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Eur. Phys. J. B 72, 323359 (2009)DOI:10.1140/epjb/e2009-00378-9

Colloquium

THE EUROPEANPHYSICALJOURNAL B

Front-end process modeling in silicon

L. Pelaza, L.A. Marques, M. Aboy, P. Lopez, and I. Santos

Departamento de Electronica, E.T.S.I. de Telecomunicacion, Universidad de Valladolid, 47011 Valladolid, Spain

Received 5 August 2009 / Received in final form 25 September 2009Published online 7 November 2009 c EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009

Abstract. Front-end processing mostly deals with technologies associated to junction formation in semicon-ductor devices. Ion implantation and thermal anneal models are key to predict active dopant placementand activation. We review the main models involved in process simulation, including ion implantation,

evolution of point and extended defects, amorphization and regrowth mechanisms, and dopant-defect in-teractions. Hierarchical simulation schemes, going from fundamental calculations to simplified models, areemphasized in this Colloquium. Although continuum modeling is the mainstream in the semiconductorindustry, atomistic techniques are starting to play an important role in process simulation for devices withnanometer size features. We illustrate in some examples the use of atomistic modeling techniques to gaininsight and provide clues for process optimization.

PACS. 02.70.-c Computational techniques; simulations 61.72.Cc Kinetics of defect formation andannealing 61.72.J- Point defects and defect clusters 61.72.uf Ge and Si

1 Introduction

The International Technology Roadmap for Semiconduc-tors (ITRS) provides the guidelines for the developmentof semiconductor devices in different aspects [1]. A chap-ter devoted to modeling and simulation indicates its rel-evance in the semiconductor industry. As many othertechnological fields, the fabrication and design of new in-tegrated circuits (ICs) makes intensive use of TechnologyComputer-Aided Design tools. Although experimentaltests are essential for the development and optimizationof new technologies, modeling and simulation are becom-ing fundamental to reduce the development times andcost. Simulation tools are used in the different steps ofthe fabrication chain from the equipment-related issues to

the material properties. Front-end process modeling dealswith the simulation of the physical effects of manufac-turing steps used to build transistors, and it is mostlyconcerned with junction formation. This information canthen be used as input for device simulation to predict theelectrical device behavior. The intrinsic electrical deviceperformance goals drive the design of the junctions andin turn the requirements on dopant placement and activa-tion. Therefore, the accurate prediction of dopant distri-bution and activation during the fabrication processes isone of the main goals of process modeling.

The most common technique used to selectively intro-duce dopants in the Si substrate and define junctions is

ion implantation because it allows a precise control of thea e-mail: [email protected]

amount and distribution of dopants [2]. However, as theenergetic incoming ions penetrate into the substrate the

Si crystal lattice is damaged. Damage consists of Si self-interstitials (Is), vacancies (Vs) or agglomerates of thesedefects [35]. But the lattice can be completely amor-phized if the implant dose is high enough [6]. Generallythe as-implanted dopant atoms do not lie in substitu-tional lattice sites and they are electrically inactive. Sub-sequent thermal anneals are required to heal the crystaldamage and to place the dopants in substitutional sites.Dopants and intrinsic lattice defects, Is and Vs, interactleading to mobile dopant species and to dopant-defect ag-glomerates that prevent their electrical activation. Sincea large defect supersaturation (defect concentration com-pared to that in equilibrium) is created during ion im-

plantation, the effects resulting from dopant-defect inter-actions are boosted. Thus, dopant diffusivity is enhancedand dopant activation is reduced compared to equilibriumvalues [715]. These effects are transient because defectsupersaturation evolves toward equilibrium.

Device scaling allows a higher level of integration inICs with improved cost/performance ratio. However, thisevolution entails a number of technological challenges. Atthe junction level, the need to control the Short ChannelEffects (SCE) in Metal-Oxide Semiconductor Field EffectTransistor (MOSFET) devices has motivated the progres-sive reduction of junction depth at the source and drain(S/D) extensions. This requires not only the shallow intro-duction of dopants but also the control of their diffusivityto avoid the profile broadening. The maximization of drivecurrent for higher switching speed involves a minimization
http://www.epj.org/http://dx.doi.org/10.1140/epjb/e2009-00378-9http://-/?-http://-/?-http://dx.doi.org/10.1140/epjb/e2009-00378-9http://www.epj.org/


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324 The European Physical Journal B

of access resistance. This requisite demands a high dopantactivation and has also driven the development of straintechnologies to enhance carrier mobility [16]. The mini-mization of off-state currents, especially for low-power ap-

plications, entails the removal of lattice defects as theymay induce energy states in the gap responsible for addi-tional leakage [17]. Dopant activation and defect removalgenerally require large thermal budgets, which cause thespreading of the junction depth, and therefore, deteriorateSCE control. The difficult trade-off between shallow andabrupt junction formation, maximum dopant activationand defect removal becomes more and more challengingas device dimensions shrink[1].

Thermal budgets with higher temperature and re-duced times offer a better compromise for dopant diffu-sion and activation[18]. Thus, for S/D extension forma-tion traditional furnace anneals have been substituted by

other anneal schemes such as spike rapid thermal processes(temperatures around 1000 C for a few seconds) or morerecently, by the so-called millisecond anneals[19] (the sub-strate is exposed to the burst of flash-lamps [20] or to asub-melting laser pulse [21] for about 1 ms with ultra fasttemperature ramp-up and ramp-down). The use of SolidPhase Epitaxial Regrowth (SPER) of amorphous Si layers(a-layers) has been proved to result in a high dopant ac-tivation and with minimal diffusion[2224]. However, thestable residual damage that remains beyond the amor-phous/crystalline (a/c) interface is a pressing concern be-cause millisecond anneals may not be able to completelyremove them. Excimer laser annealing has also been con-sidered for the healing of crystal damage and activationof dopants in particular device structures [2527]. To beable to maintain a good SCE control while fulfilling otherdevice requirements, the ITRS foresees a transition fromtraditional planar Si MOSFET to Ultra-Thin-Body FullyDepleted devices in the near future, and to alternativedevice architectures, such as Fin Field Effect Transistors(FinFETs), at the 22 nm node and beyond [1]. The incor-poration of materials with very high carrier mobility, suchas Ge and IIIV compounds, to substitute Si in the chan-nel of advanced device architectures, is being consideredfor further technologies[1,28,29].

Ion implantation and thermal annealing are key steps

to define dopant distribution and are expected to playan important role also in future technologies. Physicalmodels that include appropriate parameter setting are re-quired to describe all the mechanisms involved in theseprocesses. Models for ion implantation should provide notonly the spatial dopant distribution but also the generateddefects. Description of amorphization and regrowth is alsoneeded. Much attention should be given to the evolutionof defects since they control the point defect superstat-uration, which in turn affect dopant redistribution [12].Interactions among defects and dopants must be mod-eled in detail as they greatly determine dopant diffusionand activation through the diffusion of mobile species andthe formation and dissolution of various types of impu-rity clusters [11]. The presence of high stresses to enhancecharge mobility [30] should be also considered in process

simulation since it affects the equilibrium concentrationof point defects, the effective dopant diffusivities and alsothe dopant solid solubilities[3134]. Excimer laser anneal-ing requires a complex modeling that includes the coupled

simulation of the electromagnetic field, for the calculationof the heat source distribution, and the simulation of thethermal, phase and impurity fields, for the prediction ofthe material modification[3537].

Over the last decades, a great effort has been madeto quantitatively model the kinetics of these processesand to determine the parameters that govern the interac-tions in Si. Dedicated experiments and fundamental cal-culations have been performed to extract parameters. De-tailed models are generally computationally too expen-sive in terms of memory and time. Empirical approachesand fitting parameters are sometimes used in computa-tionally efficient models, although there is no guarantee oftheir validity outside the range of conditions where they

have been fitted. Simplified models can be also extractedfrom the understanding of the most relevant mechanismsthrough a multi-scale approach or hierarchical scheme, inwhich more fundamental simulation techniques are usedto extract physical parameters and mechanisms that arethen implemented in less detailed models [38]. Continuummodels continue being the mainstream in industry becausethey are fast and can be easily coupled to device simula-tors. Nevertheless, the reduction of device dimensions hasrevealed the power of atomistic techniques to provide thebasis of simplified physical models implemented in contin-uum simulators and even to directly perform simulationsof actual Si processing.

In this paper we review the most relevant modelsinvolved in the prediction of dopant distribution and ac-tivation, including the determination of the importantparameters. Hierarchical simulation schemes, going fromfundamental calculations to simplified models, are empha-sized. This paper is organized as follows. In Section2 webriefly review simulation techniques used in Si front-endprocessing, from continuum to atomistic methods. Sec-tion 3 is specifically devoted to the modeling of ion im-plantation. In Section 4 we center our attention on theproperties and simulation of point and extended defects,while Section5 is focused on the modeling of amorphiza-tion and recrystallization processes. Section 6 is devoted

to the analysis and modeling of dopant-defect interactions.Some practical applications of front-end process modelingare presented in Section 7. Finally, in Section8we drawsome conclusions and give future perspectives about thistopic.

2 Simulation techniques

With further reduction of the devices feature size on eachnew technology generation, new effects or effects that wereneglected so far become relevant and compromise the re-liability of the manufacturing process. Their experimentalcharacterization is a complex task, firstly because the re-alization of test lots results extremely expensive, and sec-ondly because these effects usually occur simultaneously


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L. Pelaz et al.: Front-end process modeling in silicon 325

which makes the interpretation of measurements very dif-ficult. In this situation, the use of predictive process sim-ulation techniques becomes almost imperative. The devel-opment of reliable physically based models is essential to

predict at least the correct trends during technology de-velopment, where new process and materials are explored,and they have not been previously calibrated. Neverthe-less, sometimes best models available may be too compli-cated or too slow when used in multidimensional contin-uum simulations. Simplified phenomenological models areoften developed for design applications, where speed andaccuracy are primary requirements, while predictability inuncalibrated regimes is secondary.

Modeling of the dopant and damage profiles resultingfrom ion implantation is the first step in front-end pro-cessing simulations. Different approaches can be followedfor that purpose, and they will be described in more de-tail in Section3.2.The resulting profiles are the input forother methods that describe the dynamics of the system.

2.1 Continuum methods

Most process simulators used in industrial applicationsare based on continuum methods, as it is the case ofFLOOPS [39]. In this kind of simulators the physics ofthe system is formulated as a series of partial differentialequations for each particle type considered to be relevantin the process[40,41]. Typically they are continuity equa-tions. As an example, we will consider the interaction oftwo different particles, A and B, to form a mobile species,

C: A+BC. The gain or loss of species C is formulated interms of the normalized concentrations as in the followingpartial differential equation,

[C]

t =Kf[A] [B] Kb[C] (DC [C]) (1)

whereKf and Kb are the forward and backward reactionrates, respectively, and DC the diffusivity of species C.These parameters are related to the capture volume ofinteracting species, diffusivities of mobile species, stabilityof species formed, etc. They need to be explicitly definedin order to solve the equation.

In the simulation of Si front-end processing, a rela-tively high number of coupled partial differential equationshave to be considered, since a lot of different interactingatomic species (dopants, intrinsic defects, defect clusters,dopant-defect complexes, impurities, etc.) are usually in-volved. The numerical solution of this set of partial differ-ential equations requires spatial and temporal discretiza-tion to reduce the derivatives into algebraic differences.The problem is converted to a large, nonlinear system ofcoupled equations, which are solved using standard nu-merical methods. Usually some approximations, based onempirical fitting, have to be introduced in order to reducethe number of equations and decouple them.

Continuum simulators are fast and allow the consid-eration of big systems by adjusting the grid used for thespatial discretization. However, this advantage is reduced

as the device size shrinks to nanometric scale. The atom-istic nature of the material arises and complex physicalinteractions show up. The use of a very refined grid andthe addition of more equations to include such new effects

is computationally expensive, which slows down the res-olution of the problem using continuum methods. Thenatomistic simulation techniques become a good alterna-tive even for industrial applications [4245].

2.2 Kinetic Monte Carlo

The dynamics of the system can also be simulated froman atomistic point of view by the use ofKinetic MonteCarlo (KMC) techniques. This method allow the simula-tion of device structures at a macroscopic scale, providingan atomic description of the material and allowing a fastdevelopment of new models. KMC simulates the kinetics

of defects and dopants by modeling their diffusion and in-teractions [11,46]. In non-lattice KMC models, atoms inthe perfect lattice are not simulated, and consequently sys-tem sizes of hundreds of nanometers can be treated usingaverage computers. The most popular non-lattice KMCcode for front-end process modeling is DADOS[46].

In a KMC simulator, reactions such as A+BC aremodeled by performing interactions (forward reaction)and events (reverse reaction). When particles A and Bare within the capture volume of each other they inter-act leading to the formation of C. On the other hand, theprobability of species C breaking-up is controlled by itsevent rate. Other type of events are, for example, diffu-

sion hops of mobile species, or emission of point defectsfrom clusters.Both interactions and events are determined by sev-

eral parameters (capture volumes, activation and migra-tion energies, etc.) whose values must be defined a pri-ori. All events simulated in KMC are thermally activatedprocesses. The probability of exceeding an energy barrier,Eact, follows a Boltzmann distribution,

P exp

EactkT

(2)

being T the temperature and kthe Boltzmann constant.Therefore, the frequency (or probability per unit time) at

which a particular eventi takes place can be expressed as,

i = 1

i =i0exp

EiactkBT

(3)

where i0 is the prefactor. In a system with Ni particles

which can undergo the event i, the total event rate is

R=

Nii . (4)

In Figure1 we show a schematic of the simulation flow.In each simulation loop, a particular event and an atomare randomly selected according to the event frequencydistribution. Then, the selected atom performs a jump ina random direction in order to simulate diffusion or emis-sion of mobile species from clusters. If the moved atom


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explicitly designed to properly describe the system understudy. This implies choosing an adequate mathematicalexpression and a subsequent fitting process to a wide rangeof experimental results, ab initio and TB calculations.

As in the case of semi-empirical TB methods, empiricalpotentials can only describe situations they have been de-signed for. If a new system is studied, a new parametriza-tion or even a new mathematical expression must be de-veloped, which is not an straightforward task (speciallythe latter case). Furthermore, no electronic structure in-formation is obtained through them. Thus empirical po-tentials are not suitable for studying electronic propertiesor processes affected by charge states. The first potentialsdeveloped for Si, Stillinger-Weber and Tersoff, appeared inthe mid eighties[8790]. Since then, new potentials havebeen proposed in an attempt to improve the descriptionof Si. Some are modifications of Stillinger-Weber [91,92]or Tersoff[93,94] original potentials, others are based on

modifications ofembedded-atom schemes typical of met-als [9597], and there are also new formulations such asthe environment-dependent interatomic potential [98]. Anextensive comparison of the results obtained with thesepotentials can be found in references [99101].

With empirical potentials it is possible to run MD sim-ulations with system sizes of several million atoms andto reach simulation times of about the nanosecond. Forexample, they have been used in Si for energetic andstructural characterization of self-interstitials [102105],vacancies [103105], small interstitial clusters [106], aswell as for studying self diffusion [102,105]. More com-plex situations, such as recrystallization of amorphous

regions (a-regions) [101,107109], structural transforma-tions leading to planar {113} defects[110], and ion implan-tation[3,111114], can also be simulated using empiricalpotentials.

2.4 Hierarchical scheme

As we have shown, each atomistic simulation techniquegives information at a different scale level, sometimes com-plementary, so all of them have to be used in a hierarchicalor multi-scale scheme to achieve full Si front-end process-ing simulation (see Fig.2). Fundamental techniques such

asab initioand TB can be employed to study defect con-figurations and energetics, material and electronic proper-ties, and to optimize empirical force potentials. MD sim-ulations using empirical potentials can be used in turn todetermine interaction and diffusion mechanisms involvingdefects, or to study the damage morphology obtained fromindividual implantation cascades as well as its annealingbehavior. The parameters extracted from these detailedatomistic techniques, or even from experimental measure-ments, along with the mechanisms of defect interactionsand diffusion, will define the relevant events to be consid-ered in the KMC simulator.

Ion implantation models are used to generate dopantand defect distributions which are fed to the KMC simu-lator at time intervals determined by the dose-rate. Dur-ing the inter-cascade time, defect diffusion and interaction

Parameter

fittingParameter

fittin

AbinitioTightbinding

Experiments

Defectconfigurationsandenergetics

Materialand

Empirical

PotentialsDirect

electronic

properties

InteractionanddiffusionmechanismsDamage

characterization

Dopantanddamagepofiles KMCAnnealing

IonImplantation

DoserateFig. 2. Multi-scale hierarchical simulation scheme.

events occur at a rate that depends on the sample temper-ature, thus accounting for the dynamic anneal during theimplantation process. This procedure is repeated until thespecified dose is reached. Afterward, subsequent annealscan be simulated. It is worth to note that only the loopof implantation-annealing represents the actual front-endprocessing simulation, while the rest of techniques (funda-mental simulation methods and experiments) are carriedoff-line.

3 Ion implantation modeling

Ion implantation is the technique preferred nowadays tofabricate junctions of devices since it is well establishedand provides a precise control of the distribution and con-centration of the dopants in the Si substrate [2]. In thistechnique dopant atoms are first ionized, then acceler-ated through an electric field, and finally the resultingbeam of ions is oriented toward the region to be doped.When the energetic ions penetrate into the substrate, theystart to collide with its atoms until they come to rest.These collisions can produce permanent displacements ofthe substrate atoms from their perfect lattice positions.

If the energy transferred to target atoms is high enough,they can initiate a subcascade leaving behind a vacancyand generating a Si interstitial where they stop, in ad-dition to possible displacements during the subcascade.Furthermore, extensive damage regions can also be formedby the impact of heavy ions [35,111,115] and molecularimplants [114,116118]. During annealing treatments, theinteractions of dopants with the excess of Is and Vs (gen-erated during the implantation, or released by extendeddefects) can enhance dopant diffusion[8,119] and the for-mation of dopant clusters at concentrations much lowerthan their equilibrium solid solubility [120,121]. Therefore,for the fabrication of devices it is of critical importance notonly to know the final dopant profiles, but also to quantifythe amount and morphology of the damage since it willinfluence the final performance of the junctions.


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Fig. 3. (a) Schematic representation of a dopant profile, and variation of (b)RPand (c)RPwith implantation energy (fromRef. [122]).

3.1 Analytic distributions

For practical applications in the industry, it is convenientto obtain a fast estimation of dopant and damage profilesfor different implant parameters (ion, energy, dose, . . . )rather than carrying out test experiments or time consum-ing computational evaluations. Many analytic functionshave been proposed to describe the final distribution ofimplanted dopants within the substrate. The simpler oneis the Gaussian distribution given by

n(R) =n0exp

(R RP)2

2R2P

, (5)

wheren0is the distribution maximum,RPis the projectedrange of the implanted ions, and RPis the longitudinalstraggling, related to the width of the profile. In the caseof a Gaussian distribution,RPcoincides with the depth ofthe maximum. These parameters determine the depth andabruptness of the junction, and depend on the implantedion and its energy, as represented in Figure 3.

Gaussian distributions only provide an adequate fitto profiles around the projected range. The Pearson-IV distribution [123,124] accounts for the asymmetryand sharpness of the dopant profiles by considering twomore new parameters, the skewness and the kurtosis.Pearson IV distributions are especially appropriate todescribe dopant profiles in amorphous or amorphized ma-terials. For crystalline materials, there are other alterna-tives, such as joining half Gaussian or Pearson IV func-tions with an exponential tail[125], or double-Pearson IVfunctions[126,127].

Analytic distributions allow a fast and accurate de-scription of dopant profiles without statistical noise. Theyare frequently used as the input for continuum simulators.Nevertheless, it is necessary to calibrate the parametersinvolved for each set of implantation conditions (ion type,energy, incident angle . . . ). Thus, they cannot be used topredict the dopant profiles for implantations that have notbeing previously tabulated.

3.2 Binary collision approximation

The Binary Collision Approximation (BCA), on the ba-sis of the Monte Carlo method, is used to describe in-dividual collisions during an implant cascade. The basicidea of BCA is that the moving ion only interacts withits closest target atom through a repulsive pair potentialV(r) (being r the distance between them). The result-ing collision can be numerically solved by considering themomentum and energy conservation laws of classical me-chanics [128130] to evaluate the energy transferred to thetarget atom, which constitutes the elastic energy losses ofthe ion.

In the BCA description of cascade evolution, a targetatom is displaced from its lattice position when its energyafter the collision with a moving atom exceeds the dis-placement energy threshold,Ed[129]. For Si, experimentaland theoretical estimations for this energy range from 10to 30 eV[131135], but for most BCA simulators it is con-ventionally taken as 15 eV[136138]. For energy transfersabove Ed, the target atom can create a subcascade, butfor energy transfers below Ed it is not displaced and theenergy is assumed to be lost to phonons. A moving atomis considered to stop when its kinetic energy falls belowa certain value, in the order of Ed, since it will not beable to permanently displace more atoms through colli-

sion events [129].Several aspects need to be considered to physicallymodel the collisions. Both the projectile and the targetatom are under the influence of a repulsive potential V(r).Accurate potentials for the colliding atoms can be ob-tained fromab initiocalculations[139]. Nevertheless, uni-versal functional forms are usually considered in order tohave a wider range of application. They are defined as ascreened Coulomb potential

V(r) =ZPZTe2

40r (r), (6)

whereZP and ZTare the atomic numbers of the ion andthe target, respectively,0is the vacuum permittivity, and(r) is the screening function which takes into account


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that the inner electrons shield some of the nuclear chargeby reducing the repulsive Coulomb potential [130]. Bymodeling the electronic density of the atoms and analyzingthe repulsive interaction for many different pairs of atoms,

several screening functions have been obtained, such asTomas-Fermi [140], Moliere [141], Lenz-Jensen[142], andZiegler-Biersak-Littmark (ZBL) [143]. Among them, themost used is probably the last one.

In addition to nuclear collisions, ions also interactwith the electrons of the target, causing the so-called in-elastic energy losses. Traditionally, both elastic and in-elastic energy losses are treated separately, which sim-plifies considerably the analysis of the collisions. In thetypical irradiation conditions during implantation in ac-tual MOSFET fabrication, S/D extensions are gener-ated through implants with an energy that may vary be-tween 300 eV and 30 keV, and a dose in the range of

5 1013

2 1015

cm

2

[144]. For these conditions, onlyelastic collisions are considered as responsible for damagegeneration during ion implantation, while electronic inter-actions diminish the energy of moving ions[130]. Damageproduced via electron-phonon interactions in Si has beenobserved only for molecular implants and deposited en-ergies above 28 keV/nm, which corresponds to implantenergies of the order of MeV[145].

The inelastic energy losses are usually divided into twocontributions. One arises from a local inelastic stoppingpower as a consequence of the momentum transfer be-tween the electrons of the projectile and those of targetatoms during collisions. The other one is associated to

a non-local electronic stopping power since moving ions(charged particles) travel between consecutive collisionswithin a medium with electrons. The first contribution iscommonly evaluated using the Firsovs model [146,147],which considers a viscous force derived from a velocitydependent potential acting on the particles during thecollision. For the second contribution, the classical non-local model of Lindhard, Scharff and Schiott assumes thatthe energy losses due to electronic stopping are propor-tional to the ion velocity[148,149]. The Brandt-Kitagawamodel[150] is also commonly used, in which the inelasticenergy losses of a proton moving with the same velocity asthe ion are evaluated. These results are scaled taking intoaccount the effective charge of the ion due to its degree ofionization (which in general is a function of ion velocityand the charge density of the target). Subsequent modi-fications [151,152] resulted in a model with only one ad-justable parameter for each implant species, related to theeffective electron density of the target, depending on theion-target combination. This model has been used bothin BCA simulators [138,151] and in MD with empiricalpotentials[151153].

All these physical models provide BCA simulationswith great capability for simulating almost any implanta-tion condition without additional calibrations. A detailedatomistic description of the whole cascade is obtained byfollowing the trajectories of the implanted ion and of thegenerated recoils (full cascade BCA). The result of thesimulation of the implantation cascade is the position of

40

20

0

20

0 20 40 60 80 100 120 140

c) 1 keV Ge > Si

VacanciesSi interstitialsImplanted ion

40

20

0

20b) 1 keV Si > Si40

20

0

20

40a) 1 keV B > Si

Depth () direction

Lateraldistancetoimpac

tpoint()

Fig. 4. Atomistic description of 1 keV of (a) B, (b) Si, and(c) Ge into Si provided by a typical full cascade BCA simu-lation.

the implanted ion and the generated Is and Vs, calledFrenkel pairs (FPs), which can be the input for KMCcodes [154156]. As an example, the position of the ionand the FPs generated by typical 1 keV B, Si and Gecascades into Si simulated with BCA are represented inFigure4.This approach adequately reproduces the dilutedamage generated by light ions. However, it is not able toreproduce more complex damage structures such as amor-phous pockets (a-pockets), which have been observed ex-perimentally after heavy ion implantation [4,115,157].

In order to accelerate the calculations, instead of sim-ulating all the recoils, BCA simulators may follow onlythe trajectory of the implanted ion. Within this approach,damage is calculated according to the modified Kinchin-Pease (KP) formula[158]. When a certain amount of en-ergy Edep is deposited within a given target volume, thenumber n of displaced atoms generated is

n= 0.42EdepEd

. (7)

Nevertheless, not all the generated defects may effectively

contribute to increase the damage because some of themmay recombine rapidly or may correspond to atoms pre-viously displaced. Thus, the net increase of point defectsafter recombination,n, is given by

n= fsurv n

1

N

Nsat

, (8)

whereNis the previous local defect density in the regionwhere energy is deposited,Nsat is the local defect densitynecessary to reach amorphization, and fsurv is the frac-tion of point defects surviving recombination within onerecoil cascade [159,160], or surviving both intracascadeand intercascade recombinations [138,161]. This param-eter depends on implant conditions (temperature, dose,implant energy, etc.) so that the model can account for


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Fig. 5. Comparison of profiles of a 2 keV B implantation into(100)Si (tilt = 7, rotation = 30), simulated with 2000 ionswith and without using rare event algorithms (from Ref.[138]).

dynamic annealing [162]. When implanting ions in a crys-talline target, damage generated by previous ions can ac-cumulate and the position of target atoms may differ fromthose of a perfect crystal. In order to account for thedamage accumulation effect, the degree of amorphizationof a region, N/Nsat, is used to define a probability (i)to displace the target atoms randomly from their posi-tion [159,160], (ii) to perform a random rotation of thecrystal lattice [138], (iii) for the ion to collide with a tar-get atom of the considered volume [161]. In any case, theeffect of the accumulated damage can be regarded as the

gradual transition from crystal to amorphous in the con-sidered volume, which is also reflected in a gradual tran-sition in the number of scattering events suffered by theions. When N reaches Nsat within a given volume, thelattice is assumed to transform into the amorphous stateand the target positions turn random.

Within this approach, ion profiles are rapidly evalu-ated and in good agreement with experiments, but at theexpense of losing the atomic description of damage. Thedepth distribution of dopants and defects is obtained bysimulating a large number of implant cascades. To reducethe statistical noise of the dopant profiles, specially rel-evant in the tails, several algorithms, called rare eventalgorithms, are used [138,153,163,164]. These algorithmsconsist of splitting the ion trajectory at certain points,which are assigned a statistical weight to the overall pro-file, depending on the number of times that the originaltrajectory was divided. As an example, dopant profiles ob-tained with a BCA simulation of low energy B implantsinto Si are represented in Figure5.

There are different codes available that implement theBCA approximation for simulating ion implantation. In1974 Robinson and co-workers developed one of the firstatomistic BCA simulators, known as marlowe [129], inwhich the target material is considered to be crystalline.A different simulation scheme was followed in 1985 byZiegler, Biersack and Littmark for their BCA simulator,trim [143], in which an amorphous target is consideredrather than crystalline, and the projectile trajectory is sta-

tistically followed by randomly selecting a target atom, animpact parameter and a distance (mean free path). Thismodel works well for amorphous targets, but it can not beemployed for crystalline ones. In order to overcome this

limitation, a combination of both marlowe

and trim

,known as crystal-trim[165], was developed. Other BCAsimulators, such as imsil [159,160], ut-marlowe [137],uva-marlowe[166,167] and iis [138]) allow choosing be-tween crystalline and amorphous targets.

3.3 Classical molecular dynamics

MD calculations with empirical potentials, usually knownas Classical MD simulations (CMD), can also be used tomodel implantation cascades. For that purpose the atomto be implanted is set in motion toward the substrate with

the desired energy and angle. The substrate consists of asimulation cell with a free surface with the appropriateorientation in the direction of the implantation, while pe-riodic boundary conditions are commonly applied on theother spatial directions. This simulation cell must be bigenough to encompass the full cascade evolution avoidingthe self interaction due to periodic boundary conditions.Its final dimensions depend on both the mass of the ionconsidered and on its energy: lighter ions and high energiesresult in deeper projected ranges. For instance, Caturlaet al. used simulation cells with 106 atoms for simulat-ing 3 keV B and 15 keV As cascades into Si, which entailsa huge computational cost[3].

Empirical potentials employed in CMD to describe theatomic interactions are originally fitted to account for theequilibrium properties of the materials they model. Nev-ertheless, during the cascade evolution high energy colli-sions occur, and they must be properly described as well.For this purpose, a spline at short distances to repul-sive pair potentials, such as Moliere[141] or ZBL[143], isdone[168]. In addition, the interaction of the ion with theelectrons of the target is not considered within empiricalpotentials, and it is necessary to model it to account forthe associated energy losses. In a simple approximation,the effect of the electrons can be regarded as a velocitydependent frictional force that arises from the movement

of a charged particle in a material with an homogeneouselectron density [3,112,169]. Nevertheless, more sophisti-cated electronic stopping models developed for BCA canalso be implemented in CMD simulations [151153].

CMD simulations of monatomic ion implantation re-veal that light ions generate dilute damage in the form ofisolated FPs or small defect clusters similar to the descrip-tion provided by BCA[3,5]. However, heavier ions can alsogenerate bigger defect clusters and a-pockets [3,5] whosedescription is not appropriately captured by traditionalBCA models. A detailed analysis of the energy depositionconditions indicates that a-regions are generated directlyfrom a cascade in a melting-like process [111,170], andthey have a deficit of atoms while the surrounding regionscontain isolated Si interstitials or small interstitial clusters[5,112].


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Fig. 6. Lateral snapshots showing the typical damage config-urations obtained after (a) monatomic B and (b) B18 clusterimplantations with 500 eV per B ion in both cases. Solid linesindicate B atom trajectories. Inset in (b) is a 12-A-thick slicetaken around the cluster impact point showing the generatedcrater (from Ref. [114]).

Cluster and molecular implantations have been stud-ied using CMD simulations[114,116118,171173]. Thesetype of implantations are of interest for surface modifi-cation processes (such as sputtering, surface smoothingand cluster ion assisted thin film deposition), and for thefabrication of ultra shallow junctions since they allow ac-cessing to lower effective implant energies and are self-amorphizing [174,175]. CMD simulations have provideda relationship between the cluster size and the implantenergy to maximize the surface sputtering or to achievecluster deposition without sputtering[171]. They have alsobeen used to characterize the damage generated by clusterimpacts[114,118]. A comparison of the generated damagefor monatomic B and B18 cluster implants with the sameenergy per implanted ion is shown in Figure 6. The av-erage number of atoms disordered by each monatomic Bcascade is 32, while for the B18 cascade, each B atom dis-orders 108 atoms. This is is a consequence of the differentdamage generation mechanisms present in each case. Formonatomic B implants ballistic collisions dominate andmainly isolated point defects and small defect clusters aregenerated. For the cluster implantation a high local energydeposition at the impact point causes the melting of thesubstrate and its subsequent amorphization by quench-ing[114,116].

3.4 Linking BCA and CMD

Despite the good description of generated damage by im-plant cascades obtained with CMD simulations, the highcomputational cost of this technique limits its use to thestudy of individual cascades rather than a full implanta-tion process (i.e. thousand of cascades on the same targetwith dynamic annealing during the intercascade time in-terval). Most efforts made in the past to accelerate CMDcalculations[153,169,176178] were about evaluating for-ces not on all the atoms of the simulation cell but onlyon those atoms that (i) interact with the ion [153,169],(ii) are under strong forces [177], (iii) are set in motionduring the cascade[176], or (iv) fulfill some energetic crite-rion[178]. However, either the introduced approximations

were at the expense of lacking a correct description of lat-tice damage, or the computational time reduction was notenough to replace BCA with CMD for the simulation ofthe full implantation in process simulators.

Another alternative is the use of multi-scale modeling,where several simulation techniques are applied at differ-ent time and size scales. One of the approaches uses BCAto simulate the evolution of high-energy atoms and, whentheir energy is lower than a certain threshold (500 eV[179]or 100 eV [180]), CMD is used to simulate the final partof the cascade. However, in this approach the most time-consuming stage of the cascade is still simulated withCMD, which implies a low computational gain factor.

Other approach consists of using CMD simulationsfor the evaluation and characterization of the damagegenerated under different energetic conditions. The infor-mation extracted from these studies is then fed to BCAsimulators by means of improved damage generation mod-

els with appropriate parameters [180184]. In addition tothe FPs generated by the ballistic mechanism for energiestransferred above Ed, Hobler et al. [181183] and Santoset al. [184] simulated the generation ofa-regions with BCAby accounting for the energy transfers below Ed that oc-cur during the cascade evolution. From CMD simulationsit was determined that the final size of an a-region de-pends on the total energy deposited [181] and also on thedeposited energy density [170], and its generation is theresult of the competition between the melting of the lat-tice and the energy out-diffusion[170]. In Hoblers model,to generate the damage by the melting mechanism, theheat diffusion equation together with a melting criterion

is solved by using finite differences with a grid of points co-incident with the crystallographic lattice sites [182]. Theyalso included a lattice collapse model, which requires aminimum density of melted atoms to convert a regioninto amorphous[183]. In Santos approach, an analyticalexpression similar to the modified KP formula was ob-tained [184]. The final number of additional disorderedatoms as a function of the initial energy deposition condi-tions, NDA, is given by

NDA = N N ET(N)

DC(N) , (9)

where is the initial energy density (in eV/atom), andN

is the initial number of atoms that receive energy belowthe displacement threshold. ET(N) and DC(N) are thethreshold energy density for damage production and thedamage generation cost, respectively. Unlike the modifiedKP formula where Ed is constant, in Santos model bothET and DC vary with N, accounting for the non lineareffects that appear in molecular implants. This model hasbeen implemented in a BCA code and successfully used tosimulate molecular implantations [184], as shown in Fig-ure7 for a B18 cascade.

4 Point and extended defects

Native point defects in Si have been an important fieldof both theoretical and experimental research for several


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0

1

2

34

5

6

7

8-5 -4 -3 -2 -1 0 1 2 3 4 5 6

Depth(nm

)

Lateral distance to impact point (nm)

Aditional DAVacancies

Si interstitialsB ions

Fig. 7. Depth projection of damage generated by a B18 cas-cade obtained with the improved BCA. A conventional BCAcode only provides the position of the FPs generated by theB ions. The improved damage model adds the disordered atomsgenerated through multiple interactions, which constitute themain volume of the a-region and can not be obtained only bythe superposition of individual cascades. The dimensions ofthe obtaineda-region are in good agreement with the resultsof CMD simulations (see Fig.6).

decades. The interest in its study continues today due totheir role in a large variety of phenomena, especially inthose related to the fabrication of ICs. Native Si defectsaffect the microstructure evolution of the material dur-ing several of the manufacturing steps, and thus can alter

the final performance of the device[185]. The most funda-mental building blocks for microdefect formation in crys-talline Si are the self-interstitial and the vacancy. Thesetwo species are the mediators for impurity diffusion andclustering [185,186].

Point defects interact among them giving rise to ag-gregates or clusters [187189]. Generally, the formationenergy of such clusters decreases with size, and conse-quently their population in the Si lattice is controlled byan Ostwald ripening process: larger clusters grow at theexpense of point defects freed from smaller and less sta-ble agglomerates[12,190]. The study of the properties ofsmall clusters is difficult because they are too small to bevisible in Transmission Electron Microscopy (TEM) im-ages. Moreover, they show a great variety in their atomicconfigurations which complicates their analysis using sim-ulation techniques. At sizes of several hundred point de-fects these aggregates start becoming visible in TEM; theyusually show regular atomic structures, and are generallyknown as extended defects [191]. Understanding their be-havior and properties is important in order to developpredictive atomistic simulators for the design of new ICgenerations[154].

4.1 Vacancy defects

The vacancy is the simplest intrinsic point defect in Si:its basic form is just a missing Si atom in an otherwise

tetrahedrally coordinated lattice[192]. While at cryogenictemperatures the vacancy shows an Electron Paramag-netic Resonance (EPR) signal [193], at processing tem-peratures they are not directly detectable due to their

high mobility, and so their association with phenomena atsuch temperatures is not straightforward [194]. The for-mation energy of the vacancy has been estimated usingboth theory and experiment. There is considerable un-certainty in the actual value, with various experimentalestimates lying in the range of 2 to 4 eV [193,195,196].Using ab initio techniques, calculated formation energiesfor the vacancy range from 3 to 6 eV (see Ref. [192] andreferences therein). The theoretical difficulties arising forthe vacancy are related, at least in part, to the subtlereconstruction of the dangling bonds, where some contro-versy still remains [197,198]. The diffusivity of vacancieshas been characterized experimentally by various meth-ods: at low temperatures directly by EPR, and at high

temperatures indirectly via their effect on the diffusion ofdopants and metals. While vacancy migration energies de-termined experimentally range from 0.3 to 4 eV, ab initiocalculations predict values in the lower end, between 0.3and 0.4 eV [192].

Vacancy aggregation in Si has been studied exten-sively because large vacancy clusters (voids) are knownto be harmful to microelectronic device yield and reli-ability, particularly gate-oxide integrity [199,200]. How-ever, the introduction of voids in the Si lattice has beenproposed as a way to reduce the interstitial supersatura-tion[201204]. This controlled injection of voids, part ofa more generic concept of defect engineering, allows the

reduction of the anomalous diffusion of dopants such asB. Positron annihilation experiments have been used todetermine the lifetime of vacancy clusters, being around400 ps for sizes between V3 and V10 [192]. Voids are muchmore stable, and have been observed directly by TEM toorganize into octahedral structures aligned almost exclu-sively along the {111} crystallographic planes of the Silattice[205]. This phenomenon has been explained by thelow energy of the Si(111) surface relative to other orienta-tions [206]. The thermodynamics and binding propertiesof these vacancy clusters have been studied using quan-tum and classical simulation techniques[194,207209]. Inparticular, for small vacancy clusters it has been found

that certain sizes show greater stability, as it is the case ofthe V6, V8 and V12 clusters (see Fig.8), due to particularbond reconstructions in the Si lattice [207,209211]. Forlarger sizes, binding energies tend to a value of around3 eV, in agreement with Sb diffusion and Au labeling ex-periments[212,213].

4.2 Interstitial defects

The self-interstitial, one extra Si atom in the crystal lat-tice, is the natural counterpart of the vacancy. The studyof the Si self-interstitial properties is of particular impor-tance in Si processing. Self-interstitials have been impli-cated as the origin of rodlike defects observed in Czochral-sky single-crystal growth, which can ultimately produce


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4

5

6

y(eV)

Bongiorno et al.

Staab et al.

Hastings et al.

Lee and Hwang

2

3

Binding

ene

rg

0

1

0 5 10 15 20 25 30 35

Cluster size

Fig. 8. Binding energy for vacancy clusters as a function ofcluster size (from Refs. [207,209211]).

the degradation of the manufactured Si devices [214].

On the other hand, during the implantation step a largeconcentration of excess interstitials is introduced in thelattice. They interact with dopants and cause the so-called Transient-Enhanced Diffusion(TED) which altersthe junction depth [215].

Due to its importance in Si processing, a great num-ber of theoretical studies have been devoted to deter-mine the configuration and energetics of the Si self-interstitial, as well as its diffusive behavior. These includeab initio [48,56,58,64,216220], TB [74,7680,221], andempirical potential calculations [104,107,222230]. How-ever, even when using the same calculation techniques,different authors come to different conclusions regarding

the Si self-interstitial properties. The discrepancies aremainly related to the determination of the lowest for-mation energy configuration and to the microscopic de-scription of the interstitial-mediated diffusion mechanism.At least four different interstitial configurations have beenidentified: tetrahedral (T), dumbbell (D), hexagonal (H)and extended (E). Within the first-principle framework,most authors coincide that the lowest energy configura-tion is the D interstitial, with 2.23.4 eV [48,56,58,64],while Leung et al. determine that the lowest energy cor-respond to the H interstitial with 4.8 eV[219], and Needsstates that D and H are degenerated configurations hav-ing both a formation energy of 3.3 eV [220]. In the case

of TB simulations, some authors claim that the low-est formation energy configuration is the D interstitialwith 3.85.6 eV [74,76], while others suggest it is theT configuration with formation energies between 3.8 and4.4 eV [77,78,221]. When using empirical potentials thediversity in results is even higher [222,223,225,227230].There is not a clear agreement either regarding the micro-scopic description of interstitial-mediated diffusion, wherevery different diffusion mechanisms have been proposed,with migration energies ranging from 0.1 to 1.9 eV (seeRef. [192] and references therein). In spite of such a diver-sity of results, recently Marques et al. demonstrated us-ing CMD techniques that all self-interstitial configurationscoexist in Si but with different concentrations, and diffu-sion occurs through transitions among them [102]. Themacroscopic behavior for self-interstitial diffusion can be

2

3

(eV)

1

Binding

energy

Cowern et al.

Colombo

Chichkine et al.

0

0 5 10 15 20

Cluster size

ar n- raga o e a .

Fig. 9. Binding energy for interstitial clusters as a functionof cluster size (from Refs.[12,231233]).

modeled by a simple description based on a unique inter-

stitial species with an effective formation energy of 3.8 eVand a migration barrier of 0.8 eV, in very good agreementwith experiments [12,196]. The exact numbers do not cor-respond to any of the particular interstitial configurationsor diffusion mechanisms, but are the result of the averagedbehavior of all of them. These findings help to explainwhy it is not straightforward to justify the experimen-tal measurements on interstitial diffusion, related to themacroscopic behavior, resorting to a particular interstitialconfiguration and diffusion path theoretically determinedusing ab initio or TB techniques.

Due to their implications in Si technology, self-inters-titial aggregation in Si has attracted much attention in

the literature. Large interstitial clusters formed after im-plantation act as a reservoir of Si self-interstitials thatare slowly released during subsequent thermal treatmentscausing the TED of interstitial-diffusing dopants such asB [8]. Moreover, interstitial extended defects show pho-toluminiscent signals [234], and have been proposed tofabricate optical emitters compatible with the standardand well-established IC technology. They introduce localstrain fields in the Si lattice, which modify the band struc-ture and provide spatial confinement of the charge carriersallowing room-temperature electroluminescence[235].

From B diffusion experiments, Cowern et al. deductedthe formation energy of small interstitial clusters using

the concept of Ostwald ripening and the fact that the Siinterstitial supersaturation, and therefore B diffusion, isrelated to their stability[12]. Results for the binding en-ergy of such clusters are shown in Figure 9, along withcalculations carried out by other authors using ab ini-tio [232], TB[231] and fitting to experiments [233]. Themost important finding is that oscillations of the bindingenergies occur for small, discrete, magic sizes (as withvacancies). These more stable sizes correspond to configu-rations where atoms remain four-fold coordinated [236].For larger sizes, of around one hundred atoms, {113}defects start to form. Their atomic structure was deter-mined by Takeda using TEM [237]. {113} defects consistof large interstitial chains along the 110 direction, packedtogether along the {113} plane, which gives this defectits name. It has been experimentally shown that {113}


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gy(eV/at)

Small

{113}100

mationener us ers

FLDs10-1

s

Number of atoms

10-2For

101 102 103 104 105 106 107100

Fig. 10. Formation energy of the different types of interstitialagglomerates as a function of size (from Ref.[238]).

defects grow in length along the 110 direction [191]. Inthe process, their formation energy decreases from 0.8 to0.65 eV [238]. Apart from the experimental character-

ization of these extended defects, many theoretical in-vestigations on their structure, energetics and inducedstrain fields have been published [8284,110,239,240]. If{113} defects grow up to a certain size, they can trans-form into dislocation loops, perfect (PDLs) and faulted(FDLs) [241]. This transformation has been proposed tobe due to some unfaulting reactions, as it has been shownrecently by using ab initio simulation techniques [242].The formation energy of FDLs tends to 0.027 eV withincreasing size, while it tends to 0 for PDLs [238]. In Fig-ure 10 we show the formation energy of interstitial ag-glomerates as a function of size. This energy landscapedetermines the microstructural evolution of the material.

For example, at a size of 1000 interstitials, a {113} wouldact as a sink for self-interstitials released by PDLs and asa source of self-interstitials for FDLs.

4.3 KMC and continuum modeling of defect evolution

The modeling of defect evolution is key to determine theannealing required to completely remove the damage andfor an adequate prediction of dopant profiles. Althoughsome calculations suggest that small clusters (di-intersti-tials, tri-interstitials, etc.) may be mobile[233,243], mostmodels only consider Si interstitials and vacancies as the

diffusing species. Therefore, the reactions that account forthe interactions among defects leading to the formation ofagglomerates of n Si interstitials (In) or vacancies (Vn) are:

I+V 0 (10)

In+I In+1 (11)

Vn+V Vn+1 (12)

In+V In1 (13)

Vn+I Vn1. (14)

Since defect size may extend to large numbers, a huge setof reactions should be modeled (n = 1 ). AtomisticKMC simulators can track each and every of the generatedIs and Vs and the different cluster sizes. The key param-eters for these reactions are the point defect diffusivities

a)

b)

Fig. 11. KMC simulations of the time evolution of (a) thetotal number of Si self-interstitials and vacancies, and (b) thenumber of interstitial hops, for a 150 keV Si implant to a doseof 7 1013 cm2, annealed at 800 C (from Ref.[254]).

and the capture and emission probabilities, which are de-pendent on defect stability (binding energy). For smallclusters magic numbers are used [12], while for extendeddefects generally analytical expressions relate the bindingenergy to defect size [244,245]. In continuum simulators,a large number of equations derived from reactions (10)to (14) forn = 1 is prohibitive due to the high com-putational cost. Simplified models have been developedwith the aim of reducing the number of equations to besolved[246253].

KMC simulations show that most FPs quickly recom-

bine during the implant itself and the initial stages of theannealing. However, when implanted ions become substi-tutional they generate excess Si interstitials that have novacancies to recombine with, and survive for a long timeuntil they are annihilated at the surface. This is evidencedin Figure11a, which shows the number of generated de-fects per implanted ion as obtained from KMC simula-tions. After 1 s, almost all vacancies have recombined,but the excess Si self-interstitials remain several minutes.As can be observed in Figure 11b, the number of inter-stitial hops per lattice site (which is proportional to theinterstitial supersaturation) after vacancy recombination(V = 0) is minimal compared to the I-hops after inter-stitial annihilation (I= 0). This proves that dopant dif-fusion and deactivation is mainly driven by the excess Siinterstitials remaining after FP recombination. Based on


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Fig. 12. Energy dependence of the plus factor for differentspecies implanted with a dose of 3 1014 cm2. Symbols rep-resent numerical data whereas lines correspond to an analyticalfitting (from Ref. [253]).

this observation, it is usually assumed that the amountof residual damage after implant is approximately the im-planted dose (each ion contributes as one excess Si intersti-tial), which is known as the +1model[252]. This simplemodel provides similar results than full-cascade simula-

tions (see Fig.11).The +1 model relies on an efficient FP recombina-tion and it is valid for light ions implanted at mediumenergies and doses. For heavy ions[255], low doses or highimplant temperatures [256], FP recombination is not soefficient, and generated Si interstitials and vacancies maysignificantly contribute to dopant diffusion before their re-combination. In this case, the +1model underestimatesthe amount of effective Si interstitials. For this reason, aneffective plus factoror +n was defined to take into ac-count the effect of implant parameters [253,255,257]. The+n model determines the effective number of excess Siinterstitials (n) per implanted ion according to the ionmass, implant energy and dose. Figure 12 shows the +nfactors for several ion species and implant energies.

The +n model allows to reduce the number of in-teractions to consider by neglecting the role of vacancies(Eqs. (10), (12) and (14) can be omitted) and reducingthe number of interstitials to +n. It provides good re-sults for dopant diffusion and defect evolution at a macro-scopic scale. However, some dopants undergo clusteringduring the implant or the initial stages of the annealing,before significant I V recombination has occurred. Inthese cases, dopant deactivation may be underestimatedif the initial high interstitial and vacancy concentrationsare not simulated [258].

The surface plays an important role on defect evolu-tion. Its efficiency as a sink for point defects is modeledthrough the recombination probability (in KMC mod-

Fig. 13. Simulated time evolution during annealing at 750 Cof the atomic and defect dose of Si interstitials in defects for asimulated 40 keV Si implant to a dose of 1014 ions/cm2. Thetime evolution of the mean size of the defects is also plotted

(from Ref. [260]).

els) or the recombination length (in continuum simula-tors) [259]. Although the exact values of these parametersare not known, it is generally assumed that the probabilityof a defect that approaches the surface being annihilatedis close to unity. After vacancy recombination, the surfaceis the only sink for excess interstitials. Initially, when thereare many small defects, free Si self-interstitials are morelikely to be captured by a defect than to reach the sur-face. Ostwald ripening takes place, leading to the growthof bigger defects at the expense of the smaller ones. As

a result, defects increase their size but the total numberof defects is reduced (see Fig. 13). When the separationamong them is comparable to the distance to the surface,similar probabilities exist for an interstitial to reach thesurface or to be trapped by a defect, and defects quicklydissolve [260].

The +n model serves as a starting point in con-tinuum models, but additional simplifications need to bedone because interstitial defect evolution would require aset of differential equations, each equation describing thetime evolution of a defect size (n ). Some models con-sider all clusters sizes, from small clusters of a few intersti-tials to defects with hundred of particles, but are compu-

tationally too expensive [12,261,262]. Several works haveproposed reduced models with only a few equations todescribe small interstitial clusters, without sacrificing ac-curacy[246249]. Cowern et al. showed that small clustersshould not be neglected, because they act as precursors of{113} defects and control the Si interstitial supersatura-tion at the early stages of the annealing [12]. A simplemodel considering a stable cluster (I3) and two adjacentless stable clusters (I2, I4), has been proved to correctlyreproduce the contribution of small interstitial clusters toTED[246].

The biggest effort in defect modeling has been devotedto {113} defects, since they have been identified as themain source of interstitials for TED [8]. As the bindingenergy of {113} defects is a smooth function of defectsize [261], moment-based models can be used, in which


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Fig. 14. Measured (symbols) and simulated (lines) interstitialsupersaturation after 40 keV Si implantation (from Ref.[246]).

the cluster equations are replaced by differential equa-tions for the moments of the cluster distribution [248].The first proposed model only considered the concentra-tion of interstitials trapped in clusters (first moment ofthe cluster distribution), neglecting the effects related tocluster growth [263]. Improved models do include theseeffects by considering not only the concentration of in-terstitials in {113} defects (C113), but also their density(D113), by means of two [244,246,247,250] or even threemoments [248].

In a two-moments approach, the concentration of inter-stitials in {113} defects increases when the biggest clusterconsidered (I4 in this example) incorporates an interstitial

and becomes a new {113}, or when an existing {113} trapsan interstitial. On the contrary, the concentration reduceswhen a {113} releases an interstitial or when it evolves toa dislocation loop. All these events are captured by thefollowing equation[247],

dC113dt

= 5kfCICI4+kaCID113 kbD113 k113DLC113

(15)where CIis the free interstitial concentration and CI4 isthe concentration of small interstitial clusters with size 4.The density of{113} defects increases when a small clusterevolves to a {113}, and reduces when {113} defects ripen

to larger sizes or become dislocation loops,

dD113dt

=kfCICI4 kbD113C113

D113 k113DLkD113D113

(16)kf,ka,kb,k113DL,kD113are the reaction rates that con-trol the kinetics of the system[247]. These equations havebeen tested with experimental values of supersaturation(Fig.14), concentration of interstitials trapped in defectsand defects size [246,247].

Under certain conditions, particularly for high-doseimplants,{113}defects can evolve into dislocation loops.To simulate this transformation, the most simple modelsconsider that the evolution occurs when the {113} de-fects reach a certain size, around 300 interstitials (lessthan 20 nm in leght). However, TEM images indicate that

{113} defects can reach sizes for which the dislocationloop configuration is energetically more favorable [238].Improved models consider a transformation rate that iscontrolled by a size-dependent energy barrier[264]. In con-

tinuum models, evolution of dislocation loops can be alsodescribed by a two-moment system in a similar way to{113} defects[247,250,251,265].

5 Amorphization and recrystallization

Since the beginning of the use of ion implantation for thefabrication of Si devices, ion-beam-induced amorphizationand SPER in Si have been the subject of a great num-ber of studies. There is a renewed interest in the model-ing of these processes because of their technological rel-evance for the semiconductor industry to achieve highdopant activation with minimal diffusion. For example,high implant doses of As required to achieve high levelsof carrier concentration induce amorphization of the Silattice. For B doping with monatomic beams, preamor-phizing Ge implants are used to take advantage of thehigh dopant activation achieved by SPER[266]. Implantsof BF2 or heavy molecules, such as decaborane or octade-caborane [267,268], are self-amorphizing[114]. Damage re-covery is different in amorphizing and sub-amorphizingimplants because in the former the SPER of the a-layeroccurs at a relatively low temperature, and leaves a free-of-defects zone. The amount of residual damage beyondthe a/c interface, which in turn affects dopant diffusionand activation, depends on the a-layer depth [269]. Ion-

induced amorphization and recrystallization models, com-patible with process simulators, must be able to predictthe onset of amorphization and its dependence on implantparameters.

5.1 Fundamental studies

In order to develop models that describe amorphizationand recrystallization processes at atomic level it is es-sential to identify the defect or defects that can act asamorphous embryos, and those that induce recrystalliza-tion. In the literature, several defects have been proposed

to be relevant in the amorphization and/or recrystalliza-tion mechanisms: vacancies [270,271] and vacancy com-plexes [272274], self-interstitial clusters [275277], pairsof di-vacancies and di-interstitials[278], dangling bonds inthe amorphous phase[279], kinks along [110] ledges in thea/cinterface[280,281], and bond defects[74,282]. Amongall of them, the bond defect appears to be the most ade-quate to describe amorphization and recrystallization.

Tang and coworkers encountered the bond defect whenstudying self-diffusion and I Vrecombination in Si us-ing TB techniques[74]. They found that when a vacancyapproaches a 110 dumbbell interstitial, a metastable de-fect structure is generated instead of having immediateI V recombination. For this reason the bond defect isalso known as IV pair. This defect, represented in Fig-ure15,consists of a local rearrangement of bonds in the


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Fig. 15. Atomic structure of the IV pair. Dashed lines repre-sent atoms and bonds in the perfect lattice. Atoms A and A

move along the directions indicated by the arrows and switchtheir bonds with atoms B and B, giving rise to the IV pair.

crystal with no excess or deficit of atoms. It introduces inthe Si lattice the five- and seven-membered rings typical

of the amorphous phase while maintaining perfect four-fold coordination. Information regarding the structure andenergetics of this defect has been extracted using ab ini-tio [59,283], TB [74] and CMD techniques [284]. The IVpair formation energy, 3.26 eV[59], is of the order of thecorresponding to typical point defects in Si. Its equilib-rium concentration should then be comparable or evenlarger than that of the other defects. However, the IVpair has not yet been detected experimentally, maybe be-cause perfect four-fold coordination (which means no un-paired electrons) precludes its detection in standard ex-periments such as electronic paramagnetic resonance andelectron nuclear double resonance. It has also been shown

by ab initio techniques that the IV pair hardly disturbsthe band structure of Si, which makes it undetectable inthe deep level transient spectroscopy technique[283]. Us-ing CMD techniques, Stock and coworkers observed thatthe bond defect can be generated not only by incompleteIVrecombination, but also as a result of a pure ballisticprocess [282]. Thus the IV pair can be a primary defectgenerated by irradiation, with no need of pre-existing in-terstitials and vacancies in the lattice for its formation.They showed as well that the IV pair is a characteristicstructural feature of the a/c interface [282,285].

The IV pair annihilates by the reverse movement ofatoms A and A of Figure 15 toward the perfect lattice

positions, switching again their bonds with atoms B andB. Using CMD simulations, Marques et al. showed thatthe IV pair lifetime follows an Arrhenius behavior overa wide temperature range with an activation energy of0.43 eV[284]. At room temperature, the IV pair lifetimeis about 3 s, very short in comparison with the charac-teristic inter-cascade time at typical dose-rates. This indi-cates that the IV pair as an individual defect is not stableenough to accumulate and promote amorphization. How-ever, the stability of the IV pair is affected by the prox-imity of other IV pairs in the Si lattice. CMD simulationsindicate that IV pairs randomly scattered in the Si latticeand separated from each other by at least 4 A recombinewith an activation energy of 0.44 eV, i.e. as if they wereisolated[286]. However, when the same number of IV pairsare arranged in a compact sphere with a radius of 1.2 nm,

the activation energy for recombination is 0.86 eV, be-ing the recrystallization dynamics slower. If IV pairs arearranged as to form a planar a/c interface, the activationenergy is even higher, 2.44 eV, in good agreement with the

experimental measurement of 2.7 eV [287]. This behaviorindicates that IV pairs surrounded by crystalline atomseasily rearrange into the perfect crystal lattice structure,while IV pairs surrounded by other IV pairs have a largereffective energy barrier to rearrange.

According to CMD calculations, when IV pairs arepresent in the Si lattice to a given concentration (around25%), amorphization of the Si lattice takes place [284].The resulting structure is identical to the correspondingto a pure amorphous Si matrix. Besides, the amorphouszones created by IV pair accumulation and those by di-rect irradiation show the same features, as far as energycontent, internal structure and recrystallization dynamicsare concerned [108]. All these results indicate that amor-phization can be achieved without the intervention of anyadditional defect, and also that amorphous pocket char-acterization can be studied by IV pair accumulation. Thisprocedure of accumulating IV pairs can serve then as acontrolled way to introduce damage in the Si lattice.

5.2 Atomistic KMC models

The results obtained from CMD and more fundamentalmethods set the basis of atomistic models implementedin KMC simulators that can reach actual Si processing

scales. A unifying and consistent view of amorphizationand recrystallization processes is provided by the atom-istic amorphization model based on the IV pair [288]. Itconsiders a-regions as agglomerates of IV pairs. To takeinto account the effect of the spatial distribution on thestability of IV pairs, each IV pair is locally characterizedby the number of neighboring IV pairs[288,289]. Thus, therecombination rate of the IV pair decreases as the numberof neighboring IV pairs increases reflecting the difficultyof amorphous atoms to properly rearrange when theyhave fewer neighboring atoms in crystalline positions.

IV pair recombination is a thermally activated processwhose activation energy depends on the number of neigh-

boring IV pairs. An activation energy for recombinationof 0.43 eV is given to the isolated IV pair (0 neighbors), asobtained from CMD calculations [284]. IV pairs at a pla-nara/c interface are assigned the activation energy corre-sponding to that of the experimental recrystallization ve-locity, 2.7 eV[287]. IV pairs embedded into an amorphousmatrix (completely surrounded by neighboring IV pairs,and thus with full coordination) have an activation energyof 5 eV, coincident with the experimentally observed acti-vation energy for crystal nucleation in amorphous Si [290].Other coordination numbers have intermediate activationenergies, as shown in Figure16, whose values have beenobtained by the calibration of the model with experimen-tal results [291].

The SPER velocity of a planar a/c interface and a-pockets has been experimentally observed to depend on


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5

6

(eV)

y(eV) 5

6

2

3

ctivationen

ergy

. .5eV(Exp.Crystal

Nucleation)ivationener

2

3

0

1

0 5 10 15 20 25

A

0.43eV(MolecularDynamics)Act

0 5 10 15 20 250

1

Number ofneighborsFig. 16. Activation energies for IV pair recombination as afunction of the number of neighboring IV pairs.

the presence of impurities [292,293]. Dopants usually en-

hance SPER whereas non-dopant impurities, such as F,generally decrease it. The impurity effect on SPER istaken into account in the model by making the recom-bination probability of IV pairs to depend, not only onthe number of neighboring IV pairs, but also on the lo-cal impurity concentration, according to the experimentaldata of Olson et al.[287,292].

The local characterization of the disordered atoms al-lows the model to capture any damage morphology thatmay arise from irradiation cascades, as well as the charac-teristic regrowth behavior observed in experiments [289].Figure17is a schematic that shows a continuous a-layerand several a-pockets; for some IV pairs a dotted circle

is used to highlight their neighboring IV pairs. IsolatedIV pairs and those belonging to small a-pockets have fewneighbors and recombine sooner. IV pairs at the cornersor fingers of a-pockets, and therefore belonging to con-vex regions (like A and B in Fig. 17), have fewer neigh-boring IV pairs and recombine faster, shortening the life-time of irregulara-pockets compared with more compactstructures [108,289]. IV pairs at an a/c interface (C) havefewer neighboring IV pairs than those embedded withinthe a-region (F), and therefore recrystallization (IV pairrecombination) will start at the interface. A continuousa-layer with a perfectly planara/c interface is just a par-ticular case ofa-region, where all IV pairs at the interface

have the same number of neighboring IV pairs and fol-low the same activation energy for regrowth, 2.7 eV. Therecombination of the first IV pair at the planar interface(C) starts a triggering mechanism in which its neighbors(D and E) are the most likely ones to be recombined nextsince they have lost a neighbor (the one that has recom-bined first). This mechanism leads to the complete recrys-tallization of the whole monolayer. A new triggering eventis required to start the regrowth of the next monolayer.

This simple model based on the IV pair encompassesand unifies homogeneous and heterogeneous mechanismsfor amorphization. The nucleation of the amorphous phaseconsists of the formation of IV pair structures with enoughnumber of IV neighbors to be stable. At the appropri-ate temperatures, these amorphous embryos can be ei-ther small IV pair complexes generated in dilute cascades

D

C A

F

B

Fig. 17. Schematic of a damage distribution in which a con-tinuous a-layer and several a-pockets are shown. IV pairs arerepresented by solid circles. For some IV pairs a dotted circleindicates their neighboring IV pairs.

(homogeneous nucleation) or dense IV pair agglomeratesformed in the cascades of heavy ions (heterogeneous amor-phization). Preexistinga-pockets or planar a/c interfacescan act as nucleation sites [294,295]. Isolated IV pairs maynot be able to survive by themselves but, if they interactwith a preexisting a-region, the generated IV pairs willhave more IV neighbors and become stable. IV pairs arethus added to the amorphous zone producing its growth.The superlinear behavior of the accumulated damage ver-sus dose [296] is also a consequence of the increased sta-bility of the IV pairs with the number of IV neighbors.

The adequate modeling of dynamic annealing in pro-cess simulators is very important since it may stronglyaffect damage accumulation during the implant. Implanttemperature, beam current and ion mass may determinewhether most of generated damage anneals out during theimplant or it accumulates leading to the formation ofa-regions. In this model, implant temperature directly af-fects the stability of IV pairs, since the recombination ofIV pairs is a thermally activated process. At low temper-atures most generated IV pairs are stable, while at hightemperature isolated IV pairs and those with few neigh-boring IV pairs quickly recombine, and only IV pairs be-longing to largea-regions survive [289]. Beam current sets

the temporal separation between cascades and therefore,the time during which the damage generated by one cas-cade undergoes dynamic annealing, before another cas-cade arrives at the same region increasing the damage.Damage topology is affected by ion mass. Light ions gen-erate dilute damage that easily anneals out, while heavyions produce large a-pockets that may be stable enoughto survive dynamic annealing[297].

The a-regions generated during an implant cascademay contain a local excess (Is) or deficit (Vs) of atomswithin the amorphous matrix. When an isolated a-regioncompletely recrystallizes, the unbalanced atoms are re-leased to the crystal and they appear as Si interstitialsor vacancies, as reported by CMD simulations of the re-growth ofa-pockets[111]. On the contrary, the regrowthof a continuous a-layer that extends to the surface results


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Fig. 18. Simulations of the annealing at 800 C of a sampleimplanted with 5 keV 1014 cm2 Si (sub-amorphizing) and5 keV 1015 cm2 Si (amorphizing). IV pairs are plotted asgreen circles, whereas Si Is and Vs are indicated by blue and red

circles, respectively. Several snapshots during recrystallizationare shown. Top images represent the as-implanted profiles.

in a regrown layer free of defects, because the local ex-cess or deficit of atoms are swept toward the surface andannihilate there. In this model, the energy of the localexcess or deficit of atoms decreases with the number ofneighboring IV pairs, which implies that they tend to bewithin thea-region. As IV pairs recombine and the inter-face advances, the local excess or deficit of atoms is likelyto be moved to inner regions of the a-layer. In Figure18it is shown the as-implanted damage profiles and several

snapshots during the recrystallization of sub-amorphizingand amorphizing implants [298]. In the sub-amorphizingsample the as-implanted profile shows crystalline islandssurrounded by largea-regions. Upon annealing isolateda-regions recrystallize, releasing the unbalanced atoms theycontain. Finally, after recombination, defects are presentalong the damage profile but they accumulate around themean projected range of the implant. The amount of SiIs retained in defects is approximately the implanted dose(+1 model [252]). On the contrary, simulations showhow defects contained in the continuous a-layer are swepttoward the surface as the interface advances and are elim-inated there. A band of extended defects is formed onlyat the end-of-range (EOR) region, the dose of Is stored indefects being lower than the implanted dose.

The detailed atomistic amorphization model describedhandles and stores all IV pairs generated during the im-plant process, up to the limit imposed by the amorphousmaterial density. As each IV pair consists of two latticeatoms, a fully amorphous Si region represents an IV pairconcentration of 2.5 1022 cm3. This high defect con-centration translates into significant CPU time and mem-ory requirements. This limitation has been overcome insome KMC models that implement a less detailed de-scription of the amorphous phase, using the a-pocket asthe basic amorphous structure [299]. A-pockets are con-sidered as three-dimensional, irregular shape agglomeratesof an arbitrary number of interstitials and vacancies andtrapped impurities. A-pockets can grow by incorporating

additional particles as the implant proceeds or shrink dueto annealing. Their recrystallization is a thermally acti-vated process whose activation energy depends on the ef-fective size of the a-pocket, without taking into account

the morphology[300]. The size is considered as the mini-mum number of Is or Vs it contains, i.e. the overall numberof IV pairs. This number can be rather large because itrefers to the whole size of the a-pocket, and it is not lim-ited to the local neighboring distribution. By fitting thesimulation results to experiments[291], the activation en-ergy for the shrinking of an a-pocket from size s to size(s1) was estimated to be 0.7 eV fors= 1 and saturatingat 2.7 eV fors >225.

To optimize CPU time and memory consumption, thespace is divided in small boxes of about 1 nm3. When thedefect (I+ V) concentration associated to the a-pocketsin one of these boxes exceeds the amorphization threshold(1.5 1022 cm3), the box is labeled as a-region[299,300].When an a-region is formed, only the number of unbal-anced atoms and impurities it contains is stored, but thepositions of the particles are not kept. Recrystallization ofeach of these boxes occurs as a whole at a rate given bythe experimental regrowth velocity of a planar a/c inter-face and, if needed, the unbalanced atoms are transferedto neighboring boxes or released to the bulk.

5.3 Implementation in continuum methods

Continuum methods usually simulate the crystalline toamorphous transition induced by ion implantation by me-

ans of a critical energy/defect density model [301,302].BCA calculations are typically used to evaluate the nu-clear energy transferred to the lattice during the collisioncascade. The critical energy density model states that theimplanted region turns amorphous when an energy den-sity threshold is exceeded, usually considered as approx-imately 5 1023 eV/cm3 [6,303305]. The deposited nu-clear energy is responsible for the formation of defectsin the crystalline lattice, according to the KP formula(Eq. (7)). Therefore, a critical defect concentration (CDC)can be used instead of an energy density to determine thecrystalline to amorphous transition, typically in the orderof 1.15 1022 cm3 [306]. When the theoretical damage

profile is compared with the position of the a/c interfaceexperimentally measured, the CDC value required to ob-tain a good fitting is not unique. Variations of more thanone order of magnitude have been reported [269], and ithas been found to depend on ion mass, dose rate and im-plant temperature, among others. These variations intro-duce uncertainty in the position of the a/c interface. Asit is shown in Figure 19,a small change in the position ofthe interface may result in a large change in the amountof residual damage remaining beyond the interface afterSPER, which may reach up to 50%[307]. The dependenceof the CDC value on implant parameters is due to the ef-fects of dynamic annealing on damage accumulation. Thiseffect can be taken into account by considering that afraction of generated defects recombines during implant,as indicated in equation(8)[162].


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a/cinterfaceCriticalDefectConcentration

n

Residual

damagecentrati

lpairco

Frenke

ept

Fig. 19. Schematic that shows the a-layer depth and theamount of residual damage associated to two different CDCvalues.

Since SPER is much faster than the dissolution ofextended defects, for process modeling purposes the re-crystallization of a-regions is sometimes assumed to oc-cur instantaneously, leaving no defects in the recrystal-lized region, and only the residual defects beyond thea/c interface remain. Nevertheless, this is a very simplis-tic approach that does not account for the three dimen-sional (3D) topology of a-regions formed in the fabrica-

tion of transistors. In a general sense, SPER is a threedimensional process, but it can be treated as two dimen-sional, considering than the width of the transistor is verylong. Recently, Morarka et al. have developed a model forthe evolution of the SPER in two dimensional structures,based on level set methods [308]. In this model, the localvelocity of the regrowth front is considered to depend onthe crystallographic orientation of the interface [309] andthe local interfacial curvature[310],

v(, k) =v[001]f()(1 +Ak), (17)

where v[001] is the regrowth velocity along [001], is the

substrate orientation angle from [001] to [110],A is a con-stant (2.0 107 cm) and kis the local interfacial curva-ture.f() is a fit to the normalized regrowth velocity as afunction of measured by Csepregi et al. as shown in Fig-ure20, using a least-squares fifth-order polynomial [308].

According to equation(17), if a portion of crystalline Siis encompassed by amorphous Si (convex or negative cur-vature) SPER should be retarded. On the contrary, SPERis enhanced when a region of amorphous Si is encompassedby crystalline Si (concave or positive curvature). The effectof the interface curvature on regrowth rate is analogous tothe dependence of the IV pair recombination rate on thenumber of neighboring IV pairs. Those IV pairs located ata concave interface are mostly surrounded by crystallineSi (they have few neighboring IV pairs) and will undergo afast recrystallization. If the curvature increases, the num-

Fig. 20. The normalized SPER velocity, f(), as a functionof the substrate orientation angle from [001] toward [110], ,

as measured by Csepregi et al. [309]. The inset shows a cross-section TEM micrograph of a typical 2D structure where theSPER process occurs (from Ref. [308]).

ber of neighboring IV pairs is reduced, which will enhancethe recrystallization rate [289].

6 Dopant-defect interactions

During some of the process steps in the fabrication of

Si-based ICs, impurities or dopant atoms are selectivelyintroduced into the Si substrate to modify its electricalconductivity and form n-type and p-type re

hp epjb front-end process

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