Monte Carlo Simulation Submicron MOSFET Including ...downloads.hindawi.com/journals/vlsi/1998/026390.pdf · transport in normal operation conditions for different submicron channel

VLSI DESIGN1998, Vol. 6, Nos. (1--4), pp. 287-290Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in India.

Monte Carlo Simulation of a Submicron MOSFETIncluding Inversion Layer QuantizationJ.B. ROLDAN*, E GAMIZ, J.A. LOPEZ-VILLANUEVA and J.E. CARCELLER

Departamento de Electr6nica y Tecnologfa de Computadore, Universidad de Granada, Facultad de Ciencias, Avd.Fuentenueva s/n.

18071 Granada, Spain

A Monte Carlo simulator of the electron dynamics in the channel, coupled with a solution ofthe two-dimensional Poisson equation including inversion-layer quantization and drift-diffu-sion equations has been developed. This simulator has been applied to the study of electrontransport 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 methodis held to contain a more rigorous description of devicephysics than models based on the solution of funda-mental balance equations 1-2]. A quantum descriptionof the confined inversion layer should also be included[3-4]. Knowledge of electron mobility dependence onvariables such as the transverse and longitudinal elec-tric fields is important for simulation and modelling ofMOSFETs. Monte Carlo simulations are essential to

study these electron mobility dependencies needed forcircuit simulators of state-of-the-art MOSFETs.

II. DEVICE SIMULATION

We have simplified the solution of all the equationsinvolved in the description of electron transport in a

submicron MOSFET with the procedure describedbelow. To have a departure point we took the follow-

ing steps: The quasifermi levels as well as their sepa-ration were assumed to be constant in the transverse

direction but were allowed to vary along the longitu-dinal direction. The total variation of the quasifermilevels 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. Accordingto this hypothesis, the device is divided (by settingN-1 points inside the channel) into an appropriatenumber N of smaller channels (subchannels) ofunknown length Li, where EL L, L is the effectivechannel length. The one-dimensional Schroedingerand Poisson equations are then selfconsistently solvedat both ends of each subchannel, taking into account

the pseudofermi level separation at each point. Eachsubchannel is then described as a sheet layer of chargelocated inside the semiconductor bulk at a distance zfrom the silicon-oxide interface. This position z is themean transverse position of the inversion electron dis-

* Corresponding author. Tel: 34-58-243227. Fax: 34-58-243230. Email: [email protected] or [email protected]

287

288 J.B. ROLDAN et al.

tribution. The current in the subchannel is obtained byadding drift and diffusion contributions. If the voltagedrop across the subchannel is very small, the follow-ing expression for the drain current (IDs) is obtained:

[DS qW/i(E[],E+/-)[_l(lrain tiource)Li

-t-*t (NI,drain Nl,source)] (1)

where * is the electrostatic potential in z, N theelectron density in the subchannel ends, NI the aver-age density of electrons in the subchannel, and qt thethermal voltage. The electron mobility depends onboth the longitudinal (Eli) and transverse (E+/-) elec-tric fields. An approximated dependence on EI isassumed for the first time, while an accurate expres-sion obtained by one-electron Monte Carlo simulationis used for the low-field mobility and its dependencieswith the transverse-electric field and the temperature,which is crucial for reproducing with accuracy exper-imental results. The length of each subchannel isobtained by applying Expression according to aniterative 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)0X2 -+- (2)0y2 es

where "x" is the parallel and "y" the perpendicularco-ordinates to the channel. An adaptive grid has beenset in the whole structure. The two-dimensional prob-lem considered in (2) has been decomposed into None-dimensional problems"

d2)i (Y) i(Y) (3)dY2 ;s

where

i(Y) p(xi, y) +)2)0(x,y)

x--x(4)

o(X,y) is the solution obtained previously, and x is a

grid column in the channel. With this effective chargedensity i(y), we can use the procedure explainedabove to solve the Poisson and Schroedinger equa-tions, repeated until a convergence criterium isreached.

Once the actual potential distribution, longitudinaland transverse fields, and inversion and depletioncharge concentrations along the channel have beencalculated, the electron dynamics are simulated by theMonte Carlo method. The grid is chosen to be thin

enough that a constant value of the different transportmagnitudes can be assumed in each grid interval (i.e.each grid interval was characterised by a constant

value of the longitudinal- and transverse-electricfields, surface potential, inversion and depletion-charges, electronic subband minima, etc.). Taking intoaccount 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 [4-6]. Tocontinue in Monte Carlo simulation, a great numberof electrons are introduced, one by one, into the chan-nel from the source. The longitudinal electric field ineach grid zone modifies the electron wavevector

according to the semiclassical model during a freeflight whose length is calculated according to a stand-ard Monte Carlo procedure by generating a randomnumber and using the maximum of the total scatteringrate along the whole channel. The time an electronspends in each grid zone, the electron mean-velocityand mean-energy in each interval are recorded. Theelectron velocity distribution along the channel canthus 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 channellength MOSFETs have been simulated. The bulk dop-ing concentration was NA 4x1017 cm-3, the gate-oxide thickness tox 5.6 nm and the source junctiondepth xj 0.1 gm. Figures and 2 show the 2D elec-trostatic potential distribution and the 3D electronconcentration for the 0.2 lam channel MOSFET simu-lated for VGS 2.4 V and VDS V. The short-chan-

MONTE CARLO SIMULATION OF A SUBMICRON MOSFET 289

0.1

0.2

0.50.0 0.2 0.4 0.6

x m)FIGURE 2D electrostatic potential plot for the MOSFET simu-lated. VGS 2.4 V, VDS V, xj 0,1 l.tm, NA 4x1017 cm-3,tax 5.6 nm, Left 0.2 l.tm

3

NA=4x 107 crn-’

2

0.0 0.2Channel Position (/,m)

FIGURE 3 Electron velocity versus channel position. Electron20_ velocity overshoot can be observed near the drain. VGS 2.4 V,

VDS V, xj 0.1 txm, NA 4x1017 cm-3, tax 5.6 nm,Left 0.2 pm

’ o

0 NA=4x 1017 cm-3

"O

FIGURE 2 3D electron density distribution plot for the MOSFET .,>, .5

simulated. VGS 2.4 V, VDS V, xj 0.1 l.tm, Na 4x10]7 cm-3,tax 5.6 nm, Left 0.2 l.tm

onel 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 electronsalong the channel is always below 0.5 eV, in the rangeof external voltages covered in this work. Therefore,according to Laux and Fischetti [1], a simplifieddescription of the silicon band structure is justified.Nevertheless, we have also represented the effects ofnon-parabolicity on the electron dynamics in thechannel. The inversion charge distribution along thechannel is shown in Figure 4, Monte Carlo results(symbols) and drift-diffusion results (solid line) arecompared. Both curves are in good agreement alongthe channel, and the same agreement was observedfor the rest of transistors simulated. Figure 3 shows

0.0 0.2

Channel Position (#m)FIGURE 4 Electron density along channel obtained by MonteCarlo method (symbols) and by drift-diffusion method (solid line).VGS 2.4 V, VDS V, xj 0.1 l.tm, NA 4x1017 cm-3,tax 5.6 nm, Left 0.2 I.tm

the electron drift velocity along the channel obtainedby the MC method. Noticeable velocity overshoot canbe observed near the drain. Fig 5 shows the electronmobility along the channel for the same MOSFET.Making use of the Monte Carlo results obtained weare able to model electron mobility accounting forelectron velocity overshoot and other effects observedin very short channel length MOSFETs and use themin simple drift-diffusion simulators.

290 J.B. ROLDAN et al.

"E 25O

200

5o

100

NA=4xl 07 cm--]

0.0 0.1

Channel Position (/m)FIGURE 5 Electron mobility versus channel position. VGS 2.4 V,VDS= V, xj=0.1 l.tm, NA=4xl017 crn-3, tox=5.6 nm,Left 0.2 l.tm

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 todescribe electron transport in a MOSFET channel onthe transverse and longitudinal electric fields alongthe channel, has been made possible using a MonteCarlo simulator of the electron dynamics in the chan-nel, coupled with a solution of the two-dimensionalPoisson equation including inversion-layer quantiza-tion and drift-diffusion equations.

References[1] S.E. Laux, M.V. Fischetti, "Monte Carlo simulation of sub-

micrometer Si N-MOSFET’s at 77 and 300 K", IEEE Elec-tron Device Lett., 9, p. 467 Sep. (1988)

[2] G. Baccarani, M.R. Wordeman, "An investigation of steady-state velocity overshoot in silicon", Solid-State Electron., 28,p. 407 (1985)

[3] M.V. Fischetti, S.E. Laux, "Monte Carlo study of ElectronTransport in Silicon Inversion Layers" Physical Review B,48, p. 2244 (1993)

[4] F. Gamiz, J.A. Lopez-Villanueva, J.A. Jimenez-Tejada,I. Melchor, and A. Palma, "A Comprehensive Model forCoulomb scattering in inversion layers", Journal Of AppliedPhysics, 75(2), p. 924 (1994)

[5] F. Gamiz, J.A. Lopez-Villanueva, J. Banqueri, J.E. Carceller,and P. Cartujo., "Universality of Electron Mobility Curves inMOSFETs: A Monte Carlo study" IEEE Trans. Electr.Devices, ED-42, p. 258, (1995)

[6] F. Gamiz, J.A. Lopez-Villanueva, J. Banqueri, Y. Ghailan,and J.E. Carceller, "Oxide Charge Space Correlation ininversion layers II. Three-dimensional oxide charge distribu-tion.", Semicond. Sci. Technol. 10, p. 592-600 (1995)

Biographies

Juan B. Roldan graduated with a degree in physics in1993 at Granada University. Since 1993 he has beenworking on the MOS device physics including 2Dtransport, non-local effects, electron mobility depend-encies on transverse and longitudinal electric fields byusing a Monte Carlo MOSFET simulator. He is a

Teaching Assistant at the University of Granada.Francisco Gamiz graduated with a degree in phys-

ics in 1991, and received the Ph.D. in 1994 from theUniversity of Granada. Since 199,1 he has been work-ing on the characterization of scattering mechanismsand their influence on the transport properties ofcharge carriers in semiconductor structures. He hasstudied electron mobility in silicon inversion layersby Monte Carlo method. His current research interestincludes the effects of many-carriers on the electronmobility and the theoretical interpretation of the influ-ence of high longitudinal electric fields have on theelectric properties of MO transistors. He is an Associ-ate Professor at the University of Granada.Juan A. Lopez-Villanueva graduated with a

degree in physics in 1984, and received the Ph.D. in1990 from the University of Granada with a thesis onthe degradation of MOS structures by Fowler-Nord-heim tunneling. Since 1990 he has been working onthe MOS device physics including 2D transport,effects of nonparabolitity, quantum effects, scatteringmechanisms and Monte Carlo simulation of chargetransport. His current research interest includes thecharacterization, simulation and modelling of electrondevices. He is an Associate Professor at the Univer-

sity of Granada.Juan E. Carceiler graduated with a degree in

physics in 1975, and received the Ph. D. degree in1979 from the University of Barcelona. He hasengaged in the research and characterization of deeplevels in semiconductors. His current research interestincludes degradation of MOS structures and charac-terization of electron mobility in the channel of MOStransistors. He has been a Professor at the Universitiesof Barcelona and Granada.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Documents

Monte Carlo Simulation Submicron MOSFET Including ...downloads.hindawi.com/journals/vlsi/1998/026390.pdf · transport in normal operation conditions for different submicron channel