Diffusion Fundamentals in Materials Science

• Materials Science and Engineering is the context for the study notes

Diffusion Fundamentals

• Diffusion refers to mass transport by atomic motion.
• Interdiffusion/Impurity Diffusion involves atoms of one metal diffusing into another
• Self-Diffusion involves all atoms exchanging positions are of the same type.

Diffusion Mechanism

• Diffusion involves the stepwise migration of atoms from lattice site to lattice site.
• For an atom to move, there must be an empty adjacent site.
• The atom must have sufficient energy to break bonds with its neighbors, causing lattice distortion.
• This energy is vibrational in nature

Types of Diffusion

• Vacancy Diffusion involves the interchange of an atom from a normal lattice position to an adjacent vacant lattice site
• Vacancy Diffusion applies to substitutional impurities atoms.
• The rate of Vacancy Diffusion depends on the number of vacancies and activation energy to exchange.
• Interstitial Diffusion involves atoms migrating from an interstitial position to a neighboring empty one.
• Host or substitutional impurity atoms rarely form interstitials and do not normally diffuse via this mechanism.

Fick's First Law

• Diffusion is the quantity of an element that is transported within another and is a function of time and called the rate of mass transfer.
• J = M/A, where J is diffusion flux, M is mass diffusing per unit time A, and A is cross-sectional area.
• The unit of J is (kg/m2-s or atoms/m2-s).
• Diffusion flux is proportional to the negative of the concentration gradient
• J = -D(dC/dx) where D is the diffusion coefficient, dC is the concentration gradient, and dx is the change in position
• This equation is Fick's first law, where D is the diffusion coefficient expressed in m2/s
• The negative sign indicates diffusion occurs down the concentration gradient from high to low concentration.

Fundamental Concepts

• Steady-State Diffusion occurs when the mass of diffusing species entering the high-pressure side equals the mass exiting the low-pressure surface, resulting in no net accumulation in the plate.
• Concentration Profile refers to the resulting curve when concentration C is plotted versus position within the solid.
• Concentration Gradient is the slope of the concentration profile curve at a specific point.
• concentration gradient = dC/dx = ΔC/Δx = (CA - CB)/(XA - XB)

Fick's Second Law

• For nonsteady-state diffusion, there is a net accumulation or depletion of diffusing species, and the flux depends on time.
• ∂C/∂t = ∂/∂x (D ∂C/∂t)
• If the diffusion coefficient is independent of composition, the equation simplifies to: ∂C/∂t = D ∂²C/∂x²

Important Assumptions for Fick's Second Law

• Before diffusion, any diffusing solute atoms in the solid are uniformly distributed with a concentration of C0
• The value of x at the surface is zero and increases with distance into the solid.
• Time is taken to be zero at the instant before the diffusion process begins.

Initial and Boundary Conditions for Fick's Second Law

• Initial Condition: For t = 0, C = C₀ at 0 ≤ x ≤ ∞
• Boundary Conditions:
  • For t > 0, C = Cₛ at x = 0 (constant surface concentration)
  • For t > 0, C = C₀ at x = ∞
• Application yields: (Cx - C₀) / (Cs - C₀) = 1 - erf(x / (2√(Dt)))
  • Note: erf is the error function table

Factors Influencing Diffusion

• Diffusing Species: The significant difference between self-diffusion and interdiffusion provides a contrast between rates of diffusion via vacancy and interstitial mode.
• Temperature: D = D₀exp(-Qd/RT), where:
  • D₀ = temperature-independent preexponential (m²/s)
  • Qd = the activation energy for diffusion (J/mol or eV/atom)
  • R = the gas constant, 8.31 J/mol-K or 8.62 x 10⁻⁵ eV/atom-K
  • T = absolute temperature (K)
• Activation Energy: The energy required to produce the diffusive motion of one mole of atoms. A large activation energy results in a relatively small diffusion coefficient.

Diffusion in Semiconducting Materials

• Two Heat Treatments are used: pre-deposition and drive-in diffusion.
• Pre-deposition step: Impurity atoms are diffused into silicon, often from a gas phase, with maintained constant partial pressure
• Surface composition of the impurity remains constant over time and is carried out between 900C and 1000C typically less than 1 hour
• Drive-in diffusion: is used to transport impurity atoms farther into silicon for a suitable concentration distribution and is carried out at a higher temperature than pre-deposition up to about 1200C

Assuming Impurity Atoms are Introduced

• C(x, t) = Q₀ / √(πDₐt) * exp(-x² / 4Dₐt)
• Q₀ is the total amount of impurities in the solid introduced during predeposition with the same parameters as previously
• Q₀ = 2Cₛ√(Dₚtₚ / π)
• Cₛ is the surface concentration for the predeposition step, which was constant, Dₚ is the diffusion coefficient, and tₚ is the predeposition treatment time.

Junction Depth

• Represents the background concentration of the impurity in silicon (CB).
• xj = √[(4Ddatad)ln(Q0/Cdatad√πDdtd)]
• Dd and td represent the diffusion coefficient and time for the drive-in treatment, respectively.
• Integrated Circuits are normally made of aluminum—instead of metals that have higher electrical conductivities.
• At high-temperature heat treatments, interconnect metal atoms diffuse into the silicon.

Other Diffusion Paths: Atomic Migration

• Atomic Migration also occurs along dislocations, grain boundaries, and external surfaces called short-circuit diffusion paths
• Contributions to the overall diffusion flux from short circuits are insignificant due to their small cross sectional areas.