1. Model Implementation in BSIM

The gate leakage current model is based on the inelastic trap-assisted tunneling process (ITAT). The model describes electrons tunneling to deep-lying trap states and, immediately, released to positions deeper than the original position, and subsequently tunneling to the gate ⁽¹⁵⁾. In terms of trap-related parameters, $X_{t}$, $\phi _{t}$, ${CCS}$, and ${NOID}$, defined in Table 1, the expressions for the gate leakage current density with the drain-to-source

voltage $V_{ds}$ = 0 $V$ are rewritten as ⁽¹⁵⁾

with

where ${q}$ is the electron charge, $\mathrm{\hslash }$ is the reduced Planck constant, $\varepsilon _{ox}$ is the oxide dielectric constant, $\phi _{b}$ is the Si-SiO$_{2}$ barrier height, $m_{ox}$ is the effective mass of tunneling carriers, $t_{ox}$ is the oxide thickness, $E_{loss}$ is the energy loss, $F_{ox1,2}$ is the local electric field, and ${C}$ is the correction function. The trap position$X_{t}=0.5t_{ox}$under uniform distribution extracted from a fitting procedure is consistent with the contour plot of experimental current density. The equations include the effects of charge stored in the trap states and the dependence of the energy loss on the oxide electric field ⁽¹⁵⁾.

The dependence of gate leakage current on the drain-source voltage is described by the source-drain partition model which has been already implemented in BSIM6 ⁽¹⁶⁾. The gate tunneling current is given by

with

where ${W}$ is the channel width, ${L}$ is the channel length, $P_{igcd}$ is a fitting parameter with a default value of 1, $V_{T}$ is the threshold voltage, and $V_{gs}$ is the gate-to-source voltage.

One possible mechanism for $1/f^{\gamma }$ noise contribution is for traps to affect locally the barrier height or shape due to thermal noise of the frequency-dependent conductance of the oxide slow traps ⁽¹⁷⁾. This barrier height fluctuation (BHF) in turn modulates the tunneling transmission, and causes the fluctuation in the tunneling current. The $1/f^{\gamma }$ noise model of gate leakage current is implemented into BSIM6 with three assumptions, 1) The oxide slow traps are uniformly distributed over the energy and space, 2) the parallel trap conductance and capacitance is constant over low frequencies, and 3) the carrier tunneling is effective for trap levels around the quasi-Fermi level. Thus, the $1/f^{\gamma }$ noise model of gate leakage current can be simplified as ^(4,7)

with

where $\lambda _{t}=2\sqrt{2m_{ox}\phi _{b}}/\mathrm{\hslash }$, ${f}$ is the frequency, $C_{ox}$ is the oxide capacitance, $V_{ox}$ is the oxide voltage, $\psi _{s}$ is the surface potential, $V_{FB}$ is the flat-band voltage, and $V_{poly}$ is the polysilicon voltage drop.

The mechanism for shot noise associated with generation-recombination process can be explained by carrier transport based on trap-assisted tunneling ^(7,18,19). The generation rate, i.e., the transition rate from the substrate ($g_{st})$ and the gate ($g_{gt})$ to the unoccupied trap in the oxide, and recombination rate, i.e., the transition rate from the occupied trap to the substrate ($r_{ts})$ and the gate ($r_{tg})$, are caused by the statistical occupancy fluctuation of fast oxide traps in the oxide. The time constant $\tau _{fast}$ is associated with the rate equation for carrier number fluctuation $\Delta N$ with $\tau _{fast1}$ and $\tau _{fast2}$being the time-constants related to the variation of the generation-recombination rate due to the carrier number fluctuations through the substrate and gate, respectively ⁽⁷⁾. Making a Fourier analysis of the fluctuations $\Delta r_{tg}$ in the recombination rate and $\Delta g_{st}$ in the generation rate and Langevin equation for $\Delta N$, the Lorentzian-modulated(LM) shot noise model is given by ⁽⁷⁾

with

where $F^{Fano}$ is the Fano factor, $\tau _{fast1,2}$ is the local time constants, ${N}$ is the total number of carriers, and ${g}$ and ${r}$ are the uniform generation and recombination rate, respectively. The signs of Eq. (8c) and (8d) depend on

the positive or negative correlation between tunneling current components. As illustrated in Fig. 1, for example, different signs of the time constants lead to enhanced shot noise. The Fano factor for nitrided oxide nMOSFETs was simulated with ${f}$ = 10 kHz, $t_{ox,eq}=2.2\,\,\mathrm{nm}\,, $$\varepsilon _{ox}=5.7\varepsilon _{0}, $$\phi _{b,ox}=2.6\,\,\mathrm{eV}\,, $$m_{ox}=0.4m_{0}$, and $NOIE=2\times 10^{12}\,\,\mathrm{cm}^{- 2}\mathrm{eV}^{- 1}$at ${T}$ = 300 K.

Due to the quantized energy levels in the inversion layer, the peak charge density is situated at a distance away from the Si-SiO$_{2}$ interface ⁽⁴⁾. This results in the enhancement in the oxide thickness and the reduction in the oxide voltage. Therefore, the quantum effects are taken into account by using the correction for the oxide thickness, $t_{ox}^{QM}$, and the oxide voltage, $V_{ox}^{QM}$, as follows

where $x_{n}^{QM}$ is the average centroid position from the interface and $\psi _{s}^{QM}$ is the QM corrected surface potential.

2. Benchmark Tests

For the practical application of the compact noise model of gate leakage current, the models are implemented into BSIM6 in public domain, as shown in

Fig. 2. The basic device parameters are also used for the current and noise models, but the trap-related parameters are newly introduced into simulators. The validation for the compact model is derived from the comparisons of simulation results and measurements. nMOSFETs used for measurements were fabricated with using a manufacturable remote plasma nitride oxide (RPNO) process ⁽¹⁵⁾. The gate oxide was grown with both furnace and rapid-thermal based processes. Nitridation was performed by exposing the gate oxide to a short, high density, remote helicon-based nitrogen discharge. Following N$_{2}$-plasma nitridation and post-nitridation anneal of the gate oxide, an undoped poly-Si gate layer was deposited. For n+ gates, P was implanted, and rapid-thermal annealing was then performed in N$_{2}$ for different durations. All tests are to be done using a test circuit of a single MOSFET and an input deck in SPICE format is provided in Fig. 3(a). The models have been implemented through b6noi.c file of BSIM6 which is C code with higher efficiency. As described in Fig. 3(b), several functions are provided to calculate the gate leakage current, 1/${f}$ gate current noise, gate shot noise, and 1/${f}$ drain current noise. Fig. 4(a) shows the comparison of the test results and measured data of gate leakage current as a function of drain-source voltage, and the current density versus gate-source voltage for the inset. The test for nitrided oxide nMOSFETs was performed by using $W=10\mu \mathrm{m}\,, $ $L=10\mu \mathrm{m}\,, $ $t_{ox,eq}=$ 2.2 nm, $\varepsilon _{ox}=5.7\varepsilon _{0}$, $\phi _{b,ox}=2.6\,\,\mathrm{eV}$, $m_{ox}=0.4m_{0}$, and $NOID=NOIE=2\times 10^{12}\,\,\mathrm{cm}^{- 2}\mathrm{eV}^{- 1}$at ${T}$ = 300 K. For different gate oxide processing conditions, the trap

density parameters ($NOID,NOIE$) are technology dependent. The thermal oxide devices have a trap density level one to two orders of magnitude lower than that of nitrided oxide devices ⁽²⁰⁾. It is observed that the

simulation results have good agreement with measurements. Also, Fig. 4(b) illustrates the noise characteristics as a function of frequency for different gate-source voltages. It is also verified that the compact noise model can accurately predict the noise behavior over the frequency and bias ranges.

1. Drain Current Noise Model

In order to explain the $1/f^{\gamma }$ noise of drain current in ultrathin oxide MOSFETs, the tunneling assisted-thermally activated (TATA) time constant model is employed, which is physically consistent with the gate leakage tunneling process ^(7,15,21). Since the noise is strongly affected by trap-related properties, the noise model accounts for a realistic description of trap distribution. In addition, the model should include the effects of the two-dimensional electric fields, the parasitic resistance, the doping profile, and the mobility degradation on the noise behavior ⁽¹³⁾. The QM correction noise model can be obtained by accounting for the QM effects through the in the oxide capacitance, the threshold voltage, and the drain current in the charge density parameters as well as the gate oxide thickness ⁽⁴⁾. The total noise power spectral density at low frequencies is obtained by calculating the correlated fluctuations in the channel carrier number and mobility and performing the integration over the channel length ^(4,12,13). The QM correction model for $1/f^{\gamma }~ $drain current noise in the linear region, accounting for all effects mentioned above, is given by ^(4,13)

where $I_{d}^{QM}$ is the QM corrected drain current, $V_{T0}$ is the threshold voltage with $V_{ds}$= 0 V, $k_{bulk}$ is the bulk charge parameter, ${R}$ is the parasitic resistance, $\beta _{1,2}$ are the mobility parameters, $\mu _{n0}$is the maximum mobility in the inversion layer, $\tau _{eff0}$ is an effective attempt time constant, and $\lambda _{th,eff}$ and $\lambda _{tun,eff}$ are an exponential factor of thermal activation process and tunneling process, respectively. $N_{TA}$, $N_{TB}$, and$N_{TC}$are trap-related parameters, which are attributed to the noise sources for$1/f^{\gamma }$drain current noise. These parameters are different from the oxide trap density (${NOID}$) contributing to$1/f^{\gamma }$gate current noise, resulting from the partial correlation of the low frequency noise sources between the drain and the gate ^(7,14). Unlike BSIM6 noise model, $A^{*}$, $B^{*}$, and $C^{*}$ are not empirical constants but are analytical expressions as a function of $V_{gs}$ and $V_{ds}$⁽¹³⁾.

In the saturation region, the noise power spectral density is expressed as the sum of the noise components due to the triode region, $L- L_{sat}$, and the pinch-off region, $L_{sat}$, and is given by integrating over each region ^(4,12,13)

where $V_{dsat}$ is the drain saturation voltage, and $A_{Lsat}^{*}$, $B_{Lsat}^{*}$, and $C_{Lsat}^{*}$ are analytically expressed as a function of $V_{gs}$ and $V_{dsat}$.

2. Comprehensive Noise Model

Since the gate conductance in ultrathin gate oxide MOSFETs can be substantial due to the oxide traps at low frequencies, it is observed that a partial correlation between the drain and the gate current noise sources exists ^(7,14). In order to accurately predict the noise behavior at low frequencies, the comprehensive noise performance is evaluated by taking into account the cross correlation coefficient, $C^{LF}\left(f\right)$, between the drain and the gate current noise, and is expressed as a function of the current ratio, $\lambda _{p}=I_{g}/I_{d}$, and the noise ratio, $S_{I_{g}}^{LF}\left(f\right)/S_{I_{d}}^{LF}\left(f\right)$, as follows ⁽¹⁴⁾

In addition to the influence of the correlation on the low frequency noise performance, the shot noise due to the gate leakage current, $S_{I_{g}}^{shot}\left(f\right)=2qI_{g}F^{Fano}$, has an impact on the noise behavior, and thus should be incorporated into the minimum noise figure $F_{min}^{LF}$ as follows ^(14,22,23)

where $g_{mg}=\partial I_{d}/\partial V_{gs}$ is the transconductance, $g_{g}=$ $\partial I_{g}/\partial V_{gs}$ is the gate conductance, and $C_{gs}$ is the gate-to-source capacitance.

3. Simulation Results

From the viewpoint of compact noise modeling, the model should provide an accurate prediction of noise behavior, describing all the important physical effects without an increase of parameters. For nMOSFETs with ultrathin oxide (=2.2 nm), the $1/f^{\gamma }$ noise of drain current can be accurately predicted by the expressions of Eq. (10)-(13). Fig. 5 shows the comparison of measured data and simulation results carried out with single trap-related parameter under the assumption of uniform trap distribution in the oxide. The simulation was carried out with $t_{ox,eq}=2.2\,\,\mathrm{nm}\,,$ $V_{T0}=0.45\,\,\mathrm{V},$ $k_{bulk}=0.625,$ $R=10\,\,\Omega \,,$ $\mu _{n0}=550\,\,\mathrm{cm}^{2}\mathrm{V}^{- 1}\mathrm{s}^{- 1}$and$N_{TA}=1\times 10^{18}$ $\mathrm{cm}^{- 3}\mathrm{eV}^{- 1}$at ${T}$ = 300 K. Excellent agreements between measurement and simulation are observed over the frequency and in both the linear and saturation regions for two channel lengths, as shown in Fig. 5(a) and (b), respectively.

For the overall noise evaluation of advanced CMOS devices, the compact noise models at low frequencies should include the correlation effects between the gate and the drain, together with the gate and drain current noise models. Unlike the capacitive coupling at high frequencies, the low frequency correlation is mainly attributed to the trap-assisted tunneling process, which is

a common origin of the gate and drain current noise. With the help of an analytical expression of correlation coefficient, it is possible to evaluate the minimum noise figure, as illustrated in Fig. 6. The simulation was performed by using $W=10\mu \mathrm{m}\,,$ $L=10\mu \mathrm{m}\,,$ $t_{ox,eq}=$ $2.2\,\,\mathrm{nm}\,,$ $N_{TA}=1\times 10^{18}\,\,\mathrm{cm}^{- 3}\mathrm{eV}^{- 1}$and$NOID=2\times 10^{12}$ $\mathrm{cm}^{- 2}\mathrm{eV}^{- 1}$at ${T}$ = 300 K. Fig. 6(a) and (b) show the characteristics of minimum noise figure as a function of frequency for different gate-source voltages $V_{gs}$ and as a function of the gate-source voltage for different drain-source voltages $V_{ds}$, respectively. The increasing minimum noise figure with increasing $V_{gs}$ and decreasing $V_{ds}$ can be interpreted as being due to the gate leakage current and its noise components, greatly affecting the overall noise performance at low frequencies.