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Nanosystems Physics Chemistry Mathematics

MULTI PARTICLE STATES CALCULATIONS AND PARTICLES STORAGE IN PERTURBED NANOLAYERS

Sergey I. Popov

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
Matrix eigenvalue problems arise in several areas of science and engineering. Particle physicists, in particular, are interested in solving eigenvalue problems that describe relativistic subatomic interactions. Quantum field theory provides a formal mathematical approach for specifying the eigenvalue problems, but it also introduces some complications. A particularly problematic complication is accounting for an infinite number of particles. It is not necessary to deal practically with infinities: the system maybe truncated to a discrete number. In the case of strong coupling, however, it is inaccurate to truncate at small particle numbers. Scalable computational methods are vital to the exploration of quantum field theory since they facilitate inclusion of increasing numbers of particles.
The study of phase transitions and critical phenomena is an area of great interest in science. Liquid to gas, ferromagnetic to paramagnetic, fluid to superfluid, insulator to conductor are a few examples of physical systems exhibiting phase transition and critical phenomena. Classical phase transitions have thermal fluctuations as the main driving force for the transition. In statistical mechanics phase transitions are associated with singularities in the free energy. These singularites only occur in the thermodynamic limit where the volume (V) and particles (N) go to infinity with the density held constant (N/V). By examining the partition function for a finite system, the function is a sum of a set number of terms. The partition function then would be analytical. It is only when an infinite number of terms are added is there a singularity in the partition function. The subject of Finite Size Scaling theory is the relation of the phenomena in a finite systems to the true phase transition of an infinite system. Finite Size Scaling theory provides a numerical method to obtain accurate results for infinite systems simply by studying corresponding small systems. The focus of this thesis is on transitions of a different nature. Quantum phase transitions (transitions that occur at absolute zero) have Heisenberg’s uncertainty principle as the driving force for the transition. In quantum mechanics, the finite size problem occurs when looking at the critical behaviour of the Hamiltonian as a function of a set of parameters. The interest is where the bound states energies become non-analytical. The size of the system is related to the number of elements in a complete basis set used to expand the exact wave function. In this thesis we present the use of the Finite Element Method (a numerical technique commonly used in engineering problems to solve partial differential equations or integral equations). Previous finite size scaling studies in quantum mechanics employed a basis set expansion of the wave function and the problem is solved variationally. Slater type basis sets were used for this purpose. In the effort to apply finite size scaling calculations for larger systems Gaussian type functions were employed however the a larger number of Gaussian functions are needed to obtain accurate results. The use of standard electronic structure packages was also explored, however, the lack of consistently increasing size for basis sets limits the application for finite size scaling purposes. We first apply the finite element element method (FEM) to solve for the short range Yukawa potential and obtained the critical parameters for the model potential. We develop the framework necessary to use FEM for finite size scaling. As mentioned before the 'size' of the system is the number of basis functions needed in the expansion of the wave function. However, using FEM the size of the system is the number of elements used. We then studied the Hulthen potential in comparison to the basis set results and also compare the results to the analytical solutions for the both methods. Finally we explore the use of FEM in combination with Hartree-Fock theory to solve 2 to 4 electron systems. The use of Hartree-Fock theory allows for the implementation of finite size scaling to larger systems. We used the FEM Hartree-Fock to calculate for the critical parameters of 2 to 4 electron systems. Although the results are not exact (as to be expected due to the use of Hartree-Fock theory) we get an estimate to the critical points of these systems. FEM can be combined with other electronic structure theories, which would allow for the study of larger systems exhibiting critical phenomena.