The document discusses several topics related to electronic properties of materials including: 1. Insulators, charge transport theory, localized and delocalized wave functions, and excitons. 2. Energy bands in semiconductors, doping, holes, and band structure diagrams. 3. Density functional theory, the Hohenberg-Kohn theorem, Kohn-Sham scheme, and applications to molecular dynamics. 4. Additional sections cover amorphous semiconductors, transport between isolated molecules, and excitons. One-dimensional systems and the Peierls distortion are also mentioned.

1. Section II.
. Insulators
Charge Transport Theory,
narrow bands
Delocalized (Bloch) Wave
Functions
Localized Wave Functions
Excitons
Peirels Distortion (1D systems)

2. Review of Electronic Properties of
Solids
Free Electron Fermi Gas

9. Energy Bands, Semiconductors,
Doping

10. Hydrogen Molecule

18. Charge Transport Theory
General

25. Figure 9.11. (a) Energy levels in an isolated silicon atom and (b) in a
silicon crystal of N atoms, illustrating the formation of energy bands. The
valence band contains 4N states and can accommodate all 4N valence
electrons.

26. Figure 7.1. Schematic plot of the single particle energy spectrum in a bulk
semiconductor for both the electron and hole states on the left side of the
panel with appropriate electron (e) and hole (h) discrete quantum states
shown on the right. The upper parabolic band is the conduction band, the
lower the valence.

27. Holes

30. Figure 9.12. A valence electron jumping across the energy gap in pure
silicon resulting in the generation of a free electron and hole in the crystal:
(a) energy band model, (b) bond model.

31. Figure 9.13. Extrinsic n-type silicon doped with P donor atoms. (a) Energy
band diagram and (b) Bond model.

32. Figure 9.14. Extrinsic p-type silicon doped with B acceptor atoms. (a)
Energy band diagram and (b) Bond model.

34. 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.

35. Linear Combination of Atomic
Orbitals
LCAO

37. Density Functional Theory

38. 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.

40.  (r1 , s1 ,..., rN , sN )

41. 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

43. Amorphous Semiconductors

47. Let’s estimate the mean free path, lmax=l. Let
vmax=v.
The carrier mobility is defined as
μ= eτ/m*, where τ is the carrier relaxation
time.
Consider for mfree carrier=m*=m.
We take l=v τ .
and ½mv2=3/2 kT, all of which gives
l=√(3kTm/e2 ) μ
or l=0.7 μ
where l is in angstrom units and μ is in
cm2/v sec

48. Transport between “isolated”
molecules

60. Excitons

64. A peculiarity in some One-
dimensional systems:
Peierls Distortion