1. Band structure: Semiconductor
Band gap
E (eV)
Density of states Wave vector
Empty lattice bands
[111] [100] [110]

2. “Empty lattice solution” vs. Experimental bands
E(k) = ħ2/2me  (k+Ghkl)2 + V Experiment + full theory (lines)
E (5eV)
E (2eV) Ge
(200)
Photoemission
Inverse
photoemission
(111)
Vacuum level EV
EV = 0
(-111)
CBM
(-200)
VBM
(1-1-1)
Ghkl = (-1-1-1)  (2/a)
(000) V0 V0
 k L
=(000) k L=(½½½)

3. Band structure: Noble metal
s,p-band (like a free electron) Cu 1 3 4 2
d-bands (like atomic energy levels)
d-bands
  
[111] [110]

4. Band structure: Transition metal
s,p-band (like a free electron) Ni Cu
Ni Fermi level
d-bands (like atomic energy levels)
Rigid band model
(one less e in Ni)

5. Delocalized s-electrons vs. localized d- electrons
Wave functions of the outer electrons versus distance (in Å = 0.1 nm) for nickel, a transition metal. The 4s electrons are truly metallic, extending well beyond the nearest neighbor atoms ( at r1, r2, r3 ). The 3d electrons are concentrated at the central atom and behave almost like atomic states. They carry the magnetism in Ni.
(r)
r (Å)

6. Two-dimensional -bands of graphene
Occupied
Empty
EFermi
E [eV]
K = M K
Empty *
Occupied 
kx,y M K 
In two dimensions one has the quantum numbers E, kx,y .
This plot of the energy bands shows E vertically and kx , ky in the horizontal plane.

7. “Dirac cones” in graphene
A special feature of the graphene -bands is their linear E(k) dispersion near the six corners K of the Brillouin zone (instead of the parabolic relation for free electron bands).
In a plot of E versus kx,ky one obtains cone-shaped energy bands at the Fermi level.

8. Topological Insulators
A spin-polarized version of a “Dirac cone” occurs in “topological insulators”. These are insulators in the bulk and metals at the surface, because two surface bands bridge the bulk band gap. It is impossible topologically to remove the surface bands from the gap, because they are tied to the valence band on one side and to the conduction band on the other.
The metallic surface state bands have been measured by angle- and spin-resolved photoemission.