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MIT6_012F09_lec11_drain - 6.012 Microelectronic Devices and...

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6.012 - Microelectronic Devices and Circuits Fall 2009 MOSFET Drain Current Modeling In the Gradual Channel Model for the MOSFET we write the drain current, i D , as the product of q N * (y) , the inversion layer sheet charge density at position y along the channel; s y (y), the net drift velocity of the inversion layer carriers there (electrons in the n-channel device we are modeling), and W, the channel width: i D = - q N * (y) s y( y) W with ε ox dv CS (y) q N * (y) = - [v GB - V T (y)] and s y (y) = - µ e E y = µ e dy t ox Substituting these expressions yields: dv CS (y) i D = W µ C ox * [v GB - V T (y)] dy where we have identified the gate capacitance per unit area, C ox * , as ε ox /t ox and where the threshold voltage is given by V T (y) = V FB + |2 φ p | +v CB (y) + 1 2 ε Si qN A [|2 φ p | + v CB (y) C ox * Defining the body factor, γ , as 2 ε Si qN A /C ox * , and writing v CB (y) as v CS (y) - v BS, we can rewrite this as V T (y) = V FB + |2 φ p | +v CS (y) - v BS + γ |2 φ p | + v CS (y) - v BS and thus we can write i D as dv CS (y) i D = W µ e C * [v GS -V FB - |2 φ p |- v CS (y) - γ |2 φ p | + v CS (y) - v BS ] dy ox
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