Influence Interaction on Electron in Short Si-MOSFETs ... · VLSIDESIGN 2001, Vol. 13, Nos. 1-4,pp. 175-178 Reprints available directly fromthe publisher Photocopyingpermitted bylicense

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 175-178Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Influence of Electron-Electron Interactionon Electron Distributions in Short Si-MOSFETs

Analysed Using the Local IterativeMonte Carlo Technique

T. MIETZNERa’*, J. JAKUMEITb and U. RAVAIOLI

all. Phys. Inst., Universitht Kbln, ZiJlpicher Str. 77, 50937 Kbln, Germany," blnstitute for Algorithm and ScientificComputing (SCAI), GMD-German National Research Center for Information Technology, Schlofl Birlinghoven,53754 Sankt Augustin, Germany; CBeckman Institute and Department of Electrical and Computer Engineering,

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

The effects of electron-electron interaction on the electron distribution in n-channelmetal-oxide-semiconductor field-effect transistors (MOSFETs) are studied using theLocal Iterative Monte Carlo (LIMO) technique. This work demonstrates that electron-electron scattering can be efficiently treated within this technique. The simulation resultsof a 90 nm Si-MOSFET are presented. We observe an increase of the high energy tail ofthe electron distribution at the transition from channel to drain.

Keywords: Monte Carlo; Device simulation; Electron-electron interaction; Local Iterative MonteCarlo

1. INTRODUCTION

New generations of MOSFETs are generallycharacterized by the presence of high electric fieldsand high energy electrons (hot electrons) thatcreate reliability problems for the transistor. Hotcarriers can get trapped in the oxide or createinterface traps at the Si/SiO2 interface. In shortchannel MOSFETs with bias voltages far belowthe SiO2 barrier height of 3.2 eV electron-electron

(EE) interaction may still heat electrons to energieswhich allow a damage of the device. In this workthe Local Iterative Monte Carlo (LIMO) techni-que [1, 2] is used as an numerical efficient tool toinvestigate the influence of EE interaction on theelectron energy distribution. Besides the numericalefficiency LIMO gives the possibility to calculateEE scatterings rates selfconsistently using FermisGolden rule, since the electron distribution isalways accessible.

* Corresponding author. Tel.: +49 221 470-5766, Fax: +49 221 470-2980, e-mail: [email protected]

175

176 T. MIETZNER et al.

2. LIMO TECHNIQUE

The LIMO technique can be divided into twomain parts, the short local Monte Carlo (MC)simulation steps and the iteration process. For theshort MC steps, test particles are simulated inthe same way as in a conventional MC-simulationfor a short time of free flight. A number of suchMC-steps are performed for each initial state of adiscretization of the device region and energyspace. In an iteration process the electron dis-tribution in the whole device is calculated byexchanging local charge densities due to the in-formation of the MC steps. It can be shown thatthe results converge towards a solution of theBoltzmann transport equation. Due to the piece-wise nature of the LIMO technique the computa-tion time is very efficiently distributed over phasespace in contrast to standard MC algorithms,where the computational effort is proportionalto the electron density if no statistical enhance-ment (or variance reduction) techniques are used.Therefore the LIMO technique is especially suitedto the study of hot electron effects in semiconduc-tor devices.The LIMO algorithm used in this work is based

on a full band structure model and the ensembleconstant time technique for the time flight genera-tion. The full band structure is calculated using theempirical pseudopotential model of Cohen andBergstresser [3], with form factors taken from [4].The scattering mechanisms include acoustic pho-non scattering, f- and g-type X-X and X-Loptical phonon scattering, ionized impurity scat-tering (Ridley’s model [5, 6]), and impact ioniza-tion (Kane’s model [7]).The simulation of electron-electron (EE) scat-

tering events can be treated differently in theLIMO technique compared to conventional MCprograms, since the the electron distribution isalways accessible. The scattering rate for short-range EE process, which depends on the theelectron distribution itself, can be self-consistentlyrecalculated at any time during the iteration proc-ess. A special technique (Direct Exchange (DE)

technique) has been developed for the treatment ofthe partner electron in a short-range EE events.The long-range part of the EE is treated as anadditional inelastic scattering process since theLIMO approach does not include self-consistentsolution of the Poisson equation. The electric fieldprofile was calculated with the MOCA programfrom the University of Illinois and it remainedunchanged during the simulation [8]. The LIMOapproach might be able to show additional aspectsof EE effects and, in this way, contribute to the stillcontroversial discussion on the impact of EE onhot electron effects in modern short channelMOSFETs.

3. ELECTRON-ELECTRON MODEL

For the calculation of EE scattering rates theEE interaction has been split into two compo-nents. One component is associated with themotion of individual electrons. The other is as-sociated with collective oszillation, the so-calledplasmons. The separation occurs at roughly theDebye length [9]. The scattering rates for SEE(short-range electron-electron) and PEE (long-range electron-plasmon) scattering have been cal-culated by first order perturbation theory (Fermi’sGolden Rule) [11, 12]. We have used the momen-tum relaxation scattering rate for description ofthe SEE and PEE interaction which simplifies thesearch for a final full band wave vector [10]. TheSEE and PEE scattering has been incorporatedin the same way as the other scattering mecha-nisms. Details of the model used will be publishedelsewhere.Once a PEE event is stochastically determined

during a MC step according to the momentum re-laxation rate, the new state k can be chosen iso-tropy in k-space with new energy E(k) + hap [10].The SEE scattering rate has been calculated as a

function of the relative vector between the twoinvolved electrons in the center of mass system. Inorder to obtain the SEE scattering rate as functionof the investigated electron state k one has to

ELECTRON-ELECTRON INTERACTION 177

integrate the scattering rate over all possiblescattering partners, weighted by the electrondistribution. Once a SEE scattering event occurs

during a MC step we use a special procedure tohandle the partner electron (see Fig. 1). In contrastto the MC electron, for which the SEE is justanother scattering event during the short flightsimulation, the electron density represented by thescattering partner is immediately exchanged in theelectron distribution fd in order to gain energyconservation. Therefore we call the method DirectExchange (DE) technique. The scattering partnerelectron is chosen by a rejection technique thatweights the scattering partner electrons by themomentary distribution of electrons. In order toavoid an overestimation of the SEE rate due to adouble-count during a MC step, we have multi-plied the SEE rate by a factor 1/2 [14].

one electron MC

El(Etart (g,x)

to

final

partner._AE

Mc + AE 0

4. RESULTS

The LIMO technique has been applied to 1-Dsimulations of a 90nm n-channel Si-MOSFET.We refer to Ref. [15] for a detailed description ofthe device topology. In such short devices EEscattering may play an important role for hotelectron phenomena like injection of electrons intogate oxide and parasitic bipolar action [8]. EEeffects may heat electrons over the SiO2 barriereven if the voltage drop across the device is below3.2 eV.

Figures 2 and 3 show the electron distributionsin a Si-MOSFET with an effective channel lengthof 90nm for drain and gate bias values of 1.5 Vand 3 V, respectively. Four distributions are plot-ted in the figure showing, from cold to hot dis-tributions, the results obtained with only PEE, no

EE scattering, the full model (PEE+SEE) andonly SEE. The transition from channel to drain isat 200 nm. PEE acts as an additional energy relax-ation channel and leads to a colder distributionover the whole energy range. SEE has no effect atlow energies but yields a significant increase of thehigh energy tail by pushing the electron distribu-tions to be more Fermi like. Both effects increaseinside the drain region (starting at 200nm) withincreasing carrier density, but the increase is larger

ee ee

..’,’"" : i noEE

fd 104.... , PEE

....

V----’ -- oL.’ k " """ i50

energy [eV]

FIGURE Schematic illustration of the DE technique. Theelectron distribution fd is directly modified due to the energychange of the partner electrons involved in the short-rangescattering event. Total energy Etot E,+EMC and momentumhave to be conserved.

FIGURE 2 Influence of EE scattering on the energy distri-bution of electrons in a 90nm n-channel Si-MOSFET forVg Va= 1.5V. The interesting region at the transition fromchannel to drain is plotted. The transition from channel todrain is at approximately 200 nm.

178 T. MIETZNER et al.

1014

:07

EE

....’.’’" SEE

.,.. PEE

-’-..,, ".,--,

--s’X k" ""i90

"10 *v

energy leVI

FIGURE 3 Same as Figure 2 for Vg Vd= 3 V.

for the SEE contribution. Three effects increaseSEE scattering at the transition from channel todrain. Beside the increase of the electron density,the electron temperature has its maximum (about8000K), leading to a lower screening, and thedistribution is far from a Fermi like behavior. At1.5 V at drain and gate the tail of the distributionis significantly increased just at the SiO2 barrierenergy of 3.2 eV. This should be important for thegate current and related hot electron degradationphenomena.

5. CONCLUSIONS

A special full-band MC technique, the LIMOtechnique, has been used to study the influence ofelectron-electron interaction on electron distribu-tions in n-channel Si-MOSFETs. Since the elec-tron distribution is always accessible in the LIMOtechnique, the long-range plasmon as well as theshort-range direct interaction could be calculatedusing Fermi’s Golden rule. The scattering rateswere updated during the simulation to achieve aself-consistent calculation. To handle the secondpartner in a direct interaction event the DirectExchange (DE) technique was developed.

An 1-D model was then used to investigate theinfluence of electron-electron scattering on the en-ergy distributions in n-channel Si-MOSFETs. Ourresults show that electron-electron (EE) scatteringcan lead to significant increase of the high energytail due to short-range EE scattering. For a 90 nmMOSFET with 1.5 V at drain and gate, SEE leadsto a increase of the distribution just at the SiO2 bar-rier energy of 3.2 eV, which may have a significanteffect on hot electron degradation phenomena.

Acknowledgments

This work was partially supported by the Semi-conductor Research Corporation, contract 98-SJ-406, by the National Center for ComputationalElectronics (NCCE) through National ScienceFoundation grant ECS 95-26127, and the DeutscheForschungsgemeinschaft GK Scientific Computing.

References

[1] Jakumeit, J., Sontowski, T. and Ravaioli, U., In: Proc. 6thInternational Workshop of Computational Electronics,Osaka, 1998.

[2] Jakumeit, J. and Ravaioli, U., submitted to IEEE Trans.ED.

[3] Cohen, M. L. and Bergstresser, T. K. (1996). Phys. Rev.,141, 789.

[4] Chelikowsky, J. R. and Cohen, M. L. (1976). Phys. Rev.,14, 556.

[5] Ridley, B. K. (1977). J. Phys. C, 10, 1589.[6] Van de Roer, T. G. and Widdershoven, F. P. (1986).

J. Appl. Phys., 59, 813.[7] Kane, E. O. (1967). Phys. Rev., 159(3), 624.[8] Duncan, A., Ravaioli, U. and Jakumeit, J. (1998). IEEE

Trans. Elec. Dev., 45(4).[9] Pines, D. and Bohm, D. (1952). Physical Review, 85(2),

338.[10] Kosina, H. (1999). IEEE Trans. Elec. Dev., 46(6), 1196.[11] Singh, J. (1996). Physics of Semiconductors and Their

Heterostructures, McGraw-Hill.[12] Ridley, B. K. (1988). Quantum Processes in Semicon-

ductors. Oxford Science Publications, Clarendon Press.[13] Jacoboni, C. and Reggiani, L. (1983). Reviews of Modern

Physics, 55(3), 645.[14] Fischetti, M. V., Laux, S. E. and Crabb6, E. (1995).

J. Appl. Phys., 78(2), 1058.[15] see http://www.mtl.mit.edu:80/Well

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Influence Interaction on Electron in Short Si-MOSFETs ... · VLSIDESIGN 2001, Vol. 13, Nos. 1-4,pp. 175-178 Reprints available directly fromthe publisher Photocopyingpermitted bylicense