Hi Dr M.A. Hadi,

I think it is better if I add something. My preceeding answer was a bit uncomplete.

If you need to know the strength of interatomic bonds (Eb) of a crystal from the DOS then you need first of all to derive the fundamental Eg energy bandgap (smallest one) because it is this energy which reflects more or less the bonding strength Eg=Eb.

To determine the energy bandgap from the DOS calculted e.g. from ab initio computations then you need to know where are your antibonding conduction band states and where are your bonding valence band states.

Sometimes it is given directly by the software including the value of the energy gap.

If not then you need to compute the energy band structure and then it is easy to determine electron-like states with positive E(k) energy dispersion (positive second derivative of the E(k) energy dispersion as a function of wave vector k or a parabola directed upwards in energies) from hole-like states with the reverse situation means negative E(k) energy dispersion (negative second derivative of the E(k) energy dispersion as a function of wave vector k or a parabola directed downwards in energies).

Then all the E(k) energy curves showing electron-like behavior constitute the conduction band, and the E(k) curves showing hole-like behavior constitute the valence band.

Then in the Brillouin zone, it is easy to find the highest valence states (means the highest maximum) and the lowest conduction states (the lowest minimum) and the energy difference between these two points is the fundamental energy bandgap Eg you are looking for.

To be sure, you must plot then the E(k) energy band structure and the DOS on the same diagram with E(k) on a left diagram and E(N) where N is the density of states on a right diagram. Both diagrams have same ordinate as the energy and different abscissas: the abscissa is k (Brillouion zone wave vector) in the case of E(k) variations and it is the DOS N in the case of E(N) variations.

Once done, it is easy to see that energy bandgap in the energy band structure (left diagram) is almost exactly the same as for the DOS on right diagram. This is the proof that you derive correctly the fundamental energy bandgap Eg.

Now as to the shape of DOS as a function of energy.

There are usually several peaks in the conduction band and several other peaks in the valence band with weighted contribution of s, p, d and f states (if any).

- The |s> states are of spherical symmetry and are delocalized states in real space. Correponding wavefunctions are strongly localized in reciprocal k space.

The DOS peak corresponding to such states will be rather broadened (large width and small peak) because of strong ovelap of neighboring s-states.

- Conversly the |d> or |f> states are strongly localized in real space i.e. strongly delocalized in k-space, and corresponding DOS peaks will be rather narrow (very thin and long) because of very small overlap of neighboring such d- or f-states.

- the |p> states are intermediate between s- and d- states in terms of localization but closer to s- than to d-states.

In the simplest materials, usually:

- The |s> states give the dominant contribution to antibonding conduction states and the lowest conduction band DOS peak is rather a smeared peak (broadned peak).

-The |p> states give the dominant contribution to bonding valence states and the highest valence band DOS peak is a bit less broadened than the conduction band one.

However, this simple picture which is the fundamental one may change with some new features as strong hybridization of sp states with each other and with |d> and |f> states on the other hand.

This hold in covalent materials with increasing energy bandgaps with mismatched atoms (a mixture of heavy and light atoms) .

This is even more the case in insulating materials with large energy bandgaps and ionic bondings.

I hope this could help you.

It is long but I could not do it shorter.

Best regards,

Professor A. Kadri

Hi Dr M.A. Hadi,

- The position of the peak of the DOS gives the information about the energy involved in the studied process eg. here the bonding strength,

- The height of the peak gives the information about the number of states involved in the DOS in the Dirac peak limit (i.e.highly resolved peak with no broadening effects due to scattering or disorder, or inhomogeneities),

- When the peak is broadened, at each energy there is a finite number of states contributing and the total number of states is found by summing over all energies, i.e. by the surface derived from integrating the DOS curves over the energies,

- The peak/width ratio gives a measure of the broadening, or the resolution of the peak i.e. how much scattering, or disorder, or inhomogeneities are contributing.

I hope this could help;

Eid El Fitr Moubarak once again,

Best regards,

Prof. A. Kadri

Dear Prof. Dr. Abd,errahmane Kadri,

Thank you very much for your nice and fruitful discussions.

Eid mobarok and stay safe.

Best regards,

M.A. Hadi

You are welcome,

Eid Mabrouk and stay safe too.

Best regards,

A. Kadri

Hi Dr M.A. Hadi,

I think it is better if I add something. My preceeding answer was a bit uncomplete.

If you need to know the strength of interatomic bonds (Eb) of a crystal from the DOS then you need first of all to derive the fundamental Eg energy bandgap (smallest one) because it is this energy which reflects more or less the bonding strength Eg=Eb.

To determine the energy bandgap from the DOS calculted e.g. from ab initio computations then you need to know where are your antibonding conduction band states and where are your bonding valence band states.

Sometimes it is given directly by the software including the value of the energy gap.

If not then you need to compute the energy band structure and then it is easy to determine electron-like states with positive E(k) energy dispersion (positive second derivative of the E(k) energy dispersion as a function of wave vector k or a parabola directed upwards in energies) from hole-like states with the reverse situation means negative E(k) energy dispersion (negative second derivative of the E(k) energy dispersion as a function of wave vector k or a parabola directed downwards in energies).

Then all the E(k) energy curves showing electron-like behavior constitute the conduction band, and the E(k) curves showing hole-like behavior constitute the valence band.

Then in the Brillouin zone, it is easy to find the highest valence states (means the highest maximum) and the lowest conduction states (the lowest minimum) and the energy difference between these two points is the fundamental energy bandgap Eg you are looking for.

To be sure, you must plot then the E(k) energy band structure and the DOS on the same diagram with E(k) on a left diagram and E(N) where N is the density of states on a right diagram. Both diagrams have same ordinate as the energy and different abscissas: the abscissa is k (Brillouion zone wave vector) in the case of E(k) variations and it is the DOS N in the case of E(N) variations.

Once done, it is easy to see that energy bandgap in the energy band structure (left diagram) is almost exactly the same as for the DOS on right diagram. This is the proof that you derive correctly the fundamental energy bandgap Eg.

Now as to the shape of DOS as a function of energy.

There are usually several peaks in the conduction band and several other peaks in the valence band with weighted contribution of s, p, d and f states (if any).

- The |s> states are of spherical symmetry and are delocalized states in real space. Correponding wavefunctions are strongly localized in reciprocal k space.

The DOS peak corresponding to such states will be rather broadened (large width and small peak) because of strong ovelap of neighboring s-states.

- Conversly the |d> or |f> states are strongly localized in real space i.e. strongly delocalized in k-space, and corresponding DOS peaks will be rather narrow (very thin and long) because of very small overlap of neighboring such d- or f-states.

- the |p> states are intermediate between s- and d- states in terms of localization but closer to s- than to d-states.

In the simplest materials, usually:

- The |s> states give the dominant contribution to antibonding conduction states and the lowest conduction band DOS peak is rather a smeared peak (broadned peak).

-The |p> states give the dominant contribution to bonding valence states and the highest valence band DOS peak is a bit less broadened than the conduction band one.

However, this simple picture which is the fundamental one may change with some new features as strong hybridization of sp states with each other and with |d> and |f> states on the other hand.

This hold in covalent materials with increasing energy bandgaps with mismatched atoms (a mixture of heavy and light atoms) .

This is even more the case in insulating materials with large energy bandgaps and ionic bondings.

I hope this could help you.

It is long but I could not do it shorter.

Best regards,

Professor A. Kadri

Dear Prof. Dr. Abderrahmane Kadri

Thank you very much. This is a very excellent discussion. I hope it will be helpful for a large number of researchers in the RG platform. Thanks again.

Best regards,

M.A. Hadi