Ion Implantation and Synthesis of Materials - Studium

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46 4 Cross-SectionCmd σ ( E) = d T,m 1+mE T(4.19)where m is a variable that is dependent on the reduced energy, ε, and the constantC m is given byCm22mZZ e2 M1TF⎜ a ⎟TFM2π ⎛2 1 2⎞ ⎛ ⎞= λma⎜ ⎟ ⎜ ⎟ ,2 ⎝ ⎠ ⎝ ⎠m(4.20)where λ m is an additional fitting variable.If χ(r) is taken as the Thomas-Fermi screening function, Winterbon et al.(1970) have shownλ = 1.309; λ = 0.327; λ = 0.5.1/3 1/2 1(4.21)Winterbon et al. (1970) recommend the following values of m for various regionsof ε:m = 1/3 for ε ≤ 0.2,m = 1/2 for 0.08 ≤ ε ≤ 2,m = 1 (Rutherford scattering) for ε ≥ 10.As an example, consider the case of 100 keV As implanted into Si (Z 1 = 33;Z 2 = 14; M 1 = 75; M 2 = 28). For this case, a TF = 0.105 nm and ε = 1.37. For thisvalue of reduced energy, m = ½ and C m = 1.74 × 10 −7 . Applying these values to(4.19), with energy in eV units, the energy-transfer cross-section for transferringan energy T = T M = 44.6 × 10 3 eV is dσ/dT = 5.8 × 10 −17 cm 2 eV −1 .The probability of transferring energy between T M + dT is given byd σ ( E)PT ( ) = ,σ(4.22)where σ can be derived from (4.19) using (4.18a) and assuming a constant valueof m (m = 1/2) over the energy interval between T minand T M . Applying (4.22) to ourexample of 100 keV As implanted into Si, taking dT = 1 eV and T min = 1 eV, theprobability for transferring an energy of T = T M = 44.6 × 10 3 eV is 5 × 10 −8 .
Problems 47ReferencesNastasi, M., Mayer, J.W., Hirvonen, J.K.: Ion–Solid Interactions Fundamentals and Applications.Cambridge University Press, Cambridge (1996)Winterbon, K.B., Sigmund, P., Sanders, J.B.: Spacial distribution of energy deposited byatomic particles in elastic collisions. Mat. Fys. Medd. Dan. Vidensk. Selsk. 37(14)(1970)Suggested ReadingFrench, A.P.: Newtonian Mechanics. W.W. Norton, New York (1971)Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading, MA (1959)Johnson, R.E.: Introduction to Atomic and Molecular Collisions. Plenum Press, New York(1982)Sigmund, P.: Collision theory of displacement damage, ion ranges and sputtering. Rev.Roum. Phys. 17, pp. 823, 969 and 1079 (1972)Symon, K.R.: Mechanics. Addison-Wesley, Reading, MA (1953)Torrens, I.M.: Interatomic Potentials. Academic, New York (1972)Weidner, R.T., Sells, R.L.: Elementary Modern Physics, 3rd edn. Allyn & Bacon, Boston(1980)Problems4.1 Derive the formula for the total cross-section, σ (E), using the power-lawapproximation to energy-transfer differential cross-section.4.2 Using the formula derived in problem 4.1, calculate the probability P(E),(4.10), for 50 and 100 keV B incident on a Si film of thickness 10 and100 nm. (Are there two films of varying thickness? If yes, use Si films.)4.3 Using the formula derived in problem 4.1, calculate the probabilityP(E, T ), (4.11) for 50 and 100 keV B incident on a Si film of thickness 10and 100 nm.4.4 If the projectile and target atoms interact like colliding billiard balls (elastichard-spheres), the interatomic potential that represents this condition iscalled a hard-sphere potential. For a hard-sphere potential, the power-lawcross-section parameter m in (4.19) is equal to 0. Derive the total crosssection,σ (E), for a hard-sphere potential.4.5 Using the formula derived in problem 4.4, calculate the probability, P(E),(4.10), for 50 and 100 keV B incident on a Si film of thickness 10 and100 nm. Compare your results to those obtained in problem 4.2.4.6 Using the formula derived in problem 4.4 calculate the probability,P(E, T ), (4.11), for 50 and 100 keV B incident on a Si film of thickness 10and 100 nm. Compare your results to those obtained in problem 4.2.
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M. Nastasi J.W. MayerIon Implantati
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To our loved ones
Page 7 and 8: xContents4 Cross-Section ..........
Page 10 and 11: Contentsxiii14.3.2 Punch-Through St
Page 12 and 13: 2 1 General Features and Fundamenta
Page 20 and 21: 10 1 General Features and Fundament
Page 22 and 23: 12 2 Particle InteractionsThe restr
Page 24 and 25: 142 Particle Interactions(a)V(r)r 0
Page 26 and 27: 162 Particle InteractionsHowever, a
Page 28 and 29: 182 Particle Interactionswhere the
Page 30 and 31: 202 Particle Interactionsa0.8853a0L
Page 33 and 34: 3 Dynamics of Binary Elastic Collis
Page 35 and 36: 3.3 Kinematics of Elastic Collision
Page 37 and 38: 3.4 Center-of-Mass Coordinates 27Fi
Page 39 and 40: 3.4 Center-of-Mass Coordinates 29an
Page 41 and 42: 3.5 Motion under a Central Force 31
Page 43 and 44: 3.5 Motion under a Central Force 33
Page 45 and 46: Problems 35The reduced energy for 1
Page 47 and 48: 4 Cross-Section4.1 IntroductionIn C
Page 49 and 50: 4.2 Scattering Cross-Section 39unit
Page 51 and 52: 4.2 Scattering Cross-Section 41Rdθ
Page 53 and 54: 4.3 Energy-Transfer Cross-Section 4
Page 55: 4.4 Approximation to the Energy-Tra
Page 59 and 60: 5 Ion Stopping5.1 IntroductionWhen
Page 61 and 62: 5.3 Nuclear Stopping 51MeV mg −1
Page 63 and 64: 5.3 Nuclear Stopping 531−m1−2mC
Page 65 and 66: 5.4 ZBL Nuclear Stopping Cross-Sect
Page 67 and 68: 5.5 Electronic Stopping 575.5.1 Hig
Page 69 and 70: 5.5 Electronic Stopping 59Table 5.1
Page 71: Problems 61Problems5.1 Calculate th
Page 74 and 75: 64 6 Ion Range and Range Distributi
Page 88 and 89: 78 7 Displacements and Radiation Da
Page 104 and 105: 94 8 Channeling1.00.8Non-aligned Im
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96 8 ChannelingMeV ION BEAMSUBSTITU
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98 8 ChannelingThe channeling effec
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100 8 Channeling4.0Silicon2.0P (mic
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102 8 ChannelingdσσD( ψc) = ∫
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104 8 ChannelingDechanneled fractio
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106 8 ChannelingProblems8.1 Calcula
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108 9 Doping, Diffusion and Defects
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128 10 Crystallization and Regrowth
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144 11 Si Slicing and Layer Transfe
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160 12 Surface Erosion During Impla
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180 13 Ion-Induced Atomic Intermixi
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194 14 Application of Ion Implantat
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15 Ion Implantation in CMOS Technol
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15.2 Implanters Used in CMOS Proces
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15.3 Low Energy Productivity: Beam
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15.5 Angle Control 233Fig. 15.13. O
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15.5 Angle Control 235Fig. 15.15. I
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References 237No. of implants504540
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Appendix ATable of the Elementselem
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Appendix A 241elementatomicnumber(Z
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Appendix A 243element atomicnumber(
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Appendix BPhysical constants, conve
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IndexAlpha particle 8amorphization
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Index 259differential cross-section
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Index 261layer transfer 143Lennard-
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Index 263thermodynamiceffect ion be
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