Control of nanostructures through electric fields

通过宏观力量操纵纳米结构很可能成为许多纳米技术应用的关键因素。因此，了解和控制外场对纳米尺度表面特征形状演化的影响是相当重要的。作为这个方向的第一步，我们最近研究了外加电场对晶体表面单层岛形状演化的影响[1]，发现了显著丰富的动力学行为。因此，我们认为，晶体表面的微观形状演化可以通过宏观电场来控制，这将产生很大的技术影响。从数学上讲，这导致了自由边界问题的最优控制，其中自由边界由表面(例如单层岛的边缘)上的原子高度步长给出，外部电场是控制参数。我们的目标是对这一最优控制问题进行解析研究，并提供有效的数值方法。利用相场近似，我们将考虑解的存在唯一性，推导出最优性条件，并用自适应有限元数值求解状态方程和伴随方程。自适应网格细化和粗化的广泛使用--在独立适应的网格上减少状态变量和伴随变量--将显著降低计算成本，并且这些概念将延续到一大类其他优化问题。

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.