# KP Theory Bandstructure Calculations

1. Apr 27, 2006

### Modey3

Hello,

I was wondering if anybody has any experience doing these calculations? I understand the basic concepts ( eg. k is considered far away from the BZ edge and thus is small and that the Bloch function is the linear combination of the individual Bloch functions at k=0 ). Also, the inner products are done over the primative unit cell. These individual band Bloch functions are constructed by a basis of S and P orbitals.

My question is how do we take into account he geometry of the system. Since the Hamiltonian matrix elements are taken just over the primative unit cell how do we figure in the size effects for quantum dot calculations ? Thanks.

Regards

Modey3

2. Apr 27, 2006

### Dr Transport

Remember that in a quantum well, the individual bands are separated due the confinement, i.e., the degeneracy of the heavy-hole, light-hole and spin-orbit bands is removed so we have non-degenerate energy levels. I have to think about how the system is changed when we get higher levels of confinement as in a quantum-dot. Look at Chuang's book, Physics of Optoelectronic Devices, the first couple of chapters will help. Also Bastard's book, Wave Mechanincs Applied to Semiconductor Heterostructures will also be of use although it is advanced.

One thing you'll have to remember, in calculating the wave functions for a quantum well, the effective mass is not constant, and you have to match wave functions at the boundaries. Thus making the calculation more difficult. Madarsz and Szmulowiscz did some really fine work back in the late 80's to mid 90's on graded well structures which can give you an idea where to go. I'm sure that there has been alot of work done lately on this subject.

I'll have to do some poking around.

3. Apr 27, 2006

### Gokul43201

Staff Emeritus

In a 2DEG (quantum well), the square well confining potential V(z), gives rise to corresponding wavefunctions along z. This seems to be a not-at-all terrible approximation. The x- and y- terms in the wavefunction are still Bloch states. In a quantum wire, you have Bloch states along only x. And in a quantum dot, the wavefunction looks nothing like a Bloch state in any direction. I would imagine it looked more like the particle in a (finite) box states.

So based on that, it would seem to me that you can't use the k.p approximation* for quantum dots.

PS : Shamefully, I may be completely out of the ballpark here. Please let me know if I am.

* Edit : I'm only aware of the k.p approximation in the context you've described above, ie: when dealing with Bloch states. This would imply that you are using the boundary conditions that give rise to the Bloch states (ie: Born von Karman). It's possible to have other forms of the wavefunction where you end up with a k.p term in the SE. My conclusion doesn't apply to any such case.

Last edited: Apr 27, 2006
4. Apr 28, 2006

### Modey3

Hello,

This stuff is very new to me also (never heard of KP THeory until 3 weeks ago). Nontheless, my area of study is Computational Materials Science so I should have some familiarity with the semi-empirical methods of band structure calculations. My group has just done DFT calculations to study the atomic structure of quantum dots, but unfortunatley DFT predicts band gaps that are larger than experimentally measured. Predicting the stable atomic structure is just one step in nanostructure design. Bandstructure properties also need to be determined for those stable structures. Thanks.

Best Regards

Modey3

5. Apr 28, 2006

### Dr Transport

www.sst.nrel.gov/images/MRS98-ES%20of%20Quantum%20Dots.pdf[/URL]

Last edited by a moderator: Apr 22, 2017
6. May 1, 2006

### Modey3

Hello,

Thanks Dr Transport, but that article isn't strictly an article on KP Theory. It's more about an alternative to KP Theory approaches to bandstructure calcs. However, this article does mention that the KP parameters are a function of crystal size and this being a a big drawback to KP Theory approaches. I wonder if there is a correlation between the parameters and the particle size of the quantum dot. Thanks

Regards

Modey3

7. May 5, 2006

### Dr Transport

I remember adjusting the Luttinger parameters in my $$\vec{k} \cdot \vec{p}$$ calculation if that is what you mean by adjusting the parameters, but the adjustment wasn't dependent on the crystal size but the direct band gap of the material.