Physical integrated diffusion-oxidation model for implanted nitrogen in siliconLahir Shaik Adam, Mark E. Law, Omer Dokumaci, and Suri Hegde Citation: Journal of Applied Physics 91, 1894 (2002); doi: 10.1063/1.1430537 View online: http://dx.doi.org/10.1063/1.1430537 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modeling of B diffusion in the presence of Ge J. Vac. Sci. Technol. B 24, 478 (2006); 10.1116/1.2151905 Oxidation-enhanced diffusion of boron in very low-energy N 2 + -implanted silicon J. Appl. Phys. 97, 113534 (2005); 10.1063/1.1927687 Anomalies in modeling of anisotropic etching of silicon: Facet boundary effects J. Vac. Sci. Technol. A 20, 1927 (2002); 10.1116/1.1513790 Isotopic substitution of N, O, and Si in the thermal oxidation of nitrogen-deposited silicon Appl. Phys. Lett. 74, 1872 (1999); 10.1063/1.123697 Understanding scanned probe oxidation of silicon Appl. Phys. Lett. 73, 271 (1998); 10.1063/1.121777

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Physical integrated diffusion-oxidation model for implanted nitrogenin silicon

Lahir Shaik Adama) and Mark E. Lawb)

Department of Electrical Engineering, Software for Advanced Materials Processing (SWAMP) Center,University of Florida, Gainesville, Florida 32611

Omer Dokumaci and Suri HegdeSemiconductor Research and Development Center, IBM Corporation, East Fishkill, New York 12533

~Received 2 July 2001; accepted for publication 1 November 2001!

Scaling the gate oxide thickness is one of many process development challenges facing deviceengineers today. Nitrogen implantation has been used to control gate oxide thickness. By varying thedose of the nitrogen implant, process engineers can have multiple gate oxide thicknesses in the sameprocess. Although it has been observed that nitrogen retards gate oxidation kinetics, the physics ofhow this occurs is not yet well understood. Since the retardation in oxide growth is due to thediffusion of nitrogen and its subsequent incorporation at the silicon/silicon oxide interface, the studyof the diffusion behavior of nitrogen in silicon becomes important. Further, it is also necessary tostudy how this diffusion behavior impacts oxide growth. Models have been developed to explorethese issues. The diffusion model is based onab initio results and is compared to experimentalresults at two temperatures. The oxide reduction model is based on the diffusion of nitrogen to thesurface. The surface nitrogen is coupled to the surface reaction rate of silicon and oxygen tomoderate oxide growth. ©2002 American Institute of Physics.@DOI: 10.1063/1.1430537#

I. INTRODUCTION

Nitrogen implantation has been used to control the gateoxide thickness.1 This implies that by controlling the doseand energy of the nitrogen implant, one can vary the thick-ness of the gate oxide across the wafer. Therefore, the nitro-gen implantation process is especially useful for processesthat require multiple gate oxide thicknesses for example,System On A Chip technologies. Although it has been ob-served that nitrogen retards gate oxidation kinetics, the phys-ics of how this occurs is not yet well understood. Since theretardation of oxide growth is due to the diffusion of nitrogento the surface and its subsequent incorporation at the silicon/silicon–oxide interface, the study of the diffusion behaviorof nitrogen becomes important. Furthermore, it is also impor-tant to study how this diffusion behavior impacts oxidegrowth. This article discusses models that deal with theseissues. Section II of this article summarizes the experimentalresults previously reported by us. These experimental resultsform the basis of the diffusion model. The diffusion model isdescribed in detail in Sec. III. As mentioned above, the mo-tivation behind implanting nitrogen in silicon is to retard gateoxide growth. Therefore, if the diffusion model can becoupled to an oxidation model to moderate the gate oxidegrowth, then this integrated diffusion–oxidation model canbe of considerable use in gate oxide thickness optimization.

Such an oxidation model is described in detail in Sec. IV.Finally, the central points and conclusions of this article aresummarized in Sec. V

II. PREVIOUS EXPERIMENTAL RESULTS

Experimental results of nitrogen implanted at 531013

N21/cm2, 40 keV have been reported previously in Ref. 2.

The key observations in Ref. 2 are as follows: Nitrogen dif-fuses toward the surface rapidly with time at 750 °C. How-ever, the nitrogen barely moved at 650 °C. Furthermore, thepeaks of the nitrogen implants shifted toward the surfacewith anneal time. No extended defects were observed in anyof the annealed samples.

The last observation mentioned above is of particularinterest. This is because silicon implants at this dose andenergy annealed at 750 °C show large numbers of extendeddefects.3 However, transmission electron microscopy did notreveal any extended defects for the nitrogen implants. Thismotivates the fact that any explanation of the diffusion be-havior of implanted nitrogen in silicon cannot rely on ex-tended defects. A physical model that relies only on submi-croscopic defects and nitrogen complexes has been used toexplain the observed diffusion behavior. This will be ex-plained in detail below.

III. DIFFUSION MODEL

A. Theory

The following reactions have been used in this model:

NS1I 5Ni , ~1!

Ni1V5NS , ~2!

a!Author to whom correspondence should be addressed; present address:Texas Instruments, 13560 N. Central Expressway, MS 3735, Dallas,TX 75243; electronic mail: lahir@ti.com

b!Electronic mail: law@tec.ufl.edu

JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 4 15 FEBRUARY 2002

18940021-8979/2002/91(4)/1894/7/$19.00 © 2002 American Institute of Physics

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NS1V5NV, ~3!

NV1I 5NS , ~4!

I 1I 5I 2 , ~5!

V1V5V2, ~6!

Ni1V25NV, ~7!

I 1V50. ~8!

In the above equations NS , Ni , NV, I, V, I 2 , and V2

refer to substitutional nitrogen, interstitial nitrogen, nitrogenvacancy pairs, self-interstitials, vacancies, di-interstitials, anddivacancies, respectively. The surface flux is determined ac-cording to the expression

JS52krCX50, ~9!

whereJS corresponds to the flux of the mobile species at theinterface,kr is the surface recombination velocity of the mo-bile species, andCX50 corresponds to the concentration ofthe mobile species at the interface. TheI and V profiles havebeen obtained through aUT-MARLOWE4 simulation where thesimulation implant energies were scaled according to the bo-ron to nitrogen mass ratio.

The process of determining the reaction rates will bediscussed generally for any dopant–defect pair. This is in-tended only as a brief review and the reader is invited to readthe seminal paper by Fahey and Plummer5 for an extensivereview on the subject. This formalism can then be applied toany of the reactions mentioned above.

Let us consider a reaction of the form

A1X5AX. ~10!

In Eq. ~10!, A refers to a substitutional dopant atom,Xrefers to a point defect~I or V!, andAX refers to a dopant–defect pair. Assuming local equilibrium, the concentration ofthe dopant–defect pairCAX can be written as

CAX5~kf /kr !CACx . ~11!

In Eq. ~11!, kf corresponds to the forward reaction rate,kr corresponds to the reverse reaction rate, andCA and CX

correspond to the concentrations of the substitutional dopantand point defect, respectively.

In thermodynamic equilibrium,CAX can also be writtenas

CAX5uAX exp~2SAX /k!~CACX /CL!exp~EbAX/kT!.

~12!

In Eq. ~12!, uAX refers to the degree of freedom of thedopant defect pair~for example, ifAX is a dopant–vacancypair, then, there are four equivalent positions where a va-cancy can be placed anduAV54!. SAX corresponds to theentropy of reaction~10!. CL is the lattice concentration~531022/cm3 in silicon!, Eb

AX refers to the binding energy ofthe dopant–defect pair,k is the Boltzmann constant, andTrefers to the absolute temperature in °K.

Equating Eqs.~11! and~12!, we can express the ratio ofthe forward and the reverse reaction rates as

~kr /kf !5~CL /uAX!exp~SAX /k!exp~2EbAX/kT!. ~13!

However, in order to determine unique reaction rates, theforward reaction has been determined assuming diffusionlimited kinetics.5 The forward reaction rate can be expressedas:

kf54padX exp~2DE/kT!. ~14!

In Eq. ~14!, a refers to the capture radius,dX refers to thediffusivity of the point defect, andDE refers to the barrierheight. The other symbols have their usual meanings.

Once the reaction rates have been determined, the dop-ant evolution can be determined numerically by solving a setof differential equations. For example, the evolution of Ni inEq. ~1! can be determined as

]CNi /]t5~]/]X!~dNi~]CNi /]X!!1kf 1CNSCI2kr1CNi

2kf 2CNiCV1kr2CNS2kf 7CNiCV21kr7CNV .

~15!

In Eq. ~15!, kf i represents the forward reaction rate ofthe i th reaction andkri represents the reverse reaction rate ofthe i th reaction. For example,kf 1 is the forward reaction rateof reaction~1! andkr1 is the reverse reaction rate of reaction~1!. kf i and kri are determined using the formalism devel-oped in Eqs.~10!–~14!. All the other symbols have theirusual meanings as described above. Similar equations can bewritten for the other reactions. The diffusivity of theimpurity–defect pair can be expressed as

d5d0 exp@2~Ef1Em!/kT#. ~16!

In Eq. ~16!, d refers to the diffusivity of the mobileimpurity–defect specie andEf andEm refer to their forma-tion and migration energies, respectively. To avoid any con-fusion of notation, it must be noted that the diffusivityd asmentioned in Eq.~16! is the diffusivity of the impurity–defect pair and NOT the aggregate diffusivityD of the im-purity itself.

In this model, the binding, migration, and formation en-ergies are determined throughab initio calculations6,7 orthrough experimental observation.8 It must also be men-tioned that in Ref. 6, it has been reported that the exchangebarrier for NV is 4.4 eV. This means that the energy requiredto exchange a nitrogen atom with a vacancy at the firstneighbor site is 4.4 eV. This energy barrier is high. There-fore, the probability of the nitrogen–vacancy pair being mo-bile is very low and hence is considered immobile in thismodel. Thus, the nitrogen interstitials are the only nitrogenrelated mobile species in the set of reactions considered. Theeffect of entropy was neglected~assumed to be 0! as there isno consensus in the literature on the entropy values. Thecapture radiusa was assumed to be one lattice spacing~5.43Å! and the barrier heights were extracted to fit the measureddata.

The parameter set is shown in Table I. The only fittingparameters in this model are the reaction barriers. Althoughthere are seven fitting parameters as shown in Table I, it hasto be appreciated that if every parameter related to nitrogenwere used as a fitting parameter in the model, then onewould end up with about 35 fitting parameters for this model~8 reactions32 reaction directions32 temperatures used to

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simulate the diffusion as shown later1diffusivity values!.Therefore, the number of fitting parameters have been dras-tically reduced. Further, although the values of the reactionbarriers have been used to fit measured data, still, the conceptof the reaction barrier is physical and the same parameter sethas been used at all temperatures. As can be seen from TableI, the values of the reaction barriers are quite low and wellwithin physical realms. NS is used as the reference state inthe energy calculations. Care has also been taken to ensurethat all reaction rates are energetically self-consistent. For anexample, please refer to the Appendix.

The surface site density was fixed at 131015/cm2 in thesimulations.9 The surface bonding configuration of implantednitrogen in silicon is not yet known. Therefore, it was de-cided not to arbitrarily introduce another species at the sur-face. Hence, it was decided to accumulate the nitrogen inter-stitials that reach the surface simply as nitrogen atoms at theSi/SiO2 interface. It is valid to accumulate nitrogen at theSi/SiO2 interface as x-ray photoelectron spectroscopy studieson nitrogen implanted silicon show that nitrogen is either ator very close to the Si/SiO2 interface upon annealing.1 How-ever, the nitrogen atoms that accumulate at the interface haveto occupy the surface sites. Therefore, as the nitrogen accu-mulates at the interface, the number of available sites forfurther nitrogen incorporation decreases. This can be simplyrepresented by the equation below:

dCN interface/dt5ks~10152CN interface!CNi . ~17!

In Eq. ~17!, ks corresponds to the surface recombinationvelocity, 1015 is the surface site density,CN interface corre-sponds to the concentration of nitrogen at the interface, andCNi refers to the concentration of nitrogen interstitials.

B. Qualitative description of the diffusion model

The as-implanted profile was read as Ni . The profileswere then evolved based on a room temperature simulationfor 1 day, which allowed all of the room temperature diffu-sion to settle.10 As explained above, any nitrogen thatreaches the surface is assumed to be trapped there.I 2 and V2

serve as the storage clusters that storeI and V.11 Since theas-implanted profiles were read as Ni , the Ni at room tem-perature is quickly converted to NS and NV through reac-tions ~2! and ~3!. At diffusion temperatures, the NV thusformed is then converted to NS through reaction~4! and theNS to Ni through the kickout reaction, reaction~1!. The self-interstitials released fromI 2 provide the necessary self-interstitials for reactions~1! and~4! to take place at diffusion

temperatures. The reaction rates were such that they pro-duced a nitrogen interstitial gradient toward the surface overthe depth range of the total nitrogen profile. Since nitrogeninterstitials are the only nitrogen related mobile species inthe set of reactions considered, the total nitrogen profileshifts toward the surface with time.

One could argue that varying the surface recombinationvelocity of the self-interstitials or nitrogen at the surface canalso produce a gradient toward the surface. However, vary-ing the surface recombination velocity varies the gradient ofthe self-interstitials and/or the nitrogen interstitials only overthe first 5–10 nm of the total nitrogen profile. Hence, asimple change in this surface recombination velocity at theSi/SiO2 interface is not enough to produce the necessary gra-dients of the nitrogen interstitials that are necessary to movethe nitrogen toward the surface as observed through second-ary ion mass spectrometry. The reaction rates have to bemanipulated such that the gradients of the nitrogen intersti-tials toward the surface are produced over the depth of thetotal nitrogen profile.

As shown in Fig. 1, good fits were obtained at 750 °C.The model predicts that about 90% of the implanted nitrogendose is at the interface after a 750 °C, 120 min anneal. Basedon the calculations at 750 °C, there were no adjustable pa-rameters to account for the temperature dependence at650 °C. That is, once the parameter set was decided based onthe 750 °C simulations, they were not changed subsequentlyfor the 650 °C anneal simulations. The simulations were per-formed of the measured data at 650 °C. Good fits were ob-tained as shown in Fig. 2.

IV. OXIDATION MODEL

A. Background

In order to predict the oxide thickness as a function oftime, the Deal–Grove equation has been historically used.This equation is shown below:

@X2/B#1@X/~B/A!#5t1t, ~18!

TABLE I. Parameter set used in the diffusion model.

Reaction Binding energy~eV! Reaction barrier~eV!

Ns1I 5Ni 3.5 ~ab initio! 0.55Ni1V5NS 3.1 ~ab initio! 0.07Ns1V5NV 1.73 ~ab initio! 0NV1I 5Ns 4.87 ~ab initio! 0.202I 1I 5I 2 2.38 ~ab initio! 0.18V1V5V2 3.5 ~deep level transient spectroscopy! 0.06Ni1V25NV 1.33 ~ab initio! 1.25

FIG. 1. SIMS vs simulations comparisons at 750 °C for 531013 N21 , 40

keV.

1896 J. Appl. Phys., Vol. 91, No. 4, 15 February 2002 Adam et al.

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In Eq. ~18!, X represents the oxide thickness andt representsthe time.t is a parameter that has the units of time and takesinto account the initial oxide thickness~native oxide!. In Eq.~18!, B is associated with the quadratic term and is hencecalled the parabolic rate constant. Similarly, (B/A) is asso-ciated with the linear term and is hence called the linear rateconstant. When the oxide thickness is small, the linear reac-tion rate (B/A) term is dominant and as the thickness in-creases, the parabolic rate constant term gradually takes over.

As device dimensions were scaled, the gate oxide thick-ness also decreased. It was found that the linear regime of theDeal–Grove model12 @characterized by theB/A term in Eq.~18!# underpredicted the oxide thickness when the oxidethickness was less than about 100 Å. In order to model thethin oxide regime accurately~i.e., when the oxide thicknessis ,100 Å!, Massoud and Plummer13 proposed a model forthe initial rapid oxidation. The heart of this model can bebriefly summarized by the following equation:

X5X01E a0 exp~2DE/kT!exp~2X/L !dt. ~19!

In Eq. ~19!, X0 corresponds to the thickness as calculatedfrom the Deal–Grove model characterized by Eq.~18!above,a0 andDE represent an Arrhenius relationship that istemperature dependent, andL represents a decay length.a0

has units of s21. The limits of the integral in Eq.~19! arefrom t50 to the time under consideration. As can be seenfrom Eq. ~19!, as the oxide thicknessX increases, the effectof the second term in Eq.~19! reduces, and the Deal–Grovemodel gradually takes over. It must also be mentioned that itis generally believed that molecular oxygen is the primaryoxidant species in the Deal–Grove regime of the oxidegrowth while atomic oxygen is the primary oxidant speciesin the initial rapid oxidation regime.

B. Theory

A semiempirical model that couples the diffusion modelhas been developed to predict the oxide thickness. The oxide

thickness measured~explained in detail in Sec. IV C! fall inthe thin oxide regime and we model the initial rapid oxida-tion present with the Massoud model as described in Sec.IV A. The value of the parameters used in the Massoudmodel are as shown in Table II. The model developed foroxidation includes two different factors for the reduction inoxidation rate due to nitrogen. The first applies only to theregularB/A term in the Deal–Grove equation, and the sec-ond applies to the rapid oxidation effect. This is quite rea-sonable, since the rapid oxidation is generally thought tooccur through the diffusion of atomic oxygen as mentionedabove. Since a different species is involved in accounting forthe rapid oxidation, it is quite possible that it can be effecteddifferently by nitrogen at the surface. An empirical relation-ship has been developed that predicts the oxide thickness asa function of implanted nitrogen dose. Based on the amountof nitrogen incorporated at the interface using the diffusionmodel explained in Sec. III, the surface reaction rate of theoxidant kS and the oxide growth velocity have been scaled~reduced! continuously with time according to Eq.~20! in theDeal–Grove parameter regime. In the Massoud parameterregime, the oxide growth velocity has been scaled accordingto Eq. ~21!. Equations~20! and ~21! are shown below:

B/A reduction factor:12~dose~Ninterface!/131015!, ~20!

Massoud reduction factor:

@b/~b1~dose~Ninterface!/131015!!#m. ~21!

In Eqs.~20! and~21!, b andm are fitting parameters. Thefactor of 131015 corresponds to the surface site density anddose(Ninterface) corresponds to the nitrogen dose incorporatedat the interface through the diffusion model described in Sec.III. From Eq. ~20! and ~21!, we can see that as the nitrogeninterfacial dose increases, the growth velocity and the sur-face reaction rate reduces. As described in the diffusionmodel in Sec. III, as nitrogen accumulates at the interface, itfills up available lattice sites at the surface. Therefore, thenumber of available surface sites for the oxidant to attachitself to reduces and hence the oxide thickness decreases.Since the accumulation of nitrogen at the interface is limitedto the surface site density (131015/cm2) in the diffusionmodel, Eq.~20! has a lower limit of zero. Furthermore, Eq.~20! scales linearly with dose (Ninterface). This adds furtherstrength to the model because Eq.~20! impacts the lineargrowth regime of the Deal–Grove equation. Figure 3 showsthe plots of Eq.~20! and ~21! with dose. The best fits wereobtained withb50.25 andm51.8. The same values ofb andm were used for all the oxide simulation cases that will bediscussed in Sec. IV C.

FIG. 2. SIMS vs simulations comparisons at 650 °C for 531013 N21 , 40

keV.

TABLE II. Parameter set used in the oxidation model.

Parameter Value

a0 0.35/60 s21

DE 1.51 eVL 13 Å

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C. Simulation predictions and comparison toexperiment

Simulations have been performed and the predictionscompared to measured oxide thickness. The measured oxidethickness has been reported in Ref. 14 and we have alsomeasured some data at 900 °C. Figure 4 shows the oxidethickness as a function of furnace oxidation time at 900 °C.We obtained some of the data in Fig. 4 and the rest wasobtained from Ref. 14. These are the conditions that wereprimarily used to develop the parameters in the model. Allfits are within a 5 Åmeasurement error or about 15%, which-ever is higher. Figure 5 shows the oxide thickness as a func-tion of nitrogen implant dose at 900 °C for three differenttimes. Figure 6 shows the fits of the oxide thickness as afunction of nitrogen implant dose data at 900 °C as publishedin Ref. 14. Accurate predictions of the oxide thickness areobtained in both figures.

Figure 7 shows the oxide thickness as function of im-planted nitrogen dose at 800 °C for three different oxidationtimes. Figure 8 shows the oxide thickness as a function of

rapid thermal oxidation~RTO! time for two different im-planted nitrogen doses at 1050 °C. Good predictions havebeen obtained in both figures. Therefore, we can concludethat the oxidation model predicts the oxide thickness over areasonably wide range of temperature-from 800 °C furnaceanneals to 1050 °C RTO anneals. In addition, the fact that thesame diffusion model can be used effectively to predict theoxide thickness for various doses shows that the diffusionmodel is quite robust.

It may be beneficial to point out that the oxidation modelas presented in this article, consists only of two equationsand two fitting parameters. One equation@Eq. ~19!# decreasesthe linear growth rate of the Deal–Grove equation in a linearfashion while the other equation@Eq. ~20!# moderates thegrowth rate of the initial rapid oxidation. Using just thesetwo equations and two fitting parameters accurate predictionsof oxide thickness data from different sources have been ob-tained in the wide temperature range of 800 °C furnace an-

FIG. 3. Plot of oxide growth reduction withI /I N1 dose.

FIG. 4. Plot of oxide thickness vs time at 900 °C forI /I N1 doses. Wemeasured some of the data and the rest were obtained from Ref. 14.

FIG. 5. Oxide thickness vs implanted nitrogen dose at different times at900 °C.

FIG. 6. Oxide thickness vs implanted nitrogen dose at different oxidationtimes at 900 °C. The oxide thickness data are as published in Ref. 14.

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neals to 1050 °C RTO. Hence this oxidation model is verysimple and therefore it is envisaged that it could be veryusable in integrating into process simulators to model retar-dation in oxide growth subsequent to nitrogen implantation.

V. CONCLUSIONS

A model for nitrogen diffusion has been presented. Thismodel is compared to the experiment and is based onabinitio calculations. A set of reactions has been manipulatedsuch that they produce a nitrogen interstitial gradient towardthe surface. Since nitrogen interstitials are the only nitrogenrelated mobile species in this diffusion model, the entire ni-trogen profile shifts toward the surface with time as observedexperimentally. In addition, the diffusion model is coupled toa model for oxide growth reduction. This model is developedusing two growth reduction factors. One factor impacts thelinear growth regime of the Deal–Grove equation while theother modifies the oxide growth velocity in the initial rapidoxidation regime. The reduction is based on the amount of

nitrogen that diffuses to the surface as predicted by the dif-fusion model. This model offers accurate predictions in awide temperature range of 800 °C furnace anneals to 1050 °Crapid thermal oxidation anneals using the same set of param-eters. This integrated diffusion–oxidation model will allowprediction and optimization of multithickness gate oxide pro-cess using nitrogen diffusion.

APPENDIX

According to thermodynamic theory, the energy gainedor lost by the system should depend only on the initial andfinal states of the system and not upon the path taken fromthe initial to the final state. This concept can be used todetermine the energetics of the reactions that are not withrespect to the reference state NS . This will be illustrated withan example below:

step 1 Ns1V1I 01313.656.6 eV,

step 2 Ni1V 6.623.553.1 eV,

step 3 Ns 0.

In the above reaction steps, 3 and 3.6 eV correspond to theformation energies of the vacancy and the self-interstitial,respectively, as built into the Florida Object Oriented ProcessSimulator. The value of 3.5 eV corresponds to the bindingenergy of the nitrogen interstitial as obtained throughab ini-tio calculations. To retrace steps 1 and 2, when the nitrogenatom is at a substitutional site and the interstitial and vacancyare separated from each other, the total energy of the systemis given by the sum of the individual formation energies ofthe respective species. This amounts to 6.6 eV. In step 2, thenitrogen substitutional atom and the self-interstitial react toform the nitrogen interstitial Ni . Therefore, in this step, thesystem energy will decrease by an amount equal to the bind-ing energy of the nitrogen interstitial. Hence the system en-ergy at the end of step 2 will be the system energy at the endof step 1 minus the binding energy of the nitrogen interstitial.This amounts to 3.1 eV. In step 3, Ni and V recombine toform NS . Since NS is the reference state of the system, thesystem energy at the end of step 3 will be 0. Therefore, inproceeding from step 2 to step 3 the system gains 3.1 eV.Hence, the binding energy for the Frank–Turnbull reactionNi1V5NS is 3.1 eV. The above discussion is an examplefor how the binding energies are determined for reactionsthat are not with respect to the reference state. Similar rea-soning can be used for the other reactions in the system thatare not with respect to the reference state. Such a methodol-ogy ensures that the system is closed in itself, thereby con-forming to the principles of thermodynamics.

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4UT-MARLOWE, version 5.0, Manual, University of Texas at Austin.5P. M. Fahey and J. D. Plummer, Rev. Mod. Phys.61, 289 ~1989!.6J. S. Nelson, P. A. Schultz, and A. F. Wright, Appl. Phys. Lett.73, 247~1998!.

FIG. 7. Oxide thickness vs implanted nitrogen dose at different times at800 °C. The oxide thickness data are as published in Ref. 14.

FIG. 8. Oxide thickness vs RTO time for different implanted nitrogen dosesat 1050 °C. The oxide thickness data is as in Ref. 14.

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7P. A. Schultz~private communication!.8Venezia, MRS Meeting, April 1999.9M. E. Law and S. M. Cea, Comput. Mater. Sci.12, 289 ~1998!.

10S. Coffa and S. Libertino, Appl. Phys. Lett.73, 3369~1998!.11Florida Object Oriented Process Simulator~FLOOPS! manual, 2000 re-

lease and references therein.~Manual can be obtained from the website:www.swamp.tec.ufl.edu!

12B. E. Deal and A. S. Grove, J. Appl. Phys.36, 3770~1965!.13H. Z. Massoud and J. D. Plummer, J. Appl. Phys.62, 341 ~1987!.14C. T. Liu, et al., Tech. Dig. - Int. Electron Devices Meet.98, 592 ~1998!.

1900 J. Appl. Phys., Vol. 91, No. 4, 15 February 2002 Adam et al.

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