperature.
