  The nonlinear Hall effect (NLHE) with time-reversal symmetry constitutes the
appearance of a transverse voltage quadratic in the applied electric field. It
is a second-order electronic transport phenomenon that induces frequency
doubling and occurs in non-centrosymmetric crystals with large Berry curvature
-- an emergent magnetic field encoding the geometric properties of electronic
wavefunctions. The design of (opto)electronic devices based on the NLHE is
however hindered by the fact that this nonlinear effect typically appears at
low temperatures and in complex compounds characterized by Dirac or Weyl
electrons. Here, we show a strong room temperature NLHE in the centrosymmetric
elemental material bismuth synthesized in the form of technologically relevant
polycrystalline thin films. The ($1\,1\,1$) surface electrons of this material
are equipped with a Berry curvature triple that activates side jumps and skew
scatterings generating nonlinear transverse currents. We also report a boost of
the zero field nonlinear transverse voltage in arc-shaped bismuth stripes due
to an extrinsic geometric classical counterpart of the NLHE. This electrical
frequency doubling in curved geometries is then extended to optical second
harmonic generation in the terahertz (THz) spectral range. The strong nonlinear
electrodynamical responses of the surface states are further demonstrated by a
concomitant highly efficient THz third harmonic generation which we achieve in
a broad range of frequencies in Bi and Bi-based heterostructures. Combined with
the possibility of growth on CMOS-compatible and mechanically flexible
substrates, these results highlight the potential of Bi thin films for THz
(opto)electronic applications.
