Quantum corrections in the simulation of decanano MOSFETs

Solid-State Electronics 47 (2003) 1141–1145www.elsevier.com/locate/sseQuantum corrections in the simulation of decanano MOSFETsA. Asenov * , A.R. Brown, J.R. WatlingDevice Modelling Group, Department of Electronics and Electrical Engineering, University of Glasgow, G12 8LT Glasgow, Scotland, UKAbstractQuantum mechanical confinement and tunnelling play an important role in present and future generation decanano(sub-100 nm) MOSFETs and have to be properly taken into account in the simulation and design. Here we present asimple approach of introducing quantum corrections in a 3D drift–diffusion simulation framework using the densitygradient (DG) algorithm. We discuss the calibration of the DG approach in respect of quantum confinement effects incomparison with more comprehensive but computationally expensive quantum simulation techniques. We also speculateabout the capability of DG to describe source-to-drain tunnelling in sub-10 nm (nano) MOSFETS. The applicationof the DG approach is illustrated with examples of 3D statistical simulations of intrinsic fluctuation effects indecanano and nano-scale double-gate MOSFETs.Ó 2003 Elsevier Science Ltd. All rights reserved.1. IntroductionMOSFETs scaled down to 15 nm gate lengths havebeen successfully demonstrated [1] and 9 nm MOSFETsare expected in mass production in 2016 according thenew edition of the roadmap. There is a consensus,however, that scaling around and below 10 nm will requirea departure from the traditional MOSFET architecture.Among the most promising nanometre scaletransistor candidates are double-gate MOSFETs [2].The combination of thin gate oxides and heavy dopingin the conventional MOSFETs, and the thin siliconbody of the double-gate structures, will result in substantialquantum mechanical (QM) threshold voltageshift and transconductance degradation [3]. Below 10nm gate-lengths direct source-to-drain tunnelling willrapidly became one of the major limiting factors forscaling [4]. Computationally efficient methods to includeQM effects are required for the purpose of practicalcomputer aided design of this generation of devices.First order quantum corrections based on density gradients(DGs) have already been introduced in 2-D [5]and 3-D [6] drift–diffusion simulations. In this paper wediscuss the implementation of DG quantum corrections* Corresponding author. Fax: +44-141-330-4907.E-mail address: a.asenov@elec.gla.ac.uk (A. Asenov).in a 3-D drift–diffusion simulation network designed tostudy intrinsic parameter fluctuations introduced by thediscreteness of charge and atomicity of matter. We startwith the calibration of the DG approach in respect ofquantum confinement effects in comparison with moresophisticated quantum simulations [2]. We also investigateto what extent the DG approach can describe, atleast semi-quantitatively, the expected source-to-draintunnelling in nano-scale devices. Finally we illustrate theapplication of DG corrected drift–diffusion simulationsin the analysis of various sources of intrinsic fluctuationsin decanano double-gate MOSFETs.2. The density gradient approachWe are motivated by the need to include quantumcorrections in the atomistic simulation of intrinsic fluctuationeffects in decanano MOSFETs introduced byatomicity of charge and matter [6–8]. The investigationof intrinsic fluctuation effects involves statistical 3-Dsimulations of large samples of macroscopically identicalbut microscopically different devices and is verycomputationally expensive. Therefore the computationalefficiency of the quantum correction approach isof great importance and the use of the DG approachbecomes an attractive option.0038-1101/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0038-1101(03)00030-3

1142 A. Asenov et al. / Solid-State Electronics 47 (2003) 1141–1145The DG approach may be derived from the oneparticle Wigner function [9]:"of ðk; r; tÞ2þ v r r f ðk; r; tÞ V ðrÞ sinhr ! #r r kf ðk; r; tÞoth 2 of ðk; r; tÞ¼ð1ÞotcollQuantum effects are included through the inherentlynon-local driving potential in the third term on the lefthandside. Expanding to first order in h, so that only thefirst non-local quantum term is considered, has beenshown to be sufficiently accurate to model non-equilibriumquantum transport and also for the inclusion oftunnelling phenomena in particle based Monte Carlosimulators [10,11]. The additional, non-classical, quantumcorrection term may be viewed as a modification tothe classical potential and acts like an additional quantumforce term in the particle simulations, similar inspirit to the Bohm interpretation.The DG approximation may be derived in a mannersimilar to that for deriving the drift–diffusion approximationfrom the Boltzmann transport equation and resultsin a quantum potential correction term in thestandard drift–diffusion flux [5]. pffiffir 2 nF n ¼ nl n rw D n rn þ 2lr b n pffiffiffið2Þnwhere b n ¼ h=ð12qm nÞ, and all other symbols have theirusual meaning. To avoid the discretisation of fourthorder derivatives in (1) in multidimensional numericalsimulations a generalised electron quasi-Fermi potential/ n is introduced as follows:F n ¼ nl n r/ nð3ÞThus the unipolar drift–diffusion system of equationswith QM corrections, which in many cases is sufficientfor MOSFET simulations, becomes:rðerwÞ ¼ qðp n þ N þ DN AÞ ð4Þpffiffir 2 n2b n pffiffi¼ /n n w þ kT q ln n ð5Þn iFig. 1. Threshold voltage shift due to quantum effects versussubstrate doping. Results for DG and EP are compared tothose obtained from PS.simulations. Fig. 1 shows the QM threshold voltage shiftfor DG as a function of substrate doping compared withthe results presented by Jallepalli [2]. Simulation resultsobtained using the recently developed effective potential(EP) quantum correction approach [12] are also presentedfor comparison in this and the following twofigures. Utilising a single value of the electron effectivemass of 0:18m 0 , obtained from matching the PS resultsat doping concentration 10 18 cm 3 the DG simulationsfollow precisely the PS results. Fig. 2 shows typicalcarrier concentration profiles obtained from the 1-Dsimulations. Both the DG and the EP simulations showa peak in the concentration away from the Si/SiO 2 interface,although the EP produces a sharper drop-off atthe Si/SiO 2 interface compared to PS and DG.Fig. 3, shows an I D –V G characteristic for a 30 nm · 30nm n-MOSFET obtained from our 3-D quantum sim-rðnl n r/ n Þ¼0ð6ÞThe system of equations (4)–(6) is solved self-consistentlyusing standard techniques.3. CalibrationWe have carefully calibrated the DG approachagainst the results of a full-band 1-D Poisson–Schr€odinger (PS) solver [2]. Although PS simulations aremore accurate they are not yet practical for 3-D deviceFig. 2. Electron carrier concentration as a function of distancefrom the interface, for substrate doping of 5 · 10 17 cm 3 . Allhave the same net sheet density.

A. Asenov et al. / Solid-State Electronics 47 (2003) 1141–1145 1143Fig. 3. I D –V G characteristic obtained from both classical andquantum simulators for a 30 nm 30 nm n-MOSFET, withV D ¼ 0:01 V and a substrate doping of 5 10 18 cm 3 .Fig. 5. Quantum (DG) electron concentration profile throughthe centre of a 30 30 1.5 nm double-gate MOSFET. Theoxide thickness is 1.5 nm.ulator. The threshold voltage shift between the classicaland the quantum simulations and the overall shape ofthe current–voltage characteristics obtained using theDG and EP approaches are very similar.4. Source-to-drain tunnellingIt still remains unclear to what extent the approximationsinvolved in deriving the DG approach removeits ability to model the direct source-to-drain tunnellingexpected in nanometre channel length MOSFETs. Insearch of a qualitative answer to this question we simulatea set of double-gate MOSFETs with genericstructure illustrated in Fig. 4.Fig. 5 shows the electron concentration distributionnormal to the gate obtained from the DG simulations ofa30 30 1.5 nm double-gate MOSFET. As expected,for such a thin Si body quantum confinement effectsproduce a peak in the distribution in the middle of thechannel.Fig. 6 illustrates the current–voltage characteristicsfor a set of double-gate transistors with channel lengthsFig. 4. Schematic representation of the double-gate MOSFETstructure considered in this work.Fig. 6. I D –V G characteristics for a double-gate structure, withgate lengths ranging from 30 down to 6 nm, obtained from ourclassical and DG simulations. V D ¼ 0:01 V and V G is applied toboth top and bottom gate contacts.in the range from 30 to 6 nm. In the DG simulations thesub-threshold slope degrades significantly as the channellength is decreased, while in the classical simulations thesub-threshold slope remains nearly constant with channellength. The degradation in the sub-threshold slope inthe DG simulations is consistent with the more elaborateQM simulations performed by others [13]. These observationsprovide an indication that source-to-draintunnelling is included, to some extent, in the DG simulations.Further evidence can be gained by looking at thetemperature dependence of the sub-threshold slope illustratedin Fig. 7. In the classical drift–diffusion simulationsthe sub-threshold slope, essentially thermionic innature, depends linearly on temperature. However, anycurrent due to tunnelling will have a much weakertemperature dependence [14].

1144 A. Asenov et al. / Solid-State Electronics 47 (2003) 1141–1145Fig. 9. Electron equiconcentration surface in a 30 30 5nmdouble-gate atomistic MOSFET with location of dopantsshown.Fig. 7. I D –V G characteristics for an 8 nm channel length doublegatestructure from classical and DG simulations, for a range oftemperatures. V D ¼ 0:01 V and V G is applied to both top andbottom gate contacts.5. Random discrete dopantsTheoretically the double-gate MOSFETs do not requirechannel doping to operate and therefore are consideredto be inherently resistant to random dopantinduced parameter fluctuations. Here we investigate towhat extent the random dopants in the source and drainregion introduce intrinsic parameter fluctuations in suchdevices.Figs. 8 and 9 illustrate the impact of the unavoidablediscrete dopants in the source/drain region on the potentialand the electron distribution in a 30 30 5nmdouble-gate MOSFET. The carrier concentration in thesource and drain region is modulated by the potentialfluctuations associated with individual discrete dopants.The maximum in the carrier concentration is in themiddle of the channel.As shown in Table 1 the corresponding effectivechannel length fluctuations introduce small thresholdvoltage fluctuations which increase from 0.66 to 1.07 mVas the device is scaled from 30 to 10 nm. The on-statecurrent, however, does exhibit significant fluctuations,particularly for the shorter channel length device. This isassociated with the effective channel length variationFig. 8. Electrostatic potential in a 30 30 5 nm double-gateatomistic MOSFET at threshold.Table 1Intrinsic parameter fluctuations in 10 and 30 nm double-gateMOSFETs due to atomistic doping in the source and drainChanneldimensions(L W T )[nm]along the channel introduced by the random discretedopant distribution in the source and the drain regions.6. ConclusionsThe DG approach provides computationally efficientmeans for incorporating quantum corrections in multidimensionaldevice simulations. It agrees well with theavailable data from PS simulations. In the simulation ofsub-10 nm double-gate MOSFETs the DG approachshows behaviour qualitatively consistent with sourceto-draintunnelling.ReferencesThresholdvoltagefluctuationsrV T [mV]Off-currentfluctuationsrI D [%]10 30 1.5 1.07 9.56 7.1330 30 5 0.66 3.28 1.93The threshold voltage is 200 mV.On-currentfluctuationsrI D [%][1] Thompson S, Alavi M, Argavani R, Brand A, Bigwood R,Brandenburg J, et al. IEDM Tech Digest 2001:257–61.[2] Hisamoto D. IEDM Tech Digest 2001:429–32.[3] Jallepalli S, Bude J, Shih WK, Pinto MR, Maziar CM,Tasch Jr AF. IEEE Trans Electron Dev 1997;44:297–303.[4] Ren Z, Venugopal R, Datta S, Lundstrom M, JovanovicD, Fossum J. IEDM Tech Digest 2001:107–10.[5] Rafferty CS, Biegel B, Yu Z, Ancona MG, Bude J, DuttonRW. In: De Meyer, Biesemans, editors. SISPADÕ98; 1998.p. 137–40.[6] Asenov A, Slavcheva G, Brown AR, Davies JH, Saini S.IEDM Tech Digest 1999:535–8.[7] Asenov A, Kaya S, Davies JH. IEEE Trans Electron Dev2002;49:112–9.[8] Asenov A, Slavcheva G, Brown AR, Davies JH, Saini S.IEEE Trans Electron Dev 2001;48:722–9.

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