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 12 SEMICONDUCTORS 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Last time 1. Periodic potential: atomic + pertubation 2 BlBloch sums of l f localili zedd orbitbitalls (atomiic, or LCAO)2. h (t LCAO) 3. Tight-binding formulation (in the case only one orbital has significant overlap) 4. From flat atomic “bands” to dispersive cosines 5. Bandwidths 6. Tigght-bindingg vs. emppirical ppseudoppotential ((i.e. a perturbation of the free electron gas) 7. Band structure (DETAILED) of a semiconductor 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 12 Ferroelectric perovskites Image removed due to copyright restrictions. Please see: Fig. 3 inKing-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Ferroelectric perovskites Image removed due to copyright restrictions. Please see: Fig. 4 inKing-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Silicon Lead Images removed due to copyright restrictions.Image removed due to copyright restrictions.Please see Fig. 2.24 and in Yu, Peter Y., and Cardona, Manuel.Fundamentals of Semiconductors: Physics and Materials Properties."Please see any band gap diagram of lead, such asNew York, NY: Springer, 2001.http://www.bandstructure.jp/Table/BAND/band_png/pb4800b.ps.png3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Copper -0.939-0.539X W LΓ12Γ25'Γ1Γ Κ∆1Ζ3Ζ2Α1Σ1Σ2K2K3K1K1K4Σ4Σ1Σ3Σ1Α1Α3Α3Α1Ζ1Ζ3Ζ4Ζ1∆2∆5∆2'∆1533332'2'1111'2Q_Q+Q+Q+Q_Q_Figure by MIT OpenCourseWare. Silver Image removed due to copyright restrictions.Please see and band gap diagram of silver, such ashttp://www.bandstructure.jp/Table/BAND/band_png/ag39275a.ps.png3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 3 "Platinum Gold 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Band structure of graphene 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 4 Courtesy Hongki Min. Used with permission.Used with permission.Band structure of graphene Images removed due to copyright restrictions. Please see: Fig. 2.4 and 2.6 inMinot, Ethan. "Tuning the Band Structure of Carbon Nanotubes." PhD dissertation, Cornell University, 2004.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Carbon nanotubes Image from Wikimedia Commons, http://commons.wikimedia.org 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 5•The independent-electron gas • HamiltonianHamiltonian • Eigenvalues and eigenfunctions 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 6 Zone folding: Band structure of nanotubes (8,0) semiconducting (5,5) metallic 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) KBBAKBFigure by MIT OpenCourseWare.••The independent-electron gas • BvKBvK boundary conditionsboundary conditions 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) The independent-electron gas • Counting the statesCounting the states 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 7 2 /Lx2 /LykykxImage removed due to copyright restrictions.Please see any diagram of free electron band gaps, such as http://leung.uwaterloo.ca/CHEM/750/Lectures%202007/SSNT-5-Electronic%20Structure%20II_files/image008.jpg.••The independent-electron gas • Particle densityParticle density 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) The independent-electron gas • Energy densityEnergy density 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 8∫Density of states (for any solid) rr rr 1 n () = 2∫ 8π 13 δ ε −εn ( )) k dk g ε ( 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Band structure of graphene 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 9 Courtesy Hongki Min. Used with permission.Massive vs massless bands Dimensions d=1 d=2 d=3 Massless (E≈k)Massless (E≈k) const const EE E2 E2 Massive (E≈k2) 1/sqrt(E) const sqrt(E) g ε 1 1() = 2∫ 8π 3 r dS n ∇εn ()k • S goes as kd-1 , where d is the dimensionality , where d is the dimensionalityS goes as k • 1 r for a band that has kl dispersions goes as k-(l-1),()∇ε k • the integral goes as kd-l • energy is proportional to kl, the integral goes as ε(d-l)/l 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Statistics of classical and quantum particles 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 10Probability and Partition Function 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Chemical potential 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 11Fermi-Dirac distribution 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 12 Images from Wikimedia Commons,
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