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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 10 TIGHT-BINDING 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 3.23 Fall 2007 – Lecture 10 TIGHT-BINDING 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 1 Image removed due to copyright restrictions. Please seehttp://www.helloboston.com/Images/People/5292005Houdini_Harvard_Bridge_1.jpgImages removed due to copyright restrictions. Please seehttp://www.helloboston.com/Images/People/5292005Houdini_Harvard_Bridge_2.jpg•Last time 1. Explicit solution for the Bloch orbitals 22. FFree ellecttrons 3. Band structure of free electron vs. silicon 4. Band edges 5. Ψnk (r) is not a momentum eigenstate 6. Group velocity, effective mass 77. Fermi energy Fermi surface Fermi energy, Fermi surface 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Study • ChapChap. 4 Singleton4 Singleton 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 2= ⋅Plane wave expansion ψ r (r r ) ur (r r)exp ik r r r )ψ (r) = u(r)exp (( ik⋅ r)nk nk periodic u is expanded in planewaves, labeled according to the reciprocal lattice vectors r rur() = c exp(iG⋅r)nk r ∑r nG r k rr G Explicit solution for the Bloch orbitals 2⎛ h ( qG− ′) 2 ⎞ ⎜ EC− V′′ CqG = 0− ⎟ qG′+ ∑ G G−′ −′′⎜ 2m ⎟ G′′⎝⎝ ⎠⎠ ⎛ h2 ⎞ ⎜ 2m( q− 2G) 2 V− GV−2GV−3GV−4G ⎟ ⎜ ⎟ ⎜ h2 ( q G−) 2 V ⎟⎛C− 2 ⎞ ⎛Cq G2 ⎞⎜ VG 2mV−G −2GV−3G ⎟⎜ q G⎟ ⎜ − ⎟⎜ 2 ⎟⎜ CqG− ⎟ ⎜ CqG− ⎟⎜ V V h (( q)) 2 V V ⎟⎜ C ⎟ = E⎜ C ⎟ ⎜⎜ 2G G 22m − G −2G ⎟⎟⎜⎜ q ⎟⎟ ⎜⎜ q ⎟⎟ ⎜ 2 CqG+ ⎟ Cq G⎟⎟⎜ ⎜ +⎜ V V h + 2 V ⎟⎟⎜ 2 ⎟ ⎜C ⎟⎠⎜ 3GV2G G 2m( q G)− G ⎝Cq G+ ⎠ ⎝ q G+ 2 ⎜ ⎟ ⎜ h22 ⎟⎜ V4GV3GV2G VG ( q+ 2G) ⎟⎝ 2m ⎠ 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 3•What choice for a basis ? • For molecules: often atomic orbitals, or localized functions as Gaussians • For solids periodic functions such as sinesFor solids, periodic functions such as sines and cosines (plane waves) The plane waves basis set • Systematic improvement ofSystematic improvement of completeness/resolution • Huge number of basis elements – only possible because of pseudopotentials • Allows for easy evaluation of gradients and Laplacian • Kinetic energy in reciprocal space, potential in real space • Basis set does not depend on atomic positions: there are no Pulay terms in the forces 4Hamiltonian in the Bloch representation 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Energy Bands 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 5 -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.Brillouin Zone (fcc) 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) The Fermi surface 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 6 LUWKXQkxkzkyΓΣ∆Σ'ΛFigure by MIT OpenCourseWare.Image from the Fermi Surface Database. Used with permission.Please see: http://www.phys.ufl.edu/fermisurface/jpg/K.jpg,http://www.phys.ufl.edu/fermisurface/jpg/Cu.jpg.The Fermi surface3.23 Electronic, Optical and Magnetic Properties of Materials Nicola Marzari (MIT, Fall 2007)http://www phys ufl edu/fermisurface/http://www.phys.ufl.edu/fermisurface/ -Energy of a collection of atoms ˆ ˆ ˆ ˆHTV =+ +V +V−e e−e e−N NN 1 2ˆ ⎡ rr ⎤ 1Tˆ =−∑∇i VN =∑∑ V(R−ri)⎥Vˆ e =∑∑ re e− ⎢I e− r2 i i⎣ I ⎦ i j>i| ri− rj| • Te: quantum kinetic energy of the electrons • Vee-ee: electron-electron interactions • Ve-N: electrostatic electron-nucleus attraction (electrons in the field of all the nuclei) • VN-N: electrostatic nucleus-nucleus repulsion 3.23 Fall 2006 7 Images from the Fermi Surface Database. Used with permission.ˆ Molecules and Solids: Electrons and Nuclei rHψ (r ,..., r , R ,..., R ) = E ψ (r ,..., r , RR ,..., RR ))H ( r 1 rr nR r 1 R r N ) Etot (rr 1 rr n r 1 r N • We treat only the electrons as quantum particles, in the field of the fixed (or slowly varying) nuclei • This is generically called the adiabatic or Born-Oppenheimer approximation • “Adiabatic” means that there is no coupling between different electronic surfaces; “B-O” implies there is no influence of the ionic motion on one electronic surface 3.23 Fall 2006 Complexity of the many-body Ψ “…Some form of approximation is essential, and this would mean the construction of tables. The tabulation function of one variable requires a page, of two variables a volume and of three variables a library; but the full specification of a single wave function of neutral iron is a function of 78 variables. It would be rather crude to restrict to 10 the number of values of each variable at which to tabulate this function, but even so, full tabulation would require 1078 entries.” 3.23 Fall 2006 8•Mean-field approach • Independent particle model (Hartree): each Independent particle model (Hartree): each electron moves in an effective potential, ⎤ ⎥⎥⎦representing the attraction of the nuclei and the average effect of the repulsive interactions of the other electrons • This average repulsion is the electrostatic repulsion of the average charge density of all other electrons 3.23 Fall 2006 Hartree Equations The Hartree equations can be obtained directly from the variational principle, once the search is restricted to the many-body wavefunctions that are written – as above – as the product of single orbitals (i.e. we are working with independent electrons) ψ (r r 1,..., r r n ) =ϕ1(r r 1)ϕ2(r r 2)Lϕn (r r n ) ϕ rr r 1 r( )2 dr r r ∑∫ 2 ϕ εϕ V (R ) | | (ri ) =− ∇ − + + ri r (ri )r r i I j j j i iI | | − r rij≠i j ∑ ⎡ ⎢⎢⎣ 3.23 Fall 2006 9 2 1•The


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