準彈性反射電子模擬與分析

Abstract

本論文在探討移動電子對固體做運動時而發生的彈性及非彈性交互作用，引用或研究相關的理論模式，並對從固體準彈性反射出來的電子做出模擬與分析。 在彈性交互作用方面，我們利用非相對論性及相對論性彈性交互作用模式，並配合自由原子位能及固體原子位能，來計算電子的微分彈性散射截面(elastic differential cross section: EDCS)，計算的結果，針對非相對論性及相對論性模式做出比較。利用所求得的微分彈性散射截面，可以得到電子在固體中移動時的彈性平均自由行徑(elastic mean free path: EMFP)，我們也計算了相對論性模式的彈性平均自由行徑，並對其做出分析。另外，化合物固體在電子的交互作用研究當中，也扮演重要角色，故而，我們根據化合物固體裡個別元素的彈性散射截面或彈性平均自由行徑，演示了化合物固體的總彈性散射截面及總彈性平均自由行徑的理論計算方法，並且也了解，電子在化合物固體裡碰撞個別元素的機率，此碰撞機率計算方式對後續的電子彈性波峰研究有所幫助。除此之外，由於電子在彈性碰撞過程中，還伴隨著反衝能量損失(recoil energy loss)，此反衝能量損失，可以透過Rutherford彈性碰撞模型而求得，我們也演示其理論計算式，此計算式對於電子準彈性反射能譜的研究有所助益。 在非彈性交互作用方面，我們的目的在推導一個可以適用於電子以任意方向入射或出射固體的非彈性交互作用模式。我們應用介電理論(dielectric response theory)，在滿足邊界條件之下，去解Poisson 方程式，可以得到電子相對固體做運動時，而產生的感應電場(induced electric field)，利用作用在電子本身的感應電場，可以求得電子所受到的阻擋本領(stopping power)，透過此阻擋本領，並在動量積分上採用球座標系統，從而推導出了電子的非彈性微分倒數平均自由行徑(differential inverse inelastic mean free path: DIIMFP)、非彈性平均自由行徑(inelastic mean free path: IMFP)以及表面激發參數的計算式(surface excitation parameter: SEP)。配合extended Drude介電函數的使用，我們根據所推式子而計算出，與電子位置、能量及運動方向有關的非彈性微分倒數平均自由行徑和非彈性平均自由行徑，以及與電子能量及運動方向有關的表面激發參數；並且發現表面激發參數遵守一個簡單的通式，在我們的研究當中，也對此通式的相關係數做出模擬，以適用於電子入射及出射不同固體的表面激發參數。由於所推導的非彈性交互作用模式，並不能包含電子在極高速運動之下而顯現出來的延遲效應(retardation effect)，故而，我們另針對極高速電子而推導一個含延遲效應之非彈性交互作用模式。同樣利用介電理論，去解Maxwell方程式，以類似於不含延遲效應之非彈性交互作用模式的推導過程，推導出了適用於極高速電子平行於固體表面運動時的阻擋本領、非彈性微分倒數平均自由行徑以及非彈性平均自由行徑，並與用傳統模式計算出的結果做出比較。 根據對彈性交互作用及非彈性交互作用的理論研究，我們利用Monte Carlo方法並記錄反衝能量損失，從而模擬理想狀態下的電子準彈性反射能譜。另外，我們也考慮電子束能量分佈、電子能譜測量儀器的能量解析度以及原子受熱能而振盪之三種效應對反彈能譜的影響，其最後結果與實驗值做出比較，並探討其中的差異。另外，電子的非彈性平均自由行徑，可以藉由對電子彈性波峰(electron elastic peak)的分析而得，我們根據自己的Monte Carlo方法，去分析GaAs的電子準彈性波峰，並藉由其而得出GaAs的非彈性電子平均自由行徑。

In this thesis, the models dealing with the elastic interactions between a moving electron and a solid were described. The inelastic interactions between a moving electron and a solid were studied theoretically. Based on the elastic-scattering and inelastic-scattering models, the simulation and analysis of electrons backscattered from solids surfaces were made. In the work for the elastic scattering, to calculate the elastic differential cross section (EDCS) of an electron moving in a solid, the applied non-relativistic and relativistic models with free-atom and solid-atom potentials were described. The results calculated using the non-relativistic and the relativistic models were compared. Through the EDCS, the elastic mean free path (EMFP) of the electron can be determined. Since compounds have been widely used in many studies, the methods adopted to determine the total elastic cross sections and total EMFPs for compounds were also described. The method for the calculation of the probability of an electron scattered by each element in a compound was further given. Moreover, the Rutherford-type recoil energy loss in elastic scattering was discussed. In the aspect of inelastic scattering of electrons with solids, our purpose was to derive an inelastic-scattering model that was suitable for electrons incident into and escaping from solid surfaces in arbitrary moving directions. Based on the dielectric response theory, the induced electric field can be obtained by solving the Poisson equation. Through the induced electric field with the adoption of spherical coordinates in the momentum integration, stopping power, differential inverse inelastic mean free path (DIIMFP), inelastic mean free path (IMFP) and surface excitation parameter (SEP) can be determined. The spherical coordinates was adopted in the momentum integration to satisfy energy and momentum conservations completely. Using the extended dielectric functions in this derived model, the crossing-angle-dependent and energy-dependent DIIMFPs and IMFPs and SEPs were shown. It was found that the calculated results of SEPs followed a simple formula. The fitted values of parameters in the simple formula for solid surfaces of Au, Ag and Cu were also listed in tables. Due to the exclusion of retardation effect of the induced electric fields in this derived inelastic-scattering model, it is inadequate for high-speed electrons interacting with solids. Hence, we further constructed a model that can determine the stopping power, DIIMFP and IMFP for high-speed electrons moving parallel to solid surfaces by solving the Maxwell equation instead of Poisson equation in the dielectric response theory. The results calculated using the models without and with the retardation effect were shown and compared. Applying the results of elastic and inelastic interactions of electrons with solids, the energy distribution of the elastic peak of electrons backscattered from solid surfaces can be simulated by recording the recoil energy loss in the Monte Carlo (MC) method. The effects of the energy distribution of the electron beam, the energy resolution of the spectrometer and the thermal motion of atoms in solids were also included. Comparisons of the calculated energy spectra with the experimental data were also made. On the other hand, the electron effective IMFP can be extracted from the electron elastic peak. This method was also discussed in the present thesis.