Abstract

The applicability of a general transfer-matrix method for optical analysis of multilayersreported earlier [Katsidis and Siapkas, Appl. Opt. 41, 3978 (2002)] is being extended so as to simulate asymmetric implantation doping profiles using distributions with four moments. The sensitivity of infrared reflectance spectra regarding the variation of the first four moments of a Pearson free carrier distribution is demonstrated. Experimental data of 1.5  MeV as well as 2.5  MeV As implantation into p-Si followed by annealing at 1100°C are presented, suggesting the need to use two joined Pearson IV distribution segments in the simulation of annealed profiles. A twin peak observed in the 1.5  MeV case is confirmed by Rutherford backscattering analysis.

© 2008 Optical Society of America

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  1. D. Pramanik and A. N. Saxena, "MeV implantation for VLSI," Nucl. Instrum. Methods Phys. Res. B 10-11, 493-497 (1985).
    [CrossRef]
  2. M. Takahasi, J. Nakata, and K. Kajiyama, "High energy As+ ion implantation into Si-arsenic profiles and electrical activation characteristics," Jpn. J. Appl. Phys. 20, 2205-2209 (1981).
    [CrossRef]
  3. H. Wong, E. Deng, N. W. Cheung, P. K. Chu, E. M. Strathman, and M. D. Strathman, "Profile studies of MeV ions implanted into Si," Nucl. Instrum. Methods Phys. Res. B 21, 447-451 (1987).
    [CrossRef]
  4. G. Lulli, M. Bianconi, A. Parisini, S. Sama, and M. Servidori, "Damage profiles in high-energy As implanted Si," J. Appl. Phys. 88, 3993-3999 (2000).
    [CrossRef]
  5. S. Saito, S. Shishiguchi, K. Hamada, and T. Hayashi, "Dopant profile and defect control in ion implantation by RTA with high ramp-up rate," Mater. Chem. Phys. 54, 49-53 (1998).
    [CrossRef]
  6. Y. Li, C. Tan, Y. Xia, J. Zhang, C. Xue, U. Xu, and P. Liu, "2.0-MeV Er+ implanted in silicon: depth distribution, damage profile and annealing behaviour," Appl. Phys. A 71, 689-693 (2000).
    [CrossRef]
  7. D. I. Siapkas, N. Hatzopoulos, C. C. Katsidis, T. Zorba, C. L. Mitsas, and P. L. F. Hemment, "Structural and compositional characterization of high energy separation by implantation of oxygen structures using infrared spectroscopy," J. Electrochem. Soc. 143, 3019-3032 (1996).
    [CrossRef]
  8. N. Hatzopoulos, W. Skorupa, and D. I. Siapkas, "Double SIMOX structures formed by sequential high energy oxygen implantation into silicon," J. Electrochem. Soc. 147, 354-362 (2000).
    [CrossRef]
  9. G. Lulli, E. Albertazzi, M. Bianconi, R. Nipoti, M. Cervera, A. Camera, and C. Cellini, "Stopping and damage parameters for Monte Carlo simulation of MeV implants in crystalline Si," J. Appl. Phys. 82, 5958-5964 (1997).
    [CrossRef]
  10. J. F. Ziegler, "SRIM-2003," Nucl. Instrum. Methods Phys. Res. B 219-220, 1027-1036 (2004).
    [CrossRef]
  11. P. Normand, D. Tsoukalas, N. Guillemot, and P. Chenevier, "A pile-up phenomenon during arsenic diffusion in silicon-on-insulator structures formed by oxygen implantation," J. Appl. Phys. 66, 3585-3589 (1989).
    [CrossRef]
  12. A. K. Robinson, K. J. Reeson, and P. L. F. Hemment, "Redistribution and electrical activation of implanted arsenic in silicon on insulator substrates formed by oxygen ion implantation," J. Appl. Phys. 68, 4340-4342 (1990).
    [CrossRef]
  13. C. C. Katsidis, D. I. Siapkas, A. K. Robinson, and P. L. F. Hemment, "Formation of conducting and insulating layered structures in Si by ion implantation. Process control using FTIR spectroscopy," J. Electrochem. Soc. 148, G704-G716 (2001).
    [CrossRef]
  14. J. J. Hamilton, B. Colombeau, J. A. Sharp, N. E. B. Cowern, K. J. Kirkby, E. J. H. Collart, M. Bersani, and D. Giubertoni, "Effect of buried Si/SiO2 interface on dopant and defect evolution in preamorphizing implant ultrashallow junction," J. Vac. Sci. Technol. B 24, 442-445 (2006).
    [CrossRef]
  15. R. Gwilliam, S. Gennaro, G. Claudio, B. J. Sealy, C. Mulcahy, and S. Biswas, "Ultra shallow junction formation and dopant activation study of Ga implanted Si," Nucl. Instrum. Methods Phys. Res. B 237, 121-125 (2005).
    [CrossRef]
  16. A. Satta, T. Janssens, T. Clarysse, E. Simoen, M. Meuris, A. Benedetti, I. Hoflijk, B. De Jaeger, C. Demeurisse, and W. Vandervorst, "P implantation doping of Ge: diffusion, activation, and recrystallization," J. Vac. Sci. Technol. B 24, 494-498 (2006).
    [CrossRef]
  17. L. Romano, A. M. Piro, V. Privitera, E. Rimini, G. Fortunato, B. G. Svensson, M. Foad, and M. G. Grimaldi, "Mechanism of deactivation and clustering of B in Si at extremely high concentration," Nucl. Instrum. Methods Phys. Res. B 253, 50-54 (2006).
    [CrossRef]
  18. J. W. Mayer and S. S. Lau, Electronic Materials Science: For Integrated Circuits in Si and GaAs (Macmillan, 1990).
  19. J. Lindhard, M. Scharff, and H. E. Schiott, "Range concepts and heavy ion ranges," K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33, 1-41 (1963).
  20. M. D. Giles, "Ion implantation," in VLSI Technology, S.M.Sze, ed. (McGraw-Hill, 1988), pp. 327-374.
  21. R. B. Fair, "Diffusion and ion implantation in silicon," in Semiconductor Materials and Process Technology Handbook for Very Large Scale Integration (VLSI) and Ultra Large Scale Integration (ULSI), G. E. McGuire, ed. (Noyes, 1988), pp. 455-540.
  22. H. Ryssel, G. Prinke, K. Haberger, K. Hoffmann, K. Müller, and R. Henkelmann, "Range parameters of boron implanted into silicon," Appl. Phys. 24, 39-43 (1981).
    [CrossRef]
  23. F. Jahnel, H. Ryssel, G. Prinke, K. Hoffmann, K. Müller, J. Biersack, and R. Henkelmann, "Description of arsenic and boron profiles implanted in SiO2, Si3N4 and Si using Pearson distributions with four moments," Nucl. Instrum. Methods 181-183, 223-229 (1981).
    [CrossRef]
  24. D. G. Ashworth, R. Oven, and B. Mundin, "Representation of ion implantation profiles by Pearson frequency distribution curves," J. Phys. D 23, 870-876 (1990).
    [CrossRef]
  25. A. Barthel, J. Lorenz, and H. Ryssel, "Two-dimensional simulation of ion implanted profiles using a personal computer," Nucl. Instrum. Methods Phys. Res. B 37-38, 312-316 (1989).
    [CrossRef]
  26. L. Gong, S. Bogen, L. Frey, W. Jung, and H. Ryssel, "Simulation of high energy implantation profiles in crystalline silicon," Microelectron. Eng. 19, 495-498 (1992).
    [CrossRef]
  27. C. C. Katsidis and D. I. Siapkas, "General transfer-matrix method for optical multilayer systems with coherent, partially coherent and incoherent interference," Appl. Opt. 41, 3978-3987 (2002).
    [CrossRef] [PubMed]
  28. C. C. Katsidis, "Study of the effects of ion implantation on the optical, structural and electrical properties of silicon and SIMOX structures using fast Fourier transform spectroscopy in the infrared," Ph.D. dissertation (Aristotle University of Thessaloniki, 2002).
  29. W. Karstens, D. Bobela, and D. Y. Smith, "Impurity and free-carrier effects on the far-infrared dispersion spectrum of silicon," J. Opt. Soc. Am. A 23, 723-729 (2006).
    [CrossRef]

2006 (4)

J. J. Hamilton, B. Colombeau, J. A. Sharp, N. E. B. Cowern, K. J. Kirkby, E. J. H. Collart, M. Bersani, and D. Giubertoni, "Effect of buried Si/SiO2 interface on dopant and defect evolution in preamorphizing implant ultrashallow junction," J. Vac. Sci. Technol. B 24, 442-445 (2006).
[CrossRef]

A. Satta, T. Janssens, T. Clarysse, E. Simoen, M. Meuris, A. Benedetti, I. Hoflijk, B. De Jaeger, C. Demeurisse, and W. Vandervorst, "P implantation doping of Ge: diffusion, activation, and recrystallization," J. Vac. Sci. Technol. B 24, 494-498 (2006).
[CrossRef]

L. Romano, A. M. Piro, V. Privitera, E. Rimini, G. Fortunato, B. G. Svensson, M. Foad, and M. G. Grimaldi, "Mechanism of deactivation and clustering of B in Si at extremely high concentration," Nucl. Instrum. Methods Phys. Res. B 253, 50-54 (2006).
[CrossRef]

W. Karstens, D. Bobela, and D. Y. Smith, "Impurity and free-carrier effects on the far-infrared dispersion spectrum of silicon," J. Opt. Soc. Am. A 23, 723-729 (2006).
[CrossRef]

2005 (1)

R. Gwilliam, S. Gennaro, G. Claudio, B. J. Sealy, C. Mulcahy, and S. Biswas, "Ultra shallow junction formation and dopant activation study of Ga implanted Si," Nucl. Instrum. Methods Phys. Res. B 237, 121-125 (2005).
[CrossRef]

2004 (1)

J. F. Ziegler, "SRIM-2003," Nucl. Instrum. Methods Phys. Res. B 219-220, 1027-1036 (2004).
[CrossRef]

2002 (1)

2001 (1)

C. C. Katsidis, D. I. Siapkas, A. K. Robinson, and P. L. F. Hemment, "Formation of conducting and insulating layered structures in Si by ion implantation. Process control using FTIR spectroscopy," J. Electrochem. Soc. 148, G704-G716 (2001).
[CrossRef]

2000 (3)

G. Lulli, M. Bianconi, A. Parisini, S. Sama, and M. Servidori, "Damage profiles in high-energy As implanted Si," J. Appl. Phys. 88, 3993-3999 (2000).
[CrossRef]

Y. Li, C. Tan, Y. Xia, J. Zhang, C. Xue, U. Xu, and P. Liu, "2.0-MeV Er+ implanted in silicon: depth distribution, damage profile and annealing behaviour," Appl. Phys. A 71, 689-693 (2000).
[CrossRef]

N. Hatzopoulos, W. Skorupa, and D. I. Siapkas, "Double SIMOX structures formed by sequential high energy oxygen implantation into silicon," J. Electrochem. Soc. 147, 354-362 (2000).
[CrossRef]

1998 (1)

S. Saito, S. Shishiguchi, K. Hamada, and T. Hayashi, "Dopant profile and defect control in ion implantation by RTA with high ramp-up rate," Mater. Chem. Phys. 54, 49-53 (1998).
[CrossRef]

1997 (1)

G. Lulli, E. Albertazzi, M. Bianconi, R. Nipoti, M. Cervera, A. Camera, and C. Cellini, "Stopping and damage parameters for Monte Carlo simulation of MeV implants in crystalline Si," J. Appl. Phys. 82, 5958-5964 (1997).
[CrossRef]

1996 (1)

D. I. Siapkas, N. Hatzopoulos, C. C. Katsidis, T. Zorba, C. L. Mitsas, and P. L. F. Hemment, "Structural and compositional characterization of high energy separation by implantation of oxygen structures using infrared spectroscopy," J. Electrochem. Soc. 143, 3019-3032 (1996).
[CrossRef]

1992 (1)

L. Gong, S. Bogen, L. Frey, W. Jung, and H. Ryssel, "Simulation of high energy implantation profiles in crystalline silicon," Microelectron. Eng. 19, 495-498 (1992).
[CrossRef]

1990 (2)

D. G. Ashworth, R. Oven, and B. Mundin, "Representation of ion implantation profiles by Pearson frequency distribution curves," J. Phys. D 23, 870-876 (1990).
[CrossRef]

A. K. Robinson, K. J. Reeson, and P. L. F. Hemment, "Redistribution and electrical activation of implanted arsenic in silicon on insulator substrates formed by oxygen ion implantation," J. Appl. Phys. 68, 4340-4342 (1990).
[CrossRef]

1989 (2)

P. Normand, D. Tsoukalas, N. Guillemot, and P. Chenevier, "A pile-up phenomenon during arsenic diffusion in silicon-on-insulator structures formed by oxygen implantation," J. Appl. Phys. 66, 3585-3589 (1989).
[CrossRef]

A. Barthel, J. Lorenz, and H. Ryssel, "Two-dimensional simulation of ion implanted profiles using a personal computer," Nucl. Instrum. Methods Phys. Res. B 37-38, 312-316 (1989).
[CrossRef]

1987 (1)

H. Wong, E. Deng, N. W. Cheung, P. K. Chu, E. M. Strathman, and M. D. Strathman, "Profile studies of MeV ions implanted into Si," Nucl. Instrum. Methods Phys. Res. B 21, 447-451 (1987).
[CrossRef]

1985 (1)

D. Pramanik and A. N. Saxena, "MeV implantation for VLSI," Nucl. Instrum. Methods Phys. Res. B 10-11, 493-497 (1985).
[CrossRef]

1981 (3)

M. Takahasi, J. Nakata, and K. Kajiyama, "High energy As+ ion implantation into Si-arsenic profiles and electrical activation characteristics," Jpn. J. Appl. Phys. 20, 2205-2209 (1981).
[CrossRef]

H. Ryssel, G. Prinke, K. Haberger, K. Hoffmann, K. Müller, and R. Henkelmann, "Range parameters of boron implanted into silicon," Appl. Phys. 24, 39-43 (1981).
[CrossRef]

F. Jahnel, H. Ryssel, G. Prinke, K. Hoffmann, K. Müller, J. Biersack, and R. Henkelmann, "Description of arsenic and boron profiles implanted in SiO2, Si3N4 and Si using Pearson distributions with four moments," Nucl. Instrum. Methods 181-183, 223-229 (1981).
[CrossRef]

1963 (1)

J. Lindhard, M. Scharff, and H. E. Schiott, "Range concepts and heavy ion ranges," K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33, 1-41 (1963).

Appl. Opt. (1)

Appl. Phys. (1)

H. Ryssel, G. Prinke, K. Haberger, K. Hoffmann, K. Müller, and R. Henkelmann, "Range parameters of boron implanted into silicon," Appl. Phys. 24, 39-43 (1981).
[CrossRef]

Appl. Phys. A (1)

Y. Li, C. Tan, Y. Xia, J. Zhang, C. Xue, U. Xu, and P. Liu, "2.0-MeV Er+ implanted in silicon: depth distribution, damage profile and annealing behaviour," Appl. Phys. A 71, 689-693 (2000).
[CrossRef]

J. Appl. Phys. (4)

G. Lulli, E. Albertazzi, M. Bianconi, R. Nipoti, M. Cervera, A. Camera, and C. Cellini, "Stopping and damage parameters for Monte Carlo simulation of MeV implants in crystalline Si," J. Appl. Phys. 82, 5958-5964 (1997).
[CrossRef]

G. Lulli, M. Bianconi, A. Parisini, S. Sama, and M. Servidori, "Damage profiles in high-energy As implanted Si," J. Appl. Phys. 88, 3993-3999 (2000).
[CrossRef]

P. Normand, D. Tsoukalas, N. Guillemot, and P. Chenevier, "A pile-up phenomenon during arsenic diffusion in silicon-on-insulator structures formed by oxygen implantation," J. Appl. Phys. 66, 3585-3589 (1989).
[CrossRef]

A. K. Robinson, K. J. Reeson, and P. L. F. Hemment, "Redistribution and electrical activation of implanted arsenic in silicon on insulator substrates formed by oxygen ion implantation," J. Appl. Phys. 68, 4340-4342 (1990).
[CrossRef]

J. Electrochem. Soc. (3)

C. C. Katsidis, D. I. Siapkas, A. K. Robinson, and P. L. F. Hemment, "Formation of conducting and insulating layered structures in Si by ion implantation. Process control using FTIR spectroscopy," J. Electrochem. Soc. 148, G704-G716 (2001).
[CrossRef]

D. I. Siapkas, N. Hatzopoulos, C. C. Katsidis, T. Zorba, C. L. Mitsas, and P. L. F. Hemment, "Structural and compositional characterization of high energy separation by implantation of oxygen structures using infrared spectroscopy," J. Electrochem. Soc. 143, 3019-3032 (1996).
[CrossRef]

N. Hatzopoulos, W. Skorupa, and D. I. Siapkas, "Double SIMOX structures formed by sequential high energy oxygen implantation into silicon," J. Electrochem. Soc. 147, 354-362 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

D. G. Ashworth, R. Oven, and B. Mundin, "Representation of ion implantation profiles by Pearson frequency distribution curves," J. Phys. D 23, 870-876 (1990).
[CrossRef]

J. Vac. Sci. Technol. B (2)

J. J. Hamilton, B. Colombeau, J. A. Sharp, N. E. B. Cowern, K. J. Kirkby, E. J. H. Collart, M. Bersani, and D. Giubertoni, "Effect of buried Si/SiO2 interface on dopant and defect evolution in preamorphizing implant ultrashallow junction," J. Vac. Sci. Technol. B 24, 442-445 (2006).
[CrossRef]

A. Satta, T. Janssens, T. Clarysse, E. Simoen, M. Meuris, A. Benedetti, I. Hoflijk, B. De Jaeger, C. Demeurisse, and W. Vandervorst, "P implantation doping of Ge: diffusion, activation, and recrystallization," J. Vac. Sci. Technol. B 24, 494-498 (2006).
[CrossRef]

Jpn. J. Appl. Phys. (1)

M. Takahasi, J. Nakata, and K. Kajiyama, "High energy As+ ion implantation into Si-arsenic profiles and electrical activation characteristics," Jpn. J. Appl. Phys. 20, 2205-2209 (1981).
[CrossRef]

K. Dan. Vidensk. Selsk. Mat. Fys. Medd. (1)

J. Lindhard, M. Scharff, and H. E. Schiott, "Range concepts and heavy ion ranges," K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33, 1-41 (1963).

Mater. Chem. Phys. (1)

S. Saito, S. Shishiguchi, K. Hamada, and T. Hayashi, "Dopant profile and defect control in ion implantation by RTA with high ramp-up rate," Mater. Chem. Phys. 54, 49-53 (1998).
[CrossRef]

Microelectron. Eng. (1)

L. Gong, S. Bogen, L. Frey, W. Jung, and H. Ryssel, "Simulation of high energy implantation profiles in crystalline silicon," Microelectron. Eng. 19, 495-498 (1992).
[CrossRef]

Nucl. Instrum. Methods (1)

F. Jahnel, H. Ryssel, G. Prinke, K. Hoffmann, K. Müller, J. Biersack, and R. Henkelmann, "Description of arsenic and boron profiles implanted in SiO2, Si3N4 and Si using Pearson distributions with four moments," Nucl. Instrum. Methods 181-183, 223-229 (1981).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (6)

A. Barthel, J. Lorenz, and H. Ryssel, "Two-dimensional simulation of ion implanted profiles using a personal computer," Nucl. Instrum. Methods Phys. Res. B 37-38, 312-316 (1989).
[CrossRef]

D. Pramanik and A. N. Saxena, "MeV implantation for VLSI," Nucl. Instrum. Methods Phys. Res. B 10-11, 493-497 (1985).
[CrossRef]

H. Wong, E. Deng, N. W. Cheung, P. K. Chu, E. M. Strathman, and M. D. Strathman, "Profile studies of MeV ions implanted into Si," Nucl. Instrum. Methods Phys. Res. B 21, 447-451 (1987).
[CrossRef]

J. F. Ziegler, "SRIM-2003," Nucl. Instrum. Methods Phys. Res. B 219-220, 1027-1036 (2004).
[CrossRef]

L. Romano, A. M. Piro, V. Privitera, E. Rimini, G. Fortunato, B. G. Svensson, M. Foad, and M. G. Grimaldi, "Mechanism of deactivation and clustering of B in Si at extremely high concentration," Nucl. Instrum. Methods Phys. Res. B 253, 50-54 (2006).
[CrossRef]

R. Gwilliam, S. Gennaro, G. Claudio, B. J. Sealy, C. Mulcahy, and S. Biswas, "Ultra shallow junction formation and dopant activation study of Ga implanted Si," Nucl. Instrum. Methods Phys. Res. B 237, 121-125 (2005).
[CrossRef]

Other (4)

J. W. Mayer and S. S. Lau, Electronic Materials Science: For Integrated Circuits in Si and GaAs (Macmillan, 1990).

M. D. Giles, "Ion implantation," in VLSI Technology, S.M.Sze, ed. (McGraw-Hill, 1988), pp. 327-374.

R. B. Fair, "Diffusion and ion implantation in silicon," in Semiconductor Materials and Process Technology Handbook for Very Large Scale Integration (VLSI) and Ultra Large Scale Integration (ULSI), G. E. McGuire, ed. (Noyes, 1988), pp. 455-540.

C. C. Katsidis, "Study of the effects of ion implantation on the optical, structural and electrical properties of silicon and SIMOX structures using fast Fourier transform spectroscopy in the infrared," Ph.D. dissertation (Aristotle University of Thessaloniki, 2002).

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Figures (16)

Fig. 1
Fig. 1

Role of skewness, γ, in the symmetry of a Pearson IV distribution. For a symmetric distribution (dashed curve): γ = 0 .

Fig. 2
Fig. 2

Calculated IR reflectance spectra of doped Si corresponding to the free carrier concentration profiles of Fig. 1.

Fig. 3
Fig. 3

Variation of kurtosis, β, in a Pearson IV distribution.

Fig. 4
Fig. 4

Calculated IR reflectance spectra of doped Si corresponding to the free carrier concentration profiles of Fig. 3.

Fig. 5
Fig. 5

Variation of straggle, Δ R p , in a Pearson IV distribution.

Fig. 6
Fig. 6

Calculated IR reflectance spectra of doped Si corresponding to the free carrier concentration profiles of Fig. 5.

Fig. 7
Fig. 7

Variation of peak position, R max , in a Pearson IV distribution.

Fig. 8
Fig. 8

Calculated IR reflectance spectra of doped Si corresponding to the free carrier concentration profiles of Fig. 7.

Fig. 9
Fig. 9

After annealing 2.5 MeV As+ implanted Si. The best-fit free carrier concentration profile (solid curve) is given by the combination of two Pearson IV distributions (PIV1 and PIV2). A Gaussian having the straggle of PIV1 is also shown for comparison.

Fig. 10
Fig. 10

Refractive index profiles corresponding to the combination of PIV1 and PIV2, to PIV1 and to the Gaussian carrier concentration profiles of Fig. 9, respectively.

Fig. 11
Fig. 11

Experimental IR reflectance spectrum of 2 .5   MeV As+ implanted Si after annealing and its best fit as calculated using the combination of the PIV1 and PIV2 distributions of Fig. 9.

Fig. 12
Fig. 12

Calculated reflectance spectra corresponding to the best-fit combination, the PIV1, and the Gaussian free carrier profiles of Fig. 9, respectively.

Fig. 13
Fig. 13

After annealing 1.5 MeV As+ implanted Si. The best fit free carrier concentration profile (solid curve) is given by the combination of two Pearson IV distributions, which yields a twin-peak depth profile.

Fig. 14
Fig. 14

Experimental IR reflectance spectrum of 1.5   MeV As + implanted Si after annealing and its best fit as calculated using the twin-peak distribution of Fig. 13.

Fig. 15
Fig. 15

Real (top of figure) and imaginary (bottom) part of the refractive index profile, which corresponds to the twin-peak distribution of Fig. 13, calculated at three different wavenumbers to reveal dispersion.

Fig. 16
Fig. 16

After annealing 1.5   MeV As + implanted Si. Comparison of the free carrier concentration profile obtained by FTIR analysis with the impurity concentration profile obtained by RBS measurements. The impurity concentration profile before annealing is also illustrated.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

N ( x ) = N max exp [ ( x R p ) 2 2 Δ R p 2 ] c m 3 .
N ( x ) = N max exp [ ( x R p ) 2 2 Δ R p 2 ] + N t exp [ K ( x x t ) ] cm 3 .
d f ( s ) d s = ( s a ) f ( s ) b 0 + b 1 s + b 2 s 2 ,
a = γ Δ R p ( β + 3 ) / A ,
b 0 = Δ R p 2 ( 4 β 3 γ 2 ) / A ,
b 1 = a ,
b 2 = ( 2 β 3 γ 2 6 ) / A ,
β 3 + 6 [ 7 γ 2 + ( γ 2 + 4 ) 3 / 2 8 32 γ 2 ] = β min .
f ( x R p ) = C N [ b 0 + b 1 ( x R p ) + b 2 ( x R p ) 2 ] 1 / 2 b 2 × exp [ b 1 / b 2 + b 1 D tan 1 ( 2 b 2 ( x R p ) + b 1 D ) ] ,
N ( x ) = N max f ( x R p ) cm 3 ,
C N = [ b 0 + b 1 2 + b 2 b 1 2 ] 1 / 2 b 2 × exp [ b 1 / b 2 + b 1 D tan 1 ( 2 b 2 b 1 + b 1 D ) ] .
ε ˜ = k Δ ε k ω TO k 2 ω TO k 2 ω 2 i γ k ω ω p 2 ω 2 + i γ p ω + ε .
n s , r , d 2 = A s , r , d + B s , r , d / ( Ω s , r , d 2 ω 2 ) .
ε ˜ = ω p 2 ω 2 + i γ p ω + A r + B r Ω r 2 ω 2 = ω p 2 ω 2 + i γ p ω + f r ( A s + B s Ω s 2 ω 2 ) .
N C j = N C s + ( N C max N C s ) f [ j δ x R p ] ,
ω p j 2 = 4 π N C j e 2 m * , γ p j = e m * μ j ( N C j ) a μ ,
ε j = n j 2 k j 2 = ω p j 2 ω 2 + γ p j 2 + f r ( A s + B s Ω s 2 ω 2 ) ,
ε j = 2 n j k j = γ p j ω p j 2 ω ( ω 2 + γ p j 2 ) .
n j 2 = [ ε j + ( ε j 2 + ε j 2 ) 1 / 2 ] / 2 ,
k j 2 = [ ε j + ( ε j 2 + ε j 2 ) 1 / 2 ] / 2 .

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