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

• dnvlgm
In summary, the conversation revolves around the calculation of electronic bands in solids, specifically in semiconductors. The numbers 2, 3, and 6 refer to the number of electronic bands in a solid, with 2 representing the conduction band, 4 representing the valence bands (light and heavy hole), and 6 representing either the valence bands (light, heavy, and spin-orbit) or the conduction bands with light and heavy hole bands. The more bands used, the better the approximation to a real solid. The conversation also delves into the calculation of band gaps and the neglect of electron-hole interactions in an exciton state. There is also discussion about the coupling of Wannier and Fren
dnvlgm
Can anybody tell me what they mean by 2, 3, 6 or any band hamiltonians. What does it even mean?

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.

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

Yexciton=yelectron*yhole

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

yelectron=F(relectron,phi,z)

yhole=F(rhole,phi,z)*Gmj*u

where Gmj 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.

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.

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=AnlJl(bnlr)exp(-i*l*phy)*Sin(kz)

where Anl is the normalization constant, Jl(bnlr) is the l thorder bessel function and bnl 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.

You lost me here:
dnvlgm said:
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

Yexciton=yelectron*yhole
How do you get to neglect the electron-hole interaction when you are dealing with an exciton?

Gokul43201 said:
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

dnvlgm said:
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?

Gokul43201 said:
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?

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

Gokul43201 said:
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

## What is a 2-band Hamiltonian?

A 2-band Hamiltonian is a mathematical model used in quantum mechanics to describe the energy levels and behavior of a system with two energy bands. It takes into account the interactions between particles and the external potential energy.

## What is a 3-band Hamiltonian?

A 3-band Hamiltonian is a more complex version of a 2-band Hamiltonian that includes a third energy band. This additional band allows for a more accurate description of the system's behavior, particularly in cases where the two bands may overlap.

## What is a 6-band Hamiltonian?

A 6-band Hamiltonian is a further extension of the 3-band Hamiltonian, including a total of six energy bands. This type of Hamiltonian is often used in more complex systems, such as semiconductors, to accurately describe the interactions between electrons and holes in the material.

## What is the purpose of using a Hamiltonian?

The Hamiltonian is a fundamental concept in quantum mechanics that allows scientists to accurately describe the energy levels and behavior of a system. It is used to calculate the time evolution of a quantum system and is essential in understanding the dynamics of particles and their interactions.

## How are the bands in a Hamiltonian determined?

The bands in a Hamiltonian are determined by the energy levels of the particles in the system and the external potential energy. The mathematical equations used to describe the interactions between particles and their energy levels are based on principles of quantum mechanics and are solved to determine the bands in the Hamiltonian.

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