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Band theory (Read 3919 times)
Gerrit-Jan Linker
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Band theory
17.04.10 at 21:20:11
 
Band theory
 
The electronic structure of solids is usually discussed in terms of band theory.
 

  • Orbitals and bands in one dimension
    Consider a growing lineair chain of H atoms.  
    With 2 H atoms there are 2 MO's: a bonding and anti-bonding pair of the 1s functions.
    With N H atoms there are N MO's that make more and more a continuous band the larger N gets.
  • Bloch functions, crystal orbitals and band structures

  • Band structures = Electronic band structures
    E(k) vs k
    The ranges of energy an electron is allowed to have
  • Band width
    Dispersive band vs flat band
  • Density of states (DOS)
    Number of states (levels), n, between E and E + dE
  • Fermi level: εF
    The Fermi level εF in solids is equivalent to a HOMO in a molecule.
  • Folding bands
    More than one elementary unit in the unit cell.
  • Peierls distortion
    Peierls distortion in a solid is equivalent to the Jahn-Teller effect in molecules.
    ref: http://en.wikipedia.org/wiki/Peierls_transition
  • Metal, semi-conductor or insulator

    • A partially filled band. A conductor, a metal.  
      Resistivity rises (linearly) with rising temperature.  
    • Two bands separated by a band gap. The valence band is completely filled. The conduction band is not populated (is empty). This material is a semi-conductor or an insulator.  
      Resistivity falls with rising temperature.
    • Half metals have a band gap for one spin orientation and a partially filled band for the other spin.
    Semi-conductors with a small band gap can get populate the conduction band at a certain temperature leaving a hole in the valence band. This will lead to a metal.
    Resistivity falls initially with rising temperature, just as with a semi-conductor.
    At a particular point the resistivity rises with rising temperature as with a metal.
  • 2 dimensions
      Direct lattice
    • Reciprocal lattice
    • Brillouin zone
      http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1250678083
    • Bloch orbitals
      http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?virboard=theochem;num=1247827363
    • Band structure
    • Partially filled bands: Fermi surface
      • Fermi surface = boundary surface separating "occupied" and "unoccupied" wave vectors:
        a point (1D), a line (2D), a surface (3D)  
      • Fermi surfaces dictate the dimensionality of metallic properties.
        1D metal  
        pseudo 1D metal
        2D anisotropic metal
        2D isotropic metal
      • Fermi surface nesting
        • A Fermi surface is nested by a vector q when a section can be moved by such a vector to be superimposed on another section.  
        • A metallic system with a nested Fermi surface is subject to a metal-insulator phase transition.
          Interaction of occupied and vacant crystal orbitals of the same symmetry of E=E(kF) related by q.
        • If the superposition is complete it leads to a destruction of the Fermi surface.
        • If the superposition is incomplete it leaves hole and/or electron pockets.



  • Computational aspects

 
See also:
http://en.wikipedia.org/wiki/Electronic_band_structure
 
Source:
First France-Japan Advanced School of Chemistry and Physics of Molecular Materials (Rennes, 2006)
Introduction to electronic band structure of solids: application to molecular materials
http://www.google.nl/url?sa=t&source=web&ct=res&cd=3&ved=0CBMQFj AC&url=http%3A%2F%2Fcpmm.univ-rennes1.fr%2FCours%2FJ.-F.%2520Halet.pdf&rct=j&q=Introduction+to+electronic+band+structure+of+solids%3A+application+to+molecular+materials&ei=mBXKS8DgGdD4_AaIkL3xBQ&usg=AFQjCNHkDtV0jwmM4pF8G3j-erig0Kr2Cg&sig2=BO80F1dxaxsuZh4XG2cs3Q
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« Last Edit: 18.05.12 at 10:08:05 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Gerrit-Jan Linker
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Posts: 75
QC methods and band structure
Reply #1 - 07.01.11 at 08:17:38
 
QC methods and band structure
 
Hartree Fock calculations tends to localise electrons and give rise to insulators. The metal Al for example is an insulator in HF. Is this due to the fact that HF can only use a single determinant and that not multiple determinants with fractional occupation numbers can be used to simulate the system.
 
Pure DFT tends to give rise to metals. For instance, the semi-conductor NiO is a metal in DFT. This is due to the fact that DFT suffers from self interaction. The term <ii||ii> does not vanish as in HF theory and therefore the electron 'sees itself'. It leads to a higher electron repulsion and hence a higher delocalised behavior.
 
 
Hybrid DFT functionals tend to give the correct behavior as correlation effects change the wavefunction sufficiently to give the correct description of the electrons. It gives rise to correct band structures. Mostly.
 
See also:
HF band gap
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1198498430/0#0
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« Last Edit: 23.03.11 at 07:09:55 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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