1. Electromagnetically biased Self-assemblyby
Dr. Narendra Karmarkar
Laboratory for Computational Mathematics

2. 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 ?

3. 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

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

5. 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.

6. 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.

7. 57 point geometry

8. 91 point geometry

9. 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)

10. 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

11. 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

12. 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

13. 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.

14. 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.

15. 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
For general description see
Video of California Institute of Technology or youtube