I. Introduction

High resistivity (HR) partially depleted (PD) silicon-on-insulator (SOI) MOSFETs have a great advantage in RF SoC fabrication due to low cross-coupling, low cross-talk noise, and low leakage current. When the drain-source voltage $V_{DS}$ becomes higher in floating body (FB) PD SOI n-MOSFETs, the impact ionization hole current $I_{imp}$ generated in the pinch-off region flows into the grounded source, causing an increase in the internal body voltage $V_{Bi}$ as shown in Fig. 1(a). This rise in $V_{Bi}$ leads to a decrease in the threshold voltage $V_{th}$, resulting in an increase in the channel current $I_{ch}$, so called ``kink effect'' ^[1-5].

Fig. 1. The cross-section of (a) FB PD-SOI n-MOSFET; (b) BCT PD-SOI n-MOSFET.

The DC kink voltage $V_{kink}$, which indicates the $V_{DS}$ value at which the kink effect starts to occur, decreases as the channel length decreases due to the reduction in the drain-source saturation voltage $V_{DSAT}$, and the sub-bandgap $V_{kink}$ decreases to $0.6V$ below $E_{g}/q$ of $1.12V$ in Si as the mask gate length $L_{g}$ is scaled down to $0.13\mu m$ ^[6].

PD-SOI n-MOSFETs used in this study have the body doping concentration of $6\times 10^{17}$cm$^{-3}$, the Si body thickness of 80 nm, and the gate oxide thickness of 2.8~nm. In order to reduce the parasitic gate resistance for the purpose of improving RF and noise performances, the devices use multi-finger geometry with multiple polysilicon gate fingers. In Fig. 2, which shows the $I_{DS}-V_{DS}$ curve of a multi-finger FB PD-SOI n-MOSFET with the gate length $L_{g}$ of 0.1 ${\mu}$m, unit gate finger width $W_{u}$ of 10 ${\mu}$m, and the number of gate finger $N_{f}$ of 16, $V_{kink}$ is measured to be approximately $0.55V$ at $V_{GS}=0.5V.$

Fig. 2. Comparison between measured $I_{DS}-V_{DS}$ curves of FB and BCT PD-SOI n-MOSFETs.

This reduction in $V_{kink}$ may be a significant issue when designing low-power ICs that require a lower operating voltage. The significantly increased breakdown current at $V_{DS}>1.7V$ shown in Fig. 2 appears to be caused by the punch-through leakage current that arises from merged source-body and drain-body depletion regions at the high $V_{DS}$range.

To mitigate the non-linearity caused by the kink effect, body contact (BCT) PD-SOI devices in Fig. 1(b) have been utilized ^[7-16]. However, as the $W_{u}/L_{g}$ ratio of gate finger increases, an increase in $I_{DS}$ still occurs at a much higher $V_{DS}$ ($V_{kink}=1.2V$ at $V_{GS}=0.5V$) than that in FB devices as shown in Fig. 2. This phenomenon, similar to the kink effect, is caused by an increase in the internal body voltage ($V_{Bi}=I_{imp}R_{body})$ across the body resistance $R_{body}$ that linearly increases with the $W_{u}/L_{g}$ of BCT devices in Fig. 1(b) ^[14]. The increase in $V_{Bi}$ leads to a decrease in $V_{th}$, resulting in the rise of $I_{ch}$, so called a substrate current-induced body effect (SCBE) ^[14].

Recently, in the BCT device, it has been demonstrated that an anomalous RF inductive effect originating from negative capacitance by SCBE appears at the drain when a higher $V_{DS}$ than $V_{kink}$ is applied ^[14]. Due to this effect, the $S_{21}$ and $S_{22}$-parameters rotate clockwise in the lower and upper semicircles on the Smith chart, respectively, as the frequency increases. Also, research has been conducted to analyze this effect physically and to model it using a small-signal equivalent circuit ^[14]. In addition, research has been conducted on how to effectively extract these parameters using a simple RF inductive model with an RLC resonant circuit in BCT devices ^[15,16].

Fig. 3 shows the measured $S_{22}$-parameters of our FB and BCT PD-SOI n-MOSFETs with the $I_{DS}-V_{DS}$ characteristics in Fig. 2. In Fig. 3, similar RF inductive effects are observed in both FB and BCT devices. As shown in Fig. 3, the starting point at the minimum measurement frequency ($10MHz$) in the rotating trajectory shifts to the left as $V_{DS}$ increases. Additionally, this trajectory rotates in a clockwise direction and its radius increases. As verified previously in a BCT device ^[14], this RF inductive effect in the FB device is also caused by the negative capacitance shown in Fig. 4, where the output drain-source capacitance of $C_{out}$ is determined from $\left(1/\omega \right)\text{Imag}\left(Y_{22}\right)$ converted from measured $S$-parameters. It is revealed that this negative capacitance arises from impact ionization, using an output AC equivalent circuit of Fig. 5 that is similar to one in a BCT device ^[14], with the only difference being that $R_{body}$ becomes infinite in the FB device.

Fig. 3. Admittance Smith chart for the measured $S_{22}$ -parameters of (a) FB PD-SOI n-MOSFET; (b) BCT PD-SOI n-MOSFET at $V_{GS}$=0.5V , with varying $V_{DS}$ in the frequency range of 0.01∼20 GHz.

In the case of a BCT device with $L_{g}=0.25\mu m$, it has been previously reported that the rotational trajectory of $S_{22}$-parameter apparently begins to appear at the DC $V_{kink}$ of around $1.7V$ ^[14-16]. In our BCT device with $L_{g}=0.1\mu m$ in Fig. 2, it is also observed that the RF inductive effect starts from the DC $V_{kink}$ of around $1.2V$ in Fig. 3(b). This is very reasonable because the negative capacitance originates from impact ionization leading to the occurrence of the kink effect.

However, in the FB device, the rotational locus of $S_{22}$-parameter from 10 MHz is observed in Fig. 3(a) only when $V_{DS}\geq 1.2V,$ which is significantly higher than the DC $V_{kink}$ of $0.55V$ in Fig. 2. Unlike the BCT devices where the starting voltage of the inductive effect is about same as DC $V_{kink}$, this $V_{DS}$ dependent RF inductive effect in the FB device, which occurs at about two times higher $V_{DS}$ than DC $V_{kink}$, is anomalous because the negative capacitance caused by the kink effect still exists down to $0.6\,\mathrm{V}$ in Fig. 4. This anomalous effect may be advantageous in the design of low-power RF ICs because the inductive locus in Fig. 3 disappears up to the operating voltage of $1.1V\,.$ However, the physical reason for the discrepancy between DC $V_{kink}$ and the voltage at which the RF inductive effect begins to appear has not been studied yet.

Fig. 4. Measured curve of $C_{OUT}$ versus frequency at different $V_{DS}$ for an FB PD-SOI n-MOSFET.

Therefore, in this paper, we newly analyze the $V_{DS}$-dependent RF inductive effect to reveal the origin of this anomalous discrepancy in the FB device. We focus on the analysis of the rotation trajectory in the $S_{22}$-parameter, based on the pole frequency $f_{p}$and the maximum magnitude of the output susceptance.

II. RF Inductive Effect

Fig. 5 shows the physical output equivalent circuit that considers the impact ionization and parasitic BJT ^[14] for an FB PD-SOI MOSFET, where $C_{gd}~$ is the gate-drain capacitance, $g_{dso}$ is the drain-source output conductance, $C_{box}$ is the buried oxide coupling capacitance ^[17], $C_{bd}$ is the body-drain junction capacitance, $C_{bs}$ is the body-source capacitance, $g_{bs}$ is the dynamic body-source conductance, $g_{mb}$ is the body transconductance, $g_{mp}$ is the transconductance of parasitic BJT, and $g_{mi}$ is the conductance for impact ionization current. The parasitic resistances ($R_{s}$, $R_{d}$) are omitted in Fig. 5, because these have a negligible frequency effect on the $Y_{22}$-parameter within a few $GHz$ in the Fig. 7 and 8.

Fig. 5. A physical output equivalent circuit for an FB PD-SOI MOSFET with kink effect in the saturation region. Since the body is floating, $R_{body}$ is removed from the output equivalent circuit for a BCT PD-SOI MOSFET [14].

Fig. 6. A simple output equivalent circuit for modeling the RF inductive effect.

Fig. 7. Measured $G_{out}$ versus frequency curves at different $V_{DS}$ for an FB PD-SOI n-MOSFET.

After deriving the negative drain-source capacitance $C_{dsb}$ equation from Fig. 5, it is converted into the equivalent drain-source inductance $L_{dsb}$ using $-1/\left(\omega ^{2}C_{dsb}\right)$and approximated at $f\gg g_{bs}/\left[2\pi \left(C_{bs}+C_{bd}\right)\right]$ as follows ^[14]:

In order to analyze the $V_{DS~ }$- dependence of the RF inductive effect in the FB device effectively, we uses a simple RLC output equivalent circuit ^[15,16] in Fig. 6, where $L_{k}$ is the effective inductance defined by (1), $R_{k}$ is the kink resistance, $g_{dso}$ is the non-kink drain-source conductance, and $C_{tot}$ is the total output capacitance in Fig. 5.

The output admittance $Y_{22}$ of Fig. 6 can be expressed as follows:

As the frequency increases, the output conductance $G_{out}$ of (2) rapidly decreases beyond the pole frequency $f_{p}$, which is calculated as $R_{k}/\left(2\pi L_{k}\right)$. After that, $G_{out}$ becomes a constant $g_{dso}.$ Similarly, the output susceptance $B_{out}$ of (3) decreases in a negative direction and then increases again in a positive direction. These trends agree well with the frequency-dependent curves shown in Fig. 7 and 8.

On the other hand, in the admittance Smith chart plot shown in Fig. 3, the $S_{22}$-parameter is represented by the intersection of the constant conductance circle and the constant susceptance circle. As the frequency increases, the conductance and susceptance change exhibits a rotational locus on the Smith chart. In Fig. 7 and 8, it is observed that as the frequency increases, $G_{out}$ decreases rapidly and then becomes a constant, while the magnitude of negative $B_{out}$($\left| B_{out}\right|$) increases and reaches its peak at $f_{min}\,.$ According to the frequency dependencies of $G_{out}$ and $B_{out},$ the $S_{22}$-parameter moves in a clockwise direction on the upper side of the real axis, as shown in Fig. 3.

As the value of $G_{out}$ decreases in Fig. 7, $Real\left(S_{22}\right)$ shifts toward the right. When $B_{out}$ reaches zero, $G_{out}$ is almost minimized, and $Real\left(S_{22}\right)$ stops moving at the right end. Thus, the frequency trace of the $S_{22}$-parameter in Fig. 3 is shown as a clockwise-rotating curve.

To determine the $V_{DS}$ dependence of this RF inductive effect of the FB device shown in Fig. 3(a), we need to consider the frequency-dependence of $G_{out}$ and $B_{out}$. Specifically, we utilize the magnitude of the minimum $B_{out}$($\left| B_{out\left(min\right)}\right|$) and the corresponding frequency $f_{min}$. This is because the rotation radius of the $S_{22}$-parameter trajectory in Fig. 3 is determined by $\left| B_{out\left(min\right)}\right|$, and the rotation angle is determined by $f_{p}$. As shown in Fig. 3, the rotation radius and angle for the $S_{22}$-parameter decrease as $V_{DS}$ decreases. As a result, the RF inductive effect does not appear at $V_{DS}\leq 1.1V$ on the Smith chart.

Fig. 8. Measured $B_{out}$ versus frequency curves at different $V_{DS}$ for an FB PD-SOI n-MOSFET.

Firstly, to determine $\left| B_{out\left(min\right)}\right|$ in Fig. 8, we need to find $f_{min}$ at which $dB_{out}/df=0$ and substitute it into (3). However, this process is quite complex and difficult to derive. To avoid this problem, we can simplify (3) by ignoring $C_{tot}\,,$ which is much smaller than $L_{k}/\left({R_{k}}^{2}+\omega ^{2}{L_{k}}^{2}\right)$. The $f_{min}$ obtained in this way is approximated by $f_{p}$ in (2):

Accordingly, substituting (4) into (3) yields:

In Fig. 7, the kink effect that causes the increase in $G_{out}$ disappears at high frequencies (HFs) where $f\gg f_{p}$, and $G_{out\left(HF\right)}\approx g_{dso}$ as shown in (2). Therefore, the low-frequency (LF) kink conductance $G_{k\left(LF\right)}\left(=1/R_{k}\right)$ is extracted by subtracting $G_{out\left(HF\right)}$ from the $G_{out\left(LF\right)}$ value at the minimum measurement frequency of $10MHz$. However, the measured $G_{out}$ at $10MHz$ below $V_{DS}=1.4V$ is lower than the DC value. As a result, the extracted $G_{k\left(LF\right)}$ value is inaccurate at $V_{DS}<1.4V$. However, accurate $G_{k\left(LF\right)}$ values can be extracted at $V_{DS}=1.5\sim 1.9V$.

Fig. 9 displays the extracted values of $-G_{k\left(LF\right)}/2$ compared with $B_{out\left(min\right)}$ measured from Fig. 8. In Fig. 9, $-0.5G_{k\left(LF\right)}$ shows good agreement with the measured $B_{out\left(min\right)}$, with an error rate within 10%. This indicates that the term of $(R_{k}/L_{k})C_{tot}$ in (5) is much smaller than $-0.5G_{k\left(LF\right)}\,.$ Thus, we can approximate $B_{out\left(min\right)}$ as $-G_{k\left(LF\right)}/2$.

Fig. 9. Measured $B_{out(min)}$ and extracted -$G_{k(LF)}$ as a function of $V_{DS}$ for an FB PD-SOI n-MOSFET.

Since $\left| B_{out\left(min\right)}\right|$ is progressively reduced at lower $V_{DS}$ in Fig. 9, the rotation radius of the $S_{22}$-parameter trajectory also continuously decreases in Fig. 3. To explain the anomalous disappearance of the $S_{22}$-parameter trajectory for the FB device as $V_{DS}$ decreases in Fig. 3(a), we will conduct a physical analysis of the dependencies of $V_{DS}$ on $G_{k\left(LF\right)}$ and $f_{p}$ in the next section.

III. Analysis of Bias-dependence

The primary purposes of this section are to analyze the $V_{DS~ }$- dependence of $G_{k\left(LF\right)}$ and $f_{p}$ in order to explain the invisibility of the $S_{22}$-parameter trajectory of the FB device at the bias region of $0.6\,\mathrm{V}\leq V_{DS}\leq 1.1V$ in Fig. 3(a) using the rotation radius and angle for the $S_{22}$- parameter.

1. Kink Conductance

From Fig. 5, $G_{k}$ is derived as a function of frequency as follows ^[14]:

(6)

$G_{k}=\left(g_{mb}+g_{mp}\right)\left[\frac{g_{mi}g_{bs}+\omega ^{2}C_{bd}\left(C_{bs}+C_{bd}\right)}{{g_{bs}}^{2}+\omega ^{2}\left(C_{bs}+C_{bd}\right)^{2}}\right]$.

In (6), $f_{p}$ is given by:

(7)

$f_{p}=\frac{g_{bs}}{2\pi \left(C_{bs}+C_{bd}\right)}$.

In (6), $G_{k\left(LF\right)}$ in the low-frequency region where $f\ll f_{p}$ is approximated as:

(8)

$G_{k\left(LF\right)}\approx \frac{\left(g_{mb}+g_{mp}\right)g_{mi}}{g_{bs}}$.

According to the kink effect, which causes the RF inductive effect, the impact ionization body current $I_{B}(I_{imp}),$ generated by $V_{DS}$ in the pinch-off region, flows to the internal body-source junction in the FB device. Due to the impact ionization process ^[5,18], $I_{B}$ is defined as $I_{DS}\left(M-1\right),$ where the low-voltage thermally-assisted impact ionization multiplication factor $M$ is expressed by the following equation ^[6].

(9)

$\mathrm{M}-1=M_{0}\exp \left\{\frac{\left[q\left(V_{DS}-V_{DSAT}\right)-E_{g}\right]}{mkT}\right\}$

where $M_{0}$ is the value of $M-1$ at $V_{DS}=V_{DSAT}+E_{g}/q$, $E_{g}$ is the energy gap in Si, and $m$ is the ideality factor of impact ionization.

Using (9), $g_{mi}$ under the condition of $qI_{DS}/\left(mkT\right)\gg g_{dso}$ is defined as:

(10)

$g_{mi}=\frac{dI_{B}}{dV_{DS}}=\frac{d[I_{DS}\left(M-1\right)]}{dV_{DS}}$

$=g_{dso}\left(M-1\right)+I_{DS}\frac{d\left(M-1\right)}{dV_{DS}}\approx \frac{qI_{B}}{mkT}$.

The I-V characteristic equation of the body-source junction where $I_{B}$ flows is expressed as:

(11)

$I_{B}=I_{bo}\left[\exp \left(\frac{qV_{Bi}}{\eta kT}\right)-1\right]$

where $I_{bo}$ is the reverse saturation base current and $\eta$ is the ideality factor of the body-source junction.

Using (11), $g_{bs}$ is defined as

(12)

$g_{bs}=\frac{dI_{B}}{dV_{BS}}\approx \frac{qI_{B}}{\eta kT}$.

Using (9), (10) and (12), the following equation is obtained:

(13)

$g_{mi}=\frac{\eta }{m}g_{bs}$

$=\frac{q}{mkT}I_{DS}M_{0}\exp \left\{\frac{\left[q\left(V_{DS}-V_{DSAT}\right)-E_{g}\right]}{mkT}\right\}$.

Since $g_{mp}$ is much smaller than $g_{mb}$ in the kink bias ^[14], $G_{k\left(LF\right)}$ in (8) can be approximated as $\left(\eta /m\right)g_{mb}$ using (13). This $g_{mb}$ can be obtained using the following formula in the saturation region:

(14)

$I_{DS}=\frac{\mu _{n}C_{ox}W_{u}N_{f}}{2L_{g}}\left(V_{GS}-V_{th}\right)^{2}\left(1+cV_{DS}\right)$

where $\mu _{n}$ is the electron mobility, $c$ is the channel length modulation parameter, and the threshold voltage $V_{th}$ is expressed as ^[19]:

(15)

$V_{th}=V_{FB}+2\Psi _{B}+\sqrt{2\epsilon _{s}qN_{ch}\left(2\Psi _{B}-V_{Bi}\right)}/C_{ox}$

where $V_{FB}$ is the flat-band voltage, $\Psi _{B}$ is the surface potential, $\epsilon_{s}$ is the permittivity of Si, $N_{ch}$ is the channel doping concentration, and $C_{ox}$is the gate oxide capacitance per unit channel area.

Using (14) and (15), $g_{mb}$ is defined as follows:

(16)

$g_{mb}=\left(dI_{DS}/dV_{th}\right)\left(dV_{th}/dV_{Bi}\right)$

$=\frac{g_{m}}{C_{ox}}\sqrt{\frac{\epsilon _{s}qN_{ch}}{2\left(2\Psi _{B}-V_{Bi}\right)}}$

where $g_{m}$ is the MOSFET transconductance given by:

(17)

$g_{m}=\frac{\mu _{n}C_{ox}W_{u}N_{f}}{L_{g}}\left(V_{GS}-V_{th}\right)\left(1+cV_{DS}\right)$.

As $V_{Bi}$ increases due to $I_{B}$ generated at high $V_{DS}$, $g_{mb}$ also increases in (16). In order to accurately calculate the $V_{DS}$-dependent effect of $g_{mb}$ instead of $V_{Bi}$, a relational expression between $V_{Bi}$ and $V_{DS}$ is required.

Since $I_{DS}\left(M-1\right)$ is equal to (11), the following expression at $V_{DS}>V_{kink}$ is defined as ^[6]:

(18)

$V_{DS}=\frac{mV_{Bi}}{\eta }+\frac{mkT}{q}\ln \left(\frac{I_{bo}}{I_{DS}M_{0}}\right)+\frac{E_{g}}{q}+V_{DSAT}$

where $m$, $\eta$, $I_{b0}$, $M_{0}$, and $V_{DSAT}$ are independent of $V_{DS}$.

As $V_{DS}$ increases in the kink region, $I_{DS}$ increases and the second term of (18) involving a logarithmic function decreases. However, this decrease is much smaller compared to the increase of $V_{Bi}$ in the first term. Therefore, (18) can be expressed as a linear function of $V_{DS}\approx aV_{Bi}+b$. Substituting this linear function back into (16), we obtain the $V_{DS}$-dependent equation of $g_{mb}=a\left(1+cV_{DS}\right)\left(b-V_{DS}\right)^{-0.5}$. According to this equation, the value of $G_{k\left(LF\right)}\approx \left(\eta /m\right)g_{mb}$ is reduced as $V_{DS}$ decreases, as shown in Fig. 10.

Fig. 10. Extracted $G_{k(LF)}$ data from Fig. 7 as a function of $V_{DS}$ for an FB PD-SOI n-MOSFET.

Since $\left| B_{out\left(min\right)}\right| \approx 0.5G_{k\left(LF\right)}$ in (5), it is verified that the reduction of $\left| B_{out\left(min\right)}\right|$ with decreasing $V_{DS}$ in Fig. 9 is primarily due to the $V_{DS}~$- dependence of $g_{mb}$ in (16). Accordingly, the $g_{mb}$ equation provides a theoretical explanation for the decrease in the rotation radius of the $S_{22}$-parameter trajectory. Based on $G_{k\left(LF\right)}$ in Fig. 10, it is confirmed that the rotation radius decreases largely from $1.8V$ to $1.7V,$ and then gradually decreases further below $1.6V$.

2. Pole Frequency

Meanwhile, $G_{out}$ in Fig. 7 is measured from the minimum measurement frequency of $10MHz$ in our vector network analyzer. If $f_{p}$ is less than $10MHz$, the $G_{out}$ measured at $10MHz$ is much smaller than the $G_{k\left(LF\right)}$ value at DC, and the $B_{out\left(min\right)}$ at $f_{p}$ can’t be measured. Thus, the measured $\left| B_{out}\right|$ at $10MHz$, which is much less than $\left| B_{out\left(min\right)}\right|$, only represents the latter part of the entire rotation trajectory of the $S_{22}$-parameter in Fig. 3. Therefore, in order to accurately understand the $V_{DS}$-dependency of the $S_{22}$-parameter rotation trajectory, $f_{p}$ should be measured, and it is necessary to analyze the $V_{DS~ }$-dependency on $f_{p}$ in (7).

To determine the accurate values of $f_{p}$, we utilize a novel curve-fitting method ^[20] based on the simple frequency-dependent $G_{out}$ equation derived from (6) :

(19)

$G_{out}=G_{k}+~ g_{dso}=\frac{H}{1+\left(f/f_{p}\right)^{2}}+K$.

where

(20)

$\mathrm{H}=G_{k\left(LF\right)}\left[1-\left(f_{p}/f_{z}\right)^{2}\right]$

(21)

$\mathrm{K}=\frac{\left(g_{mb}+g_{mp}\right)C_{bd}}{C_{bs}+C_{bd}}+g_{dso}$

where $f_{z}$ is the zero frequency of $G_{k}$ in (6) and is expressed as:

(22)

$f_{z}=\frac{1}{2\pi }\sqrt{\frac{g_{mi}g_{bs}}{C_{bd}(C_{bs}+C_{bd})}}$.

Under the kink bias, $C_{bd}\ll C_{bs},$ because the body-drain junction is reverse-biased and the body-source junction is forward-biased in the parasitic BJT. Under the condition of $C_{bd}\ll C_{bs}\,,$ it is satisfied that $g_{dso}\gg \left(g_{mb}+g_{mp}\right)C_{bd}/\left(C_{bs}+C_{bd}\right)$ in (21), thus resulting in $G_{out\left(HF\right)}=K~ \approx g_{dso}\,.$ Since $H~ \approx ~ G_{k\left(LF\right)}$ at $f_{p}\ll f_{z}$ in (20), $G_{k\left(LF\right)}\approx G_{out\left(LF\right)}-G_{out\left(HF\right)}$ in Fig. 7.

Since H and K at the fixed bias are constant values that are independent of frequency, the $V_{DS}$-dependent $f_{p}$ data is easily extracted by fitting (19) to match with the $G_{out}$ versus frequency data in Fig. 7. When $V_{DS}\leq 1.1V$, the extraction of $f_{p}$ data is impossible because it is less than $10MHz$. Thus, only $f_{p}$ data extracted at voltages above $1.2V$ is shown in Fig. 11. The log$(f_{p})$ increases linearly when $V_{DS}>1.2V$ and gradually saturates after $V_{DS}$ reaches $1.5V$.

Under the kink bias region, (7) can be expressed as $f_{p}\approx g_{bs}/C_{bs}$, where $C_{bs}$ is the sum of the depletion capacitance $C_{js}$ and the diffusion capacitance $C_{diff}\,.$ Under forward bias, $C_{js}$ and $C_{diff}$ are expressed as:

(23)

$C_{js}=C_{jso}\left(1+M_{js}V_{Bi}/V_{built\_ in}\right) \\ $

(24)

$C_{diff}=\frac{q\beta _{F}I_{bo}\tau _{F}}{kT}\exp \left(\frac{qV_{Bi}}{kT}\right)$

where $C_{jso}$ is the value of $C_{js}$ when $V_{Bi}=0V$, $M_{jd}$ is the junction grading coefficient, $V_{built\_ in}$ is the built-in potential, $\beta _{F}$ is the common-emitter(source) current gain, and $\tau _{F}$ is the forward transit time.

In Fig. 11, when $V_{Bi}$ is lower than $V_{built\_ in}$ of the body-source junction, $C_{bs}$ is primarily influenced by $C_{js}$ because $C_{diff}\ll C_{js}\,.$ Since $V_{DS}$ and $V_{Bi}$ have a linear relationship in (18), $C_{js}$ in (23) is also a linear function of $V_{DS}\,.$ Additionally, $g_{bs}$ exponentially increases due to impact ionization as $V_{DS}$rises in (13). Thus, $ln\left(f_{p}\right)$ is expressed as $ln(g_{bs})-ln\left(C_{js}\right)\approx CV_{DS}-Dln\left(V_{DS}\right)+E.$ Since$ln(g_{bs})$ increases more rapidly than $ln\left(C_{js}\right)$ with rising $V_{DS}$, the slope in Fig. 11 becomes approximately linear at $V_{DS}\leq 1.5V$. However, when $V_{Bi}$ is higher than $V_{built\_ in}$, $C_{diff}$ in (24) becomes dominant ($C_{diff}\gg C_{js}$). Therefore, the value of $f_{p}\approx g_{bs}/C_{diff}$ becomes saturated at very high $V_{DS}$ values.

In Fig. 11, it is evident that $f_{p}$ is less than $10~ MHz$ for $V_{DS}$ values lower than $1.1V$. Accordingly, the measured $G_{k}$ at $10MHz$ becomes negligible at $V_{DS}\leq 1.1V$ in Fig. 7, as $G_{out}$ decreases rapidly for $f>f_{p}$. Thus, the values of $\left| B_{out}\right|$ at frequencies above $10MHz$ at $V_{DS}\leq 1.1V$ are much lower than $\left| B_{out\left(min\right)}\right|$ shown in Fig. 9. Since $f_{p}$$\leq$$10MHz$ at $V_{DS}\leq 1.1V$, the rotation angle of the frequency trajectory in the $S_{22}$-parameter in Fig. 3 is substantially reduced at $V_{DS}\leq 1.1V$ compared to $180^{\circ}$ at very high $V_{DS}$.

Fig. 11. Extracted $f_{p}$ data from Fig. 7 using (19) as a function of $V_{DS}$ for an FB PD-SOI n-MOSFET.

IV. Conclusion

In order to reveal the physical origin of the anomalous $V_{DS}~$- dependence in the RF inductive effect of FB PD-SOI n-MOSFETs for the first time, the variation of the rotation trajectory of the $S_{22}$-parameter on the Smith chart at different $V_{DS}$ values is newly analyzed. This new analysis is based on the frequency-dependent equations of $G_{out}$ and $B_{out}$, which are derived from an output equivalent circuit. The accuracy of $\left| B_{out\left(min\right)}\right| \approx G_{k\left(LF\right)}/2$, derived from the simple RLC circuit, is verified using the $G_{k\left(LF\right)}$ extracted from the measured $G_{out}$ data. We also physically proved that $G_{k\left(LF\right)}\approx \left(\eta /m\right)g_{mb}\,.$ Using the physically derived equation for $V_{DS}$-dependent $g_{mb}$, it is confirmed that the decreased turning radius of the $S_{22}$-parameter at lower $V_{DS}$ is caused by the reduction of $g_{mb}$. In addition, it has been found that $f_{p}$ decreases below the minimum frequency of $10MHz$ when $V_{DS}\leq 1.1V$. This reduction in $f_{p}$ is caused by the exponential decrease in $g_{bs}$ as $V_{DS}$ decreases. Due to the reduction in $g_{mb}$ and $f_{p}$ at lower $V_{DS}$, the RF inductive effect in the $S_{22}$-parameter of FB devices does not appear at $V_{DS}=0.6\sim 1.1V$, which is larger than the DC $V_{kink}$, even though negative capacitance exists.