Ion Implantation and Activation - Volume 1 [1 ed.] 9781608057818, 9781608057825

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Ion Implantation and Activation (Volume 1) Authored By

Kunihiro Suzuki Fujitsu limited Minatoku kaigan 1-11-1 Tokyo Japan

Bentham Science Publishers

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CONTENTS Foreword

i

Preface

iii

Acknowledgements

vi

CHAPTERS 1.

Selection of Impurities

3

2.

Physics of Ion Implantation

8

3.

Monte Carlo Simulation

4.

Analytical Model for Ion Implantation Profiles

121

5.

Pearson Function Family

168

6.

The Other Analytical Model for Ion Implantation Profiles: Edgeworth Polynomial 223

7.

Parameter Extraction for Analytical ion Implantation Profile Model

241

8.

Lateral Distribution

266

9.

LSS Theory

302

92

10. Quasi Crystal LSS Theory

376

11. Simplified LSS Theory

399

Subject Index

452

i

FOREWORD Process Modeling—the prediction of how fabrication steps affect the electrical performance of integrated circuit devices—emerged in the late 1970’s, commensurate with rapid progress in MOS scaling and Moore’s Law that predicted an exponential increase in functional density of integrated circuits. By the mid-1980’s self-aligned, silicon gate CMOS technology had overtaken earlier MOS technologies—Metal Gate and Enhancement-Depletion Mode NMOS—as the dominant digital MOS technology; a position it still holds after more than three decades. The reasons for the dominance of self-aligned CMOS are several. Most importantly, the alignment of source and drain junctions to the gate, which is the lithographic feature that controls the scaling limit of MOS devices. Moreover, the use of a silicon gate, which can be simultaneously doped, allows the gate work function to also be controlled by the junction doping step. Ion implantation of dopants for virtually all integrated circuit device technologies—MOS, bipolar, silicon and compound semiconductors—is unquestionably the method of choice, owing to its precision in controlling dose as well as the ability to create steep lateral and vertical profiles. The connected process steps of activating the dopant and reducing the risk of damage to the substrate material have also kept pace with scaling requirements. This three-volume series not only addresses the physics and technology of ion implantation, activation and substrate annealing, it also provides a robust discussion of the modeling and simulation techniques that are essential for the process and device engineers to design and scale integrated circuit devices. It is not surprising to me that Kunihiro Suzuki has prepared this comprehensive work. We first met on a very cold winter evening in Kyoto in 1988 during my one-year sabbatical leave from Stanford University. It was clear even then, in the early years of his very productive industrial career, that Suzuki-san was destined to master and expand the boundaries of process modeling. Over the ensuing quarter century, he has published cutting edge research papers in process

ii

modeling, pushed the limits of all available commercial computer-aided technology (TCAD) modeling tools and applied the result in innovative ways for the design of advanced integrated circuits. It is certainly my great pleasure to see this excellent treatise and “tour de force” in the discussion of semiconductor doping, using ion implantation that supports ongoing engineering applications as well as informs deeper understanding of the materials and processing of advanced electronic devices.

Robert W. Dutton Palo Alto, California USA

iii

PREFACE Ion implantation is a standard technology, by which impurities are doped into Si substrates in very large scale integration (VLSI) processes in modern times. Defects are unintentionally generated in the substrate by the ion implantation simultaneously since impurities are introduced physically to the lattice, i.e., in non-thermal equilibrium. Hence, a subsequent thermal process is indispensable to recover the damage and activate the introduced impurities. Intensive studies on predicting the ion implantation profiles and activation processes have been done, and they are still under investigations. Many books on fundamental ion implantation technology and detailed physics related to the ion implantation have been published. Furthermore, database for practical ion implantation profiles is also established and it is valid even for modern technologists and students. However, theoretical models and systematic fundamental experimental data are not closely related to each other, and how accurately the models reproduce the experimental data is not clear. We have been collecting systematic experimental data to evaluate and develop the models for a long time. We also try to compare the data with related models in this eBook. Details about the derivation process of fundamental and advanced models are desribed step by step, and their related assumptions and approximations are clarified. If some mathematical and physical background is necessary, related appendixes will be added to make this eBook self-contained. Models that are not established well and under investigations are discussed. Although it is not ensured to be accurate, knowing the detail of models under investigations is interesting for furthering the models. The author aims at the readers who are experts and non-experts in various fields associated with the ion implantation, hoping that various members can cover some knowledge to collaborate with one another. It is not easy for non-expert members to understand this eBook, but it is believed that they can do it with time and efforts.

iv

This eBook consists of three volumes, and this volume treats the following. Various ions are implanted into various regions of devices. How the impurities are selected is briefly reviewed. The interaction between incident ion and substrate atoms is classical physical subjects, where the target energy range is more than MeV. Ion implantation gradually becomes a standard technology to dope Si substrates, and the technology-oriented studies have been intensively investigated, where the energy range is less than MeV. We can use the established interaction between ion and substrate electron cloud, and nuclei with fundamental atomic properties, such as atomic number and mass. The detail derivation process of nuclei and electron stopping power models is also shown. The physics can be directly implemented into Monte Carlo (MC) simulation. The accuracy of MC is verified by comparing systematic experimental data with wide ion implantation conditions via tuning electron stopping power. Channeling and damage accumulation related to ion implantation are analyzed, but their impact on ion implantation profiles is not treated. It is indispensable to predict ion implantation profiles in crystal substrates since the substrates are used in practical VLSI processes. Systematic experimental data have been accumulated to construct related database. However, the database does not play a role for predicting ion implanted profiles for any ion implantation condition as it is. Analytical models used to describe the profiles are invoked to predict the profiles to do that. The ion implantation database is a table of parameter values of these functions, and an ion implantation profile for any ion implantation condition can be generated by interpolating or extrapolating the parameter values. The analytical models, such as Gaussian, joined half-Gauss, Pearson IV, dual Pearson IV and tail function, have also been developed. Details about these functions will be described. Furthermore, details realted to Pearson function family, where Pearson IV is one of them, and also Edgeworth polynomial, will also be presented. How parameters are extracted from experimental and MC data for the analytical functions is also an important subject. The uniqueness of extracting parameters values is an essential subject, which is also studied.

v

Lateral straggling is very important to predict two-dimensional profiles. However, the lateral resolution of experimental data is poor to obtain accurate lateral straggling. We show how they can be extracted by combining with a theory and experimental data. This eBook also treats Lindhart, Scharf, and Schiott (LSS) theory based on the same physics as MC. LSS theory cannot predict the profile itself but its moments. Therefore, we can generate profiles by using analytical functions which are related to the moments. Probability functions related to various ranges are introduced instead of tracing the trajectories of the ions in the LSS theory, and we can obtain the results instantaneously. The LSS theory needs some approximations to solve the related differential equations. The approximations influence accuracy of the LSS theory. Performing perturbation approximation, we improve the accuracy step by step and obtain the results which are similar to those of MC. Simplification of the LSS theory is done to obtaining thumb of a rule for ion implantation profiles. Moreover, this eBook treats the profiles in crystal substrates theoretically by dealing with channeling phenomenon empirically in the LSS theory. It is not complete enough but gives rough shape of the profiles. This model is applied to profiles in Si1 xGex substrates, where x is a component ratio of Ge in the substrate. CONFLICT OF INTEREST The author(s) confirm that this chapter content has no conflict of interest.

Kunihiro Suzuki Fujitsu limited Minatoku kaigan 1-11-1 Tokyo Japan E-mail: [email protected]

vi

ACKNOWLEDGEMENTS Many members in various fields contributed to complete this eBook. Ken-ichi Okabe, Takehiro Hisaeda, Syuji Watanabe, Dr. Ken-ichi Goto, Dr. Toshihiko Miyashita, and Syuji Kudo helped me to get various systematic SIMS data. Takayuki Aoyama also gave me invaluable advices to use experimental equipment and to set experimental conditions. I collaborated with Prof. Shin-ichi Takagi, Dr. Keiji Ikeda and Yoshimi Yamashita on Ge transitors, and got information and data on processes for Ge and SiGe substrates. I want to give special thanks to Masanori Nagase with whom I started to establish systematic database for ion implantation profiles. His support substantially contributed to establish systematic vast ion implantation database. Tsutomu Nagayama helped me to obtain many ion implantation samples with ion implantation machine in his company, and Susumu Nagayama helped me to obtain many SIMS data. Yasuharu Fujimori helped me to obtain various carrier profiles. Masatoshi Yoshihara and Syuichi Kojima contributed to integrate models into a system which is used by many members. I thank Prof. Fichtner to invite me to his laboratory, where I started studying process modeling and enjoyed the collaboration with Dr. Alexander Hoefler, Dr. Thomas Feudel, and Dr. Nobert Strecker. Dr. Christoph Zechner gave cutting edge information on process modeling, and I enjoyed collaboration with him in this field. He also gave me invaluable discussions, comments, and corrected my English. Hiroko Tashiro and Ritsuo Sudo helped me to simulate ion implantation and diffusion profiles. Toyoo Miyajima provided me two-dimensional views of Si lattice using CrystalMaker.

vii

Yoko Tada and Dr. Yuji Kataoka contributed to obtain accurate ion implantation profiles in low energy region. Dr. Yoshiharu Tosaka and Dr. Takahiro Yamazaki taught me basic physics associated with ion implantation. I owe Dr. Min Yu a great deal for his invaluable discussions and correcting my English. His devotion made it possible to complete this eBook. Hiroyo Miyamoto helped me to set format of word file. Prof. Seijiro Furukawa and Prof. Hiroshi Ishihara directed me to become a researcher. I also want to thank Prof. Robert W. Dutton who always encouraged me from the beginning of my career as a researcher. Finally, I want to thank to my family Kyoko, Yuji, Saori, Takayuki, Misato, Nahoko, Ryohei, and my parents Fukushi and Yae, and Kyoko’s ex-father Haruo Abe, and mother Chiyoko Abe with whom I enjoy comfortable life.

Kunihiro Suzuki Fujitsu limited Minatoku kaigan 1-11-1 Tokyo Japan

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 3-7 3

CHAPTER 1 Selection of Impurities Abstract: Various impurities are used in VLSI processes. The key factors for the selection of the impurities are their solid solubility and diffusion coefficient. B, As, P are commonly used as doping impurities due to their high solid solubility. In and Sb are sometimes used to realize shallow junctions due to their low diffusion coefficients. We briefly showwhere various impurities are used in two distinguished devices of bipolar transistors and MOS FET’s.

Keywords: Solid solubility, diffusioncoefficient, boron, arsenic, phosphorous, antimony, indium, bipolar, MOS, emitter, base, collector, source, drain, gate, extension, pocket. INTRODUCTION We should select ions and ion implantation conditions depending on the purpose of each process. The important characteristics of impurities for the selection of are solid solubility and diffusion coefficient. SOLID SOLUBILITY We should be careful about solid solubility. One is associated with precipitation, and the other associated with cluster formation. The reported values are mixture of these two types of solid solubility [1], and the former is higher than the latter [2]. The latter is directly related to the device characteristics and we mean the latter when we say solid solubility hereafter. Physical understanding of solid solubility is not well established, and the reported values are scattered significantly especially at low temperatures [3]. Fig. 1 shows the rough extraction of solid solubility reported in [3]. One rough critical value of electrical solid solubility is 1020 cm-3. B, As, and P have high solid solubility values and Sb and In have rather low onesin the practical process temperature range of around 1000oC. DIFFUSION COEFFICIENT Fig. 2 summarizes the diffusion coefficient of various impurities [2, 4, 5]. Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

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Kunihiro Suzuki

Electrical solid solubility (cm -3)

Temperature ( oC) 21 1300

1100 1000 900

10

20

10

800

700

600

P

As Sb

19

10

18

10

B

17

10

In

16

10

7

8

9

10 11 12 1/kBT(1/eV)

13

14

Figure 1: Electrical solid solubility of various impurities [3]. o

Intrinsic diffusion coefficient (cm 2/s)

Temperature ( C) -12

1100

10

1000

900

800

700

B As P Sb In

-13

10

-14

10

-15

10

-16

10

-17

10

-18

10

8

9

10 11 1/(kBT) (1/eV)

12

Figure 2: Diffusion coefficient of various impurities [2, 4, 5].

Diffusion coefficients of Sb and As are relatively low compared with that of P in n-type dopant. Diffusion coefficient of In is relatively low compared with that of B, but they are both much higher than those of As and Sb. SELECTION OF IMPURITIES Fig. 3 shows the structure of bipolar transistors. In bipolar transistors, emitter concentration is more than 1x 1020 cm-3 to improve emitter efficiency and to reduce emitter resistance, base concentration is of the order of 1x 1018 cm-3 to improve emitter efficiency. The base concentration is also expected to be high to reduce base

Selection of Impurities

Ion Implantation and Activation, Vol. 1

5

resistance. The base concentration is determined under the trade-off between emitter efficiency and base resistance, and also base transit time in which it degrades with increasing the concentration. The collector concentration is of the order of 1 x 1016 cm-3 to ensure break down voltage. The collector concentration is also respected to be high to reduce collector transit time. Therefore, the concentration is determined under the trade-off between breakdown voltage and transit time.

Figure 3: Device structure of npn bipolar transistor.

In npn bipolar transistors, As or P is used for emitter, B is used for base, and P is used for collector. In pnp bipolar transistors, B are used for emitter, P is used for base, and B is used for collector.

Figure 4: Device structure of MOSFET.

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In MOS transistors as shown in Fig. 4, source/drain concentrations is more than 1020 cm-3 to reduce resistance. Channel concentration is less than 1018 cm-3. The channel concentration and profiles are determined under the tradeoff between short channel immunity and high on-current. High concentration of more than 1020 cm-3 with shallow junction is required in the extension region. Pocket ion implantation is introduced to improve short channel immunity of scaled MOSFETs and the corresponding concentration is of the order of 1018 cm-3. Polycrystalline is commonly used for gate, and it should be doped as high as possible. In nMOSFETs, P, or As are used for source/drain region, As is preferably used for extension due to lower diffusivity than that of P. B is used for channel doping and also In is available to realize sharp profiles. B and In are used for pocket region. Table 1: Selection of impurities for various devices Transistor nMOSFET

pMOSFET

npn Bipolar

pnp Bipolar

Region

Impurity

Source/Drain

As, P

Channel

B, In

Extension

As

Pocket

B, In

Well

B

n+ poly-Si Gate

As, P

Source/Drain

B

Channel

P, As, Sb

Extension

B

Pocket

As, P, Sb

Well

P

p+ poly-Si Gate

B

Emitter

As, P

Base

B

Collector

P

Emitter

B

Base

P, As

Collector

B

Selection of Impurities

Ion Implantation and Activation, Vol. 1

7

In pMOSFETs, B is used for source/drain and extension regions. P is used for channel doping and also Sb is available to realize sharp profiles. As, P, and Sb are all available for pocket region. Table 1 summarizes typical impurities used for each region. REFERENCES [1] [2] [3] [4] [5]

F. A. Trumbore, “Solid solubility of impurity elements in germanium and silicon,” Bell Syst. Tech. J. vol. 39, p. 205, 1960. S. K. Ghandhi, VLSI fabrication principles, Jhon Wiley & Sons, 1994, New York. P. Pichler, Intrinsic Point Deffects, Impurities, and Their Diffusion in Silicon, Springer2004, Wien. K. Suzuki, H. Tashiro, and T. Aoyama,"Diffusion coefficient of indium in Si substrates and analytical redistribution model,"Solid-State Electronics、vol. 43, pp. 27-31, 1999 K. Suzuki, H. Tashiro, and T. Aoyama,"Sb diffusion in heavily doped Si substrates,"J. Electrochem. Society, vol. 146, pp. 336-338, 1999.

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 8-91

8

CHAPTER 2 Physics of Ion Implantation Abstract: The interactions between incident ion and substrate atoms are classical physical subjects. Ions lose its initial accelerated energy through the interaction between electron cloud and nuclei. The process is described with fundamental atomic properties such as atomic number and mass. Therefore, the model accommodates any combination of incident atom as well as the substrate composed of various kinds of atoms. We show the detail derivation process of nuclei and electronic stopping power models.

Keywords: Ion implantation, nuclear stopping power, electronic stopping power, two-body collision, laboratory system, center of mass system, elastic collision, Bohr speed, Bethe model, Firsov, Lindhard model, Thomas-Fermi potential, Poisson equation, Coulomb potential, channeling, damage, amorphous layer. INTRODUCTION Accelerated ions lose energy through the interaction with substrate atoms and finally settle down at a certain depth. One ion reaches deeper region and one ion in the shallow region depending on the process of the interaction. The interaction depends on the potential between ions and the substrate atoms. We briefly review theoretical background of the interaction [1-7] in this chapter. Accelerated ions interact with ions, lose energy, and the energy loss per distance dz is described by



dE  dE   dE       dz  dz n  dz e

(1)

The first term corresponds to the interaction with nuclear atom and the second the interaction with the electrons. Focusing on the one atom in the substrate, the energy loss is also expressed by 

dE  N  S n  E   Se  E   dz

(2)

where N is the density of the substrate atoms. Therefore, Sn  E  and Se  E  are expressed by

Sn  E  

1  dE  1  dE     , Se  E      N  dz n N  dz e Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

(3)

Ph hysics of Ion Im mplantation

I Implantation Ion n and Activation n, Vol. 1

9

Sn  E  is callled the nucleear stopping power, Se  E  the electrronic stoppinng power.

Fiigure 1: Nucleear interaction in laboratory system. s

Fiigure 2: Nucleear interaction in center of maass system.

N NUCLEAR STOPPING S G POWER We treat the nuclei interraction as classical two--body collisiion. Fig. 1 shows W s the cooordinate off the nuclearr interaction in laboratorry system annd Fig. 2 in center of m mass system which signiificantly sim mplifies the related r subjeect. An ion with w mass

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number M 1 is ion implanted into the substrate of which atomic mass number is M2. The collision parameter was denoted by b . We assume that incident ion with initial velocity of v1i and atomic mass number of M 1 collides with a substrate atom with initial velocity of 0 and atomic mass number of M 2 . The velocity of incident atom and substrate atom after the collision is v1 f , and v2 f , respectively. The conservation of moment in laboratory system in Fig. 1 yields p  M 1 v1i  M 1 v1 f  M 2 v 2 f

(4)

In center of mass system, we select the axis so that the total momentum is zero. We assume the axis system that moves with the velocity of vc with respect to the laboratory system. We denote each momentum in center of mass system with ', that is, p '  M 1 v '1i  M 2 v '2 i  M 1 v '1 f  M 2 v '2 f  0

(5)

where

v '1i  v1i  vc

(6)

v '2i  v 2i  vc   vc

(7)

v '1 f  v1 f  v c

(8)

v '2 f  v 2 f  v c

(9)

Substituting v '1i and v '2i into Eq. 5, we obtain as M 1  v1i  v c  v '1i  M 2 v c  0

vc can then be obtained as vc 

M1 v1i M1  M 2

Therefore, v '1i and v '2i are

(10)

Physics of Ion Implantation

v '1i 

Ion Implantation and Activation, Vol. 1

M2 v1i M1  M 2

(11)

M1 v1i M1  M 2

v '2 i  

11

(12)

We then obtain v '2 i  

M1 v '1i M2

(13)

M1 v '1 f M2

(14)

and

v '2 f  

from the latter part of Eq. 5. Therefore, the two atoms are always parallel and opposite direction in center of mass system. The ratio of the magnitude of the velocity is M1 M 2 . We assume elastic collision. The energy conservation in laboratory system yields 1 1 1 M 1 v1i 2  M 1 v1 f 2  M 2 v 2 f 2 2 2 2

(15)

and that in center of mass system yields 1 1 1 1 M 1 v '1i 2  M 2 v '2i 2  M 1 v '1 f 2  M 2 v '2 f 2 2 2 2 2

(16)

Substituting Eqs. 13 and 14 into Eq. 16, we obtain

M1  M1  M 2  2 M1  M1  M 2  v '1i  v '1 f 2 2M 2 2M 2 that is v '1i  v '1 f

(17)

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From Eqs. 13 and 14, we also obtain v '2 i  v '2 f

(18)

Therefore, the magnitude of the velocity is invariant in center of mass system. We evaluate the relationship of scattering angle in laboratory and center of mass system  and  which are shown in Figs. 1 and 2. We start with the equation v1 f  v '1 f  v c

(19)

Inspecting Fig. 3, and expressing the component of the vector of Eq. 19 into each scalar component, we obtain v1 f sin   v '1 f sin 

(20)

v1 f cos   v '1 f cos   vc

(21)

Dividing Eq. 20 by Eq. 21, we obtain

tan  

sin  v cos   c v '1 f

(22)

From Eqs. 10 and 11, we obtain

vc M  1 v '1 f M 2

(23)

Substituting Eq. 23 into Eq. 22, we obtain the relationship between  and  as tan  

sin  M cos   1 M2

(24)

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1

13

Geometrical relationship between incident ion and recoiled substrate atom depends on the magnitude of M 1 and M 2 as shown in Fig. 3. When M1  M 2 , the incident atom can be scattered over the all angle. When M1  M 2 , the incident atom is apt to be scattered forward, and the maximum scattered angle max exists and is given by sin  max 

M2 M1

(25)

Figure 3: Geometrical relationship between incident ion velocity vector in laboratory and center of mass systems.

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The transferred energy to the substrate atom in laboratory system T2 f can be evaluated as 1 M 2 v2 f 2 2 1  M 2 v '2 f  v ' 2 i 2 1  M 2  v '2 f 2  v '2i 2  2v '2 f v '2i cos   2  M 2 v '2i 2 1  cos  

T2 f 

(26)

  2 M 2 v '2i 2 sin 2   2

Normalizing this with respect to incident ion energy T1i , and utilizing Eq. 12, we obtain

 2 M 2 v '2i 2 sin 2   T2 f 2  1 T1i M 1v1i 2 2 2

 M1  v1i   4M 2  M 1  M 2    sin 2   2 M1 v1i 2 4 M 1M 2  sin 2    2 2  M1  M 2 

(27)

When T2 f is larger than a certain energy Ed , the substrate atom is escaped from the lattice cite, and interact with other substrate atoms. We call Ed as displacement energy. Inspecting the geometrical relationship of recoiled atom as shown in Fig. 4, the recoiled angle  2 of the substrate atom, we obtain the relation of v2 f sin  2  v '2 f sin 

(28)

v2 f cos  2  vc  v '2 f cos 

(29)

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1

15

Figure 4: Geometrical relationship of recoiled atom and incident ion.

Dividing each side of Eqs. 28 and 29, we obtain

tan  2 

vc

v '2 f

sin 

(30)

vc  cos  v '2 f

can be further simplified as vc vc M v M M   2 c  2 1 1 M1 v '2 f M1 M 2 v '1 f M 1 v '1 f M2

and Eq. 30 is reduced to

tan  2 

sin  1  cos 

(31)

Therefore, the initial condition of the recoiled atom is the energy of T2 f  Ed , and the angle of  2 .

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Figure 5: Collision of atoms in a reduced one body system.

Next, we derive the relationship between T2 f and collision parameter b . We assume the potential only depends on distance between the incident ion and the substrate atom. The two body problem can then be reduced to one body problem using reduced mass as shown in Appendix A. Considering the general case and the case that the distance between the incident atom and the target atom is very far, we obtain from the conservation of angular momentum as

 L  M c r 2  M c vc b

(32)

where M c is the reduced mass and is given by 1 1 1   M c M1 M 2

(33)

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Ion Implantation and Activation, Vol. 1

17

Therefore, the angular speed is obtained as

  vcb  r2

(34)

The total energy E in the center of mass system is then given by 1  2   V  r  M c  r 2   r     2  2  vc b 2  1  M c  r   r 2    V  r  2  r   

E

(35)

 2  vcb 2  1  M c  r   V r  2 r 2  

  0 , and we In the distant case, we can approximate as V  r   0 ,   0 , and  obtain the energy as E  Ec 

1 M c v1i 2 2

(36)

From Eqs. 35 and 36, we obtain  2  vc b 2  V  r   r   r 2  Ec  2 1  2  vc b   V  r   2  r   vc  r 2  Ec

1 Mc 1 2 Ec

Solving this equation with respect to r , we obtain r   vc 1 

V  r  b2  2 Ec r

(37)

The sign is negative when r decreases and positive when r increases. The derivative of  with respect to r is

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

d  d  dt  dr dt dr vb 1   c2 r V  r  b2 vc 1   2 Ec r

(38)

b

 r2 1

V  r  b2  2 Ec r

Therefore, the relationship between  and b is given by 

rmin

  b b  dr   dr   2 2   V r V r     b b   2 2  2  2  r 1  rmin r 1  Ec r Ec r 

 b dr  2 2  V r   b  2  2 rmin r 1  Ec r

(39)

where rmin can be obtained from dr d  0 , that is

V  rmin  b 2 1  0 Ec rmin 2

(40)

Therefore, the scattering angle in the center of mass system  is given by

   

 b    2 dr  V  r  b2  2  2 rmin r 1  Ec r Substituting Eq. 41 into Eq. 27, the transferred energy is given by

(41)

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1

      T2 f b 4 M 1M 2  2   dr  sin    2 2 T1i  M 1  M 2  2  V r  b  2    2 rmin r 1    E r c  



4 M 1M 2

 M1  M 2 

2

     b  2  dr  cos  2  V r  b 2   r 1   2  r  min  E r c  

19

(42)

We consider the beam areal density  and assume the substrate atom is far distant each other. We first consider the case that the ion beams irradiated into the area between the collision parameter is b and b  db . The total amount of beam in this area is 2 bdb . The total transferred energy related to these ions is 2 bdbT2 f . Therefore, the total transferred energy with respect to one substrate atom for total incident beam is given by

  2 bT2 f db Strictly speaking, the integral region is between 0 and lattice constant. However, the interaction becomes negligible when b becomes rather big. We can then integrate area of 0 and infinity and have total transferred energy of 

  2 bT2 f db 0

When incident beam trace the distance of z in the substrate, the ion beam meets the number of substrate atom of N Z , where N is the atomic density of the substrate. Therefore, the total transferred energy is 

N z  2 bT2 f db 0

The number of incident ion per unit area is  . Therefore, the energy loss of one incident atom during the path z is given by

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki



E  N z  2 bT2 f db 0

(43)

Finally, the nuclear stopping power is given by

Sn 

 E   2 bT2 f db N z 0

(44)

Substituting Eq. 42 into Eq. 44, we obtain 

       4 M 1M 2 b   Sn  T  cos 2   dr  2 bdb 2 1i  V  r  b2  M1  M 2   2   r 1   2   r   min  E r c   0

(45)

Introducing differential collision cross section of d   2 bdb

(46)

Eq. 46 is simply reduced to 

Sn   T2 f d 0

(47)

We should know potential V  r  to evaluate Sn . Various potential have been proposed for V  r  , and ZBL (Ziegler-Litmark-Biersak) potential is frequently employed among them which are empirical model that is fitted to vast experimental data. ZBL potential is given by

V r  

q 2 Z1Z 2  r  f  4 0 r  aU 

(48)

where  0 is the permittity of vacuum q is the electron charge. The latter term f in Eq. 48 is screening function and is given by f     0.1818e 3.2   0.5099e 0.9423   0.2802e 0.4028   0.02817e 0.2016 

(49)

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21

where  is the normalized distance given by



r aU

(50)

aU is the screening distance given by aU 

0.8854 aB Z10.23  Z 2 0.23

(51)

where aB (0.052917 nm) is the Bohr radius. We then try to obtain universal form for Sn , that is, obtain separated term that does not explicitly depend on Z1 , Z 2 , M 1 , and M 2 . Normalizing collision parameter b by screening length aU , we define  as



b aU

(52)

We further normalize the energy by the Coulomb potential with the distance of screening length as



Ec q Z1Z 2 4 0 aU 2



4 0 aU 1 M 1M 2 v1i 2 2 q Z1Z 2 2 M 1  M 2



4 0 0.8854aB M2 T1i  eV  2 0.23 0.23 q Z1Z 2 Z1  Z 2 M 1  M 2

(53)

where we used unit of eV for T1i . Inspecting Appendix C, we can further calculate Eq. 53, and obtain 1 M2 T eV  Z Z  Z 0.23  Z 0.23  M  M 1i   1 2 1 2 1 2

  3.25  10 2  1 eV  

(54)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

It should be noted that  is a non-unit number. Similarly, we can obtain scattering angle with universal form as 

   au au d    2 2 q Z1Z 2  f  2  au  4 0  au    2  min  au   1  2 q 2 Z1Z 2  au    4 0 au

(55)



  d  2 2  f      2  2 min  1 





 ( o)

Fig. 6 shows the dependence of  on reduced energy. When the reduced collision parameter  is large, the collision angle becomes small, and when  is larger than 1, the collision angle becomes significantly small, which means that screening length well expresses the collision. The collision angle becomes small when the reduced energy increases. This means the time for interaction becomes short. 180 160 140  = 0.001 120 0.01 100 0.1 80 60 1 2 40 20 5 10 0 -2 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 ε

Figure6: The dependence of collision angle on reduced energy.

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Ion Implantation and Activation, Vol. 1

23

Nuclear stopping power is also expressed by the normalized parameters as 

       4 M 1M 2  2   Sn  T cos  d   2 bdb 2 1i 2   f    M1  M 2  2     min  1     2    0 

       4 M 1M 2   2 2    aU T  cos  d   d  2  2 1i 2  f    M1  M 2   2     2  1    min     0   

(56)

 aU 2 T1i Sn    

 8.462 1015 Z1Z 2 M 1  S n     eV 0.23 0.23  atom 2   M M Z Z  1 2  1  2  cm





   

where  is defined as

 

4M 1M 2

 M1  M 2 

2

(57)

S n    is universal nuclear stopping power given by 

        2   Sn      cos  d   d  2    f   2 2    1   2     min    0  

(58)

Substituting f    of Eq. 49 into Eq. 58, we obtain energy dependence of S n    on  . The analytical expression fitted to numerical result is given by [1]

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

ln 1  1.1383    0.21226  0.19593 0.5   2    0.01321 Sn      ln   2

for   30

(59) for   30

which is shown in Fig. 7.

10

0

Numerical Analytical

-1

n

S ()

10

-2

10

-3

10

-5

10

10

-3

-1

1

10 10 Reduced energy ()

10

3

Figure 7: Dependence of universal S n    on reduced energy.

When the energy increases, the transferred energy increases. However, as the energy further increases, the transferred energy then decreases. This is the time for interaction becomes decreases as is pointed out for collision angle. The second order energy straggling  n 2 is also defined and corresponding universal one is given by

Physics of Ion Implantation

n  2





0

Ion Implantation and Activation, Vol. 1

25

T2 f d 2



    2     4M 1M 2  2  4   2 bdb   cos  T d  1i 2  2     f     M1  M 2   2  1        2   0   min



    2     4M 1M 2  2  2 4  2   aU  T1i  cos  d   d   2  2     f     M1  M 2    2    2 1      0   min



 aU

2

 T 

2

n

1i



2

2

 

15



8.462  10 Z1 Z 2 M 1

M

1



 M 2  Z1

0.23

 Z2

0.23





Z1 Z 2  M 1  M 2  Z 1

0.23

 Z2

13

32.53  10 M 2

 2  2 Z1 Z 2 M 1  2  eV     M M  n atom   1 2  2  cm





4M 1M 2

M

3

2

 2.601  10

0.23



   

1

 M2 

2

n

2

  (60)

where the universal n 2    is given by 

        2 2 4  cos  d   d  2  n      2   f   2     min  1     2  0  

(61)

Similar to S n    , substituting f    of Eq. 49 into Eq. 61 and an analytical expression to fit the result is given by [1]

n 2    

1 4  0.197

as shown in Fig. 8.

1.6991

 6.584 1.0494

(62)

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Ion Implantation and Activation, Vol. 1

10

Kunihiro Suzuki

0

Numerical Analytical

-1

10

-2

n

 2 ()

10

-3

10

-4

10

-5

10

-6

10

-7

10

-5

10

10

-3

-1

1

10 10 Reduced energy ()

3

10

Figure 8: Dependence of universal second order energy straggling  n 2    on reduced energy.

The third order energy straggling is also evaluated as 

 n 3   T2 f 3d 0



    3     4M M  3   6 1 2 T1i  cos   d   2 bdb  2 2   f   2   M 1  M 2       min  1     2    0 

    3     4M M  3  2 6  1 2  T1i d   d  2  cos   aU  2   2   f     M 1  M 2   2   1       min   2   0

 a 2  T   U 3 1i  n 3     3

 1.599 10

11

Z

0.23 1

 Z2

0.23

3  3  2Z1Z 2 M 1  3   M  M   n   eV atom  1 2  cm 2 





   

(63)

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Ion Implantation and Activation, Vol. 1

27

The universal  n 3    is given by 

        3 3 6  cos  d   d  2   n     2   f   2    1   2    min     0  

(64)

Substituting f    of Eq. 49 into Eq. 64, the dependence of  n 3    is given by 1

0.5 0.5 0.5  1 1     3  n         2.4145    0.12377      0.33495  

(65)

as shown in Fig. 9. 5

10

3

10

Numerical Analytical

1

n

 3 ()

10 10

-1

10

-3

10

-5

10

-7

10

-9 -5

10

-3

10

-1

1

10 10 Reduced energy ()

3

10

Figure 9: Dependence of universal third order energy straggling  n 3    on reduced energy.

Here, we express the collision angle of  as a function of incident and transferred energies. From Eqs. 26 and 27, we obtain

28

Ion Implantation and Activation, Vol. 1

 M1  M 2  cos   1 

2

M 1M 2

Kunihiro Suzuki

T2 f

(66)

2T1i

Modifying Eq. 24, we obtain 1  tan 2   1 

1  cos 2   M1   cos    M2  

2

2

  M 1  M 2 2 T2 f  1  1   M 1M 2 2T1i    1 2   M 1  M 2 2 T2 f M   1 1  M M T M 2  2 1 2 1i  1 

T2 f T1i

T2 f   1  1    2T  1i  

2



1 cos 2 

Therefore, we obtain a final form as

cos  

1  1    1

T2 f 2T1i

T2 f

(67)

T1i

where  is defined as



M2 M1

(68)

ELECTRONIC STOPPING POWER

Fig. 10 shows schematic electron interaction. Incident ion interacts with electron cloud of substrate atom, and the direction of incident atom is almost invariant.

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Ion Implantation and Activation, Vol. 1

29

There are two treatments associated with charge state of the incident ion. One is that the ion is naked and it is assumed to be charged of Z1 . The interaction is then expressed by the excitation of electron state of substrate atom. The other is that the incidention has its electron cloud and the interaction is expressed by the moment loss of the electron cloud of the incident ion.

Figure 10: Schematic expression of interaction between incident ion and electron.

The critical point whether each model becomes dominant can be evaluated by comparing the velocity of the incident ion and the electron velocity of orbit of the incident ion. The velocity of the orbit electron of the incident ion is evaluated by [5] vc  Z1 3 vB 2

where vB is the Bohr speed given by vB 

q2 1 1  1.44  107  eVcm   2.19  108  cm / s  34 1.054  10 4 0   eVs  1.602  10 19

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Assuming the atomic mass number of incident ion as M 1 , and the energy as E , the critical energy can be evaluated as

E



M1 2 3 Z1 v0 2



2

Applying this energy to B ion implantation, we obtain 11 103 2 kg   2 23   6 6.02  10 3 E  5  2.19  10  m / s   2    3.75  1013  J 

4.38  10 14 eV  1.602  10 19  2.34  106  eV  

 2.34  MeV 

We can easily evaluate the other ions’ critical energy base on the critical energy of B of EB as 4

M  Z1  3 E   EB M B  Z1 B 

The critical energy of P is about 10 MeV, and that of As is about 30 MeV. In the common VLSI processes, the energy region of around few keV is used, and the naked ion assumption is not available. However, some cases such as well formation of CMOS, few MeV ion implantations are used. In that case, the naked ion assumption becomes valid. Therefore, we should cover both models in general. Here, wefirst introduce Bethe’s electronic stopping power model [8], where naked ion is assumed. This model treats the detail of the electron excitation of substrate atom. We then describe Firsov and Lindhard’s electronic stopping power model. This model empirically expresses the interaction of electron cloud of incident and

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Ion Implantation and Activation, Vol. 1

31

substrate atom. This model is very important because it covers almost all energy region used in practical VLSI process. We further introduce model that covers the whole energy region, where the Bethe and Lindhard model are combined. BETHE’S ELECTRONIC STOPPING POWER MODEL

In Bethe’s electronic stopping power model, the electron stopping that is the energy loss per unit length is expressed by

Sn  E  

1  dE    N  dx e  2

 q 2  2 M 1Z12 Z 2  4me E  ln     me  IM 1   4 0  where N: substrate atom density Z1:charge number of incident ion Z2: charge number of substrate atom

 n   0 : electron excitation energy of substrate atom   0  n  : collision cross section from the energy state of  0 to  n . v: ion velocity

me : mass of electron I: average electron excitation energy of substrate atom We will show the derivation of Eq. 69 [9]. Wave functions for incident and scattered ions are given by

(69)

32

Ion Implantation and Activation, Vol. 1

in  k , 0 

1

Kunihiro Suzuki

ei  r0  r1 , r2 , , rz 

3 2

(70)

L

1

out  k ', n 

3 2

ei ' rn  r1 , r2 , , rz 

(71)

L

Equations 70 and 71 describe the wave function for before the collision and after the collision, respectively. Equation 70 expresses propagation of the incident ion with the plane wave of eik r and substrate electron not exited withthe energy level of  0 . Equation 71 expresses the propagation of the scattered ion with the plane wave of eik ' r and substrate electron with excited energy level of  n . The scattering probability  is given by for 2 k ', n V k , 0  

2

3

k ' M1  L    d  2  2 

(72)

from the Fermi’s golden rule (Appendix D). On the other hand,  can also be expressed by the probability flux J in and differential scattering cross section d d  as

  J in

d d d

(73)

The probability flux is given by (Appendix E)

J in 

 I m m*in*  M1 

 1  I m  3 e ik r ikeik r  M1  L  k  M 1 L3 

(74)

Assuming that the scattering angle is negligibly small, we only consider the incident ion direction component. We can then regard J in as scalar. Therefore, we obtain

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Ion Implantation and Activation, Vol. 1

33

d   d  J in d  3

2 k 'M  L  2 1 k ', n V k , 0   d   2  2   k d M 1 L3

k ' L3 2M 1  k ', n V k , 0 k 4  2

(75)

2

The potential can be expressed by V 

z1 z2 e 2 z e2  1 r i r  ri

(76)

The first term correspond to the Coulomb interaction between incident ion and nuclei of the substrate atom. The second term corresponds to the interaction between the incident ion and electrons of substrate atom with the number of Z2. Equation 76 uses the cgs unit system. It is used for convenience here. We can easily 2 convert the unit system to SI unit system by replacing of e2  q 4 0 . In SI unit system, the permittivity of vacuum is usually expressed by  0 which is the same for the energy level of  0 . The electron charge is usually expressed by q which is also used to express the change of momentum later here. Therefore, we derive the model in cgs unit system, and final model form (which do not contain energy level of  0 and momentum change q ), and then convert the model to the SI unit system in the end. Substituting Eq. 76 into Eq. 75, we obtain k ', n 

z1 z2 e 2 z e2  1 k, 0 r i r  ri



z1 z2 e 2 z1e 2 1  i  k  k '  r k ',   e n  i r  r k , 0 dr L3  r i 



1  iq r  z1 z2 e 2   1 e     n 0 dr  3 n 3  L  r   L 



 e i



iq  r

z1e 2 dr 0 r  ri

(77)

34

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

where

q  k k'

(78)

and 0  0  r1 , r2 , , rz 

(79)

n  n  r1 , r2 , , rz 

(80) 2

The first term  Z1Z2e r can be put outside of the bracket, and the integral in the second term can be put inside of the bracket in the derivation process in Eq. 77.

Figure 11: The relationship between the vector q and wave vector k and k ' .

Figure 12: The axis system for integration.

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Ion Implantation and Activation, Vol. 1

35

Figure 13: Volume element in spherical system.

We now consider the integration of the first term in Eq. 77. Defining  as shown in Fig. 11, eiq r can be expressed by

eiq r  eiqr cos On the other hand, we can select any system for integration with respectto r . Therefore, we can select x-axis with same direction of vector q as shown in Fig. 12 without losing generality. The angles  associated with q and r and the  in the spherical system are the same. Inspecting Fig. 13, the integration can be performed as 

iq  r  e dr  2     r 0

  eiqr cos 2  r sin  d  dr  0 r 

Converting the variable as

(81)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

cos    The integration region becomes

 :0   :1  1 and differential displacement becomes

 sin  d  d Utilizing these relations in Eq. 81, we obtain iq  r



 e dr  2   1 reiqr d  dr     0  1  r 

 eiqr  e  iqr  2  r dr iqr 0

(82)

 sin  qr   4  dr q 0 

We can further perform this integration as  4  sin  qr  4  dr  lim I m   e   r eiqr dr   q q  0  0 0 

  e   iq r     0  lim I m      iq   0    1  lim I m    0     iq 



4 q



4 q



   iq  4 lim I m  2 q  0    q 2 

4 q lim 2 0   q   q2 4  2 q 

(83)

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37

where I m denotes imaginary part. We then consider the integration of the second term in Eq. 77. Converting the integration dummy variable to related to each electron in each orbital of the substrate atom as

si  r  ri The integration becomes

 eiq  si ri   e iq  r i  r  r dr  i  s dr   i i  eiq si   eiq ri  dr i  si 4  2  eiq ri q i

(84)

From Eqs. 83 and 84, we then obtain

 z1 z2 e 2 z1e 2 4 e2   k ', n  k , 0  3 2   z1 z2 e2 n 0  z1 n  eiq ri 0  r Lq  i r  ri i  

4 z1 z2 e   n 0  Fn  q   L3 q 2  2

(85)

where Fn  q  is called form factor and defined by

Fn  q  

n  eiq  ri 0 i

(86)

z2

When q is 0, this becomes

lim Fn  q   q 0

z2 n 1 0 z2

  n0

(87)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

It may be interesting to consider the meaning of

e

iq  ri

here. The electron density

i

can be described as

  r      r  ri 

(88)

i

 is the Dirac’s delta function (Appendix F). The Fourier transformation can be expressed by

  q      r  eiq r dr      r  ri eiq r dr  i

(89)

  eiq ri i

Therefore,

e

iq  ri

is the Fourier transformation of the electron density.

i

Finally, the differential collision cross section is expressed by d k ' L3 2 M 1  k ', n V k , 0 d  k 4  2

2

k ' L3 2 M 1 4 Z1Z 2 e 2    n 0  Fn  q   k 4  2 L3 q 2  k ' 2 M 1 Z1Z 2 e 2    n 0  Fn  q   k 2 q2 

2

(90)

2

We consider the electron interaction and hence n  0 , and d k ' 2 M 1 Z1Z 2 e 2 Fn  q   d  k 2 q2

2

(91)

This considers the interaction between incident ion and the all binded electrons. We want to know the conversion

d d  d dq

(92)

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Ion Implantation and Activation, Vol. 1

39

to perform integration with respect to q for obtaining all cross section later. Inspecting Fig. 11, we obtain the relationship between q , k , and k ' as

q2  k  k '

2

(93)

 k 2  k '2  2kk 'cos  Therefore, we obtain

dq 

kk ' sin  d q

(94)

Incremental cubic angle is given by

d   sin  d d

(95)

We assume that the phenomenon is independent on  , and Eq. 95 is reduced to d   2 sin  d

(96)

We then obtain d d  d  2 sin  d d  q dq 2 sin  kk 'sin  kk ' d  2 q dq

(97)

Therefore, the differential cross section is given by d 2 q d 0  n  dq kk ' d  2 q k ' 2 M 1 Z1Z 2 e 2  Fn  q  kk ' k  2 q2 

8  Z1Z 2 e 2   q 2

2

3

2

Fn  q 

2

2

(98)

40

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

where



k M1

(99)

In the above discussion, we assume certain q , that is, the one case of with q differential collision cross section. We should therefore integrate for various q . In the collision, we should be careful about the energy and momentum conservations. This requirement may limit the number of q . We evaluate the limit of q , but tentatively set minimum and maximum values of q as qmin and qmax , respectively. The electron stopping power can then be evaluated as 

dE  N   n   0   0  n  dx e n

 d  N   n   0   dq  dq n  8  Z1Z 2 e 2  2 Fn  q  dq  N   n   0   2 2 3   q n  2

2

N

N

8  Z1Z 2 e 2   2 2 8  Z1e 2   2

2

2

2

n e

qmax

 1  n  n   0   q3  qmin

0

i

dq

Z 22

qmax

 n  n   0   qmin

iq  ri

2

n e i

iq  ri

0

dq q3

(100)

We then evaluate qmin . The energy of incident ion E can be expressed by

 2 k 2 M 1 2  E 2M1 2

(101)

where we use the velocity of Eq. 99. We can assume q is small which correspond to the small scattering angle. Therefore, we can assume   1 , and  n   0  E . Therefore, we can also assume k  k '  k . Then, we obtain

Physics of Ion Implantation

n  0 

Ion Implantation and Activation, Vol. 1

41

 2  k 2  k '2  2M1   k  k ' k  k '  2

 

(102)

2M1  2 k  k  k ' M1

 v  k  k ' 

Further, when   1 , we can approximate as

cos   1 

2

(103)

2

We can then modify the relationship of q and k of Eq. 93 as  2  q 2  k 2  k '2  2kk ' 1   2     k  k '   k  2

2

(104)

2

2       n 0    k   v 

q with small scattering angle can then be evaluated as 2

2    q   n 0    k   v 

(105)

Finally, the minimum q can be evaluated as

qmin 

n  0 v

(106)

It should be noted that it depends on  n . We then evaluate maximum value of q denoted by qmax . The energy of incident ion is given by

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

M 1v 2 E 2

(107)

Assuming M1v  q , the energy after the interaction can be expressed by E

1 2 M 1 v  q 2M1



1  M12v 2  2M1vq  2M1



1 M 1v 2  v q 2

(108)

Therefore, the energy loss E is given by

E  vq

(109)

This is the energy that the electron of substrate atom gains, and hence is also expressed by

 q  E 

2

2me

 v q  vq cos 

(110)

We then obtain q

2me v cos  

(111)

Therefore, qmax is given by qmax 

2me v 

(112)

Before, we substitute qmin of Eq. 106 and qmax of Eq. 112 into Eq. 100, we introduce tentative q of q1 . q1 is required that it is independent on  n and can be approximated as   1 . This requirement is rather vague, but q1 will be vanished in the last form. Therefore, it is only required that this kind of value exist. Equation 100 can be expressed by

Physics of Ion Implantation

8  Z1e 2  dE  N dx e 2v 2 N

2

8  Z1e 2  2 2

v

2

Ion Implantation and Activation, Vol. 1 q1

 n  n   0   n i eiq ri 0 qmin qmax

 n  n   0   q1

2

2

n

e i

iq  ri

0

43

dq q3

(113) dq q3

We can expand eiq ri of portion of first term denoted by K0 into Taylor series since q can be assumed to be small, and obtain q1

 K0    n   0   n  1  iqri  0  n i qmin q1

    n   0   n   iqri  0  n i qmin

2

2

dq q3

(114) dq q3

where we utilize n 0  0 .

Figure 14: Schematic expression of interaction between incident ion and electron of the substrate atom. The wave vector variation is also shown.

Let us consider the vector product of qr j after inspecting Fig. 14. The interaction of incident ion and electron is small in the attractive direction. Therefore, the direction variation is towards the nuclei of the substrate atom. The variation is

44

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

small, and hence the direction of q is almost the same as that of ri . Therefore, we can further perform calculation of Eq. 114 as q1

 K0    n   0   n   qri  0  n i qmin q1

     n   0   q 2 n  ri 0  n i qmin   

1 e2

  n   0  

1 e2

 

1 e2

  n   0 

q1

qmin

n

n

n dp 0 2

 0  n d p 0

n

2

n dp 0

n

2

2

2

dq q3 dq q3

dq q

 q  ln  1   qmin 

(115)

 q1v  ln    n  0 

2 1    0  n d p 0 ln  q1v  2  n e n 2 1  2    n   0  n d p 0 ln   n   0  e n



where

d p  e ri

(116)

i

and is the dipole moment of electron. According to the sum rule shown in Appendix G, we obtain

 

n

 0  n d p 0

n

2

 e  

2

2 me

Z2

(117)

Therefore, the first term of Eq. 115 becomes

 e 

2

2me

Z 2 ln  q1v 

(118)

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Ion Implantation and Activation, Vol. 1

45

We cannot apply sum rule to the second term in Eq. 115 since it has the term ln   n   0  . Therefore, we replace it with its average one. We introduce a variable of N on 

2me

 e 

  0  2  n

n dp 0

2

(119)

Summing up this with respect to n, we obtain

 Non  n



2me

 e 

  0  2  n

2

n

2me

 e 

n dp 0

2

 e 

2

2me

(120)

Z2

 Z2 Therefore, the sum becomes the number of electron binded to the substrate atom. Therefore, the average value of ln   n   0  can be expressed by ln   n   0   ln I 

N n

on

ln   n   0 

N



N

on

ln   n   0 

n

on

Z2

(121)

n

and the second term of Eq. 115 can be expressed by

2 Z 2 ln I 2me

K0 is then given by K0 

2  q v  Z 2 ln  1  2me  I 

(122)

Let us then consider the second term of Eq. 113. Utilizing the different sum rule as shown in Appendix G given by

46

Ion Implantation and Activation, Vol. 1

 

n

 0  n e

iq  ri

0

 q  

2

Kunihiro Suzuki

2

(123)

2me

n

and hence 2

  n

n

 0  n  e

iq  ri

 Z2

0

 q 

i

2

(123’)

2me

where Z 2 comes from the number of ri . We evaluate the term associated with the second term of Eq. 113 which we denote K1 as qmax

 K1     n   0    n q1

2

n e

iq  ri

i

0

dq q3

  q  dq   Z2 2me q 3 q1 qmax

2

(124)

q  2 Z 2 ln  max  2me  q1   2me v 2  2  Z 2 ln   2me  q1v  

Therefore, the final form for the electronic stopping power is given by

8  Z1e dE  N 2v 2 dx e

 K

2 2

0

 K1  (125)

4 Z 2 Z12 e4  2me v 2  ln  N  me v 2  I 

We then convert the expression in cgs unit system to SI one by the conversions of

e2  q

2

4 0

.

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Ion Implantation and Activation, Vol. 1

47

and use the relationship of

M1v 2 E 2 Equation 125 is modified as Se  E  

1  dE    N  dx e  2

 q 2  2 M 1Z 2 Z12 e 4  4me E  ln     4  m E 0 e  IM 1   

(126)

In this expression, I is called average excitation energy, and also evaluated experimentally and fitted analytical function [10] is

11.2  11.7 Z 2 I  eV    52.8  8.71Z 2

for Z 2  13 for Z 2  13

(127)

When Z 2 is large, the form I  eV   10 Z 2

(128)

is also used for rough estimation.

FIRSOV AND LINDHARD’S ELECTRONIC STOPPING POWER MODEL When the ion energy decreases, the assumption of naked incident ion becomes invalid. The energy range commonly used is around few 10 keV. We should consider the interaction of electron cloud of incident ion and that of substrate atoms. Firsov assumed that incident ion transfers the moment of its electron cloud and derived his electronic stopping power model [4]. Firsov assumed that the cloud distribution is determined based on Thomas-Fermi potential. Therefore, we first derive Thomas-Fermi potential.

48

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

There are  L 2   levels in momentum space like in the case of free electron. Assuming spherical symmetry, the number of energy level of the energy of 3

p0 2 E0  2m

(129)

is 4  L  2   p0 3   3  2  

3

where we consider the spin and multiply the associated factor of 2. Therefore, the electron density at r n  r  is given by

n r  

p0  r 

3

(130)

3 2 3

On the other hand, the potential energy and kinetic energy is assumed to be related to

E0  q  r  

p0  r 

2

2m

(131)

The potential and n  r  should hold Poisson equation of

  r   

qn  r 

(132)

0

Thomas Fermi potential is derived from keeping the relation of Eqs. 130, 131 and 132 as follows. Substituting Eq. 130 into Eq. 132, we obtain qp  r    r    02 3 3  0  3

(133)

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49

We further substitute Eq. 131 into Eq. 133, and expand  , and obtain the differential equation with respect to  as 2 1 d  r    r   r dr 2 3



q  2mq  r   2

(134)

3 2 0 3 3

3 q  2mq  2  r       2 3 2 0 3 

We assume that  has the form of the product of Coulomb potential and screening function   r  , that is

 r   

Zq 4 0 r

 r 

(135)

Substituting Eq. 135 into the first and third terms of right side of Eq. 134, we obtain 3

2 q  2mq  2 Zq 1 d   r     4 0 r dr 2 3 2 0 3

3

 Zq 2 r       4 0 r 

This can be arranged as 1

r2

3 d 2 r  2m3 q 6 Z  r 2  2 5 3 6 dr 9  0 

(136)

Introducing a normalized variable defined by x

r a

The related relation is

(137)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

d2 1 d2  dr 2 a 2 dx 2 and 1

r2

1 1 d 2 1 d 2 1 12 d 2  2 2   a x x 3 dr 2 a 2 dx 2 dx 2 2 a

Therefore, if we set a as 1

 9 5 03 6  3 a  3 6  2m q Z  2

1  3  3 4 0  2    1 2  4  mq 2 Z 3

(138)

2

1  3  3 a    B1 2 4  Z 3 where aB is the Bohr radius given by

4 0 2 aB  mq 2

(139)

The differential Eq. 136 can then be reduced to 3 d 2 x  2 dx 2 1 2

(140)

This differential equation can be solved numerically (Appendix H) under the boundary conditions of

  0   1,      0,  '     0

(141)

The numerical data is approximately fitted by various analytical expressions [3]. Moliere fitted as

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1

  x   0.35e 0.3 x  0.55e 1.2 x  0.10e 6.0 x

51

(142)

and Lindhard as

  x  1

x

(143)

3  x2

Fig. 15 shows the comparison of numerical results with analytical models of Moliereand Lindhard. Moliere’s analytical model well reproduces the numerical data. Lindhard model slightly deviated from the numerical data in the small x , but its form is simple.

1.0 Moliere

0.8

Lindhard Numerical



0.6 0.4 0.2 0.0

0

2

4

x

6

8

10

Figure 15: Thomas-Fermi screening function.

Firsov assumed that the electrons of incident ion cross through Firsov plane shown in Fig. 16 and transfer their momentum to the substrate atom. The force associated with the interaction is expressed by the variation of the moment per unit time, and is given by

52

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

v  F  mR  ne e ds S 4

(144)

Figure 16: Firsov electronic stopping power model.

The moment is related to the velocity of the incident ion ve and is related to the orbit velocity. ve is assumed to be identical in the direction. The velocity component perpendicular to the Firsov has ratio of cos  and is evaluated as





2 0

1

cos  2 sin  d  2  cos  d  cos     0

On the other hand the total volume angle is 4 . Therefore, the factor is 144.

1

4

in Eq.

The work W associated with this force related to the collision is then given by

W   FdR S

(145)

v  m  vdR   ne e ds S 4

 . Electron velocity v is related to the electron density n as where v  R e e 1

pe  r   3 2 3 ne  r   3  mve

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53

ve is then given by 1  2 ve  3 ne  r   3 m

(146)

The potential  and concentration is related as pe 2 2m 2 2 2 3 n 3    e 2m

q  r   

ne is then given by ne 

1 3 2 

3

2mq  2 3 

(147)

Therefore, the product is reduced to 1 1 1  ne ve  ne 3 2 ne  r   3 4 4 m 4 1  2 3   3  ne 3 4m 4

1 3 3 1   2 3  2 3 2  mq        2 3   4m  3 

(148)

mq 2 2  3 2 3 4 0 2   3 aB 

The work W is then given by

W

4 0 m vdR   2 ds  S 3 aB

(149)

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Assuming small angle scattering we approximate as v  dR  vdx , and substitute it into Eq. 149, and obtain

W

4 0 mv   2   ds dx  3 aB x   S

(150)

The potential with the distant t from the Firsov plane as shown in Fig. 17 is given by

 Zq  r        4 0 r  a  

2

2

  r     2 2    4 0  r   a   Z 2q2





Z 2q2

 4 0 

2

Z 2q2

 4 0 

2

2

  R 2      t 2 1   2 2   a R 2    t   2  

      

2

2   b 2 x             t2 1   2 2 2 2   a b x     2       t   2 2  

Therefore, we obtain

(151)

      

2

Physics of Ion Implantation





x 0

Ion Implantation and Activation, Vol. 1

55

  2 ds dx  S  

       Z 2q2 1   2 2 2   4 0   b    x   t 2         2 2  x  t 0 

     2 2   t 4 Z q    2 2 2  4 0     b    x   t 2       2 2   x 0 t 0

2   b 2       x   t 2   2 2   a      

2   b 2       x   t 2   2 2   a      

2       dx tdt 2        

2         dt dx       

(152)

Converting variable as



x t ,  2a a

The integration of Eq. 152 is modified as 

    ds dx x 0

2

S



      8 aZ 2 q 2    2 2  4 0     b    2   2   0  2a     0 

 2 2          b    2   2   d d       2a      

(153)

The integration with respect to  and  can be converted to the integration with respect to corresponding polar axis system of y ,  , that is

  y cos     y sin  The integration of Eq. 153 is then performed as

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Figure 17: Schematic explanation of Firsov plane. 

  2 8  aZ q y sin   2 x0  S  ds dx   4 2   0 y02  y 2 0  y 0  2





2

8 aZ 2 q 2

2  4 0  



2  0



 



y2  sin  d  2 2  y 0 y0  y



y2  8 aZ 2 q 2    2   4 0   y 0 y0 2  y 2 



 2  y0  y yd dy    2

 

2



 



2

y0 2  y 2  dy 

(154)

2

y0 2  y 2  dy 

where y0 

b 2a

Therefore, the work is given by

(155)

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1 

4 mv 8 aZ 2 q 2  y2  W 0  2 2 3 aB  4 0 2   y 0 y0  y  2









5 y   4  mq Z 3 v    2 2 2  3  4 0   y 0 y0  y

2

1

2



y2   4  3 1 53  Z v    2 2  3  aB  y 0 y0  y 



2

y0 2  y 2  dy 

4 mv 8 Z 2 q 2 1  3  3 aB  y2  0   1 2 2 3 aB  4 0 2 2  4  Z 3   y 0 y0  y 1 3

57

 

 









2

y0 2  y 2  dy  (156) 2

y0 2  y 2  dy 



2

y0 2  y 2  dy 

0.6 Numerical

0.5

Fit

Integral

0.4 0.3 0.2 0.1 0.0

0

1

2

y0

3

4

5

Figure 18: Dependence of integral on y0.

Fig. 18 shows the numerical results of integral in Eq. 156, where we use Moliere’s  . We can fit the results as 

y2    y 2  y 2    y 0 0





2 0.577 y0 2  y 2  dy   1  0.4 y

0

5



0.577 b    1  0.4  2a  

5

(157)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Therefore, W is given by 1

0.577  4 3 1 53 W  Z v  5  3  aB b    1 0.4   2a  

(158)

Integrating W with respect to collision factor b , we obtain electronic stopping power Se as 

Se   2 bW  b  db 0



 b  4  5 3 v  b db  0.5772   Z 5 aB    3  b  0 1    5a  1 3

1

 4  3 5 v 25 2 a  0.5772   Z 3 aB 12  3  2   0.57725  4  5 3 v  1  3  3 aB     Z   1 aB  2  4  Z 3  6  3    2 2 q 4 0 0.57725 ZaB vaB  32 4 0 q 2 1 3

(159) 2

Bohr speed vB given by

vB 

q2 4 0 

(160)

Equation 159 is then expressed as 0.57725 2 q2 v Se  8 ZaB 328 4 0 vB q2 v  0.1778 ZaB 4 0 vB

(161)

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Ion Implantation and Activation, Vol. 1

59

The factor is slightly different from original one [4]. This difference comes from the fitting of the integration and Firsov uses a certain correction factor of 3 4 in the derivation process. Later, we multiply a fitting factor to the model, and hence the difference is not matter. Thomas-Fermi potential model, the atomic number Z in the Thomas-Fermi potential model is ambiguous since it is also influenced by the nuclei of the substrate atom. Firsov treated it empirically the electron is binded by the nuclei with effective atomic number of Z1  Z2 , and is hence

Z  Z1  Z2 On the other hand, Lindhard derived a similar model without detailed derivation process [5]. Lindhard’s model can be obtained by replacing Z in Firsov model by Z 6 Z1  Z 2 Z 1 0.177 Z 2 3  Z 2 3 1 2 1





3

2

Lindhard model is hence almost the same as the Firsov model but only modified its ambiguous physical parameter. Assuming Z1  Z 2  10 , the values of each model factor are 0.177  Z1  Z 2   3.54  Firsov  1

Z1 6

Z

1

Z1  Z 2 2

3

 Z2

2

3



3

2

 5.1  Lindhard 

The values are different but the order is same. We need a fitting parameter to obtain better fit to the experimental data as shown later, and multiply a factor re as

Se  re 0.1778  Z1  Z 2  aB

q2 v : Firsov 4 0 vB

(162)

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Ion Implantation and Activation, Vol. 1

S e  re 8 Z1 6 1

Z

1

Z1  Z 2 2

3

 Z2

2

3



3

2

aB

Kunihiro Suzuki

q2 v : Lindhard 4 0 vB

(163)

Inspecting Appendix C as is the case of nuclei stopping power, the expressions are reduced to convenient form using energy E as

Se  re 2.14 1017  Z1  Z 2 

Se  re1.211016 Z1 6 1

Z

1

1 M1  g 

Z1  Z 2 2

3

 Z2

2

3



E  eV  eV cm 2  : Firsov

1 3

2

M1  g 

(164)

E  eV  eV cm2  : Lindhard (165)

COMBINED MODEL As we mentioned, Bethe’s model is valid in high energy region and Lindhard in low energy region. Therefore, we try to combine these models to make a model valid for entire energy region [11]. We first modify Bethe’s model before combination. The maximum of the Bethe’s model can be evaluated from 2

Se  q 2  2 M 1Z 2 Z12   1  E     ln     me E  4 0  E  E  Er   E 1  ln   2 2 2  q  2 M 1Z 2 Z1 E  Er   0   2 me E  4 0 

(166)

and we obtain the energy at the maximum Se is

E  eEr

(167)

where Er 

M1 I 4me

(168)

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Ion Implantation and Activation, Vol. 1

61

and we should note that e in Eq. 167 is e  Exp 1  2.71828 . We then modify the Bethe’s model Se _ mB  E  as

 q 2 2 2 M Z Z 2 1 2 1   m 4   e eEr 0  Se _ mB  E    2  q 2  2 M 1Z 2 Z12  E  ln     me E  Er   4 0 

for E  eEr (169)

for E  eEr

We can alternatively use 2

 q 2  2 M 1Z 2 Z12  E Se _ mB  E    ln  1    me E  Er   4 0 

(170)

which Biersack proposed [12]. This model becomes Bethe’s model when E  Er . When E  Er

 E ln 1   Er

 E   Er

and the model is reduced to 2

 q 2  2 M 1Z 2 Z12 Se _ mB  E     me Er  4 0 

(171)

This is identical to Eq. 169 for low energy region. We propose the combined model as  LH  1 1  1       Se  re SeL   Se _ mB 

 LH

  

   

1

 LH

(172)

This model becomes Lindhard model in low energy region, and Bethe model in high energy region.

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

4

10

B in Si LH

LH =2 = 1.45

Lindhard

 LH =1 Modified Bethe

3

10

e

S (keV/m)



2

10

2

3

10

4

5

10 10 Energy (keV)

10

Figure 19: Dependence of Se on energy with various  LH .

 LH controls the transition from Lindhard model to Bethe model. As  LH increases, the transition becomes sharp as shown in Fig. 19. Ziegler utilized the linear response method and treated the transition from Lindhard’s model to Bethe’s model more universally [1]. Although the Ziegler’s model is physical one, we cannot obtain a good agreement as it is [11] and tune its parameters, which is not easy to handle.

Stopping power (keV/m)

10

4

Se (As)

Si substrate  = 1.65

10

3

10

2

10

1

Se (P)

Sn (As)

LH

Sn (P) Se (B)

Sn (B)

0

10 -1 10

0

10

1

2

3

10 10 10 Energy (keV)

Figure 20: Stopping powers of B, P, and As in Si substrate.

10

4

5

10

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63

Fig. 20 shows the dependence of stopping powers on energy. We do not need to care about this subject of the limitation of Lindhard’s Se model for P and As since eEr values are much larger than 5 MeV for P and As. However, if we use much higher energies for these ions or light ions such as B, we should find optimal values of  LH for each combination of ion and substrate since the model is empirical. 10

21

Concentration (cm -3)

As: 40 keV 1 x 1015 cm-2

10

20

10

19

10

18

10

17

10

16

a-Si c-Si

0

50

100 150 Depth (nm)

200

Figure 21: Comparison of As ion implantation profile in crystalline Si substrate with one in amorphous Si. The data are form the database in FabMeister-IM [13].

CHANNELING We implicitly assume amorphous substrate. However, the substrate used in VLSI processes is crystalline. The prominent phenomenon associated with crystalline is channeling. A lucky ion in a crystalline substrate is always far from nuclei, and reaches a deep position. Fig. 21 shows the As ion implantation profiles in amorphous and crystalline substrates, where the data are in the database of FabMeister-IM [13]. We can observe the channeling tail in the low concentration region. It should be noted that the profile in the peak region is almost the same. This is because the crystalline substrates are damaged by the ion implantation and the substrate becomes amorphous when the dose is high.

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Kunihiro Suzuki

Figure 22: Schematic description of channeling.

Si atoms are arranged periodically in crystalline Si, and an ion with low incident angle may go far in the substrate with less interaction with nuclei, which is called channeling phenomenon. The critical angle for channeling is empirically described by the following. Fig. 22 shows a schematic description of channeling. We assume the distance of the lattice is d . We denote the velocity component parallel to the lattice as v , and vertical one as v . The channeling ion gradually changes its direction through the interaction with atom nuclei, that is, the ion passes through many substrate atoms. The minimum distance between incident ion and substrate atom rmin exists and it can be evaluated by incident ion energy and potential. The time that the ion near rmin go outside the lattice column is about rmin v . On the other hand, the time the ion go the distance of lattice size of d is d v . If d v is shorter than rmin v , the ion direction in modified by the next ion so that the ion is inside the lattice column. Therefore, the critical situation is expressed by

rmin d  v v Therefore, rmin is related to the critical angle c as

(173)

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rmin  d

Ion Implantation and Activation, Vol. 1

v  d tan c  dc v

65

(174)

The vertical energy component E is given by E  E sin 2 c  Ec 2

(175)

On the other hand, Lindhard assumed the potential at the distance of r is given by [14] e 2 Z1Z 2   3aU U r  ln 1   d   r 

Assuming



3au

r



2

  

2

   

(176)

 1 , Eq. 176 is reduce to

e2 Z1Z 2  3aU U r   d  r

  

2

(177)

Equating the energy component and U  rmin  , we obtain

e2 Z1Z 2  3aU  d  rmin

2

 2   Ec 

(178)

Substituting Eq. 174 into Eq. 178, we obtain

c 

4

3e 2 aU 2 Z1Z 2 d 3E

(179)

2

3e 2 Z1Z 2  0.8854aB  1 4   d 3  Z10.23  Z 2 0.23  E

Converting to the expression in SI unit, we obtain 2

c 

4

q 2 3Z1Z 2  0.8854aB  1   4 0 d 3  Z10.23  Z 2 0.23  E

(180)

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Kunihiro Suzuki

10

Critical angle (o)

Si substrate d = 0.543 nm

B P As

5

0

0

100

200 300 400 Energy (keV)

500

Figure 23: Dependence of critical channeling angle on energy.

This model shows that the critical angle increases with decreasing energy as shown in Fig. 23. c is around 3 for the energy region more than 100 keV, and it increases significantly when the energy is less than 50 keV. It should be denoted that the critical angle increases with increasing atomic number. 10

21

10

20

-3

Concentration (cm )

As: 40 keV c-Si

10

19

10

18

10

17

10

16

0

50

SIMS 1 x 10

13

cm

-2

SIMS 1 x 10

14

cm

-2

SIMS 1 x 10

15

cm

-2

100 150 Depth (nm)

200

Figure 24: Dependence of As ion implantation profile on dose. Data are from FabMeister-IM [14].

DAMAGE The profiles linearly depend on dose in amorphous layer substrate. However, it is not true in crystalline substrate. Fig. 24 shows the dependence of As ion

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67

implantation profiles on dose. The profile is shifted almost parallel from the dose of 11013 cm2 to 11014 cm2 . The channeling tail profile is invariant if we increase the dose to 11015 cm2 , while the profile near the peak region increases linearly. This indicates that amorphous layer is formed and channeling is suppressed. The damage should be related to the nuclei stopping power. Therefore, the portion of the energy loss is dQE and is given by dQE 

Sn  E  dE Sn  E   Se  E 

(181)

The damage accumulation QE is then expressed by Sn  E   QE   dE 0 S n  E   Se  E  E

(182)

Therefore, the number of displaced substrate atom  is simply expressed by



QE Ed

(183)

This neglects the recombination and the dynamic trajectory of the displaced atom, and the energy less than Ed . Since the amorphous layer formed by ion implantation is important, it will be treated in more detail later.

REFERENCES [1] [2] [3] [4] [5]

J. F. Ziegler, J. P. Biersack, and U. Littmark, The stopping and range of ions in solid, Pergamon, 1885. SRIM-2003: http://www.srim.org/ H. Maes, W. Vandervorst, and R. Overstraeten, Impurity Profile of Implanted ions in Silicon, Chapte 8 in Impurity Doping edited by F. F. Y. Wang, North-Holland Publishing Company, 1981. O. B. Firsov, "A quantitative interpretation of the mean electron excitation energy in atomic collisions," Sov. Phys. JETP, vol. 8, number 5, pp. 1076-1080, 1959. J. Lindhard and M. Scharff, “Energy dissipation by ions in the keV region,” Physical Review, vol. 124, pp. 128-130, 1961.

68

[6] [7] [8] [9] [10] [11] [12] [13] [14]

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Kunihiro Suzuki

J. F. Gibbons, “Ion implantation in semiconductors-Part I range distribution theory and experiments,” Proc. of IEEE, vol. 56, No. 3, pp. 295-319, 1968. D. K. Brice, “Recoil contribution to ion implantation energy deposition distributions,” J. Appl. Phys., vol. 46, pp. 3385-3394, 1975. H. A. Bethe, “Zur Theorie des durchgangs schneller korpuskukasrahlen durch materie,” Ann. Phys. (Leipzig) 5 (1930) 325. Y. Tosaka, private communication. P. Dalton and J. E. Turner, “New evaluation of mean excitation energies for use in radiation dosmetry,” Health Physsics Pergamon Press, vol. 15, pp. 257-262, 1968. K. Suzuki, Y. Tada, Y. Kataoka, and T. Nagayama,“Monte Carlo Simulation of Ion Implantation Profiles Calibrated for Various Ions over Wide Energy Range,”J. Semiconductor Technology and Science, vol.9, No. 1, pp. 67-74, 2009. J. P. Biersack, and L. G. Haggmark, “A Monte Carlo computer program for the transport of energetic ions in amorphous targets,” Nuclear Inst. And Meth., vol. 174, pp. 257-269, 1980. FabMeister IM http://www.mizuho-ir.co.jp/science/ion/index.html J. Lindhard, Kgl. DanskeVidenskab. Selskab., Mat.-Fys.Medd., vol. 34, No.14, 1965.

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69

APPENDIX A TWO-BODY PROBLEM Consider two-body problem with masses of M 1 and M 1 they interact each other under the potential that only dependent on distance r. Newton theory for each mass are described by

d 2r1 M1 2  F12 dt M2

d 2r2  F21 dt 2

(A-1)

(A-2)

r1 and r2 are the location vector for each mass, and F12 is the force that mass 2 influenceｓ to mass 1, and F21 is the force that mass １ influenceｓ to mass 2. From Newton’s law, we assume F21  F12 , and obtain d 2r1 d 2r2 d 2  M 1r1  M 2r2  M1 2  M 2 2   M1  M 2  2  0 dt dt dt  M 1  M 2 

(A-3)

Location vector related to center of mass system R is given by R

M 1r1  M 2r2 M1  M 2

(A-4)

Therefore, Eq. A-3 reduces to

d 2R  M1  M 2  2  0 dt

(A-5)

Therefore, center of mass moves linear with constant speed, and total momentum is conserved. We use the center of mass system. We define the relative location vector r given by

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r  r1  r2

Kunihiro Suzuki

(A-6)

From Eqs. A-1 and A-2, we obtain

d 2r1 d 2r2 F12 F21  2   dt 2 dt M1 M 2 Using r this equation can be modified as

d 2r F12 F12 M 1  M 2    F12 dt 2 M 1 M 2 M 1M 2 Therefore, we obtain

Mc

d 2r  F12 dt 2

(A-7)

where Mc 

M 1M 2 M1  M 2

(A-8)

This is called reduced mass. The scattered angle in this reduced one body system is the same as the scattered angle of mass 1 in the center of mass system. Once we obtain r  t  by solving Eq. A-7, we can evaluate r1  t  and r2  t  from Eqs. A-4 and A-6 as r1  t   R 

M2 r t  M1  M 2

(A-9)

r2  t   R 

M1 r t  M1  M 2

(A-10)

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71

B NEWTON LAW IN POLAR SYSTEM If we treat system where the force is only dependent on distance, the polar system  r ,  is more convenient than Cartesian system  x, y  , which is shown in Fig. B-1. We derive Newton law in polar system.

Figure B-1: Cartesian and polar systems.

Newton’s law is

m

d 2r F dt 2

(B-1)

The vector components in Cartesian system are given by

m

d 2x d2y  F m  Fy , x dt 2 dt 2

(B-2)

We denote the unit vectors in x-direction and y-direction as e x , e y , and location and force vectors are expressed by r  xe x  ye y

(B-3)

F  Fx e x  Fy e y

(B-4)

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Kunihiro Suzuki

The location vector can be expressed in polar system  r ,  as r  r cos  e x  r sin  e y

(B-5)

Velocity vector v , and acceleration vector a are given by dr dt dr d   cos  e x  sin  e y   r   sin  e x  cos  e y  dt dt

v

(B-6)

d 2r dt 2 d 2r dr d  2  cos  e x  r sin  e y     sin  e x  cos  e y  dt dt dt dr d d 2   sin  e x  cos  e y   r 2   sin  e x  cos  e y   dt dt dt

a

2

 d  r    cos  e x  r sin  e y   dt  2  d 2r  d    2 r    cos  e x  r sin  e y   dt dt      d 2 dr d  r 2  2    sin  e x  cos  e y  dt dt   dt

(B-7)

The unit vector in the r -direction e r and one in the direction perpendicular to r opposite to the clock rotation direction el can be expressed by e r  cos  e x  r sin  e y

(B-8)

el   sin  e x  r cos  e y

(B-9)

Therefore, the vector v , acceleration a , and force F are expressed by e r and el as v

dr d er  r el dt dt

(B-10)

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73

2  d 2r  d 2 dr d   d    a  2 r e r 2 2    el r   dt  dt dt dt dt      

(B-11)

F  Fr er  Fl el

(B-12)

The Newton’s law in polar system is then given by 2  d 2r  d   m 2  r     Fr  dt  dt   

(B-13)

 d 2 dr d  m r 2  2   Fl dt dt   dt

(B-14)

In this force system, Fl  0 , and hence

 d 2 dr d  m r 2  2 0 dt dt   dt

(B-15)

This is identical to d  2 d   mr 0 dt  dt 

(B-16)

This expresses the conservation of angular momentum L given by L  mr 2

d dt

(B-17)

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Kunihiro Suzuki

C NOTATION ABOUT UNIT SYSTEM Various unit systems are used, in this section, and we sometimes have difficulties in obtaining quantitative values of the models. The conversion for convenient unit system can be done as the following. Starting with q  1.60218 1019 C 

(C-1)

F 

 0  8.854 1012   m

(C-2)

We can calculate the following quantities. 1.60218  1019   mC   q2  4 0 4 8.854  10 12  F  2

 2.307  10 28  Joul m 

(C-3)

2.307  10 28   100  eV cm  1.60218  10 19  1.44  107  eV cm 

aB 

4 0  2 q2m

4 8.854 1012  F  1.055 1034  J  s   m  2 1.60218 1019 C  9.11031  kg 

2

(C-4)

 5.30 1011  m   5.30 109  cm  5.30  109  cm  4 0 aB 8.854  8.854 1.44  107  eV cm  q2  1   3.25  102    eV 

(C-5)

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aB 

Ion Implantation and Activation, Vol. 1

q2  5.30 109  cm1.44 107  eV cm 4 0

75

(C-6)

 7.63 1016 eV cm2  Let us consider v and vB . vB 

q2 4 0 

1.60218 10 C  19



2

4  0  8.854 1012  F  1.055 1034  J  s   m m  2.19 106   s  cm   2.19 108    s 

(C-7)

The relationship between v and energy E is then given by

1 M1  g  2 v  E  J   qE  eV  2 N av 1000

(C-8)

v is then expressed with the energy as v 

2 N av 1000qE  eV  M1  g 

26.02204 1023 10001.60218  1019 E  eV  M1  g 

 1.39 104

E  eV   m  M 1  g   s 

 1.39 106

E  eV   cm  M 1  g   s 

(C-9)

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Kunihiro Suzuki

Therefore, the ratio is

v  vB

1.39 106

E  eV  cm   M1  g   s 

2.19 108 cm   s

 6.35 103

(C-10)

E  eV 

M1  g 

The factor is given by E  eV  q2 v 8 aB  8 5.30  109  cm 1.44  107  eV cm 6.35  103 4 0 vB M1  g   1.21 1016

E  eV 

 eV cm 2  M1  g  

(C-11)

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77

D FERMI’S GOLDEN RULE

An electron starts from one state to the other state through an interaction. Fermi’s golden rule gives a probability of the transition per unit time. We denote Schrodinger equation without interaction. i

  H 0 t

(D-1)

We assume that the corresponding wave functions are known, and a wave function with energy eigenvalue of En is expressed by

n  En ,   n

(D-2)

 denotes the other parameter that characterizes the state. We assume that any wave function is expressed by the sum of these wave functions and is given by  iE t    t    Cn  t  n exp   n     n

(D-3)

We assume that the interaction is weak, and denote the interaction potential as V .  is introduced to clarify the order of incremental value. Schrodinger equation is then expressed by i

   t    H 0  V    t  t

(D-4)

Substituting Eq. D-3 into Eq. D-4, we obtain i

  iE t   iE t  Cn  t  n exp   n    H 0  V   Cn  t  n exp   n   t n       n

This can be expanded to dC  t   iE t   iE t  exp   n   i n n exp   n  dt       n n  iE t   iE t    En Cn  t  n exp   n    Cn  t  V n exp   n     n    n

 E C t  n n

n

(D-5)

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Kunihiro Suzuki

and obtain

i n

dCn  t   iE t   iE t  n exp   n    Cn  t  V n exp   n  dt    n    f

Operating

form the left side of Eq. D-6, and considering we obtain

f n   fn i

dC f  t  dt

(D-6)

(D-7)

 i  E f  En  t    Cn  t   f V n exp    n  

(D-8)

We assume that Cn  t  is expressed by

Cn  t   Cn

0

 t   Cn1  t    2Cn 2  t   

(D-9)

Substituting Eq. D-9 into Eq. D-8, we obtain

i

d C f

0

 t   C f1  t    2C f2  t    dt

  Cn

 0

t   

n

2

Cn  t    Cn 1

3

 2

i E  E t   t    f V n exp  f n    

(D-10)

Comparing the both sides of components associated of 0 order with respect to  , we obtain

i

dC f0  t  dt

0

(D-11)

Therefore, the 0-th order is invariable with respect to time. If the initial state is 0 , we obtain  C0 0   t   1   0 Cn  0  t   0

(D-12)

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79

Comparing the both side of components associated of first order with respect to  , we obtain

i

dC f1  t  dt

 i  E f  En  t    Cn 0  t  f V n exp    n    i  E f  E0  t   f V 0 exp     

(D-13)

Assume that the initial time as  T 2 , differential equation of Eq. D-13 can be solved as  i  E f  E0  t  f V 0  dt C f t    exp   i  T     t

1

(D-14)

2

Therefore, the probability of the state f 2

T  C f1    2

f V 0 

at the time of

  i  E f  E0  t   dt  exp  T      t

2

2

T

2

is given by

2

(D-15)

2

The variation rate  ' can be evaluated as

' 1  T

 Cf   2  T 

f V 0 2

2

(D-16) 2

 i  E f  E0  t  dt 1   i  E f  E0  t  1    dt  exp   exp  2  T   TT           t

2 

t

2

2

Taking long T , the term reduced to delta function of

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 i  E f  E0  t  dt 1  lim     E f  E0   exp  T  2      T   t

2

Therefore, we can set E f  E0 in the last term of Eq. D-16, and obtain  i  E f  E0  t  1  dt  1  exp   TT     t

2

Finally, Eq. D-16 reduces to

'

2 

f V 0

2

  E f  E0 

(D-17)

This means that the transition never occur unless the energy is conserved. We only consider energy up to now. Incident ion’s wave vector becomes k ' after the interaction in the direction of volume angle element of d . We should include this phenomenon to Eq. D-7. The energy corresponding k ' is assumed to be E ' and related energy density of   E ' , the number of quantum state is expressed by

  E ' dE '  4 n 2 dn

d  n 2 dnd  4

(D-18)

We now need to know the relationship between E and the number density of energy level n . In the free electron approximation theory, k is given by k

2 n L

Energy is then related to k as

(D-19)

Physics of Ion Implantation

2k 2  2  2 E   2M 1 2M 1  L

Ion Implantation and Activation, Vol. 1

 n 

81

2

(D-20)

We then obtain 2

 2 k 2  2  2  dE     ndn 2M 1 M 1  L 

(D-21)

Substituting Eqs. D-19 and D-21 into Eq. D-18, we obtain

  E ' dE '  nndnd  M  L 2 L  k' 1  dE ' d  2  2  2  k ' M  L 3 1    dE ' d  2   2 

(D-22)

Therefore, the density where the ion is scattered to k’ is given by

  E ' 

3

k ' M1  L    d  2  2 

(D-23)

Finally, we obtain the transition probability per unit time is given by

   '   E ' 

2 

f V 0

2

  E f  E0 

3

k ' M1  L    d  2  2 

(D-24)

This is called by Fermi’s golden rule. If we assume energy conservation from the first, and express the state as wave vector, Eq. D-23 is expressed by



2 k'V k 

2

3

k ' M1  L    d  2  2 

We use this expression in the text.

(D-25)

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E FLUX DENSITY OF PROBABILITY

We consider one-dimensional flux with axis x. We denote the flux for x  a, b as fa and f b , respectively. fb  f a corresponds to the decreased element of probability  * in the region  a, b  , that is, fb  f a  

d b *   dx dt a b

  d * d    *     dx dt  a  dt

(E-1)

On the other hand, Schrodinger is given by i

  H t

(E-2)

The complex conjugate of the equation is given by

i

 *  H * t

(E-3)

Substituting Eqs. E-2 and E-3 into Eq. E-1, we obtain b

  1  1  f b  f a      H  *     *  H    dx i i      a   i b     H  *     *  H    dx  a b

  2  2 *   2  2 * *      V    dx V       2 2   2m x   2m x

b

2   2 * *      dx  2 x 2   x

i    a 

i  2m  a

b b    *   *    i    * *         dx     x  a  x x x x   2m   x   a   b



i   *    *   x  a 2m  x

(E-4)

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Therefore, the flux at any location is expressed by

f 

i   *    *   x  2m  x

(E-5)

and is called by probability of flux. This can be modified as *  i   *  *  f        2m  x  x 

 

i    2i Im  *  2m  x 

(E-6)

    Im  *  m  x 

Considering the other direction, the vector flux is given by f

 Im  *  m

(E-7)

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F DIRAC’S DELTA FUNCTIONδ Dirac’s delta function  is defined as

for a  a0

0 1

  a  a0   

for a  a0

(F-1)

A simple expression for  is given by

a = 1 2

 -

e iaxdx

(F-2)

Consider a function f in which the integration region is finite of g in Eq. F-2, that is, f a = 1 2

g -g

e iaxdx

iag -iag = 1 e -e ia 2 sin ag g =  ag

(F-3)

Integrating Eq. F-3 with respect to a, we obtain  -

g f a da = 

 -

1 =

sin ag ag da 

-

sin s s ds

=1



Values for a  0,  are g g f 0 = f ± g =0

(F-4)

(F-5) (F-6)

Therefore, the contribution of f to the integration is at 0 in the limitation of g of ∞. Therefore, Eq. F-2 satisfies the definition of δ function of Eq. F-1.

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85

G SUM RULE

We denote Hamiltonian H as

pˆ 2 H V 2m

(G-1)

and assume that the eigen vector n constitutes complete normalized one, and its corresponding eigen value is  n , that is H n  n n

(G-2)

On the other hand,

n

n 1

(G-3)

n

is generally valid. We hope to prove the following Eqs. G- 4 and G-5.

  n   0 

n eiqx 0

  n   0 

nx0

2



n

n

2



2 q 2 2m

2 2m

(G-4)

(G-5)

We first prove the exchanging relationship of

2 q 2   H , eiqx  , eiqx       2m

(G-6)

In general,

f  x   pˆ x , f  x    f  x  pˆ x  i  f  x  pˆ x x f  x   i x

(G-7)

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Kunihiro Suzuki

is valid, and we have eiqx  pˆ x , eiqx   i x iqx  qe

(G-8)

The exchanging relationship between pˆ x 2 and eiqx is  pˆ x 2 , eiqx   pˆ x  pˆ x , eiqx    pˆ x , eiqx  pˆ x  q  pˆ x eiqx  eiqx pˆ x 

(G-9)

We assume one-dimensional potential V , and the Hamiltonian is

H

pˆ x 2 V 2m

(G-10)

Therefore, the exchanging relationship is given by q  H , eiqx   pˆ x eiqx  eiqx pˆ x   2m

(G-11)

where V does not operate to the function, and vanishes in this exchanging relationship. Further, we can obtain exchanging relationship between  H , eiqx  and eiqx as q   H , eiqx  , e iqx    ˆ iqx iqx ˆ  iqx     2m  px e  e px , e  q  pˆ x eiqx , e  iqx   eiqx pˆ x , e  iqx   2m q pˆ x  e  iqx pˆ x eiqx  eiqx pˆ x e  iqx  pˆ x    2m q  e iqx  eiqx pˆ x  qeiqx   eiqx  e iqx pˆ x  qe  iqx  2m







 q   2m



2

(G-12)

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This is also expressed by   H , eiqx  , e  iqx       H , eiqx  e  iqx  e  iqx  H , eiqx 

(G-13)

 Heiqx e  iqx  eiqx He  iqx  e  iqx Heiqx  e  iqx eiqx H    Heiqx n n e  iqx  eiqx H n n e  iqx  e  iqx H n n eiqx  e  iqx n n eiqx H  n

Operating 0 from both sides of Eq. G-13, we obtain 0   H , eiqx  , e  iqx  0   0 Heiqx n n e  iqx 0 n

 0 eiqx H n n e  iqx 0 n

 0 e  iqx H n n eiqx 0 n

  0 e  iqx n n eiqx H 0 n

   0 0 eiqx n n e  iqx 0 n

  n 0 eiqx n n e  iqx 0 n

  n 0 e  iqx n n eiqx 0 n

   0 0 e  iqx n n eiqx 0 n

 2   n   0  n eiqx 0 n

2

(G-14)

Therefore, we obtain

  n

n

 0  n e

iqx

0

2

 q   2m

When q is small, we obtain

2

(G-15)

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Ion Implantation and Activation, Vol. 1

n eiqx 0

2

 n 1  iqx 0

2

 n 0  iq n x 0  iq n x 0  q2 n x 0

Kunihiro Suzuki

2

2

(G-16)

2

Therefore, we obtain

  n   0 

2

n x0



n

2 2m

(G-17)

Let us extend this one dimensional analysis to three-dimensional case. The exchanging relationship is given by

 pˆ x , e

iq  r

  i

e



i qx x  q y y  q z z

x



 qx eiq r

(G-18)

The other moment operator associated with the direction is similar and denote them as  , and is given by  pˆ  , e iq  r   q e iq  r

(G-19)

Further, we obtain  pˆ  2 , e iq  r   q  pˆ  e iq  r  e iq  r pˆ  

Therefore, we obtain

(G-20)

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89

 H , eiq r  q     pˆ  eiq r  eiq r pˆ    2m q      pˆ  eiq r , eiq r   eiq r pˆ  , e iq r    2m q      pˆ  eiq r , eiq r   eiq r pˆ  , e iq r    2m q      pˆ  eiq r eiq r  eiq r pˆ  eiq r   eiq r pˆ  eiq r  eiq r eiq r pˆ     2m q     eiq r  q eiq r  eiq r pˆ    eiq r  q eiq r  eiq r pˆ     2m  

 q 

2

2m



 q  

2

(G-21)

2m

Consequently, we obtain the same expression as for one-dimensional as

  n   0  n eiqr 0 n

  n

n

 0  n r 0

2

2



2 q 2 2m

2  2m

(G-22)

(G-23)

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Kunihiro Suzuki

H NUMERICAL EVALUATION OF THOMAS FERMI POTENTIAL

We solve the differential equation given by

x1 2

3 d 2 = 2 2 dx

(H-1)

under the boundary conditions of  0 = 1,   = 0, '  = 0

(H-2)

The boundary conditions of Eq. H-2 is not used as they are. The boundary condition at 0 is unstable. We hence move the origin from zero, and use  0.00000000001 = 1

(H-3)

We could not use boundary condition at ∞. We use the boundary condition of ' 0.00000000001 = c

(H-4)

That satisfies the boundary conditions of  =0 '  = 0

(H-5)

c influences the results sensitively, and we obtainthe appropriate results with c of -1.588071 as shown in Fig. H-1.

Physics of Ion Implantation

Ion Implantation and Activation, Vol. 1

1.0 c = -1.56



- 1.58

0.5

0.0

- 1.588071

- 1.60

0

5 x

10

Figure H-1: Dependence of numerical results for Thomas-Fermi potential on c.

91

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92

CHAPTER 3 Monte Carlo Simulation Abstact: The physics for the ion implantation can be directly implemented in Monte Carlo (MC) simulation, and hence MC simulation is widely used for predicting ion implantation profiles. Here, we compared MC simulation results with a vast database of ion implantation secondary ion mass spectrometry (SIMS), and showed that the Monte Carlo data sometimes deviated from the experimental data. We modified the electronic stopping power model, calibrated its parameters, and reproduced most of the database. We also demonstrated that the Monte Carlo simulation can accurately predict profiles in a low energy range of around 1 keV once it is calibrated in the higher energy region.

Keywords: Ion implantation, low energy ion implantation, Monte Carlo, nuclear stopping power, electronic stopping power, channeling, damage, collision parameter, transferred energy, scattering angle, energy loss, cross section, SIMS, As, P, B, Ge, HfO2. INTRODUCTION The physics discussed in chapter 2 can be directly implemented in Monte Carlo (MC) simulation [1-6]. Here, we show the MC calculation procedure [5, 6], and evaluate its accuracy by comparing the results with vast experimental data used in [5, 6]. The channeling and damage accumulation should be precisely described to predict profiles in crystal substrates. We treat physics related to the channeling, but do not treat resultant profiles. CALCULATION PROCEDURE Once we set the collision parameter b , everything needed for MC can be evaluated. The transferred energy T2 f is given by

T2 f T1i



4 M 1M 2

 M1  M 2 

2

 sin 2   2

(1)

where T1i is the energy of incident ion. The scattering angle  is determined when collision parameter b is set. In the previous chapter, we implicitly assume integration region for b from 0 to infinity. However, a finite value of b must be used for each collision in MC. Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

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93

When b increases T2 f becomes small. One can define a threshold value bmax . For b larger than bmax , T2 f becomes negligible. We set this T2 f as T2 fc . In this case, the scattering angle is small and is denoted as  c . Equation 1 in this case can be expressed by (sin(x) ≈ x for small x), and is reduced to T2 fc T1i



c2 4

4 M 1M 2

 M1  M 2 

2

(2)

Therefore, we obtain

c 

 M1  M 2  M 1M 2

2

T2 fc T1i

(3)

The collision parameter bmax corresponding to  c can be evaluated from 

 bmax dr  c    2 2  V r b    2  max2 rmin r 1  Ec r

(4)

It is time consuming to evaluate bmax by evaluating this integration for each collision. Therefore, we want to evaluate the relationship of Eq. 4 numerically, and want to obtain analytical fit. Equation 4 can be expressed by a universal form as 

   c    2 d  f   2  2  2 min  1  Ec 

(5)

where



bmax aU

(6)

Ion Implantation and Activation, Vol. 1

 (radian)

94

10

1

10

0

10

Kunihiro Suzuki

Numerical Fit  = 0.001 0.01 0.1

-1

10

-2

10

-3

10

-4

10

5

2

1

20

10

50

-5

10

-3

10

-1



10

1

10

3

10

5

Figure 1: Dependence of the scattering angle  on the reduced energy  for different values of the normalized collision parameter . The values calculated from Eq. 5 (dots) are compared to the analytic fit expressed in Eq. 7. The fit is good for small values of 

3

10

Extract Fit

1

10

-1

10

-3

10

-5

10

-7

a

10

-3

10

10

-2

-1

10

 (a)

10

0

1

10

2

10

Monte Carlo Simulation

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95

2

10

 Fit

1

10

0



10 10

-1

10

-2

10

-3 -7

10

10

-5

10

-3

-1

a (b)

10

1

10

10

3

Figure 2: Relationship between a and (a) Dependence of a on (b) Dependence ofon a.

 can be fitted by

1  1     0.98     a    

1

(7)

as shown in Fig. 1. The fitting is poor for  > 0.1, but very good for smaller values of , typical for  c . Focusing on the low scattering angle, we can use a simple expression:   a    0.98

(8)

where a   can be fitted as   1 1 1 a       1 0.98982 2.6983 5 7.22433  1.6587 2.1872 10   9.53110   2.1872 105 7.2243  1  2.2984 105 6.23448  1.3186 105 4.526

1

(9)

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Kunihiro Suzuki

It is convenient to express  which is shown in Fig. 2(a) as a function of a , which is also fitted as

1 1 1      a   0.125 0.4 1.01  2.8a 0.96a  7.0a  1.01 0.96a  1  0.1371a 0.885  0.343a 0.61

1

(10)

The cross section associated with which is shown in Fig. 2(b) nuclear interactions  n can be given by

 n   bmax 2

(11)

We can define the mean free path L as

LN n  1

(12)

where N is the atomic density, and we can consider that incident ions collide once when travelling the distance L . Therefore, we can obtain L

1 N n

If the mean free path L is smaller than the average atomic distance Lmin  we should set

L  Lmin

(13) 1 N

1

3

,

(14)

In that case the collision cross section is given by

Lmin N n  1

(15)

and the maximum collision parameter bmax can be calculated from Eq. 11, and we obtain

bmax

n 1 Lmin 3 L     min   NLmin  Lmin 

(16)

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

It is also possible that we always set as Lmin 

1 N

1

3

97

, and bmax is always given by Eq.

16. This treatment is simple, and is sometimes used in commercial simulators. However, the nuclear interaction is evaluated even when we do not expect significant nuclear interaction. Fig. 3 shows the evolution of the mean free path for ten trajectories of ions implanted at 40 keV. L is evaluated using Eqs. 13 and 14. L decreases with decreasing energy caused by collisions. L is much larger than Lmin for light ion of B, and it approaches to Lmin after many collision events. For the heavier As ions L does not exceed 2.4 Lmin for 40 keV implants. Fig. 4 shows MC results with variable L and with L fixed to Lmin . The calculated result is the same for both treatments of L , while the computation time is shorter by a factor of 5 with variable L . Now, let’s see how the calculation is performed in MC for amorphous materials. We only consider electronic stopping power during the path L . The ion loses the energy of Ee  Se  E  L

(17)

15 Lmin = 0.27 nm

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

B 40 keV

L/L min

10

5

0

0

50

100 Event (a)

150

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Kunihiro Suzuki

5 Lmin = 0.27 nm P 40 keV

L/Lmin

4 3 2

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

1 0

0

50

Event (b)

100

150

3 Lmin = 0.27 nm

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

As 40 keV

L/Lmin

2

1

0

0

50

100

150

Event (c) Figure 3: Dependence of L on number of collision event. (a) B, (b) P, (c) As.

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

99

21

Concentration (cm-3)

10

B 40 keV 1 x 1015 cm-2 aSi sub

20

10

SIMS Monte Carlo variable L Monte Carlo Lmin

19

10

18

10

17

10

16

10

0

100

200 300 Depth (nm)

400

500

Figure 4: Comparison of MC with variable L and fixed L of Lmin .

where Se  E  is the electronic stopping power, and is given by  LH  1 1  1       Se  re SeL   Se _ mB 

 LH

  

   

1

 LH

(18)

The model consists of Lindhart model SeL for low energy region and modified Bethe model S e _ mB for high energy region. The experimental data and the theory are tried to fit by changing a fitting parameter re later. We select a random number Rand  n  , which is between 0 and 1. The collision parameter b is given by

b  bmax Rand  n 

(19)

Note that we can provide uniform probability on circle with radius bmax if we use Rand  n  as shown in Fig. 5.

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Figure 5: A ring regions which have the same area.

Once b is determined, the transferred energy can be evaluated as

T2 f 

4 M 1M 2

 M1  M 2 

2

     b  2  cos  dr  T1i 2  V r  b 2     2 1 r  r min   E r c  

(20)

and the energy loss associated with the nuclear interaction En is En  T2 f

(21)

The scattering angle is then given by 

 b dr     2 2  V r   b  2  2 rmin r 1  Ec r

(22)

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Ion Implantation and Activation, Vol. 1

101

Figure 6: Selection of  .

The angle  as shown in Fig. 6 can be randomly selected as

  2 Rand  n 

(23)

We can then trace the trajectory with the next length of L and with new energy E  Ee  En and the direction determined from Eqs. 22 and 23. In the above procedure, the integral in Eq. 22 induces high calculation cost. Ziegler proposed a Magic formula and solved the integral with a simple numerical protocol [1]. Systematic table for the integration is provided and the integral based on the table is evaluated in the commercial process simulator Sentaurus Process [4]. One simple treatment is using an analytical expression of the integral such as Eq. 7. However, it is inaccurate in the transition region as shown in Fig. 1. For the purpose of determining  min it was enough because we only need accurate when  is small in Eq. 7. However, we now need accurate one over the entire range of  . One can add one more terms to Eq. 7 to improve the accuracy, for example:

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

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Kunihiro Suzuki

1 3 3 3  2   2   2 1 1 1         0.98  0.2         a g          

2

(24) 3

where

 1 1  g       0.46 9.0 1.3   4.0

1

(25)

Equation 24 reproduces  well, as shown Fig. 7. SUBSTRATE THAT CONSISTS OF VARIOUS ATOMIC SPECIES

We consider the case of a substrate that consists of various atomic species in this section. We assume a substrate that consists of atoms A with number of atoms per unit volume of nA and atom B with a number of nB in one unit cell, and the concentration of the unit cell is N . The critical scattering angle related to the atom A and B are described by

 cA, B 

M

 M 2 A, B  T2 fc 2

1

M 1 M 2 A, B

T1i

(26)

where weuse the same T2 fc . The related maximum collision parameter bmax A and bmax B can be evaluated as is the same way as for pure substrates. The total cross section is given by

 n  nA nA  nB nB

(27)

We can evaluate the mean free path length L defined by

LN n  1 When L is shorter than the average distance between atoms

(28)

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

10

103

1

 (radian)

Numerical Fit

10

 = 0.001

0

0.01 0.1

-1

5

10

10

2 1

20

-2

10

10

4 10

-5

10

-3

10

-1

 (a)

10

1

10

3

10

5

0

 (radian)

Numerical Fit

3 10

0

2 10

0

 = 0.001 0.1 2

1 10

0.01

1

5

0

10 20

0

0 10 -5 10

-3

10

-1

10

 (b)

10

1

10

3

10

5

Figure 7: Analytical expression of dependence of  on reduced energy . (a) Log scale (b) Linear scale.

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Lmin 

Kunihiro Suzuki

1  N  nA  nB  

1

(29) 3

L should be set to Lmin .

When L  Lmin , the probability whether incident atoms collide with A or B is given by nA A nB B , nA A  nB B nA A  nB B

(30)

We select atom A or B by generating random number related to the ratio. When energy decreases L decreases monotonically, and one that satisfies Eq. 28 becomes less than L . Therefore, we should consider the treatment where we set L  Lmin independent of ion energy for the substrate that consists of various atomic species assumed above. Introducing r , we use the relationship given by N  rnA nA  rnB nB  Lmin  1

(31)

and obtain r as r

1

(32)

NLmin  nA nA  nB nB 

The corresponding bmax a and bmax b are bmax A 

r nA



, bmax B 

r nB



(33)

We can extend this procedure to the substrate that consists of more than two kinds of atoms. We assume a substrate with i atom and the number is ni . The critical angle for each atom is given by  ci 

 M 1  M 2i  M 1M 2i

2

T2 fc T1i

(34)

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

105

The related maximum collision parameter bmax i can be evaluated in the same procedure as for single atom substrate, and we obtain related cross section  ni , and the total cross section is given by

 n   ni ni

(35)

i

We can evaluate the mean free path length L defined by

LN n  1

(36)

The probability for a collision to take place with an atom of species i is given by ni ni  n j nj

(37)

j

We select atom i by generating random number related to the probability. When L , which is evaluated with Eq. 36, is shorter than Lmin of

Lmin 

1    N  n j nj   j 

1

3

(38)

L should be set to Lmin . We can evaluate r as

r

1

NLmin  ni ni

(39)

i

and then bmax i is given by bmax i 

r ni



(40)

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Kunihiro Suzuki

CALIBRATION OF MC PARAMETERS

Concentration (cm-3)

Fig. 8 shows the comparison of SIMS and MC data with a fitting parameter related to electronic stopping power re , which is described in the chapter 2, of 1 for B, P, and As implantation. We use MC simulator implemented in FabMeister-IM [6] in this chapter, and related models are described in [5]. We compared MC data to SIMS profiles in amorphous Si substrates. We obtained good agreement between MC and SIMS data for As profiles, and close agreement for P profiles, but significant deeper profiles are predicted by MC than SIMS data for B as shown in Fig. 8. This may mean that we cannot expect predictive results from MC simulation as it is.

10

21

10

20

10

19

10

18

10

17

10

16

80 keV 1 x 10 15 cm-2

SIMS B SIMS P SIMS As Monte Carlo re = 1.0

B As

0

P

100 200 300 400 500 600 Depth (nm)

Figure 8: Comparison of SIMS B, P, and As profiles and MC with re  1 .

Fig. 9 compares SIMS and MC results for B and As data with various re . The profile becomes shallower with increasing re . We can obtain close agreement of peak position as well as the overall shape of the profile with re of around 1.5 for B and 1.0 for As.

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

21

10

B in Si

Concentration (cm-3)

15

SIMS r = 0.5

-2

80 keV, 1 x 10 cm Monte Carlo

20

10

e

r = 1.0 e

r = 1.5 e

r = 2.0 e

19

10

18

10

17

Concentration (cm -3)

10

0

10

22

10

21

10

20

500 Depth (nm) (a)

As in Si 80 keV, 1 x 10 15 cm-2 Monte Carlo

1000

SIMS r = 0.5 e

re = 1.0 re = 1.5 r = 2.0 e

10

19

10

18

10

17

0

50

100 150 Depth (nm) (b)

200

Figure 9: Comparison of SIMS B and As profiles and MC with various re .

107

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Kunihiro Suzuki

Various energy-dependent SIMS profile data are compared with the MC with optimized re in Si substrate (Fig. 10) and in Ge substrate (Fig. 11). In some data, we have crystalline Si substrates. We verified that the profiles are almost the same as those in amorphous data near the peak region and the surface region. Therefore for these data we focus on these regions. It is noteworthy that we can fit the data with a single value of re over a wide energy range. We obtain similar results for Mo, HfO2, and photo-resist substrates by tuning corresponding re that is valid for various energies, which is shown in [5]. Table 1 summarizes optimized re . The values of re are not far from 1. In many cases they are equal to 1. We can predict profiles in an amorphous layer using a MC simulation with a default re taken from Table 1 and can further improve the accuracy if we tune re with some experimental data. Table 1:

re for various incident ion and substrate atoms Z2

Z1

01:H

06:C

08:O

14:Si

32:Ge

42:Mo

72:Hf

05:B

1.0

1.7

0.8

1.55

1.0

1.0

3.0

06:C

---

---

---

1.5

---

---

---

07:N

---

---

---

1.4

---

---

---

09:F

---

---

---

1.0

---

---

---

14:Si

---

---

---

1.25

1.0

---

---

15:P

1.0

1.0

1.0

1.2

1.0

1.0

1.0

32:Ge

---

---

---

1.0

---

---

---

33:As

---

---

0.5

1.0

1.0

---

1.0

49:In

---

---

---

1.0

---

---

---

51:Sb

---

---

---

1.0

---

---

---

Monte Carlo Simulation

10

Ion Implantation and Activation, Vol. 1

21

Concentration (cm-3)

B: 1 x 1015 cm-2

10

20

10

19

10

18

10

17

10

aSi substrate re = 1.55

0

200

400 600 Depth (nm) (a)

800

1000

21

C: 1 x 1015 cm-2

Concentration (cm-3)

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

10

20

10

19

10

18

10

17

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo cSi substrate re = 1.5

0

200

400 600 Depth (nm) (b)

800

1000

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10

Kunihiro Suzuki

21

Concentration (cm-3)

N: 1 x 1015 cm-2

10

20

10

19

10

18

10

17

10

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo cSi substrate re = 1.4

0

200

400 600 Depth (nm) (c)

800

1000

21

F: 1 x 1015 cm-2

Concentration (cm-3)

110

10

20

10

19

10

18

10

17

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo cSi substrate re = 1.0

0

200

400 600 Depth (nm) (d)

800

1000

Monte Carlo Simulation

10

Ion Implantation and Activation, Vol. 1

21

Concentration (cm-3)

Si: 1 x 1015 cm-2

10

20

cSi substrate re = 1.25

10

19

10

18

10

0

100

200 300 Depth (nm) (e)

400

500

21

P: 1 x 1015 cm-2

Concentration (cm-3)

SIMS 20 keV SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

10

20

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

10

19

aSi substrate re = 1.2

10

18

10

17

0

100 200 300 400 500 600 Depth (nm) (f)

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Ion Implantation and Activation, Vol. 1

10

22

10

21

10

20

10

19

10

18

10

17

Kunihiro Suzuki

Concentration (cm-3)

Ga: 20 keV 1 x 10 15 cm-2

10

SIMS Monte Carlo cSi substrate re = 1.0

0

50 100 Depth (nm) (g)

150

21

Ge: 1 x 1015 cm-2

Concentration (cm-3)

112

10

20

10

19

10

18

10

17

SIMS 20 keV SIMS 40 keV SIMS 80 keV Monte Carlo cSi substrate re = 1.0

0

100 Depth (nm) (h)

200

Monte Carlo Simulation

Concentration (cm-3)

10

Ion Implantation and Activation, Vol. 1

21

As: 1 x 1015 cm-2

10

20

SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

10

19

aSi substrate re = 1.0

10

18

10

17

10

0

100

200 300 Depth (nm) (i)

22

In 1 x 10 15 cm-2

Concentration (cm-3)

400

10

21

10

20

10

19

10

18

10

17

SIMS 10 keV SIMS 20 keV SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

cSi substrate re = 1.0

0

50

100 150 Depth (nm) (j)

200

113

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Kunihiro Suzuki

22

10

SIMS 10 keV SIMS 20 keV SIMS 40 keV SIMS 80 keV SIMS 160 keV Monte Carlo

Sb: 1 x 10 15 cm-2

21

Concentration (cm-3)

10

20

10

cSi substrate re = 1.0

19

10

18

10

17

10

0

50

100 150 Depth (nm) (k)

200

Figure 10: Comparison of Monte Carlo and SIMS data in Si substrate. (a) B, (b) C, (c) N, (d) F, (e) Si, (f) P, (g) Ga, (h) Ge, (i) As, (j) In, (k) Sb.

22

10

Ge 120 keV 1 x 10 15

Concentration (cm )

-2

cm

SIMS

-2

Monte Carlo

B: 5 keV 1 x 10 cm

21

-3

15

10

a-Ge substrate re = 1.0

20

10

19

10

18

10

17

10

0

50 Depth (nm) (a)

100

Monte Carlo Simulation

10

Ion Implantation and Activation, Vol. 1

21

Concentration (cm-3)

Si: 1 x 1015 cm-2

10

20

SIMS 20 keV SIMS 40 keV SIMS 80 keV Monte Carlo

10

19

c-Ge substrate re = 1.0

10

18

10

17

10

0

100

200 300 Depth (nm) (b)

22 Ge 120 keV 1 x 1015 cm-2

Concentration (cm-3)

400

10

21

10

20

10

19

10

18

10

17

15

P 40 keV 1 x 10 cm

SIMS

-2

Monte Carlo

a-Ge substrate re = 1.0

0

100 Depth (nm) (c)

200

115

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10

Kunihiro Suzuki

22

Concentration (cm-3)

As :1 x 1015 cm-2

10

21

10

20

10

19

10

18

10

17

SIMS 10 keV SIMS 20 keV SIMS 40 keV SIMS 80 keV Monte Carlo c-Ge substrate re = 1.0

0

50 100 Depth (nm) (d)

150

Figure 11: Comparison of Monte Carlo and SIMS data in Ge substrate. (a) B, (b) Si, (c) P, (d) As.

Low-energy ion implantation of around 1 keV is frequently used to realize shallow junctions. However, SIMS has resolution problems in the energy region of around 1keV. Recently, high resolution SIMS measurement techniques have been developed [7, 8]. Here, we compare the SIMS data with MC. There is no critical point at this energy region from the standpoint of physics, and MC also well predicts the data around these energy regions as shown in Fig. 12. As we pointed out in chapter 2, the Lindhard’s Se model becomes invalid in high-energy region, especially for B in the practical high-energy region of around MeV. Fig. 13 compares B SIMS data with the MC simulation using Lindhard’s Se model. SIMS and MC results agree well at 400 keV. However, Lindhard’s model predicts much shallower B profiles at 1200 and 2000 keV.

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

10

22

Concentration (cm -3)

B 1 x 10 15 cm-2

10

SIMS 0.3 keV SIMS 0.5 keV SIMS 1 keV SIMS 3 keV Monte Carlo

21

cSi substrate re = 1.55 10

20

10

19

0

10

20 30 Depth (nm)

40

50

(a)

22

10

-3

Concentration (cm )

As 1 x 1015 cm-2

SIMS 1 keV SIMS 3 keV Monte Carlo

21

10

cSi substrate re = 1.0 20

10

19

10

18

10

0

10

20

Depth (nm) (b) Figure 12: Comparison of low-energy B and As SIMS profiles with MC.

117

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18

Concentration (cm-3)

10

B 1 x 1013 cm-2

 LH =1.65 1200 keV

400 keV

2000 keV

SIMS This work Lindhard

17

10

cSi substrate re = 1.55

16

10

0

1000

2000 3000 Depth (nm)

4000

Figure 13: Comparison of high-energy B SIMS data with MC using Lindhard and combined Se models.

Fig. 13 also compares SIMS and MC results with combined Se model of Eq. 8. We applied  LH of 1.65 to the other energies and ions and obtained good agreement. We therefore can predict ion implantation profiles over the energy region from 0.5 keV to more than 2000 keV with the MC simulation code. DAMAGE

We can evaluate the energy transferred to the substrate atom of T2 f . The substrate atoms are displaced from their lattice site if T2 f is larger than the critical displacement energy Ed . We then trace trajectories of the recoiled atom with the energy of T2 f  Ed and count up the vacancies generated by the recoiled atom. To save computational time, a modified Kinchin-Pease model is commonly applied to the primary recoiled atoms instead of tracing the recoiled substrate atoms [1]. The number of vacancies is evaluated using an analytical formula as a function of

Monte Carlo Simulation

Ion Implantation and Activation, Vol. 1

119

T2 f , and it is recorded at the location of the primary recoiled position. In this treatment, the damage (vacancy) can be expressed by [1]

 T2 f 0.4 Ed   T    2f  Ed 0  

for T2 f  2.5 Ed for Ed  T2 f  2.5Ed

(41)

for T2 f  Ed

The transferred energy is not completely consumed by nuclear interactions. Some energy is also consumed by electronic stopping during many collisions. Therefore, the term of 0.4 is added in the first term of Eq. 41. Ed is an important parameter that controls the radiation damage, but the experimental methods show widely varying results for Ed of Si substrate in the range of 10-30 eV, and theoretical evaluation shows that it depends on the direction of recoiled atom in the lattice.It is in the range between 12 and 36 eV and has an average value of around 24 eV [9]. I 50 keV I 100 keV I 200 keV

21

10

Monte Carlo 22

10

20

21

10

20

10

19

10 500

10

19

10

18

10

10

P concentration (cm -3)

Recoiled Si (I) concentration (cm -3)

23

10

P 50 keV P 100 keV P 200 keV

17

0

100

200 300 400 Depth (nm)

Figure 14: P ion implanted profiles and generated recoiled Si concentration profiles.

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Fig. 14 shows ion implanted P profiles and the concentration of generated interstitial Si (identical to vacancy) concentration evaluated by Eq. 41 using Ed of 25 eV. The peak of the interstitial Si (I) profile is slightly shallower than the P ion planted profile. The interstitial Si profile is rather broad up to the peak position, and rapidly decreases after then. We will discuss the damage in more detail later. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

J. F. Ziegler, J. P. Biersack, and U. Littmark, The stopping and range of ions in solid, Pergamon, 1885. SRIM-2003: http://www.srim.org/ S. Tian, “Predictive Monte Carlo ion implantation simulator from sub-keV to above 10 MeV”, J. Appl. Phys., vol. 93, No. 10, pp. 5893-5904, 2003. Sentaurus Process http://www.synopsys.com/Tools/TCAD/ProcessSimulation/Pages/SentaurusProcess.aspx K. Suzuki, Y. Tada, Y. Kataoka, and T. Nagayama”, Monte Carlo Simulation of Ion Implantation Profiles Calibrated for Various Ions over Wide Energy Range”, J. Semiconductor Tech. and Sci., vol.9, No. 1, pp. 67-74, 2009. FabMeister IM http://www.mizuho-ir.co.jp/science/ion/index.html Y. Kataoka and T. Itani”, Ultra-shallow depth profiling by using SIMS and Ion Scattering Spectroscopy”, Surf. Interface Anal., vol. 39, pp. 826-831, 2007. Y. Tada, K. Suzuki, and Y. Kataoka, “Segregation under low-energy oxygen bombardment in the near-surface region”, Applied Surface Science, vol. 255, pp. 1320-1322, 2008. E. Holmstrom, A. Kuronen, and K. Nordlund”, Threshold defect production in silicon determined by density functional theory molecular dynamics simulation”, Physical Review B, vol. 78, 045202, 2008.

Send Orders for Reprints to [email protected] Ion implantation and activation, Vol. 1, 2013, 121-167 121

CHAPTER 4 Analytical Model for Ion Implantation Profiles Abstract: In the design of VLSI devices, accurate prediction of the doping profiles resulting from ion implantation, a standard method for doping impurities in VLSI (very-large-scale integrated circuit) processes, is essential. This is done by obtaining analytical expressions for the SIMS (secondary ion mass spectrometry) data of ion implantation profiles, and these analytical formulas are used to compile an ion implantation profile database. The profiles of arbitrary implantation conditions can be generated using interpolated parameter values. Various analytical models have been developed for expressing ion implantation profiles. The functions used to express these ion implantation profiles include Gaussian, joined half Gaussian, Pearson, and dual Pearson functions. In addition to these, a tail function was proposed. This tail function has fewer parameters than the dual Pearson function, and it is better able to specify an arbitrary profile using a unique set of parameters.

Keywords: Ion implantation, skewness, kurtosis, Gaussian, joined half Gaussian, Pearson, Pearson IV, Pearson family, dual Pearson, tail function, SIMS, database, Monte Carlo, projected range, dose, amorphous substrate, damage, differential channel dose. INTRODUCTION Ion implantation profiles can be evaluated with Secondary Ion Mass Spectrometry (SIMS), and a database can be established by accumulating the data. However, it is impossible to build a database which has entries for all possible conditions for ion implantation. We therefore need an analytical model to fit the profiles, which enables us to calculate the profiles with any ion implantation conditions by interpolating or extrapolating the parameter values corresponding to the analytical model. The profiles can also be calculated with Monte Carlo for any ion implantation condition. However, Monte Carlo is time consuming, and hence it is useful to have also an analytical model for ion implantation. The moment parameters for the analytical model can be extracted either from Monte Carlo simulation results or from SIMS data. MOMENTS OF PROFILES Moments of the profile are the fundamental characteristics and can be automatically calculated once the concentration profiles N  x  are given. Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

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The projected range R p is given by 

Rp 

 

 

xN  x  dx



N  x  dx

(1)

which corresponds to the average depth of the ions and is called projected range. The i-th moment is defined as follows

  x  R  N  x  dx   N  x  dx 

i



i

p



(2)



Obviously, 0  1 and 1  0 . We can calculate any order moments according to Eqs. 1 and 2. A calculation up to forth moment is sufficient to express the profiles as shown later.  2,  3, and  4 are not used as their forms, but normalized one with respect to R p as shown below.

R p =

2

(3)

=

3 R 3p

(4)

=

4 R 4p

(5)

R p simply expresses the average depth of the ions as illustrated in Fig. 1. It should be noted that R p is equal to depth of the peak R pm only when the profile is symmetrical. In general, R p is different from R pm . R p corresponds to the standard deviation which expresses the width of the peaks illustrated in Fig. 1. The concentration at the depth Rp +∆Rp is approximately half the peak concentration.

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123

21

Concentration (cm -3)

10

Rp = 100 nm =0 =3

20

10

Rp = 25 nm 19

10

18

10

0

50

100 150 Depth (nm)

200

Figure 1: Explanation of R p and R p of ion implantation profile. 21

Concentration (cm-3)

10

=0  = 0.2  = 0.4  = 0.6

20

10

Rp = 100 nm

19

10

Rp = 25 nm =3

18

10

17

10

0

100 200 Depth (nm)

300

Figure 2: Dependence of ion implantation profile on  .

 expresses asymmetry of the profile as defined before.  is negative if the distribution is skewed in front of R p , positive if it is skewed behind R p , and zero if

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the distribution is symmetrical. The dependence of the distribution on  is shown in Fig. 2. A larger value of  results in a more asymmetric distribution. We used positive values of  in Fig. 2. The distributions for negative values can be obtained by reflecting the distributions at an axis positioned at R p . 21

Concentration (cm-3)

10

=2 =3  = 10

Rp = 100 nm Rp = 25 nm

20

=0

10

19

10

18

10

0

50

100 150 Depth (nm)

200

Figure 3: Dependence of ion implantation profile on  .

 controls the shape of the profiles as shown in Fig. 3. Let us consider how we can obtain large  based on Eq. 5. We can increase  by increasing 4 or by

decreasing R p . We should increase the concentration in the tail region far from the peak location to increase 4 and sharpen the shape around peak concentration region to decrease R p . Therefore, profiles with large values of  have a shape of Mt. Fuji, and those with small values of  have the shape of a box. GAUSSIAN PROFILE Using the moments of R p and R p , Gaussian profile is expressed by N x = N m exp -

x - Rp 2R 2p

2

(6)

where N m is the maximum concentration at x  R p . The dose  is evaluated by

Analytical Model for Ion Implantation Profiles 

=

N m exp -

=

 -

x - Rp 2R 2p

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125

2

dx

2 R pN m exp - s 2 ds

s=

x - Rp 2 R p

= 2 R pN m

(7) 21

10

Concentration (cm -3)

P 40 keV 1 x 10 15 cm-2 20

SIMS Gauss

10

Rp = 47 nm

19

Rp = 25 nm

10

18

10

17

10

0

50 100 Depth (nm)

150

Figure 4: Gaussian profile which is expressed with R p and R p .

The integration in Eq. 7 is called Gauss integration and can be solved as shown in Appendixes A, B, and C. Therefore, N m is expressed by

Nm =

 2 R p

(8)

Ion implanted P profiles in amorphous Si substrates can be approximated by this Gaussian profile as shown in Fig. 4. A Gaussian distribution is convenient to describe the profile since its parameters can be easily related to the shape of the profile. Therefore, this profile is frequently used despite of its limitation for accuracy.

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It is also interesting to evaluate moments of Gaussian profile. The first moment is given by 

 exp - x - R p 2 R p 2R 2p

x - Rp 1 =

-

 -  -

=



=

-

2 R p

2

dx

N x dx

 s exp - s 2 ds 2 R p 

1 s exp - s 2 ds 

=0

(9)

This integrationvalue of 0 is obvious due to the symmetry of the function of s and exp   s 2  . The second moment is 

x - Rp 2 =

2

-

 exp - x - R p 2 R p 2R 2p  -



=

-

 2 R p

= R 2p 2  = R



2

dx

N x dx 2

2 R p s 2 exp - s 2

2 R pds

 s 2 exp - s 2 ds

-

2 p

which is the expected value. The third moment is obviously zero and hence  of this profile is zero.

(10)

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127

The forth moment is 

   x  R 2  4  p  dx   x  Rp  exp    2R p 2   2 R p   4    2   x  R    p  dx  exp    2R p 2   2 R p     R p 4  s 4 exp   s 2  ds 

(11)



 3R p 4

Therefore, the Gaussian profile is characterized by R p and R p , with  of 0 and  of 3. JOINED HALF GAUSSIAN PROFILE

A Gaussian profile is symmetrical with respect to R p . However, ion implantation profiles are asymmetrical in general. Gibbons proposed a joined half Gaussian profile where two Gaussian profiles with different standard deviations are joined at the peak position [1]. The function is given by    x  R 2  pm  N exp    m   2R pf2     N  x   2  x  R    pm   N m exp   2   2 R   pb   

for x  R pm

(12) for x  R pm

N m is the peak concentration, and R pm is the peak position as are mentioned before. R pf is the standard deviation associated with surface region, and R pb is the standard deviation associated with the deeper side. N m can be evaluated from the integration of =

 -

N x dx

(13)

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Kunihiro Suzuki

where  is the dose. This integration can be performed as shown in Appendix D, and we obtain  2  R + R pf pb

Nm =

(14)

R pf and R pb can be evaluated form the integrations 

R pm

2

x - R p N x dx

-

R 2pf =

R pm -

; R 2pb =

R pm

N x dx

2

x - R p N x dx  R pm

N x dx (15)

This model can cover B profile skewed toward surface and As profiles skewed away from the surface as shown in Fig. 5. The parameters associated with joined half Gaussian are related to the moment parameters. Substituting Eq. 12 into Eqs. 1 and 2, we obtain (see Appendix D)

2

R p  R pm 



 R

pb

 R pf 

 2 R p 2  1    R pb  R pf   2

 R 

R p 3  

4





 

20 

 R p   3  4

9 

pb



2

(16)

 R pb R pf

 4   R pf    1  R pb  R pf   

2  2   R  R 4    pb pf    

(17)



2

  R pb R pf  

(18)

 3

2 2   R pb  R pf  R pb R pf  R pb R pf

 

2

(19)

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Ion Implantation and Activation, Vol. 1

21

10 Concentration (cm-3)

B a-Si 15

40 keV 1 x 10 cm

20

10

SIMS Analytical

-2

Joined half Gauss

19

10

Rpm = 145 nm Rpf = 55 nm

18

10

Rpb = 33 nm

17

10

Concentration (cm-3)

10

0

100 200 Depth (nm) (a)

300

21

As a-Si

10

20

10

19

80 keV 1 x 10 15 cm-2

SIMS Analytical Joined half Gauss

Rpm = 53 nm

10

18

10

17

Rpf = 19 nm Rpb = 25 nm

0

50

100 150 Depth (nm) (b)

200

Figure 5: Ion implantation profile expressed by using joined half Gaussian (a) B, (b) As.

129

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Kunihiro Suzuki

4 3

 



2 1 0

Joined half Gauss

-1

0

5 r (a)

10



4.0

3.5

Joined half Gauss

3.0 0.0

0.5

2

 (b)

1.0

1.5

Figure 6: Joined half Gauss profile: (a) dependence of  and  onthe r (ratio between left and right hand side standard deviation) (b) relationship of    2 .

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131

The expressions for Eqs. 18 and 19 are rather complicated. We define r as R pb R pf

r

(20)

 and  are uniquely determined by r from Eqs. 18 and 19 as 2





 4   2  1  r  1  r     

 r  1 

 2   2 1     r  1  r    



(21)

3 2

2  4 20  4 2 2   2 3   3     r  1   9    r  1  3r         

 2   2 1     r  1  r    

2

(22)

Fig. 6(a) shows the dependences of  and  on r .  and  have a limited range of values. This can be evaluated as 24  1     lim     0.995272 3 r 0 2  2 1     24  1     lim    0.995272 3 r   2 2 1    

lim  

r  0, 

2  4 2  3   3         

 2 1    

2

 3.86918

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Kunihiro Suzuki

Therefore, the limitation is expressed by

24  1     0    0.995272,3    3 2  2 1    

2  4 2  3 3              3.86918 2  2 1    

(23)

Concentration (cm-3)

 and  are related to each other through the parameter r as shown in Fig. 6 (b).

10

20

10

19

10

18

10

17

B 80 keV 1 x 10 15 cm-2 in a-Si

Pearson Gauss Joined half Gauss SIMS

0

100

200 300 Depth (nm)

400

500

Figure 7: Expression of a SIMS B profile with Gauss, joined half Gauss, and Pearson function.

PEARSON IV FUNCTION

The SIMS profiles of B ion implanted into an amorphous Si substrate at 80 keV is shown in Fig. 7. A Gaussian distribution cannot express these asymmetric profiles. Although a joined half Gaussian can provide a good representation of the peak depth and the neighborhood of the peak, it cannot represent profiles that decrease

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133

exponentially towards the surface. A strong limitation of joined half Gaussian profiles is that the values of exponential terms vary with the square of the distance. Pearson IV is a flexible function to express various profiles measured after ion implantation [2, 3], and can express the profile as shown in Fig. 7. The Pearson IV function is one of the Pearson function family which is generated from the differential equation

s  a h dh  ds b0  b1s  b2 s 2

(24)

where s is s  x  Rp

(25)

h is the normalized concentration, that is, 

 h  s  ds  1

(26)



The parameters of a , b0 , b1 , and b2 are related to the moments of h as follows: Modifying Eq. 24 we obtain

b

 b1s  b2 s 2 

0

dh  s  a h ds

(27)

Multiplying s n to both sides of Eq. 27 and integrating with respect to s gives   b s n  b s n 1  b s n  2 dh ds    s  a  s n hds   0  ds  1 2 

(28)

Performing the integration, we obtain

 b0 s n  b1s n 1  b2 s n  2  h     nb0 s n 1   n  1 b1s n   n  2  b2 s n 1  hds       







s

n 1

 as  hds n



(29)

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Kunihiro Suzuki

We impose that lim s n  2 h  0

(30)

s 

This means that h should be decreased more rapidly than s n  2 . We can then eliminate the first term of left side of Eq. 29 and obtain  -

nb 0s n - 1 + n + 1 b 1 - a s n + n + 2 b 2 + 1 s n + 1 hds

= nb 0

 -

s n - 1hds + n + 1 b 1 - a

 -

s nhds + n + 2 b 2 + 1

 -

s n + 1hds

=0

(31)

Using the definition of the moments, Eq. 31 is expressed by nb 0 n - 1 + n + 1 b 1 - a  n + n + 2 b 2 + 1  n + 1 = 0

(32)

Substituting n = 0, 1, 2, 3 into Eq. 32, we obtain following. n = 0: b1 - a = 0

(33)

n = 1: b 0 + 3b 2 + 1  2 = 0

Expressing 2 with R p , we obtain b 0 + 3R p 2b 2 = - R p 2

(34)

n = 2: 3b 1 - a  2 + 4b 2 + 1  3 = 0

Expressing 2 and 3 with R p and  , and using a  b1 , we obtain 2b 1R 2p+ 4b 2 + 1 R 3p = 0

Analytical Model for Ion Implantation Profiles

Ion Implantation and Activation, Vol. 1

2b 1 + 4R pb 2 = - R p

135

(35)

n = 3: 3b 0 2 + 4b 1 - a  3 + 5b 2 + 1  4 = 0

Expressing 2 , 3 and 4 with R p ,  , and  we obtain 3b 0 + 3R pb 1 + 5R 2pb 2 = - R 2p

(36)

The parameters b0 , b1 , and b2 are determined from Eqs. 34-36 as

b0 = -

4 - 3 2 2 R p A

(37)

b1 = -

+3 R p A

(38)

b2 = -

2 - 3 2 - 6 A

(39)

where A = 10 - 12 2 - 18

(40)

The differential equation 24 generates various types of functions called Pearson families depending on the value of the parameters. Equations 33-40 are valid for all Pearson functions. However, some combinations of  and  generates unrealistic profiles. Therefore, some limitations for combination of  and  exist. We briefly discuss the limitations for Pearson IV functions here. A more detailed discussion will be presented in the next chapter related to Pearson function family. Only bell-shaped functions are needed for ion implantation. These functions have only one peak and decrease with distance from the peak position. We can get some information about the shape of h from Eq. 24 without solving it. The gradient is zero for s  a , that is, it has a peak for x = R p + a . We can also observe dh ds  0 for h  0 .

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A mathematical expression to ensure bell shape is then given by  dh  ds  0   dh  0  ds

for s  a

(41) for s  a

This condition is satisfied when denominator of Eq. 24 is negative, that is, b 0 + b 1 s + b 2s 2 < 0

(42)

The equation has no real root, and the profile has bell shape. This is identical to b2 < 0 b 22 - 4b 0b 2 < 0

(43)

Converting the parameter to moment parameters in Eq. 43, we obtain 2 - 3 2 - 6 >0 10 - 12 2 - 18 32 -  2  2 - 6 13 2 + 16 + 9 2 4 2 + 7 > 0

(44) The former gives  > 3  2 + 3 or  < 6  2 + 9 5 5 2

(45)

We assume 32-2> 0, which is valid for practical ion implantation profiles, and the latter in Eq. 44 gives

  1 or    2

(46)

where 1 and  2 are the roots of the latter part of Eq. 44 regarded as to be zero and are given by 3

3 13 2 + 16 - 6  2 + 4 2 3 13 2 + 16 + 6  2 + 4 1 = ,  = 2 32 -  2 32 -  2

3 2

(47)

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137

We should add one more condition that ensures that h is positive in the entire region. It is given by

 > 1 + 2

(48)

The corresponding derivation will be shown in the next section. Consequently, the final limitation of Pearson IV is given by

  2

(49)

This region is covered by Pearson IV functions as shown in Fig. 8. Gaussian profile is one limiting case of   0 and   3 . Strictly speaking, we do not clarify the definition of Pearson IV among the Pearson function family. A more rigorous treatment will be done in the next chapter. Joined half Gauss is outside the Pearson IV region as shown in Fig. 8. We will see many experimental data and MC data breaks this Pearson IV region. If  breaks the Pearson IV region,  is usually modified to 2. A more detailed analysis of Pearson function family is described in the next chapter. 15

Pearson IV



10

5 Gauss 0

Joined half Gauss 0

1

2

3

4

5

2

Figure 8:  2   plane covered by Pearson IV function.   2 ,     0,3 is Gauss. The allowable line in the plane for joined half Gaussis also shown and is outside of the Pearson IV region.

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Under the condition of Eq. 49, the differential equation 24 can be solved as follows. We first modify the original form of thedifferential equation and integrate it as

sa  1 dh ds   ds   2  h ds  b0  b1s  b2 s

(50)

This can be solved as a logarithmic form as

ln h 

1 ln b0  b1s  b2 s 2 2b2

 2b s  b  b  2 2 1  1  a tan 1  2 2   2b2  4b2b0  b1  4b2b0  b1

   ln K  

(51)

We can then obtain

h  s   K b0  b1s  b2 s 2

1 2 b2

  b  2b s  b  2 2 1 exp    1  a  tan 1  2 2    2b2  4b2b0  b1  4b2b0  b1 

   (52)  

where K is determined by the normalization condition. Fig. 9 shows the comparison between analytical and SIMS profiles. Pearson IV functions readily reproduce the B SIMS profiles skewed towards the surface, symmetrical P profiles, and As profiles skewed towards deeper region. Pearson IV functions cover whole ion implanted profiles in amorphous substrates, and hence arethe standard functions used for analytic implant models. DUAL PEARSON IV FUNCTION

Damage is accumulated in the Si substrate with increasing dose. The damage region increases linearly with dose at low dose region, overlaps, and finally continuous amorphous layers are formed at high dose region. Therefore, the interaction between ions and substrate atoms are quite different at low and high dose regions especially with respect to ion channeling along channels in the silicon crystal. Therefore, we can easily imagine that the profiles do not change linearly with dose. This means that we cannot get parameters related to the energy as we can for the amorphous substrates.

Analytical Model for Ion Implantation Profiles

-3

Concentration (cm )

10

21

a-Si 15

10

20

10

19

10

18

10

17

10

Ion Implantation and Activation, Vol. 1

B 1 x 10 cm

0

100

Analytical SIMS 20 keV SIMS 40 keV SIMS 60 keV SIMS 80 keV

-2

200 300 Depth (nm) (a)

-3

500

21

a-Si 15

Concentration (cm )

400

P 1 x 10 cm

10

20

10

19

10

18

10

17

0

50

-2

100 150 Depth (nm) (b)

Analytical SIMS 20 keV SIMS 40 keV SIMS 60 keV SIMS 80 keV

200

250

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17

10

16

Kunihiro Suzuki

a-Si As 1 x 10

0

15

cm

-2

Analytical SIM S 20 keV SIM S 40 keV SIM S 60 keV SIM S 80 keV

100 Depth (nm) (c)

200

Figure 9: Ion implantation profile in amorphous Si. (a) B, (b) P, (c) As.

-3

Concentration (cm )

Fig. 10 shows the dependence of P profiles on dose. If we fit the profile at 3 x 1014 cm-2, and assume that the profile linearly depends on dose, the analytical model underestimates the profile tail at lower dose and overestimates it at higher dose. This means that we should fit all the profiles changing all parameters with dose even when the energy is the same, which requires vast matrix of the database. 10

22

10

21

10

20

10

19

10

18

10

17

10

16

P 30 keV SIMS 1 x 1013 cm-2 SIMS 3 x 1014 cm-2 SIMS 1 x 1016 cm-2 N(3 x 1013 cm-2) x (1 x1016/ 3 x 1014)

N(3 x 1013 cm -2) x (1 x1013/ 3 x 1014)

0

100

Figure 10: Dose dependence of P profiles.

200 300 Depth (nm)

400

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Tash proposed a dual Pearson to alleviate this problem [4, 5]. Dual Pearson functions are composed of two parts: the amorphous part ha and the channeling part hc as shown in Fig. 11. Both have corresponding four Pearson IV parameters and are normalized with dose. The dependence of the profile on dose is expressed by changing the ratio of the corresponding dose. Therefore, the profile is expressed by N x =  - chan h a x + chanh c x

(53)

Concentration (cm-3)

where  is the dose and chan is the channeling dose. Note that ha and hc are not exactly related to amorphous and channeling, respectively. ha corresponds to the part where it depends on the doses in high dose region, and the subtraction of ha from the total is described by hc .

10

22

10

21

10

20

10

19

10

18

10

17

10

16

P 30 keV

SIMS Ntotal

3 x 1013 cm-2

Namo

Nchan

0

100

200 300 Depth (nm)

400

Figure 11: P profiles expressed by dual Pearson.

A dual Pearson function can readily express the dose dependence of the ion implantation profiles as shown in Fig. 12. It covers ion implantation profiles for a large range of doses in crystalline substrates and is a standard function today.

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22

10

P 30 keV

-3

Concentration (cm )

21

10

13

-2

14

-2

SIMS 1 x 10 cm

20

SIMS 3 x 10 cm

19

SIMS 1 x 10 16 cm-2 Ntotal

10 10

ha

18

hc

10

17

10

16

10

0

100

200 300 Depth (nm)

400

Figure 12: Expression of dose dependence of P profiles with dual Pearson function.

TAIL FUNCTION

Dual Pearson can cover whole ion implantation profiles as mentioned in the previous section. However, it has problems associated with the uniqueness of parameter values. Fig. 13 shows the comparison of SIMS BF2 profiles with analytical model based on dual Pearson. We can obtain good agreement with SIMS data with quite different values of Rp 2 of 0 and 100 nm. The resultant the other parameter values of R p 2 and  chan are also quite different as shown in Table 1. This arbitrariness of available parameter set is a severe problem for establishing a vast ion implantation database. Table 1: BF2 profile moment parameters Case

Rp(nm)

Rp(nm)

(a)

38.1

34.4

(b)



-1.25



11

Rp2(nm)

Rp2(nm)

0

90

100

56

2

2

-0.94

6.21

chan(cm-2)

4x1014 4x1012

One pragmatic solution for this problem is that we fix the relationship between R p and Rp 2 such as R p  R p 2 . We should always set larger value of R p 2 than the

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value of R p in this method. This means the profile in the surface region is also influenced by hc when channel dose ratio is high. However, the profile shape before the peak region basically does not depend on the dose, but only the deeper region depends. We expect hc that expresses the channeling dependence without influencing the profile shape in the surface region. We may be able to find good relationship of R p  rR p 2 where r is larger than 1. However, the selection of r is rather arbitrary. Tail function was proposed to alleviate this problem on uniqueness of parameter value set [6-9]. This function is also composed of an amorphous part na  x  and a channeling part nc  x  and dose dependence is expressed with the same manner as dual Pearson as N x =  - chan n a x + chann c x

(54)

na  x  is a Pearson IV function and hence is expressed as hma like in the case of dual Pearson, that is, na  x   hma  x 

(55)

nc  x  is expressed by a Pearson IV function and a tail function hTc as follows

hmc  x  nc  x      hmc  x   hTc  x  

for x  xT for x  xT

(56)

hma and hmc have the same moment parameters. However, R p for hmc is changed when we observe double peaks. Therefore, the moment of the function is expressed by h ma = h ma R p, R p, ,  h mc = h mc R p2, R p, , 

(57)

where xT  R p 2  R p

(58)

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21

Concentration (cm-3)

10

BF2: 60 keV 1 x 10 15 cm-2

20

Rp2 = 0 nm

10

SIMS Total na

19

10

nc

18

10

17

10

16

Concentration (cm -3)

10

10

21

10

20

10

19

10

18

10

17

10

16

0

100 200 Depth (nm) (a)

300

BF2: 60 keV 1 x 10 15 cm-2 Rp2 = 100 nm SIMS Total na nc

0

100 200 Depth (nm) (b)

300

Figure 13: Expression of BF2 profiles with dual Pearson function with different moment parameters (a) Rp 2  0 nm , (b) Rp 2  100 nm .

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The peak concentration position x p for channeling part is given by x p  R p 2  b1

(59)

The tail function was introduced to express the channeling profiles in the low concentration region given by    x  xp   hTc  x   hmc  x p  exp   ln       L  

(60)

Concentration (cm-3)

 is arbitrary, but we use the constant value of 1000. The value of  is related to the meaning of L , that is, L is the distance where the concentration is 1/1000 of the peak concentration.  controls the shape of the tail profile.  is 1 for exponential, and 2 for Gaussian profile. Depending on the value of , the shape changes as shown in Fig. 14. This variation of the shape is limited, but can cover whole channeling tail profiles.  is determined by the continuity condition for x  xT , that is 10

20

10

19

10

18

L = 200 nm  = 1000  = 100 10 2

10

17

10

16

10

15

1 0.5 0.1 0.01

0

100 200 Depth (nm)

300

Figure 14: Dependence of hTc on .

hmc  xT     hmc  xT   hTc  xT  

(61)

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It should be noted that the tail function is not switched in the region of x  xT , but is defined from x p , and used with Pearson function in the region x  xT . This treatment ensures that profile becomes single Pearson function when L (channeling length) approaches to zero, which is a plausible feature. Finally, one profile is expressed by the two components as shown in Fig. 15. This function can cover whole ion implantation profiles with an unambiguous parameter value set and can be used for large ion implantation databases [9]. 21

10

15

-2

Concentration (cm-3)

B 40 keV 1 x 10 cm

SIMS Analytical

20

10

19

10

18

10

nc hc

17

10

na

16

10

0

100

200 300 Depth (nm)

400

500

Figure 15: Expression of an ion implantation impurity profile with the tail function.

APPROXIMATE TREATMENT OF GAUSS, JOINED HALF GAUSS FUNCTION

Gauss function and joined half Gauss function cannot accommodate whole ion implantation profiles as we mentioned. However, these functions are simple and we can easily obtain physical intuition from these functions. Furthermore, this treatment enables us to obtain tilt dependent Pearson profiles as shown in chapter 8. Therefore, it is convenient to generate these functions based on the database of Pearson functions which is commonly used.

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APPROXIMATE EXPRESSION OF GAUSS FUNCTION

We propose to set R p of Gaussas the peak position of Pearson, and the same R p as that of Pearson, that is R p Pearson IV + a Pearson IV  R p Gauss

(62)

R p Pearson IV  R p Gauss

(63)

APPROXIMATE EXPRESSION OF JOINED HALF GAUSS

Peak position of joined half Gauss R pm can be simply related to the peak position of Pearson function given by R p Pearson IV + a Pearson IV  R pm Joined half Gauss

(64)

The relationship between peak position and projected range is given by R pm  R p  a : Pearson

(65)

Further, the relationship between R pf , R pb , R pm , and R p in joined half Gauss are given by

R pm  R p 

2



 R

pf

 Rpb  : Joined half Gauss

(66)

From these, we obtain R pf - R pb =

a 2

(67)

Substituting Eq. 67 into Eq. 17, we obtain

R 2p = R pf R pb +  - 1 a 2 2

(68)

From Eqs. 67 and 68, we obtain R pf =

R 2p - 3 - 1 a 2 + 8

a 8

(69)

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R pb =

Kunihiro Suzuki

a 8

R 2p - 3 - 1 a 2 8

(70)

It is invoked that the inside of the root in Eqs. 69 and 70, and hence R 2p > 3 - 1 a 2 8

(71)

Further, R pf and R pb themselves are positive, that is R 2p - 3 - 1 a 2 >  a 2 8 8

(72)

We then obtain as R 2p >  - 1 a 2 2

(73)

Equation 73 is more severe than Eq. 71, and Eq. 73 is the final limitation. Fig. 16 shows comparison of Pearson function evaluated by FabMeister-IM [10] and approximate expression of Gauss and joined half Gauss functions. The B profile is asymmetrical and hence cannot be expressed by Gaussian but well by joined half Gauss. The profiles become symmetrical for P and As, and then the profiles can be better expressed by Gauss. 21

10

15

B 1 x 10 cm

-3

Pearson (LSS) Gauss Joined half Gauss

Concentration (cm-3)

10 keV 20 keV

20

10

40 keV

19

10

18

10

0

100 200 Depth (nm) (a)

300

Analytical Model for Ion Implantation Profiles

10

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10

20

10

19

10

18

10

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10 keV 20 keV

P 1 x 1015 cm-3 40 keV

Concentration (cm-3)

149

0

Pearson (LSS) Gauss Joined half Gauss

100 Depth (nm) (b)

200

22

10

15

Concentration (cm-3)

As 1 x 10 cm

21

10

-3

Pearson (LSS) Gauss Joined half Gauss

10 keV 20 keV 40 keV

20

10

19

10

18

10

0

50 Depth (nm) (c)

100

Figure 16: Approximate expressions for Pearson functions with Gauss and Joined half Gauss functions.

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DIFFERENTIAL DOSE

Four-direction multiple ion implantation is frequently used for the VLSI process to ensure symmetric device characteristics. Each ion implantation process step is treated independently in usual process simulators; therefore, a one time ion implantation profile is simply multiplied by four in such simulations. This treatment is good for low-dose ion implantation; however, the ion implantation dose for an extension region is high and the profile no longer increases linearly with multiple ion implantations [11]. To address this issue, we introduced a differential channel dose that enables us to obtain accurate ion implantation profiles. Consider that the ion implantation is performed n times and the final total dose is . Therefore, the dose for each ion implantation is  n . The plausible total ion implantation profiles after each ion implantation are expressed by   N1 x =  n - chan n n a x + chan n n c x   N2 x = 2 n - chan 2 n n a x + chan 2 n n c x

･････････・・・・・・・・・・・・・・・   Ni x = i n - chan i n n a x + chan i n n c x   Ni + 1 x = i + 1  n - chan i + 1 n n a x + chan i + 1 n n c x

･････････・・・・・・・・・・・・・・・・

  Nn x = n n - chan n n n a x + chan n n n c x =  - chan  n a x + chan  n c x

(74)

However, each ion implantation is treated independently in the process simulation, and the profile associated with each multiple ion implantation is the same and is

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given by N1  x  in Eq. 74. Therefore, the profile after each ion implantation profile in the simulation is   N1 x =  n - chan n n a x + chan n n c x   N2 x = 2  n - chan n n a x + 2chan n n c x

･････････・・・・・・・・・・・・・・・   Ni x = i  n - chan n n a x + ichan n n c x   Ni + 1 x = i + 1  n - chan n n a x + i + 1 chan n n c x

･････････・・・・・・・・・・・・・・・

  Nn x = n  n - chan n n a x + nchan n n c x = nN 1 x

(75)

22

10 -3

Concentration (cm )

14

21

10

-2

As 2.5 x 10 cm x 4 times

5 keV 40 keV

SIMS Differential dose Conventional

20

10

19

10

18

10

17

10

0

50 100 Depth (nm)

150

Figure 17: Comparison of SIMS profiles with analytical model. Solid lines correspond to our method, and dotted lines correspond to the former method.

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Impurity concentration profiles for four implantations of 2.5 x 1014 cm-2 are regarded as the profile of 2.5 x 1014 cm-2 multiplied by 4. The corresponding profiles are shown in Fig. 17 as dotted lines. This conventional method always predicts deeper junction depths for As, while it also predicts deeper junction depth at 20 keV but predicts similar junction depth at 3 keV for B. The profile at the (i+1)-th ion implantation, N i 1  x  , should be N i 1  x   N i  x  in Eq. 75 that is N i + 1 x = N i + 1 x - N i x   = i+1  n - chan i + 1 n n a x + chan i + 1 n n c x   - i n - chan i n n a x + chan i n n c x

(76)

Therefore, the profile associated with the (i+1)-th ion implantation has a dose of  chan i + 1  n - chan i n

Obviously, this treatment reproduces plausible ion implantation profiles as shown in Fig. 17. This methodology can be easily extended to multiple ion implantations, where a dose is divided by any amount. REFERENCES [1] [2] [3] [4] [5]

J. F. Gibbons, S. Mylroie, “Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions,” Appl. Phys. Let., Vol. 22, p.568, 1973. W. K. Hofker, “Implantation of boron in silicon,” Philips Res. Rep. Suppl., vol. 8, pp. 1-121, 1975. D. G. Ashworth, R. Oven, and B. Mundin, “Representation of ion implantation profiles by Pearson frequency distribution curves,” Appl. Phys. D., vol. 23, pp. 870-876, 1990. A. F. Tasch, H. Shin, C. Park, J. Alvis, and S. Novak, “An improved approach to accurately model shallow B and BF2 implants in silicon,” J. Electrochem. Soc., vol.136, pp. 810-814, 1989. C. Park, K. M. Klein, and A. L. Tasch, “Efficient modeling parameter extraction for dual Pearson approach to simulation of implanted impurity profiles in silicon,” Solid-State Electronic, vol. 33, pp. 646-650, 1990.

Analytical Model for Ion Implantation Profiles

[6] [7] [8] [9] [10] [11]

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K. Suzuki, Ritsuo Sudo, and T. Feudel,"Simple analytical expression for dose dependent ion-implanted Sb profiles using a jointed half Gaussian and one with exponential tail", Solid-State Electronics, vol. 42, pp. 463-465, 1998. K. Suzuki, R. Sudo, Y. Tada, M. Tomotani, T. Feudel, and W. Fichtner”, Comprehensive analytical expression for dose dependent ion-implanted impurity concentration profiles”, Solid-State Electronic, vol. 42, pp. 1671-1678, 1998. K. Suzuki, R. Sudo, T. Feudel, and W. Fichtner”, Compact and comprehensive database for ion-implanted As profile”, IEEE Trans. Electron Devices, ED-47, pp. 44-49, 2000. K. Suzuki and R. Sudo”, Analytical expression for ion-implanted impurity concentration profiles”, Solid-State Electronics, vol. 44, pp. 2253-2257, 2001. Mizuho Information & Research Institute, Inc.: Overview of the FabMeister-IM ion implantation profile simulator. http: //www.mizuho-ir.co.jp/solution/research/semiconductor/fabmeister/ion/index.html K. Suzuki and H. Tashiro, "Multiple ion implantation profile using differential channel dose”, IEEE Trans. Electron Devices, ED-50, pp. 1701-1705, 2003.

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APPENDIX A. GAUSS INTEGRATION

Gauss integration is given by 

exp - ax 2 dx = 1 2

I= 0

 a

This derivation process is as follows. We extend the integration region form  0,   to  ,   given by 

I' =

exp - ax 2 dx

-

(A-1)

Changing a variable from x to y, we also express the integration given by 

I' =

exp - ay 2 dy

-

(A-2)

Multiplying the both equation, we obtain 



2

I' =

exp - a x 2 + y 2 dxdy -

-

(A-3)

We changethe integration in Cartesian axis system to that in polar axis system, and obtain 

I' 2 =

exp - ar 2 2rdr 0 

exp - au 2r 1 du 2r  =  - 1a exp - au 0 = a =

u  r 2, du = 2rdr

0

(A-4)

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Therefore, we obtain the value of the integration

I' =

 a

(A-5)

I is related to I' given by 

I' =



exp - ax dx = 2 2

-

0

exp - ax 2 dx = 2I

(A-6)

Therefore, the Gauss integration is given by I = I' = 1 2 2

 a

(A-7)

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B. MODIFIED GAUSS INTEGRATION

The integration related to the Gauss integration is given by 

In a =

0

=

x 2n exp - ax 2 dx

2n - 1 !! 2n + 1

 a 2n + 1

for 2n - 1 !! = 1  3  5  7    2n - 1 

I0 a =

0



I1 a =

0 

I2 a =

0 

I3 a =

0

exp - ax 2 dx = 1 2

 a

x 2 exp - ax 2 dx = 12 2

 a3

x 4 exp - ax 2 dx = 33 2

 a5

 x 6 exp - ax 2 dx = 3 45 2

 a7

This derivation process is as follows. We start with Gauss integration given by 

I0 a =

0

exp - ax 2 dx = 1 2

 a

(B-1)

We regard this as a function of a . Differentiating this equation with respect to a , and obtain 

dI 0 a = da

0

3 - x 2 exp - ax 2 dx = 1  - a - 2 2

(B-2)

Therefore, we obtain I1  a  as 

I1 a =

0

x 2 exp - ax 2 dx = 12 2

 a3

(B-3)

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We differentiate this further and obtain 

dI 1 a = da

0

5 - x 4 exp - ax 2 dx = 12  - 3 a - 2 2 2

(B-4)

Therefore, we obtain I 2 a 

I2 a =

 a5

x 4 exp - ax 2 dx = 33 2

0

(B-5)

We differentiate this further and obtain 

dI 2 a = da

0

7 - x 6 exp - ax 2 dx = 33  - 5 a - 2 2 2

(B-6)

We obtain 

I3 a =

0

 a7

 x 6 exp - ax 2 dx = 3 45 2

(B-7)

Following the process, we obtain a general form as 

In a =

0

x 2n exp - ax 2 dx

3  5  7    2n - 1 2n + 1 2n - 1 !!  = 2n + 1 n+1 a 2 =

a



2n + 1

for 2n - 1 !! = 1  3  5  7    2n - 1

(B-8)

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C. MODIFIED GAUSS INTEGRATION 2

The integration related to the Gauss integration is also given by 

In a =

x 2n + 1exp - ax 2 dx = 0

n! 2a n + 1

I0 a = 1 2a 

I1 a = 0 

I2 a = 0 

I3 a = 0

x 3 exp - ax 2 dx = 1 2 2a x 5 exp - ax 2 dx = 2 3 2a  x 7 exp - ax 2 dx = 2 34 2a

This derivation process is as follows. We start with n=0 given by 

I0 a =

x exp - ax 2 dx 0 

x exp - au 1 du 2x

= 0

u = x 2, du = 2xdx



=1 2

exp - au du 0

= - 1 exp - au 2a 1 = 2a

 0

(C-1)

We regard this as a function of a . Differentiating this equation with respect to a , and obtain dI 0 a = da

 0

- x 3 exp - ax 2 dx = 1 - a - 2 2

(C-2)

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159

We obtain 

x 3 exp - ax 2 dx = 1 2 2a

I1 a = 0

(C-3)

We differentiate this further and obtain 

dI 1 a = da

0

- x 5 exp - ax 2 dx = - 23 2a

(C-4)

Therefore, we obtain 

x 5 exp - ax 2 dx = 2 3 2a

I2 a = 0

(C-5)

We further differentiate Eq. C-5 with a and obtain 

dI 2 a = da

0

 - x 7 exp - ax 2 dx = - 2 43 2a

(C-6)

We obtain 

I3 a = 0

 x 7 exp - ax 2 dx = 2 34 2a

(C-7)

Following the process, we obtain a general form as 

In a =

x 2n + 1 exp - ax 2 dx 0       = 2 3 4n +1 n 2a n! = n +1 2a

(C-8)

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D. DERIVATION OF MOMENTS OF JOINED HALF GAUSS

We describe the detail process of joined half Gauss moments. D-1 Dose    Rpm   x  R 2    x  R 2    pm pm   dx   exp    dx    Nm  exp   2 2      2R pf 2R pb          Rpm 

(D-1)

We denote the first and second term in Eq. D-1 as I 01 , I 02 and they are derived as 

   x  R 2  pm  dx I 02   exp    2R pb 2   Rpm     x  R pm 2  2R pb  e  s ds   for s  0  2R pb  

   

(D-2)

2 R pb 2

Considering the symmetry, I 01 can be given by I 01 

2 R pf 2

(D-3)

Therefore, the dose is given by

 2  2   Nm  R pf  R pb  2  2  

 2

 R

(D-4)

 R pb  N m

pf

Therefore, the maximum concentration N m is given by Nm 



 2

 R

pf

 R pb 

(D-5)

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161

D-2 Rp

Rp

 





xN  x  dx 

  R   x  R pm 2    x  R pm 2      x exp    dx   x exp    dx  2 2 2R pf  2 R pb           R   R pf  R pb  pm

1



(D-6)

pm

2

We denote the first and second term in Eq. D-6 as I11 , I12 and are given by 

   x  R 2  pm  dx I12   x exp   2    R 2  pb   Rpm 



0





2R pb s  R pm e  s 

2

(D-7)

2R pb ds 

 2R pb 2  s e s ds  2R pb R pm  e  s ds 2

0

0



 R pb 2 

2

2

R pb R pm

Considering the symmetry, we obtain

I11  R pf 2 

 2

R pf R pm

(D-8)

Therefore, we obtain       2 R pb R pm    R pf 2  R pf R pm   R pb  2 2    Rp     Rpf  Rpb  2  R pm 

2



 R

pb

 R pf



D-3 Rp

The second moments R p can be evaluated as

(D-9)

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Kunihiro Suzuki

R p

 x  R   

2

p



N  x  dx

(D-10)

 

R pm



    x  R 2    x  R 2  2 2 pm pm  dx    x  R p  exp    dx   x  R p  exp   2  2R pf   2R pb 2     Rpm    

 2

 R

pf

 R pb 

We denote the first and second term in Eq. D-9 as I 21 , I 22 and are given by 

   x  R 2  2 pm  dx I 22    x  R p  exp    2R pb 2   Rpm   

  x  R 2   2 pm  dx    x  R pm    R pm  R p   exp    2R pb 2   Rpm   

   x  R 2  2 pm  dx    x  R pm  exp    2R pb 2   Rpm   

   x  R 2  pm  dx 2  R pm  R p    x  R pm  exp    2R pb 2     Rpm 

  R pm  R p 





2R pb

   x  R 2  pm  dx  exp    2R pb 2     Rpm

2

 3



0

 R  R   2  R pm  R p  2

pm



 2

p

2

s 2 e  s ds 2R pb 2R pb

  se ds   e ds 2



 s2

0



 s2

0

R pb 3  2  R pm  R p  R pb 2 



R 2

 R p  R pb 2

pm

(D-11)

Analytical Model for Ion Implantation Profiles

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Considering the symmetry, we obtain



I 21 

R pf 3  2  R pm  R p  R pf 2 

2



R 2

 R p  R pf 2

pm

(D-12)

Consequently, we obtain a final form for R p of 1

R p 

 2

 R

pf

 R pb 



  2  R pf 3  2  R pm  R p  R pf 2  R pm  R p  R pf   2  2

  

(D-13)

    2    R pb 3  2  R pm  R p  R pb 3  R pm  R p  R pb    2  2   2  2  R pf R pb  1    R pb  R pf   

D-4 

The third moment 3 is given by

 x  R   

3



3

p

N  x  dx

 

R pm



   x  R 2     x  R 2  3 3 pm pm  dx    x  R p  exp    dx (D-14)   x  R p  exp   2 2  2R pf   2R pb         Rpm

 2

 R

pf

 R pb 

We denote the first and second term in Eq. D-14 as I 31 , I 32 and are given by

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Kunihiro Suzuki



   x  R 2  3 pm  dx I 32    x  R p  exp    2R pb 2   Rpm   

   x  R 2  3 pm  dx    x  R pm    R pm  R p   exp    2R pb 2   Rpm   

   x  R 2  3 pm  dx    x  R pm  exp    2R pb 2   Rpm   

3  R pm  R p 

2

   x  R 2  pm  dx   x  R pm  exp    2R pb 2   Rpm   

  x  R 2   2 pm  dx 3  R pm  R p    x  R pm  exp    2R pb 2   Rpm   

  R pm  R p  



2R pb

   x  R 2  pm  dx  exp    2R pb 2   Rpm  

3

 4



0

  s e ds 3  R  R   2R   se ds   R  R   2R   e ds 3  R pm  R p 



2

s 3e  s ds 2R pb

p

 2R pb 4  3 



R 2

pb



R 2



 s2

0

 R p  R pb 3  3  R pm  R p  R pb 2 2

pm

 R p  R pb 3

pm

(D-15)

 s2

0



p

2  s2

0

pb

3

pm



2

2

pm

3

Considering the symmetry, we obtain

Analytical Model for Ion Implantation Profiles



R 2

I 31  2R pf 4  3

pm

3  R pm  R p  R pf  2

2

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165

 R p  R pf 3

 2

R

(D-16)

 R p  R pf 3

pm

Consequently, we obtain 1

3 

 2

 R

 R pb 

pf



 2 3   4 R pm  R p  R pf 3  3  R pm  R p  R pf 2  R pm  R p  R pf    2R pf  3 2 2 

  

(D-17)

   2 3     2R pb 4  3 R pm  R p  R pb 3  3  R pm  R p  R pb 2  R pm  R p  R pb     2 2     4   R pf    1  R pb  R pf   

2

 R 



pb



2

  R pf R pb  

D-5 

The forth moment 4 is

 x  R   

4



4

p

N  x  dx

 

R pm



   x  R 2     x  R 2  4 4 pm pm  dx    x  R p  exp    dx (D-18)   x  R p  exp   2 2  2R pf   2R pb     Rpm    

 2

 R

pf

 R pb 

We denote the first and second term in Eq. D-18 as I 41 , I 42 and are given by

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Kunihiro Suzuki



   x  R 2  4 pm  dx I 42    x  R p  exp    2R pb 2     Rpm 

  x  R 2   4 pm  dx    x  R pm    R pm  R p   exp    2R pb 2   Rpm   

   x  R 2  4 pm  dx    x  R pm  exp    2R pb 2   Rpm   

   x  R 2  pm  dx 4  R pm  R p    x  R pm  exp    2R pb 2     Rpm 

6  R pm  R p 

2

4  R pm  R p 

3

  x  R 2   2 pm  dx   x  R pm  exp    2R pb 2     Rpm 

  x  R 2   3 pm  dx   x  R pm  exp    2R pb 2     Rpm 

  R pm  R p  



2R pb

  x  R 2   pm  dx  exp    2R pb 2     Rpm

4

 5



0

  s e ds 6  R  R   2R   s e ds 4  R  R   2R   se ds   R  R   2R   e ds 4  R pm  R p 



2

s 4 e  s ds 2R pb

p

pm

p

2

3

 2

pb

2  s2



 s2

0



p



0

pb

4

3  s2

0

pb

3

pm



3

2

pm

4

 s2

0

R pb 5  8  R pm  R p  R pb 4  3 2  R pm  R p  R pb 3 2

4  R pm  R p  R pb 2  3



R 2

 R p  R pb 4

pm

(D-19)

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167

Considering the symmetry, we obtain I 41  3

 2

R pf 5  8  R pm  R p  R pf 4  3 2  R pm  R p  R pf 3 2

4  R pm  R p  R pf  3

2

 2

R

 R p  R pb

(D-20)

4

pm

Consequently, we obtain

4 

I 41  I 42

 2

 R

pf

 R pb 

2  4 4 2   3   3     R pb  R pf        2 2 20     9    R pb  R pf  R pf R pb  3  R pf R pb    

(D-21)

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168

CHAPTER 5 Pearson Function Family Abstract: Pearson IV is one function among Pearson function family, where various Pearson functions exist. Each Pearson function is characterized by its related  2   plane. The moments of real profiles frequently deviate from the plane for the Pearson IV function. Therefore, we need to extend the function to whole Pearson function family. We described the detail derivation of the functions.

Keywords: Ion implantation, Pearson, Pearson function family, Pearson I, Pearson II, Pearson III, Pearson IV, Pearson V, Pearson VI, Pearson VII, skeweness, kurtosis, moment, projected rage. INTRODUCTION SIMS or MC data are sometimes outside of the  2   plane for Pearson IV.  is usually forced to the Pearson IV region in such cases. Pearson function family consists of Pearson IV and the others. If we use all Pearson functions, we need not to modify  and simply use a function that corresponds to the given moment parameters. Here, we describe Pearson function family [1-4]. MOMENTS OF PROFILES Pearson function is not defined in an infinite plane in general. We assume that the function h  x  is defined in the region  xa , xb  . The function is assumed to be normalized in this region, that is, xb

 h  x  dx  1

(1)

xa

The projected range R p is given by xb

R p   xh  x  dx

(2)

xa

We set the origin at R p and introduce a dummy variable s given by

s  x  Rp

(3) Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

Pearson Function Family

Ion Implantation and Activation, Vol. 1

169

We also introduce variables of sa  xa  Rp and sb  xb  R p . The i-th moment is defined as follows sb

i   s i h  s  ds sa

(4)

It is obvious, 0  1 and 1  0 . R p ,  , and  are defined as sb

R p   s 2 h  s  ds sa

 

sb

sa

s 3 h  s  ds R 3p

 

sb

sa

s 4 h  s  ds R p4

(5)

(6)

(7)

RELATIONSHIP BETWEEN PEARSON FUNCTION PARAMETERS AND MOMENTS

Pearson function family is generated from the same differential equation as Pearson IV given by

dh  s   s  a  h  s   ds b0  b1s  b2 s 2

(8)

Parameters of a , b0 , b1 , and b2 in Eq. 8 are related to the moments of h  s  as follows: Modifying Eq. 8, we obtain

b

0

 b1s  b2 s 2 

dh  s   s  a hs ds

(9)

Multiplying s n and integrating with respect to s in the region of  sa , sb  , we obtain

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sb  b s n  b s n 1  b s n  2 dh ds  sb s n 1  as n hds   0  1 2 sa  sa ds

(10)

Performing the integration, we obtain b 0s n + b 1s n + 1 + b 2s n +2 h =

sb sa

sa sa

-

sb sa

nb 0s n - 1 + n + 1 b 1s n + n + 2 b 2s n +1 hds

s n + 1 - as n hds

(11)

We impose that lim  b0 s n  b1s n 1  b2 s n  2  h  s   0

(12)

s  sa , sb

that is, lim s n  2 h  s   0

(13)

s  sa , sb

Equation 11 is then reduced to sb

sb

sb

sa

sa

sa

nb0  s n 1hds   n  1 b1  a   s n hds   n  2  b2  1  s n 1hds  0

(14)

Using the definition of the moment of Eq. 4, Eq. 14 is expressed by

nb0 n 1   n  1 b1  a  n   n  2  b2  1 n 1  0

(15)

This expression is the same as the one in the previous chapter. Substituting n = 0, 1, 2, and 3 in Eq. 15, we also obtain the followings: b0 = -

4 - 3 2 R p 2 A

(16)

b1 = -

+3 R p A

(17)

Pearson Function Family

b2 = -

Ion Implantation and Activation, Vol. 1

2 - 3 2 - 6 A

a  b1

171

(18) (19)

where A  10  12 2  18

(20)

These are exactly the same as those in the previous chapter where only the integration region is different. DIVISION OF PEARSON FUNCTION FAMILY AND THEIR SHAPE

Ion implantation profile has a bell shape in general, that is, the concentration increases, reachesits maximum value, and then decreases with increasing depth. The concentration should be always positive in the defined region. We modify Eq. 8 as

d ln h  s  sa  ds b0  b1s  b2 s 2

(21)

The denominator of the right side of Eq. 21 becomes 0 for certain values of s in some cases. Therefore, the profiles are defined in whole region when the denominator has no root, and defined in a limited region when the denominator has one or two root. Based on the above, we can divide the Pearson function family and draw the corresponding shape qualitatively [4]. The Pearson function is classified as follows: Denominator is constant  b2  b1  0  →Gauss

Denominator is first order  b2  0  →Pearson III

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Denominator is second order and has two roots  b2  0, D  0  →Pearson I, II    0  , VI

Denominator is second order and has one double root  b2  0, D  0  →Pearson V

Denominator is second order and has no root  b2  0, D  0  →Pearson IV, VII    0 

D is the discriminate of the denominator in Eq. 21 given by D  b12  4b0b2

(22)

The roots of the denominator in Eq. 21 are important in the classification of the Pearson function family, and hence we inspect them first. When the denominator is first order, the root  s is

s  

b0 b1

(23)

When the denominator is second order and has two roots, the corresponding roots 1 and  2 are

1 

b1  D 2b2

(24)

2 

b1  D 2b2

(25)

When the denominator has one double root, the corresponding root  D is given by

D  

b1 2b2

(26)

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173

We can evalu W uate the sign of the Eqq. 21 by insppecting b1 , roots, and hence h the grradient of prrofiles in alll region, andd draw the shape. s We can c then seleect proper caases for ion implantationn profile, andd know some constraintss of the profi files. D Denominator is constant

 b1  b2  0  : Gauss

Inn this case, Eq. E 21 reduces to

d ln h  s  ds



s b0

(27)

Assuming b0  0 , we cann evaluate thhe sign of eaach componnents, and thee gradient A off the profile based on thee signs of thhe componennts as follows. l h  s  for Gaauss Table 1: Sign of parameteres and a increase orr decrease of ln

The shape of the profile then T t becomees as shown in Fig. 1, whhich can be generated baased on Tab ble 1. This is Gauss function. fu Whhen b0 is positive, p the shape is uppside down,, and hence is not propeer for ion im mplantation profile. p Therrefore, the coorresponding g constraintss are given by b b1  b2  0, b0  0 D Denominator is first order  b2  0  ：Pearson P III Inn this case, Eq. E 21 reduces to

(28)

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d ln h s  b1  ds b1  s   s 

(29)

Fiigure 1: Shapee of Gauss funcction associated with Table 1. 1

We can evalu W uate the sign of each com mponent, andd the gradiennt of the profile based onn the signs of o the compoonents as folllows considdering variouus orders with respect too b1 and  s .

(ii) b1   s A Assuming b1  0 , we obtaain the follow wings. l h  s  for Peaarson III with b1   s Table 2: Sign of parameteres and a increase orr decrease of ln

Pearson Function Family

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175

Figure 2: Shape of Pearson III associated with Table 2.

The shape of the profile then becomes as shown in Fig. 2, which can be generated based on Table 2. The profile is defined in the region of  ,  s  with respect to s . This profile is called as Pearson III. When b1  0 , the profile is upside down, and is not appropriate for ion implantation profile. The corresponding constraints are b2  0, 0  b1   s

(30)

(ii) b1   s Assuming b1  0 , we obtain the followings. The shape of the profile then becomes as shown in Fig. 3,which can be generated based on Table 3. It is not appropriate for ion implantation profile. (iii) b1   s Assuming b1  0 , we obtain the followings.

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Table 3: Sign of parameteres and increase or decrease of ln h  s  for Pearson III with b1   s

Figure 3: The shape of Pearson III associated with Table 3.

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177

l h  s  for Peaarson III with b1   s Table 4: Sign of parameteres and a increase orr decrease of ln

Fiigure 4: Shapee of Pearson IIII associated wiith Table 4.

The shape of the profile then T t becomees as shown in Fig. 4, whhich can be generated baased on Tab ble 4. The prrofile is defined in the reegion  s ,   with resppect to s . T profile is also calledd as Pearsonn III. When b1  0 , the shape This s is upsiide down, annd it is no ot appropriatte for ion implantationn. Thereforee, the correesponding coonstraints fo or this profilee are b2  0,  s  b1  0 D Denominator is second orrder and has two compleex roots: Peaarson IV, VIII

(31)

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Assuming b1  0 , where the denominator is always negative, and we obtain the followings. Table 5: Sign of parameteres and increase or decrease of ln h  s  for Pearson IV

Figure 5: Shape of Pearson IV associatedwith Table 5.

The function is defined in all regions and the shape is shown in Fig. 5, which can be generated based on Table 5. This is called Pearson IV. When b2  0 , the shape

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179

iss upside dow wn, and it is not n appropriiate for ion implantation i n profile. Whhen b1  0 , thhe profile is i symmetryy, and this special caase is calledd Pearson VII. The coorresponding g constraintss for the proffile are expressed by D  0, b2  0

(32)

D Denominator is second orrderand has one double root: r Pearsonn V Inn this case, Eq. E 21 reduces to

s  b1 d ln h  2 ds b2  s   D 

(33)

We can evaluate the siggn of each components, W c , and the grradient of thhe profile baased on the signs of thee componentts as followss considering various orrders with reespect to b1 an nd  D . l h  s  for Peaarson V with b1   D Table 6: Sign of parameteres and a increase orr decrease of ln

(ii) b1   D A Assuming b2  0 , we obtaain the follow wings. The shape of the profile then T t becomees as shown in Fig. 6, whhich can be generated baased on Tablle 6. The profile is defined in the regioon of  ,  D  with resppect to s . T profile iss called as Peearson V. When This W b2  0 , the profile is i upside dow wn, and is

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not appropriate for ion implantation profile. The corresponding constraints for the profile are D  0, b2  0, b1   D

(34)

(ii) b1   D

Figure 6: Shape of Pearson V associated with Table 6.

Assuming b2  0 , the denominator is always negative or zero and we obtain the followings. Table 7: Sign of parameteres and increase or decrease of ln h  s  for Pearson V with b1   D

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181

Fiigure 7: Shapee of Pearson V associated witth Table 7.

The shape of the profile then T t becomees as shown in Fig. 7, whhich can be generated baased on Table 7. This is not approprriate for ion implantation i n profiles. (iiii) b1   D Assuming b2  0 , the denominator A d r is alwayss negative and we obbtain the foollowings. Table 8: Sign of parameteres and a increase orr decrease of ln l h  s  for Peaarson V with b1   D

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Figure 8: Shape of Pearson V associated with Table 8.

The shape of the profile then becomes as shown in Fig. 8, which can be generated based on Table 8. The profile is defined in the region of  D ,   with respect to s . This profile is also called as Pearson V. When b2  0 , the profile is upside down, and is not appropriate for ion implantation profile. The corresponding constraints for the profile are D  0, b2  0, b1   D

(35)

Denominator is second order and has two real roots: Pearson I, II, VI We denote the different two roots as 1 and  2 , and also denoted the smaller one as  a and the other as b . The relationship between 1 ,  2 and  a , b is decided by the sign of b2 .

1   a   2  b

for b2  0

(36)

for b2  0

(37)

and

1  b   2   a

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183

Inn this case, Eq. E 21 reduces to

d ln h s  b1  ds b2  s   a  s  b 

(38)

We can evalu W uate the signn of each component and a the graddient of the profile as foollows. We can c evaluate five cases depending d onn the order of o  a , b and a b1 . (ii) b1   a  b A Assuming b2  0 , and heence, 1  b   2   a , we w obtain thhe followingss. Table 9: Sign off parameteres and a increase orr decrease of ln h  s  for Peaarson VI with b1   a  b

The correspo T onding shapee is shown in Fig. 9, which w can bee generated based on T Table 9. The profile is deefined in thee region of  ,  a  2   with resppect to s . T This profile is also called as Pearsson VI. Whhen b2  0 , the profile is upside doown, and iss not appropriate for ion implantaation profilee. The correesponding coonstraints fo or the profilee are D  0, b2  0, b1   2 (iii) b1   a  b A Assuming b2  0 , and heence  a   2 , b  1 , wee obtain as following. fo

(39)

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Figure 9: Shape of Pearson VI associated with Table 9.

Figure 10: Shape of Pearson VI associated with Table 10.

Kunihiro Suzuki

Peearson Function n Family

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185

Table 10: Sign n of parameteres and incrrease or decreease of ln h  s  for Pearsoon VI with b1   a  b

The correspo T onding shapee is shown in i Fig. 10, which w can be generated based on T Table 10. Thiis is not apprropriate for ion i implantaation profile.. (iiii)  a  b1  b A Assuming b2  0 , and heence,  a  1 , b   2 , we w obtain as following. f Table 11: Sign of parameteress and increase or o decrease of ln h  s  for Pearson I with  a  b1  b

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Figure 11: Shape of Pearson I associated with Table 11.

The corresponding shape is shown in Fig. 11, which can be generated based on Table 11. The profile is defined in the region of  a 1  , b  2   with respect to s . This profile is also called as Pearson I. When b1  0 , the profile is symmetrical with respect to s  0 , and the profile with this special case is called Pearson II. When b2  0 , the profile is upside down, and is not appropriate for ion implantation profile. The corresponding constraints for the profile are D  0, b2  0, 1  b1   2

(40)

(iv)  a  b1  b Assuming b2  0 , and hence,  a  1 , b   2 we obtain as following. The corresponding shape is shown in Fig. 12, which can be generated based on Table 12. This shape is not appropriate for ion implantation profile. (v)  a  b  b1 Assuming b2  0 , and hence,  a   2 , b  1 , we obtain following.

Peearson Function n Family

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187

Table 12: Sign n of parameteres and incrrease or decreease of ln h  s  for Pearsoon VI with

 a  b1  b

Fiigure 12: Shap pe of Pearson VI V associated with w Table 12.

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Kun nihiro Suzuki

Table 13: Sign n of parameteeres and incrrease or decreease of ln h  s  for Pearsoon VI with

 a  b  b1

Fiigure 13: Shap pe of Pearson VI V associated with w Table 13.

The correspo T onding shapee is shown in i Fig. 13, which w can be generated based on   with resppect to s . T Table 13. The profile is defined d in thhe region off b 1  ,  T profile iss called as Pearson This P VI. When W b2  0 , the profile is upside down, d and iss not approp priate for ionn implantatioon profile. The T correspoonding consttraints for thhe profile aree

Pearson Function Family

Ion Implantation and Activation, Vol. 1

D  0, b2  0,  2  1  b1

189

(41)

 2   PLANE FOR PEARSON FUNCTION FAMILY Pearson function family has corresponding unique regions in   2 ,   plane. We first discuss some constraints that ion implantation profile should hold. POSITIVE VALUE CONDITION FOR h(s)

We suppose that the normalized ion implantation profile h s must be positive inthe defined region. We consider the integration I including arbitrary three variables x1 , x2 , and x3 given by

I=

2

x1 + x2s + x3s 2 h s ds

(42)

Since the term  x1  x2 s  x3 s 2  is always positive with respect to s for arbitrary combination of x1 , x2 , and x3 , where the variables do not hold x1  x2  x3  0 . This gives us a constraint that the moments should hold to ensure h  s  is positive in the defined region. 2

Expanding Eq. 42, we obtain I=

x1 2 + x2 2s 2 + x3 2s 4 + 2x1 x2s + 2x2 x3s 3 + 2x3 x1s 2 h s ds

= x1 2 + x2 2 2 + x3 2 4 + 2x2 x3 3 + 2x3 x1 2

(43)

Utilizing the relationship of  0 = 1,  1 = 0, we modify Eq. 43 as I = x1 +  2 x3 -  2 2 x3 2+ x2 2 2 + x3 2 4 + 2x2 x3 3 2

3 +  4 -  2 2 x3 2 2 2   2 2 = x1 +  2 x 3 +  2 x 2 2 + 3 x 3 - 3 x 3 2 +  4 -  2 2 x 3 2 2 2 2   2 2 = x1 +  2 x 3 +  2 x 2 2 + 3 x 3 +  4 -  2 2 - 3 x 3 2 2 2 = x1 +  2 x3 +  2 x2 2 + 2x2 x3 2

(44)

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Since 2 is positive, the first and the second term in Eq. 44 are positive. Therefore, we can impose the factor for x3 2 is positive to ensure I is positive, that is

4 > 22 +

32 2

(45)

This can be converted to

 > 1 + 2

(46)

More general treatment is shown in Appendix which is shown in Fig. 14.  limit

10

 D2



 b2 3 A

5

2  limit  D1

0

0

1

2

2

3

 b0 1

4

5

Figure 14: Region that h  s  should hold to ensure positive value.

The ion implantation profile should have bell shape and the concentration should be positive in the defined region. This imposes another constraint. Fig. 15 shows schematic asymmetrical bell shape profile where b1 corresponds to peak position. Inspecting the figures, we can impose  is negative when b1 is positive, and vice versa. This implicitly assumes that h  s  is positive in the defined region.

Pearson Function Family

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191

We describe again the expression for b1 as b1 = -

+3 R p A

(47)

  3 and R p are always positive. Therefore, A must be positive to ensure that  is negative when b1 is positive. On the other hand, A must also be positive to ensure that  is positive when b1 is negative. Consequently, A must always be positive to ensure positive value of h  s  . We denote  that holds A  0 as  A that is given by A 

12 2 18   10 10

(48)

Figure 15: The relationship between b1 , R p , and sign of  

Finally,  should hold

  A

(49)

to ensure positive value for h  s  in whole defined region. It should be noted that we use A for Pearson function family, and hence the constraint is limited to Pearson function family and cannot be applied to general cases. The constraints is more severe than positive condition of Eq. 44, and hence we can regard this for final constraint for Pearson function. This region is shown in Fig. 16.

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Kunihiro Suzuki

A+

10

 D2



 b2 3 A

5

2  limit  D1

0

0

1

2

2

3

 b0 1

4

5

Figure 16: The region where A is positive. This is the positive value constraints for Pearson function family.

BELL SHAPE CONDITION

We assume that the function has bell shape. This can be mathematically expressed by the second derivative is negative for the peak position of s  b1 . Differentiating Eq. 8 twice with respect to s , we obtain

2

d h ds 2 s b

1

dh   h s b b0  b1s  b2 s 2    s  b1  h  b1  2b2 s       1   s   2  b0  b1s  b2 s 2  s  b1

  s  b1  h  b  b s  b s 2  s  b h b  2b s   1  1 2  2  h   s  b1   0 1 b0  b1s  b2 s 2    2  b0  b1s  b2 s 2 

(50) s  b1

 

h b0  b1s  b2 s 2 h  b1  b0  b12  b1b2

s b1

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193

We assume that the peak concentration h  b1  is positive, and hence this can be regarded as denominator is negative, that is, b0  b12 1  b2 

   3  2 R 2 1  2  3 2  6  4   3 2  R p 2   p  A A2 A   2

  4   3 2 10   12 2  18    2    3  8  9 2  12  2



(51)

2

A3

R p

2

B R p  0 A A 2

where B is defined as

B    4  3 2 10   12 2  18    2    3  8  9 2  12  2

2

(52)

We also utilize the positive condition of A  0 , and hence Eq. 51 can be regarded as B  0 . We regard B  0 as a function of  , that is, it is third order equation with respect to  . The factor associated with  3 is 8  50   2  . In practical ion implantation profiles, we can assume  2  50 , and hence the factor is assumed to be negative. Therefore, we have three roots of 1 ,  2 , and  3 in order as schematically drawn in Fig. 17.

Figure 17: Shape of B    .

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We evaluated these roots numerically and showed them in  2   plane （Fig. 18). The bell shape condition for B A  0 generates three yellow regions. Combining the positive value condition, we finally impose Pearson function family as

  3

(53)

Since  3 is important for Pearson function, its approximate analytical expression was proposed [2] and is frequently used. It is given by   9 6 9  6 2  5     8 4  25   2  1  16  3  2    2 50  

(54)

Bell shape

10

 D2



 b2 3  A

5

2  limit  D1

0

0

1

2

2

3

Figure 18: Bell shape condition in  2   plane.

Pearson function family region in  2   plane

 b0 1

4

5

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195

Here, we assign each Pearson function region in  2   plane considering its corresponding constraints. GAUSS PROFILE

The constraints for Gauss profile are b1  b2  0, b0  0 . Therefore, they are expressed by b1 = -

+3 R p = 0 A

(55)

b2 = -

2 - 3 2 - 6 =0 A

(56)

This means that

  0    3

(57)

We further consider b0  0 . b0 is given by b0  

4   3 2 R p 2 10   12 2  18

 R p

(58)

2

Therefore, b0  0 is always held. PEARSON III FUNCTION

The constraints for Pearson III profile are b2  0, 0  b1   s , or b2  0,  s  b1  0 . We denote  that holds b2  0 as  b 2 that is given by

3 2

b 2   2  3 The parameters of b1 and b2 with this  are given by

(59)

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b1  

b 2  3 R p 10  b 2  12 2  18

3 2     3  3 2    R p 3 2  2 10    3   12  18 2  1   R p 2 b0  

Kunihiro Suzuki

(60)

4b 2  3 2 Rp 2 10 2  12 2  18

3  4   2  3   3 2 2   Rp 2 3 2  10    3   12 2  18 2  2  Rp

(61)

Therefore,  s is given by

s  

b0 b1

R p 2  1  R p 2 2R p 

(62)



The ratio of b1 with respect to  s is given by

1  R p b1 2 2   2R p s 4 

(63)



In the former constraint of b2  0, 0  b1   s , b1  0 means   0 , and it then also means  s  0 . Therefore,  2  4 to ensure 0  b1   s . In the latter

Pearson Function Family

Ion Implantation and Activation, Vol. 1

197

constraint of b2  0,  s  b1  0 , b1  0 means   0 , and it then also means  s  0 . Therefore,  2  4 to ensure  s  b1  0 . Consequently,  2  4 always holds. Finally, the corresponding region are given by

  b 2 and  2  4

(64)

PEARSON IV FUNCTION

The constraint for Pearson IV are D  0 and b2  0 . The discriminate D is given by D = b 21 - 4b 0b 2 2

2 +3 2 4 - 3 2 2 2 2 - 3 - 6 =  R 4 R p p A A A2 2 R p 2 2 =   + 3 - 4 4 - 3 2 2 - 3 2 - 6 2 A R p 2 2 =  - 32  2 + 78 2 + 96  -  2 36 2 + 63 2 A

(65)

We assume  2  32 , and denote the two roots of the second order equation in Eq. 65 as  D1 and  D 2 which are given by 3

 D1 

3 13 2  16   6   2  4  2 32   2

(66)

3

 D1 

3 13 2  16   6   2  4  2 32   2

(67)

The corresponding region for D  0 is then given by

   D1 or    D 2 The former is invalid since it does not hold   3 . The corresponding region for b2  0 under the condition of    D 2 is

(68)

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Ion implantation and activation, Vol. 1

Kunihiro Suzuki

  b 2

(69)

This is weaker restriction than    D 2 . Finally the corresponding region for Pearson IV is

  D2

(70)

PEARSON VII FUNCTION

Pearson VII is a special case for Pearson IV where   0 , and hence, the corresponding region is

   D 2 and   0

(71)

PEARSON V FUNCTION

The constraints for Pearson V are D  0, b2  0, b1   D or D  0, b2  0, b1   D . From D  0 , we obtain

   D1 or    D 2

(72)

From the general constraint of   3 ,    D1 is invalid. Under the condition of    D 2 , b2  0 corresponds to

  b 2

(73)

This automatically holds. Next, we consider the former condition of b1   D . Note that

D  

b1 2b2

Therefore, b1   D is expressed by

b1  

b1 2b2

2b1b2  b1

b1  2b2  1  0

(74)

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Ion Implantation and Activation, Vol. 1

199

When b1  0 , that is,   0 , Eq. 74 reduces to 2b2  1  0

(75)

This is modified to 2

2   3 2  6 1  0 10   12 2  18

2   3 2  6    5  6 2  9  7   9 2  15 9 7

  2 

(76)

15 7

where we use A  10  12 2  18  0 under the condition of    D 2 . Equation 76 breaks the Pearson function limitation, and is hence invalid. When b1  0 , that is,   0 , Eq. 74 reduces to 2b2  1  0

(77)

This is modified to

9 7

  2

15 7

(78)

and is always valid. Therefore, the constraints of D  0, b2  0, b1   D correspond to the region of

  D2 ,   0

(79)

Similarly, from the latter restriction of D  0, b2  0, b1   D , we obtain

  D2 ,   0

(80)

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Kunihiro Suzuki

PEARSON I FUNCTION

The constraints for Pearson I are D  0, b2  0, 1  b1   2 . From D  0 and b2  0 , we obtain

 D1     D 2

(81)

and

  b 2

(82)

Considering the general constraint of Pearson function of   3 , we obtain

3    b 2

(83)

1  b1   2 gives no additional constraint. PEARSON II FUNCTION

Pearson II function is the special case of Pearson I with   0 , and hence the correspond region is expressed by  3    b 2    0

(84)

PEARSON VI FUNCTION

The constraints for Pearson VI are D  0, b2  0, b1   2 or D  0, b2  0, b1  1 . From D  0 , we obtain

 D1     D 2

(85)

From b2  0 , we obtain

  b 2 or    A

(86)

Pearson Function Family

Ion Implantation and Activation, Vol. 1

201

Considering the general constraint of Pearson function of   3 , we obtain

b 2     D 2

(87)

The constraints of b1   2 or b1  1 can be relate to  as b1   2    0;  ,  2 

(88)

b1  1    0;  2 ,  

(89)

10  D2 VI VII

IV



V

 b2 3

III

5

A 2

I Gauss

0

II

0

 D1

1

2

2

3

1

4

5

Figure 19: Regional map for each Pearson function in  2   plane.

Summarizing above, we obtain the regional area for each Pearson function as shown in Fig. 19. It should be noted that the constraint of   3 is related to Pearson function. The constraint for positive value is    A . As we showed before, the region associated with joined half Gauss is in between  2    3 . It should also be noted that we assume

 2  32

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Kunihiro Suzuki

in the evaluation process. If this treatment breaks, we use the following treatment.



 0.9 32 

for   0.9 32

   D 2  2 

(90) (91)

ANALYTICAL EXPRESSION FOR PEARSON FUNCTIONS

We derive analytical expression for Pearson function by solving a corresponding differential equation. GAUSS FUNCTION

b2  b1  0 in Gauss function and the corresponding differential equation is d ln h s  ds b0

(92)

This can be solved as

ln h 

s2  ln K 2b0

(93)

Arbitrary constant K is determined by the normalization condition. It is also the case of the other functions. This can also expressed by  s2  h  s   K exp    2b0 

(94)

In Gauss function,

  0    3 Therefore, b0 is evaluated as

(95)

Pearson Function Family

Ion Implantation and Activation, Vol. 1

4 - 3 2 R p 2 10 - 12 2 - 18 = - 12 R p 2 30 - 18 = - R p 2

203

b0 = -

(96)

We then obtain  s2  h  s   K exp    2R p 2   

(97)

The normalization condition is expressed by 

K

exp -

s 2 ds = K 2 R = 1 p 2R p 2

(98)

We obtain K and the final expression is as we expected and is given by hs =

2 1 exp - s 2 2 R p 2R p

(99)

PEARSON III FUNCTION

In this function, b2  0 and the corresponding differential equation is given by d ln h = s - b 1 b 0 + b 1s ds s - b1 = 1 b1 b s+ 0 b1 b0 b0 s+ - b1 b1 b1 1 = b1 b s+ 0 b1 b0 + b1 b = 1 1- 1 b1 b s+ 0 b1

(100)

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Kunihiro Suzuki

This can be solved as b b ln h = 1 s - 1 + 02 ln s + 0 + ln K b1 b1 b1

(101)

PEARSON IV FUNCTION

In this function, the corresponding restrictions are D  0 and b2  0 . The discriminate D is negative and hence  D  4b0b2  b12  0

(102)

The denominator of the differential equation is modified as

b 0 + b 1s + b 2s 2 = b 2 s 2 + = b2 = b2

b b1 s+ 0 b2 b2 b 2 b1 2 b0 s2 + 1 + 2b 2 2b 2 b2 b 2 4b 0b 2 - b 1 2 s2 + 1 + 2b 2 4b 2 2

The differential equation is then modified as d b 0 + b 1 s + b 2s 2 b d ln h = 1 ds - b1 + 1 2b 2 2b 2 b 0 + b 1s + b 2s 2 ds

This can be solved as

1 b2

b s + 1 2b 2 2

2

4b 0b 2 - b 1 2 + 4b 2 2

(103)

Pearson Function Family

ln h 

Ion Implantation and Activation, Vol. 1

205

1 ln b0  b1s  b2 s 2 2b2

  b  s2  1  1 2b2  tan 1    ln K 2 2  b2 b b b 4b0b2  b1 4  0 2 1   2 2   4b2 4b2   b 2b1  1  2b s 2  b b2 1 1 2 2 1  ln b0  b1s  b2 s  tan  2 2  2b2 4b0b2  b1  4b0b2  b1 b1 

b1 2b2

(104)

   ln K  

In the derivation process of Eq. 104, we use that b2 is negative and the detailed process is as follows.

 b  s2  1 1 2b2 tan 1  4b0b2  b12 4b0b2  b12   4b2 2 4b2 2 

 b   s2  1  1 2b2  tan 1   1  1 4b0b2  b12 4b0b2  b12    2b2  2b2   2b s 2  b  2b2 2 1  tan 1    2 2   4b0b2  b1 4  b b b 0 2 1    2b s 2  b  2b2 2 1  tan 1   2 2   4b0b2  b1 4  b b b 0 2 1  

     

PEARSON VII FUNCTION

In this function, the restrictions are D  0 , b2  0 b2, and b1  0 , that is, this function is a special case of Pearson IV with   0 . We therefore, obtain the analytical expression by substituting b1  0 to Eq. 104 and obtain ln h 

1 ln b0  b2 s 2  ln K 2b2

(105)

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Kunihiro Suzuki

PEARSON V FUNCTION

The corresponding restrictions are D  0 and b2  0 . Discriminate D  0 and the corresponding differential equation is modified as d b 0 + b 1s + b 2s 2 b d ln h = 1 ds - b1 + 1 2 2b 2 b 0 + b 1s + b 2s 2b 2 ds

1 b2 s +

b1 2b 2

2

(106)

This can be solved as ln h = 1 ln b 0 + b 1s + b 2s 2 + 2b 2

= 1 ln b 0 + b 1s + b 2s 2 2b 2

b1 +

b1 2b 2

b2

1 s+

b1 2b 2

+ ln K

b1 b2 + + ln K 2b 2s + b 1 2b 1+

(107)

PEARSON I FUNCTION

The corresponding constraints are D  0 and b2  0 . The differential equation can be modified as s - b1 d ln h = ds b 0 + b 1s + b 2s 2 1 b + 2b s - b - b 1 2 1 2b 2 1 2b 2 = 2 b 0 + b 1s + b 2s b 1 + 2b 2s b 1 = 1 - b1 + 1 2b 2 b 0 + b 1s + b 2s 2 2b 2 b 0 + b 1s + b 2s 2 d b 0 + b 1s + b 2 s 2 b ds 1 = 1 - b1 + 1 2b 2 b 0 + b 1s + b 2s 2 2b 2 b 0 + b 1s + b 2s 2

(108)

The discriminate D is positive and hence the denominator can be modified as

Pearson Function Family

Ion Implantation and Activation, Vol. 1

1 1  2 b0  b1s  b2 s b2  s   a  s  b 

207

(109)

where  a and b are the tow real roots of b0  b1s  b2 s 2  0 . Equation 109 can be modified as 1

b2  s   a  s  b 



r  1  r1  2   b2  s   a s  b 

(110)

The arbitrary constants can be determined as follows. Multiplying

 s   a  s  b 

on the both sides of Eq. 110, we obtain

1  r1  s   a   r2  s  b    r1  r2  s   r1 a  r2b  r1 and r2 should hold  r1  r2  0   r1 a  r2b  0

(111)

From  a  1 , b   2 , we obtain r1  

1 b   a



b2

r 2=

(112)

b12  4b0b2 b2 2

b 1 - 4b 0b 2

(113)

The differential equation is

d ln h 1  ds 2b2

d  b0  b1s  b2 s 2  ds b0  b1s  b2 s 2

 b1   b1   2b2   1 1       2 b1  4b0b2  s   a s  b 

(114)

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Kunihiro Suzuki

We can save the differential equation as  b1   b1   2b2  s  b 1  2  ln K ln h  ln b0  b1s  b2 s  ln s  a 2b2 b12  4b0b2

(115)

From  a  1 , b   2 , we obtain  b1  b12  4b0b2 b1  s  b1   2b2  1 2b2  2 ln h  ln b0  b1s  b2 s  ln  ln K 2b2 b12  4b0b2 b1  b12  4b0b2 s 2b2

(116)

 b1   b1   s  b1  b12  4b0b2 2b2  1  2 ln b0  b1s  b2 s  ln 2b2   ln K 2b2 b12  4b0b2 2b2 s  b1  b12  4b0b2

PEARSON II FUNCTION

This function is a special case of Pearson I with b1  0 , and hence we can obtain corresponding analytical expression by substituting b1  0 into Eq. 116 and obtain ln h 

1 ln b0  b2 s 2  ln K 2b2

(117)

PEARSON VI FUNCTION

The corresponding constraints are D  0 and b2  0 .These are the same as Pearson I and the expression is also the same as one. The difference is the definition region. SUMMARY OF PEARSON FUNCTION FAMILY

Here, we summarize the discussions above.

Peearson Function n Family

Ioon Implantation and Activation,, Vol. 1

209

Fig. 20 show ws the correspponding reggion in  2   plane, annd various conditions c inn Table 14.

10  D2 VI VIII

IV



V

 b2 3

IIII

5

 lim mit

I Gauss

0

III

0

1

2

Fiigure 20: Pearrson function inn   2 ,   planne. Table 14: Summ mary of Pearsoon parameters

2

3

4

5

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Ion implantation and activation, Vol. 1

Kunihiro Suzuki

Differential equation and relationship between moments and related to coefficients. s-a h dh = ds b 0 + b 1s + b 2s 2

(118)

b0 = -

4 - 3 2 R p 2 A

(119)

b1 = -

+3 R p A

(120)

b2 = -

2 - 3 2 - 6 A

(121)

a  b1

(122)

A = 10  - 12  2 - 18

(123)

Prameters associated with defined region

s  

b0 b1

(124)

b1  b12  4b0b2 1  2b2

(125)

b1  b12  4b0b2 2  2b2

(126)

D  

b1 2b2

(127)

Parameters associated with   2 ,   plane 3

D2 

3 13 2  16   6   2  4  2 32   2

(128)

Pearson Function Family

Ion Implantation and Activation, Vol. 1 .

3 2

b 2   2  3

211

(129)

  9 6   8 4  25   2  1  9 6 2  5  16  3   2 50  

(130)

 lim it   2  1

(131)

Analytical expressions for Pearson function family Gauss:   0,   3;  ,  

ln h  s  

s2  ln K 2b0

(132)

Pearson III:    b 2 ;  ,  s  for   0,  s ,   for   0 ln h  s  

 b  b 1 s  1  02  ln s  0  ln K b1 b1  b1 

(133)

Pearson IV:    D 2 ;  ,  

ln h  s  

2b1 

b1 b2

 2b s 2  b 1 2 1 ln b0  b1s  b2 s 2  tan 1  2 2  2b2 4b0b2  b1  4b0b2  b1

   ln K  

(134)

Pearson VII:    D 2 ;  ,   (Pearson IV with   0 、that is b1  0 ) ln h  s  

1 ln b0  b2 s 2  ln K 2b2

(135)

Pearson V:    D 2 ;  ,  D  for   0,  D ,   for   0 b1 b2 1  ln K ln h  s   ln b0  b1s  b2 s 2  2b2 2b2 s  b1 2b1 

(136)

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Kunihiro Suzuki

Pearson I:  3     b 2 ; 1 ,  2 

ln h  s  

b1 

2b  b  b12  4b0b2 1 ln b0  b1s  b2 s 2  ln 2 1  ln K (137) 2b2 4b0b2  b12 2b2  b1  b12  4b0b2

Pearson II: ln h  s  

 3     b 2 ; 1 ,  2 

(Pearson I with.., that is, b1  0 )

1 ln b0  b2 s 2  ln K 2b2

Pearson VI:

ln h  s  

b1 2b2

(138)

b 2     D 2 ;  , 1  for   0, 2 ,  for   0 b1 

b1 2b2

2b  b  b12  4b0b2 1 ln b0  b1s  b2 s 2  ln 2 1  ln K 2b2 4b0b2  b12 2b2  b1  b12  4b0b2

(139)

REFERENCES [1] [2] [3] [4]

W. K. Hofker, “Implantation of boron in silicon,” Philips Res. Rep. Suppl., vol. 8, pp. 1-121, 1975. D. G. Ashworth, R. Oven, and B. Mundin, “Representation of ion implantation profiles by Pearson frequency distribution curves,” Appl. Phys. D., vol. 23, pp. 870-876, 1990. S. Selberherr, Analysis and simulation of semiconductor devices, Springer-Verlag, Wien New York, 1984. HySyProSUsers’ manual http: //www.selete.co.jp/?lang=EN&act=Research&sel_no=103: Semiconductor Leading Edge Technology.

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Ion Implantation and Activation, Vol. 1

213

APPENDIX A. POSITIVE VALUE CONDITION

The necessary condition for positive h(s) is I= =

2

x1 + x2s + x3s 2 h s ds x1 2 + x2 2s 2 + x3 2s 4 + 2x1x2s + 2x2x3s 3 + 2x3x1s 2 h s ds

= x1 2 0 + x2 2 2 + x3 2 4 + 2x1x2 1 + 2x2x3 3 + 2x3x1 2 > 0

(A-1)

This integration is identical to the matrix of 0 1 2 I = x 1, x 2, x 3  1  2  3 2 3 4

x1 x2 x3

(A-2)

We can verify it by expanding Eq. A-2 as x1 0 + x2 1 + x3 2 x1 1 + x2 2 + x3 3 x1 2 + x2 3 + x3 4

I = x1, x2, x3

= x1 2 0 + x1 x2 1 + x1 x3 2 + x1 x2 1 + x2  2 + x2 x3 3 + x3 x1 2 + x3 x2 3 + x3  4 (A-3) 2

2

2

2

= x1 2 0 + x2  2 + x3  4 + 2x1 x2 1 + 2x2 x3 3 + 2x1 x3 2

This is identical to Eq. A-1 as expected. We consider symmetrical matrix a 11 a 12 a 13 A = a 21 a 22 a 23 for a ij = a ji a 31 a 32 a 33

(A-4)

The related positive value condition for x t Ax > 0

is expressed by

(A-5)

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Ion implantation and activation, Vol. 1

Kunihiro Suzuki

a 11 a 12 a 13 a a a 11 > 0 and a 11 a 12 > 0 and a 21 a 22 a 23 > 0 21 22 a 31 a 32 a 33

(A-6)

 x1  where x   x2  . This can be proved as follows. x   3

a 11x1 + a 12x2 + a 13x3 I = x1, x2, x3 a 12x1 + a 22x2 + a 23x3 a 13x1 + a 23x2 + a 33x3 =a 11x1 2 +a 22x2 2 + a 33x3 2 + 2a 12x1 x2 + 2a 13x1x3 + 2a 23x2 x3

(A-7)

The term including x1 can be modified as a 11x1 2 + 2a 12x1x2+ 2a 13x1 x3 a a = a 11 x1 2 + 2 12 x1 x2+ 2 13 x1x3 a 11 a 11 2 a 13 2 a 12 a a 12 = a 11 x1 + x2+ x3 x2+ 13 x3 a 11 a 11 a 11 a 11 a 13 2 a a 12 a 12 = a 11 x1 + x2+ x3 - a 11 x2+ 13 x3 a 11 a 11 a 11 a 11

2

(A-8)

The sum of the later term and terms in I not including x1 is given by 2 a a 12 x2+ 13 x3 a 11 a 11 2 a 2 a a a = a 22 x2 2 + a 33 x3 2 + 2a 23 x2 x3 - 12 x2 2 + 13 x3 2 + 2 12 13 x2 x3 a 11 a 11 a 11

a 22 x2 2 + a 33 x3 2 + 2a 23 x2 x3- a 11

=

a 11a 22 - a 12 2 a 11

x2 + 2 2

a 11a 23 - a 12a 13 a 11

x2 x3 +

a 11a 33 - a 13 2

The terms including x2 can be modified as

a 11

x3 2

(A-9)

Pearson Function Family

a 11a 22 - a 12 2 a 11 = = =

x2 2 + 2

a 11a 22 - a 12 2

a 11a 23 - a 12a 13 a 11

x2 2 + 2

a 11 a 11a 22 - a 12 2

x2 +

a 11 a 11a 22 - a 12

Ion Implantation and Activation, Vol. 1

x2 +

a 11

x2 x3

a 11a 23 - a 12a 13 a 11a 22 - a 12 2

a 11a 23 - a 12a 13 a 11a 22 - a 12

2

a 11a 23 - a 12a 13

2

215

a 11a 22 - a 12 2

x2 x3 2

x3

-

a 11a 23 - a 12a 13 a 11a 22 - a 12

2

2

x3 2 2

2

1 a 11a 23 - a 12a 13 x3 2 a 11 a 11a 22 - a 12 2

x3

(A-10)

The sum of the latter term and terms in I not including x1, x2 is given by a 11a 33 - a 13 2 a 11 = = =

2

1 a 11a 23 - a 12a 13 x3 x3 2 a 11 a 11a 22 - a 12 2 2

a 11a 33 - a 13 2 a 11a 22 - a 12 2 - a 11a 23 - a 12a 13 a 11 a 11a 22 - a 12 2

2

x3 2

a 11 2a 33a 22 - a 11a 33a 12 2 - a 13 2a 11a 22 + a 13 2a 12 2 - a 11 2a 23 2 - a 12 2a 13 2 + 2a 11a 23a 12a 13 a 11 a 11a 22 - a 12 2 a 11 2a 33a 22 - a 11a 33a 12 2 - a 13 2a 11a 22 - a 11 2a 23 2 + 2a 11a 23a 12a 13 a 11 a 11a 22 - a 12

2

a 11a 33a 22 - a 33a 12 - a 13 2a 22 - a 11a 23 2 + 2a 23a 12a 13

x3 2

x3 2

2

=

a 11a 22 - a 12 2

x3 2

(A-11) Therefore, I is given by

I = a 11

a 11 a 12 a 21 a 22 a 13 2 a 12 x1 + x2+ x3 + a 11 a 11 a 11

a 11 a 12 a 13 a 21 a 22 a 23 2 a 11a 23 - a 12a 13 a 31 a 32 a 33 2 x2 + x + 3 a 11 a 12 x3 a 11a 22 - a 12 2 a 21 a 22 (A-12)

Therefore, the condition that is positive is given by Eq. A-6.

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B.GENERAL TREATMENT FOR POSITIVE CONDITION

More general positive condition for positive h  s  is given by I    x1  x2 s  x3 s 2   xn s n 1  h  s  ds 2

 n n    xi x j s i  j  2 h  s  ds  i 1 j 1 n

n

n

n

(B-1)

    s i  j  2 h  s  ds  xi x j   i 1 j 1   aij xi x j  0 i 1 j 1

where

aij   s i  j  2 h  s  ds  i  j  2

(B-2)

 a ji This can give the conditions that high order moments should hold to ensure positive h  s  . This canbe expressed by a matrix form given by

I   x1

x2

x3

 a11   a21  xn   a31    a  n1

a12 a22 a32  an 2

a13  a1n   x1    a23  a2 n   x2  a33  a3n   x3          an 3  ann   xn 

(B-3)

Selecting terms that include x1 in I , we obtain a11 x12  2a12 x1 x2  2a13 x1 x3    2a1n x1 xn

(B-4)

I should be positive for any xi . Setting x1  1, x2  x3    xn  0 , we obtain

Pearson Function Family

Ion Implantation and Activation, Vol. 1

a11  0

217

(B-5)

Rearranging Eq. B-4, we obtain   a a a a11  x12  2 12 x1 x2  2 13 x1 x3    2 1n x1 xn  a11 a11 a11   2

  a  a a a a a  a11  x1  12 x2  13 x3    1n xn   a11  12 x2  13 x3    1n xn  a11 a11 a11  a11 a11    a11

2

(B-6)

We convert the variables as X 1  x1 

a a a12 x2  13 x3    1n xn a11 a11 a11

(B-7)

The other variables are invariant but change the expression as X r  xr

(B-8)

Summing the latter term in Eq. B-6 and the terms not including x1 in I , I is then expressed by n

n

I  a11 X 12   bij X i X j

(B-9)

i 2 j 2

Selecting terms including X 2 , we obtain b22 X 2 2  2b23 X 2 X 3  2b24 X 2 X 4    2b2 n X 2 X n

(B-10)

In this case, we set X 2  1, X 3  X 4    X n  0 , and obtain b22  0

Rearranging terms including X 2 , we obtain

(B-11)

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Kunihiro Suzuki

  b b b b22  X 2 2  2 23 X 2 X 3  2 24 X 2 X 4    2 2 n X 2 X n  b22 b22 b22     b b b  b22  X 2  23 X 3  24 X 4    2 n X n  b22 b22 b22   b  b b b22  23 X 3  24 X 4    2 n X n  b22 b22  b22 

2

(B-12)

2

We convert the variable as Y2  X 2 

b23 b b X 3  24 X 4    2 n X n b22 b22 b22

(B-13)

The other variables are invariant but change the expression as Yr  X r

(B-14)

Then the coefficient associated with Y1 is b22 and the other parameters are changed and express cij as n

n

I  a11Y12  b22Y2 2   cijYY i j

(B-15)

i 3 j 3

Repeating the same procedure, and obtain I  a1112  b22 2 2  c3332     nn n 2

(B-16)

where

1  x1 

a a a12 a x2  13 x3  14 x4    1n xn a11 a11 a11 a11

(B-17)

 2  x2 

b23 b b x3  24 x4    2 n xn b22 b22 b22

(B-18)

Pearson Function Family

3  x3 

Ion Implantation and Activation, Vol. 1

c34 c x4    3n xn c33 c33

219

(B-19)

---------

 n  xn

(B-20)

Summarizing them, we obtain

I  1  2 3

 a11 0   0 b22  n   0 0      0 0 

0



0  c33    0 

0   1    0   2  0   3         nn    n 

(B-21)

The positive condition is given by a11  0, b22  0, c33  0, ,  nn  0 , and they can be expressed by

a11  0,

a11

0

0

b22

a11

0

 0, 0

b22

0

0

0

a11 0 0 b22

0  0, , 0  c33 0

where variable are converted as

0  0

0 0

c33  0

 0  0  0 0     nn

(B-22)

220

Ion implantation and activation, Vol. 1

 1  1     0  2    3          0     n   0

a12 a11 1 0  0

a13 a   1n  a11 a11   x1  b23 b2 n      x b22 b22   2  x  c3n   3  1    c33    x     n   0  1 

Kunihiro Suzuki

(B-23)

 x1     x2   L  x3      x   n

where  1   0 L  0    0

a12 a11 1 0  0

a13 a   1n  a11 a11  b23 b2 n   b22 b22  c  1  3n  c33       0  1 

Therefore, we obtain

(B-24)

Pearson Function Family

I   x1

x2

x3

 1  2 3

  x1

x2

x3

Ion Implantation and Activation, Vol. 1

 a11 a12 a13   a21 a22 a23  xn   a31 a32 a33      a  n1 an 2 an 3 0  a11 0   0 b22 0 0 c33  n   0       0 0 0   a11 0   0 b22 t 0  xn  L  0      0 0 

 a1n   x1     a2 n   x2   a3n   x3          ann   xn   0   1     0   2   0   3           nn    n  0  0   x1     0  0   x2  c33  0  L  x3           0   nn   xn 

221

(B-25)

We then obtain a11 a21 a31 

a12 a22 a32 

a13 a23 a33 

an1

an 2

an 3

    

a1n a11 0 a2 n 0 b22 a3n  L 0 0   

0 0 c33 

ann

0

0

0

 0  0  0     nn

(B-26)

Therefore, the positive condition is given by  

a11 0 0 b22

0 0

0 0

0 

0 

0

0

c33  0  0    0   nn

(B-27)

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Kunihiro Suzuki

This is identical to  a1n  a2 n  a3n  0  

a11 a21 a31 

a12 a22 a32 

a13 a23 a33 

an1

an 2

an 3  ann

(B-28)

Setting xn as zero, we obtain the positive condition as

a a11  0, 11 a21

a11 a12  0, a21 a22 a31

a12 a22 a32

a13 a23 a33

a11 a21  0, , a31 

a12 a22 a32 

a13 a23 a33 

an1

an 2

an 3

    

a1n a2 n a3n  0  ann

(B-29)

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 223-240 223

CHAPTER 6 The Other Analytical Model for Ion Implantation Profiles: Edgeworth Polynomial Abstract: Pearson function is commonly used to express ion implantation profiles based on its four moments. Pearson function is not the unique one that uses the given four moments. Edgeworth polynomial is a function that is also generated using the four moments or more. We explain the model in this section, and express some experimental data with this model.

Keywords: Ion implantation, Edgeworth polynomial, Hermete polynomial, kurtosis, moment, characteristic function, exponential function, Hermete polynomial, Gaussian function, central limit theorem, cumulants, standard deviation, SIMS, Sb, Pearson function. INTRODUCTION We can generate ion implantation profiles using Pearson function with four moment parameters. Pearson function is not unique function for given four moments. Edgeworth polynomial is the other function for the given moments, which is sometimes used former days [1, 2]. Furthermore, Edgeworth polynomial can further utilize additional higher moments. MOMENT

 th moment of a distribution function f  x  is defined by 

   x f  x  dx

(1)



where 

 0   f  x  dx  1 



1   xf  x  dx  Rp 

(2) (3)

This is also expressed by 

  E  x    x f  x  dx  using operator E which is associated with expectation value. Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

(4)

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Kunihiro Suzuki

The other moment  with respect to R p is defined by      E  x  R p      x  Rp  f  x  dx







(5)

Each  is related to  as

0  1

(6)

1  0

(7)

2   2  R p 2

(8)

3   3  3R p 2  2 R p 3

(9)

4   4  4 R p 3  6 R p 2 2  3R p 4

(10)

CHARACTERISTIC FUNCTION

Characteristic function of a distribution function is given by 

  t   E eitx    eitx f  x  dx 

(11)

eitx is expanded with

eitx  1  itx 

1 1 2 3  itx    itx   2! 3!

(12)

Substituting Eq. 12 into Eq. 11, we obtain 

1 1 2 3      t    1  itx   itx    itx    f  x  dx 2! 3!      1 2    f  x  dx   it   xf  x  dx   it   x 2 f  x  dx      2!  1    E  x   it   0  !  1      it   0  !

(13)

The Other Analytical Model for Ion Implantation Profiles

Ion Implantation and Activation, Vol. 1

225

The characteristic function is related to the moments as follows. Differentiating Eq. 13 with respect to t, we obtain  d  i  eitx xf  x  dx  dt

(14)

d 2 2  itx 2  i  e x f  x  dx  dt 2

(15)

 d        t   i  eitx x f  x  dx   dt

(16)

Therefore, the  -th differential factor is related to  as 

    0   i  x f  x  dx  i 

(17)

EXPONENTIAL EXPRESSION OF CHARACTERISTIC FUNCTION

If the characteristic function has a form of exponential expression, the corresponding characteristic functions is simply expressed by the sum of the terms in the exponential function. Therefore, we modify the polynomial expression of 

1

  t      it   0  !



(18)

to exponential expression. It can be done when we express ln   t   as polynomial expression. Inspecting a well-known expansion of

x x 2 x3 x 4 ln 1  x       1 2 3 4 we substitute  in to 1  x in Eq. 19, and obtain

(19)

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Kunihiro Suzuki

ln   t    1   ln     it    0  !   1    ln 1     it     1  ! 

(20) 2

3

4

1  1 1  1 1  1  1          it        it        it        it       1  !  2   1  !  3   1  !  4   1  ! 

We express this polynomial with respect to it as  1  ln   t       it   1  !

(21)

where  is called cumulant. Comparing Eqs. 20 and 21 with respect to  it  , we can relate  to  as follows. 

Considering   1

1 1



1 1

and we obtain

 1  1

(22)

Considering   2

2 2



2

1        1 1  2  1 2 2 1 1  2 2

2

and we obtain

 2   2  12 Considering   3

(23)

The Other Analytical Model for Ion Implantation Profiles

3 3!



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227

3 1       1         3   1 2  2 1    1    3  1 2  1 3! 2  1 2! 2! 1  3  1   3! 2 3

3

and we obtain

 3   3  31 2  213

(24)

Considering   4 1          1 3  2 2  3 1 4! 4! 2  1 3! 2! 2! 3! 1  1            4   1 1 2  1 2 1  2 1 1 1 3  1 1 2! 1 2! 1 2! 1 1  4

4



4

and we obtain

 4   4  41 3  3 2 2  1212 2  614

(25)

These are also expressed by the form of moment  as

1  R p

(26)

 2  2

(27)

 3  3

(28)

 4   4  3 2 2

(29)

Therefore, we obtain two expressions for the characteristic function as 

1

  t      it   0  !



 1   exp     it     1  ! 

(30)

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Kunihiro Suzuki

The conversion from the exponential expression to the polynomial expression can be done as 2

1 1  1  1    exp     it    1      it       it       1  !    1  !  2!   1  ! 

(31)

In the polynomial expression, the low expansion term is related to low order moments. However, the low order moment of the exponential expression is included as a form of exponential function and can be expanded to high order expansion terms. Therefore, we can utilize low order moment information more effectively in this form. EXPANSION FUNCTION

OF

DISTRIBUTION

FUNCTION

USING

GAUSS

We want to express the distribution function using Gaussian and related Gaussian functions, which can be done using Hermete polynomial [3]. Hermete polynomial is defined by the equation given by n

 s2   s2  n d    exp      1 H n exp     ds   2  2

(32)

Performing the differentiation in Eq. 32, we obtain H n as follows

H0  1

(33)

H1  s

(34)

H2  s2 1

(35)

H 3  s 3  3s

(36)

H 4  s 4  6s 2  3

(37)

・・・

The Other Analytical Model for Ion Implantation Profiles

Ion Implantation and Activation, Vol. 1

229

Hermete polynomial has orthogonal relationship as 

 s2   1 n ! H s H s exp       ds   m n  2  2   0

m  n m  n

(38)

Using the normalized Gaussian function

g s 

 s2  1 exp    2  2

(39)

we obtain it’s differentiate form given by g

v



d  s     g  s   dx 

(40)

where

 s2  1 exp    2  2

(41)

g (3)  s    3s  s 3  g  s 

(42)

g (4)  s    3  6 s 2  s 4  g  s 

(43)

g (5)  s    15s  10 s 3  s 5  g  s 

(44)

g (6)  s    15  45s 2  15s 4  s 6  g  s 

(45)

g (7)  s   105s  105s 3  21s 5  s 7  g  s 

(46)

g (8)  s   105  420 s 2  210 s 4  28s 6  s 8  g  s 

(47)

g (9)  s    945s  1260 s 3  378s 5  36 s 7  s 9  g  s 

(48)

g s 

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Kunihiro Suzuki

g (10)  s    945  4725s 2  3150 s 4  630 s 6  45s 8  s10  g  s 

(49)

We can then express the distribution function given by

f  s   C0 g  s  

C C1 1 C g  s   2 g  2  s   3 g  3  s   1! 2! 3!

(50)

This is also expressed by f  s   C0 g  s   s2  C1 1   1 H1 exp    1!  2 

 s2  C2 2  1 H 2 exp    2!  2



 s2  C3 3  1 H 3 exp    3!  2

(51)



Using the orthogonal relationship of H , the factor C is evaluated as







H  s  f  s  ds 

C    1  !   1 C !

(52)

We can easily evaluate C as

C0  



f  s  ds  1





C1    sf  s  ds  0 

C2  





C3   

s





2

s

(53) (54)

 1 f  s  ds  1  1  0

(55)

 3s  f  s  ds  

(56)

3

The Other Analytical Model for Ion Implantation Profiles

C4  





s

4

Ion Implantation and Activation, Vol. 1

 6s 2  3 f  s  ds    6  3    3

231

(57)

Therefore, we can express the distribution function as f s  g s   g s 

 3!

g

3

s 



 3 4!

g

4

 s 

 3 3s  s  g  s     3  6s 3! 4! 3

2

 s4  g  s   

(58)

 3     g  s  1   3s  s 3   3  6s 2  s 4   4!  3!  We neglect the terms which are related to the moments of more than five in Eq. 58, which are generally crucial approximation. Therefore, the other expression Edgeworth polynomial was investigated. CHARACTERISTIC FUNCTION RELATED TO GAUSSIAN FUNCTION

The product of characteristic function and e as

t2

2

is related to Hermete polynomial



 t2    t2  exp     t    exp   its  f  s  ds 2 2      H s     it  f  s  ds   0  ! 

 it    0  ! 



  0









 it 

(59)

H  s  f  s ds



!

C

We then obtain

it  it   t2   t2   t2      t   exp     C3 exp     C4 exp      3! 4!  2  2  2 3

4

(60)

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Kunihiro Suzuki

Comparing this with Eq. 50, and making equation with respect to the same C , we obtain

 t2     itx g x dx it   exp exp           2 

(61)

Therefore, we obtain simple conversion rule of

 t2      it exp        1 g  s   2

(62)

This means that once we obtain e 2   t  as a polynomial with respect to  and we can convert them to probability function with the sum of g    s  . t2

 it  ,

EDGEWORTH POLYNOMIAL FUNCTION

We regards the distribution function f  x  as the sum of n same probability functions, which means that f  x  approaches to Gaussian with increasing n based on central limit theorem. The deviation of f  x  from Gaussian profile is expressed by n and the third and fourth moments of each function. The moment of f  x  can include n, but the moments as a function of n is regarded as the moments of f  x  and n vanishes in the final expression. The variable x in f  x  consists of n variables of the same group given by

x  x1  x2    xn

(63)

The characteristic function   t  for f  x  is then expressed as

  t   1  t  2  t  n  t 

(64)

where 1  t  , 2  t  , , n  t  are characteristic functions for x1 , x2 , , xn , respectively. The moments for each i are expressed by suffixes 0. R p and standard deviation are given by R p  nR p 0

(65)

The Other Analytical Model for Ion Implantation Profiles

Ion Implantation and Activation, Vol. 1

 2  n 0 2

233

(66)

The cumulants are given by

  n 0

(67)

We regard 0  t  as a characteristic function of a variable  0  R p 0  . We regard   t  as a characteristic function of a variable   Rp  . Since   n 0 ,   t   and 0  t  are expressed by

  t    t   0      n 0  

n

(68)

 0   it     1  !  



0  t   exp  

(69)

Substituting Eq. 69 into 68, we obtain

      t   exp   0    1  !       exp  n 0   1  ! 



 it     n 0 



 it     n 0 

   

n

(70)

   

where average of 0  t  is 0, and hence  01 is zero, which leads to 01  0 . The standard deviation of 0  t  is  0 . t2

Multiplying e 2 to Eq. 70, we obtain t2   t2   exp     t   exp   n 0 2  1  ! 2 



 it     n 0 

   

The term in exponential function is described as

(71)

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  t2  n 0 2  1  !

Kunihiro Suzuki

    it 2   t 3   it  t2 02 03     n         2 3!  n 0   2!  n 0    n 0    2 3   t    2  it  t2 03   n 0    n      2 2!  n 0   3!  n 0     3   t   03  n      3!  n 0    

(72)

That is the terms associated with   1, 2 vanish, and we obtain     t2  exp     t   exp  n 0 !  2   3



 it     n 0 

   

We modify this equation as follows to expand it with respect to

    t2  exp     t   exp  n 0   3  ! 2 



 it     n 0 

(73)

it

n

.

   

   2  it   0 n   exp      n     3  !    0  it     exp   2    3  ! n     2   0  it   2   exp  it    2    3  ! n  

 

 

 

  2    0  2  it   exp  it       1   2  !  n    

We introduced a variable of

(74)

The Other Analytical Model for Ion Implantation Profiles

0 

Ion Implantation and Activation, Vol. 1

 0  0

235

(75)

and expand Eq. 74 as    t2  1  2  0  2  it   exp     t   1    it       1   2  !  n   h 1 h !  2    it    0  2  it     1    h !   1   2  !  n   h 1 2h



h

(76)

h

We focus on the term  it  associated with Eq. 76, and consider up to fourth 10 moments and the derivative of Gauss up to g    s  

 it 

3

When h  1 , we extract a term associated with   1 in  it 

2



0  2  it     .  1   2  !  n  

The corresponding term is

03 3! n We have no available term with h of more than 1.

 it 

4

When 

h 1 ,



extract

a

term

associated



 it   . The corresponding term is n

 it   0  2   1   2  !  2

we

04 4!n

We have no available term with h of more than 1.

with

 2

in

236

 it 

Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

5

When h  1 , we extract a term associated with   3 in  it 

2





 4  3

in

0  2  it      1   2  !  n 

which exceeds the acceptable order of moment. We have no available term with h of more than 1.

 it 

6

When 

h 1 ,



we  it   n

 it   0  2   1   2  !  2

extract

a

term

associated

with



which exceeds the acceptable order of moment.

When h  2 , we extract product of terms associated with   1

 it 



in

2

  0  2  it     2!   1   2  !  n   . The corresponding term is 4

1  03  2  3! n 

2

We have no available term with h of more than 2.

 it 

7



  it  When h  1 , we extract a term associated with   5 in  it   0  2    1   2  !  n  2



which exceeds the acceptable order of moment. When h  2 , we extract product of terms associated with   1 and   2 in

 it 

2

  0  2  it     2!   1   2  !  n   . The corresponding term is 4

The Other Analytical Model for Ion Implantation Profiles

Ion Implantation and Activation, Vol. 1

237

1  03   04  2 2!  3! n   4!n  We have no available term with h of more than 2.

 it 

8



0  2  it  When h  1 , we extract a term associated with   6 in  it     2 !    n   1  2



which exceeds the acceptable order of moment. When

 it 

h  2 , we extract product of terms associated with   2 in 2

  0  2  it     2!   1   2  !  n   . The corresponding term is 4

1  04    2!  4!n 

2

We have no available term with h of more than 2.

 it 

9



0  2  it  When h  1 , we extract a term associated with   7 in  it     2 !    n   1  2



which exceeds the acceptable order of moment. We have no available term when h  2 . When h  3 , we extract product of three terms associated with   1 in

 it 

3

   0  2  it    0  2  it  2   it      . The corresponding term is 3!   1   2  !  n    1   2  !  n  6

1  03  3!  3! n 

3

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Kunihiro Suzuki

We have no available term with h of more than 3.

 it 

10

We have no term when h are 1 and 2. When h  3 , we extract product of three terms associated with   1 ,   1 , and

 it 

3

  0  2  it      2 in 3!     2 !    n   . We have three cases for this term, and the   1  corresponding term is 6

2

1  03  04 3 3!  3! n  4!n We have no available term with h of more than 3. Consequently, we obtain f s  g s 

1 03 3 1 04  4 g s  g s 3! n 4! n

2 2  1   2  6  1  04  8 1  03  04 10 03  g s g s g s               2  2  2 2! 4!  n  2! 3! 4!  n  n  2  3!  n   3  1   7  1  03   9 03 04    g s  g s     4   3!  n   3!4! n n 

(77)

where

3

03

1  03 1 n   3  3     3 3 3 3 n n 0 n      n 

04

(78)

4

2 1  04 1 n   4  4  32  4  3    3   4 n n  04 n   4 4 4    n 

(79)

The Other Analytical Model for Ion Implantation Profiles

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239

Therefore, Eq. 77 is then expressed with moments as f s  g s 

 6

g  3  s  

   3 g  4 24

s

2   2  6    3  8  2    3 10    g s  g s  g  s  1152 1728  72       3  7    3 9  g s  g  s    1296  144 

(80)

This is called as Edgeworth polynomial. The first three terms is the same as ones in Eq. 58. This expression has more terms associated with moments up to four, and also reduces to Gaussian profile when   0,   3 . The depth x is related to s as

s

x  Rp

(81)



Therefore, the impurity profile is converted as

N  x 





f s

(82)

where  is the dose. However, we select finite term of Eq. 80 which consists of infinite terms. Therefore, the integration of Eq. 80 deviated from the dose. Therefore, we need to multiply a correlation factor K to Eq. 82, which is determined from

K





f  s ds  1

(83)

The Edgeworth polynomial well expresses the non-symmetrical SIMS Sb data [4] as shown in Fig. 1, where a Pearson function is also shown. Therefore, it is an alternative one to express the profile. However, it has sometimes negative value, and also have peculiar kink. Therefore, Pearson function is preferred to use in these days.

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20

Concentration (cm-3)

10

Sb 80 keV 1 x 1014 cm-2 aSi

19

10

SIMS Pearson Edgeworth Rp = 0.046 m Rp = 0.145 m

18

10

 = 0.6  = 3.5

17

10

16

10

0

50 100 Depth (nm)

150

Figure 1: Comparison of SIMS Sb profile with Edgeworth polynomial. The Pearson function is also shown.

REFERENCES [1] [2] [3] [4]

E. M. Baroody, “Influence of anisotropic scattering on stopping by elastic collisions,” J. Apply. Phys., vol. 36, No. 11, pp. 3565-3573, 1965. S. Furukawa and H. Ishiwara, “Range distribution theory on energy distribution of implanted ions,” J. Apply. Phys., vol. 43, No. 3, pp. 1268-1273, 1972. H. Cramer, Mathematical Methods of Statistics, Princeton University Press, U. S. A.1999. K. Suzuki, R. Sudo, and T. Feudel,"Simple analytical expression for dose dependent ion-implanted Sb profiles using a jointed half Gaussian and one with exponential tail,"Solid-State Electronics, vol. 42, pp. 463-465, 1998.

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 241-265 241

CHAPTER 7 Parameter Extraction for Analytical Ion Implantation Profile Model Abstract: Experimental data and Monte Carlo data provide data sources for constructing ion implantation profile database. We need to extract parameters for various analytical functions from the profile data. We showed that, for the expression of ion implantation profiles, there are many local minimum values sets for the third and fourth moment parameters of  and  for the Pearson function that comprises the standard dual Pearson and tail functions. It was proposed to use a joined tail function as a mediate function to extract  and  , and demonstrated that this enables us to extract the parameters uniquely. Other parameters associated with channeling phenomena can also be simply and uniquely extracted by the procedure.

Keywords: Ion implantation, Pearson function, dual Pearson, tail function, database, Gauss, joined half Gauss, SIMS, electron stopping power, Monte Carlo, low energy, shallow junctions, parameter extraction, B, As, joined tail function, mediate function, channeling. INTRODUCTION Functions to express ion implantation profiles have been developed by using Gauss, joined half Gauss [1], Pearson [2, 3], dual Pearson [4, 5], and tail functions [6-9]， as we described in chapters 4 and 5. We can extract moment parameters from Secondary Ion Mass Spectrometry (SIMS) data for ion implantation profiles and establish a corresponding database. Monte Carlo (MC) simulation can predict ion implantation profiles theoretically without SIMS data. We showed that this is possible if we tune electron stopping power value by comparing few SIMS data as shown in chapter 3. A Large number of ion trajectories are traced to suppress statistical errors, and profiles are made up by the end points of the trajectories in MC approach, which costs long time. Therefore, the moments of MC results are extracted and used to generate analytical functions, which is similar as the extract ion of moments from SIMS data. The database based on MC is also available. We can generate profiles using the analytical functions for various ion implantation conditions by interpolating the moment parameter values. The Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

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procedure to extract the parameter values from SIMS or MC data somewhat seems simple. However, it is shown that Pearson function profiles using raw moment values evaluated from MC or SIMS do not reproduce the source profile data of MC and SIMS [12]. We show the procedure to extract parameters of the tail function in this chapter. PARAMETER EXTRACTION PROBLEMS ASSOCIATED SHALLOW JUNCTIONS Both tail function and dual Pearson have a Pearson function for expressing amorphous part of the profile. First, we focus on the parameters associated with this amorphous part. Low energy ion implantation is used to realize shallow junctions. We suffer influence of surface in extracting parameters in this case. Fig. 1 shows SIMS and MC data for B ion implantation profiles. The substrates were pre-amorphized by Ge ion implantation. When ion implantation energy decreases, the profiles become shallow, and the concentration at the surface increases. The experimental concentration is zero in the negative plane even in these cases. On the other hand, Pearson function is defined in infinite plane region, and is implicitly assumed not to be zero even in negative plane. Therefore, the direct extraction of the moments may induce deviation of analytical model from these data. We use MC data to analyze this subject since SIMS suffers resolution limit and the profiles near the surface region are not so accurate. Fig. 2 compares the MC B data ion implanted at 0.5 keV and Pearson function generated from the moments of MC data. Pearson function is forced to express the abrupt decrease at the surface, and the agreement is not so good. MC tracing to the virtual negative plane was proposed to alleviate this problem as shown in Fig. 3 [10, 11]. The negative plane is virtually set, and the trajectories of ions back scattered from the substrate to the air are traced as if they are still in the substrate. This enables us to extract moment parameters for generating accurate Pearson function.

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Concentration (cm-3)

There are no clear difference between the results using MC tracing to the negative plane and the ones using the conventional MC in the positive plane as shown in Fig. 2. The moment parameters from the negative plane MC enable us to generate accurate Pearson function. The significant difference of the moments for both MC calculation modes is  . The conventional MC is cut at the surface as we mentioned before, and this induces large  . The ratio of dose in positive plane to that in negative plane is significantly different for both Pearson profiles, which is much higher for conventional MC. This is the reason why Pearson profile generated from conventional MC moment parameters has high concentration near the peak concentration region. 10

22

10

21

10

20

10

19

10

18

10

17

Pre-Amo. : Ge 60 keV 1 x 10 15 cm-2 B 1 x 10 15 cm-2

SIMS 0.5 keV SIMS 1.0 keV SIMS 3.0 keV Monte Carlo re = 1.55

0

10

20 30 Depth (nm)

40

50

Figure 1: Boron ion implantation profiles at low energy. SIMS and MC data are shown.

We cannot obtain data in the negative region for SIMS data. This means that we should monitor deviations of the profile of SIMS and Pearson by changing four moment parameters. Furthermore, the SIMS data are inaccurate in low concentration region, and it influences higher order moment values, that is,  and  . We should select target region to use for fitting, and optimize the moment values to fit the data. Therefore, we cannot directly use the raw moment values of practical SIMS data in low energy region. The subject becomes pure fitting procedure as presented in [5]. It is, therefore, invoked to fit the MC to SIMS data. We then utilize data in negative plane as we discussed here.

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22

10 Concentration (cm -3)

B 0.5 keV 1 x 10 15 cm-2 Monte Carlo Pearson

21

10

20

10

-5

0

5

10

Depth (nm) (a)

Concentration (cm -3)

10

22 Monte Carlo (Negative plane) Monte Carlo Pearson (Negative Plane) Pearson

10

21

Negative plane MC Conventional MC

Rp : 3.042 nm ; 3.415 nm Rp: 2.338 nm ; 2.088 nm

10

20

-5

 

: 0.303 : 2.92

0

; 0.607 ; 3.00

5

10

Depth (nm) (b) Figure 2: Comparison of MC data with Pearson functions. (a) Using moment parameters extracted from conventional MC. (b) Using moment parameters extracted from negative plane MC.

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Figure 3: MC tracing to the negative plane.

EXTRACTION OF  AND  OF PEARSON FUNCTION In this section, we discuss an extraction procedure of  and  for the Pearson function that is used for expressing profiles in amorphous substrates. Fig. 4 compares SIMS data and Pearson function with optimized moment parameters. The Pearson functions readily reproduce SIMS data. Fig. 5 shows extracted parameters. R p and R p are almost linearly dependent on energy.   is set constant at -1.5, which will be explained later. This negative value of  means that the profiles are asymmetrical and the concentration near

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the surface region is high compared to the deeper region.  is around 9 and decreases with increasing energy.

-3

Concentration (cm )

10

21

B 1 x 1015 cm-2

10

20

10

19

10

18

10

17

0

100

200 300 Depth (nm)

Analytical SIMS 20 keV SIMS 40 keV SIMS 60 keV SIMS 80 keV

400

500

Figure 4: Concentration profiles of B ions implanted in amorphous Si. Pearson function using raw moment parameters, and moment parameters of MC tracing to the negative plane are almost the same in this energy region.

300 B a-Si

Rp, Rp (nm)

R

p

Rp

200

100

0

0

50 Energy (keV) (a)

100

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0.0

247

10

-0.5 -1.0

9

-1.5 -2.0







 0

50 Energy (keV) (b)

8 100

Figure 5: Extracted moment parameters for B profiles. (a) R p , R p , (b)    

The analytical model readily reproduces the SIMS data as shown in Fig. 4. However, let us examine the profiles and extracted parameters in more detail. The B profiles in Fig. 4 look comparatively symmetrical at low energies and become asymmetrical with increasing energy. Therefore,  should be near 0 at low energy region, and become more negative with increasing energy. However, the extracted  value is constant at -1.5, which does not express the qualitative trend of the profiles above. This is a consequence of the extraction method, as shown below. Fig. 6 shows that various combinations of   ,   can reproduce the SIMS profile, which means that the parameter set of   ,   lacks uniqueness. This represents a fatal problem if we want to establish a robust database. It may be possible to extract decreasing  with increasing energy, as would be expected. However, it is not clear what value of  we should choose. This is why we use constant  , as shown in Fig. 5(b). In this way, we force the  values to be constant as far as possible, and reproduce the profile by correspondingly tuning  values.

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Kunihiro Suzuki

21

Concentration (cm-3)

15

10

20

10

19

10

18

10

17

B 60 keV 1 x 10 cm Rp = 183.8 nm

SIMS Pearson A ( = -1.0  = 5.0 ) Pearson B ( = -1.5  = 9.0 ) Pearson C ( = -1.8  = 13.0)

-2

Rp = 69.1 nm

0

100

200 300 Depth (nm)

400

500

Figure 6: Comparison of the SIMS B profile with Pearson functions with various   ,   .

15 VII

V

IV

 D2

VI

10 

 b2 III

5 Gauss

I

3

II

B 60 keV

Joined half Gauss

0

Figure 7: Extracted



2

0

1

2

2

3

4

5

,   for B ion implantation profiles in the   2 ,   plane.

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Fig. 7 shows plots of   ,   in the   2 ,   plane for the data in Fig. 6. It is expected that the   ,   sets on the line are able to reproduce the SIMS profiles of B ion implanted at 60 keV. This means that the parameters  and  interact strongly with each other, which induces various local minimum set of   ,   , and hence, it is difficult to obtain a unique   ,   . PARAMETER FUNCTION

EXTRACTION

PROCEDURE

USING

MEDIATE

Joined half Gauss has limited   2 ,   as shown in Fig. 7, where only limited combinations of  and  are allowed on one line, while it is allowed on two dimensional plane for Pearson function. Therefore, the combination must be selected from one point on the line for joined half Gauss, and the uniqueness of the parameter is ensured. However, joined half Gauss cannot cover the whole ion implantation profile as mentioned in chapter 4. A joined tail function is proposed as a mediate function to extract moment parameters, which is given by [12]

  1  R  x 1   N m exp    pm     2  R p1      N  x   2  1 xR    pm   N exp  m     2  R p 2     

x  R pm (1) x  R pm

The function is simple and directly related to the profile shape. R p1 and R p 2 are straggling of former and rear part of the profile, respectively. 1 and  2 controls the shape of the profile, which is always 2 in joined half Gauss. Therefore, the function can be expected to cover the whole ion implantation profiles in amorphous target. We first evaluate the allowable   2 ,   for joined tail function.

 and  for this function are evaluated by the moments of Eq. 1. Once a ratio r defined by

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Kunihiro Suzuki

R p1

(2)

R p 2

is set,  and  are uniquely given by

 r  

f 3  r   3 f1  r  f 2  r   2 f1  r   f 2  r   f1  r     2

 r  

3

(3)

3 2

f 4  r   4 f1  r  f3  r   6 f1  r  f 2  r   3 f1  r  2

 f 2  r   f1  r 2   

2

4

(4)

where

f1  r  

f2  r  

f3  r  

 2  2          1    2  r2

1

2

  1 1   1     1   r  1   2 

(5)

 3  3       1     2  r 3

1

2

  1  1   1     1   r  1   2 

(6)

 4  4          1    2  r4

1

2

  1  1   1     1   r  1   2 

(7)

Parameter Extraction for Analytical Ion

f4  r  

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251

 5  5       1     2  r 5

1

2

(8)

  1  1   1     1   r  1   2  5

IV

VII

4

 D2

V VI III

 b2

 = 1.5



 = 2.0

3

 = 2.5 II

2

0

I

3

Joined half Gauss

Gauss

Extracted

1 2

2

Figure 8: Dependence of  on  2 with 1   2 and 1  1.5 , 2.0, and 2.5.

Fig. 8 shows the dependence of  on  2 with 1 equal to 1.5, 2 and 2.5, where we assume  2  1 for simplicity. Once 1 is set, the allowable region is expressed by one limited line, and length of the line becomes short with increasing 1 . The limiting values of the lines are also evaluated as 2

 2 min

  3    1 3 1 1 2 1 f f f f   1  2  1    0 2   1   3  3    f 2 1  f1 12  2    

 min   1

(9)

(10)

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 2 max    0  

Kunihiro Suzuki

2

(11)

max    0 

(12)

as shown in Fig. 9.

 2, 

5 4

 max

3

 min

2  2max

1 0 1.5

2.0 1(=2)

2.5

Figure 9: Dependence of minimum and maximum values of  2 and  .

21

10

15

Concentration (cm-3)

B a-Si 1 x 10 cm

-2

20

10

SIMS 20 keV SIMS 40 keV SIMS 60 keV SIMS 80 keV Joined tail Pearson

19

10

18

10

17

10

0

100

200 300 Depth (nm)

400

500

Figure 10: Comparison of SIMS data with joined tail function, and Pearson function generated by the moment parameters of the joined tail function.

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253

R p1 and R p 2 in Eq. 1 correspond to the distance where the concentration is 1 e2 times peak concentration. Therefore, it is related to the distance very near the peak region, and hence it is awkward to use for fitting the data. We therefore use the modified relations just for convenience of parameter extraction as 1    R pm  x    N m exp   ln  1        L1   N  x   2    x  R pm    N m exp   ln  2     L  2     

x  Rpm (13)

x  R pm

L1,2 is the distance from the peak position where the concentration is Nm 1,2 . Therefore, we can easily fit the theory to the data by inspecting the profiles. L1,2 can be related to R p1,2 as

R p1,2 

1  2 ln  1,2    

1

L1,2

(14)

1,2

Fig. 10 shows the joined tail function fitted to SIMS B profiles. We also evaluate the moment parameters from the joined tail function and generate a Pearson function using the same moments. It should be noted that the joined tail function and Pearson function slightly deviate from each other at high energy conditions. Therefore, we should, at the same time, focus on the Pearson function. This means that raw moment parameters of joined tail function induce error for Pearson function in low concentration region at high energy. This is one fundamental problem and we will discuss the subject in more detail later. Fig. 11 shows the dependence of the parameters of the joined tail function on energy. R pm and L1 increases energy as the energy increases. L2 also increases linearly but the dependence is weak compared with R pm and L1 . 1 decreases with increasing energy, which means that the shape moves from Gaussian like to exponential like.  2 is almost constant. These parameters are directly related to the shape of the profiles, and only limited values are available.

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Rpm, L1, L 2 (nm)

500 B a-Si 1 =  = 1000

400

2

R

pm

L

1

L2

300 200 100 0

0

3

50 Energy (keV) (a)

100

B a-Si 1 =  = 1000 2

 1,  2

2

1 1 2

0

0

50 Energy (keV) (b)

100

Figure 11: Extracted parameters of joined tail function (a) R pm , L1 , and L2 . (b) 1 and  2 .

Fig. 12 shows the dependence of the moment parameters of the Pearson function on the energy extracted from the joined tail function, acting as a mediate function. R p and R p increase linearly with increasing energy and R p is around 1/3 of R p . The

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weak dependence of R p on energy can be explained by the weak dependence of L2 on energy. The resultant  decreases with increasing energy, which is what we would expect. Conversely,  increases with increasing energy, which shows the trend opposite to that in Fig. 2.

Rp, Rp (nm)

300

B a-Si 1 = 2 = 1000 Rp

200

R

p

100

0

0

50 Energy (keV) (a)

0.0

100

5 

-0.5 -1.0





4



3

-1.5 -2.0

0

50 Energy (keV)

2 100

(b) Figure 12: Extracted moment parameters evaluated using joined tail function.

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Kunihiro Suzuki

 AT HIGH ENERGY REGION Moment parameters of joined half tail function cannot reproduce the SIMS data at high energy region with Pearson function as shown in Fig. 10. This is a fundamental problem related to the limitation of Pearson function, which has been discussed for longtime [3]. We use profiles evaluated with MC tracing to the negative plane to evaluate clearly the availability of Pearson function. 10

21

Concentration (cm -3)

B 1 x 10 15 cm-2

10

20

10

19

10

18

10

17

0

100

MC 20 keV MC 40 keV MC 80 keV Pearson (MC moments)

200 300 Depth (nm)

400

500

Figure 13: MC simulation of B ion implantation profiles in Si substrates in the moderate-energy region between 20 and 80 keV. Pearson function using raw moment parameters, and moment parameters of MC tracing to the negative plane are almost the same in this energy region.

MC results for B profiles ion implanted at 20, 40, and 80 keV agree well with Pearson function using its moment parameters as shown in Fig. 13. Therefore, using raw data of moment parameters gives accurate Pearson function.



,   extracted from Monte Carlo data of ion implantation profiles in Si and Ge substrates with the energy up to 320 keV are almost on the line of 2

 f  2.661  1.852 2

(15)

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257

as shown in Fig. 14(a).   2 ,   is always outside   2 ,   plane region for Pearson IV, but it is in the region of Pearson family limitation of 3 . 10 

9



8

IV



7

D2

 b2

V

VII

III

6

B in Si As in Si P in Si B in Ge As in Ge P in Ge

3

5 4

I

3 2

II

0

200

f

VI

1

2 2 (a)

3

4

B in Si 

As in Si

160

P in Si

120

IV



u

D2

V

VI



B in Ge As in Ge

80

 3(Pearson limit)

P in Ge



VII  b2

40

f

 limit

0

0

10

2 (b)

20

30

Figure 14:  2   relationship for various ion and Si and Ge substrates. (a) Energy region less than 360 keV. (b) Energy up to 5000 keV for B.

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Concentration (cm-3)

Concentration (cm-3)

We further evaluated relationship between  2 and  of B profiles in high energy region where energy reaches up to 5000 keV as shown Fig. 14(b).  deviates from  f of Eq. 15 in this high energy region. It is noteworthy that  2 ,   breaks the limitation of Pearson function, which means that the simple use of moment parameters for Pearson function is invalid. B 0.5 keV 1 x 10 15 cm-2

10

22

10

21

Monte Carlo Pearson Pearson IV

10

20

Pearson ( u)

10

19

10

18

10

17

10

21

10

20

10

19

10

18

10

17

0

5

10 15 Depth (nm) (a)

20

B 80 keV 1 x 10 15 cm-2 Monte Carlo Pearson Pearson IV Pearson ( u)

0

100

200 300 Depth (nm) (b)

400

500

Parameter Extraction for Analytical Ion

Concentration (cm-3)

10

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259

15 -2 21 B 160 keV 1 x 10 cm

10

20

10

19

10

18

10

17

10

16

Monte Carlo Pearson Pearson IV Pearson ( u)

0

200

400 600 Depth (nm) (c)

800

15 -2 20 B 1000 keV 1 x 10 cm

Concentration (cm-3)

10

Monte Carlo Pearson Pearson IV Pearson ( )

19

10

u

18

10

17

10

0

500

1000 1500 Depth (nm) (d)

2000

Figure 15: MC simulation of B ion implantation profiles in Si substrates. Pearson function using raw moment parameters, and moment parameters of MC tracing to the negative plane are also shown. (a) 0.5 keV, (b) 80 keV, (c) 160 keV, and (d) 1000 keV.

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Kunihiro Suzuki

We evaluate the availability of using Pearson IV forcing  to  D 2 by comparing with the MC data as shown in Fig. 15. We also show the Pearson profiles using raw moment parameters in the figure. We can observe systematic trend for deviation of Pearson function from MC data as the following in Fig. 15. At energy of 0.5 keV, the Pearson function using raw moment parameters well reproduces the MC data, while the Pearson IV profile deviates from the MC data in the low tail concentration region, as shown in Fig. 15(a). At energy of 80 keV, the Pearson profile using raw moment parameters become to deviate from the MC data in the low tail concentration region, and MC data is between the Pearson and Pearson IV profiles, as shown in Fig. 15(b). At energy of 160 keV, the Pearson function that uses raw moment parameters clearly deviates from the MC data in the low concentration tail region, while the Pearson IV profile well reproduces the MC data, as shown in Fig. 15(c). At energy of 1000 keV, the Pearson function that uses raw moment parameters clearly deviate from the MC data in the low concentration tail region and further the peak concentration region, while the Pearson IV profile well reproduces the MC data, as shown in Fig. 15(d). Consequently, the Pearson function using raw moment parameters well reproduces the MC data in low energy region and deviate from MC in high energy region, while Pearson IV reproduces the MC data in high energy region and deviate from MC in low energy region. The energy region can be related to the symmetry of the profiles, that is,  of the profile is small in low energy region and increases with increasing energy. It should be noted that Pearson function uses only four moment parameters, although any order moments can be defined for the profile. Therefore, Pearson function neglects the influence of the higher order effects on the shape of the profiles.

Parameter Extraction for Analytical Ion

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261

The above results mean that the first four moments are enough for expressing the profiles when the profile is symmetrical. Therefore, the Pearson function using raw moment parameters well reproduces the MC data, while Pearson IV that uses different moment parameters form real ones deviate from the MC data. When the profile becomes asymmetrical, we need higher order moments and analytical model using the moments to accurately reproduce the asymmetrical profiles. However, we can only use the first four moments in Pearson function. Therefore, we should use moments different from the raw data to reproduce the MC data if we limit ourselves to Pearson function family. The results that Pearson IV becomes accurate with increasing energy indicate a way how we should modify the moment parameters. We can improve accuracy by increasing  although we do not know the mathematical reason. An empirical form for  was proposed as [12]

u 

25  2.661  1.852 2  25   2

(16)

and the Pearson function using u accurately expressed MC over the whole energy region as shown in Fig. 15. Finally, extraction procedure is recommended as follows. (i) Fit the joined tail function to SIMS or MC data. (ii) Evaluate the moment parameters of the joined tail function. (iii) Generate the Pearson function using the moment parameters. (iv) Change  to improve the fitting accuracy, or use Eq. 16. L AND 

In the previous section, we showed that the parameters of the main part of the tail function, that is, the Pearson function, can be robustly extracted. Now, let us move to the channeling part of the tail function.

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Concentration (cm-3)

10 10

Kunihiro Suzuki

21 As 160 keV 1 x 10

10 10

18

10

17

10

16

cm-2

 1 = 10, 2 = 100 L1 = 70.6 nm, L2 = 140.6 nm  1 = 2.38,  2 = 1.93

B

20

19

15

Rp = 112.3 nm Rp = 39.7 nm

A

L1 L2

nm

C

 = 0.35  = 3.01 L = 676.2 nm  = 1.01

SIMS Pearson Tail L-Tail Total Fix point

chan = 1.73 x 1013 cm-2

nm0

0

100

200 300 Depth (nm)

400

500

Figure 16: Comparison of SIMS As data with Pearson function generated using moment parameters form joined tail function as a mediate function.

The tail function itself is expressed by hTc _ nm 0  x     x  xp   hTC _ nm 0  x   nm 0 exp    ln        L  

(17)

This is modified to implement in the final tail function. However, we simply use this function to fit to the SIMS data. The fitted line is denoted as L-tail as shown in Fig. 16. We extracted nm 0 , L and  only focusing on the tail region. Equation 17 is not used as it is. The tail function is implemented as follows. The total concentration is expressed by

N  x      chan  na  x    chan nc  x 

(18)

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263

na  x  is a Pearson function and hence is expressed as hma as was the case of dual Pearson, that is, na  x   hma  x 

(19)

nc  x  is expressed by Pearson IV function and tail function hTc  x  as follows nc x =

h mc x  h mc x + h Tc x

for x < xT for x > xT

(20)

where xT  R p  R p

(21)

We can use L and  as the final parameters, but nm0 is a tentative parameter, which should be converted to nm . We impose that the tail function of Eq. 18 and implemented tail function of Eq. 20 gives the same value in the yellow region in Fig. 16. We can evaluate  ,



hmc  xT  hmc  xT   hTc  xT 

(22)

which is determined from L and  independent of nm . We express the implemented tail function as hTc  x  and express its peak concentration as nm . In the deep x region, the following should hold

 hTC _ n  x   hTC _ n m

m0

 x

(23)

We can obtain nm as

nm 

nm 0



(24)

Integrating this over depth, we can obtain the channel dose  chan   nc  x  dx

(25)

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Ion Implantation and Activation, Vol. 1

Kunihiro Suzuki

Now let us see how we can fit the SIMS As profile in Fig. 16 with tail function from the beginning. We extract the parameters as follows. The reliable dynamic range of the SIMS profile in the region from the peak towards the surface is about one order, and hence we set  1  10 . The distance from the peak position to a concentration of 1/10 of peak concentration is simply evaluated as 70.6 nm. Once we set L1 , the red circle point A near the surface region and B at the peak concentration is fixed during 1 tuning. We can easily determine 1 to be 2.38 and fit the profile in the surface side of the joined tail function. A channeling tail emerges where the concentration is 1/100 of the peak concentration, and hence we set  2  100 , and L2 is easily determined to be 140.6 nm. Once we set L1 , the red circle point C in deeper region B at the peak concentration is fixed during  2 tuning. We can easily determine  2 to be 1.93 and fit the profile in the rear side of the joined tail function. We can thereby generate the Pearson profile. We then focus on the profile deeper than 350 nm, and fit the profile with the single tail function (denoted by L-tail in the figure) by tuning L,  , and nm0 . The corresponding parameters for construction of the tail function can then be calculated automatically and the whole profile can be generated as shown in the Fig. 16. The evaluated  chan was 2.75 103 cm2 . The other parameters are also shown in the figure. REFERENCES [1] [2] [3] [4] [5]

J. F. Gibbons, S. Mylroie, “Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions,” Appl.Phys.Let.,Vol. 22, p.568, 1973. W. K. Hofker, “Implantation of boron in silicon,” Philips Res. Rep. Suppl., vol. 8, pp. 1-121, 1975. D. G. Ashworth, R. Oven, and B. Mundin, “Representation of ion implantation profiles by Pearson frequency distribution curves,” Appl. Phys. D., vol. 23, pp. 870-876, 1990. A. F. Tasch, H. Shin, C. Park, J. Alvis, and S. Novak, “An improved approach to accurately model shallow B and BF2 implants in silicon,” J. Electrochem. Soc., vol.136, pp. 810-814, 1989. C. Park, K. M. Klein, and A. L. Tasch, “Efficient modeling parameter extraction for dual Pearson approach to simulation of implanted impurity profiles in silicon,” Solid-State Electronic, vol. 33, pp. 646-650, 1990.

Parameter Extraction for Analytical Ion

[6] [7] [8] [9] [10]

[11] [12]

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K. Suzuki, R. Sudo, and T. Feudel,"Simple analytical expression for dose dependent ion-implanted Sb profiles using a jointed half Gaussian and one with exponential tail,“Solid-State Electronics, vol. 42, pp. 463-465, 1998. K. Suzuki, R. Sudo, Y. Tada, M. Tomotani, T. Feudel, and W. Fichtner,"Comprehensive analytical expression for dose dependent ion-implanted impurity concentration profiles,“Solid-State Electronic, vol. 42, pp. 1671-1678, 1998. K. Suzuki, R. Sudo, T. Feudel, and W. Fichtner,"Compact and comprehensive database for ion-implanted As profile,"IEEE Trans. Electron Devices, ED-47, pp. 44-49, 2000. K. Suzuki and R. Sudo,"Analytical expression for ion-implanted impurity concentration profiles, “Solid-State Electronics, vol. 44, pp. 2253-2257, 2001. K. Suzuki, Y. Tada, Y. Kataoka, and S. Kojima,"Robust boron ion implantation profile database with an energy range of 0.5 to 2000 keV based on accurate SIMS data and calibrated Monte Carlo simulation tracing to virtual negative plane,”18th Ion Implantation Tech.,P2-24, 2010. Mizuho Information & Research Institute, Inc.: Overview of the FabMeister-IM ion implantation profile simulator. http: //www.mizuho-ir.co.jp/solution/research/semiconductor/fabmeister/ion/index.html K. Suzuki and S.Kojima,“Parameter extraction procedure for ion implantation profiles to establish robust database based on tail function,” J. Semiconductor Tech. and Science, vol. 10, Number 4, pp. 251-259, 2011.

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 266-301

266

CHAPTER 8 Lateral Distribution Abstract: An analytical model with lateral straggling parameters was developed to describe the tilt dependence of ion-implantation profiles. There are three parameters associated with depth-dependent lateral straggling into the model. On the basis of comparison between experimental and analytical data, a database of ion-implantation profiles that includes lateral-straggling parameters have established. The data with tilt 0o were evaluated using off angle substrates.

Keywords: Ion implantation, lateral straggling, effective gate length, short channel effects, MOSFETs, lateral penetration, lateral resolution, SIMS, tilt, tail function, Pearson function, Gaussian function, joined half Gauss, error function, amorphous layer, channeling. INTRODUCTION Implanted ions penetrate under a pattern edge of a mask as shown in Fig. 1. This penetration significantly influences effective gate length and hence short channel effects in the scaled MOSFETs. The lateral penetration can be evaluated if we know lateral straggling Rpt of the profiles. The lateral resolution of Secondary Ion Mass Spectrometry (SIMS) is a few m, while we need a resolution of afew nm to design devices. Therefore, we cannot directly evaluate Rpt experimentally. SIMS profiles of ions implanted with high tilt angle include the information of Rpt , and Rpt can be evaluated by combining the SIMS data with a related model for ion implanted profile with high tilt angle [1-4].

Figure 1: Schematic two-dimensional profile in gate patterned substrate. Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

Lateral Distribution

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267

LATERAL STRAGGLING WITH TILT 7o We denote profiles along ion beam path using a tail function that consists of na and nc , and corresponding lateral distribution as g a and g c . We use axis system shown in Fig. 2, where  x, y  is related to the wafer surface, and  t , s  is the beam direction. When tilt angle is 0o, both systems are identical. Tilt 7o is frequently used in practical ion implantation. This is not zero, but can be approximately zero.

Figure 2: Axis system (a) Tilt 0o, (b) Tilt .

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Kunihiro Suzuki

Let us consider a case of tilt 0o as shown in Fig. 2(a). The net contribution of impurities ion implanted into the region of  xi , xi  dxi  to the region  x, y  is given by dN x, y =  - chan n a y g a x - xi, y dxi + chann c y g c x - xi, y dxi

(1)

na , nc and g a , g c are normalized, that is,  -



g a x - xi, y dxi =

-

g c x - xi, y dxi = 1

(2)

The concentration N  x, y  can be evaluated as the sum of any increment of dN  x, y  for various xi and is given by N x, y =  - chan

 -

n a y g a x - xi, y dxi + chan 

=  - chan n a y

-

 -

g a x - xi, y dxi + chann c y

n c y g c x - xi, y dxi  -

g c x - xi, y dxi

=  - chan n a y + chann c y

(3)

We can take na  y  and nc  y  outside from the integral in Eq. 3, and perform the integral. The final expression of Eq. 3 is the same as the vertical one as expected, and it has no information of Rpt . Let us consider a case of tilt angle of  as shown in Fig. 2(b). Ion implantation profiles along the beam path are assumed to be identical independent of tilt angle. The net contribution of impurities implanted into the region of ti , ti  dti  to the concentration at  t , s  can be expressed by N  t , s       chan  





 chan 





s tan 

s tan 

na  s  ti tan   g a  t  ti , s  ti tan   dti

nc  s  ti tan   g c  t  ti , s  ti tan   dti

(4)

Lateral Distribution

Ion Implantation and Activation, Vol. 1

Figure 3: Relationship between

 x, y 

269

and  t , s  system.

Inspecting Fig. 3, we relate systems of (x, y) and (t, s) to each other:

t  x cos   y sin    s   x sin   y cos 

(5)

Substituting Eq. 5 into 4, we obtain N  x, y       chan 







 x sin   y cos 

na   x sin   y cos   ti tan  

tan 

 g a  x cos   y sin   ti ,  x sin   y cos   ti tan   dti   chan







 x sin   y cos  tan 

nc   x sin   y cos   ti tan  

 g c  x cos   y sin   ti ,  x sin   y cos   ti tan   dti

(6)

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Kunihiro Suzuki

N  x, y  should not depend on x since we use plane substrate. Introducing a variable of

k

 x sin   ti tan  tan 

(7)

we modify Eq. 6 as N  x, y       chan  



y

 chan 



y

cos  tan 

cos  tan 

na  y cos   k tan   g a  y sin   k , y cos   k tan   dk

nc  y cos   k tan   g c  y sin   k , y cos   k tan   dk

(8)

This is the one independent of x as we expected. Equation 8 is valid for any lateral function of g a and g c , but we assume normalized Gaussian functions given by g a x, y =

x2 1 exp , g c x, y = 2R 2pta y 2 R pta y

x2 1 exp 2R 2ptc y (9) 2 R ptc y

Rpt depends on depth y as shown in the chapter 3 of MC, and can be expressed by  R pta  y   R pt 0  ma  y  R p 

Rpt 0  ma  y  Rp  Rptc  y     Rpt 0  mc  y  Rp 

(10)

for y  Rp for y  Rp

(11)

Rpt 0 is the lateral distribution at the depth y  R p . It should be noted that we use Rp in Eq. 11 instead of Rp 2 since Rpt 0 is related to the lateral distribution of amorphous part. It should also be noted that Rpt becomes negative or significant large value for y far from Rp if we apply Eqs. 10 and 11 to the whole region. This is due to the simple form of Eqs. 10 and 11, and validity of them for such y are questionable. The concentration for such y is significantly low and the value of Rpt is not important. However, we should avoid undue value of Rpt and set its limit given by

Lateral Distribution

Ion Implantation and Activation, Vol. 1

 R pt 0  Rpt  y    10 10Rpt 0 

for R pt 

271

R pt 0 (12)

10 for Rpt  10Rpt 0

Fig. 4 compares the results from the model of Eq. 8 and SIMS measurements of the tilt-dependent impurity concentration profiles. The dashed lines show the profiles generated by the model with constant lateral-straggling Rpt parameters. The results from the analytical model agree well with the experimental data. In terms of constant lateral straggling, the model profiles at large tilt angles deviate from the SIMS profiles in the low concentration regions. This means that the lateral straggling depends on depth and the analytical forms of Eqs. 10 and 11 are necessary for accurate expression of the profiles. We can extract ma from SIMS data in amorphous layer or MC data, and we then evaluate mc using the data in crystalline Si substrate.

10

19

B 80 keV 1 x 10 14 cm-2

SIMS (tilt = 7o)

-3

Concentration (cm )

Analytical o

SIMS (tilt = 20 )

10

18

SIMS (tilt = 40o) o

SIMS (tilt = 60 ) Analytical (Rpt0)

10

17

10

16

0.0

0.5 Depth (m) (a)

1.0

Ion Implantation and Activation, Vol. 1

10

Kunihiro Suzuki

As 40 keV 1 x 1014 cm-2

20

Analytical

-3

Concentration (cm )

o

10

SIMS (tilt = 7 ) SIMS (tilt = 20o)

19

SIMS (tilt = 40o) o

10

18

10

17

10

16

10

SIMS (tilt = 60 ) Analytical (Rpt0)

0.0

20

0.1 0.2 Depth (m) (b)

0.3

P 40 keV 1 x 1014 cm-2 Analytical

-3

Concentration (cm )

272

SIMS (tilt = 7o)

10

19

o

SIMS (tilt = 20 ) SIMS (tilt = 40o) o

10

18

10

17

10

16

0.0

SIMS (tilt = 60 ) Analytical (Rpt0)

0.1

0.2 0.3 Depth (m) (c)

0.4

0.5

Lateral Distribution

Ion Implantation and Activation, Vol. 1

10

14

273

-2

Sb 40keV 1 x 10 cm

20

-3

Concentration (cm )

Analytical o

SIMS (tilt = 7 )

10

19

o

SIMS (tilt = 20 ) SIMS (tilt = 40o) o

10

18

10

17

10

16

SIMS (tilt = 60 ) Analytical (R ) pt0

0.0

0.1 Depth (m) (d) In 40 keV 1 x 10

20

10

14

cm

0.2

-2

-3

Concentration (cm )

Analytical SIMS (tilt = 7o)

19

10

SIMS (tilt = 25o) SIMS (titl = 25o) SIMS (tilt = 60o) Analytical (Rpt0)

18

10

17

10

16

10

0

0.1 Depth (m) (e)

0.2

Figure 4: Comparison of analytical model with tilt dependent SIMS profiles (a) B, (b) As, (c) P, (d) Sb, (e) In.

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The B SIMS profiles at large tilt angles are lower than those given by model with constant Rpt 0 parameters. This means that Rpt decreases with an increase in depth, that is, mc is negative. The SIMS concentration profiles for the other impurities at large tilt angles are higher than those given by the analytical model with constant lateral-straggling parameters. This means that Rpt increases with an increase in depth, that is, a positive value for mc . Table 1 summarizes the extracted parameters associated with lateral straggling as well as the other parameters. Table 1: Extracted parameters E (keV)



L (nm)



Rp(nm)

Rpt0(nm)

ma

mc

18.9

0.5

3.5

155.0

1.0

9.55

9.0

0.1

0.75

40

32.3

0.5

3.5

278.0

1.0

15.80

13.0

0.1

0.65



80

59.1

0.5

3.5

480.0

1.0

26.60

21.0

0.1

0.50



160

112.0

0.5

3.5

829.0

1.0

44.80

40.0

0.1

0.32

20

75.7

-0.4

3.9

222.0

3.0

37.40

34.1

-0.1

0.00

40

145.0

-0.4

3.9

301.0

3.1

55.60

51.5

-0.1

-0.01



80

279.0

-0.4

3.9

375.0

3.5

85.70

77.4

-0.1

-0.12



160

493.0

-0.4

3.9

461.0

4.0

115.0

96.1

-0.1

-0.14

20

29.5

0.7

4.6

239.0

1.0

29.50

10.1

0.1

0.40

40

58.9

0.7

4.6

371.0

1.0

58.90

17.8

0.1

0.35



80

118.0

0.7

4.6

563.0

1.0

118.00

30.0

0.1

0.30



160

238.0

0.7

4.6

902.0

1.0

238.00

54.4

0.1

0.25

20

17.2

0.7

4.5

108.0

1.0

5.90

5.0

0.2

1.00

40

28.2

0.7

4.5

211.0

1.0

9.53

7.0

0.2

1.00

80

46.7

0.7

4.5

373.0

1.0

15.70

12.0

0.2

1.00

160

83.0

0.7

4.5

791.0

1.0

27.30

22.0

0.2

1.00

20

17.4

0.5

3.5

110.6

1.0

6.03

5.0

0

0.33

40

27.9

0.5

3.5

172.2

1.0

9.40

7.5

0

0.33

80

45.1

0.5

3.5

307.7

1.0

15.66

11.0

0

0.33

160

81.0

0.5

3.5

546.0

1.0

28.25

20.0

0

0.33

As



B



P



Sb 

In



20





Rp(nm)

Lateral Distribution

Ion Implantation and Activation, Vol. 1

275

Figure 5: Definition of tilt and rotation angles.

VALIDITY OF THE MODEL

We assume that the profile generated along the beam axis is identical independent of tilt angle in deriving Eq. 8. However, a certain combination of tilt and rotation angles, of which definition is shown in Fig. 5, induces significant channeling in crystalline Si at the certain combination of rotation and tilt angles. We define R p* as the projected range perpendicular to the wafer surface. If the profiles generated along the ion-beam incident axis are independent of the tilt angle, R p* should be related to Rp as

R*p  R p cos 

(13)

Fig. 6 shows the dependence of R p * on the tilt angle which is estimated by a Monte Carlo simulation, Crystal-TRIM [5]. Since we assume that the profiles generated along the path of the ion beam do not depend on the tilt angle, our analysis is valid when Eq. 13 holds. The deviation can be expressed as:

Deviation 

R p*  R p cos  R p cos 

(14)

The deviation has peak values at tilt angles of 20, 35, and 55o as shown in Fig. 6. The analysis is valid for regions where the deviation is small.

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Kunihiro Suzuki 14

-2

B 10 keV 1 x 10 cm

100

100

R cos p

50

50

p

R* (nm)

p

0

0

10

20

30 40 o Tilt ( )

50

Abs[(R*p-Rp cos)/Rp cos] (%)

R*

0 60

Figure 6: Dependence of Rp on tilt angle.

Fig. 7 shows the two-dimensional view evaluated with Crystal Maker [6], which looks each incident ion. We use ten cells in the depth direction with tilt 0o and rotate and then tilt them. The two dimensional view of various tilt with rotation 0o, which corresponds to Fig. 6 is shown in Fig. 7(a). We can see vacant space at tilt of 0, and 35o, which well correspond to large error points in Fig. 6. It is also clear that atom rows are overlapped with certain tilt angles and we can expect the suppression of channeling. However, we cannot vanish plane channeling. If we rotate the wafer with 20o and then tilt the wafer with 10o as shown in Fig. 7(e), we can expect perfect filling of the two-dimensional spaces. This is called amorphous ion implantation. Fig. 7 shows a map of two-dimensional view of crystal structure [7].

Laateral Distributiion

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277

2778

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Laateral Distributiion

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281

Fiigure 7: Two-dimensionalview evaluated with w CrystalMaaker. (a) Rotatiion 0o, (b) Rotation 5o, (c) o o R Rotation 10 , (d)) Rotation 15 , (e) Rotation 20 2 o, (f) Rotationn 25o, (g) Rotaation 30o, (h) Rotation R 35o, o o (i)) Rotation 40 , (j) Rotation 45 .

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We evaluated the accuracy for whole tilt rotation combination using Eq. 14 as shown in Fig. 8. The analysis is valid in the dark green region.

Figure 8: Deviation from a function of R p cos  . (a) 3-D view, (b) 2-D view.

Lateral Distribution

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283

LATERAL STRAGGLING WITH TILT 0o

The ion implantation condition of tilt 0o is frequently used in VSLI processes. However, the method described here cannot be applied to this case because we cannot assume identical situation with tilt 0o for the other tilt angles. It was proposed to use off angle substrates to evaluate Rpt with tilt angle 0o as shown in Fig. 9 [8]. Using 30o and 60ooff angle substrates, ions were implanted with tilt angle of 0, 30, and 60o, respectively. It is obvious that we can assume identical profile along the beam axis.

Figure 9: Experimental system in this work. The tilt angle was set to the same as the off-angle cut of the substrate. Off angle substrate for extracting Rpt 0 with tilt0o.

Fig. 10 compares the analytical model with SIMS measurements of the tilt-dependent B concentration profiles. The dashed lines show the implantation profiles generated by the model with constant lateral-straggling parameters, Rpt 0 . The analytical model agrees well with the experimental data. In terms of constant lateral straggling, the model profiles at large tilt angles deviate from the SIMS profiles in the low concentration regions. The figure shows that Rpt associated with the channeling portion of the distribution is very small and decreases with depth, that is, mc is the negative. The profile at a tilt angle of 60o deviates from the SIMS data in the deep side of the profile near the peak region. The theory predicts a more broad profile. We do not know the reason for this deviation. The surface is not perfect Si, but covered with native oxide, a surface transition region, or surface roughness with an approximate thickness of 1 nm. When implanting with a tilt angle of 60o, the ion beam encounters twice the surface transition region thickness as compared with that encountered using a tilt angle of 0o. Channeling phenomenon is sensitive to the surface condition, and suppressed by the surface non-crystalline layer. When the

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channeling is suppressed,  chan decreases in our model, and the profile becomes close to that for the amorphous part. Fig. 10(b) shows the amorphous part for a tilt of 60o. The experimental data is between the amorphous part profile and our predicted overall profile.

-3

Concentration (cm )

10

20

B 14

10

19

10

18

10

17

SIMS Tilt 0 o SIMS Tilt 30o SIMS Tilt 60o Theory Theory (m = 0)

R

pt0

= 34 nm

ma = - 0.1 m = - 0.2 c

10

16

10

0

0.1

0.2 0.3 Depth (m) (a)

0.4

0.5

20

B 40 keV

Concentration (cm -3)

-2

20 keV 1 x 10 cm

10

19

10

18

10

17

14

1 x 10 cm

SIMS Tilt 0o -2

SIMS Tilt 30o SIMS Tilt 60o Theory Theory (m = 0) Theory (a-part Tilt 60o) R

pt0

= 51 nm

m = - 0.1 a

m = - 0.2 c

10

16

0

0.2

0.4 0.6 Depth (m) (b)

0.8

Lateral Distribution

Ion Implantation and Activation, Vol. 1

-3

Concentration (cm )

10

285

20

B 80 keV

10

19

10

18

14

SIMS Tilt 0

SIMS Tilt 30

-2

1 x 10 cm

17

10

16

o o

SIMS Tilt 60 Theory Theory (m = 0)

R

10

o

pt0

= 77 nm

m = - 0.1 a

m = - 0.2 c

0.0

0.2

0.4 0.6 Depth (m) (c)

0.8

1.0

Figure 10: Tilt dependence of B ion-implantation profiles. Data from the analytical model and SIMS are compared. Analytical model with constant lateral straggling is also shown with dashed lines. (a) 20 keV, (b) 40 keV, (c) 80 keV.

Rpt of the B ions decreases with depth, that is, mc is the negative as shown in Fig. 11. Lateral straggling associated with the main portion of the distribution is almost the same as tilt 7o cases. We used - 0.2 for mc which was 0 for a tilt angle of 7o. This means that the lateral straggling Rpt associated with channeling is much smaller for tilt 0o cases. This is plausible since that the channeling ions are less scattered. The tentative treatment for tilt 0o is as follows.  2mc  tilt 7   mc  tilt 0    1  2 mc  tilt 7 

for mc  tilt 7   0 for mc  tilt 7   0

(15)

Although we use this off-angle substrate to extract Rpt with tilt angle of 0o, we can use these for arbitrary tilt angle of  .

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Kunihiro Suzuki

We use tilt  for 0o-off angle substrate, and tilt   30 for 30o-off angle substrate, and tilt   60 for 60o-off angle substrate. 100

160 keV

Tilt 7

80 keV

Rptc (nm)

Tilt 0o o

40 keV

50

20 keV

0

0

100 200 y - yp (nm)

300

Figure 11: Extracted Rptc from B ion implanted profiles.

LATERAL DISTRIBUTION FUNCTION

Equation 8 accommodates any lateral distribution function as we mentioned before. We used Gauss functions of Eq. 9 up to here. This may be a crucial approximation although the model succeeds in reproducing the experimental data used here. More general treatments have been proposed using depth dependent Rpt and further lateral kurtosis t that also depends on depth [8, 9]. Lorents et al. proposed the forms [9] 2  R 1 a b y R c y R           0 pt p p  R pt  x    2 R pt 0 exp  p  q  y  R p   r  y  R p     

for y  R p for y  R p

(16)

Lateral Distribution

Ion Implantation and Activation, Vol. 1

2 t  y   1  exp  L1  L2  y  Rp   L3  y  R p  



287

(17)



where a, b, c, p, q, r , L1 , L2 , L3 were determined by more basic theoretical calculation. Once, we obtain Rpt and t , we can use Pearson function for lateral distribution. On the other hand, Hobler proposed the forms [10] 1  R pt  y   R pt  ln  e a1 p1  e a2 p2    a1 

(18)

1 ln  ea1 p1  ea2 p2  a1

(19)

t  y  

where a1 , p1 , p2 are evaluated using results of MC simulation [10]. He also proposed to use lateral distribution function for





 K exp  by p   1 g  y   2 b2  K 1  b2 y 2  b0

t  3 (20)

t  3

Hobler et al. further proposed a three-dimensional response function where specific channeling direction dependence is expressed [11]. CALCULATION ALGORITHM IMPLANTATION PROFILE

FOR

HIGH

TILT

ANGEL

ION

We obtained a general form for high tilt angle ion implantation profiles of Eq. 8. However, it takes long time as shown the following. Fig. 12 schematically shows the calculation of concentration at depth y denoted by N  y  . We must summarize the contribution of each beam along B0 to obtain N  y  . We need to perform similar calculations at differenty, and finally obtain the depth-dependent N  y  . Therefore, we need an n-th integration calculation for the

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Kunihiro Suzuki

n-th data point for N  y  , which takes a long time. Here, we describe algorithms that enable us to calculate the tilt dependent profile instantaneously.

Figure 12: Schematic explanation of calculating the concentration at the depth of s. The concentration is evaluated by summing all of the beams. The contribution of each beam is evaluated by the depth to the line B0 and the distance between each beam and beam A0.

NUMERICAL ALGORITHM

Fig. 13 shows proposed mesh configuration [12]. The distance between neighboring beams at the surface is set to d sin  . Using this configuration, all the mesh points are identical for all beams, although the distance and depth value are different for each beam at a given mesh point. Let us consider again the integration along axis B0 to evaluate the concentration at the point O as shown in Fig. 14. The solid lines correspond to the universal mesh for any beam, and the dashed lines correspond to the mesh associated with beam A0 . We should add the concentration n1 associated with beam A1 , n2 associated with beam A2 , n1 associated with beam A1 , n2 associated with beam A2 , and so on. These notations are solid line circled to clarify that these are related to the concentration at the mesh point O associated with each beam.

Lateral Distribution

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289

Figure 13: The proposed mesh configuration.

Figure 14: The identical concentration of each beam contribution at O can be found in the concentration on the mesh associated with beam A0. Solid line circled concentrations correspond to the one at mesh point O associated with each beam, and dashed line circled concentration correspond to the one at the circled mesh point associated with the beam A0.

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Kunihiro Suzuki

Since the concentration is uniquely determined from the depth and the distance from the beam line, n1 is identical to the concentration of dashed line circled n1_ A0 associated with beam A0 , n2 with dashed line circled n2 _ A0 , and so on. Therefore, we can find the identical concentration with beam A0 , and further, the integration along B0 is identical to the integration of the same depth associated with beam A0 . Therefore, we can obtain depth profiles only using the A0 distribution by summarizing the concentration at the same depth as shown in Fig. 15.

Figure 15: The concentration at depth s is identical to the summation of the concentration of beam A0 two-dimensional concentration at the same depth.

The amount of concentration associated with mesh  i, j  can be evaluated with 2   d    j   1 tan    d n  id   exp     2R pt 2  tan  2 R pt    

(21)

where i corresponds to the mesh number associated with the depth along the ion beam, and j corresponds to the mesh number associated with the distance from the main beam axis. However, we know that the integration of a Gaussian function is an error function, and hence it can be evaluated more accurately as

Lateral Distribution j

d



Ion Implantation and Activation, Vol. 1 d

 tan  2 tan  n  id   j d  d tan  2 tan 

 1 x2  exp   dx 2 2 R pt  2R pt 

 d d   n  id   Erf  j   tan  2 tan  

291

(22)

d d       Erf  j    tan  2 tan   

Fig. 16 shows the tilt-dependent implantation profiles for In ions. The new algorithm reproduces the SIMS profiles as was done in [3] with the conventional integration algorithm. The CPU time with this algorithm is smaller than that with the conventional integration algorithm by about two orders [12]. Therefore, this algorithm enables us to calculate the profile quickly without loss of accuracy.

10

In 40 keV 1 x 10

20

14

cm

-2

-3

Concentration (cm )

Analytical

10

SIMS (Tilt = 7o)

19

SIMS (Tilt = 25o) SIMS (Tilt = 45o)

10

18

10

17

10

16

SIMS (Tilt = 60o)

0

0.1 Depth (m)

0.2

Figure 16: Tilt-dependent In concentration profiles. The profiles calculated using this algorithm matched the SIMS data.

ANALYTICAL MODEL FOR HIGH TILT ANGLE ION IMPLANTATION PROFILE

We further derive an analytical model for profile of impurities ion implanted with high tilt angle. We can derive rigorous analytical one if we assume Gaussian profile

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Kunihiro Suzuki

for vertical profile. We cannot obtain the one if we use vertical Pearson profile which is commonly used as we described before. Performing some approximations, we can obtain analytical model for it. GAUSSIAN PROFILE

The axis system for the analysis here is the same as that in Fig. 2. The axis system regarding the beam direction  t , s  , and that regarding the wafer surface  x, y  are also defined. The concentration at  t , s  can be evaluated as the summation of the contribution ions implanted in the region between t  ti and t  ti  dti . We can integrate from ti of  s tan  to ∞ is given by 

   s  t tan   R 2    t  ti 2  1 1 i p    N t, s    exp  exp    dti   2 R pt 2R p 2 2R pt 2   2 R p      s tan   1 1    erf 2 2 

 1   2R p R pt

    2 cos  2 2      s R t R cos  R R sin  p p p pt     tan     

  s  R cos   t sin   2   p  cos    exp    2   2 2  

(23) where

 2  Rp 2 cos 2   Rpt 2 sin 2  .

(24)

The system of  t , s  is related to the one of  x, y  as shown in Eq. 5. Equation 23 is then converted to   1 1 N  x, y     erf 2 2 

 yRp 2   Rp Rpt 2 sin      y  R cos  2   p   cos  tan     exp     2 2 2Rp Rpt 2       

Note that this depends on y, but not on x as is expected.

(25)

Lateral Distribution

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293

The first term in Eq. 25 is 1 for large y, and close to 1 2 for small y. This term expresses that the contribution of the left side become less near the surface. We treat the profile for certain depth, that is, yR p 2 tan 

 R p R pt sin 

yR p 2

2

2R p R pt

 

tan  2R p R pt yR p tan  2R pt

(26) 1

This means that we focus on the concentration at the depth of y  tan  2

R pt R p

(27)

Equation 25 reduces to

  y  R cos  2   cos  p  N  y  exp     2 2 2  

(28)

This can be regarded as a Gaussian profile projected to the y-axis. The projected profile of the Gaussian profile is obtained by using the projection of the moment parameters as

   cos    R p  R p cos   2 2 2 2 R p  R p cos   R pt cos      0      0    3    3     

(29)

Since we assume Gaussian lateral distribution, the projections of  and  are invariable.

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Kunihiro Suzuki

EXTENSION TO PEARSON FUNCTION

If we use the Pearson function along the beam axis instead of the Gaussian function, the projection of the moments for the Pearson function N  y  may be as follows    cos   R  R cos  p  p  2 2 2 2 R p  R p cos   R pt cos     ?   ? 

(30)

We expect the same conversions for , Rp , and Rp , but cannot expect the same ones for  and  since those for the Pearson function are different from those of the lateral-distribution function of Gauss of   ,     0,3 in general. Although we do not know what their values are, we can expect those in the limiting case of 90o to be the same as those of Gaussian profile of lim   0, limo   3

  90 o

(31)

  90

It is impossible to derive the dependence of  on the tilt-angle dependence using the Pearson function, since we cannot perform integration as was done in [1] with the Pearson function. We propose using the joined half Gauss [13] as a mediate function and approximately obtaining the angle dependent  as the following. Joined half Gauss is expressed by [13]    y  R 2  pm  N exp    m   2R pf 2     N  y   2  yR    pm   N m exp   2   R 2   pb   

for y  R pm (32) for y  R pm

N m is the peak concentration, Rpm is the peak position, Rpf and Rpb are the straggling of the surface part and the rear part of the joined half Gauss, respectively.

Lateral Distribution

Ion Implantation and Activation, Vol. 1

295

We approximately express the Pearson function with this joined half Gauss, focusing on the peak region as shown before, which is given by.

1   3  a R pf  R p 2    1 a 2  2 2  8 

(33)

1   3  a Rpb  R p 2    1 a 2  2 2  8 

(34)

 is uniquely determined by the ratio of Rpf to Rpb in the joined half Gauss, R that is r  pf Rpb is given by [14] 2

 JHG 



 4   2  1  r  1  r     

 r  1 

3

 2  2 2 r r 1   1           

(35)

We propose that Rpf and Rpb for tilt  are expressed by R pf    R pf 2 cos 2   R pt 2 cos 2 

(36)

R pb    R pb 2 cos 2   R pt 2 cos 2 

(37)

We can then obtain the r for tilt  as r    R rJHG   

pf

 

 R pb  

and

R pf  

(38)

R pb  

 JHG   for tilt  is given by substituting r   in Eq. 35. We set  for Pearson with tilt  as    , and propose relating it to  JHG   as

  JHG



    JHG   

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and finally obtain tilt angle dependent  for the Pearson function as

   

  JHG

 JHG  

(39)

We used a simple expression for β given by

   3        3  2 2

(40)

which holds  in the limited cases of   0, 90o . However, Eq. 40 is invalid when   0 . We then use

       3 cos n   3

(41)

In this case, we used n of 1 as a default value.

Rpt depends on depth in general. The depth y corresponds to beam, and hence  y  Rpt  y   Rpt 0  m   Rp   cos  

y

cos

along the

(42)

We use this Rpt in Eq. 6. Fig. 17 compares the tilt-dependent B and As ion implantation profiles. The profiles become to close to Gaussian with an increasing tilt angle. The analytical model reproduces the numerical data and readily expresses the change in the shapes. DUAL PEARSON AND TAIL FUNCTION

We should use a dual Pearson or tail function to express the ion implantation profiles in crystalline Si (cSi). If we use a dual Pearson, we can simply apply this procedure to each Pearson function. Therefore, we treat the tail function here.

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21

10

15

Concentration (cm-3)

B 40 keV 1 x 10 cm a-Si

-2

SIMS Tilt 7o SIMS Tilt 20o

20

10

SIMS Tilt 40o SIMS Tilt 60o

19

10

SIMS Tilt 80o Analytical R p = 143.4 nm

18

Rp = 50.8 nm

10

R = 52.1 nm pt

 = - 0.88  = 4.1

17

10

0

100

200 300 Depth (nm) (a)

400

500

21

Concentration (cm-3)

10

As 40 keV 1 x 10 15 cm-2 a-Si Numerical Tilt 7 o

20

10

Numerical Tilt 20 o Numerical Tilt 40 o Numerical Tilt 60 o

19

Numerical Tilt 80 o Analytical

10

Rp = 32. 4 nm Rp = 11.6 nm

18

10

R = 9.1 nm pt

 = 0.03  = 2.7

17

10

0

50

100 150 Depth (nm) (b)

200

Figure 17: Comparison of numerical and analytical tilt dependent ion implantation profiles. (a) B, (b) As.

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Kunihiro Suzuki Numerical Tilt 7 o

21

Numerical Tilt 20 o

Concentration (cm-3)

15

B 40 keV 1 x 10 cm c-Si

10

20

10

19

-2

Numerical Tilt 60 o Numerical Tilt 80 o Analytical Rp = 136.7 nm Rp = 56.4 nm R = 51.4 nm pt

10

 = -0.7  = 5.5 L = 333.8 nm  = 2.75

18

chan = 1.00 x 10 14 cm-2

10

10

17

0

100

200 300 Depth (nm) (a)

21 15

10

20

10

19

400

500

Numerical Tilt 7 As 40 keV 1 x 10 cm c-Si

Concentration (cm-3)

Numerical Tilt 40 o

-2

o

Numerical Tilt 20

o

Numerical Tilt 40

o

Numerical Tilt 60 o Numerical Tilt 80 Analytical

o

Rp = 32. 4 nm R p = 13.6 nm R pt = 13.0 nm

10

18

10

17

 = 0.5  = 3.5 L = 274.4 nm  = 0.95 chan = 1.56 x 10 13 cm -2

0

50

100 150 Depth (nm) (b)

200

Figure 18: Comparison of analytical model for tail functions with numerical one. (a) B, (b) As.

There is no mathematical discussion, but from a geometrical guess, we propose using parameters for projected nc  x  with these modified parameters

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Ion Implantation and Activation, Vol. 1

 xT  R p  R p   R p  R p  cos    L  L cos     

299

(43)

Although the treatment of Eq. 43 provides practical accuracy, it lacks a mathematical base. One simpler treatment is to evaluate the moments of nc  x  , and using the moment of Rp , Rp ,  , and  , we can generate the Pearson profile, and the procedure is available for this case. We call it the tail Pearson. Fig. 18 compares the tail function and tail Pearson. Both agree well for B and As. 7KHFRUUHVSRQGLQJ SDUDPHWHUYDOXHVDUHVKRZQLQ7DEOHVIn this case, we can simply apply this procedure to each Pearson function as was for dual Pearson. Table 2: B (Tail function) Energy

Rp

Rp

(keV)

(nm)

(nm)







L

chan (cm-2)

(nm)

20

73.4

41.1

-0.7

5.5

252.4

2.60

1.9 x 1014

40

136.8

56.4

-0.7

5.5

333.8

2.75

1.0 x 1014

80

253.3

77.1

-0.7

5.5

421.2

3.00

8.0 x 1013

21

10

15

Concentration (cm -3)

B: 1 x 10 cm

-2

Tail 20 keV Tail 40 keV Tailt 80 keV Tail Pearson

20

10

19

10

18

10

17

10

0

100 200 300 400 500 600 Depth (nm) (a)

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21

10

15

-2

Concentraton (cm -3)

As: 1 x 10 cm

Tail 20 keV Tail 40 keV Tail 80 keV Tail Pearson

20

10

19

10

18

10

17

10

0

100 200 Depth (nm) (b)

300

Figure 19: Pearson function generated from tail function. Table 3: B (Tail Pearson) Energy (keV)

Rp2 (nm)

Rp2 (nm)





20

91.9

56.4

0.28

4.0

40

162.0

76.7

0.24

4.0

80

285.5

102.0

0.15

3.9

Table 4: As (Tail function) Energy (keV)

Rp (nm)

Rp (nm)







L (nm)

chan (cm-2)

20

19.4

8.9

0.5

3.5

156.8

0.95

2.2 x 1013

40

32.5

13.5

0.5

3.5

274.4

0.95

1.6 x 1013

80

59.5

23.3

0.5

3.5

487.0

0.95

1.5 x 1013

Table 5: As (Tail Pearson) Energy (keV)

Rp2 (nm)

Rp2 (nm)





20

25.6

18.9

2.56

13.8

40

44.6

33.2

2.64

14.0

80

81.7

59.1

2.65

14.0

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REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8]

[9] [10] [11] [12] [13] [14]

T. Miyashita and K. Suzuki "Experimental evaluation of depth-dependent lateral standard deviation for various ions in a-Si from one-dimensional tilted implantation profiles," IEEE Trans. Electron Devices, ED-46, pp. 1824-1828, 1999. K. Suzuki, R. Sudo, and M. Nagase, "Estimating lateral straggling of impurity profiles of ions implanted into crystalline silicon" IEEE Trans. Electron Devices, ED-48, pp. 2803-2807, 2001. K. Suzuki, R. Sudo, and H.Tashiro,“Estimating lateral straggling of indium implanted into crystalline silicon," IEEE Trans. Electron Devices, ED-49, pp. 1312-1314, 2002. K. Suzuki, K. Tanahashi, S. Nagayama, C. W. Magee, T. H. Buyuklimanli, and E. Iwamoto,“Estimating lateral straggling of boron profiles ion-implanted into crystalline silicon with a tilt angle of 0o using off-angle substrates,”IEEE Trans. Electron Devices, vol. ED-53, pp. 1262-1265, 2006. Crystal Maker: http: //www.crystalmaker.com/ M. Posselt, B. Schmidt, C. S. Murthy, T. Feudel, and K. Suzuki, “Modeling of damage accumulation during ion implantation during ion implantation into single-crytalline silicon,” J. Electrochem. Soc., vol. 144, pp. 1495-1504, 1997. T. Miyajima, private communication. K. Suzuki, K. Tanahashi, S. Nagayama, C. W. Magee, T. H. Buyuklimanli, and E. Iwamoto,“Estimating lateral straggling of boron profiles ion-implanted into crystalline silicon with a tilt angle of 0o using off-angle substrates,”IEEE Trans. Electron Devices, vol. ED-53, NO. 5, pp. 1262-1265, 2006. J. Lorenz, R. J. Wierzbicki, and H. Ryssel, “Analytical modeling of lateral implantation profile,” Nucl. Instrum. Meth. Phys. Res. B. Beam Interact. Mater. At., vol. 96, pp. 168-172, 1995. G. Hobler, “Monte Carlo simulation of two-dimensional implanted dopant distributions at mask edges,” Nucl. Instrum. Meth. Phys. Res. B. Beam Interact. Mater. At., vol. 96, pp. 155-162, 1995. G. Hobler, H. H. Vuong, J. Bevk, A. Agarwal, H. J. Gossmann, D. C. Jacobson, M. Ford, A. Murrell, and Y. Erokin, “Modeling of ultra-low energy boron implantation in silicon,” in IEDM Tec. dig., 1997, pp. 489-492. K. Suzuki and S. Kojima, “Efficient calculation algorithm for one-dimensional ion Implantation profiles with high tilt angles,”IEEE Trans. Electron Devices, vol. ED-55, NO.8, pp.2263-2267, 2008. J. F. Gibbons, S. Mylroie, “Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions,” Appl.Phys.Let., Vol. 22, p.568, 1973. S. Selberherr, ”Analysis and simulation of semiconductor devices” Springer-Verlag, Wien New York, 1984.

Send Orders for Reprints to [email protected] Ion Implantation and Activation, Vol. 1, 2013, 302-375

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CHAPTER 9 LSS Theory Abstract: Ion implantation profiles are expressed by the Pearson function with first, second, third and fourth moment parameters of R p ,  R p ,  , and  We can derive an analytical model for these profile moments by solving a Lindhard-Scharf-Schiott (LSS) integration equation using perturbation approximation. This analytical model reproduces Monte Carlo data which were well calibrated to reproduce a vast experimental database. The extended LSS theory is vital for instantaneously predicting ion implantation profiles with any combination of incident ions and substrate atoms including their energy dependence.

Keywords: Ion implantation,LSS theory, amorphous, Monte Carlo, nuclear stopping power, electron stopping power, range, projected range, skewness, kurtosis, lateral straggling, Thomas-Fermi potential, ZBL potential, nuclear cross section, electron cross section, cross section. INTRODUCTION We can generate profiles of ion implantation for any combination of incident ion and substrate atom by using Monte Carlo (MC) simulation as shown in chapter 3. Furthermore, we can expect accurate results if we tune electron stopping power implemented in MC. However, it will take a long time since it traces more than ten thousands ion trajectories. Lindhart, Scharf, and Schiott developed a theory that predicts the moment of the profiles by using the nuclear and electronic stopping powers Sn and Se , respectively, which was called LSS theory [1]. This model has been developed intensively and excellent reviews of the theory are described in [2-5]. LSS theory provides integral equation, so we can evaluate any order moment by solving the equation in principle. However, we must expand some terms into Taylor series to convert the integral equations to differential equations. We can then obtain analytical forms of the moment parameters. LSS theory gives energy dependence of the range parameters instantaneously. Using the moments, we can generate profiles of ion implantation by analytical model to express the profiles described in chapter 4. The accuracy of LSS theory depends on the order of the Taylor expansion. The order of the differential equation is determined by the order of the expansion associated with energy loss due to nuclear and electronic stopping powers. In the Kunihiro Suzuki All rights reserved-© 2013 Bentham Science Publishers

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original LSS theory, the terms associated with energy loss were expanded to the first order, which can be solved analytically, and gave analytical expressions for R p and Rp . Gibbons used the analytical LSS theory to analyze R p and Rp for B, N, Al, P, Ga, As, In, and Sb implanted into Si substrate and obtained some good agreement with SIMS data [2]. However, its original form has a mathematical problem, that is, it has singular points and hence we must be careful about alleviating this problem, which is shown in Appendix A. Furukawa et al. solved the LSS integration equation numerically [3]. They encountered conversion problem and performed detailed analyses by using power law potential for nuclear stopping power instead of more general one like Tomas Fermi or ZBL potentials. They clarified the accuracy of analytical LSS theory for various cases by using power low potential. The analytical model for R p is accurate, but that for R p is inaccurate in some cases. Numerical evaluation of the third order moment has been developed [4, 5]. Gibbons et al. established a systematic database for ion implantation profile moments used even these days. They also mentioned the conversion problem and the fatal problem for initial condition, tried to alleviate the numerical problem by using power law potential in some cases and made up the table database with various numerical techniques. Therefore, we should tackle conversion problem when we face a new combination of incident ion and substrate atom. Computer power has been significantly developed and we can obtain MC results within a few minutes, while LSS theory is neither stable and nor accurate. Therefore, only MC is used to predict profiles in new combination of incident ion and substrate atoms, and no one uses LSS theory these days. Recently, LSS integral equation with higher order Taylor expansion has been solved with a perturbation approximation [6], which demonstrates that the LSS with the third order approximation gives the same accuracy as MC, and gives the results instantaneously for any energy as well as a given energy. It was also demonstrated that we could also obtain lateral straggling and its depth dependence with the theory. The extended LSS theory is free from the conversion problem. We mentioned the conversion problem in former analytical model in Appendix A. The extended LSS theory was implemented into a system FabMeister-IM [7], and we could generate ion implantation profiles instantaneously for any combination of incident ion and substrate atoms and further generate two-dimensional profiles.

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Fiigure 1: Trajecctory of an ionn implanted imppurity.

R RANGE R In ntegral Equ uation for R Ioons implanteed with an innitial energyy of E inteeracted with substrate attoms, lose thheir energy and settle inn the certainn location inn the substraate. Fig. 1 shows s the scchematic traajectory of an a implantedd ion. The total t distance between the t places w where an imp planted ion starts its travel and the onne where it comes c to resst is called raange R . Thee projection of o this distannce onto the direction off incident is called the prrojected ran nge R p . The R is deefined as thee straggling perpendicuular to the diirection of in ncident. x is the laterral stragglingg projected to t the x-axiss, and y iss the lateral straggling s prrojected to y-axis. y T probabiliity that the ioons on the way The w of the trajectory witth energy E settle at R is denotedd as P  E, R  . Thereforre, P  E, R  should hold 

 P  E, R dRR  1 0

(1)

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We define the m-th order moment of the range as 

R m  E    R m P  E, R dR 0

(2)

Figure 2: Schematically expression of the interaction between ion and substrate atoms. Nuclear and electronic stopping power is expressed by related collision cross sections of  n and  e , respectively.

The ions move from R to R  R , interact with the substrate atoms and lose energy through the nuclear or electronic stopping power. These interactions are schematically shown in Fig. 2. The probability of the interaction can be expressed by the corresponding collision cross section, i.e., nuclear cross section  n and electronic cross section  e . The probability that the ions interact during ions move from R to R  R is given by N R  d  n  N R  d  e

(3)

We assume that the lost energy after the nuclear and electronic interactions are Tn and Te , respectively. The probability that the ions settle at R after the interaction is given by

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Kunihiro Suzuki

N R  P  E  Tn , R  R  d n N R  P  E  Te , R  R  d e

(4)

The probability that the ions do not interact with atom or electron during R is given by



1  N R  d n  N R  d e



(5)

Ions do not lose energy in this process, and the probability that the ions settle at R after the interaction is given by





1  N R d n  N R d e  P  E , R  R     

(6)

Therefore, the equation for P  E, R  is given by

P  E , R   N R  P  E  Tn , R  R  d n  N R  P  E  Te , R  R  d e



(7)



 1  N R  d n  N R  d e  P  E , R  R    Equation 7 is schematically expressed by Fig. 3. Considering the limiting case that R approaches to zero in Eq. 7, we obtain dP  E , R   N   P  E  Tn , R  d n   P  E , R  d n  dR  N   P  E  Te , R  d e   P  Ee , R  d e 

(8)

Multiplying R m to the Eq. 8 and integrate them from 0 to ∞ with respect to R , we obtain    m dP  E , R  dR  N  R m   P  E  Tn , R  d n   P  E , R  d n dR  R 0 dR 0 

 N  R   P  E  Te , R  d e   P  Ee , R  d e dR 0 m

(9)

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Figure 3: Schematically expression of probability of ion and substrate atom interaction.

This can be modified as 



 R m P  E , R    m  R m 1 P  E , R  dR 0 0    N    R m P  E  Tn , R  dRd n    R m P  E , R  dRd n   0  0    N    R m P  E  Te , R  dRd e    R m P  E , R  dRd e  0  0 

(10)

The first term on the left side of Eq. 10 is approximated to be zero, that is, we assume that P  E, R  is zero for infinite R . We define a probable value of R as 

R m  E    R m P  E , R  dR

(11)

0

and obtain from Eq. 10 as m R m1  E   N   R m  E  d n   R m  E  Tn  d n     N   R m  E  d e   R m  E  Te  d e   N    R m  E   R m  E  Tn    N    R m  E   R m  E  Te  

 d n     d e   

(12)

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This is the m-th integral equation with respect to R . Setting m  1 in Eq. 12, we obtain

1  N    R  E   R  E  Tn   d n   

(13)

 N    R  E   R  E  Te   d e    ANALYTICAL MODEL FOR R First-Order Analytical Model for R

Assuming Tn, Te