MOSFET MODELING FOR VLSI SIMULATION - Theory and Practice Episode 8 potx

256 6 MOSFET DC Model Though accurate, this is a complicated expression and not suitable for CAD models. However, the following simplified form of Eq. (6.76) has been used in the drain current model [28] (6.77) This approximation, though accurate, has 6 as a function of the variable V; so 6 must be calculated for each V. The effect of approximating the function F(V, V,) with different 6 expressions is most sensitive at zero Vsb. Therefore, a comparison is made between the exact and approximate functions at zero vsb by calculating the relative errors between them using different 6 expressions. The results are shown in Figure 6.12 where the error E, is defined as Fexact - Fapprox Fexact E, = 100 x where Fexac, and Fapprox are values of F given by Eqs. (6.68) and (6.69), respectively. Note from this figure that the simplest approximation for 6 [cf. Eq. (6.71)] has maximum error, therefore this approximation will underestimate the depletion layer charge Qb the most. However, the result- ing error in Ids calculations is not usually significant because Qb is much smaller than Qi. In fact, for Id, calculations, any of the 6 functions discussed above can be used depending upon the desired accuracy and speed of calculation. However, accuracy in 6 approximations are important for CURVE SAPPRO)?\ 1 #= 1 Ea. (6.71) 0.0 L.0 8.0 V Fig. 6.12 Error between the exact and approximate square-root function F(V, Vo) for different 6 approximations 6.4 Piece-Wise Drain Current Model for Enhancement Devices 251 MOSFET capacitance calculations, where small error in Qb can cause large errors in the capacitances. For this reason Eqs. (6.73) or (6.74) are most appropriate for circuit design, although these expressions can create problem in the capacitance calculations as we shall see in Chapter 7. 6.4.4 Drain Current Equation with Square-Root Approximation With the square-root approximation (6.69), Eq. (6.63) for Qh(y) becomes QdY) = - coXl"6 v(Y) + d-1 while Eq. (6.64) for Qi(y) reduces to (6.78) where we have made use of Eq. (6.46) for Vfh and a is defined as I a=1+6y.( (6.80) Note the similarity of Eqs. (6.79) and (6.45); the only difference being the presence of the a term which takes into account variations in the bulk charge Qb along the channel. Using the above value of Qi(y) in Eq. (6.41) and integrating we get the current in the linear region as (6.81) Comparing this equation with Eq. (6.65) we see that just by approximating the square root term in Qb we could get a much simpler expression for Zds. It is this current equation which is used in most of the newly developed MOSFET models for circuit simulation. For example, SPICE MOS Level 3 [23] and Level 4 [25] use Eq. (6.81) for Zds; however, Level 3 uses 6 given by Eq. (6.71), while Level 4 (BSIM model) uses 6 given by Eq. (6.73). Differentiating Eq. (6.81) and equating the resulting expression to zero gives the following simple expression for V,,,,, namely 'gs - 'th a ',sat = (6.82) Substituting this equation into Eq. (6.8 1) yields the saturation region current, without CLM, as 258 6 MOSFET DC Model To summarize, we now have a more accurate drain current model that takes into account the bulk charge variation along the channel region and is represented by the following set of equations: 0 Vgs 5 vtfl (cutoff region) P(Vgs - Vth - 0.5aVds)Vds (linear region) CVgs - vth)2 (saturation region). 1: (6.84) It is worth pointing out that the charge-sheet model, discussed in section 6.3, can also be simplified using the square root-approximation. In this case, the final equation for I,, in the linear region looks similar to Eq. (6.81). This can be seen as follows: replacing the square-root dependence of Qb in Eq. (6.27) with a linear approximation [cf. Eq. (6.69)] we get Q~Y) - YCA& + 6(4,(~) - 4so)l (6.85) where 6 is given by any of the expressions discussed earlier in section 6.4.3. Using this value of Qb(y) in Eq. (6.28) yields Vgs > vth, v,, I V,,,, Vgs > vh, Vd, > Vd,,, Ids = (6.86) where Vn = ~1, + ~64sO - Y& and ct is given by Eq. (6.80). Substituting Qi from Eq. (6.86) in Eq. (6.33) and carrying out the integration under the boundary conditions given in Eq. (6.34) yields [18] Ids = Ids1 + Ids2 = Plvgb - vn + avt - 0.5a(4sL - ~SO)](~SL - 4~0). (6.87) Note that unlike Eq. (6.84), the above equation is continuous in all the regions of device operation (subthreshold, linear and saturation). Compar- ing Eq. (6.87) with Eq. (6.84) in the linear region we see that there is an extra term ctVt(4,, - 4so) in Eq. (6.87). This is due to the diffusion compo- nent of the current that is neglected in Eq. (6.81). Figure 6.13 shows comparison of the calculated I,, - Vds characterstics using the charge-sheet model, the rudimentary (first order) model and the bulk-charge model. The model parameters used are those shown in Table 6.1. Note that the piece-wise models (rudimentary and bulk-charge models) overpredict the drain current compared to the charge-sheet model. This can be explained as follows. In deriving the piece-wise drain current models in the previous sections we assumed that in strong inversion the potential drop 4s across the silicon was pinned at 24f + v&. In reality this is not 6.4 Piece-Wise Drain Current Model for Enhancement Devices 259 Fig. 6.13 Comparison of the MOSFET output characteristics using (a) charge-sheet model, (b) bulk-charge model and (c) rudimentary (classical model). The classical model overpredicts current true and indeed the potential does increase by few times the thermal voltage (- 4Vt) as was discussed in chapter 5. This shows that piece-wise models underestimate 4, and hence the bulk charge Qb. For a given gate voltage, underestimating 4, means overestimating V,,, the voltage across the oxide [cf. Eq. (4.16)]. Overestimating V,, leads to an overestimation of silicon charge Q,, which in turn means overestimating Qi because Qb is being underestimated. The overestimation of inversion charge Qi in the channel results in an overestimation of drain current. Indeed it has been found that the piece-wise multisection model overestimates the drain current by 15-20% [ 111. In spite of its inaccuracy, the multisection (piece-wise) model [cf. Eq. (6.84)] is the one used in today’s widely used circuit simulators because of its simplicity. 6.4.5 Subthreshold Region Model While deriving I,, Eqs. (6.62) and (6.84), it was assumed that the current flow is due to drift only (assumption 6). This resulted in I, = 0 for Vgs < Vth, that is, there is no current flow for V,, below threshold. In reality this is not true and I,, has small but finite values for V,, < Vrh. For the device shown in Figure 6.5 this current is of the order A when V,, approaches Vr, and then decreases exponentially below Vth. In fact the A to 260 6 MOSFET DC Model device behavior changes from square law to exponential when V,, approa- ches Vth. This current below V,, is called the subthreshold or weak inversion current and occurs when V,, < Vth, or 4s> 4,> 24s. Unlike the strong inversion region where drift current dominates, the subthreshold region conduction is dominated by diflusion current. It should be emphasized that the transition from weak to strong inversion is not well defined, as was discussed in chapters 4 and 5. This region of device (subthreshold) current is important in that it is a leakage current that affects dynamic circuits and determines CMOS standby power. In this region of operation, Eqs. (6.62) or (6.84) are no longer valid. In the subthreshold region of operation, the surface potential 4,, or the band bending, is nearly constant from the source to the drain end because the inversion charge density Qi is several orders of magnitude smaller than the bulk charge density Qb (cf. section 4.2). This means that we can replace 4,(y) in subthreshold region by some constant value, say $J~,. With this assumption, the bulk charge Qb can be expressed as Qb = - cuxY = - cuxY (6.88) Further, since Qi << Qb, we have Q, z Qb, so that Eq. (6.19) becomes Qb Vgb = Vfb + 4ss - CUX Solving Eqs. (6.88) and (6.89) for 4,s we get (6.89) or (6.90) This shows that 4s, is nearly linearly dependent on VgS. It should be emphasized that the surface potential 4,s in the subthreshold region is constant from source to drain only for a long channel device. As the channel length become shorter, 4s, no longer remains constant over the whole channel length. Because @,, is constant, the electric field by is zero. Hence, the only current that can flow is diffusion current as can be seen from Eq. (6.2) and is given by JJdiffusion) = qD,- (6.91) Integrating this equation across the channel of thickness t,, and making use of Eq. (6.13) we can write the drain current I,, (due to diffusion) in the dn dY 6.4 Piece-Wise Drain Current Model for Enhancement Devices 26 1 subthreshold region as dQ i dY (6.92) where we have made use of the Einstein relation D, = p,Vt [cf. Eq. (2.34)] and made the assumption that dp,/dx = 0. Integrating the equation above from y = 0 to y = L we get I,, = p,WVt- (subthreshold region) (6.93) where Qis and Qid are the inversion charge densities at the source and the drain end respectively when the device is in the subthreshold or weak inversion region. Following the MOS capacitor case, the inversion charge density Qi(y) in weak inversion [cf. Eq. (4.43)] is given by (6.94) where we have replaced 4, by q5s, and have made use of Eq. (6.23) for y and Eq. (6.22) for Vq!. Remembering that Vcb(y = 0) = V,b and Vcb(y = L) = V,b+ V,,, the inversion charge Qis and Qid at the source and drain ends, respectively, can be written as (6.9 5a) Qi,(drain end) = & I/,e(d’ss- 26f- vsb ~ Vds)lVt. Using these values of Qis and Qid in Eq. (6.93) yields I- psWCoxy I/:e(6”’-26ff-V,b)/V,(1 - e- Vds/Vf). (6.9 5 b) 26 (6.96a) Above equation takes the following form, after eliminating 4J using Eq. (2.15) and making use of Eq. (6.50) for B, ds- 2LJZ (6.96b) This is the current equation for the subthreshold region. For each Vgs we first calculate q5ss from Eq. (6.90), which in turn is used to calculate I,, from 262 6 MOSFET DC Model m a- - - - - - 1.5 3 GATE VOLTAGE,V,, CVI Fig. 6.14 Typical device I,, - V,, characteristics in the subthreshold or weak inversion region for two different back bias Eq. (6.96). The following conclusions about subthreshold conduction can be drawn from Eq. (6.96): 0 The subthreshold current increases exponentially with the surface potential 4ss and hence Vgs [cf. Eq. (6.90)]. This is evident from Figure 6.14 where measured Id, is plotted against Vgs for different values of V,, and Vd, for a nMOST fabricated using 1 pm CMOS technology. The current is dependent upon an exponentially decreasing term which for Vd, larger than 4Vt (- 100mV) is negligible, becoming independent of Vds. It should be pointed out that this is true only for long channel devices. In fact for short channel devices, this region of drain current exhibits a significant dependence on the drain voltage as we will see in section 6.9. The subthreshold current is strongly dependent on temperature due to its dependence on the square of the intrinsic carrier concentration ni, resulting in steeper slopes at low temperatures. The temperature depen- dence of subthreshold current is discussed in section 6.9. Often Eq. (6.96) is written in terms of the surface concentration n, as6 (6.97) Equation (6.97) can be derived as follows: The inversion channel is confined by the potential well created by the oxide to the silicon interface on one side and on the other side by the perpendicular electric field gS at the surface in the substrate. Since Qi << Qb in weak inversion, is equal to the depletion field, that is (Continued next page) 6.4 Piece-Wise Drain Current Model for Enhancement Devices 263 Most of the expressions reported in the literature for Id, in weak inversion region are variations of Eq. (6.96) [4], [29]-[32]. For circuit simulation models, often a simplified form of this equation is used. Since Qb is a weak function of 4,,, we can expand Qb using Taylor series around 4so which lies between 4f and 24r. Retaining the first two terms of the Taylor series, we get From Eq. (6.88) we get (6.98) (6.99) where Cd is called the depletion region ~apacitance.~ Combining Eqs. (6.98) and (6.99) with (6.89) yields (6.100) For calculating Ids, it is more appropriate to take 4so in the middle of the subthreshold region (i.e., = 1.54f + V,b) because 4ss lies between 24f + V,, and 4J + V,b. However, by assuming = 24f + l/,b, the condition for the onset of strong inversion, we arrive at an expression for Id, that is often used in circuit models. Thus, assuming 4so = 24r + VSb, Eq. (6.100) becomes 4s - 24f - (6.101) The average thermal energy of the carriers for motion perpendicular to the surface is kT. The average thickness t,,, of the weak inversion channel is, therefore, given by pfst,, = k7 Solving these two equations for &, by eliminating &s, and then combining Eqs. (6.96a) and Eq. (6.10) results in the desired equation. ' Rewriting Eq. (6.99) in the following form c,- YCOX -Jy-'.'.X 2Jz Xd, - EOEOX - thickness of the depletion region under the channel clearly shows Cd as the depletion layer capacitance. 264 6 MOSFET DC Model where we have made use of Eq. (6.46) for Vth, and Y 1+ (6.102) Typical values of r] range from 1 to 3. Physically, r] signifies the capacitive coupling between the gate and silicon surface. If there is a significant interface trap density, the capacitance Ci, associated with this trap is in parallel with the depletion layer capacitance cd, and therefore Eq. (6.102) becomes (6.103) This is the equation for r] used in SPICE model Levels 2 and 3. In this equation Cif, called the surface state capacitance, is normally regarded as an adjustable parameter through qo and is used to fit the value of q to measured characteristics. Combining Eqs. (6.96), (6.99) and (6.101) yields or where I,, = b(cd/c,,)V: = b(r] - l)V:, is a prefactor term. This is the most commonly used drain current equation for the subthreshold region of device operation. It clearly shows that the subthreshold current ( V,, < Vth) increases exponentially with Vgs and for V,, larger than about 3Vr, the current becomes independent of Vds. Further, since the parameter is inversely proportional to the square root of Vsb, the subthreshold slope becomes steeper at higher values of Vsb. This indeed is the case as can be seen from Figure 6.14 which is a plot of log(Zds) versus Vg, for an experimental device. Note that the curve is linear (on the log scale) until the device starts to turn on. When V,, approaches Vfh, Eq. (6.104) is no longer valid and the current will increase either linearly (linear region) or as the square of (V,, - Vrh) (saturation region) depending upon the value of Vd,. Very often the following simpler version of Eq. (6.104) is used for circuit models [34] (6.105) 6.4 Piece-Wise Drain Current Model for Enhancement Devices 265 where Vd, dependence is ignored because its effects on I,, is negligible for VdS > 3Vf. The parameter rn is inserted to correct for various approximations made in the derivation of Eq. (6.104) and is calculated in the same way as qo, that is, by curve fitting the experimental data. Subthreshold Slope. An important parameter characteristic of the sub- threshold region is the gate voltage swing required to reduce the current from its ‘on’ value to an acceptable ‘off’ value. This gate voltage swing, also called the subthreshold slope S, is dejined as the change in the gate voltage V,, required to reduce subthreshold current Ids by one decade. For a device to have good turn-on characteristics, S should be as small as possible. Clearly, S is a convenient measure of the turnoff characteristics of a MOSFET. By this definition S= dvgb = 2.3 [ *] (Vldecade) (6.106) where the factor 2.3 accounts for the conversion from “log” (logarithm to the base 10) to “ln” (logarithm to the base e). Strictly speaking, S varies with the current level. However, this variation is small over one decade of current so that S can be taken as gate swing per decade [32]. We can rewrite Eq. (6.106) for S as d(log Ids) Ids) Differentiating Eq. (6.89) we get where taking where (6.107) (6.108) we have made use of Eq. (6.99) for C,. Assuming vd, > 31/, and the logarithm of both sides of Eq. (6.96b), we get I, = p,C,,y v,- . 2L ( :J2 Now differentiating Eq. (6.109), we get as vr 24s Vf (6.109) (6.1 10) (6.111) where again we have made use of Eq. (6.99) for C,. Substituting Eqs. (6.108) [...]... widely used in simulators [27] (6.153) - For n-channel devices, the value of 8, is usually small ( 0.005 V - I), and is often neglected For p-channel devices improves the current-voltage data fit The SPICE MOSFET Levels 3 and 4 models use 0 , = 0 in Eq (6.153) for both p - and n-channel devices resulting in the following equation for p, Ps = (6.154) PO + e(vgs - Vth) Physically speaking, this is an... capacitance and not by decreased mobility for thin oxides However, they suggested the following empirical formulation, which is slightly different from Eq (6.140) w (6.141) The parameters and v are given in Table 6.3 [62] 6.6 Effective Mobility 219 x ' ' 3 i \ 0 0 x + +-ox* 50i 89 8 16 98 # tox: 'Xx.0 1 + ' 1 588 43 68 # ' 's, 5312 89 a + -1 - 100/100 pn n M S T N,j 6 x i O " ~ m - 3 pMOST N a I 3 x 10i6crn -3 WiL... < 1) is a short-channel factor This value of Qb when used in Eq (6.43) yields Qi(Y) = - + CoxCVp - v f b - 24f - FlYJ24f + Vs, - (1 + Fly 6) v(Y)I Replacing (1 F,y6) by a and recalling that the short-channel threshold voltage Vr, is [cf Eq (5.69)] Vrh = ‘ f b + ’4.f + FfyJ24f + vsb we see that Q,(y) for short-channel device becomes Q d Y ) = - C o x C v g s - Vt, - av(~)l (6.1 68) 6 MOSFET DC Model... mobility data for n-channel devices However, for p-channel devices it was found by Arora and Gildenblat [59] and later confirmed by other workers [60 ]-[ 61] that when the factor 0.5 in Eq (6.147) is replaced by 0.3, the resulting l/ps vs &eff curve is independent of back bias It is thus more appropriate to write &eff as (6.1 48) where 5 = 0.5 for n-channel devices, and 5 = 0.25 - 0.30 for p-channel devices."... parameter The SPICE MOSFET Level 2 model uses the following equation for ps, (6.155) which is similar in form to Eq (6.140) Here u, is a fitting parameter whose value lies between 0 and 0.5 and represents the contribution to the field due to drain voltage The exponent v is approximately 0.25 for n-channel and 0.15 for p-channel devices [SO] For devices in strong inversion, 284 6 MOSFET D C Model Eq... mobiluy for (a) electrons and holes at highfields (After Takagi et al [65]) Gauss' law it is easy to see that - &xx2 = Qi ~ (6.145) EOEsi and (6.146) 6 MOSFET DC Model 282 Solving Eqs (6.145) and (6.146) for cYxl and cYX2 and substituting in Eq (6.144) yields (6.147) Thus is related to the bulk depletion charge Qb and to the inversion charge Q i This equation for cYeff when used in Eqs (6.140 )-( 6.143)... depletion device fabricated using an NMOS 1 o I 0 .8 o,6 - - E - - ( 0 0.4 - I I I MEASURED .SIMULATED w d m= ~ v,,= 2 5 0.1 v - r - L Vlb=O v v,,=7v 1 v -3 0 2 6 4 vds v 8 (v) Fig 6.20 Measured and calculated transfer and output characteristics at different back bias for an n-channel depletion device (After Divekar and Dowell [42]) 276 6 MOSFET DC Model process with W J L , = 5012.5 Solid lines are... expression €c Usat = - (6.1 58) P S The reported value of usa,, for the MOSFET inversion layer, varies over a wide range [72 ]-1 791 For electrons usat varies between 6-9 x 106cm/s, while for holes it is between 4 -8 x 106cm/s The use of Eq (6.157) in Eq (6.42) leads to an expression for the current which cannot be solved in the general case By making some approximations, Eq (6.157) has been used for the current... calculation of n-channel MOSFET s 1341, [77 ]-[ 82 ], though the final I,, expression is too complicated to be suitable for circuit models Various expressions, which closely approximate Eq (6.157) for electrons (v = 2), have been proposed These are of the general form 1 281 , [82 ]-[ 83 ] U= Ps gy (6.159) 1 + 60(gy/&c) where 6, is usually a function of the field.’ However, the final expression for the current... &f , f , I 8 lo5 1 I I 2 4 (V / cm) Fig 6.21 Inversion layer electron mobility data for silicon at 300K for two different substrate dopings N , , = 1.25 x l O " ~ m - ~ , N , , = 1.33 x 10'6cm-3 (After Sun and Plummer [ 5 2 ] ) value for electrons is 40 0-7 00cm2/(V.s) and holes is 10 0-3 00 cm2/(V.s) go is the critical electric field below which ps = p o and above which ps begins to decrease and v is an . ( 0- r - 0.4 - L - Vlb=O v v,,=7v __ -1 v 0 2 4 6 8 -3 v vds (v) Fig. 6.20 Measured and calculated transfer and output characteristics at different back bias for an n-channel. Qb = - cuxY = - cuxY (6 .88 ) Further, since Qi << Qb, we have Q, z Qb, so that Eq. (6.19) becomes Qb Vgb = Vfb + 4ss - CUX Solving Eqs. (6 .88 ) and (6 .89 ) for 4,s. [ 151. 6.4 Piece-Wise Drain Current Model for Enhancement Devices 269 n MOST V* =QV 1 5 10.5 v) n - 5 1 0-6 3 0 1 0-7 z a 1 0 -8 n > [r 1 0-9 1 0-1 0 0.0 1 .o 2.0