Electron dispersion and bound states in the vicinity of nanoestructures

Abstract

The controlled manufacture of nanometric scale structures poses important challenges to the theoretical description of the physical phenomena involved. Comprehending them facilitates the design of new technologies based in manipulating matter at the atomic or molecular levels, in order to develop applications relevant to industrial processes. This thesis addresses two concrete problems, one in the realm of electron dispersion, and another being the existence of electrons bound to a nanostructure as a whole instead of, as usual, being bound to one of the atoms. In this work, specific emphasis is placed on the problem of electron dispersion, where novel aspects of modeling the phenomenon at nanometric scale are taken into account: 1.- The motion of electromagnetic waves in reduced spaces so minuscule that their dimensions are smaller than the length of the wave through which they propagate. These are called "sub-wavelength electromagnetic phenomena." 2.- The motion of electron packets in spaces that are too close together: the emission source, obstacles that get in their way, and the region where the phenomenon of dispersion is detected. For the case of the motion of electrons in minuscule spaces, two problems are outlined: A) the algebraic di_culty in treating the phenomenon of dispersed outgoing waves analytically, which is complex given that nanometric scales do not allow for one to impose the usual boundary conditions. The other is the existence of interactions between the wave packet and obstacles in their way, where it becomes important to make the distinction if they are dielectric or conductors. Considering all of that, an important phenomenon is the double-slit experiment, carried out by electrons moving across semiconductors. It was carried out for the first time using electrons in a vacuum in 1961 and has been repeated many times since. The other modality is that when the electrons move through a semiconductor which has further semiconductors inserted on it, that have a larger band gap. Thus, the energy of the moving electrons is not enough to move through an array of semiconductors, with a gap in between them. In this kind of dispersion phenomena involving electrons, the material of the obstacles that are utilized to represent the slits matters. If it is dielectric, we can see an interaction between the electron charge and the electric dipoles that corresponds to the presence of walls interacting with moving charges. On the other hand, if the material is a conductor, one can see that the negative charges tend to move away from the electron cloud as it travels near them. In this case we see the rise of charge-charge interaction. The relevant aspect of this latter interaction is that it is five times more energetic than a dipole-dipole interaction. Also, the strength of the former has a longer range and loses intensity in the form, 1/r2 whereas the latter does so at 1/r6 . In this last case, writing the Hamiltonian can be done without sacrificing considerable precision. However, when dealing with charge-charge interactions it is necessary to account for a term that includes an excess of positive charges in the conductor, that agglutinate near the slits. The situation becomes more complex when choosing the direction of an electron pulse because the positive charges of the conductive material get close to the slits and then return to reestablish electrostatic equilibrium. The model being posed here has significance in the design of electronic devices at the nanometric scale and the focus of the work on this thesis is the simplest form of the problem, which is with electrons moving through a vacuum. With this problem solved, it becomes feasible to tackle the complications mentioned previously, one by one. That is the purpose of Chapter 1.The first chapter of this thesis is titled "Spatial and temporal description of electron diffraction through a double slit at the nanometer scale,"[8] where the time dependent Schrödinger equation is used to solve the double-slit problem, that is represented as an electron pulse traveling towards a dielectric wall with two long,parallel slits. The solution is obtained from an initial condition of a function (~r; t = 0), which is a gaussian probability amplitude.The standard analytical framework [6] is not applicable in this case due to the following reasons: the slits are not infinitely thin. Their width plays a role in the phenomenon. the emission source is in such close proximity to the slits that it cannot be represented as a plane wave. Hence, what reaches the slits is a wavefront that still presents signifiECUcant curvature and thus the hypothesis of the magnitude of the wave being the same in all points of the slit is unfullled. the detectors that make it possible to measure the diffracted pulse are so close to the slits that it is not possible to make the approximation of the wave length being too small compared to the distances involved between the slits and the detectors. It is necessary to mention various aspects before explaining that we choose a specific physical system in the problem of electrons bound to a nanostructure because they are taken to be clusters of atoms that have been manufactured with diferent techniques. When nucleation takes place in gaseous phase, atoms aggregate as they move randomly throughout the gas. Under this circumnstance it is possible that, before a cluster can be bound to an atom in direct manner, it is possible that it binds a single electron. Hence we are dealing with a kind of super atom where the electron follows the laws of quantum mechanics and has properties that can be studied. This thesis focuses on the case of a cluster with electric dipole moment in which one of its atoms has lost an electron. In this case, the work starts from a classical problem calculated by Sergio Gutierrez Lopez [22], who showed that one of these physical systems has an associated constant of motion, and, if an electric charge moves in its vicinity, then it can have stable orbits. Therefore, the quantum treatment of the problem is outlined as follows: What are the properties of a nanostructure after it became ionized and bound an electron to its periphery? The theoretical classical result opens the possibility that there exist quantum states of electrons bound to nanoparticles. If so, its energy spectra would give raise to electronic transitions with emission and absorption properties that can be observed so long as the half-lives of those states allow for it. In case of existing, the detection of an absorption-emission spectrum of light must depend on its intensity and, thus, on the quantity of ionized nanostructures, but this last aspect is left outside of the scope of this work. The results are located in Chapter 2 of this thesis and correspond to the published paper titled "Bidimensional bound states for charged polar nanoparticles."[9] We also have obtained conclusive results for the tridimensional case. The second objective consisted in generalizing the previous work to three dimensions. We obtained the classical form of a new constant of motion and found the quantum operator associated with it. We also solved the Schrödinger equation separating the angular part to demonstrate that the probability density slides towards the angle ϴ<π/2We studied the radial part and obtained the energies for the ground state and two excited states. We plotted the radial component of the wave function for a small electric dipole such as the cluster (GaAs)3 and for very large electric dipoles such as fullerenes RbC60 and LiC60. The results were published in the academic paper titled "Behavior of an electron in the vicinity of a tridimensional charged polar nanoparticle through a classical and quantum constant of motion,"[10] in the Journal of Nanoparticle Research.