  Owing to the outstanding properties provided by nontrivial band topology,
topological phases of matter are considered as a promising platform towards
low-dissipation electronics, efficient spin-charge conversion, and topological
quantum computation. Achieving ferroelectricity in topological materials
enables the non-volatile control of the quantum states, which could greatly
facilitate topological electronic research. However, ferroelectricity is
generally incompatible with systems featuring metallicity due to the screening
effect of free carriers. In this study, we report the observation of memristive
switching based on the ferroelectric surface state of a topological semimetal
(TaSe4)2I. We find that the surface state of (TaSe4)2I presents out-of-plane
ferroelectric polarization due to surface reconstruction. With the combination
of ferroelectric surface and charge-density-wave-gapped bulk states, an
electric switchable barrier height can be achieved in (TaSe4)2I-metal contact.
By employing a multi-terminal grounding design, we manage to construct a
prototype ferroelectric memristor based on (TaSe4)2I with on/off ratio up to
10^3, endurance over 10^3 cycles, and good retention characteristics. The
origin of the ferroelectric surface state is further investigated by
first-principles calculations, which reveals an interplay between
ferroelectricity and band topology. The emergence of ferroelectricity in
(TaSe4)2I not only demonstrates it as a rare but essential case of
ferroelectric topological materials, but also opens new routes towards the
implementation of topological materials in functional electronic devices.
