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9 Appendix

9.1 Transconductance as a function of the gate bias overdrive

The transconductance in the linear regime is calculated by differentiating Eq. 2.1 by (math image):

\{begin}{align*} g_\mathrm {m,lin} = \frac {\partial {I_\mathrm {D,lin}}}{\partial {V_\mathrm {GS}}} &\approx I_\mathrm {D,lin} \left (\frac {1}{V_\mathrm {GS}-V_\mathrm {TH}} + \frac {1}{\mu _\mathrm {eff}} \frac {\partial {\mu _\mathrm {eff}}}{\partial {V_\mathrm {GS}}}\right ), \\ &\approx I_\mathrm {D,lin} \left (\frac {1}{V_\mathrm {GS}-V_\mathrm {TH}} - \frac {\Theta }{1+\Theta (V_\mathrm {GS}-V_\mathrm {TH})} - \frac {N_\mathrm {it}}{1+\alpha N_\mathrm {it}} \frac {\partial {\alpha }}{\partial {V_\mathrm {GS}}}\right ), \\ &\approx I_\mathrm {D,lin} \left (\frac {1}{(1+\Theta (V_\mathrm {GS}-V_\mathrm {TH})) (V_\mathrm {GS}-V_\mathrm {TH})} + \frac {\alpha N_\mathrm {it}}{2 (1+\alpha N_\mathrm {it}) (V_\mathrm {GS}-V_\mathrm {TH})}\right ), \\ &\approx -\frac {W}{L} \frac {\mu _\mathrm {0} C_\mathrm {OX} V_\mathrm {DS}}{(1+\alpha N_\mathrm {it})} \left (\frac {1}{(1+\Theta (V_\mathrm {GS}-V_\mathrm {TH}))^2} + \frac {\alpha N_\mathrm {it}}{2 (1+\alpha N_\mathrm {it}) (1+\Theta (V_\mathrm {GS}-V_\mathrm {TH}))}\right ).\\ \\ \noalign {\mbox {The transconductance in the saturation regime is calculated by differentiating Eq.~\ref {e:idsat} by \gls {V_GS}:}} g_\mathrm {m,sat} = \frac {\partial {I_\mathrm {D,sat}}}{\partial {V_\mathrm {GS}}} &\approx I_\mathrm {D,sat} \left (\frac {2}{V_\mathrm {GS}-V_\mathrm {TH}} + \frac {1}{\mu _\mathrm {eff}} \frac {\partial {\mu _\mathrm {eff}}}{\partial {V_\mathrm {GS}}}\right ), \\ &\approx I_\mathrm {D,sat} \left (\frac {2}{V_\mathrm {GS}-V_\mathrm {TH}} - \frac {\Theta }{1+\Theta (V_\mathrm {GS}-V_\mathrm {TH})} - \frac {N_\mathrm {it}}{1+\alpha N_\mathrm {it}} \frac {\partial {\alpha }}{\partial {V_\mathrm {GS}}}\right ), \\ &\approx I_\mathrm {D,sat} \left (\frac {2+\Theta (V_\mathrm {GS}-V_\mathrm {TH})}{(1+\Theta (V_\mathrm {GS}-V_\mathrm {TH})) (V_\mathrm {GS}-V_\mathrm {TH})} + \frac {\alpha N_\mathrm {it}}{2 (1+\alpha N_\mathrm {it}) (V_\mathrm {GS}-V_\mathrm {TH})}\right ), \\ &\approx -\frac {W}{L} \frac {\mu _\mathrm {0} C_\mathrm {OX} (V_\mathrm {GS}-V_\mathrm {TH})}{(1+\alpha N_\mathrm {it})} \left (\frac {2+\Theta (V_\mathrm {GS}-V_\mathrm {TH})}{(1+\Theta (V_\mathrm {GS}-V_\mathrm {TH}))^2} + \frac {\alpha N_\mathrm {it}}{2 (1+\alpha N_\mathrm {it}) (1+\Theta (V_\mathrm {GS}-V_\mathrm {TH}))}\right ). \{end}{align*}

The linear transconductance (\( g_\mathrm {m,lin} \)) decreases gradually in the triode region of the device (\( |V_\mathrm {GS}| \geq |V_\mathrm {TH}| \)) due to enhanced field dependent scattering (\( \Theta \)). The saturation transconductance (\( g_\mathrm {m,sat} \)) increases more or less linearly in the deep inversion region of the device. Due to field enhanced scattering (\( \Theta \)), a positive curvature is superimposed.

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