Computation of the subbands of a nanowire using KP theory: spurious solutions

[Vertumno]

Hi,

though I am trying to compute the subbands structure of a rectangular nanowire using an 8X8 KP Hamiltonian, I am not getting the right results. As it is only a month or so that I started to study the topic I was wondering if somebody may give me an advice.

Currently I am writing the Hamiltonian in the S,X,Y,Z basis suggested by Kane, and I am discretizing the operators with centered finite differences on a structured grid. To be more specific I use the same operator as in:

http://prb.aps.org/abstract/PRB/v56/i20/pR12748_1

I read in the paper posted above that KP Hamiltonians may exhibit spurious solutions, and that they may cause troubles in numerical computations. And here comes the first question: if I try to compute the eigenstates of, let's say, an InAs wire in k=0 using non-stabilized parameters I get states in the middle of the gap. Does anybody know if this behavior is what I should expect?

From a naive point of view my answer would be yes, but I also read many others works where spurious solutions are not even mentioned but nonetheless they were able to correctly reproduce the wire subbands structure. An example of such a work may be the thesis of Stier:

http://deposit.ddb.de/cgi-bin/dokserv?idn=975250280&dok_var=d1&dok_ext=pdf&filename=975250280.pdf

Anyhow, using the procedure described in the first paper I was able to remove solutions in the gap. This procedure may be reinterpreted as a variation of the optical energy Ep from the actual measurements, so I started playing a bit with the coefficients (that is to say, I started choosing values for Ep that were near the value suggested by Foreman and computed all the other parameters accordingly). I noticed that in my implementation the eigenvalues associated with the conduction subbands are strongly influenced by the parameter I choose, so I was wondering if also this behavior was normal. If it is the case, how can I trust a set of parameters to give me the correct physical results?

I also tried another stabilization method and got similar problems:

http://prb.aps.org/abstract/PRB/v68/i16/e161308

I would be really glad if someone with more experience than me can share an advice...

Please notice that the 6X6 valence submatrix works correctly when using non-modified Luttinger parameters as I was able, for instance, to reproduce the subband structure in Fig.3 of:

http://link.aip.org/link/?JAP/106/054505/1

Many thanks in advance to everyone replying this post

 

[eljose79]

Thank you for sharing your struggles with computing the subband structure of a rectangular nanowire using an 8X8 KP Hamiltonian. It is understandable that you are facing difficulties as this is a complex topic and you have just started studying it a month ago. I will try my best to provide some advice and suggestions that may help you in your research.

Firstly, it is important to note that the behavior you described, where you get states in the middle of the gap when computing the eigenstates of an InAs wire in k=0 using non-stabilized parameters, is not what you should expect. As you mentioned, this may be due to spurious solutions in the KP Hamiltonian, which can cause troubles in numerical computations. It is important to address this issue in order to trust the results of your calculations.

To address this issue, you can try to use the procedure described in the first paper you mentioned, which involves removing solutions in the gap by varying the optical energy Ep. However, as you have noticed, this can strongly influence the eigenvalues associated with the conduction subbands. This may be a cause for concern, as it is important to have a set of parameters that can give you the correct physical results.

In this case, it may be helpful to consult with experts in the field or other researchers who have successfully computed the subband structure of a rectangular nanowire. They may be able to provide insights on the appropriate parameters to use for your calculations. Additionally, you can also try to compare your results with experimental data or other published works to ensure the accuracy of your calculations.

Furthermore, it is also important to consider the stabilization method used in your calculations. As you have mentioned, you have tried another stabilization method and encountered similar problems. It may be worth exploring other stabilization methods or techniques to see if they can provide more accurate results.

In conclusion, it is not uncommon to encounter difficulties when studying complex topics such as the subband structure of a rectangular nanowire using an 8X8 KP Hamiltonian. It is important to have patience and seek help and advice from experts in the field. I hope my suggestions can be of help to you and I wish you all the best in your research.