Mixed layers: effective-medium approximation
A co-evaporated emissive layer, a sputtered film with voids, metal nanoparticles in a dielectric, a rough surface — the optical constants of a layer that is not one material are built from its constituents, their fractions and an assumption about the microstructure. The result behaves as an ordinary layer in plane-wave, ellipsometry, OLED, solar-cell and optimisation calculations.
Refractive indices are not averaged
Mixing takes place in the dielectric function ε = N², not in the refractive index. Every rule in Phazic mixes the ε of the constituents and converts back to n, k. Averaging n directly is close only when the indices are similar and both phases are nearly transparent.
The composition alone does not determine the answer. Two films with the same ratio can have different optical constants because the phases are arranged differently in space; the choice of rule is a statement about that arrangement.
Six mixing rules
- Bruggeman — self-consistent, with no phase as the host. The default for co-deposited films, voids and roughness, and able to represent percolation. Closed form for two phases; solved numerically (on the passive branch) for three or more.
- Maxwell Garnett — dilute guests in a continuous host: lightly doped films, metal nanoparticles in a matrix.
- Lorentz–Lorenz — mixing at the molecular scale with no phase boundary: co-evaporated organics that do not phase-separate, solutions.
- Looyenga — dense random mixtures of modest contrast: porous silicon and anodic alumina, co-sputtered oxides.
- Wiener bounds (parallel / series) — lamellae along and across the field. Every microstructure lies between them; they are the two limits of form birefringence.
What the editor handles
- Up to six components, each a constant, an n, k file or a dispersion model.
- Shape factor L — sphere (1/3), needle (0), lamella (1) — for the inclusion shape in Bruggeman and Maxwell Garnett.
- Volume, mass or mole fractions — every rule is a volume-fraction rule, so mass and mole ratios are converted with the densities (and molar masses) supplied. Without them the editor refuses rather than substituting a plausible density.
- Spread between rules — the distance between the two extreme microstructures at that composition. When it is small the choice of rule does not matter; when it is large, an index quoted without naming the rule is not a number.
- The recipe (components, fractions, rule) is stored on the layer and saved with the project, so a fraction can be changed later and the layer recomputed.
Ellipsometry roughness layers
The surface-roughness layer of an ellipsometry fit is a Bruggeman layer of film and ambient mixed 50:50; for a film of n = 1.80 under air it comes out at n ≈ 1.38. The same principle explains why a measured index below the bulk value is a density measurement.
Applications
- Two-host co-evaporated OLED emissive layers and doped transport layers
- Porosity and density of reactively sputtered and evaporated films, inferred from the measured index
- Metal–dielectric cermets, metal nanoparticles and quantum dots in a matrix
- Low-index layers and sensors based on porous silicon and anodic alumina
- Surface roughness, interfacial mixing layers and form birefringence of sub-wavelength laminates
Try it
Mixed layers are created directly in the layer editor. Phazic runs on Windows 10 and 11 and is available from the Microsoft Store.
Related: Anisotropy (Berreman 4×4) · Thickness-tolerance Monte Carlo · Tandem OLED