What is the relation between lattice constant and energy gap of the magnetic or non magnetic materials? Is any formula for that "a" and "Eg"?
[A]Eg is band gap and a [ in fact a bold] is the lattice constant.
[B]The two are related as:
Eg ----------- ħ ^2 π^2 /2m*a(bold)
m *= associative effective mass
m∗ = mU/2Λ
U = potential.
Eg decreases with increasing [a(bold)] value.
[C] The lattice constant(a bold) depends upon Six parameters [three sides(a,b,c) and three angles(α,β,γ ) which have different values in 7 types of crystals having 14 unit cells).It is experimentally determined either by Bragg’s or X-Xay differaction or atomic force microscope*.
*Atomic force microscopy or scanning force microscopy is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit.
Oh how I wish that condensed matter physics were as simple as that! To calculate band gap (if your E_g means that) we need to calculate the electronic structure. That needs more information than just "lattice constant a". Your lattice parameter a does not even describe unit cell for most of the materials we encounter in practice. It is valid only for cubic substance. I recommend that you read a bit of solid state physics for which there are plenty of text books available.
see ..
if you are changing the lattice constants (a1, a2, a3) in real space .. the reciprocal constants will also be change .. (b1,b2,b3)..
..
now think over brillioun zone schemes ..the limit of BZs depends upon lattice constants ..
ie ..zone scheme will be different now ..and that will change Eg also ..
I am looking for a copy of Hubbard I paper. Can anyone provide me a pdf file?
Dear Behnam, thanks a lot. This is already quite formidable! I know that there exist other five such Hubbard Proc. Roy. Soc. papers!! Are they also important? Do you have them?
Please send me privately to the address [email protected]
Well Anderson had everything: Nobel prize and a long life. Poor Hubbard died at the age of 49 and never had any prize at all for his work!! I read that Hubbard did not at first wish to publish his work knowing that others have done the same thing by other methods and got the same results. His boss at Harwell, Flowers came to know about this work, ordered him to publish it. Then came all these six Hubbard paper.
Constant of the lattice represents the distance between molecules or atoms which construct the lattice. So it is logical considering the constant of the lattice as a important parameter for energy levels and energy gaps for the lattice. Of course, in my mind there is no simple analytical formula for it. But we could calculate Eg by simulation of the lattice using its constants.
I agree fully with T. Chatterji. Upon calculating the band structure, the cell parameters (not only a) appear in the dispersion relation in a complicated manner, even for simple structures such as bcc.
While I agree that the band gap E_g depends on a variety of things besides the lattice constant a, it does stand to reason that the gap would usually increase as a decreases, possibly as 1/a^2 in many cases, simply because the zone boundary (in the direction a is measured) will occur at p=h/a and most band structure related energies at the zone boundary will scale like p^2/2m. But this would only be a "trend", not a "rule". (And I could be completely wrong; I'm an experimentalist! :-)
Generally, for a familiy of semiconductors the optical energy gap is proportional to the molecular weigth and then to the volume.
@ Amodini
Your relation is not even dimensionally correct. Where from did you get this relation for energy gap.
Dear Saravanan,
At first i would like to congratulate you on the number of letter a contained in your name.
welcome,
May i came very late to this forum. However as it is a basic question i will give a simplified qualitative answer to it that may be useful.
It is so as the lattice constant decreases the interatomic distance will be reduced. As a consequence binding forces between the valence electrons and and the parent atoms will increase. These valence electrons will occupy the valence band. Since the valence electrons are bound, they have to be supplied with energy to make them moving free inside the material and become conduction electrons. The minimum energy that must be given to valence electrons to become a conduction electron is the energy gap.
So, as the valence electrons get more bound by decreasing the interatomic distance, the more energy required to make them free in the conduction band.
So, A direct consequence of decreasing the lattice constant is the increase in the energy gap. As a rule of thumb the energy gap is inversely proportional to the interatomic distance.
An other factor affecting the energy gap is the dielectric constant, which depends on the density of atoms and their polarizability. The dielectric constant is proportional to N the density of atoms per cm^3 and the alpha the polarizability. The polarizability depends on the electronic structure of the atom. It is so that the energy gap is inversely proportional to the dielectric constant which in turn is inversely proportional to the inter-atomic distance.
Therefore, there are to competing effects to the lattice constant on the energy gap.
the binding energy and the dielectric screening.
Best wishes
ya. Actually i should thank to you professor for your nice answer. Thank you so much for all.
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