Of bonds and bands How to understand MO theory for extended solids?

What does this mean?

Linear chain of hydrogen atoms Polyene

Energy The strongest attraction is found for the configuration with the smallest number of nodes. The distances between the nodes is the reciprocal of their number. If there are no nodes, the distance is infinite. If there is a node between every atom the distance is a.

E No nodes, k=0 Nodes between all atoms, k=  /a k=  /2a

 0  1  2  3  4  5  6  7  8 Linear chain of hydrogen atoms a  n exp(ikna)  n - What is this?

 k  n exp(ikna)  n - what is this?  n are basis functions, orbitals for H k is an index related to the number of nodes, or rather  times the reciprocal of the distance between the nodes. If there are no nodes k=0. If there are nodes between all atoms, k=  /a

 k  n exp(ikna)  n  0  n  n  0 +  1 +  2 +  2 +… Strongly bonding No nodes, k=0

  /a  n exp(i  /a na)  n   n exp(i  n)  n (alternating signs)   /a  0 -  1 +  2 -  2 +… Strongly anti-bonding Nodes between all atoms, k=  /a

E /a/a k  /2a E(k)

Band width If the hydrogen atoms are at large distances, they do not interact: a=5Å E /a/a k  /2a

E /a/a k  a=0.5Å

A stack of square planar platinum PtL 4

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Monomer E Pt PtL 4 L 4 psdpsd 4L x 2 -y 2 z z 2 yz xz xy

Dispersion – z 2 Strongly bonding –strongly antibonding

Dispersion – z Strong bonding –antibonding

Dispersion – z Strong bonding – antibonding

Dispersion – xz, yz Intermediate bonding – antibonding

Dispersion – x 2 -y 2 Weak bonding – antibonding

Polymer E x 2 -y 2 z z 2 yz xz xy     

Polymer E x 2 -y 2 z z 2 yz xz xy     

Polymer E x 2 -y 2 z z 2 yz xz xy     

Polymer E      Pt is d 8 EFEF k

EFEF In oxidised systems, the Pt- Pt distance shortens. Why?

BS DOS COOP

Linear chain of hydrogen atoms E a

E k Dispersion a

Peierls distortion - H 2 E k a-  a+   /a  /2a

Peierls distrotion E k  /2a

The Brillouin zone The Brillioun zone is the primitive cell of the reciprocal lattice. Special points in the Brillioun zone have particular properties and are therefore given special symbolms

Special points of the Brillouin zone

Two dimensions - Graphene Face center Body centre Edge centre Face centre

All P z orbitals in-phase, , Strongly  -bonding

All P z orbitals out-of-phase, , Strongly anti  -bonding

Two dimensions - Graphene Face center Body centre Edge centre Face centre

 

Pz, , K: non- bonding

Pz,  *,  : non- bonding

Pz, ,  : bonding

Pz, ,  : anti- bonding

 bands –no gap at  gap at 

Px, ,  : strongly bonding, weakly anti-bonding

Px,  *,  : strongly anti-bonding, weakly bonding

Px, ,  : strongly bonding, weakly bonding

Px,  *,  : strongly anti-bonding, weakly anti- bonding

 interactions in graphene  bands run down away from .  *bands run up away from 

What’s the use? Bonding and electronics. Graphene is strongly bonded. It is a zero bandgap semiconductor.

Copper – A Metal DOS E EFEF e-e- e-e- e-e-

Si has four valence electrons and achieves octet by bonding to four neighbours. All electrons are taking part in bonding and the electronic conductivity is low Silicon –A semiconductor DOS E EFEF

Si Semiconductor Fermi-Dirac: f(E) =[e (E-E F )/kT +1] -1 k≈8.6*10 -5 eV/K E g in silicon ≈1eV f(E g +E f ) 300K ≈ [e 1/ ] -1 ≈ e -40 ≈ 4*10 -18

Silicon – Extrinsic (K, ) excitation DOS E EFEF Excited electrons Hole

e-e- Silicon - Doping DOS E EFEF