  The spin Hall effect (SHE) is one of the promising phenomena to utilize a
spin current as spintronics devices, and the theoretical understanding of its
microscopic mechanism is essential to know how to control its response.
Although the SHE in multiorbital systems without inversion symmetry (IS) is
expected to show several characteristic properties due to the cooperative roles
of orbital degrees of freedom and a lack of IS, a theoretical understanding of
the cooperative roles has been lacking. To clarify the cooperative roles, we
study the spin Hall conductivity (SHC) derived by the linear-response theory
for a $t_{2g}$-orbital tight-binding model of the $[001]$ surface or interface
of Sr$_2$RuO$_4$ in the presence of dilute nonmagnetic impurities. We find that
the band anticrossing, arising from a combination of orbital degrees of freedom
and a lack of IS, causes an increase of magnitude and a sign change of the SHC
at some nonmagnetic impurity concentrations. Since a similar mechanism for
controlling the magnitude and sign of the response of Hall effects works in
other multiorbital systems without IS, our mechanism provides an ubiquitous
method to control the magnitude and sign of the response of Hall effects in
some multiorbital systems by introducing IS breaking and tuning of the
nonmagnetic impurity concentration.
