INTRODUCTION
The DG MOSFET (DGMOS) has emerged as the most promising device for circuit design in nanometer scale. This is essentially due to its excellent control of the short channel effects: Drain Induced Barrier Lowering (DIBL), subthreshold current, tunnel current and better ‘ON’ current (Park and Colinge, 2002; Pati et al., 2013; Sarkar et al., 2008).
To continue the downward scaling, dielectrics with higher dielectric constant are being suggested as a solution to achieve the same transistor performances while maintaining a relatively thick physical thickness in order to minimize the leakage tunneling current. A lot of studies has been done in this area (Tripathi et al., 2012; Chang and Shin, 2002). Hafnium oxide (HfO_{2}) and Zirconium oxide (ZrO_{2}) have a good potential for present and future CMOS application (Lim and Armstrong, 2007). On the other hand, the use of DG MOSFET in Radio frequency (RF) applications is more and more considered due to their interesting high cutoff frequency (f_{t}) and high maximum oscillation frequency (f_{max}) which could be getting (VideivicMisic and Jevtic, 2004; Varadharajan and Kaya, 2005; Liang et al., 2008). The Voltage Controlled Oscillator (VCO) is a typical example of this applications, it is an essential part of many electronic systems such as image rejection demodulator in wireless transceivers (Li and Afshari, 2010).
The phase noise in RF oscillators is one of the most critical performance parameters for signal generators and integrated transceivers. Many works had been done using Spicelike simulations (Leeson, 1966; McNeill, 1994; Hajimiri and Lee, 1998, 1999; Hajimiri et al., 1999; Demir et al., 2000; Lee and Hajimiri, 2000; Navid et al., 2005). Hajimiri and Lee (1998) have proposed a time variant model (LTV) based on the socalled Impulse Sensitivity Function (ISF) to predict this phase noise. This technique provides insight into the design of oscillator. The ISF function characterizes only the stationary results of a perturbation. The purpose of this study is the modeling and the optimization of DG MOSFET for Radio frequency application. For this, a numerical modeling of this device using a 2D PoissonSchrodinger self consistent approach was presented, taking into account the effects to the aggressive scaling of the device (SCE effects, I_{off}, I_{Gtunnel}). The obtained results can give helpful design guidelines for DG MOS devices for RF applications. Indeed, we ‘built’ an LC Colpitts oscillator around the considered device and then, present the method to calculate Impulse Sensitivity Function. These results are favorably compared with a developed analytical model.
MATERIALS AND METHODS
Device mathematical model: The mathematical model consists on a 2D solution of Poisson equation coupled to a 1D Schrödinger equation (Bella et al., 2011). Usually, 2D Poisson equation describes electrostatic transport and 1D Schrodinger equation is used to handle the quantum transport. The coupling of Poisson and Schrödinger equations is necessary when the oxide thickness is decrease and the thickness of the channel takes values close to the wavelength of the electrons. This approach is justified by several results presented by Bescond et al. (2004) and Colinge (2008).
The Poisson (Eq. 1) and Schrodinger (Eq. 2) are given by the following equations:
where, V(x,y) is the electrostatic potential, ρ(x,y) is the charge density, ε is the permittivity of the dielectric material constant, m* is the electron effective mass, q is the electron charge, h is the Planck’s constant and ψ(y) is the wave function corresponding to the eigenvalue E.
It is clearly seen that the Eq. 1 and 2 are coupled. It is therefore self consistency in their resolution. The self consistent system can be illustrated by the following one:
where, the functions S[V(y)] and P[ρ(y)] represent the Schrödinger and Poisson equations. To solve these later, the concept of the finite difference method was used (Bella et al., 2011). The system equations obtained is then numerically solved by the NewtonRaphson method. The results are obtained with a precision within 10^{9}.
Structure device: The DG MOSFET structure under analysis and related parameters values are shown in Fig. 1 and Table 1.

Fig. 1:  Schematic of the double gate MOS transistor 

Fig. 2:  Considered Colpitts oscillator 
Table 1:  Main parameter used in this study 

The threshold voltage of the DG MOSFET is adjusted to be 0.15 V using a gate work function (φ_{m}) equals to 4.25 eV corresponding to SiO_{2} material. The voltages Vgs applied to the two gates are identical.
Mixed mode simulation (DGMOS oscillator)
Oscillator: An RF oscillator is ‘Built’ around the considered DG MOSFET. A mixedmode analysis involve, applying a 2D PoissonSchrodinger model to the device, whereas, the rest of the circuit used is governed by Kirchhoff’s laws using the DESSIS tool in the ISETCAD software (ISE., 2002) (Fig. 2). In this fact, the simulator combines the device and circuit equations in one signal equation system.
In order to get a stable oscillation with a frequency oscillation f_{0}, the system must satisfy the Barkhausen criteria:
where, G(ω_{0}) is the gain of the amplifier (DGMOS) and β(ω_{0}) is the attenuation introduced by the LC tank.

Fig. 3(ab): 
Consequence of the injection of a (a) Maximum noise amplitude and (b) Zerocrossing 

Fig. 4:  Model of the phase impulse response 
Then the oscillation frequency is majority determined by the LC tank resonant frequency:
With:
where, C is the gate substrate capacitance which is very small compared to C1/C2.
Linear TimeVariant (LTV) phase noise model: This model provides a technique for calculating the phase noise based on the study of the sensitivity of the phase of an oscillator according to the time (Hajimiri et al., 1999). The idea of this model is to determine the impulse response of the phase excited by injection of low amplitude assimilated to current noise oscillator.
Usually the output of an oscillator can be written as:
where, f is periodic function, A(t) is variation of the amplitude, φ(t) is phase variation.
For characterizing the phase noise of the oscillator, Hajimiri and Lee (1998) defined the Impulse Sensitivity Function (ISF). It indicates that the time (t) according to which the pulse is injected current, the phase response of the oscillator is different.
In fact, if the pulse is applied at the instant when the output signal of the oscillator has a maximum (zerocrossing), the phase variation is maximal. Conversely, if the pulse is applied at the moment the output signal has a zero, the change phase is zero (Fig. 3).
The purpose of the Hajimiri’s model is to determine the impulse response of the phase hφ(t, τ) an oscillator when applying a current impulse i (t) at time τ (Fig. 4).
So, the phase noise can be characterized by an impulse response (Hajimiri et al., 1999) as:
where, U(tτ) is the unit step function, q_{max} the maximum charge displacement, Γ(ω_{0}τ) is the Impulse Sensitivity Function (ISF). The LTV phase noise model can predict the phase noise but it is difficult to define the ISF function by simulation.
RESULTS AND DISCUSSION
DC characteristic: In this section, the principal static characteristics (DC) of the considered DGMOS were described. They are obtained by selfconsistent solution of PoissonSchrodinger system. ShortChannel Effects (SCE) is monitored in several ways such as leakage current (I_{off}) and the DrainInduced Barrier Lowering (DIBL).
Firstly, the electrical characteristic obtained by selfconsistent model and SILVACOTCAD software were compared successfully (Fig. 5). This later executes quantum numerical simulation using a full 2D PoissonSchrodinger code.
Drain leakage current: One of the important problems mechanisms in offstate MOSFET’s is OFFleakage current. Several papers addressing leakage current have been published (Chang et al., 1995; Roy et al., 2003; Orouji and Kumar, 2006).
Figure 6 present the drain leakage current I_{off} for different gate work function. This current decrease consequently with the increased of the gate work function, this reduction of the current is related to the increase of the barrier potential. In order to maintain I_{off} very low, it is necessary to consider gate metal with higher corresponding metal work function.

Fig. 5:  Comparison of the output characteristics between present model and simulation with SILVACOTCAD software 

Fig. 6:  Leakage current vs. gate work function 
DIBL: Analysis of the characteristics Id (Vds) has identified the DIBL for different channel lengths for low and high drain polarizations.
Figure 7 represents the effect of varying the channel length on the DIBL for low (Vds = 0.1 V) and high (Vds = 1.5 V) drain bias. It can be observed that when the gate work function decreases from 4.254.5 eV, the DIBL is reduced. Moreover, the DIBL is lightly affected by gate work function, for the channel length greater than 12 nm.
Gate tunneling leakage current: Another parameter characterizing the short channel performance is the tunnel current. In MOS structures, the tunneling effect is characterized by the passage of the carriers through the oxide. The direct tunneling current is the dominant type of conduction in ultrathin oxide MOS structures (<3 nm) (Lee and Hu, 2001).
Modeling of the direct tunneling current, the WentzelKramersBrillouin (WKB) approximation have been largely used.

Fig. 7:  DIBL vs. channel length for gate work functions 

Fig. 8:  Gate current vs. drain voltage for oxide thickness 
The direct tunneling current is depending exponentially on the electric field across the oxide (Mukhopadhyay et al., 2007). It can be expressed by Lee and Hu (2001) in the following equation:
where, V_{ox} is oxide voltage, T_{ox} is oxide thickness, m_{ox} is effective mass in the oxide, Φ_{b} is the barrier height of the electron.
Figure 8 presents the gate tunneling current versus the drain voltage for different oxide thickness. Increasing the oxide thickness reduces the gate to channel tunneling. This is, because the reduction of gate oxide thickness results in an increase in the field across the oxide.

Fig. 9:  Impact of the permittivity of the material to the output current 

Fig. 10: 
Leakage current vs. gate work function for different permittivity 
The high electric field coupled with low oxide thickness results in tunneling of electrons (tunnel current) from channel to gate.
The most promising technology to reduce the tunneling effect is Highk technology. This later is based on the change of the gate oxide SiO_{2} by another insulate with a higher permittivity.
The two highk materials widely studied are Hafnium (HfO_{2}) and Zirconium (ZrO_{2}) oxides, due to their dielectric constant evaluated between 20 and 25. Figure 9 represents the drain current versus the drain voltage for different permittivity values (SiO_{2}, ZrO_{2} and HfO_{2}) for T_{SiO2} = 1.5 nm and EOT_{HfO2} = 7.7 nm, EOT_{ZrO2} = 9.6 nm. It can be observed that when the permittivity increases, the drain current also increases. This is because the gate tunneling current through the gate decrease (Fig. 10).
Output signal of the oscillator: The circuit generates after a very short transient regime (<1 nsec), a sinusoidal signal with a frequency oscillation in the order of 3 GHz corresponding to a period T = 335.8 ps (Fig. 11).

Fig. 11: 
A typical response of the DGMOSFET oscillator (mixedmode simulation) 

Fig. 12: 
Injection of an impulse at t = τ_{0} 
Analyse of the ISF function
Determination of the shift phase: Based on the LTV model and in order to determine the phase noise performance of the oscillation, we are interested in determining Φ ΔΦ. For this, a current pulse (i(t)) was injected at time τ of period T and the oscillator response for a few periods after the injection was analysed. By measuring the deviation Δt of the moment of zero crossing, it can measure the phase difference ΔΦ = 2πΔt /T, produced by the injection (Fig. 12).
The phase shift of the output signal depends on the moment where a disturbance is injected. This relationship between the phase shift and the timing of the injection pulse can be characterized using the sensitivity function (ISF). This function is used to study the sensitivity of the elements (L, C1 and C2) of the Colpitts oscillator to a current disturbance. In order to have sufficiently of measuring points to trace the ISF function, we chose a step 6.74 psec corresponding to 50 measuring point on the period T = 335.8 psec.
Figure 13ac represents the transient simulated output waveform obtained by ISETCAD simulator. It can observed that the:
• 
ISF function depends on injection node 
• 
ISF function is sinusoidal with a period of 2.97 GHz which is the nearly same period of the output signal 

Fig. 13(ac): 
ISF function for current pulses in (a) Inductance (L), (b) Capacitance (C1) and (c) Capacitance (C2) 
Analytical model: Now, we would like to extend the discussion to any value of the Φ phase (Φ E [0, 2π]), not only for a small charge injection. Gontrand et al. (2009) have established that if we consider a parasitic pulse injection (a “dirac”) at τ time, the global output voltage of an oscillator can be considered as the superposition of the permanent voltage and the response to the parasitic injection, because any time or phase shifted solution remains a solution of the harmonic oscillator:
where, V_{T} is total output voltage, Vs is output voltage of the free oscillator, C is global capacity of the Colpitts oscillator (Fig. 2).
Equation 8 can be reformulated as:
With:
Now, from these general formulas and assuming a limited expansion of φ and considering only the first order, can be reformed by:

Fig. 14:  Phase shift (output of transistor) versus injected charges 
If the latter Eq. 12 compared to the appendices of Hajimiri and Lee (1998), the well known results can be found out: Γ = sin(ω_{0}t) (Eq. 11) of Hajimiri and Lee (1998) with qmax = CV_{0}; this result is valid but for q very small.
The mixedmode simulations that we have realized, have identified each time the phase deviation ΔΦ, considering CV_{0} = 100fc. Raised these variations is reported in Fig. 14 and favorably compared with the analytical model. A good linearity is observed.
CONCLUSION
In this study, the analysis of the RF oscillator (Colpitts oscillator) built around the nanoscale DGMOS has been presented. Firstly, the SCEs (leakage currents and DIBL) in the DGMOS device was analysed; the results show that the impact of metal gate work function on leakage current and DIBL is important and must be considering to minimize SCEs. It have also note that above 12 nm, the DIBL is relatively small, but for greater channel length, significant effect of DIBL appears and must be take into account. Furthermore, the Double Gate transistor with Highk gate dielectric was analysed in order to reduce current leakage tunneling while maintaining the same value of capacitance.
Secondly, we go further on phase noise theory of such oscillator; starting with mixed mode analysis of an LCtype oscillator; some analytical solutions for the phase noise were derived, through a new lecture of the Impulse Sensitivity Function (ISF). First numerical simulation raw results are very consistent with such a numerical and analytical models; the phase shift of the output signal depends on the moment where a disturbance is injected. The method in this study don’t require specific ‘Artificial’ noise source to be introduced at some nodes (cf. Spicelike simulations) and it is its principal originality.
ACKNOWLEDGMENT
This study was supported by UPM (Union pour la méditéranée).