Electrothermal Compact Drift Model of TiO$_2$ Memristors: Verilog-A Implementation and Dimensionless Regime Mapping
Compact models of titanium-dioxide memristors used in circuit simulation commonly follow the drift formulation of Strukov \textit{et al.} and assume a constant ionic mobility, although oxygen-vacancy migration is thermally activated and Joule self-heating is unavoidable. We present ATDM (Arrhenius Thermally Activated Drift Model), a minimal electrothermal compact model that couples the drift equation to a lumped heat balance through an Arrhenius mobility. It adds one thermal state while preserving the electrical state variable, the relation $v=iR(x)$, and the isothermal limit. The model is implemented in \textsc{Verilog-A}, compiled with OpenVAF, and simulated as a device in \texttt{ngspice}, where it reproduces an independent reference implementation within $0.009\,\%$ of the peak voltage and recovers the isothermal excursion at zero activation energy. A dimensionless formulation introduces a thermal lag $\varepsilon$ and an effective switching number $Θ$. Across $2.5\times10^{5}$ simulations, $Θ$ orders the state-variable excursion over the tested quasi-static domain with a robust relative scatter of $4\,\%$ and no fitted constant, and the $(\varepsilon,Θ)$ regime map quantifies when the quasi-static thermal reduction remains valid. Under current drive, the stroboscopic map of the reduced model is proved to be strictly increasing, which excludes period-doubling and chaos in that reduction. Within the sampled ranges, a variance-based sensitivity analysis identifies the effective thermal resistance as a first-order contributor to self-heating, while separate capacitance sweeps show a weak influence of the thermal capacitance in the quasi-static regime. The results characterize the specified model with representative, uncalibrated parameters.