  Amorphous insulators have localized wave functions that decay with the
distance $r$ following exp($-r/\zeta$). Since nanoscale conduction is not
excluded at $r<\zeta$, one may use amorphous insulators and take advantage of
their size effect for nanoelectronic applications. Voltage-regulated nanoscale
conductivity is already utilized in metal-insulator-metal devices known as
memristors. But typically their tunable conductivity does not come from
electrons but from migrating ions within a stoichastically formed filament, and
as such their combined resistor-memory performance suffers. Here we demonstrate
amorphous-silicon-based memristors can have coherent electron wave functions
extending to the full device thickness, exceeding 15 nm. Remarkably, despite
the large aspect ratio and very thin thickness of the device, its electrons
still follow an isotropic, three-dimensional pathway, thus providing uniform
conductivity at the nanometer scale. Such pathways in amorphous insulators are
derived from overlapping gap states and regulated by trapped charge, which is
stabilized by electron-lattice interaction; this makes the memristor exhibit
pressure-triggered insulator$\rightarrow$metal transitions. Fast, uniform,
durable, low-power and purely electronic memristors with none of the
shortcomings of ion-migrating memristors have been fabricated from a variety of
amorphous silicon compositions and can be readily integrated into silicon
technology. Therefore, amorphous silicon may provide the ideal platform for
building proximal memories, transistors and beyond.
