  We consider the physical effects of dimensional quantization in conductive
filaments that underlie operations of some modern electronic devices. We show
that, as a result of quantization, a sufficiently constricted filament acquires
a positive charge. Several applications of this finding include describing the
local host material polarization, the stability limit of filament
constrictions, equilibrium filament radius, effects of polarity in device
switching, and quantization of conductance.
