Tightbinding (LCAO) Approach to Bandstructure Theory

Bandstructure Theories Another Qualitative Discussion for a while! REMINDER: Computational methods fall into 2 general categories which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier discussion).

Bandstructure Theories 2 general categories of theories, 1. “Physicist’s Viewpoint”: Start from the “almost free” e- & add a periodic potential in a highly sophisticated, self-consistent manner.  Pseudopotential Methods 2. “Chemist’s Viewpoint”: Start with the atomic & molecular picture & build up the periodic solid from atomic e- in a sophisticated, self-consistent manner.  Tightbinding/LCAO Methods Now, we’ll focus on this 2nd method.

“A Chemists Viewpoint” 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 the periodic interatomic interaction V turns on. 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!

The Tightbinding Method My Personal Opinion The Tightbinding / LCAO method gives a much clearer physical picture (than the pseudopotential method does) of the causes of the bands & the gaps. In this method, the periodic potential V is discussed 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.

Bound States in Atoms: Qualitative Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. . The Coulomb potential energy of an electron a distance r from a positively charge nucleus of charge q is V(r) E2 E1 E0

Bound & “Free” States in Solids: Qualitative 1D The 1D Coulomb potential energy of an electron due to an array of nuclei of charge q separated by a distance R is V(r) E2 E1 E0 V(r) Solid Where n = 0, +/-1, +/-2 etc. This is shown as the black line in the figure. r + R Nuclear positions

Band of allowed energy states. Energy Levels & Bands In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms. Lower binding electron states become bands of allowed states. We will find that only partially filled bands conduct Band of allowed energy states. E Electron level similar to that of an isolated atom + + + + + position

Bond Center Equilibrium Position First: Qualitative Diatomic Molecule Discussion Some Quantum Chemistry! Consider a 2 atom molecule AB with 1 valence e- per atom, & a strong covalent bond. Assume 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  Ψ- = (ψA - ψ B)/(2)½ & An Antibonding State: Empty: Ψ+ = (ψA + ψ B)/(2)½ - & + are qualitatively as shown. Antibonding State Ψ+ + -  Bonding State Ψ- 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 the bonding levels in the molecular picture. Conduction bands: Unoccupied  Correspond to the antibonding levels in the molecular picture.

Energy Band Theory Solid State N~1023 atoms/cm3 2 atoms 6 atoms

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 a0  Material Lattice Constant  s-like Antibonding States  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 (α-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 

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