simulation of statistical variability in nano-cmos transistors...

J Comput Electron (2009) 8: 349–373DOI 10.1007/s10825-009-0292-0

Simulation of statistical variability in nano-CMOS transistorsusing drift-diffusion, Monte Carlo and non-equilibrium Green’sfunction techniques

Asen Asenov · Andrew R. Brown · Gareth Roy · Binjie Cheng · Craig Alexander ·Craig Riddet · Urban Kovac · Antonio Martinez · Natalia Seoane · Scott Roy

Published online: 19 September 2009© Springer Science+Business Media LLC 2009

Abstract In this paper, we present models and tools devel-oped and used by the Device Modelling Group at the Uni-versity of Glasgow to study statistical variability introducedby the discreteness of charge and matter in contemporaryand future Nano-CMOS transistors. The models and tools,based on Drift-Diffusion (DD), Monte Carlo (MC) and Non-Equilibrium Green’s Function (NEGF) techniques, are en-capsulated in the Glasgow 3D statistical ‘atomistic’ devicesimulator. The simulator can handle most of the knownsources of statistical variability including Random DiscreteDopants (RDD), Line Edge Roughness (LER), ThicknessFluctuations in the Oxide (OTF) and Body (BTF), granu-larity of the Poly-Silicon (PSG), Metal Gate (MGG) andHigh-κ (HKG), and oxide trapped charges (OTC). The re-sults of the statistical simulations are verified with respect tomeasurements carried out on fabricated devices. Predictionsabout the magnitude of the statistical variability in futuregenerations of nano-CMOS devices are also presented.

Keywords Semiconductors · MOSFET · Numericalsimulation · Variability

A. Asenov (�) · A.R. Brown · G. Roy · B. Cheng · C. Alexander ·C. Riddet · U. Kovac · A. Martinez · N. Seoane · S. RoyDevice Modelling Group, Department of Electronics andElectrical Engineering, The University of Glasgow, Glasgow,G12 8LT, UKe-mail: [email protected]

N. SeoaneDept. Electronics and Computing Science, Univ. Santiagode Compostela, Santiago de Compostela, 15782, Spain

1 Introduction

The years of ‘happy scaling’ are over and the fundamen-tal challenges that the semiconductor industry faces at thetechnology and device level will deeply affect the design ofthe next-generation integrated circuits and systems. The pro-gressive scaling of CMOS (Complementary Metal-Oxide-Semiconductor) transistors to achieve faster devices andhigher circuit density has fuelled the phenomenal success ofthe semiconductor industry—captured by Moore’s famouslaw. Silicon technology has entered the nano-CMOS erawith 35 nm MOSFETs in mass production in the 45 nmtechnology generation. However it is widely recognisedthat the increasing variability in the device characteristicsis among the major challenges to scaling and integrationfor the present and next generation of nano-CMOS tran-sistors and circuits. Variability of transistor characteristicshas become a major concern associated with CMOS tran-sistor scaling and integration [1, 2]. It already critically af-fects SRAM (Static Random Access Memory) scaling [3],and introduces leakage and timing issues in digital logic cir-cuits [4]. Variability is the main factor restricting the scal-ing of the supply voltage, which for the last four technologygenerations has remained virtually constant, adding to thelooming power crisis.

This paper focuses on the simulation of statistical vari-ability, which has become a dominant source of variabilityfor the 45 nm technology generation and which cannot bereduced by tightening process control. While in the case ofsystematic variability the impact of lithography and stresson the characteristics of an individual transistor can be mod-elled or characterised and therefore factored into the designprocess, in the case of statistical variability only the statisti-cal behaviour of the transistors can be simulated or charac-terised. Two adjacent macroscopically identical transistors

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350 J Comput Electron (2009) 8: 349–373

can have characteristics from the two distant ends of the sta-tistical distribution.

Figure 1 shows that MOSFETs are becoming truly atom-istic devices. The conventional way of describing, design-ing, modelling and simulating such semiconductor devices,illustrated in Fig. 1(a), assuming continuous ionised dopantcharge and smooth boundaries and interfaces, is no longervalid. The granularity of the electric charge and the atom-icity of matter, as illustrated in Fig. 1(b), begin to introducesubstantial variation in individual device characteristics. Thevariation in number and position of dopant atoms in theactive region of decanano MOSFETs makes each transis-tor microscopically different and already introduces signif-icant variations from device to device. In addition, the gateoxide thickness becomes equivalent to several atomic lay-ers with a typical interface roughness of the order of 1–2atomic layers. This will introduce a localised variation inthe oxide thickness resulting in each transistor having a mi-croscopically different oxide thickness or body thicknesspattern. The granularity of the photoresist, together withother factors, will introduce unavoidable line edge rough-ness (LER) in the gate pattern definition and statistical vari-ations in geometry between devices. The granularity of thepoly-silicon or the metal gate and the granularity of the high-κ dielectric introduced at the 45 nm technology generationare other prominent sources of statistical variability.

In this paper, we present models and tools developed overmore than a 10 year period and used by the Device Mod-elling Group at the University of Glasgow to study statisticalvariability introduced by the discreteness of charge and mat-ter in contemporary and future nano-CMOS transistors. Themodels and tools, based on Drift-Diffusion (DD), MonteCarlo (MC) and Non-Equilibrium Green’s Function (NEGF)techniques, are encapsulated in the Glasgow 3D statistical‘atomistic’ device simulator.

Section 2 describes the models, tools and limitations ofthe DD module of the Glasgow ‘atomistic’ simulator, mainlytargeting the electrostatic impact of the variability sources.The MC module of the simulator, developed to capture thetransport variability and its specific impact on the currentvariability, is described in Sect. 3. The fully 3D NEGF mod-ule, designed for the accurate study of variability in 10 nmchannel length transistors and used for the validation andcalibration of the DD and MC modules, is described inSect. 4. Finally the conclusions are drawn in Sect. 5.

2 Drift diffusion simulation

In the presence of statistical variability the aim of the nu-merical simulation shifts from predicting the characteristicsof a single device towards estimating the mean values andthe variance of basic design parameters, such as threshold

Fig. 1 Transition from continuous towards ‘atomistic’ device concepts(a) The traditional approach to semiconductor device simulation as-sumes continuous ionised dopant charge and smooth boundaries andinterfaces; (b) Sketch of a 20 nm MOSFET corresponding to 22 nmtechnology generation. There are less than 50 Si atoms along the chan-nel. Random discrete dopants, atomic scale interface roughness, andline edge roughness introduce significant intrinsic parameter fluctua-tions; (c) Sketch of a sub 10 nm MOSFET expected in mass productionin 2020. There are a handful of Si atoms along the channel. The devicebecomes comparable to biologically important molecules such as ionicchannels

J Comput Electron (2009) 8: 349–373 351

voltage, subthreshold slope, transconductance, drive current,etc. for a whole ensemble of microscopically different de-vices in the system [5]. It must be emphasised that even themean values obtained from, for example, statistical atomisticsimulations are not identical to the values corresponding tocontinuous charge simulation [6]. The simulation of a sin-gle device in the presence of statistical variability sourcesrequires a 3D solution with fine-grain discretisation. The re-quirement for statistical simulations transforms the probleminto a four-dimensional one where the fourth dimension isthe size of the statistical sample. This dictates that a fast sim-ulation technique is used for which the drift-diffusion tech-nique, with quantum corrections, is the best candidate. Sim-ulations of bulk MOSFETs with a fine discretisation mesh(approximately 1,000,000 nodes) can be performed in a mat-ter of a few hours on a single processor.

3D Drift Diffusion (DD) simulations are typically usedto study statistical threshold voltage variability introducedby individual and combined variability sources [7, 8]. Al-though much faster compared to 3D MC or NEGF statisticalsimulations, the DD approximation does not capture non-equilibrium carrier transport effects and therefore underesti-mates the on-state drain current in decanano and nanometrescale devices. However it is perfectly adequate for calculat-ing the threshold voltage and its variations based on a cur-rent criterion in the subthreshold region when the Poissonequation is decoupled from the current continuity equation,the electrostatics dominate the device behaviour and the cur-rent density depends exponentially on the surface potentialand its fluctuations. We do not taken into account Fermi-Dirac statistics as it has been shown that in Si MOSFETswith high DOS this has negligible effect on the results ofnumerical simulations. Additionally, in the DG formulation,as presented in the original paper of Ancona [9], the DOS isconsistent with Boltzmann statistics.

2.1 Density gradient quantum corrections

Density gradient (DG) quantum corrections for both elec-trons and holes are implemented in the simulator to accountfor quantum confinement effects and to allow the fine grainresolution of individual discrete charges [10] which will bediscussed in the next section. Typically in the DG simula-tion of, for example, n-channel MOSFETs it is sufficientto solve, self-consistently [11], the Poisson (1) and DensityGradient (2), using a modified Gummel approach:

∇ · (ε∇ψ) = −q(p − n + N+D − N−

A ) (1)

where ψ is the electrostatic potential, ε is the dielectric con-stant of the material, q is the electronic charge, p is thehole concentration, n is the electron concentration, ND is

the donor concentration and NA is the acceptor concentra-tion, and

2b∗n

Sn

(1

mnx

∂2Sn

∂x2+ 1

mny

∂2Sn

∂y2+ 1

mnz

∂2Sn

∂z2

)

= φn − ψ + kBT

qln(S2

n) (2)

where φn is the quasi-Fermi potential for electrons, Sn =√n/ni (ni is the intrinsic carrier concentration) and b∗

n =�

2/4qr (r is a variable parameter). Equation (2) is theanisotropic density gradient equation [12] giving differenteffective mass components in the transport (longitudinal) di-rection, mnx , and in the confinement (transverse) directions,mny and mnz. These effective masses are treated as fittingparameters [13].

The effective quantum-corrected potential is then calcu-lated from

ψeff = ψ + 2b∗n

Sn

(1

mnx

∂2Sn

∂x2+ 1

mny

∂2Sn

∂y2+ 1

mnz

∂2Sn

∂z2

)

= φn + kBT

qln(S2

n) (3)

and is then used as the driving potential for the current con-tinuity equation,

∇ · Jn = 0 (4)

where

Jn = −qnμn∇ψeff + qDn∇n (5)

is the current density which is solved using a standardScharfetter-Gummel discretisation based on the effectivequantum-corrected potential, ψeff .

In the simulation of n-channel MOSFETs this takescare of problems associated with the atomistically dopedsource/drain regions. In order to avoid problems associatedwith the atomistically doped substrate the DG hole equation-of-state,

2b∗p

Sp

(1

mpx

∂2Sp

∂x2+ 1

mpy

∂2Sp

∂y2+ 1

mpz

∂2Sp

∂z2

)

= ψ − φp + kBT

qln(S2

p) (6)

has to be added to the above system but without solving thehole current continuity equation. The systems of [(1), (2)and (6)] and [(5)] are solved self-consistently until conver-gence.

Similarly to the NEGF approach described in Sect. 4,Neumann boundary conditions (NBC) are used in the sourceand drain, as the potential adjusts the electron injection topreserve charge neutrality [14]. The NBC work better in


conjunction with Density Gradient than Dirichlet boundaryconditions, allowing the electron concentration in the sourceand drain to follow the quantum mechanical distribution.Following Jin et al. [15] the gradient of the electron con-centration perpendicular to the Si/SiO2 interface depends onthe relative effective masses in the silicon and the SiO2, ac-counting for penetration of the electron wave function intothe oxide.

2.2 Introduction of variability sources

In this section we describe the techniques and the modelsused to introduce some of the most important variabilitysources in the DD simulator. The impact of these variabil-ity sources on the device electrostatics also propagates inthe corresponding MC and NEGF simulations.

2.2.1 Random discrete dopants

The best way of introducing RDD in variability simula-tions is to use output from atomic scale process simula-tions like the one illustrated in Fig. 2(a). However this isvery time consuming and, in most of the cases, random dis-crete dopants are generated based on the continuous dopingprofiles obtained from conventional TCAD process simula-tors. In our DD simulator, following the methodology de-scribed in [16], all sites of the silicon lattice covering thesimulated device are scanned one by one. Dopants are in-troduced randomly in the sites with a probability given bythe corresponding dopant-to-silicon concentration ratio us-ing a rejection technique. The charge of each dopant is as-signed to the eight surrounding mesh nodes using the cloud-in-cell (CIS) technique commonly used in Monte Carlo sim-ulations. A typical 3D potential distribution reflecting theimpact of random discrete dopants in a typical 35 nm n-channel MOSFET [17] is illustrated in Fig 2(b).

The resolution of individual charges in ‘atomistic’ DDsimulations using a fine mesh creates problems [18]. Due tothe use of Boltzmann or Fermi-Dirac statistics in the clas-sical drift-diffusion approach the electron concentration fol-lows the electrostatic potential, gained from the solution ofthe Poisson equation. As a result, a significant amount ofmobile charge can become trapped (localised) in the sharplyresolved Coulomb potential wells created by discrete dopantcharges assigned to a fine mesh. Such trapping is unphysicalsince, quantum mechanically, confinement keeps the groundelectron state high in the Coulomb well. The artificial chargetrapping increases the resistance of the source/drain regionsand modifies the depletion layer resulting in artificial low-ering of the threshold voltage. Another detrimental effectof this charge trapping in classical simulations is the strongsensitivity of the quantity of trapped charge to the mesh size.If a finer mesh is used, better resolving the singular Coulombpotential well, the amount of trapped charge increases.

Fig. 2 (a) The position of discrete random dopants obtained from theoutput of an atomic scale process simulation and (b) a typical 3D po-tential distribution reflecting the impact of random discrete dopants ina typical 35 nm n-channel MOSFET

Attempts to correct these problems in ‘atomistic’ simula-tions have been made by charge smearing [19] or by splittingof the Coulomb potential into short- and long-range compo-nents based on screening considerations [18]. The chargesmearing approach is however purely empirical and can re-sult in a loss of resolution with respect to ‘atomistic’ scaleeffects. The splitting of the Coulomb potential into short-and long-range components also suffers from drawbacks in-cluding the arbitrary choice of the cut off parameter andthe potential double counting of the mobile charge screen-ing. The depth of the well in the long range potential at thepoint charge is also much larger than the ground state inthe coulomb well and still could result in substantial charge


trapping. The use of DG quantum corrections capture accu-rately not only the confinement effects in the channel but theconfinement in the Coulomb potential well associated withindividual discrete dopants. By adjusting the effective massparameters it is possible to calibrate the Density Gradientsimulations against rigorous 3D NEGF simulations [13] andwe were able to reproduce accurately the quantum mechani-cal charge distribution around a single dopant in the channelof a nanowire MOSFET shown in Fig. 3 [14].

The reduction in mesh sensitivity when using DG cor-rections in the DD simulation is illustrated in the simula-tion of an ‘atomistic’ silicon resistor with donor concen-tration NA = 1 × 1020 cm−3, representative of the MOS-FET source and drain regions. Figure 4 illustrates the meshsize dependence of the statistically simulated ohmic current-voltage characteristics of the resistor. In the purely classical‘atomistic’ simulations the resistance increases with the re-

Fig. 3 Electron concentration along the centre of a nanowire MOS-FET with a dopant in the middle of the channel for different gate volt-age, VG, comparing density gradient simulations with non-equilibriumGreen’s functions

duction of the mesh spacing while in the DG simulations theresistance is practically mesh spacing independent, althoughslightly higher than the resistance corresponding to contin-uous doping simulations. This slight increase in the resis-tance, associated with some remaining, but mesh size inde-pendent, degree of charge trapping, could be compensatedfor by adjustment of the doping concentration dependenceof the mobility.

For the accurate ‘atomistic’ simulation of a MOSFET it isalso important to consider the trapping of holes in the chan-nel region when ‘atomistic’ acceptors are considered. Thistrapping leads to a change in both the size and shape of thedepletion region, which in turn alters the characteristics ofthe simulated device by reducing, for example, the thresh-old voltage. The smoothing effect of the DG approximationcan be seen in Fig. 5 which shows the potential profile, takenfrom the interface down through the bulk of a 30 × 30 nmn-channel MOSFET. Figure 5(a) shows the classical po-tential in which the sharply resolved potential of individ-ual dopants can be observed, Fig. 5(b) shows the smoothedquantum potential obtained from DG for the same device.

2.2.2 Line edge roughness

Line edge roughness (LER) illustrated in Fig. 6(a) caused bytolerances inherent to materials and tools used in the litho-graphy processes is yet another source of fluctuations thatneeds close attention. It will be increasingly difficult to re-duce LER below the current level of approximately 5 nm,which is limited by the molecular dimensions in the pho-toresist used in the 193 nm lithography systems, and there-fore will be an increasingly important source of ‘intrinsic’parameter fluctuations in the future [20].

The method used to generate random junction patternsis based on a one-dimensional (1-D) Fourier synthesis that

Fig. 4 I-V characteristics of a30 × 20 × 20 nm resistorcomparing continuous dopingdistribution with atomisticaverages for classical and DGsimulations for different meshspacing


Fig. 5 Vertical 2D profile of the classical electrostatic potential in a30 nm ‘atomistic’ MOSFET with (a) drift-diffusion and (b) densitygradient for electrons and holes

generates gate edges from a power spectrum correspondingto a Gaussian autocorrelation function. The parameters usedto describe this gate edge are the correlation length, �, andthe rms amplitude, �. The rms amplitude can be thought ofas standard deviation of the x-coordinate of the gate edge ifwe assume that the gate edge is parallel to the y-direction.In most cases the value quoted for LER is traditionally de-fined as three times the rms amplitude (i.e. 3�). The corre-lation length is obtained by fitting a particular type of auto-correlation function to the gate edge line. The algorithm forgenerating a random line creates a complex array of N ele-ments whose amplitudes are determined by the power spec-trum obtained from a Gaussian autocorrelation function. SG

as shown in (7) is the power spectrum for a Gaussian auto-

Fig. 6 (a) Typical LER in photoresist (Sandia Labs.) and (b) potentialdistribution in a 35 nm MOSFET subject to LER

correlation function where k = i(2π/N,dx) is the discretespacing used for the line and 0 ≤ i ≤ N/2.

SG(k) = √πδ2�e−k2�2/4 (7)

The phases of each of the elements is selected at randomwhich makes each line unique, however only (N/2)-2 ele-ments are independent while the rest are selected throughsymmetry operations so that after an inverse Fourier trans-form the resulting height function, H(x) will be real. An ex-ample of the impact of LER on the potential distribution ofthe 35 nm transistor used also as an example in Fig. 2(b) isillustrated in Fig. 6(b).


2.2.3 Polysilicon granularity

The polycrystalline granular structure of the polysilicon gatehas also been identified as an important source of intrinsicparameter fluctuations [21–24]. Enhanced diffusion alongthe grain boundaries and localised penetration of dopantsthrough the gate oxide into the channel from the high dopingregions in the gate are potential sources of variability [25].However, the most significant source of fluctuations withinpolysilicon gates is likely to be Fermi level pinning at theboundaries between grains due to the high density of defectstates [26].

In order to introduce a realistic random grain structurein the simulations, a large AFM image of polycrystallinesilicon grains [27], shown in Fig. 7(a), has been used as atemplate. The grain boundaries in this image were tracedin black in Adobe Illustrator leaving the grains white. Thepolysilicon grain size distribution depends strongly on thedeposition and annealing conditions. In our simulations theimage is scaled so that the average grain diameter can repli-cate an experimentally observed average diameter, and thena rasterised template image is saved in a format readable bythe simulator. The simulator imports a random (in both lo-cation and orientation) section of the grain template imagethat corresponds to the gate dimensions of the simulated de-vice. It pins the Fermi level along any grain boundaries, de-fined by black pixels in the image, detected in the templatesection. The typical device dimensions simulated are muchsmaller than the dimensions of the template image, meaningthat a large number of completely independent grain pat-terns can be extracted and used for the simulation of eachdifferent device in the statistical ensemble. The energy posi-tion of the Fermi level pinning can be varied in the simula-tions. Figure 7(b) illustrates the impact of the surface poten-tial pinning on the potential distribution in the same 35 nmMOSFET previously used as an example.

2.2.4 Other sources of variability

The introduction of high-κ /metal gate technology improvesthe RDD-induced variability due to the reduction in theequivalent oxide thickness and removes the PSG effects. Atthe same time it introduces high-κ granularity illustrated inFig. 8(a) and variability due to work-function variation dueto the metal gate granularity illustrated in Fig. 8(b) [28]. Inextremely scaled transistors atomic scale interface rough-ness illustrated in Fig. 8(c) [29] and corresponding bodythickness variations [30] can become an important sourceof statistical variability.

2.3 Validation of the drift-diffusion simulationmethodology

The validation of our simulation technology is done in com-parison with measured statistical variability data in 45 nm

Fig. 7 (a) An SEM micrograph of typical PSG [27] and (b) Potentialdistribution in a 35 nm MOSFET subject to PSG

LP CMOS transistors [31]. The simulator was adjusted toaccurately match the carefully calibrated TCAD device sim-ulation results of devices without variability by adjusting theeffective mass parameters involved in DG formalism, andthe mobility model parameters. The calibration results areshown in Fig. 9, where ‘low’ and ‘high’ drain bias stand for50 mV and 1.1 V respectively.

The simulation results for the standard deviation of thethreshold voltage introduced by individual and combined


Fig. 8 (a) Granularity in HfON high-κ dielectrics (Sematech),(b) metal granularity causing gate work-function variation and (c) in-terface roughness (IBM)

sources of statistical variability are compared with the mea-sured data in Table 1. In the n-channel MOSFET case theaccurate reproduction of the experimental measurements ne-

cessitates the assumption that, in addition to RDD and LER,the PSG related variability has to be taken into account.Good agreement has been obtained assuming that the Fermilevel at the n-type poly-Si gate grain boundaries is pinnedin the upper half of the bandgap at approximately 0.35 eVbelow the conduction band of silicon. However, in the p-channel MOSFET case the combined effect of just the RDDand LER is sufficient to reproduce accurately the experimen-tal measurements. The reason for this is the presence of ac-ceptor type interface states in the upper half of the bandgapwhich pin the Fermi level in the case of n-type poly-Si, andthe absence of corresponding donor type interface states inthe lower part of the bandgap which leaves the Fermi levelunpinned in the case of p-type poly-Si [32].

2.4 Predicting the future

In order to foresee the expected magnitude of statistical vari-ability in the future we have studied the impact of RDD,LER and PSG on MOSFETs with 35 nm, 25 nm, 18 nm,13 nm and 9 nm physical gate length. The scaling of the sim-ulated devices is based on a 35 nm MOSFET published byToshiba [33] against which our simulations were carefullycalibrated. The scaling closely follows the prescriptions ofthe ITRS in terms of equivalent oxide thickness, junctiondepth, doping and supply voltage. The intention was also topreserve the main features of the reference 35 nm MOSFETand, in particular, to keep the channel doping concentrationat the interface as low as possible. Figure 10 shows the struc-ture of the scaled devices. More details about the scaling ap-proach and the characteristics of the scaled devices may befound in [34]. Figure 11(a) compares the channel length de-pendence of σVT introduced by random dopants, line edgeroughness and poly-Si grain boundaries with Fermi levelpinning. The average size of the polysilicon grains was keptat 40 nm for all channel lengths. Two scenarios for the mag-nitude of LER were considered in the simulations. In thefirst scenario the LER values decrease with the reduction ofthe channel length following the prescriptions of the ITRS.In this case the dominant source of variability at all chan-nel lengths are the random discrete dopants. The variabilityintroduced by the polysilicon granularity is similar to thatintroduced by random discrete dopants for the 35 nm and25 nm MOSFETs, but at shorter channel lengths the randomdopants take over. The combined effect of the three sourcesof variability is also shown in the same figure. In the sec-ond scenario LER remains constant and equal to its currentvalue of approximately 4 nm the results for the 35 nm andthe 25 nm MOSFETs are very similar to the results withscaled LER but below 25 nm channel length LER rapidlybecomes the dominant source of variability.

Figure 11b is analogous to Fig. 11(a) exploring the sce-nario when the oxide thickness, which is difficult to scale


Fig. 9 Top: structure of the simulated 45 nm LP technology tran-sistors; Middle: Agreement between the commercial TCAD and theGlasgow ‘atomistic’ simulator results; bottom potential distribution in

one of the simulated 200 microscopically different characteristics inthe presence of RDD, LER and PSG


Table 1 σVT introduced byindividual and combinedsources of statistical variability

n-channel MOSFET p-channel MOSFET

σVT [mV] σVT [mV] σVT [mV] σVT [mV]

(VDS = 0.05 V) (VDS = 1.1 V) (VDS = 0.05 V) (VDS = 1.1 V)

RDD 50 52 51 54

LER 20 33 13 22

PSG 30 26 – –

Combined 62 69 53 59

Experimental 62 67 54 57

Fig. 10 Examples of realistic conventional MOSFETs scaled froma template 35 nm device according to the ITRS requirements for the90 nm, 65 nm, 45 nm, 32 nm and 22 nm technology generations, ob-tained from process simulation with Taurus Process

further, even with the introduction of high-κ gate stacks willremain stagnated at 1 nm. This will lead to an explosionin the threshold voltage variability for bulk MOSFETs withphysical channel length below 25 nm.

Thin-body SOI transistors tolerate very low channel dop-ing and therefore are resilient to the main source of statis-tical variability in bulk MOSFETs, the RDD. At the sametime very good electrostatic integrity and the correspond-ing reduction of the threshold voltage sensitivity on chan-nel length and drain voltage also reduces their susceptibil-ity to LER induced variability. However the introductionof a high-κ gate dielectric and the corresponding relativelyhigh density of fixed and trapped charge (FTC) introducesunwanted variability, which can neutralise the benefits ob-tained from low channel doping and reduced short channeleffects.

Figure 12 illustrates the impact of FTC with differentareal density on the potential distribution in 32 nm ultra-thin body (UTB) silicon-on-insulator (SOI) MOSFETS de-scribed in detail elsewhere [35]. The simulation results ofthe impact of RDD, LER and FTC with different areal den-sities on threshold voltage variation are summarised in Ta-ble 2.

3 Monte Carlo simulations

3.1 ‘Ab initio’ ionised impurity scattering

Monte Carlo (MC) simulations are needed when studyingcurrent variability in nanoscaled MOSFETs for two rea-

Fig. 11 (a) Channel length dependence of σVT introduced by ran-dom dopants, line edge roughness and poly-Si granularity: (A) LERscales according ITRS; (B) LER = 4 nm and (b) Channel length de-pendence of σVT introduced by random dopants, line edge roughnessand poly-Si granularity: (A) tox scales according ITRS; (B): tox = 1 nm

sons. Firstly, the DD simulation cannot properly representthe non-equilibrium, quasi-ballistic transport in contempo-rary sub-50 nm MOSFETs. Secondly, the DD simulations


Fig. 12 Typical potential profiles corresponding to trap charge withsheet density at (a) 1 × 1011 cm−2, (b) 5 × 1011 cm−2, and(c) 1 × 1012 cm−2

Table 2 Summary of simulation results with trapped charges

32 nm σVT (mV) 22 nm σVT (mV)

Vds Vds Vds Vds

(50 mV) (1.0 V) (50 mV) (1.0 V)

Trap (1e11 cm−2) 11 11 5.1 4.8

Trap (5e11 cm−2) 18 17 13 12

Trap (1e12 cm−2) 26 23 18 17

accurately capture the electrostatic impact of the differentvariability sources but have conceptual problems in resolv-ing their impact on transport and therefore on the on-currentvariability. Transport variation may be naturally includedwithin 3D MC via the propagation of carriers within theunique potential landscapes resulting from random deviceconfigurations. Scattering is then no longer treated as aninstantaneous event, as is traditionally the case when con-sidering scattering rates within MC, but is instead recov-ered from particle trajectories extended over time. Such an‘ab initio’ scattering approach will necessarily include dy-namic screening effects, simultaneous scattering from mul-tiple sources of variability as well as scattering from inter-actions within the carrier ensemble itself. (Note that ‘ab ini-tio’ is used here in the general sense of the term to describethe scattering in MC, and should not be confused with thespecific meaning adopted for use with respect to electronicstructure calculations.) The electrostatic modulation in car-rier density captured within DD simulations will be recov-ered in MC. However, accuracy demands that both the in-teraction potential and the carrier propagation be accuratelyresolved.

3.1.1 Implementation of short range corrections

In the case of ionised impurity scattering, which representsthe source of largest variation within conventional (bulk)MOSFETs [17], the interaction potential is the unscreenedCoulomb potential of the RDD. The long-range componentof this potential is accurately determined by the mesh-basedsolution of Poisson’s equation in which ionised impuritiesare represented as point charges. Due to aliasing in this so-lution however, the potential within a few mesh spacings of apoint charge is under represented and needs to be corrected.This is also needed in order to avoid the artificial carrier-trapping in Coulomb wells due to the classical treatment ofthe particles, as discussed in Sect. 1. Short-range correctionscould be introduced via a molecular-dynamics-like approachin which short-range particle-particle interactions are evalu-ated directly, the so-called P3M algorithm [36]. Using thismethod, efficient evaluation of long-range forces is obtainedfrom the mesh-based potential. This additionally includescontributions from all applied external biases as well as ef-fects associated with changes in dielectrics throughout the


device structure. A short-range correction for all interactionswithin a certain radius then accurately accounts for the to-tal interaction. Care must be taken to avoid double count-ing the short-range interaction as it is partly resolved by themesh. It is therefore necessary to approximate and removethis mesh-based contribution [36, 37]. With minimal effortspent in sorting the simulated particles, searching for nearestneighbour interactions may be optimised [36].

For the short-range interactions we adopt a well-testedanalytical model, (8) [36], defining the interaction at sepa-ration r from a point charge. It agrees with the long-rangeCoulomb field at large separation but reaches a maximum atcharacteristic radius r = rc. At radii smaller than this cut-offradius the field decreases to zero, removing the rapidly vary-ing short-range component and the potential singularity andreducing the artificial carrier trapping.

E(r) = qr

4πε0εr (r2 + 2r2c )3/2

(8)

Increasing rc improves numerical integration but reducesthe contribution of the short-range Coulomb scattering.However, resolving short-range collisions is necessary forthe accurate direct treatment of ionised impurity scatteringand so choosing rc is a compromise between maximisingthe short-range interaction while minimising propagationerrors. As a means of quantifying the impact of this short-range correction approach, simulation of Rutherford scatter-ing of electrons was studied using values of rc in the rangeof 0–2 nm. Figure 13 shows the corresponding scatteringangle dependence upon impact parameter for propagation

Fig. 13 Scattering angle dependence upon impact parameter for vari-ous values of rc

with �t = 0.1 fs. The repulsive interaction with the nega-tive charge is again well reproduced over all cutoff radii andclosely follows the Rutherford result, while the reduction inmaximum scattering angle with increasing rc is seen in theattractive interaction. Based upon this, we adopt rc = 0.5 nmto maximise scattering with a propagation time step of 0.1 fsto maximise efficiency.

The reproduction of the field concentration dependentbulk mobility in Silicon is a reasonable validation test forthe ‘ab initio’ ionised impurity scattering approach. Sim-ulation of a series of simple n+−n-n+ diodes containingaround 15,000 donors within the central ‘bulk’ region wasperformed in this regard and the size of the simulation re-gion was changed accordingly. The resultant potential dis-tributions for donor concentrations of 1015, 1016, 1017 and1018 cm−3 are shown in Fig. 14 and clearly indicate themesh resolved potential fluctuation associated with the ran-dom distribution of donors. Mobility estimated from the ra-tio of average velocity to average field within the central‘bulk’ region is plotted in Fig. 15 together with results ofsimilar MC device [38] and molecular dynamics [39] simu-lations. It is seen that the electron-donor interaction is well

Fig. 14 Potential distribution from ‘ab initio’ MC simulation ofn+−n-n+ diodes


Fig. 15 Simulated bulk mobility compared with experimental results

reproduced and that it compares well with the other reportedresults.

3.1.2 Impact on on-current variability

Drain current variation associated with RDD was esti-mated via statistical DD and both frozen-field (FF) and self-consistent (SC) ‘ab initio’ MC at both low and high drainbias for a series of realistic, scaled devices. The standarddeviation of the drain current distribution as a function ofdevice channel length is shown in Fig. 16. In each case,and consistently with previously published results [17], thedistribution in drain current variation increases with the re-duction of the gate length. It is also seen that the MC simula-tion with ‘ab initio’ ionised impurity scattering consistentlyyields greater variation compared with corresponding DDresults. This is attributed to the additional position depen-dent variation in transport accompanying the electrostaticvariation in electron density from device to device.

At low drain bias, VD = 0.01 V, the frozen-field and self-consistent MC results are in good agreement, reproducingthe same trend and justifying the use of the frozen-field ap-proximation in this regime. The frozen field results consis-tently show a slightly larger current variation but the differ-ence does not exceed more than 6–7% over the whole chan-nel length range. This difference may be related to the over-estimation of the Coulomb potential screening in the staticscreening inherent to the DD simulations used to deliver thepotential landscape for frozen field MC simulations.

Since DD simulation only captures the electrostatic vari-ation, the contribution of transport variation to the total draincurrent variation can be inferred if it is assumed the electro-static contribution within MC simulation is well reproduced

Fig. 16 Percentage drain current variation as a function of channellength from DD simulation and frozen field and self-consistent ‘ab ini-tio’ MC. The contribution from transport variation within MC is shownin the inset

by DD simulation. This approximation fails if the dynamicscreening of impurities by electrons within the ‘ab initio’MC simulation significantly differs from the static screeningwithin DD. Within frozen-field simulation the electrostaticpotential is identically that from DD and the simulated elec-tron dynamics do not alter the electron distribution aroundacceptor ions and hence the screening of their Coulomb po-tential. Within self-consistent simulations, the electron dis-tribution around acceptor ions is determined by the carrierdynamics and the corresponding dynamic screening is cap-tured by the self-consistent solution of the Poisson equation.The percentage contribution of transport variation to the to-tal drain current variation from MC simulation is then de-termined and plotted as an inset to Fig. 16. While results ofcurrent variation from MC simulation are always larger thanthat of DD, their comparison differs at low and high drainbias and will be discussed separately.

The correlation of percentage drain current variationfrom self-consistent ‘ab initio’ MC simulations and DD sim-ulations for each device within a statistical ensemble of 100devices is plotted in Fig. 17. The correlation coefficient isalso indicated in the figure and is rather high (correlationcoefficient on the order of 0.98) at the largest gate lengths,while decreasing very slightly with decreasing gate length.Correlation is to be expected since the electrostatic influenceof the random acceptor configuration is the same within bothDD and MC simulations. The spread in data reflects the in-fluence of the acceptor configuration on electron transport.While some impurity configurations give rise to similar elec-trostatic variation in current, their impact on scattering mayvary greatly. The slight reduction in correlation at smaller


Fig. 17 Correlation of DD and MC estimated drain current variationat low drain over the entire device ensembles

gate lengths indicates that scattering can become increas-ingly sensitive to variation in dopant positions.

Figure 16 also reveals that at a high drain bias of VD =0.8 V the current variation estimated by DD simulationsare larger compared with DD results at low drain bias. TheMC simulations again reveal more variation compared to theDD simulations, but show less variation compared to thelow drain bias MC results. In this case the self-consistentMC simulation yields larger variation compared to frozenfield simulations, though again both recover the same trendand the results are comparable. The increase of electrostatic-associated variation in the DD simulations and the reduction

of combined variation in the MC simulations indicate thatthe percentage of the total variation associated with trans-port in the high drain bias case is less than at low drain bias.This is highlighted in the inset of Fig. 16 where transportvariation accounts for, at most, around 25% of the total. Thereason for this decrease of the transport related variability istwofold. At higher drain bias the average energy of the car-riers in the channel is higher and the ionised impurity scat-tering, the source of the transport variation, is less effective.

3.2 Robust density gradient quantum corrections

Quantum confinement is increasingly important in contem-porary MOSFETS and has to be accurately captured notonly in DD but also in MC simulations [40]. Several tech-niques have been implemented before with varying success.We have opted to use in the MC module the same Den-sity Gradient (DG) approach already implemented in the DDmodule of the Glasgow ‘atomistic’ simulator.

3.2.1 Implementation

Density Gradient quantum corrections [9] have been in-corporated within the 3D MC module following three ap-proaches, each introducing an increasing degree of self-consistency between transport, field and the quantum cor-rections. These approaches are illustrated in Fig. 18. In the‘frozen field’ approach (Frozen Field Monte Carlo—FFMC)the DG effective quantum potential solution from 3D quan-tum corrected DD is used in the MC module as the soledriving force for the particles and is never updated. Thisapproach targets low drain voltage (VD) MOSFET opera-tion in the linear regime where non-equilibrium transporteffects are negligible. The greatest benefit of FFMC is itscomputational efficiency allowing for transport variabilityto be studied over large statistical samples. Simulation athigher VD , aiming to capture non-equilibrium transport ef-fects, requires the field to be updated self-consistently withthe carrier distribution over the course of the MC simula-tion. This particle-field self-consistency is at the heart ofthe ‘frozen quantum corrections’ approach (Frozen Quan-tum Monte Carlo—FQMC). In this approach a quantumcorrection term, ψqc , defined as the difference between theclassical and effective quantum potential from an initial DDsolution, is calculated for each mesh point. During self-consistent MC simulation, the quantum corrected drivingforce Fq is then calculated based upon the classical poten-tial, ψcl , using:

Fq = −∇(ψcl + ψqc) (9)

In the third, fully self-consistent approach (Self-ConsistentQuantum Monte Carlo—SCQMC), the above quantum cor-rection term is updated periodically during the course of the


Fig. 18 Flowchart showing thecomputational steps needed forFFMC, FQMC and SCQMC

self-consistent simulation. Time averaging is used to smoothout the inherently noisy carrier distribution that usually re-stricts the use of DG corrections in MC. The quantum cor-rection is updated by first solving the modified DG equation[41] for the quantum density, nq :

2b∗n

Sn

(1

mnx

∂2Sn

∂x2+ 1

mny

∂2Sn

∂y2+ 1

mnz

∂2Sn

∂z2

)

= φn − 〈ψ〉t + kBT

qln(S2

n) (10)

Again, S = √nq/ni , the other symbols have their usual

meaning, and 〈. . .〉t denotes a time-averaged value. Equa-tion (10) is discretised using a finite box method; the corre-sponding system of equations is linearised and solved usinga Red-Black Successive Over-Relaxation (SOR) iterativescheme, which is amenable to parallelisation. A Maxwell-Boltzmann equation of state is assumed [42], and ψn is up-dated using:

φn = 〈ψcl + ψqc〉t − kBT

qln

( 〈nmc〉tni

)(11)

where nmc is the electron distribution obtained from a cloud-in-cell assignment of the MC particles to the mesh. From the

new distribution of nq , obtained by solving equation (10), anew quantum correction term is calculated using:

ψqc = φn + kBT

qln

(nq

ni

)− 〈ψcl〉t (12)

This may then be applied within self-consistent MC as in(9). As shown in Fig. 18, the update of ψqc is carried outself-consistently with Poisson’s equation and the propaga-tion/scattering routines in the MC engine throughout thecourse of the simulation. This implementation is similar inspirit to the Schrödinger-based quantum correction schemein [42]. The quantum correction is not updated during theinitial transient period to avoid excessive noise in the calcu-lation and, due to the averaging, the SCQMC technique isnot suitable for transient simulations. The quantum correc-tion is typically updated every few 100-fs.

For the self-consistent FQMC and SCQMC simulations,careful implementation of the Ohmic particle contacts inthe source and drain regions is essential in order to avoidinstabilities and a build-up of a space-charge region thatcan destabilise the simulation. Carriers reaching the contactboundary are removed and replaced in order to maintain lo-cal charge neutrality and to ensure that the difference in ψn

between the source and drain contacts is equal to the applieddrain bias. In the case of classical MC simulations, differ-ent approaches to carrier injection have been implemented,from sophisticated techniques to the use of reservoir con-tacts in the place of planar ones. However, the inclusion of


Fig. 19 Comparison of ID–VG

characteristics from DD, FFMC,FQMC and SCQMC atVD = 1 mV and VD = 0.7 V inthe absence of interfaceroughness showing an excellentagreement for all, except forFFMC at low VD whichunderestimates ID

Table 3 Dimensions and doping of the double gate device studied,with channel doping of 1014 cm−3

Architecture Double Gate

Channel Length (Lchan) 20 nm

Device Width 20 nm

Silicon Thickness (tSi ) 3.3 nm

Oxide Thickness (tox) 1.05 nm

Source/Drain Doping 2×1020 cm−3

Channel Doping 1014 cm−3

quantum corrections alters the confined carrier distribution,shifting the peak away from the semiconductor/oxide inter-face, and can complicate carrier injection when the contactis adjacent to a confined system. Therefore, Neumann BCs[43, 44] have been implemented at the Ohmic contact re-gions similar to the techniques used in NEGF simulations[14], which help maintain stability in the simulation.

The validity of the three self-consistency schemes in-troduced above is evaluated in comparison with DD sim-ulations for the device with dimensions shown in Table 3.Figure 19 compares the ID–VG characteristics at low andhigh VD obtained using all simulation methodologies. Here,FFMC fails to accurately reproduce the value of ID achievedby the other MC methods and the calibrated DD simula-tion, particularly at high drain voltage. The explanation ofthis can be found in Fig. 20 which compares the velocitydistribution along the channel from the three different MCapproaches, along with the quantum corrected potential ob-tained from the FQMC for VD = 0.7 V, VG = 0.8 V. Thelack of self-consistency in the FFMC simulations results inthe incorrect velocity at the source end of the channel andunderestimation of the current at high drain voltage. At thesame time, the good agreement between FQMC and SC-QMC suggests that the self-consistency of the quantum cor-

Fig. 20 Velocity profiles at VD = 0.7 V for all versions of the MCsimulator, along with the quantum corrected potential from FQMC. InFFMC the carriers fail to react to changes in the field, and thus thevelocity profile is incorrect

rections in this case is not important, and the VT shift and thequantum carrier distribution at the source end of the chan-nel, which do not change significantly with the applied drainvoltage, are the main effects that need to be captured [45].As the FFMC simulation does not involve the self-consistentsolution of Possion’s equation the simulation time is rela-tively short (a few hours). The self-consistent FQMC andSCQMC simulations take substantially longer (up to oneweek).

3.2.2 Application to ‘ab-initio’ impurity scattering

The incorporation of DG quantum corrections in the MCmodule creates interesting opportunities for consistent han-dling of the ‘ab-initio’ impurity scattering. The effectivequantum potential from the density gradient solution sur-rounding a point charge is compared in Fig. 21 with the po-tential corresponding to the short-range force model used in


Fig. 21 Comparison of the density gradient solution surrounding apoint charge with the analytical short-range model used in ‘ab initio’ionised impurity scattering

Fig. 22 Comparison of simulated Rutherford scattering using analyti-cal short-range correction and density gradient mesh solution

the ‘ab initio’ ionised impurity scattering. The two are in avery close agreement for mesh spacing below 1 nm, whichimplies that the derivative of the effective quantum potentialcan be used as a particle driving force instead of (8). Thisassumption needs validation similar to the validation of theshort-range correction model.

Figure 22 shows results of the scattering angle depen-dence for thermal electrons interacting with both a posi-

Fig. 23 Potential distribution throughout the simulated ‘bulk’ resistorstructure comparing the classical potential (top) with the density gra-dient resolved potential (bottom)

tive and negative central ion indicating a close agreementbetween the effective quantum potential and the analyticalshort-range interaction model. In both models the scatter-ing angle closely agrees with the Rutherford model over thecomplete range of impact parameters in the case of a re-pulsive interaction with a negative ion, while significant un-derestimation of the scattering angle is seen at small impactparameters for attractive interactions.

Further validation requires the reproduction of experi-mental concentration-dependent bulk mobility using the pre-viously described simulation of atomistically doped resis-tor structures. However, such simulations using the effectivequantum potential are limited by practicality to small, highlydoped simulation domains due to the upper limit of 1 nm onthe mesh spacing. We therefore extend previous results ofthe self-consistent simulation of an n+–n-n+ diode with cen-tral ‘bulk’ doping of 2 × 1018 cm−3, relevant to the channelof modern nanoscale bulk MOSFETs. Mobility results wereobtained by averaging 10 simulations with unique randomdopant configurations. The classical electrostatic potentialis compared with the effective quantum potential from den-sity gradient for one such simulated structure in Fig. 23. Theeffective quantum potential is seen to smooth the peaks as-sociated with the donor impurities and limit their interac-tion with electrons as expected. The simulated mobility isplotted in Fig. 24 and compared with experimental data andprevious ‘ab initio’ results obtained using the short-rangecorrection. Good agreement is seen with experimental dataand with the continuation of the trend of the prior simulationresults.

4 Non-equilibrium Green’s function simulations

When the nano-CMOS devices reach sub-10 nm dimen-sions, not only quantum confinement effects but also source-


Fig. 24 Simulated bulk mobility using density gradient mesh resolvedinteraction alone. Previous results using an analytic short-range correc-tion are shown for comparison

to-drain tunnelling will start to affect their performance. Al-though the DG approach described in the previous two sec-tions can cope successfully with the impact of quantum con-finement on the operation and the performance of contem-porary CMOS transistors it conceptually cannot accuratelymodel source-to-drain tunnelling. Therefore for sub-10 nmtransistors the development of a full-scale quantum transportsimulator becomes a necessity. The non-equilibrium Green’sFunction (NEGF) formalism is a well-established techniquefor quantum transport simulation. It has certain advantagesif scattering also needs to be included in the realistic simu-lation of nano-CMOS transistors.

4.1 Implementation

In the quantum transport module of the Glasgow ‘atom-istic’ simulator the carrier transport is described using theNEGF approach, which is a generalisation of Landauer’sformalism [46] to treat many body systems at room temper-ature in the context of the one particle Green’s function. TheHamiltonian used in the discretisation of the NEGF equa-tions is an effective-mass Hamiltonian that folds the fullcrystal interaction into the electron effective masses. The ef-fective masses of the transport valleys are extracted fromTight Binding calculations that capture the dependence ofthe electron band structure on the nanowire diameter [47].

Sources of incoherent scattering such as phonon interac-tion, and the corresponding self-energies, are not currentlyincluded in the NEGF module. We calculate the correlationmatrix, G<, using the recursive algorithm described in [48].

Fig. 25 Flow chart of the NEGF simulation process

From the correlation matrix, the electron and current densi-ties are calculated by the following equations:

n(E,x) = iG<(E,x, x) (13)

J (E,x) = −iqh

2m(∇ − ∇′)G<(E,x, x′)

∣∣∣∣x=x′

(14)

The boundary conditions of the Green’s function equationsat the contacts, which are given through the contact self-energies, are defined using the algorithm described in [49].

Figure 25 shows the flow chart of our simulator illustrat-ing the computational procedure used to solve the coupledPoisson-NEGF equations. The electrostatic potential and theelectron density obtained from a density gradient (DG) so-lution of the Drift-Diffusion (DD) equations [50] serve as aninitial condition for the Poisson-NEGF cycle. The DD solverhas Neumann boundary conditions for Poisson’s equation inthe source and drain instead of the Dirichlet boundary condi-tions usually used in the DD formalism [14], which matcheswell with the Green’s function boundary conditions. Suchclose initialisation of the potential distribution at the begin-ning of the Poisson-NEGF loop drastically reduces the num-ber of NEGF iterations.


After the first Poisson-NEGF iteration the change in elec-tron concentration from the initial DG solution to the newNEGF solution is moderated by damping. A gradual changein the electron density prevents oscillatory behaviour of thesolution around the impurities leading to divergence. Thesolution instabilities are associated with the extreme sen-sitivity of the quantum density to the shape of the attrac-tive potential. This is related to the discrete nature of thequasi-bound states and their energy sensitivity to the shapeof the potential. We have found that solving the non-linearPoisson equation results in a much more stable convergenceof the Poisson-NEGF system than if the linear version isused. Once a new electron density is obtained from theNEGF solver, a quasi-Fermi level, fn, is calculated usingthe new density and the old potential [51]. This quasi-Fermilevel is used to update the electron concentration and theJacobian when solving the non-linear Poisson equation it-eratively (see Fig. 25). Adaptive damping is used after thesolution of the Poisson equation to limit the change in po-tential and to improve convergence. The alternate solutionsof Poisson and NEGF are iterated until density and currentconverge. This fully-3D self-consistent simulation is com-putationally very expensive and a single device simulationcan take from a few days to over a week to run on a singleprocessor depending on bias conditions and the number ofenergy levels used.

4.2 Simulation of statistical variability in nanowiretransistors

We have studied the impact of different variability sourceson Si nanowire transistors (NWT). The simulated deviceshave a 6 nm undoped channel with 2.2 × 2.2 nm2 cross-section, 0.8 nm SiO2 oxide and 10 nm S/D regions dopedat 1020 cm−3. The transport in the nanowire occurs in the〈100〉 direction. The nanowire diameter-dependent effectivemasses are extracted from sp3d5 second-neighbour-basistight-binding calculations [47]. All the simulations in thiswork have been done at room temperature.

The oxide thickness is 0.8 nm and is not included in theNEGF solution region but is included in the electrostatic po-tential calculation. We do not consider charge penetrationin the oxide. Nevertheless the electrostatic potential at theinterface (which includes the conduction band offset of theoxide) is given as a boundary potential in the calculationof the electron density using the NEGF. The electron con-centration is not forced to zero at the discretisation pointat SiO2 interface, but in the next discretisation point insidethe oxide, providing a softer boundary condition, which al-lows the electron wave function to interact with the inter-face.

4.2.1 Impact of single donors

Firstly, we study a single donor located at the middle ofthe channel of a nanowire MOSFET transistor at differentscreening conditions. The importance of studying a singledonor is twofold: it is relatively easy to isolate the reso-nances coming from a single donor compared to the caseof several dopants, which produce complex interferencepatterns and common resonances; secondly, the screeningby conducting electrons around the single impurity understrong quantisation conditions, and also the self-consistentelectrostatic potential, can be investigated in isolation fromthe influence of other fixed charge sources. All simulationsin this subsection were carried out close to equilibrium atVD = 1 mV in order to study more easily the charge densityaround the impurity.

Figure 26 maps the 3D self-consistent potential and elec-tron density distribution along the wire at VG = 0.4 Vshowing also the equipotential/equiconcentration lines. Theequipotential contours around the dopant are denser alongthe channel direction showing a higher degree of screen-ing in this direction compared to the orthogonal direction.The screening electron concentration extends for severalnanometres from the impurity centre along the channel di-rection. This is formed by resonances (extended states) con-necting the source and drain.

The 1D electrostatic potential and the electron con-centration distribution along the channel in the middle ofthe cross-section, for different gate voltages, are shown inFig. 27. Note the inverse ‘sombrero’ shape of the potentialin Fig. 27(a), which is a combination of the Coulomb po-tential of the ionised donor and the potential associated withthe electron screening charge. This double barrier shape pro-duces quasi-bound states similar to those that appear in res-onant tunnelling structures. At low gate bias the electronsare repelled from the channel, therefore the electron density

Fig. 26 3D potential and density for the device with the donor at themiddle


Fig. 27 Self-consistent electrostatic potential (a) and electron density(b) along the wire at different VG

around the impurity is very low. At high gate bias the po-tential barrier of the channel decreases. As a consequence,the electrons penetrate the channel and therefore contributeto the screening of the impurity. At gate voltages lower than0.3 V the electron concentration at the impurity position isnegligible compared to the source/drain (S/D) doping. This,however, does not indicate an absence of the resonances,as they can be clearly seen in the transmission coefficientsillustrated in Fig. 28 at different gate bias conditions. Thetransmission coefficient contains one resonance (or quasi-bound state) at the onset of each sub-band’s conductance.This resonance can be clearly seen in the density of statesplot in Fig. 29 along the middle of the wire for the firstsub-band. The energy resolution of this plot is 0.1 meV.The impurity potential of Fig. 27(a) introduces transmissionchannels, which increase the current in this device whencompared with the impurity-free device.

Fig. 28 Transmission coefficients at different VG

Fig. 29 Density of states along the middle of the wire for the firstsub-band

4.2.2 Discrete dopants in the source and drain

In this section we study the effect of the discreteness ofdopants in the S/D on the performance of the Si nanowiretransistor. The discrete dopants are located in a 4 nm exten-sion of the S/D regions preserving the same average dopingas in the previously homogeneously doped source/drain re-gions. The total length of the wire simulated is (6 + 4 +6 + 4 + 6) nm = 26 nm, where the five numbers representthe length of the different regions into which the device issubdivided. The 6 nm regions on the far left (source) andright (drain) represent regions of continuous doping, whichguarantee smooth contact injection self-energies and chargeneutrality on the contacts. The 4 nm regions are where thediscrete dopants are located. The central 6 nm region is thegate or channel region.


Fig. 30 ID–VG characteristics, on linear and logarithmic scales, forthe smooth device and the three different dopant configurations

We have studied three cases with different spatial dopantarrangements denoted Ra, Ch and Cr for ‘randomly’ ar-ranged, channel aligned and cross-section aligned donorplacements respectively. The Ch case has been chosen toenhance the injection of electrons in the channel. In the Crcase, the channel injection diminishes because electrons arenaturally repelled from the interface by quantum confine-ment effects. The simulations are done at VD = 0.05 V. Fig-ure 30 shows the I–V characteristics for the smooth (con-tinuous doping), Ra, Ch and Cr devices. There is an averagethreshold voltage shift of approximately 20 mV due to thediscrete dopants. The sub-threshold slope is worse in the Chcase due to poor gate control of the charge which is con-centrated, in this case, along the middle of the wire. The Racase has almost identical subthreshold slope compared to thesmooth case.

The effect of the discreteness of the impurities on the on-current is more dramatic. Compared to the smooth devicethe on-current falls by approximately 70% in the Cr case,and by approximately 27% in the Ch case. The detailed ex-planation of this behaviour is based on an analysis of thecorresponding transmission coefficient as a function of en-ergy illustrated in Fig. 31 for the four devices consideredin this work at VG = 0.3 V. The presence of resonances inthe transmission coefficients is typical for devices with dis-crete donor dopants, since there are electrons partially re-flected by the impurity potential, decreasing the total trans-mission. This partial reflection of the electrons can be con-sidered as ‘coherent scattering’ from the impurity potential.The integrated transmission is quite similar for the Ch andCr cases, and therefore the current is almost the same. Thetransmission of the Ra configuration shows less backscat-tering, higher transmission and therefore produces a highercurrent compared to the other two discrete cases.

Fig. 31 Transmission probability at VG = 0.3 V, for the smooth deviceand the three different dopant configurations

4.2.3 Random dopants in source and drain

The random dopants have been introduced in a 4 nm re-gion of the S/D leads between the channel and the contin-uously doped S/D regions next to the contacts. Each Si lat-tice site in these regions is considered and whether this sitehas a dopant or not is determined using the same rejectiontechnique used for the DD simulations. The total number ofdopants in the discrete dopant regions closely follows a Pois-son distribution. The charge of each dopant is distributed tothe surrounding nodes of the discretisation mesh using thecloud-in-cell technique. The region with continuous dopingbetween the discrete dopant regions and the contacts guar-antees a homogeneous injection into the source/drain fromthe reservoirs.

Due to the significant computational burden associatedthe 3D NEGF approach, and its slow convergence in thepresence of attractive impurity potentials, the statistical sim-ulation study has been restricted to a small statistical sampleof 30 randomly generated device configurations. All simu-lations were carried out at VD = 1 mV in the linear modeof device operation. Figure 32 shows, on a linear and log-arithmic scale, the current-voltage characteristics of the 30microscopically different NWTs with different RDD (Ran-dom Discrete Dopant) configurations. The nanowire withcontinuous doping (labelled as ‘smooth’ in the figure), andone with no dopants at all in the random dopant regions,are shown for comparison. At VG < 0.3 V, which marks thetransition between the subthreshold and the linear region ofthe transistor operations, the device configurations with ahigh concentration of discrete dopants located close to thechannel lower the gate barrier potential, leading to a highercurrent than in the smooth case. At VG > 0.3 V the smoothdevice always delivers a higher current than in the RDD de-vices. This is mainly associated with the coherent impurity


Fig. 32 ID–VG characteristics of 30 microscopically differentnanowires with different random dopant configurations

scattering in RDD devices, which reduces the current dueto partial reflection from the impurity potentials. The degreeof backscattering, for a particular RDD configuration, de-pends strongly on the gate voltage. As a consequence, theon-current in ‘unlucky’ RDD devices is reduced to as lowas 20% of the current for the smooth device. A particularconfiguration may have low scattering at low VG and highscattering at high VG, or vice versa, relative to other con-figurations. This sensitivity to the gate voltage is a result ofthe relative proximity of the discrete dopants to the tail ofthe channel potential barrier and lead to a crossing of theID–VG curves as can be seen in Fig. 32.

There is a variation in the sub-threshold slope of theID–VG curves produced by the different atomistic configu-rations. Devices with configurations of dopants close to thecentral axis of the wire, far from the SiO2 interfaces, willhave relatively poor electrostatic control compared with de-vices with dopants distributed closer to the interfaces and tothe channel/source and channel/drain junctions. In additionto the subthreshold slope variation due purely to electrostat-ics, which have been observed in drift-diffusion simulations,there is an additional contribution from the varying degreesof source-to-drain tunnelling.

Figure 33 shows the 3D self-consistent electrostatic po-tential for the lowest, middle and highest current in the lot,illustrating the localised potential features created by thediscrete dopants. The trend observed in the figure is an in-crease in the current when the impurities are closer to themiddle of the channel as they weaken the channel barrierpotential.

In the presence of RDD in the access regions the averagemagnitude of the on-current drops by 48% compared to theuniformly doped purely ballistic device. More importantly,despite the absence of channel doping the standard deviationof the on-current is extremely high at 38%. This is beyondthe variability level of tolerance for the present circuit designpractices. Either the design practices need to be changed to

Fig. 33 3D potential for the low, median and high current (from top tobottom) of the random dopant devices at VG = 0.1 V

cope with such levels of variability or Schottky source draincontacts have to be considered to remove the doping granu-larity issues from the extensions.

4.2.4 Interface roughness

In the study of interface roughness the random rough inter-face between Si and SiO2 is introduced using the approachdescribed in [52]. Essentially the interface is modelled asa randomly generated surface with exponential autocorre-lation function and specified rms amplitude and correlationlength selected based on results reported in [53]. The corre-lation length of the order of 2 nm is similar to the one typ-ically used in Monte Carlo simulations used to reproducethe universal mobility curve [54]. The generated ‘analogue’random surface is interdigitised on the scale of one inter-atomic layer steps. We have assumed rough interface onlyin the channel and have ignored any roughness between theoxide and the gate material.

In order to study the statistical effect associated withthe interface roughness the ID–VG characteristics of thirtynanowire transistors with different randomly generated in-terface roughness patterns were simulated and analysed. Thecollection of individual ID–VG characteristics of the differ-ent surface roughness device configurations are shown inFig. 10. The devices with a smooth interface and with thesmallest possible cross-section (all the interface roughnesssteps are in the direction that decreases the body thicknessof the channel) are also presented in Fig. 34 for comparison.The three surface roughness cases selected for analysis pro-duce the lowest, median and the second highest current atboth low (VG = 0.1 V) and higher (VG = 0.4 V) gate biases.At 2.2 nm body thickness even a small intrusion of the inter-face into the channel is sufficient to raise the ground state of


Fig. 34 ID–VG characteristics of 30 microscopically differentnanowires with different rough interfaces

Fig. 35 3D potential for the low, median and high current (from top tobottom) of the surface roughness devices

the transverse wavefunction resulting in the step increasesin transmission occurring at higher energies. This effect, to-gether with the roughness induced scattering, means that thecurrent of the rough devices are generally lower comparedto that of the smooth device. It is possible to have a roughdevice, with a higher current than that of the smooth deviceif there is no overall narrowing of the channel, but the prob-ability for this is very low. The body thickness confinementvariation causes a large spread in threshold voltage which(for the 30 devices simulated here) spans 124 mV. In theextreme random roughness case there is a decrease in theon-current by more than 25% compare to that of the smoothdevice if the threshold voltages are aligned.

The 3D distribution of the self-consistent electrostatic po-tential for the three selected devices at low gate (Vg = 0.1 V)are shown in Fig. 35 with the lowest current device at the

top and the highest current device at the bottom. The impactof the reductions in channel body thickness on the potentialdistribution can be seen in the figure. This together with thecorresponding up-shift in the ground-state sub-band result-ing from the increased confinement leads to a decrease inthe electron current.

5 Conclusions

In this paper we have presented the most comprehensivecurrently available 3D tool for statistical simulation of vari-ability in contemporary and future nano-CMOS transistorsincluding drift diffusion, Monte Carlo and non-equilibriumGreen’s function transport modules. The tool allows the sim-ulation of all of the important sources of statistical variabil-ity introduced by discreteness of charge and matter. The DDresults accurately reproduce the measured threshold voltagevariability in real devices. MC simulations with ‘ab initio’ionised impurity scattering are needed to capture properlythe transport variability introduced by the random numberand position of scatterers in the devices. In the sub-100 nmchannel-length range, full-scale quantum transport simula-tions will be needed to reflect potential contributions fromsource-to-drain tunnelling.

Predictive simulations carried out with the developedtools show that the statistical variability introduced by dis-creteness of charge and matter has become one of the majorconcerns for the semiconductor industry. More and more thestrategic technology decisions that the industry will be mak-ing in the future will be motivated by the desire to reduce sta-tistical variability. The useful life of bulk MOSFETs, froma statistical variability point of view, can be extended belowthe 20 nm technology mark only if the LER and the equiv-alent oxide thickness (EOT) could be successfully scaled tothe required values. The introduction of fully-depleted SOIMOSFETs, and perhaps FinFETs, will mainly be motivatedby the necessity to reduce the statistical variability. This,however, might be jeopardised by other sources of variabil-ity associated with the introduction of the high-κ/metal gatestack and increased statistical reliability problems.

References

1. Bernstein, K., Frank, D.J., Gattiker, A.E., Haensch, W., Ji, B.L.,Nassif, S.R., Nowak, E.J., Pearson, D.J., Rohrer, N.J.: IBM J. Res.Develop. 50, 433 (2006)

2. Brown, A.R., Roy, G., Asenov, A.: IEEE Trans. Electron Devices54, 3056 (2007)

3. Cheng, B.-J., Roy, S., Asenov, A.: In: Proc. 30th European Solid-State Circuits Conference (ESSCIRC), Leuven, p. 219 (2004)

4. Agarwal, A., Chopra, K., Zolotov, V., Blaauw, D.: In: Proc. 42ndDesign Automation Conference, Anaheim, p. 321 (2005)


5. Asenov, A., Brown, A.R., Davies, J.H., Saini, S.: Hierarchicalapproach to ‘atomistic’ 3D MOSFET simulation. IEEE Trans.Comput.-Aided Des. Integr. Circuits Syst. 18(11), 1558–1565(1999)

6. Wong, H.-S., Taur, Y.: Three dimensional ‘atomistic’ simulationof discrete random dopant distribution effects in sub-0.1 µm MOS-FET’s. In: Proc. IEDM’93. Dig. Tech. Papers, pp. 705–708

7. Vasileska, D., Gross, W.J., Ferry, D.K.: Modeling of deepsub-micrometer MOSFETs: Random impurity effects, threshold volt-age shifts and gate capacitance attenuation. In: Extended abstractsIWCE-6, Osaka, Japan, 1998, IEEE Cat. No. 98EX116, pp. 259–262

8. Asenov, A.: Random dopant induced threshold voltage loweringand fluctuations in sub 0.1 _m MOSFETs: A 3-D ‘atomistic’ simu-lation study. IEEE Trans. Electron Devices 45, 2505–2513 (1998)

9. Ancona, M.G., Iafrate, G.J.: Quantum correction to the equationof state of an electron gas in a semiconductor. Phys. Rev. B 39,9536–9540 (1989)

10. Roy, G., Brown, A.R., Asenov, A., Roy, S.: Bipolar quantum cor-rections in resolving individual dopants in ‘atomistic’ device sim-ulations. Superlattices Microstruct. 34, 327–334 (2004)

11. Asenov, A., Brown, A.R., Cheng, B., Watling, J.R., Roy, G.,Alexander, C.: Simulation of nano-CMOS devices: from atoms toarchitecture. In: Korkin, A., Labanowski, J., Gusev, E., Luryi, S.(eds.) Nanotechnology for Electronic Materials and Devices, pp.257–303. Springer, New York (2006)

12. Sampedro-Matarín, C., Gámiz, F., Godoy, A., García Ruiz, F.J.:The multivalley effective conduction band-edge method for MonteCarlo simulation of nanoscale structures. IEEE Trans. ElectronDevices 53, 2703–2710 (2006)

13. Brown, A.R., Martinez, A., Seoane, N., Asenov, A.: Comparisonof density gradient and NEGF for 3D simulation of a nanowireMOSFET. In: Proc. Spanish Conference on Electron Devices(CDE), Feb. 11–13, Santiago de Compostela, Spain, pp. 140–143(2009)

14. Brown, A.R., Martinez, A., Bescond, M., Asenov, A.: NanowireMOSFET variability: a 3D density gradient versus NEGF ap-proach. In: Silicon Nanoelectronics Workshop, 10–11 June, Ky-oto, Japan, pp. 127–128 (2007)

15. Jin, S., Park, Y.J., Min, H.S.: Simulation of quantum effects in thenano-scale semiconductor device. J. Semicond. Technol. Sci. 4,32–40 (2004)

16. Frank, D.J., Taur, Y., Ieong, M., Wong, H.-S.P.: In: Symposium onVLSI Circuits Digest of Technical Papers, p. 171 (1999)

17. Roy, G., Brown, A.R., Adamu-Lema, F., Roy, S., Asenov, A.:Simulation study of individual and combined sources of intrin-sic parameter fluctuations in conventional nano-MOSFETs. IEEETrans. on Electron Devices 53(12), 3063–3070 (2006)

18. Sano, N., Matsuzawa, K., Mukai, M., Nakayama, N.: On discreterandom dopant modelling in drift-diffusion simulations: physicalmeaning of ‘atomistic’ dopants. Microelectronics and Reliability42(22), 189–199 (2002)

19. Asenov, A., Jaraiz, M., Roy, S., Roy, G., Adamu-Lema, F.,Brown, A.R., Moroz, V., Gafiteanu, R.: Integrated atomisticprocess and device simulation of decananometre MOSFETs. In:Proc. SISPAD, IEEE cat. No. 02TH8621, pp. 87–90 (2002)

20. Kaya, S., Brown, A.R., Asenov, A., Magot, D., Linton, T.: Analy-sis of statistical fluctuations due to line edge roughness. in sub-0.1 µm MOSFETs. In: Proc. Simulation of SemiconductorProcesses and Devices, Athens, Greece, 2001, p. 78

21. Asenov, A., Saini, S.: Polysilicon gate enhancement of the ran-dom dopant induced threshold voltage fluctuations in sub 100 nmMOSFETs with ultrathin gate oxide. IEEE Trans. Electron De-vices 47(4), 805–812 (2000)

22. Difrenza, R., Vildeuil, J.C., Llinares, P., Ghibaudo, G.: Impact ofgrain number fluctuations in MOS transistor gate on matching per-

formance. In: Proc. International Conference on MicroelectronicTest Structures, Monterey, CA, USA, pp. 244–249 (2003)

23. Okayama, Y., Saito, T., Nakajima, K., Taniguchi, S., Ono, T.,Nakayama, K., Watanabe, R., Oishi, A., Eiho, A., Komoda, T.,Kimura, T., Hamaguchi, M., Takegawa, Y., Aoyama, T.,Iinuma, T., Fukasaku, K., Morimoto, R., Oshima, K., Oono, K.,Saito, M., Iwai, M., Yamada, S., Nagashima, N., Matsuoka, F.:Suppression effects of threshold voltage variation with Ni FUSIgate electrode for 45 nm node and beyond LSTP and SRAM De-vices. In: Symp. on VLSI Technol., pp. 118–119 (2006)

24. Watanabe, H.: Statistics of grain boundaries in polysilicon. IEEETrans. Electron Devices 54(1), 38–44 (2007)

25. Tuinhout, H.P.: Impact of parametric mismatch and fluctuationson performance and yield of deep-submicron CMOS technologies.In: Proc. 32nd European Solid-State Device Research Conference(ESSDERC), Firenze, Italy, pp. 95–101 (2002)

26. Cathignol, A., Rochereau, K., Ghibaudo, G.: Impact of a singlegrain boundary in the polycrystalline silicon gate on sub 100 nmbulk MOSFET characteristics—implication on matching proper-ties. In: Proc. 7th European Workshop on Ultimate Integration ofSilicon (ULIS), Grenoble, France, pp. 145–148 (2006)

27. Uma, S., McConnell, A.D., Asheghi, M., Kurabayashi, K., Good-son, K.E.: Temperature-dependent thermal conductivity of un-doped polycrystalline silicon layers. Int. J. Thermophys. 22, 605–616 (2001)

28. Watling, J.R., Brown, A.R., Ferrari, G., Babiker, J.R., Bersuker,G., Zeitzoff, P., Asenov, A.: J. Comput. Theor. Nanosci. 5, 1072(2008)

29. Asenov, A., Kaya, S., Davies, J.H.: IEEE Trans. Electron Devices49, 112 (2002)

30. Brown, A.R., Asenov, A., Watling, J.R.: IEEE Trans. Nano-technol. 1, 195 (2002)

31. Cathignol, A., Cheng, B., Chanemougame, D., Brown, A.R.,Rochereau, K., Ghibaudo, G., Asenov, A.: Quantitative evaluationof statistical variability sources in a 45 nm technological node LPN-MOSFET. IEEE Electron Device Lett. 29(6), 609–611 (2008)

32. Asenov, A., Cathignol, A., Cheng, B., McKenna, K.P., Brown,A.R., Shluger, A.L., Chanemougame, D., Rochereau, K.,Ghibaudo, G.: Origin of the asymmetry in the statistical variabilityof n- and p-channel poly Si gate bulk MOSFETs. IEEE ElectronDevice Lett. 29(8) (2008)

33. Inaba, S., Okano, K., Matsuda, S., Fujiwara, M., Hokazono, A.,Adachi, K., Ohuchi, K., Suto, H., Fukui, H., Shimizu, T., Mori,S., Oguma, H., Murakoshi, A., Itani, T., Iinuma, T., Kudo, T., Shi-bata, H., Taniguchi, S., Takayanagi, M., Azuma, A., Oyamatsu,H., Suguro, K., Katsumata, Y., Toyoshima, Y., Ishiuchi, H.: Highperformance 35 nm gate length CMOS with NO oxynitride gatedielectric and Ni salicide. IEEE Trans. Electron Devices 49(12),2263–2270 (2002)

34. Roy, G., Adamu-Lema, F., Brown, A.R., Roy, S., Asenov, A.: In-trinsic parameter fluctuations in conventional MOSFETs until theend of the ITRS: a statistical simulation study. In: Proc. 7th Inter-national Conference on New Phenomena Mesoscopic Systems and5th International Conference on Surfaces and Interfaces in Meso-scopic Devices (NPMS/SIMD), 27 November–2 December, Maui,HI, USA, pp. 35–36 (2005)

35. Cheng, B., Roy, S., Brown, A.R., Millar, C., Asenov, A.: Evalua-tion of statistical variability in 32 and 22 nm technology genera-tion LSTP MOSFETs. Solid-State Electron. 53, 767–772 (2009)

36. Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Par-ticles. McGraw-Hill, New York (1981)

37. Vasileska, D., Gross, W.J., Ferry, D.K.: Monte Carlo particle-based simulations of deep-submicron n-MOSFETs with real-space treatment of electron-electron and electron-impurity inter-actions. Superlattices Microstruct. 27, 147–157 (2000)

38. Gross, W.J., Vasileska, D., Ferry, D.K.: A novel approach for in-troducing the electron-electron and electron-impurity interactions


in particle based simulations. IEEE Trans. Electron Devices Lett.20, 463–465 (1999)

39. Ezaki, T., Ikezawa, T., Hane, M.: Investigation of realistic dopantfluctuation induced device characteristic variation for sub-100 nmCMOS by using atomistic 3D process/device simulator. In: Proc.IEDM 2002, pp. 311–314 (2002)

40. Vasileska, D., Khan, H.R., Ahmed, S.S., Ringhofer, C.,Heitzinger, C.: Quantum and coulomb effects in nanodevices. Int.J. Nanosci. 4(3), 305–361 (2005)

41. Rafferty, C.S., Biegel, B., Yu, Z., Ancona, M.G., Bude, J., Dut-ton, R.W.: Multi-dimensional quantum effect simulation using adensity-gradient model and script-level programming techniques.In: International Conference on Simulation of SemiconductorProcesses and Devices (SISPAD), pp. 137–140, September 2–41998

42. Winstead, B., Ravaioli, U.: A quantum correction based onSchrodinger equation applied to Monte Carlo device simulation.IEEE Trans. Electron Devices 50, 440–446 (2003)

43. Ancona, M.G., Yergeau, D., Yu, Z., Biegel, B.A.: On ohmicboundary conditions for density-gradient theory. J. Comput. Elec-tron. 1, 103–107 (2002)

44. Riddet, C., Brown, A.R., Roy, S., Asenov, A.: Boundary condi-tions for density gradient corrections in 3D Monte Carlo simula-tions. J. Comput. Electron. 7(3), 231–235 (2008)

45. Bude, J.D.: MOSFET modeling into the ballistic regime. In: Inter-national Conference on Simulation of Semiconductor Processesand Devices (SISPAD), pp. 23–26, September 2000

46. Landauer, R.: Electrical resistance of disordered one-dimensionallattices. Philos. Mag. 21, 863 (1970)

47. Nehari, K., Cavassilas, N., Autran, J.L., Bescond, M., Munteanu,D., Lannoo, M.: Influence of band structure on electron ballis-

tic transport in silicon nanowire MOSFET’s: an atomistic study.Solid-State Electron. 50, 716 (2006)

48. Svizhenko, A., Anantram, M.P., Govindan, T.R., Biegel, B., Venu-gopal, R.: Two dimensional quantum mechanical modelling ofnanotransistors. J. Appl. Phys. 91, 2343 (2002)

49. Venugopal, R., Ren, Z., Datta, S., Jovanovic, D., Lundstrom, M.S.:Simulating quantum transport in nanoscale transistors: Real versusmode-space approaches. J. Appl. Phys. 92, 3730 (2002)

50. Asenov, A., Slavcheva, G., Brown, A.R., Davies, J.H., Saini, S.:Increase in the random dopant induced threshold fluctuations andlowering in sub-100 nm MOSFETs due to quantum effects: a 3-Ddensity-gradient simulation study. IEEE Trans. Electron Devices48(4), 722–729 (2001)

51. Anantram, M.P., Svizhenko, A.: Multidimensional modelling ofnanotransistors. IEEE Trans. Electron Devices 54, 2100 (2007)

52. Martinez, A., Bescond, M., Barker, J.R., Svizhenko, A.,Anantram, M.P., Millar, C., Asenov, A.: A self-consistent 3D-fullreal space NEGF simulator for studying nonperturbative effect innano-MOSFETs. IEEE Trans. Electron Devices 54(9), 2213–2222(2007)

53. Goodnick, A.M., Ferry, D.K., Wilmsen, C.W., Lilienthal, A.,Fathy, D., Krivanek, O.L.: Surface roughness at the Si (100)-SiO2 interface. Phys. Rev. B, Condens. Matter 32(12), 8171–8186(1985)

54. Takagi, S., Toriumi, A., Iwase, M., Tango, H.: On the universal-ity of inversion layer mobility in Si MOSFET’s: Part I-effects ofsubstrate impurity concentration. IEEE Trans. Electron Devices41(12), 2357–2362 (1994)

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