ON THE SENSITIVITY OF THE TUNNELING CURRENT TO ELECTRIC FIELD IN A MOSFET WITH TWO GATES

A theoretical model to evaluate the sensitivity of the tunneling current to the electric field in an n-channel MOSFET with two gates is proposed. This sensitivity is calculated in a real situation.


INTRODUCTION
It is well-known that MOSFETs with control gate and floating gate play an important role in EEPROM for low-voltage microcontrollers.This role arises mainly from the physical electronics involved in the above devices; in particular, the tunneling current through the oxide layer constitutes a relevant feature of these devices.This current is very sensitive to the electric field in the oxide layer so that evaluation of this sensitivity is very useful in order to estimate quantitatively the performance of the devices.The aim of this paper is to establish a parameter to estimate the above sensitivity; at this respect, we think that our mod- el improves really the state of the art since to date not much has been done with respect to theoretical research on MOSFETs performance.At any rate, with respect to this research, we can mention Refs.[1][2][3].*Corresponding author.M.A. GRADO-CAFFARO AND M. GRADO-CAFFARO

THEORETICAL MODEL
First of all, we will consider the mathematical expression of the tunneling current density through the oxide layer of an n-channel MOSFET with two gates: the control gate and the floating gate.We will assume an A1-SiO2-Si device so that the magnitude of the current density in question is given by the Fowler-Nordheim model, namely [1][2][3][4] where e is the electron charge, E denotes strength of the electric field in the oxide layer, mo stands for the electron rest-mass, m, is the tun- neling electron effective mass, h is the reduced Planck's constant and Eg is the barrier height of Si to SiO2 [1-4].Now we define the fol- lowing quantity: s j-ldJ/dE so that by formula (1) one obtains: By using the numerical values of the parameters involved in Eq. ( 2), including m, .1.1mo and Eg 4.35eV (room temperature) [3,5], it follows: s ,.E-l(2 + 6.48 x 101E-1) (3 where E is expressed in V/m and s in m/V; we can conceive s as a sen- sitivity parameter which measures the sensitivity of J to E (in Fig. 1, s is depicted as a function of E for a range of E-values of interest).
Notice that s decreases as E increases; a typical value of E is 3 x 108V/m which corresponds to Vox 3 V and tox 100 , where Vo, stands for the voltage drop across the oxide layer and to is the oxide thickness (see previous references).The above numerical values correspond obviously to a uniform electric field E, Vo,/to which in practice constitutes a reasonable approximation.

FIGURE
Plot of s versus E for a range of interest.

CONCLUDING REMARKS
The model described previously represents a useful approach to estimate the sensitivity of J to E; our method may be regarded as a technique extrapolable to other situations in the context of high-speed electronics.In particular, by examining the E-dependence of s, it is very easy to see that for E<< 3 x l08V/m s varies sharply with E although the situation corresponding to E << 3 x 10 8 V/m is irrelevant in practice; in contrast, E-values near 3 x 108 V/m are relevant with a relatively remarkable variation of s in terms of E; on the other hand, values between 3 x 108V/m and 12 x 108V/m present some interest.