nanomanufacturing by bottom-up growthacg.cs.tau.ac.il/courses/computational-geometry/... ·...

Nanomanufacturing by Bottom-up Growth

An Application of Voronoi Diagrams

Karl F Böhringer University of Washington

Seattle, WA, USA

Lectures on Computational Geometry Tel Aviv University

Tel Aviv, Israel, 26 March 2012

Manufacturing

• “Top-down” subtractive manufacturing:

– Milling, drilling, turning, boring, broaching, sawing, shaping, cutting, planing, reaming, tapping, grinding, etching, molding, …

• “Bottom-up” additive manufacturing

– Assembly, synthesis, growth, …

26 March 2012 © Karl F Böhringer

Some Useful Terms and Definitions

• MEMS: microelectromechanical systems – tiny mechanisms made of silicon, such as accelerometers in cell phones and cars. – milli: 10–3 , micro: 10–6 , nano: 10–9

• Electrodeposition: common technique to coat objects with metals; dissolved metals (positive ions) deposit on negatively charged electrodes from a solution (electrolyte).

• Isotropic: having properties that are identical in all directions. 26 March 2012 © Karl F Böhringer

Desired Object Seed Planting

Software Seed Positions

Output Object Electrodeposition

Growth Electron Beam Lithography Patterning

20 μm

Bottom-up Nanomanufacturing

Automated synthesis of product:


Orchestrated Structure Evolution

Automated OSE process:

A. Enter desired pattern (CAD)

B. Generate seeds (algorithms, models)

C. “Plant” (by electron beam or dip-pen) and grow seeds (electrolyte)


How Does Growth Proceed?

Positive ions move towards negative electrodes


Top View of Growth Regions

• Voronoi diagram: decomposition of space into cells; a Voronoi cell is the set of all points whose distance to a given object is not greater than their distance to the other objects – Note: distance metric does not

need to be Euclidean

Image source: wikipedia.org


Growth of Simple Structures by Electrodeposition

• Disks and squares

• Place seeds across areas

• Grow structures until areas are covered

• How accurate is coverage?


Bottom-up Nanomanufacturing

Orchestrated structural evolution – results:

- 10m


0%

5%

10%

15%

20%

25%

30%

e are

a

Pitch ratio

C 1:2 1:5 1:10

Time and quality trade-off

Deposited at -350mV vs. SEC in 0.5M CuSO4 and 0.5M H2SO4

Growth error (area)

Pitch ratio (Seed size : Pitch distance)

0%

5%

10%

15%

20%

25%

30%

e pe

rim

ete

r

Pitch ratio C 1:2 1:5 1:10

Growth error (perimeter)

0

5

10

15

20

25

C 1:2 1:5 1:10

Tim

e (

s)

Growth Time

Write Time

We can reduce process time with small quality tradeoff

Conventional

No CMP

No CMP

Orchestrated Structure Evolution

Growth models for OSE:

Structure evolution may be limited by

• charge transfer (surface)

• mass transfer (bulk)


OSE Design Challenges

• Forward problem: what is end result of OSE?

– Look at Voronoi diagram!

• Inverse problem: how to choose seeds and process conditions, given a desired outcome?

– Solution for simplified case see [Thiyanaratnam et al. SIAM’09]


Anisotropic growth

Space-filling growth

Solution for Simplified Problem


P. Thiyanaratnam et al., SIAM J. APPL. MATH. Vol. 69, No. 4, pp. 1043–1064

General Solution of Seed Layout Problem

• Solution by P. Thiyanaratnam et al. assumes isotropic growth with constant growth rate

• In practice, growth rate is not isotropic and time-varying

• Seeds compete for ions during electrodeposition ( Voronoi regions)

• General solution is inverse design problem with complex underlying physics

• Numerical solutions: S. Abbasi et al. Nanotech. 2011


nanomanufacturing by bottom-up growthacg.cs.tau.ac.il/courses/computational-geometry/... ·...

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