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$$m$ thickness, consisting of an ordered array of vertically oriented nanotubes with outer diameter of 75 nm. $Zr/ZrO_2/Au$ sandwich structures have been fabricated by mask magnetron deposition. The effects of resistive switching in the $Zr/ZrO_2/Au$ memristors in unipolar and bipolar modes have been investigated. The resistance ratios $geq3cdot10^4$ between high-resistance (HRS) and low-resistance (LRS) states have been evaluated. It has been founded the conductivity of LRS is quantized in a wide range with minimum value of $0.5G_0=38.74$ $mu$$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.
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