Cohesive energy is the amount of energy (normalized per atom, per electron, per unit cell, etc., so that it is intensive) that a system gains by being in a solid phase (with atoms at their equilibrium positions) in comparison with the phase in which atoms are so far apart that their electronic wavefunctions do not overlap. The binding energy is therefore a measure of how electronic wavefunctions deform with respect to what they are in the atomic state as atoms form solids. Assuming that band picture is valid, band gap is a characteristic of a limited set of k points of the Brillouin zone of the system (always of measure zero in the thermodynamic limit with respect to the totality of k points). The question arises as to the contribution of the states at these points (or better, in the neighbourhood of these points) to the cohesive energy. If there is only minor change in these states with respect to what they are in the atomic state, then naturally there is little correlation between the band gap and the cohesive energy. This is exactly what you seem to have observed in your calculations. One extreme example concerns a system comprised of closed-shell atoms. For these systems to leading order the ground state of the solid consists of the ground-state configurations of the underlying atoms. Clearly, here to leading order excitation energies (including the value of the band gap) do not come into view at all. In reality they come, however theoretically the contribution of the atomic excited-states configurations can be minimized by increasing the gap energies in the atomic states.

In The name Of GOD

Hi!

I think we are discussing about two subjects. Consider an empty space. It has maximum symmetry. Now, assign a geometry, such Diamond lattice (i.e., two FCC lattice),  to the space. Its symmetry decreases. Then assign Ga to one FCC and As to another FCC lattice. For doing that, i.e., for putting the atoms in the points of space, you should consume energy. In the other words you should bring the atoms from infinity to the specific point of space and put them beside each others and for doing that, you should consume energy. The energy is stored in lattice, which is called cohesive energy. Finally you have GaAs lattice.

Now, please put Nitrogen instead of As in the lattice . You will have GaN lattice. Ga, As, and N are different kinds of elements with different molecular mass, electron configuration and energy eigenvalues (levels) and therefore you will see different cohesive energy.

Also, since the energy levels of As-element is different from N-element and interaction between Ga and As is different from Ga and N you will find different energy dispersion relation , E(k). If you note to the existing differences between two E(k)s, you can see their band gaps is not same.

Therefore although both of them have Diamond lattice structure, but they have not similar cohesive energy, E(k), and band gap energy.

Finally, depends on the assigned geometry to space, type of elements which are assigned to the geometry and the electrical properties of elements and interaction between elements, you may see different properties.

Best Regards

H.Simchi

Hi,

You can read the paper of Akira MORITA about Theory of Cohesive Energies and Energy-Band Structures. http://ptp.oxfordjournals.org/content/19/5/534.full.pdf

Essentially, the band gap is the minimum energy needed to move an electron from a bound non-conducting state (i.e. the "valence band") to a "free" conducting state (the "conduction band") --- the ability to have charge move/not move in the conduction/valence bands is strictly true only at zero kelvin for insulators and intrinsic semiconductors. But an electron in a conduction band is still within the solid. The specifics of the band gap depend of course on the crystal structure, and the elements and/or molecules in the basis of that structure.

The cohesive energy is the energy you need to completely dissociate the solid into its neutral atomic components (see Charles Kittel, "Introduction to Solid State Physics", John Wiley & Sons, Inc. the second paragraph of the introduction to Chapter 3 in the 6th or 7th edition, which are the copies I have) --- i.e. the whole lot no longer being a solid but pretty much a rarefied gas of neutral atoms.

These two energies, although they pertain to a specific compound or element, they refer to completely different characteristics of the material, so they are at best very tenuously related.

Put in other words, the exact same molecule when "forced" (e.g. by applying high pressures) to adopt a different crystal lattice than the one formed at NTP (normal temperatures and pressures), IF it is still an insulator or semiconductor it will have a different band gap and most likely a different cohesive energy also, but their relationship is not expected to be very well correlated.

Band gap is proportional to bond energy at equilibrium;

Cohesive energy is the sum of bond energy over all coordinates of an atom.