  The areal footprint of memristors is a key consideration in material-based
neuromorophic computing and large-scale architecture integration. Electronic
transport in the most widely investigated memristive devices is mediated by
filaments, posing a challenge to their scalability in architecture
implementation. Here we present a compelling alternative memristive device and
demonstrate that areal downscaling leads to enhancement in memristive memory
window, while maintaining analogue behavior, contrary to expectations. Our
device designs directly integrated on semiconducting Nb-SrTiO$_3$ allows
leveraging electric field effects at edges, increasing the dynamic range in
smaller devices. Our findings are substantiated by studying the microscopic
nature of switching using scanning transmission electron microscopy, in
different resistive states, revealing an interfacial layer whose physical
extent is influenced by applied electric fields. The ability of Nb-SrTiO$_3$
memristors to satisfy hardware and software requirements with downscaling,
while significantly enhancing memristive functionalities, makes them strong
contenders for non-von Neumann computing, beyond CMOS.
