Determining non-quasi-static small-signal equivalent circuit of a RF silicon MOSFET

Solid-State Electronics 45 (2001) 359±364

Determining non-quasi-static small-signal equivalent circuit of a RF silicon MOSFET Seonghearn Lee a,*, Hyun Kyu Yu b a

b

Department of Electronic Engineering, Hankuk University of Foreign Studies, San 89, Wangsan-ri, Mohyun-myun, Yongin, Kyungki-do 449-791, South Korea Micro-Electronics Technology Laboratory, Electronics and Telecommunications Research Institute, Yusong-gu, Taejon 305-606, South Korea Received 6 January 2000; received in revised form 18 July 2000

Abstract We present an accurate non-quasi-static small-signal MOSFET model incorporating distributed channel and substrate resistances to extend the frequency limit of the model validity to higher frequencies. The model parameters are accurately determined using a new semi-analytical extraction technique combining analytical and optimization approaches as an e€ective way to attain a global minimum. The validity of the model and parameter extraction method is justi®ed by observing excellent agreements between the measured and modeled S-parameters in the wide range of frequency. Ó 2001 Published by Elsevier Science Ltd. Keywords: MOSFET; Parameter extraction; Non-quasi-static; Modeling; S-parameters

1. Introduction Recently, RF communication systems are going toward the wider bandwidth and higher-frequency range. In RF/microwave IC applications [1], a low-cost silicon MOSFET is rapidly emerging as a core device [2]. Accurate small-signal equivalent circuit modeling of a silicon MOSFET becomes very important for designing RF ICs and characterizing processes and devices. In most papers [3±6], a simple quasi-static small-signal equivalent circuit in Fig. 1 has been widely used for MOSFET modeling and parameter extraction. However, it was recently pointed out that the connection of drainbulk junction capacitance Cds to the internal source in Fig. 1 is not physically acceptable and produces severe errors in extracting model parameters at high frequencies [7]. In order to eliminate this problem, an improved

*

Corresponding author. Tel.: +82-31-330-4117; fax: +82-31330-4120. E-mail address: [email protected] (S. Lee).

model in Fig. 2 has been proposed and its physical accuracy has been demonstrated [7]. In general, the Si well/substrate region that possesses the lossy dielectric property acts as the distributed resistive network. This substrate e€ect becomes increasingly important to predict RF performance in the highfrequency region [8±16], but is not accounted for the conventional small-signal models discussed above. Recently, several substrate models with a distributed resistive network [8±10] have been developed for a SPICE BSIM3v3 model [17]. As a simpli®ed version, a compact model with a single series resistance is also proposed [12±16]. This simple model is more useful to enhance the accuracy of parameter extraction than the distributed network [15]. On the other hand, the accuracy of these previous quasi-static models is considerably limited to lower frequencies below the cuto€ frequency (fT ) [18±20]. Physically, the inversion channel region is distributed in the lateral direction under the gate oxide, and this distributed nature results in the channel propagation delay generated by RC transmission line e€ect [18±21]. Recently, this channel delay e€ect has been successfully

0038-1101/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 0 3 8 - 1 1 0 1 ( 0 1 ) 0 0 0 0 6 - 5

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Fig. 1. A simple quasi-static small-signal equivalent circuit model for a RF Si MOSFET gm ˆ gmo exp… jxs†.

Fig. 2. A physically acceptable quasi-static small-signal equivalent circuit model for a RF Si MOSFET.

modeled as a bias-dependent RC series element [18±23]. This non-quasi-static e€ect due to the channel delay plays an important role when the operating frequency gets close to fT . Thus, the non-quasi-static e€ect should be incorporated to extend the frequency limit of the model validity to higher frequencies. Together with this model development, intense research of the accurate parameter extraction of the models should be carried out as an essential step in RF MOSFET modeling. Therefore, in this paper, we present an accurate non-quasi-static small-signal model accounting for the distributed channel and substrate e€ects in RF Si MOSFET, and develop a new semi-analytical parameter extraction technique combining analytical and optimization approaches for enhancing the extraction accuracy.

2. A non-quasi-static model Fig. 3 shows a non-quasi-static small-signal MOSFET model that is constructed by adding the distributed channel and substrate resistances into Fig. 2. In this model, bias-dependent RC distributed channel e€ects

Fig. 3. A non-quasi-static small-signal equivalent circuit model for a RF Si MOSFET.

are considered by connecting non-quasi-static channel resistances …Rgsi and Rgdi † in series with channel capacitances …Cgsi and Cgdi †, respectively [18±22]. In addition, the high-frequency substrate region is modeled by connecting the series bulk resistance …Rbk † to Cds [12±16]. For the purpose of enhancing the model accuracy, overlap and fringing components …Cgso and Cgdo † are separated from channel ones [23]. The parameter of Rg models the e€ective gate resistance [24] accounting for the distributed transmission line e€ect [25]. The model parameters in a conventional quasi-static model are usually determined from the measured Sparameters using several direct methods [3±6], but these methods are not applicable to extract the non-quasistatic model due to the large number of unknown parameters as well as di€erent circuit topology. In order to remove this drawback, a global optimization technique is traditionally used to ®t the model to the measured Sparameters, but this optimization may su€er uncertainties in ®nding an unique solution due to the large number of unknowns. Therefore, in this paper, the following semi-analytical procedure combining analytical and optimization techniques has been developed as an ecient way to reduce unknown parameters.

3. Model parameter extraction N-MOSFETs with multiple n‡ poly-gate ®ngers with 0.8 lm length and 10  10 lm total width were fabricated on p-type 2 kX cm high-resistivity Si wafers using a standard twin-well CMOS process [26]. S-parameters were measured under the grounded bulk con®guration and probe pad parasitics are eliminated from the measured ones using an ``open'' test structure [27,28]. Since the non-quasi-static model in Fig. 3 can be simpli®ed as a quasi-static model in Fig. 2 at low frequencies well below fT , all quasi-static parameters can be directly determined. Then, non-quasi-static parameters are extracted using the numerical optimization while

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keeping all quasi-static parameters constant. As the ®rst step of this semi-analytical extraction technique, a direct extraction technique is developed to determine the quasi-static parameters as follows. Because gm and Rds are neglected in Fig. 2 at zero bias, the following equation is derived [3]: Rd ˆ Re …Z22 Z12 †. Thus, Rd is extracted to be 24 X from Fig. 4 by taking a frequency-independent value in the low-frequency range. Also, the value of Rs ˆ 24 X is determined by plotting the frequency response of Re…Z22 Z12 † for a test MOSFET where source and drain are interchanged. As shown in Fig. 5, Ld is determined to be 112 pH by using the curve ®tting of the following equation [4,5] in the wide range of frequency: 1 Im…Z22 x

Z12 † ˆ Ld

Ed x2 ‡ B

…1†

where B, Ad and Ed are expressed as functions of intrinsic parameters, and are constant values at ®xed bias because the intrinsic ones are independent of frequency. Eq. (1) is originally derived from the quasi-static model, but is not strongly a€ected by non-quasi-static and bulk elements in Fig. 3. Thus, the Ld extraction using Eq. (1) is likely to be valid for the non-quasi-static model. The value of Cds is approximated by the following simple expression: Cds 

1 Im…Y22A ‡ Y12A † x

…2†

where the Y A parameters are obtained by subtracting parasitic resistances from the measured Z-parameters at low frequencies where inductances can be neglected. In this subtraction, we used the value of Rg that was ex-

Fig. 5. The measured data (s, n) and ®tted curves (Ð) versus frequency for measured equation (1) and corrected equation (5).

tracted from the following simple equation [3] derived at zero bias: Rg ˆ Re…Z11 Z12 †. Using Eq. (2), Cds is directly extracted to be 35 fF. In order to determine the rest of parameters more accurately, corrected Z c -parameters are obtained by subtracting extracted values of Rd , Ld , and Cds sequentially from the measured Z-parameters. Using the corrected Z c -parameters, the new value of Rg is extracted to be 16 X by ®nding a constant value in Fig. 4 after abrupt fall-o€ of the frequency dependent term in the following equation [4,5]: c Re…Z11

c Z12 † ˆ Rg ‡

Ag x2 ‡ B

…3†

The values of Ls ˆ 50 pH, and Lg ˆ 86 pH are extracted by ®nding constant terms through the curve-®tting process of the following equations (4) and (5) in the wide frequency range, respectively [4,5]: 1 c † ˆ Ls Im…Z12 x 1 c Im…Z11 x

Fig. 4. The frequency response curves of measured Re…Z22 Z12 † at zero bias and corrected Re…Z11 Z12 † at VGS ˆ 2 V and VDS ˆ 3 V.

Es x2 ‡ B

c Z12 † ˆ Lg

…4† Eg ‡B

x2

Fg 2 x …x2 ‡

B†

…5†

where Ag , Es , Eg , and Fg are functions of the intrinsic parameters, and are independent of frequency. As shown in Fig. 5, Lg is extracted using the measured data in the high-frequency region where Fg =‰x2 …x2 ‡ B†Š is neglected compared to Eg =…x2 ‡ B†: This high-frequency ®tting process results in the correct extraction of Lg , because Eq. (5) is approximately valid for a non-quasistatic model in Fig. 3. Fig. 5 shows a good correspondence between the measured data and ®tted curves of

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Eqs. (1) and (5) in the wide range of frequencies up to 39.5 GHz, which demonstrates the accuracy of the extracted values. After Rs , Ls , Rg , and Lg are sequentially subtracted from the corrected Zc -parameters, intrinsic parameters are extracted using the following Yi -parameter equations [3,5] for an intrinsic MOSFET in a dotted box of Fig. 2: Cgs ˆ

1 Im…Y11i ‡ Y12i † x

…6†

Cgd ˆ

1 Im…Y12i † x

…7† Fig. 7. The s and Rds versus frequency.

1 Rds ˆ Re…Y22i †

…8†

gmo ˆ jY21i

…9†

sˆ

Y12i j

1 phase…Y21i x

Y12i †

…10†

After these intrinsic parameters are plotted as a function of frequency in Figs. 6 and 7, uniform data in the low-frequency range are taken as their quasi-static parameter values. After all, it was found that Cgs ˆ 126 fF, Cgd ˆ 22:5 fF, Rds ˆ 745 X, gmo ˆ 12:5 mS, and s ˆ 3:8 ps. In order to verify the inadequacy of the quasi-static model in the high-frequency region, the modeled S-parameters for Fig. 2 with the above extracted parameters are compared with the measured ones in Fig. 8. The deviation between the measured and modeled S-parameters starts to be noticeable from about 7 GHz, and becomes larger at higher frequencies. Fig. 8. Comparison of measured S-parameters (}) with modeled ones using quasi-static equivalent circuit of Fig. 2 (


) and non-quasi-static equivalent circuit of Fig. 3 (Ð) from 0.5 to 32.5 GHz.

Fig. 6. The frequency response curves of Cgs , Cgso , Cgd and gmo .

This discrepancy is originated from non-quasi-static channel and bulk e€ects. In order to remove the discrepancy, a non-quasistatic model in Fig. 3 is used in this work. At zero bias …VGS ˆ VDS ˆ 0 V†, Cgs is reduced to Cgso owing to the absence of channel capacitance. Using …1=x† Im …Y11 ‡ Y12 †, Cgso of 45 fF is obtained with higher accuracy from the measured S-parameters at zero bias. Thus, Cgsi is extracted by Cgsi ˆ Cgs Cgso ˆ 81 fF. Since Cgdi disappears in the saturation region …VGS ˆ 2 V; VDS ˆ 3 V†, Cgdo ˆ Cgd ˆ 22:5 fF. However, when a device operates in the linear region, Cgdi should be determined by Cgdi ˆ Cgd Cgdo . Next, Rgsi and Rbk were optimized to obtain the closest possible ®t to be measured

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S-parameters, while all quasi-static parameters were ®xed at previously extracted values. From this optimization, it was obtained that Rgsi ˆ 25 X and Rbk ˆ 25 X. In Fig. 8, a good agreement is observed between the measured and modeled S-parameters using Fig. 3, demonstrating the accuracy of the non-quasi-static and semianalytical extraction method. 4. Conclusions An accurate small-signal model accounting for nonquasi-static and substrate e€ects has been proposed for high-frequency circuit simulation. This non-quasi-static model is developed to solve the serious problem that a quasi-static model is inaccurate in the high-frequency region. In order to identify the model parameters, all quasi-static parameters are directly extracted from derived formulations, and the non-quasi-static channel and bulk resistances are next obtained using optimization. This new semi-analytical approach provides an ecient way to prevent the optimization from trapping into a local minimum. The validity of this model and extraction method is veri®ed over the wide range of frequency. The non-quasi-static model will be successfully used to design RF ICs operating at higher frequencies.

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