Analysis and modeling of arsenic activation and deactivation in silicon

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Analysis and modeling of arsenic activation and deactivation in silicon
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Includes bibliographical references (leaves 126-133).
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ANALYSIS AND MODELING OF ARSENIC ACTIVATION
AND DEACTIVATION IN SILICON














By


HACI OMER DOKUMACI

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997














ACKNOWLEDGEMENTS


I would like to thank my advisor, Dr. Mark E. Law, for his support, guidance, and patience throughout my studies in the University of Florida. His great sense of humor, wide knowledge of processing physics, and excellent teaching abilities make him an ideal professor to work with. I would like to thank Drs. Gijs Bosman, Jerry G. Fossum, and Robert M. Fox for their guidance as members of my doctoral committee. I am grateful to Dr. Kevin S. Jones for coming up with several valuable ideas during the course of this work.

I am very grateful to Viswanath Krishnamoorthy, Sushil Bharatan, Jinning Liu, and Brian Beaudet for their help in the TEM work. I would like to thank James Chamblee and Steve Schein for helping me in my experiments, and Mary Turner for her extensive administrative help.

I was very fortunate to share my time at the university with several nice people: Srinath Krishnan, Chih-Chuan Lin, Jonathan Brodsky, Samir Chaudhry, Stephen Cea, Ming-Yeh Chuang, Susan Earles, David Zweidinger, Doug Weiser, and Glenn Workmann. I was lucky to have two very friendly office-mates: Ahmed Ejaz Nadeem and Hernan Rueda.

This dissertation would have been impossible without the unconditional and unending support of my parents, Osman Dokumaci and Esin Dokumaci, and my sister, Merva Dokumaci. I am very grateful to have been blessed with such a good family.



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Finally, I would like to thank all my friends in Gainesville for making this period of my life an enjoyable and meaningful one.


















































111















TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ............................................................... ii

A B ST R A C T ...................................................................................... ..................... vi

CHAPTERS

I INTRODUCTION ........................................ ......................... 1

1.1 Dopant Activation/Deactivation .......................................................... 3
1.1.1 Electrically Active Dopant Concentration .............................................5
1.1.2 Dopant D iffusion .......................................... ............ ................. 9
1.1.3 Point Defect Injection ......................................... .............. 12
1.1.4 D islocation Loops .................................... ........................................ 16
1.2 Organization........................................................17

H TRANSMISSION ELECTRON MICROSCOPY ANALYSIS OF
EXTENDED DEFECTS IN HEAVILY ARSENIC DOPED, LASER,
AND THERMALLY ANNEALED LAYERS IN SILICON ............................... 20

2.1 Introduction ...............................................................20
2.2 Experim ent ............................................................ 22
2.3 TE M R esults ...................................................................................... 24
2.4 C onclusions ............................................................................................... 33

III HIGH DOSE ARSENIC IMPLANTATION INDUCED TRANSIENT
ENHANCED DIFFUSION............................................................................. 34
3.1 Introduction ................................................................. ....................... 34
3.2 Experim ent ....................................................................................... 36
3.3 Results and Discussion ....................................................... ............... 37
3.4 C onclusions............................................................. ............................ 43

IV INVESTIGATION OF VACANCY POPULATION DURING ARSENIC
ACTIVATION IN SILICON ........................................................ ................ 45
4.1 Experim ental D etails....................................................... .................... 46
4.2 Results and Discussion ...................................................... ............... 48
4.3 Vacancy Population in the Partial Absence of Dislocation Loops ........ 55 4.4 Conclusions......................................................... ............................ 58





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V A KINETIC MODEL FOR ARSENIC DEACTIVATION......................... 60

5.1 The Physical Structure of Inactive Arsenic ..................................... 60
5.2 Previous Models for Inactive Arsenic................................. ........ 62
5.3 A Kinetic Model for Extended Defects ............................................ 65
5.4 Sim ulation Results ............................................................................. 72
5.5 Conclusions........................................................79

VI A COMPARISON OF VARIOUS NUMERICAL METHODS FOR
THE SOLUTION OF THE RATE EQUATIONS IN EXTENDED
DEFECT SIM ULATION............................................................................ 80

6.1 Introduction ..............................................................80
6.2 Rediscretization................................................ 83
6.2.1 Continuous Form for the Rate Equations.............................. ..... 83
6.2.2 Linear Discretization........................................... 87
6.2.3 Logarithmic Discretization ........................................ .......... 88
6.3 Interpolation ............................................................. 90
6.4 Comparison of Numerical Methods ...................................... ....... 92
6.4.1 Comparison of Size Distributions ....................................... ..... 94
6.4.2 Comparison of Active Concentrations ...................................... 99
6.5 Conclusions ........................................ 102

VII AN ARSENIC DEACTIVATION MODEL INCLUDING THE
INTERACTION OF ARSENIC DEACTIVATION WITH
INTERSTITIALS AND DISLOCATION LOOPS ........................................ 104

7.1 Introduction ........................................ 104
7.2 Model for the Inactive Arsenic-Vacancy Complexes .......................... 106
7.3 Model for Dislocation Loops ..................................... 109
7.4 Point Defect Continuity Equations .............................................. 111
7.5 Simulation Parameters ..................................... 112
7.6 Comparison with Experiments ............................ 115
7.7 Conclusions.............................. 120

VIII CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK....... 122

REFERENCES ........................................ 126
BIOGRAPHICAL SKETCH ..................................... 134














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

ANALYSIS AND MODELING OF ARSENIC ACTIVATION AND DEACTIVATION IN SILICON

By

HACI OMER DOKUMACI

May 1997

Chairman: Dr. Mark E. Law
Major Department: Electrical and Computer Engineering

Heavily doped layers are one of the basic components of semiconductor device technology. As device dimensions shrink, higher electrically active dopant concentrations are required to fabricate devices with desirable properties. However, clustering and precipitation limit the obtainable active dopant concentration. Because of its high solubility and low diffusivity, arsenic is the most commonly used dopant for the fabrication of n+ layers in silicon. The electrical activation and deactivation process of arsenic needs to be understood to assess its effects on the electrical properties of devices.

The focus of this thesis is analysis and modeling of phenomena related to electrical activation and deactivation of arsenic. The properties of the dislocation loops formed during arsenic deactivation were investigated with transmission electron microscopy. The loops are confined inside the arsenic layer, suggesting a strong link between loop formation and inactive arsenic. The reduction in buried boron layer diffusivity can be explained by the increase in the number of atoms bound by the loops. Transient enhanced


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diffusion after high dose arsenic implantation was quantified using buried boron layers. Dislocation loops were also studied in these samples. The enhancement in boron diffusivity decreases at higher arsenic doses. The effect of arsenic activation on vacancy population was investigated with antimony marker layers. There is not any observable vacancy injection during arsenic activation in the presence of dislocation loops. Small enhancements in antimony diffusion were observed after the loops had dissolved.

An arsenic deactivation model that takes into account the size distribution of inactive arsenic structures was developed in FLOOPS. The model successfully reproduces the initial rapid deactivation of arsenic and the slow deactivation after the first few minutes. Various numerical methods were evaluated for the numerical solution of the rate equations that describe the size evolution of extended defects. The logarithmic rediscretization method was found to be the most accurate and stable technique for reducing the number of rate equations. The inclusion of the interactions between the arsenic-vacancy complexes, interstitials and dislocation loops led to a more general and physical arsenic deactivation model which can account for several experimental trends.





















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CHAPTER I
INTRODUCTION


Heavily doped layers are one of the basic components of semiconductor device technology. As semiconductor devices are scaled down to sub-micron dimensions, the design and the development of these regions become increasingly complex. Figure 1.1 illustrates various heavily doped layers in a typical BiCMOS process. The source/drain regions of the MOS transistors are doped with a high concentration of dopants in order to obtain a low sheet resistance. The buried layers below the MOS devices are used to minimize latch-up. For the bipolar transistor, heavy doping is utilized in the emitter to increase the current gain. The heavily doped extrinsic base and buried collector regions reduce the parasitic resistances associated with the bipolar transistor.

As device dimensions shrink, the absolute value of the threshold voltage of short-channel MOS transistors decreases. One way to reduce this effect is to fabricate shallower source/drain junctions. The resulting increase in the source/ drain sheet resistance should be countered by increasing the electrically active dopant concentration. As the device dimension is reduced, the metal/silicon contact resistance also increases due to the reduction in the area of the contact. The contact resistance can be decreased by utilizing higher active dopant concentrations. However, the active dopant concentration does not increase indefinitely as the




I






2







NMOS PMOS Bipolar





+ P Well P+ N Well P + n+

N Well
P
p n n n






Figure 1.1 A cross-section of the active layers in a typical BiCMOS process.


dopant concentration is increased, because it is limited by the precipitation and clustering processes. These processes determine the electrical activation/ deactivation behavior of dopants. They need to be understood in order to obtain lower sheet and contact resistances and to assess their effects on other electrical properties of the devices.

Arsenic is the most commonly used dopant in the fabrication of n-type layers because of its high electrical activation and low diffusivity, making it a suitable candidate for shallow junction technology. In this chapter, a review of literature on dopant activation/deactivation in silicon will be provided with an emphasis on arsenic. Some of the phenomena related to activation/deactivation and their effects on silicon devices will be discussed. The reasons of why a further understanding








and modeling of arsenic activation/deactivation processes is needed will be explained. Finally, the organization of this dissertation will be given.



1.1 Dopant Activation/Deactivation



Semiconductor technology relies on the ability to fabricate two types of electrically different layers: n-type and p-type. These layers are obtained by doping the semiconductor with electrically active donor or acceptor atoms. An electrically active dopant atom provides a free carrier to the conduction or valence band. When a dopant atom occupies a substitutional site, it creates an energy level that is very close to one of the bands, making it electrically active.

There are several ways of introducing dopant atoms into the semiconductor lattice. Ion-implantation is the most widely used technology for this purpose because of its excellent controllability and reproducibility which are required by today's integrated circuit technology.

During ion-implantation into silicon, the dopant atoms are accelerated to a certain energy and targeted to the silicon substrate. They penetrate through the surface, colliding with many host atoms before losing all their energy and coming to rest. These collisions disturb the crystalline nature of silicon and create disorder in the lattice in the form of point defects and amorphous zones. Most of the implanted dopant atoms do not occupy substitutional sites in such a disordered state. In order to transfer the dopants to substitutional sites and activate them, the substrate is subjected to a temperature treatment at several hundreds degrees Celsius.





4










o





1020




700 800 900 1000 1100
T (C)

Figure 1.2 The temperature dependence of the equilibrium active concentration of arsenic [1].


The electrically active dopant concentration in high concentration layers is determined by the activation/deactivation processes during thermal anneal. Generally speaking, it is a function of the anneal temperature and time, the dopant, the chemical dopant concentration, and the state of the surface layer just after the ion-implantation (amorphous or non-amorphous). Under equilibrium conditions, the active concentration of dopant atoms in silicon, such as arsenic and boron, has been found to exhibit an Arrhenius dependence on temperature. For example, Figure 1.2 shows the equilibrium active concentration of arsenic as a function of temperature [1]. Today's integrated circuit processing conditions usually keep the silicon-dopant system away from equilibrium. Therefore, it is not simply possible to accurately model the active dopant concentration just as a function of temperature.





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The following sections will describe various phenomena which are related to dopant activation/deactivation in silicon.



1.1.1 Electrically Active Dopant Concentration



When the ion-implantation damage exceeds a certain level, an amorphous layer is formed. Dopants in amorphous layers can be electrically activated up to a concentration many times higher than their equilibrium activation levels. During thermal annealing, arsenic has been observed to activate up to 5x1020 cm-3 at 5600C [2] even though the equilibrium active concentration of arsenic at 7000C is around lx1020 cm-3 (Figure 1.2). The carrier concentration increases until the amorphous layer crystallizes completely. Further annealing causes arsenic to deactivate rapidly [2,3]. Non-equilibrium activation is also observed for boron [4,5] and antimony [6] in amorphized layers. In the case of boron, a preamorphization step is done since impractical high doses of boron are necessary to get an amorphous layer during boron implantation.

The higher activation effect is totally reversed when the sample is not amorphized by the implant. There are several reports indicating that active boron concentration remains much below its equilibrium activation level after ionimplantation and a low temperature anneal [7-9]. As shown in Figure 1.3, the tail of the boron profile undergoes an enhanced diffusion due to ion-implant damage. The peak is immobile, because boron at the peak region is either in the form of clusters or precipitates which can not diffuse through the lattice. The suppressed activation





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Equilibrium active concentration Immobile peak


O Implanted profile
Annealed
profile Active concentration profile









Depth

Figure 1.3 Representative boron profiles after ion-implantation and diffusion. The profile is not completely electrically active although the peak concentration is much below the equilibrium active concentration.


of boron at the peak has been attributed to the large supersaturation of interstitials just after ion-implantation [9-11]. If boron atoms are assumed to form clusters through a reaction with silicon interstitials, a high supersaturation of interstitials will increase the rate of formation of boron clusters, thus decreasing the electrically active boron concentration. Precipitation has been another explanation for boron deactivation in implanted samples [7].

Arsenic and phosphorus have been shown to exhibit a phenomenon called reverse annealing [12-14] (Figure 1.4). After being annealed at 6500C, high concentration arsenic samples have been subsequently annealed at 7500C [12].





7





6500C 7500C














Equilibrium active concentration


Time


Figure 1.4 Illustration showing the reverse annealing phenomenon in arsenic and phosphorus doped samples.


Although the carrier concentration at the end of the 6500C anneal is above the equilibrium active concentration of arsenic at 7500C, it still increases at the beginning of the 7500C anneal to a maximum. Normally, one would expect a faster rate of deactivation upon increasing the temperature, since the electrically active concentration is above the equilibrium value at that temperature. This phenomenon has been claimed to be a solid proof that the electrically inactive dopant is in the form of precipitates rather than a cluster.

In terms of activation/deactivation, arsenic has been the most studied dopant in silicon because of its peculiar properties. Various studies [15-17] have reported incomplete arsenic deactivation for concentrations below the equilibrium activation





8


level. The deactivation is observed both at the tail and the peak of the profiles. It was suggested that inactive arsenic is in the form of arsenic-point defect pairs [15]. Another explanation has recently been given: Although the electrically active arsenic concentration is below its equilibrium value and no deactivation is expected, significant deactivation will still take place if the free energy of formation of very small inactive arsenic clusters is small enough [18]. The electrically active arsenic concentration is also dependent on the rate of ramp-down at the end of a thermal anneal. A slower ramp-down rate has been observed to increase the arsenic sheet resistance by 30% [19]. In a BiCMOS process, it was demonstrated that the sheet resistance of arsenic doped polysilicon increases upon doubling the ramp-down time [20]. It was also shown in the same study that a subsequent 15 second RTA step significantly reduces n+ silicon and polysilicon sheet resistances and the poly/mono silicon contact resistance by dissolving the precipitates formed during ramp-down.

Laser annealing is another way of activating dopants in semiconductors. Although it is not widely used by industry, it has proved to be a very useful research tool. During laser annealing, a laser beam is directed to the silicon surface, melting the surface layer to a depth which is dependent on the incident laser power. The melt layer regrows very rapidly, incorporating nearly all of the dopant atoms onto substitutional sites. Therefore, up to a certain concentration, dopants can be totally activated with laser annealing. For example, phosphorus has been shown to activate up to a concentration of 5x1021 cm-3 and arsenic up to 3x1021 cm-3 [21]. Laser annealing also wipes out implantation damage and prevents the formation of extended defects like dislocation loops and stacking faults.





9


Because of this macroscopically clean state of silicon just after laser annealing, laser annealing and subsequent thermal annealing have been used numerous times to investigate the deactivation kinetics of dopants [3,5,6,12,13,2123]. Subsequent thermal annealing deactivates dopants at temperatures as low as 3000C [12]. Since laser annealing activates all of the dopants, there is no uncertainty concerning the initial activation conditions. In one of the studies, boxshaped profiles have been created by repeated laser pulse annealing [23]. This allows one to interpret the electrical measurements as characteristic of a single doping concentration. This is not possible with simple thermal annealing since the dopant concentration is a function of depth in that case.



1.1.2 Dopant Diffusion



Substitutional dopants in silicon, such as arsenic, phosphorus and boron, are believed to diffuse via interactions with point defects: interstitials and vacancies [24]. These point defects can exist in various charge states. A chemical reaction between the substitutional dopant and point defects can be envisioned that converts the immobile substitutional dopant to a mobile dopant-defect pair. For a donor atom, this reaction can be written as:




A+ + Xc A+Xc 1-1





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where A+ represents the substitutional donor atom, Xc represents a point defect with charge c. Since the point defects are charged, their concentration will depend on free carrier concentrations. For example, the concentration of negatively charged point defects with respect to their intrinsic values is given by [24]:




S n- I 1-2 (Cx-) ni




where Cx-, denotes the concentration of X-c defects, (Cx-c)' their concentration in intrinsic silicon, n the electron and ni the intrinsic electron concentration. This equation has been written under the assumptions that the environment is inert and the reactions between point defects and electrons are so fast that they may be considered in chemical equilibrium.

The diffusivity of a dopant is proportional to the concentration of the mobile dopant-defect pairs. Since the number of dopant-defect pairs is proportional to the number of point defects and the number of point defects depend on the free carrier concentration, the diffusivity depends on the free carrier concentration under extrinsic conditions. Actually, if the dopant-defect reactions and the defects are in equilibrium, the diffusivity for a donor atom can be written as:




D = Do+D (- +D (ni)... 1-3





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where DO is the diffusivity due to uncharged point defects, D_ due to negatively charged defects, etc. At high concentrations and under the condition of charge neutrality, for a single dopant:



n = CA 1-4




where C A is the active dopant concentration. So, ultimately the diffusivity depends on the active dopant concentration under extrinsic conditions.

In so-called isoconcentration studies, the dopant under study is diffused in a region which is homogeneously doped with either another dopant or an isotope of the same dopant at a much higher concentration. This way, the carrier concentration is controlled independently of the diffusing species. The isoconcentration experiments have shown that arsenic [25], phosphorus [26], and antimony [27,28] diffusion are enhanced with increasing electron concentration. Also, boron [29,30] diffusivity is observed to increase with hole concentration.

Recent isoconcentration study by Larsen et al. [31] has revealed an even stronger dependence of arsenic and antimony diffusion on donor concentration than that expressed by Equations 1-3 and 1-4. The diffusivities of arsenic and antimony have been found to be proportional to the third to fifth power of the background phosphorus donor concentration for phosphorus concentrations greater than 2x1020 cm-3. At the highest donor concentrations, the diffusion coefficients are identical for arsenic, antimony and tin, although these dopants have different diffusivities at lower electron concentrations.





12


All the results mentioned above indicate that dopant diffusivities get increasingly sensitive to the amount of electrically active dopant concentration as the concentration is increased. Therefore, high concentration dopant diffusion is tightly coupled to the dopant activation/deactivation phenomena which determine the electrically active dopant concentration. From a modeling point of view, even small errors in the prediction of the electrically active dopant concentrations can result in a big error in dopant diffusivities and make the diffusion simulations unreliable. This point may become more and more important in the future as the IC technology requires higher carrier concentrations and therefore higher temperatures in the sub-micron regime.



1.1.3 Point Defect Injection



A recent study by Rousseau et al. [32] confirmed the interaction between point defects and dopant activation/deactivation (Figure 1.5). Boron buried layers were used as markers of interstitial supersaturation in the bulk. These layers were obtained by first growing a boron doped silicon epi-layer on silicon and then growing an undoped silicon layer on top of it. Arsenic was implanted and the wafers were laser annealed. The laser annealing achieved full electrical activation of arsenic and a flat arsenic concentration, and destroyed the implant damage by melting the surface layer. For control purposes, another wafer was doped with germanium in exactly the same way. Germanium is similar to arsenic in size and mass. Next, the wafers were annealed at 7500C in an inert ambient for 15s and 2 h.





13






- - Electrically active arsenic, as-lased S- - - Electrically active arsenic, after 7500C Boron as-lased


Interstitial injection Boron after 7500C a Arsenic/
S/ Boron




Depth


Figure 1.5 Representative arsenic and boron profiles in Rousseau et al.'s experiment. Boron diffusivity is enhanced up to a factor of 460 when arsenic is deactivated.



Normally, these anneals will not cause any significant diffusion of boron. But, the diffusivity of the boron buried layer was enhanced by a factor of up to 460 for the 2h anneal. Arsenic deactivation occurred in parallel to the enhancements in boron diffusivity.

Extended X-ray Absorption Fine Structure (EXAFS) measurements [33] and theoretical calculations [34] suggest that arsenic deactivates in the form of arsenicvacancy complexes. Since the equilibrium concentration of vacancies is much smaller than the concentration of inactive arsenic, a huge amount of vacancies should be generated in the bulk. These vacancies have been postulated to come from arsenic deactivation assisted interstitial-vacancy pair generation. The generated





14


vacancies are incorporated in the inactive arsenic-vacancy complexes and the interstitials diffuse towards the surface and the bulk, and enhance boron diffusion.

Another study has shown the importance of this phenomenon in bipolar transistors [35]. After arsenic was implanted to form the emitter of the device, it was activated at 11000C for 10s. The wafer was then cut in two, with one half undergoing a deactivation anneal at 7500C for 2h. The two halves were then annealed at 10000C for 15s so that the final active arsenic concentration in both halves would be equal. There should not be any difference in the electrical characteristics of these two transistors since the inert diffusion of boron or arsenic is negligible at 7500C with respect to 10000C or 11000C. However, the electrical measurements show that the Gummel number is considerably reduced in the transistors that have received the 7500C anneal, because the interstitial injection during arsenic deactivation enhances the diffusion of both arsenic and boron. Therefore, the devices exhibit higher beta, lower output resistance and earlier punchthrough breakdown (Figure 1.6).

Shibayama et al. [36] were the first to observe the diffusivity enhancement underneath a high concentration arsenic layer. In their study, arsenic was diffused into silicon from an arsenosilicate glass at 10000C. The diffusivity of both arsenic and boron were enhanced upon a low temperature anneal between 500-8000C.





15




As & B without the deactivation anneal S- -- As & B with the deactivation anneal


Emitter \ Base


/ I


/ \

/ I


Depth


No deactivation anneal / /




IrB
- - Deactivation afineal / /

/ /











Collector emitter voltage Figure 1.6 Schematic representation of the emitter and base profiles, and electrical characteristics of an npn bipolar transistor with and without a deactivation anneal.





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1.1.4 Dislocation Loops



A dislocation loop is an extra layer of silicon atoms having a disc shape. Dislocation loops form when there is a high supersaturation of silicon interstitials during thermal annealing. The major cause of excess interstitials and dislocation loops is ion-implantation. Ion-implantation induced dislocation loops have been extensively studied [37]. Dislocation loops act as sinks and sources for interstitials and vacancies, affecting point defect concentrations and dopant diffusion as a result [38]. They increase the p-n junction leakage current by gettering metallic impurities along their peripheries. The stress fields that they generate in silicon also affect dopant redistribution [38]. The analysis of dislocation loops is therefore crucial in understanding and interpreting the effect of processing conditions on device electrical characteristics.

Since arsenic deactivation injects a huge amount of excess interstitials, one may expect dislocation loop formation as a result. In fact, several Transmission Electron Microscopy (TEM) observations have revealed dislocation loops in laser and then thermally annealed high concentration arsenic samples [21,22,39-42]. These loops have been found to be composed of silicon atoms [42]. It was suggested in the same work that arsenic clustering injects the excess silicon interstitials needed to form the loops. Nearly all studies have concluded that the number of atoms bound by the loops or any other extended defects is much smaller than the inactive arsenic dose. When an electron beam was used for initial annealing instead of a laser, dislocation loops were still observed after thermal annealing. In Chapter





17


II, the dislocation loops formed after laser and subsequent thermal annealing will be discussed in more detail.

On the other hand, in implanted and thermally annealed high concentration arsenic samples, a band of arsenic-related precipitates and dislocation loops were detected [37,44,45]. These defects lie at the projected range of the implant where arsenic concentration is at its maximum. They form when arsenic concentration exceeds its equilibrium active concentration. The defects that form at the peak of the implanted profile have been classified as type-V defects [37]. It is interesting to note that type-V dislocation loops are observed only in arsenic-implanted samples [37].

Upon high temperature annealing (>9000C), end-of-range (type-II) dislocation loops dissolve much faster in the presence of gallium, phosphorus and arsenic if their concentrations exceed their solid solubility [46]. During enhanced elimination of type-II loops, the precipitates were also observed to be dissolving.



1.2 Organization



The focus of this thesis is analysis and modeling of the phenomena related to the arsenic activation/deactivation process. Chapter II presents the extended defects that form as a result of the interstitial supersaturation during arsenic deactivation. Understanding the mechanism of these defects is important for both defect engineering and an evaluation of their effect on arsenic deactivation induced interstitials and, therefore, dopant diffusion near high concentration arsenic layers. Chapter II reports the results of TEM observations on samples which were doped





18


with different arsenic doses above the equilibrium active concentration and received a laser anneal to activate all of the dopant, followed by thermal annealing. Various characteristics of the observed defects are evaluated and comparisons with published data are made.

Chapter III presents the results of boron enhanced diffusion due to high dose arsenic implantation in silicon. This study is made to investigate the possible effects of arsenic deactivation on implant damage enhanced diffusion. The behavior of both type-V and type-II loops is studied with TEM. Reduction in enhanced diffusion is observed with increasing arsenic dose at three different temperatures. The possible explanations for this reduction are also included in Chapter III.

Chapter IV contains an investigation of the vacancy population during arsenic activation. Since arsenic is believed to deactivate through the formation of arsenic-vacancy complexes, these complexes are expected to dissolve and generate free vacancies during arsenic activation. The vacancy population is monitored using antimony buried layers.

Chapter V presents a general kinetic model for extended defects. This model is applied to arsenic deactivation in silicon. The model calculates the evolution of the arsenic precipitate size density. It reproduces various trends and a good agreement is obtained between the simulations and the experiments.

In Chapter VI, various numerical methods are evaluated for the solution of the rate equations in extended defect simulation. The derivations of these methods are presented. The accuracies of these methods are evaluated using the arsenic deactivation model in Chapter V.





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Chapter VII presents an arsenic deactivation model including dislocation loop formation and the interaction of the interstitials with the inactive arsenic structures. The model shows quantitative agreement with the arsenic deactivation and boron enhancement data. It is also in qualitative agreement with the properties of the loops in the deactivated layer. Finally, Chapter VIII presents the conclusions of this dissertation and recommendations for future experimental and modeling efforts.














CHAPTER II
TRANSMISSION ELECTRON MICROSCOPY ANALYSIS OF EXTENDED DEFECTS IN HEAVILY ARSENIC DOPED, LASER, AND THERMALLY ANNEALED LAYERS IN SILICON


2.1 Introduction



Arsenic is the most commonly used dopant for creating n' layers in silicon, such as the source/drain regions in a MOSFET and the emitter of a bipolar transistor. In order to get high conductivity, arsenic is often incorporated into these layers in excess of its equilibrium active concentration. Subsequent thermal annealing gives rise to inactive arsenic. As mentioned in Chapter I, deactivation of arsenic is accompanied by silicon interstitial injection and the enhancement of the boron layers underneath the arsenic layer [32]. This reaction can be written as:



mAs + nSi A sm Vn + nI 2-1



where As represents a substitutional arsenic atom, AsnV, the inactive arsenic complex, V a vacancy and I an interstitial.The reaction can be modified accordingly if the inactive arsenic complex also includes silicon.

Several transmission electron microscopy (TEM) observations have revealed precipitate-like defects, rod-shaped structures and/or dislocation loops in laser and



20





21


thermally annealed samples which have been doped with arsenic in excess of its equlibrium active level [21,22,39-42]. Lietoila et al. [22] have suggested that rodlike defects may be arsenic precipitates, whereas Armigliato et al. [40] have reported that all defects observed by TEM cannot explain the amount of electrically inactive arsenic in their experiments. No extended defects were detected by Lietoila et al. for a much smaller arsenic dose. Parisini et al. [41] measured the number of atoms bound by various types of interstitial-type defects at different annealing temperatures and confirmed the large discrepancy between the concentration of inactive arsenic and the concentration of atoms in observable defects. They proposed that the extended defects are formed as a result of agglomeration of silicon interstitials which are created during laser annealing. In a later work [42], based on double-crystal x-ray diffractometry (DCD) and extended x-ray absorption fine structure analysis (EXAFS) measurements, they suggested that deactivation of arsenic is the cause of excess interstitials. They also found that the dislocation loops are composed of silicon atoms. When an electron beam was used for annealing instead of a laser, similar defects were observed upon subsequent thermal treatment between 600-9000C [43].

On the other hand, a band of arsenic related precipitates and dislocation loops were detected after solid phase epitaxy of arsenic layers doped in excess of its equilibrium active level [37,44,45]. These defects were shown to lie at a depth corresponding to the projected range of the implant. In addition, half-loop dislocations that are located near the surface were observed to grow during arsenic precipitate dissolution even after 72h at 9000C [37]. Jones et al. found that the





22


elimination of end-of-range damage (category-II dislocation loops) is enhanced in the presence of a high concentration arsenic layer [46]. Both half-loop formation and enhanced elimination of category-II loops occur when arsenic peak concentration exceeds its solid solubility.

Understanding the mechanism of these defects is important for both defect engineering and an evaluation of the effects of arsenic deactivation on point defects and, therefore, dopant diffusion underneath high concentration arsenic layers. This chapter reports the results of TEM observations on samples which were doped with different arsenic doses above the equlibrium active concentration, received a laser anneal to activate all of the dopant and finally thermally annealed.



2.2 Experiment



Arsenic was implanted into <100>, 10 Q-cm p-type silicon substrates at doses ranging from 4xl015 to 3.2x1016 cm-2 with an energy of 35 keV. Completely active, box-shaped profiles were obtained by repeated laser pulse annealing (308 nm XeCl, 35 ns FWHM pulses, silicon melt duration 75 ns). The melt-region thickness of about 200 nm was considerably larger than the depth of the implant; thus any major implant damage in the as-implanted layer was effectively annealed out. The samples then received additional rapid thermal anneals at 700 or 7500C for durations that resulted in no appreciable diffusion. Further experimental details about the preparation of these samples can be found in Rousseau et al. [13] and Luning et al.[23].






23


The defects were studied by TEM using both bright field and weak beam conditions. Both plan-view and cross-section samples were analyzed with a JEOL 200 CX electron microscope operating at 200 keV. All micrographs were taken with a g220 condition.

The quantification of the defects on the TEM pictures were done by counting them and measuring the longest dimension of the defect. The density of the defects can be found from the following expression:



NM2
D NM2 2-2 "p



where D is the areal density of the defects, N is the number of defects in the area of interest, M is the magnification, and Ap is the analyzed area on the picture. The loops are assumed to lie on [111] planes where the picture shows the defects through <100> direction. The areal density of the atoms bound by the loops was found by measuring the largest dimension of the loops and using the following formula:




N Ld d 2-3 bound A p11




where r is the radius of a loop and is equal to half of the largest dimension, and dll1 is the areal density of silicon on the [111] plane. The F3 factor projects the





24


analyzed area onto the [111] plane. So, Equation 2-3 effectively calculates the areal density of the bound atoms on the [111] plane.



2.3 TEM Results



The as-lased samples were completely free of any visible defects. Subsequent thermal annealing revealed a very strong dose dependence of the defect structure in the arsenic doped layer. This phenomenon can be seen in Figure 2.1 to Figure 2.4 which show plan-view TEM micrographs of samples after a thermal treatment at 7500C for 2h. Density and size information about the defects are listed in Table 2.1 along with the inactive arsenic dose.

In contrast to the absence of any extended defects at the lowest dose (4x1015 cm-2), large dislocation loops and rod-like defects are observed upon increasing the dose by just a factor of two. For the next two doses, only dislocation loops are detected. Upon increasing the arsenic dose, the density of the loops increases dramatically. Especially, the 1.6x1016 cm-2 sample exhibits almost fifty times more loops than the 8x1015 cm-2 sample. The concentration of atoms bound by the loops was found to be around 30-50 times smaller than the inactive arsenic concentration and is insufficient to directly account for most of the inactive arsenic.

Figure 2.5 and Figure 2.6 show the plan-view micrographs of the samples which were doped with a dose of 1.6x1016 cm-2 and annealed at 7000C for 15s (short-time) and 100 min (long-time). Very dense fine particles appear after the short-time anneal whereas the long-time sample exhibits a high density of loops.





25

























Figure 2.1 Plan-view TEM micrograph of the sample implanted with 4x1015 cm-2 and annealed at 7500C (bright field).























Figure 2.2 Plan-view TEM micrograph of the sample implanted with 8x015 cm-2 and annealed at 7500C (weak beam dark field).





26

























Figure 2.3 Plan-view TEM micrograph of the sample implanted with 1.6x1016 cm-2 and annealed at 7500C (weak beam dark field).























Figure 2.4 Plan-view TEM micrograph of the sample implanted with 3.2x1016 cm-2 and annealed at 7500C (weak beam dark field).





27
























Figure 2.5 Plan-view TEM micrograph of the sample implanted with 1.6x1016 cm-2 and annealed at 7000C for 15s (weak beam dark field).























Figure 2.6 Plan-view TEM micrograph of the sample implanted with 1.6x 1016 cm-2 and annealed at 7000C for 100 min (weak beam dark field).






28




Table 2.1: Summary of TEM observations made on the samples annealed at 7500C for 120 minutes.

Density of Inactive Surface
Dose Defect Size Density Bound As
Conc.
(cm-2) Type (AO) (cm2) Atoms Dose
(cm3)
(cm-2) (cm2)

4x1015 2.3x1020 None 2.2x1015

Dislocation
200- 1200 3.3x109 1.2x1014 loops

8x1015 4.5x1020 4.9x1015

Rod-like
400- 2800 1.8x109 defects

Dislocation
1.6x1016 9.1x1020 50- 350 1.3x1011 3.5-4.5x1014 1.3x1016 loops

Dislocation
3.2x1016 1.9x1021 100 400 > 2.5x1011 > 6x1014 2.9x1016 loops




Further analysis showed that the defects in the short-time sample are dislocation

loops with a density of around 6x1011 cm-2 and average size of 90 A. Recently,

Rutherford back-scattering (RBS) measurements were made on the same samples

[47]. It was demonstrated that the backscattered angular-scan spectra for silicon

have the same minimum in both the as-lased and short-time samples whereas the

minimum yield of arsenic increases. Furthermore, silicon minimum yield was






29


shown to increase appreciably in the long-time sample, and both arsenic and silicon angular-scan spectra were characteristic of a large degree of dechanneling.

Figure 2.7 to Figure 2.10 show the cross-section views of the samples implanted with doses of 8x1015 and 1.6x1016 cm-2. For the higher dose, the defects lie uniformly in a region from the surface down to a depth of about 180 nm, which coincides well with the junction depth. The uniformity of the defects in the arsenic layer is in contradiction with an earlier TEM work where dislocation loops were observed to lie at the amorphous-crystalline interface for a similar surface concentration [41,42]. Except for the intrusion of rod-like defects to a depth of 230250 nm in the 8x1015 cm-2 sample, the confinement of the defects to the arsenic layer suggests that inactive arsenic complexes reduce the formation energy of the loops.

In a similar study [42], it has been found that the loops are composed of silicon atoms. It has been suggested in the same work that arsenic clustering injects the excess silicon interstitials needed to form the loops. As mentioned before, buried boron layers show enhanced diffusion as a result of arsenic deactivation, suggesting that arsenic deactivation is accompanied by interstitial injection [32]. TEM observations in this study involve exactly the same samples for the 7500C, 2h anneal. The enhancements for these samples are shown in Table 2.2. There is a large increase in boron diffusivity at an arsenic concentration of 4.5x1020 cm-3 and the enhancement keeps decreasing as the dose is increased. The TEM results show that the number of atoms bound by the loops increases with arsenic concentration. Therefore, the loops seem to be responsible for the reduction in boron diffusivity.





30
























Figure 2.7 Cross-section TEM micrograph of the sample implanted with 1.6x1016 cm-2 and annealed at 7000C for 15 s (weak beam dark field).






















Figure 2.8 Cross-section TEM micrograph of the sample implanted with 1.6x1016 cm-2 and annealed at 7000C for 100 min (weak beam dark field).





31

























Figure 2.9 Cross-section TEM micrograph of the sample implanted with 8x1015 cm-2 and annealed at 7500C for 2h (weak beam dark field).























Figure 2.10 Cross-section TEM micrograph of the sample implanted with 1.6x1016 cm-2 and annealed at 7500C for 2h (weak beam dark field).





32




Table 2.2: Enhancement of buried boron layer diffusivity for various arsenic doses [32].

Surface Concentration Inactive As Dose (cm-2) Enhancement
(cm-3) dose (cm-2) 4x1015 2.3x1020 30 2.2x1015 8x1015 4.5x1020 460 4.9x1015 1.6x 1016 9.1x1020 230 1.3x1016 3.2x1016 1.9x1021 110 2.9x1016



The dislocation loop density gets higher as the arsenic dose is increased, although the interstitial supersaturation decreases. From nucleation theory, the loop density is expected to decrease as the interstitial supersaturation gets smaller, which seems to contradict these experimental observations. But, since loop nucleation occurs at the very early stages of the deactivation anneal, the density of the loops is determined by the interstitial supersaturation at the beginning of the anneal. Instead of the time averaged interstitial supersaturation, it is enough that the initial interstitial supersaturation be higher at higher arsenic concentrations for the loops to be denser.

At the lowest arsenic concentration (2.3x1020 cm-3), the enhancement is 15 times smaller than the one at the next higher concentration (4.5x1020 cm-3), although the inactive arsenic doses are comparable. Also, no dislocation loops are observed at the lowest concentration. This suggests that another deactivation mechanism may be dominant in this sample, such as the simultaneous formation of arsenic-interstitial pairs.





33


2.4 Conclusions



The properties of the extended defects that are formed during arsenic deactivation were investigated. The density of the dislocation loops depends on the chemical arsenic concentration. As the arsenic concentration increases, more interstitials are injected during deactivation, and, therefore, a higher density of loops form. The number of atoms bound by the loops is much smaller than the inactive arsenic dose. The loops are confined in the arsenic layer, indicating that the inactive arsenic reduces the formation energy of the loops. Finally, the increase in the number of the atoms bound by the loops explains the reduction in the buried boron layer diffusivity at higher arsenic concentrations.














CHAPTER III
HIGH DOSE ARSENIC IMPLANTATION INDUCED TRANSIENT ENHANCED DIFFUSION


3.1 Introduction



Ion implantation of dopants into silicon introduces damage in the form of interstitials and vacancies. If the damage is above a certain level, an amorphous layer forms. Upon thermal annealing, the amorphous layer grows back into crystalline silicon. If arsenic has been implanted at high doses, end-of-range loops appear below the original amorphous/crystalline (a/c) interface, whereas type-V loops form at the projected range (Figure 3.1). When a dopant layer is present beyond the a/c interface, its diffusion is enhanced because of the ion-implant damage. It has been previously suggested that the damage beyond the a/c interface is the main cause of the enhanced diffusion of dopants after amorphizing implants [48,49].

As mentioned in Chapter II, arsenic deactivation is accompanied by interstitial injection. Buried boron layers have been observed to exhibit enhanced diffusion when arsenic is deactivated at the surface [32]. In Chapter II, the high level of interstitial injection is also confirmed by TEM studies of type-V dislocation loops formed during arsenic deactivation in initially defect-free laser annealed samples [39].



34





35



After ion implantation After annealing





End-of-range
implant O
damage










a/c interface type-V loops end-of-range loops

Figure 3.1 Schematic representation of damage and loop formation in high dose arsenic implanted silicon.



Although arsenic deactivation creates excess interstitials and causes dislocation loop formation. it is not yet clear how much effect it has on the enhanced diffusion caused by high dose arsenic implantation. While excess interstitials should contribute to the implant damage enhanced diffusion, type-V loops may decrease the amount of enhanced diffusion by absorbing the interstitials. In this work [50], arsenic was implanted into silicon at various doses with the same energy. Buried boron layers were used as markers of interstitial supersaturation. Most of the chosen doses give rise to peak arsenic concentrations above the equilibrium active concentration. The density of type-V loops is a very strong function of the arsenic concentration, and the interstitial supersaturation during arsenic deactivation





36


increases abruptly above a certain arsenic concentration. Hence, an abrupt change in the diffusivity of the buried boron layer can be expected if arsenic deactivation has an appreciable effect on damage enhanced diffusion.



3.2 Experiment



The boron buried layers were prepared in Texas Instruments. After depositing 200 A of oxide on <100> silicon to reduce ion channeling, boron was implanted at 10 keV with a dose of 3x1012 cm-2. The oxide was then etched with an HF solution and an approximately 0.6 gtm thick epi layer was grown. Arsenic was implanted at 50 keV at 70 tilt with doses ranging from 4x1014 to 4x1015 cm-2. The peak arsenic concentrations determined from SIMS are shown in Table 3.1 along with the implanted doses. The arsenic concentrations were normalized to the implanted dose.The samples were annealed at three different times temperatures: 7500C 2h, 9000C Imin, and 10500C 15s. It has been previously reported that the damage enhanced diffusion due to interstitial cluster dissolution is complete for boron implants within the anneal times selected for each temperature [7,51 ].

Arsenic and boron profiles were obtained by secondary ion mass spectrometry (SIMS) using an oxygen beam. Since the boron dose was low, the sample was tilted at an angle so that the boron yield was higher during sputtering. The defects were studied by transmission electron microscopy with a JEOL 200 CX electron microscope operating at 200 keV. Plan-view and cross-section samples were analyzed in g220 and gill conditions.





37



Table 3.1: Peak arsenic concentration vs. arsenic implant dose

Dose (cm-2) 4x1014 8x1014 1.6x1015 2.4xl015 4x1015 Peak Concentration (cm-3) 1.1x1020 2x1020 4x1020 6.4x1020 lx1021



3.3 Results and Discussion



The annealed profiles of the boron buried layer are shown in Figure 3.2 for different arsenic doses at 7500C. The doses of the original boron SIMS profiles were normalized to that of the unannealed sample. The damage enhanced diffusion of the boron buried layer is reduced as the arsenic dose is increased. This trend is also observed after the 9000C and 10500C anneals. The reduction in the diffusivity is also supported by the arsenic SIMS profiles. After the 7500C anneal, the highest dose arsenic profile moves 200 A less than the 8x1014 cm-2 profile. The enhancements in the boron diffusivity can be seen in Figure 3.3 to Figure 3.5. The enhancement was found by finding the diffusivity that best matches the experimental profile and dividing it by the reference inert diffusivity which is given by:




D = 0.757ex( 3.46) 3-1




At both 7500C and 9000C, the boron diffusivity drops nearly by a factor of 2 from the lowest to the highest arsenic dose. The cross-sectional TEM (XTEM)






38





1 0 18I ... .. I .. . . .
None
G---- 8x1014 cm-2 E 3-0 2.4x1015 cm-2
0 4x1015 cm-2 C 1017

(D
C)
C
o 1016 1015



10 15 . . . , ,
0.4 0.5 0.6 0.7 0.8 0.9 Depth (ptm)


Figure 3.2 Buried boron profiles for different arsenic doses after the 7500C, 2hr. anneal.


1 8 0 0 . . . . . . . . .. . . . . . . .



1600



1400

w
1200



1000



8 0 0 . . . . ..
Oe+00 le+15 2e+15 3e+15 4e+15
Arsenic Dose (cm-2) Figure 3.3 The enhancement of boron diffusivity as a function of arsenic dose at 7500C.





39


5 5 0 . . . . . . . . . . . . . .

500

C
0 450
E
C
cz 400
-c
w
350

= 300 250

2 0 0 . . . . .., ..
Oe+00 le+15 2e+15 3e+15 4e+15
Arsenic Dose (cm-2) Figure 3.4 The enhancement of boron diffusivity as a function of arsenic dose at 9000C.


3 5 . . . . . . . . . .. . . . .





30





8 25





2 0 .. . .. .
Oe+00 le+15 2e+15 3e+15 4e+15
Arsenic Dose (cm-2) Figure 3.5 The enhancement of boron diffusivity as a function of arsenic dose at 10500C.





40























(a)






















(b)


Figure 3.6 Weak beam-dark field cross-section TEM micro raphs of the sample implanted with an arsenic dose of (a) 1.6x 1015 cm-2, (b) 2.4x 1015 cm- and annealed at 7500C.





41


analysis of the samples reveals that a high density of type-V loops forms when the peak arsenic concentration is 6.4x1020 cm-3 at both 7500C and 9000C. On the other hand, only a very low density of type-V loops form at a peak concentration of 4x1020 cm-3 (Figure 3.6). These results are consistent with the dislocation loop results in Chapter II. The diffusivity data shows that the abrupt appearance of a high density of type-V loops does not have a drastic effect on damage enhanced diffusion. It has been previously suggested that the enhanced diffusion after amorphizing implants is mainly caused by the damage beyond the a/c interface [48,49]. Since type-V loops form at the projected range, they may not absorb an appreciable amount of the interstitials that are beyond the a/c interface and blocked by the end-of-range loops (Figure 3.1). Moreover, XTEM micrographs show that there is not any intrusion of the type-V loops into the end-of-range loop layer even at the highest dose, suggesting a lack of strong interaction between type-V loops and end-of-range damage. Still, type-V loops may be contributing to the decrease in the enhanced diffusivity, although the decrease in the enhanced diffusivity starts before their formation. It is interesting that the buried layer diffusivity does not increase at the arsenic peak concentration of 4x 1020 cm-3 although a great amount of interstitial supersaturation is expected [52] and nearly no type-V loops form at this concentration. The end-of-range loops should be acting as very efficient sinks for these excess interstitials and preventing them to contribute to the enhanced diffusion.

The density and the number of atoms bound by both the type-V and the endof-range loops are plotted in Figure 3.7 for 7500C and 10500C. Both the density and





42


the number of bound atoms show an increase as the arsenic dose is increased, suggesting that the loops are responsible for the reduction in boron diffusivity. A low density of rod-like defects (6x108 cm-2) was observed at the dose of 8x1014 cm 2 along with the loops. For the 10500C anneal, no type-V loops were observed in the XTEM micrographs for all the doses. So, the abrupt increase in the loop density at this temperature is only due to end-of-range loops.

Two mechanisms may exist if the reduction in the enhanced diffusivity is caused only by the dose dependence of the implant damage below the a/c interface. The depth of the amorphous layer has been observed to increase with dose [53]. More of the implant damage beyond the a/c interface may be incorporated into the amorphous layer at higher doses. This mechanism may reduce the enhanced diffusion. However, a possible decrease in the implant damage below the a/c interface is inconsistent with our experimental loop data where both the loop density and the number of bound atoms increase even when only end-of-range loops exist. So, the damage and therefore the excess interstitials beyond the a/c interface are actually increasing with higher arsenic doses. This may not necessarily mean a larger interstitial supersaturation during the whole enhanced diffusion. A higher interstitial supersaturation at the beginning of the anneal can cause a higher density of end-of-range dislocation loops, which in turn can capture more interstitials during the anneal, decreasing the enhanced diffusion. This statement is supported by the results of other experiments, such as the decrease in enhanced diffusion at higher arsenic concentrations during arsenic deactivation in laser annealed samples [52] and the reduction in boron diffusion with increasing boron dose [8]. Arsenic





43


4 e + 14 I......... ...... 5 e + 10 ....... ......... ......... ........


-E -- 7500C 7500C
3 10500C 4e+10 [3-. 10500C 3e+14

> E
0 3e+10
E
2e+14 a. 2e+10
o o &Z le+14
S le+10


Oe+00 S.....F Oe+00
Oe+00 le+15 e+15 3e+15 4e+15 Oe+00 le+15 2e+15 3e+15 4e+15
Arsenic Dose (cm-2) Arsenic Dose (cm2)

(a) (b) Figure 3.7 Arsenic dose dependence of (a) the density of the bound atoms by loops, and
(b) the density of the loops.



deactivation induced interstitials may also contribute to the supersaturation and increase the density of end-of-range loops, besides nucleating the type-V loops.



3.4 Conclusions



Buried boron layers were used to quantify the transient enhanced diffusion after high-dose arsenic implantation. The enhancement in boron diffusivity decreases with increasing arsenic dose at 750, 900 and 10500C. At the same time, the number of atoms bound by the loops increase, suggesting that the dislocation





44


loops are responsible for the reduction in boron diffusivity. Finally, arsenic deactivation induced interstitials do not increase the enhanced diffusion.














CHAPTER IV
INVESTIGATION OF VACANCY POPULATION DURING ARSENIC ACTIVATION IN SILICON


Recent experimental investigations have shown that electrical deactivation of arsenic in silicon creates excess silicon interstitials [52]. This has been attributed to the formation of arsenic-vacancy clusters and generation of silicon interstitials during this process. As pointed out in Chapter II, this high level of interstitial injection is also confirmed by TEM studies of type-V dislocation loops formed during arsenic deactivation in the same initially defect-free laser annealed samples [39].

The formation of a large number of arsenic-vacancy clusters has been confirmed with positron annihilation measurements [54]. Furthermore, extended xray absorption fine-structure (EXAFS) results combined with Rutherford BackScattering (RBS) measurements also indicate that the deactivation of arsenic proceeds through the formation of arsenic vacancy complexes below 7500C [55]. Upon electrical activation of arsenic, these complexes are expected to dissolve and generate free vacancies. In this work [56], antimony doping superlattice (DSL) structures were used to detect any possible vacancy injection into the bulk during the activation of arsenic. Since antimony diffuses predominantly through a vacancy mechanism, its diffusion is enhanced when there is a supersaturation of vacancies [24].



45





46


4.1 Experimental Details



The DSL structures used in the experiment contained six narrow antimony buried marker layers with 10 nm widths, peaks spaced 100 nm apart and doped to a concentration of 1.5x1019 cm-3. They were grown by low temperature molecular beam epitaxy on Si(100) floatzone substrates [57]. The samples were split into three during ion-implantation: no implant, arsenic implant, and germanium implant. Since germanium is similar to arsenic in mass and size, it allows us to monitor any possible effects of similar ion-implantation damage on antimony diffusion if the damage is not completely wiped out during the damage anneal. Arsenic was implanted at 50 keV at 70 tilt with doses of 3x1015 cm-2 and 8x1015 cm-2, while germanium was implanted at only 8x1015 cm-2. All the samples were capped with an approximately 2000 A layer of oxide and then a nitride layer to prevent the evaporation of arsenic. They were annealed at 11500C for 5s in order to eliminate the implantation damage. After the samples were subjected to a deactivation anneal at 7500C for 2h, some of them were further annealed either at 8500C for 4 h or 9500C for 30 min to electrically activate some of arsenic. The unimplanted, arsenic and germanium doped samples were annealed very close to each other during each thermal cycle so that there would not be any thermal variations between these splits. Figure 4.1 shows a flowchart of the experimental steps.

Chemical arsenic and antimony concentrations were measured by SIMS. Plan-view samples were studied by TEM with a JEOL electron microscope





47




Starting material: Silicon
with DSLs




As implant Ge implant No implant 50 keV 50 keV 3x1015 & 8x1015 cm2 8x1015


Prevent arsenic evaporation: Oxide and nitride deposition



Eliminate implant damage: Anneal at 1150'C for 5s


S Deactivation anneal: 750oC for 2 hr.



Activation anneals: 8500C & 9500C



As & Sb SIMS As Electrical Measurements
As TEM


Figure 4.1 A flowchart of the experiment.





48


operating at 200 keV. Spreading and sheet resistance measurements were made on arsenic doped samples.

In order to find the antimony diffusivity during the activation anneals, the following procedure was employed: For each spike, the SIMS profile after the deactivation anneal was supplied as an initial profile to the process simulation program FLOOPS. The antimony diffusivity was assumed to have the form DSb = aD}rf, where a is an enhancement factor and the reference diffusivity DW is the default inert antimony diffusivity in FLOOPS, and is given by:




DSg = 0.21exp- ) +(n/ni)15exp 4-1




The diffusion of the initial profile was simulated with FLOOPS for different values of a until the error between the simulated and the experimental profiles was minimized. The enhancement or the retardation in antimony diffusivity in the arsenic or germanium doped samples was found by dividing the diffusivity in these samples to the diffusivity extracted from the unimplanted samples which were annealed under the same conditions.



4.2 Results and Discussion



Figure 4.2 shows the antimony SIMS profiles in the as-deposited DSL sample, and in the high dose (8x 1015 cm-2) arsenic sample which has been subjected to the 9500C,





49


1020

As-deposited E 8x1015 cm2 As, 9500C 30 min

o 1019

C 0


E C


10.17.. L .........
0.00 0.10 0.20 0.30 0.40 0.50 0.60 Depth (m)
Figure 4.2 Antimony SIMS profiles in the as-deposited DSL sample and the sample implanted with 8x 1015 cm-2 arsenic and annealed at 9500C for 30 min after the damage and deactivation anneals.


0 8x1015 cm-2, 7500C 13---- 8X1015 cm-2, 7500C + 9500C 30 min 1021 0 3x1015 cm-2, 7500CO &--A 3x1015 cm-2, 7500C + 9500C 30 mi


1020





10



0.00 0.10 0.20 0.30 Depth (gm)
Figure 4.3 Arsenic SIMS profiles. The temperatures in the legends represent the anneals after the damage anneal at 11500C.






50


2.5
(0--- 8x1015 cm-2, 8500C 4h
3--- 8x1015 cm2, 9500C 30min
2.0 3x1015 cm2, 8500C 4h
SA-- 3x1015 cm-2, 9500C 30min E 1.5
-4


S1.0


c 0.5


0.0 I I I
0.2 0.3 0.4 0.5 Depth ( m )
Figure 4.4 Enhancement in the antimony diffusivity as a function of depth in the arsenic doped samples. The thermal cycles shown in the legends represent only the activation anneals.


30 min activation anneal. Antimony inside the arsenic layer exhibits considerable amount of more diffusion than antimony outside the arsenic layer. This can be attributed to an increase in the equilibrium concentration of free vacancies inside the arsenic layer. Figure 4.3 presents the arsenic profiles in the both the high (8x1015 cm-2) and low (3x1015 cm-2) dose samples after the deactivation and the 9500C, 30 min activation anneal. A significant amount of arsenic diffusion takes place during the activation anneal.

Figure 4.4 shows the enhancement in the antimony diffusivity during the activation anneals in the arsenic doped samples. This enhancement was calculated by dividing the antimony diffusivity in the arsenic doped samples to the inert antimony diffusivity extracted from the unimplanted samples. The inert antimony diffusivity was found to be around 40% less than the default diffusivity in FLOOPS





51


(Equation 4-1) at 8500C and 5% less at 9500C. The enhancements in antimony diffusivity are very close to unity, indicating that there is no observable vacancy supersaturation during arsenic activation in these samples. The activation of arsenic was confirmed by both sheet resistance (Table 4.1) and spreading resistance measurements (Figure 4.5). The spreading resistance data shows that diffusion of arsenic into the bulk as well as the higher activation level of arsenic at 9500C contribute to the amount of more electrically active arsenic.

The positron annihilation experiments show that the average number of vacancies per inactive arsenic atom is between 1/2 and 1/4 in an arsenic doped silicon sample which has been laser annealed and then thermally annealed at 7500C (concentration of arsenic = 8x1020 cm-3) [54]. RBS and EXAFS measurements have led Brizard et al. to propose the existence of arsenic-vacancy clusters which have a vacancy/arsenic ratio of around 1/3 [55]. Thus, it is unexpected that there is not any significant enhancement in the arsenic doped samples since a very large amount of free vacancies is expected to be released upon the dissolution of these clusters during the activation anneals.

To investigate the possibility that the generated vacancies may be absorbed by the extrinsic extended defects either left over from the damage anneal or created by the subsequent deactivation anneal, a plan-view TEM study of the arsenic doped samples was undertaken (Table 4.1). Dislocation loops were observed in the samples after the deactivation anneal, and they completely dissolve during the activation anneals with the exception of the 8500C high dose sample where only a very small amount of loops has survived the activation anneal.





52



Table 4.1: Summary of the electrical and TEM measurements on the arsenic doped samples.

Density of The total Anneal sequence
Arsenic after the Sheet atoms bound by amount of after the
dose damage anneal Resistance dislocation electrically
(1015 cm-2 ) (f /sq ) loops active arsenic
2 OC ) -2
(O) ( cm-2) ( cm-2 ) 8 750 71.8 1.1x1014 1.1x1015
8 750 + 850 4h 48.7 Very few loops
8 750 + 950 30min 30.0 No loops 3x1015 3 750 76.4 2.3x1013 1.1x1015 3 750 + 850 4h 63.4 No loops 1.5x1015 3 750 + 950 30min 50 No loops 2x1015




1021
0 7500C
---E 7500C + 9500C 30 min
E
c 1020
0

a)




10
C.)

C 1019
0
U



1018
0.00 0.10 0.20 Depth (gm)
Figure 4.5 Spreading resistance measurements of the arsenic doped layer for the high (8x1015 cm-2) arsenic dose. The temperatures in the legends represent the anneals after the damage anneal at 11500C.





53


Table 4.1 also shows the total amount of electrically active arsenic obtained by integrating the electron concentration from the spreading resistance measurements with depth. In the high dose sample, the ratio of the density of atoms bound by the loops to the activated amount of arsenic during the 9500C activation anneal is roughly 1/17, whereas in the low dose sample this ratio is 1/40. These ratios are much smaller than the vacancy/arsenic ratio in the clusters calculated from either the positron annihilation or EXAFS measurements. Therefore, even if the loops are annihilated by only absorbing the vacancies generated by the activation process, this mechanism alone is not enough to explain the lack of vacancy injection into the bulk assuming that vacancies are generated with a ratio indicated by the positron annihilation and EXAFS measurements.

The unaccounted vacancies may be either recombining at the surface or with the interstitials created during the deactivation anneal and possibly trapped by an impurity such as arsenic or carbon. In fact, carbon has been observed to be an efficient sink for excess interstitials. At high enough doses, carbon can mostly eliminate the transient enhanced diffusion [58] which is believed to be caused by excess interstitials. The carbon profiles have also been measured along with arsenic and antimony. Figure 4.6 shows the carbon profiles after the deactivation anneal for the unimplanted and high dose arsenic samples. In the unimplanted sample, a rapidly decreasing carbon concentration can be observed at the surface. This is believed to be caused by carbon-related particles residing on top of the surface. On the other hand, in the high dose arsenic sample, carbon profile exhibits a peak around 80 nm and a deeper penetration at the surface, suggesting that the observed





54



1021

SG-: Unimplanted
E
o2 _-E0 8x1015cm-2 As
1020


~ 1019
C


.4


O
C

C 1018


1017
0.00 0.10 0.20 0.30 0.40 0.50 Depth (gpm)
Figure 4.6 Carbon SIMS profiles in the unimplanted and high dose arsenic samples after the deactivation anneal.



carbon in this sample is actually inside silicon, rather than at the top of the surface. This high concentration of carbon may be capturing the excess interstitials injected during the deactivation cycle and creating a sink for excess vacancies.

Another explanation for the unaccounted vacancies is a possible increase in the ratio of the concentration of clusters that have a higher vacancy/arsenic ratio, such as As2V, to the concentration of higher order clusters which have a smaller vacancy/arsenic ratio, such as As4V. This can especially happen in the regions where chemical arsenic concentration is not very high (i.e. diffused regions) and the formation of arsenic rich clusters is kinetically limited. Therefore, instead of being injected into the bulk, the excess vacancies may be recaptured by the relatively more vacancy rich arsenic-vacancy complexes.





55


7.0 I ,

> 6.0 G- Ge, 8500C 4h
a 6.0
-E Ge, 9500C 30min
5.0

E 4.0

._ 3.0 E 2.0



0.0
0.2 0.3 0.4 0.5 Depth ( pm )

Figure 4.7 Enhancement in the antimony diffusivity as a function of depth in the germanium doped samples. The thermal cycles shown in the legends represent only the activation anneal.


Figure 4.7 shows the enhancement in the antimony diffusivity in germanium doped samples during the activation anneals. Unlike arsenic, antimony diffusion is enhanced by a factor of 2 to 4 in these samples.



4.3 Vacancy Population in the Partial Absence of Dislocation Loops



A second set of samples was prepared in order to find out whether the enhancement in antimony diffusivity increases after the dislocation loops dissolve completely. These samples were prepared like the previous ones (Figure 4.1) except that the activation anneals were done without any oxide or nitride on the surface. The activation anneal conditions were 16 hours at 8500C, and 1 and 2 hours at





56


9500C. These anneal times were longer than the time when all or most of the dislocation loops had completely dissolved (Table 4.1). Therefore, the vacancy population in the partial absence of the loops can be determined from these samples. The anneals involved either unimplanted or high dose (8x1015 cm-2) arsenic samples.

Figure 4.8 and Figure 4.9 compare the antimony profiles of the unimplanted and high dose arsenic samples at the fourth peak after the deactivation anneal and the activation anneals of 8500C, 16h and 9500C, lh. Although these profiles are quite similar after the deactivation anneal, the high dose arsenic sample exhibits more antimony diffusion than the unimplanted sample during the activation anneals. This indicates that antimony diffusion is enhanced in the high dose arsenic samples. Figure 4.10 shows the enhancements in the antimony diffusivity which are between

1.5 and 2 at both temperatures.

As discussed before, these enhancements may have been caused by the vacancies injected during the arsenic activation. Non-equilibrium diffusion of arsenic may also have created this enhancement. Since dopant atoms are believed to diffuse by pairing with point defects, rapid dopant diffusion can carry a large number of dopant-defect pairs to the bulk. These dopant-defect pairs can create excess point defects upon dissolution if the recombination of the vacancies and interstitials is not as fast as their generation through dopant diffusion. This effect becomes more significant as the dopant concentration and diffusivity increase. It has been proposed as the mechanism responsible for the enhanced diffusion observed in the tail of the high concentration phosphorus profiles [59-62]. Although the inert





57



1020 _-O Unimplan'ed, 7500C
0 8x1015 cm-2 As, 7500C 0--1 Unimplanted, 7500C + 8500C 16h
E -- 8x1015 cm-2 As, 7500C + 8500C 16h
c 1019
o0


101
o


E 1017



1016
0.30 0.35 0.40 Depth (gm)
Figure 4.8 Comparison of the antimony profiles of the unimplanted and high dose arsenic samples after the deactivation and activation (8500C, 16h) anneals. During the activation anneal, antimony diffusion is enhanced in the arsenic sample.

1020 _--) Unimp laned, 7500C
0 8x10 5 cm-2 As, 7500C T a -- El Unimplanted, 7500C + 9500C 1h
E H--- 8x10 5 cm-2 As, 7500C + 9500C 1h
a 1019
o


Q 101
0
o


E 1017



1016 e Al
0.30 0.35 0.40 Depth (gm)
Figure 4.9 Comparison of the antimony profiles of the unimplanted and high dose arsenic samples after the deactivation and activation (9500C, lh) anneals. During the activation anneal, antimony diffusion is enhanced in the arsenic sample.





58


2.5


2.0

0
E 1.5

0~- 8500C 16h
S1.0
a [3--1 9500C lh
E
S0.5 9500C 2h
a 0.5
-C
C
0.0 I I
0.2 0.3 0.4 0.5 Depth ( Lm )

Figure 4.10 Enhancement in the antimony diffusivity as a function of depth in the high dose (8x1015 cm-2) arsenic doped samples. The thermal cycles shown in the legends represent only the activation anneals.


arsenic diffusivity is smaller than that of phosphorus, non-equilibrium diffusion of arsenic may still be responsible for the small enhancements in these samples.



4.4 Conclusions



The effect of arsenic activation on vacancy population has been studied using antimony buried layers. The antimony diffusivity has been found to be very close to its inert diffusivity during arsenic activation in the presence of dislocation loops, indicating that there is no observable vacancy injection under these conditions. The density of the atoms bound by the loops are not sufficient to absorb all the vacancies which are expected to be generated in an amount indicated by the positron






59

annihilation and EXAFS measurements. Other possible mechanisms that can explain the lack of vacancy injection in the presence of the loops include surface recombination, recombination with trapped interstitials generated during the deactivation anneal and absorption of vacancies by relatively more vacancy rich arsenic defects, such as As2V clusters. On the other hand, antimony diffusion is enhanced for the same anneals when germanium is present at the surface.

In the partial absence of the loops, antimony diffusion is enhanced by a factor of 1.5-2 during arsenic activation. Vacancy injection during arsenic activation as well as non-equilibrium diffusion may explain this enhancement.














CHAPTER V
A KINETIC MODEL FOR ARSENIC DEACTIVATION


5.1 The Physical Structure of Inactive Arsenic



In the arsenic-silicon phase diagram, monoclinic SiAs phase is the closest one to a dilute arsenic-silicon (solute-solvent) mixture [63]. In this phase, arsenic has three nearest neighbors, whereas silicon has four [64]. The monoclinic phase has also been observed in arsenic implanted silicon at very high arsenic concentrations (-lx1022 cm-3) [65,66]. Nobili et al. have determined the solid solubility of arsenic associated with the monoclinic phase [67]. The solid solubility is defined as the equilibrium concentration of the solute (arsenic) when the solid mixture is in contact with an infinitely large film of the second phase. Figure 5.1 shows the solid solubility value along with the equilibrium active concentration of arsenic. The solid solubility of arsenic is about an order of magnitude larger than its equilibrium active level.

At equilibrium, monoclinic SiAs precipitates start to form above solid solubility, whereas arsenic is substitutionally dissolved in silicon below the equilibrium active concentration. The inactive arsenic structure between these two limits has been the subject of much recent research. RBS [55] and X-ray standing wave measurements [68] show that this inactive arsenic structure is coherent with




60





61



1022
G-e Equilibrium active concentration
I0-E3 Solid solubility


E



0 4




1 o


7.0 7.5 8.0 8.5 9.0 9.5 10.0 104/Tr ( K1)
Figure 5.1 The temperature dependences of the solid solubility associated with the monoclinic SiAs phase and the equlibrium active concentration.



the silicon lattice. This result suggests that the inactive arsenic does not have the crystal structure of the monoclinic SiAs phase, since the monoclinic phase, having a crystal structure different from silicon, would create incoherence in silicon [40,52]. EXAFS measurements on high concentration arsenic samples demonstrate that the number of the nearest neighbors of an arsenic atom decreases from four silicon atoms in laser annealed samples to an average of 2.5 to 3.5 in deactivated samples. This result can be explained by the formation of arsenic-vacancy complexes (AsmVn) in which arsenic has less than four neighbors. Recent positron annihilation experiments have also found evidence of a high density of vacancies related to inactive arsenic [54]. Moreover, theoretical calculations indicate that the formation of arsenic-vacancy complexes is an energetically favorable process [34,69]. The





62


interstitial injection observed during arsenic deactivation also indirectly supports the existence of arsenic- vacancy complexes [32].

Very small (-15 A) precipitates have been identified by TEM in deactivated arsenic samples [40]. These precipitates have been proposed to have a zinc-blende structure. In this structure, every alternating silicon atom is replaced by an arsenic atom and both silicon and arsenic have four nearest neighbors of the other species. The aforementioned EXAFS measurements show that the number of silicon first neighbors of arsenic tends back to four atoms at high temperatures (>7500C) [33]. This observation has been connected to the co-existence of arsenic-vacancy complexes and zinc-blende type AsSi precipitates.

In summary, the inactive arsenic structure that is formed between the limits of solid solubility and equilibrium active concentration is believed to be arsenicvacancy complexes. At high temperatures, the vacancy content seems to be decreasing with vacancies being replaced by silicon atoms.



5.2 Previous Models for Inactive Arsenic



Clustering and precipitation have been proposed to explain the formation of electrically inactive arsenic in silicon. In clustering models, multiple arsenic atoms are assumed to come together and form a new defect which is electrically inactive at room temperature. These clusters may also contain point defects. For example, an As2V cluster can be formed by the following reaction:





63


2 As + V As2V 5-1



On the other hand, precipitation models consider the formation of much larger structures that may contain thousands of dopant atoms. The precipitates may form at any size whereas clusters are assumed to have on the order of a few atoms. The size distribution of precipitates depend on the initial supersaturation of dopant atoms, the time and the temperature of the anneal and other kinetic factors such as diffusivity and reaction rates.

Many TEM observations have concluded that extended defects in high concentration arsenic doped samples are not sufficient enough to account for most of the inactive arsenic [39-41]. This evidence favors the clustering explanation such that most of the inactive arsenic may be in the form of clusters which are too small to be observed by TEM. However, it is also possible that very small coherent silicon-arsenic precipitates can be responsible for the unobserved inactive arsenic.

As mentioned earlier, Armigliato et al. [40] have identified very small precipitates in heavily arsenic doped samples. It is well known that the electrically active arsenic concentration at equilibrium (i.e. for long anneal times) is very insensitive to the chemical arsenic concentration [12]. This has been put forward as suggestive evidence for precipitation [12] since cluster models predict that the active arsenic concentration depend on the chemical concentration at equilibrium. However, the equilibrium electron concentration can be made independent of the chemical concentration if one assumes that the clusters are charged at the annealing temperature [70]. As mentioned in Chapter I, reverse annealing [12,13] has been





64


shown to occur in arsenic implanted samples, suggesting the existence of precipitates having a distribution of sizes of different free energy. But, reverse annealing can also be explained by the existence of multiple clusters [13]. It certainly rules out the single cluster model. All of the experimental data suggests that the most reasonable assumption for inactive arsenic is the coexistence of clusters and precipitates [24,42,55]. The clusters can form at the beginning of the deactivation cycle and act as embryos for larger precipitates.

Most of the previous quantitative models for inactive arsenic has been single cluster models. Sheet resistances and electron concentrations in arsenic doped layers have been fit with various equilibrium cluster models [71-73]. Dynamic clustering models have been used to fit the initial stages of deactivation [42] and the effect of ramp-down on the sheet resistance [19]. Luning et al. [23] have pointed out that single clustering models can not at the same time account for the rapid arsenic deactivation at the beginning and the slow one at long times that they observe in their experiments.

Some of the quantitative precipitation models that have been proposed in the literature solve for the whole size distribution of precipitates. This kind of model has been demonstrated for antimony precipitation [74], arsenic and phosphorus precipitation [18], and oxygen precipitation [75]. The dopant precipitation model by S. Dunham [18] has been shown to exhibit reverse annealing. However, it has not been applied extensively to arsenic and does not account for interstitial injection and dislocation loop formation. The oxygen precipitation model includes the interstitial injection and bulk stacking fault growth observed during oxygen





65


precipitation. A recently proposed precipitation model solves for the first three moments of the precipitate size distribution, making it computationally less intensive [76].

The next section will present a general extended defect model which can be applied to arsenic deactivation as a precipitation model [77]. The model calculates the evolution of the precipitate size distribution and is able to account for various phenomena related to arsenic activation and deactivation. The interaction of arsenicvacancy complexes with interstitials and dislocation loops will be included into the model in Chapter VII.



5.3 A Kinetic Model for Extended Defects



In the literature, the evolution of extended defects has usually been described by two phases: a nucleation and a growth phase. The nucleation theories study the formation of stable nuclei in a supersaturated solution while the growth theories try to determine the growth rate of these particles after they are formed. This is a somewhat artificial distinction since the nucleation and the growth of an extended defect is a continuous event. However, it makes the modeling problem more tractable and less computation intensive.

The classical nucleation theory was first formulated by Volmer and Weber 70 years ago [78]. It was later developed by several authors [79-82]. In this approach, the defects are assumed to grow or shrink by gaining or losing one atom at a time.





66



Ei.1 Ei









Ei Eijl









Figure 5.2 Schematic representation of the reactions taking place during extended defect formation. The defects are assumed to grow or shrink by gaining or losing one atom at a time.


This process, as shown in Figure 5.2, can be written as a set of reactions in the following form:



Ei- + El Ei Ei + E, Ei+1 5-2



where Ei represents a defect containing i solute atoms and El is a single solute atom. El may be an arsenic atom or a silicon interstitial, where Ei is then an arsenic precipitate (or cluster) or a dislocation loop. In this formulation, the collision and fusion of defects are ignored, as well as the fission of defects into two or more other defects. The formulation may not be valid in non-dilute solutions where the concentration of the solute is comparable to the concentration of solvent.





67


According to reactions 5-2, the change in the density of the i-sized defect is given by the following equation:




fi
t = (Pi- fi- 1 aifi) (Pifi Oi+ fi + 1), i 2 2 5-3




where i- is the forward and a t is the reverse reaction rates of the first reaction in 5-2. Equation 5-3 can be put in a more convenient form:




i i- i -J i>, i2 5-4




where Ji is the rate at which defects of size i become defects of size i+l and is defined by:




i = ifi i + fi + 1 5-5




Equation 5-3 does not apply to the case of i=l, i.e. the single solute atoms. The concentration of the single solute atoms can be found from the mass conversation equation:





68



fl + ifi = Ct 5-6 i=2



where Ct is the total solute concentration and is a given.

The forward reaction rate can be found by a treatment similar to the one given by Turnbull and Fisher [83]. The free energy diagram of the right hand side of reaction 5-2 is shown in Figure 5.3. Here Gi represents the non-mixing component of the free energy of defect Ei when El is taken as the reference phase. Agf is the reaction barrier between EI+Ei and the activated complex which turns into Ei+1. The forward reaction rate can be written as:




i= Aidnt dexp(Agf 5-7




where Ai is the surface area of the size-i defect, Xd is the lattice spacing around the defect, Cn" is the concentration of El atoms at the interface of the defect, vd is the vibration frequency and Agf is the energy barrier. Ai dCint is the number of El atoms that are around each Ei defect and are ready to react with Ei. vdexp(- kT is the number of times that this reaction takes place per unit time.

The diffusivity of the El atom in the solvent can roughly be written as:




D = v exp -_AgD 5-8 k T





69


Free
Energy i gb



ggf
Energy---------------------Gi+ - - ---/- ---
Activated
Gi ---Complex I
EI+E, I Eil
I I I Configuration

Figure 5.3 Schematic representation of free energies related to the following reaction between the extended defects and a single atom: Ei + El Ei+1.



where k is the lattice spacing, v is the vibration frequency and AgD is the migration energy. If the lattice spacing and the vibration frequency do not change near a defect and, most importantly, the migration energy, AgD, is equal to the reaction barrier, Agf, then




=i = AiCi"nt 5-9




The interface solute concentration around a defect, C"nt, is not necessarily equal to the far-field solute concentration, fi, because the transport of the solute atoms to the defect may not be fast enough to resupply all the solute atoms that have reacted with the defect. So, the solute may be depleted around the defect. For





70


convenience the diffusion of the solute atoms is assumed to be fast enough so that Cjnt = fl. So the forward reaction rate becomes:




Pi = AiD f 5-10




The reverse reaction rate can be found from the forward reaction rate and the thermal equilibrium condition, as suggested by Katz [81]. At thermal equilibrium, the detailed balance condition requires all fluxes to be equal to zero. So, from Equation 5-5:



fieq
a li+ = iq 5-11




where fieq is the density of size-i defects at equilibrium.

Upon minimizing the Gibbs free energy of a system consisting of defects of all sizes, the solute and solvent atoms, one can obtain the following expression for the density of the defects at thermal equilibrium:




1 = f exp Wi-Wi+l) 5-12 fieq C kT





71


where Wi is the interfacial free energy of a single i-sized defect, C,,sol is the solubility, k is Boltzmann's constant and T is the absolute temperature. Previously, it has been incorrectly assumed that the solute concentration is equal to its solubility at thermal equilibrium for this system [82,84] although Equation 5-12 is the only result of applying thermodynamics to this system. If the mixture was in touch with an infinitely big film of the second phase of the solute-solvent system, then the solute concentration would reach to its solubility at thermal equilibrium.

The interfacial free energy of a size-i defect, Wi, depends on the size of the defect. In the classical nucleation model, it is proportional to the surface area of the defect if the defect is large:



Wi = cAi 5-13



where Y is the interfacial surface energy per area and is defined by this equation. If the defect is spherical:




(367c 1/3.2/3
Ai = C2 5-14




where Cp is the density of the solute atoms in the defect. For small defects, the interfacial free energy is expected to deviate from the size dependence expressed in Equation 5-13. Terms that are proportional to the linear dimension of the defect plus





72


other terms may become more important. An empirical relation relating the interfacial free energy to defect size can be used [18]:





Wi = Clia' + c2ia2 + ... 5-15





where a1, a2,... shapes the size dependence of the interfacial free energy and c1, c2... are constants. Since the interfacial free energy is proportional to the surface area for large defects, a, can be set to 2/3 for spherical defects, to 1/2 for planar ones, etc. If the defect creates stress in the lattice, the strain energy term should also be included in Wi.



5.4 Simulation Results



Equations 5-3, 5-6, 5-10, 5-11, 5-12, 5-14, and 5-15 were implemented into FLOOPS. The interaction between the precipitates and point defects was not included in the model. The simulated experimental data was taken from Luning et al. [23]. In that work, arsenic was implanted into silicon at various doses, followed by laser annealing. Then, the samples were subjected to thermal annealing at various temperatures. The laser annealing activated all of the dopant, thus giving a clear initial condition for the simulations, i.e. the active concentration fl, is equal to the total dopant concentration, Cr at the beginning. The profiles were box-shaped and no appreciable diffusion of arsenic was observed; therefore a single chemical dopant concentration could





73


be used instead of the whole arsenic profile for each simulation. The discrete rate equations were solved without any rediscretization. The initial density of all precipitates was assumed to be very small. As the boundary condition, the density of an arbitrarily large precipitate was taken to be zero.

The arsenic diffusivity used in the simulations has the following form [18]:




DAs Do + D (n)[1 + (fl] 5-16




where Do and D. are the neutral and negative components of inert arsenic diffusivity, n is the electron concentration and ni is the intrinsic carrier concentration. The term, [ + ( takes into account the experimentally observed power law dependence of the arsenic diffusivity on the active concentration [31]. Do, D_ and CO have Arrhenius dependences on temperature.

The simulation parameters are given in Table I. The solubility, Csol,is taken to be equal to the equilibrium active concentration of arsenic. The only parameters that were fitted during the simulations are the interfacial energy coefficients c1 and c2, and the reference concentration Co.

Figure 5.4 to Figure 5.7 show a comparison of simulations with the arsenic deactivation data obtained between 500 and 8000C for chemical arsenic concentrations of lx1021 cm-3 and 4.4x1020 cm-3. The simulations successfully reproduce the experimental data. All of the simulations exhibit a very fast deactivation of arsenic at the beginning of the thermal anneal just like the





74


Table 5.1: Parameter values used for simulations of arsenic deactivation Activation
Parameter Pre-exponential Activation Energy (eV)

Do 0.0666 cm2/s 3.44 D 12.8 cm2/s 4.05 CO 3x1023 cm-3 0.631

m 3.5 a, 2/3

C1 0.13 eV

a2 1/3

C2 0.115 eV X 2.7x10-8 cm CP 2.5x 1022 cm-3

Csol 2.2x 1022 cm-3 0.47



experiments. This is due to the rapid formation of small clusters which have relatively small formation energies. The slow deactivation after around a few minutes is also replicated by the simulations. This slow deactivation could not be reproduced by single size cluster models [23]. Figure 5.8 shows the simulated size distributions of the 1x1021 cm-3 sample after 5 and 124 minutes at 7000C. The distribution becomes broader for the longer anneal, suggesting that the slow deactivation is due to further precipitation at larger sizes.





75



5e+20


E 0 Experiment
a 4e+20
c Simulation

2 3e+20


o 2e+20 0 0 0 0Sle+20


Oe+00
0 20 40 60 80 100 120 140 160 180 200 Time (min)
(a)

5e+20


E 0 Experiment
0 4e+20
S- Simulation



0 o oooo
o 2e+20




Oe+00

0 20 40 60 80 100 120 140 160 180 200
Time (min)

(b)

Figure 5.4 Comparison of experiments and simulation at 500C at a chemical arsenic concentration of (a) lx1021 cm-, (b) 4.4x1020 cm-





76


5e+20


E O Experiment
o 4e+20
a c- Simulation


a)
3e+20

0
o 2e+20
000000000

Sle+20


0e+00
0 20 40 60 80 100 120 140 160 180 200 Time (min)

(a)

5e+20 I


E 0 Experiment
o 4e+20
c ,- Simulation
0

c 3e+20


0
C 0
o 2e+20


a le+20


Oe+00 I I I I I
0 20 40 60 80 100 120 140 160 180 200 Time (min)

(b)

Figure 5.5 Comparison of experiments and simulation at 6000C at a chemical arsenic concentration of(a) lx1021 cm-3, (b) 4.4x 1020 cm-3.





77



5e+20 I


E O Experiment
o 4e+20
r Simulation
0

3e+20
o
O
o 2e+20
(

1 e+20


Oe+00 I I I I I I
0 20 40 60 80 100 120 140 Time (min)
(a)

5e+20 I I I


E O Experiment
o 4e+20
c e Simulation
0

C 3e+20

o
o 2e+20


1 e+20


Oe+00
0 20 40 60 80 100 120 140 Time (min)

(b)

Figure 5.6 Comparison of experiments and simulation at 7000C at a chemical arsenic concentration of (a) lx1021 cm-3, (b) 4.4x 1020 cm-3.





78


5e+20 I C,?
E O Experiment
a 4e+20
C Simulation

E 3e+20

CD




le+20
1 e+20



Oe+00 I
0 5 10 15
Time (min)

(a)

5e+20


E O Experiment
S4e+20
c- Simulation
0

3e+20
Q

o 2e+20 0O


) le+20


Oe+00 ' I , I
0 5 10 15
Time (min)

(b)

Figure 5.7 Comparison of experiments and simulation at 8000C at a chemical arsenic concentration of (a) lx1021 cm-3, (b) 4.4x1020 cm-3.





79





1020
~- 5 min S-El 124 min
E

0 1019


0

C 1018




1o17
0 5 10 15 20 Size

Figure 5.8 Simulated defect size distributions for the 1x1021 cm-3 sample after 5 min and 124 min at 7000C.



5.5 Conclusions



An arsenic deactivation model that takes into account the size distribution of inactive arsenic structures was presented. The rate equations that describe the evolution of the size distribution were derived from the kinetic theory and thermodynamics. The model is in good quantitative agreement with the experimental arsenic deactivation data, and successfully reproduces the rapid deactivation at the beginning and the slow deactivation after a few minutes.














CHAPTER VI
A COMPARISON OF VARIOUS NUMERICAL METHODS FOR THE SOLUTION OF THE RATE EQUATIONS IN EXTENDED DEFECT SIMULATION


6.1 Introduction



Extended defects in silicon play an important role in the final electrical characteristics of silicon devices. Precipitation/clustering of dopant atoms determine the carrier concentration in heavily doped regions. Dislocation loops affect the population of point defects and therefore dopant diffusion. { 311 } defects have recently been identified as an interstitial source during transient enhanced diffusion [85,86]. In general, extended defects form during phase transitions and as a result of aggregation of impurities such as dopants and point defects. The extended defects have been widely studied in the literature. Nucleation of clusters in gases, nucleation of vacancy voids [87,88], crystal nucleation in glasses [89] and crystal nucleation in amorphous silicon [90] are just a few examples.

In Chapter V, arsenic deactivation was simulated by solving the size distribution of electrically inactive arsenic precipitates. The rate equations describing the size distribution were directly solved without the necessity of a numerical method to reduce the number of equations. A small number (-50) of rate equations was enough to get a good fit to the experimental data. In addition, the




80





81


experimental arsenic profile was homogenous, so the rate equations which were solved at only one point on the arsenic profile were representative of all the other points.

Other extended defects such as dislocation loops and oxygen precipitates may contain millions of atoms. It is impossible to solve those many rate equations in a reasonable time with today's computer technology. Moreover, the impurity profiles are almost always inhomogenous, creating the necessity of solving different rate equations at each point in space. For these two reasons, the number of equations should be reduced to a reasonable level with an appropriate numerical technique [91].

The size evolution of the extended defects are described by the rate equations (also known as the discrete birth and death equations) which are introduced in Chapter V:




afi
= Ji- 1 Ji, i 2 2 6-1



i = ifi i + 1 fi + 1 6-2




where fi is the density of i-sized defect, Pi is the forward and ai+ 1 is the reverse reaction rates. As detailed in Chapter V, the reverse reaction rate can be found from the forward reaction rate and the thermal equilibrium condition:





82


feq
ei+ I qifq 6-3 I+1




where fieq is the density of i-sized defects at equilibrium. Upon substituting Equation 6-3 into Equation 6-2, an alternate form for Ji can be obtained:




J eqr fi fi+ 1 6-4 Ji = Pifi 64ffq





Several researchers [82,84,87,92] have worked on the solution of either the rate equations or its continuous form which is referred as the Fokker-Planck equation:




8 a a 2
-f(s, t) = -[A(s)f(s, t)] + [B(s)f(s, t)] 6-5 at T-7




where s is the continuous size variable, A and B are the drift and diffusion coefficients respectively. In spite of all this work, there still exists the need for a simple numerical method which can reproduce the solution of the rate equations with a sufficient degree of accuracy for large defect sizes (>1000). In this chapter, various numerical methods will be derived and they will be evaluated using the arsenic deactivation model of Chapter V. Two general techniques will be investigated: rediscretization and interpolation.





83


6.2 Rediscretization



In this approach, the rate equations are first put into a continuous form and then rediscretized. After finding a suitable continuous form, the equations are discretized using two different methods: linear and logarithmic discretization.



6.2.1 Continuous Form for the Rate Equations



The first continuous form for the rate equations has been given by Zeldovich [93] and Frenkel [80]:




f(s, t) J(s, t) 6-6 t Ts





J(s, t) = -P(s)feq() 6-7




These equations can be obtained by expanding Equations 6-1 and 6-4 into a Taylor series about i and keeping only the first two terms (the constant and the first derivative). This approach was later criticized by Goodrich [82]. He approximated the rate equations at the mid-points, rather than at the end-points. However, he used Equation 6-2 instead of Equation 6-4 in his derivation. Therefore, his analysis led to an equilibrium condition that is different from that of the discrete rate equations





84


(Equation 6-3). In contrast, the Frenkel-Zeldovich equation automatically contains the exact equilibrium conditions, because the starting point is Equation 6-4 which already includes the discrete equilibrium conditions. Shizgal and Barrett [84] suggested another methodology to approximate the rate equations with a continous form and compared all three methods to the exact solution. They found that their and Goodrich's approaches are more accurate than Frenkel's.

In order to approximate the rate equations with a continous form, the mathematical procedure suggested by Goodrich will be followed. First, the right
1
hand side of Equation 6-1 is expanded into a Taylor series around i- I up to the
2
second term. The resulting equation is:




a a 6-8
-f(s) J(s)





where s is the continous size variable. The implicit dependences off and J on t are not shown in order to make the derivation more clear. The same mid-point expansion is then applied to Equation 6-4, rather than Equation 6-2. As discussed in the previous paragraph, Equation 6-4 already contains the exact equilibrium conditions, therefore its continous form will also preserve them. Rewriting Equation
1
6-4 by substituting i- 2 in place of i, the following expression is obtained:





85



f I 7 1 6-9 J I~ - feql 2 6-9 -- 1+f
2 I




Expanding 3, feq, and separately into Taylor series around i, and keeping the feq
first two terms for p and f, and the first three terms for feq, the following feq
equation is obtained:




J(s) -[P(s) p(sa eq s)- feq eq s) + 6-10 2as J[s2as 8 a2s
2


s \q e




Substituting this equation into Equation 6-8, the following differential equations can be written for the discrete rate equations:




a-f(s) =-J(s) 6-11









feq was expanded up to the second derivative for the following reason: As mentioned in Chapter V, feq can be written as:





86


feq(s+ 1) fl (W(s) -W(s + 1)
-=- exp 6-13 feq() Csol kT



for s = i. fl represents the solute concentration, Csoi is the solubility and W is the interfacial free energy. From this iteration, feq can be found to be an exponential function of size s. If only the first two terms in the Taylor expansion are kept for eq feq_ I afeq
fe, namely fe e, this term will become negative if:
2as



feq < feq 6-14 2s



After finding an expression for feq from Equation 6-13 and inserting it into this inequality, the condition above can be rewritten as:




In i I T W(s) > 2 6-15 (Cs0l kTas



This condition can be met at large supersaturations or slowly changing interfacial free energy values. Then, the flux J(s) will be positive, which is completely nonphysical. The sign of the flux should only be decided by the ") term, as in the discrete rate equations. A similar analysis shows that the inclusion of the second derivative term in the expansion of feq guarantees that the approximate Taylor series for feq will always be positive as long as feq is expressed as similar to Equation 6-13 and W similar to 5-15.





87


Now that a continous form has been derived for the rate equations, the next step will be to discretize it for an arbitrary size spacing. Figure 6.1 shows an arbitrary segment of the size space for which Equations 6-11 and 6-12 will be discretized. For size 1, the right hand side of Equation 6-11 can be approximated as:




at Jk' 6-16




where 1' and k' are the mid-points in the respective interval and A is the size difference between the mid-points. The task is now reduced to finding an appropriate discrete expression for J at 1' and k'. Two discretization schemes will be investigated for this purpose: linear and logarithmic.



6.2.2 Linear Discretization



In this method, the function to be discretized, y(s), is assumed to be linear in size at each interval [k,l]:



y(s) = as + b 6-17



The value of this function and its derivative at the mid-point, k', can be approximated with the values of the function at the end-points, k and 1:





88






k k' I l' m
I I I Size





Akl AIm


Figure 6.1 An arbitrary interval in the size space for which the continuous rate equations are discretized.



Yk + 6-18 Yk' = 6-18 Yl-Yk 6-19 s k' Ak




If p, feq, and are assumed to be linear functions of size, one can obtain a
feq

discrete equation for Jk' by substituting these relations into Equation 6-12. Repeating the same procedure in the interval [1,m], a similar expression can be derived for J1r.



6.2.3 Logarithmic Discretization



In this scheme, the function to be discretized is assumed to be an exponential function of size:





89


y(s) = aebs 6-20



Similar to linear discretization, the value of the function and its derivative at the mid-point can be evaluated in terms of the values of the function at the end-points:




Yk' = 6-21


ak- kYIn -; 6-22 s k1 Yk


2 In
= [ kIj1 6-23





If p, feq, and f are assumed to be exponential in size, a discrete equation can be
feq
obtained for Jk' by substituting these relations into Equation 6-12.

There is one important problem with the logarithmic discretization method. The derivative of with respect to size is approximated as:
feq




(I= 1 f n (ff 6-24 Seq klf fekq Aklf kf7q




As one of the densities, such as ft, decreases, the derivative also decreases, eventually approaching zero for very small values of f1. This causes the flux, Jk', to





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get smaller, which in turn decreases the rate of change of ft (Equation 6-16). This result is both counter-intuitive and incompatible with the original discrete rate equations. In the rate equations, the rate of change of fI increases as fI gets smaller (Equations 6-1 and 6-4).

This difficulty can be overcome by exploiting a technique that has been widely used in diffusion simulations and FLOOPS. This method works by mixing a small part of fk into fl and vice versa:




fk -'ofk +(1 a)f 6-25 flt afl + (1 )f k 6-26



where a is close to unity. If these substitutions are put into Equation 6-24, it can be found that Equation 6-24 is now compatible with the rate equations in the limit of small fl. In this work, the natural logarithm term in the derivative has not been changed in order to preserve the equilibrium conditions.



6.3 Interpolation



Figure 6.2 shows a segment of the size space where a solution for the discrete rate equations is desired. The rate equation for size I can be rewritten as:




af t f ft g feq 1+ 6-27
-t =eq Jeq q eq
_1 l I f1+ 1





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k 1 m
I I I Size 1-1 1+1


Figure 6.2 An arbitrary interval in the size space. In the interpolation method, the density fl- is interpolated with fk andft and the density f ,+ with, andfm.



These equations include the densities for sizes 1- 1 and 1+ 1, i.e. fl-1 and fl+l" If only the solutions for fk, fI and f,, are desired, f-l_ and f1+l can be interpolated using these densities. Then, the rate equations will have only the densities whose solutions are desired.

In the linear interpolation case, the densities are assumed to be linear in size at each interval such as [k,l]. ft- I can be interpolated as:




S(1- k 1)ft+fk 6-28
1 -k



A similar expression can be obtained for fi+1. If these interpolations are substituted in Equation 6-27, the rate equations will only have fk, fl and fm as unknowns.

The exponential interpolation is similar to the linear one, except the densities are assumed to be exponential in size at each size interval. The exponential interpolation expression for fl 1 is given by:






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(-k
fl-1 = (f(lkftk) 6-29




6.4 Comparison of Numerical Methods



The numerical techniques described in the previous sections were implemented in FLOOPS. Their accuracies in the solution of the rate equations were evaluated using the arsenic deactivation model in Chapter V. The chemical arsenic concentration was taken to be lx1021 cm-3. Table 6.1 shows the simulation parameters that are different than the ones used in Chapter V.

Figure 6.3 (a) shows the exact solution of the original discrete rate equations with these parameters. The equations have been solved up to size 2000. The exact solution will be a reference point in comparing the different numerical techniques. Figure 6.3 (b) shows the integrated dopant concentration in the precipitates. This concentration has been calculated by taking the sum of the dopant concentration up to the size shown in the x-axis. Most of the dopant atoms are located between the


Table 6.1: Simulation parameters that are different from the ones in Chapter V.

Activation
Parameter Pre-exponential Energy (eV) Energy (eV)

Co 4.8x1021 cm-3 0.4

c/ 0.3 eV c2 0.0 eV





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1020 . .... . I

1019

1018

E
E 1017

4
r 1016

1015

1014 1013
1 10 100 1000 Size

(a) 1e+21

E
c 8e+20
0
o

C
S6e+20

O
C 4
CL
4e+20
0
o





Oe+00 I

Size

(b)

Figure 6.3 (a) Exact solution of the rate equations for the simulation described in the text. (b) The integrated dopant concentration for the exact solution.