Temperature-dependent indirect gaps for two-dimensional bismuth oxychalcogenides probed by spectroscopic ellipsometry

In-plane optical properties of two-dimensional bismuth oxychalcogenides Bi2O2X (X = S, Se, and Te) are reported for a wide spectral range of 0.73–6.42 eV and at temperatures of 4.5–500 K by spectroscopic ellipsometry. At room temperature, Bi2O2S, Bi2O2Se, and Bi2O2Te exhibit an indirect band gap of 1.18 ± 0.02, 0.95 ± 0.01, and 0.60 ± 0.01 eV, respectively. As the temperature decreases, the indirect absorption edge of Bi2O2S undergoes a blueshift, while the indirect band gap of Bi2O2Se shows a redshift, and Bi2O2Te remains independent of temperature. The chalcogenide-dependent behavior as a function of temperature may be relevant to electron–phonon interactions in Bi2O2X materials. The observed pseudo-isotropic complex dielectric function and optical absorption coefficient by spectroscopic ellipsometry are directly compared with the first-principles calculations with a hybrid functional approach.

At room temperature, Bi 2 O 2 X has an indirect band gap ranging from 0.6 to 1.2 eV, whose value decreases with increasing atomic number of X ions, Z X [1,2,11,[15][16][17].When the band gap decreases from X = S to Se, the effective masses of both an electron and a hole decrease [19], too, resulting in higher electron mobility for Bi 2 O 2 Se (e.g.450 and 3 × 10 5 cm 2 V −1 s −1 at 300 and 1.9 K, respectively), with both a sufficiently high current on/off ratio of 10 6 , and excellent environmental stability [5,7].Furthermore, since Bi 2 O 2 Te exhibits a low thermal conductivity of 0.91 W m −1 K −1 at room temperature, a thermo-electric figure-of-merit value gives a relatively high value of ZT = 1.27 at 600 K [17].These unique properties make Bi 2 O 2 X materials attractive for use in thermoelectric devices, photocatalysts, photodetectors, photovoltaic devices, and field-effect transistors.Since the ionic interactions of the X 2− ions modify the electronic and phonon properties as a function of Z X and temperature, measuring the optical properties for a wide region of photon energy is essential for understanding the physical properties of the Bi 2 O 2 X.
Spectroscopic ellipsometry is a suitable method for such the purpose of obtaining the optical properties of Bi 2 O 2 X (X = S, Se, and Te) thin films at temperatures of 4.5-500 K with a spectral range of 0.73-6.42eV.Spectroscopic ellipsometry is a standard optical technique used to extract the complex dielectric function of 2D materials for a wide range of photon energies.In spectroscopic ellipsometry, two ellipsometric parameters, denoted by Ψ and ∆, are measured [21], in which Ψ and ∆ denote, respectively, the amplitude ratio and phase difference of the reflected lights for incident p-and spolarized lights to surface of material [22].The ellipsometric parameters directly provide both the real and imaginary parts of the dielectric function and the optical absorption spectra without Kramers-Kronig transformation, which is frequently used for 2D materials such as graphene and transition-metal dichalcogenides [23][24][25].Temperature-dependent optical properties may serve as a reference for evaluating the self-heating performance of devices since high temperatures considerably affect device efficiency, reliability, and overall functionality of semiconductor devices [26,27].It is pointed out, however, that the previous experiments on Bi 2 O 2 X materials have predominantly been conducted at room temperature and within a limited frequency range [1-3, 11, 15-17].
Here, we first confirm that 2D Bi 2 O 2 X materials have indirect band gaps ranging from 0.6 (X = Te) to 1.2 eV (X = S) at room temperature.The observed band gaps are directly compared using the firstprinciples calculations, in which a hybrid functional is used to obtain correct band gap values.When we measure the indirect absorption edge with increasing temperature from 4.5 to 500 K, the observed absorption edge shows a redshift (Bi 2 O 2 S), blueshift (Bi 2 O 2 Se), and no change (Bi 2 O 2 Te), which is puzzling.The X-dependent indirect absorption edge is discussed by the electron-phonon interactions.
This paper is organized as follows.In section 2, we describe both the experimental and theoretical methods.In section 3, we describe the results from spectroscopic ellipsometry.The pseudoisotropic complex dielectric function is compared with the first-principles calculations.We also discuss temperature-dependent indirect absorption edges through electron-phonon interactions.In section 4, we summarize our findings.
To confirm the in-plane epitaxial relationship between the thin film and the substrate, we perform the azimuthal ϕ scan of the x-ray intensity for the thin films.To obtain structural details and assess the crystallinity of the thin films, the thicker films (200 nm) are fabricated, ensuring a sufficiently strong signal for the XRD and the following transmission electron microscopy (TEM) analysis.In the azimuthal ϕ scan, the Bi Beyond characterizing the in-plane relationship, the quality of the Bi 2 O 2 S and Bi 2 O 2 Se thin films is determined by the ω scans (figure 1(e)).In the ω scans, the detector is positioned at a specific Bragg angle while the sample is incrementally tilted by an angle ω.For a high-quality thin film, a very sharp peak is observed as a function of ω, only when the crystallographic direction aligns parallel to the diffraction vectors.The full width at half maximum of the peaks for Bi 2 O 2 S (004) and Bi 2 O 2 Se (004) is found to be both 0.08 • as shown in figure 1(e), whose value is comparable with that of single-crystal, indicating the reasonable quality of the thin films.Based on the observation from XRD and TEM analyses, we can confirm that the high-quality Bi 2 O 2 X films with an area of 5 mm 2 and thickness of about 100 nm have been grown on the single-crystalline substrates, in which the thickness is observed by spectroscopic ellipsometry.
Spectroscopic ellipsometry is measured first at room temperature within a spectral range of 92.0 ± 0.1 135.0 ± 0.1 -Substrate (mm) LAST (0.5) SrTiO3(0.5)NdScO3(0.5)0.73-6.42eV for obtaining the thickness of the samples.The samples used for optical measurements and those for TEM analysis are from different batches and have varying thicknesses.Measurements are conducted using an M-2000U ellipsometer (J. A. Woollam, Lincoln, NE, USA) at two incident angles of 60 • and 70 • .For the analysis, the raw ellipsometric parameters of Ψ and ∆ for three different substrates are measured and their complex optical constants are derived, as depicted in supplementary figure S2.
The obtained complex optical constants are used in the stacked-layer model for spectroscopic ellipsometry analysis.A stacked-layer model, which consists of a substrate, a thin film, a rough surface, and an air ambient structure, is used to fit the raw ellipsometric parameters of Ψ and ∆ [23].S4.This similarity indicates an isotropic in-plane optical response, aligning with the XRD analysis that confirms their tetragonal structures.However, under the experimental conditions, the thickness of the thin film is 100 nm, much smaller than both the 5 mm sample size.Furthermore, in our experimental configuration, the reflection mode exhibits greater sensitivity to the dielectric function's in-plane component compared to the out-of-plane component.To accurately measure the out-of-plane component of the dielectric function, ε zz , adopting smaller incident angles of light, ranging from 25 • , 30 • , to 35 • , is required, which, however, leads to weaker reflected light signals.Given these constraints, accurately determining the out-of-plane component of the dielectric function, ε zz , through the current spectroscopic ellipsometry setup is challenging.Therefore, we are focusing on the in-plane optical properties of these 2D bismuth oxychalcogenides and the calculations of the anisotropic dielectric function in this study.Temperature-dependent spectroscopic ellipsometry is measured at an incident angle of 70 • over a temperature range of 4.5-500 K.The sample is mounted inside a Janis ST-400 cryostat designed for continuous-flow liquid helium.The ultrahigh vacuum is achieved with a base pressure of 10 −8 torr [24,25].

Theoretical model
We adopt Quantum ESPRESSO to obtain the electronic band structure from the first-principles based on density functional theory (DFT) [28].Optimized norm-conserving Vanderbilt pseudopotentials [29] with a Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional [30] are used to examine the atoms of Bi, O, and X.A cutoff energy of 60 Ry for a plane wave and a k-point mesh of 9 × 9 × 3 for Brillouin zone integration are used to calculate the band structure.Because the band gap value is underestimated by the PBE functional [31], we use a hybrid functional based on the Heyd-Scuseria-Ernzerhof (HSE) approach [32] to correct it.A 3 × 3 × 1 qmesh with a Fock operator is used to calculate the HSE.To interpolate the band structure from the HSE calculation, the Wannier interpolation is performed by using Wannier90 [33].As shown in figure 1(a), the crystal structures of the 2D Bi 2 O 2 X materials are optimized using a Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm [31], with a force of less than 10 −6 Ry Bohr −1 and a pressure of less than 0.05 kbar.The results indicate that the optimized lattice constants of Bi 2 O 2 S, Bi 2 O 2 Se, and Bi 2 O 2 Te are (a, c) = (3.841Å, 11.961 Å), (3.878 Å, 12.140 Å), and (3.959 Å, 12.457 Å), respectively, which are consistent with previous DFT [17] and experimental values [4,12].To calculate the complex dielectric functions and optical absorption coefficient of Bi 2 O 2 X materials, the epsilon.xcode in Quantum ESPRESSO is used with independent particle approximation [28,31], in which a large k-points mesh of 30 × 30 × 10 is used to calculate the non-self-consistent field in order to achieve convergence in the complex dielectric function.

Results and discussion
In figure 2(a), we plot the observed in-plane real (ε 1 ) and imaginary (ε 2 ) parts of the pseudo-isotropic complex dielectric function of Bi 2 O 2 X (X = S, Se, and Te) at room temperature as a function of photon energy, as obtained through spectroscopic ellipsometric analysis.The ε 1 of Bi 2 O 2 S has a positive value up to 4.8 eV, followed by a negative value from 4.8 to 6.0 eV.In figure 2(b), the observed in-plane optical absorption coefficient α is plotted as a function of photon energy.Since large α is observed in the photon energy region near 5 eV with ε 1 = 0 and ε 2 > 1, the optical absorption is not due to intersubband plasmon, because the resonance condition of the inter-subband plasmon is given by ε 1 = 0 and ε 2 ≪ 1 [34].As shown in figure 2(a), the photon energy of ε 1 = 0 systematically redshifts from Bi 2 O 2 S to Bi 2 O 2 Te.This is consistent with the fact that the absorption peaks of Bi 2 O 2 Se and Bi 2 O 2 Te redshift to 2.7 and 1.8 eV from 3.3 eV for Bi 2 O 2 S as shown in figure 2(b).The ε 2 spectrum of Bi 2 O 2 S has a peak at 3 eV, accompanied by several relatively small peaks at higher photon energies.
The optical absorption coefficient α is expressed by the complex dielectric function as follows where c is the speed of light in vacuum and ω is the angular frequency of light.At ε 2 > 1 and ε 1 ∼ 0, equation ( 1) transforms into Hence, the peak of ε 2 corresponds to that of α for ε 2 > 1 and ε 1 ∼ 0. When photon energy increases up to 2.0 eV, the α value of Bi 2 O 2 S gradually increases, followed by a sharp increase beyond 2.0 eV, and finally reaches its first peak at 3.4 eV.The gradually and sharply increasing regions correspond to, respectively, the absorption around the indirect and direct energy gap, which we discuss below.Additional absorption peaks are observed at 4.1, 4.5, and 5.2 eV, which correspond to joint-density-of-state peaks.It can be pointed out that this relatively high optical absorption coefficient above 2.0 eV (in the order of 10 6 cm −1 ) is advantageous for optical devices, such as photodetectors and photocatalysts.By contrast, when the photon energy exceeds 3.0 eV, the α value of both Bi 2 O 2 Se and Bi 2 O 2 Te decrease relative to Bi 2 O 2 S, and their absorption edges shift to lower photon energies of 1.2 and 0.6 eV compared with those of Bi 2 O 2 S.
In a typical three-dimensional (3D) solid, the optical absorption coefficient depends on the contributions of both direct and indirect band gap transitions.In order to obtain the values of direct and indirect energy gap, E g,dir , and E g,ind , respectively, from α (E), we adopt the following formulas a function of photon energy E [35]: where A and B are constants, and E ph denotes absorbed (emitted) phonon energy.As shown in supplementary figures S5 and S6, E g,dir and E g,ind can be obtained as the intersection with the x-axis when (αE) 2 and (αE) 0.5 are plotted as a function of E and the plots are extrapolated to the E-axis [36,37].In this study, we use equation ( 3) to obtain an approximation of (αE) 2 ∼ A 2 (E-E g,dir ) and (αE) 0.5 ∼ B 0.5 (E-E g,ind ∓ E ph ).The analysis of the indirect band gap reveals that phonon emission occurs at both high and low temperatures, whereas phonon absorption only happens at high temperatures and should be absent at low temperatures.For simplicity, we estimate the extrapolated value as an indirect band gap energy.
In table 2, we list the values of E g,dir and E g,ind for Bi 2 O 2 X (X = S, Se, and Te) at room temperature, which is compared with the theoretical results of the first-principles calculations, indicating that all Bi 2 O 2 X materials are indirect band gap semiconductors.These materials hold promise for photovoltaic applications, facilitating photon capture in the spectral region above indirect and direct band gaps.Overall, our indirect band gap results for both Bi 2 O 2 S and Bi 2 O 2 Se are consistent with the values reported for the powder, nanocrystal, and single crystal forms of Bi 2 O 2 S [1-3], and with thin films of Bi 2 O 2 Se [11].However, the magnitude of the indirect band gap of Bi 2 O 2 Te (0.60 eV) is larger than that observed in its powder form (0.23 eV) [12].Now let us discuss the temperature dependence of the indirect energy gap. Figure 3 presents the temperature-dependent in-plane real (ε 1 ) and imaginary (ε 2 ) parts of the pseudo-isotropic complex dielectric function of Bi 2 O 2 X (X = S, Se, and Te).As the temperature decreases, the ε 1 value of Bi 2 O 2 S also decreases for photon energies below 2.5 eV, whereas the ε 1 value increases between 2.5 and 3.0 eV.The relationship between ε 1 and temperature for Bi 2 O 2 Se and Bi 2 O 2 Te is opposite to that for Bi 2 O 2 S. For ε 2 , the peak intensity becomes increasingly pronounced for all samples with decreasing temperature, which corresponds to an increase in the peak value of the optical absorption coefficient α (figures 4-6).
In figure 4(a), we plot a (E) of Bi 2 O 2 S for several temperatures.In figure 4(b), a (E) of Bi 2 O 2 S at 4.5 K are fitted to several Lorentzian functions.The pseudo-isotropic complex dielectric function can be represented using the Lorentzian model as follows: where ω j , γ j , and ω pj are the frequency, damping, and oscillator strength of the jth Lorentzian  contribution.ε ∞ is the high-frequency limit of ε (ω), which includes the interband transitions at frequencies above the measured range.The supplementary table 1 lists the fitting parameters of the Lorentzian model.Based on previous first-principles calculations [20], we assign the absorption peaks observed at 2.6 and 3.0 eV to the charge-transfer excitation from the S 3p orbital to the Bi 6p orbital, and we assign the peaks In the following paragraphs, we discuss the characteristic changes observed in the in-plane absorption edge as a function of temperature.Figure 4(c) shows a plot of (αE) 0.5 for Bi 2 O 2 S as a function of photon energy E at several temperatures, near the indirect absorption edge.As discussed above, because spectroscopy ellipsometry is conducted using data obtained at only 0.73-6.42eV, the indirect absorption edge (E g,ind ) is provided as an extrapolated value on the horizontal axis.As the temperature decreases, the value of (αE) 0.5 above the indirect absorption edge monotonically decreases, and E g,ind undergoes a blueshift.This temperature dependence can be explained by phonon-assisted optical absorption, which is commonly observed in silicon and germanium [38][39][40].In figure 4(d), we plot E g,ind for Bi 2 O 2 S as a function of temperatures.Below 100 K, E g,ind remains almost constant, but it monotonically decreases with increasing temperature.The observed temperature dependence of E g,ind in indirect semiconductors can be described using the Bose-Einstein model as follows [41]: where E g (0) is the band gap energy at 0 K, a B is the strength of the electron-phonon interactions, and Θ B is the average phonon temperature.When  1 summarizes the fitted parameters.Figures 5(c

Summary
In this paper, we use spectroscopic ellipsometry to examine the in-plane optical properties of Bi 2 O 2 X (X = S, Se, and Te) materials.We analyze their optical transitions using the observed and calculated pseudo-isotropic complex dielectric function, the optical absorption coefficient obtained from spectroscopic ellipsometry, and the first-principles calculations.Our results indicate that the indirect band gaps of Bi 2 O 2 S, Bi 2 O 2 Se, and Bi 2 O 2 Te are 1.18 ± 0.02, 0.95 ± 0.01, and 0.60 ± 0.01 eV, respectively, at room temperature, which are reproduced by the hybrid functional calculations of 1.34, 0.95, and 0.40 eV, respectively.When the temperature decreases, the indirect band gap of Bi 2 O 2 X shows blueshifts (Bi 2 O 2 S), redshifts (Bi 2 O 2 Se), and unchanged (Bi 2 O 2 Te).This unconventional temperature dependence implies the presence of intricate electron-phonon interactions in Bi 2 O 2 X materials.

Figure 4 .
Figure 4. (a) Temperature-dependent in-plane optical absorption coefficient spectra of Bi2O2S.(b) Fitting results of the spectrum obtained at 4.5 K with the Lorentzian model.(c) Temperature-dependent indirect absorption edge of Bi2O2S.(d) Temperature-dependent indirect band gap energy of Bi2O2S.The solid line represents the fitting results obtained using the Bose-Einstein model.

Figure 5 .
Figure 5. (a) Temperature-dependent in-plane optical absorption coefficient spectra of Bi2O2Se.(b) Fitting results of the spectrum obtained at 4.5 K with the Lorentzian model.(c) Temperature-dependent indirect absorption edge of Bi2O2Se.(d) Temperature-dependent indirect band gap energy of Bi2O2Se.The solid line represents the fitting result obtained using the Bose-Einstein model.
this equation is fitted to the data, the parameters are E g (0) = 1.23 ± 0.01 eV, a B = 18 meV, and Θ B = 168 K.The a B of Bi 2 O 2 S is lower than the values reported for monolayer transition metal dichalcogenides (26 ∼ 37 meV) [23].In figures 5(a) and 6(a), respectively, we plot the temperature-dependent in-plane optical absorption coefficient of Bi 2 O 2 Se and Bi 2 O 2 Te.In figures 5(b) and 6(b), we show the fitted results of α (E) obtained at 4.5 K by the Lorentzian functions.Supplementary table ) and 6(c) illustrate the temperature-dependent indirect absorption edge of Bi 2 O 2 Se and Bi 2 O 2 Te, respectively.As the temperature decreases, the absorption intensity of Bi 2 O 2 Se increases, and the indirect absorption edge redshifts.

Figure 5 (
d) shows the temperature-dependent E g,ind value of Bi 2 O 2 Se, indicating that E g,ind redshifts with decreasing temperature.This behavior is opposite to that observed in Bi 2 O 2 S (figure 4(d)).The fitting results of the Bose-Einstein model give E g (0) = 0.88 ± 0.01 eV, a B = −59 meV, and Θ B = 396 K for Bi 2 S 2 Se, indicating an unconventional negative value of a B for Bi 2 S 2 Se.

Figure 6 .
Figure 6.(a) Temperature-dependent in-plane optical absorption coefficient spectra of Bi2O2Te.(b) Fitting results of the spectrum obtained at 4.5 K with the Lorentzian model.(c) Temperature-dependent indirect absorption edge of Bi2O2Te.(d) Temperature-dependent indirect band gap energy of Bi2O2Te.

Figure 7 .
Figure 7. (a) Electronic band structures of (a) Bi2O2S, (b) Bi2O2Se, and (c) Bi2O2Te, calculated using the HSE (solid lines) and PBE (dashed lines) functionals.The Fermi energy levels (dotted lines at zero energy) were set at the highest occupied states.(d), (e) The calculated real (ε1) and imaginary (ε2) parts of in-plane εxx = εyy(solid lines) and out-of-plane εzz (dashed lines) complex dielectric function and (f) in-plane (solid lines) and out-of-plane (dashed line) optical absorption coefficient α of Bi2O2X are plotted as a function of photon energy.
2 O 2 S and Bi 2 O 2 Se samples are set and inclined to 83.93 • and 72.32 • with respect to the zaxis, to fulfill the diffraction conditions of Bi 2 O 2 S (013) and Bi 2 O 2 Se (101), respectively.After inclination, the sample rotates 360 • around the z-axis.

Table 1 .
Fitting parameters of the stacked-layer model for the ellipsometric spectra of Bi2O2X materials.

Table 1
• , 45 • , and 90 • .These results for Ψ and ∆ in Bi 2 O 2 S, Bi 2 O 2 Se, and Bi 2 O 2 Te thin films are almost identical, as depicted in supplementary figure lists the fitting parameters of the model.The obtained thicknesses of Bi 2 O 2 S and Bi 2 O 2 Se are 92.0 ± 0.1 and 135.0 ± 0.1 nm, respectively.However, because of the opaque properties of Bi 2 O 2 Te, we are unable to determine the thickness of Bi 2 O 2 Te.As shown in supplementary figureS3, the two ellipsometric spectra that are obtained by independent measurements for two incident angles fit well with the modeled curves.It is noted that all thin films, including Bi 2 O 2 S, exhibit tetragonal structures.As a result, there is one in-plane dielectric function, ε xx , associated with the tetragonal structure.We also measure the raw ellipsometric parameters of Ψ and ∆ at a 70 • incident angle for the same films by rotating their azimuthal orientation through angles of 0

Table 2 .
Direct and indirect band gap values of Bi2O2X materials at room temperature, compared with the theoretical results of the first-principles calculations.