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

1. Jun 14, 2010

### 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 [Broken]

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: