Journal of Modeling and Simulation of Microsystems, Vol. 2, No. 1, Pages 51-56, 1999. PHYSICS-BASED THRESHOLD VOLTAGE MODELING WITH REVERSE SHORT CHANNEL EFFECT K-Y Lim, X. Zhou, and Y. Wang School of Electrical & Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798. Abstract This paper presents a physic-based reverse short channel effect (RSCE) model for threshold voltage (V th ) modeling of deep submicron MOSFETs. Unlike those conventional empirically-based RSCE models, the proposed model is derived based on two Gaussian pile-up profiles located at the source and drain edges of a MOSFET. The model has a simple compact form that can be utilized to characterize the advanced halo-implant MOSFETs. A detailed comparison of the proposed RSCE model with the previously proposed model is also presented. The analytical model has been applied to, and verified with, experimental data of a 0.25-µm CMOS process for ten different gate lengths as well as various drain and substrate bias conditions. Keywords: RSCE, lateral non-uniform profile, threshold voltage, compact model, MOSFET 1. Introduction Advanced MOSFETs are non-uniformly doped as a result of complex process flow. Basically, non-uniform doping profiles can be categorized into vertical non-uniformity and lateral non-uniformity, as shown in Fig. 1. The vertical nonuniformity can be due to additional implantation for threshold voltage adjustment or for punchthrough prevention. On the other hand, lateral non-uniformity may be due to the intended pocket implantation for deep submicron technology or the unavoidable transient enhanced diffusion of boron impurity in nmos. Therefore, one of the key factors to model threshold voltage accurately is to model its non-uniform doping profile. Currently, there are many V th models [1-6] that are able to model the vertical non-uniform doping profile of a MOSFET. However, all the V th models for lateral non-uniform doping profiles for reverse short channel effect (RSCE) [7-12] are empirical, which are normally modeled by simply adding exponential functions to its long channel V th expression. Hence, the focus here is to transform the lateral 2-D pile-up profile across the channel to an effective doping expression that can be applied directly to the compact V th expression [7] for efficient circuit simulation. Despite that RSCE observed in V th - L g curve is solely due to the lateral doping nonuniformity, the complete V th model of advance MOSFETs should also include the other critical effects, such as drain induced barrier lowering, charge sharing effect as well as the effect of vertical non-uniformity. In the V th model that is presented in this paper, all short channel effects, except for the RSCE, are modeled as in a recent published article [7]. The model for vertical non-uniform doping profile of MOSFET has been explained in [1]. Fig. 1: MOSFET with both vertical and horizontal non-uniform doping profiles. In Section 2 of this paper, formulation of the proposed model is presented. Comparison of the model with the empirical hyperbolic cosine RSCE model [7] is discussed in Section 3, together with experimental verification of the proposed model. Finally, a brief summary is given in Section 4. exzhou@ntu.edu.sg Computational Publications, ISSN 1524-2021.
JMSM, Vol. 2, No. 1, Pages 53-56, 1999. 2. Model RSCE in nmosfets is mainly caused by the boron dopant pile-up phenomenon at the edge of the source and drain regions. Therefore, the basis of the model is to assume two Gaussian profiles at the source and drain edges. The Gaussian profile is expressed as: N p ( y) N pile exp[ ( y l β ) 2 ] (1) where y represents the distance across the channel and is the metallurgical channel length of the MOSFET. N pile and I β are the peak pile-up doping concentration and the characteristic length, respectively. Since the pile-up profile is due to boron redistribution along the channel after the post- LDD annealing process or due to direct pocket implant at the source and drain side, it can be assumed symmetrical for both the source and drain sides. With these conceptual pile-up profiles, as shown in Fig. 1, the profiles are summed up mathematically along the entire metallurgical channel length of the MOSFET. It is then divided by the metallurgical channel length to obtain an averaged effective concentration expression, as follows: Fig. 2: MEDICI simulated doping profiles across MOSFET channel for L g 0.24, 0.34, 0.5 and 1 µm. Eq. (2) merely gives an effective value of the non-uniform channel doping as a function of. However, RSCE is not quantified through its effective channel doping, but through the threshold voltage. Thus, to fully characterize RSCE, it is necessary to substitute the derived effective channel doping expression (2) into a physical short-channel V th model. The V th model used in this work is given as [13]: N eff y N exp pile --- l β 2 y + exp ---------------- l β 2 + N s dy 0 --------------------------------------------------------------------------------------------------------------------------------- λζ V th V FB + γ s + φ s0 V bs 1 -------- φ L s v bs + ------------------ eff v bs φ s v bs (3) N s l β + -------- N pile exp( t 2 ) dt exp( t 2 ) dt l β 0 0 l β (2) ζ 2ε si ------------- qn eff (3a) πerf ( L N s N eff l β ) + pile ------------------------------------- ( ) l β φ s0 2kT N eff --------- q ln ---------- n i (3b) where N s is the effective vertical non-uniform substrate doping without considering the lateral pile-up charge [1]. Notice that the pile-up terms at the source and drain edges are only differentiated by an translation term. The final effective concentration expression is an error function of its metallurgical channel length as well as the peak value and characteristic length of its pile-up profile. φ s φ s0 φ s 1 z φ s ------------------------- ( V cosh L -------- eff 2l α bi φ s0 ) -- 2 iv sinh z -------- -- ------- ds 2l α 2 cosh + --------------------------------- 2 sinh L -------- eff 2l α (3c) (3d) Fig. 2 shows the MEDICI-simulated doping profiles across the surface channel for four different devices with increasing channel length starting from 0.24 µm up to 1 µm. As shown in Fig. 2, when the channel length is long, such as in the case of L g 0.5 µm and 1 µm, their center channel doping profiles overlap each other, thus producing a threshold voltage independent of the channel length. But this is not true for the case when L g 0.24 µm, where its center channel doping profile is higher than its long-channel counterpart. Its center channel profile is higher because the two pile-up profiles at the source and drain edges overlap each other, thereby causing RSCE to be observed as the device shrinks. l α αφ ( s0 V bs ) 0.25 l β βφ ( s0 V bs ) 0.25 V bi φ s0 + V ds z ln ------------------------------------- V bi φ s0 (3e) (3f) (3g) The threshold voltage expression consists of three main terms, namely, the flat-band voltage, the surface potential and the voltage across the insulated gate dielectric. V FB, γ, n i, and 52
Physics-Based Threshold Voltage Modeling with Reverse Short Channel Effect V bi are the flat-band voltage, body-effect factor, intrinsic doping concentration, and built-in potential, respectively. φ s0 and φ s are the surface potentials without and with considering the barrier-lowering effect. φ s is the surface potential barrier lowering based on quasi-2d formulation, with l α being the characteristic length of the non-uniform surface potential profile. The square root term in the V th expression is based on charge sharing formulation. N pile, α, β, λ, i, and j are processdependent fitting parameters in the V th model. However, there are only two parameters that determine the RSCE, which are used to characterize the lateral non-uniform Gaussian profiles. They are the characteristic length l β (which determines the lateral spread of the pile-up) and the peak concentration N pile (which determines the amount of the pileup). As the channel length of MOSFET reduces, its effective doping value increases exponentially, causing a roll-up characteristic of the V th - L g curve at decreasing, which is known as the RSCE. 3. Discussion There exist various empirical RSCE models, which are introduced based on the anomalous RSCE trend observed in advanced MOSFETs. One of the empirical models that is very similar to the proposed model is explained in [7], and rewritten below: N pile N eff N + ------------------------------------------ s (4) cosh( ( 2l β )) Similar to the proposed effective channel-doping model, this empirical model has two important terms, namely, the effective vertical doping (N s ) and the lateral non-uniform pile-up. The pile-up term has two process-dependent fitting parameters, N pile and l β, which are same as the proposed model. However, the proposed model and the empirical model have different pile-up functionality. The proposed model is an error function for the horizontal non-uniformity whereas (4) employs a hyperbolic cosine function. Unlike the proposed model that assumes two Gaussian-shaped pile-up profiles at the source and drain edges, the empirical model is borrowed from the quasi-2d potential barrier lowering solution [7]. Fig. 3 compares the two N eff models. The solid lines illustrate the characteristic of the proposed N eff model for three increasing l β values (0.08; 0.1; 0.12 µm), whereas the dashed lines represent the characteristic of the hyperbolic cosine N eff model for the same set of l β. The N s terms in (2) and (4) are both fixed to 6x10 17 cm -3, and N pile is fixed to 2x10 17 cm -3. As seen from Fig. 3, the roll-up behavior of the proposed model begins at a longer channel length. Its roll-up rate is gradual with smaller initial roll-up rate followed by steeper roll-up rate when L g is below 1 µm. As for the empirical hyperbolic cosine model, it shows no roll-up until L g is below 1 µm, and a abrupt roll-up is observed. Therefore, the hyperbolic cosine model tends to give a more abrupt V th roll-up as compared to the proposed model. The parameters l β and N pile control the roll-up part of the V th - L g curve. When l β or N pile increases, larger RSCE is observed. The effect of l β and N pile on V th roll-up is illustrated in Figs. 6a and 6b. The two parameters contribute differently to the V th roll-up characteristic. When l β changes, there is no change at the roll-off portion of the V th - L g curve. The roll-off portion will only shift when N pile changes, as shown in Fig. 4b. This is because l β represents the characteristic length of the pile-up profile, which determines the spread of the pile-up profile. When l β increases, the two pile-up profile will meet at a longer channel length, thereby causing the onset of V th rollup occuring at a longer channel length. Once the two profiles meet, the amount additional pile-up charges will remain the same as long as N pile is unchanged, so the roll-off characteristic will be the same based on the principle of charge sharing. On the other hand, if N pile increases, there will be more pile-up charges, thereby causing a consistent shift in the roll-off portion of the V th - L g curve, but the onset of roll-up will remain the same. In Fig. 4a, the onset of V th roll-up for different l β is not clearly separated due to the intrinsic nature of the proposed model, which has a gradual roll-up characteristic. If similar l β changes are applied to the empirical RSCE model (4), its onset roll-up is clearly related to the l β value, as seen in Fig. 5a. Fig. 5a and Fig. 5b are reproduced from Fig. 2d and Fig. 2e of [7], with the same respective l β and N pile changes as in Figs. 4a and 4b. It is clearly seen from Fig. 5b that when N pile changes, the onset of roll-up is not altered, which confirms the physical representations of l β and N pile. Fig. 3: Effective channel doping against channel length for three different characteristic lengths. Solid lines: proposed RSCE model; Dashed lines: hyperbolic cosine model [7]. 53
JMSM, Vol. 2, No. 1, Pages 53-56, 1999. The derived model employs an error function instead of the empirical hyperbolic cosine function as proposed in [7]. Although it is less computationally efficient for error function as compared to hyperbolic cosine function, the proposed model has shown more accurate and physical results. Fig. 6a and Fig. 6b plot the same set of MEDICI-simulated V th data for three different characteristic lengths (l β 0.08, 0.1, and 0.12 µm) in three different symbols. The lines in Fig. 6a represent the newly proposed model, whereas the lines in Fig. 6b are the empirical V th model [7]. Although both can model the V th roll-up behavior, the new model shows a better match as compared to the hyperbolic cosine function, especially when the characteristic length is large. This is because the formulation of the model is based on the average of the individual local profiles, which is more accurate as the pile-up profile becomes more gradual. The empirical model is limited to pile-up profile that is abrupt with small characteristic length. Fig. 4: The proposed RSCE threshold voltage against channel length for l β variation and N pile variation. Fig. 6: Threshold voltage against channel length for three different characteristic lengths. proposed RSCE model; hyperbolic cosine model [7]. Symbols: MEDICI data, Lines: model data. Fig. 5: The empirical hyperbolic cosine RSCE threshold voltage against channel length for l β variation and N pile variation. Fig. 7a and Fig. 7b further verify the proposed RSCE V th model by comparing to the experimental data for a 0.25-µm CMOS technology with ten different channel lengths from 10 µm down to 0.2 µm. Four different Vbs conditions with low and high drain bias conditions are compared as shown in Figs. 54
Physics-Based Threshold Voltage Modeling with Reverse Short Channel Effect 7a and 7b, respectively. As is clearly shown, the proposed model can accurately model the actual experimental V th data. Fig. 7: Threshold voltage for various substrate biases. Symbols: experimental data, Lines: model data. Vds 0.1V; Vds 2.5V. 4. Conclusion In conclusion, the well-known RSCE has been characterized and modeled through the proposed N eff model. The model is developed based on two gradual Gaussian pile-up profiles and further reduced to a useful compact expression. It is more robust as compared to the previous proposed hyperbolic cosine model [7]. It is relatively easy to use and has good value to technology development and device modeling. References [1] K. Y. Lim, X. Zhou, "Modeling of Threshold Voltage with Non-uniform Substrate Doping," Proc. of the 1998 IEEE International Conference on Semiconductor Electronics (ICSE'98), Malaysia, 1998, pp. 27-31. [2] N. D. Arora, "Semi-empirical model for the threshold voltage of a double implanted MOSFET and its temperature dependence," Solid-State Electron., vol. 30, pp. 559-569, 1987. [3] C. Lallement, M. Bucher, and C. Enz, "Modelling and characterization of non-uniform substrate doping," Solid-State Electron., vol. 41, pp. 1857-1861, 1997. [4] W. Zhang and Z. Yang, "A new threshold voltage model for deep-submicron MOSFET's with non-uniform substrate dopings," 1997 Hong Kong Electron Devices Meeting, pp. 39-41, 1997. [5] P. Ratnam, C. Andre, and T. Salama, "A new approach to the modelling of non-uniformly doped short channel MOSFET's," IEEE Trans. Electron Devices, vol. 31, pp. 1289-1298, 1984. [6] D. A. Antoniadis, "Calculation of threshold voltage in non-uniformly doped MOSFET's," IEEE Trans. Electron Devices, vol. 31, pp. 303-307, 1984. [7] X. Zhou, K. Y. Lim, and D. Lim, "A General Approach to Compact Threshold Voltage Formulation base on 2- D Numerical Simulation and Experimental Correlation for Deep-Submicron ULSI Technology Development," IEEE Trans. Electron Devices, vol. 47, pp. 214-221, Jan. 2000. [8] N. D. Arora, and M. S. Sharma, "Modeling the Anomalous Threshold Voltage Behavior of Submicrometer MOSFET's," IEEE Electron Device Lett., vol. 13, pp. 92-94, Feb. 1992. [9] M. K. Khanna, M. C. Thomas, R. S. Gupta, and S. Haldar, "An Analytical Model for Anomalous Threshold Voltage Behavior of Short Channel MOSFET's," Solid-State Electron., vol. 41, pp. 1386-1388, 1997. [10] B. Yu, C. H. J. Wann, E. D. Nowak, K. Noda, and C. Hu, "Short-Channel Effect Improved by Lateral Channel-Engineering in Deep-Submicrometer MOSFET's," IEEE Trans. Electron Devices, vol. 44, pp. 627-633, Apr. 1997. [11] Y. Cheng, T. Sugii, K. Chen, and C. Hu, "Modeling of Small Size MOSFETs with Reverse Short Channel and Narrow Width Effects for Circuit Simulation," Solid-State Electron., vol. 41, pp. 1227-1231, 1997. [12] Y. Cheng, T. Sugii, K. Chen, Z. Liu, M. C. Jeng, and C. Hu, "Modeling Reverse Short Channel and Narrow Width Effects in Small Size MOSFET's for Circuit Simulation," SISPAD'97, pp. 249-252, 1997. [13] K. Y. Lim, X. Zhou, and D. Lim, "A predictive lengthdependent saturation current model based on accurate threshold voltage modeling," Proc. MSM99, Puerto Rico, Apr. 1999, pp. 423-426. Received in Cambridge, MA, USA, 19th February 2000 Paper 2/01779 55
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