336 7 Dynamic Model -+-=o aQs aQB (7.34 b) aVgd avgd (7.34c) It can easily be verified that the above equations can be rewritten as (7.35) C,B + CsB = 0. Together with the reciprocity law (Cij = Cji), this leads us to c,S = - CBs = CsB = c,, = CB, = CS, = - CDS. (7.36) This is only possible if all derivatives in Eq. (7.34) are zero, which indeed is in contradiction with experiments and physical intuition. For example, bulk charge is a function of the source voltage and therefore C,, can not be zero. Furthermore, Eq. (7.36) implies that the channel charge must be separated in a part Qs( V,,) and QD( Vgd). Since the channel charge depends non-linearly upon both voltages, this separation is not possible. Thus, charge nonconservation and reciprocity are mutually exclusive properties of a MOSFET charge model. Now if the expression for the charges as a function of terminal voltages are available, the integration of Eq. (7.27) can be carried out in the following way, which avoids all problems of charge nonconservation. Note that in general6 dQj(t) ij(t) = at (7.37) By integrating from t, to t, we get Jt:ijdi = Qj(t2) - Qj(tl) = f(v(t2)) - f(v(t~)). (7.38) Since f(v(t,)) will be evaluated at the new time point t2, one can approxi- mate it by performing a Taylor series expansion about the voltage at the last iteration to obtain the companion model used in the Newton-Raphson iteration. The integration on the left hand side of Eq. (7.38) can easily be carried out using either trapezoidal or the Gear integration formula. Note The subscript j stands for G, S, D or B for the charge Q and capacitance C, as we are now dealing with the total charge or total capacitance. However, for current and voltage, the subscript j represents g, s, d and b. 7.2 Charge-Based Capacitance Model 337 that changing variables of integration from C( V) to Q( V) reduces numerical errors (not eliminate them), although mathematically they appear to be the same. 7.2 Charge-Based Capacitance Model Since the terminal charge Qj (j = G, S, D, B) in general is a function of terminal voltages Vg, V,, V, and Vb, we can write the terminal current ij as (7.39) From this equation it is evident that each terminal has a capacitance with respect to the remaining three terminals. Thus, a four terminal device will have 16 capacitances, including 4 self capacitances corresponding to its 4 terminals. Excluding the self capacitances, there will be 12 intrinsic capaci- tances which in general are nonreciprocal. The 16 capacitances form the so called indejinite admittance matrix (IAM). Each element Cij of this capacitance matrix describes the dependence of the charge at the terminal i with respect to the voltage applied at the terminal j with all other voltages held constant. For example, CGs specifices the rate of change of QG with respect to the source voltage V, with voltages at the other terminals (V,, V, and V,) held constant. Thus, in general, (7.40) where the signs of the Ciis are chosen to keep all of the capacitance terms positive for well-behaved devices, i.e., devices for which the charge at a node increases with an increase in the voltage at that node and decreases with an increase in the voltage at any other node. All 16 capacitances of the matrix C,,, shown below, are not independent (7.41) Each row must sum to zero for the matrix to be reference-independent, and each column must sum to zero for the device description to be charge- conservative, which is equivalent to obeying KCL. One of these four 338 7 Dynamic Model capacitances, corresponding to each terminal of the device, is the self capacitance which is the sum of the remaining three capacitances. Thus, for example, the gate capacitance C,, is cGG = cGS cGD f cGB, (7.42) The twelve internodal or intrinsic capacitances (excluding self capacitances C,,, CDD, C,, and Cnn) of a MOSFET are also called the transcapacitances. Further, these capacitances are non-reciprocal. Thus, for example, CD, and CGD differ both in value and physical interpretation. Note that of the 12 transcapacitances only 9 are independent. Therefore, if we choose to evaluate C,,, CGS, C,,, C,,, C,,, c,D, CDG, CDs, CD, then the other three capacitances C,,, C,,, C,, can be determined from the following relations cSC = cGB + cGD + cGS - cBG - cDG cSD = ‘BG + ‘BD + ‘BS - ‘GB - cDB cSB = cDG + cDB + cDS - cGD - cBD. For the sake of comparison, the corresponding Cij matrix for the Meyer model is shown below. (7.43) C,D + CGs + C,n - C,D - C,s CGD 0 [ :::: 0 cGS - CGB 0 0 cGL3 Thus, we see that a MOSFET has capacitances that are much more complex than the Meyer model assumes. It is thus evident from Eq. (7.40) that to calculate MOSFET intrinsic capacitances we need to calculate the charges Qc,QD,Qs and Q, as a function of node voltages, and if we take these charges as independent variables then charge conservation will be guaranteed. It should be pointed out that though the Meyer model represents an inaccurate approximation of MOSFET capacitances, it is reported to predict the high frequency capacitances more accurately than the charge based reciprocal capacitance model to be discussed in sections 7.3 and 7.4. This is because a network with non-reciprocal capacitances based on quasi-static operation can generate infinite power at infinite frequency [20]. For this reason models based on quasi-static approximation fail at very high frequencies (see section 7.5). Channel Charge Partition. The gate and bulk charges, Q, and QB res- pectively, can easily be obtained by integrating the corresponding charge per unit area over the area of the active gate region as is given by Eqs. (7.6) and (7.7). However, calculation of the source and drain charges Q, and QD, respectively, can only be determined from the channel charge Qr, 7.2 Charge-Based Capacitance Model 339 because both source and drain terminals are in intimate contact with the channel region. It is thus necessary to partition the channel charge into a charge QD associated with the drain terminal and a charge Qs associated with the source terminal, such that (7.44) Although this partition of Q, into Qs + QD is not accurate physically [l], nonetheless it does leads to MOSFET capacitance model which agrees with the experimental results. Various approaches have been used in the literature to partition Q, into Qs and QD [3]-[12], some of these are discussed by Yang [ll]. These different approaches vary from an equal division of Q, across both terminals (Qs = QD = 0.5QI) [6] to a QI multiplied by a ‘linear partioning’ or ‘weighted function’ [3]. The approach which can rigorously be shown to be correct and which agrees with the experimental results is that proposed by Ward [3] and is based on the l-D continuity equation. Neglecting recombination in the channel region, the l-D continuity equation is given by Qr = Qs + QD. (7.45) Integrating the above equation along the channel from the source (y = 0) to an arbitrary point y along the channel yields: or (7.46) Integrating again Eq. (7.46) along the whole length of the channel results in: The right hand side of the above equation can be rewritten by taking the time derivative outside the integral and integrating by parts. We finally obtain (7.48) We now have an expression for the current at the position y = 0 in the channel for any time t, that is, the total current flowing through the source 340 7 Dynamic Model contact. The first term on the right hand side is the average transport current in the channel at time t; this is the DC current under quasi-static operation. If we compare Eq. (7.48) with (7.4a), it is easy to see that the charge Qs associated with the source is Qs= -W[oL(l-t)eidy. (7.49a) A similar expression can be derived for the drain current, where the charge QD associated with the drain is given by (7.49 b) Note that Qs and QD sum up to the total inversion charge QI in the channel. It is this charge partioning scheme represented by Eq. (7.49) which is commonly used. This approach has been criticized on the ground that it predicts non-zero drain charge in the saturation region [7]. It is argued that since the drain is insulated from rest of the device, it should have zero charge in saturation. However, this is inconsistent because in saturation it is still possible for a charging current to flow through the channel via the drain. We will now derive the charge expressions first for the long channel devices, and then modify those charge expressions for short-channel devices. While deriving the charge expressions, both assumptions of the Meyer model are removed. The information required for calculating the charge expressions is normally available from any model used to calculate the steady-state (DC) current in a MOSFET. Thus, we can use Qi and Qb from the charge- sheet model 122,233. However, we will compute the terminal charges using the piece-wise DC current model because that is the model commonly used in SPICE. This is discussed in the next section. 7.3 Long-Channel Charge Model In this section we will compute the terminal charges using the piece-wise DC current model discussed in section 6.4.4. The charge model, similar to the DC model, will thus have different charge equations for different regions of device operation. Strong Inversion. The channel charge density Qi for a long-channel device was derived as [cf. Eq. (6.79)] (7.50) 7.3 Longchannel Charge Model 341 while the bulk charge density is given by [cf. Eq. (6.78)] QdY) = - coxY[16V(Y) 4- J-1. (7.51) Since the total charge in the system must be zero, i.e., Q, + Qi + Qb = 0, the gate charge density Q, becomes (7.52) where V,, is given by Eq. (6.45), and a = (1 + y6) [cf. Eq. (6.80)]. Equations (7.50)-(7.52) can be used to calculate the terminal charges using Eqs. (7.6)-(7.7) and (7.49). Let us first calculate Qs and QD using Eq. (7.49). Since Qi(y) is known as a function of V, we first change the variable of integration 'dy' in Eq. (7.49) to 'dV using Eq. (7.13). This yields Q&) = cox[Vp - Vfb - 24~- - v(Y)l (7.53a) (7.5 3 b) To express y in the above equations in terms of Vds, we integrate Eq. (7.13) from y = 0 to an arbitrary point in the channel. This yields At the drain end y = L, and V = Vd,, so that we have Now combining Eq. (7.53) with Eqs. (7.50) and (7.54) and carrying out the integration, we get after lengthy algebra the following expression for QD and Qs in the linear region of device operation QD = - cox~[~vgt - iaT/ds + dg] (7.55a) Q S- C ox't ['V 2 gt -1 GaVds+ 8(1-g)1 (7.5 5b) where (7.56a) Cr2V;s d= 12(Vgt - 0.5aVdS) (7.56b) 342 7 Dynamic Model and Vqr = V,, - Kh and Cox, = WLC,,. When V,, = 0, we find that Qs = Q, = 0.5Cox,V,, as is expected from symmetry. The total gate charge Qc can be obtained by integrating the gate charge density Q, over the area of the active gate region as (7.57) where we have replaced the differential channel length 'dy' with the corre- sponding differential potential drop 'dV using Eq. (7.13). Substituting Qi and Q, from Eqs. (7.50) and (7.52), respectively, and carrying out the integration results in the following expression for the charge Qc Qc=Cox,[ Vqs- vfb-24f-0.5Vds+-d . (7.58) a 'I Similarly, the total bulk charge QB can be written as (7.59) Substituting Qi and Qb from Eqs. (7.50) and (6.78), respectively, and carrying out the integration yields where 3 vgt - 2c( vd, 9= 6(Vgt - 0.5aVd,)' (7.60) (7.60aj Note that the bulk charge consists of two terms. The first term gives the total bulk charge due to the back bias V,, and is related to the threshold voltage. The second term describes additional charge induced by the drain bias. As expected, it reduces to zero when Vd, = 0. In terms of Vrh, one can write QB as QB = - Coxt[vth - Vfb - 26f + (a - 1)Vds91. It is easy to verify that the sum of Qc, Qs, QD and QB is zero. Equations (7.59, (7.58) and (7.60) are charges for the linear region of the device operation. The corresponding charges in the saturation region are obtained by replacing vd, in these equations with V,,/c() [cf. Eq. (6.82)], resulting in the following expressions for Qs, Q,, Qc and QB in the saturation region (7.6 1 a) QD = & Cox, vg 7.3 Long-Channel Charge Model 343 (7.6 1 c) (7.6 1 d) Adding Eqs. (7.61a) and (7.61b) we find inversion charge in saturation region as (7.62) which is the same result as obtained in the Meyer model [cf. Eq. (7.18)] assuming QB = 0. Note from Eqs. (7.61) that none of the charges in saturation depends upon Vd,. This is because in saturation, due to the pinch-off, the drain has no influence on the behavior of the device. Also note that the mobility degradation factor 8 due to the gate field does not appear in the charge expressions. This is because of the global way of modeling the mobility, which cancels out while deriving the charges. In fact 2-D device simulators confirm the analytical results that mobility degradation has little effect on the charges [lE]. The model proposed by Yang et al. [7] and Sheu et al. [12] uses the same charge expressions as discussed above; except that in their model a, is replaced by u, which is not a simple body factor term, but is rather effective gate voltage dependent [cf. Eq. (6.171)]. Figure 7.4 shows Qs and Qo, as a function of V,, for different Vgs( > Vih), for a MOSFET with parameters shown in Table 7.1. It is clear that drain and source charges generally behave the same, except that the drain charge saturates to a smaller absolute value than the source charge. This is because the potential difference between the gate and channel decreases when going from source to drain. The bulk charge as a function of Vd, for different V,,(> Vth) are shown in Figure 7.5a while the gate charge as a function of V,, is shown in Figure 7.5b. QI = Qs + QD = - $Cox, vgt Weak Inversion Region. Although mobile charge at the interface is small when the device is in weak inversion, still these charges are important for the simulation of switching behavior of a MOSFET. Further, in this region bulk charge behaves differently as compared to the strong inversion condition because it is now not screened from the channel. In order to arrive at the expression for the terminal charges in the weak inversion, we will assume that current transport occurs by diffusion only as was the case while deriving the subthreshold drain current expression [21]. Indeed this is a good approximation for low gate voltages. For higher gate voltages (> Vth), the diffusion current saturates and drift transport becomes more and more important, as discussed in Chapter 6. From 344 7 Dynamic Model -2.01 I I I I 0 1 2 3 4 "h 0 Fig. 7.4 The normalized source and drain charges Qs and QD, respectively, as a function of V,, for different V,, in strong inversion. The normalization factor is total gate oxide capacitance Cox, = Cox WL Table 7.1. nMOST parameter ualues used .for Figures 7.6-7.9 Parameter Parameter symbol Parameter description value Units L Effective channel length 50 Pm W Effective channel width 50 Pm to* Gate oxide thickness 150 A A Channel mobility 600 cm2/V.s Flat band voltage - 0.8 V v, h Threshold voltage 0.6 V N, Substrate concentration 3 x 1OI6 cm-3 Eq. (6.92) the drain current (due to diffusion) at any point y along the surface is given by (7.63) which on integration yields y=- Vt(Qi - Qis) (7.64) where V, = kT/q is the thermal voltage and Qis is the mobile charge density at the source end [cf. Eq. (6.95)]. At the drain end Qi = Qid. Id, 7.3 Long-Channel Charge Model 345 -0.55- I I I - 0.60 - v,,=o v - Fig. 3.5 3.0 - - - I I I I 012345 Vgs (V) (a) 0.0 7.5 The normalized (a) gate charge QG as a function of V,, for different V,,, (b) bulk charge QB as a function of V,, for different V,, in strong inversion Let us first calculate the source and drain charge QD and Qs, respectively. Application of Eqs. (7.63) and (7.64) with Eq. (7.49) results in (7.65) which on integration yields, after using Eq. (6.93) for Ids, QD = iWL(2Qid + Qis). (7.66) We can now relate charge densities Qis and Qid using Eq. (6.95), resulting in the following equation for QD Q D -__ - A wLcox(q - l)vt exp ( vg~tvth)(2~pvds/vt + 1) (7.67) where q = (1 + Cd/C,,) [cf. Eq. (6.103)]. Similar procedures can be used for calculating the source charge Qs and is found to be Q s -_- - A W~Co,(r1 - 1)V exp (vg$tvth)(e-vds/vt + 2). Note that when V,, = 0, and Vgs = T/rh, we have QD = Qs = - 0.5C,,,(q - 1)v. From Eq. (7.67) it is evident that Vd, dependence on Qs and QD is rather weak because for Vds greater than a few V,, the terms involving vd, become negligible and we find Qs = 20,. Figure 7.6 shows drain and source charges [...]... quasi-static and non-quasi-static capacitance models for the four terminal MOSFET, ’ IEEE Electron Device Lett., EDL-8, pp 37 7-3 79 (1987) M H Bagheri and Y Tsividis, ‘A small signal dc-to-high frequency nonquasistatic model for the four-terminal MOSFET valid in all regions of operation’, IEEE Trans Electron Devices, ED-32, pp 238 3-2 391 (1985) P Mancini, C Turchetti, and G Masetti, ‘A non-quasi-static... MOSFETYs’, Solid-state Electron., 32, pp 5 7-6 3 (1989) [l5] R Gharabagi and A El-Nokali, ‘A charge-based model for short-channel MOS transistor capacitances’, IEEE Trans Electron Devices, ED-37, pp 106 4-1 072 (1990) [16] C Turchetti, P Prioretti, G Masetti, E Profumo, and M Vanzi, ‘A Meyer-like approach for the transient analysis of digital MOS ICs’, IEEE Trans Computer-Aided Design, CAD-5, pp 49 9-5 06 (1986)... This is shown in Figure 7.14 for Qs and QD with and without velocity saturation, 0.0 I - ' -0 .3 ,,,,(,,,,,,, ,,,,,,, ,,,,(,, W - o* 2 d -0 .8 v,,.= 0 VJ QD ,, ( , _ Qs /I-== - 1 1 I - -1 .0 - No Velocity Saturation ._ -2 .0 I I I I Fig 7.14 The normalized source and drain charges, Qs and Qd, respectively, as a function of Vds for different Vgs, with and without velocity saturation... V,, = 4.6 and 3.8 V and V,, = OV for a typical MOSFET Circles are experimental points for an n-channel LDD device ( L = 0.77 pm and to, = 150A).Solid, dashed and dotted lines are 3 different model equations (see text) (After Arora and Sharma [18]) Fig 8.2 Plot of log[Zb/Z,(Vds- V,,,,)] versus 1/(Vds - V,,,,) for different effective channel lengths L and gate voltage Vgsfor both p - and n-channel devices... MOSFET terminal capacitances’, Tech Digest, IEEE Custom Integrated Circuits Conference, CICC-87, pp 40 0-4 04 (1987) B J Sheu and P K KO,‘Measurement and modeling of short-channel MOS transistor gate capacitances’, IEEE J Solid-state Circuits, SC-22, pp 46 4-4 72 (1987) Y T Yeow, ‘Measurement and numerical modeling of short channel MOSFET gate capacitances’, IEEE Trans Electron Devices ED-35, pp 251 0-2 519... Hsu, and P K KO, ‘An MOS transistor charge model for VLSI design’, IEEE Trans Computer-Aided Design, CAD-7, pp 52 0-5 27 (1 988) [13] H Masuda, Y Aoki, J Mano, and 0 Yamashiro, ‘MOSTSM: A physically based charge conservative MOSFET model’, IEEE Trans Computer-Aided Design, CAD-7, pp 122 9-1 235 (1988) [14] R Gharabagi and A El-Nokali, ‘A model for the intrinsic gate capacitances of short-channel MOSFETYs’,... long-channel MOST valid in all regions of operation,’ IEEE Trans Electron Devices, ED-34, pp 32 5-3 35 (1987) P 5 V Vandeloo and W M C Sansen, Modeling of the MOS transistor for high frequency analog design’, IEEE Trans Computer-Aided Design, CAD-8, pp 71 3-7 23 (1989) L J Pu and Y Tsividis, ‘Small-signal parameters and thermal noise of the fourterminal MOSFET in non-quasistatic operation,’ Solid-state... transistors’, Solid-state Electron., 35, pp 4 5-4 9 (1 992) W Budde and W H Lamfried, ‘A charge-sheet capacitance model based on drain current modeling , IEEE Trans Electron Devices, ED-37, pp 167 8-1 687 (1990) H J Park, P K KO, and C Hu, ‘A charge sheet capacitance model of short channel MOSFET s for SPICE‘, IEEE Trans Computer-Aided Design, CAD -1 0, pp 37 6-3 89 (1991) C T Yao, I A Mack, and H C Lin, ‘Short-channel... Eindhoven, Eindhoven, The Netherlands K A Sakallah, Y T Yen, and S S Greenberg, ‘A first order charge conserving MOS capacitor model’, IEEE Trans Computer-Aided Design, CAD-9, pp 9 9-1 08 (1990) J J Paulos and D A Antoniadis, ‘Limitations of quasi-static capacitance models for the MOS transistor’, IEEE Electron Device Lett., EDL-4, 22 1-2 24 (1983) A Afzali-Kushaa and A El-Nokali, Modeling subthreshold capacitances... simulation , IEEE Trans Computer-Aided Design, CAD-1, pp 15 0-1 56 (1982) [7] P Yang, B D Epler, and P Chatterjee, ‘An investigation of the charge conservation problem for MOSFET circuit simulation , IEEE J Solid-state Circuits, SC-18, pp 12 8-1 38 (1983) [8] K Y Tong, ‘AC model for MOS transistors from transient-current computation’, TEE Proc., Vol 130, Pt I, pp 3 3-3 6 (1983) [9] J G Fossum, H Jeong, and . Qs W - - /I-== ___________________ -0 .8 - -1 .0 - Fig. 7.14 The normalized source and drain charges, Qs and Qd, respectively, as a function of Vds for different Vgs, with and without. and 352 0.61 I I I I 7 Dynamic Model Fig. 7.12 Normalized plot of the drain-to-source and source-to-drain capacitance C,, and C,,, respectively, for two different expressions for. (6.95)]. At the drain end Qi = Qid. Id, 7.3 Long-Channel Charge Model 345 -0 .5 5- I I I - 0.60 - v,,=o v - Fig. 3.5 3.0 - - - I I I I 012345 Vgs (V) (a) 0.0 7.5 The