Table 4.

The state variable equations presented in an improved form

Model	Original state variable dynamics (f2)	Modified using concepts from Wang and Roychowdhury100
Linear ion drift [3]	f2 = μv × Ron × f1(vpn, s)	DC hysteresis not present. Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1
Non-linear ion drift [46, 68]	f2 = a × vpnm	DC hysteresis not present. Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1
Simmons tunneling barrier [70–72]	$f_2=\left\{\begin{array}{c}{c}_{\mathrm{off}}\times sinh\left(\frac{i}{i_{\mathrm{off}}}\right)\times exp\left(-exp\left(\frac{s-a_{\mathrm{off}}}{w_c}-\frac{i}{b}\right)-\frac{s}{w_c}\right),\text{if}i\ge 0\\ c_{\mathrm{on}}\times sinh\left(\frac{i}{i_{\mathrm{on}}}\right)\times exp\left(-exp\left(\frac{a_{\mathrm{on}}-s}{w_c}+\frac{i}{b}\right)-\frac{s}{w_c}\right),\text{otherwise}\end{array}\right.$ where i = f1 (vpn, s)	No DC hysteresis present. Consists of fast growing functions. sinh is changed to safesinh(), exp to safeexp(). Smoothing is performed and bounds for s, so that 0 ≤ s ≤ 1
TEAM/VTEAM [75–77]	$f_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{\mathit{vpn}}{v_{\mathrm{off}}}-1\right)}^{a_{\mathrm{off}}},\text{if}\mathit{vpn}>v_{\mathrm{off}}\\ k_{\mathrm{on}}\times {\left(\frac{\mathit{vpn}}{v_{\mathit{on}}}-1\right)}^{a_{\mathrm{on}}},\text{if}\mathit{vpn}<v_{\mathrm{on}}\\ 0,\text{otherwise}\end{array}\right.$	The equation is redesigned as: $f_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{\mathit{vpn}-v^{\ast }}{v_{\mathrm{off}}}\right)}^{a_{\mathrm{off}}},\mathit{\text{if}}\mathit{vpn}>v_{\mathrm{off}}\\ k_{\mathrm{on}}\times {\left(\frac{\mathit{vpn}-v^{\ast }}{v_{\mathrm{on}}}\right)}^{a_{\mathrm{on}}},\text{otherwise}\end{array}\right.$ where v∗ = (1 − s) × voff + s × von, Such that when s = 1 and s = 0, it is equivalent to the VTEAM equation in the vpn > voff and vpn < von regions, respectively. The functions are also smoothened by: $f_{2p}=k_{\mathrm{off}}.{\left(\mathit{vpn}-v^{\ast }/v_{\mathrm{off}}\right)}^{{\alpha }_{\mathrm{off}}}$, $f_{2n}=k_{\mathrm{on}}.{\left(\mathit{vpn}-v^{\ast }/v_{\mathrm{on}}\right)}^{{\alpha }_{\mathrm{on}}}$, f2 = smoothswitch(f2n, f2p, vpn − v∗, smoothing) The bounds for s are set using clipping techniques.
Yakpocic [73, 74]	f2 = g(vpn) × f(s), where $g\left(\mathit{vpn}\right)=\left\{\begin{array}{c}{A}_p\times \left(exp\left(\mathit{vpn}\right)-exp\left(V_p\right)\right),\mathit{\text{if}}\mathit{vpn}>V_p\\ -A_n\times \left(exp\left(-\mathit{vpn}\right)-exp\left(V_n\right)\right),\mathit{\text{if}}\mathit{vpn}<-V_n\\ 0,\text{otherwise},\end{array}\right.$ and $f\left(s\right)=\left\{\begin{array}{c}exp\left(-{\alpha }_p\times \left(s-x_p\right)\right),\mathit{\text{if}}s\ge x_p\\ exp\left({\alpha }_n\times \left(s-1+x_n\right)\right),\mathit{\text{if}}s\le 1-x_n\\ 1,\text{otherwise}\end{array}\right.$	The equations are designed to get proper DC hysteresis: $g\left(\mathit{vpn}\right)=\left\{\begin{array}{c}{A}_p\times \left(exp\left(\mathit{vpn}\right)-exp\left(v^{\ast }\right)\right),\mathit{\text{if}}\mathit{vpn}>v^{\ast }\\ -A_n\times \left(exp\left(-\mathit{vpn}\right)-exp\left(V_n\right)\right),\mathit{\text{otherwise}}\end{array}\right.$ where v∗ =  − Vn × s + Vp × (1 − s) Also exponential function is changed to safeexp(). The whole function is made smooth. Clipping is used to set bounds for s.
ASU/Stanford [78–81]	$f_2=-v_0\times exp\left(-\frac{q\times E_a}{k\times T}\right)\times sinh\left(\frac{\mathit{vpn}\times \gamma \times a_0\times q}{k\times T\times t_{\mathit{ox}}}\right)$ where γ = γ0 − β0 × Gap3	The d/dt Gap is converted to d/dt s: $f_2=\left(\mathit{\text{maxGap}}-\mathit{\text{minGap}}\right)\times v_0\times exp\left(-\frac{q\times E_a}{k\times T}\right)\times sinh\left(\frac{\mathit{vpn}\times \gamma \times a_0\times q}{k\times T\times t_{\mathit{ox}}}\right)$ Also exp is changed to safeexp() and sinh tosafesinh (). Clipping is used to set bounds for s.