Yes, it is valid to calculate lattice mismatch using d-spacing values, provided that the d-spacings correspond to the same crystallographic planes in both the film and the substrate. This approach is commonly used in transmission electron microscopy (TEM) studies, especially in HRTEM and SAED analyses, where direct measurement of lattice constants (a, b, c) may not be feasible, but interplanar spacings (d-values) can be measured with high precision.

Why It’s Valid

Lattice mismatch is generally defined as:

Lattice Mismatch (%)=(𝑎 film−𝑎 substrate/ 𝑎 substrate)× 100

However, for specific crystallographic directions, the lattice constant a can be replaced by the d-spacing of a given (hkl) plane, as long as:

The same crystallographic orientation is considered for both film and substrate (e.g., both are measured along the (100) plane).

The crystal systems are compatible or properly accounted for (hexagonal vs. cubic).

Any strain effects are considered if the d-spacings are measured from thin films (they might be slightly off the bulk values due to residual stress).

In TEM-based techniques like selected area electron diffraction (SAED) and high-resolution TEM (HRTEM), the measured d-spacing is often the most directly accessible metric for calculating mismatch, especially for epitaxial or oriented growth.

How to Use d-Spacing for Lattice Mismatch

If your AlN films and substrates are aligned along specific planes (e.g., (100) or (002)), and you measure those same interplanar spacings in both, then:

Mismatch =(𝑑film−𝑑substrate/𝑑substrate)×100

This is particularly useful when working with polycrystalline substrates or when there's a known epitaxial orientation relationship.

Cautions

• Ensure that plane indexing is correct, especially if your substrate and film are different crystal systems (e.g., cubic Si vs. hexagonal AlN).
• If there's strain in the film, the measured d-spacing might reflect strain rather than actual lattice parameter—this should be discussed or corrected if you're comparing with relaxed substrate values.
• If possible, cross-check with XRD to confirm relaxed lattice parameters for consistency.

For a monocrystalline film, this can be done. Shweta Dwivedi explained everything in detail. However, for a polycrystalline film, the planes of the contacting crystals can have different orientations. And in this case, it is more correct to use the ratio of the lattice parameters.

Calculating lattice mismatch using d-spacing values instead of lattice constants (a) is valid but requires careful consideration of crystallographic orientation and material symmetry. For cubic crystal systems, d-spacing values directly relate to lattice parameters through the plane-specific relationship dₕₖₗ = a/√(h² + k² + l²), making d-spacing-based mismatch calculations mathematically equivalent to using lattice constants when comparing the same crystallographic planes between materials. The lattice mismatch percentage δ can be calculated as δ = |d₂ - d₁|/d₁ × 100%, which provides identical results to the lattice constant formula δ = |a₂ - a₁|/a₁ × 100% for cubic systems when using equivalent planes. However, for non-cubic systems like hexagonal or tetragonal crystals, d-spacing values depend on multiple lattice parameters and Miller indices, meaning mismatch calculations become orientation-dependent - for example, the (001) plane spacing in hexagonal crystals relates only to the c-axis parameter (d₀₀₁ = c/2) and won't reflect in-plane a-axis mismatch. This anisotropy requires explicit specification of the crystallographic planes being compared and recognition that different planes may yield different mismatch values within the same material pair. Practically, d-spacing measurements from X-ray diffraction (XRD) are often preferred for experimental studies of epitaxial growth since they account for real-world strain effects in thin films that may alter local lattice parameters from bulk values. However, researchers must be aware that strained layers can exhibit modified d-spacings that don't correspond to relaxed lattice constants, potentially complicating mismatch interpretation. For complete characterization, especially in non-cubic systems, combining d-spacing analysis with full structural refinement to determine all lattice parameters (a, c) provides the most comprehensive understanding of interfacial compatibility, though for many cubic material systems and specific epitaxial orientations, d-spacing comparisons alone offer a valid and experimentally convenient approach to quantify lattice mismatch.

A single-crystal film grown on a substrate of a different composition (heteroepitaxy) usually has a distortion with loss of symmetry.

That is, instead of a cubic structure, either a tetragonal lattice for growth on the 100 plane or a hexagonal or monoclinic structure on a 111 substrate can grow.

Moreover, there can be a continuous film with a distorted lattice, and a multilayer structure with a variable layer-by-layer structure can also form, starting from the first interface and the next interface, and only on the surface itself can a free (unstressed) layer be formed.

Moreover, these interfaces can be discrete compositions formed as a result of spinodal decomposition.

For example, they tried to grow GaP on a Si substrate, but individual peaks of the SixGaP1-x composition were observed on the X-ray scattering curve, but in individual technological processes a continuous series of solid solutions was discovered.

You can use d-spacing to calculate lattice mismatch if:

• You’re comparing the same (hkl) planes

• The materials have similar crystal structures

• The film and substrate are oriented the same way

In that case, using d-spacing is equivalent to using lattice constants.

Avoid using d-spacing if:

• The planes or crystal structures differ

• The materials are not aligned sim