Researches On Phonon Modes In Several Quantum Systems Of Complex Shapes

Currently, many fabricating technologies such as metal-organic chemical vapour de-position, molecular beam epitaxy, and molecular self-assembly have been developed forfabricating nanostructures with a wide range of sizes, shapes, and dielectric environ-ments. It is of great interest to research the e?ect of shape on their physical properties,because nanomaterials of complex shapes have di?erent crystallographic facets and di?er-ent fraction of surface atoms on their corners and edges, which contributes significantlyto modulating their physical properties. On the other hand, the optical phonon modesand electron-phonon interaction play an important role in many physical properties of thepolar crystals such as the binding energy of impurities, carrier transportation, linear andnon-linear optical properties, especially in low-dimensional materials. So, it is of practicalinterest to research polar optical phonon modes and the electron-optical-phonon interac-tion Hamiltonian for quantum systems of complex shapes. The dissertation includes sixchapters.In the first chapter, the general characters and applications of low-dimensional ma-terials are summarized brie?y and a simple introduction to the current research of polaroptical phonon modes and the electron-optical-phonon interaction Hamiltonian in low-dimensional materials is given. At last, the research directions of the thesis are putforward.In the second chapter, under the dielectric continuum model, we start from the elec-trostatic equations with the electrostatics boundary conditions. Then, the eigenfunctionsand polarization vectors for the confined bulklike LO phonons in equilateral triangularquantum dot (wire), hemi-equilateral triangular (30?-60?-90?) quantum dot (wire) andisosceles right triangular (45?-45?-90?) quantum dot (wire) are deduced. The confined LOpolarization vectors form an orthonormal and complete set. Second, we dynamic equa-tions of motion of the crystal lattice and derive the Hamiltonian of the free vibration.Then, the Hamiltonian can be expanded in terms of the complete set of orthonormalpolarization modes. What's more, we derive the Hamiltonian operator for confined LOphonon modes and the corresponding Fro¨hlich electron-phonon interaction Hamiltonian.Finally, the potential applications of these results are discussed.In the third chapter, under the dielectric continuum model and separation of vari-ables, the IO (interface optical) phonon modes and electron-optical-phonon interaction inrectangular quantum wire and quantum dot embedded in a nonpolar matrix are studied. We found there exist various types of IO phonon modes in rectangular nanostructures.The IO phonon modes in rectangular quantum wire include IO-PR (IO-propagating) andIO-IO hybrid phonon modes, while the IO phonon modes in rectangular quantum dotcontain IO-IO-PR and IO-PR-PR hybrid phonon modes. The results of numerical calcu-lation show that these hybrid phonon modes contain CO (corner optical) phonon modesand EO (edge optical) phonon modes. The potential applications of these results are alsodiscussed.By the theoretical scheme of the equilateral triangular quantum dot, we derive theLO and IO phonon modes and the corresponding Fr¨ohlich electron-phonon interactionHamiltonian of the hemispherical quantum dot, fan-shaped quantum dot and quantumwire in the fourth and fifth chapters. Study on optical phonon modes and phonon-assistedphysical processes in quantum systems of complex shapes is of fundamental interest inphysics.In the last chapter, a brief summary of this thesis is given. Then, we discuss the opensubjects which can be further researched. What's more, we further discuss the shortcom-ing of the dielectric continuum model and the possible future directions of research andimprovement.