Electromagnetically biased Self-assembly by Dr. Narendra Karmarkar Laboratory for Computational Mathematics narendrakarmarkar@yahoo.com

Towards Fabrication of nano-scale and finer structures Current Status : many nano-scale devices demonstrated in Labs but, However, unlike CMOS, no method of assembling them on large scale on a substrate in reliable, reproducible way. Top Down : Optical lithography – current practice, resolution limited by diffraction effects Mask-less methods – E-beam, FIB – accurate but low throughput Interference lithography – produces simple periodic patterns at nano-scale Bottom-Up : Self-assembly – fast, high throughput - some structures self-assemble in nature, but not the structures we want Can we make more general, customized, complex, sparse patterns ?

Role of Global Optimization Our approach – based on insights derived from our advances in interior point theory of global optimization for situations having multiple global optima Stable structures -- minimum energy configuration Optimization in Nature : Nature solves optimization problems all the time, laws of nature optimality conditions. - minimum energy principle Presence of Symmetry : Optimal configurations often have high degree of symmetry

Symmetries in naturally occurring structures Example – Crystal Structure has Translational Symmetry Potential energy function governing electron motion is periodic in 3D

Mathematical Analysis of the structure 3D translational symmetries of the structure form a group 3-space can be partitioned exactly into Wigner-Seitz Cells Motion of electrons in the resulting structure can be analyzed theoretically. Theory of semiconductors based on such analysis, -- preceded invention of transistors Even simple translational symmetry existing in nature has led to many useful electronic devices. Can we go beyond naturally occurring structures by artificially creating symmetries designed with purpose ? e.g. Choose symmetries governing “perfect patterns” – a powerful abstract concept for parallel computing.

Example of Artificially Created Symmetry This structure is based on Finite projective geometry Behavior of electrons in systems with such symmetries can be analyzed theoretically, just as crystals with natural translational symmetry permit theoretical analysis The structure allows powerful method of organizing parallel computation.

57 point geometry

91 point geometry

Other Examples of Artificially Created Symmetric Structures Photonic Crystals : Symmetry Group : periodic structure, occures in nature Material : artificially created Our approach is to go one step beyond : Symmetry Group : custom desiged for application Advantages : Structures based on automorphisms of projective geometry allows us to overcome major limitation of CMOS for parallel computing by exploiting parallelism inherent in Quantum Mechanics for communication , while doing logic in silicon ( Quantum computing proposes to use it for logic, in future)

Self-assembly of customized complex patterns First, we have to solve a global optimization problem very different from traditional. Traditional optimization theory deals with “convex” problems essentially unique global minimum However, optimization problems, with multiple, global minima, arise in many contexts - In nature - In engineered systems They lead to number of inter-related problems Finding Optimal Solution Proving Optimality Inverse Problem Synthesis

Synthesis Given specification of desired global minima in terms of their - value - locations - or shape of energy landscape around the minima, - how do we design a system with appropriate energy function, subject to further constraints? - mixture of “direct” and “inverse” problems in the same context. We solve the synthesis problem to create Energy Landscape With Multiple Global Minima based on insights derived from our advances in interior point theory of global optimization

Electromagnetically biased self-assembly The Next step is to create Electromagnetic bias favoring structure we would like to be self - assembled To enable this, we have designed an “Electromagnetic Cavity Machine” In previous applications to microwave devices, lasers, or particle accelerators walls of electromagnetic cavities were just passive reflectors. In contrast, in our design, cavity walls are lined with arrays of electrodes used for path length modulation. Actual physical mechanism used for modulation such as MEMS, inverse piezoelectric effect etc. varies with energy range Our Inverse Optimization Algorithm finds control signals to be applied to control cavity wall electrodes control wiggler shape in case source is a free electron laser

Electromagnetic Cavity Machine Just as operation of a laser tweezer to trap and move atoms can be understood in terms of optical potential with unique global minimum, Operation of Electromagnetic Cavity Machine can be understood Using theory of global optimization with multiple global minima Boundary Value Problem The 2-D boundary values applied to the cavity walls uniquely determine the effective 3-D optical potential in cavity volume The 3-D potential controls evolution of multiple global optima in space-time to deposit atoms trapped in those minima into desired structure. This machine for manufacturing self-assembled customized nanostructures can be explained as a perfectly co-ordinated orchestra of millions of laser tweezers.

Symmetry and Stability Any random perturbation deviating from symmetry increases energy, pulling the structure back towards restoring symmetry. i.e. Symmetry has self-stabilising effect. Even after the electromagnetic field leading to the structure is withdrawn, the structure remains ( upto some finite temperature) The main role of the electromagnetic field is to provide an artificially created pathway towards the structure.

Summary The proposed approach can significantly enlarge the class of structures that can be assembled This includes some symmetric structures with powerful properties. For more information/ papers : Follow links on wikipedia page about author email : narendrakarmarkar@yahoo.com For general description see Video of California Institute of Technology or youtube