lec12 - 6.012 - Electronic Devices and Circuits Lecture 12...

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Unformatted text preview: 6.012 - Electronic Devices and Circuits Lecture 12 - MOSFET Basics - Outline Announcements C On web site - 2 write-ups on MOSFET models (n-MOS example) Inversion Depletion vGC + G SiO2 Review - MOS Capacitor Accumulation vBC < 0 vBC + n+ p-Si v GC B Flat-band voltage: VFB vGB such that f(0) = fp-Si; VFB = fp-Si fm Threshold voltage: VT vGC such that f(0) = fp-Si; VT(vBC) = VFB 2fp-Si + [2eSi qNA(|2fp-Si| vBC)]1/2/Cox* (vGC at threshold) Inversion layer sheet charge density: qN* = Cox*[vGC VT(vBC)] V FB 0 VT The MOSFET - qualitative explanation Definition of structure: cross-section (Example: n-channel MOSFET) Gate action (creating a channel) and channel current (drift) Quantitative modeling - the Gradual-Channel Approximation The gate and substrate currents: iG(vGS, vDS,vBS), iB(vGS, vDS,vBS) The drain current: iD(vGS, vDS,vBS) 1. the in-plane problem (relating voltage drop along channel to channel charge) 2. the normal problem (relating gate voltage to channel charge) 3. the full drain current expression (quadratic approximation) Lecture 12 - Slide 1 Clif Fonstad, 10/03 Characteristics and regions of operation C vGC + G SiO2 An n-channel MOSFET capacitor Reviewing the results of Lecture 11 vBC < 0 vBC + n+ p-Si B Flat - band voltage : VFB vGB at which f (0) = f p-Si VFB = f p-Si - f m Threshold voltage : VT vGC at which f (0) = - f p-Si + v BC 1/ 2 1 VT (v BC ) = VFB - 2f p-Si + * 2eSi qN A 2f p-Si - v BC Cox { [ } Accumulation Depletion Inversion v GC V FB 0 VT * q* = -Cox [vGC - VT (v BC )] N * q* = -Cox [vGB - VFB )] P Inversion layer sheet charge density : Accumulation layer sheet charge density : Clif Fonstad, 10/03 Lecture 12 - Slide 2 An n-channel MOSFET S n+ vGS G + iG vDS D iD n+ p-Si vBS + B iB Clif Fonstad, 10/03 Lecture 12 - Slide 3 An n-channel MOSFET showing gradual channel axes S n+ 0 vGS 0 G + iG vDS D iD x n+ p-Si vBS + B iB L y Extent into plane = W Clif Fonstad, 10/03 Lecture 12 - Slide 4 Gradual Channel Approximation i-v Modeling (n-channel MOS used as the example) Restrict voltage ranges: vBS 0, vDS 0, so iG(vGS, vDS,vBS) 0, iB(vGS, vDS,vBS) 0 The issue is the drain current (and only when vGS > VT, otherwise iD = 0): The in-plane problem: Looking at electron motion in the channel we write the drain current as dv cs (at moderate E - fields) dy This can be integrated from y = 0 to y = L, and vCS = 0 to vCS = vDS, to get iD(vGS, vDS,vBS), but first we need qn*(y) iD = - W sey (y)q* (y) = - W me q* (y) n n The normal problem: The channel charge at y is qn*(y), which is: * * q* (y) = - Cox [vGC (y) - VT (y)] = - Cox [vGS - vCS (y) - VT (y)] n with VT (y) = VFB - 2f p-Si + 2eSiqN A 2f p-Si - v BS + vCS (y) { [ } 1/ 2 * Cox Before proceeding it is desirable to linearize the vCS(y) dependence of VT(y). This is easier to do if we first simplify the expression by introducing g (2eSi qNA)1/2 /Cox*. With this substitution we have: 1/ 2 VT (y) = VFB - 2f p-Si + g 2f p-Si - v BS + vCS (y) [ Clif Fonstad, 10/03 (derivation continues on next foil) Lecture 12 - Slide 5 The normal problem, cont: Next we factor out (|2fp-Si| vBS )1/2 : VT (y) = VFB - 2f p-Si + g 2f p-Si - v BS [ 1/ 2 vCS (y) 1+ 2f p-Si - v BS 1/ 2 Then, using the approximation (1 + x)1/2 1 + x/2 if x << 1, we have 1/ 2 vCS (y) VT (y) VFB - 2f p-Si + g 2f p-Si - v BS 1+ 2 2f p-Si - v BS Multiplying the factor (|2fp-Si| vBS )1/2 back through again we have 1/ 2 g vCS (y) VT (y) VFB - 2f p-Si + g 2f p-Si - v BS + 1/ 2 2 2f p-Si - v BS [ [ [ [ This expression has the linear dependance on vCS we seek, but before proceeding, it is convenient to introduce two new factors. We define the first three terms as VT(vBS), and we introduce the factor d: 1/ 2 -1/ 2 f V (v ) V - 2 + g 2f -v and d 2 2f -v T BS FB p-Si [ p-Si BS [ p-Si BS yielding the final result we will work with: Clif Fonstad, 10/03 VT (y) VT (v BS ) + d g vCS (y) (derivation continues on next foil) Lecture 12 - Slide 6 The normal problem, cont: With the linearization of the vCS(y) dependence of VT complete, and we are now ready to use this expression in the formula for qn*(y). Substituting the linearized approximation to VT into our earlier formula for qn*(y) we obtain * q* (y) - Cox {vGS - VT (v BS ) - vCS (y) [ 1+ d g ]} n Defining (1 + dg) as a, we arrive at the final form of the expression for the channel charge, qn*(y) ,that we will use to calculate the MOSFET current:, * q* (y) - Cox {vGS - VT (v BS ) - a vCS (y)} n The drain current expression: Putting our result for channel charge into the drain current expresthe sion we obtained from considering the in-plane problem, we find: dv dv * iD = -W me q* (vCS ) CS W me Cox {vGS - VT (v BS ) - a vCS (y)} CS n dy dy This expression can be integrated with respect to dy for y = 0 to y = L. On the left-hand the integral with respect to y can be converted to one with respect to vCS, which ranges from 0 at y = 0, to vDS at y = L: v DS L * iD dy = W me Cox {vGS - VT (v BS ) -a vCS (y)} dvCS 0 0 Clif Fonstad, 10/03 (derivation continues on next foil) Lecture 12 - Slide 7 , The drain current expression, cont: Next we perform the definite integrals on each side obtaining: v * iD L = W me Cox vGS - VT (v BS ) - a DS v DS 2 Isolating the drain current, we have, finally: iD (vGS ,v ,v BS ) = DS W v * me Cox vGS - VT (v BS ) - a DS v DS L 2 Plotting this equation for iD increasing values of vGS we see that it traces inverted parabolas as shown to the right. The decrease in iD after the peak (dashed lines) is not physical, however, and in reality iD saturates at the peak value for larger values of vDS (solid lines). Clif Fonstad, 10/03 inc. vGS vDS Lecture 12 - Slide 8 (discussion continues on next foil) The drain current expression, cont: The point at which iD reaches its peak value and saturates is easily found. Taking the derivative and setting it equal to zero we find: iD 1 =0 when v DS = [vGS - VT (v BS )] v DS a What happen physically at this voltage is that the channel at the inversion layer at the drain end of the channel disappears: * q* (L) - Cox {vGS - VT (v BS ) - a v DS } n 1 = 0 when v DS = [vGS - VT (v BS )] a For vDS > [vGS-VT(vBS)]/a, all the additional drain-to-source voltage appears across the high resistance region at the drain end of the channel where the mobile charge density is very small, and iD remains constant independent of vDS: 1 W 1 2 * iD (vGS ,v DS ,v BS ) = me Cox [vGS - VT (v BS )] for v DS > [vGS - VT (v BS )] 2a L a Clif Fonstad, 10/03 (discussion concludes on next foil) Lecture 12 - Slide 9 The gate and substrate currents, iG and iB: The oxide under the gate is an insulator, so the steady-state gate current, iG, is zero, and we restrict the model to the ranges vBS 0 and vDS 0, to insure that the source-substrate and drain-substrate diodes are always reverse biased and thus that the steady-state substate current, iB, is also zero: iG (vGS ,v DS ,v BS ) = 0 and iB (vGS ,v DS ,v BS ) = 0 The full model: The complete Gradual Channel Approximation model for the MOSFET is summarized below (n-channel example): Valid for v BS 0, and v DS 0 : iG (vGS ,v DS ,v BS ) = 0 and iB (vGS ,v DS ,v BS ) = 0 1 0 for [vGS - VT (v BS )] < 0 < v DS a 1 W 1 2 * iD (vGS ,v DS ,v BS ) = me Cox [vGS - VT (v BS )] for 0 < [vGS - VT (v BS )] < v DS 2a L a 1 W m C * v - V (v ) - a v DS v DS for 0 < v DS < [vGS - VT (v BS )] e ox GS T BS L 2 a 1/ 2 1 with VT (v BS ) VFB - 2f p-Si + * 2eSiqN A 2f p-Si - v BS Cox 1/ 2 1 eSiqN A e a 1+ * C* ox ox Cox 2 2f p-Si - v BS t ox Clif Fonstad, 10/03 Lecture 12 - Slide 10 { [ } [ The operating regions of MOSFETs and BJTs: Comparing an n-channel MOSFET and an npn BJT MOSFET iG iD D + Linear iD or Triode G+ vGS vDS Saturation (FAR) iD K [vGS - V T(vBS)]2/2a BJT iB iC C + S Cutoff vDS B+ vBE vCE i Saturation iC B E FAR iB IBSe qV BE /kT vCE > 0.2 V vBE Forward Active Region iC bF iB Cutoff Clif Fonstad, 10/03 0.6 V 0.2 V Cutoff vCE Input curve Output family Lecture 12 - Slide 11 6.012 - Electronic Devices and Circuits Lecture 12 - MOSFET Basics - Summary Qualitative operation - the MOSFET as a switch and transistor S n+ vGS G + iG vDS D iD n+ iD iG D + vDS + iB p-Si vBS + B iB G+ vGS B "Off" when vGS < VT "On" when vGS > VT vBS Quantitative modeling - the Gradual-Channel Approximation Restrict voltage ranges: vBS 0; vDS 0 (n-channel MOS used as the example) No gate and substrate currents: iG(vGS, vDS,vBS) 0, iB(vGS, vDS,vBS) 0 The drain current: iD(vGS, vDS,vBS) ( 0 if vGS < VT, so we model assuming vGS > VT) S 1. the in-plane problem: iD = W e qn*(y) dvCS/dy; this is integrated from y = 0 to y = L, and vCS = 0 to vCS = vDS to get iD(vGS, vDS,vBS) 2. the normal problem: qn*(y) Cox*[vGS VT(vBS) avCS(y)], where VT(vBS) VFB 2fp-Si + g(|2fp-Si| vBS)1/2 , g (2eSi qNA)1/2 /Cox* , and a 1 + g/2(|2fp-Si| vBS)1/2 (many texts set a = 1) 3. the full drain current expressions: 0 for (vGS VT) 0 avDS (cutoff) *(v 2 /2a iD (W/L)eCox GS VT) for 0 (vGS VT) avDS (saturation) *(v (W/L)eCox GS VT avDS/2)vDS for 0 avDS (vGS VT) (linear) { Clif Fonstad, 10/03 with VT = VFB 2fp-Si + [2eSi qNA(|2fp-Si| vBS)]1/2/Cox* and a = 1 + [(eSi qNA/2(|2fp-Si| vBS)]1/2 /Cox (frequently 1) Lecture 12 - Slide 12 ...
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This note was uploaded on 07/20/2009 for the course CSAIL 6.012 taught by Professor Prof.cliftonfonstadjr. during the Fall '03 term at MIT.

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