Diffusion in Materials

Diffusion Introduction

• Diffusion is mass transport by atomic motion
• It involves the movement of particles from high to low concentration areas

Diffusion Mechanisms

• In gases and liquid diffusion is random(Brownian) motion
• In solids diffusion can be vacancy or interstitial

Types of Diffusion

• Interdiffusion occurs in alloys when atoms migrate from high to low concentration regions
• Self-diffusion occurs in elemental solids where atoms also migrate

Vacancy Diffusion

• A stepwise migration of atoms from lattice site to lattice site
• The 2 conditions for this are an available empty adjacent site, and atoms with sufficient energy to break bonds and cause lattice distortions during displacement
• Atoms exchange with vacancies
• It applies to substitutional atoms
• The rate depends on the number of vacancies and the activation energy to exchange

Interstitial Diffusion

• Smaller atoms diffuse between atoms
• It is more rapid than vacancy diffusion

Processing using Diffusion

• Case hardening diffuses carbon atoms into the host iron atoms at the surface
• Case hardening is an example of interstitial diffusion
• It results in making the iron(steel) harder with the presence of carbon atoms
• Doping is processing silicon with phosphorus for n-type semiconductors
• In this process phosphorous rich layers are placed on the surface on silicon, and then heat is applied to get doped semiconductor regions

Understanding Flux

• Flux is the rate at which mass, charge, or energy traverses through a unit area across time

Diffusion as a Time-Dependent Process

• Diffusion is a time-dependent process
• Flux helps quantifies the amount or rate of diffusion
• Flux is measured empirically by making a thin film of known surface area, imposing a concentration gradient, and measuring how fast atoms/molecules diffuse
• Diffusion flux (J) is the mass (M) diffusing through and perpendicular to the unit cross-sectional area of solid per unit time, either in mol/cm²s or kg/m²s

Steady-State Diffusion

• Diffusion flux does not change with time
• Steady state is reached when the diffusing species entering the high-pressure side equals the mass exiting from the low-pressure side
• No accumulation of diffusing species occurs in the plate
• The slope of the concentration profile at a particular point is the concentration gradient

Fick's First Law of Diffusion

• The rate of diffusion is independent of time
• D is a coupling coefficient between force and flux
• D quantifies the relative ability of a species to respond to a given force with movement
• D also quanitifies the effective strength of the driving force
• The net flow of atoms is equal to D times the concentration gradient dC/dx
• D also depends on the property of the species that is moving and the material through which it is moving
• D = diffusion coefficient
• The minus sign indicates that the flux travels from high to low concentration

Factors Affecting Diffusion

• Diffusion depends on diffusion mechanism, temperature of diffusion, the type of crystal structure(BCC > FCC), crystal imperfections, and concentration

Chemical Protective Clothing (CPC) Example

• Methylene chloride is an ingredient of paint removers that can be absorbed and irritate the skin
• Butyl rubber gloves are used to protect the skin when using paint remover

Methylene Chloride Example (cont)

• When assuming a linear conc. gradient, it can be determined that approximately 1.16x10-5 g/cm²s goes through the gloves given the data: D = 110 x 10-8 cm²/s, C1 = 0.44 g/cm³, C2 = 0.02 g/cm³, x2 - x1 = 0.04 cm

Non-Steady State Diffusion Characteristics

• The concentration is a function of both time and position, with a net accumulation or depletion of the diffusing species
• The diffusion flux and concentration gradient vary with time
• Fick's Second Law is used in this case

Fick's Second Law simplified

• If the diffusion coefficient is independent of composition, then ∂C/∂t = D(∂²C/∂x²)
• At this point the rate of compositional change is equal to the product of the diffusion coefficient and the change of concentration gradient
• In semi-infinite solids where surface concentration is constant, an example would be copper diffusing into a bar of aluminum

Solving for Non-Steady State Diffusion

• C(x,t) = Conc. at a pint x at a time t
• erf = mathematical error function
• Z = x / 2√Dt

Factors Affecting Diffusion

• The magnitude of the diffusion depends on the rate at which atoms diffuse, the atoms, and the host material
• Increasing temperature increases the the diffusion coeffecient

Diffusion Coefficient Increase Formula

• D =D₀exp(-Qd/RT)
• D = diffusion coefficient [m²/s]
• D₀ = a temperature independent pre-exponential [m²/s]
• Q_d = activation energy for diffusion of one mole of atom [J/mol or eV/atom]
• R = gas constant [8.314 J/mol-K]
• T = absolute temperature [K]

Taking Natural Logarithm

• When log D is plotted against the reciprocal of the absolute temperature, a straight line comes out having a slope and intercept = -Qd 2.3

Diffusion in Semiconductors

• Semiconductor Integrated Circuits (ICs) are typically 6mm x 6mm x 0.4mm
• The base of ICs are made of single crystalline silicon
• There is a precise concentration of Impurity
• Two step heat process: (1) Predeposition and (2) Drive-in

Heat Treatment Process

• In the Predeposition step, impurity atoms are diffused into the silicon from a gas phase while maintaining constant partial pressure of the gas
• The surface composition remains the same over time, at 900-1000°C
• The Drive-in step is a treatment done at a higher temperature (1200°C) in an oxidizing environment

Measuring Semiconductor Diffusion

• Fick's second law shows how the drive in diffusion works
• Assuming that impurity atoms are confined to a thin layer during pre-deposition, Qo is the total amount of impurities introduced during pre-deposition
• Junction depth xj measures the depth (x) at which the diffusing impurity concentration is equal to the background concentration

Diffusion Rate

• Diffusion is faster for open crystal structures, materials with secondary bonding, smaller diffusing atoms, and lower density materials
• Diffusion is slower for close-packed structures, materials with covalent bonding, larger diffusing atoms, and higher density materials