Lattice Mismatch Calculator

Example data table

Example lattice constants for quick testing. Always verify values for your specific temperature, alloy fraction, and crystal polytype.

Formula used

• Domain matching: L_film=m·a_film, L_sub=n·a_sub.
• Thermal option: a(T)=a₀(1+αΔT).
• Cubic plane option: d_hkl=a/√(h²+k²+l²).

How to use this calculator

1. Enter the film and substrate lattice parameters using the same units.
2. Keep m=n=1 for standard mismatch on lattice constants.
3. Use domain matching when supercells better represent alignment.
4. Enable thermal adjustment if α and ΔT are known and relevant.
5. Enable cubic plane spacing only for cubic materials and valid (h k l).
6. Press Calculate to see results above the form and under the header.
7. Use the export buttons to download CSV or a PDF report.

1) What lattice mismatch means

Lattice mismatch compares repeat lengths of a film and its substrate. In epitaxial growth, mismatch sets the elastic strain needed for registry and influences dislocation formation. It is commonly reported near room temperature for fast screening. The calculator reports mismatch in percent and ppm for practical decisions.

2) Choosing lattice parameter and units

Use lattice parameters for the same phase, composition, and reference conditions. Angstroms are common for crystal constants, while nanometers appear in device literature. Because mismatch is a ratio, unit choice does not change results, but consistent units prevent entry mistakes and confusing documentation.

3) Domain matching and supercells

Some interfaces align over multiple unit cells. Domain matching evaluates a supercell relationship, m·a_film versus n·a_sub. This helps when a near-integer ratio reduces effective mismatch and can improve interface coherence for rotated or reconstructed domains. It is also useful for matching in-plane lattice vectors.

4) Plane spacing for oriented growth

For growth on a specific plane, the relevant spacing may be d_hkl rather than the bulk lattice constant. For cubic materials, d_hkl=a/√(h²+k²+l²). Enabling the plane option compares oriented repeat distances using the same indices for both materials.

5) Thermal strain in real processes

Films are deposited hot and then cooled. If film and substrate thermal expansion differ, additional mismatch develops on cooldown. The thermal option uses a(T)=a₀(1+αΔT) so you can estimate temperature-driven strain shifts for a chosen ΔT and evaluate post-growth stress risk.

6) Interpreting sign and strain state

Positive mismatch means the film is effectively larger than the substrate, typically giving compressive strain if clamped in-plane. Negative mismatch suggests tensile strain. Sign matters for band shifts, piezoelectric response, cracking risk, and wafer bow, especially for thick or brittle layers.

7) Practical mismatch ranges and outcomes

As a rule of thumb, |mismatch| below about 0.1% is excellent, 0.1–0.5% is often manageable, 0.5–1% needs careful thickness control, and above 1% usually needs buffers, grading, or different substrates. Outcomes also depend on critical thickness and relaxation kinetics. The ppm output helps compare close candidates.

8) Reporting and traceability

Record your inputs, options, and assumptions. Note whether values are room-temperature or growth-temperature, whether plane spacing was used, and any domain ratio. CSV supports lab notebooks and spreadsheets, while the PDF snapshot supports quick sharing in design reviews. Include references for constants and measurement methods where possible.

FAQs

1) What does a negative mismatch mean?

A negative value means the film’s effective length is smaller than the substrate’s. If the film locks to the substrate in-plane, it is typically under tensile strain at that interface.

2) Can I use this for alloy compositions?

Yes. Enter the alloy lattice parameter you expect at the chosen composition and temperature. If you estimate with Vegard’s law or literature fits, note the method in your report.

3) What if my materials are not cubic?

The base mismatch works for any comparable in-plane length you enter. The d_hkl option is cubic-only. For hexagonal systems, use the appropriate a-plane spacing value directly.

4) Why show ppm as well as percent?

PPM is convenient when mismatch is very small. Since 1% equals 10,000 ppm, ppm highlights fine differences that may matter for precision lattice matching and strain budgets.

5) When should I use domain matching m:n?

Use it when the interface repeats over multiple unit cells or a coincidence lattice is expected. Try small integers first and check whether the ratio reflects a physically plausible epitaxial relationship.

6) How do I pick α and ΔT?

Use thermal expansion coefficients from reliable material data and set ΔT as growth temperature minus the reference temperature of your lattice constants. If data varies with temperature, use an average over your range.

7) Does zero mismatch guarantee a defect-free film?

No. It reduces misfit strain, but defects can still arise from impurities, thermal stress, surface steps, polarity, and growth kinetics. Use mismatch as one key screening metric, not the only one.