Lecture II.

. Insulators

Charge Transport Theory, narrow bands

Delocalized (Bloch) Wave Functions

Localized Wave Functions

Excitons

Peirels Distortion (1D systems)

Review of Electronic Properties of Solids

Free Electron Fermi Gas

•Electron in a box: example

Electrons are sufficiently delocalized in

•conjugated molecules that they can be

•considered as an electron box

Electronic absorption of β-carotene

0.294nm

Energy Bands, Semiconductors, Doping

Hydrogen Molecule

Charge Transport Theory General

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.

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.

Holes

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.

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.

1 1( , ,..., , )N Ns s r r

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

Amorphous Semiconductors

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

Transport between “isolated” molecules

Excitons

A peculiarity in some One-dimensional systems:

Peierls Distortion