1. Diffusion ‏ This animation illustrates the process of diffusion in which particles move from a region of higher concentration to a region of lower concentration by random motion  Related LOs: > Prior Viewing - > Future Viewing -  Course Name: VLSI Technology Level(UG/PG):UG  Author(s) : Raghu Ramachandran  Mentor: Prof Anil Kottantharayil *The contents in this ppt are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 2.5 India license

2. Learning objectives After interacting with this Learning Object, the learner will be able to: Explain the process of diffusion 5 3 2 4 1

3. Master Layout 5 3 2 4 1 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020 Particles Container Before Diffusion After Diffusion

4. Definitions and Keywords 1 Diffusion is the process of motion of particles from regions of higher concentration to regions of lower concentration by random motion 2 Container could be a vessel containing gas molecules, or a piece of semiconductor containing particles such as electrons or dopant atoms 3 Particles could be dopant atoms, gas molecules, electrons etc. 5 3 2 4 1

5. Step 1: Audio Narration (if any) ‏ Instructions to Animator Initially the particles are collected together inside a rectangular box moving randomly and undergoing collisions amongst themselves and with the walls of the container. There is an outer boundary that is not visible to the user that encloses the box. For User  Particles collected together in a rectangular box moving randomly and undergoing collisions amongst themselves and with the walls of the container. T1: Random Motion of Particles in a Box ‏ 5 2 1 4 3 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020

6. Step 1: Audio Narration (if any) ‏ Instructions to Animator Continue with random particle motion in the box while text on the left is displayed For User  In a gas, pressure is caused by collision of the molecules with the walls of the container  In a gas or a semiconductor, the mean free path of a molecule or a charge carrier is the average distance travelled between collisions with another similar moving particle T1: Random Motion of Particles in a Box ‏ 5 2 1 4 3 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020

7. Audio Narration (if any) ‏ Instructions to Animator Define rule for particles colliding with each other and particles colliding off the walls of the container. Particles are assumed to undergo specular reflection in both cases. Set a counter that increments every time a collision occurs. For User 5 2 1 4 3

8. Step 2: Audio Narration (if any) ‏ Instructions to Animator After the collision counter reaches a suitable value, remove the walls of the rectangular box. Undergoing collisions with each other the particles move outward beyond their region of confinement. Stop the animation when any particle hits the outer boundary. For User  Upon removal of the barrier the particles spread out into space resulting from collisions and random motion. T1: Remove Rectangular Boundary ‏ 5 2 1 4 3 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020 1 23 4 5 6 7 8 9 1010 1 1212 1313 1414 1515 1616 1717 1818 1919 2020

9. Step 3: v Audio Narration (if any) ‏ Instructions to Animator Atoms vibrate randomly about their mean positions. For User  In a perfect lattice with no defects the atoms vibrate about their stable positions due to thermal energy.  Since there are no defects the atoms cannot move from one site to another. T1: Diffusion Mechanisms in Solids ‏ 5 2 1 4 3

11. Step 4: Audio Narration (if any) ‏ Instructions to Animator Atoms vibrate randomly about their fixed positions, however, two atoms are missing from the lattice For User  In a real material there are defects such as the vacancies. Atoms can then move within the structure from one atomic site to another. T1: Substitutional Diffusion due to Vacancy ‏ 5 2 1 4 3

12. Step 5: Audio Narration (if any) ‏ Instructions to Animator Atom shifts into the vacant site For User  There is an energy barrier for the movement of atoms into the vacant sites.  At a high enough temperature some of the atoms gain enough energy to move into the vacant sites.  The rate of diffusion is determined by the density of vacancies. T1: ‏ Substitutional Diffusion due to Vacancy 5 2 1 4 3

13. Step 6: Audio Narration (if any) ‏ Instructions to Animator Only change in text displayed to user For User The probability of any atom in a solid to move is the product of:  The probability of finding a vacancy in an adjacent lattice site and  The probability of thermal fluctuation necessary to overcome the thermal barrier T1: ‏ Substitutional Diffusion due to Vacancy 5 2 1 4 3

14. Step 7: Audio Narration (if any) ‏ Instructions to Animator Only change in text displayed to user For User The diffusion coefficient which is a measure of mobility of the diffusing species is given by T1: ‏ Substitutional Diffusion due to Vacancy 5 2 1 4 3 where Qd is the activation energy for diffusion (J/mol) where Q d is the activation energy for diffusion (J/mol)

15. Step 8: Audio Narration (if any) ‏ Instructions to Animator An atom occupies an interstitial position and all atoms vibrate randomly about their mean positions For User  When the atom is not on a lattice site but on an interstice, it is free to move to an adjacent unoccupied interstitial position.  The probability of an atom in an interstitial to diffuse is controlled by the probability of its overcoming the thermal barrier, since there are a lot of vacant interstitial sites T1: Interstitial Diffusion 5 2 1 4 3

16. Step 9: Audio Narration (if any) ‏ Instructions to Animator The interstitial atom pushes the lattice atoms out of the way to move to the adjacent interstitial position For User  The atom on the interstice ‘pushes’ the lattice atoms out of the way to move to the adjacent vacant interstitial position, thus overcoming the energy barrier T1: Interstitial Diffusion 5 2 1 4 3

17. Step 10: Audio Narration (if any) ‏ Instructions to Animator The atom moves into an adjacent interstitial position For User  The atom moves into an adjacent interstitial position T1: Interstitial Diffusion 5 2 1 4 3

18. Step 11: constant concentration constant concentration Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User For a flux of diffusing particles (atoms, molecules or ions) diffusing in one dimension Fick’s first law can be written as where D is the diffusivity, J x is the flux of particles (diffusion flux) and C is their number density (concentration). Fick’s first law can only be used to solve steady state diffusion problems T1: Fick’s First Law 5 2 1 4 3 Flux J x

19. Step 12: constant constant concentration concentration Low Temp HighTemp Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User In the above graph, diffusivity changes as Fick’s first law holds even if diffusivity changes with position T1: Fick’s First Law 5 2 1 4 3 Flux J x

20. Step 13: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User In one dimension Fick’s second law is Consider a semi-infinite bar with a small fixed amount of impurity diffusing in from one end. T1: Fick’s Second Law: Finite Source 5 2 1 4 3 ImpurityBar

21. Step 14: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User Boundary Conditions: 1. Since the amount of impurity in the system must remain constant where B is a constant and C is the concentration of impurity in the bar. 2. The initial concentration of impurity in the bar is zero, therefore T1: Fick’s Second Law: Finite Source 5 2 1 4 3 Impurity Bar Impurity Bar Before Diffusion After Diffusion

22. Step 15: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User The solution for C(x,t) is The diffusion of impurity material into the bar with time is shown above. T1: Fick’s Second Law: Finite Source 5 2 1 4 3

23. Step 16: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User Consider a semi-infinite bar with a constant concentration of impurity material diffusing in from one end. Boundary conditions: 1. Since there are no impurity atoms in the bar at the start 2. Since the concentration at x=0 is constant T1: Fick’s Second Law: Infinite Source 5 2 1 4 3 Infinite Source Bar Infinite Source Bar Before DiffusionAfter Diffusion

24. Step 17: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User The solution for C(x,t) is The diffusion of impurity material into the bar with time is shown above. T1: Fick’s Second Law: Infinite Source 5 2 1 4 3

25. Step 18: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User Consider a bar of length ‘L’ such that impurity material enters at one end of the bar and leaves at the other end. The concentrations at either end of the bar are held constant. The boundary conditions are: 1.Since there are no impurity atoms in the bar at the start 2. Since the concentration at x=0 is constant T1: Fick’s Second Law: Non-Zero Boundary Conditions 5 2 1 4 3 Before Diffusion After Diffusion Infinite SourceBarInfinite Sink

26. Step 19: Graph Here Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User 3. Consider the case where the concentration at x = L is  Since there are no sources or sinks in the bar and the concentration at the ends is fixed, the concentration distribution eventually stabilizes and no longer depends on time. This is shown above. T1: Fick’s Second Law: Infinite Source 5 2 1 4 3

27. Step 20: Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User Consider a bar of length ‘L’ such that impurity material enters at one end of the bar and leaves at the other end. The concentrations at either end of the bar are held constant. The boundary conditions are: 1.Since there are no impurity atoms in the metal at the start 2. Since the concentration at x=0 is constant T1: Fick’s Second Law: Non-Zero Boundary Conditions 5 2 1 4 3 Before Diffusion After Diffusion Infinite SourceBarInfinite Sink

28. Step 21: Graph Here Audio Narration (if any) ‏ Instructions to Animator Display the figure and text for user For User 3. Consider the case where the concentration at x = L is  Since there are no sources or sinks in the bar and the concentration at the ends is fixed, the concentration distribution eventually stabilizes and no longer depends on time. This is shown above. T1: Fick’s Second Law: Infinite Source 5 2 1 4 3

29. Questionnaire: 1.For an ideal gas at constant volume and pressure, how does the mean free path of gas molecules change with increasing temperature? a) Increases b) Decreases c) Remains the same 2.For an ideal gas at constant volume and temperature, how does the mean free path of gas molecules change with increasing pressure? a) Increases b) Decreases c) Remains the same 3.For a doped solid semiconductor, how does the mean free path of electrons change with increasing temperature? a) Increases b) Decreases c) Increases and then decreases APPENDIX 1

30. Questionnaire: 4.The ‘minus’ sign in Fick’s first law signifies the fact that the flux is a) in the direction of decreasing impurity concentration b) in the direction of increasing impurity concentration 5.Pre-deposition and drive-in are cases of a) diffusion from finite and infinite sources respectively b) diffusion from infinite and finite sources respectively APPENDIX 1

31. Links for further reading Reference websites: Books: James D. Plummer, Michael D. Deal, Peter B. Griffin, “Silicon VLSI Technology” Ben G. Streetman, Sanjay Banerjee, “Solid State Electronic Devices” Research papers: APPENDIX 2

32. Summary APPENDIX 3 Diffusion is a process in which particles move from a region of higher concentration to a region of lower concentration by random motion Impurities diffuse through solids by substitutional diffusion due to the presence of vacancies or interstitial diffusion where the impurity atom occupies an interstitial position For diffusing particles, Fick’s first law is and Fick’s second law is