  The upper critical field $H_{c2}$ of polycrystalline samples of
$Ln$O$_{0.5}$F$_{0.5}$BiS$_{2}$ ($Ln$ = La, Nd) at ambient pressure (tetragonal
structure) and high pressure (HP) (monoclinic structure) have been investigated
via electrical resistivity measurements at various magnetic fields up to 8.5 T.
The $H_{c2}$($T$) curves for all the samples show an uncharacteristic concave
upward curvature at temperatures below $T_c$, which cannot be described by the
conventional one-band Werthamer-Helfand-Hohenberg theory. For the
LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ sample under HP, as temperature is decreased, the
upper critical field $H_{onset}$, estimated from the onset of the
superconducting transitions, increases slowly between 4.9 and 5.8 T compared
with the slope of $H_{onset}$($T$) below 4.9 T and above 5.8 T. This anomalous
behavior reveals a remarkable similarity in superconductivity between
LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ samples measured under HP and synthesized under
HP, although the crystal structures of the two samples were reported to be
different. The experimental results support the idea that local atomic
environment, which can be tuned by applying external pressure and can be
quenched to ambient pressure via high temperature-pressure annealing, is
possibly more essential to the enhancement of $T_c$ for BiS$_2$-based
superconductors than the structural phase transition. On the other hand, such
anomalous behavior is very subtle in the case of NdO$_{0.5}$F$_{0.5}$BiS$_{2}$
under HP, suggesting that the anisotropy of the upper critical field in the
$ab$-plane and the possible lattice deformation induced by external pressure is
weak. This explains why the pressure-induced enhancement of $T_c$ for
NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ is not as large as that for
LaO$_{0.5}$F$_{0.5}$BiS$_{2}$.
