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MIT 6 012 - MOSFET Drain Current Modeling

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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, iD, as the product of qN* (y) , the inversion layer sheet charge density at position y along the channel; sy(y), the net drift velocity of the inversion layercarriers there (electrons in the n-channel device we are modeling), and W, the channel width: iD = - qN*(y) sy(y) W with εox dvCS(y)qN*(y) = - [vGB - VT(y)] and sy(y) = - µe Ey = µe dy tox Substituting these expressions yields: dvCS(y)iD = W µ Cox * [vGB - VT(y)] dy where we have identified the gate capacitance per unit area, Cox * , as εox/tox and where the threshold voltage is given by VT(y) = VFB + |2φp| +vCB(y) + 1 2εSi qNA [|2φp| + vCB(y)Cox * Defining the body factor, γ, as 2εSi qNA /Cox * , and writing vCB(y) as vCS(y) -vBS, we can rewrite this as VT(y) = VFB + |2φp| +vCS(y) - vBS + γ |2φp| + vCS(y) - vBS and thus we can write iD as dvCS(y)iD= W µeC* [vGS -VFB - |2φp|- vCS(y) - γ |2φp| + vCS(y) - vBS ] dy ox To proceed we integrate both sides for y = 0 to y = L, recognizing that the right-hand integral is equivalent to integrating with respect to vCS(y) from 0 to vDS: 1L vDS iD⌡⌠ dy = W µeC* ⌠⌡[vGS - VFB - |2φp|- vCS - γ |2φp| + vCS - vBS ] dvCSox 00 The left-hand integral is iDL, so we can write iD as vDS iD = K ⌡⌠ [vGS -VFB - |2φp|- vCS - γ |2φp| + vCS - vBS ] dvCS 0 where K is defined as (W/L) µe Cox * . It is not hard to do the integral on the right-hand side of this equation, andyou may want to do it as an exercise (it is done in the text and the result is givenin Equation 10.9). The resulting expression is awkward and, most importantly,the threshold voltage, VT, is hard to identify in the expression and the role it plays in the current-voltage relationship is hard to understand; the result is not very intuitive. It will not be obvious to you until you get much more experiencewith MOSFETs, but it is very desirable from a modeling standpoint to dosomething to simplify the result and to get an expression that gives us more useful insight. Many texts simply ignore the vCS factor under the radical and write vDS iD ≈ K ⌡⌠ [vGS -VFB - |2φp|- vCS - γ |2φp|- vBS ] dvCS 0 which we can simplify as vDS iD = K ⌠⌡[vGS - VT' - vCS ] dvCS 0 with VT' defined to be VFB + |2φp| + γ |2φp| - vBS . Doing the integral we get vDS2 iD = K [(vGS - VT')vDS -2 ] A more satisfying approach is to not ignor the vCS factor, but rather to try to linearize the dependance on it. The troublesome term is |2φp| + vCS(y) - vBS which can be written as 2vCS |2φp| + vCS - vBS = 1 + |2φp| - vBS |2φp| - vBS ≈ |2φp| - vBS [1 + vCS 2(|2φp| - vBS) ] vCS |2φp| - vBS + 2 |2φp| - vBS ≈ With this approximation, we next define 1/2 |2φp| - vBS to be δ and (1 + γδ) to be α, and write iD as vDS iD ≈ K ⌠⌡[vGS -VFB - |2φp|- α vCS - γ |2φp|- vBS ] dvCS 0 Using our earlier definition for VT', this becomes vDS iD ≈ K ⌠⌡[vGS - VT' - α vCS ] dvCS 0 and doing the integral yields vDS2 iD = K [(vGS - VT')vDS - α 2] In saturation, which now occurs for vDS > (vGS - VT')/α, we have iD =2K α (vGS - VT') 2 These results are the same as those we obtained after ignoring vCS under the radical, except that we now have a factor of α appearing. To the extent that α is very near one, our earlier approximation is correct, inspite of it being rather adhoc. Collecting all the factors in α, we find it is α = 1 + 2 Cox * 2εSi qNA |2φp| - vBS 3Typically this is near 1, and it can be approximated as such. On the other hand, it is easy to leave α in the expression for iD since is such a minor complication. To summarize, our expressions for the drain current, when we retain α are iD = 0 for (vGS - VT')/α < 0 < vDS (Cutoff) iD =2K α (vGS - VT') 2 for 0 < (vGS - VT')/α < vDS (Saturation) vDS2 iD = K [(vGS - VT')vDS - α 2 ] for 0 < vDS < (vGS - VT')/α (Linear region) with K, VT’, γ, and α defined as K ≡ (W/L) µe Cox * VT' ≡ VFB + |2φp| + γ |2φp| - vBS 2εSi qNAγ ≡ Cox * α ≡ 1 + 2 Cox * 2εSi qNA |2φp| - vBS 4MIT OpenCourseWarehttp://ocw.mit.edu 6.012 Microelectronic Devices and Circuits Fall 2009 For information about citing these materials or our Terms of Use, visit:


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MIT 6 012 - MOSFET Drain Current Modeling

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