2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm ,

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1 2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm , Opticl Wves in Crystls A. Yriv nd P. Yeh (John Wiley, New Jersey, 2003) Week Chpter Week Chpter Mr. 03 * Bsics of Crystl Structures Apr. 26, 29 Periodic Medium II Mr. 08, 10 Chp. 1: EM Fields Chp. 3: Polriztion of Light My 03 Chp. 7: Electro-Optics I Mr. 15, 17 Chp. 4: EM Propgtion I My 10, 12 Electro-Optics II Mr. 22, 24 EM Propgtion II My 17, 19 Chp. 8: EO Devices Mr. 29, 31 Chp. 5: Jones Clcultion I My 24, 26 Tke-ome Exm (Midterm II) Apr. 05, 07 Midterm Exm I My 31, Jun. 2 Chp. 8: EO Devices Apr. 12, 14 Jones Clcultion II Jun. 07, 09 Chp. 11: Guided Optics Apr. 19, 21 Chp. 6: Periodic Medium I Jun. 14 Term Pper (Finl)

2 Chp. 0 Bonding nd Types of Solids 0.1 Molecules nd Generl Bonding Principles - Net force = ttrctive nd repulsive Molecule r o r = + Seprted toms + Force Attrction 0 Repulsion r o F A = Attrctive force F N = Net force Intertomic seprtion, r F R = Repulsive force Potentil Energy, E(r) Repulsion 0 Attrction E o E R = Repulsive PE E = Net PE r o E A = Attrctive PE r () Force vs r (b) Potentil energy vs r Fig. 1.3: () Force vs intertomic seprtion nd (b) Potentil energy vs intertomic seprtion. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

3 0.2 Covlently Bonded Solids: 2, C 4, dimond Covlent bond C L shell K shell C covlent bonds () (b) C (c) Fig. 1.5: () Covlent bonding in methne, C4, involves four hydrogen toms shring electrons with one crbon tom. Ech covlent bond hs two shred electrons. The four bonds re identicl nd repel ech other. (b) Schemtic sketch of C4 on pper. (c) In three dimensions, due to symmetry, the bonds re directed towrds the corners of tetrhedron. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

4 0.3 Metllic Bonding: electron gs or cloud (collective shring of electrons) Positive metl ion cores Free vlence electrons forming n electron gs Fig. 1.7: In metllic bonding the vlence electrons from the metl toms form "cloud of electrons" which fills the spce between the metl ions nd "glues" the ions together through the coulombic ttrction between the electron gs nd positive metl ions. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

5 0.4 Ioniclly Bonded Solids: slt (ction-nion) N Cl 3s 3s 3p Closed K nd L shells () Closed K nd L shells Cl- Cl- N+ F A F A 3s 3p N+ r (b) r o (c) Fig. 1.8: The formtion of n ionic bond between N nd Cl toms in NCl. The ttrction is due to coulombic forces. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

6 - Potentil energy per ion pir in solid NCl Potentil energy E(r), ev/(ion-pir) nm Cohesive energy Cl - Cl r = 1.5 ev r = N + N Seprtion, r Cl - N + r o = 0.28 nm Fig. 1.10: Sketch of the potentil energy per ion-pir in solid NCl. Zero energy corresponds to neutrl N nd Cl toms infinitely seprted. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

7 0.5 Secondry Bonding : hydrogen bonds (polr), vn der Wls bonds (induced dipolr) O () (b) Fig. 1.12: The origin of vn der Wls bonding between wter molecules. () The 2O molecule is polr nd hs net permnent dipole moment. (b) Attrctions between the vrious dipole moments in wter gives rise to vn der Wls bonding. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

8 Time verged electron (negtive chrge) distribution Closed L Shell Ne Ionic core (Nucleus + K-shell) Instntneous electron (negtive chrge) distribution fluctutes bout the nucleus. A vn der Wls force B Synchronized fluctutions of the electrons Fig. 1.13: Induced dipole-induced dipole interction nd the resulting vn der Wls force. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

9 0.6 The Crystlline Stte Type of Crystls : periodic rry of points in spce - lttice Lttice Crystl Bsis 90 Unit cell () (b) (c) Unit cell (d) Bsis plcement in unit cell (0,0) y (1/2,1/2) x Fig. 1.70: () A simple squre lttice. The unit cell is squre with side. (b) Bsis hs two toms. (c) Crystl = Lttice + Bsis. The unit cell is simple squre with two toms. (d) Plcement of bsis toms in the crystl unit cell.

10 UNIT CELL GEOMETRY CUBIC SYSTEM = b = c α = β = γ = 90 Mny metls, Al, Cu, Fe, Pb. Mny cermics nd semiconductors, NCl, CsCl, LiF, Si, GAs TETRAGONAL SYSTEM = b - c α = β = γ = 90 In, Sn, Brium Titnte, TiO 2 Simple cubic Body centered cubic Simple tetrgonl Fce centered cubic Body centered tetrgonl ORTOROMBIC SYSTEM - b - c α = β = γ = 90 S, U, Pl, G (<30 C), Iodine, Cementite (Fe 3 C), Sodium Sulfte Simple orthorhombic Body centered orthorhombic Bse centered orthorhombic Fce centered orthorhombic EXAGONAL SYSTEM = b - c α = β = 90 ; γ = 120 Cdmium, Mgnesium, Zinc, Grphite exgonl ROMBOEDRAL SYSTEM = b = c α = β = γ - 90 Arsenic, Boron, Bismuth, Antimony, Mercury (<-39 C) Rhombohedrl MONOCLINIC SYSTEM - b - c α = β = 90 ; γ - 90 TRICLINIC SYSTEM - b - c α - β - γ - 90 α Selenium, Phosphorus Lithium Sulfte Tin Fluoride Simple monoclinic Bse centered monoclinic Potssium dicromte Triclinic Fig. 1.71: The seven crystl systems (unit cell geometries) nd fourteen Brvis lttices.

11 - Fce-Centered Cubic (FCC) Structure: () FCC Unit Cell 2R (b) (c) Fig. 1.30: () The crystl structure of copper is Fce Centered Cubic (FCC). The toms re positioned t well defined sites rrnged periodiclly nd there is long rnge order in the crystl. (b) An FCC unit cell with closed pcked spheres. (c) Reduced sphere representtion of the FCC unit cell. Exmples: Ag, Al, Au, C, Cu, γ-fe (>912 C), Ni, Pd, Pt, Rh

12 - Body-Centered Cubic (BCC) Structure: b Exmples: Alkli metls (Li, N, K, Rb), Cr, Mo, W, Mn, α-fe (< 912 C), β-ti (> 882 C). Fig. 1.31: Body centered cubic (BCC) crystl structure. () A BCC unit cell with closely pcked hrd spheres representing the Fe toms. (b) A reduced-sphere unit cell. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

13 - exgonl Closed Pcked (CP) Structure: Lyer B () Lyer A Lyer B Lyer A Lyer A (b) Lyer A c (c) (d) Exmples: Be, Mg, α-ti ( < 882 C ), Cr, Co, Zn, Zr, Cd Fig. 1.32: The exgonl Close Pcked (CP) Crystl Structure. () The exgonl Close Pcked (CP) Structure. A collection of mny Zn toms. Color difference distinguishes lyers (stcks). (b) The stcking sequence of closely pcked lyers is ABAB (c) A unit cell with reduced spheres (d) The smllest unit cell with reduced spheres.

14 - Dimond & Zinc Blende Cubic Structure: C S Zn Fig. 1.33: The dimond unit cell is cubic. The cell hs eight toms. Grey Sn (α-sn) nd the elementl semiconductors Ge nd Si hve this crystl structure. Fig. 1.34: The Zinc blende (ZnS) cubic crystl structure. Mny importnt compound crystls hve the zinc blende structure. Exmples: AlAs, GAs, GP, GSb, InAs, InP, InSb, ZnS, ZnTe. From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

15 Tble 1.3 Properties of some importnt crystl structures Crystl nd R Structure (R is the rdius of the tom). Coordintion Number (CN) Number of toms per unit cell Atomic Pcking Fctor Exmples Simple = 2R None cubic BCC = 4R/ Mny metls: α-fe, Cr, Mo, W FCC = 4R/ Mny metls Ag, Au, Cu, Pt CP = 2R c = Mny metls: Co, Mg, Ti, Zn Dimond = 8R/ Covlent solids: Dimond, Ge, Si, α-sn. Zinc blende Mny covlent nd ionic solids. Mny compund semiconductors. ZnS, GAs, GSb, InAs, InSb NCl 6 4 ctions 4 nions CsCl 8 1 ction 1 nion 0.67 (NCl) Ionic solids such s NCl, AgCl, LiF MgO, CO Ionic pcking fctor depends on reltive sizes of ions. Ionic solids such s CsCl, CsBr, CsI From Principles of Electronic Mterils nd Devices, Second Edition, S.O. Ksp ( McGrw-ill, 2002)

16 0.6.2 Three Phses of Crbon Covlently bonded lyer Cubic crystl Covlently bonded network of toms Lyers bonded by vn der Wl bonding Covlently bonded lyer exgonl unit cell () Dimond unit cell (b) Grphite The FCC unit cell of the Buckminsterfullerene crystl. Ech lttice point hs C 60 molecule (c) Buckminsterfullerene Fig. 1.42: The three llotropes of crbon. Buckminsterfullerene (C 60 ) molecule (the "buckybll" molecule)

17 Chp. 1. Electromgnetic Fields 1.1. Mxwell's Equtions nd Wve Equtions - E nd : the electric nd mgnetic field vectors (E, ) <-> (D, ) In MKS units, with constitutive equtions - Poynting's Theorem nd Conservtion Lw: U = the energy density of the EM fields (joules/m 3 ) S = the Poynting vector, energy flow (joules/m 2.sec) = the net EM power flowing out of unit volume. [omework] Conservtion of liner momentum of the EM fields: Problem 1.4 (p. 18)

18 1.4. Wve Equtions - From Mxwell's equtions, Differentiting the bove eqn. wrt to time nd eliminting, Use the vector identities Since, Similrly,

19 - Inside homogeneous nd isotropic medium, the refrctive index of the medium 1.5. Propgtion of Lser Pulse; Group Velocity - Denoting A(k) = the mplitude of the plne-wve component with k, with the Fourier spectrum - For lser pulse, (see the next pge) then,

20 with the envelope function - The lser pulse trvels long undistorted in shpe with velocity ; the group velocity

21 - In optics,, the phse velocity

22 Chp. 3. Polriztion of Light Wves 3.2. Polriztion of Monochromtic Plne Wves - In complex-function representtion, where In the x-y plne, where - Let x' nd y' be new set of xes long the principl xes of the ellipse:, with the rottion ngle where nd

23 3.2.1 Liner nd Circulr Polriztion - Linerly polrized: stright lines, nd - Circulrly polrized: right-hnded nd left-hnded ; - Ellipticity of polriztion: ±, (+) when the rottion of E is right-hnded.

25 3.3 Complex-Number Presenttion - The mplitudes nd the phse ngles of the x nd y components of E: - The inclintion ngle nd the ellipticity ngle of the polriztion ellipse: nd 3.4 Jones Vector Presenttion , R. C. Jones, with normliztion - Mutully orthogonl polriztion:, ;

27 - For generl ellipticl polriztion,

Solid State Electronics EC210 Arab Academy for Science and Technology AAST Cairo Spring 2016 Lecture 1 Crystal Structure

Solid State Electronics EC210 Arab Academy for Science and Technology AAST Cairo Spring 2016 Lecture 1 Crystal Structure Solid Stte Electronics EC210 AAST Ciro Spring 2016 Lecture 1 Crystl Structure Dr. Amr Byoumi, Dr. Ndi Rft 1 These PowerPoint color digrms cn only be used by instructors if the 3 rd Edition hs been dopted