Section II. . Insulators Charge Transport Theory, narrow bands Delocalized (Bloch) Wave Functions Localized Wave Functions ExcitonsPeirels Distortion (1D systems)
Review of Electronic Properties of Solids Free Electron Fermi Gas
Energy Bands, Semiconductors, Doping
Charge Transport Theory General
Figure 9.11. (a) Energy levels in an isolated silicon atom and (b) in asilicon crystal of N atoms, illustrating the formation of energy bands. The valence band contains 4N states and can accommodate all 4N valence electrons.
Figure 7.1. Schematic plot of the single particle energy spectrum in a bulksemiconductor for both the electron and hole states on the left side of the panel with appropriate electron (e) and hole (h) discrete quantum statesshown on the right. The upper parabolic band is the conduction band, the lower the valence.
Figure 9.12. A valence electron jumping across the energy gap in puresilicon resulting in the generation of a free electron and hole in the crystal: (a) energy band model, (b) bond model.
Figure 9.13. Extrinsic n-type silicon doped with P donor atoms. (a) Energy band diagram and (b) Bond model.
Figure 9.14. Extrinsic p-type silicon doped with B acceptor atoms. (a) Energy band diagram and (b) Bond model.
Electronic structures of Organic Molecules(1) Core electrons.(2) s electrons, localized between two atoms.(3) n electrons, located at a particular heteroatom, usually have high orbital energy and could be promoted easily.(4) p electrons, delocalized over an array of atoms, usually have high MO energy and could be promoted easily.
Linear Combination of Atomic Orbitals LCAO
Density Functional Theory
Background• 1920s: Introduction of the Thomas-Fermi model.• 1964: Hohenberg-Kohn paper proving existence of exact DF.• 1965: Kohn-Sham scheme introduced.• 1970s and early 80s: LDA. DFT becomes useful.• 1985: Incorporation of DFT into molecular dynamics (Car-Parrinello) (Now one of PRL’s top 10 cited papers).• 1988: Becke and LYP functionals. DFT useful for some chemistry.• 1998: Nobel prize awarded to Walter Kohn in chemistry for development of DFT.
(r1 , s1 ,..., rN , sN )
Lecture 8:Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003
Let’s estimate the mean free path, lmax=l. Letvmax=v.The carrier mobility is defined asμ= eτ/m*, where τ is the carrier relaxationtime.Consider for mfree carrier=m*=m.We take l=v τ .and ½mv2=3/2 kT, all of which givesl=√(3kTm/e2 ) μor l=0.7 μwhere l is in angstrom units and μ is in cm2/v sec
Transport between “isolated” molecules
A peculiarity in some One- dimensional systems: Peierls Distortion