The transistor is modeled generically by a heavily simplified virtual-source (short-channel) MOSFET model (Khakifirooz 2009). Although this model was first defined for Silicon transistors, it has been successfully adapted to numerous other contexts, including Graphene (Wang 2011) and Gallium Nitride devices, both HEMTs (Radhakrishna 2013) and MOSHEMT+VO\({}_{2}\) HyperFETs (Verma 2017). Following Khakifirooz (Khakifirooz 2009), the drain current \(I_{D}\) is expressed

\begin{equation} \frac{I_{D}}{W}=Q_{ix_{0}}v_{x_{0}}F_{s}\\ \end{equation}where \(Q_{iz_{0}}\) is the charge at the virtual source point, \(v_{x_{0}}\) is the virtual source saturation velocity, and \(F_{s}\) is an empirically fitted ”saturation function” which smoothly transitions between linear (\(F_{s}\propto V_{DS}/V_{DSSAT}\)) and saturation (\(F_{s}\approx 1\)) regimes. The charge in the channel is described via the following semi-empirical form first proposed for CMOS-VLSI modeling (Wright 1985) and employed frequently since (often with modifications, eg (Khakifirooz 2009, Radhakrishna 2013)):

\begin{equation} Q_{ix_{0}}=C_{\mathrm{inv}}nV_{\mathrm{th}}\ln\left[1+\exp\left\{\frac{V_{GSi}-V_{T}}{nV_{\mathrm{th}}}\right\}\right]\\ \end{equation}where \(C_{\mathrm{inv}}\) is an effective inversion capacitance for the gate, \(nV_{th}\ln 10\) is the subthreshold swing of the transistor, \(V_{GSi}\) is the transistor gate-to-source voltage, \(V_{T}\) is the threshold voltage, and \(V_{\mathrm{th}}\) is the thermal voltage \(kT/q\).

For precise modeling, Khakifirooz includes further adjustments of \(V_{T}\) due to the drain voltage (DIBL parameter) and the gate voltage (strong vs weak inversion shift), as well as a functional form of \(F_{s}\). For a first-pass, we will ignore these effects, employ a constant \(V_{T}\), and assume the supply voltage is maintained above the gate overdrive such that \(F_{s}\approx 1\). However, we will add on a leakage floor with conductance \(G_{\mathrm{leak}}\). Altogether, the final current expression (for the analytical part of this analysis) is

\begin{equation} \label{eq:transistor_iv}\frac{I_{D}}{W}=nv_{x_{0}}C_{\mathrm{inv}}V_{th}\ln\left[1+\exp\left\{\frac{V_{\mathrm{GSi}}-V_{\mathrm{T}}}{nV_{th}}\right\}\right]+\frac{G_{\mathrm{leak}}}{W}V_{\mathrm{DSi}}\\ \end{equation}The phase-change material is included by a similarly generic and brutally simple model. As done with the transistor, the goal is to capture only the most relevant feature: here, an abrupt change in resistance. However, for a concrete example, the material most frequently used in HyperFET research (Pergament 2013, Shukla 2015) is Vanadium Diozide (VO\({}_{2}\)), which features an S-style (ie current-controlled) and hysteretic negative differential resistance (NDR) region (Pergament 2016, Zimmers 2013) due to an insulator-metal transition (IMT), the underlying mechanism of which has been a source of long-running controversy (Pergament 2013). Though the literature contains numerous examples of voltage-swept I-V curves (Shukla 2015, Zimmers 2013, Radu 2015, Yoon 2014), proper modeling of a current-controlled NDR device in a circuit requires a current-swept I-V, examples of which can be found in (Zimmers 2013, Kumar 2013, Pergament 2016). The cleanest of these is Figure 1(b) of Kumar (Kumar 2013), which is suggested to the reader as a concrete realization of the model used herein.

The phase-change resistor (PCR) will be described by a hysteretic piecewise-linear model:

\begin{equation} \label{eq:PCR_iv}V_{R}=\left\{\begin{array}[]{llr}I_{R}R_{\mathrm{ins}}&,&I_{R}<I_{\mathrm{IMT}}\\ V_{\mathrm{met}}+I_{R}R_{\mathrm{met}}&,&I_{R}>I_{\mathrm{MIT}}\\ \end{array}\right\}\\ \end{equation}where we require \(I_{\mathrm{MIT}}\leq I_{\mathrm{IMT}}\) to ensure that the model is defined for all values of the current; \(I_{\mathrm{MIT}}=I_{\mathrm{IMT}}\) would be the case of zero hysteresis. For convenience, we define voltage thresholds, \(V_{\mathrm{IMT}}=I_{\mathrm{IMT}}R_{\mathrm{ins}}\) and \(V_{\mathrm{MIT}}=I_{\mathrm{MIT}}R_{\mathrm{met}}+V_{\mathrm{met}}\). Finally, we require \(V_{\mathrm{met}}+I_{IMT}R_{\mathrm{met}}<V_{\mathrm{IMT}}\) and \(I_{\mathrm{MIT}}R_{\mathrm{ins}}>V_{\mathrm{MIT}}\) to ensure that the absolute resistance of the metallic state is lower than that of the insulating state wherever they are both defined.

When the PCR is attached in series with the source of the transistor, the total device satisfies the above equations with the additional matching \(I=I_{D}=I_{R}\) and \(V_{\mathrm{GSi}}=V_{GS}-V_{R}\) where \(I\) is the current through the device and \(V_{GS}\) is the voltage between HyperFET gate (the transistor gate) and the HyperFET source (the exterior node of the resistor). We can immediately solve for several regions of the HyperFET model. For this section, it is assumed that the transistor and PCR are scaled such that the hysteretic region is entirely contained within subthreshold, and above the leakage floor; these choices will be discussed in the next section.

When the transistor is completely off, only the leakage term of (\ref{eq:transistor_iv}) remains, and combines with the PCR off-state resistance, leading to

\begin{equation} I=G_{\mathrm{off}}V_{DS},\quad G_{\mathrm{off}}^{-1}=R_{\mathrm{ins}}+1/G_{\mathrm{leak}}\\ \end{equation}For the lower branch (in the region above the leakage floor), we plug \(V_{\mathrm{GSi}}=V_{\mathrm{GS}}-IR_{\mathrm{ins}}\) into the transistor I-V (\ref{eq:transistor_iv}), and take the subthreshold limit: \(\ln(1+e^{x})\approx e^{x}\) for \(-x\gg 1\).

\begin{equation} \label{eq:insbranch_preW}\frac{I}{W}=nC_{\mathrm{inv}}v_{x_{0}}V_{th}\exp\left\{\frac{V_{\mathrm{GS}}-IR_{\mathrm{ins}}-V_{\mathrm{T}}}{nV_{th}}\right\}\\ \end{equation}This can be rearranged and solved in terms of the Lambert \(\mathcal{W}\) function

\begin{equation} \label{eq:insbranch}I=\frac{nV_{th}}{R_{\mathrm{ins}}}\mathcal{W}\left[WC_{\mathrm{inv}}v_{x_{0}}R_{\mathrm{ins}}\exp\left\{\frac{V_{\mathrm{GS}}-V_{\mathrm{T}}}{nV_{th}}\right\}\right]\\ \end{equation}For the upper branch, we plug in \(V_{\mathrm{GSi}}=V_{\mathrm{GS}}-V_{\mathrm{met}}-IR_{\mathrm{met}}\), and follow the same procedure to find

\begin{equation} \label{eq:metbranch}I=\frac{nV_{th}}{R_{\mathrm{met}}}\mathcal{W}\left[WC_{\mathrm{met}}v_{x_{0}}R_{\mathrm{met}}\exp\left\{\frac{V_{\mathrm{GS}}-V_{\mathrm{met}}-V_{\mathrm{T}}}{nV_{th}}\right\}\right]\\ \end{equation}Note that if the metal-state resistance is small \(IR_{\mathrm{met}}\ll nV_{th}\), we can approximate

\begin{equation} \frac{I}{W}\approx nV_{th}C_{\mathrm{met}}v_{x_{0}}\exp\left\{\frac{V_{\mathrm{GS}}-V_{\mathrm{met}}-V_{\mathrm{T}}}{nV_{th}}\right\}\\ \end{equation}