量子与经典方法研究粒子与固体的相互作用
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摘要
电子显微技术以及电子能谱技术已成为材料表征特别是定量分析的重要工具。作为这些技术的物理基础,电子与固体相互作用的研究对定量解释实验电子显微成像或电子能谱起着至关重要的作用,成为凝聚态物理研究的一个非常重要的研究领域。本论文分别采用经典Monte Carlo方法、波动力学方法和玻姆力学方法,从不同角度对电子与固体的相互作用过程进行了理论研究。
Monte Carlo方法通过模拟电子在固体中的随机性散射和信号产生和发射过程,数值处理十分容易,已经成为电子与固体相互作用最主要的研究工具。但此方法原理上属于经典力学范畴,忽略了电子的相干散射,因此只能适用于非晶或多晶材料。当电子的德布罗意波长与晶体材料的原子间距相当时,电子的相干散射(衍射效应)将会十分明显。因此必须改进Monte Carlo轨迹模拟方法以处理单晶中电子的相干散射问题。
由德布罗意和玻姆发展起来的玻姆力学(量子轨迹)理论既具有传统量子力学的精确预见性,又具有直观解释物理现象的能力,提供了有别于量子力学哥本哈根学派诠释的另一种概念框架。根据玻姆力学,粒子轨迹由波函数控制,粒子的分布则反映波函数分布,因此其统计预测结果与传统的量子力学完全一致。量子轨迹理论不仅为我们提供了一条新的途径去解释量子现象,而且提供了一种新的计算方法,因此量子轨迹理论近年来受到越来越多的关注。
第一章介绍了电子能谱和电子显微技术的发展和现状。能谱方面介绍了俄歇电子能谱、X光电子能谱、电子能量损失谱等；在电子显微技术方面,介绍了扫描电子显微镜、透射电子显微镜、扫描透射电子显微镜等。随着像差矫正技术的发展,像差矫正的扫描透射电子显微镜已经达到了0.1nm的分辨率。基于相差矫正技术,原子级分辨率在电子能量损失谱和扫描电子显微镜中也得以实现。对于晶体材料,衍射效应对电子能谱也有着显著的影响。本章还介绍了用于电子能谱和电子显微技术计算模拟的Monte Carlo方法、波动力学计算方法等。
第二章介绍了玻姆力学的发展历程、单粒子和多体玻姆力学基本方程以及基本假定。Bell理论和实验证实量子力学中的隐变量模型必须是非局域的,因此我们讨论了玻姆力学非局域性特性以及它的经典极限。玻姆力学中的量子势是一个全新的物理量,它和玻姆力学的非局域特性相关联,我们对它做了详细讨论。关于量子轨迹的测量问题,我们介绍了利用弱测量技术测量的单光子量子轨迹以及如何通过纠缠光子对观测玻姆轨迹非局域性的实验。还比较了Feynman路径与量子轨迹的异同。最后介绍了三类计算量子轨迹的方法。
第三章利用Monte Carlo模拟方法系统地研究了俄歇电子能谱定量分析中的重要修正因子——背散射因子。计算的能量范围从电离能阈值到30keV,入射电子入射角度以及俄歇电子发射角度的变化范围均为0—89°。共计算了28种元素的主要俄歇跃迁信号的背散射因子。计算采用了新的普适的背散射因子定义式、更为精确的Casnati内壳层电离截面、最新的电子散射Monte Carlo模型,并且通过模拟大量电子轨迹以减少统计误差。计算中还分别考虑了两种电子探测器(轴半球分析仪和筒镜分析器)的几何构置。计算得到的背散射因子能够很好地描述文献中报道的俄歇电子信号强度随入射电子能量以及入射角度的实验变化关系。计算出的背散射因子全部存于一个开源在线数据库中。
第四章采用量子力学Bloch波方法,并结合倒易原理描述电子在晶体中的输运性质,以计算表面电子能谱中信号电子的发射深度分布函数(EMDDF)。明确考虑了晶体结构以及电子相干散射的影响,这在经典Monte Carlo模拟方法中都是被忽略的。电子的非弹性散射通过光学势来表征,我们使用量子力学方法计算了俄歇电子从Au(001)表面发射的EMDDF。它对发射方向以及晶体结构的关系表明,俄歇电子从晶体表面的发射强度与发射方向强烈有关,对晶体结构十分敏感。平均逃逸深度(MED)表征俄歇电子能谱的表面灵敏度,通过当前计算的EMDDF能够导出MED。结算结果表明,晶体样品中的MED是Monte Carlo计算的非晶样品中的MED值的大约一半。
第五章利用玻姆量子轨迹理论研究晶体中电子衍射。首先研究了电子在薄晶样品中的散射行为,用不含时相互作用势场来描述二维和三维模型晶体。通过数值求解含时薛定谔方程计算波函数,然后计算电子运动的量子轨迹,由此得到表征电子衍射行为的实空间几率密度函数和量子轨迹。量子轨迹提供了电子衍射的一个直观认识。对于晶体中的高能电子衍射,我们采用Bloch波方法计算得到了高能电子在晶体中衍射的波函数,量子轨迹则显示了电子在晶体中通道的形成过程。这些计算表明,在电子与固体／表面／纳米结构的相互作用研究中,量子轨迹理论有着巨大的应用前景。如果通过光学势将电子非弹性散射也包含在其中,那么该方法也能同时应用于电子能谱技术中。
第六章采用玻姆量子轨迹理论研究三维实空间中的氦原子在模型表面上的散射。氦原子散射是一种能获得表面结构、无序以及声子谱等信息的表面分析工具。用氦原子从W(112)表面散射的二维平面模型得到的三维模型相互作用势构造模型表面。氦原子散射的动力学过程以玻姆粒子的复杂量子轨迹的形式展示,玻姆力学也成为研究原子与物质相互作用现象的一条新路径。
第七章采用Bloch波方法和倒易原理模拟背散射电子衍射花样。背散射电子衍射花样可以提供非常重要的高分辨晶向信息。我们精确重现了衍射花样的相对强度分布。为模拟大视角的实验衍射花样,需要考虑大量的反射面。我们模拟得到的Mo(001)面背散射电子衍射花样与实验观察结果吻合得非常好,精确再现了实验中的晶带轴精细结构、高角Laue环等特征以及相对强度分布。
第八章是全文的总结。
Electron microscopy and electron spectroscopy techniques have been important tools for material characterization particularly on quantitative analysis. Study of elec-tron-solid interaction, as the physical basis of these techniques, is vital to quantitative interpretation of experimental electron microscopic image and electron spectra. It has been a significant research field of condensed matter physics. This thesis has theoret-ically studied the electron-solid interaction from different perspectives by employing classical Monte Carlo simulation method, wave mechanical method and Bohmian me-chanical method.
Monte Carlo simulation method treats the random scattering process of electrons in a solid and the associated signal generation and emission process. It is easy for numerical calculation and becomes the most powerful tool in study of electron-solid interaction. However, the method is in principle of classical mechanics and omits elec-tron coherent scattering; application is then limited to amorphous and polycrystalline solids. When de Broglie wave length of electrons is comparable with lattice constant of crystalline the coherent scattering (diffraction effects) becomes obvious. Therefore, Monte Carlo simulation method of electron trajectory must be improved to take account of coherent scattering in single crystals.
Bohmian mechanics is a theoretical formalism combining both the accuracy of the predictions of quantum mechanics and the capability of providing an intuitive inter-pretation of the physical phenomena involved. This alternative formalism of quantum mechanics presents significant conceptual differences from the conventional Copen-hagen interpretation. By Bohmian mechanics the particle trajectory is controlled by wave function, and the particle distribution follows that of wave function so that all statistical predictions agree exactly with those of standard quantum mechanics. In ad-dition to its interpretative importance, there has recently been growing interest from the computational point of view for its feature as a new calculation method.
Chapter One reviews the development and present status of electron spectroscopy and electron microscopy, such as, Auger electron spectroscopy (AES), X-ray pho-toelectron spectroscopy (XPS), electron energy loss spectroscopy (EELS), scanning electron microscopy (SEM), transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM). With development of aberration correction technique, aberration-corrected STEM instruments now routinely achieve a resolution of better than0.1nm. Atomic resolution was demonstrated in EELS and STEM based on aberration correction. For the crystal, the diffraction effects on the electron spec-troscopy are also significant. We also introduce Monte Carlo simulation method and wave mechanical method for simulation study in electron spectroscopy and electron microscopy.
Chapter Two reviews the history of Bohmian mechanics. The fundamental equa-tions of motion of Bohm particles for single particle systems and for many-body sys-tems are outlined. We also summarize the basic postulates of Bohmian mechanics. It has been demonstrated by experiment and Bell theorem that only non-local hidden variable model can be compatible with standard quantum mechanics. We present the non-locality of Bohmian mechanics and its classical limit. We also discuss the quantum potential, which is a new physical quantity and connected to non-locality in Bohmian mechanics. The experiments to measure the quantum trajectory of single photon and to observe the nonlocality of Bohmian trajectories with entangled photons are outlined. Feynman path integrals are compared with Bohmian particle paths from various as-pects. Finally we present three numerical methods for calculation of Bohmian quantum trajectories.
Chapter Three studies systematically with a Monte Carlo simulation method an important correction factor, the backscattering factor, in quantitative analysis by Auger electron spectroscopy. The primary energy ranges from the threshold energy of inner-shell ionization to30keV. The incident angle of primary beam and emission angle of Auger electrons are within0-89°. Principal Auger transitions in28pure elements are considered. The calculation employs a novel and general definition of backscattering factor, Casnati's ionization cross section, up-to-date Monte Carlo model of electron scattering, and a large number of electron trajectories are simulated to reduce statistical error. Both the configuration geometries of concentric hemispherical analyzer (CHA) and the cylindrical mirror analyzer (CMA) for Auger electron detection are considered in the calculation. The calculated backscattering factors are found to describe very well an experimental dependence of Auger electron intensity on primary energy and on incident angle in literature. The calculated numerical values of backscattering factor are stored in an open and on-line database.
Chapter Four employs the quantum mechanical Bloch-wave method and the reci-procity principle to describe electron transport in crystalline solids for determining the emission depth distribution function (EMDDF) of signal electrons in surface electron spectroscopy. The crystal structure and electron coherent scattering are explicitly taken into account while they are neglected in classical Monte Carlo methods. While electron inelastic scattering is treated with an optical potential, the EMDDF for Auger-electron emission from an Au (001) surface is calculated by the quantum method. The depen-dence of the EMDDF on emission angle and crystal structure has indicated that Auger electron emission intensity strongly depends on the emission direction and is sensitive to crystal structure. Comparisons made between the present results and classical Monte Carlo simulation results show pronounced differences due to electron-diffraction ef-fects. The mean escape depth (MED) which describes the surface sensitivity of Auger electron spectroscopy can also be evaluated. The present calculation shows that the MED for a crystalline solid is about half that for an amorphous solid from a Monte Carlo calculation.
Chapter Five employs Bohmain mechanics to study electron diffraction in crys-tal. An investigation of electron scattering in a thin crystal has been performed. A time-independent interaction potential was employed to model a2D and3D crystal. Quantum trajectories were calculated through wave function by a numerical solution of the time-dependent Schrodinger equation. The probability density functions and quan-tum trajectories representing electron diffraction in real space were obtained from the calculation. The quantum trajectories provide an intuitive dynamics of the the elec-tron diffraction. For high energy electron diffraction in a crystal, we obtain the wave function by Bloch wave method. The quantum trajectories then present the formation process of electron channeling in crystal. Such example calculations show that the quantum trajectory method has great potential application to theoretical study of elec-tron interaction with solids, surface and nanostructures. Once electron inelastic scat-tering is included via an optical potential, the method can also be applied to electron spectroscopy techniques.
Chapter Six studies helium atom scattering from a model surface in3D real space by using the Bohmian quantum trajectory theory. Helium atom scattering by solid surfaces is a well established technique for obtaining information about the structure, disorder and phonon spectra of the surface. The model surface was described by a3D interaction potential based on the model for the in-plane scattering of helium from a W(112) surface. The dynamic process of helium atom scattering has been presented by quantum trajectories of Bohmian particles. Bohmian mechanics becomes thus a new way to study atom-matter interaction.
Chapter Seven presents a simulation of electron backscattering diffraction (EBSD) patterns with Bloch wave theory and reciprocity principle. EBSD pattern can provide important crystallographic information with high spatial resolution. For the simulation of an experimental image with a large field of view, a large number of reflecting planes are taken into account. Very good agreement is obtained for the simulated and experi-mental EBSD patterns of Mo (001) surface. Experimental features like zone-axis fine structure and higher-order Laue zone rings as well as relative intensity distribution are accurately reproduced.
Chapter Eight is the summary of the thesis.
Monte Carlo方法通过模拟电子在固体中的随机性散射和信号产生和发射过程,数值处理十分容易,已经成为电子与固体相互作用最主要的研究工具。但此方法原理上属于经典力学范畴,忽略了电子的相干散射,因此只能适用于非晶或多晶材料。当电子的德布罗意波长与晶体材料的原子间距相当时,电子的相干散射(衍射效应)将会十分明显。因此必须改进Monte Carlo轨迹模拟方法以处理单晶中电子的相干散射问题。
由德布罗意和玻姆发展起来的玻姆力学(量子轨迹)理论既具有传统量子力学的精确预见性,又具有直观解释物理现象的能力,提供了有别于量子力学哥本哈根学派诠释的另一种概念框架。根据玻姆力学,粒子轨迹由波函数控制,粒子的分布则反映波函数分布,因此其统计预测结果与传统的量子力学完全一致。量子轨迹理论不仅为我们提供了一条新的途径去解释量子现象,而且提供了一种新的计算方法,因此量子轨迹理论近年来受到越来越多的关注。
第一章介绍了电子能谱和电子显微技术的发展和现状。能谱方面介绍了俄歇电子能谱、X光电子能谱、电子能量损失谱等；在电子显微技术方面,介绍了扫描电子显微镜、透射电子显微镜、扫描透射电子显微镜等。随着像差矫正技术的发展,像差矫正的扫描透射电子显微镜已经达到了0.1nm的分辨率。基于相差矫正技术,原子级分辨率在电子能量损失谱和扫描电子显微镜中也得以实现。对于晶体材料,衍射效应对电子能谱也有着显著的影响。本章还介绍了用于电子能谱和电子显微技术计算模拟的Monte Carlo方法、波动力学计算方法等。
第二章介绍了玻姆力学的发展历程、单粒子和多体玻姆力学基本方程以及基本假定。Bell理论和实验证实量子力学中的隐变量模型必须是非局域的,因此我们讨论了玻姆力学非局域性特性以及它的经典极限。玻姆力学中的量子势是一个全新的物理量,它和玻姆力学的非局域特性相关联,我们对它做了详细讨论。关于量子轨迹的测量问题,我们介绍了利用弱测量技术测量的单光子量子轨迹以及如何通过纠缠光子对观测玻姆轨迹非局域性的实验。还比较了Feynman路径与量子轨迹的异同。最后介绍了三类计算量子轨迹的方法。
第三章利用Monte Carlo模拟方法系统地研究了俄歇电子能谱定量分析中的重要修正因子——背散射因子。计算的能量范围从电离能阈值到30keV,入射电子入射角度以及俄歇电子发射角度的变化范围均为0—89°。共计算了28种元素的主要俄歇跃迁信号的背散射因子。计算采用了新的普适的背散射因子定义式、更为精确的Casnati内壳层电离截面、最新的电子散射Monte Carlo模型,并且通过模拟大量电子轨迹以减少统计误差。计算中还分别考虑了两种电子探测器(轴半球分析仪和筒镜分析器)的几何构置。计算得到的背散射因子能够很好地描述文献中报道的俄歇电子信号强度随入射电子能量以及入射角度的实验变化关系。计算出的背散射因子全部存于一个开源在线数据库中。
第四章采用量子力学Bloch波方法,并结合倒易原理描述电子在晶体中的输运性质,以计算表面电子能谱中信号电子的发射深度分布函数(EMDDF)。明确考虑了晶体结构以及电子相干散射的影响,这在经典Monte Carlo模拟方法中都是被忽略的。电子的非弹性散射通过光学势来表征,我们使用量子力学方法计算了俄歇电子从Au(001)表面发射的EMDDF。它对发射方向以及晶体结构的关系表明,俄歇电子从晶体表面的发射强度与发射方向强烈有关,对晶体结构十分敏感。平均逃逸深度(MED)表征俄歇电子能谱的表面灵敏度,通过当前计算的EMDDF能够导出MED。结算结果表明,晶体样品中的MED是Monte Carlo计算的非晶样品中的MED值的大约一半。
第五章利用玻姆量子轨迹理论研究晶体中电子衍射。首先研究了电子在薄晶样品中的散射行为,用不含时相互作用势场来描述二维和三维模型晶体。通过数值求解含时薛定谔方程计算波函数,然后计算电子运动的量子轨迹,由此得到表征电子衍射行为的实空间几率密度函数和量子轨迹。量子轨迹提供了电子衍射的一个直观认识。对于晶体中的高能电子衍射,我们采用Bloch波方法计算得到了高能电子在晶体中衍射的波函数,量子轨迹则显示了电子在晶体中通道的形成过程。这些计算表明,在电子与固体／表面／纳米结构的相互作用研究中,量子轨迹理论有着巨大的应用前景。如果通过光学势将电子非弹性散射也包含在其中,那么该方法也能同时应用于电子能谱技术中。
第六章采用玻姆量子轨迹理论研究三维实空间中的氦原子在模型表面上的散射。氦原子散射是一种能获得表面结构、无序以及声子谱等信息的表面分析工具。用氦原子从W(112)表面散射的二维平面模型得到的三维模型相互作用势构造模型表面。氦原子散射的动力学过程以玻姆粒子的复杂量子轨迹的形式展示,玻姆力学也成为研究原子与物质相互作用现象的一条新路径。
第七章采用Bloch波方法和倒易原理模拟背散射电子衍射花样。背散射电子衍射花样可以提供非常重要的高分辨晶向信息。我们精确重现了衍射花样的相对强度分布。为模拟大视角的实验衍射花样,需要考虑大量的反射面。我们模拟得到的Mo(001)面背散射电子衍射花样与实验观察结果吻合得非常好,精确再现了实验中的晶带轴精细结构、高角Laue环等特征以及相对强度分布。
第八章是全文的总结。
Electron microscopy and electron spectroscopy techniques have been important tools for material characterization particularly on quantitative analysis. Study of elec-tron-solid interaction, as the physical basis of these techniques, is vital to quantitative interpretation of experimental electron microscopic image and electron spectra. It has been a significant research field of condensed matter physics. This thesis has theoret-ically studied the electron-solid interaction from different perspectives by employing classical Monte Carlo simulation method, wave mechanical method and Bohmian me-chanical method.
Monte Carlo simulation method treats the random scattering process of electrons in a solid and the associated signal generation and emission process. It is easy for numerical calculation and becomes the most powerful tool in study of electron-solid interaction. However, the method is in principle of classical mechanics and omits elec-tron coherent scattering; application is then limited to amorphous and polycrystalline solids. When de Broglie wave length of electrons is comparable with lattice constant of crystalline the coherent scattering (diffraction effects) becomes obvious. Therefore, Monte Carlo simulation method of electron trajectory must be improved to take account of coherent scattering in single crystals.
Bohmian mechanics is a theoretical formalism combining both the accuracy of the predictions of quantum mechanics and the capability of providing an intuitive inter-pretation of the physical phenomena involved. This alternative formalism of quantum mechanics presents significant conceptual differences from the conventional Copen-hagen interpretation. By Bohmian mechanics the particle trajectory is controlled by wave function, and the particle distribution follows that of wave function so that all statistical predictions agree exactly with those of standard quantum mechanics. In ad-dition to its interpretative importance, there has recently been growing interest from the computational point of view for its feature as a new calculation method.
Chapter One reviews the development and present status of electron spectroscopy and electron microscopy, such as, Auger electron spectroscopy (AES), X-ray pho-toelectron spectroscopy (XPS), electron energy loss spectroscopy (EELS), scanning electron microscopy (SEM), transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM). With development of aberration correction technique, aberration-corrected STEM instruments now routinely achieve a resolution of better than0.1nm. Atomic resolution was demonstrated in EELS and STEM based on aberration correction. For the crystal, the diffraction effects on the electron spec-troscopy are also significant. We also introduce Monte Carlo simulation method and wave mechanical method for simulation study in electron spectroscopy and electron microscopy.
Chapter Two reviews the history of Bohmian mechanics. The fundamental equa-tions of motion of Bohm particles for single particle systems and for many-body sys-tems are outlined. We also summarize the basic postulates of Bohmian mechanics. It has been demonstrated by experiment and Bell theorem that only non-local hidden variable model can be compatible with standard quantum mechanics. We present the non-locality of Bohmian mechanics and its classical limit. We also discuss the quantum potential, which is a new physical quantity and connected to non-locality in Bohmian mechanics. The experiments to measure the quantum trajectory of single photon and to observe the nonlocality of Bohmian trajectories with entangled photons are outlined. Feynman path integrals are compared with Bohmian particle paths from various as-pects. Finally we present three numerical methods for calculation of Bohmian quantum trajectories.
Chapter Three studies systematically with a Monte Carlo simulation method an important correction factor, the backscattering factor, in quantitative analysis by Auger electron spectroscopy. The primary energy ranges from the threshold energy of inner-shell ionization to30keV. The incident angle of primary beam and emission angle of Auger electrons are within0-89°. Principal Auger transitions in28pure elements are considered. The calculation employs a novel and general definition of backscattering factor, Casnati's ionization cross section, up-to-date Monte Carlo model of electron scattering, and a large number of electron trajectories are simulated to reduce statistical error. Both the configuration geometries of concentric hemispherical analyzer (CHA) and the cylindrical mirror analyzer (CMA) for Auger electron detection are considered in the calculation. The calculated backscattering factors are found to describe very well an experimental dependence of Auger electron intensity on primary energy and on incident angle in literature. The calculated numerical values of backscattering factor are stored in an open and on-line database.
Chapter Four employs the quantum mechanical Bloch-wave method and the reci-procity principle to describe electron transport in crystalline solids for determining the emission depth distribution function (EMDDF) of signal electrons in surface electron spectroscopy. The crystal structure and electron coherent scattering are explicitly taken into account while they are neglected in classical Monte Carlo methods. While electron inelastic scattering is treated with an optical potential, the EMDDF for Auger-electron emission from an Au (001) surface is calculated by the quantum method. The depen-dence of the EMDDF on emission angle and crystal structure has indicated that Auger electron emission intensity strongly depends on the emission direction and is sensitive to crystal structure. Comparisons made between the present results and classical Monte Carlo simulation results show pronounced differences due to electron-diffraction ef-fects. The mean escape depth (MED) which describes the surface sensitivity of Auger electron spectroscopy can also be evaluated. The present calculation shows that the MED for a crystalline solid is about half that for an amorphous solid from a Monte Carlo calculation.
Chapter Five employs Bohmain mechanics to study electron diffraction in crys-tal. An investigation of electron scattering in a thin crystal has been performed. A time-independent interaction potential was employed to model a2D and3D crystal. Quantum trajectories were calculated through wave function by a numerical solution of the time-dependent Schrodinger equation. The probability density functions and quan-tum trajectories representing electron diffraction in real space were obtained from the calculation. The quantum trajectories provide an intuitive dynamics of the the elec-tron diffraction. For high energy electron diffraction in a crystal, we obtain the wave function by Bloch wave method. The quantum trajectories then present the formation process of electron channeling in crystal. Such example calculations show that the quantum trajectory method has great potential application to theoretical study of elec-tron interaction with solids, surface and nanostructures. Once electron inelastic scat-tering is included via an optical potential, the method can also be applied to electron spectroscopy techniques.
Chapter Six studies helium atom scattering from a model surface in3D real space by using the Bohmian quantum trajectory theory. Helium atom scattering by solid surfaces is a well established technique for obtaining information about the structure, disorder and phonon spectra of the surface. The model surface was described by a3D interaction potential based on the model for the in-plane scattering of helium from a W(112) surface. The dynamic process of helium atom scattering has been presented by quantum trajectories of Bohmian particles. Bohmian mechanics becomes thus a new way to study atom-matter interaction.
Chapter Seven presents a simulation of electron backscattering diffraction (EBSD) patterns with Bloch wave theory and reciprocity principle. EBSD pattern can provide important crystallographic information with high spatial resolution. For the simulation of an experimental image with a large field of view, a large number of reflecting planes are taken into account. Very good agreement is obtained for the simulated and experi-mental EBSD patterns of Mo (001) surface. Experimental features like zone-axis fine structure and higher-order Laue zone rings as well as relative intensity distribution are accurately reproduced.
Chapter Eight is the summary of the thesis.
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