Control of nanostructures through electric fields

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The manipulation of nanostructures by macroscopic forces is likely to become a key ingredient in many nanotechnology applications. Understanding and controlling the influence of external fields on the shape evolution of nanoscale surface features is therefore of considerable importance. As a first step in this direction we recently investigated the effects of an external electric field on the shape evolution of a single-layer islands on a crystalline surface [1], discovering a remarkable richness of dynamical behavior. We therefore believe, that the microscopic shape evolution of crystalline surfaces may be controlled through a macroscopic electric field, which would have large technological impact.Mathematically this leads to the optimal control of a free boundary problem, where the free boundaries are given by atomic height steps on the surface (e.g. the edge of a single layer island) and the external electric field is the control parameter. Our goal is to investigate this optimal control problem analytically and to provide efficient numerical methods. Using a phase-field approximation, we will consider existence and uniqueness, derive the optimality conditions and numerically solve the system of state and adjoint equations using adaptive finite elements. 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, and these concepts will carry over to a large class of other optimization problems.