There are different numerical methods to solve the k.p Hamiltonian for multi quantum well structures such as the ultimate method which is based on a quadrature method (e.g. doi: 10.1016/0039-6028(94)90904-0), or one based on a finite-different method (e.g. doi: 10.1063/1.342118). In both, so-called spurious solutions can emerge and different methods exist to eliminate them (e.g. doi: 10.1063/1.3689821). However, I am searching for a comprehensive overview of the different numerical models and solutions regarding the spurious solutions to find advice on which numerical model and solution should be preferred.
Adding to the question, how to handle tunneling modes and modes which pop-off from the well on applying external field, bending the potential profile. I looked for tunneling and unconfined modes by integrating the wavefunction square in well and surrounding regions while studying QCSE using effective mass model. In another attempt, I restricted myself to lowest two solutions which would be the only significant solutions while operating away from band-edge. Even that works only for deep wells. Shallow wells will have eigen values popping-out and causing steps in calculated refractive indices or absorption coefficients while band-bending for ex. under external field. Using the continuum solutions is often recommended, I wish to avoid that.
Regarding spurious solutions in kp, related to the lack of ellipticity that some sets of coefficients provide to the coupled differential equations (i.e. to the Hamiltonian) see e.g. Ratko G. Veprek, Sebastian Steiger, and Bernd Witzigmann, Phys. Rev. B 76 (2007) 165320.
Dear Phillips,
As far as quantum well is concern the fundamental finite difference scheme is useful. Once we involve the non-parabolicity factor, mismatches in material parameters and some peculier dielectric functions, once can get apparent answers from it.
The problem which i am facing is the grid points setting. It directly affects the results over few magnitude.
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