Title:
Mathematical modeling of quantum dots with generalized envelope functions approximations and coupled partial differential equations
Type:
Masters Thesis
Status:
Defended
Source:
WLU Theses and Dissertations (Comprehensive). 959.

By introducing the concept of an electronic “bandstructure” for an ideal crystalline solid in the late 1920’s Bloch [24] presented a revolution in the world of physics, hitherto dominated by research on atoms. In atoms, the energies of bound electrons are discrete and prede- fined within the limit of Heisenberg uncertainty relation. In solids, the electron energy is a multivalued function of the momentum resulting in energy bands, continuous densities of states, and gaps. The wavefunctions become completely delocalised in a real space. Central assumptions of the Bloch’s theory are based on an infinite extension of the regular array of lattice points in all three dimensions of space. Soon after this theoretical contribution, band phenomena were detected in bulk crystal materials with finite dimensions. The ex- perimental picture followed the predictions of Bloch’s theory, even for the structures with a few micrometers in size, as they were still very large compared to next-neighbor distances. The next era in applications of quantum mechanics had come in the 1950s. At that time it became possible to fabricate the ultra-thin layer materials, sufficient to observe the quantization effects in one dimension and having the solid-like behavior in other two. The main object for the research was thin films of semi-metals (e.g. Be) on silicate substrates obtained by vacuum deposition or so-called quantum wells [60]. For these films, not only the presence of the quantization effects was directly shown, but also the dependency between the size of the fundamental bandgap and the film thickness was deduced. Such pioneering experiments encouraged the creation of a new branch of experimental and theoretical sci- ence, concentrated on material modeling, where methods and tools of applied mathematic plays a key role. The rigorous mathematical models together with an increased experi- mental progress in this field, have made a major contribution into the development of the integrated circuits technology [59]. This branch is recognized now as a basic element for any electronic device, from consumer electronics to extra sharp physical instrumentation. At that time several important theoretical discoveries were made, during attempts to extend the Bloch’s theory beyond its application to bulk materials only. Thus, in 1962, author of [89] studied the motion of electrons in a crystal with a superimposed periodic potential having the period much larger than the lattice constant. He came up with the prediction of minizones with the negative differential resistance. This phenomenon was later named as a resonant tunneling effect [81],[163]. Truly fascinating results came from the optical applications of quantum wells and their closely packed arrays (superlatices). In 1971 scientists [87] theoretically considered the possibility of creating a unipolar long-wavelength laser using radiative transitions between electron size quantization subbands in superlatices. Such devices were finally constructed in 1994 [37]. Along with that, the scientists directly observed how the step like behavior of the absorbtion spectrum related to the two dimensional character of density of states in GaAs quantum wells. This fact marked the beginning of a new period in the quantum science, where a complete reduction of two remaining macro-dimensions of a quantum well to atomic values leads to the structures with equally small sizes in all three dimensions. Ever since, such three dimensional nanostructures, with size on order of the de Broglie wavelength, came to known as quantum dots (QD’s). Their size causes the carrier localization in all vn dimensions and induces the complete breakdown of classical continuous dispersion of energy into the set of discrete levels. Along with that, a profound size-dependant change of all other physical properties occurs, as compared to the macroscopic case. The study of single quantum dots and their arrays presents a new frontier in theoretical and mathematical physics. This field of quantum theory is rich in the important multi- disciplinary applications, such as medical (bioimaging, drug delivery), hi-tech engineering (nano-machines, revolutionary phase-changing materials, nanosize integrated circuits), envi- ronmental sciences (selective absorbtion membranes). At the same time, many applications of QD’s have arisen as a further evolution of earlier developments in quantum wells. Some of the most successful among them are the applications in electronic, optoelectronic and photogalvanic areas. Here, the main theoretical discoveries had been done by 60’s when the successful modeling of quantum wells were performed [19] using a combination of a solid state physics methodology and quantum mechanical approach [111], [55], [22]. In spite of the fact that the developed model is valid for three dimensional nanostructures as well, there are still many unanswered questions in its application to the modeling of QD’s. The mathematical model for the one dimensional problem was initially formulated [111], [55], [22] in terms of the Fourier transform of the original Schrodinger equation, making use of the k • p term [177] leading to what is now known as the k • p model. In this form it satisfied the needs of the theoretical and experimental physics, at that time. Furthermore, since the experimental area was focused on the applications of quantum wells there was no need to challenge the mathematical aspect of this k • p model in more than one dimension. The main problem connected with an extension beyond the one dimensional case is due to the increased complexity of the model resulting from the symmetry properties of the crystalline solid. The practical side of this issue is closely connected with the experimental measurements of the main electronic properties for different materials. In accordance to the theory, these properties are spatially anisotropic, geometry dependant and mutually influence each other. The variety of inimitable measurement techniques sometimes leads to situations, where the values of these main electronic properties fluctuate as high as 50% from one measurement technique to another one (e.g., see the comparison for Si [142] and Ge [38]). While for some materials, in addition, the measured properties are susceptible to arbitrarily gentle perturbation of a certain external force, such as a magnetic field. Thus, the measurement problem has not been settled yet despite the joint scientific efforts and technological leap in this area over the last two decades. Finally, when it comes to a fabrication of QD’s the ultra small size problem, as predicted theoretically, emerges. The typical QD is composed from 102 atoms (for stand-alone dots) to 105 atoms (for dots in array) [30]. For such tiny structures there are striking differences in their physical characteristics which occur simply because of the small number of atoms. Even in thermal equilibrium, there is a root-mean-square volume change d V/V, which, for a dot of about 100 atoms at room temperature, is around 1% (i.e. about one atomic layer) [30]. All it transforms to the size and geometry deviations during the fabrication procedure results in a serious obstacle for the industrial applications. Only recently it became possible to simultaneously grow many QD’s with mostly uni- vm form predescribed geometry(on average) using a self-organized (also referred to as self- assembled) Stranski-Krastanow growing technique [153]. This kind of growth is based on the strain induced QD structure deposition on the subtract, corrected afterwards by the strain relaxation of QD interface to the surrounding matrix and possible size enlargement. As a result, such highly strained QD’s are packed into the structure formed by substrate material. In this case QD and the surrounding substrate form a nano-size two-component compound, which is conventionally called a QD heterostructure. All crucial electronic properties of QD become interdependent with properties of the substrate, and due to the grow technique will depend on the intrinsic strain or strain assisted effects. Therefore they must be considered as coupled [104], [119], [135]. Different approaches have been developed to study the electronic properties of QD’s heterostructures in the coupled form described above. They varies from continuum ap- proaches, based on the extension one dimensional k • p model mentioned before, such as effective mass [73], [73] and k - p [22], [12], [135] approximations to atomistic models, e.g. pseudopotential [170], [171] and tight-binding [151], [152] approaches. In atomistic models, precise for arbitrarily small systems, the computational effort grows with the number of atoms in the QD heterostructure. The continuous models, derived via averaging procedures, are free of that computational burden. Their accuracy, however, decreases when the QD characteristic dimensions reach the length scale of the atomic bonds. On the other hand, the accuracy of continuum models are not susceptible to a maximum size of the structure. In the scope of this work we will analyze one of such continuous-type models, namely, the k • p approach to bandstructure modeling. This model stems from the same classical works [111], [55], but instead of Schrodinger equations in the Fourier space it considers a Schrodinger equation in the differential form. This alternative point of view allows to incor- porate crucial non symmetric effect in to the resulting approximation, such as arbitrarily external fields including strain and strain-induced effects. In addition this approach was further generalized from bulk material to the QD heterostructure case [34], [36], and named subsequently as an envelope function approximation. Our goals are the following: Firstly we intend to justify the mathematical aspects of the k • p approach for certain, application important, bulk materials, where it coincides with more general envelope function approximation. Then we will show how to extend the k • p model to the envelope function approximation in realistic QD heterostructure and preserve all important physical and mathematical properties of the former model in bulk. Secondly we will propose an effective technique for the realistic modeling of QD’s using finite element method (FEM). The composition of the industrial standard FEM meshing software and effective parallel linear eigenvalue solvers offers a flexible and rather general computational approach, which can be extended to the modeling of arrays of QD’s, superlatices, etc. To achieve these goals we will employ the modern partial differential equations (PDE) technique to examine the models received as a result of k • p approach. A consecutive math- ematical insight into that subject requires us to use an apparatus of quantum mechanics and the theory of solids, briefly summarized in the first chapter. Equipped with the full IX theoretical power of general mathematical apparatus of quantum mechanics we will proceed to the next chapter, where all main theoretical aspects are stated. In Chapter 2, first we present the main concepts of k • p approach and illustrate its application to derive a 2-band model. After that we state the basic principles of pertur- bation theory needed to derive realistic multiband models. In the following subsections we consider three most common of them, based on Kane [84], Luttinger-Kohn[110] and Bir- Pikus [22] Hamiltonians. For these models we further analyze the principal properties of Hamiltonians, based on the real values of material parameters, and associated PDE’s. It turns out that most of them don’t satisfy the crucial ellipticity conditions, which leads to the nonphysical solutions, encountered in the applications over the last two decades [67], [166], [103]. These, new results, guide us to the last section, where we present a mathemat- ically rigorous way to modify the above mentioned models and reestablish their ellipticity, material-wise. These modifications remain completely within the framework of the k • p approximation and therefore preserve physically important symmetry properties. Chapter 3 serves two purposes. It presents an extension for one of the bandstructure models [84], [110], [22], described before, to include the piezoelastic effects. The resulting coupled models, along with the realistic QD geometries, reflect the contemporary level of physical applications. Following this, we present the numerical evidence that demonstrates the effectiveness of the previously stated ellipticity conditions and the modification tech- nique using as an example pyramidal QD heterostructures. The computational scheme used for simulations presented in this chapter, is discussed in Chapter 4. In Chapter 4, we will describe computational difficulties of multiscale simulations for QD’s caused by limitations of the finite element software package (COMSOL) we used. Then we will show the ways how to overcome these limitations by applying a parallel eigenvalue solvers package known as SLEPc. The combination of these two software packages provides a basis for the developed parallel software framework. The details of its implementation and the simulation algorithm itself are discussed in the rest part of this chapter. Chapter 5 of the thesis is devoted to the inverse problem of bandstructure modeling in the bulk case. Here we will give a description of an inverse problem for the multiband Hamiltonian operators [26], [27] and review the theoretical aspects and limitations of the existing solution technique based on the Marchenko integral equation [6]. Then we will present a robust numerical method for the solution of this basic equation as well as an example of the inversion procedure together with the a posteriori error analysis. Conclusions and the future directions are given in Chapter 6

Last updated on 19:56, 06 December 2024
Tags: kp theory, schrodinger equation, inverse problem.