3.23 Fall 2007 – Lecture 8 PERIODICPERIODICPERIoDICPERIODICPERLast timeStudyExamples of reciprocal latticesPeriodic potentialBloch TheoremBloch TheoremBloch TheoremPeriodic boundary conditions for the electrons: Born – von KarmanExplicit proof of Bloch’s theoremΨnk (r) is not a momentum eigenstateMIT 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 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)3.23 Fall 2007 – Lecture 8PERIODICPERIODICPERoDICPERIODICPERImage removed due to copyright restrictions. Please see M. C. Escher. "Ascending and Descending." 1960.3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)Last time1. Newtonian, Lagrangian, and Hamiltonian formulations2. 1-dim monoatomic and diatomic chain. Acoustic and optical phonons.3. Bravais lattices and lattices with a basis4. Point groups and group symmetries5. Primitive unit cell, conventioanl unit cell, periodic boundary conditions6. Reciprocal latticeStudy• Chapter 2 of Singleton textbook – “Band theory and electronic properties of solids”• Start reading Chapter 3• Problem sets from same book are excellent examples of “Exam Material”3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Examples of reciprocal latticesDirect lattice Reciprocal latticeSimple cubic Simple cubicFCC BCCBCC FCCOrthorhombic Orthorhombic()2321123aabaaaπ×=⋅×Periodic potential3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Bloch TheoremBloch Theorem• n, k are the quantum numbers (band index and crystal momentum), u is periodic• From two requirements: a translation can’t change the charge density, and two translations must be equivalent to one that is the sum of the twoBloch Theorem()()()expnk nkrR ikR rΨ+= ΨiCrystal momentum k (in the first BZ)Periodic boundary conditions for the electrons: Born – von Karman3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Explicit proof of Bloch’s theorem3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Ψnk(r) is not a momentum eigenstate3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall
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