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# MIT6_012F09_lec11_gradual - 6.012 Microelectronic Devices...

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6.012 - Microelectronic Devices and Circuits Fall 2009 The Gradual Channel Approximation for the MOSFET: We are modeling the terminal characteristics of a MOSFET and thus want i D (v DS , v GS , v BS ), i B (v DS , v GS , v BS ), and i G (v DS , v GS , v BS ). We restrict our model to v DS 0 and v BS 0, so the diodes at the source and drain are always reverse biased; in this case i B 0. Because of the insulating nature of the oxide beneath the gate, we also have i G = 0, and our problem reduces to finding i D (v DS , v GS , v BS ). The model we use is what is called the "gradual channel approximation", and it is so named because we assume that the voltages vary gradually along the channel from the drain to the source. At the same time, they vary quickly perpendicularly to the channel moving from the gate to the bulk semiconductor. In the model we assume we can separate the problem into two pieces which can be worked as simple one-dimensional problems. The first piece is the x-direction problem relating the gate voltage to the channel charge and the depletion region; this is the problem we solved when we studied the MOS capacitor. The second piece is the y-direction problem involving the current in, and voltage drop along, the channel; this is the problem we will consider now. To begin we assume that the voltage on the gate is sufficient to invert the channel and proceed. Notice that i D (v DS , v GS , v BS ) is the current in the channel; this is a drift current. There is a resistive voltage drop, v CS (y), along the channel from v CS = v DS at the drain end of the channel, y = L, to v CS = 0 at the source end of 1

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the channel, y = 0. At any point, y, along the channel we will have: i D = -q N * (y) s y (y) W The current is not a function of y, -q N *
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