  Divergent density of states (DOS) can induce extraordinary phenomena such as
significant enhancement of superconductivity and unexpected phase transitions.
Moreover, van Hove singularities (VHSs) are known to lead to divergent DOS in
two-dimensional (2D) systems. Despite the recent interest in VHSs, only a few
controllable cases have been reported to date. In this work, we investigate the
electronic band structures of a 2D VHS with angle-resolved photoemission
spectroscopy and control transport properties by utilizing an atomically
ultrathin SrRuO$_3$ film. By applying electric fields with alkali metal
deposition and ionic-liquid gating methods, we precisely control the 2D VHS,
and the sign of the charge carrier. Use of a tunable 2D VHS in an atomically
flat oxide film could serve as a new strategy to realize infinite DOS near the
Fermi level, thereby allowing efficient tuning of electric properties.
