Monte Carlo Simulation of a Submicron MOSFET Including Inversion Layer Quantization

A Monte Carlo simulator of the electron dynamics in the channel, coupled with a solution of the two-dimensional Poisson equation including inversion-layer quantization and drift-diffusion equations has been developed. This simulator has been applied to the study of electron transport in normal operation conditions for different submicron channel length devices. Some interesting non-local effects such as electron velocity overshoot can be observed.


I. INTRODUCTION
Non-local effects are becoming more and more promi- nent as MOSFET dimensions shrink to deep-submi- crometer regimes.In order to study these effects, non- local models must be used to accurately describe elec- tron transport in MOSFETs.The Monte Carlo method is held to contain a more rigorous description of device physics than models based on the solution of fundamental balance equations [1][2].A quantum description of the confined inversion layer should also be included [3-4].Knowledge of electron mobility dependence on variables such as the transverse and longitudinal elec- tric fields is important for simulation and modelling of MOSFETs.Monte Carlo simulations are essential to study these electron mobility dependencies needed for circuit simulators of state-of-the-art MOSFETs.

II. DEVICE SIMULATION
We have simplified the solution of all the equations involved in the description of electron transport in a submicron MOSFET with the procedure described below.To have a departure point we took the follow- ing steps: The quasifermi levels as well as their separation were assumed to be constant in the transverse direction but were allowed to vary along the longitu- dinal direction.The total variation of the quasifermi levels along the channel is given by the drain-to- source bias voltage and is broken down into a sequence of steps spread out among several still unde- fined spatial locations along the channel.According to this hypothesis, the device is divided (by setting N-1 points inside the channel) into an appropriate number N of smaller channels (subchannels) of unknown length L i, where EL L, L is the effective channel length.The one-dimensional Schroedinger and Poisson equations are then selfconsistently solved at both ends of each subchannel, taking into account the pseudofermi level separation at each point.Each subchannel is then described as a sheet layer of charge located inside the semiconductor bulk at a distance z from the silicon-oxide interface.This position z is the mean transverse position of the inversion electron dis- tribution.The current in the subchannel is obtained by adding drift and diffusion contributions.If the voltage drop across the subchannel is very small, the follow- ing expression for the drain current (IDs) is obtained: where * is the electrostatic potential in z, N the electron density in the subchannel ends, NI the aver- age density of electrons in the subchannel, and qt the thermal voltage.The electron mobility depends on both the longitudinal (Eli) and transverse (E+/-) elec- tric fields.An approximated dependence on EI is assumed for the first time, while an accurate expression obtained by one-electron Monte Carlo simulation is used for the low-field mobility and its dependencies with the transverse-electric field and the temperature, which is crucial for reproducing with accuracy experimental results.The length of each subchannel is obtained by applying Expression according to an iterative procedure.The two-dimensional Poisson equation is solved using the solution obtained previ- ously as a starting point" )2,(x, y) 02,(x, y) p(x, y) where "x" is the parallel and "y" the perpendicular co-ordinates to the channel.An adaptive grid has been set in the whole structure.The two-dimensional problem considered in (2) has been decomposed into N one-dimensional problems" o(X,y) is the solution obtained previously, and x is a grid column in the channel.With this effective charge density i(y), we can use the procedure explained above to solve the Poisson and Schroedinger equations, repeated until a convergence criterium is reached.
Once the actual potential distribution, longitudinal and transverse fields, and inversion and depletion charge concentrations along the channel have been calculated, the electron dynamics are simulated by the Monte Carlo method.The grid is chosen to be thin enough that a constant value of the different transport magnitudes can be assumed in each grid interval (i.e. each grid interval was characterised by a constant value of the longitudinal-and transverse-electric fields, surface potential, inversion and depletioncharges, electronic subband minima, etc.).Taking into account these values, the scattering rates are evalu- ated in each grid zone.Phonon, surface-roughness, and Coulomb scattering have been considered, fol- lowing the procedure in a previous work [4][5][6].To continue in Monte Carlo simulation, a great number of electrons are introduced, one by one, into the chan- nel from the source.The longitudinal electric field in each grid zone modifies the electron wavevector according to the semiclassical model during a free flight whose length is calculated according to a stand- ard Monte Carlo procedure by generating a random number and using the maximum of the total scattering rate along the whole channel.The time an electron spends in each grid zone, the electron mean-velocity and mean-energy in each interval are recorded.The electron velocity distribution along the channel can thus be evaluated.Taking into account expression Vdrif g(x)Eil we can define a local electron mobil- ity in the channel to be used in Expression to calcu- late the new drain current and subchannel lengths.The whole procedure is solved again until a conver- gence criterium is reached.

III. RESULTS
0.1 gm, 0.2 gm, 0.25 gm, 0.5 gm, and gm channel length MOSFETs have been simulated.The bulk dop- ing concentration was N A 4x1017 cm-3, the gate- oxide thickness tox 5.6 nm and the source junction depth xj 0.1 gm.Figures and 2 show the 2D elec- trostatic potential distribution and the 3D electron concentration for the 0.2 lam channel MOSFET simu- lated for VGS 2.4 V and VDS V.The short-chan- x m) FIGURE 2D electrostatic potential plot for the MOSFET simu- lated.VGS 2.4 V, VDS V, xj 0,1 l.tm, N A 4x1017 cm-3, tax 5.6 nm, Lef t 0.2 l.tm .,>,.5 simulated.VGS 2.4 V, VDS V, xj 0.1 l.tm, N a 4x10 ]7 cm-3, tax 5.6 nm, Lef t 0.2 l.tm o nel effects are quite evident even for this bias; however, we have seen that the two-dimensional cur- rent corrections to the one-dimensional model are not important due to the high bulk-doping concentration.
The average kinetic-energy distribution for electrons along the channel is always below 0.5 eV, in the range of external voltages covered in this work.Therefore, according to Laux and Fischetti [1], a simplified description of the silicon band structure is justified.Nevertheless, we have also represented the effects of non-parabolicity on the electron dynamics in the channel.The inversion charge distribution along the channel is shown in Figure 4, Monte Carlo results (symbols) and drift-diffusion results (solid line) are compared.Both curves are in good agreement along the channel, and the same agreement was observed for the rest of transistors simulated.Figure 3 shows 0.0 0.2 Channel Position (#m) FIGURE 4 Electron density along channel obtained by Monte Carlo method (symbols) and by drift-diffusion method (solid line).
VGS 2.4 V, VDS V, xj 0.1 l.tm, N A 4x1017 cm-3, tax 5.6 nm, Lef t 0.2 I.tm the electron drift velocity along the channel obtained by the MC method.Noticeable velocity overshoot can be observed near the drain.

IV. CONCLUSION
The study of non-local effects such as velocity over- shoot, as well as modelling of the dependence of elec- tron mobility and other magnitudes needed to describe electron transport in a MOSFET channel on the transverse and longitudinal electric fields along the channel, has been made possible using a Monte Carlo simulator of the electron dynamics in the chan- nel, coupled with a solution of the two-dimensional Poisson equation including inversion-layer quantiza- tion and drift-diffusion equations.

FIGURE 3 FIGURE 2
FIGURE 3 Electron velocity versus channel position.Electron