# Evaluation of the Thickness in Nanolayers Using the Transfer Matrix

Authors: Juan E. González-Ramírez; Juan Fuentes; Luis C. Hernández; Luís Hernández
Journal: Research Letters in Physics
 (2009)

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## Abstract

The reflectivity spectra have been traditionally used to determine the thicknesses in semiconductor films. However, thicknesses of nanofilms are not easy to evaluate because the interference fringes are not visible in the transparent region. In this paper, we present a computed method based on the transfer matrix (TM) which is used to match the calculated and experimental room temperature reflectivity spectra of the ZnTe/GaAs films and to determine its thickness film values afterwards. The TM method needs only to know refraction indices and absorption coefficients as a function of wavelength for the film and the substrate. The thickness nanofilms evaluated by our method are in agreement with the values measured by ellipsometry, Rutherford backscattering spectroscopy and transmission electron microscopy techniques. The present procedure extends the application of the standard spectral reflectance technique to determine semiconductor nanolayer thicknesses.


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## Body

## 1. Introduction

Semiconductor
nanolayers are profusely used in the modern technology, therefore to determine
their thicknesses becomes an essential theme. The spectral reflectance has been
extensively applied to measure the semiconductor layer thicknesses as it is
carried out quickly: it is a nondestructive technique, and
a special surface preparation of the sample is not necessary. These measurements
are usually taken in the photon energy region, where the semiconductor layers
are transparent, and distinct interference fringes are clearly observed.
However, in the very thin semiconductor films, the spectral reflectance
technique is rejected because the interference fringes do not become visible;
there is no evidence of maxima or minima intensity: only broad modulations are
observed.The transfer matrix (TM) method has
been widely used to determine the values of the optical constants and thickness
of thin films from their reflectance spectra in the visible wavelength range
[1–5]. The photon transport through the multilayered structure is usually
described using the TM method. In this approximation, it is assumed that each
individual layer can be characterized by a well-defined thickness and
refractive index which depends on wavelengths. The general features of the
reflection and transmission coefficients are theoretically reproduced by this
method.The TM method is critically tested in this
work to obtain the computed reflection spectra which are matched with
experimental data of ZnTe/GaAs nanolayers in order to determine their
thicknesses. Using the minimum squares method, the thickness value uncertainty
of each layer is found out. To verify our results, a commercial program is used
to compare with the TM reflection spectra showing a good agreement.Likewise, we compare the ZnTe/GaAs film
thicknesses obtained by reflectivity using the TM method with the thicknesses
reported for the same films measured by ellipsometry, transmission electron microscopy
(TEM), and Rutherford backscattering spectroscopy (RBS) techniques. The present
procedure is very useful because it extends the application of the standard
spectral reflectance technique to determine semiconductor nanolayer
thicknesses.


## 2. Model Details

The TM method is shortly
presented below. Stationary regimes of electromagnetic monochromatic plane
waves impinge on a homogeneous and isotropic thin dielectric film between two
semi-infinite transparent media. Taking into account the boundary conditions
for electric (E) and magnetic (B) fields, a matrix equation is
obtained which relates the fields at two adjacent boundaries:
(1)[EaBa]=[cosδisenδ/γ1iγ1senδcosδ][EbBb],where a and b indices denote
the two layer boundaries, δ=2πn1tcosχ/λ0 is the electromagnetic wave phase change
arising from path length in the layer, n1 is the layer refraction index, t is
the layer thickness, and χ is
the refraction angle. The γ1 coefficient takes two values: (2)γ1=n1ε0μ0/cosχ,γ1=n1ε0μ0cosχ,   when electric field is
parallel or perpendicular to the plane of incidence, respectively. The ε0andμ0 constants have the usual
physical meaning. The second-order matrix in (1) is the TM, and it is commonly
denoted as (3)M=[m11m12m21m22].Using the above
scheme, the amplitude coefficients of reflection (ρ) and transmission (τ)
are derived as a function of the matrix components: (4)ρ=γ0m11+γ0γsm12−m21−γsm22γ0m11+γ0γsm12+m21+γsm22,(5)τ=2γ0γ0m11+γ0γsm12+m21+γsm22, where the γ0 and γs coefficients are identical to
the reported ones by Hecht [6], and they characterize  the optical properties in the film and in the
substrate, respectively. To find either (ρ)
or (τ) for any configuration of
layers, it is only needed to compute the transfer matrices for each layer,
multiply them, and then substitute the resulting matrix components into the
above equations. Multiplying (ρ) by
its complex conjugate leads to the reflectance coefficient R: (6)R=ρ×ρ*.The present
procedure consists in matching the computed reflectance coefficient obtained by
(6) with experimental data. In order to determine the film thickness, the t parameter is varied in (5), and the value
that better matches
computed and experimental reflectance coefficients is chosen. This better value of t parameter is found by minimum squares
method: (7)dif=Rexp(λi)−Rcom(λi), where Rexp and Rcom are the experimental and computed reflectance coefficients, respectively, and the difference dif is taken at the same wavelength λi. The layer thickness that does
the minimum sum of the
quadratic difference for each wavelength is chosen. The uncertainty σ in
the thickness value determination was evaluated by the well-known equation (8)σ=∑i=1n(Rexpi(λi)−Rcomi(λi))2(n−2).


## 3. Results and Discussions

The reflectance spectra were
recorded on ZnTe/GaAs samples which were grown onto n-type GaAs (100) substrates
in an isothermal close space configuration using elemental Zn and Te elemental
sources [7]. Zinc telluride is a promising semiconductor material of II–VI groups for fabrication of
high efficiency thin film solar cells and other optoelectronic devices due to
its suitable intrinsic energy gap, 2.26 eV [8]. The reflectance measurements in
the visible range were measured by employing an UV-Vis unicam photo-spectrometer.
The wavelength accuracy is 0.4 nm, and the accuracy of the photometric measurements
is 0.3%. A tungsten halogen lamp was employed as light source in the visible
range. This simple equipment allows, as it would be exposed, a great accuracy
in the determination of the optical parameters of the samples measured.The goal of the
work is to determine the thickness in very thin films, where the light is not
completely absorbed. For this reason, it is necessary to include in the
reflectance spectra calculations, the ZnTe and GaAs refraction indices and the absorption coefficients as a
function of wavelength, which its values were taken from [9, 10].Figure1 shows
the ZnTe/GaAs reflectance spectra calculated by the TM method, for two
different thicknesses: (a) thin film, t = 500 nm, where several interference fringes are observed, and (b) nanofilm, t = 50 nm, where there are no interference fringes in the transparent photon
energy region, only broad intensity modulation. Figure 1 also displays the
reflectance spectra computed by a commercial program [11], showing a good
agreement with TM spectra in the maximum, minimum, and critical point
positions.The ZnTe/GaAs
reflectance spectra calculated by the TM method and by “SCOUT 98” commercial program, for two
different thicknesses: (a) thin film,t = 500 nm, and (b) very thin film, t = 50 nm.
(a)(b)The
experimental and computed reflectance spectra are shown in Figure2 for the
best fit: (a) t = (22 ± 4) nm and (b) t = (59 ± 5) nm. The uncertainty
values were calculated by the minimum squares method. Note the good agreement between
both spectra, in particular the coincidence in the wavelength values for the
critical points.The experimental and
computed reflectance spectra for the best fit: (a)t = (22 ± 4) nm, and (b) t = (59 ± 5) nm. The uncertainty values were calculated by the minimum squares
method.
(a)
t = (22 ± 4) nm(b)
t = (59 ± 5) nmTo verify our
results, ZnTe/GaAs film thickness values are compared with those measured by
other techniques for the same layers [12]. Table 1 displays the good concordance
between the film thickness values determined by reflectivity, using the TM
method, and the values measured by ellipsometry, TEM, and RBS techniques.
Similar results in the thickness film values were reported for very thin films
[13]. However, these authors used a method based on the model dielectric
function to fit the room temperature reflectivity spectra. This model makes use
of numerous optical constants becoming troublesome, avoiding the evaluation of
the uncertainty in the thickness determination.Table 1
Thickness evaluation using ellipsometry, RBS, TEM, and reflectivity spectra
(present work) techniques.
Thickness measurement (nm)(22 ± 4)*23**(59 ± 5)*60†(104 ± 6)*125‡*Reflectivity; **Ellipsometry; †TEM; ‡RBS.The TM method
using usual optical constants, refraction index, and absorption coefficient
gives a simple procedure to determine, by the reflectivity spectra, the
thicknesses of very thin films when the interference fringes in the transparent
region are not observable.


## 4. Conclusions

A simple method to determine
the thicknesses of very thin films by reflectivity spectra was described here.
A procedure based on the TM is used to match the computed and experimental room
temperature reflectivity spectra of the ZnTe/GaAs films and to determine its
thickness film values afterwards. The TM method needs only to know usual
optical constants, that is, refraction indices and absorption coefficients as a
function of wavelength for the film and the substrate. The thickness films
evaluated by our method are in agreement with the values measured by ellipsometry,
RBS, and TEM techniques. The present procedure extends the application of the
standard spectral reflectance technique to determine semiconductor nanolayer
thicknesses.



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*Source: 594175-2009-04-23.xml*