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
Nodes between all atoms, k=P/a
k=P/2a
No nodes, k=0
Linear chain of hydrogen atoms
c0 c1 c2 c3 c4 c5 c6 c7 c8
a
Y= Sn exp(ikna) cn - What is this?
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 k=0. If there are nodes
between all atoms, k=P/a
No nodes, k=0
Yk= Sn exp(ikna) cn
Y0= Sn cn = c0 + c1 + c2 + c2 +…
Strongly bonding
Nodes between all atoms, k=P/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
E
E(k)
0
P/2a
P/a k
E
Band width
If the hydrogen atoms are at large
distances, they do not interact:
a=5Å
0
P/2a
P/a k
E
a=0.5Å
0
P/2a
P/a k
A stack of square planar platinum PtL4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Monomer
E
p
z
x2-y2
s
z2
yz
xz
xy
d
Pt
PtL4
4L
L4
Dispersion – z2
Strongly bonding –strongly antibonding
Dispersion – z
Strong bonding –antibonding
Dispersion – z
Strong bonding – antibonding
Dispersion – xz, yz
Intermediate bonding – antibonding
Dispersion – x2-y2
Weak bonding – antibonding
Polymer
E
s
z
d
x2-y2
z2
yz
xz
xy
p
d
s
Polymer
E
s
z
d
x2-y2
z2
yz
xz
xy
p
d
s
Polymer
E
s
z
d
x2-y2
z2
yz
xz
xy
p
d
s
s
Polymer
E
d
Pt is d8
p
s
EF
d
k
In oxidised systems, the PtPt distance shortens. Why?
EF
BS
DOS
COOP
Linear chain of hydrogen atoms
E
a
Linear chain of hydrogen atoms
E
Dispersion
a
k
Peierls distortion - H2
E
a+d
a-d
k
P/2a
P/a
Peierls distrotion
E
k
P/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 Pz orbitals in-phase, G,
Strongly p-bonding
All Pz orbitals out-of-phase,
G, Strongly anti p-bonding
Two dimensions - Graphene
Face center
Body centre Edge centre Face centre
K
M
Pz, p, K: nonbonding
Pz, p*, K: nonbonding
Pz, p, M: bonding
Pz, p, M*: antibonding
p bands –no gap at K, gap at M
Px, s, G: strongly
bonding, weakly
anti-bonding
Px, s*, G: strongly
anti-bonding,
weakly bonding
Px, s, K: strongly
bonding, weakly
bonding
Px, s*, K: strongly
anti-bonding,
weakly antibonding
s interactions in graphene
s bands run
down away
from G.
s*bands run up
away from G
What’s the use?
Bonding and electronics. Graphene is
strongly bonded. It is a zero bandgap
semiconductor.
Copper – A Metal
E
eeEF
eDOS
Silicon –A semiconductor
E
Si has four valence
electrons and
achieves octet by
bonding to four
neighbours.
E
All electrons are F
taking part in
bonding and the
electronic DOS
conductivity is low
Si Semiconductor
Fermi-Dirac: f(E) =[e(E-EF)/kT+1]-1
k≈8.6*10-5 eV/K
Eg in silicon ≈1eV
f(Eg+Ef)300K ≈ [e1/0.025+1]-1 ≈ e-40 ≈ 4*10-18
Silicon – Extrinsic (K,n) excitation
E
Excited
electrons
EF
Hole
DOS
Silicon - Doping
E
e-
EF
DOS