1. Summer School for Integrated Computational Materials Education 2017 Kinetics Module Review
Katsuyo Thornton,1 Edwin Garcia,2 Larry Aagesen,3
Mark Asta4, Jonathan Guyer5
Department of Materials Science & Engineering, University of Michigan
Purdue University
Idaho National Laboratory
University of California, Berkeley
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 5-16, 2017

3. Concepts Illustrated Through Kinetics Module
Diffusion
Driving Force
Fick’s Law
Mass Conservation
Semiconductor Processing
Computational Kinetics
FiPy
Part 1
Part 2
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

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 =
Concentration
High conc.
Low conc. x
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

5. Fick’s First Law The flux is proportional to the driving force.
The proportionality constant is the diffusion coefficient.
high
concentration
low J dc J dx
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

6. Solution to the Diffusion Equation
For a fixed concentration on one end of semi-infinite domain, an analytical solution exists.
Cs = the surface concentration
C0 = initial condition
Cs = C(x=0,t)
...
Co = C(x, t=0)
Retype equaitons
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

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 5-16, 2017

8. Semiconductor Device Processing
active devices
(transistors, etc.)
metallic conductors
oxide passivation
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 5-16, 2017

9. Photolithography Minimum feature size depends on wavelength of “light”
Visible light: ~ 1 µm
Ultraviolet light: ~ 0.1 µm
Electrons, x-rays 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 5-16, 2017

10. Doping Add electrically active species Simple method
dopant
distribution
ions
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
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

11. Doping: The Chemical Distribution
diffusion
ion implantation
laser anneal c x
dopant
distribution
laser
light
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
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

12. Overlay to Create Junctionsp n
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 5-16, 2017

13. Part 2. Introduction to Computational Kinetics
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

17. What is FiPy? Simply put: In more detail:
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 5-16, 2017

18. FiPy Resources FiPy Manual (tutorials and useful examples)
FiPy Reference (what every single command does)
Mailing List:
You can also the coauthors:
John Guyer:
Dan Wheeler:
FiPy Website
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017

19. 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 5-16, 2017

20. Step-By-Step Walk-Though Follows
Summer School for Integrated Computational Materials Education Ann Arbor, MI, June 5-16, 2017