Using computer algebra to diagonalize some Kane matrices
Claude-Pierre Jeannerod, Nicolas Maillard · Journal of Physics A Mathematical and General · 2000
In semiconductor theory, applying the kp -method to the monodimensional Schrödinger equation leads to a symmetric perturbed eigenvalue problem (Kane E O 1967 The kp method Semiconductors and Semimetals (New York: Academic) p 75), i.e. to the diagonalization of a matrix A ( ) depending on a small parameter , symmetric . The eigenelements of A ( ) admit expansions in fractional powers of (Puiseux series). Usually, physicists solve this problem by using Schrödinger perturbation formulae under some restrictive conditions, which make perturbed eigenvector symbolic approximation impossible. This is illustrated by the modified Kane matrix (Fishman G 1997 Quasi-cube et Würtzite: Application au GaN (Montpellier: École thématique du CNRS)). To solve this problem completely from a symbolic computing point of view, we consider the symmetric perturbed eigenvalue problem in the case of analytic perturbations (Baumgärtel H 1985 Analytic Perturbation Theory for Matrices and Operators (Basle: Birkhauser), Kato T 1980 Perturbation Theory For Linear Operators (Berlin: Springer)). We first review the classical characteristic polynomial approach, showing why it may be not optimal. We also present a direct matricial algorithm (Jeannerod C P and Pflügel E 1999 Int. Symp. on Symbolic and Algebraic Computation (Vancouver, Canada, July 1999) (New York: ACM) pp 121-8): transforming the analytic matrix A ( ) into its so-called q -reduced form allows to recover the information we need for the eigenvalues. This alternative method, as well as the classical one, can be described in terms of the Newton polygon. However, our approach uses only a finite number of terms of A ( ) and is more suitable for large matrices and a low approximation order. Besides, we show that the q -reduction process can simultaneously provide symbolic approximations of both the perturbed eigenvalues and eigenvectors. The implementation of this algorithm in MAPLE is used to diagonalize the modified Kane matrix up to a given order.