**Project Heads**

Patricio Farrell, Annegret Glitzky, Matthias Liero, Barbara Zwicknagl

**Project Members**

Grigor Nika

**Project Duration**

01.01.2021 − 31.12.2022

**Located at**

WIAS

Mechanical strain has a strong impact on the electronic and optical properties of semiconductor materials. Therefore, strain engineering is crucial in novel optoelectronic device designs e.g. for Germanium microbridges or quantum dots. Also elastomeric polymer LEDs emit significant light if they are exposed to strains in the range of 120%.

Considerable strain is reached in quasi lower-dimensional structures like nano- and microwires or thin-film (organic) LEDs and solar cells. Our partners at Paul-Drude Institute (PDI) observe a strong influence of strain on the mobility in nanowires [4]. Whereas in practice such phenomena are often described by small-strain elasticity as input for charge-carrier transport and no backward coupling is taken into account, in this project we consider the full coupling in the finite-strain setting.

The project follows three main topics: (i) We derive nonlinear PDE models for the coupling of charge-carrier transport and finite-strain elasticity. We plan a consistent modeling using the GENERIC framework (see [1,6,7]) and want to seek the consistency with the small-strain setting. (ii) Our analytical investigations concern the isothermal situation with realistic mixed boundary conditions. For different temporal regimes we study the existence of solutions and their properties. Analytical tools are time discretization, Galerkin approximation, and Schauder’s fixed-point theorem. Moreover, we assume second-grade non-simple materials as in [8,9]. Additionally, we derive effective models for thin wires, where the elasticity is reduced to a 1D rod model, see [2], and the charge transport remains fully 3D. (iii) The final aim is develop and approve new numerical techniques for specific applications. We derive structure preserving discretizations based on finite-volume methods and generalized Scharfetter-Gummel schemes that include mechanical (small-) strain. Here, we combine the existing ddfermi solver for charge-transport with a linear elasticity solver. Additionally, we include the backward coupling to elasticity and perform simulations for single nanowire investigated at our partners at PDI.

**External Website**

**Event**s

**Related Publications
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- M. Grmela and H. Chr. Öttinger: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E (3) 56.6 (1997), pp. 6620–6632.
- R. V. Kohn and E. O’Brien: On the bending and twisting of rods with misfit, Journal of Elasticity, 130 (2018) pp. 115–143.
- M. Kručík and T. Roubíček: Mathematical methods in continuum mechanics of solids, Cham: Springer, 2019.
- R. B. Lewis, P. Corfdir, H. Küpers, T. Flissikowski, O. Brandt, and L. Geelhaar: Nanowires bending over backward from strain partitioning in asymmetric core-shell heterostructures, Nano Letters, 18 (2018) pp. 2343–2350.
- A. Mielke: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity 24.4 (Mar. 2011), pp. 1329–1346.
- A. Mielke: Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn. 23.3 (2011), pp. 233–256.
- H. Chr. Öttinger and M. Grmela: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E (3) 56.6 (1997), pp. 6633–6655.
- T. Roubíček and A. Mielke: Thermoviscoelasticity in Kelvin–Voigt rheology at large strains, Arch. Ration. Mech. Anal., 238 (2020)
- T. Roubíček and G. Tomassetti: Dynamics of charged elastic bodies under diffusion at large strains, DCDS-B 25.4 (2020) pp. 1415–437.

**Related Pictures
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