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 DOT1 Mathematical Modeling, Analysis, and Optimization of Strained GermaniumMicrobridges is to optimize the design of a strained germanium microbridge with respect to the light emission. It is a joint project with the HumboldtUniversität 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 (LeibnizInstitute for Innovative High Performance Microelectronics, Frankfurt Oder).
A crosssection 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 tightbinding calculations developed at IHP in collaboration with Pisa University [R2]. These results are included into van Roosbroecktype 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 quasiFermi 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 bandstructure and optical gain g.

For details about equations (1a1f) 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 bandgap 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 phasefield approach is chosen, i.e., the fraction of each material within Ω is given by the corresponding component of the phasefield vector φ.

Doping optimization: The goal is to find an optimal doping profile C, for which the (negative) netgain G=lg containing gain g and losses l is minimal, so that we achieve the largest possible netgain. 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 xdirection 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.


Publications
[12] 
S. Hittmeir, J. Haskovec, P.A. Markowich, A. Mielke, Decay to equilibrium for energy reactiondiffusion systems,
2016. Submitted, (Matheon preprint, WIAS preprint 2233).

[11] 
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.

[10] 
M. Hintermüller and T. Surowiec, On the directional dierentiability of the solution mapping
for a class of variational inequalities of the second kind. Submitted to Set Valued Var. Anal., 2015.

[9] 
G. Capellini, M. Virgilio, Y. Yamamoto, L. Zimmermann, B. Tillack, D. Peschka, M. Thomas,
A. Glitzky, R. Nürnberg, K. Gärtner, Th. Koprucki, Th. Schroeder, Modeling of an edgeemitting strainedGe laser.
In Advanced Solid State Lasers, Art.no. ATu2A19, Optical Society of America, 2015.

[8] 
M. Hintermüller and T. Surowiec, A bundlefree 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, NikaidoIsodabased pathfollowing methods, and applications. SIAM Journal on Optimization, 25(3):18261856, 2015.

[6] 
D. Peschka, Numerics of contact line motion for thin films. IFAC, 48(1):390393, 2015.

[5] 
D. Peschka, N. Rotundo, and M. Thomas,
Towards doping optimization of semiconductor lasers.
To appear in:
Journal of Computational and Theoretical Transport, 2016, (WIASPreprint 2180, Matheon preprint).

[4] 
T. Koprucki, D. Peschka, M. Thomas,
Towards the Optimization of Onchip 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 EdgeEmitting strainedGe laser.
In: Proceedings of "Advanced Solid State Lasers 2015", OSA Publishing. 13(1):537540.

[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 edgeemitting 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 edgeemitting lasers based on tensile strained germanium microstripes.
IEEE Photonics Journal, 7 (2015) Art.Id 1502115 

Further references
[R1] 
Bandelow, Gajewski, and Hünlich. FabryPerot lasers: Thermodynamicsbased modeling. Optoelectronic Devices. Springer New York (2005). 
[R2] 
Virgilio, Manganelli, Grosso, Pizzi, and Capellini, Radiative recombination and optical gain spectra in biaxially strained ntype 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 metaloxidesemiconductor
process. Optics Express (2014) 

Presentations
 June 1317, 2016, ECMI 2016, Santiago de Compostela;
D. Peschka, talk: Towards the optimization of Ge microbridges

March 0304, 2016, 9th Annual Meeting Photonic Devices, FU Berlin;
D. Peschka, talk: Towards the optimization of onchip germanium lasers.

February 20, 2016, Workshop on Variational Analysis and Optimization, Marianska, Czech Republic;
L. Adam, invited talk: A multimaterial phasefield approach for the optimal design of a GeonSi microlaser.
 February 1617, 2016, WIASDays, TU Berlin;
M. Thomas, talk:
Modeling, analysis, and optimization of germaniumlasers.

December 28, 2015, Shenyang Institute Of Automation,
Chinese Academy Of Sciences, China;
L. Adam, invited talk: Optimal design of a germaniumonsilicon laser.
 September 0711, 2015, SAMM 2015: Materials with Discontinuities, U Stuttgart;
L. Adam, poster: Optimal design of a GeonSi laser aperature via a multimaterial phase field approach.
 August 1014, 2015, ICIAM 8th International Congress on Industrial and Applied Mathematics, Beijing (China);
M. Thomas, invited talk in minisymposium (MSMoD56):
Analysis for edgeemitting 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: NonSmooth CahnHilliard/NavierStokes Problems.
 March 1113, 2015, AMaSiS, WIAS Berlin;
M. Thomas, talk:
Modeling of edgeemitting 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 1718, 2015, WIASDays;
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 CahnHilliard System with NonSmooth 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 projectrelated minisymposia, schools, and workshops
 7th ECM 2016: Minisymposium Mathematical methods for semiconductors at TU Berlin,
organized by N. Rotundo, P. Farrell, D. Peschka (all WIAS).
 The 19th European Conference on Mathematics for Industry 2016: Minisymposium Charge transport in semiconductor materials: Emerging and established mathematical topics in Santiago de Compostela (Spain),
organized by N. Rotundo, P. Farrell, D. Peschka (all WIAS).

November 30December 4, 2015: PDE2015  Theory and Applications of Partial Differential Equations,
WIAS Berlin,
organized by H.C. Kaiser, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci (all WIAS), D. Knees (U Kassel).

July 0711, 2015 SIAM Conference on Control and Its Applications, CT15: Minisymposium MS4 Optimal Control of Multiphase Systems,
organized by D. Wegner (HUB) and M. Hintermüller.
