Advanced processing of differential phase contrast data: Distinction between different causes of electron phase shifts

Differential phase contrast, in its high resolution modification also known as first moment microscopy or momentum resolved STEM [1-7] , basically measures the lateral momentum transfer to the electron probe due to the beam interaction with either electrostatic and/or magnetic fields, when the probe transmits the specimen. In other words, the result of the measurement is a vector field p→(x,y) which describes the lateral momentum transfer to the probe electrons. In the case of electric fields, this momentum transfer is easily converted to the electric field E→(x,y) causing the deflection, and from ϱ=ɛ₀∇⋅E→ the local charge density can be calculated from the divergence of the electric field. However, from experimental data it is known that also the calculation of the vector field's curl ∇→×p→ in general yields non-zero results. In this paper, we use the Helmholtz decomposition (Wikipedia contributors, 2022), also known as the fundamental theorem of vector calculus, to split the measured vector fields into their curl-free and divergence-free components and to interpret the physical meaning of these components in detail. It will be shown, that non-zero curl components may be used to measure geometric phases occurring from irregularities in crystal structure such as a screw dislocation.

Properties of the Geometric Phase in Electromechanical Oscillations of Carbon-Nanotube-Based Nanowire Resonators

The geometric phase is an extra phase evolution in the wave function of vibrations that is potentially applicable in a broad range of science and technology. The characteristics of the geometric phase in the squeezed state for a carbon-nanotube-based nanowire resonator have been investigated by means of the invariant operator method. The introduction of a linear invariant operator, which is useful for treating a complicated time-dependent Hamiltonian system, enabled us to derive the analytical formula of the geometric phase. By making use of this, we have analyzed the time behavior of the geometric phase based on relevant illustrations. The influence of squeezing parameters on the evolution of the geometric phase has been investigated. The geometric phase, in large, oscillates, and the envelope of such oscillation increases over time. The rate of the increase of the geometric phase is large when the parameters, such as the classical amplitude of the oscillation, the damping factor, and the amplitude of the driving force, are large. We have confirmed a very sharp increase of the geometric phase over time in the case that the angular frequency of the system reaches near the resonance angular frequency. Our development regarding the characteristics of the geometric phase is crucial for understanding the topological features in nanowire oscillations.