6.012 - Microelectronic Devices and Circuits Fall 2009 The Gradual Channel Approximation for the MOSFET: We are modeling the terminal characteristics of aMOSFET and thus want iD(vDS, vGS, vBS), iB(vDS, vGS, vBS),and iG(vDS, vGS, vBS). We restrict our model to vDS ≥ 0 and vBS ≤ 0, so the diodes at the source and drain are alwaysreverse biased; in this case iB ≈ 0. Because of the insulatingnature of the oxide beneath the gate, we also have iG = 0, and our problem reduces to finding iD(vDS, vGS, vBS). The model we use is what is called the "gradualchannel approximation", and it is so named because weassume that the voltages vary gradually along the channelfrom the drain to the source. At the same time, they varyquickly 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 beworked as simple one-dimensional problems. The first piece is the x-direction problem relating the gate voltageto the channel charge and the depletion region; this is theproblem we solved when we studied the MOS capacitor.The second piece is the y-direction problem involving thecurrent in, and voltage drop along, the channel; this is theproblem we will consider now. To begin we assume that the voltage on the gate is sufficient to invert the channeland proceed. Notice that iD(vDS, vGS, vBS) is the current in the channel; this is a drift current. There is a resistive voltage drop, vCS(y), along the channel from vCS = vDS at the drain end of the channel, y = L, to vCS = 0 at the source end of 1the channel, y = 0. At any point, y, along the channel we will have: iD = -qN* (y) sy(y) W The current is not a function of y, -qN* (y) is the channel sheet charge density at y, * (y) *-qN = - Cox [vGB - VT(y)] *with Cox ≡ εo/to, and sy(y) is the net velocity of the charge carriers in the y-direction at y, which for modestelectric fields is linearly proportional to the field: - dvCS(y)sy(y) = - µeEy(y) = - µe dy The current is then dvCS(y)*iD = W µe Cox [vGB - VT(y)] dy To proceed, we must examine the factor [vGB - VT(y)].We are referencing our voltages to the source so we firstwrite vGB = vGS - vBS. Next we look at VT(y); why is it a function of y? To answer this question we must note that the picture is a bit different in the MOSFET than it wasbefore with the isolated MOS structure because now the channel (inversion layer) can have a voltage relative to thesubstrate. It is reverse biased by an amount -vCB(y) and so now the potential drop across the depletion region is - 2φp+ vCB(y). Thus in our expression for VT, - 2φp is replaced by - 2φp + vCB(y). We have: VT(y) = VFB - 2φp + vCB(y) + 1 2 εSi qNA[-2φp + vCB(y)]*Cox 21It is common practice to name the factor Cox 2 εSi qNA* the body factor, and call it γ, so we can then write VT(y) as VT(y) = VFB - 2φp + vCB(y) + γ [-2φp + vCB(y)] Using this in the factor [vGB - VT(y)] in the iD expression, we have [vGB - VT(y)] = vGB - VFB + 2φp - vCB(y) -γ [-2φp + vCB(y)] which, after using vGB = vGS - vBS and vCB = vCS - vBS, and rearranging terms somewhat, is [vGB - VT(y)] = vGS - vCS(y) - VFB + 2φp - γ [-2φp + vCS(y) - vBS] The vCS(y) factor under the square root turns out tocomplicate the subsequent mathematics annoyingly and ithas been found that it is better (and possible) to linearizethis term before proceeding. We write the term as [-2φp + vCS(y) - vBS] = 1 + vCS(y)[-2φp - vBS] [-2φp - vBS] and approximate the factor involving vCS by expanding it and retaining only the first (linear term): ≈ [-2φp - vBS] (1 + 2 [-v2φCSp(-yv) BS]) which upon multipying becomes vCS(y)= [-2φp- vBS] + 2 [-2φp - vBS] 3Finally, giving the factor 1/2 [-2φp - vBS] the symbol δ,we write our linear approximation to the troublesometerm as: [-2φp + vCS(y) - vBS] ≈ [-2φp - vBS] + δ vCS(y) Making this replacement, we have [vGB - VT(y)] ≈ vGS - (1 + γδ) vCS(y) - VFB + 2φp- γ [-2φp - vBS] Defining VT(vBS) as, VT(vBS) ≡ VFB - 2φp+ γ [-2φp - vBS] and giving the factor (1 + γδ) the symbol α, we can write [vGB - VT(y)] ≈ [vGS - VT(vBS) - α vCS(y)] Putting this back into our expression for iD, we find: iD = εo µe W [ vGS - VT(vBS) - α vCS(y)] dvCS(y)to dy Multiplying both sides by "dy" yields * iDdy = W µeCox [ vGS - VT(vBS) - α vCS] dvCS We can now integrate both sides from y = 0 and vCS = 0 to y = L and vCS = vDS. We have LL ⌠⌠⌡ iDdy = iD ⌡dy = iDL 00 and 2 vDS α vDS⌠⌡ [ vGS - VT - α vCS] dvCS = [(vGS - VT)vDS -2 ]0 Setting these two integals equal, and dividing both sidesby L yields the expression for iD we are looking for: 42W* α vDSiD(vDS, vGS, vBS) = L µe Cox [{vGS - VT(vBS)} vDS -2 ] It is worth reminding ourselves that arriving at this resultwe assumed that vGS > VT; otherwise iD is zero because their is no channel. We also specified vDS ≥ 0 and vBS ≤ 0. If we plot this expression for iD verses vDS for fixed values of vGS and vBS, we find that iD varies linearly with vDS when vDS is small, but increases sub-linearly as vDS increases further, i.e., the curve bends over to the right. Physically, the amount of inversion decreases toward the drain end of the channel and the resistance of the channel increases. Still, iD continues to increase until vDS = (vGS - VT)/α, at which point the equation says iD starts to decrease. What happens physically is that the channel disappears near the drain when vDS = (vGS - VT)/α, i.e., the region under the gate is no longer inverted near the drain. For larger values of vDS the current does not decrease, but stays saturated at the peak value. We find 1 W *iD(vDS, vGS, vBS) = 2α L µe Cox [vGS - VT(vBS)]2 for vDS ≥ (vGS - VT)/α and vGS > VT. This completes the gradual channel approximationmodel for the MOSFET. Summarizing the results, wehave a model valid for vDS ≥ 0 and vBS ≤ 0, and it says thatthe gate and substrate currents are zero for this entirerange, i.e., iG(vDS, vGS, vBS) = 0 and iB(vDS, vGS, vBS) = 0 5The drain current has three regions: Cutoff: iD(vDS, vGS, vBS) = 0 for (vGS - VT)/α ≤ 0 ≤ vDS Saturation: KiD(vDS, vGS, vBS) = 2α [vGS - VT(vBS)]2 for 0 ≤ (vGS - VT)/α ≤ vDS Linear (or triode): 2 iD(vDS, vGS, vBS) = K [{vGS - VT(vBS)} vDS --α v2DS] …
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