  Memristive switching devices, candidates for resistive random access memory
technology, have been shown to switch off through a progression of states with
quantized conductance and subsequent non-integer conductance (in terms of
conductance quantum $G_0$). We have performed calculations based on density
functional theory to model the switching process for a Pt-HfO$_2$-Pt structure,
involving the movement of one or two oxygen atoms. Oxygen atoms moving within a
conductive oxygen vacancy filament act as tunneling barriers, and partition the
filament into weakly coupled quantum wells. We show that the low-bias
conductance decreases exponentially when one oxygen atom moves away from
interface. Our results demonstrate the high sensitivity of the device
conductance to the position of oxygen atoms.
