Lattice mismatch and critical thickness calculator
Device physics and crystals. Everything is calculated in your browser; nothing you enter is sent to a server.
KEY POINTS
Lattice mismatch is f = (a_epi − a_sub)/a_sub: about 1.0% for Si₀.₇₅Ge₀.₂₅ on Si and about −17% for GaN on Si(111). Beyond the critical thickness, misfit dislocations form and relax the strain. Embedded SiGe source/drain regions and the Si/SiGe superlattices of GAA transistors are designed so composition and thickness stay within this limit.
Inputs
Calculated with Poisson's ratio ν = 0.28. Matthews–Blakeslee gives the equilibrium critical thickness; People–Bean is an empirical fit to the thickness at which dislocations start to appear in low-temperature SiGe growth. Real values lie between the two.
Ge fraction and critical thickness for SiGe/Si(100)
SiGe composition and lattice mismatch
| Composition | Lattice constant | Mismatch to Si | hc (M–B) | hc (P–B) |
|---|---|---|---|---|
| Si0.9Ge0.1 | 5.4513 Å | 0.37 % | 32.4 nm | 1.25 µm |
| Si0.8Ge0.2 | 5.4721 Å | 0.76 % | 13.5 nm | 243.9 nm |
| Si0.7Ge0.3 | 5.4934 Å | 1.15 % | 7.8 nm | 89.5 nm |
| Si0.6Ge0.4 | 5.5153 Å | 1.55 % | 5.2 nm | 42.5 nm |
| Si0.5Ge0.5 | 5.5377 Å | 1.97 % | 3.8 nm | 23.1 nm |
| Si0.0Ge1.0 | 5.6580 Å | 4.18 % | 1.1 nm | 2.2 nm |
Lattice mismatch and strain engineering
When a crystal with a different lattice constant is grown on a substrate, a thin epitaxial layer stretches or compresses in plane to match the substrate and grows strained but defect-free (coherent growth). As the film thickens, strain energy builds up, and beyond a certain thickness (the critical thickness) misfit dislocations form and relax the strain. The larger the mismatch, the thinner the critical thickness.
Strained silicon technology puts this strain to work. Embedding SiGe with about 20 to 40% Ge in pMOS source and drain regions compresses the channel and raises hole mobility. GAA nanosheets start from a superlattice of alternating Si and SiGe layers, and the SiGe is selectively removed to open the space between sheets, so composition and thickness are designed to keep every layer within its critical thickness.
FAQ
What is critical thickness?
The thickness at which a strained epitaxial layer starts to relax by forming misfit dislocations. Larger mismatch means a thinner critical thickness.
Why do Matthews-Blakeslee and People-Bean differ?
Matthews-Blakeslee is the equilibrium limit; People-Bean fits the thickness at which dislocations appear in low-temperature growth, where metastable layers can exceed equilibrium. Real values fall between them.
Related equipment and companies
More calculators
Device physics and crystals
- All semiconductor calculators
- MOSFET threshold voltage (Vth) calculator
- Miller index, d-spacing and plane angle calculator
- Sheet resistance and resistivity calculator (four-point probe, doping)
- Depletion width, built-in potential and junction capacitance calculator
- Intrinsic carrier concentration and Fermi level calculator
- MOSFET drain current (I–V characteristics) calculator
- Diode I–V characteristics and forward voltage calculator
Last updated: October 23, 2026