Three-beam electron diffraction for structural investigations
thesis
posted on 2017-03-15, 05:28 authored by Yueming GuoTransmission electron
microscopy and diffraction can provide structural information of materials at
high spatial resolutions. Due to the strong interaction between electrons and
matter, the phase information of structure factors, which is lost in kinematic
diffraction, is preserved in dynamic electron diffraction. Nevertheless, it is
still difficult to use dynamically diffracted intensities to solve an unknown
crystal structure [1].
This work has developed a method for the measurement of the three-phase invariant, which is the summation of three structure factors phases (ϕ = φg+ φ-h+ φh-g), in noncentrosymmetric crystals from convergent-beam electron diffraction (CBED) patterns. CBED patterns are taken in special crystal orientations known as three-beam conditions, where two reflections, g and h, satisfy their Bragg conditions simultaneously. Unlike direct methods, which derive probability distributions of the cosine phase invariants from kinematically diffracted intensities [2], three-beam CBED allows for physical measurements of three-phase invariants (including the signs) from dynamically diffracted intensities. It has been shown that replacements of the randomly assigned values of three-phase invariants with the measured ones as input to the direct methods can greatly improve phasing [3]. Therefore, it may be expected that three-beam electron diffraction may play a significant role in solving a crystal structure.
The research on three-beam electron diffraction was initiated several decades ago [4-19]. However, there are still some fundamental problems that need to be tackled. In the case of centrosymmetric crystals (where ϕ=0 or π), an simple inversion of three-beam dynamic diffraction has been completed [10, 11, 20], which enables the determination of the three-phase invariants by just inspection of the three-beam CBED patterns [17, 18, 21]. In the case of noncentrosymmetric crystals (where ϕ can be any value between 0 and 2π), previous analytical theories of three-beam electron diffraction have included some approximations, which are based on perturbing kinematic [14] or two-beam dynamic diffraction [6, 15, 22], for inverting three-phase invariants. Due to the limitations of these approximations, phase measurements are limited by the applicable range of these approximations. Based on reduction of the exact solution to three-beam electron diffraction, the current work has developed a new method which allows for the determination of three-phase invariants to within 45° only by inspection of the three-beam CBED patterns without the necessity of knowing the specimen thickness or the structure factor magnitudes. This thesis has also implemented large-angle rocking beam electron diffraction (LARBED) [23] to demonstrate the experiments for the new method.
In addition, an analytical theory of three-beam electron diffraction given here (which is developed from [24]) has inspired a novel approach for local composition measurement in a technically important semiconductor, InxGa1-xAs. This approach can provide simultaneous yet independent measurements of composition, thickness and possibly strain from three different parts of a single CBED pattern which is recorded in a specific three-beam condition. The composition measurement does not require any sophisticated procedures like refining the intensities in CBED but needs only a simple comparison of a certain intensity ratio in the CBED pattern to a pre-calculated look-up table based on Bloch wave calculations of many-beam diffraction. The composition measurement is not only simple but also has the potential to be very accurate and precise when compared to existing methods of composition measurement. Further simulations which contain the finite element method and multislice calculations of CBED have suggested that the current approach can be used in practical specimens, such as cross-sectional specimen of InxGa1-xAs/GaAs quantum wells.
This work has developed a method for the measurement of the three-phase invariant, which is the summation of three structure factors phases (ϕ = φg+ φ-h+ φh-g), in noncentrosymmetric crystals from convergent-beam electron diffraction (CBED) patterns. CBED patterns are taken in special crystal orientations known as three-beam conditions, where two reflections, g and h, satisfy their Bragg conditions simultaneously. Unlike direct methods, which derive probability distributions of the cosine phase invariants from kinematically diffracted intensities [2], three-beam CBED allows for physical measurements of three-phase invariants (including the signs) from dynamically diffracted intensities. It has been shown that replacements of the randomly assigned values of three-phase invariants with the measured ones as input to the direct methods can greatly improve phasing [3]. Therefore, it may be expected that three-beam electron diffraction may play a significant role in solving a crystal structure.
The research on three-beam electron diffraction was initiated several decades ago [4-19]. However, there are still some fundamental problems that need to be tackled. In the case of centrosymmetric crystals (where ϕ=0 or π), an simple inversion of three-beam dynamic diffraction has been completed [10, 11, 20], which enables the determination of the three-phase invariants by just inspection of the three-beam CBED patterns [17, 18, 21]. In the case of noncentrosymmetric crystals (where ϕ can be any value between 0 and 2π), previous analytical theories of three-beam electron diffraction have included some approximations, which are based on perturbing kinematic [14] or two-beam dynamic diffraction [6, 15, 22], for inverting three-phase invariants. Due to the limitations of these approximations, phase measurements are limited by the applicable range of these approximations. Based on reduction of the exact solution to three-beam electron diffraction, the current work has developed a new method which allows for the determination of three-phase invariants to within 45° only by inspection of the three-beam CBED patterns without the necessity of knowing the specimen thickness or the structure factor magnitudes. This thesis has also implemented large-angle rocking beam electron diffraction (LARBED) [23] to demonstrate the experiments for the new method.
In addition, an analytical theory of three-beam electron diffraction given here (which is developed from [24]) has inspired a novel approach for local composition measurement in a technically important semiconductor, InxGa1-xAs. This approach can provide simultaneous yet independent measurements of composition, thickness and possibly strain from three different parts of a single CBED pattern which is recorded in a specific three-beam condition. The composition measurement does not require any sophisticated procedures like refining the intensities in CBED but needs only a simple comparison of a certain intensity ratio in the CBED pattern to a pre-calculated look-up table based on Bloch wave calculations of many-beam diffraction. The composition measurement is not only simple but also has the potential to be very accurate and precise when compared to existing methods of composition measurement. Further simulations which contain the finite element method and multislice calculations of CBED have suggested that the current approach can be used in practical specimens, such as cross-sectional specimen of InxGa1-xAs/GaAs quantum wells.
