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A new improved model for subthreshold slope for submicron MOSFETs
MEJ 668
Microelectronics Journal Microelectronics Journal 31 (2000) 105–111 www.elsevier.com/locate/mejo
A new improved model for subthreshold slope for submicron MOSFETs S.B. Thakare 1, A.K. Dutta* Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208 016, India Accepted 3 May 1999
Abstract A new improved analytical model for the subthreshold slope (ss) for submicron MOSFETs, suitable for analog circuit CAD work is presented in this paper. An existing ss model (independent of drain voltage) is taken up and is modified in order to include the effect of the drain voltage. Additionally, it accounts for the effects of the effective channel length and the body voltage on the ss. An attempt is also made to represent the fudge factor, used widely in existing literature in the expression for the characteristic length, by a more physical representation than was done before. This ss model has been put in an existing drain current model in order to obtain the dc characteristics. The simulated results are compared with those reported experimentally for MOSFETs having channel lengths less than hundred nanometers, and the results show an excellent match between the two. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Subthreshold slope; Submicron MOSFETs; DIBL parameter
1. Introduction With the rapid progress made in the design of low-power mixed-signal circuits in recent times, an accurate modeling of the subthreshold region of operation for submicron MOSFETs, which fully exploits their potential to be operated in low-power mode, along with an accurate modeling of the subthreshold slope (ss) h have become extremely crucial. For long channel MOSFETs, h can be taken to be a constant. However, with rapid shrinking of device sizes, several second-order effects take place and in reality it becomes a function of several parameters, e.g. the effective channel length Leff, the drain voltage VDS, the body bias VSB, etc. Several 1D and 2D models for submicron MOSFETs have been presented in literature in recent times [1–6], however, none of these present an accurate model for h . Recently, a ss model is proposed by Kang et al. [7], which has been developed from the capacitor equivalent circuit of short channel MOSFETs in the subthreshold region of operation. In [7], due to some invalid assumptions made in the derivation for h , it comes out to be independent of VDS. However, from experimental results, it has been observed that h is a strong function not only of VDS (increasing with an increase in VDS), but also of VSB and Leff. In this work, we have analytically modeled the effects of these * Corresponding author. Tel.: 1 91-512-597661; fax: 1 91-512-590063. E-mail address: [email protected], [email protected] (A.K. Dutta) 1 Present address: Alliance Semiconductor (India) Pvt. Ltd., 39 Langford Road, Bangalore 560 025, India.
parameters on h , and have developed a new and improved analytical expression for it. This ss model is used in the drain current model of Deshpande and Dutta [8] to obtain the DC characteristics. The simulated results have been compared with the experimentally obtained ones reported in literature for MOSFETs having Leff ranging from 1.63 to 0.075 mm, and a good match is observed between the two. The model development is discussed in Section 2, the results are presented in Section 3, and Section 4 highlights the summary and conclusion.
2. Model development In this work, the following expression for the threshold voltage VTs for short-channel MOSFETs, as given by BSIM [1] is used p VTs VFB 1 2fF 1 g 2fF 1 VSB 2 G 2fF 1 VSB 2 sVDS ;
1
where VFB is the flatband voltage, f F the bulk potential, g the body effect coefficient, G the nonuniform doping effect parameter, and s the DIBL parameter (function of Leff and VSB). The capacitance equivalent circuit for short-channel MOSFETs at subthreshold proposed by Kang et al. [7] introduces two new capacitances Csc and Cdc (in addition to the oxide and the depletion region capacitances), which couple the source and the drain to the channel, respectively, given
0026-2692/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(99)00096-8
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p p Fig. 1. Plot of (DVTs/DVDS)/ 2fF 1 VSB versus 2fF 1 VSB for the four devices (obtained from the data of Lee and Min [9]).
by Csc 1Si k=x0
and
Cdc 1Si k= Leff 2 x0 ;
2
where 1 Si is the permittivity of Si, k a proportionality constant, and x0 the position along the channel (measured from the source) where the surface potential is minimum. The values of these capacitances increase as Leff is reduced, and so does the value of s . This implies that the DIBL effect is more pronounced in short-channel MOSFETs, which is obvious. This makes it important to model this effect accurately using the parameter s , which has been expressed by Kang et al. [7] as p s s 2fF 1 VSB 2 S 2fF 1 VSB ; 3 where s and S are the fitting parameters. The final expression for the ss h obtained by Kang et al. [7] using the capacitance
coupling theory is given by " # g h 1 1 p 2 G 1 1 2:7s: 2 1:5fF 1 VSB
4
Although x0 is known to be a complex function of Leff and VDS; however, in the derivation of Eq. (4) [7], it has been assumed that for large values of VDS, x0 converges rapidly to Leff/3 and becomes independent of VDS, which makes h independent of VDS. This is incorrect, as has been proved by experimental data on short channel devices reported in the literature, which show changes in h with a change in VDS. In reality, x0 is also a function of VSB and Leff. Thus, if these variations can be effectively modeled in the expression for x0, then the values of Cdc and Csc would also be altered, with each having a specific dependence on VDS,
Fig. 2. Simulated results of the ideality factor model of Kang et al. [7] (Eq. (4)) at VDS 0:1 V for Leff 0:33 mm. The measured values [9] are also shown.
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111
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Fig. 3. Plot of x0/Leff versus Leff for VSB 0 V and VDS 3 V.
Fig. 4. Plot of h f versus VSB for Leff 0:33 mm. The measured values [9] also shown.
VSB and Leff. As h is a function of Cdc and Csc [7], it would also have an explicit dependence on these three parameters, which the previous authors have failed to model. In order to determine the variation in the surface potential as a function of position in the channel, the expression given Table 1 The extracted values of the parameters used in the ss model given by Eq. (4) for the four devices Leff (mm)
g (V 1/2)
G
s (V 21/2)
S (V 21)
1.43 0.63 0.43 0.33
0.550498 0.590389 0.604011 0.509722
20.027129 20.005396 0.024737 0.023842
0.014307 0.029623 0.045488 0.067303
0.004690 0.007481 0.009397 0.015221
by Liu et al. [10] is used in this work, where the characteristic length l is defined by [10] q l 1Si Tox Xdep = 1ox hf ;
5
where Tox and 1 ox are the thickness and permittivity of the oxide, respectively. The depletion layer thickness Xdep used in the above expression has been assumed to be uniform along the channel [10]. However, in reality, it is a function of VDS and Leff, more so for short channel devices. Therefore, in order to compensate for this invalid assumption (of constant Xdep), another factor h f has been introduced in the expression for l (Eq. (5)), which is basically a fitting parameter referred to as the fudge factor. The term Xdep/h f
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Fig. 5. Simulated results of the improved ideality factor model (Eq. (8)) at VDS 0:1 V for Leff 0:33 mm. The measured values [9] are also shown.
may be considered as an average of the depletion layer thickness along the channel. It has been observed that at subthreshold, the surface potential is almost uniform along the channel for long channel devices. Hence, the value of h f can be taken to be equal to unity. However, for short channel devices (having submicron channel lengths), the portion of the channel where the surface potential stays uniform is quite small, and hence, the value of h f is correspondingly higher. As the surface potential profile along the channel is a function of both Leff and VSB, then h f also is a function of both these parameters. It has been observed in this work that the value of h f decreases drastically with increasing VSB, which implies that the depletion width can be modulated significantly by the body voltage. This is a true physical picture for short channel MOSFETs. It has also been observed that h f is a
monotonically decreasing function of Leff. Thus, in this work, the dependence of h f on VSB and Leff is modeled by the following equation:
hf hf0 exp 2k 0 VSB ;
6
where k 0 and h f0 are fitting parameters. The parameter h f0 is the value of h f for VSB 0 for a particular channel length, which strongly depends upon Leff (decreasing with an increase in Leff). The minimum surface potential position x0 in the channel has been expressed as [10] L 1 V 2 csL 1 VDS x0 eff 2 ln bi ; 7 2 2 Vbi 2 csL where Vbi is the built-in voltage of the drain/source– substrate junction, and c sL the long channel surface potential [10]. Eq. (7) explicitly shows the dependence of x0 on
Fig. 6. Simulated results of the improved ideality factor model (Eq. (8)) at VDS 3 V for Leff 0:33 mm. The measured values [9] are also shown.
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111 Table 2 The extracted values of the parameters used in the model of the fudge factor h f (Eq. (6)) for the four devices Leff (mm)
h f0
k 0 (V 21)
1.43 0.63 0.43 0.33
1.09 1.15 1.54 2.92
0.75 0.74 0.73 0.75
Leff and VDS. Also, the dependence of x0 on VSB is implicit through the term l (Eq. (5)) (through the parameter h f [Eq. (6)]). Thus, this expression for x0 clearly accounts for its dependence on all the three parameters, i.e. Leff, VDS and VSB. Following the algorithm of Kang et al. [7], the following improved expression for the ss h has been developed in this work: " # g x h 1 1 p 2 G 1 1 4 1 2 0 s ; Leff 2 1:5fF 1 VSB 8 where x0 is given by Eq. (7). This expression for the ss h has been used in the drain current model developed by Deshpande and Dutta [8] in order to obtain the dc transfer characteristics. 3. Results and discussion We have the ID 2 VGS characteristics for four MOSFETs having Leff of 1.43, 0.63, 0.43 and 0.33 mm, substrate doping ˚ for VSB 0; 1; 2; 3 and of 1:6 × 1017 cm 23 and Tox 100 A 4 V, and for VDS 0:1 and 3 V [9]. The threshold voltages are extracted from these characteristics using the gm,max/3 method for VDS 0:1 V, and the constant current threshold method for VDS 3 V. From the threshold voltage data, the
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values of the parameters s, S, g and G are extracted, which yield the value of the parameter s , and this is used to determine the ss from Eq. (4) (the model of Kang et al. [7], which is independent of VDS). In order topfind the variation in the value of s with respect to the term 2fF 1 VSB for different channel lengths, the term DVTs =DVDS (i.e. the change in VTs with respect to the change in VDS) obtained from the ID 2 VGS p characteristics [9] is plotted with respect to 2fF 1 VSB in Fig. 1 for the four devices. The values of parameters s and S have been obtained from the curve fitting of Eq. (3) with Fig. 1. It has been observed that the value of s (and ultimately that of s ) increases (which implies that the DIBL effect becomes more pronounced) as the channel length is reduced. The behavior of DVTs follows the same pattern as suggested by Liu et al. [10]. The values of parameters g and G have been found by the curve fitting of Eq. (1) with VTs data for VDS 0:1 V. The extracted values of the model parameters g , G , s and S for the four devices are listed in Table 1, and these are used in Eq. (4) to calculate the ideality factor h as a function of VSB. This characteristic is shown in Fig. 2 for Leff 0:33 mm, along with the measured values [9] for VDS 0:1 V. The simulated results are observed to grossly overestimate the measured results. Similar mismatch was found for the other three devices as well. This is expected as the dependence of h on VSB was not modeled properly by the earlier authors [7]. The assumption made by Kang et al. [7] that x0 converges rapidly to Leff/3 for large values of VDS (which makes h independent of VDS) does not quite hold true for short channel devices, because for these cases the applied drain voltage modulates the surface potential profile quite markedly, and hence, x0 may not converge to Leff/3, but instead becomes a strong function of VDS. In order to validate this claim, the surface potential calculated from the expression
Fig. 7. Comparison of the ss calculated from our model (Eq. (8)) with that obtained from the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device.
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Fig. 8. Comparison of the ID 2 VGS characteristics in the subthreshold region calculated from our model with the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device.
given by Liu et al. [10] was plotted as a function of position in the channel for VDS 3 V, and it was observed that for relatively longer channel MOSFETs (e.g. for Leff 1:43 mm), it was almost uniform along the channel. This implies that x0 is closer to Leff/2 than to Leff/3. Values of x0/Leff for the four devices are obtained from the expression given by Eq. (7), and these are plotted as a function of Leff in Fig. 3 for VDS 3 and VSB 0 V. From this figure, it is obvious that x0 does not converge to Leff/3 for large values of VDS for any of the devices considered, which validates the claim made earlier. As has been explained in the previous section, the parameter h f appearing in the expression for the characteristic length l (Eq. (5)) is a function of both Leff and VSB. From a comparison between the results obtained from the ss model (Eq. (8)) and the measured data [9], the values of h f have been found for each Leff and VSB, and the values of the fitting parameters h f0 and k 0 are extracted for each of the four devices, which are given in Table 2. The parameter h f0 is a strong function of Leff, and takes on different values for different channel lengths, ranging from 1.09 (for Leff 1:43 mm) to 2.92 (for Leff 0:33 mm). The coefficient k 0 is almost constant with an approximate value of 0.75 for all devices. The values of the parameter h f are plotted as a function of VSB in Fig. 4 for Leff 0:33 mm, along with the measured values [9]. An excellent match between the two is seen from the figure. Similar match has been found for the other three devices as well. This model for h f is used in the expression for l in order to calculate x0. The simulated results for the ss obtained from our model (Eq. (8)) are plotted as a function of VSB for Leff 0:33 mm in Figs. 5 and 6 for VDS 0:1 and 3 V, respectively, along with the experimentally measured data [9] for the sake of comparison. From these figures, it can be clearly seen that our model gives a much better fit to the
measured data than the original model [7], which can be found by comparing these figures with Fig. 2. Thus, our earlier assertion that x0 and thus the ideality factor h are functions of VDS, VSB and Leff are substantiated. Incidentally, good match was observed between the simulated and the measured results for the other three devices as well, however, these are not presented here for brevity. In order to obtain the dc characteristics, the ss model developed in this work has been used in the drain current model of Deshpande and Dutta [8]. The simulated results of our ss model and the ID 2 VGS characteristics were compared with the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device, having ˚ for VDS 0:05 and 1.55 V. The values of the Tox 35 A model parameters g , G , s, and S are extracted from the threshold voltage data obtained from these characteristics, and these are used to compute the ss, which are plotted in Fig. 7 along with the experimentally obtained values [11] for the sake of comparison. It is observed that the match between the two is very good. The ID 2 VGS characteristics simulated in the subthreshold region for the same device are shown in Fig. 8, where the experimentally obtained results [11] are also shown for comparison, which also shows an excellent match between the two. Simulated results have also been obtained for another sub-0.1 mm MOSFET ˚ ) reported by (channel length 0:075 mm and Tox 70 A Rittenhouse et al. [12], which also showed an excellent match with the experimental data, however, these are not presented here for brevity.
4. Summary and conclusion A new improved analytical model for the ss, suitable for analog circuit CAD work, has been proposed in this work,
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111
which accounts for its dependence on VDS, VSB and Leff. An attempt has also been made to express the fudge factor h f appearing in the expression for the characteristic length l by a more physical representation than was done before. The ss model developed in this work was used in the drain current model of Deshpande and Dutta [8], and the ID 2 VGS characteristics for different submicron devices were obtained. These were compared with the experimental data, and the results showed an excellent match between the two. Acknowledgements
[4]
[5]
[6] [7]
[8]
We are extremely grateful and indebted to Drs K. Lee and K.S. Min for sharing their experimental data with us.
[9] [10]
References
[11]
[1] B.J. Sheu, D.L. Scharfetter, P.K. Ko, M.C. Jeng, BSIM: Berkeley short-channel IGFET model for MOS transistors, IEEE J. Sol. St. Circ. 22 (1987) 558–565. [2] M. Shur, T.A. Fjeldly, T. Ytterdal, K. Lee, Unified MOSFET model, Sol. St. Elect. 35 (1992) 1795–1802. [3] N.D. Arora, R. Rios, C.L. Huang, K. Raol, PCIM: a physically based
[12]
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continuous short channel IGFET model for circuit simulation, IEEE Tran. Elect. Dev. 41 (1994) 988–997. A. Chatterjee, C.F. Machala III, P. Young, A submicron DC MOSFET model for simulation of analog circuits, IEEE Tran. Comp.-Aided Des. 14 (1995) 1193–1207. Y.A. El-Mansy, A.R. Boothroyd, A simple 2D model for IGFET operation in the saturation region, IEEE Tran. Elect. Dev. 24 (1977) 254–262. S. Hariharan, A.K. Dutta, An improved quasi two-dimensional model for short-channel MOSFETs, Sol. St. Elect. 38 (1995) 861–866. S.W. Kang, K.S. Min, K. Lee, Parametric expression of subthreshold slope using threshold voltage parameters for MOSFET statistical modeling, IEEE Tran. Elect. Dev. 43 (1996) 1382–1386. D.R Deshpande, A.K. Dutta, A new unified model for submicron MOSFETs, Microelectronics J. 29 (1998) 565–570. K. Lee, K.S. Min, private communication. Z.H. Liu, C. Hu, J.H. Huang, T.Y. Chan, M.C. Jeng, P.K. Ko, Y.C. Cheng, Threshold voltage model for deep submicrometer MOSFETs, IEEE Tran. Elect. Dev. 40 (1993) 86–94. Y. Mii, S. Rishton, Y. Taur, D. Kern, T. Lee, K.A. Jenkins, D. Quinlan, T. Brown Jr., D. Danner, F. Sewell, M. Polcari, Experimental high peroformance sub-0.1 mm channel nMOSFETs, IEEE Elect. Dev. Lett. 15 (1994) 28–30. G.E. Rittenhouse, W.H. Mansfield, A. Kornbilt, R.A. Cirelli, D. Tomes, G.K. Celler, Sub-0.1 mm NMOS transistors fabricated using laser-plasma point-source X-ray lithography, IEEE Elect. Dev. Lett. 16 (1995) 322–324.
Microelectronics Journal Microelectronics Journal 31 (2000) 105–111 www.elsevier.com/locate/mejo
A new improved model for subthreshold slope for submicron MOSFETs S.B. Thakare 1, A.K. Dutta* Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208 016, India Accepted 3 May 1999
Abstract A new improved analytical model for the subthreshold slope (ss) for submicron MOSFETs, suitable for analog circuit CAD work is presented in this paper. An existing ss model (independent of drain voltage) is taken up and is modified in order to include the effect of the drain voltage. Additionally, it accounts for the effects of the effective channel length and the body voltage on the ss. An attempt is also made to represent the fudge factor, used widely in existing literature in the expression for the characteristic length, by a more physical representation than was done before. This ss model has been put in an existing drain current model in order to obtain the dc characteristics. The simulated results are compared with those reported experimentally for MOSFETs having channel lengths less than hundred nanometers, and the results show an excellent match between the two. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Subthreshold slope; Submicron MOSFETs; DIBL parameter
1. Introduction With the rapid progress made in the design of low-power mixed-signal circuits in recent times, an accurate modeling of the subthreshold region of operation for submicron MOSFETs, which fully exploits their potential to be operated in low-power mode, along with an accurate modeling of the subthreshold slope (ss) h have become extremely crucial. For long channel MOSFETs, h can be taken to be a constant. However, with rapid shrinking of device sizes, several second-order effects take place and in reality it becomes a function of several parameters, e.g. the effective channel length Leff, the drain voltage VDS, the body bias VSB, etc. Several 1D and 2D models for submicron MOSFETs have been presented in literature in recent times [1–6], however, none of these present an accurate model for h . Recently, a ss model is proposed by Kang et al. [7], which has been developed from the capacitor equivalent circuit of short channel MOSFETs in the subthreshold region of operation. In [7], due to some invalid assumptions made in the derivation for h , it comes out to be independent of VDS. However, from experimental results, it has been observed that h is a strong function not only of VDS (increasing with an increase in VDS), but also of VSB and Leff. In this work, we have analytically modeled the effects of these * Corresponding author. Tel.: 1 91-512-597661; fax: 1 91-512-590063. E-mail address: [email protected], [email protected] (A.K. Dutta) 1 Present address: Alliance Semiconductor (India) Pvt. Ltd., 39 Langford Road, Bangalore 560 025, India.
parameters on h , and have developed a new and improved analytical expression for it. This ss model is used in the drain current model of Deshpande and Dutta [8] to obtain the DC characteristics. The simulated results have been compared with the experimentally obtained ones reported in literature for MOSFETs having Leff ranging from 1.63 to 0.075 mm, and a good match is observed between the two. The model development is discussed in Section 2, the results are presented in Section 3, and Section 4 highlights the summary and conclusion.
2. Model development In this work, the following expression for the threshold voltage VTs for short-channel MOSFETs, as given by BSIM [1] is used p VTs VFB 1 2fF 1 g 2fF 1 VSB 2 G 2fF 1 VSB 2 sVDS ;
1
where VFB is the flatband voltage, f F the bulk potential, g the body effect coefficient, G the nonuniform doping effect parameter, and s the DIBL parameter (function of Leff and VSB). The capacitance equivalent circuit for short-channel MOSFETs at subthreshold proposed by Kang et al. [7] introduces two new capacitances Csc and Cdc (in addition to the oxide and the depletion region capacitances), which couple the source and the drain to the channel, respectively, given
0026-2692/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(99)00096-8
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p p Fig. 1. Plot of (DVTs/DVDS)/ 2fF 1 VSB versus 2fF 1 VSB for the four devices (obtained from the data of Lee and Min [9]).
by Csc 1Si k=x0
and
Cdc 1Si k= Leff 2 x0 ;
2
where 1 Si is the permittivity of Si, k a proportionality constant, and x0 the position along the channel (measured from the source) where the surface potential is minimum. The values of these capacitances increase as Leff is reduced, and so does the value of s . This implies that the DIBL effect is more pronounced in short-channel MOSFETs, which is obvious. This makes it important to model this effect accurately using the parameter s , which has been expressed by Kang et al. [7] as p s s 2fF 1 VSB 2 S 2fF 1 VSB ; 3 where s and S are the fitting parameters. The final expression for the ss h obtained by Kang et al. [7] using the capacitance
coupling theory is given by " # g h 1 1 p 2 G 1 1 2:7s: 2 1:5fF 1 VSB
4
Although x0 is known to be a complex function of Leff and VDS; however, in the derivation of Eq. (4) [7], it has been assumed that for large values of VDS, x0 converges rapidly to Leff/3 and becomes independent of VDS, which makes h independent of VDS. This is incorrect, as has been proved by experimental data on short channel devices reported in the literature, which show changes in h with a change in VDS. In reality, x0 is also a function of VSB and Leff. Thus, if these variations can be effectively modeled in the expression for x0, then the values of Cdc and Csc would also be altered, with each having a specific dependence on VDS,
Fig. 2. Simulated results of the ideality factor model of Kang et al. [7] (Eq. (4)) at VDS 0:1 V for Leff 0:33 mm. The measured values [9] are also shown.
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111
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Fig. 3. Plot of x0/Leff versus Leff for VSB 0 V and VDS 3 V.
Fig. 4. Plot of h f versus VSB for Leff 0:33 mm. The measured values [9] also shown.
VSB and Leff. As h is a function of Cdc and Csc [7], it would also have an explicit dependence on these three parameters, which the previous authors have failed to model. In order to determine the variation in the surface potential as a function of position in the channel, the expression given Table 1 The extracted values of the parameters used in the ss model given by Eq. (4) for the four devices Leff (mm)
g (V 1/2)
G
s (V 21/2)
S (V 21)
1.43 0.63 0.43 0.33
0.550498 0.590389 0.604011 0.509722
20.027129 20.005396 0.024737 0.023842
0.014307 0.029623 0.045488 0.067303
0.004690 0.007481 0.009397 0.015221
by Liu et al. [10] is used in this work, where the characteristic length l is defined by [10] q l 1Si Tox Xdep = 1ox hf ;
5
where Tox and 1 ox are the thickness and permittivity of the oxide, respectively. The depletion layer thickness Xdep used in the above expression has been assumed to be uniform along the channel [10]. However, in reality, it is a function of VDS and Leff, more so for short channel devices. Therefore, in order to compensate for this invalid assumption (of constant Xdep), another factor h f has been introduced in the expression for l (Eq. (5)), which is basically a fitting parameter referred to as the fudge factor. The term Xdep/h f
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Fig. 5. Simulated results of the improved ideality factor model (Eq. (8)) at VDS 0:1 V for Leff 0:33 mm. The measured values [9] are also shown.
may be considered as an average of the depletion layer thickness along the channel. It has been observed that at subthreshold, the surface potential is almost uniform along the channel for long channel devices. Hence, the value of h f can be taken to be equal to unity. However, for short channel devices (having submicron channel lengths), the portion of the channel where the surface potential stays uniform is quite small, and hence, the value of h f is correspondingly higher. As the surface potential profile along the channel is a function of both Leff and VSB, then h f also is a function of both these parameters. It has been observed in this work that the value of h f decreases drastically with increasing VSB, which implies that the depletion width can be modulated significantly by the body voltage. This is a true physical picture for short channel MOSFETs. It has also been observed that h f is a
monotonically decreasing function of Leff. Thus, in this work, the dependence of h f on VSB and Leff is modeled by the following equation:
hf hf0 exp 2k 0 VSB ;
6
where k 0 and h f0 are fitting parameters. The parameter h f0 is the value of h f for VSB 0 for a particular channel length, which strongly depends upon Leff (decreasing with an increase in Leff). The minimum surface potential position x0 in the channel has been expressed as [10] L 1 V 2 csL 1 VDS x0 eff 2 ln bi ; 7 2 2 Vbi 2 csL where Vbi is the built-in voltage of the drain/source– substrate junction, and c sL the long channel surface potential [10]. Eq. (7) explicitly shows the dependence of x0 on
Fig. 6. Simulated results of the improved ideality factor model (Eq. (8)) at VDS 3 V for Leff 0:33 mm. The measured values [9] are also shown.
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111 Table 2 The extracted values of the parameters used in the model of the fudge factor h f (Eq. (6)) for the four devices Leff (mm)
h f0
k 0 (V 21)
1.43 0.63 0.43 0.33
1.09 1.15 1.54 2.92
0.75 0.74 0.73 0.75
Leff and VDS. Also, the dependence of x0 on VSB is implicit through the term l (Eq. (5)) (through the parameter h f [Eq. (6)]). Thus, this expression for x0 clearly accounts for its dependence on all the three parameters, i.e. Leff, VDS and VSB. Following the algorithm of Kang et al. [7], the following improved expression for the ss h has been developed in this work: " # g x h 1 1 p 2 G 1 1 4 1 2 0 s ; Leff 2 1:5fF 1 VSB 8 where x0 is given by Eq. (7). This expression for the ss h has been used in the drain current model developed by Deshpande and Dutta [8] in order to obtain the dc transfer characteristics. 3. Results and discussion We have the ID 2 VGS characteristics for four MOSFETs having Leff of 1.43, 0.63, 0.43 and 0.33 mm, substrate doping ˚ for VSB 0; 1; 2; 3 and of 1:6 × 1017 cm 23 and Tox 100 A 4 V, and for VDS 0:1 and 3 V [9]. The threshold voltages are extracted from these characteristics using the gm,max/3 method for VDS 0:1 V, and the constant current threshold method for VDS 3 V. From the threshold voltage data, the
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values of the parameters s, S, g and G are extracted, which yield the value of the parameter s , and this is used to determine the ss from Eq. (4) (the model of Kang et al. [7], which is independent of VDS). In order topfind the variation in the value of s with respect to the term 2fF 1 VSB for different channel lengths, the term DVTs =DVDS (i.e. the change in VTs with respect to the change in VDS) obtained from the ID 2 VGS p characteristics [9] is plotted with respect to 2fF 1 VSB in Fig. 1 for the four devices. The values of parameters s and S have been obtained from the curve fitting of Eq. (3) with Fig. 1. It has been observed that the value of s (and ultimately that of s ) increases (which implies that the DIBL effect becomes more pronounced) as the channel length is reduced. The behavior of DVTs follows the same pattern as suggested by Liu et al. [10]. The values of parameters g and G have been found by the curve fitting of Eq. (1) with VTs data for VDS 0:1 V. The extracted values of the model parameters g , G , s and S for the four devices are listed in Table 1, and these are used in Eq. (4) to calculate the ideality factor h as a function of VSB. This characteristic is shown in Fig. 2 for Leff 0:33 mm, along with the measured values [9] for VDS 0:1 V. The simulated results are observed to grossly overestimate the measured results. Similar mismatch was found for the other three devices as well. This is expected as the dependence of h on VSB was not modeled properly by the earlier authors [7]. The assumption made by Kang et al. [7] that x0 converges rapidly to Leff/3 for large values of VDS (which makes h independent of VDS) does not quite hold true for short channel devices, because for these cases the applied drain voltage modulates the surface potential profile quite markedly, and hence, x0 may not converge to Leff/3, but instead becomes a strong function of VDS. In order to validate this claim, the surface potential calculated from the expression
Fig. 7. Comparison of the ss calculated from our model (Eq. (8)) with that obtained from the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device.
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Fig. 8. Comparison of the ID 2 VGS characteristics in the subthreshold region calculated from our model with the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device.
given by Liu et al. [10] was plotted as a function of position in the channel for VDS 3 V, and it was observed that for relatively longer channel MOSFETs (e.g. for Leff 1:43 mm), it was almost uniform along the channel. This implies that x0 is closer to Leff/2 than to Leff/3. Values of x0/Leff for the four devices are obtained from the expression given by Eq. (7), and these are plotted as a function of Leff in Fig. 3 for VDS 3 and VSB 0 V. From this figure, it is obvious that x0 does not converge to Leff/3 for large values of VDS for any of the devices considered, which validates the claim made earlier. As has been explained in the previous section, the parameter h f appearing in the expression for the characteristic length l (Eq. (5)) is a function of both Leff and VSB. From a comparison between the results obtained from the ss model (Eq. (8)) and the measured data [9], the values of h f have been found for each Leff and VSB, and the values of the fitting parameters h f0 and k 0 are extracted for each of the four devices, which are given in Table 2. The parameter h f0 is a strong function of Leff, and takes on different values for different channel lengths, ranging from 1.09 (for Leff 1:43 mm) to 2.92 (for Leff 0:33 mm). The coefficient k 0 is almost constant with an approximate value of 0.75 for all devices. The values of the parameter h f are plotted as a function of VSB in Fig. 4 for Leff 0:33 mm, along with the measured values [9]. An excellent match between the two is seen from the figure. Similar match has been found for the other three devices as well. This model for h f is used in the expression for l in order to calculate x0. The simulated results for the ss obtained from our model (Eq. (8)) are plotted as a function of VSB for Leff 0:33 mm in Figs. 5 and 6 for VDS 0:1 and 3 V, respectively, along with the experimentally measured data [9] for the sake of comparison. From these figures, it can be clearly seen that our model gives a much better fit to the
measured data than the original model [7], which can be found by comparing these figures with Fig. 2. Thus, our earlier assertion that x0 and thus the ideality factor h are functions of VDS, VSB and Leff are substantiated. Incidentally, good match was observed between the simulated and the measured results for the other three devices as well, however, these are not presented here for brevity. In order to obtain the dc characteristics, the ss model developed in this work has been used in the drain current model of Deshpande and Dutta [8]. The simulated results of our ss model and the ID 2 VGS characteristics were compared with the experimental data reported by Mii et al. [11] for a 0.09 mm channel length device, having ˚ for VDS 0:05 and 1.55 V. The values of the Tox 35 A model parameters g , G , s, and S are extracted from the threshold voltage data obtained from these characteristics, and these are used to compute the ss, which are plotted in Fig. 7 along with the experimentally obtained values [11] for the sake of comparison. It is observed that the match between the two is very good. The ID 2 VGS characteristics simulated in the subthreshold region for the same device are shown in Fig. 8, where the experimentally obtained results [11] are also shown for comparison, which also shows an excellent match between the two. Simulated results have also been obtained for another sub-0.1 mm MOSFET ˚ ) reported by (channel length 0:075 mm and Tox 70 A Rittenhouse et al. [12], which also showed an excellent match with the experimental data, however, these are not presented here for brevity.
4. Summary and conclusion A new improved analytical model for the ss, suitable for analog circuit CAD work, has been proposed in this work,
S.B. Thakare, A.K. Dutta / Microelectronics Journal 31 (2000) 105–111
which accounts for its dependence on VDS, VSB and Leff. An attempt has also been made to express the fudge factor h f appearing in the expression for the characteristic length l by a more physical representation than was done before. The ss model developed in this work was used in the drain current model of Deshpande and Dutta [8], and the ID 2 VGS characteristics for different submicron devices were obtained. These were compared with the experimental data, and the results showed an excellent match between the two. Acknowledgements
[4]
[5]
[6] [7]
[8]
We are extremely grateful and indebted to Drs K. Lee and K.S. Min for sharing their experimental data with us.
[9] [10]
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