EF3 – Model-Based Imaging



Model-Based Geometry Reconstruction from TEM Images

Project Heads

Thomas Koprucki, Karsten Tabelow

Project Members

Anieza Maltsi (WIAS) 

Project Duration

01.01.2019 – 31.12.2021

Located at



Semiconductor quantum dots are nanostructures that form a technological path to innovative optoelectronic and photonic devices (Bimberg 2006). Among them single quantum dots are promising candidates for single and entangled photon sources which are of importance for future quantum technologies such as quantum information processing, quantum cryptography, and quantum metrology (Santori et al. 2010; Buckley et al. 2012).


Quantum dots (QDs) are composed of two materials: the quantum dot material, e.g., InAs, and the surrounding crystal (GaAs). The lattice mismatch between both materials causes a mechanical strain field that strongly influences the QD electronic states (Schliwa 2007). QDs can be produced by a self-organized epitaxial growth process (Stranski-Krastanow growth), where shape, size and density of the QDs can be controlled by process parameters (Bimberg et al. 1999). With the buried stressor approach it is even possible to nucleate single QDs in prescribed spatial regions (Strittmatter et al. 2012).

The growth of QDs with desired electronic properties would highly benefit from the assessment of QD geometry, distribution, and strain profile in a feedback loop between growth and analysis of their properties. One approach to assist the optimization of QDs consists in imaging bulk-like samples (thickness 100-300 nm) by transmission electron microscopy (TEM) instead of high resolution (HR) TEM of capped samples (thickness 10 nm). The sample preparation for HRTEM is much more time-consuming, strongly modifies the strain field and potentially destroys the QDs. However, a direct 3D geometry reconstruction from TEM of bulk-like samples by solving the tomography problem is not feasible due to its limited resolution (0.5-1 nm) and strong stochastic influences, e.g., detector noise, spatially correlated events on the detector array.

In this project, we will therefore develop a novel 3D model-based geometry reconstruction (MBGR) of QDs. This will include

  • an appropriate model for the QD configuration in real space,
  • a characterization of corresponding simulated TEM images as well as
  • a statistical procedure for the estimation of QD properties and classification of QD types based on acquired TEM image data.
Recent years have seen tremendous progress in the design and application of functional data analysis (FDA) especially in the field of computer vision (Turaga & Srivastava 2016). An effective and elegant approach for FDA on surface data is elastic shape analysis (Kurtek & Dira 2015; Kurtek et al. 2016), see (Jermyn et al. 2012) for the special case of images. It enables statistical data analysis, see e.g. Fletcher & Zhang 2016; Kurtek & Dira 2015 (or Srivastava & Klassen 2016) for functional principal component analysis in the special case of contours) that we will use for MBGR.

The MBGR approach will enable a high-throughput characterization of QD samples by TEM via QD geometry, distribution and strain field. Furthermore, it will provide a guiding example for mathematically enhanced microscopy for the reconstruction of other nanoscale objects in different applications.

Simulation workflow of TEM images
To simulate TEM images of quantum dots we need to solve the elasticity problem and the relativistic Schrödinger equation for dynamic electron scattering (DeGraef 2003).

The WIAS-pdelib software is used to find the displacement and the pyTEM software, from TU Berlin, to simulate the corresponding TEM image.

Model Based Geometry Reconstruction
Model-based geometry reconstruction requires the simulation of TEM images and methods like deep learning or elastic shape analysis to solve the inverse problem.

Project Webpages


Selected Publications

  • Koprucki, T., Maltsi, A. and Mielke, A.. On the Darwin–Howie–Whelan equations for the scat- tering of fast electrons described by the Schr ̈odinger equation. SIAM Journal on Applied Math- ematics, 81(4):1552–1578, 2021. doi:10.1137/21M139164X.
  • Maltsi, A., Niermann, T., Streckenbach, T. et al. Numerical simulation of TEM images for In(Ga)As/GaAs quantum dots with various shapes. Opt Quant Electron 52, 257 (2020). https://doi.org/10.1007/s11082-020-02356-y
  • Maltsi, A., Koprucki, T., Niermann, T., Streckenbach, T., Tabelow, K. and Polzehl, J. (2018), Model‐based geometry reconstruction of quantum dots from TEM. Proc. Appl. Math. Mech., 18: e201800398. doi:10.1002/pamm.201800398
  • T. Koprucki, A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow and J. Polzehl, “Towards Model-Based Geometry Reconstruction of Quantum Dots from TEM,” 2018 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Hong Kong, 2018, pp. 115-116, doi: 10.1109/NUSOD.2018.8570246

Selected Pictures

Simulated TEM images of four different shapes

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