  The electronic properties of the surface of $\beta$-FeSi$_2$ have been
debated for a long while. We studied the surface states of $\beta$-FeSi$_2$
films grown on Si(001) substrates using scanning tunnelling microscopy (STM)
and spectroscopy (STS), with the aid of density functional theory (DFT)
calculations. STM simulations using the surface model proposed by Romanyuk et
al. [Phys. Rev. B 90, 155305 (2014)] reproduce the detailed features of
experimental STM images. The result of STS showed metallic surface states in
accordance with theoretical predictions. The Fermi level was pinned by a
surface state that appeared in the bulk band gap of the $\beta$-FeSi$_2$ film,
irrespective of the polarity of the substrate. We also observed negative
differential conductance at $\sim$0.45 eV above the Fermi level in STS
measurements performed at 4.5 K, reflecting the presence of an energy gap in
the unoccupied surface states of $\beta$-FeSi$_2$.
