Книги+1 / 2013 [Chandan_Kumar_Sarkar]_Technology_CAD

Consider an infinitesimally small section of the noiseless channel of a MOS transistor of length dy. Let the resistance of this small section be dR and the channel voltage produced by this resistance is dVCS. For a channel current IDS, these are related as

From (7.139) it follows that

The noise spectral density due to the thermal noise generated by this small resistance dR is given by

The power spectral density for the elemental noise voltage is from (7.141)

From this elemental noise voltage, the elemental noise current power spectral density is

The conductance for the elemental channel segment is determined as follows:

Integrating over the entire channel length, the total noise current power spectral density is given by

In (7.146), QINV = QinvWL represents the total inversion charge under the gate. The thermal noise power spectral density is often expressed in the following manner, referred to as the Klaassen-Prins equation for thermal noise [19]:

It is to be noted that (7.146) is used in a BSIM compact model with appropriate substitution of QINV.

7.8.2  Characterization of Flicker Noise in MOS Transistor

The flicker noise in the drain current or gate voltage of a MOS transistor is important to characterize precisely because it deteriorates the signal-to-noise ratio of several analog circuits. It also increases the phase noise of oscillators in RF applications. For proper characterization of flicker noise, the underlying physical mechanism of the flicker noise must be understood. This is briefly discussed below.

7.8.2.1  Physical Mechanisms of Flicker Noise

The conductivity of a conductor due to drift motion of the carriers is given by

In (7.148), n represents the carrier concentration, and μ represents the carrier mobility. It appears from (7.148) that any fluctuation in the carrier density or mobility leads to fluctuation of the current flowing through the conductor. There are several different theories for explaining the physical cause of flicker noise. These are broadly classified into three different categories [20]:

(1) carrier density fluctuation model, (2) mobility fluctuation model, and (3) correlated carrier and mobility fluctuation model.

According to the carrier density fluctuation model [20], the flicker noise is caused by random trapping and de-trapping of mobile carriers by the interface traps at the Si-SiO2 interface. The interface traps dynamically exchange

carriers with the channel causing fluctuation in the surface potentials, giving rise to fluctuation in the inversion charge density. The carrier density fluctuation model is observed to successfully explain the flicker noise spectrum in n-channel MOS transistors. According to the mobility fluctuation model [20], on the other hand, the flicker noise is caused due to fluctuation in the carrier mobility, caused due to phonon scattering. The mobility fluctuation model successfully explains the flicker noise spectrum in p-channel MOS transistor. According to the correlated carrier and mobility fluctuation model [21,22], also referred to as the unified flicker noise model, when an interface trap captures an electron from the inversion layer, it becomes charged and reduces the carrier mobility due to Coulombic scattering. Thus according to this model, both the carrier number and the carrier mobility fluctuate due to trapping and de-trapping of the carriers by the interface traps. The unified model shows good matching with experimental results.

7.8.2.2  Empirical Approach for Characterization of Flicker Noise

The power spectral density of the flicker noise spectrum is given by [21]

In (7.149), KF is the flicker noise coefficient, and AF is the flicker noise exponent. The value of the parameter AF lies in the range of 0.5 to 2. The constant KF is proportional to the interface trap density, which is technology-specific. The lack of systematic approach in determining the empirical parameters limits the use of this model. However, two significant observations are made. First, the flicker noise is dominant at low frequency. Because of its dependence on frequency as (1/f), flicker noise is sometimes referred to as the (1/f) noise. At frequencies above 100 MHz, the flicker noise spectrum becomes negligible compared to that of the thermal noise. Second, the flicker noise spectrum reduces as the gate area is increased. Third, for PMOS transistors, it has been found that the value of the flicker noise coefficient is smaller compared to NMOS transistors; therefore, PMOS transistors are used in designing low noise circuits, at least at the first stage.

7.8.2.3  Characterization of Flicker Noise through Physics-Based Model

In (7.150), μs is the carrier mobility, q is the electron charge, N is the number of channel carriers per unit area, and ξy is the lateral channel field. Fluctuation of local drain current is given by [21,22]

traps per unit area, and N is the inversion carrier density. The ± sign in the mobility term of (7.151) denotes whether the trap is neutral or charged when filled.

Let us first evaluate the first term on the right-hand side of Equation (7.151). The ratio of fluctuations in carrier number to fluctuations in occupied trap number R = δ N/δ Nt is close to unity in strong inversion but assumes a smaller value in other bias conditions. A general expression of R is therefore

In (7.152a), Cinv , Cdm , and Cit are inversion layer, depletion layer, and interface trap capacitances, respectively. A more concise form of R is as follows:

In (7.152b), N* = (kT/q2 )(Cox + Cdm + Cit ) and the typical value of this quantity is 1–5E10/cm–2.

Let us now evaluate the first term on the right-hand side of Equation (7.151). The carrier mobility is related to the oxide trap density as follows:

In (7.153), µCit = 1/αsc Nt is the mobility limited by Coulombic scattering of the mobile carriers at trapped charges near the Si-SiO2 interface, and µB ,µSR ,µPh

represents the mobility limited by ionized impurity scattering, surface roughness scattering, and phonon scattering, respectively. The scattering coefficient αsc is a function of the local carrier density due to the screening effect as well as the distance of the trap from the interface. From experimental

results, it has been found that μCit increases with the inversion carrier density due to the screening effect. The relationship is given as follows [23]:

However, in the original unified mobility model [21,22], the scattering parameter is considered to be independent of the inversion carrier density. The reduction of αsc with an increase of N is understood as follows. As the inversion carrier density increases, the screening length and the scattering cross section due to the screening by minority carriers reduce and hence the scattering parameter increases. In a weak inversion region, screening due to minority carriers becomes less significant compared to that by majority carriers. Because the majority carrier concentration does not change much in the weak inversion region, the scattering cross section remains almost constant with inversion carrier density. Consequently, in the weak inversion region, αsc saturates to a particular value and (7.153b) is no longer valid [23]. By differentiating (7.153) and substituting in (7.151), we arrive at

This can be written as

The power spectrum density of the local current fluctuation is obtained from (7.155b) as follows:

In (7.156), S Nt (y, f ) is the power spectrum density of the fluctuations in the number of occupied traps over the area W y and is given by