Introduction

In recent years, pseudo-binary chalcogenide (GeTe)x(Sb2Te3)1−x alloys (GSTs) have attracted much attention from the fundamental and applied points of view. GSTs relate to the class of phase-change materials and possess a rather unconventional set of properties. Optical and electrical properties inherent to amorphous and crystalline states of GSTs exhibit significant contrast1,2,3,4. In GTSs the optical dielectric constant is 70–200% larger for the crystalline than for the amorphous phases3. GSTs provide evidence for a disorder-driven metal–insulator transition5. Furthermore, crystallization in these materials can be extremely rapid and can occur in less than 100 ns6. These aspects may explain the interest to the properties of the crystalline phase of these compounds.

GST alloys are among the most promising materials for data storage applications. They are already successfully employed in rewriteable optical data storage (compact discs (CD), digital versatile discs (DVD) and Blue-ray Discs (BD))7. In addition, GSTs also have a unique potential as emerging non-volatile electronic memories. The phase change random access memory (PCRAM) is a possible successor for current memory devices such as dynamic random access memory (DRAM), static random access memory (SRAM) and flash memory8. Among its attractive features is the simple cell structure with high scalability; it is non-volatile, has a relatively high read/write operation speed (<10 ns), and a long cycle life (>1012 operations).

As GST alloys exhibit the contrast in optical properties between amorphous in crystalline states, they are also promising for non-volatile photonic applications, such as all-photonic memories, colour-rendering, and nano-pixel displays9. These applications employ the property contrast between amorphous and the crystalline states of GSTs, the long-term stability of the amorphous phase plays one of the crucial roles in reliability of corresponding devices.

In the recent work10, we studied the compositional dependences in the vibrational properties of amorphous GSTs using Raman spectroscopy at ambient temperature. It was found that the systematic compositional dependences in the intensities of characteristic Raman bands correlate with evolution of the concentration of different structural units present in (GeTe)x(Sb2Te3)1−x alloys. To gain deeper insight into vibrational properties of amorphous and crystalline GSTs, the investigation of the temperature-dependent behaviour of phonon modes was performed11, which directly revealed a correlation between anharmonicity, vacancy concentration, and ordering.

While modeling the dielectric function of some phase-change alloys in the amorphous state12, it was shown that quite small deviations in ε2 can cause pronounced differences at the calculated reflectance spectra. Introduction an additional constant ε2 to the dielectric function could significantly improve the quality of the fit of the experimental data. It was assumed that additional constant ε2 corresponds to the contribution of absorption present in the amorphous samples studied. Analysis of the decrease of the additional constant ε2 upon cooling and annealing was interpreted as shrinking absorption by defect states, caused by a non-reversible decrease in the number of defect states upon relaxation. However this approach does not give one a chance to evaluate the width of the Urbach-Martienssen tails formed by localized and additional states. Furthermore, there is a lack of the information on the compositional dependences in the Urbach-Martienssen tails in amorphous phase-change alloys. The latter might bring insight to the understanding of the influence of the disordering on the optical properties of the amorphous state in phase-change alloys.

This study is aimed to reveal the compositional dependences in parameters that govern the optical properties of (GeTe)x(Sb2Te3)1−x amorphous alloys in the wide spectral range from the phonons to the inter-band electronic transitions. By focusing on the absorption edge fluctuations that are linked to the variations of the bandgap Eg, Tauc parameter B1/2, and the width of Urbach-Martienssen tails EU in amorphous (GeTe)x(Sb2Te3)1−x alloys, we expect to reveal the compositional trends in the influence of the structural and thermal disordering on the absorption processes in studied amorphous alloys.

Results and Discussion

In the spectral range between phonons and below the absorption edge, real part of the dielectric function (DF) is governed by polarizability of bonded electrons and therefore can be described by the constant εinf. In the case of covalently bonded compounds the dielectric constant satisfies the Clausius-Mosotti relation3,13 which expresses the dielectric constant εinf of material in terms of the atomic polarizabilities of the material’s constituent atoms.

The DF of studied compounds shown in Fig. 1 was obtained from the fit of the measured spectra. The real part of the DF remains constant below the optical bandgap. The imaginary part of the DF of studied amorphous GSTs possesses nearly zero values at the energies below the bandgap, and then ε2 increases due to the electronic inter-band transitions with the range around the optical bandgap. Obtained dielectric constants εinf of studied samples follow the trend line estimated by the Clausius-Mossotti relation as shown in Fig. 1(I): εinf decreases while GeTe contents increases. For the calculation of the trend line we used the following values of the atomic polarizabilities3: αGe = 5.07 × 1040 Cm2 V−1, αSb = 7.82 × 1040 Cm2 V−1, and αTe = 7.66 × 1040 Cm2 V−1. As in the range between phonons and the absorption edge, εinf is determined by the polarizabilities of the constituent atoms, the compositional evolution of the εinf is explained by the difference in atomic polarizabilities of Ge, Sb, and Ge atoms, whose concentration changes along the compositional line. In amorphous GSTs the majority atoms show covalent bonding and the coordinations around the majority atoms of Ge, Sb, and Te satisfy 8-N rules3.

Figure 1
figure 1

The dielectric function of amorphous (GeTe)x(Sb2Te3)1−x samples: () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe. Dashed lines correspond to the ε1, while solid lines correspond to the ε2. The dielectric constants εinf were determined as values of ε1 at 0.07 eV. The positions of the Eg are marked as vertical dashed lines and were determined by applying the α10k-criterion. Inset: The dielectric constants εinf of studied alloys. Solid line is calculated by applying the Clausius-Mossotti relation.

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Except for evolution of the εinf, there are some more messages coming from Fig. 1. In the spectral range around 1.6 eV, the amplitude of the imaginary part of the dielectric function decreases as Ge content increases. The changes in the dispersion of ε2 in the range 0.6–1 eV indicate the shift of the optical absorption edge. These findings are in the agreement with theoretical and experimental data shown in3,14. The amorphous phase of GSTs is characterized by locally ordered motifs10, and long-range disorder with the volume expansion of 6% compared with the crystalline phases14. Altogether visually recognizable changes in the ε2 (Fig. 1) might indicate the evolution of the ordering that results in the changes of band-structure in the amorphous GSTs, which we now consider in more details.

The absorption coefficient α can be obtained from the equation α = 4πk/λ, where k is the extinction coefficient and λ is a wavelength13. From the perspective of fundamental material science the dispersion of the absorption coefficient can be divided into several spectral ranges. High-absorption region (α > 104 cm−1) involves optical transitions between the valence and conduction band states, whereas the spectral range with 102 < α < 104 cm−1 is called the Urbach’s exponential tail region. Due to the lack of atomic long-range order that is caused by disordering the bond angles and lengths, amorphous materials do not have as sharp bandgap as crystalline materials have13. Therefore, the band structure of amorphous materials has to be complemented by localized states at the band edges and additional defect states inside the bandgap forming co-called the Urbach-Martienssen tails15. In the Urbach’s exponential tail region most of the optical transitions take place between localized tail states and extended band states. The region (α < 102 cm−1) involves low-energy absorption which occurs due to the optical transitions between the localized states. The range around and slightly below the absorption edge in amorphous GSTs is of our interest. In the Urbach’s tail range, the absorption coefficient α shows an exponential dependence on photon energy E, and obeys the relation15,16:

$$\alpha ={\alpha }_{0}\,\exp \,(\frac{E-{E}_{0}}{{E}_{U}}),$$
(1)

where α0, is a constant, E is energy, E0 corresponds to the energy close to that of the bandgap at low temperature, and EU is the Urbach energy, i.e. the width of the band tail of the localized states in the bandgap. Eq. 1 implies that logarithms of the absorption coefficient plotted as a function of the energy can be approximated by a straight line for the energies just below the fundamental absorption edge.

Figure 2 displays the absorption coefficients in amorphous GSTs which were derived from the data shown in Fig. 1. We show the linear approximation of the absorption coefficient as a function of the energy, for 1.5 × 103 cm−1 < α < 5 × 103 cm−1 in Fig. 2, where the value of EU corresponds to the inverse slope of the approximating straight line. The value of Urbach energy EU for amorphous Ge2Sb2Te5 obtained in the present study is consistent with corresponding value obtained in17,18. The evolution of the inverse slope of the approximating straight line reflects the systematic increase of the width of band tails of the localized states in the bandgap in amorphous materials along the Sb2Te3-GeTe line, as shown in the Inset of the Fig. 2.

Figure 2
figure 2

Dispersion of the absorption coefficients α in amorphous (GeTe)x(Sb2Te3)1−x samples: () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe. Straight dashed lines correspond to the linear approximation by using the Eq. 3. Inset: the compositional dependence in the Urbach energy EU in (GeTe)x(Sb2Te3)1−x samples. Straight trend line is shown for the visualization of the compositional dependence.

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To get the deeper insight into the nature of increase of the width of band tails in amorphous (GeTe)x(Sb2Te3)1−x, we now analyze the higher energies range above the absorption edge (α > 104 cm−1), where the absorption is associated with inter-band electronic transitions, and the absorption coefficient obeys Tauc relation19:

$$\alpha E=B{(E-{E}_{G})}^{n},$$
(2)

where B is the Tauc parameter, E is photon energy, Eg is the optical bandgap and n is the number which relates to the mechanism of transition process. It was shown in14,20,21 that in amorphous (GeTe)x(Sb2Te3)1−x alloys the photon energy is used for non-direct transition, that according to22 corresponds to n = 2. The Tauc parameter B represents the measure of the disorder in the system23,24. It includes the information on convolution of the valence band and conduction band states and on the matrix element of optical transitions, that reflects the k selection rule and the disorder-induced spatial correlation of optical transitions between the valence band and conduction band25. A decrease in B1/2 indicates an increase in the disorder20,26.

Figure 3 displays the plots of αE as a function of photon energy in the higher absorption range. The dashed lines were obtained from the Eq. 2. The interceptions of these lines with abscise correspond to the values of Eg, which are in good agreement with values determined by α10 k criterion (the optical gap Eg that was also identified as the energy at which the absorption exceeds the value of 104 cm−1, as shown in Fig. 1). The obtained results revealed the compositional dependence in the Eg, as studied GST are mixtures of parent binary compounds, they inherit their properties in the corresponding proportion27. The values of the dielectric constant and the optical bandgap of Ge1Sb2Te4, Ge2Sb2Te5, and GeTe are in the quantitative consistency with data previously reported in3, where the samples have been prepared in the same manner using magnetron sputtering method. Trend in the evolution of the band-gap energies upon increasing GeTe content correlates with theoretical analysis of band-structure evolution in amorphous GSTs14. Obtained values of the optical band gap are consistent with results presented in other works. The previously reported values of optical bang-gap in amorphous Ge1Sb2Te4 vary from 0.69 eV to 0.76 eV3,26, in amorphous Ge2Sb2Te5 vary from 0.7 eV to 0.75 eV18,26,28,29, in amorphous GeTe they vary from 0.85 eV to 0.9 eV12,30. These minor discrepancies can be explained by the difference in the quality of samples obtained using different techniques. It is worth to mention, that the increase of the bandgap shown in the Fig. 3(II), is accompanied by the decrease of the dielectric constant εinf (Fig. 1, inset) in GeTe-rich compounds. This finding is reasonable, as both bandgap Eg and the dielectric constant εinf are linked via the empirical Moss rule in the following way31:

$${\varepsilon }_{\infty }^{2}{E}_{g}\approx const.$$
(3)
Figure 3
figure 3

Plots of (αE)1/2 vs E for amorphous (GeTe)x(Sb2Te3)1−x samples: () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe. Dashed lines were calculated using the Eq. 1. Inset I: the compositional dependence in the Tauc parameter B1/2 in the (GeTe)x(Sb2Te3)1−x samples. Straight trend line demonstrates the compositional dependence the Tauc parameter B1/2 (modification in short range order in GeTe-rich amorphous samples). Inset II: The optical bandgap Eg of the alloys. (*) the value of Eg of amorphous Sb2Te3 was taken from14. Trend line is shown for the visualization of the compositional dependence.

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The slopes of the dashed lines in Fig. 3 are related to the corresponding values of B1/2 20. We observe the decrement in B1/2 which reflects the increment in the randomness in atomic arrangement in the GeTe-rich amorphous materials from the studied part of the pseudo-binary Sb2Te3-GeTe line. This trend found to be in agreement with evolution of the widths of Urbach’s tails. It is worth to mention that obtained value of the Tauc constant for amorphous Ge2Sb2Te5 are higher than that reported in20 (for example, 831 vs. 516 at 300 K). This discrepancy can be explained by the different quality of the samples, as in20 samples were prepared using thermal evaporation method. However, both Urbach energy Eu and Tauc parameter B1/2 follow the compositional trends which are quite similar to those obtained in Sn-Sb-Se system with increase of concentration of Sn32.

Optical and electrical properties of crystalline GSTs are sensitive to the presence of the disorder in the samples5,33,34,35,36,37. The presence of any kind of defects in material shortens the phonon lifetime. As the linewidth of a line in the spectrum is inversely proportional to the phonon lifetime, the presence of the disorder leads to the broadening of the linewidth. Raman spectra of amorphous (GeTe)x(Sb2Te3)1−x samples, which are displayed in the Fig. 4, exhibit the similar pattern. In the previous work10 we analyzed the compositional dependencies in the intensities of bands in Raman spectra of amorphous (GeTe)x(Sb2Te3)1−x samples. After the Gaussian decomposition procedure38, we focus on the evolution of the halfwidth FWHM in Raman spectra amorphous (GeTe)x(Sb2Te3)1−x samples. The systematic increment in the average halfwidth <FWHM> (Fig. 4, Inset) results from the shortening of lifetime of Raman-active phonons caused by their increasing scattering due to the increasing imperfectness of the material.

Figure 4
figure 4

Raman spectra of amorphous (GeTe)x(Sb2Te3)1−x samples: () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe10. Dashed lines correspond to Gaussian decomposition of Raman spectrum of amorphous Ge1Sb2Te4 with positions of Gaussians marked10. Inset: The compositional increase of the average halfwidth <FWHM> in studied amorphous (GeTe)x(Sb2Te3)1−x samples.

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Parameters that govern optical properties of the studied (GeTe)x(Sb2Te3)1−x alloys at 300 K (Table 1) exhibit compositional evolutions. Increase in the width of localized states EU that reflects in the widening of the optical bang-gap Eg is consistent with evolution of the Tauc parameter B1/2 and the average halfwidth <FWHM> of Raman bands. The compositional changes in B1/2 and <FWHM> evidence the increase of the disordering in the GeTe-rich amorphous (GeTe)x(Sb2Te3)1−x alloys.

Table 1 Optical parameters of the studied (GeTe)x(Sb2Te3)1−x alloys, 300 K.
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To obtain further and deeper insight into the reasons that cause the changes in EU, we now try to determine the contributions of the structural (static) and thermal-induced (dynamic) disorder on the optical properties of amorphous (GeTe)x(Sb2Te3)1−x samples by the analysis of the temperature-dependent data. An analysis of the contributions of the structural and thermal disorder into the variation of the optical absorption has been performed in15,39. The concept of equivalence of structural and compositional disorder has explained the relationship between the optical bandgap and Urbach energy39. An attempt to analyze the model of39 has been made40 to better understand the role of topological and thermal disorder in determining the optical bandgap in amorphous solids.

In Fig. 5, one can clearly distinguish the blue-shift of the absorption edge in amorphous Ge3Sb2Te6 at low temperatures, that reflects the significant widening of the bandgap in the studied temperature range (17–19%) shown in Inset in Fig. 5. Variation of the energy bandgap with temperature arises from the shift in the relative position of the conduction and valence bands due to dilatation of the lattice and due to temperature-dependent electron-lattice interaction41,42,43,44. Due to thermal dilatation of lattice, an increased interatomic spacing decreases the potential seen by electrons in the material, which reduces the width of the energy bandgap. To analyze the temperature dependence of the optical bandgap we used the following expression39:

$${E}_{g}(T)={E}_{g}(0)-\beta {(exp(\theta /T)-1)}^{-1},$$
(4)

where T is the sample temperature, Eg(0) is the bandgap energy at T = 0 K, β is the material constant linked to the second-order deformation potential15,39, and Θ is characteristic Einstein temperature.

Figure 5
figure 5

Temperature dependence of the absorption coefficient in amorphous Ge3Sb2Te6 with linear approximation according to Eq. 3. Horizontal dashed lines determine the range of the approximation. Vertical dashed lines indicate the position of the optical Eg at the extreme temperatures. Inset: Optical bandgap Eg(T) as a function of EU(T) for studied (GeTe)x(Sb2Te3)1−x alloys: () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe. This dependence demonstrates correlation between Eg(T) and EU(T).

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The blue-shift of the absorption edge and the corresponding dependence Eg(T) is not the only feature notable in Fig. 5. We approximated the temperature evolution of the absorption coefficient by straight lines according to Eq. 3 for 1.5 × 103 cm−1 < α < 5 × 103 cm−1. These lines for various temperatures are not parallel and they converge at a point (E0,α0), called the “converging point”45. As the inverse slope of the approximating straight line corresponds to the Urbach energy EU, this fact evidences the temperature dependence EU(T). The similarity between the temperature dependences of Eg and EU is illustrated in the Inset in Fig. 5 where Eg(T) is plotted against EU(T), with temperature as a parameter. The linear relationship between Eg and EU for all studied compounds (Table 2) confirms that Eg(T) and EU(T) have the same functional form, as shown15.

Table 2 Temperature dependences of the Urbach energy and the optical bandgap in amorphous (GeTe)x(Sb2Te3)1−x alloys. (The errors in determining the values of the Urbach energy and the optical bandgap are ±1 meV and ±0.05 eV, correspondingly).
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In order to estimate the temperature dependence of the Urbach tails, we applied the following expression15:

$${E}_{U}(T)={E}_{U(st.)}+{E}_{U(dyn.)}=\frac{{k}_{B}\theta }{{\sigma }_{0}}(\frac{1+X}{2}+\frac{1}{exp(\theta /T)-1}),$$
(5)

where EU(st.) is contribution of structural disorder; EU(dyn.) is the contribution of thermally-induced disorder, kB is Boltzmann constant, Θ is characteristic Einstein temperature, X is a measure of structural disorder normalized to the zero-point uncertainty in the atomic positions15, and σ0 is Urbach edge parameter of order unity, which is inversely proportional to the strength of the coupling between electrons and phonons45. For the purpose of this study, the characteristic Einstein temperature is a good approximation to the temperature corresponding to the average phonon energy, as reported20.

As shown in Fig. 6(A), Eg(T) dependence exhibits the parabolic shape at low temperatures (below the Debye temperature) and linear shape at higher temperatures. The Debye temperatures of these alloys are in the range of 110–125 K46,47. Before applying fitting procedures to the experimental dependences Eg(T) and EU(T) we evaluated them by calculating ΔEU/EU(5K) and ΔEg/Eg(5K). These values shown in the Table 2 demonstrate following compositional trends: GeTe-rich amorphous (GeTe)x(Sb2Te3)1−x alloys exhibit less pronounced temperature dependences of the optical bandgap and Urbach energy, the latter may serve as an additional evidence of the increase of the disordering in these compounds. We found and show in the Fig. 6 that the fitting Eg(T) and EU(T) by Eqs 4 and 5 using set of parameters shown in the Table 3 is quite satisfying for tracing the experimental data.

Figure 6
figure 6

(A) Optical bandgap Eg as a function the temperature. Dots correspond to the values of Eg obtained by applying the α10k-criterion to the dispersion of the absorption coefficient. Solid lines were calculated using Eq. 4. (B) Urbach energy EU as a function the temperature. Dots correspond to the values of EU obtained from Eq. 3, solid lines were calculated using Eq. 5. () Ge1Sb2Te4, () Ge2Sb2Te5, () Ge3Sb2Te6, () GeTe.

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Table 3 Fit parameters of EU (T) and Eg(T) in (GeTe)x(Sb2Te3)1−x alloys.
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The contribution of thermally-induced disorder EU(dyn.) is controlled by thermal lattice vibrations, and, hence it increases upon increasing temperature. Ternary (GeTe)x(Sb2Te3)1−x alloys in the amorphous state possess equal average phonon energies, and they do not exhibit any significant compositional trends in the degree of phonon anharmonicity as shown by results of temperature-dependent IR and Raman spectroscopy10,11, and therefore we can conclude that the contribution of the thermally-induced disorder EU(dyn.) to EU(T) should be alsmost equal for all studied amorphous (GeTe)x(Sb2Te3)1−x alloys without any compositional dependences.

The contribution of structural disorder EU(st.) arises from the lack of the long range order in amorphous (GeTe)x(Sb2Te3)1−x alloys. Therefore, the variation of the disorder in studied samples should reflect in evolution of EU(st.). The fit of EU(T) with Eq. 5 yields the clear compositional increment in the fit parameter X, as shown in the Table 3. While moving along the pseudo binary line towards GeTe, the concentration of the vacancies reduces, latter leads to the broader distributions of bond lengths and bond angles48 in corresponding amorphous (GeTe)x(Sb2Te3)1−x alloys.

Conclusions

In summary, we observed systematic trends in the optical properties of amorphous (GeTe)x(Sb2Te3)1−x alloys in the IR. The dielectric constant εinf obeys the Clausius-Mossotti relation and can be determined by the polarizabilities of the material’s constituent atoms. The evolution of the dielectric function around the fundamental absorption edge reveals the systematic changes in the optical bandgap width Eg and in the width of the localized electronic states EU. The compositional increment in EU can be explained by increasing of the disorder in GeTe-rich amorphous (GeTe)x(Sb2Te3)1−x alloys. This conclusion is supported by corresponding trends in the Tauc parameter B1/2 and in the average halfwidth <FWHM> of Raman bands. The low-temperature data enabled us to analyze the contributions of the structural and thermal-induced disorder into the temperature dependence of EU and Eg. While no compositional trends in the contribution of thermal-induced disorder can be traced, there is a systematic increase of the structural disorder in GeTe-rich amorphous (GeTe)x(Sb2Te3)1−x alloys. This might be due to the fact that reducing the concentration of vacancies in GeTe-rich GSTs leads to the broader distributions of bond lengths and bond angles that worsen long-range ordering in studied amorphous (GeTe)x(Sb2Te3)1−x alloys. Obtained results may enable one to optimize the tailored optical properties of amorphous (GeTe)x(Sb2Te3)1−x alloys. The latter is quite important for their technological applications which require the long-term stability and the absence of the drift effects.

Methods

Samples preparation

In this study we have chosen several phase-change alloys along the (GeTe)x(Sb2Te3)1−x compositional line: Ge1Sb2Te4 (x = 0.5), Ge2Sb2Te5 (x = 0.66), Ge3Sb2Te6 (x = 0.75), and GeTe (x = 1). To prepare the samples for optical measurements, the 150 nm Al layer was deposited onto a glass substrate. The phase-change films (1,000 nm) were d.c. sputtering-deposited with LS 320 von Ardenne system (background pressure 4 × 10−7 mbar, 20 s.c.c.m. Ar flow, deposition rates 0.1–0.2 nms−1, operating in the constant power mode by using 20–25 W) with stoichiometric targets of 99.99% purity. In each sputter session 4 samples of a certain compound have been prepared: 3 samples were used for the optical measurements to provide the repeatability of the data, one reference sample w used to determine the thickness of the phase-change layer by using a profilometer.

Optical measurements

Reflectance spectra were measured within the energy range from 0.05 up to 1 eV with a resolution of 2 meV, using Bruker IFS 66 v/s spectrometer equipped with the continuous flow Cryovac cryostat, in the sample temperature range from 5 up to 400 K. As a reference sample, a glass substrate coated with the 150 nm Au layer was used, it acted as an almost ideal mirror in the studied energy range. Reflectance spectra of the reference sample and the studied one were measured subsequently to exclude drift effects. For normalization, the final spectrum was obtained by dividing the measured spectrum by the reference one. The angle of incidence of the incoming beam was kept constant at 10° with respect to the surface normal. To extend the studied spectral range we measured ellipsometry spectra in the range from 0.7 to 2 eV using a Woolam M-2000 UI with angles of incidence of 60 and 69 at room temperature. The ellipsometer was equipped with a InGaAs diode array, as well as silicon charge-coupled device camera that served as detectors in the NIR and the VIS/UV ranges correspondingly. The resolution was of about 7 meV. The light sources were deuterium and halogen lamps.

The relative measurement error for the optical measurements was 0.2% in the used wavelength range.

Modeling the spectra

Optical spectra were analyzed using the SCOUT and CoRa software. The model of the dielectric function ε(ν) for the studied samples was composed of the εinf, a real constant that describes the polarizability of bonded electrons, which plays a major role in the energy range above the phonons. The inter-band electron transitions were described by the OJL model49 which includes tail states exponentially decaying into the band gap. The dielectric constant εinf was determined at 0.1 eV.