Summer School for Integrated Computational Materials Education 2018

Summer School for Integrated Computational Materials Education 2018 Kinetics Module Review Katsuyo Thornton, 1 Edwin Garcia, 2 Larry Aagesen, 3 Mark Asta 4, Jonathan Guyer 5 1. 2. 3. 4. 5. Department of Materials Science & Engineering, University of Michigan Purdue University Idaho National Laboratory University of California, Berkeley National Institute of Standards and Technology

Purposes of Kinetics Module • Develop deeper understanding of diffusive transport through hands-on exercises. • Learn how computational tools can be used to determine concentration profiles during diffusion. • Demonstrate the technological importance of diffusion through an application to a semiconductor processing problem. Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 2

Concepts Illustrated Through Kinetics Module 1. Diffusion – – – Driving Force Fick’s Law Mass Conservation 2. Semiconductor Processing 3. Computational Kinetics 4. Fi. Py Part 1 Part 2 Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 3

Driving Force for Diffusion • Consider 1 D diffusion. • The atoms are randomly hopping right and left. • Half the atoms are moving toward right, and the other half is moving to left. Concentration • Below, left side has more atoms than right. • Net flux toward the low concentration. • Driving force = High conc. Low conc. Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 x 4

Fick’s First Law • The flux is proportional to the driving force. • The proportionality constant is the diffusion coefficient. high concentration J dc J dx low concentration Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 5

Solution to the Diffusion Equation • For a fixed concentration on one end of semiinfinite domain, an analytical solution exists. • Cs = the surface concentration • C 0 = initial condition Co = C(x, t=0) . . . Cs = C(x=0, t) Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 6

Mass Conservation • Mass must be conserved. • Difference in flux will lead to change in concentration (accumulation or depletion). • Mass conservation equation: • In 1 D: Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 7

Semiconductor Device Processing oxide passivation metallic conductors active devices (transistors, etc. ) silicon chip • Manufacture millions of devices simultaneously on a “chip” • Steps in manufacture (simplified) – Crystal growth and dicing to “chip” – Photolithography to locate regions for doping – Doping to create n-type regions (can in some cases be done during growth) – Overlay to create junctions – Metallization to interconnect devices – Passivation to insulate and isolate devices – Higher level “packaging” to interconnect chips Based on figures from MSE 201 course notes of J. W. Morris, Jr. , University of California, Berkeley Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018

Photolithography • Minimum feature size depends on wavelength of “light” – – Visible light: ~ 1 µm Ultraviolet light: ~ 0. 1 µm Electrons, x-rays 0. 1 -1 nm New and exotic methods • Must have photoresist suitable to the “light” – Or use “light” to cut through oxide directly Based on figures from MSE 201 course notes of J. W. Morris, Jr. , University of California, Berkeley Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018

Doping dopant ions • Add electrically active species • Simple method • More precise: Ion implantation dopant distribution – Coat surface and diffuse – Expose surface to a vapor and allow interdiffusion – Diffusion field is electrically active – Accelerate ions of the electrically active species toward surface – Ions embed to produce doped region Based on figures from MSE 201 course notes of J. W. Morris, Jr. , University of California, Berkeley Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5 -16, 2017

Doping: The Chemical Distribution implantation laser anneal c laser light dopant distribution diffusion x • Initial distribution is inhomogeneous • Can homogenize by “laser annealing” – Diffusion produces gradient from surface – Ion implantation produces concentration at depth beneath surface – Use a laser to melt rapidly, locally – Rapid homogenization in melted region – Rapid re-solidification since rest of body is heat sink Based on figures from MSE 201 course notes of J. W. Morris, Jr. , University of California, Berkeley Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5 -16, 2017

Overlay to Create Junctions n n p • Once primary doping is complete – – Re-coat Re-mask Re-pattern Dope second specie to create desired distribution of junctions Based on figures from MSE 201 course notes of J. W. Morris, Jr. , University of California, Berkeley Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018

Part 2. Introduction to Computational Kinetics Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 13

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What is Fi. Py? • Simply put: – Is a set of python libraries to solve PDEs • In more detail: – Provides a numerical framework to solve for the finite-volumes equation – The emphasis is on microstructural evolution Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 17

Fi. Py Resources • Fi. Py Manual (tutorials and useful examples) • Fi. Py Reference (what every single command does) • Mailing List: fipy@nist. gov • You can also email the coauthors: • John Guyer: guyer@nist. gov • Dan Wheeler: daniel. wheeler@nist. gov • Fi. Py Website http: //www. ctcms. nist. gov/fipy/ Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 18

A PDE is Solved in Five Steps • Variables Definitions • Equation(s) Definition(s) • Boundary Condition Specification • Viewer Creation • Problem Solving Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 19

Step-By-Step Walk-Though Follows Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 4 -15, 2018 20