I. INTRODUCTION

Recently, high-resistivity (HR) partially-depleted (PD) silicon-on-insulator (SOI) MOSFETs have been frequently used in RF IC fabrication. These devices exhibit enhanced RF performance compared to bulk MOSFETs because the combination of a thick buried oxide (BOX) layer and an HR substrate structure reduces leakage current and substrate loss, enabling RF IC design ^{[1, 2]}. For RF switch design using body-contact (BCT) PD-SOI MOSFETs ^{[3, 4]}, accurate equivalent circuit models capable of representing RF on-state and off-state characteristics are required.

In conventional approaches, the off-state model parameters of PD-SOI MOSFETs have been directly extracted using a simple equivalent circuit ^[5] in a common source-body (CSB) configuration shown in Fig. 1, where $C_{jdo}$ is the drain-body junction capacitance, $C_{bo}$ is the drain-source coupling capacitance through the BOX layer, and $C_{gb}$ is the gate-body capacitance. The extrinsic resistances ($R_G, R_D, R_S$) and inductances ($L_G, L_S, L_D$) can be neglected in the low-frequency (LF) region because they are much smaller than the impedance of capacitances.

However, unlike the bulk MOSFET model ^[6], this simple model cannot be fully extracted with LF $S$-parameter measurement data alone, because $C_{bo}$ is added to the bulk device model. For a symmetric source and drain device structures at zero-bias conditions ($V_{GS} = V_{DS} = 0$ V), the assumption of $C_{gdo} = C_{gso}$ is valid. Based on this symmetry, $C_{gb}$ can be extracted using $(1/\omega)Im(Y_{11} + 2Y_{12})_{LF}$. However, it becomes impossible to extract $C_{bo}$ and $C_{jdo}$ individually, because $(1/\omega)Im(Y_{22} + Y_{12})_{LF} \approx C_{bo} + C_{jdo}$.

To overcome this limitation, an additional LF $S$-parameter measurement using a common gate-body (CGB) test pattern has been employed to separately extract $C_{bo}$ ^[5]. However, this method has the drawback that the extraction process becomes complex, because it requires both CSB and CGB test patterns. Furthermore, the model in Fig. 1 is highly simplified to enable direct extraction, which limits its ability to accurately reflect the physical characteristics of the off-state device. This simplification also reduces the accuracy of parameter extraction in the high-frequency (HF) region.

In particular, the body resistance $R_b$ in series with $C_{gb}$ becomes significantly larger in PD-SOI MOSFETs than in bulk ones due to the thin body region above the BOX layer, resulting in increased sheet resistance. Thus, the simple model, neglecting $R_b$, exhibits insufficient accuracy. In the depletion region where $V_{GB}$ is greater than the flat-band voltage $V_{FB}$, the small value of $C_{gb}$ suppresses the effect of $R_b$ at LFs. However, at HFs, the influence of $R_b$ becomes non-negligible. In the accumulation region ($V_{GB} < V_{FB}$), $C_{gb}$ increases significantly, and thus the effect of $R_b$ becomes pronounced even at LFs. Consequently, the accuracy of a simple model that neglects $R_b$ is degraded significantly in the accumulation bias.

Therefore, a more complex physical off-state equivalent circuit model ^[7] shown in Fig. 2 was used instead of the one in Fig. 1. In this model, $R_b$ and $C_{jso}$, which are not negligible within the high frequency range, are added to the simple equivalent circuit. In this work, to account for the body-distributed effect resulting from the high aspect ratio of the body region, $C_{gb}$ is divided into intrinsic gate-body capacitance $C_{gbi}$ and extrinsic gate-body capacitance $C_{gbe}$.

This physical model has a larger number of parameters than the simple one, and the input and output parameters of the model are cross-linked via $C_{gbi}$ in Fig. 2, making derivation of the $Y$-parameter equation for direct extraction much more difficult. To overcome this difficulty of the direct extraction, numerical optimization is typically utilized, but may be complicated and produce uncertainties in finding a unique solution.

To solve this problem without optimization process, a direct method using a simplified model by treating the large $C_{gbi}$ under accumulation bias as a short circuit at HFs in Fig. 2 was previously reported ^[7]. However, $C_{gbi}$, $C_{jdo}$ and $C_{jso}$ could not be directly extracted through this simplification. To determine these parameters, further $S$-parameter measurements at zero bias ($V_{GS} = 0$ V) and inversion bias ($V_{GS} > V_{TH}$) are used, making this previous method more complex. Moreover, because the model is simplified for the accumulation region, this direct extraction can not apply in the depletion region.

To overcome the limitations of existing simple models and complex extraction methods, in this paper, we propose a novel technique for directly extracting all parameters of a physical off-state equivalent circuit using only $S$-parameters measured at a single bias, and verify its accuracy in various biases.

II. PARAMETER EXTRACTION AND ANALYSIS

Multi-finger HR PD-SOI N-MOSFETs with a gate length $L_g$ of 0.25 µm, a unit finger width $W_u$ of 5 µm, the number of gate fingers $N_f$ of 16 and a T-shaped single body contact (BCT) layout were fabricated in the CSB configuration. $S$-parameters were measured from 100 MHz to 40 GHz and de-embedded to eliminate the parasitic components of “on-wafer” RF probe pads and interconnections using open and short patterns ^{[8, 9]}.

In Fig. 2, the extrinsic resistances and inductances affecting HF $Y$-parameters are extracted under each off-state bias using the y-intercepts of the following HF Z-parameter equations versus $\omega^{-2}$ ^[10]:

After de-embedding the extracted extrinsic resistances and inductors from Fig. 2, the $Y^i$-parameters of the intrinsic equivalent circuit can be expressed as follows:

where $A = R_b C_{gbi}^2$, $B = R_b(C_{jdo} + C_{jso} + C_{gbi})$, $C = R_b(C_{jdo} + C_{jso})$, $D = (C_{gso} + C_{gbo} + C_{gdo})$, $E = R_b C_{jdo}^2$, $F = R_b(C_{gbi} + C_{jso})$, $G = (C_{bo} + C_{gdo})$, and $H = R_b C_{jdo} C_{gbi}$.

First, it is assumed that $C_{gdo} = C_{gso}$ in symmetric source and drain structures at $V_{DS} = 0$ V. This assumption may not hold under process variations or asymmetric S/D layouts in which the extraction method for $C_{gso}$ is required.

Using the LF approximation of Eq. (9), the value of $C_{gdo} = C_{gso}$ can be directly extracted from the LF data of $-(1/\omega)Im(Y^i_{12})$.

Since Fig. 2 contains more unknown parameters than Fig. 1, and the feedback $C_{gbi}$ is connected between the input and output, the expressions for the $Y^i$-parameters shown in Eqs. (7)-(9) become more complex, making it impossible to extract all parameters directly using only LF approximated equations. Therefore, we derive newly simplified equations based on HF approximations, enabling complete direct extraction. Unlike conventional methods that require additional test patterns or multiple bias measurements, this novel method allows all off-state equivalent circuit parameters to be extracted directly from a single $S$-parameter measurement, making the method significantly simpler. Using this novel approach that combines both LF and HF approximations, the remaining parameters are directly extracted as follows:

To extract $C_{gbi}$ and $C_{jdo}$, the differences ($m$, $k$) between the LF and HF approximations derived from the $Y^i$-parameter equations in Eqs. (7)-(9) are expressed by the following equations:

where

By combining Eqs. (10) and (11), we can derive the following new expressions for extracting $C_{gbi}$ and $C_{jdo}$ ($= C_{jso}$):

where the assumption of $C_{jdo} = C_{jso}$ is valid in multi-finger SOI devices at $V_{DS} = 0$ V because the junction capacitance is confined to the sidewall due to the n$^+$ source and drain contact region just above the BOX layer.

In Figs. 3 and 4, $m$ and $k$ are determined from the LF-HF difference values of Eqs. (12) and (13), respectively. The values of $C_{jdo}$ ($= C_{jso}$) and $C_{gbi}$ are then extracted by substituting $m$ and $k$ into Eqs. (14) and (15).

To extract $C_{bo}$, the following new equation approximated from Eqs. (8) and (9) in the HF region is used:

Substituting the HF value of $(1/\omega)Im(Y^i_{22} + Y^i_{12})$ from Fig. 4 and the previously extracted ones of $C_{jdo}$ ($= C_{jso}$) and $C_{gbi}$ into Eq. (16), $C_{bo}$ can be extracted.

To extract $C_{gbe}$, the following new equation approximated at HFs from Eq. (7) is used:

$C_{gbe}$ is extracted by substituting the HF value of $(1/\omega)Im(Y^i_{11})$ from Fig. 5 and the extracted ones of $C_{jdo}$ ($= C_{jso}$), $C_{gbi}$, and $C_{gdo}$ ($= C_{gso}$) into Eq. (17).

Applying the proposed extraction method at mmWave frequencies, it becomes possible to select more accurate HF values that fits Eqs. (12)-(17) at frequencies higher than 40 GHz in Figs. 3-5. Therefore, the validity of the HF approximation equation in Eqs. (12)-(17) is improved in the mmWave range, leading to improved extraction accuracy.

Typically, it is better to select the lowest frequency for the LF value and the highest frequency for the HF value in the measured frequency range. However, in this work, the LF extraction is performed using the 500 MHz data, because the lowest frequency of 100 MHz seems to have some measurement uncertainty. For the HF extraction, the average value of the 38-40 GHz measurement range is used, considering the fluctuation observed in the measured HF data.

To extract $R_b$, the following equation derived from Eq. (8) is employed:

At LFs where $\omega^2 R_b^2 (C_{jso} + C_{jdo} + C_{gbi})^2 \ll 1$, Eq. (18) simplifies to $Re(Y^i_{22}) \approx R_b C_{jdo}^2 \omega^2$. Since $R_b C_{jdo}^2$ is the LF slope of the $Re(Y^i_{22})$ versus $\omega^2$ curve, $R_b$ is calculated by dividing the slope by the extracted $C_{jdo}^2$.

Table 1 shows the sensitivity analysis of the extracted parameters in the case of shifting LF or HF points. As shown in Figs. 3-5, the slope of the decreasing trend in the LF region is much steeper than that in the HF region. Consequently, the variation in the extracted parameter values according to the frequency points selected for determining the LF and HF values in Eqs. (12)-(18) is more pronounced in LF than in HF. When LF values are selected at the measurement frequency point of 900 MHz, the average magnitude of percentage change in the extracted parameters compared to the novel method is approximately 19.8%, as shown in Table 1.

To examine the change in the extracted parameters based on the selected HF points, an additional parameter extraction is carried out using the average value in the 28-30 GHz range, which is lower than the 38-40 GHz frequency range selected for the HF values in Eqs. (12)-(18) using the novel method. Although the frequency shift in HF extraction is much larger than LF one in Table 1, the resulting average magnitude of % change in the extracted parameters based on those using the novel method is about 9.8%, which is considerably lower than that of the LF extraction. Therefore, we conclude that the proposed extraction method exhibits higher parameter sensitivity to LF value selection than HF one, with $R_b$ showing the largest variation among all parameters. Furthermore, Table 1 shows that the LF point shift strongly affects $C_{jdo}$ ($= C_{jso}$), $C_{gbi}$ and $R_b$, while the HF point shift largely affects $C_{gbi}$, $C_{gbe}$ and $R_b$.

Fig. 6 shows the extracted model parameters in Fig. 2 as a function of $V_{GB} = V_{GS}$, using the proposed direct method. The extracted parameters show physically consistent trends: As $V_{GB}$ decreases, the channel depletion depth $X_d$ of the active region becomes shallower. Since the gate oxide capacitance $C_{OX}$ and channel depletion capacitance $C_{de}$ are connected in series between the gate and body, the total gate-body capacitance $C_{gbi} + C_{gbe}$ increases in the accumulation region due to the increase in $C_{de}$ ^[5]. Therefore, $C_{gbe}$, which is separated to model the body-distributed effect, exhibits a similar voltage dependence to $C_{gbi}$, but is smaller than $C_{gbi}$. Additionally, the reduction in $X_d$ leads to an increase in the gate-edge sidewall junction area between the lightly doped drain (LDD) and the internal body, resulting in an increase in $C_{jdo}$ and $C_{jso}$ as $V_{GB}$ decreases ^[11]. As $V_{GB}$ decreases, $R_b$ shows little change in the depletion region, but sharply decreases at $V_{GB} < -0.5$ V in the thin body because the effective body thickness increases due to the decreased channel depletion depth.

III. VERIFICATIONS

In Fig. 7, simulated $S$-parameters from a simple model (Fig. 1) are compared to a physical one (Fig. 2) together with the measured data. For the simple model simulation, the extracted resistances and inductances are de-embedded from measured $S$-parameters, and the following equations are then used to extract parameters in Fig. 1:

As shown in Fig. 7, the physical model using the novel direct method shows good agreement with the measured $S$-parameters in a wide frequency range (100 MHz ~ 40 GHz) and various bias conditions. To quantitatively verify the superior extraction accuracy of the proposed direct method, the error rates of the simple and physical models with respect to the measured data are evaluated using the following equation:

where $N$ represents the number of measured frequency points, $\sum_{i,j=1}^{2}$ denotes the sum of the errors for each $S_{ij}$, and $\sum_{f}$ denotes the summation for all measured frequency points.

The modeling error rates under various bias conditions for the simple and physical models are shown in Fig. 8. The physical model exhibits $S$-parameter error rates at various biases, which are considerably lower than those of the simple model, demonstrating superior accuracy under off-state bias conditions. However, as $V_{GB}$ decreases, the simple model shows a larger error rate than the physical one, because $R_b$ is neglected in the simple one. In Fig. 7, the error in $S_{11}$-parameter for the simple model is larger than that in $S_{22}$-parameter, and this error increases as $V_{GB}$ decreases. This $S$-parameter error occurs for the following reasons: In the input circuit of Fig. 2, $R_b$ is in series with large $C_{gbi}$, leading to a more significant frequency dependence on $S_{11}$-parameter, whereas in the output circuit, $R_b$ is in series with small $C_{jdo}$, which has a relatively minor frequency effect on $S_{22}$-parameter. Furthermore, the increase in $C_{gbi}$ in the accumulation region as $V_{GB}$ decreases results in a larger error in the $S_{11}$-parameter due to the shorted $R_b$ in the simple model.

Table 2 compares the main extraction parameters of Fig. 2 obtained from the previous complex method ^[7] and the proposed method in the accumulation region. The previous method extracts $C_{jdo}$ at $V_{GS} = V_{GB} = 0$ V, which fails to reflect the physical increase in capacitance as the gate voltage decreases ^[11]. Thus, an unphysically smaller $C_{jdo}$ than that obtained using the proposed method is extracted using the previous one. For $C_{gbi}$, the previous method extracts the value from the difference between the on-state ($V_{GS} = 2.5$ V) and the zero-$V_{GS}$ condition. Since the depletion region exists inside the n$^+$-doped polysilicon gate in the inversion region ^[12], the previous method incorrectly extracts a smaller $C_{gbi}$ than the value obtained in the accumulation region using the proposed method.

In the previous method ^[7], $C_{bo}$ and $C_{gbe}$ are extracted using the $Y$-parameter equations derived from the simplified equivalent circuit, assuming that the large value of $C_{gbi}$ is shorted in the HF region. However, this assumption is inaccurate because extra fractional terms composed of multiple capacitance components are added in Eqs. (16) and (17). Thus, the previous method yields extracted values of $C_{bo}$ and $C_{gbe}$ that are unphysically larger than those obtained using the novel method. In addition, the novel model exhibits an $S$-parameter error rate of 3.99% in the range of 100 MHz to 40 GHz, which is considerably lower than the 9.35% of the previous complex method at $V_{GS} = V_{GB} = -2.5$V, verifying the superior accuracy of the proposed simple method.

Therefore, the novel extraction method enables direct extraction of all parameters in the complex off-state equivalent circuit using only a single $S$-parameter measurement. Compared to previous methods requiring complex extraction processes, our method is much simpler and shows significantly improved accuracy, especially in the accumulation region.

IV. CONCLUSIONS

A novel extraction method is proposed for directly extracting all parameters of a physical off-state BCT PD-SOI MOSFET equivalent circuit without numerical optimization. By utilizing not only LF but also HF measurement data, this method enables complete parameter extraction using only a single $S$-parameter measurement in both accumulation and depletion regions. New $Y$-parameter equations are derived from LF and HF approximations of the off-state equivalent circuit, allowing direct extraction of all parameters. This approach significantly simplifies the extraction process compared to conventional direct methods that require additional test patterns or measurements at different biases. The accuracy of the novel method is validated by ensuring that the modeled $S$-parameters closely match the measured data at various biases up to 40 GHz.