Self-consistent computation of electronic and optical properties of a single exciton in a spherical quantum dot via matrix diagonalization method

Cataloged from PDF version of article.In this study, we develop and demonstrate an efficient self-consistent calculation schema that computes the electronic structure and optical properties of a single exciton in a spherical quantum dot (QD) with an interacting pair of electron and hole wave functions. To observe modifications on bands, wave functions, and energies due to the attractive Coulomb potential, the full numeric matrix diagonalization technique is employed to determine sublevel energy eigenvalues and their wave functions in effective mass approximation. This treatment allows to observe that the conduction and valance band edges bend, that the electron and hole wave functions strongly localize in the QD, and that the excitonic energy level exhibits redshift. In our approach for the Coulomb term between electron and hole, the Poisson-Schrodinger equations are solved self-consistently in the Hartree approximation. Subsequently, exciton binding energies and associated optical properties are computed. The results are presented as a function of QD radii and photon energies. We conclude that all of these numerical results are in agreement with the experimental studies. (C) 2009 American Institute of Physics

Optical Properties in a ZnS/CdS/ZnS Core/Shell/Shell Spherical Quantum Dot: Electric and Magnetic Field and Donor Impurity Effects

A theoretical analysis of optical properties in a ZnS/CdS/ZnS core/shell/shell spherical quantum dot was carried out within the effective mass approximation. The corresponding Schrödinger equation was solved using the finite element method via the 2D axis-symmetric module of COMSOL-Multiphysics software. Calculations included variations of internal dot radius, the application of electric and magnetic fields (both oriented along z-direction), as well as the presence of on-center donor impurity. Reported optical properties are the absorption and relative refractive index change coefficients. These quantities are related to transitions between the ground and first excited states, with linearly polarized incident radiation along the z-axis. It is found that transition energy decreases with the growth of internal radius, thus causing the red-shift of resonant peaks. The same happens when the external magnetic field increases. When the strength of applied electric field is increased, the opposite effect is observed, since there is a blue-shift of resonances. However, dipole matrix moments decrease drastically with the increase of the electric field, leading to a reduction in amplitude of optical responses. At the moment impurity effects are activated, a decrease in the value of the energies is noted, significantly affecting the ground state, which is more evident for small internal radius. This is reflected in an increase in transition energies

Introduction von Neumann Algebras and II_1 Factors

Denote by B.(H) the algebra of bounded linear operators on the Hilbert space H. Recall that B(H) is naturally endowed with an involution x 7 ⟼ x^* associating with an operator x its adjoint x^*. The operator norm ∥x∥ of an element of B(H) is defined by ∥x∥ = sup {∥xξ∥ : ξ є H, ∥ξ∥ ≤ 1}. Endowed with this norm, B(H) is a Banach algebra with involution satisfying the identity ∥x^*x∥ = ∥x∥^2 (C^*-identity) i.e. a C^*-algebra. The weak operator topology on B(H) is the weakest topology making the map x ⟼ (xξ,η

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