1. Summer School for Integrated Computational Materials Education 2015 Kinetics Module Review Katsuyo Thornton, 1 Edwin Garcia, 2 Larry Aagesen, 1 Mark Asta 3, Jonathan Guyer 4 1.Department of Materials Science & Engineering, University of Michigan 2.Purdue University 3.University of California, Berkeley 4.National Institute of Standards and Technology

2. 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 15-26, 2015 2

3. Concepts Illustrated Through Kinetics Module 1.Diffusion –Driving Force –Fick’s Law –Mass Conservation 2.Semiconductor Processing 3.Computational Kinetics 4.FiPy Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 3 Part 1 Part 2

4. Driving Force for Diffusion Consider 1D diffusion. The atoms are randomly hopping right and left. Half the atoms are moving toward right, and the other half is moving to left. Below, left side has more atoms than right. Net flux toward the low concentration. Driving force = Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 4 Concentration x High conc. Low conc.

5. Fick’s First Law dc dx high concentration low concentration J J The flux is proportional to the driving force. The proportionality constant is the diffusion coefficient. 5Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015

6. Solution to the Diffusion Equation... C o = C(x, t=0) C s = C(x=0,t) For a fixed concentration on one end of semi- infinite domain, an analytical solution exists. C s = the surface concentration C 0 = initial condition 6Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015

7. Mass Conservation Mass must be conserved. Difference in flux will lead to change in concentration (accumulation or depletion). Mass conservation equation: In 1D: Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 7

8. Semiconductor Device Processing 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 active devices (transistors, etc.) metallic conductors oxide passivation silicon chip Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 Based on figures from MSE 201 course notes of J. W. Morris, Jr., University of California, Berkeley

9. 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 Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 Based on figures from MSE 201 course notes of J. W. Morris, Jr., University of California, Berkeley

10. Doping Add electrically active species Simple method –Coat surface and diffuse –Expose surface to a vapor and allow interdiffusion –Diffusion field is electrically active More precise: Ion implantation –Accelerate ions of the electrically active species toward surface –Ions embed to produce doped region dopant distribution dopan t ions Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 Based on figures from MSE 201 course notes of J. W. Morris, Jr., University of California, Berkeley

11. Doping: The Chemical Distribution Initial distribution is inhomogeneous –Diffusion produces gradient from surface –Ion implantation produces concentration at depth beneath surface Can homogenize by “laser annealing” –Use a laser to melt rapidly, locally –Rapid homogenization in melted region –Rapid re-solidification since rest of body is heat sink diffusion ion implantation laser anneal c x dopant distribution laser light Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 Based on figures from MSE 201 course notes of J. W. Morris, Jr., University of California, Berkeley

12. Overlay to Create Junctions Once primary doping is complete –Re-coat –Re-mask –Re-pattern –Dope second specie to create desired distribution of junctions p nn p n n Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 Based on figures from MSE 201 course notes of J. W. Morris, Jr., University of California, Berkeley

13. Part 2. Introduction to Computational Kinetics Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 13

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17. What is FiPy ? 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 17Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015

18. FiPy Resources FiPy Manual (tutorials and useful examples) FiPy 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 FiPy Website http://www.ctcms.nist.gov/fipy/ 18Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015

19. A PDE is Solved in Five Steps Variables Definitions Equation(s) Definition(s) Boundary Condition Specification Viewer Creation Problem Solving 19Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015

20. Step-By-Step Walk-Though Follows Summer School for Integrated Computational Materials Education Ann Arbor, MI June 15-26, 2015 20