Ellipsometry

Ellipsometry reflects a collimated, polarized beam from a sample at an oblique angle, typically 55–75°, and measures how the reflection changes the relative amplitude and phase of the s- and p-polarized components. Linear light generally comes back elliptically polarized, which gives the method its name. The result is a pair of angles, Ψ and Δ, defined by the ratio of the complex reflection coefficients:

$$\rho = \frac{r_p}{r_s} = \tan\Psi \; e^{i\Delta}.$$

Here $\tan\Psi = |r_p|/|r_s|$ and $\Delta = \delta_p - \delta_s$ is the phase difference introduced by the reflection. Because a ratio is measured, the result does not depend on the absolute intensity of the source or on an absolute reflectance calibration, and Δ in particular is sensitive to layers far thinner than a wavelength. It is used for the thickness and index of films on silicon, the complex refractive index of absorbing materials, and in-situ monitoring of deposition.

Worked values for oxide on silicon

The reflection coefficients follow from the Fresnel equations, extended for a film by the multiple-beam sum of thin-film interference. For silicon at 633 nm (n = 3.882, k = 0.019) at 70° incidence:

The 2 nm native-oxide case shows the sensitivity: Ψ barely moves while Δ falls by 2.9° per nanometer, so an instrument that resolves Δ to 0.01° resolves a change of about 0.0035 nm in average film thickness. Thicker films move both angles around a closed trajectory that repeats with a thickness period

$$d_\text{p} = \frac{\lambda}{2\sqrt{n_1^2 - \sin^2\theta_0}},$$

284 nm for SiO₂ (n₁ = 1.457) at 633 nm and 70°, so a single-wavelength measurement determines thickness only modulo this period; a second angle or wavelength removes the ambiguity.

Model fitting

Ψ and Δ are not the film parameters. They are inputs to a fit: the operator builds a layer model (substrate, each film with a thickness, and a dispersion law for each material), computes Ψ and Δ for it, and adjusts the free parameters to minimize the mean squared difference from the data. Transparent films are usually described by the Cauchy relation $n(\lambda) = A + B/\lambda^2 + C/\lambda^4$; absorbing ones by oscillator models (Lorentz, Tauc-Lorentz for amorphous semiconductors, Drude for free carriers) that keep $n$ and $k$ consistent with the Kramers-Kronig relations. For a bare, smooth, thick substrate the inversion is direct and gives the pseudo-dielectric function

$$\langle\varepsilon\rangle = \sin^2\theta_0\left[1 + \tan^2\theta_0 \left(\frac{1-\rho}{1+\rho}\right)^2\right],$$

which returns 15.07 for the silicon example and is the starting point for identifying a material. Any overlayer or roughness makes it a "pseudo" value.

Spectroscopic and variable-angle instruments

Spectroscopic ellipsometers measure Ψ and Δ over a wide band, commonly from the ultraviolet near 200 nm into the near infrared, often at several angles. The extra data constrain the dispersion model and break correlations between thickness and index. Common designs are the rotating analyzer, the rotating compensator and the photoelastic phase modulator; the rotating-analyzer type cannot determine the sign of Δ and loses precision near 0° and 180°, which is exactly where thin films on silicon lie, and a compensator or modulator design recovers it. Measurements are most sensitive near the substrate's Brewster angle, about 75.6° for silicon at 633 nm, where $|r_p|$ is small.

The comparison with the prism coupler, minimum-deviation and fringe methods is made in How to Measure Refractive Index.

Pitfalls

• Correlated parameters: for films thinner than about 10 nm, thickness and index trade off against each other, and one of them must be fixed or measured independently.
• Overfitting: a model with more free parameters than the data support can match the data with unphysical values; a good fit is not proof of a correct model.
• Unmodeled layers: native oxide, surface roughness, interface intermixing and backside reflections from transparent substrates all change Ψ and Δ and must be included or suppressed.
• Sign conventions for Δ and for $k$ differ between textbooks and software, so values from different sources must be converted before comparison.

Common questions

What does ellipsometry measure directly?

Two angles per wavelength and angle of incidence: Ψ, from the amplitude ratio of p to s reflection, and Δ, their phase difference. Thickness, refractive index and extinction coefficient are derived quantities that depend on the model.

How thin a film can ellipsometry measure?

Sub-nanometer changes in average thickness are detectable, because Δ moves by about 2.9° per nanometer of oxide on silicon at 633 nm. Separating the thickness from the index of a film that thin is not reliable without additional information.

Is ellipsometry better than reflectometry?

It measures a phase as well as an amplitude ratio and is insensitive to source intensity, so it is more sensitive to very thin films and gives $n$ and $k$ independently. Normal-incidence reflectometry is simpler and adequate for films thick enough to show several fringes.

References: R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007); H. G. Tompkins and E. A. Irene (eds.), Handbook of Ellipsometry (William Andrew, 2005); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).