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#1 2021-12-18 18:57:31

Hannibal lecter
Registered: 2016-02-11
Posts: 392


what is microfabrication related to mathematics?
if anyone have ideas, comments, opinion, on that topic please talk here

Last edited by Hannibal lecter (2021-12-18 18:58:29)

Wisdom is a tree which grows in the heart and fruits on the tongue


#2 2021-12-18 20:31:54

Jai Ganesh
Registered: 2005-06-28
Posts: 47,159

Re: microfabrication


Microfabrication is the process of fabricating miniature structures of micrometre scales and smaller. Historically, the earliest microfabrication processes were used for integrated circuit fabrication, also known as "semiconductor manufacturing" or "semiconductor device fabrication". In the last two decades microelectromechanical systems (MEMS), microsystems (European usage), micromachines (Japanese terminology) and their subfields, microfluidics/lab-on-a-chip, optical MEMS (also called MOEMS), RF MEMS, PowerMEMS, BioMEMS and their extension into nanoscale (for example NEMS, for nano electro mechanical systems) have re-used, adapted or extended microfabrication methods. Flat-panel displays and solar cells are also using similar techniques.

Miniaturization of various devices presents challenges in many areas of science and engineering: physics, chemistry, materials science, computer science, ultra-precision engineering, fabrication processes, and equipment design. It is also giving rise to various kinds of interdisciplinary research. The major concepts and principles of microfabrication are microlithography, doping, thin films, etching, bonding, and polishing.

Fields of use

Microfabricated devices include:

* integrated circuits (“microchips”) (see semiconductor manufacturing)
* microelectromechanical systems (MEMS) and microoptoelectromechanical systems (MOEMS)
* microfluidic devices (ink jet print heads)
* solar cells
* flat panel displays (see AMLCD and thin-film transistors)
* sensors (microsensors) (biosensors, nanosensors)
* power MEMS, fuel cells, energy harvesters/scavengers


Microfabrication technologies originate from the microelectronics industry, and the devices are usually made on silicon wafers even though glass, plastics and many other substrate are in use. Micromachining, semiconductor processing, microelectronic fabrication, semiconductor fabrication, MEMS fabrication and integrated circuit technology are terms used instead of microfabrication, but microfabrication is the broad general term.

Traditional machining techniques such as electro-discharge machining, spark erosion machining, and laser drilling have been scaled from the millimeter size range to micrometer range, but they do not share the main idea of microelectronics-originated microfabrication: replication and parallel fabrication of hundreds or millions of identical structures. This parallelism is present in various imprint, casting and moulding techniques which have successfully been applied in the microregime. For example, injection moulding of DVDs involves fabrication of submicrometer-sized spots on the disc.

Mathematical modeling of focused ion beam microfabrication:


A mathematical model for sputtering a shape or cavity with an arbitrary cross-sectional profile has been developed for focused ion beam milling. The ion beam is assumed to have a Gaussian intensity distribution and a submicron width. The model solves for ion beam dwell times on a pixel grid which yields the desired feature depth as a function of the pixel (x,y) coordinate. The solution is unique and accounts for the ion beam flux contribution at any point from all other pixels in the address matrix. A semiempirical sputter yield treatment allows for a very wide range of ion beam/solid combinations and for yield variations with ion energy and angle of incidence. Solutions have been obtained for parabolic surfaces of revolution, a parabolic trench (with a plane of symmetry) and a hemispherical pit. Either a square or a circular pixel matrix was used for the parabolic shapes. Correspondence between the predictions of the model and experimental 20 keV Ga+ sputtering of a parabolic cross-section trench in Si(100) was within the limits of the accuracy of the experimental control.

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