The Tightbinding Bandstructure Theory or LCAO Approach To Bandstructure Theory

Bandstructures Another qualitative discussion for a while Recall the beginning of our discussion about band calculations: Bandstructure Theories are Highly computational! The theories fall into 2 general categories, which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier): “Physicist’s View”: Start from an “almost free” e- & add the periodic potential in a highly sophisticated, self-consistent manner.  Pseudopotential Methods “Chemist’s View”: Start with atomic e- & build up the periodic solid in a highly sophisticated, self-consistent manner.  Tightbinding or LCAO Methods Now, we’ll focus on the 2nd method.

Atomic Orbitals (LCAO) method. Method #2 (Qualitative Physical Picture #2) “A Chemists Viewpoint” Start with the atomic/molecular picture of a solid. The atomic energy levels merge to form molecular levels, & merge to form bands as periodic interatomic interaction V turns on. The Tightbinding or Linear Combination of Atomic Orbitals (LCAO) method. This method gives good bands, especially valence bands! The valence bands are ~ almost the same as those from the pseudopotential method! Conduction bands are not so good!

QUESTION ANSWER How can 2 (seemingly) completely different approaches (pseudopotential & tightbinding) lead to essentially the same bands? (Excellent agreement with valence bands; conduction bands are not too good!). ANSWER (partial, from YC): The electrons in the conduction bands are ~ “free” & delocalized. The electrons in the valence bands are ~ in the bonds in r space.  The valence electrons in the bonds have atomic-like character. (So, LCAO is a “natural” approximation for these).

The Tightbinding Method My personal opinion The Tightbinding / LCAO method gives a much clearer physical picture (than pseudopotential method does) of the causes of the bands & the gaps. In this method, the periodic potential V is discussed as in terms of an Overlap Interaction of the electrons on neighboring atoms. As we’ll see, we can define these interactions in terms of a small number of PHYSICALLY APPEALING parameters.

First: A Qualitative Diatomic Molecule Discussion Consider a 2 atom molecule AB with one valence e- per atom, & a strong covalent bond. Assume that the atomic orbitals for A & B, ψA & ψB, are known. Now, solve the Molecular Schrödinger Equation as a function of the A-B separation. The Results are: A Bonding State (filled, 2 e-. Spin-up  & Spin-down ) & An Antibonding State (empty) Qualitatively like  Antibonding State Ψ+ = (ψA + ψ B)/(2)½   Bonding State Ψ- = (ψA - ψ B)/(2)½ Bond Center (Equilibrium Position)

The bonding & antibonding states broaden to become bands. Tightbinding Method “Jump” from 2 atoms to 1023 atoms! The bonding & antibonding states broaden to become bands. A gap opens up between the bonding & the antibonding states (due to the crystal structure & the atom valence). Valence bands: Occupied  Correspond to bonding levels in the molecular picture. Conduction bands: Unoccupied  Correspond to antibonding levels in the molecular picture.

Schematic: Atomic Levels Broadening into Bands In the limit as a   the atomic levels for the isolated atoms come back  p-like Antibonding States  p-like Bonding States  s-like Antibonding States a0  material lattice constant  s-like Bonding States a0

Schematic: Evolution of Atomic-Molecular Levels into Bands p antibonding  p antibonding  s antibonding   Fermi Energy, EF p bonding   Fermi Energy, EF  Isolated Atom s & p Orbital Energies s bonding   Molecule Solid (Semiconductor) Bands The Fundamental Gap is on both sides of EF!

Schematic Evolution of s & p Levels into Bands at the BZ Center (Si) Lowest Conduction  Band EG    Fermi Energy  Highest Valence Band Atom Solid

Schematic Evolution of s & p Levels into Bands at the BZ Center (Ge) Lowest Conduction Band EG  Fermi Energy Highest Valence Band Atom Solid

Schematic Evolution of s & p Levels into Bands at the BZ Center (-Sn) EG = 0 Highest “Valence Band” Lowest “Conduction Band” Fermi Energy Atom Solid

Schematic Dependence of Bands & Gaps on Nearest-Neighbor Distance (from Harrison’s book) Atom Semiconductors Decreasing Nearest Neighbor Distance 

Dependence of Bands & Gap on Ionicity (from Harrison’s book) Schematic Dependence of Bands & Gap on Ionicity (from Harrison’s book) Covalent Bonds Ionic Bonds Metallic Bonds