MOSFET MODELING FOR VLSI SIMULATION - Theory and Practice Episode 6 pptx

176 5 Threshold Voltage 3.0 2.0 - > w 0 z 0 20 =! u 2 1.0 u) >- -I 2 -1.0 2 -2.0 +c a f > -3.0 I 014 4.0 - 3.0 3 w 0 2 0 2 i 2 2.0 1.0 ?I 0 +* 2 -1.0 >- 0 a. K L SUBSTRATE DOPING, Nb ( ~m-~ ) Fig. 5.5 Calculated threshold voltage V,, for n- and p-channel MOSFETs as a function of substrate doping N, for n+ polysilicon gate (left scale) and p+ polysilicon gate (right scale) for three different oxide thickness. (After Sze [5]) curves are based on the assumption that Qo = 0, a reasonable assumption for modern VLSI processes. The temperature through the 4f term, the higher the temperature, the lower the Vlh (for details see section 5.4) The body bias Vsb; the higher the Vsb, the higher the Vfh. The increase in Vth due to an increase in V,, can be obtained from Eq. (5.16) as (5.17) Avfh = vfh - vTO = Y[Jm - 61. Body-EfSect. The variation of v,h with v,, is often called the substrate bias sensitivity or body-efSect. Differentiating Eq. (5.14) with respect to V,, we get (5.18) where the + and the - signs are for n- and p-channel devices, respectively. This equation shows that the body-effect increases as the body factor y Y f dvfh - "sb 2Jm 5.2 Nonuniformly Doped MOSFET 111 increases and body bias V,b decreases. For circuit design, it is often desirable to lower the body effect,3 which means the body factor y should be reduced. From Eq. (5.1 1) it is evident that y can be reduced with lower doping con- centration N, and/or lower oxide thickness tax. However, lowering N,, for example, conflicts with the scaling rule (cf. section 3.4). In fact, the choice of process or circuit parameters is a trade off between various parameters involved in device design. SPICE Implementation. Note that Eq. (5.14) becomes invalid for v,b I - 24,., i.e., when the S/D diodes become forward biased by an amount 24, Although during normal operation of the device the S/D diodes will not be forward biased; however, in SPICE, during Newton-Raphson iterations, it is possible to encounter V,, < - 24, This is just an artifact of the iteration solution process, and convergence to a proper solution requires the model to behave well even in such invalid operating regions. Therefore, to use Eq. (5.14) or (5.16) in the forward biased region of S/D junction, some sort of smoothing function is used to limit the value of (V,, + 24f) such that it is always positive. The smoothing function assures a smooth transition with- out any discontinuity. In SPICE, the transition point is chosen as V,, = - 4, Thus, when V,, + 4J 2 0, Eq. (5.14) is used and when V,, + 4,. < 0, the term Jm is replaced by 2&/(1 - Vsb/4f) such that V,,, and its first derivative are continuous at V,, = - 4f in the forward biased S/D region. 5.2 Nonuniformly Doped MOSFET In the previous section we have seen that for a given gate material the threshold voltage V,, of a MOSFET depends upon the substrate doping concentration Nb and the gate oxide thickness Lox. Therefore, in principle, V,,, could be set to any value by proper choice of Nb and fox (see Figure 5.5). However, considerations like the body-effect, source-drain junction capaci- tances and breakdown voltages often dictate desirable values of these parameters. In practice this is achieved by ion implanting a shallow layer of dopant atoms into the substrate in the channel region. Thus, by adjusting the channel surface concentration (using ion implantation) any desired value of V,, can be achieved. In fact, in VLSI devices, more than one implant is often used in the channel region-one to adjust the threshold voltage and another to avoid the punchthrough effect-as was discussed ~ 3 During circuit operation, in NMOS circuits, the MOSFET source voltage often increases which results in higher V,, thereby causing V,, to increase. This results in a decrease in the drain current I,, [see Eq. (3.4)], consequently the circuit runs at a lower speed and might not even function properly. For this reason, it is desirable to reduce the change in V,, due to increase in V,,, that is reduce body-effect. 178 5 Threshold Voltage in section 3.5.2. The fact that the surface is no longer uniformly doped, due to the channel implant, means Eq. (5.14) is generally not valid. Recall that in n+ polysilicon gate CMOS technology, an nMOST has channel implant dopants (boron) which are of the same type as that of the substrate (p-type), while pMOST (compensated p-device) has shallow channel implant dopants, which are of the type opposite to that of the substrate (cf. section 3.5.2). Since compensated pMOST has shallow channel implant, the surface layer is depleted at zero gate bias. When Vys > I/th, the current flows at the surface. Therefore, these compensated devices are usually modeled in the same way as nMOST, so far as the drain current modeling is concerned; however they have a different threshold voltage model as we will see later. In a recent submicron CMOS technology, pMOSTs are being fabricated with p+ polysilicon gate while nMOSTs are with n+ polysilicon gates. With p+ polysilicon gate, pMOST has channel implant of the same type as substrate and therefore, from a modeling point of view, these devices are similar to nMOST. In the depletion type devices the channel implant, which is of opposite type to that of the substrate, is deep so that significant current flows even at V,, = OV. These (depletion) devices are referred to as normally-on buried channel (BC) MOSFET as against the compensated devices, which are also referred to as normally-of buried channel (BC) MOSFETs. The two BC MOSFETs result in entirely different V,, behavior due to the different potential distributions associated with the built-in pn junctions in the channel region. This can easily be seen from their energy band diagrams as shown in Figure 5.6 for a p-channel device with a p-type buried layer in n-type substrate and an n+ polysilicon gate. While for the normally-off BC MOSFET (Figure 5.6a) the energy band bending of the bulk junction extends to the channel surface, the depletion device (normally-on) has an (hole) energy minimum (Figure 5.6b). It was pointed out earlier (cf. section 3.5.2) that in practice p-channel depletion devices are not usually made. It Ec E f Ei n E" (a) (b) Fig. 5.6 Energy band diagram for a p-type buried-channel MOSFET in (a) the surface channel mode and (b) the buried channel mode (depletion device) 5.2 Nonuniformly Doped MOSFET 179 is the n-channel depletion device, with negative threshold voltage, which is more important and thus modeled here. There is an extensive literature on threshold voltage models for ion implanted devices [ll-[36]. However, we will discuss and develop only those models which are suitable for circuit simulators. We will first discuss enhancement mode devices and then depletion mode devices. 5.2.1 Enhancement Type Device When ions are implanted into the channel, the implanted profile can be fairly accurately approximated by the following Gaussian distribution function (see Figure 5.7, also see Appendix H, Eq. (H.5)) (5.19) where No = Di/(ARp&) is the maximum concentration and occurs at x = R,, R, = projected range (average penetration depth), Di = dose, i.e., number of implanted ions per unit area. x = the depth measured from the oxide-silicon interface, ARp = straggle (standard deviation) The channel implant dose Di is typically of the order of 10" - 1012 cm-2 while the implant energy varies from 10-200 KeV. Following the implantation process, devices go through various high-temperature fabrication steps, which change the final profile. Figure 5.8 shows the final channel implant profiles for nMOST and pMOST for a typical 2pm CMOS technology with n+ polysilicon gate. Fig. 5.7 Gaussian doping profile in the channel region of a VLSI MOSFET 180 5 Threshold Voltage 0 lo1’ - ASSUMED DOPING PROFILE Nb 20 1014 00 05 10 15 DEPTH INTO SILICON, X (pm) (a) - - , P-N JUNCTION - 1014 I I!/:[ I I I I I I I I I I I I I I 00 05 10 15 20 DEPTH INTO SILICON, X (pm) (b) Fig. 5.8 Vertical doping profile of channel implanted region under the gate for a typical 2 prn CMOS process for (a) nMOST and (b) pMOST The result of the channel implant in an otherwise uniformly doped substrate is to change the threshold voltage. The extrapolated threshold voltage V,, as a function of Jm for different channel implant dose is shown in Figure 5.9 [6]. One can see from this figure that the slope of the V,, versus V,, curve changes from single dope at low doses to two distinct slopes at higher doses. This shows that a simple square root dependence, which relates V,, to Vsb, is not correct for channel implanted devices with high doses. However, for these devices Eq. (5.7) is still valid provided we use appropriate value of Vfbr 4si, and Qb(4si). We will now consider each of these terms and see how they are modified for implanted devices. 5.2 Nonuniformly Doped MOSFET 181 6 4 c > u 2 f > 0 vsb(v) 0.5 0 .2 .4 .6 .8 I I I I I1 I1 -2 DOSE = 12 XI0 C cm 1 SUBSTRATE P-Si <I00 > I I I Fig. 5.9 Threshold voltage dependence on body bias for different channel implants. (After Kamoshida [6]) Flat Band Voltage Vfb. As has been discussed earlier (cf. section 4.7), the concept of the flat band voltage is strictly applicable to a uniformly doped substrate. However, being an important reference voltage, it has been redefined for nonuniformly doped substrates as that gate voltage which causes the overall space charge to be zero [cf. Eq. (4.79)]. Whatever definition is used, for circuit modeling vfb is treated as a model parameter to be determined for a given process. Surface Potential at Strong Inversion (4si). Like the uniformly doped case, different criteria for strong inversion have been suggested for non-uniformly doped substrates [2], [33]-[36]. Some of these are: I. The classical criterion given by Eq. (5.12) is still used for non-uniformly doped substrate [33], although strictly speaking it is valid only for implanted channels with low dose. Others [ 1 I] have used this criterion by replacing Nb (bulk concentration) with N, (surface concentration) in the df term, that is, c$~~ = 24f(surface) = 2Vt In 2 . (ti 1 (5.20) I82 5 Threshold Voltage Compare Eq. (5.20) with the corresponding Eq. (5.12) for uniform doped substrate. 2. The minority carrier concentration at the surface is equal to majority carrier density at the boundary of the depletion region 171, 191, [31] that is, (5.21) where N(X,,) is the dopant density at the edge of the depletion region of width X,,. Note that, the condition defined by Eq. (5.21) reduces to Eq. (5.12) when N(X,,) = N,, is. when the boundary of the depletion region is located in the uniformly doped part of the profile. In real devices this will be the case for shallow implants or higher values of the back bias. 3. The variation in the inversion and depletion charge densities Qi and Qb, respectively, with respect to the surface potential $s are equal [34]-[36], that is, This criterion is equivalent to (5.22) where N is the average concentration given by It can easily be seen that this criterion is equivalent to the classical criterion for uniformly doped substrate. Again, these different criteria result in slightly different values of threshold voltages. In fact, the above three criteria lead to threshold voltages that are about 0.2 V apart. A detailed comparison of the threshold voltage shift as a function of implant dose for boron implanted MOS structures, based on both 2-D numerical solution and depletion approximation, has been studied by Demoulin and Van De Wiele [2]. It has been found that agreement between the criterion 3 and experimentally measured V,, is fairly good, while the classical condition 1 is not valid for heavy implant doses. In spite ofthis inadequacy ofthe classical criterion [cf. Eq. (5.20)], it is still usedfor circuit models because of its simplicity. 5.2 Nonuniformly Doped MOSFET 183 Bulk Charge Qb. Under the depletion approximation, the bulk charge Qb for implanted channels can be obtained from the following equation4 [cf. Eq. (4.29)] Xdm Qb = - 4 J,, N(x)dx. (5.23) Therefore, Eq. (5.14) for implanted devices becomes (5.24) Assuming that the implanted profile is Gaussian as given by Eq. (5.19), many authors [9], [27] have calculated Qh using Eq. (5.23). The resulting expression for Qb is fairly complex, involving error functions. These expres- sions have predictive capabilities so that, for example, one can know how the change in the implant dose Di will affect the bulk charge and hence threshold voltage. However, such complex models are not suitable for use in circuit simulators. For this reason they are not discussed here and details of the equations for Qb and V,, are left to the interested reader. The fact that the threshold voltage is determined by the integral of the doping profile rather than by its actual shape, and the desire to get tractable equations for Vth have led to the replacement of the exact profiles by idealized step profiles of concentration N, and width Xi, as shown by dotted lines in Figure 5.8a, such that (5.25) We choose N, and Xi such that the total charge under the exact profile is the same as that under the step profile. Rearranging Eq. (5.25) yields the following expression for the surface concentration N, of the step profile Di Xi N, = - + N, (cm ~ ’). (5.26) Although one can express N, and Xi in terms of implant parameters Equation (5.23) assumes that quasi-neutrality holds at every point outside the depletion region of width X,,,. This in general is not true and concentration gradient causes a built- in field which has to be taken into account when integrating Poisson’s equation. Therefore, strictly speaking Eq. (5.23) needs to be modified as [9] Jo where €(X,,) is the electric field at the boundary X,, of the depletion region 184 5 Threshold Voltage (R,,AR, and Di as given in Eq. (5.19)) [29], for circuit models it is more appropriate to use N, and Xi as model parameters. These parameters are then chosen to make the resulting threshold voltage model match the experimental data. Shallow Implant Model. In many devices, a very shallow implant is used to modify V,,,. The limiting case would be an infinitely thin sheet, approxi- mately a delta function, of ionized charge 40, localized at the Si-SiO, interface. This is equivalent to modifying the flat band voltage by an amount qDi/Co, resulting in the following equation for V,, [8] Cox V,,, = Vfb + 24f + + y,/- (shallow implant). (5.27) Thus, a shallow implant increases V,, without increasing the depletion width xdm. Deep Implant Model. The threshold model described by Eq. (5.27) is fairly good for shallow channel implants. However, the model becomes inaccurate when the implant becomes deep. In such cases the channel doping profile is often replaced by an idealized step profile (see Figure 5.10a). Depending upon the depth of the channel depletion width Xdm, in relation to the depth Xi of the step profile, two cases will arise: Case I. When the back gate bias V,, is such that the depletion depth X,, is less than the depth of the implant Xi (i.e. X,, < Xi), the surface can be considered to be uniformly doped with concentration N, given by Eq. (5.26). In this case V,, is obtained simply by replacing Nb in Eq. (5.14) with N,, that is, (5.28) where 4si is given by equation Eq. (5.20) and (5.29) In fact, for low values of V,, (0-1 V), the slope of the V,, versus V,, curve could be used to calculate N,. Case 11. When V,, is such that X,, lies outside Xi (i.e. X,, > Xi), V,, is no longer given by Eq. (5.28) because X,, has now to be determined from the high-low step doping profile. In this case the bulk charge Qb is given by (see Figure 5.10a, shaded area) (5.30) - Qb = qNsXi + qNb(Xdm - xi). 5.2 Nonuniformly Doped MOSFET i 185 f (a) (b) Fig. 5.10 (a) Step doping profile for an n-channel MOSFET, (b) Doping transformation pro- cedure for calculating the equivalent concentration N,, and width X,, of the transformed box Thus, Qh can be determined provided Xdm is known. The latter can be obtained by solving the Poisson’s equation (2.41) under the depletion approximation in the two regions subject to the following doping distribution: Ns forxIXi Nb forx>Xi N(x) = and satisfying the following two boundary conditions: the electric field &(x) is continuous at x = Xi, the field &(x) = 0 at x = xdm. This yields, after some algebraic manipulation, (5.31) (5.32) Combining Eqs. (5.30) and (5.32) and using the resulting value of Qb in Eq. (5.7) yields the following expression for the threshold voltage’ Often Eq. (5.33) is written in terms of dose Di as where we have made use of Eq. (5.25). Compare this with Eq. (5.27) for shallow implants. [...]... the depletion region, and is given by (5 .64 ) where Xd, is given by Eq (5.8) From Figure 5.14b it can easily be seen, using triangle ABC, that x c = x j ( J 1t ,d m Xx 2- 1) (5 .65 ) which leads to -" '(Jl L+L' - L + ( L - 2 X c ) - 1 +2x, ,- l) 2L 2L L Xi This equation when combined with Eq (5 .64 ) yields (5 .66 ) If we define (5 .67 ) then Eq (5 .66 ) reduces to Qb = qNbXdmFf = YCoxFl Jm (5 .68 ) 5.3 Threshold Voltage... then the MOSFET is 5.3 Threshold Voltage Variations with Device Length and Width >, 0.0 - 1 ' 1 ' 1 ' 1 ' 1 ~ 1 ' 1 195 1 I ~ ' - Nb = 2 X 10'"cm-3 >- 0.7 " u W Q ! i 0 .6 0 - - > 4 0.5 m "A&ING 0 W=20 5 I 0.4 - f"\L - lY I bO.3 I I 1 I 1 I I I I I I I I I I Fig 5.13 Threshold voltage variation with channel length L (curve A) and width W (curve) B) based on 2-d devices simulation. (From akers and scanchez[41])... small compared to an n-chann'el device, the effect of roll-off due to short-channel effects is small Using Yau's approach, one can easily calculate the charge sharing factor F , for compensated p-devices as ( /- F I - 1 - 2 X 1 - - xj = - 1) (5.78) L L Comparing this equation with F , calculated for an enhancement device [cf Eq (5 .67 )] it is easy to see that the correction factor for buried channel devices... the factor 2 accounts for the charge at both the source and drain ends This shift in the threshold voltage due to the fixed charge is independent of the back bias, and therefore, can be absorbed in the flat-band voltage VIb.This means that V f bbecomes length dependent Thus, modeling 0.9 1 , EXPERIMENTAL - MODEL 0 u - 0 s$5 A 0 - - 950 .6 5-% 9s 3 ’ CT z 0.4 I Fig 5.19 measured and modeled threshold... doping transformation procedure of modeling n-channel threshold voltage works very well for present day MOS technologies On the other hand Eq (5. 46) seems to work well for p-channel devices 5.2.2 Depletion Type Device As was pointed out earlier, depletion type MOSFETs (normally-on BC MOSFETs) conduct even at V,, = 0 V A cross-section of an n-channel depletion mode MOSFET is shown schematically in Figure... Length and Width 197 p-SUBSTRATE (Nb - " b s (a) Or- 1 E _- ' (b) (C) Fig 5.14 Yau charge sharing model (a) for calculating threshold voltage V,, in a short channel MOSFET and (b) calculation of X , from the triangle ABC, (c) condition when source and drain depletion boundaries meet and depletion width X,, reaches maximum value X i , where we have made use of Eqs (5.8) and (5.11) Now substituting Q6 for. .. pMOST and nMOST devices, a general expression for the threshold voltage can be written as (5 .63 ) + where the and - signs are for n- and p-channel devices respectively, and AV,, is the threshold voltage shift due to the channel implant of depth Xi term Vo(Ns, Xi) a correction term due to the threshold voltage The N,, is implant For a uniformly doped substrate (unimplanted channels), AV,, = V, = 0 For. .. equation for p-channel Vth6 [1 5- 161 where (5.44) Note that when NJ, = N,(X,, -Xi), the depletion charge at the surface (p-type) just balances the depletion charge in the substrate (n-type) Under this so called compensation condition, V,, = V,, - +Ai = VlhC.When V,, < VIhcwe have a surface channel device, we however, when V,, > VIhc have a buried channel device For n-well CMOS p-channel devices, VIhc - 1.0... 10 cm x Nb=15 6 x 1Ol6cm3 N 4 - 2 5 x 1 0'6cm-3 0 .6 - - 0.5> > 0.L -3 .0 2 6 10 I& I8 WIDTH (pm) (b) 2 Fig 5.21 (a) Aker's model for calculating threshold voltage of narrow width MOSFETs fabricated using the LOCOS isolation (b) Threshold voltage versus width for various channel doping Continuous lines are model based on Eq (5.85) while symbols are data points (After Akers et al [61 ]) 5.3 Threshold Voltage... Eqs (5.73 )-( 5.75) In order to obtain the best fit between the experimental data and calculated Vrh, a To arrive at Eq (5.74),we use the equation of an ellipse x z / a : + yZ/b: = 1, where Za, and 2b1 are given by Eq (5.72) 200 5 Threshold Voltage 06 I I I I w- z+ go n J> V,b'IV / - - , ,a0 03 - // -1 / VSb= ov / a 1 - (a) - $2 w (I)> LT kI 2.Or I I 0.0 , I , I to, = 150 A Wm=12.5pm I I , I ' - 1 7 LL . both pMOST and nMOST devices, a general expression for the threshold voltage can be written as (5 .63 ) where the + and - signs are for n- and p-channel devices respectively, and AV,,. + and the - signs are for n- and p-channel devices, respectively. This equation shows that the body-effect increases as the body factor y Y f dvfh - "sb 2Jm 5.2 Nonuniformly. criteria for strong inversion have been suggested for non-uniformly doped substrates [2], [33 ]-[ 36] . Some of these are: I. The classical criterion given by Eq. (5.12) is still used for non-uniformly