AMaSiS 2018 Workshop: Abstracts

Poster Computing TEM images of semiconductor nanostructures

Anieza Maltsi${}^{\mathbf{\text{(1)}}}$, Thomas Koprucki${}^{\mathbf{\text{(1)}}}$, Tore Niermann${}^{\mathbf{\text{(2)}}}$, Timo Streckenbach${}^{\mathbf{\text{(1)}}}$, Karsten Tabelow${}^{\mathbf{\text{(1)}}}$, Jörg Polzehl${}^{\mathbf{\text{(1)}}}$

(1) Weierstrass Institute for Applied Analysis and Stochastics, Berlin

(2) Technische Universität Berlin, Institut für Optik und Atomare Physik

Transmission electron microscopy (TEM) of semiconductor nanostructures such as quantum dots (QDs) or quantum wells embedded in bulk-like samples (thickness 100-300 nm) is an imaging modality with a spatial resolution of 0.5-1 nm. It is characterized by a highly non-linear behavior of the dynamic electron scattering, non-local effects due to strain and strong stochastic influences due to uncertainties in the experiment. Thus, the interpretation and analysis of TEM images typically renders difficult and currently relies on understanding the imaging process for model geometries by simulation.

Here, we present a mathematical model for the numerical simulation of TEM images. This includes elasticity theory to obtain the strain profile coupled with the Darwin-Howie-Whelan equations [1], which describe the propagation of the electron wave through the sample. With the availability of such simulations for semiconductor nanostructures the influence of specimen properties on the image formation can be examined. This enables new reconstruction methods, e.g., the model-based geometry reconstruction (MBGR) for geometric properties of QDs, from TEM images as proposed in [2].

The dynamic electron scattering in crystalline solids, e.g., semiconductor nanostructures, is influenced by spatial variations in the chemical composition and by local deformations of the lattice due to elastic strain. To model the elastic relaxation of the misfit-induced strain we employ continuum mechanics and follow the concept of Eshelby’s inclusion [3]. An anisotropic elastic stiffness tensor is then used to obtain the displacements in the equilibrium configuration. Further, the propagation of the electron wave through the specimen is governed by the relativistic Schrödinger equation [1]. A multi-beam ansatz for the wave function in combination with the column approximation finally leads to the Darwin-Howie-Whelan equations [1]. They describe the propagation of the coupled beams through the sample depending on the variations in the chemical composition and the lattice deformations given by the displacement field.

We demonstrate the non-local and nonlinear behavior of the image formation by numerical results on simulated TEM images for a specific class of semiconductor nanostructures, namely InAs/GaAs QDs. For the computation of the elastic deformation a FEM-based elasticity solver implemented in the WIAS-pdelib has been used. A simulated TEM image is generated by propagating the beams through the sample for every pixel and numerically solving the DHW equations with pyTEM, a software by TU Berlin, using the displacement fields from the elasticity solver. This accurate mathematical model can, e.g., be used to generate a database of simulated TEM images for different configurations, which is a prerequisite for a geometry reconstruction approach like MBGR [2].

Acknowledgments: This research is carried out in the framework of MATHEON supported by Einstein Foundation Berlin (project OT7).