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Abstract
SrTiO3 is the model perovskite compound and has demonstrated technological relevance from being the foundation of oxide electronics to a promising candidate as a high-temperature hydrogen sensor. Here, we fully investigate the temperature-dependent UV-Vis optical constants of pristine and doped SrTiO3 in the spectral range 0.73-5.90 eV using spectroscopic ellipsometry and first-principles. The oscillator model is applied to analyze and quantify the temperature and doping effect. After heating SrTiO3 from 300 to 873 K, the amplitude of the dominant absorption peak at 4.752 eV is reduced by 11% and the shoulder of optical spectrum at 3.708 eV gradually disappears. By doping SrTiO3 with Nb element from 0.05 to 0.7% wt, extra momentum is provided to enable indirect absorption and an unconventional Urbach tail is observed due to the increased crystalline disorder. Moreover, the first-principles calculations considering the many-body excitonic effect and phonon-assisted indirect absorption are performed, which help to determine the physics origin of absorption peaks and explain the temperature influence. This work sheds light on temperature and doping dependence of UV-Vis dielectric functions of cubic SrTiO3, aiming to elucidate the novel fundamental optical phenomena of SrTiO3-based heterostructures or devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As the paradigmatic perovskite material, strontium titanate (SrTiO3) has intrigued the scientific community for more than half a century [1–5]. SrTiO3 exhibits intriguing physical properties such as quantum phase transition [6], phonon hydrodynamics [7], and high mobility of charge carriers [8,9]. Consequently, SrTiO3 demonstrates great promise in applications spanning from being the ideal substrate for high-temperature superconducting films [10,11] to the preferred template for creating two-dimensional electron gas at oxide interfaces [12]. With such usefulness and prominence, SrTiO3 has become the foundation of the emerging field of oxide electronics [12].
SrTiO3 crystallizes in the cubic phase at normal conditions, transits to a tetragonal phase as temperature decreases below 105 K and experiences a phase transition to a quantum paraelectric state for temperature lower than 37 K [13,14]. The cubic SrTiO3 exhibits high-temperature stability and finds new application as the platform for hydrogen sensing at/over 600-800 ℃ [15]. A thorough understanding of temperature effect on optical response of SrTiO3 is thus critical to improve its performance as the high-temperature gas sensor. Kok and co-workers [16] applied the optical transmission method to measure the optical absorption edge and near infrared absorption of SrTiO3 from 4 K to 1703K. They observed the redshift of absorption edge and excited free carrier absorption in the infrared range with increasing temperature. The redshift was interpreted by the temperature-renormalized band gap from first-principles calculations which account for electron-phonon coupling using harmonic and anharmonic phonon modes [17]. Yet, the accuracy of the above optical transmission method suffers from uncertainties originated from the surface roughness, uneven thickness and self-radiation. On the other hand, spectroscopic ellipsometry (SE) is a non-invasive optical technique which measures the polarization state change of light as it is reflected from sample and can obtain sample’s optical response with a high accuracy [18–22]. Gogoi et al. [23] reported the complex dielectric function of pristine bulk SrTiO3 between 4.2 K and 300 K within the energy range of 0.6-6.5 eV using SE. Chernova et al. [24] measured the room-temperature optical constants of pristine as well as 0.7% wt Nb-doped SrTiO3 over a spectral range from 0.74 eV to 8.8 eV. Benthem et al. [25] based on the vacuum ultraviolet spectroscopy and SE to measure the dielectric function of pristine and 0.14% wt Fe-doped SrTiO3 at 300 K over the energy spectral range 1.7-35 eV. Yet, no high-temperature or doping-dependent dielectric functions of cubic SrTiO3 measured by SE have been reported.
In the present work, we use SE to measure optical constants of pristine as well as doped SrTiO3 at high temperatures up to 873 K in the spectral range 0.73-5.90 eV. The measured optical constants provide valuable information about the microscopic electronic band structure. The wide spectral range allows characterization of critical points which originate from electronic interband transitions from top valence bands to low-lying conduction bands, while the broad temperature range provides more data to fully investigate lattice vibration’s influence on dielectric functions. Recently, cubic SrTiO3 with doping have shown exquisite properties due to the fascinating interactions of quasiparticles [26,27]. Thus, the temperature-dependent optical constants of Nb- and Fe-doped SrTiO3 are measured. Detailed analysis on the change trend of absorption peaks with varying temperature and dopants is conducted. To explore the underlying physics of optical absorption peaks, we apply the first-principles method which accounts for the many-body excitonic effect and phonon-assisted indirect absorption (Inabs) process.
This work is structured as follows: Sec. 2 presents the details about SE experiments and computational methodology. In Sec. 3, the measured optical constants of pristine cubic SrTiO3 at room-temperature by SE are presented and fitted with the oscillator models. The first-principles calculations considering excitonic effect and phonon-assisted indirect absorption process are performed to explain the physics origin of absorption peaks. Then, by comparing the pristine and doped-SrTiO3 at varying temperatures, the influence of temperature and doping on optical constants is discussed.
2. Experiment and methodology
2.1 Spectroscopic ellipsometry
SE is a state-of-the-art optical technique that can precisely determine thin film thickness as well as optical constants. Figure 1 depicts the work principle of SE that measures the polarization change of incident light upon reflection from a sample to extract optical constants. The involved ellipsometric relationship is described by [18]
where ρ is the ratio of the Fresnel reflection coefficient r for the p- over s- polarized light, and the ellipsometric parameters Ψ and Δ are the collected quantities that determine optical constants/or thin film thickness upon an optical model analysis.We performed the SE measurements of cubic SrTiO3 at temperatures of 300-873 K over the energy spectral range 0.73-5.90 eV using the RC2 ellipsometer which is purchased from the J.A. Woollam Co., Inc. The RC2 ellipsometer adopts the dual rotating compensators to achieve high accuracy and fast measurement speed, and the light source consisting of a deuterium lamp and a halogen lamp provides a continuous spectrum of incidence radiation. A Linkam heating and freezing stage system combined with the use of liquid nitrogen help control the temperature of sample with the precision of ±0.1 K. When heating samples, the pure nitrogen gas was continuously injected into the stage system to avoid high-temperature oxidation. The incidence angle was kept at 70° for all SE measurements. The cubic SrTiO3 crystals are single-side polished with the size of 10×10×0.5 mm3 and the surface roughness less than 0.5 Å.
2.2 Computational methodology
Dielectric function is a relevant physical quantity that reflects the intrinsic light-matter interaction [28,29]. The incident light consisting of high-energy photons excites ground state electrons near the top valence bands and induces electronic interband transitions, thus leading to optical absorption. Following the Fermi’s golden rule, the imaginary part of dielectric function ε2 arising from the intrinsic electron-photon interaction is expressed as [30]
We implemented the Vienna ab initio simulation package (VASP) [31] to perform electronic band structure and dielectric function calculations. The exchange and correlation term was described by the generalized gradient approximation (GGA) with the Perdew-Becke-Erzenhof (PBE) [32,33] functional. The choice of GGA potential normally underestimates band gap and thus the HSE hybrid exchange-correlation functional calculations were performed to obtain accurate electronic band structure [34]. Furthermore, previous study [35] has identified the existence of strong excitonic effect that greatly affects optical spectrum of SrTiO3. To include excitonic effect, a model Bethe-Salpeter equation (BSE) method [36,37] was adopted which could reproduce the main features of optical spectrum at a much lower cost compared to the GW + BSE calculation [38]. To perform first-principles calculations, a 5-atom unit cell was chosen and the relaxed lattice parameter with the HSE hybrid functional [34] was 3.902 Å, in excellent agreement with experimental data of 3.905 Å [39]. The energy cutoff was 520 eV and the k-mesh was set as 12×12×12. To apply the model BSE method, an energy scissor of 1.46 eV was chosen, which was estimated by the difference in the band gap predicted by the HSE and PBE calculation.
Besides the strong excitonic effect, cubic SrTiO3 owns an indirect band gap where the top of valence band locates at the R point and the lowest conduction band is at the Γ point in the Brillouin zone [23]. In the indirect-band-gap semiconductors, the phonon-assisted indirect absorption is of crucial importance to determine the onset of absorption [40]. The phonon-assisted absorption coefficient is expressed as [41]
3. Results and discussions
3.1 Dielectric function and oscillator model
Using the RC2 ellipsometer, the ellipsometric parameters Ψ and Δ of pristine SrTiO3 are firstly measured and analyzed to obtain the optical constants. Figure 2 presents the measured dielectric function ε2 of pristine SrTiO3 at 300 K, in excellent agreement with literature data [25]. Upon the oscillator model analysis, the dielectric function ε2 is perfectly reproduced by four Gaussian and one Psemi-Tri oscillators with the mean square error (MSE) less than 1.
Fig. 2. The measured dielectric function ε2 of pristine SrTiO3 with the RC2 ellipsometer at 300 K, compared with literature data [25] and fitted with four Gaussian and one Psemi-Tri oscillators. The inset is a zoomed-in view.
The number of oscillators confirmed the five electron-hole transitions theoretically predicted by the BSE fatband calculations [35]. Of particular interest, the Psemi-Tri oscillator locates at 3.708 eV and well describes the shoulder of optical spectrum. Previous studies show that excitons (electron-hole pair) mainly contribute to the shoulder and participate in the first transition [23,35]. Theoretically calculations [35] showed that first transition between the top of O 2p valence bands and the bottom of conduction bands contributed by Ti t2g states occurs at 3.85 eV, consistent with the position of the Psemi-Tri oscillator. Thus, it can conclude that the Psemi-Tri oscillator describe the excitonic contribution to optical absorption. Moreover, the other four Gaussian oscillators locate at 4.199 eV, 4.752 eV, 5.140 eV and 6.419 eV, respectively, in excellent agreement with the theoretically predicted positions 4.07 eV, 4.76 eV, 5.20 eV and 6.58 eV of the second, third, fourth and fifth electron-hole transitions [35].
The occurrence of electron-hole transitions can be clearly explained by the electronic band structure. Figure 3 displays the electronic band structure of cubic SrTiO3 obtained with both the HSE and PBE functional. The calculations show that the valence band maximum (VBM) is at R, whereas the conduction band minimum (CBM) is at Γ in the first Brillouin zone. The indirect (R - Γ) and direct (Γ - Γ) band gap predicted by PBE is 1.792 eV and 2.146 eV, respectively, 45% and 43% smaller than the experimental data of 3.25 eV and 3.75 eV [25]. For HSE the band gap values of 3.289 eV and 3.623 eV are close to experimental values [25]. Upon a combined analysis of electronic band structure, BSE fatband calculations [35] and SE experiments, it finds that the first transition (Psemi-Tri oscillator at 3.708 eV) is localized at Γ and corresponds to the direct band (Γ - Γ) transition. The second transition (Gaussian oscillator at 4.199 eV) involves the energy states along Γ – X and contributes to the first absorption peak. The dominant absorption peak corresponds to the third transition (Gaussian oscillator at 4.752 eV) between the next lower lying set of O 2p states near the VBM and Ti t2g states at CBM localized strongly at Γ. The fourth transition (Gaussian oscillator at 5.14 eV) is relatively weak and occurs along Γ – M. The fifth transition (Gaussian oscillator at 6.419 eV) involves the O 2p states at VBM and the Ti eg states, and is highly delocalized along Γ – X.
Fig. 3. The electronic band structure of pristine SrTiO3 calculated by the HSE and PBE functional. The arrows indicate the Inabs process across the indirect band gap. The dashed arrows denote the electron-phonon scattering events, while the solid arrows show the electronic interband transitions.
To verify the above analysis, we predict the dielectric function ε2 of pristine SrTiO3 using several first-principles calculations that consider random phase approximation (RPA), many-body excitonic effect and phonon-assisted indirect absorption process, respectively. In Fig. 4, the RPA calculation [46] using the PBE functional can reproduce the onset and absorption peaks at 4.752 eV and 6.149 eV, but fail to predict the shoulder of optical spectrum at 3.708 eV where the excitonic effect dominates. Upon including many-body excitonic effect, the model BSE (mBSE) calculation is performed and it can reproduce this shoulder with the amplitude and slope variation more close to the SE data. Though the absorption peak at 4.199 eV can be predicted, yet its amplitude is largely underestimated. Considering that cubic SrTiO3 is an indirect band gap, the phonon-assisted indirect absorption process would influence optical absorption. That is because that phonons provide extra momentum to get more states involved in the intraband and interband transitions. So, we perform Inabs calculation and evaluate phonons’ contribution to optical absorption. The Inabs calculation needs to compute the intrinsic electron-phonon scattering and predict optical absorption. The involved theory is complex and computational cost is very huge, so the Inabs calculation was performed in the framework of independent particle approximation and the excitonic effect cannot be considered. The predicted optical spectrum of Inabs is similar to RPA except the enhanced absorption near 4.199 eV, suggesting that the phonon-assisted process plays a nontrivial role in determining this absorption peak.
Fig. 4. The theoretically calculated dielectric function ε2 of pristine SrTiO3 using RPA, mBSE and Inabs method, in comparison with the SE data. The inset is a zoomed-in view.
3.2 Temperature dependence
After heating the SrTiO3 crystal with the Linkam stage system, the temperature-dependent dielectric function ε2 are measured with the RC2 ellipsometer. In Fig. 5, it observes that the absorption peaks are greatly influenced by temperature. As temperature increases from 300 to 873 K, the shoulder at 3.708 eV gradually disappears and the amplitude of the dominant absorption peak at 4.752 eV decreases by 11% and the width broadens.
Fig. 5. The measured UV-Vis dielectric function ε2 of pristine SrTiO3 by RC2 ellipsometer between 300 and 873 K. The inset is a zoomed-in view.
Based on previous work [47], the lattice vibration would strengthen as temperature increases. Thus, the electron-phonon scattering increases and leads to the reduced electronic lifetime. Consequently, less electrons participate in the interband transitions, inducing the decreased amplitude of absorption peak. Moreover, the linewidth in the electronic band structure will broaden, which can explain the broadened absorption peak. To quantify the temperature effect, we analyze the change of critical parameters amplitude (ΔAm), position (ΔEn) and broadening width (ΔBr) describing the four Gaussian and one Psemi-Tri oscillators (see Fig. S1 in Supplement 1). It observes a non-uniform change trend for those oscillator parameters. The locations of Gaussian oscillators 2-4 describing electronic transitions at 4.752 eV, 5.14 eV and 6.149 eV, respectively, redshift to lower values with increasing temperature, which coincidences with the decrease of band gap [23]. Interestingly, temperature has a very strong influence on the transition at 6.149 eV, with a large redshift of 0.45 eV and reduced broadening width of 0.68 eV from 300 to 873 K. As the broadening width is largely reduced, the absorption peak becomes sharp, indicating that delocalized Ti eg states along Γ – X are relocalized by lattice vibration. In contrast, the positions of the Psemi-Tri and first Gaussian oscillator corresponding to the electronic transition at 3.708 eV and 4.199 eV, respectively, blueshift to higher values as temperature increases. The first transition is dominated by excitons and the blueshift suggests that the electron-hole pairs are scattered by lattice vibration and their contributions to optical absorption are decreased. This is consistent with the decreased amplitude and broadened width of the Psemi-Tri oscillator. As for the first absorption peak at 4.199 eV, theoretical calculations show that it originates from the many-body excitonic effect and phonon-assisted indirect absorption process. Though excitonic contribution decreases as temperature increases, yet more phonons would participate in the electronic transitions and thus strengthen optical absorption. This explanation is further confirmed by the increased amplitude and blueshift of the first Gaussian oscillator.
3.3 Doping dependence
To investigate the doping effect, we measure the dielectric function ε2 of Nb-doped and Fe-doped SrTiO3 with the RC2 ellipsometer. In Fig. 6, it is observed that dopants introduce more free carriers and greatly alter optical absorption. In the low-energy range below 3.2 eV, the Urbach tail [23] exists and the slope of optical absorption consistently decreases as Nb dopant concentration increases from 0.05% wt to 0.5% wt. However, as the Nb dopant concentration further increases to 0.7% wt, the slope increases and thus an unconventional Urbach tail behavior is observed. The unconventional Urbach tail was proposed as a consequence of the crystalline disorder [23]. Moreover, as the dopant concentration increases, an extra Gaussian oscillator locating around 0.1 eV emerges. The amplitude of this oscillator is relatively small (0.66 for the 0.7% wt Nb-doped SrTiO3), and it is supposed to arise from the free carrier absorption. Furthermore, dopants would introduce an extra momentum to participate in the phonon-assisted indirect absorption process, thus leading to the enhanced absorption peak at 4.199 eV. Yet, for the dominant absorption peak at 4.752 eV, the amplitude decreases with increasing dopant concentration. Previous first-principles calculations [35] show that this absorption peak arises from the transitions from O 2p states at the VBM to the Ti t2g states. By replacing Ti atoms with Nb or Fe atom, the density of Ti t2g states reduces and thus leads to the weakened absorption peak.
Fig. 6. The UV-Vis dielectric function ε2 of pristine, Nb-doped and Fe-doped SrTiO3 at 300 K measured by RC2 ellipsometer. The inset is the zoomed-in view.
We also study the temperature effect on the optical absorption of doped-SrTiO3. Take the 0.5% wt Nb-doped SrTiO3 for instance, the change trend of the absorption peaks with increasing temperature is similar to that of pristine SrTiO3 (see Fig. S2 in Supplement 1). The shoulder gradually disappears, the first absorption peak strengthens and the dominant absorption peak weakens as temperature increases. The main difference is that the small Gaussian oscillator locating around 0.1 eV redshifts to lower value and the absorption peak becomes sharp with increasing temperature. This can be explained by the fact that high temperature excites more free carriers to participate in the intraband transitions and enhance optical absorption.
4. Conclusion
To summarize, the influence of doping and temperature on dielectric functions of cubic SrTiO3 is investigated by the combined ellipsometric measurements and first-principles calculations. The measured dielectric function ε2 is fitted by the oscillator model, and the temperature and doping effect can be quantified upon analyzing the change of oscillator parameters. As temperature increases up to 873 K, the lattice vibration enhances and the electron-phonon scattering strengthens, which results in the reduced amplitude and broadened width of the dominant absorption peak at 4.752 eV. After performing the mBSE and Inabs calculations, it finds that the many-body excitonic effect and phonon-assisted indirect absorption process determine the shoulder at 3.708 eV and first absorption peak at 4.199 eV, respectively. At high temperatures, excitons are scattered by lattice vibration and the excitonic contribution to optical absorption is reduced. When doping SrTiO3 with Nb or Fe atoms, dopants provide an extra momentum to participate in the indirect absorption process and leads to the enhanced absorption peak at 4.199 eV. As the Nb dopant concentration increases from 0.5 to 0.7% wt, an unconventional Urbach tail is observed due to the crystalline disorder. Upon heating the doped crystal, similar temperature influence on dielectric function is observed. This work provides an updated and deep insight into the doping and temperature-dependent UV-Vis optical constants of SrTiO3, which is crucial to advance the applications in the fields of gas sensing and oxide electronics.
Acknowledgements
J.-Y. Y. is grateful for the support from Shandong University (Qilu Young Scholar 89963031). W. Z. appreciates the support by the National Natural Science Foundation of China (Grant no. 52006127) and the fundamental research funds of Shandong University. L. L. acknowledges the support by the National Natural Science Foundation of China (Grant no. 52076123).
Disclosures
The authors declare no conflicts of interest.
Supplemental document
See Supplement 1 for supporting content.
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