Table 3 Improved I-V relationships of the various models

Linear ion drift [3]

f₁ = (R_ON × s + R_off × (1 − s))⁻¹ × vpn

Can have division by zero error when s = R_off/(R_on/R_off).

Modified equation:

y = smoothclip(s − R_off/(R_on − R_off), smoothing) + R_off/(R_on − R_off)

Then,

f₁ = (R_on × y + R_off × (1 − y))⁻¹ × vpn

Non-linear ion drift [46, 68]

I = sⁿβ sinh(α × vpn) + χ(exp(γ × vpn) − 1)

sinh can be changed to safesinh(), exponential function to safeexp()

Yakopcic [73, 74]

\( I(t)=\left\{\begin{array}{c}{A}_1\times s\times \sinh (Bvpn),\kern0.5em vpn\ge 0\\ {}{A}_2\times s\times \sinh (Bvpn),\kern2.25em vpn<0\end{array}\right. \)

sinh is changed to safesinh(). The function is then smoothed.

f_1p = A₁ × s × safesinh(B × vpn, maxslope)

f_1n = A₂ × s × safesinh(B × vpn, maxslope)

f₁ = smoothswitch(f_1n, f_1p, vpn, smoothing)

TEAM/VTEAM [75,76,77]

\( v(t)={R}_{\mathrm{ON}}{e}^{\left(\lambda /{x}_{\mathrm{off}}-{x}_{\mathrm{on}}\right)\left(x-{x}_{\mathrm{on}}\right)}\times i(t) \)

The exponential function is changed to safeexp()

ASU/Stanford [78,79,80,81]

\( I\left(g,V\right)={I}_0\exp \left(\frac{-g}{g_0}\right)\sinh \left(\frac{V}{V_0}\right) \)

The gap is expressed using s:

gap = s ×  min  gap + (1 − s) × maxgap

Then sinh is changed to safesinh(), exponential function to safeexp()