ECMath OT1: Mathematical Modeling, Analysis, and Optimization of Strained Germanium-Microbridges

Project heads Michael Hintermüller ★☆, Alexander Mielke ★☆,
Thomas Surowiec ☆, Marita Thomas ★
Staff Dirk Peschka ★, Lukas Adam ☆
Institutes Humboldt-Universität zu Berlin, ★ Weierstrass Institute

Project partner Thomas Schröder, Giovanni Capellini at IHP
Collaboration (ECmath) SE2, SE4, SE13
Collaboration (WIAS) Thomas Koprucki, Annegret Glitzky, Reiner Nürnberg
Collaboration (Ext) Michele Virgilio (Pisa, IT), Klaus Gärtner (Lugano,CH)

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

Research goal and background

The goal of the Matheon project D-OT1 Mathematical Modeling, Analysis, and Optimization of Strained Germanium-Microbridges is to optimize the design of a strained germanium microbridge with respect to the light emission. It is a joint project with the Humboldt-Universität zu Berlin (M. Hintermüller, T. Surowiec) and the Weierstrass Institute (A. Mielke, M. Thomas), that also involves the close collaboration with the Department for Materials Research at IHP (Leibniz-Institute for Innovative High Performance Microelectronics, Frankfurt Oder).

A cross-section through a typical material design of a strained microbridge is shown in Figure 1. The key idea is to increase the optical gain by applying a strong mechanical strain together with high doping. The dependence of gain on strain is captured by tight-binding calculations developed at IHP in collaboration with Pisa University [R2]. These results are included into van Roosbroeck-type models for semiconductors and equations for the optical field [R1] by an upscaling procedure [1].

Further details can also be found in publication [4] from the WIAS Annual Research Report 2015.

Figure 1: schematic cross section through a germanium microbridge, indicating optical mode and current densities

Figure 2: Deformation of the microbridge due to SiN stressor layer on top

Mathematical modelling of the strained heterostructure

The macroscopic PDE model for the mechanics, the carrier transport, the optics has the following components
  • van Roosbroeck system for electricstatic ψ and quasi-Fermi potentials φ (1a),
  • waveguide equation for optical mode in semiconductor Θ (1b),
  • rate equation for the photon number S (1c),
  • small strain linear elasticity for strain e(u) of an anisotropic material (1d),

Linear elasticity (1d) can be solved independently. The strain e and doping C then influence the band-structure and optical gain g.

For details about equations (1a-1f) refer to [R1]. One important detail in the model is the dependence of the gain g on the strain, which can be found in recent works by our collaborators Virgilio et al. in [R2]. Joining both approaches by upscaling is a topic of our recent publication [1]. The biaxial strain used there matches with experiments and theory from [R3].

Mathematical optimization

Within the Matheon project OT1 we determine the optimal design of a Ge microstrip based on the optimization of optical properties (the photonic cavity and the optical mode), of mechanical strain, and of the doping profile. Due to the complexity of the underlying system, the shape optimization and the doping optimization are carried out separately.

Topology optimization: The aim here is to determine an optimal configuration of the given materials within a domain Ω such that the band-gap is minimized by maximizing a suitable strain F within a given optical cavity D while adhering to a small strain setting. For the modeling of this, a phase-field approach is chosen, i.e., the fraction of each material within Ω is given by the corresponding component of the phase-field vector φ. Doping optimization: The goal is to find an optimal doping profile C, for which the (negative) net-gain G=l-g containing gain g and losses l is minimal, so that we achieve the largest possible net-gain. We only need to achieve large gain in regions, where the optical mode Θ is localized, which is why only the overlap GΘ² needs to be maximized. However, one still needs to study the dependence on the regularization, i.e. on γ, C and the reference doping C.

Figure 3: Different device topologies (top) IHP reference (middle) optimized with equal strain weights (bottom) optimized with weights favoring strain in x-direction showing (1st column) the material distribution (2nd column) biaxial strain (3rd and 4th column) main mode of Helmholtz problem for λ=1700nm and λ=4000nm.


[13] L. Adam, M. Hintermüller, T.M. Surowiec, A semismooth Newton method with analytical path-following for the H1-projection onto the Gibbs simplex (WIAS preprint 2340, 2016)
[12] A. Mielke, D. Peschka, N. Rotundo, M. Thomas, Gradient structure for optoelectronic models of semiconductors, (WIAS preprint 2340, Matheon preprint 1115, 2016)
[11] S. Hittmeir, J. Haskovec, P.A. Markowich, A. Mielke, Decay to equilibrium for energy reaction-diffusion systems, Submitted, (Matheon preprint, WIAS preprint 2233, 2016).
[10] M. Hintermüller and T. Surowiec, A nonsmooth approach for the optimal control of variational inequalities of the second kind. Submitted to Comput. Optim. Appl., 2016.
[9] M. Hintermüller and T. Surowiec, On the directional differentiability of the solution mapping for a class of variational inequalities of the second kind. Submitted to Set Valued Var. Anal., 2015.
[8] M. Hintermüller and T. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space. 2015. To appear in Math. Prog. Ser. A.
[7] M. Hintermüller, T. Surowiec, and A. Kämmler, Generalized Nash equilibrium problems in Banach spaces: Theory, Nikaido-Isoda-based path-following methods, and applications. SIAM Journal on Optimization, 25(3):1826-1856, 2015.
[6] D. Peschka, Numerics of contact line motion for thin films. IFAC, 48(1):390-393, 2015.
[5] D. Peschka, N. Rotundo, and M. Thomas, Towards doping optimization of semiconductor lasers,
(Journal of Computational and Theoretical Transport 45, pp. 410-423, 2016), (WIAS-Preprint 2180, Matheon preprint).
[4] T. Koprucki, D. Peschka, M. Thomas, Towards the Optimization of On-chip Germanium Lasers.
In: WIAS Annual Research Report 2015.
[3] G. Capellini, M. Virgilio, Y. Yamamoto, L. Zimmermann, B. Tillack, D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, K. Gärtner,
Th. Koprucki, and Th. Schroeder, Modeling of an Edge-Emitting strained-Ge laser.
In: Proceedings of "Advanced Solid State Lasers 2015", OSA Publishing. 13(1):537-540.
[2] D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, M. Virgilio, S. Guha, T. Schröder, G. Capellini, T. Koprucki,
Robustness analysis of a device concept for edge-emitting lasers based on strained germanium.
Topical Collection on "Numerical Simulation of Optoelectronic Devices, NUSOD'15",
Optical and Quantum Electronics Journal, 48:156, 2016.
[1] D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, K. Gärtner, M. Virgilio, S. Guha, T. Schröder, G. Capellini, T. Koprucki, Modeling of edge-emitting lasers based on tensile strained germanium microstripes. IEEE Photonics Journal, 7 (2015) Art.-Id 1502115

Further references

[R1] Bandelow, Gajewski, and Hünlich. Fabry-Perot lasers: Thermodynamics-based modeling. Optoelectronic Devices. Springer New York (2005).
[R2] Virgilio, Manganelli, Grosso, Pizzi, and Capellini, Radiative recombination and optical gain spectra in biaxially strained n-type germanium. Physical Review B (2013).
[R3] Capellini, Reich, Guha, Yamamoto, Lisker, Virgilio, Ghrib, Kurdi, Boucaud, Tillack, and Schröder, Tensile Ge microstructures for lasing fabricated by means of a silicon complementary metal-oxidesemiconductor process. Optics Express (2014)


  • July 18-22, 2016: 7ECM, Berlin, Germany;
    Talk (CS-08-A): Analysis and optimization for edge-emitting semiconductor-heterostructures. (M. Thomas)
    Talk (MS-29-C): Coupling rate-independent and rate-dependent processes: Existence results. (M. Thomas)
    Talk (MS-29-C>: A free boundary problem for the motion of viscous liquids. (D. Peschka)
  • July 01-05, 2016: AIMS 2016, Orlando, Florida, USA; (M. Thomas)
    Invited Talk (SS-02): Analysis and optimization for edge-emitting semiconductor-heterostructures.
    Talk (SS-68): From adhesive contact to brittle delamination in visco-elastodynamics.
  • June 10, 2016: Oberseminar 'Mathematik in den Naturwissenschaften', University of W├╝rzburg, Germany; (M. Thomas)
    Invited Talk: Coupling rate-independent and rate-dependent processes: Delamination models in visco-elastodynamics.
  • March 07-11, 2016: Joint Annual Meeting of GAMM and DMV, Braunschweig, Germany; (M. Thomas)
    Talk: Nonsmooth PDEs in material failure -- Towards dynamic fracture.
  • Feb.22-26, 2016: ERC-Workshop on Modeling Materials and Fluids using Variational Methods (MoMatFlu), WIAS, Berlin;
    Talk: From adhesive contact to brittle delamination in visco-elastodynamics. (M. Thomas)
    Talk: A free boundary problem for the flow of viscous bilayers. (D. Peschka)
  • June 13-17, 2016, ECMI 2016, Santiago de Compostela; (D. Peschka)
    Talk: Towards the optimization of Ge microbridges
  • March 03-04, 2016, 9th Annual Meeting Photonic Devices, FU Berlin; (D. Peschka)
    Talk: Towards the optimization of on-chip germanium lasers.
  • February 20, 2016, Workshop on Variational Analysis and Optimization, Marianska, Czech Republic; (L. Adam)
    Invited talk: A multi-material phase-field approach for the optimal design of a Ge-on-Si microlaser.
  • February 16-17, 2016, WIAS-Days, TU Berlin; (M. Thomas)
    Talk: Modeling, analysis, and optimization of germanium-lasers.
  • December 28, 2015, Shenyang Institute Of Automation, Chinese Academy Of Sciences, China; (L. Adam)
    Invited talk: Optimal design of a germanium-on-silicon laser.
  • September 07-11, 2015, SAMM 2015: Materials with Discontinuities, U Stuttgart;
    L. Adam, poster: Optimal design of a Ge-on-Si laser aperature via a multi-material phase field approach.
  • August 10-14, 2015, ICIAM 8th International Congress on Industrial and Applied Mathematics, Beijing (China);
    M. Thomas, invited talk in minisymposium (MS-Mo-D-56): Analysis for edge-emitting semiconductor heterostructures.
  • June 22, 2015, International Workshop From Open to Closed Loop Control, Graz, Austria;
    M. Hintermüller, talk: Optimal Control of Multiphase Fluids and Droplets.
  • June 13, 2015, ISMP 2015, Pittsburgh, PA, USA;
    M. Hintermüller, talk: Non-Smooth Cahn-Hilliard/Navier-Stokes Problems.
  • March 11-13, 2015, AMaSiS, WIAS Berlin;
    M. Thomas, talk: Modeling of edge-emitting lasers based on tensile strained Germanium microstripes.
  • February 19, 2015, MathMod 2015, TU Vienna, Austria;
    D. Peschka, invited talk in MS10: Numerics of Contact Line Motion for Thin Films.
  • February 17-18, 2015, WIAS-Days;
    M. Thomas, talk: Germanium lasers: Towards the optimization of mechanical strains and doping.
  • January 2015, Bilevel Optimal Control, Heidelberg, Germany;
    T. Surowiec, talk: Nonsmooth analysis and implicit programming approaches for optimal control problems governed by variational inequalities of the 1rst and second kind.
  • December 08, 2014, WSA 2014, Rio de Janeiro;
    M. Hintermüller, plenary talk: Optimal Control of the Cahn-Hilliard System with Non-Smooth Homogeneous Energy Densities.
  • December 04, 2014, Seminar Dünne Filme, Freie Randwertprobleme, TU Berlin;
    D. Peschka, invited talk: Modeling of suspension flows and its benefit for undergraduate studies.
  • October 16, 2014, RICAM, Linz (Austria);
    M. Hintermüller, invited talk: Shape and Topological Sensitivity Based Methods in Tomographic Reconstruction and Image Segmentation.
  • September 09, 2014, NHOC 2014, Cascais (Portugal);
    M. Hintermüller, plenary talk: Generalized Nash Equilibrium Problems in Banach Spaces.

Organization of project-related minisymposia, schools, and workshops