Table 4 The state variable equations presented in an improved form

Linear ion drift [3]

f₂ = μ _v  × R _on  × 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 × vpn^m

DC hysteresis not present.

Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1

Simmons tunneling barrier [70,71,72]

\( {f}_2=\left\{\begin{array}{c}{c}_{\mathrm{off}}\times \sinh \left(\frac{i}{i_{\mathrm{off}}}\right)\times \exp \left(-\exp \left(\frac{s-{a}_{\mathrm{off}}}{w_c}-\frac{i}{b}\right)-\frac{s}{w_c}\right),\mathrm{if}\ i\ge 0\\ {}{c}_{\mathrm{on}}\times \sinh \left(\frac{i}{i_{\mathrm{on}}}\right)\times \exp \left(-\exp \left(\frac{a_{\mathrm{on}}-s}{w_c}+\frac{i}{b}\right)-\frac{s}{w_c}\right),\mathrm{otherwise}\end{array}\right. \)

where i = f ₁ (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 [75,76,77]

\( {f}_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{vpn}{v_{\mathrm{off}}}-1\right)}^{a_{\mathrm{off}}},\mathrm{if}\ vpn>{v}_{\mathrm{off}}\\ {}{k}_{\mathrm{on}}\times {\left(\frac{vpn}{v_{on}}-1\right)}^{a_{\mathrm{on}}},\mathrm{if}\ vpn<{v}_{\mathrm{on}}\\ {}0,\kern10.25em \mathrm{otherwise}\end{array}\right. \)

The equation is redesigned as:

\( {f}_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{vpn-{v}^{\ast }}{v_{\mathrm{off}}}\right)}^{a_{\mathrm{off}}}, if\ vpn>{v}_{\mathrm{off}}\\ {}{k}_{\mathrm{on}}\times {\left(\frac{vpn-{v}^{\ast }}{v_{\mathrm{on}}}\right)}^{a_{\mathrm{on}}},\kern2.75em \mathrm{otherwise}\end{array}\right. \)

where

v^∗ = (1 − s) × v_off + s × v_on,

Such that when s = 1 and s = 0, it is equivalent to the VTEAM equation in the vpn > v_off and vpn < v_on regions, respectively.

The functions are also smoothened by:

\( {f}_{2p}={k}_{\mathrm{off}}.{\left( vpn-{v}^{\ast }/{v}_{\mathrm{off}}\right)}^{\alpha_{\mathrm{off}}} \),

\( {f}_{2n}={k}_{\mathrm{on}}.{\left( vpn-{v}^{\ast }/{v}_{\mathrm{on}}\right)}^{\alpha_{\mathrm{on}}} \),

f₂ = smoothswitch(f_2n, f_2p, vpn − v^∗, smoothing)

The bounds for s are set using clipping techniques.

Yakpocic [73, 74]

f₂ = g(vpn) × f(s),

where

\( g(vpn)=\left\{\begin{array}{c}{A}_p\times \left(\exp (vpn)-\exp \left({V}_p\right)\right),\kern0.75em if\ vpn>{V}_p\\ {}-{A}_n\times \left(\exp \left(- vpn\right)-\exp \left({V}_n\right)\right),\kern0.75em if\ vpn<-{V}_n\\ {}0,\kern13.00em \mathrm{otherwise},\end{array}\right. \)

and

\( f(s)=\left\{\begin{array}{c}\exp \left(-{\alpha}_p\times \left(s-{x}_p\right)\right),\kern0.75em if\ s\ge {x}_p\\ {}\exp \left({\alpha}_n\times \left(s-1+{x}_n\right)\right),\kern0.75em if\ s\le 1-{x}_n\\ {}1,\kern2.5em \mathrm{otherwise}\ \end{array}\right. \)

The equations are designed to get proper DC hysteresis:

\( g(vpn)=\left\{\begin{array}{c}{A}_p\times \left(\exp (vpn)-\exp \left({v}^{\ast}\right)\right),\kern0.75em if\ vpn>{v}^{\ast}\\ {}-{A}_n\times \left(\exp \left(- vpn\right)-\exp \left({V}_n\right)\right),\kern0.75em otherwise\end{array}\right. \)

where

v^∗ =  − V _n  × s + V _p  × (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 [78,79,80,81]

\( {f}_2=-{v}_0\times \exp \left(-\frac{q\times {E}_a}{k\times T}\right)\times \sinh \left(\frac{vpn\times \gamma \times {a}_0\times q}{k\times T\times {t}_{ox}}\right) \)

where

γ = γ₀ − β₀ × Gap³

The d/dt Gap is converted to d/dt s:

\( {f}_2=\left( maxGap- minGap\right)\times {v}_0\times \exp \left(-\frac{q\times {E}_a}{k\times T}\right)\times \sinh \left(\frac{vpn\times \gamma \times {a}_0\times q}{k\times T\times {t}_{ox}}\right) \)

Also exp is changed to safeexp() and sinh tosafesinh (). Clipping is used to set bounds for s.