6.012 - Microelectronic Devices and Circuits Lecture 11 - MOSFETs II; Large Signal Models - Outline • Announcements On Stellar - 2 write-ups on MOSFET models • The Gradual Channel Approximation (review and more)MOSFET model: 0K (vGS K (vGS with K ≡ (W/L)µegradual channel approximation (Example: n-MOS) for (vGS – VT)/α ≤ 0 ≤ vDS (cutoff) iD ≈ – VT)2 /2α for 0 ≤ (vGS – VT)/α ≤ vDS (saturation) – VT – αvDS/2)vDS for 0 ≤ vDS ≤ (vGS – VT)/α (linear) Cox *, VT= VFB – 2φp-Si + [2εSi qNA(|2φp-Si| – vBS)]1/2/Cox * and α = 1 + [(εSi qNA/2(|2φp-Si| – vBS)]1/2 /Cox (frequently α ≈ 1) • Refined device models for transistors (MOS and BJT)Other flavors of MOSFETS: p-channel, depletion mode The Early Effect: 1. Base-width modulation in BJTs: wB(vCE)2. Channel-length modulation in MOSFETs: L(vDS) Charge stores:1. Junction diodes 2. BJTs 3. MOSFETs Extrinsic parasitics: Lead resistances, capacitances, and inductances Clif Fonstad, 3/18/08 Lecture 11 - Slide 1An n-channel MOSFET showing gradual channel axes p-SiBG+vGSn+Dn+S–vDSvBS+iGiBiDL0yx0Extent into plane = W Gradual Channel Approximation: - A one-dimensional electrostatics problem in the x direction is solved to find the channel charge, qN*(y); this charge depends on vGS, vCS(y) and vBS. - A one-dimensional drift problem in the y direction then gives the channel current, iD, as a function of vGS, vDS, and vBS. Clif Fonstad, 3/18/08 Lecture 11 - Slide 2Gradual Channel Approximation i-v Modeling(n-channel MOS used as the example) The Gradual Channel Approximation is the approach typically used to model the drain current in field effect transistors.* It assumes that the drain current, iD, consists entirely of carriers flowing in the channel of the device, and is thus proportional to the sheet density of carriers at any point and their net average velocity. It is not a function of y, but its components in general are: ! iD= " W # "q# nch*(y)# sey(y)In this expression, W is the width of the device, -q is the charge on each electron, n*ch(y) is sheet electron concentra-tion in the channel (i.e. electrons/cm2) at y, and sey(y) is the net electron velocity in the y-direction. If the electric field is not too large, sey(y) = -µeEy(y), and ! iD= " W # q# nch*(y)#µeEy(y)= W # q# nch*(y)#µedvCS(y)dyCont. * Junction FETs (JFETs), MEtal Semiconductor FETs (MESFETs1), and Heterojunction FETs Clif Fonstad, 3/18/08 (HJFETs2), as well as Metal Oxide Semiconductor FETs (MOSFETs). Lecture 11 - Slide 3 1. Also called Shottky Barrrier FETs (SBFETs). 2. Includes HEMTs, TEGFETs, MODFETs, SDFETs, HFETs, PHEMTs, MHEMTs, etc.! iD= W " q" nch*(y)"µedvCS(y)dyGCA i-v Modeling, cont. p-SiBG+vGSn+Dn+S–vDSvBS+iGiBiDL0yx0We have: To eliminate the derivative from this equation we integrate both sides with respect to y from the source (y = 0) to the drain (y = L). This corresponds to integrating the right hand side with respect to vCS from 0 to vDS, because vCS(0) = 0 to vCS(L) = vDS: ! iD0L"dy = W #µe# q# nch*(y)dvCS(y)dy0L"dy = W #µe# q# nch*(vCS)0vDS"dvCSThe left hand integral is easy to evaluate; it is simply iDL. Thus we have: ! iD0L"dy = iDL # iD=WL$µe$ q$ nch*(vCS)0vDS"dvCSClif Fonstad, 3/18/08 Cont. Lecture 11 - Slide 4GCA i-v Modeling, cont. The various FETs differ primarily in the nature of their channelsand thereby, the expressions for n*ch(y). For a MOSFET we speak in terms of the inversion layer charge, qn*(y), which is equivalent to - q·n*ch(y). Thus we have: ! iD= "WLµeqn*(vGS,vCS,vBS) dvCS0vDS#We derived qn* earlier by solving the vertical electrostatics problem, and found: ! qn*(vGS,vCS,vBS) = " Cox*vGS" vCS" VT(vCS,vBS)[ ] with VT(vCS,vBS) = VFB" 2#p"Si+ 2$SiqNA2#p"Si" vBS+ vCS[ ]{ }1/ 2Cox*Using this in the equation for iD, we obtain: ! iD(vGS,vDS,vBS) =WLµeCox*vGS" vCS" VT(vCS,vBS)[ ]{ }dvCS0vDS#At this point we can do the integral, but it is common to simplifythe expression of VT(vCS,vBS) first, to get a more useful result. Clif Fonstad, 3/18/08 Cont. Lecture 11 - Slide 5GCA - dealing with the non-linear dependence of VT on vCS Approach #1 - Live with it Even though VT(vCS,vBS) is a non-linear function of vCS, we can still put it in this last equation for iD: ! iD=WLµeCox*vGS" vCS"VFB+ 2#p"Si"tox$ox2$SiqNA2#p"Si" vBS+ vCS[ ]% & ' ( ) * + , - . / 0 dvCS0vDS1and do the integral, obtaining: ! iD(vDS,vGS,vBS) =WLµeCox*vGS" 2#p"VFB"vDS2$ % & ' ( ) vDS* + , +322-SiqNA2#p+ vDS" vBS( )3/2" 2#p" vBS( )3/2. / 0 1 2 3 4 5 6 The problem is that this result is very unwieldy, and difficult to work with. More to the point, we don't have to live with it because it is easy to get very good, approximate solutions that are much simpler to work with. Clif Fonstad, 3/18/08 Cont. Lecture 11 - Slide 6GCA - dealing with the non-linear dependence of VT on vCS Approach #2 - Ignore it Early on researchers noticed that the difference between VT at 0 and at y, i.e. VT(0,vBS) and VT(vDS,vBS), is small, and that using VT(0,vBS) alone gives a result that is still quite accurate and is very easy to use: ! iD(vGS,vDS,vBS) =WLµeCox*vGS" vCS"VT(0,vBS)[ ]{ }dvCS0vDS# =WLµeCox*vGS"VT(0,vBS)[ ]vDS"vDS22$ % & ' ( ) The variable, vCS, is set to 0 in VT. This result looks much simpler than the result of Approach #1, and it is much easier to use in hand calculations. It is, in fact, the one most commonly used by the vast majority of engineers. At the same time, the fact that it was obtained by ignoring the dependence of VT on vCS is cause for concern, unless we have a way to judge the validity of our approxima-tion. We can get the necessary metric through Approach #3. Clif Fonstad, 3/18/08 Cont. Lecture 11 - Slide 7GCA - dealing with the non-linear dependence of VT on vCS Approach #3 - Linearize it (i.e. expand it, keep first order term) In this approach we leave the variation of VT with vCS in, but linearize it by doing a Taylor's series expansion about vCS = 0: ! VT[vCS,vBS] " VT(0,vBS) +#VT#vCSvCS= 0$ vCSTaking the derivative and evaluating it at vCS = 0 yields: Clif Fonstad, 3/18/08 Lecture 11 - Slide 8 With this qn * is where Cont. ! VT[vCS,vBS] " VT(0,vBS) +tox#ox#SiqNA2 2$p% vBS( )& vCS! qn*(vGS,vCS,vBS) " # Cox*vGS# vCS+ VT(0,vBS) #tox$ox$SiqNA2 2%p# vBS( )& vCS' ( ) ) * + , , = #Cox*vGS#-vCS+ VT(vBS)[ ]! "# 1+tox$ox$SiqNA2 2%p& vBS( )' ( ) * + , and VT(vBS) # VT(0,vBS)GCA - dealing with the non-linear dependence of VT on vCS Using this result in the integral in
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