1 Introduction

Layered and chain structured compounds have been popular research topics in recent years. Layered crystals can be made two-dimensional (2D) due to their weak interlayer bonds. Chain structured compounds may be brought into one-dimensional (1D) since they are form from one-dimensional chains. The outstanding properties of 2D and 1D materials make them effective especially in optoelectronic applications. The ternary compound of TlInS2 has layered structure, while TlInSe2 has chain structure. In a recent paper, TlInS2 crystal grown by Bridgman method was brought into quasi-two dimensional form by cleaving crystals along the layers [1]. The result of the study showed that quasi-2D TlInS2 is a potential material for sensor applications. The electronic properties of quasi-one dimensional TlInSe2 were investigated by density functional theory [2]. The study pointed out that quasi-1D TlInSe2 compound is favorable for thermoelectric applications. Studies on thin film, bulk crystal and/or nano structures of the compounds have shown that TlInS2 and TlInSe2 may be used in spintronic, photoconductivity, photovoltaics, sensor, and thermoelectric applications [3,4,5,6,7].

As a result of the replacement of S atoms in the TlInS2 compound with Se atoms, the quaternary compounds TlIn(SxSe1-x)2 are formed. Tuning of the optical properties of the compounds formed by the S-Se exchange provides certain advantages in the design of optoelectronic devices. For example, while the bandgap of TlInS2 and TlInSe2 are 2.40 and 1.1 eV, respectively, the bandgap energies of TlIn(SxSe1-x)2 compounds vary between these two values depending on the composition [8]. The electronic and optical characteristics of TlIn(S0.75Se0.25)2 compound was reported taking into consideration the results of density functional theory calculations [9]. The morphological, structural and Raman scattering properties of TlInSSe compound were investigated in Ref. [10]. In another study, the dilatometric and optical anisotropy parameters of compounds corresponding to compositions of x = 0.75, 0.85, 0.90, 0.94, 0.98 and 1.0 were investigated [11].

TlIn(S0.25Se0.75)2 is one of the members of TlIn(SxSe1-x)2 compounds. Our research group reported the structural and optical properties of the TlIn(S0.25Se0.75)2. The indirect and direct bandgap energies were revealed as 1.96 and 2.16 eV from the analyses of transmittance and reflectance spectra [12]. Photoluminescence spectra of the TlIn(S0.25Se0.75)2 crystal was presented in low temperature region of 25–63 K [13]. The broad emission band, centered at 633 nm (1.96 eV) at T = 25 K was associated with donor–acceptor recombination. Determination of the optical properties of semiconductor materials is of great importance for technological device applications. How any semiconductor interacts with a light directly affects the working principles of the respective devices. In the design and performance of photovoltaic cells, LEDs, lasers, and optical sensors, the transmission, reflection and emission characteristics of semiconductor materials used in these devices play a role. Similarly, the investigation of the nonlinear properties of semiconductor materials is important for applications such as optical signal processing, optical switching, and frequency conversion. Taking into account these points and potential of TlIn(SxSe1-x)2 compounds in device applications, we aim in this article to obtain detailed linear and nonlinear properties of TlIn(S0.25Se0.75)2 by using spectroscopic ellipsometry measurement technique. As a result of the analysis of the ellipsometry data, the spectra of refractive index, extinction coefficient, absorption coefficient, optical conductivity were plotted. Various linear and nonlinear optical constants were determined using appropriate optical models.

2 Experimental details

TlIn(S0.25Se0.75)2 single crystal was grown by Bridgman method. Melt materials were placed inside suitable silica cylinders with diameter of 1.0 cm and length of 25 cm. The pressure was kept around 10–5 Torr and ampule was moved with a speed of 1.0 mm/h through the thermal gradient of ~ 30 °C/cm. X-ray diffraction measurement was applied by Rigaku Miniflex diffractometer (Cu-Kα, λ = 1.54 Å). The ellipsometry experiments were done using SOPRA GES-5E rotating polarizer ellipsometer. The measured spectral range was 1.25–4.50 eV. The angle of the incidence was applied as 70°. In ellipsometer measurements, it is necessary to ensure that the surface is as smooth or less rough as possible. The surfaces of the samples obtained after the growth process were not smooth enough to obtain reliable results in the ellipsometer measurements. In addition, undesired impurities can be found on the surfaces of crystals that interact with the medium for a long time. In order to get rid of such problems that would adversely affect the reflection, bulk crystals grown with the Bridgman method were cut in the middle with the help of a razor. By this way, samples with a smooth surface were obtained before the measurement.

3 Results and discussions

XRD pattern of the powder form of grown TlIn(S0.25Se0.75)2 ccompound is represented in the Fig. 1. Nine sharp and well-defined peaks around 22.70, 23.55, 25.20, 30.65, 45.85, 48.10, 50.70, 53.50 and 56.05° were observed in the XRD pattern recorded in the 20–60° range. The sharp diffraction peaks point out that the grown TlIn(S0.25Se0.75)2 compound has good crystalline structure. The crystalline structure of the compound was reported as monoclinic with lattice constants of a = 5.6367, b = 7.9060, c = 7.0674 Å, and β = 92.56° in Ref. [13]. Miller indices corresponding the observed peaks were presented above the peaks.

Fig. 1
figure 1

XRD pattern of TlIn(S0.25Se0.75)2 powder

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Detailed optical investigation of TlIn(S0.25Se0.75)2 crystal was performed using ellipsometry measurements. The spectroscopic ellipsometry presents two experimental data of amplitude ratio (ѱ) and phase shift (∆) of parallel (p) and perpendicular (s) components of the reflected light. The spectral dependences of ѱ and ∆ with energy in the 1.25–4.50 eV range were indicated in Fig. 2. These experimental parameters are utilized under the light of suitable optical models to get optical parameters of the compounds. Air/sample optical model is the well-known and most suitable model for bulk materials of sufficiently thick. In this model, complex dielectric function (ε) and complex reflectance ratio of polarized beam (ρ) which is given in terms of Ψ and Δ as \(\rho =\mathrm{tan}\Psi \mathrm{exp}(i\Delta )\) are related by following expression [14]

$$\varepsilon ={\varepsilon }_{1}+i{\varepsilon }_{2}={\mathrm{sin}}^{2}\left(\varphi \right)\left[1+{\left(\frac{1-\rho }{1+\rho }\right)}^{2}{\mathrm{tan}}^{2}\left(\varphi \right)\right],$$
(1)

where φ = 70° is the angle of incidence and, \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\) are real and imaginary components of dielectric constant, respectively. The refractive index (n) and extinction coefficient (k) are expressed in terms of \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\) as the following [14]

Fig. 2
figure 2

Spectra of ellipsometric data of ψ and Δ

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$$n={\left[\left({\varepsilon }_{1}+{\left({\varepsilon }_{1}^{2}+{\varepsilon }_{2}^{2}\right)}^{1/2}\right)/2\right]}^{1/2},$$
(2)
$$k={\left[\left(-{\varepsilon }_{1}+{\left({\varepsilon }_{1}^{2}+{\varepsilon }_{2}^{2}\right)}^{1/2}\right)/2\right]}^{1/2},$$
(3)

As a result of using the appropriate ambient-substrate optical model of the experimental parameters obtained as a result of the measurements, the energy-dependent graphs of the refractive index and extinction coefficient were obtained as seen in Fig. 3. The refractive index exhibits normal dispersion behavior in the energy region below 2.57 eV and varies between 2.51 and 2.95 in this region. A sharp change was observed around 1.97 eV in the extinction coefficient spectrum. Similar changes have been seen before in the literature for different materials [15,16,17]. The reason for this sharp change is explained by the fact that the relevant energy value coincides with the bandgap energy value. The energy of 1.97 eV detected in the extinction coefficient spectrum agrees with the previously reported bandgap energy. In the following parts of the manuscript, the bandgap energy will also be obtained as a result of the analysis of the absorption coefficient.

Fig. 3
figure 3

Spectral dependences of the refractive index and extinction coefficient for TlIn(S0.25Se0.75)2 crystals

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There are optical models that examine the energy dependence of the refractive index in the below bandgap energy region. One of these models is given by Wemple and DiDomenico as follows [18]

$${n}^{2}\left(hv\right)=1+\frac{{E}_{so}{E}_{d}}{{E}_{so}^{2}-{(hv)}^{2}}$$
(4)

where Eso and Ed are single oscillator and dispersion energies, respectively. When Eq. (4) is arranged, Eso and Ed energies can be obtained from the (n2 − 1)−1 vs. (hv)2 graph. As a result of linear fit of the related graph shown in Fig. 4, Eso and Ed energies were found as 3.82 and 18.29 eV, respectively. The relationship between the Eso value and bandgap (Eg) energy is given as Eso ≈ 2.0 Eg in the literature [19]. The ratio of the Eso = 3.82 eV to the Eg = 1.96 eV derived from the extinction coefficient graph was calculated as 1.95 which is well-consistent with proposed ratio of 2.0. As a result of substituting Eso, Ed energies and hv = 0 in Eq. (4), the zero-frequency dielectric constant and refractive index were evaluated as \({\varepsilon }_{0}={n}_{0}^{2}=5.79\) and \({n}_{0}=2.41\), respectively.

Fig. 4
figure 4

The variation of (n2 − 1)−1 with E2. Inset: The variation of n2 − k2 with λ2. The lines correspond to the linear fits

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The real part of dielectric constant expressed as ε1 = n2 – k2 is given in the high wavelengths (n2» k.2) as [20]

$${\varepsilon }_{1}={\varepsilon }_{\infty }-\frac{{\omega }_{p}^{2}}{{\omega }^{2}}= {\varepsilon }_{\infty }-\frac{{\omega }_{p }^{2 }{\lambda }^{2}}{4{\pi }^{2} {c}^{2}} = {\varepsilon }_{\infty }-a{\lambda }^{2}$$
(5)

where ε∞: high-frequency dielectric constant, c: speed of light, ωp: plasma frequency defined as \({\omega }_{p}^{2}= \frac{{e}^{2} N}{{\varepsilon }_{0} {m}_{e}^{*}}\) (e: elementary charge, N: carrier density, \({m}_{e}^{*}\): effective mass). Considering Eq. (5), when n2 – k2 vs. λ2 graph was drawn, the optical parameters given in the equation can be obtained as a result of the linear fit. Inset of Fig. 4 represents the aforementioned plot and applied linear fit. The fitting parameters presented the optical constants given in the Eq. (5) as ε∞ = 8.3, ωp = 31.1 × 1014 rad s−1 and N/\({m}_{e}^{*}\) = 33.4 × 1056 m−3 kg−1. Using the reported effective mass of \({m}_{e}^{*}=0.14{m}_{0}\) [21], the carrier density was calculated as N = 4.26 × 1026 m−3. Moreover, the high-frequency dielectric constant and high frequency refractive index were found as \({\varepsilon }_{\infty }={n}_{\infty }^{2}=\) 8.24 and \({n}_{\infty }=\) 2.87, respectively.

The absorption coefficient (α) of the crystal can be obtained by using the extinction coefficient with the help of the relation α = 4πk/λ. Using the k-spectrum given in the Fig. 2 and the related equation, the absorption coefficient spectrum was presented in Fig. 5. Absorption coefficient spectrum may be utilized to get Urbach energy (Eu) which is associated with disorders in the crystalline structure. The relation between α and Eu is given as [22]

Fig. 5
figure 5

Absorption coefficient spectrum for TlIn(S0.25Se0.75)2 crystals. Inset shows the ln(α) vs. energy plot and linear fit for Urbach energy analysis

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$$\mathrm{ln}\alpha =\mathrm{ln}{\alpha }_{0}+E/{E}_{u}.$$
(6)

Inset of Fig. 5 indicates the ln (α) vs. E graph which provides determination of the Urbach energy from the linear fit of the plot. Urbach energy was calculated as 0.68 eV from the slope equal to 1/Eu value. Urbach energies of similar crystals have been found in the literature. For example, the Urbach energy of the TlIn(S0.75Se0.25)2 thin film was found to be 0.58 eV [23], while the energy of the TlInS2 single crystal was obtained as 0.11 eV [24]. It is clearly seen that Urbach energy of the TlInS2 single crystal is considerably higher than Urbach energy of TlIn(S0.25Se0.75)2 crystal. Urbach energy is affected by various factors like crystal defect, impurity, strain, and surface roughness. However, since single crystal structures have well-ordered electronic structures, the Urbach energies of single crystals are primarily determined by defects. The fact that TlInS2 crystal has lower Urbach energy than TlIn(S0.25Se0.75)2 crystal may be attributed to the fact that TlInS2 crystal has defects and/or impurities in a lower concentration and more well-ordered crystalline structure than TlIn(S0.25Se0.75)2.

The bandgap energy of the TlIn(S0.25Se0.75)2 crystal can be determined from the spectrum of the energy-dependent derivative of the absorption coefficient. The spectrum of the dα/dE indicates a valley around the absorption edge and the minimum of the plot corresponds to the bandgap energy. Figure 6 shows the corresponding graph, and a spectral pit is observed in the high absorption region, reaching its minimum at 1.96 eV. This value represents the bandgap energy of the studied compound and agrees with the value extracted from the above-given n-k spectra.

Fig. 6
figure 6

Spectrum of dα/dE of TlIn(S0.25Se0.75)2 crystal

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The optical \(({\sigma }_{opt})\) conductivity of the TlIn(S0.25Se0.75)2 compound was obtained using the expression σopt = αnc/4π, where c is the speed of light [25]. The optical conductivity spectrum drawn using the spectra of the absorption coefficient and refractive index was shown in Fig. 7. The bandgap can also be obtained with the help of the region where the optical conductivity is linearly dependent on the energy [26]. As shown in the figure, the linear fit line applied to the region showing linear behavior intersects the energy axis at the bandgap energy of 1.97 eV. This energy value is compatible with the bandgap obtained from the energy derivative of the absorption coefficient. The optical conductivity spectrum shows the presence of two peak maxima at 3.00 and 4.16 eV. These two values correspond to the energy values of the transitions taking place in the band structure of the compound [27].

Fig. 7
figure 7

Spectrum of optical conductivity of TlIn(S0.25Se0.75)2 crystal

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Determination of nonlinear optical parameters of semiconductors is important for researchers working especially on optoelectronic device design. The nonlinear refractive index (n2), first-order (χ(1)) and third-order nonlinear susceptibilities (χ.(3)) of the TlIn(S0.25Se0.75)2 crystal, which has chain structure and has potential in the field of optoelectronic devices, were calculated using the following equations [27, 28]

$$\chi^{\left( 1 \right)} = \frac{{n_{0}^{2} - 1}}{4\pi },\quad \chi^{\left( 3 \right)} = \frac{{1.7 \times 10^{ - 10} }}{{\left( {4\pi } \right)^{4} }}\left[ {n_{0}^{2} - 1} \right]^{4} \quad {\text{and}}\quad n_{2} = \frac{{12\pi \chi^{\left( 3 \right)} }}{{n_{0} }}$$
(7)

where n0 as 2.41. The calculations resulted in optical values of χ(1) = 0.38 esu, χ(3) = 3.7 × 10−12 esu and n2 = 5.7 × 10−11 esu. When the nonlinear property study studies on similar materials were examined, a study on TlInS2 thin film was found in the literature. The third order nonlinear susceptibility, χ(3), and nonlinear refractive index, n2, of the nanostructured thin films annealed at different temperatures were found in the ranges of 0.8 − 6.0 (× 10−11) esu and 1.2 − 7.0 (× 10−10) esu, respectively [29]. When the nonlinear values of TlIn(S0.25Se0.75)2 crystal are compared with these reported values belonging to TlInS2 nanostructured thin film, it is seen that the values belonging to TlInS2 are higher. The high values of χ(3) and n2 may be associated with high refractive index value of TlInS2.

4 Conclusion

Linear and nonlinear optical parameters of the TlIn(S0.25Se0.75)2 crystal were investigated using the ellipsometry method. The spectra of the refractive index and extinction coefficient were plotted in the range of 1.25–4.50 eV. As a result of the analysis of the refractive index in the region under the band energy, Eso and Ed energies were found as 3.82 and 18.29 eV, respectively. The analysis of the real component of the dielectric function showed the high-frequency dielectric constant, plasma frequency and carrier density as ε∞ = 8.3, ωp = 31.1 × 1014 rad s−1 and N = 4.26 × 1026 m−3. The spectra of extinction coefficient, first energy derivative of absorption coefficient and optical conductivity pointed out bandgap energy of the TlIn(S0.25Se0.75)2 crystal around 1.96 eV. Urbach energy was also determined for the crystal as 0.68 eV. The nonlinear refractive index, first-order and third-order nonlinear susceptibilities were determined as χ(1) = 0.38 esu, χ(3) = 3.7 × 10−12 esu and n2 = 5.7 × 10−11 esu, respectively.