Diffusion This animation illustrates the process of diffusion

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 Author(s) : Raghu Ramachandran Mentor: Prof Anil Kottantharayil Level(UG/PG): UG *The contents in this ppt are licensed under Creative Commons Attribution-Non. Commercial-Share. Alike 2. 5 India license

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

1 Master Layout 2 1 7 1 9 1 4 3 3 2 1 0 1 3 1 1 4 4 Container 1 8 6 2 0 1 6 5 7 2 1 1 9 9 4 8 2 0 1 6 1 3 1 2 1 8 6 1 0 7 8 1 5 Particles 5 1 4 1 1 3 1 7 Before Diffusion After Diffusion 5

1 2 3 4 5 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.

1 Step 1: T 1: Random Motion of Particles in a Box 1 7 1 9 1 4 1 2 3 2 1 0 1 3 1 1 3 4 5 4 1 8 6 2 0 1 6 5 7 1 2 9 8 1 5 For User §Particles collected together in a rectangular box moving randomly and undergoing collisions amongst themselves and with the walls of the container. 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.

1 Step 1: T 1: Random Motion of Particles in a Box 1 7 1 9 1 4 1 2 3 2 1 0 1 3 1 1 3 4 §In a gas, pressure is caused 5 2 0 1 6 5 7 1 2 9 8 1 5 For User 4 1 8 6 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 Audio Narration (if any) Instructions to Animator Continue with random particle motion in the box while text on the left is displayed

1 2 3 4 5 Audio Narration (if any) For User 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.

1 Step 2: 1 7 1 9 T 1: Remove Rectangular Boundary 1 4 1 9 1 2 3 4 5 1 3 1 1 1 4 6 2 0 1 6 5 4 7 1 2 1 0 1 3 1 1 9 4 8 1 5 For User §Upon removal of the barrier the particles spread out into space resulting from collisions and random motion. 1 5 Audio Narration (if any) 1 4 1 3 2 1 8 1 7 1 9 1 3 2 1 0 1 7 1 8 6 2 0 1 6 5 7 1 2 3 2 2 0 1 6 1 3 5 1 2 1 1 9 9 8 1 8 6 1 0 4 7 8 1 5 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.

1 2 Step 3: T 1: Diffusion Mechanisms in Solids v 3 For User §In a perfect lattice with 4 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. 5 Audio Narration (if any) Instructions to Animator Atoms vibrate randomly about their mean positions.

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

1 Step 5: T 1: Substitutional Diffusion due to Vacancy 2 3 For User §There is an energy barrier 4 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 5 determined by the density of vacancies. Audio Narration (if any) Instructions to Animator Atom shifts into the vacant site

1 Step 6: T 1: Substitutional Diffusion due to Vacancy 2 3 For User 4 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 5 fluctuation necessary to overcome thermal barrier Audio Narration (if any) Instructions to Animator Only change in text displayed to user

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

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

1 Step 9: T 1: Interstitial Diffusion 2 3 For User 4 5 §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 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

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

1 2 3 4 5 Step 11: T 1: Fick’s First Law constant concentration Flux Jx constant concentration For User For a flux of diffusing particles (atoms, molecules or ions) diffusing in one dimension Fick’s first law can be written as Audio Narration (if any) Instructions to Animator Display the figure and text for user 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

1 2 Step 12: T 1: Fick’s First Law constant concentration Low Temp Flux Jx constant concentration High. Temp 3 4 For User In the above graph, diffusivity changes as Instructions to Animator Display the figure and text for user Fick’s first law holds even if diffusivity changes with position 5 Audio Narration (if any)

1 Step 13: T 1: Fick’s Second Law: Finite Source 2 3 Impurity For User 4 5 In one dimension Fick’s second law is Bar Audio Narration (if any) Instructions to Animator Display the figure and text for user Consider a semi-infinite bar with a small fixed amount of impurity diffusing in from one end.

1 Step 14: T 1: Fick’s Second Law: Finite Source 2 3 Impurity Bar Before Diffusion Impurity Bar After Diffusion For User Boundary Conditions: 4 1. Since the amount of impurity in the system must remain constant Instructions to Animator Display the figure and text for user where B is a constant and C is the concentration of impurity in the bar. 5 Audio Narration (if any) 2. The initial concentration of impurity in the bar is zero, therefore

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

1 Step 16: T 1: Fick’s Second Law: Infinite Source 2 3 Infinite Source Before Diffusion For User 4 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 5 Bar 2. Since the concentration at x=0 is constant Infinite Source Bar After Diffusion Audio Narration (if any) Instructions to Animator Display the figure and text for user

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

1 Step 18: T 1: Fick’s Second Law: Non-Zero Boundary Conditions 2 3 4 Infinite Source Bar Before Diffusion After Diffusion 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. 5 Infinite Sink Since the concentration at x=0 is constant Audio Narration (if any) Instructions to Animator Display the figure and text for user

1 Step 19: T 1: Fick’s Second Law: Infinite Source 2 Graph Here 3 For User 3. Consider the case where the concentration at x = L is 4 5 §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. Audio Narration (if any) Instructions to Animator Display the figure and text for user

1 Step 20: T 1: Fick’s Second Law: Non-Zero Boundary Conditions 2 3 4 5 Infinite Source Bar Infinite Sink Before Diffusion After Diffusion 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 Audio Narration (if any) Instructions to Animator Display the figure and text for user

1 Step 21: T 1: Fick’s Second Law: Infinite Source 2 Graph Here 3 For User 3. Consider the case where the concentration at x = L is 4 5 §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. Audio Narration (if any) Instructions to Animator Display the figure and text for user

APPENDIX 1 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 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 2 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 3 Summary 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