Effects Of The Inversion Layer Centroid On MOSFET Behavior

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 11, NOVEMBER 1997

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Effects of the Inversion Layer Centroid on MOSFET Behavior Juan A. L´opez-Villanueva, Pedro Cartujo-Casinello, Jesus Banqueri, Member, IEEE, F. G´amiz, and Salvador Rodr´ıguez

Abstract— The effects of the average inversion-layer penetration, which are termed the inversion-layer centroid, on the inversion-charge density and the gate-to-channel capacitance have been analyzed. The quantum model has been used, and a variety of data have been obtained by self-consistently solving the Poisson and Schr¨odinger equations. An empirical expression for the centroid position that is valid for a wide range of electrical and technological variables has been obtained and has been applied to accurately model the inversion-layer density and capacitance.

I. INTRODUCTION

Q

UANTIZATION of the transverse electron motion in the inversion layer of a metal-oxide-semiconductor (MOS) transistor has been widely studied in the last 30 years. An excellent review can be found in [1]. A remarkable consequence for MOSFET behavior is the fact that electron density does not reach its maximum at the oxide-semiconductor interface but instead inside the semiconductor, while it almost vanishes right at the interface. At first, the main interest on quantum effects focused on the two-dimensional (2-D) properties of the electron gas, which were apparent mainly at low temperatures since multisubband occupation is produced at 300 K. Nevertheless, the extension of the electron density inside the semiconductor and its effects on capacitance were noted early on by Pals, even at 300 K [2]. Two consequences of this fact that are important for device behavior are that 1) the electric potential value at the interface is greater than that predicted by the classical model and 2) when the gate voltage is varied, the main modification of electron charge at the semiconductor is produced away from the interface, thus leading to reduced capacitance and transconductance [3]. However, although quantum effects were known early on, they were not seriously considered by the modeling community until very recently, as classical models worked reasonably well with previous transistor generations. In those transistors, gate oxides were relatively thick, and therefore, most of the gate voltage drop was produced at the oxide, making the error in the interface potential to be relatively unimportant. Furthermore, the average distance of the electron density from the interface was small compared with the oxide thickness. The situation has obviously changed in modern transistor Manuscript received November 13, 1996; revised April 2, 1997. The review of this paper was arranged by Editor D. Verret. This work was supported by the CICYT of the Spanish Government under Contract TIC95-0511. The authors are with the Departamento de Electr´onica, Facultad de Ciencias, 18071 Granada, Spain (e-mail: [email protected]). Publisher Item Identifier S 0018-9383(97)07730-7.

generations, in which oxide thickness is comparable with the extension of the inversion layer. Inversion-layer penetration is therefore important and must be incorporated in accurate models. Quantum effects have been included in different ways. Some examples follow. 1) The classical electron density used in classical driftdiffusion simulators has been corrected in order to force it to vanish at the oxide-semiconductor interface, and suitable boundary conditions are derived [4]. 2) As energy levels below the ground subband minimum are forbidden for the electrons, an effective bandgap widening has been proposed to modify the intrinsic carrier density used in drift diffusion simulators near the semiconductor-oxide interface [5]. 3) The oxide thickness is modified by adding a constant value representing the average inversion-layer penetration [6], [7]. Flatband and threshold voltages are also modified [6]. 4) The oxide capacitance is corrected by adding a term representing the difference between the classical and quantum average penetrations, which is modeled as a function of the inversion and depletion charges [8], [9]. Nevertheless, despite increasing interest in modeling the effects of average inversion-layer penetration, there are several aspects that require further clarification, such as the correction to be applied, whether this correction is the same in current–voltage characteristics and in capacitance–voltage characteristics, and what the dependence of the average inversion layer on the surface potential and the doping level is, among other parameters. Some of the above questions are addressed here. To do so, a self-consistent solution of the Schr¨odinger and Poisson equations is obtained in a variety of conditions, and a detailed examination of the average penetration of the inversion layer into the semiconductor, which is termed the inversion-layer centroid, is performed. Some useful expressions are provided throughout the sections below. II. THE INVERSION-LAYER CENTROID To obtain the inversion-layer centroid, an accurate solution of the Poisson and Schr¨odinger equations is obtained within the Hartree approximation. After the first variational calculation by Stern and Howard [10], these equations were self-consistently solved in the pioneering work by Stern (and the unpublished one by Howard) [11], [12], where the most

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important features of quantization were shown. Several other authors have added some improvements [13], and some of the authors of this paper developed a procedure that provided a good convergence that starts from the classical solution, even in the accumulation case at room temperature [14]. This procedure has been used as the basis for a Monte Carlo simulator of low-field transport in a quantized inversion layer, which has allowed us to accurately predict the effect of charged impurities on electron mobility [15]–[17]. A complete simulator including inversion-layer quantization has also been reported by Fischetti and Laux [18]. The procedure detailed in [14] has been used here to obtain the electric potential, the inversion and depletion charges, the energy of the lowest subband minima, and the inversion-layer centroid as a function of the applied gate voltage for different doping levels and profiles and different oxide thicknesses. We chose the number of subbands considered in the calculation so that the addition of a new subband would not change the results, even in weak inversion. This is not a serious problem in our calculation as all the higher subbands are included in a three-dimensional (3-D) gas. Oxide thickness has been observed to have no significant influence on the results provided below. The electron density in the inversion layer as a function of the distance from the silicon-SiO interface is shown in Fig. 1. The total electron concentration is shown as a solid line, while the electron concentrations contained in the three lowest subbands are plotted in dashed lines. Long dashed lines correspond to the two ellipsoids perpendicular to the interface in a transistor with a (100)-oriented substrate. Short dashed lines correspond to the four parallel ellipsoids. As can be observed, the total electron density has a shape similar to the electron density contained in the ground subband, although multisubband occupation causes the total electron density to widen, and the centroid of the total density is therefore different from the ground subband centroid. This similarity in shapes is the reason for the relatively good behavior of the variational expression, which was obtained by Stern and Howard by considering only occupation of the ground subband [10], even in the multisubband occupation case at room temperature, as reported by several authors [9], [19]. According to this approximation, the centroid position depends on the depletion and inversion charges as (1) and are the modulus of the depletion and where inversion-charge densities per unit area, respectively, and is a fitting parameter. We have tested this dependence in our results and have compared it with other expressions. The most general one, due to multisubband occupation, is to assume a dependence on the effective electric field, as defined by Sabnis and Clemens [20], which is the average transverse electric field in the inversion layer, regardless of the shape of the electron density. This transverse effective electric field is given by (2) where

is the silicon permittivity.

Fig. 1. Total electron concentration as a function of the distance from the silicon-SiO2 interface (solid line) and electron concentrations contained in the three lowest subbands (dashed lines). Long dashed lines correspond to the two ellipsoids perpendicular to the interface in a transistor with a (100)-oriented substrate. Short dashed lines correspond to the four parallel ellipsoids. Doping concentration: 5 1 1017 cm03 . Electron density per unit area: 1012 cm02 .

Therefore, we can also test the suitability of the following empirical expression (3) The result of this comparison for the best and values is shown in Fig. 2, in which the inversion-layer centroid is plotted versus the electron density per unit area. Numerical results are shown in solid lines, and the best fit obtained with (1) and (3) in the text is shown in long- and mediumdashed lines, respectively. The substrate-doping concentration is cm in Fig. 2(a), and 3 10 cm in Fig. 2(b). Approximation 3 behaves better for the lowest doping case [Fig. 2(a)], whereas for the highest doping case [Fig. 2(b)], (1) and (3) behave comparably, providing a small absolute error. Equation (1) may provide a slightly better agreement with the numerical results in this case, which can be interpreted as being a consequence of the greater confinement in the ground subband. Both the and values depend on the doping concentration. Therefore, to obtain a general expression applicable across a wide range of doping levels, we propose a more general empirical approximation given by (4)

MV/cm

and are fitting parameters. We have obtained the where values of these parameters that produce the best fit for doping cm to levels ranging from cm , obtaining very good agreement with the numerical results, with a maximum error of about 1%. The resulting and values show a linear dependence with the doping level in a semilogarithmic scale and can be expressed as cm cm

nm (5)

When (5) is used in (4), the maximum error is increased to about 3% in the worst cases, which is still fairly small. In obtaining this approximation, values for the surface potential within the moderate and strong inversion regions, as defined by Tsividis [21], have been used. These regions have been

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also tested the following expression. MV cm

(a)

(b) Fig. 2. Inversion-layer centroid versus the electron density per unit area. Numerical results are shown in solid lines. The best fit obtained with (1), (3), and (6) in the text are shown in long-dashed lines, medium-dashed lines, and short-dashed lines, respectively. The substrate-doping concentration is (a) 10 17 cm03 and (b) 3 11018 cm03 .

chosen since the inversion-layer centroid produces noticeable is not negligible with respect to effects when the surface potential , as explained in the next section. At the limit between the moderate and weak inversion regions, is lower than 0.5 mV in all the cases considered here. Therefore, although the average penetration of the inversion layer rapidly increases in weak inversion when the surface potential is reduced, its effect is negligible due to the exponential decrease of the inversion charge. We also show in the next section that the inversion-layer (where centroid effects are noticeable when is the oxide permittivity and is the physical oxide thickness) is not negligible versus one. These effects are thus important mainly for thin oxides. However, according to scaling rules, thin oxides must be used in transistors with high bulk doping levels so that an empirical expression specific for high doping concentrations is also interesting. One advantage of such an expression is that the dependence of parameter on the doping concentration in (4) can be obviated, thus making it applicable to a variety of doping profiles. As the values vary from 0.38 to 0.27 in the range cm to cm , the value used in (3) is a good choice in this range. Furthermore, we values obtained for different have analyzed the different doping concentrations of the silicon bulk and have observed can also produce that a dependence on approximately a reasonable good agreement in this range. Therefore, we have

(6)

has been introduced here to express The silicon permittivity the charges in electric field units (in megavolts per centimeter). is a reference length acting as a fitting parameter. The curves obtained for the best value are also shown in Fig. 2 in short dashed lines. As seen in this figure, (6) provides a slightly better fit than (3). However, the great advantage of (6) is that it can be used for different doping levels and profiles and even for different bulk biases. In the range of doping concentrations considered now, we have observed that the constant value nm provides good agreement with numerical results (a higher value should be considered for lower doping levels, and a different value should in fact be chosen). Such a good agreement is shown in Fig. 3, where the inversion-layer centroid is represented versus electron density per unit area for different doping concentrations [Fig. 3(a)] and for different bulk biases [Fig. 3(b)]. Numerical results are shown in solid lines, whereas results obtained with (6) in the text, with nm, are shown in dashed lines. The substrate-doping concentration in Fig. 3(a) ranges from 10 cm (curve 1) to cm (curve 5), with the substrate biased at 0 V, whereas in Fig. 3(b), ,the substratedoping concentration is kept fixed at 10 cm , and the substrate bias is changed from V to V (from top to bottom). The maximum error is about 0.1 nm, which is comparable to the error of the numerical procedure itself as image effects and exchange-correlation effects have not been considered [22]. The validity of (6), with nm, has also been tested for nonuniform doping profiles. Fig. 4(a) shows the inversion-layer centroid versus the electron density per unit area for the two doping profiles depicted in Fig. 4(b). Numerical results are plotted in solid lines, and results obtained with (6), with nm, are shown in dashed lines. Curve 1 corresponds to a 10-nm low-doped epitaxial layer grown on a highly doped substrate plotted in short-dashed lines in Fig. 4(b), and Curve 2 corresponds to a Gaussian profile that peaked at 100 nm from the interface, which is depicted in long-dashed lines in the same figure. Good agreement can also be noted here. The strong influence of the doping level was already remarked on by Pals [2] and has been recently underscored by Arora et al. [8], who commented on the inaccuracy of simple models that propose a constant value independently of the technological variables. III. EFFECT ON THE INVERSION CHARGE DENSITY We have analyzed the effects of the centroid on the inversion charge as they have a direct effect on the I–V characteristics. The influence of the inversion-layer centroid on the total band bending in the semiconductor has been recognized since the pioneering papers on the subject [2], [12]. If is the coordinate perpendicular to the interface, multiplying the Poisson equation by , and integrating it, we obtain (7)

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(a)

(a)

(b) Fig. 3. Inversion-layer centroid versus electron density per unit area. Solid lines: numerical results. Dashed lines: results obtained with (6) in the text, 1:2 nm. (a) Substrate-doping concentration is (1) 1017 cm03 , with zI 0 (2) 2 1 1017 cm03 , (3) 5 1 1017 cm03 , (4) 1018 cm03 , and (5) 3 1 1018 cm03 . The substrate is biased at 0 V. (b) Substrate-doping concentration is 1017 cm03 , and the substrate bias is VSB = 0; 1; 2; 3; and 5 V, respectively, from top to bottom.

=

where modulus of the electron charge; doping profile; depletion layer width. If is the 3-D electron density in the inversion layer, the average penetration depth is defined as

(8)

[ has been assumed to vanish at and higher distances from the silicon-oxide interface]. Equation (7) can also be obtained by assuming that the inversion layer is a sheet of charge with infinitesimal width (a delta function) placed at from the silicon-oxide interface and with charge density per unit area [23]. The first term of the right-hand member can be defined as a depletion-layer band bending, which is increased by the second term in order to obtain the total surface potential. Therefore, we can define (9) Although (7) and (8) do not depend on the shape, the effect of the term containing the inversion-layer centroid is more important with the quantum model than with the classical

(b) Fig. 4. (a) Inversion-layer centroid versus the electron density per unit area for the two doping profiles shown in figure. (b) Solid lines: Numerical results. Dashed lines: Results obtained with (6) in the text, with zI 0 = 1.2 nm. Curve 1: for profile depicted in short-dashed lines. Curve 2: for profile depicted in long-dashed lines.

model as the value of the centroid provided by the former is greater. This is the reason for the higher surface potential value obtained with the quantum model. Both and are represented in Fig. 5 as a function of the applied gate voltage for two extreme values of the doping concentration, cm and cm (an -poly has been used as gate metal). is shown in solid lines and in shortdashed lines. The value , where is the distance between the Fermi level and the intrinsic level in the silicon , bulk, is shown with solid squares, and the value defined by Tsividis as the limit between the moderate and strong inversion regions [21], is shown with horizontal dashed lines. It is apparent that is used instead of to define the threshold voltage in standard MOSFET models and that is almost constant in the strong inversion region, whereas strongly diverges. Furthermore, must be used to calculate the depletion charge. With a constant doping profile, it is well known that (10) is the Boltzmann constant and is the absolute where temperature. The correction within the root symbol accounts for the effect of the majority carrier tail in the depletion layer edge. An interesting treatment of the corrections necessary for a nonconstant doping profile was provided by Baccarani

´ LOPEZ-VILLANUEVA et al.: EFFECTS OF THE INVERSION LAYER CENTROID

Fig. 5. Surface potential versus the gate voltage for two different doping-concentration values: (1) 17 cm03 and (2) 1 18 cm03 . Solid lines: F points. s . Short-dashed lines: dep . Solid squares show the s 1 kB T=q . (tox Horizontal long-dashed lines correspond to F nm).

10

3 10 2 +6

=2

= 10

et al. [24]. Equation (10) has been observed to produce a good agreement with our numerical results for constant doping profiles, with an error lower than 1%, whereas the maximum error increases to about 4% if the correction is not included. can be calculated By using the definition given in (9), according to the well-known expression (11) is the gate voltage, and is the flatband voltage, where which includes the difference between the work functions in the metal and the semiconductor and the effect of the fixed charge within the oxide. is the oxide capacitance per unit area, which is defined as (12) Equation (11) is exact regardless of the shape of the charge distributions, provided the interface-state density and poly depletion are negligible, as is obtained by simply applying Gauss’s law and imposing the continuity of the electric field times the permittivity at the oxide silicon interface. To obtain an expression for the inversion-layer charge, (9) and (11) can be used, leading to (13) with (14) The denominator of (14) can be termed the “electrical oxide thickness,” according to Arora et al. [8]. Equation (14) can also be interpreted as a series combination of and a “centroid . Although this same correction of capacitance” given by the oxide thickness has been proposed by several other authors [6], [8], [9], it is worthwhile to emphasize the following points: 1) Equation (14) is not an empirical correction, as it has been derived directly from well-known equations. This correction is a consequence of the increase in the surface potential with the inversion charge and should not be

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used together with the actual surface potential but with . 2) appearing in (14) is not the difference between the quantum and classical centroids but the actual centroid position obtained according to the chosen model (either quantum or classical). Quantum effects have not explicitly been used in (7)–(14). Nevertheless, it is expected that the quantum model provides a more realistic value for the centroid. 3) Introducing the centroid correction leads to using instead of in the definition of threshold voltage. can also be expressed according to (10) if an effective constant doping concentration is used. is divided by instead of by . The body4) effect coefficient is therefore not affected by the centroid correction. 5) As (7) and (11) have been obtained with no approximations, (13) applies even in weak inversion, provided that accurate expressions for and are used in this regime. The electron concentration per unit area as a function of is shown in Fig. 6 for the two extreme doping values 10 cm and 3 10 cm on a semilogarithmic scale [Fig. 6(a)] and on a linear scale [Fig. 6(b)]. Data obtained with the numerical procedure detailed in the previous section are shown with a solid line, whereas data obtained using (13) with the numerical values for and are shown in solid circles. These data are compared with those obtained with a simple strong-inversion model in which is assumed to be approximately equal to . is obtained with (10), and is calculated by using (6), (13), and (14), with an iterative procedure. The result is shown in Fig. 6 with solid squares, which also produces a reasonably good agreement with numerical results except in weak inversion. The shift observed in weak inversion to higher gate voltages is mainly due to the constant value used for , which overestimates the surface potential in this region. The result for the highest doping case can be improved if a correction greater than is used, according to the results in can be approximated by a straight line in strong Fig. 5. inversion, as shown in Fig. 6(b), but the value deduced for has been observed to be very sensitive to the range of the curve used to apply the least square routine, thus leading to inaccurate results. IV. EFFECT

ON THE

GATE CAPACITANCE

The correction given by (14) is not directly justified in the gate capacitance expression as it has been obtained only for the relationship. Nevertheless, it has been used empirically even for the capacitance [8], [9], [19]. In fact, Baccarani and Wordeman showed that if the centroid capacitance is compared with the actual inversion-layer capacitance, the two curves cross each other and do not merge even asymptotically (see Fig. 2 in [3]). Therefore, if a deviation capacitance is searched as (15)

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(a) Fig. 7. Inversion-layer capacitance versus gate voltage. Dots: Numerical results. Long-dashed lines: the “centroid capacitance” defined in text. Short-dashed lines: the “corrected centroid capacitance” defined in (19) in text.

(b) Fig. 6. Electron concentration per unit area in the inversion layer versus the gate voltage for the two extreme doping values: (1) 1017 cm03 and (2) 3 1 1018 cm03 . Solid lines: Data obtained with the numerical procedure. Solid circles: Results obtained by using (13) with the numerical values for zI and dep . Solid squares: Results obtained with a simple strong-inversion model but with centroid correction. Results are shown in (a) logarithmic scale and (b) linear scale. tox = 10 nm.

with

where the superindex “si” stands for “strong inversion.” , which is computed numerically according to (16), is compared with and the “centroid capacitance” in Fig. 7 as a function of the gate voltage. Numerical results are shown in dots, long-dashed lines show the “centroid capacitance,” and the “corrected centroid capacitance,” which is defined in (19), is plotted in short-dashed lines. The crossing between and observed by Baccarani and Wordeman is also produced with the quantum-model data, and the better asymptotic behavior given by is apparent. Hartstein and Albert also proposed an inverse-capacitance term proportional but multiplied by a numerical factor [22], which is to simply 2/3 with our approximation. [Instead of the factor 2/3, a factor would have been obtained if the more general (4) had been used.] By using this asymptotic behavior, we can define a deviation capacitance that does not change its sign.

given by (20) (16)

then is negative for gate-voltage values greater than that of the cross point. To obtain a suitable asymptotic expression, we used the following reasoning. Regarding (9), in the very strong inversion limit, any increase in is produced by the increase , while , and thus , remain almost constant. This in is the well-known effect of the depletion-layer screening by the inversion layer. Therefore, the derivative (17) must vanish at this limit. Here, we can use the empirical (6), which is obtained by fitting the numerical data produced by the quantum model, and consider as a parameter (which is not affected by the variations). The result is (18) In the case , the derivative of (17) approaches unity, and (18) provides (19)

is basically due to the nonnegThe physical origin of ligible influence of the depletion charges. By using (18) and (20), we can obtain (21) Although (21) cannot be easily used for modeling, it aids in . This deviation capacitance is partly the interpretation of due to the nonnegligible versus but mainly due to the nonvanishing derivative [which is defined in (17)], which is a measure of the change in the amount of the space charges in the depletion layer due to variations in the gate voltage. This derivative should decrease as the screening effect of increases. Therefore, following Takagi and Toriumi, we can assay a dependence . These authors proposed this defined above and claimed to have obtained dependence for good agreement without the change in sign predicted. This may be due to the fact that they are actually using the difference between the centroids obtained with the quantum and classical models for instead of the centroid itself. This difference may be close to . We have also obtained a good agreement to the empirical expression (22)

´ LOPEZ-VILLANUEVA et al.: EFFECTS OF THE INVERSION LAYER CENTROID

Fig. 8. Gate-to-channel capacitance versus gate voltage. Solid line: Numerical results. Symbols: Adjustment with (24). Doping concentration: 1017 cm03 (solid circles) and 3 1 1018 cm03 (solid squares). tox = 10 nm.

with being a fitting factor depending on the doping concentration, ranging from 0.774 for cm to 0.391 for cm . The error in (22) is acceptably low except for very high values, where it is not very important because dominates. We have compared the numerical gate-to-channel capacitance as defined in the split-capacitance method (23) to the value provided by the approximation: (24) where the approximation given in (6) is used for , and the correction term for the depletion-layer capacitance given by Sodini et al. [25] has been neglected because it contributes to less than 1% for higher than 10 cm Results for this comparison are shown in Fig. 8, where the gate-to-channel capacitance is plotted versus gate voltage. Numerical results are plotted as a solid line, and the adjustment with (24) is shown in symbols (solid circles for a doping concentration of cm , and solid squares for 3 10 cm ). Good agreement of the results from (24) with the numerical results is apparent for the two extreme doping concentrations. V. CONCLUSIONS The increase in the average penetration of the inversion layer, termed the inversion-layer centroid, is one of the main quantum effects on MOSFET behavior. This averaged penetration can be described as a function of the depletion and inversion charges by a compact empirical expression, which is valid across a wide range of electrical and technological variables. One important effect of the centroid on the static characteristics is the strong increase of the surface potential with the gate voltage, even in the strong inversion regime. If a “depletion-layer surface potential,” which is less dependent on the gate voltage, is defined, the centroid effect leads to a modification of the oxide thickness, resulting in an effective electrical oxide thickness that is dependent on the gate voltage itself. This effective electrical oxide thickness is not an

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empirical correction as it has been derived directly from wellknown equations. Introducing the centroid correction leads to using the “depletion layer surface potential” in the definition of threshold voltage as well. The body-effect coefficient is therefore not affected by the centroid correction. Concerning the inversion-layer capacitance, the “centroid capacitance,” which was defined in the text, is confirmed not to be a correct asymptotic value in the very strong-inversion limit. Nevertheless, it is a suitable asymptotic value if it is multiplied by a constant derived from the empirical expression for the centroid. Good agreement with the numerical results for the gate-to-channel capacitance is obtained with this expression.

REFERENCES [1] T. Ando, A. B. Fowler, and F. Stern, “Electronic properties of twodimensional systems,” Rev. Mod. Phys., vol. 54, p. 437, 1982. [2] J. A. Pals, “Experimental verification of the surface quantization of an n-type inversion layer of silicon at 300 and 77 K,” Phys. Rev., vol. B-5, pp. 4208–4210, 1972. [3] G. Baccarani and M. R. Wordeman, “Transconductance degradation in thin-oxide MOSFET’s,” IEEE Trans. Electron Devices, vol. ED-30, pp. 1295–1304, 1983. [4] W. Hansch, T. Vogelsand, R. Kircher, and M. Orlowski, “Carrier transport near the Si/SiO2 interface of a MOSFET,” Solid-State Electron., vol. 32, pp. 839–849, 1989. [5] M. J. Van Dort, P. H. Woerlee, and A. J. Walker, “A simple model for quantization effects in heavily-doped silicon MOSFET’s at inversion conditions,” Solid-State Electron., vol. 37, pp. 411–414, 1994. [6] Y. Ohkura, “Quantum effects in Si n-MOS inversion layer at high substrate concentration,” Solid-State Electron., vol. 33, pp. 1581–1585, 1990. [7] C. K. Park et al., “A unified current–voltage model for long-channel nMOSFET’s,” IEEE Trans. Electron Devices, vol. ED-38, pp. 399–406, 1991. [8] N. D. Arora, R. Rios, and D. A. Antoniadis, “Capacitance modeling for deep submicron thin gate oxide MOSFET’s,” in Proc. ESSDERC, 1995, pp. 569–572. [9] S. Takagi and A. Toriumi, “Quantitative understanding of inversionlayer capacitance in Si MOSFET’s,” IEEE Trans. Electron Devices, vol. 42, pp. 2125–2130, 1995. [10] F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev., vol. 163, pp. 816–835, 1967. [11] F. Stern, “Iteration methods for calculating self-consistent fields in semiconductor inversion layers,” J. Comput. Phys., vol. 6, pp. 56–67, 1970. [12] , “Self-consistent results for n-type Si inversion layer,” Phys. Rev., vol. B-5, pp. 4891–4899, 1972. [13] C. Moglestue, “Self-consistent calculation of electron and hole inversion charges at silicon–silicon dioxide interfaces,” J. Appl. Phys., vol. 59, pp. 3175–3183, 1986. [14] J. A. L´opez-Villanueva, I. Melchor, F. G´amiz, J. Banqueri, and J. A. Jim´enez-Tejada, “A model for the quantized accumulation layer in metal-insulator-semiconductor structures,” Solid-State Electron., vol. 38, pp. 203–210, 1995. [15] F. G´amiz, J. A. L´opez-Villanueva, I. Melchor, J. A. Jim´enez-Tejada, and A. Palma, “A comprehensive model for Coulomb scattering in inversion layers,” J. Appl. Phys., vol. 75, pp. 924–934, 1994. [16] F. G´amiz, J. A. L´opez-Villanueva, J. Banqueri, J. E. Carceller, and P. Cartujo, “Universality of electron mobility curves in MOSFETs: A Monte Carlo study,” IEEE Trans. Electron Devices, vol. 42, pp. 258–264, 1995. [17] F. G´amiz, J. A. L´opez-Villanueva, J. Banqueri, and J. E. Carceller, IEEE Trans. Electron Devices, vol. 42, pp. 999–1004, 1995. [18] M. V. Fischetti and S. E. Laux, “Monte Carlo study of electron transport in silicon inversion layers,” Phys. Rev., vol. B-48, pp. 2244–2274, 1993. [19] R. Rios and N. D. Arora, “Determination of ultra-thin gate oxide thicknesses for CMOS structures using quantum effects,” IEDM Tech. Dig., pp. 613–616, 1994. [20] A. G. Sabnis and J. T. Clemens, “Characterization of the electron mobility in the inverted h100i Si surface,” IEDM Tech. Digest, pp. 18–21, 1979.

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[21] Y. P. Tsividis, Operation and Modeling of the MOS Transistor. New York: McGraw-Hill, 1993. [22] A. Hartstein and N. F. Albert, “Determination of the inversion-layer thickness from capacitance measurements of metal-oxide-semiconductor field-effect transistors with ultrathin oxide layers,” Phys. Rev., vol. B-38, pp. 1235–1240, 1988. [23] J. Banqueri et al., “A procedure for the determination of the effective mobility in an n-MOSFET in the moderate inversion region,” Solid-State Electron., vol. 39, pp. 875–883, 1996. [24] G. Baccarani et al., “Interpretation of C –V measurements for determining the doping profile in semiconductors,” Solid-State Electron., vol. 23, pp. 65–71, 1980. [25] C. G. Sodini, T. W. Ekstedt, and J. L. Moll, “Charge accumulation and mobility in thin dielectric MOS transistors,” Solid-State Electron., vol. 25, pp. 833–841, 1982.

Juan A. L´opez-Villanueva (M’90), for photograph and biography, see p. 846 of the May 1997 issue of this TRANSACTIONS.

Pedro Cartujo-Casinello was born in 1969. He received the B.S. degree in physics in 1995 from the University of Granada, Spain. He is currently pursuing the Ph.D. degree at the Department of Electronics, University of Granada. His current research interest is focused on the study of MOSFET modeling for analog circuits.

Jesus Banqueri (M’95) was born in 1965. He graduated with a degree in physics in 1988 and received the Ph.D. degree in 1994 from the University of Granada, Spain, with a thesis on the experimental study of electron mobility in MOS transistors. Since 1990, he has been working on MOS device physics, including characterization, simulation, and modeling of MOSFET’s. His current research interests include modeling of MOSFET’s for analog applications, and in particular, modeling of the mobility and determination of the series resistance and the effective channel length in submicron MOSFET’s. He is an Associate Professor at the University of Granada, Spain.

F. G´amiz, for photograph and biography, see p. 846 of the May 1997 issue of this TRANSACTIONS.

Salvador Rodr´ıguez was born in 1971. He graduated with a degree in physics in 1994 and in electronic engineering in 1997, both at the University of Granada, Spain. He is currently pursuing his Ph.D. degree at the University of Granada, Spain, where he is a Teaching Assistant. His current research interest is the study of hole confinement in heterostructures.