  In a previous work we showed that keV scale sterile neutrino dark matter
$\nu_s$ is possible to be detected in $\beta$ decay experiment using
radioactive sources such as $^3$T or $^{106}$Ru. The signals of this dark
matter candidate are mono-energetic electrons produced in neutrino capture
process $\nu_s + N' \to N +e^-$. These electrons have energy greater than the
maximum energy of the electrons produced in the associated decay process $N'
\to N +e^- + {\bar \nu}_e$. Hence, signal electron events are well beyond the
end point of the $\beta$ decay spectrum and are not polluted by the $\beta$
decay process. Another possible background, which is a potential threat to the
detection of $\nu_s$ dark matter, is the electron event produced by the
scattering of solar neutrinos with electrons in target matter. In this article
we study in detail this possible background and discuss its implication to the
detection of keV scale sterile neutrino dark matter. In particular, bound state
features of electrons in Ru atom are considered with care in the scattering
process when the kinetic energy of the final electron is the same order of
magnitude of the binding energy.
