In the band structure of crystals the narrow or wide valence or conduction band stand for what? Is there some information about interactions between orbitals of elements in crystals?
Dear Beenish,
I think it's helpful to think about two different limits for the particles in your crystal: the tightly-bound limit; and the nearly-free limit.
a) In the tightly-bound limit, the behaviour of the particles is dominated by the potential energy. This ultimately means that the band eigenvalues become independent of k, so you get the same answer for all k. This leads to a band-structure where the bands are all completely horizontal -- in other words, they are "energy levels", exactly as you get for isolated systems (e.g. the famous Balmer series for atomic hydrogen).
In general, flat "dispersion-less" bands occur when the particles are localised.
b) In the nearly-free limit, the particles' behaviour is dominated by the kinetic energy, then we can neglect the potential energy. The eigenstates of the kinetic energy are just plane-waves of the form exp(iq.r), where q is an arbitrary wavevector and the energy depends simply on the second derivative -- i.e. |q|^2.
In a periodic lattice the translational symmetry means it is convenient to write q in terms of a reciprocal lattice vector G (which can be *any* reciprocal lattice vector, not just a primitive one) and the "remainder" k, which will always lie in the first Brillouin zone. Thus the energy may be written in terms of |G+k|^2 and so the bands form parabolae centred on the reciprocal lattice vector G (and the translational symmetry in reciprocal space means there are identical parabolae centred on each and every G).
Thus wide, parabolic-like bands indicate nearly-free-particle-like behaviour.
Plain text isn't very good for expressing the equations which underlie this qualitative description, but you might find my lecture notes from a DFT workshop helpful (attached at the bottom of this post).
Hope that helps,
Phil Hasnip
(CASTEP developer)
Hello Sir, Thanks for your comments. Its really helpful for my further research.
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