Table 4 The state variable equations presented in an improved form
From: A Collective Study on Modeling and Simulation of Resistive Random Access Memory
Model | Original state variable dynamics (f2) | Modified using concepts from Wang and Roychowdhury100 |
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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 |
f2 = a × vpnm | DC hysteresis not present. Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1 | |
\( {f}_2=\left\{\begin{array}{c}{c}_{\mathrm{off}}\times \sinh \left(\frac{i}{i_{\mathrm{off}}}\right)\times \exp \left(-\exp \left(\frac{s-{a}_{\mathrm{off}}}{w_c}-\frac{i}{b}\right)-\frac{s}{w_c}\right),\mathrm{if}\ i\ge 0\\ {}{c}_{\mathrm{on}}\times \sinh \left(\frac{i}{i_{\mathrm{on}}}\right)\times \exp \left(-\exp \left(\frac{a_{\mathrm{on}}-s}{w_c}+\frac{i}{b}\right)-\frac{s}{w_c}\right),\mathrm{otherwise}\end{array}\right. \) 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 | |
\( {f}_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{vpn}{v_{\mathrm{off}}}-1\right)}^{a_{\mathrm{off}}},\mathrm{if}\ vpn>{v}_{\mathrm{off}}\\ {}{k}_{\mathrm{on}}\times {\left(\frac{vpn}{v_{on}}-1\right)}^{a_{\mathrm{on}}},\mathrm{if}\ vpn<{v}_{\mathrm{on}}\\ {}0,\kern10.25em \mathrm{otherwise}\end{array}\right. \) | 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) × 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: \( {f}_{2p}={k}_{\mathrm{off}}.{\left( vpn-{v}^{\ast }/{v}_{\mathrm{off}}\right)}^{\alpha_{\mathrm{off}}} \), \( {f}_{2n}={k}_{\mathrm{on}}.{\left( vpn-{v}^{\ast }/{v}_{\mathrm{on}}\right)}^{\alpha_{\mathrm{on}}} \), f2 = smoothswitch(f2n, f2p, vpn − v∗, smoothing) The bounds for s are set using clipping techniques. | |
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. | |
\( {f}_2=-{v}_0\times \exp \left(-\frac{q\times {E}_a}{k\times T}\right)\times \sinh \left(\frac{vpn\times \gamma \times {a}_0\times q}{k\times T\times {t}_{ox}}\right) \) 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. |