Reflection spectroscopy

The reflectance spectroscopy is a variant of spectroscopy in which the dependence of the reflection (specifically, the reflectance ) of a material on the wavelength of electromagnetic radiation - the light is examined -. The reflection spectrum or the “ dispersion of the reflection” is measured. Through quantitative analysis of the measured curve shape, information on details of some properties of the material under investigation can be obtained.

Measuring arrangement

A reflection spectrum is usually measured in an optical spectrometer .

In order to compensate for the wavelength dependencies of the measuring arrangement itself (intensity curve of the light source, reflection spectra of the mirrors used, etc.), measurements are usually made against a sample made from a reference material with a known reflection spectrum. In the area of visible light and the adjacent areas, plane mirrors coated with aluminum are mostly used here .

Depending on the material property of interest, an optical modulator may be inserted into the measuring arrangement.

The ellipsometry drives a higher expenditure on equipment by the reflection degree is determined not only for normal incidence, but depending on the angle and for both polarization directions. In this way, more complex properties of the sample can be determined.

Theoretical calculation of a reflection spectrum, measurement evaluation

Calculated (0 ° and 60 °) and measured (approx. 5 °) reflection spectra of silver with the characteristic plasma edge ω _p and ω _s (see plasma resonance )

An incident electromagnetic wave interacts differently with the material depending on its energy (i.e. wavelength), depending on whether the material has a permitted band transition at this energy, for example , and could absorb the energy for this. There are also other absorption mechanisms that can be described theoretically. The following shows the steps in which conclusions can be drawn about the existence and parameters of such mechanisms from a measured reflection spectrum. The procedure is typically as follows:

1. The relevant mechanisms in the material that can influence its complex permittivity are determined, such as band transitions, electron densities in the conduction band, polarizabilities , etc. ${\displaystyle \varepsilon _{\mathrm {r} }=\varepsilon _{1}+\mathrm {i} \varepsilon _{2}}$
2. For each of these mechanisms, its contribution to electrical susceptibility is set as a (generally complex-valued ) frequency-dependent function. These formulas will contain various parameters that represent the actual material properties. In order for the further calculations to work, one must typically be able to provide reasonably good estimates for these parameters in order to achieve a sensible approach. These susceptibilities are finally added together to form the total permittivity number . ${\displaystyle \chi _{j}}$ ${\displaystyle \varepsilon _{\mathrm {r} }=1+\sum \nolimits _{j}{\chi _{j}}}$
3. The complex refractive index whose real part of the refractive index and the imaginary part of the extinction coefficient is related to the permittivity and the permeability through together . With the formulas determined above for the permittivity values, the refractive index and the absorption coefficient can be calculated directly for non-magnetic materials ( ): ${\displaystyle N=n+\mathrm {i} k}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle \varepsilon _{\mathrm {r} }}$ ${\displaystyle \mu _{\mathrm {r} }\,}$ ${\displaystyle N={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}}}$ ${\displaystyle \mu _{\mathrm {r} }\approx 1}$ ${\displaystyle N=n+\mathrm {i} k={\sqrt {\varepsilon _{1}+\mathrm {i} \varepsilon _{2}}}}$

   ${\displaystyle n^{2}={\frac {1}{2}}\left({\sqrt {\varepsilon _{1}^{2}+\varepsilon _{2}^{2}}}+\varepsilon _{1}\right),}$

   ${\displaystyle k^{2}={\frac {1}{2}}\left({\sqrt {\varepsilon _{1}^{2}+\varepsilon _{2}^{2}}}-\varepsilon _{1}\right).}$

4. And with this you can use the Fresnel equations to calculate the degrees of reflection and , for example, for the perpendicular incidence on an air-medium interface (with and ) (attention special case!): ${\displaystyle R_{\mathrm {s} }}$ ${\displaystyle R_{\mathrm {p} }}$ ${\displaystyle N_{\text{Luft}}=N_{1}\approx 1}$ ${\displaystyle N_{\text{Medium}}=N_{2}=n_{2}+\mathrm {i} k_{2}}$

   ${\displaystyle R_{\mathrm {s} }=R_{\mathrm {p} }={\frac {(n_{2}-1)^{2}+k_{2}^{2}}{(n_{2}+1)^{2}+k_{2}^{2}}}.}$

5. If necessary, do this for all frequencies in the part of the spectrum of interest and compare the result with the measurement curve.
6. By means of an adjustment calculation between the theoretical and measurement curve, specific values ​​for parameters can be determined, as they were introduced in step 2.

See also

• Reflectometric interference spectroscopy
• Reflectometric thin film measurement
• Ellipsometry

literature

• Gustav Kortüm (1969). Reflection spectroscopy; Basics, methodology, applications. Berlin, New York, Springer-Verlag