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Nanosystems Physics Chemistry Mathematics

Hartree-Fock approximation for the problem of particle storage in deformed nanolayer

Ivan F. Melikhov, I. Yu. Popov

Published Jan 1, 2013• 2 citations
OpenAlex

Abstract

The paper deals with the problem of quantum particle storage in a nanolayered structures. The system of a few electrons interacting via a -potential is considered. The particles are placed into a two-dimensional deformed waveguide. From a mathematical point of view, the bound state of the system means that the corresponding Hamiltonian will have eigenvalues. To treat a multi-particle problem, the Hartree-Fock approach and the finite element method are used. Three different types of the perturbation are considered: deformation of the layer boundary, a small window in a wall between two layers and a bent layer. The systems of 2‐10 particles with various total spins are studied. The dependence of the minimal deformation parameter, which keeps bound state on the number of particles, is given. Comparison of the storage efficiencies in those cases is made.

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The paper deals with the problem of quantum particle storage in a nanolayered structures. The system of a few electrons interacting via a -potential is considered. The particles are placed into a two-dimensional deformed waveguide. From a mathematical point of view, the bound state of the system means that the corresponding Hamiltonian will have eigenvalues. To treat a multi-particle problem, the Hartree-Fock approach and the finite element method are used. Three different types of the perturbation are considered: deformation of the layer boundary, a small window in a wall between two layers and a bent layer. The systems of 2‐10 particles with various total spins are studied. The dependence of the minimal deformation parameter, which keeps bound state on the number of particles, is given. Comparison of the storage efficiencies in those cases is made.
The problem of particle storage in a nanolayered structure is considered. Local perturbations of the nanolayers can lead to the appearance of eigenvalues of the corresponding one-particle Hamiltonian. To study the particle storage it is necessary to deal with a multi-particle problem. The Hartree method and the finite element method are used. The discrete spectrum of the Hamiltonian of two interacting particles is considered. Two different types of the perturbation are treated: deformation of the layer boundary and a small window in a wall between two layers. The relation between the system parameters (interaction intensity–waveguide deformation) ensuring the existence of a non-empty discrete spectrum is studied. Comparison of particle storage efficiencies in these two cases is made.
The problem of particle storage in nanolayered structures will be considered. Local perturbations of nanolayers can lead to the appearanceof eigenvalues of the corresponding one-particle Hamiltonian. To study particle storage it is necessary to deal with the multi-particle problem. This problem faces essential computational difficulties due to the great increase of the spatial dimension. Using a composite of natural physical models, analytical methods and computational approaches allows one to simplify the problem and to obtain useful results for application. Particularly, the Hartree method and Finite Elements Method (FEM) are used. The discrete spectrum of the Hamiltonian for two interacting particles is considered. Two different types of perturbation are considered: deformation of the layer boundary and a small window in a wall between two layers. The relation between the system parameters (interaction intensity - waveguide deformation) ensuring the existence of a non-empty discrete spectrum is studied. A comparison of particle storage efficiencies is made for these two cases.
Problem of particles storage in nanolayered structure is considered. Local deformations of the nanolayers leads to appearance of eigenvalues of the corresponding one-particle Hamiltonian. The cardinality of the discrete spectrum is used to estimate the maximal number of non-interacting fermions stored in the nanolayers. The discrete spectrum of the Hamiltonian of two interacting particles is considered. Relation between the system parameters (interaction intensity - waveguide deformation) ensuring the existence of non-empty discrete spectrum is studied. Hartree method and FEM are used.
Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the ‘pantographic’ case (same shape of the box through the entire process) and the case with deformation.
In treating a system of NNN antisymmetric particles, it is shown that, if the total Hamiltonian Hop{\mathcal{H}}_{\mathrm{op}}Hop​ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" O\mathcal{O}O. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.