Strained-Layer Epitaxy & Misfit Dislocations

Substrate System

Growth Parameters

Lattice mismatch f 1.5%

Film thickness h 4.0 nm

Strain State

Critical thickness h꜀

— nm

Current thickness h

4.0 nm

Residual strain

—%

Misfit dislocations

Coherently strained — the film lattice is fully compressed to match the substrate.

How it works

When a semiconductor film is grown epitaxially on a substrate with a different bulk lattice constant, the mismatch f = (a_film − a_sub)/a_sub is initially accommodated elastically: the film's in-plane lattice matches the substrate exactly ("pseudomorphic" / coherent growth), storing strain energy that grows linearly with thickness h.

Above a critical thickness h꜀, it becomes energetically cheaper to nucleate misfit dislocations at the interface — line defects whose extra half-plane of atoms locally relieves the mismatch. This is the Matthews–Blakeslee model (1974), used throughout strained-Si CMOS, SiGe heterojunction transistors, and III-V laser design:

h꜀ = b(1 − ν cos²θ) / (8π f (1+ν) cosλ) · [ln(h꜀/b) + 1]

for 60° dislocations on {111} planes: θ = λ = 60°
b = Burgers vector ≈ a₀/√2,  ν = Poisson ratio

Solving the same relation for a given h > h꜀ yields the equilibrium residual strain f_r still locked in the film — it falls roughly as 1/h, so thicker films relax further. The number of dislocations needed across the visualised width follows from the strain each one relieves, ρ = (f − f_r)/b.

Substrate system — sets the lattice constant a₀ and Poisson ratio ν for three real heteroepitaxial pairs.
Lattice mismatch f — how far the film's natural lattice constant differs from the substrate's (0.2–4%, spanning Si₁₋ₓGeₓ and typical III-V pairs).
Film thickness h — grow the layer past h꜀ and watch the interface nucleate misfit dislocations (small red markers) while the film spacing drifts away from the substrate's.
Play growth — animates h increasing continuously so you can watch the coherent → relaxed transition happen in real time.

Substrate atoms Film atoms Misfit dislocation core

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