  The development prospects of memristive elements for non-volatile memory with
use of the metal-dielectric-metal sandwich structures with a thin oxide layer
are due to the possibility of reliable forming the sustained functional states
with quantized resistance. In the paper we study the properties of fabricated
memristors based on the non-stoichiometric $ZrO_2$ nanotubes in different
resistive switching modes. Anodic oxidation of the $Zr$ foil has been used to
synthesize a zirconia layer of $1.7$ $\mu$$m$ thickness, consisting of an
ordered array of vertically oriented nanotubes with outer diameter of 75 nm.
$Zr/ZrO_2/Au$ sandwich structures have been fabricated by mask magnetron
deposition. The effects of resistive switching in the $Zr/ZrO_2/Au$ memristors
in unipolar and bipolar modes have been investigated. The resistance ratios
$\geq3\cdot10^4$ between high-resistance (HRS) and low-resistance (LRS) states
have been evaluated. It has been founded the conductivity of LRS is quantized
in a wide range with minimum value of $0.5G_0=38.74$ $\mu$$S$ due to the
formation of quantum conductors based on oxygen vacancies ($V_O$). Resistive
switching mechanisms of $Zr/ZrO_2/Au$ memristors with allowing for migration of
$V_O$ in an applied electric field have been proposed. It has been shown that
the ohmic type and space charge limited conductivities are realized in the LRS
and HRS, correspondingly. We present the results which can be used for
development of effective memristors based on functional $Zr/ZrO_2/Au$
nanolayered structure with multiple resistive states and high resistance ratio.
