1. Introduction

Nonlinear optical spectroscopy of Raman active lattice or molecular vibrations is among powerful tools and techniques with key applications ranging from characterization of semiconductor nanostructures, energetic materials and to biological cells and tissues [1,2]. The spectral selectivity connects to an inter-atomic bond or sub-lattice levels, and therefore, microscopic physical mechanisms and fundamental interactions in condensed matter can be accessed. The spectroscopy methods and techniques are based on resonant nonlinear optical interactions in condensed matter [3,4] and a variety of ultrafast nonlinear spectroscopy systems have been developed recently to demonstrate the effectiveness of the approach when it comes to investigating elementary excitations in condensed matter [5–9].

Wide bandgap perovskites have been a subject of detailed studies because of the materials’ potential for device applications. The key applications may include tunable microwave devices, phase agile electronics, capacitors, dynamic random-access memory, and even energy storage devices [10–13]. Material property changes and corresponding effects on career transport are important when going from bulk to thin film layers, as the specific lattice dynamical properties can lead to drastic changes. The doped semiconducting oxides, like BaSnO₃, SrTiO₃, can provide high mobility and retain their properties at interfaces, as this is critical for device applications. Experimental characterizations of bulk BaSnO₃ (BSO) perovskite demonstrated very high mobility at room temperatures with values within 100- 320 cm²/V s range [14,15]. However, one of the main mechanisms, such as carrier-phonon scattering, can strongly affect the mobility as the hot carriers exchange energies with optical (LO) phonons within the lattice. The mobility values can be as high as 500 cm²/V s for electron concentrations >10¹⁹cm⁻³ [16].

Knowledge of mechanisms for phonon interactions and decay routes is important as those directly affect carrier energy relaxation and scattering rates [17,18]. The phonon-damping rate (Γ) is a key constant in the coupled equations describing carrier scattering by phonons [17] and, therefore, becomes as important parameter as the vibration frequency (Ω). The homogeneous full width at half maximum (FWHM) has a direct relationship with the damping rate as the phonon/lattice vibration linewidth is Δν[cm⁻¹] = Γ/2πc. The latter relationship leads to a very useful possibility and opportunity. The precision in getting the linewidth is solely dependent on the precision of measuring phonon decay/dephasing time (to be discussed later) as Γ is an inverse of the former. While spontaneous Raman techniques have practical limit in achieving good spectral resolution, time-domain techniques can provide in practices up to an order of magnitude better equivalent spectral resolution. Therefore, ultrafast coherent Raman studies can become key, especially in cases when traditional Raman technique lacks sensitivity and spectral resolution. Raman active modes have been theoretically modeled and calculated for cubic lattice structure BSO using density functional theory (DFT) that yields vibrational spectra and dielectric function [19,20]. Experimental data of the Raman active mode decays is lacking for BSO perovskite, and no study has been performed so far to trace the decay directly in time. Regarding strontium titanate (SrTiO₃ or STO), one should consider that bulk STO is a centro-symmetric crystal (space group Pm$\bar{3}$m) at room temperature. The material transitions to the tetragonal non-ferroelectric point groups at low temperatures. Also, the most important fact is that the optical modes (TO and LO) near zone-centered $\mathrm{\Gamma }$-point are symmetry forbidden and cannot be detected in the first- order Raman scattering [21]. Nevertheless, first-order Raman active modes have been detected for the lower temperature phase transition from cubic to tetragonal in prior studies and attributed to the effect of oxygen vacancies and impurities [22–25]. Second-order Raman resonances have been interpreted in several spectroscopic studies with bulk STO at different phases [26,27]. Hyper-Raman scattering studies shed light on all eight TO and LO modes that are forbidden (not detected in traditional Raman scattering) in bulk crystal at room temperature [28,29]. First-order Raman lines have been detected however in thin film STO material at room temperatures [30]. The explanation highlights the effect of lattice stress on symmetry distortions [31]. The symmetry breakage mechanism due to oxygen vacancies have also been pointed out in recent Raman experiments [32]. As has been mentioned above, the dispersion properties of the lattice vibration modes in STO crystals could only be inferred from the hyper-Raman spectroscopy studies [29]. Detection of Raman active modes at thin STO material and bulk-air interfaces due to symmetry distortions and an ability to trace the time decay of the modes gives an opportunity to reveal the lattice vibration properties.

In this work, we present experimental data on direct time-dependent measurements of the intrinsic phonon decay in cubic BaSnO₃ and SrTiO₃ crystals. With time resolution better than 120 fs and exceptional sensitivity delivered by the three-color Coherent Anti-Stokes Raman Scattering (CARS) technique, the performed experiments provided valuable information on ultrafast decay of lattice vibrations that was not available from previous studies. Damping rates of the main Raman active resonances have been determined for both materials with a precision that cannot be attained using other experimental techniques. The results of this study are important from the standpoint of practical applications of the materials, as understanding of fundamental mechanisms that put limits in achieving high electronic mobility in device application is crucial.

2. Experimental

The experimental implementation of CARS for this study is schematically shown in Fig. 1. Briefly, the laser system relies on intrinsically synchronized femtosecond pulsed output from two independently tunable optical parametric oscillators (OPOs). OPOs are simultaneously and synchronously pumped by splitting beams from 130 femtosecond, 76 MHz repetition rate Ti:sapphire oscillator tuned to 814 nm. OPO₁ utilizes magnesium-oxide-doped periodically poled lithium niobate (MgO:PPLN) crystal grown and processed for quasi-matching interaction with Λ=21.6 µm poling period section to provide ultrashort pulses within 985-1020 nm. Non-linear optical gain material used in the second OPOs allows for quasi-phase-matched (QPM) interactions between the pump and tunable near-infrared (1035-1180 nm) signal pulses within the OPO cavity as the poling period in the gain material (periodically poled stoichiometric lithium tantalate -PPSLT) varies along one of the crystal dimensions within 19.6-20.8 mm. The OPOs produce nearly Gaussian 130-180 fs pulses that are close to transform limited while the average powers are within 280-320 mW depending on signal wavelength. CARS process can be considered as scattering of probe pulse ${E_3}(t )$ on the coherent excitation (or driven vibrations) in a material created by two fields, ${E_1}(t ),\; {E_2}(t )$, in the vicinity of Raman resonance with frequency ${\mathrm{\Omega }_R}$. The anti-Stokes wave with amplitude ${E_{as}}$ is detected at the frequency, ${\omega _{as}} = {\omega _1} - {\omega _2} + {\omega _3}$. The difference frequency, ${\omega _1} - {\omega _2}$ needs to be tuned close to phonon frequency ${\mathrm{\Omega }_R}$ so that the coherent amplitude Q [33] for the driven vibrations is maximized. Output pulsed fields from the two OPOs (tuned to frequencies ${\omega _1},{\omega _2}$) provide the driving force (${\sim} {E_1}(t )\times {E_2}(t )$ for targeted Raman active vibration modes in the perovskite crystals while a replica from Ti:S oscillator, with an average power of ∼150 mW and optical frequency ${\omega _3}$, serves as the time-delayed (t_d) probe pulse. The ${E_{1,2,3}}(t )$ pulse polarizations have been set parallel. The beams are tightly focused into samples with objective lens (NA = 1.2) to generate a CARS signal at the anti-Stokes frequency that is coming from within a 0.1 $\mu $m³ volume rotational ellipsoid with 210 nm (x-y) and 560 nm (z-) long axes [34]. Introducing a temporal delay to the probe pulse with respect to the driving pulses enables the time-resolved version of CARS. The generated signal was fed through intensity attenuating and bandpass filters to the grating monochromator and read by 2048 × 70 pixel CCD. CARS transients can be traced within ∼75 dB span. The instrument response function (time resolution limit) is ∼120 fs. The software-controlled data acquisition system simultaneously detects and saves CARS spectra at different delay times.

 

Fig. 1. TiS – mode-locked TiAl₂O₃ oscillator, OPO₁ and OPO₂ – synchronously pumped femtosecond optical parametric oscillators. DM_1,2 – dichroic mirrors, MONO – diffraction grating monochromator, CCD – charge-coupled device array, OBJ – objective lens, SMP -sample, MSL – microscope cover slide. Lower-right inset: Output spectra of OPO₁ (two blue spectra) and OPO₂ (four spectra at λ₂> 1050 nm) used to target the studied Raman active vibrations in BSO and STO crystals. Position of the objective can be adjusted with respect to the sample with positive z - direction corresponding to probing sectional planes deeper into the sample (see upper-right inset).

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To analyze CARS transients, we must note that excitation and probing processes are separated in time by a delay ${t_d}$, and the transient nonlinear polarization is expressed in terms of time-evolving coherent amplitude $Q(t )$, created in the material by the ${E_1}(t )\times {E_2}(t )$ driving force so that

The detected anti-Stokes pulse energy at a delay of ${t_d}$ is derived with the help of formula (1) and the considerations above and, as a result, it can be computed using the formula:

Gaussian shape for all three pulses is assumed with the same width $({\; {t_p}} )$. $H(t )$ and $g(t )$ functions stand for the Heaviside step and time-dependent Gaussian envelope, respectively. S_max – CARS signal maximum at zero delay time and it has been derived as a function of focused pulsed energies and material parameters in our earlier work [36]. For broadband femtosecond pulses, ultrafast electronic nonlinearity may play a significant role. In time-domain, it shows as the non-resonant peak at zero-delay time and is the time resolution limited as the electronic nonlinearity is a fast-decaying component with T₂∼10 fs. The fitting function, in this case, is a sum of the fast decaying (<10 fs) non-resonant term and the one that corresponds to a much slower resonant process due to phonon. We should note that the former may contribute stronger to the CARS signal at shorter time delays and appear as a resolution limited peak. There is also an important question on the effect of the pulse frequency modulation (chirp) that may affect the shape of S(t_d) [34]. The effect might be pronounced at short delay times but is negligible in affecting slower decay part due to resonant phonon modes.

3. Results and discussions

As it has been mentioned above, cubic BaSnO₃ (BSO) is a wide-bandgap semiconductor for which the carrier mobility can attain values as high as 500 cm²/V•s. In our experiments, we used single crystal that was grown using the PbO-based flux method. The 1.7 × 1.7 × 1 mm³ crystal orientation allowed for the pulsed beams used in CARS to be focused onto <100 > facet. We have reported some of the results to be discussed in this work [37]. Figure 2 presents (a) time- and (b) spectrally-resolved CARS signals when the main LO₃ phonon mode at ∼632 cm^-1 was driven by ${E_1}(t ),{E_2}(t )$- pulses tuned to central wavelengths of λ₁ = 1014 nm and λ₂= 1078 nm. This creates the driving force centered at ∼ 585 cm^-1. The CCD detection window was set at pixels that matched the anti-stokes signal wavelengths within 758-782 nm. The phonon dynamics was followed by detecting the peak amplitude for the spectral line at λ_as= 774 nm versus time delay (t_d). The detected signals correspond to the LO₃ phonon energy calculated by the density functional theory studies [19,20]. The phonon decay constant (T₂) is found to be 1.46 ps as this results in the best fit, produced by using formula (3), to the experimental data shown above. The decay is perfectly exponential and proceeds along orders of magnitude in the signal change. The decay constant and precision that we achieved in detecting CARS transients yield in linewidth of 7.26 ± 0.09 cm^-1 that was calculated using formula (2). Slight redshift for the detected phonon energy, compared to the values obtained from the theoretical predictions of Refs. [19,20], is attributed to the experimental error/deviations in detecting anti-Stokes spectra. The obtained phonon decay results are analyzed within the framework of theories that consider anharmonic crystal potential that leads to phonon energy splits via parametric interactions [38]. In good quality crystals, excess carriers and spatial modulation of potential are very weak and do not affect intrinsic phonon modes. Hall measurements for our BSO samples show that the estimated excess carrier density is small (<10¹⁷cm^-3), and therefore, the corresponding plasma frequency is less than 1THz, which is well below the characterized longitudinal optical (LO₃) phonon mode. Thus, the pure dephasing mechanism due to the coupling of intrinsic LO-phonon and plasmon modes is negligible.

 

Fig. 2. (a) Time-resolved CARS signal (open circles) for BaSnO₃ crystal when LO₃ phonon mode (Ω_R∼630 cm^-1) is coherently driven and probed. The best fit (solid line) corresponds to the phonon decay time ${T_2}$ = 1.46 ps. (b) CARS spectra detected at different delay times while the driving force ${|{{E_1}(\omega ){E_2}(\omega )} |^2}$ was centered at ω₁-ω₂∼586 cm^-1 by tuning OPO_1,2 wavelengths to λ₁= 1014 nm and λ₂= 1078 nm. CARS spectra at λ_as= 774 nm peak position correspond to scattering of our probe pulse (λ₃= 814 nm) on the LO₃ mode and this can be detected for delays longer than 5000 fs. The spectra have been scaled by factors indicated in the legend for a better viewing purpose.

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The energy split of a LO- or TO-phonon at the zone-center (i.e., $\vec{q} \simeq \vec{0}$) into other phonons is dependent on the third-order anharmonic term (Γ_ij) in the crystal potential [38]:

In the equation, $n({{\omega_{i,j}}} )= {({{e^{\hbar {\omega_{i,j}}({{{\vec{q}}_{i,j}}} )/{k_b}T}} - 1} )^{ - 1}}$ terms stand for occupation numbers for final phonons of lower energy at crystal temperature (T). The momentum conservation for the interacting phonons yields in ${\vec{q}_i} + {\vec{q}_j} = \vec{0}$. Due to the density of states factor (${d_{{\omega _i} + {\omega _j}}}$), zone-edge phonons (with ${\vec{q}_i} ={-} {\vec{q}_j}$) are expected to provide the highest contribution in the decay process. The DFT results of Ref. [20] show that there might be several different decay channels for the LO₃ mode to lose its energy. The first option is the overtone path that results in final phonons with equal energy (315 cm⁻¹) and the opposite wavevectors. Other routes are two combination channels. The DFT calculations of Ref. [20] suggest phonon pair involving LO- and TO- vibrations with energies around ∼ 385 cm⁻¹ and ∼240 cm⁻¹ as well as an option involving final states with phonon energies of 490 cm^-1 and 140 cm^-1. The above arguments, together with the phonon decay constant obtained from data presented in Fig. 2 for the 632 cm⁻¹ mode, yield the decay rate at zero-temperature to be within 0.63- 0.89 ps⁻¹, and the corresponding linewidths are within 2.38-3.36 cm⁻¹.

Figure 3(a) shows CARS transient when a pair of broadband ∼160-180 fs pulses have been tuned to λ₁= 1014 nm and λ₂= 1106 nm that would correspond to targeting Raman active modes with energies ∼821 cm^-1. The slow decaying part in the CARS spectra (Fig. 3(b)) was at position λ_as∼764.6 nm. This corresponds to the Raman active lattice vibration with a frequency of ∼794 cm^-1. The measurement yields in the phonon decay constant of T₂= 1.27 ps. Based on the ratio of non-resonant (t_d= 0 fs) to resonant signals (t_d∼300 fs), we believe that we rather probe two-phonon states in this case as the ratio is more than 10 dB higher compared to the LO₃ phonon case (data on Fig. 2). Indeed, the ratio factor (r) mentioned above and the DFT calculations [19–20] both indicate that the CARS transient presented in Fig. 3 is due to a two-phonon state. We conclude that the two-phonon state is formed by lower-energy phonons. For example, the position matches well with 2 × LO₂ phonon state (LO₂∼410 cm^-1 [19,20]).

 

Fig. 3. (a) Time-resolved CARS signal and (b) anti-Stokes wavelength spectra obtained from BaSnO₃ crystal under high-frequency (ω₁-ω₂∼780 cm^-1) coherent excitation. The CARS spectra have been scaled by factors indicated in the legend for a better viewing purpose.

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Regarding the strength of the resonant CARS signal, much higher differences (order of magnitude) in the Raman signal cross-sections are found between two-phonon and one-phonon spectra for semiconductors with bandgap in near-infrared. However, recent theoretical and experimental studies show the two-phonon states may have enhanced Raman scattering cross-section as those are dependent on the ratio of the material’s bandgap to the probe photon energy [39]. The origin of the observed spectral feature (double-peak) at t_d= 440 fs has been discussed in our earlier publication [36].

In our experimental characterizations of STO material, we used a 0.4 mm thick (3 × 3 mm²) crystal grown by the Vernuil technique.

The interacting light beams have been incident on the <100 > oriented facet. Zone center optical phonons (i.e., with zero-wavevector) have odd parity, and Raman active in the bulk of a quality STO crystal cannot be observed. Many previous studies reported that the forbidden first-order Raman resonances can still be observed in treated samples with vacancies within the STO lattice and that can be attributed to the local vibrational modes at density of vacancies on the order ∼10¹⁸cm^-3 and higher [22–25,30–32]. Absence of any delectable resonant Raman mode in STO crystal bulk has been well confirmed in our experiments as we have been detecting CARS transients that replicated instrument function when we probed the crystal well within the bulk. However, with our spatial resolution and method sensitivity, we can also probe planes that are close to surface and the symmetry distortions near it may result in detectable Raman modes and their decay in time. In our experiments, the sample’s position along the beam propagation axis (z-axis) can be adjusted with ∼20 nm precision. Zero position along the z-axis has been determined in a separate second harmonic generation experiment with 500 µm thick KTP crystal. Figure 4(a) clearly illustrates the effect of symmetry distortions as we progressively sample areas that are closer to the surface. The symmetry is lowered at the surface and make Raman modes allowed in our coherent scattering experiments. Figure 4(a) presents time-resolves CARS signals generated within the near-surface planes of the STO crystal at different sample positions along z-axis. The OPO_1,2 wavelengths have been set to ∼998 nm and ∼1064 nm so that the targeted ${\omega _1} - {\omega _2}$ frequency corresponds to ∼625 cm^-1 with the anti-Stokes signal maximum at ${\lambda _{as}}$ ∼ 774 nm at zero delay. The signal at z∼-0.01 $\mu $m (blue open circles) is generated primarily within the ∼200 nm surface layer of the STO crystal’s interface and from the microscope glass slide itself.

 

Fig. 4. (a) Time-resolved CARS signal and spectra (b) for STO crystal while targeting ∼538 cm^-1 vibration (TO₄ phonon) at the crystal’s surface. The resonant signal is getting smaller as the probing volume moves deeper into the sample. The CARS spectra have been scaled for a better viewing purpose.

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At t_d∼0 fs we can detect the CARS signal as high as ∼1.3${\times} {10^8}$ counts, that is mostly due to the non-resonant contribution due to ultrafast electronic nonlinearity. The non-resonant CARS signal from the glass slide decays rapidly showing resolution limited 23 dB peak above the signal at t_d> 300 fs. The CARS signal at longer than t_d> 300 fs delay time is at ∼ $1.5 \times {10^5}$ counts level and shows much slower decay. This is attributed to decay of the resonant Raman active mode. The phonon mode frequency at ${\mathrm{\Omega }_R}$∼538 ± 20 cm^-1 can be estimated from the detected CARS spectra, shown in Fig. 4(b), at delay times longer than 500 fs. The phonon energy matches well with the one of transverse optical mode (TO₄) that was calculated and observed for the STO lattice vibrations studied in earlier works on spontaneous Raman and hyper-Raman scattering [21,22,25–28]. The decay is an ideal exponential curve as can be seen in comparison with the fitting curve (dashed line) that assumes purely exponential decay. The best fit was achieved for T₂= 1.4 ± 0.05 ps. The corresponding homogenous linewidth can be estimated at Δν=7.69 ± 0.18 cm^-1 using formula (2). Probing crystal areas deeper inside the bulk show a substantial decrease in the resonant (slower decaying) part of the CARS signal as the symmetry is restored for the crystal deeper in the bulk. The CARS transient shows instrument response at z-positions above +0.3 $\mu $m. Figure 5 shows the results of probing the optical phonon mode at ∼770 cm^-1 at the near-surface plane (z∼0.05 µm).

 

Fig. 5. CARS transient for STO crystal while targeting ∼770 cm^-1 vibration (LO₄ phonon) at the near surface layer of the crystal. The LO₄ decays exponentially with T₂= 1.12 ps and the corresponding homogenous linewidth is ∼9.5 cm^-1.

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The detected phonon frequency (Fig. 5) is within a match to the frequency of the LO₄ mode reported in earlier studies [21,25–27]. As in the previous case, the decay is along a perfect exponential line with the time of T₂= 1.12 ± 0.02 ps that provides the best fit. Therefore, the homogenously broadened Raman linewidth results in Δν = 9.47 ± 0.16 cm^-1 according to formula (2). Appearance of the resonant CARS signal due to Raman modes has been explained by the presence of vacancies that can be introduced by special processing or present at the surface. The vacancies lead to a buildup of excess carrier concentrations. The intrinsic LO-phonon modes will then get modified due to the coupling to the field of carrier plasma. The typical excess carrier densities (vacancies) at the surface of $3 \times {10^{18}}$ cm^-3 produce plasma oscillation frequency, ${\nu _p}[{c{m^{ - 1}}} ]= \left( {\frac{1}{{2\pi c}}} \right){\left( {\frac{{n{e^2}}}{{{\varepsilon_0}{\varepsilon_\infty }{m_{eff}}}}} \right)^{\frac{1}{2}}}\sim 265\; c{m^{ - 1}}$. This is in the same order but still smaller frequency than for the modes we have been reporting here. If we apply small damping limit approach, the investigated modes are essentially phonon-like. The plasmon may lead to an increase for the pure dephasing rate ($T_\varphi ^{ - 1}$). The dephasing time is $T^{-1}_\varphi = \omega {^\prime}2\omega _p^2 /{\rm \Omega }_R^4 {\rm \tau }_e$ [40] where (1/${\tau _e}$) is the carrier momentum relaxation rate, and $\omega ^{\prime}$ represents LO-TO phonon frequency difference. With $\tau $_e ∼100 fs [41], $\omega ^{\prime}$∼2 THz (∼11 cm^-1) [29], we can have an estimate that the pure dephasing rate contribution to the total decay rate (1/T₂) at the above-mentioned densities is on the order of 4% of that for typical phonon decay rates (1/T₁ ∼ 0.1-1 ps^-1) measured in solids. Thus, the effect on the studied modes is negligible. Longer decay time for the TO₄ mode (∼540 ± 20 cm^-1) can be attributed to the fact that we have just a single overtone channel for the phonon energy split compared to multiple overtone and combination routes for LO₄ mode energy and momentum exchanges with other phonons.

4. Conclusion

In conclusion, we have presented time-resolved coherent Raman spectroscopy studies for two technologically important semiconducting oxides. Time decay for the lattice vibrations have been directly traced with better than 120 fs resolution and detected within a high dynamic range that resulted in precise measurements of damping rates for the main phonon modes. The intrinsic vibrations decay via mechanisms of parametric phonon interaction. In BSO perovskite, the detected LO₃ phonon (∼632 cm^-1) showed the phonon lifetime of 1.47 ps (damping rate 1.36 ps^-1). The higher frequency modes that are active in Raman scattering have been attributed to two-phonon states based on the strength of the resonant signal and theoretical calculations for Raman mode properties. For SrTiO₃ crystal, we have been able to detect and trace time decays of the main LO/TO phonon modes by selectively probing the near-surface areas where symmetry distortions make the modes detectable. The effective phonon decay rates are within 1.45-1.78 ps^-1 and rather characterize intrinsic phonon decay rates that are not affected by the presence of charged carriers due to vacancies. The obtained information is valuable both for some topics in fundamental condensed matter physics and applied research to provide a good understanding of mechanisms that put limits to achieving the high electronic mobility that is critical for device applications.