2-band, 3-band, 6-band Hamiltonian

[dnvlgm]

Can anybody tell me what they mean by 2, 3, 6 or any band hamiltonians. What does it even mean?

 

[Dr Transport]

The numbers refer to the number of electronic bands in a solid, 2 bands usually means just the conduction band, 4 bands the valence bands (light and heavy hole), 6 bands can mean either the valence bands (light, heavy and spin-orbit) or the conduction bands with the light and heavy hole bands.

For example, silicon and germanium can be modeled with 6 bands, GaAs needs 8. The more bands you use, the better the applroximation to a real solid you get.

 

[dnvlgm]

So for instance i have an excitonic state in piece of semiconductor. It occurs to me that the total wavefunction Y would be the product of the individual wavefunctions of the electron and the hole, or

Y_exciton=y_electron*y_hole

and (lets say the semiconductor is a nanorod) in cylindrical coordinates

y_electron=F(r_electron,phi,z)

y_hole=F(r_hole,phi,z)*G_mj*u

where G_mj is the angular momentum coupling part and u are the Wannier functions. Assuming this is a correct initial rough approximation, how many bands are we talking about here? Does my question even make any sense? I'd really appreciate your help.

 

[sirchasm]

I'm fairly sure you can't calculate band gaps with a single electron-hole exchange; bands in solids like semiconductors are a function of long-range order; you have to derive the geometry, the minimal lattice and its shape and modes; there's materials science, molecular bonding in solids, electronics/solid-state, there's a lot that you can look into.

But it's a function of iterated small-scale order to long-range effects.

 

[dnvlgm]

ok i guess i didnt give enough detail. The exciton is a Wannier-Mott exciton with Bohr radius much larger than the lattice constant. The geometry is considered within the boundary conditions of the electron wavefunction, such that for a cylinder of length L~300Amstrong we have

y=A_nlJ_l(b_nlr)exp(-i*l*phy)*Sin(kz)

where A_nl is the normalization constant, J_l(b_nlr) is the l thorder bessel function and b_nl is the n-th zero to the l-th bessel function. All I am asking is how many bands we have here if the hole has the same form and also includes the 3j Wiegner symbol (angular momentum coupling) and Wannier periodic functions.

 

[Gokul43201]

You lost me here:

So for instance i have an excitonic state in piece of semiconductor. It occurs to me that the total wavefunction Y would be the product of the individual wavefunctions of the electron and the hole, or

Y_exciton=y_electron*y_hole

How do you get to neglect the electron-hole interaction when you are dealing with an exciton?

 

[dnvlgm]

You lost me hereow do you get to neglect the electron-hole interaction when you are dealing with an exciton?

You do realize this is a Wannier-Mott exciton we are talking about, right? If you look at a later post I mentioned the rod's length is 300amstrong which means it is delocalized and we can neglect the coulomb interaction, for its Bohr radius is way larger than the lattice constant

 

[Gokul43201]

You do realize this is a Wannier-Mott exciton we are talking about, right? If you look at a later post I mentioned the rod's length is 300amstrong which means it is delocalized and we can neglect the coulomb interaction, for its Bohr radius is way larger than the lattice constant

How do you get the physics of a bound state if you neglect the interaction? Even in a Wannier-Mott exciton, the binding energy is typically ~100meV. What do you figure is the KE of a free electron or hole at room temperature in your material?

 

[dnvlgm]

How do you get the physics of a bound state if you neglect the interaction? Even in a Wannier-Mott exciton, the binding energy is typically ~100meV. What do you figure is the KE of a free electron or hole at room temperature in your material?

Whoa touche! that's a good point which might throw away my basic understanding of the problem. It seemed to me that I could do a rough but good enough estimate by accounting for the coupling of angular momentum but now it doesn't seem that way... what do you recommend? maybe I should not neglect the coulombic potential on the Hamiltonian but in this case I'm afraid I'm going to get something very nasty, what do you think?

 

[Gokul43201]

What exactly is it you want to achieve? That is not particularly clear. What are you trying to calculate?

 

[dnvlgm]

What exactly is it you want to achieve? That is not particularly clear. What are you trying to calculate?

Electronic structure... The idea is to obtain the wavefunction of an exciton inside a nanorod which is covered by a thin organic layer. What I expect to see is a coupling (through excitonic resonance) of the Wannier (in the nanorod) and Frenkel (in the organic molecules) excitons. The idea is that this hybrid exciton would have the properties of both (oscillator strengths, optical transitions...) since their characteristics are pretty much complementary: the weaknesses of one are the strengths of the other. Now, the coupling part appears to be easy, but I am having a hard time not only figuring out the correct form for the wavefunction of the exciton in the nanorod, but also with my fundamental understanding of the theory. So if you have any suggestions it would be nice