Envelope method

The Einhüllendenverfahren (English: Envelope Method ) is in the thin film technology , a method for determining the optical properties of a dielectric single-layer (eg. Titanium dioxide , silica or magnesium fluoride ) on a transparent substrate (e.g., quartz glass).

Beam distribution of a single layer; Part of the light is reflected at each interface. The components can interfere with one another, depending on the thickness of the layer and its refractive index.

Basic sketch of a single-beam spectrophotometer. The light from a source is broken down into its spectral components by a diffraction grating before it propagates through a slit. A wavelength range can be traversed by moving the slit or tilting the grating. A photodetector thus measures the wavelength-dependent intensity of the light.

This method is based on the spectrophotometric measurement of the transmission of this layer under normal incidence of light. The optical parameters that can be determined with this method are the refractive index , the degree of absorption and the thickness of the individual layer . The method is used to assess the quality of the applied layer (e.g. compactness) and to be able to optimize the manufacturing parameters accordingly. ${\ displaystyle n}$ ${\ displaystyle k}$ ${\ displaystyle d}$

The transmission spectrum of a transparent single layer, obtained z. B. over the visible spectral range , generates a wave pattern as a result of interference phenomena , as shown in the figure. The wave pattern is limited by an upper and a lower envelope , which gives the process its name. The amplitude of the wave trains, the position of the maxima and minima and the intensity in relation to that of the uncoated substrate are the input parameters of an algorithm which determines the desired optical parameters. The optical parameters can be determined separately for each extreme point of the wave curve. This also enables the dispersion and absorption curve of the material to be determined.

The transmission coefficient T of a single layer, plotted against the wavelength of the transmitted light. The transmission curve oscillates between two curves, the envelope. In the short-wave range, the absorption gains the upper hand, which is why the curve drops sharply there. The envelope method requires several value pairs and as input data , which are either measured or interpolated points of contact between the transmission curve and the envelope.

{\ displaystyle T_ {M}}

{\ displaystyle T_ {m}}

Working principle

Depending on the difference in refractive index, a small proportion of the light is reflected at the two interfaces between air-single layer and single-layer substrate. These partial beams can interfere both in the transmission direction and in the reflection direction. Depending on the phase position, which is determined by the layer thickness and the refractive index of the layer, the interference can be constructive or destructive. The interference condition is dependent on the wavelength. If one now carries out measurements over a frequency range, one finds varying conditions for constructive and destructive interference. If the frequency range is sufficiently wide and the layer is thick enough, a wave pattern with alternating constructive and destructive interference is obtained, as shown in the figure below. The optical properties of the thin layer can be reconstructed from this wave pattern.

Calculation path

The calculation of the optical parameters is carried out at each pair of contact points and the curve with its envelope. For each extremum (example: maximum) the opposite extremum (in the example the minimum) is also required. Since only one of the two values is a real measured variable, the other extreme value can only be obtained by spline interpolation ( cubic splines , fractional-rational or, in the case of low absorption, also linear) with its neighbors. The value pairs for and including the associated wavelengths are the output variables of the calculation process. The transmission curve of a single layer can be passed through ${\ displaystyle T_ {M}}$ ${\ displaystyle T_ {m}}$ ${\ displaystyle T_ {M}}$ ${\ displaystyle T_ {m}}$ ${\ displaystyle \ lambda}$

{\ displaystyle T = {\ frac {Ax} {B-Cx \ cos \ phi + Dx ^ {2}}}}

describe with

{\ displaystyle A = 16n ^ {2} s}

{\ displaystyle B = (n + 1) ^ {3} (n + s ^ {2})}

{\ displaystyle C = 2 (n ^ {2} -1) (n ^ {2} -s ^ {2})}

{\ displaystyle D = (n-1) ^ {3} (ns ^ {2})}

{\ displaystyle \ phi = 4 \ pi nd / \ lambda}

{\ displaystyle x = \ exp (-kd)}

Here, k is the absorption coefficient and s is the refractive index of the substrate, which can be found in the relevant data sheets. To a good approximation, the substrate can be regarded as free of absorption. The optical parameters of the layer can be determined by rearranging and evaluating this expression.

Refractive index n

The refractive index of the layer is reflected in the amplitude of the oscillating curve and results from rearranging the above expression for the transmission curve. A distinction must be made here between two cases: those with no absorption and those with little absorption. The method cannot be used for strong absorption because of the collapsing interference pattern. ${\ displaystyle n}$

Without absorption, the refractive index is given by

{\ displaystyle n = {\ sqrt {M + {\ sqrt {M ^ {2} -s ^ {2}}}}}}

in which

{\ displaystyle M = {\ frac {2s} {T_ {m}}} - {\ frac {s ^ {2} +1} {2}}}

is.

It should be noted here that, strictly speaking, there are two values for that provide identical transmission curves, one of which is larger and the other smaller than the refractive index of the substrate. However, the lower of the two arithmetical solutions can often be ruled out: either the refractive index is so low that no solid substance can accept it, or it cannot be matched with literature values for the layer material used. ${\ displaystyle n}$ ${\ displaystyle s}$

The refractive index is calculated separately for each extremum because it is subject to dispersion , i.e. i.e., it increases towards shorter wavelengths.

In the case of a slight absorption of the layer one gets modified expressions:

{\ displaystyle n = {\ sqrt {N + {\ sqrt {N ^ {2} -s ^ {2}}}}}}

With

{\ displaystyle N = 2s {\ frac {T_ {M} -T_ {m}} {T_ {M} T_ {m}}} + {\ frac {s ^ {2} +1} {2}}}

Layer thickness d

The layer thickness d results from the position of the extreme values to one another. The expression results for the layer thickness ${\ displaystyle d}$

{\ displaystyle d = {\ frac {\ lambda _ {1} \ lambda _ {2}} {2 (\ lambda _ {1} n_ {2} - \ lambda _ {2} n_ {1})}}}

Absorption coefficient k

In the absorption-free case, the maximum values of the transmission curve touch the line of transmission of the uncoated substrate, while with absorption they lie below. In the latter case, the total absorption can be calculated with

{\ displaystyle x = {\ frac {E_ {M} - {\ sqrt {E_ {M} ^ {2} - (n ^ {2} -1) ^ {3} (n ^ {2} -s ^ { 4})}}} {(n-1) ^ {3} (ns ^ {2})}}}

With

{\ displaystyle E_ {M} = {\ frac {8n ^ {2} s} {T_ {M}}} + (n ^ {2} -1) (n ^ {2} -s ^ {2})}

However, the size depends on the layer thickness. In order to be able to evaluate material properties, it is less the total absorption than the layer thickness-independent absorption coefficient that is decisive. The absorption coefficient results from the already known layer thickness ${\ displaystyle x}$ ${\ displaystyle k}$ ${\ displaystyle k}$

{\ displaystyle x = \ exp (-kd)}

The absorption coefficient k is also calculated individually for each extreme, since it is not constant over the wavelength range, but increases sharply at shorter wavelengths (e.g. in the ultraviolet).

Advantages and disadvantages

The equipment required for the measurement is moderate; all that is required is a spectrophotometric measurement. This procedure can be carried out relatively quickly; If necessary, its results can also be backed up by applying procedures or ellipsometric measurements.

Half of the points of contact between the wave curve and the envelope can only be determined by means of interpolation , which can affect the evaluation. This effect is particularly important when there is stronger absorption at shorter wavelengths, since the envelopes are more inclined or curved in these cases.

If the layer thickness is too small, the number of extremes is too small for an evaluation; you need at least four extreme values in order to be able to carry out a meaningful interpolation. In these cases, matching algorithms deliver better results.

variants

The process can be modified in order to be able to carry out a layer thickness control during the coating .
By adding reflection measurements, the inhomogeneity of the layer (variability of the refractive index over the depth in the layer) can be determined.

literature

Manifacier et al .: A Simple Method for the Determination of the Optical Constants n, k and the Thickness of a Weakly Absorbing Thin Film , J. Phys. E: Sci. Instrum. 9 (1976)
R. Swanepoel: Determination of the Thickness and Optical Constants of Amorphous Silicon , J. Phys. E: Sci. Instrum. 16 (1983)

Web links

Validity of Swanepoel's Method for Calculating the Optical Constants of Thick Films (PDF; 492 kB)
Envelope and waveguide methods (PDF; 582 kB)