A PHYSICAL MODEL FOR MOSFET DRAIN CURRENT IN NON-OHMIC REGIME USING OHMIC REGIME OPERATION

In order to characterise the velocity saturation phenomena in short channel MOSFET's, 
a simple method is proposed in this work. It is based on the comparison between transistor behaviour in ohmic and saturation regime respectively. Therefore, the MOSFET characteristic Id0(Vd). avoiding velocity saturation phenomena, can be obtained from ohmic characteristic Id(Vg) 
 and compared with the experimental 
characteristic Id(Vd).


INTRODUCTION
It's not obvious to consider drain current modelling Ia(Va), without taking into account the velocity saturation phenomena.In fact, it is physically difficult to access due to non linear dependence of drift velocity with lateral field Fy [1] in short channel devices.It is becoming so interest to dissociate physical parameters which are in the origin of velocity saturation and these which are not.In this previous work a new model of drain current Iao(Va) has been proposed.This method has allowed to avoid velocity saturation phenomena and more *Corresponding author.strongly to characterise the velocity saturation effect systematically by comparison between the model and a measured device out-put characteristics.It was shown that Ido(Vd) is not other than out-put characteristic of long channel device.

MODEL
The operation mode in non ohmic regime consisted on applying an important voltage on the drain in addition to the applied gate voltage.The inversion layer which was evenly distributed in the channel should be more pinched close the drain.At the voltage value Vd equal to Vdsat the charge at the drain level becomes very weak (Pinch Off).For the drain voltage Vd exceeding Vdsat this charge doesn't vary practically.The transistor is saturated on the length AL [2].In the case of short channel MOSFET, the drain current is limited by velocity saturation, and its saturation value depends on both Vdst value and transistor velocity saturation [3].

la-Va Characteristic
The MOSFET operation in non ohmic regime can be shared in two principal zones: 2.1.1.Non-ohmic Regime ( Vd < V,lsat) The inversion charge at y space along the channel which is being in strong inversion is written as" Qi(y) Cox(Vg-Vt-V(y)) (1 Where Cox is the oxide capacitance per unit area, Vg and Vt are, respectively, gate and threshold voltage and V(y) is the voltage on the area at y space of the channel which is being in strong inversion.The current drain takes the following expression: Where W is the channel width, v(y) is the carriers velocity on the area.In this zone the channel is not pinched close the drain.For long channel MOSFET's the carriers velocity is proportional to gradual voltage in the channel and the drain current follow the relation (3) in condition that effective mobility should be independent of Va such as: This hypothesis is not always verified as the mobility varies versus the inversion charge existing in the channel.Moreover, the channel length reduction was accompanied by a degradation of MOSFET component properties [4].Carriers velocity reduction for short channel MOSFETs is a direct consequence of channel reduction and its expression was given by Thornber model [5].It depends, so, on lateral field Fy.Con-  sequently, the expression (3) is correct if only both the attenuation coefficients and carriers velocity saturation are neglected.

Saturation Regime ( Va > Vasat)
The channel is pinched close the drain and current can not increase versus drain voltage Va.In this zone it is depending only on the applied gate voltage, the inversion charge becomes null at Vdsat--Vg-Vt, the Eq. ( 3) is rearranged as: W I,/= lasat -" CoxlZeff(Vg Vt) : (4) Therefore, the variation of pinched zone length in the ease of short channel devices implies Ia variation with Va.This leads to non neglected conductance Ga, which can be also modulated by thermal heating effects [7].If, in the first time, we neglect velocity saturation phenomena, we can write the expression of carriers drift velocity in y channel point such as: v(y) #eft -d-Y-y From relation (2) the drain current expression at every y channel point becomes: --*--V=O.It's observed so, that the expression inside the integral corresponds to dynamic conductance expression in ohmic regime and strong inversion (Fig. 1) which is given by: W Ga(V) =-#erfCox(Vg-Vt-V) (7 Where Vg was exchanged by Vg-Vt, therefore, we can obtain the drain current characteristic of long channel transistor in saturation regime for gate voltage value Vgi by simply integrating.The Ga characteristic of ohmic regime between Vgi and 0. The expression of Ia0 is given such as: t 0(v , %) (V)dV --Vd (8)

RESULTS AND DISCUSSION
In order to validate experimentally these results we have used a conventional MOSFET where its main parameters are defined as: thickness oxide tox= 6.8 nm, aspect ratio W/L= 25/0.85,mobility at low field #0=0.0392Cm2/V.s,substrate doping ranged between 105-1016Cm -. Figure 1 shows the experimental drain current Ia versus gate voltage Vg.It is found that for Va exceeding Vat the drain current varies weakly against drain voltage Va which implies a dynamic conductance Ga(V) along the channel (Eq.( 7)).The relation (8) is practically interest since it allows to access to transistor char- acteristics exempt velocity saturation phenomena which has been manifested in the experimental out-put characteristic Ia(Va) (Fig. 2).
In the absence of velocity saturation, the behaviour of simulated drain current in non ohmic regime Iao(Va), approaches a long channel device operation (Fig. 3).For short channel lengths, the drain current satu- ration is mainly due to carriers velocity saturation.In the obtained characteristic Iao(Va, Vg), the limit of integration's are conserved when the drain voltage go up from 0 voltage to Vdsat voltage, the inversion charge Qi go down from Cox(Vgi-Vt) value to Cox(Vgi-Vt-V)_ 0 value corresponding to the same limits in Ia(Vg) characteristic since gate voltage varies from Vg down to 0 value, this supposes surly that Vas,t -Vg-Vt, which is the case of long channel devices (L > 5 gm) 0,010---*--V =4.95V without velocity saturation.(Fig. 4) shows a comparison with the drain current model Ido(Vd) without velocity saturation, and experi- mental data Id(Vd).

CONCLUSION
In this work we have presented a new method which used the inte- gral of Id(Vg) characteristic in ohmic regime to reach non ohmic characteristic Id(Vd).This technique allowed to eliminate velocity saturation effects and improves powerful instrument to dissociate sim- ultaneously the parameters related to velocity saturation phenomena and these which are not.

FIGURE 2 FIGURE 3 FIGURE 4
FIGURE 2 Experimental drain current Ia versus drain voltage Va.