文章将分数维方法与平均参数方法结合，探讨GaAs/Al_xGa_1-xAs核壳纳米线中的极化子和激子现象。由于各类形态低维异质结构中类体声子、表面声子、界面声子、半空间声子等各种声子模的存在，导致异质结构中的极化子问题十分复杂。本文通过将分数维方法与平均参数方法结合，可以在仅考虑体LO声子模和系统维数的情况下，研究上述各类形态低维异质结构中的复杂问题，并在此基础上探讨了多层核壳纳米线结构中的激子问题。研究结果充分体现了分数维方法研究各类形态低维异质结构中复杂极化子和激子问题的简捷性、高效性和普适性。文章主要包括以下两方面内容： （一）概述分数维空间基础知识，用分数维方法研究各类形态低维异质结构中的极化子，给出各类形态低维异质结构中极化子物理性质随结构尺度的变化关系；阐述多层核壳纳米线结构中的电子态，为下文研究GaAs/Al_xGa_1-xAs核壳纳米线中的极化子和激子物理性质做铺垫。 （二）将分数维方法与平均参数方法相结合，探讨GaAs/Al_xGa_1-xAs核壳纳米线中的极化子和激子现象，并数值计算GaAs/Al_xGa_1-xAs核壳纳米线中极化子、重空穴激子和轻空穴激子的结合能、有效质量、结构维数等物理性质随各核壳层尺度的变化关系: （1）对于两层GaAs/Al_xGa_1-xAs核壳纳米线结构：当核半径很小时，极化子结合能随核半径的增加而增加。随后，当核半径足够大时，束缚极化子结合能随核半径的增加而减小。当核半径非常大时，束缚极化子结合能几乎不受核半径变化的影响，在整个范围内保持不变。极化子有效质量随核半径的变化关系与此相似。当壳层厚度很小时，极化子结合能随壳层厚度的增加而增加。随后，当壳层厚度足够大时，极化子结合能随壳层厚度的增加而减小。当壳层厚度非常大时，极化子结合能几乎不受壳层厚度变化的影响，在整个范围内保持不变。极化子有效质量随壳层厚度的变化关系与此相似。 （2）对于四层GaAs/Al_xGa_1-xAs核壳纳米线结构：重空穴激子和轻空穴激子结合能随壳层厚度的增加，先增加，达到最大值后，随壳层厚度的增加而减小。当壳层厚度足够大时，重空穴激子和轻空穴激子结合能递减越来越缓慢。当核半径增加时，重空穴激子和轻空穴激子结合能缓慢增加。较小的核半径比大的核半径对重空穴激子和轻空穴激子结合能影响更为显著，与核半径相比，壳层厚度的变化对重空穴激子和轻空穴激子结合能影响更为显著。

We combine the fractal dimension method and the mean parameter method to study the polaron and exciton effects in GaAs/AlxGa1-xAs core-shell nanowires. Polaron problems in heterostructures are very complicated owing to the presence of a variety of phonon modes such as bulklike phonons, interface phonons, halfspace phonons, and slab phonons. Taking all these effects into account, polaron problems in multilayered heterostructures can be solved simply by considering only the LO phonon mode when the fractal dimension method and the mean parameter method are combined. Additionally, on the basis of what have been mentioned aboved, the exciton effects in core-multishell nanowires are discussed. The results fully demonstrate the simplicity, efficiency and universality of the fractal dimension method for studying the complex polarons and excitons problems in different kinds of low dimensional heterostructures. This study mainly includes the following two aspects: I. We introduce the basic knowledge of fractal dimension space. Polaron effects in low dimensional heterostructures of various shapes are studied by the fractal dimension method. Variation relationships between polaron properties and the structure scale of various shapes are calculated. The basic knowledge of electronic state in core-multishell nanowires is introduced for the following study polaron and exciton properties in GaAs/AlxGa1-xAs core-shell nanowires. II. The fractal dimension method and the mean parameter method are combined to study the polaron and exciton effects in GaAs/AlxGa1-xAs core-shell nanowires. The variation relationships between polaron properties and the structure scale are calculated. On the basis of this, the numerical results of heavy-hole exciton binding energy, light-hole exciton binding energy and fractal dimension parameter are worked out as functions of shell width and core radius: (1) For GaAs/AlxGa1-xAs core-shell nanowires of two layers, the polaron binding energy first increase as core radius increases. Then the polaron binding energy decreases monotonically as core radius continues to increase.For large core radius the polaron binding energy stays constant over the whole range. The Variation relationship between the polaron mass shift and the core radius is similar to that of the polaron binding energy; The polaron binding energy first increases as the shell width increases and then decreases monotonously as the shell width continues to increase. For a very large shell width, the polaron binding energy stays constant over the entire range. The variation relationship between the polaron mass shift and the shell width is similar to that of the polaron binding energy. (2) For GaAs/AlxGa1-xAs core-shell nanowires of four layers, the heavy-hole exciton binding energy and light-hole exciton binding energy first increase when the shell width increases. Then the heavy-hole exciton binding energy and light-hole exciton binding energy decrease monotonously while continuing to increase the shell width further. For relatively large shell width, the heavy-hole exciton binding energy and light-hole exciton binding energy decrease more and more slowly. The heavy-hole exciton binding energy and light-hole exciton binding energy decrease gradually when the core radius increases. When the core radius is relatively small, it has more significant impact on the binding energy than a large core radius. While compared to the core radius, the change of the shell width has a more significant impact on the heavy-hole exciton binding energy and light-hole exciton binding energy.