An improved charge-based MOSFET model with

An improved charge-based MOSFET model with parameterized-logistic fitting functions Tijana Kevkić1, Vladica Stojanović1 and Ljubica Spalević2 1

Faculty of Sciences and Mathematics, K. Mitrovica 2 Faculty of Economics, K. Mitrovica 1 E-mail: [email protected], [email protected], [email protected]

Abstract

2 The charge sheet model equations

New fitting of an important parameter which is used in the MOSFET modeling has been proposed in this paper. This parameter enables the smooth transition of the inversion charge density (ICD) from depletion to strong inversion region. Fitting is achieved by using the so-called parameterized logistic (PL) model. The ICD using the proposed model show a good match with the results of the implicit charge sheet based model. Moreover, the ICD model with this new form of fitting parameter, also enables inclusion of the quantum mechanical effects which appear in actual MOSFET devices.

1 Introduction The modeling of MOSFET transistors for integrated circuits design has been driven by the needs of digital circuit simulation for many years. The interest for mixed analog-digital chips in recent years, creates a neccesity for MOSFET models appropriate for analog and RF design as well [1,2]. Charge-based modeling approach is one of the basic modeling approaches. It is based on the computation of the inversion charge density in the MOSFET channel in terms of the terminal voltages i.e. gate and drain voltages [3]. The most important advantage of this approach is its simplicity and flexibility to add features resulting from technology advancements [4]. In [5] the depletion and the inversion charge densities have been clubbed together into an equivalent carrier density. However, since these two charge densities have different functional dependence on the gate bias, hence, using the same functional form for both of these with respect to the gate bias over the entire operating range is not desirable. These two components are separated [6] and an explicit inversion charge density expression is introduced. Here, that expression is improved by using the standardized PL-functions. The inversion charge density thus modeled has a much higher opportunity for modification and adaptation compared to the classical as well as quantummechanical incorporated charge-based model.

ERK'2015, Portorož, A:15-18

15

We consider an n-type MOS transistor with gate oxide thickness tox, and the channel region homogenously doped with an acceptor concentration of NA. Under the assumptions of gradual channel and charge sheet approximations, for the usual range of an n-MOSFET operation, the electrostatic surface potential ψs is related to the externally applied gate-body voltage VG through the implicit relationship [7]:    2 F  Vch VG  V FB  s     s  uT  exp s uT 

 . (1)  

Here, VFB is the flat-band voltage, uT = kT / q is the thermal voltage, ϕF = uT ln (NA / ni) is the bulk Fermi potential, and γ = 2qε si N A / Cox is the body factor, where the gate oxide capacitance per unit area is Cox = ε ox / t ox . The channel potential Vch is defined by the difference between the quasi-Fermi potentials of the carriers forming the channel (ϕn) and that of the majority carriers (ϕp). From Eq. (1) it is clear that ψs cannot be explicitly found as a function of VG and Vch, and the equation has to be numerically solved. The inversion charge density (ICD) is given by [7]:     2F  Vch Q'1    Cox   s  uT  exp s  uT  

     s  . (2)    

The values of Q'1 obtained from Eq. (2) with numerical solutions of Eq. (1) for ψs serve as corresponding benchmark results for the ICD model.

3 Inversion charge density modeling In the weak inversion region, the ICD can be approximated in the following way [6]:  V  VT Q'1wi  n  1  C ox  u T  exp G  n  uT

.  

(3)

Here, n is the slope factor slightly dependent on the gate voltage and VT is equalibrium threshold voltage. On the other hand, for the model ICD in the region of strong inversion the following linear approximation can be used [5]: (4) Q'1si  C ox  VG  VT  .

One single expression for the ICD, valid for all regions of operation, can be obtained by using the socalled interpolation function [8]:   V  VT Q' inv   K  C ox  uT  ln 1  exp G   n U T

 .  

(5)

Eq. (5) can be used to predict the values of the ICD, starting from the depletion region all the way to the strong inversion region if the parameter K changes smoothly from K = n - 1 in the weak inversion region to K = n in the strong inversion region. Since the parameter K has an important role in the Q'inv transition from one to the other region of inversion, the following functional form for it (versus the gate-bias VG) is proposed in [6]: K m VG   n 

 1 1  2  

VG  VT

VG  VT 

2

  22

 ,   

(6)

where ε2 is the smoothing parameter which takes the values between 0.2V and 0.8V.

transition of K PL VG  values as it is shown in Fig. 1 (left diagram). On the other hand, a smoother and slower transition of KPL(VG) from the weak to the strong inversion region can be easily achieved by decreasing the value of parameter a, as is shown in Fig. 1 (right diagram). In this way, parameter a completely controls the behavior, i.e. the rate of change and the smoothness of the function KPL(VG). In both diagrams in Fig. 1, as a comparison, the graphs of function Km (VG) in two "extreme" cases, for ε2 = 0.2V and ε2 = 0.8V are also shown. As it can easily be seen, the PL-functions KPL(VG) given by Eq. (7) have a much higher opportunity for modification and adaptation compared to the model given by Eq. (6). Especially, the function KPL(VG) can also be modified to shift its values for an arbitrary point of transition (e.g. named PT). By using the simple substitution K'PL (VG) = KPL (VG - PT), the function K'PL(VG) can be obtained whose values are translatory shifted with respect to the values of KPL (VG).

n

n

n-1

n-1

Fig. 1. Graphs of the PL-fitting of K, by using the Eq. (7) (solid lines), compared to the fitting model proposed in Eq. (6) (dashed lines). However, several simulations have shown that the ICD values obtained from Eq. (5) with Km (VG) deviate significantly from those obtained from the implicit charge sheet (CS) solution especially in the weak inversion region, where the ICD exponentially depends on the gate bias. In order to avoid the constraints related to the empirical determination of the smoothing parameter and of the obtained explicit model that has a smoother and slower transition from the weak to the strong inversion region we propose a different, the socalled Parameter Logistic (PL) form of the parameter K:   a  a VG  VT  K PL VG   n  1  exp u u  T  T 

1

.

(7)

Here, a ≥1 is a parameter given in advance (e.g. in accordance to some numerical computations and/or MOSFETs technical characteristics). It has an important role in the speed of transition of the KPL (VG) values from n-1 to n. Namely, if a >> uT then there is a rapid

16

As an illustration, the right plot in the Fig. 1 shows the graphs of the functions K'PL (VG) whose values are shifted to theirs transition points PT = uT /a ln (uT /a). In this way, we get “new” transition points at the threshold voltage VT ≈ 1.5094V.

4

Results for inversion charge density

The results of Eq. (5) obtained by using Km and KPL parameters are compared in Fig. 2. The ICD values obtained from Eq. (2) are also shown in Fig. 2 as the reference values. The implicit charge sheet ICD-model was compared to the interpolation model given by Eq. (5) where first Km is used with the smoothing parameter ε2 = 0.8V (as it is proposed in [6]), and then KPL with parameter a = 0.065. In addition, K-modeled values given by Eq. (6) are shifted to the transition (saddle) point PT = 4uT , and K-modeled values from Eq. (7) are shifted to the point PT = uT /a ln (uT /a) + 4uT. As it can be seen in the same diagram, there is better fitting with

the PL-modeled K-values, obtained by using the Eq. (7), than with the K-values modeled by using Eq. (6). The following diagram in Fig. 2 (top right) shows the graphs of the absolute error functions which are defined as AEF = |Q'1 – Q'inv|. These functions describe the absolute values of the difference between the implicit CS model |Q'1| and the interpolating model |Q'inv|, obtained by using the both of the mentioned Kfitting models. It is easy to observe that PL-fitted values of the parameter K, obtained by Eq. (7), show less deviation from the reference model |Q'1|, compared to the K-fitted values where Eq. (6) is used. Finally, in the bottom two diagrams in Fig. 2 the agreement of the interpolating models |Q'inv| with the “individual” ICDapproximation given by Eq. (3) and Eq. (4), that represent separately ICD in regions of weak or strong inversion, are shown. In all of these ICD-models, the values of parameter K are fitted again using the model described by Eq. (6), as well as the PL-model given by Eq. (7). As in the case of ICD-model given by Eq. (5), faster transition from the weak to the strong inversion region occurs, for K-fitting model given by Eq. (6) when the smoothing parameter is set to ε2 = 0.2V, and for the K-fitting model given by Eq. (7) when we used a = 150.

5 Modeling the decrease in ICD due to the quantum mechanical effects The present generation of MOSFETs shows significant quantum mechanical effects primarily caused by the high substrate doping and small effective oxide thickness [9]. These two factor often cause forming of a narrow potential well on the Si side of the Si-SiO2 interface. The Si conduction band splits into discrete subbands with most of the electrons, responsibile for current transport in channel, residing in the lowest subband and described by Schrӧdinger’s wave equation [10]. Also, the charge distribution arising from these electrons and the depletion layer acceptor ions must also satisfy Poisson's equation. Thus, Schrӧdinger's and Poisson's (SP) equations are coupled here and these need to be solved simultaneously and self-consistently in order to find the electron concentration present in the lowest subband. Using the variational approach, the QM corrected surface potential ψs[qm] can be obtained [11,12]. In the following, we explain in detail the procedure of computation the QM-modified inversion charge densities Q'1[qm] and Q'inv[qm] vs. terminal voltage V G.

Eq. (6) Eq. (7)

Implicit CS model Eq. (6) Eq. (7)

Eq. (3) Eq. (6) Eq. (7)

Eq. (4) Eq. (6) Eq. (7)

Fig. 2. Inversion charge densities |Q'1| and |Q'inv| versus gate voltage VG (above left panel), absolute error function (AEF) vs. VG (above right panel) and |Q'inv| vs. VG (lower panels). It is obvious from Fig.2 that PL-model shows improvement over the explicit model [6] in the weak as well in strong inversion region. This improvement in the modeled ICD has been observed consistently for a wide range of values of and

17

The values of classical ICD model, i.e. the function Q'1 are obtained from Eq. (2), with surface potential ψs obtained by numerically solving Eq. (1). Quantum mechanical incorporated ICD model Q'1[qm] is obtained by modifying Eq. (2), as follows

   s[ qm]  2F  Vch Q'1[ qm]    Cox   s[ qm]  uT  exp uT  

.     s[ qm]     

(8)

Bellow, the ICD-model obtained by using Eq. (8) will be used as the reference model. On the other hand, we computed the interpolation QM- modified ICD model Q'inv[qm] in two ways. Firstly, it is obtained from the expression   V  VT Q' inv[ qm]   K m[ qm]  C ox  uT  ln 1  exp G   n  uT

 ,  

(9)

where QM-modified parameter Km[qm] is obtained from Eq. (6), but using QM-modified threshold voltage VT[qm], given by VT [ qm] = VFB + 2Φ F + Vch + δψs + γ 2Φ F + Vch + δψ s . (10)

In the similar way, we obtained the interpolation QMmodified ICD model   V  VT Q'inv[ qm]   K PL[ qm]  Cox  uT  ln 1  exp G   n  uT

 ,   

(11)

where KPL[qm] is the QM-modified PL-parameter obtained from Eq. (7), with the same VT[qm]. Table 1 shows the values of the cumulative absolute error (CAE), defined as the integrals of the absolute error function (AEF), of the both of interpolating QMmodified ICD-models given by Eqs. (9)-(11), which are compared to the reference ICD-model [Eq. (8)]. The values of CAE were computed individually, in regions of weak or strong inversion, and for both of the ICDmodels, obtained by using the parameters Km[qm] and KPL[qm], respectively. The last two rows of Table 1 contain the estimated values of the maximum absolute error (MAE), given as MAE = max |Q'1[qm] – Q'inv[qm]|. As it can be easily seen, in both of the inversion regions the estimated errors of the ICD-model given by Eq.(11), are less than the corresponding errors of the ICD-model given by Eq.(9). This is especially noticeable in the strong inversion region. Table 1. Estimated errors of interpolating ICD-models with different choices of parameters K(VG), compared to the reference QM-incorporated ICD- model. Q'1[qm] Errors

CAE MAE

Q'inv[qm] Eq. (9)

Weak inversion 5.434E-07

Strong inversion 7.059E-02

Eq. (11)

4.895E-07

3.402E-02

Eq. (9) Eq. (11)

8.540E-15 7.937E-15

1.328E-03 6.408E-04

Device parameters: tox = 2.5nm, NA = 5×1017 cm-3, VFB = -0.8V, Vch = 1V, 2ϕF = 1.91V, uT = 0.026V.

18

6. Conclusion An improved charge-based MOSFET model with parameterized logistic functions has been presented. The results of the inversion charge density values simulated from the proposed model have been verified extensively with the numerical results of a classical implicit charge sheet based model. A great agreement was found. This model can be extended to the case where quantum mechanical effects play a significant role in inversion mode. The incorporation of our parameterized logistic functions into the adequate relations of the charge sheet model which takes into account quantum effects also gives satisfactory values for the surface potential and particularly for the inversion charge density.

References [1] R. Van Langevelde, FM. Klaassen: An explicit surfacepotential-based MOSFET model for circuit simulation. SolidState Electron 409-18, 2000. [2] HC. Morris, H. Abebe: A compact surface potential model for a MOSFET device. Mathematical and Computer Modelling 893-900, 2010.

[3] M. Chan, X. Xi, J. He, KM. Cao, M.V. Dunga, A.M. Niknejad, P.K. Ko, C. Hu: Practical compact modeling approaches and options for sub-.1 μm CMOS technologies, Microelectronics Reliability, Journal of, 399-404, 2003.

[4] D. Foty: MOSFET Modeling with SPICE, Principles and Practice, Prentice Hall, 1997. [5] Pregaldiny, C. Lallement, D. Mathiot: Accounting for quantum mechanical effects from accumulation to inversion, in a fully analytical surface-potential-based MOSFET model. Solid-State Electron 781–787, 2004. [6] D. Basu, AK. Dutta: An explicit surface-potential-based MOSFET model incorporating the quantum mechanical effects. Solid-State Electron, 1299-1309, 2006. [7] YP. Tsividis: Operation and modeling of the MOS transistor. Second ed. USA: McGraw-Hill International; 1999. [8] HJ. Oguey, S. Cserveny: MOS modeling at low-current density. Summer Course on Process and Device Modeling. ESAT LeuvenHeverlee, Belgium, 8–11 December 1996, p. 555–8. [9] F. Stern, WE. Howard: Properties of semiconductor surface inversion layers in the electric quantum limit. Phys Rev 816-35, 1967 [10] JR. Brews, A charge-sheet model of the MOSFET. SolidState Electron 345-355, 1978. [11] T. Kevkić, D. Petković: Classical and quantum mechanical models of surface potential and MOS capacitance in strong inversion, Proc. of 53th Conference ETRAN, MO1.4, Vrnjačka banja, Serbia 2009. [12] T. Kevkić, D. Petković: A Quantum Mechanical Correction of Classical Surface Potential Model of MOS Inversion Layer, Proc. of 26th International Conference on Microelectronics (MIEL 2010), 115-118. Niš, Serbia 2010.