10. Appendix

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654 <strong>Appendix</strong> B Solution to Problem 7.6 Figure 7.26(b) shows the scattering geometry used by Henry and Hopfield in determining the polariton dispersion of GaP. (a) From this figure it is clear that the minimum value of the polariton wave vector q for a given value of kL and the scattering angle ı is: qmin kL sin ı. (b) From Pythagoras Theorem one obtains that q2 a2 q2 min .ora [q2 q2 min ]1/2 . Since ks a kL cos ı one obtains the result: ks kL cos ı a kL cos ı [q2 q2 min ]1/2 . (c) Using the result in (b) we can write: kL ks kL (kL cos ı [q 2 q 2 min ]1/2 ) kL(1 cos ı) q[1 (qmin/q) 2 ] 1/2 Since ˆL hckL and ˆS hckS (where c is the speed of light inside the crystal) the Raman frequency ˆL ˆS hc(kL ks) ˆL(1 cos ı) hcq[1 (qmin/q) 2 ] 1/2 .Forq ≫ qmin one obtains the dependence of ˆL ˆS on q as hcq. This is the reason why the series of broken curves in Fig. 7.26(a) are “nearly” parallel straight lines for different values of ı. Forq near qmin these lines are not really straight lines since ˆL ˆS depends on q as [q 2 q 2 min ]1/2 . Solution to Problem 7.8 (a) The solution to this part of the problem can already be found in Fig. 7.28. The results of translating the various Feynman diagrams into perturbation terms can be found in (7.50). The correspondence between the diagrams and the terms in (7.50) is simply: (a) ⇔ term 1; (b) ⇔ term 2; (c) ⇔ term 3 etc. Students are urged to work through each diagram to learn the “art” of translating Feynman diagrams into perturbation theory expressions. In addition, notice how some of the terms are related by permutation of the timeorder of the vertices. This is a very convenient way to generate all the diagrams by first starting with a few diagrams and then use them to generate all the remaining diagrams via permutation of the order in which the vertices occur in time. (b) To write down all the Feynman diagram contributing to a phonon-assisted optical transition in an indirect bandgap semiconductor, let us start with the schematic diagram in Fig. 6.16. The Feynman diagram corresponding to the process labeled as (1) in Fig. 6.16 is:
The diagram for the process labeled (2) is: Solution to Problem 7.8 655 These diagrams translate into perturbation terms similar to that in (6.61). In both of the above diagrams the vertex for the electron-photon interaction occurs first. We can generate other diagrams in which the electron(hole)phonon interaction occurs first. For example, the following is a diagram in which the electron-phonon interaction excites an electron from an intermediate valence band (°v ′) to the conduction band (¢c) and then the photon excites another electron from the final valence band state (°v) to fill the hole in °v ′. This diagram can also be translated into an expression similar to (6.61). However, since the electron-hole pair has to be created via emission of a phonon the energy denominator in this case has the form: Ecv ′ ˆphonon and this term is highly non-resonant. (c) The Feynman diagrams for two-phonon Raman scattering processes can be divided into three types. These are labeled (a) to (c) in the following figure.
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Appendix A: Pioneers of Semiconduct
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Ultra-Pure Germanium 555 Ultra-Pure
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References Ultra-Pure Germanium 557
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Two Pseudopotential Methods: Empiri
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The Early Stages of Band-Structures
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Cyclotron Resonance and Structure o
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References Cyclotron Resonance and
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Optical Properties of Amorphous Sem
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Optical Spectroscopy of Shallow Imp
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Page 21 and 22:
Page 23 and 24:
On the Prehistory of Angular Resolv
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The Discovery and Very Basics of th
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The Birth of the Semiconductor Supe
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Number of papers 500 200 100 50 20
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Appendix B: Solutions to Some of th
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Solution to Problem 2.6 585 transfo
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⎧ ⎫ 2apple ⎪⎨ cos (y z)
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Partial solution to Problem 2.8 589
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Solution to Problem 2.16 Solution t
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Solution to Problem 2.20 597 to the
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Therefore the result of I ′ on
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Solution to Problem 3.2(c) Solution
Page 51 and 52: Solution to Problem 3.5 603 Similar
Page 53 and 54: Solution to Problem 3.7 (b and c) 6
Page 55 and 56: u3. The above equation can be reduc
Page 57 and 58: Solution to Problem 3.8 (a, c and d
Page 59 and 60: Solution to Problem 3.11(a,b and c)
Page 61 and 62: Solution to Problem 3.16 613 (1) Al
Page 63 and 64: Solution to Problem 3.17 Solution t
Page 65 and 66: Solution to Problem 3.17 617 forces
Page 67 and 68: Solution to Problem 4.1 619 the abo
Page 69 and 70: C dz ′ z ′ i° Solution to Pro
Page 71 and 72: R -R A y E’-E y E’-E R x Soluti
Page 73 and 74: Solution to Problem 5.3(a) 625 tion
Page 75 and 76: Solution to Problem 5.5 627 To obta
Page 77 and 78: Ùm ∞ NLOq 0 2 apple dq 0 ∞
Page 79 and 80: plus Jz ·Fz Solution to Problem 5
Page 81 and 82: Solution to Problem 6.7 633 axes. F
Page 83 and 84: which can be found in standard text
Page 85 and 86: Solution to Problem 6.11 637 The co
Page 87 and 88: Solution to Problem 6.19(a) 639 if
Page 89 and 90: Solution to Problem 6.19(a) 641 k.
Page 91 and 92: Solution to Problem 6.19(a) 643 The
Page 93 and 94: Solution to Problem 6.21 645 by a s
Page 95 and 96: Solution to Problem 6.21 647 °25
Page 97 and 98: Equating the two expressions for Nn
Page 99 and 100: Solution to Problem 7.5 651 erties
Page 101: A Short Cut to the Calculation of t
Page 105 and 106: Solution to Problem 7.12 657 or sca
Page 107 and 108: Solution to Problem 8.1 Solution to
Page 109 and 110: Solution to Problem 8.9 661 of 19.5
Page 111 and 112: Solution to Problem 8.10 663 corres
Page 113 and 114: Solution to Problem 9.2 Solution to
Page 115 and 116: Solution to Problem 9.4 667 Table 9
Page 117 and 118: Solution to Problem 9.14 669 Again
Page 119: Solution to Problem 9.14 671 Note t
Page 122 and 123: 674 Appendix C A4.1.2 Historical Ba
Page 124 and 125: 676 Appendix C from the slopes of t
Page 126 and 127: 678 Appendix C Fig. A4.3 The electr
Page 128 and 129: 680 Appendix C Fig. A4.5 (a)-(d)The
Page 130 and 131: 682 Appendix C tion of alloying (se
Page 132 and 133: 684 Appendix C 17 -3 Hall Concentra
Page 135 and 136: Appendix D: Recent Developments and
Page 137 and 138: Chapter 1 689 ing up the glass to a
Page 139 and 140: Chapter 2 691 Popov, V. N., Lambin,
Page 141 and 142: Chapter 2 693 and zinc-blende group
Page 143 and 144: Chapter 3 695 Inelastic x-ray Scatt
Page 145 and 146: Chapter 4 697 results from a better
Page 147 and 148: Chapter 5 699 K. Alberi, K. M. Yu,
Page 149 and 150: Chapter 5 701 P. Ramvall, N. Carlss
Page 151 and 152: Chapter 6 Ab initio calculations of
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Chapter 6 705 Strain-induced birefr
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Chapter 7 707 S. Baroni, S. de Giro
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Chapter 7 709 W. Windl, K. Karch, P
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Chapter 8 711 Strain Shift Coeffici
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Chapter 9 713 Z. J. Zhao, F. Liu, L
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Chapter 9 715 S. F. Ren, W. Cheng,
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Chapter 9 717 V. P. Gusynin, S. G.
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720 References tions II. Misfitting
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722 References 2.30 R. M. Wentzcovi
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724 References 3.23 R. M. Martin: E
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726 References Nye J. F.: Physical
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728 References Schubert E. F.: Dopi
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730 References Ferry D. K.: Semicon
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732 References 6.28 S. Logothetidis
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734 References 6.73 H. D. Fuchs, C.
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736 References 6.116 D. E. Aspnes:
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738 References Chapter 7 7.1 W. Hei
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740 References 7.43 G. Finkelstein,
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742 References 7.82 J. R. Sandercoc
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744 References 7.122 D. C. Reynolds
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746 References 8.27 A. Goldmann, J.
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748 References Electronic and Surfa
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750 References 9.31 A. Pinczuk, G.
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752 References 9.68 J. P. Palmier:
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Subject Index A ·-Sn (gray tin) 95
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C c(2 × 8)(111) surface of Ge - di
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Dielectric function 117, 246, 253,
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Emission rate vs frequency in Ge 34
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Galena (PbS) 1 Gamma-ray detectors
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Insulators 1 Integral quantum Hall
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- - vs T 222 - temperature dependen
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- frequency 253, 306, 335 - - of va
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Resonant Raman profile 403 Resonant
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- - LA phonons 512 - - LO phonons 5
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- structure 142 - symmetry operatio
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