ECMath-Projekt OT7 "Model-based geometry reconstruction of quantum dots from TEM"

Project heads Thomas Koprucki, Karsten Tabelow

Staff Anieza Maltsi
Funding period June 1st, 2017 - December 31st, 2018
Institute Weierstrass Institute for Applied Analysis and Stochastics

Project partner Jörg Polzehl (WIAS Berlin),
Tore Niermann (TU Berlin, Institut für Optik und Atomare Physik, AG Lehmann)
Collaboration (ECMath) Project CH15 Analysis of Empirical Shape Trajectories
Collaboration (Extern) Anuj Shrivastava (Dept. of Statistics, Florida State University, Tallahassee),
Sebastian Kurtek (Dept. of Statistics, Ohio State University, Columbus)

This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin.

Background and aim of the project

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.
MBGR requires contributions from two mathematical disciplines: the simulation of TEM images by the numerical solution of partial differential equations for the propagation of the electron wave through the sample and statistical methods from functional data analysis.

The mapping of QD geometry to TEM images is governed by the relativistic Schröd-inger equation for dynamic electron scattering (DeGraef 2003).

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.


  • D. Bimberg, M. Grundmann, and N.N. Ledentsov. Quantum dot heterostructures. John Wiley & Sons, 1999.
  • D. Bimberg. Der Zoo der Quantenpunkte - Mit Halbleiter-Quantenpunkten zu neuartigen Bauelementen. Physik Journal, 5(8/9):43-50, 2006.
  • S. Buckley, K. Rivoire, and J. Vučkovć. Engineered quantum dot single-photon sources. Reports on Progress in Physics, 75(12):126503, Nov 2012.
  • M. De Graef. Introduction to conventional transmission electron microscopy. Cambridge University Press, 2003.
  • Th. Fletcher and M. Zhang. Probabilistic Geodesic Models for Regression and Dimensionality Reduction on Riemannian Manifolds, pages 101-121. In Turaga and Srivastava, 2016.
  • I.H. Jermyn, S. Kurtek, E. Klassen, and A. Srivastava. Elastic shape matching of parameterized surfaces using square root normal fields. In ECCV 2012.
  • S. Kurtek and H. Drira. A comprehensive statistical framework for elastic shape analysis of 3d faces. Computers & Graphics, 51:52-59, 2015.
  • S. Kurtek, I.H. Jermyn, Q. Xie, E. Klassen, and H. Laga. Elastic Shape Analysis of Surfaces and Images, pages 257-277. In Turaga and Srivastava, 2016.
  • C. Santori, D. Fattal, and Y. Yamamoto. Single-photon Devices and Applications. Wiley-VCH, Weinheim, 2010.
  • A. Strittmatter, A. Holzbecher, A. Schliwa, J.-H. Schulze, D. Quandt, T. Germann, A. Dreismann, O. Hitzemann, E. Stock, I. Ostapenko, S. Rodt, W. Unrau, U. Pohl, A. Hoffmann, D. Bimberg, and V. Haisler. Site-controlled quantum dot growth on buried oxide stressor layers. physica status solidi (a), 209(12):2411-2420, 2012.
  • A. Srivastava and E. Klassen. Functional and Shape Data Analysis. Springer, 2016.
  • A. Schliwa, M. Winkelnkemper, and D. Bimberg. Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)AsGaAs quantum dots. Phys. Rev. B, 76:205324, 2007.
  • P.K. Turaga and A. Srivastava, editors. Riemannian Computing in Computer Vision. Springer International Publishing, 2016.