## 2 Methods  
### Theoretical Models  
The structural configuration of the proposed thermal memristor under consideration is shown in Figure 2. The system consists of Tungsten (W)-doped vanadium dioxide (VO\({}_{2}\)) phase-change material (PCM) and a silicon carbide (SiC) material separated by a distance, d. Radiative heat transfer occurs within the system, facilitated by a time-varying sinusoidal temperature change (\(\Delta\)T), as depicted in Figure 2. The gap between the upper layer (PCM) and the lower layer (SiC) was set at 10 \(\upmu\)m to maintain the system in the far-field regime. Each material is characterized by its respective thermal conductivity. The thermal conductivity of the PCM (\(k_{1}\)) is modeled to be dependent on \(T_{av}+\Delta T(t)\), while SiC's thermal conductivity depends solely on \(T_{av}\). The PCM undergoes phase changes at temperatures \(T_{h}\) and \(T_{c}\) during the heating and cooling processes, respectively. The temperature range (\(T_{h}-T_{c}=9\) K) is significant enough to observe a hysteresis effect, illustrating that the thermal conductivity of the PCM is confined between its metallic and insulating phases (\(k_{i}\leq k_{1}\leq K_{m}\)). Analogous to an electrical memristor, a time-varying sinusoidal temperature difference (\(\Delta T(t)=\theta\sin(2\pi t/\tau)\)) is applied to the PCM, where the amplitude (\(\theta\)) equals the difference between the heating (\(T_{h}\)) and cooling (\(T_{c}\)) temperatures. This ensures that the PCM completes its hysteresis loop within one period (\(\tau\)), while the SiC material is maintained at a constant temperature. Experimental verification can be achieved using two separate Peltier cells: one set for periodic oscillation and the other held at a constant temperature. Peltier cells have been reported by [PERSON] et al. to offer a resolution of 0.1 K with a modulation frequency ranging from 0.1 to 100 Hz [28; 29]. The physical parameters (\(k_{m}\) and \(K_{i}\)) for WVO used in this paper were experimentally determined by [PERSON] et al [25], \(\beta\), \(T_{c}\), \(T_{h}\), and the SiC thermal conductivity (\(k_{2}\)) values are as shown in Table 1.  
Figure 2: Schematic diagram of the proposed thermal memristor comprised of a WVO and SiC materials separated by a vacuum gap (\(d=10\)\(\upmu\)m).  
The far-field radiative heat flux between the proposed thermal memristor is given by:  
\[Q=\frac{k_{eff}(t)\times(T_{av}+\Delta T(t))-T_{av})}{d} \tag{1}\]  
where \(k_{eff}\) is the effective thermal conductivity of the whole system and is given as:  
\[k_{eff}=\frac{1}{\frac{1}{k_{1}}+\frac{1}{k_{2}}} \tag{2}\]  
\(k_{1}\) is the thermal conductivity of the Tungsten-doped vanadium dioxide (WVO) phase-change material (PCM) with its temperature dependency modeled as:  
\[k_{1}(T)=k_{i}+\frac{k_{m}-k_{i}}{1+Exp(-\beta(T-T_{av})} \tag{3}\]  
Hence, the effective thermal conductivity as a function of time (\(k_{eff}(t)\)) is expressed in Equation (4).  
\[k_{eff}(t)=\begin{cases}k_{eff}(T_{av}+\Delta T(t),T_{h}),&\text{if }\Delta T(t)>0\\ k_{eff}(T_{av}+\Delta T(t),T_{c}),&\text{if }\Delta T(t)<0\end{cases} \tag{4}\]  
The reference temperature (\(T_{av}\)) is assumed to be the average temperatures of the metallic and insulating phases of the PCM, which is given as:  
\[T_{av}=\frac{T_{c}+T_{h}}{2} \tag{5}\]  
The thermal memristor (\(M\)) of the system illustrated in Figure 2 can be expressed as:  
\[M=\frac{\Delta T}{Q} \tag{6}\]