MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.•3.23 Fall 2007 – Lecture 23 FERMI’s GOLDEN RULE 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Study • FoxFox, Optical Properties of Solids: 3.1 to 3.6 Optical Properties of Solids: 3 1 to 3 6 (skip 3.3.5 and 3.3.6), 4.1, 4.2, and Appendix B.2 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 1⎪k k= =Boundary conditions ˆ ⋅(r r nB2 − 1 )= 0(r r nDˆ ⋅2− D1 )=σσ(= surface charge density)r r nˆ ×(2 1( E2− E)= 0(r 1 ) r r nˆ × H 2− H1 )= K (r K = surface current density) 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Snell’s law rr rr rrkr= kr′⋅ ⋅(⋅) ( )=(k r)1t t 1t t 2t t r k1z =ksinsinθθ1 =nn1 ω sinθ1 ⎫⎪⎪⎪ sinθ1 c ⎬n1sinθ1 =n2 sinθ2r k2 z = sinθ2 =n2 ω sinθ2 ⎪k2 c ⎭⎪Image from Wikimedia Commons, http://commons.wikimedia.org 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 2 kk nkk n•Energy conservation rr ∂ ∂ r r r r r r∫ J ⋅ Edv + ∫ ( E ⋅ D + H ⋅ B ) dv + ∫ ( E × H ) ⋅ ndSˆ = 0∂t 1442 4 43 142 43 total energy stored in electrical energy surface and magnetic field flux per unit area per volume r cr r S = E × H 4π 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Optical processes • Reflection and refractionReflection and refraction • Absorption • Luminescence • Scattering 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 3Optical coefficients T: ratio of transmitted vs incident power T: ratio of transmitted vs incident power R+T=1 (no absorption, scattering) Absorption: Transmission: 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Modeling Optical Constants with a Damped Harmonic Oscillator ε=(ni+ k)2 = n2 − k2 123 + i 2{nk εε1 2 2ω2 −ω 2 2ε =+14πχ+ 4πNe(0 ) i4π Neγω ε=+14πχ+ 4π − i 4πm((2 ω 2 2 − ω 2 )+γ2ω 2 ) m((ω2 ω γ20 − 2 2 0 0 0)+ ω144444424444443 14444244443) ε1 ε2 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 4Amorphous silica 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Kramers-Kronig relations 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 5 Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.Figure by MIT OpenCourseWare.ε"ε'Optical materials Image removed due to copyright restrictions. Please see: Fig. 1.4 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001. 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Infrared active modes Image removed due to copyright restrictions. Please see Fig. 1a and 2a in Giannozzi, Paolo, et al. "Ab initio Calculation of Phonon Dispersions in Semiconductors." Physical Review B 43 (March 15, 1991): 7231-7242.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 6Optical materials Image removed due to copyright restrictions. Please see: Fig. 1.7 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001. 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Optical materials Image removed due to copyright restrictions. Please see: Fig. 1.5 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001. 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 7Interband absorption Image removed due to copyright restrictions. Please see: Fig. 3.1 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001. 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Direct and indirect transitions Image removed due to copyright restrictions. Please see: Fig. 3.2 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001. 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 8Transition rate for direct absorption 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Transition rates: perturbing Hamiltonian 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 9Transition rates: perturbing Hamiltonian 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Transition rate for direct absorption 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)