Yukawa model of screening in low-dimensional excitons: diagonalization, perturbation, variation, and resummation analysis

Abstract

The Yukawa interaction describes Coulomb screening in several physical systems. We investigate bound quantum states of the Yukawa potential in low-dimensional structures as a model of e.g. excitons in semiconductor nanostructures. Diagonalization, perturbation, variation, and resummation methods are all applied to the problem and their accuracy is compared. For moderate positive screening, all methods are found to be applicable and, in particular, the variational approach is highly accurate except near the threshold for bound states. In contrast, only hypergeometric resummation captures the correct behaviour in the negative (antiscreened) regime. We also determine the dimensional dependence of the critical screening, above which bound states cease to exist. As an application, the critical screening is used to compute critical doping levels for the existence of bound excitons in quantum wells.

Critical screening parameters and critical behaviors of one-electron systems with screened Coulomb potentials

“…They obtained the most accurate predictions of the critical values in the literature for states with n 5 by the combination of two methods, i.e., the perturbation theory for s-wave states and the Lagrange mesh method for non-s-wave states. Recent work of Pedersen [6] focused on the dimensional dependence of the critical screening parameter and presented an analytical estimate of λ c (1s) by applying the Padé approximation. The author predicted this interesting phenomenon, indicating that the critical screening parameter decreases monotonically with the increasing dimensionality of the system.…”

“…to their close relationship with the symmetry breaking of electronic structure configurations, quantum phase transitions, the threshold behavior in atomic and molecular scatterings, level crossings, and molecule dissociation [1][2][3][4][5][6][7]. The variation of the energy levels of a given system as one varies the parameters in the corresponding Hamiltonian is particularly interesting near the binding threshold.…”

“…These authors have strictly and numerically proved the validity of virial and Hellmann-Feynman theorems in both one-and two-electron screened atoms, and also paid special attention to the detachment of two electrons by simultaneously varying the nuclear charge and screening parameter. Keeping in mind that critical phenomena in short-range potentials depend explicitly on the physical dimension of the system, as discussed in the celebrated work of Klaus and Simon [1] and Kais and Serra [3], the dimensional dependence of the critical screening parameters in SCP has recently been investigated by Pedersen [6]. The obtained results are shown to be very useful in the computation of critical doping levels for the existence of bound excitons in quantum wells.…”