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

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Of bonds and bandsHow to understand MO theory
for extended solids?
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What does this mean?
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Linear chain of hydrogen atoms
Polyene
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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.

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E
Nodes between all atoms, kP/a
kP/2a
No nodes, k0
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Linear chain of hydrogen atoms
c0 c1 c2 c3 c4 c5 c6 c7 c8

a
Y Sn exp(ikna) cn - What is this?
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• Yk Sn exp(ikna) cn - what is this?
• cn are basis functions, orbitals for H
• k is an index related to the number of nodes, or
  rather P times the reciprocal of the distance
  between the nodes. If there are no nodes k0. If
  there are nodes between all atoms, kP/a

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No nodes, k0

• Yk Sn exp(ikna) cn
• Y0 Sn cn c0 c1 c2 c2
• Strongly bonding

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Nodes between all atoms, kP/a

• YP/a Sn exp(i P/a na) cn
• Sn exp(iPn) cn (alternating signs)
• YP/a c0 - c1 c2 - c2
• Strongly anti-bonding

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E
E(k)
P/a
k
0
P/2a
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Band width
E
If the hydrogen atoms are at large distances,
they do not interact a5Å
P/a
k
0
P/2a
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E
a0.5Å
k
P/a
0
P/2a
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A stack of square planar platinum PtL4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Monomer
E
p s d
z
x2-y2
z2 yz xz xy
4L
Pt PtL4 L4
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Dispersion z2
Strongly bonding strongly antibonding
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Dispersion z
Strong bonding antibonding
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Dispersion z
Strong bonding antibonding
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Dispersion xz, yz
Intermediate bonding antibonding
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Dispersion x2-y2
Weak bonding antibonding
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Polymer
E
s
z
d
x2-y2
s
p
z2 yz xz xy
d
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Polymer
E
s
z
d
x2-y2
s
p
z2 yz xz xy
d
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Polymer
E
s
z
d
x2-y2
s
p
z2 yz xz xy
d
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Polymer
s
E
d
Pt is d8
EF
s
p
d
k
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In oxidised systems, the Pt-Pt distance shortens.
Why?
EF
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BS DOS COOP
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Linear chain of hydrogen atoms
E
a
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Linear chain of hydrogen atoms
E
Dispersion
a
k
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Peierls distortion - H2
E
ad
a-d
k
P/a
P/2a
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Peierls distrotion
E
k
P/2a
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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
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Special points of the Brillouin zone
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Two dimensions - Graphene
Face center Body centre Edge centre Face
centre
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All Pz orbitals in-phase, G, Strongly p-bonding
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All Pz orbitals out-of-phase, G, Strongly anti
p-bonding
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Two dimensions - Graphene
Face center Body centre Edge centre Face
centre
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K
M
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Pz, p, K non-bonding
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Pz, p, K non-bonding
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Pz, p, M bonding
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Pz, p, M anti-bonding
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p bands no gap at K, gap at M
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Px, s, G strongly bonding, weakly anti-bonding
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Px, s, G strongly anti-bonding, weakly bonding
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Px, s, K strongly bonding, weakly bonding
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Px, s, K strongly anti-bonding, weakly
anti-bonding
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s interactions in graphene
s bands run down away from G. sbands run up away
from G
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Whats the use?

• Bonding and electronics. Graphene is strongly
  bonded. It is a zero bandgap semiconductor.

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Copper A Metal
E
e-
e-
EF
e-
DOS
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Silicon A semiconductor
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
EF
DOS
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Si Semiconductor

• Fermi-Dirac f(E) e(E-EF)/kT1-1
• k8.610-5 eV/K
• Eg in silicon 1eV
• f(EgEf)300K e1/0.0251-1 e-40 410-18

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Silicon Extrinsic (K,n) excitation
E
Excited electrons
EF
Hole
DOS
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Silicon - Doping
E
e-
EF
DOS