A new approach for parameter extraction of complex models and an application for SPICE MOSFET level-3 static model

MEJ 489

Microelectronics Journal Microelectronics Journal 30 (1999) 149–155

A new approach for parameter extraction of complex models and an application for SPICE MOSFET level-3 static model Metin Yazgı*, Hakan Kuntman 1 Electronics and Communication Engineering Department, Istanbul Technical University, 80626, Maslak, Istanbul, Turkey Accepted 18 September 1998

Abstract In this work, a new approach is presented for specifying the error term used in the parameter extraction algorithms. By using the new approach, it is possible to remove the differential operations in the algorithms used for the extraction of parameters in complex models. Also, we present an iteration procedure obtained by using the new approach for the extraction of SPICE level-3 MOST static model parameters KP, VTH, TETA, VMAX and RS ( ˆ RD). As well as the triode region parameters, GAMMA, NFS and KAPPA can be found in the overall procedure. Results of the procedure have been compared with the experimental results. It is obvious from this comparison that the new approach is effective for the determination of level-3 model parameters, as for the determination of parameters of proper models. 䉷 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction After establishing the mathematical model of a device the only way of reducing its deficiencies is the optimum extraction of its parameters. Several studies have been carried out and methods developed on this subject [1–12]. For the extraction of model parameters, graphic methods or computer aided optimization methods can be used. Except for some basic parameters, graphic methods are not preferred, as they do not yield accurate results for models with a large number of parameters. For such complex models, computer aided optimization methods are used. The computer aided optimization methods [1–3,5–7] are mathematical and converge the solution step by step. In these methods, parameters are all in the vector of parameters to be optimized and the vector of parameters, which makes the error between the experimental data and the data obtained from the model minimum, is sought by the mathematical optimization algorithm. These methods adjust the vector of parameters to minimize the error. In these methods, the squared sum of the relative or the absolute error terms is minimized. In general, the models are not proper to describe these error terms. Therefore, the

* Corresponding author. Tel.: ⫹ 90-212-285-3554; Fax: ⫹ 90-212-2853679; e-mail: [email protected] 1 Tel.: ⫹90-212-285-3647; e-mail: [email protected]

optimization algorithms including differential calculations are used. In this work, a new approach is proposed for the error term. The new approach can be used for models which can be changed to polynomials as in Eq. (2). In this way, an alternative to known error terms which offers the chance to dispense with hard calculations in computer analyses has been obtained. By means of this approach, an iteration procedure has been developed for extraction of SPICE MOSFET level-3 static model parameters. In this method, gate and drain characteristics can be used separately or together. This is important, especially to obtain good results in the saturation region. At the same time, the relation between the weak and the strong inversion regions has been used in the method and this has provided a good accuracy for the weak inversion region. Therefore. good results can be obtained for all regions in the model by means of this method. In addition, the method obtained for the level-3 model has an important property that the number of data used in the extraction procedure is almost unlimited. The number of data is important only for calculations of some values which are necessary in the iteration procedure; the time which is necessary for these calculations is unimportant.

2. The new approach

0026-2692/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(98)00101-3

The squared sum of the absolute or the relative error

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terms can be easily minimized in the simple models. In that case, the optimum extraction of the parameters is easy. But, in complex models it can not be directly minimized. So, the optimum parameters, which make the squared sum of the absolute or the relative error terms minimum, are obtained by means of the algorithms including differential operations and reaching the result step by step. The model equations can be represented in the form of y ˆ f …y; X1 ; …; Xl ; P1 ; …; Pm †

…1†

In this equation y is the simulation value, Xi represents bias values, and Pj represents parameters. Some equations can be changed to nth degree polynomials. In that case, the equation takes the form of an yn ⫹ an⫺1 yn⫺1 ⫹ … ⫹ a1 y ⫹ a0 ˆ 0

…2†

In this equation, the values of ak are functions of Xi and Pj. When the experimental value, yg, is put in place of y ⫹ … ⫹ a1 y g ⫹ a0 苷 0 an yng ⫹ an⫺1 yn⫺1 g

…3†

is obtained. As is easily seen, this can be used as an error term and a relative error term can be obtained by dividing this equation by yg. This error term presents an important advantage; by using this error term and after routine steps, for many complex models we can get an extraction method which includes an iteration algorithm and does not need differential operations as in simple models. In this way, the limitation in the number of data is mostly removed.

3. The extraction method for level-3 MOST static model parameters The current equation in the triode region for the SPICE level-3 model is h 0 i 0 0 KP…W=Leff † IDS ˆ VGS ⫺ vTH ⫺ FVDS VDS 0 0 1 ⫹ u…VGS ⫺ VTH † ⫹ mVDS …4a† 0

0

0

0

VDS ˆ VDS ⫺ IDS …RS ⫹ RD †

2 ⫹ a1 IDSg ⫹ a0 苷 0 a2 IDSg

a0 ˆ bVDS …VGS ⫺ VTH ⫺ FVDS †

…7†

Eq. (7) can be accepted as an error term. If we want this error term to be relative, we divide Eq. (7) by IDSg : a …8† ei ˆ a2 IDSgi ⫹ a1i ⫹ 0i 苷 0 IDSgi In order to obtain the error function; we take the square of Eq. (8). So, X 2 …9† Eˆ ei E is the error function which depends on the parameter values and bias voltages. The subscript i represents the dependence on bias voltages. The derivative of this function with respect to each parameter is found and the equation is obtained for each parameter where the derivative equals zero [9]. Each equation depends on the other parameters and bias voltages. The optimum parameter values are obtained by performing iterations between these equations. Initial values can be found for KP, VTH and R by using conventional (graphic) methods [4]. In this work, a method has been developed to obtain the initial values of these parameters [9,10]. Also GAMMA is extracted by using a conventional method [4] and an iteration is performed between this parameter and PHI (surface potential) by means of the basic equations [9,10]. The current equation for the weak inversion region is IDS ˆ Ion exp‰…VGS ⫺ Von †=…Von ⫺ VTH †Š

…10†

Both the absolute error term and the relative error term can be obtained for this equation. The relative error term was used first, but the results were not good. Therefore, the absolute error term was preferred. So, …11†

Repeating the same procedure for Eq. (11), the error function is obtained for the weak inversion region. Taking the derivative of this error function with respect to Ion,

…5a†

Ion ˆ

IG G2

…12†

is obtained, where X   IDSgi exp …VGSi ⫺ Von †=…Von ⫺ VTH † IG ˆ

a2 ˆ uR ⫹ bR2 …2 ⫺ 4F† ⫹ 2mR

⫺ mVDS ⫺ 1

…6†

F, m 0 and Leff are known values in this method. The current values are obtained at the end of the simulation. Replacing simulation results with experimental results, we obtain

where

a1 ˆ bR…4F ⫺ VDS ⫺ 2VGS ⫹ 2VTH † ⫺ u…VGS ⫺ VTH †

1 ⫹ FB m0 ; R ˆ RS ˆ RD ; m ˆ 2 Vmax Leff

…4b†

This equation can be written as 2 ⫹ a1 IDS ⫹ a0 ˆ 0 a2 IDS

Fˆ

IDSg ⫺ Ion exp‰…VGS ⫺ Von †=…Von ⫺ VTH †Š 苷 0

where VDS and VGS are defined by VGS ˆ VGS ⫺ IDS RS

and

i

…5b†

G2 ˆ

X

  exp 2…VGSi ⫺ Von †=…Von ⫺ VTH †

…13†

i

Ion is the current which is obtained for the strong inversion

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In addition, using a conventional approach, a computer program has been derived for optimum extraction of KAPPA. KAPPA(x) is the parameter of the saturation region in the model.

Table 1 Extracted parameter values of n-type and p-type transistors Parameter

151

Extracted value

The n-type transistor GAMMA 0.3651 PHI 0.5414 KP 2.5273E⫺5 VTH 1.116 u 7.665E⫺3 VMAX 1.907E⫹5 1.8078E12 NFS KAPPA 1.21 R 2.71

4. Results and discussion It should be remembered that the error term which is intended to be minimum is the relative error term eri ˆ

The p-type transistor GAMMA 0.731 PHI 0.5773 KP 1.3769E⫺6 VTH ⫺1.123 u ⫺5.212E⫺3 VMAX 4.0154E⫹5 NFS 4.0317E⫹12 KAPPA 1.503 R 5.51

eai ˆ IDSgi ⫺ IDSi

ei ˆ 2a2 eai ⫹ a1i eri

⫹ 2Rb…VTH ⫹ FVDS † ⫺ mVDS ⫹ b…2F ⫺ 1†VDS RŠ ⫹ b…VTH ⫹ FVDS †VDS n ˆ Ion …⫺u ⫺ 2bR† ⫹ bVDS …14† are obtained. It is noteworthy that in case of the transistor being in the saturation region, VDS in Eq. (14) should be replaced with VDSsat. Finally, there are two equations (Eqs. (12) and (14)) between which iterations can be performed.

The n-type transistor Drain characteristic Gate characteristic Saturation characteristic

1.435 1.036 1.171

The p-type transistor Drain characteristic Gate characteristic Saturation characteristic

1.75 1.404 1.887

…16†

The term of IDSi represents the values given by the models. Eq. (8) can be modified to obtain the relation among absolute, relative and the new error term. So,

2 ‰uR ⫺ 2R2 b…2F ⫺ 1† ⫹ 2mRŠ ⫺ Ion ‰⫺1 ⫹ uVTH m ˆ ⫺Ion

RMS error (%)

…15†

or the absolute error term

region where VGS ˆ Von and Von depends on the weak inversion parameter NFS. We must also pay close attention to the equation for the strong inversion region, because Ion depends on it. So, using the equation in the strong inversion region: m Von ˆ n

Table 2 RMS error values of the n-type and p-type transistors

IDSgi ⫺ IDSi IDSgi

…17†

In this equation the constant 2 in front of a2 is an approximate value. Although a2 and a1i change in the iteration steps, considering Eq. (17), it can be said that the minimization of ei means approaching an optimum point for both absolute and relative error terms. The practice results given below obviously show that this is true. On the other hand, as can be seen from Eq. (4), if it is intended to directly minimize relative or absolute error terms, some algorithms including differential operations must be used for the optimization. As is known, differential calculations are hard in computer analysis. When using the new approach, we need not use differential operations during computer analysis. Because of that, as expressed above, the number of experimental data used in the extraction procedure is almost unlimited. This advantage makes the new approach important and interesting. Computer programs have been derived using C programming language to realize this procedure. The performance of this procedure has been tested by using the programs on both n and p type transistors. The measurement results for the transistors have been obtained from the HP4145 parameter analyzer. The transistors were produced by the Turkish Scientific and Technical Research Council–Semiconductor Technologies Development Laboratory. Three different responses of the model have been examined and 500 data points used for each response during the tests. These responses are drain characteristics for different values of VGS (2:5 V ⱕ VGS ⱕ 5:5 V), gate characteristics for a small value of VDS in the triode region and saturation characteristics obtained for VGS ˆ VDS. As expected, the necessary time in computer analyses has become very short in spite of the large amount of data. Table 1 gives the extracted model parameter values. Table 2 gives the RMS error values obtained from the

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Fig. 1. Drain characteristics of the n-type transistor (a) and of the p-type transistor (b).

M. Yazgı, H. Kuntman / Microelectronics Journal 30 (1999) 149–155

Fig. 2. Gate characteristics of the n-type transistor (a) and of the p-type transistor (b).

153

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Fig. 3. Saturation characteristics of the n-type transistor (a) and of the p-type transistor (b).

M. Yazgı, H. Kuntman / Microelectronics Journal 30 (1999) 149–155

iteration procedure. Three different responses of the model are illustrated in Figs. 1–3. 5. Conclusion A new approach is presented for specifying the error term used in the parameter extraction algorithms. The approach can be used for models which can be represented by polynomials in which the parameters and bias voltages are seen in coefficients as in Eq. (2). By means of the new approach, we need not use differential operations during the extraction procedure, so almost unlimited data can be used to extract the parameters of complex models. Using the new approach, we obtained an iteration procedure for the accurate extraction of SPICE level-3 MOST static model parameters. Comparison of the results given in Section 4 shows that the proposed method has a good performance in the extraction of SPICE level-3 MOST static model parameters. Therefore, it can be said that the new approach offers a advantageous way for the extraction of parameters.

[2]

[3]

[4] [5]

[6]

[7]

[8]

[9]

Acknowledgements The authors would like to thank Tuna B. Tarım for the collaborative efforts and the Turkish Scientific and Technical Research Council–Semiconductor Technologies Development Laboratory for providing the devices. References [1] D.E. Ward, K. Doganis, Optimized extraction of MOS model

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