The Gradual Channel Approximation for the MOSFET

6.012 - Microelectronic Devices and Circuits The Gradual Channel Approximation for the MOSFET: We are modeling the terminal characteristics of a MOSFET 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 always reverse biased; in this case iB ≈ 0. Because of the insulating nature 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 "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 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 1 the 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, -qN* (y) * [vGB - VT(y)] = - Cox * with Cox ≡ εo/to, and sy(y) is the net velocity of the charge carriers in the y-direction at y, which for modest electric fields is linearly proportional to the field: sy(y) = - µe Ey(y) = - µe The current is then iD = W * µe Cox - dvCS(y) dy dvCS(y) [vGB - VT(y)] dy To proceed, we must examine the factor [vGB - VT(y)]. We are referencing our voltages to the source so we first write 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 was before with the isolated MOS structure because now the channel (inversion layer) can have a voltage relative to the substrate. 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 2 1 It is common practice to name the factor C * 2 εSi qNA ox 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 to complicate the subsequent mathematics annoyingly and it has been found that it is better (and possible) to linearize this term before proceeding. We write the term as [-2φp + vCS(y) - vBS] = [-2φp - vBS] vCS(y) 1 + [-2φ - v ] p BS and approximate the factor involving vCS by expanding it and retaining only the first (linear term): ≈ vCS(y) [-2φp - vBS] (1 + 2 [-2φ - v ]) p BS which upon multipying becomes = vCS(y) [-2φp - vBS] + 2 [-2φp - vBS] 3 Finally, giving the factor 1/2 [-2φp - vBS] the symbol δ, we write our linear approximation to the troublesome term 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 = t o dvCS(y) µe W [ vGS - VT(vBS) - α vCS(y)] dy Multiplying both sides by "dy" yields * [ vGS - VT(vBS) - α vCS] dvCS iDdy = W µe Cox We can now integrate both sides from y = 0 and vCS = 0 to y = L and vCS = vDS. We have L ⌠ ⌡ iD L dy = iD ⌠⌡ dy = iDL 0 0 and 2 α vDS = [(vGS - VT)vDS - 2 ] vDS ⌠ ⌡ [ vGS - VT - α vCS] dvCS 0 Setting these two integals equal, and dividing both sides by L yields the expression for iD we are looking for: 4 2 α vDS W * [{vGS - VT(vBS)} vDS iD(vDS, vGS, vBS) = L µe Cox ] 2 It is worth reminding ourselves that arriving at this result we 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 * [vGS - VT(vBS)]2 iD(vDS, vGS, vBS) = 2α L µe Cox for vDS ≥ (vGS - VT)/α and vGS > VT. This completes the gradual channel approximation model for the MOSFET. Summarizing the results, we have a model valid for vDS ≥ 0 and vBS ≤ 0, and it says that the gate and substrate currents are zero for this entire range, i.e., iG(vDS, vGS, vBS) = 0 and iB(vDS, vGS, vBS) = 0 5 The drain current has three regions: Cutoff: iD(vDS, vGS, vBS) = 0 for (vGS - VT)/α ≤ 0 ≤ vDS Saturation: K iD(vDS, vGS, vBS) = 2α [vGS - VT(vBS)]2 for 0 ≤ (vGS - VT)/α ≤ vDS Linear (or triode): 2 α vDS iD(vDS, vGS, vBS) = K [{vGS - VT(vBS)} vDS -- 2 ] for 0 ≤ vDS ≤ (vGS - VT)/α where K, α, and VT(vBS) are defined as to 2 εSi qNA W * ,α ≡ 1+ K ≡ L µe Cox , 2 εo [-2φp - vBS] and to VT(vBS) ≡ VFB - 2φp + εo 2 εSi qNA[-2φp - vBS] One last point: It is often convenient to write VT(vBS) in terms of VT(0), and a function of vBS. We have and VT(vBS) ≡ VFB - 2φp + γ (-2φp - vBS) VT(0) ≡ VFB - 2φp + γ -2φp so the expression we want is VT(vBS) ≡ VT(0) + γ [ (-2φp - vBS) -2φp ] This will be useful when we look at linear small signal models for the MOSFET. 6