There has been significant interest in a quantum paraelectric material strontium titanate (SrTiO3), and its lattice dynamics and unusual dielectric character have been extensively studied. In low temperatures, its dielectric constant increases up to about 30 000. The dielectric constant increases extraordinarily with decreasing temperature while the paraelectric phase is stabilized by quantum fluctuations without any ferroelectric phase transition even below the classical Curie temperature Tc=37 K . In SrTiO3, a ferroelectric transition is easily induced by a weak perturbation such as an uniaxial stress , an isotopic substitution of oxygen 18 for oxygen 16 , and an impurity doping [4,5].
SrTiO3 has a perovskite structure as shown in Fig. 1(a) and is known to undergo a structural phase transition at Tc=105 K . The cubic (Oh) structure above Tc, where all phonon modes are Raman forbidden, changes into the tetragonal (D4h) structure below Tc, where Raman-allowed modes of symmetries A1g and Eg appear . The phase transition is due to the collapse of the Γ25 mode at the R point of the high-temperature cubic Brillouin zone. Below Tc, the R point becomes the Γ point of the D4h phase. The phase transition is characterized by the softening of phonons at the R point and concomitant doubling of the unit cell.
The distortion consists of an out-of-phase rotation of adjacent oxygen octahedra in the (100) planes . The order parameter for the phase transition is inferred to be the angle of rotation of the oxygen octahedra. Only a small rotation of the oxygen octahedra is involved for the transition. The rotation angle for the oxygen octahedra varies from ~2° of arc near 0 K down to zero at Tc=105 K; the transition is second order. At liquid nitrogen temperature, the rotation angle is about 1. 4° and the linear displacement of the oxygen ions about their high-temperature equilibrium positions is less than 0.003 nm. This oxygen octahedron motion can be described as a rotation only as a first approximation; the oxygen ions actually remain on the faces of each cube and therefore increase in separation from the titanium. Because the (100) planes are equivalent in the cubic phase, the distortion produces domains below Tc in which the , , or  axis becomes the unique tetragonal c axis.
According to the measurement of dielectric constants, Sr1−xCaxTiO3 undergoes a ferroelectric transition above the critical Ca concentration xc=0. 0018, where doped Ca ions are substituted for the Sr ions [8–11] as shown in Fig. 1(b). The cubic structure above the structural phase-transition temperature Tc1 changes in to the tetragonal structure below Tc1 and into the rhombohedral structure below the ferroelectric transition temperature Tc2. The structural phase transition at Tc1 is also antiferrodistortive as in pure SrTiO3 at 105 K . As in the case of impurity systems, Li-doped KTaO3 and Nb-doped KTaO3 , off-centered impurity ions are supposed. Their polarized dipole moments show a ferroelectric instability below the ferroelectric transition temperature . In the case of Ca-dopedSrTiO3, a spontaneous polarization occurs along  directions in the c plane, where the tetragonal (D4h) symmetry is lowered to C2v . With decreasing temperature a ferroelectric ordering process dominates, that is, due to the thermal growth of the polarization clouds surrounding the off-center Ca2+ dipoles . The system behaves like a super paraelectric as the ferroelectric nano-ordered regions contain disordered cluster like regions. The investigation by x-ray and neutron diffractions and first-principles calculations  suggests that polar instabilities originating from the off-center displacements of Ca2+ ions are not likely to directly polarize the host matrix by an electrostatic mechanism. Instead, the possible role of random fields in inducing the presence of disordered polar clusters was suggested, which is similar to polar nanoregions in relaxor materials.
Recently, a gigantic change in the dielectric constant by an ultraviolet (UV) illumination was discovered [13,14], and a deeper interest has been taken in SrTiO3 again. The origin of the giant dielectric constants, however, has not yet been clarified. In Ca-doped SrTiO3, it was reported that a UV illumination causes a ferroelectric peak shift of the dielectric constant toward the lower temperature side . In several ferroelectric materials such as BaTiO3 , SbSI , and oxygen-isotope-substituted SrTiO3 , the Tc reduction under a UV illumination has been observed.
The optical information on the dielectric response is usually obtained from the experiments of Raman scattering or infrared spectroscopy. The usefulness of the investigation of low-frequency dielectric response by observing coherent phonons have also been demonstrated by the time-resolved study of the dynamics of phonons  and phonon polaritons . At low frequencies this technique is very sensitive and provides a very good signal-to-noise ratio as compared to the conventional frequency-domain techniques while at higher frequencies a better performance will be achieved by using the conventional techniques. Therefore the coherent phonon spectroscopy and the conventional frequency-domain techniques can be considered to be complementary methods for the investigation of the dielectric response.
The observed signal of the Raman scattering  in SrTiO3 is very weak because the distortion from cubic structure in the low-temperature phase is very small. The intensity of the first-order Raman signal is of the same order of magnitude with many second-order Raman signals, and then a background-free signal of the first-order Raman scattering cannot be observed.
Under a UV illumination, SrTiO3 and Ca-doped SrTiO3 show a broadband luminescence in the visible region originated from a relaxed excited state . The coherent phonon spectroscopy is not sensitive to the luminescence and a powerful technique to investigate UV-illumination effects in paraelectric materials as compared to the Raman-scattering measurement because in Raman-scattering experiments, it is not easy to separate Raman-scattering signals from the luminescence.
In the present study, ultrafast polarization spectroscopy is used to observe the coherent optical phonons in pure and Ca-doped SrTiO3, which are generated by femtosecond optical pulses through the process of impulsive stimulated Raman scattering [22,23]. Time-dependent linear birefringence induced by the generated coherent phonons is detected as a change of the polarization of probe pulses. High detection sensitivity of ~10−5 in polarization change has been achieved in our detection system. Damped oscillations of coherent phonons in SrTiO3 were observed below the structural phase-transition temperature (Tc=105 K), and temperature dependences of the phonon frequency and the relaxation rate are measured . The mechanism of the phonon relaxation is discussed by using a population decay model, in which an optical phonon decays into two acoustic phonons due to an harmonic phonon-phonon coupling.
The doping-induced ferroelectric phase transition in Ca-doped SrTiO3 is investigated by observing the birefringence and coherent phonons . In the birefringence measurement, the structural and the ferroelectric phase-transition temperatures are examined. In the observation of coherent phonons, the soft phonon modes related to the structural (Tc1=180 K) and the ferroelectric (Tc2=28 K) phase transitions are studied, and their frequencies are obtained from the observed coherent phonon signals. The behavior of the softening toward each phase-transition temperature and the UV-illumination effect on the ferroelectric transition are discussed. In addition to the ferroelectric phase transition at Tc2=28 K, another structural deformation at 25 K is found. A shift of the ferroelectric phase-transition temperature under the UV illumination and a decrease in the phonon frequencies after the UV illumination are found. We show the approach in the time domain is very useful for the study of the soft phonon modes and their UV-illumination effect in dielectric materials.
The experiments are performed on single crystals of pure SrTiO3 and Ca-doped SrTiO3 with the Ca concentration of x=0.011. SrTiO3 was obtained commercially (MTI Corporation) and Ca-doped SrTiO3 was grown by the floating zone method . The thickness of the sample is 1. 0 and 0.5 mm for the pure and Ca-doped crystals, respectively. In Ca-dopedSrTiO3, the structural phase-transition (Oh→D4h) temperature, Tc1=180 K, is obtained from the result of the birefringence measurement in section 3. The ferroelectric phase-transition temperature, Tc2=28 K, was determined by the measurement of dielectric constants . The value of x was determined by the empirical relation of Bednorz and Müller .
Schematic diagram of the experiment of polarization spectroscopy is shown in Fig. 2. The change in optical anisotoropy (birefringence) is detected by a polarimeter as the change in the polarization of the probe light (ellipticity). In the birefringence measurement, the birefringence generated by the lattice deformation is detected by a continuous-wave (cw) probe beam with no pump beam. In the coherent phonon spectroscopy, the transient birefringence due to the coherent phonons generated by a pump pulse is detected by a probe pulse.
2.1. Birefringence measurement
In the birefringence measurement, the linearly polarized probe beam is provided by a Nd:YAG laser (532 nm, cw) and is perpendicular to the (001) surface of the sample.
The construction of the polarimeter is shown in Fig. 3. The polarimeter [26,27] detects the rotation of polarization plane of a light beam. A linearly-polarized beam is split by a polarized beam splitter (PBS) and incident on the two photodiodes (PD) whose photocurrents are subtracted at a resistor (R). When the polarized beam splitter is mounted at an angle of 45o to the plane of polarization of the light beam, the two photocurrents cancel. If the plane of polarization rotates, the two currents do not cancel and the voltage appears at the resistor.
In the present experiment, the birefringence generated by the lattice deformation is detected as the change in polarization of the probe beam using a quarterwave plate and the polarimeter. The birefringence generated in the sample changes the linear polarization before transmission to an elliptical polarization after transmission. The linearly-polarized probe beam is considered to be a superposition of two circularly-polarized components which have the opposite polarizations and the same intensities. The generated birefringence destroys the intensity balance between the two components. The two circularly-polarized beams are transformed by the quaterwave plate to two linearly-polarized beams whose polarizations are crossed each other, and the unbalance of circular polarization is transformed to the unbalance of linear polarization or the rotation of polarization plane. This rotation is detected by the polarimeter as the signal of the lattice deformation.
2.2. Observation of coherent phonons
Coherent phonons are observed by ultrafast polarization spectroscopy with the pump-probe technique. The experimental setup for coherent phonon spectroscopy is shown in Fig. 4. Coherent phonons are generated by femtosecond optical pulses through the process of impulsive stimulated Raman scattering [22,23], and are detected by monitoring the time-dependent anisotropy of refractive index induced by the pump pulse. The pump pulse is provided by a Ti: sapphire regenerative amplifier whose wavelength, pulse energy, and pulse width at the sample are 790 nm, 2 μJ, and 0.2 ps, respectively. The probe pulse is provided by an optical parametric amplifier whose wavelength, pulse energy, and pulse width are 690 nm, 0. 1 μJ, and 0.2 ps, respectively. The repetition rate of the pulses is 1kHz. The linearly polarized pump and probe beams are nearly collinear and perpendicular to the (001) surface of the sample, and are focused on the sample in a temperature-controlled refrigerator. The waist size of the beams at the sample is about 0.5 mm.
The induced anisotropy of refractive index is detected by the polarimeter with a quarter-wave plate as the polarization change in the probe pulse. The plane of polarization of the probe pulse is tilted by 45° from that of the pump pulse. The two different wavelengths for the pump and probe pulses and pump-cut filters are used to eliminate the leak of the pump light from the input of the polarimeter. The time evolution of the signal is observed by changing the optical delay between the pump and probe pulses. The pump pulse is switched on and off shot by shot by using a photoelastic modulator, a quarter-wave plate, and a polarizer, and the output from the polarimeter is lock-in detected to improve the signal-to-noise ratio.
The source of UV illumination is provided by the second harmonics (380 nm, 3. 3 eV) of the output from another mode-locked Ti: sapphire laser, whose energy is larger than the optical band gap of SrTiO3 (3.2 eV). Since the repetition rate of the UV pulses is 80 MHz, this UV illumination can be considered to be continuous in the present experiment, where the UV-illumination effect appearing more than one minute after is studied.
3. Birefringence measurement
Figure 5 shows the temperature dependences of the change in birefringence in SrTiO3 and Ca-doped SrTiO3 between 4.5 and 250 K, where the polarization plane of the probe light is along the and axes. In SrTiO3, a change in birefringence appears below Tc=105 K, which is the temperature of the structural phase transition, and is increased as the temperature is decreased. In Ca-doped SrTiO3, large changes in birefringence come out at Tc1=180 K, which is the temperature of the structural phase transition, and at Tc2=28 K, which is the temperature of the ferroelectric phase transition. The change in birefringence is increased as the temperature is decreased from Tc1 to Tc2, as well as the case in SrTiO3, and shows a increase in the gradient for both axes around Tc2. Below Tc2, another kind of lattice distortion is added, and the peaks of Δn due to the competition between the two kinds of lattice distortion appear. In both SrTiO3 and Ca-dopedSrTiO3, a cusp on the birefringence curve appears around Tc=105 K and Tc1=180 K, respectively, because of the fluctuation associated with the second-order structural phase transition .
4. Coherent phonon spectroscopy in SrTiO3
4.1. Angular dependence of the coherent phonon signal
Figure 6 shows the transient birefringence in SrTiO3 observed at 6 K for the 0° pumping, where the polarization direction of the pump pulse is parallel to the  axis of the crystal. Vertical axis is the ellipticity η in electric-field amplitude of the transmitted probe pulse. At zero delay, a large signal due to the optical Kerr effect, whose width is determined by the pulse width, appears. After that, damped oscillations of coherent phonons are observed as shown in Fig. 6(b), where the vertical axis is enlarged by 50 from that of Fig. 6(a). The change of the polarization for the oscillation amplitude of the coherent phonon signal is 4×10−4 of the electric-field amplitudeof the probe pulse, which corresponds to the change Δn=7×10−8 of the refractive index. In our detection system, polarization change of ~10−5 in the electric field amplitude can be detected.
Angular dependence of the coherent phonon signal in SrTiO3 observed at 6 K is shown in Fig. 7(a), where the angle between the  axis of the crystal and the polarization direction of the pump pulse is 0°, 15°, and 45°. The angle between the polarization directions of the pump and probe pulses is fixed to 45°. The 0.7 ps period signal for the 0° pumping disappears for the 45°pumping, where the 2.3 ps period signal appears. The Fourier transform of the coherent phonon signals in Fig. 7(a) is shown in Fig. 7(b). Oscillation frequency of the signal for the 0° pumping is 1.35 THz, and that for the 45° pumping is 0.4 THz. For other pumping angles both frequency components coexist in the coherent phonon signal. From the oscillation frequencies the 1.35 THz component is considered to correspond to the A1g mode, and the 0.4 THz component to the Eg mode .
In addition to the oscillation signal there exists a dc component. The dc component has a maximum amplitude for the 0° pumping and minimum amplitude for the 45° pumping. However, the creation mechanism is not clear at present. In the following we pay attention to the oscillation component.
4.2. Temperature dependence of the coherent phonon signal
The temperature dependence of the coherent phonon signal in SrTiO3 observed for the 0° pumping, which corresponds to the A1g mode, is shown in Fig. 8(a). The oscillation period and the relaxation time of coherent phonons at 10 K are 0.7 and 12 ps. As the temperature is increased, the oscillation period becomes longer and the relaxation time becomes shorter. At Tc=105 K, the phase transition point, the oscillation disappears. Above Tc no signal of coherent phonons is observed. The temperature dependence of the coherent phonon signal in SrTiO3observed for the 45° pumping, which corresponds to the Eg mode, is shown in Fig. 8(b). The oscillation period and the relaxation time of coherent phonons at 10 K are 2.3 and 45 ps. Similar behavior to that of the A1g mode was observed as the temperature was increased.
4.3. Phonon frequencies
The observed coherent phonon signal S(t) is expressed well by the damped oscillation
where is the oscillation frequency and is the relaxation rate. This sine-type function is expected for phonons induced by impulsive stimulated Raman scattering [22,23]. The temperature dependence of the oscillation frequency in SrTiO3 obtained from the observed coherent phonon signal below Tc is shown in Fig. 9. The diamonds are the oscillation frequency for the A1g mode, and the squares are that for the Eg mode. As the temperature is increased from 6 K, the oscillation frequencies decrease and approach zero at the phase transition temperature Tc for both modes. This result is consistent with the temperature dependence of phonon frequency observed by Raman scattering . The solid curves describe a temperature dependence of the form . The experimental results for the temperature region between 50 K and Tc are explained well by n=0.4 for both modes, while those below 40 K deviate from that form.
The intensity of the first-order Raman-scattering signal in SrTiO3 is very weak and is of the same order of magnitude as many second-order Raman-scattering signals because the distortion from the cubic structure in the low-temperature phase is very small. Then observation of a background-free signal of first-order Raman scattering is not easy, and information on the relaxation, or the spectral width, is not given in a study of Raman scattering . By the present method of coherent phonon spectroscopy in the time domain, on the other hand, background-free damped oscillations can be observed directly, and the oscillation frequency and the relaxation rate can be obtained accurately.
4.4. Relaxation rates
The temperature dependence of the relaxation rate in SrTiO3 obtained from the observed coherent phonon signal below Tc is shown in Fig. 10. The diamonds in Fig. 10(a) are the relaxation rate for the A1g mode and the squares in Fig. 10(b) are that for the Eg mode. The relaxation rates increase as the temperature is increased.
In general, relaxation of coherent phonons is determined by population decay (inelastic scattering) and pure dephasing (elastic scattering). In metals, pure dephasing due to electron-phonon scattering, which depends on the hot electron density, contributes to the phonon relaxation . In dielectric crystals, the relaxation process of the coherent phononis considered to be dominated by the population decay due to the anharmonic phonon-phonon coupling [30-32], rather than pure dephasing. According to the anharmonic decay model , the relaxation of optical phonons in the center of the Brillouin zone is considered to occur through two types of decay process, the down-conversion and up-conversion processes. In a down-conversion process, the initial phonon with wave vector k0 decays into two lower-energy phonons i and j, with opposite wave vectors k and −k, which belong to the i branch and the j branch of the phonon. Energy and wave-vector conservation is given by 0=ik+j−k. In an up-conversion process, the initial excitation is scattered by a thermal phonon (ik) into a phonon of higher energy (jk), where 0+ik=jk. The down-conversion process can be realized either for i= j (overtone channel), or for i≠j (combination channel), depending on the phonon band structure of the material, while the up-conversion process contains only the combination channel and has no overtone channel. The combination channel is less likely, because three frequencies of phonons and three phonon branches have to be concerned and stringent limitations are imposed by the energy and wave-vector conservation. The overtone channel, on the other hand, is more likely because two (an optical and an acoustic) phonon branches are concerned, and the energy and wave-vector conservation are necessarily satisfied by two acoustic phonons with the same frequency and opposite wave vectors, if the frequency maximum of the acoustic branch is higher than half the frequency of the initial optical phonon.
Here we consider the down-conversion process in which an optical phonon decays into two acoustic phonons with half the frequency of the optical phonon and with opposite wave vectors. Schematic diagram of this down-conversion process is shown in Fig. 11. The temperature dependence of the relaxation rate of the coherent phonon is given by [30,31]
where 0 is the frequency of the optical phonon, and kB is the Boltzmann constant.
In ordinary materials the temperature dependence of the phonon frequency is small, and a theoretical curve with a fixed value of phonon frequency fits the experimental data well. In SrTiO3, however, the phonon frequencies are changed greatly as the temperature is increased; thus a frequency change has to be considered. The solid curves in Fig. 10 show the theoretical curves including the frequency change obtained from Eq. (2) with =8. 0×1010 s−1 for the A1g mode and 1.5×1010 s−1 for the Eg mode, where the observed phonon frequencies in Fig. 9 are used for each temperature. The broken curves show the theoretical curves in the case of no frequency change. As is seen in Fig. 10, the solid curves explain well the experimental data.
Deviation of the experimental data near Tc from the solid curve may be caused by the effect of the phase transition. However, the relaxation rate just around the phase transition point cannot be obtained in the present experiment, and the relation between the temperature-dependent relaxation rate and the structural phase transition is not clear.
5. Coherent phonon spectroscopy in Ca-doped SrTiO3
5.1. Angular dependence of the coherent phonon signal
The angular dependence of the coherent phonon signal in Ca-doped SrTiO3 observed at 50 K is shown in Fig. 12(a), where the angle between the  axis of the crystal and the polarization plane of the pump pulse is 0°, 25°, and 45°. The angle between the polarization planes of the pump and probe pulses is fixed to 45°. Vertical axis is the ellipticity in electric field amplitude of the transmitted probe. At zero delay, a large signal due to the optical Kerr effect, whose width is determined by the laser-pulse width, appears. After that, damped oscillations of coherent phonons are observed. For the 0° pumping a 0.5 ps period signal appears while it disappears for the 45° pumping and a 2 ps period signal appears. TheFourier transform of the coherent phonon signals in Fig. 12(a) is shown in Fig. 12(b). The oscillation frequency of the signalfor the 0° pumping is 1. 9 THz and that for the 45° pumping is 0.5 THz. For other pumping angles both frequency components coexist in the coherent phonon signal. These phonon modes are the soft modes related to the structural phase transition at Tc1=180 K. From the oscillation frequencies the 1.9 THz component corresponds to the A1g mode, and the 0.5 THz components to the Eg mode, which are assigned from the modes of pure SrTiO3  and from that of Sr1−xCaxTiO3 (x=0.007) .
5.2. Temperature dependence of the coherent phonon signal
The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 45° pumping, which corresponds to the Eg mode, is shown in Fig. 13. At 6 K some oscillation components, which have different frequencies, are superposed. As the temperature is increased, the number of the oscillation components is decreased, the oscillation period becomes longer and the relaxation time becomes shorter. At Tc1=180 K, which is the structural phase-transition temperature obtained from the birefringence measurement, no signal of coherent phonons is observed.
The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25° pumping are shown in Figs. 14 and 15. Figure 14(a) shows the coherent phonon signals below the ferroelectric phase-transition temperature Tc2 and Fig. 15(a) shows those above Tc2. Figure 14(b) shows the Fourier transform of the coherent phonon signals in Fig. 14(a), where the peaks with arrows 1, 2, and 3 correspond to the ferroelectric phonon modes . Figure 15(b) shows the Fourier transform of the coherent phonon signals in Fig. 15(a), where modes 1, 2, and 3 disappear but the A1g and Eg modes related to the structural phase-transition remain.
5.3. Phonon frequencies
Each component of the coherent phonon signal is expressed well by damped oscillations in Eq. (1). The temperature dependence of the oscillation frequencies in Ca-doped SrTiO3 obtained from the observed coherent phonon signals below Tc1 is shown in Fig. 16. The solid circles are the oscillation frequencies for the A1gand Eg modes. The triangles are that for modes 1, 2, and 3 which are related to the ferroelectric phase transition. As for the A1gand Eg modes, as the temperature is increased from 6 K, the oscillation frequencies decrease and approach to zero at the structural phase-transition temperature Tc1 for both modes. This result is consistent with the temperature dependence of phonon frequency observed by Raman scattering  and coherent phonons in section 4.3 for pure SrTiO3 except for the phase-transition temperature. The broken curves describe a temperature dependence of the form . The experimental results for the temperature region between Tc1 and Tc2 are explained well by n=0.5 for both modes. Below the ferroelectric phase-transition temperature Tc2, another mode appears at 0.9 THz. It is considered that the doubly degenerate Eg mode is split into two components under the tetragonal-to-rhombohedrallattice distortion.
The temperature dependence of the phonon frequencies, which are related to the ferroelectric phase transition, obtained from the observed coherent phonon signals below Tc2is shown in Fig. 17, where only the frequencies of the reproducible peaks in the Fourier spectrum are plotted. While the lowest peaks at 6 and 16 K in Fig. 14, for example, may be the third mode observed in the Raman experiment , the frequencies of the peaks with poor reproducibility are not plotted in Figs. 16 and 17. Under dark illumination, two phonon modes 1 and 2 are softened toward about 25 K and degenerate into mode 3, which seems to be softened toward the ferroelectric phase-transition temperature Tc2=28 K. The lift of degeneracy for mode 3 below Tc2 suggests that another structural deformation occurs at 25 K. In the Raman-scattering experiment , three modes deriving from the TO1 mode has been observed. The three modes do not all split off at the same temperature; the highest-energy component splits off at the ferroelectric transition temperature while the other two split off at the temperature about 3 K lower. These two temperatures may correspond to the two temperatures, 28 and 25 K, observed in our experiment, and the temperature differences are nearly equal to each other. The existence of the second structural deformation at 25 K is consistent with the Raman-scattering data.
5.4. UV-illumination effect
In order to examine the UV-illumination effect, two types of measurements, under the UV illumination and after the UV illumination, are carried out. The temperature dependence of the phonon frequencies under the UV illumination is shown in Fig. 17(a), where the intensity of the UV illumination is 7 mW/mm2. As is seen in Fig. 17(a), the temperature, toward which the two modes 1 and 2 are softened and degenerate into mode 3, shifts to the lower temperature side. The temperature shift due to the UV-illumination effect is ~3 K.
Doped Ca ions behave as permanent dipoles and ferroelectric clusters are formed around Ca dipoles with high polarizability of the host crystal. The ferroelectric transition is caused by the ordering of randomly distributed Ca dipoles. The UV-illumination-induced Tc2 reduction is related to the screening of the internal macroscopic field by UV-excited carriers. Under the UV-illumination non equilibrium carriers appear, which are captured by traps and screen the polarization field. The ordering is prevented by the photo excited carriers. Thus, the screening effect due to the UV-excited carriers weakens the Coulomb interaction between dipoles, and the transition temperature is decreased.
The theoretical Tc2 reduction under the UV illumination Tc2 is given by 
where Tc2(0) is the transition temperature before the UV illumination, λ is the inverse of the screening length, a is the mean separation between the dipoles, and is the number of the nearest-neighbor dipoles. The expression of λ is presented by , where n is the carrier concentration which depends on the UV-illumination intensity and is the relative dielectric constant. In the present case, the domain is large enough and the dipole-dipole interaction is expressed as a simple Coulomb interaction. The UV-illumination-intensity dependence of the Tc2 reduction for Sr1−xCaxTiO3 (x=0. 011) was analyzed by using Eq. (3). Assuming that the carrier concentration is proportional to the UV-illumination intensity, the fitting result reproduced well the experimental result obtained in the measurement of dielectric constants . Here we estimate the inverse λ of the screening length at the UV-illumination intensity of 7 mW/mm2. As a result of the measurement of dielectric constant , the carrier concentration is n=2. 51017 cm−3 at 3 mW/mm2. From this value we can estimate the carrier concentration to be n=5. 81017 cm−3 at 7 mW/mm2. The value of the inverse of the screening length is obtained as
where aB and E1s are Bohr radius and the energy of the hydrogen atom in the 1s ground state, respectively, and we use the dielectric constant =4 in the visible region. Substituting the value of into Eq. (3), the shift of the transition temperature can be estimated to be Tc2~8 K. This estimation is not inconsistent with the observed values of the temperature shift in the experiment of coherent phonons, in which the value of the observed transition temperature shift is ~3 K.
The temperature dependence of the phonon frequencies after the UV illumination is shown in Fig. 17(b), where theUV illumination of 7 mW/mm2 is on before the coherent phonon measurement but is off during the measurement. In this case, on the other hand, the shift of the softening temperature for modes 1 and 2 is not clear. The phonon frequencies for modes 1 and 2 are decreased while the coherent phonon signal for mode 3 is not observed. The relaxation time of the UV-illumination induced carriers is on the order of milliseconds below 30 K for SrTiO3 . As is seen in Fig. 17(b), however, it is suggested that the UV-illumination effect, frequency decrease for modes 1 and 2 and disappearance of mode-3 signal, remains for at least several minutes, even if the UV illumination is switched off, although the Tc2 shifting effect disappears immediately.
We observed coherent phonons in pure and Ca-doped SrTiO3 using ultrafast polarization spectroscopy to study the ultrafast dynamics of soft phonon modes and their UV-illumination effect. Coherent phonons are generated by linearly polarized pump pulses. The time-dependent linear birefringence induced by the generated coherent phonons is detected asa change of the polarization of the probe pulses. A high detection sensitivity of ~10−5 in polarization change, which corresponds to the change n=210−9 of the refractive index for a 1 mm sample, has been achieved in our detection system.
In SrTiO3, damped oscillations of coherent phonons for the A1g and Eg modes, which contribute to the structural phase transition at 105 K, were observed. The temperature dependences of the frequency and the relaxation rate of the observed coherent phonons were measured. Softening of the phonon frequencies was observed. The phonon relaxation is explained well by a decay model of a frequency-changing phonon, in which the optical phonon decays into two acoustic phonons due to the anharmonic phonon-phonon coupling.
We observed the temperature dependences of the birefringence and the coherent phonon signal to investigate the doping-induced ferroelectric phase transition in Ca-dopedSrTiO3 with the Ca concentration of x=0.011. In the birefringence measurement, it was confirmed that the structural phase-transition temperature is Tc1=180 K. Coherent phonons were observed by using ultrafast polarization spectroscopy. The damped oscillations of coherent phonons for the A1g and Eg modes, which contribute to the structural phase transition at Tc1=180 K, and for the modes 1, 2, and3, which contribute to the ferroelectric phase transition at Tc2=28 K, were observed. The phonon frequencies were obtained from the observed signals of coherent phonons, and their softening toward each phase-transition temperature was observed. Another structural deformation at 25 K was found in addition to the ferroelectric phase transition at Tc2. It was found that the UV illumination causes the shift of the ferroelectric phase-transition temperature toward the lower temperature side, and the temperature shift is ~3 K. The decrease in the phonon frequencies after the UV illumination suggests the UV-illumination effect remains even if the UV illumination is switched off.
It was shown that the coherent phonon spectroscopy in the time domain is a very useful approach to study the soft phonon modes and their UV-illumination effect in dielectric materials.
We would like to thank Dr. Y. Koyama for experimental help and Dr. Y. Yamada and Prof. K. Tanaka for providing us the samples of Ca-doped SrTiO3.