CHAPTER 3: Unified I-V Model

Transcription

1 CHAPTER 3: Unified - Model The development of separate model expressions for such device operation regimes as subthreshold and strong inversion were discussed in Chapter 2. Although these expressions can each accurately describe device behavior within their own respective region of operation, problems are likely to occur between two well-described regions or within transition regions. n order to circumvent this issue, a unified model should be synthesized to not only preserve region-specific expressions but also to ensure the continuities of current (ds) and conductance (G x ) and their derivatives in all transition regions as well. Such high standards are met in BSM3v3. As a result, convergence and calculation efficiencies are much improved. This chapter will describe the unified natured of BSM3v3 s model equations. While most of the parameter symbols in this chapter are explained in the following text, a complete description of all - model equation parameters can be found in the Appendix A. 3.1 Unified Channel Charge Density Expression Separate expressions for channel charge density are shown below for subthreshold (Eq. (3.1.1a) and (3.1.1b)) and strong inversion (Eq. (3.1.2)). Both expressions are valid for small ds. BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-1

2 Unified Channel Charge Density Expression Q Q chsubs0 0 gs th exp( ) nvt (3.1.1a) where, Q 0 is: Q 0 qεsinch off v t exp( 2 ) φs nvt Qchs0 Cox( gs th) (3.1.1b) (3.1.2) n both Eqs. (3.1.1a) and (3.1.2), the parameters Qchsubs0 and Qchs0 are the channel charge densities at the source for very small ds. To form a unified expression, an effective (gs-th) function named gsteff is introduced to describe the channel charge characteristics from subthreshold to strong inversion: gsteff gs th 2 nvtln 1+ exp( ) nvt 2 2Φs gs th 2off 1+ 2 n COX exp( ) qεsinch 2 nvt (3.1.3) The unified channel charge density at the source end for both subthreshold and inversion region can therefore be written as: Q chs0 Coxgsteff (3.1.4) Figures 3-1 and 3-2 show the "smoothness" of Eq. (3.1.4) from subthreshold to strong inversion regions. The gsteff expression will be used again in subsequent sections of this chapter to model the drain current. 3-2 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

3 Unified Channel Charge Density Expression gsteff () gsteff Function exp[(gs-th)/n*v t ] gsth -1.0 (gs-th) gs-th () gs-th () Figure 3-1. The gsteff function vs. (gs-th) in linear scale. 4 2 Log(gsteff) 0 Log(gsteff) (gs-th) exp[(gs-th)/n*v t ] gsth gs-th () gs-th () Figure 3-2. gsteff function vs. (gs-th) in log scale. BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-3

4 Unified Channel Charge Density Expression Eq. (3.1.4) serves as the cornerstone of the unified channel charge expression at the source for small ds. To account for the influence of ds, the gsteff function must keep track of the change in channel potential from the source to the drain. n other words, Eq. (3.1.4) will have to include a y dependence. To initiate this formulation, consider first the re-formulation of channel charge density for the case of strong inversion: Qchs( y) Cox( gs th AbulkF( y) ) (3.1.5) The parameter F(y) stands for the quasi-fermi potential at any given point, y, along the channel with respect to the source. This equation can also be written as: Q Q 0+ Q chs( y) chs chs( y) (3.1.6) The term Q chs(y) is the incremental channel charge density induced by the drain voltage at point y. t can be expressed as: Q CoxA chs( y) bulk F ( y) (3.1.7) For the subthreshold region (gs<<th), the channel charge density along the channel from source to drain can be written as: 3-4 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

5 Unified Channel Charge Density Expression gs th AbulkF ( y) Qchsubs( y) Q0exp( ) nvt AbulkF ( y) Qchsubs0exp( ) nvt (3.1.8) A Taylor series expansion of the right-hand side of Eq. (3.1.8) yields the following (keeping only the first two terms): Q chsubs( y) AbulkF( y) Qchsubs 0( 1 ) nvt (3.1.9) Analogous to Eq. (3.1.6), Eq. (3.1.9) can also be written as: Q Q 0+ Q chsubs( y) chsubs chsubs( y) (3.1.10) The parameter Q chsubs(y) is the incremental channel charge density induced by the drain voltage in the subthreshold region. t can be written as: Q chsubs( y) AbulkF ( y) Qchsubs0 nvt (3.1.11) Note that Eq. (3.1.9) is valid only when F(y) is very small, which is maintained, fortunately, due to the fact that Eq. (3.1.9) is only used in the linear regime (i.e. ds 2vt). Eqs. (3.1.6) and (3.1.10) both have drain voltage dependencies. However, they are decuple and a unified expression for Qch(y) is desperately needed. To obtain a unified expression along the channel, we first assume: BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-5

6 Unified Mobility Expression Q ch( y) Q Q Q + Q chs( y) chsubs( y) chs( y) chsubs( y) (3.1.12) Here, Q ch(y) is the incremental channel charge density induced by the drain voltage. Substituting Eq. (3.1.7) and (3.1.11) into Eq. (3.1.12), we obtain: Q F( y) b Q ch ( y ) chs0 (3.1.13) where b (gsteff+n*v t )/A bulk. n order to remove any association between the variable n and bias dependencies (gsteff) as well as to ensure more precise modeling of Eq. (3.1.8) for linear regimes (under subthreshold conditions), the variable n is replaced with 2. The expression for b now becomes: b + 2vt Abulk gsteff (3.1.14) A unified expression for Qch(y) from subthreshold to strong inversion regimes is now at hand: Q ch( y) F( y) Qchs0( 1 ) b (3.1.15) The variable Qchs0 is given by Eq. (3.1.4). 3.2 Unified Mobility Expression BSM3v3 uses a unified mobility expression based on the gsteff expression of Eq Thus, we have: 3-6 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

7 Unified Linear Current Expression (Mobmod1) (3.2.1) µ o µ eff gsteff + 2th gsteff th Ua Ucbseff U ( + )( ) + b( ) TOX TOX To account for depletion mode devices, another mobility model option is given by the following: (Mobmod2) (3.2.2) µ o µ eff gsteff gsteff Ua Ucbseff U 2 1+ ( + )( ) + b( ) TOX TOX To consider the body bias dependence of Eq further, we have introduced the following expression: (For Mobmod3) (3.2.3) µ eff µ o gsteff th gsteff th U + 2 a U [ ( ) + b( ) ]( 1+ U c bseff) TOX TOX 3.3 Unified Linear Current Expression ntrinsic case (Rds0) Generally, the following expression [2] is used to account for both drift and diffusion current: BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-7

8 Unified Linear Current Expression WQ µ d( y) ch( y) ne( y) d dy F( y) (3.3.1) where the parameter u ne(y) can be written as: µ µ eff ( ) E 1+ E ne y y sat (3.3.2) Substituting Eq. (3.3.2) in Eq. (3.3.1) we get: d( y) F y µ eff WQchso( 1 ) b E 1+ E y sat d dy ( ) F( y) (3.3.3) Eq. (3.3.3) resembles the equation used to model drain current in the strong inversion regime. However, it can now be used to describe the current characteristics in the subthreshold regime when ds is very small (ds<2v t ). Eq. (3.3.3) can now be integrated from the source to drain to get the expression for linear drain current in the channel. This expression is valid from the subthreshold regime to the strong inversion regime: ds0 ds Wµ effqchs 0ds( 1 ) 2 b ds L( 1+ ) a (3.3.4) 3-8 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

9 Unified dsat Expression Extrinsic Case (Rds > 0) The current expression when Rds > 0 can be obtained based on Eq. (2.5.9) and Eq. (3.3.4). The expression for linear drain current from subthreshold to strong inversion is: ds dso R 1 + ds dso ds (3.3.5) Chapter 8 will illustrate the "smoothness" of this expression. 3.4 Unified dsat Expression ntrinsic case (Rds0) To get an expression for the electric field as a function of y along the channel, we integrate Eq. (3.3.1) from 0 to an arbitrary point y. The expression is as follows: E y dso ( WQchs 0µ eff ) Esat dso 2ds WQchs µ eff y b (3.4.1) f we assume that drift velocity saturates when EyEsat, we get the following expression for dsat: BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-9

10 Unified dsat Expression dsat Wµ Q 0E L 2 LE ( satl+ b) eff chs sat b (3.4.2) Let dsdsat in Eq. (3.3.4) and set this equal to Eq. (3.4.2), we get the following expression for dsat: dsat EsatL( gsteff + 2vt) AbulkEsatL+ gsteff + 2vt (3.4.3) Extrinsic Case (Rds>0) The dsat expression for the extrinsic case is formulated from Eq. (3.4.3) and Eq. (2.5.10) to be the following: b b ac dsat 2 4 2a (3.4.4a) where, a A W ν C R + ( 1) A λ 2 1 bulk eff sat ox DS bulk (3.4.4b) (3.4.4c) 2 b ( + 2v)( 1 ) + A E L + 3 A ( + 2 v) W ν C R λ gsteff t bulk sat eff bulk gsteff t eff sat ox DS 2 c ( + 2v) E L + 2( + 2v) W ν C R gsteff t sat eff gsteff t eff sat ox DS (3.4.4d) 3-10 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

11 Unified Saturation Current Expression λ Agsteff + A 1 2 (3.4.4e) The parameter λ is introduced to account for non-saturation effects. The parameters A1 and A2 are extracted. 3.5 Unified Saturation Current Expression A unified expression for the saturation current from the subthreshold to the strong inversion regime can be formulated by introducing the gsteff function into Eq. (2.6.15). The resulting equations are the following: ds R 1+ dso( dsat) ds dso( dsat) dsat ds dsat ds A ASCBE dsat (3.5.1) where, A Pvaggsteff Asat + ( 1+ )( + ) EsatLeff ACLM ADBLC (3.5.2) Asat (3.5.3) Abulkdsat EsatLeff + dsat + 2RDSνsatCoxWeffgsteff[ 1 2 ( gsteff + 2 vt) ] 2/ λ 1+ RDSνsatCoxWeffAbulk ACLM A E L + PCLMAbulkEsat litl bulk sat eff gsteff ( ds dsat) (3.5.4) BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-11

12 Single Current Expression for All Operational Regimes of gs and ds ADBLC θ ( gsteff + 2vt) t 1+ P rou ( DBLCB bseff ) 1 Abulkdsat Abulkdsat + gsteff + 2vt (3.5.5) L L exp( ) exp( ) P 2l l + eff eff θrout PDBLC 1 DROUT DROUT + 2 t 0 t 0 DBLC2 (3.5.6) P litl exp ds dsat 1 scbe2 scbe1 ASCBE P L eff (3.5.7) 3.6 Single Current Expression for All Operational Regimes of gs and ds The gsteff function introduced in Chapter 2 gave a unified expression for the linear drain current from subthreshold to strong inversion as well as for the saturation drain current from subthreshold to strong inversion, separately. n order to link the continuous linear current with that of the continuous saturation current, a smooth function for ds is introduced. n the past, several smooth functions have been proposed for MOSFET modeling [22-24]. For BSM3v3, the smooth function used is similar to that proposed by R. M. D. A. elghe et al [24]. The overall current equation for both linear and saturation current now becomes: ds R 1+ dso( dseff ) ds dso( dseff ) dseff ds dseff ds A ASCBE dseff (3.6.1) Most of the previous equations which contain ds and dsat dependencies are now substituted with the dseff function. For example, Eq. (3.5.4) now becomes: 3-12 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

13 Single Current Expression for All Operational Regimes of gs and ds ACLM A E L + PCLMAbulkEsat litl bulk sat eff gsteff ( ds dseff ) (3.6.2) Similarly, Eq. (3.5.7) now becomes: P litl exp ds dseff 1 scbe2 scbe1 ASCBE P L eff (3.6.3) The dseff expression is written as: 2 ( δ ( δ) 4δ ) dseff dsat dsat ds dsat ds dsat (3.6.4) The expression for dsat is that given under Section 3.4. The parameter δ is an extracted constant. The dependence of dseff on ds is given in Figure 3-3. The dseff function follows ds in the linear region and tends to dsat in the saturation region. Figure 3-4 shows the effect of δ on the transition region between linear and saturation regimes. BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-13

14 Single Current Expression for All Operational Regimes of gs and ds 1.4 dseffdsat 1.2 dseffds 1.0 dseff () gs1 gs3 gs ds () Figure 3-3. dseff vs. ds for δ0.01 and different gs. dseffdsat 2.0 dseffds dseff () δ0.05 δ0.01 δ ds () Figure 3-4. dseff vs. ds for gs3 and different δ values BSM3v3 Manual Copyright 1995, 1996, UC Berkeley

15 Substrate Current 3.7 Substrate Current BSMv3 uses the Eq. (3.7.1) to model substrate current. sub αo βo dso ( ds dseff )exp( ) Leff ds dseff R 1+ ds dso 1+ The parameters α0 and β0 refer to impact ionization current. dseff ds A dseff (3.7.1) 3.8 A Note on bs n BSM3v3, all bs terms which have appeared in Chapters 2 and 3 have been substituted with a bseff expression as shown in Eq. (3.8.1). This is done in order to set an upper bound for the body bias value during simulations. Unreasonable values can occur if this expression is not introduced. (3.8.1) bseff bc + 05.[ bs bc δ1+ ( bs bc δ1) 2 4δ1bc ] where δ The param eter bc is the maximum allowable bs value and is obtained based on the condition of dth/dbs0 for the th expression of BSM3v3 Manual Copyright 1995, 1996, UC Berkeley 3-15

16 A Note on bs 3-16 BSM3v3 Manual Copyright 1995, 1996, UC Berkeley