# Control of Nanostructures through Electric Fields

### From DFG-SPP 1253

## Project leaders

TFH Berlin, University of Applied Science

Luxemburgerstr. 10

13353 Berlin

Haus Beuth, Raum 225

Tel: +49 (0)30 4504 5225

Email: hausser@tfh-berlin.de

Homepage: http://www.tfh-berlin.de/~hausser/

Institut für Wissenschaftliches Rechnen

TU Dresden

Willersbau B217

01062 Dresden

phone: 0351 463-34187

phone: 0351 463-34266 (secr.)

fax: 0351 463-37096

email: axel.voigt@tu-dresden.de

Homepage:

## DFG funded assistant

Sandra Rasche (TU Dresden)

sandra.rasche@tu-dresden.de

## Description

This project is motivated by theoretical and experimental observations, that an external (macroscopic) electric field has large influence on the shape evolution of nanostructures on crystalline surfaces.

As a first step to solve a control problem for the evolution of a complicated surface consisting of terraces/islands separated by atomic height steps, we will consider a single island given as a closed curve. The aim is to drive the island by means of an external electric field to a predefined shape. Mathematically this leads to the optimal control of a free boundary problem.

We will investigate this optimal control problem analytically and provide efficient numerical methods. The free boundary problem will be formulated as a degenerate phase-field model where the island boundary is given as the 1/2 level set of a phase-field variable whose dynamic is derived from a Ginzburg-Landau free energy functional.

A key point in the numerical treatment will be the extensive use of adaptive mesh refinement and coarsening. Descretizing the state and adjoint variables on independently adapted meshes will significantly reduce the computational cost.