The trivalent rare earth element doped perovskite oxides La_1-xCa_xMnO₃ film exhibits rich electrical and magnetic properties. For example， in La_0.7Ca_0.3MnO₃ membranes， the balance between metallic and insulating phases can be tuned by the lattice， and extending the range of lattice control would enhance the ability to access other phases^［1-2］. A colossal magnetoresistance （CMR） phenomenon was reported in a perovskite-like La_0.67Ca_0.33MnO₃ thin oxide films on LaAlO₃ （LAO） substrates^［3-6］. It is well-known that the CMR effect appears around the metal-insulator transition temperature （T_MI）， and lots of works were done on the metal-insulator （MI） phase transition based on La_1-xCa_xMnO₃ film^［7-10］. Below the Curie temperature （T_C）， the thin film is ferromagnetic （FM） phase， and shows a higher conductivity. The film is transformed into paramagnetic （PM） phase with a good insulator above T_C^［11-12］. The FM to PM phase transition was attributed to the spin order change in 3d orbit electrons of Mn ions^［13］. In La_1-xCa_xMnO₃ film， the double exchange interaction between Mn³⁺-O-Mn⁺⁴ predicts that ferromagnetic corresponds to metal phase， and paramagnetic corresponds to insulator phase^［14-16］. By monitoring the doping proportion of rare element， both T_C and T_MI of La_1-xCa_xMnO₃ film can be tuned accordingly^［17-22］. In addition， the substrate strain also has strong influence on the magnetic transport and the metal-insulator phase transition temperature of the film^［23-30］.

It was reported that the La_0.7Ca_0.3MnO₃ （LCMO） film showed Drude conductivity when the temperature was below around 0.7 T_C， and the film conductivity was far deviated from the Drude model during the temperature range from 0.7 T_C to T_C^［31］. Some literatures reported that the T_MI temperature of LCMO epitaxial films was different from the T_C^［25］. But T_MI was usually measured by contacted electrical measurement， and the contact resistance between electrodes and film had significant influence on the measured value of T_MI. It is still controversial whether the T_MI and T_C temperature falls the same temperature point or not. Terahertz spectroscopy is contact-free， invasive， and extremely sensitive method to study the metal to insulator phase transition compared to the traditional method. With the advanced THz spectroscopy， the dynamical phase transition from ferromagnetic metal phase to paramagnetic insulator phase can be clarified^［31-32］， specifically， near the T_C and T_MI temperatures， the competitive and coexistence mechanism behind the phase transition can be clarified.

In this paper， the thermodynamics in LCMO film between ferromagnetic metal phase and paramagnetic insulator phase are investigated by the low temperature THz time-domain spectroscopy. The Curie temperature T_C is measured to be the same as the metal-insulator transition temperature T_MI. The conductivity of the film can be well reproduced with Drude model from 40 K to 200 K， while its conductivity follows the Drude-Lorentz model in high temperature range from 210 K to 290 K. The experimental results are instrumental to comprehend the electromagnetic and photoelectric properties related to phase transition， and have important application in thermoelectric device， contactless reluctance switch and so on.

The LCMO film with thickness of 200 nm （measured with a step profiler） was fabricated on a 0.5-mm， （100）-oriented LaAlO₃ single crystal substrate by pulse laser deposition method. The detail process of the film fabrication has been described elsewhere^［24］. The temperature dependent conductivity of the LCMO film in THz frequency was characterized with a THz time domain spectroscopy in transmission configuration. The output of a mode-locked Ti： sapphire laser， with pulse duration of 100 fs， centered wavelength of 800 nm， and repetition rate of 80 MHz （Mai Tai HP-1020， Spectra-Physics）， was used to generate and detect the THz transient. The THz emitter and detector are a pair of low-temperature-grown GaAs photoconductive antennas. The polarization of the THz radiation is vertical， which is perpendicular to the photoconductive antennas. The effective bandwidth of the THz spectrum is from 0.2 to 2.5 THz. The diameter of THz beam is around 3 mm at the sample position. The sample was placed in a cold finger cryostat with two THz-transparent windows， for which the temperature was tunable in the range of 40 to 300 K with best resolution of 1 K. The characterization of the LCMO film magnetization as a function of temperature was performed on a physical property measurement system （Quantum Design， PPMS-9） under vacuum condition in the temperature range from 10 K to 300 K with temperature stability better than ±0.02%. The temperature dependent magnetization measurement （M-T curve） was carried out separately from THz transmission measurement.

Figure 1（a） shows the transmitted THz electric field through the LCMO film on a 0.5 mm-LAO substrate with three selected temperatures of 100 K， 200 K， and 300 K. Apart from the main pulses localized around~19 ps， it is seen that week signals appear around 36 ps， which correspond to the second-order reflection （or say， echo pulse） of the main pulse at each temperature. With decreasing temperature from 300 K to 40 K， the amplitude of main pulse is seen to decrease， which suggests the increase of the sample conductivity. Moreover， the echo pulse of THz electric field shows in-phase with the main pulse at high temperature （for instance， 300 K）， and shows a π-phase shift at low temperature （100 K）. The echo pulse disappears completely around ~200 K. In order to exclude the contribution from LAO substrate， Fig. 1（b） shows the transmitted time-domain THz electric field of a 0.5-mm LAO single crystal， it is seen that both main pulse and echo pulse show almost no observable change with temperature. Thus the temperature dependent THz echo pulse signals come from the temperature dependent conductivity of the LCMO thin film^［33］.

Firstly， we focus on the LCMO film conductivity change with temperature. Figure 2（a） presents the transmitted THz amplitude of the sample as a function of temperature at several frequencies； an obvious turning point occurring at 260 K is seen for all selected frequencies： the transmitted THz amplitude increases near linearly with temperature in the temperature range from 40 K to 260 K， reaches a maximum value at about 260 K， and it is almost unchanged in the temperature range from 260 K to 280 K. Figure 2（b） shows magnetization of the LCMO thin film as a function of temperature for comparison. The magnetization of the film is seen to decrease rapidly with temperature in the temperature range from 260 K to 280 K. Figure 2（d） shows the dM/dT （derivation of magnetization with temperature） as a function of temperature. The minimum value of dM/dT is found to be the same with the turning point （~260 K） in the temperature dependent THz transmission. The real part of complex film conductivity in Fig. 2（c） is obtained from the THz amplitude transmission shown in Fig.2（a） with Eq.1^［34］，

here， φ and A represent the THz phase and amplitude difference between LCMO/LAO and LAO substrate， respectively. n_sub， d， and Z₀ are the refractive index of the LAO substrate， the thickness of the LCMO film， and the free space impedance （with Z₀=377 Ω）. With the thickness of the LCMO film d=200 nm， we give the real part of complex conductivity of the LCMO film， σ_r. It is also noted that the real part of complex conductivity changes as a function of temperature shows a turning point around 260 K in Fig. 2（c）. The real part of film conductivity decreases near linearly with temperature in the temperature range from 40 K to 260 K， and reaches a minimum value at 260 K. By increasing temperature further， the THz conductivity remains a constant in the temperature range from 260 K to 280 K. The turning temperature of THz conductivity occurring at about 260 K represents the metal-insulator phase transition temperature. In contrast， by looking at the temperature dependent magnetization shown in Figs. 2（b） and 2（d）， the magnetic phase transition temperature is also around 260 K. Based on Fig. 2， we can conclude that the LCMO is magnetically ordered metallic phase when temperature is lower than 260 K. When the temperature is higher than 260 K， the LCMO film lies at paramagnetic insulating phase. The metal to insulator phase transition occurs at the same temperature for the FM to PM phase transition.

Then， let’s focus on the THz pulse second order reflection occurring at the LCMO/LAO film interface. Figure 3（a） shows the THz echo pulse occurring at the interface of LCMO thin film at three selected temperatures of 198 K， 204 K， and 210 K， respectively. The echo pulse disappears completely at ~204 K， when temperature is higher than 204 K， the echo pulse show in-phase with the input main THz pulse， and when temperature is lower than 204 K， the echo pulse shows a π-phase shift with respect to the main pulse. Figure 3（b） schematically shows the THz pulse reflection occurring at the interfaces of the substrate LAO with different temperature. When temperature is below 200 K， LCMO shows a higher conductivity， and the refractive index difference between substrate and vacuum is smaller than Z₀σ_r， i.e.， n_sub-1<Z₀σ_r， THz pulse experiences reflection at the LCMO/LAO interface and reflected THz pulse has a π-phase shift with the incident pulse. As the temperature is above 210 K， we have n_sub-1>Z₀σ_r， THz pulse reflection occurs at the interface between LCMO film and vacuum， therefore the reflected THz pulse is in-phase with the incident pulse. Among 200 K and 210 K， n_sub-1≈Z₀σ_r， in which case impedance matching condition between vacuum and LAO substrate is met， thus the incident THz pulse does not experience reflection at the interface， as a result， the THz echo pulse disappears in this temperature.

To understand the carriers’ thermodynamics of the LCMO film in THz region， we employ Drude-Lorentz model as shown in Eq. 2^［35］ to reproduce the conductivity at all investigated temperatures.

here N， e， m^* and τ are the carrier density， carrier charge， effective mass of carrier and scattering time， respectively. ω₀ denotes the resonant frequency of the system. When ω₀=0， the Drude-Lorentz model is reduced to Drude model that is usually used to describe the free electron response， such as metal-like materials. Below the

Curie temperature， T_C， LCMO film behaves a metallic phase and the Drude model is expected to fit the film conductivity. However， it is found that Drude model can only reproduce the film conductivity at low temperature below 200 K （~0.77 T_C）. The solid lines in Figs. 4（a） and （b） show the fitted curves at 40 K and 200 K， respectively. Both real and imaginary parts can be reproduced with Drude model when temperature is lower than 200 K. As temperature is above 200 K and below 260 K （T_C）， although the film is still in metallic magnetic phase， it is seen that the imaginary part of the conductivity exhibits negative value as given in Fig. 4（c）， which indicates that the Lorentz parameter is not equal to zero （i.e.， ω₀≠0） at this temperature region. The solid lines in Fig. 4（c-d） are fitting lines with Drude-Lorentz model， and the fitting parameter， ω₀， as a function of temperature is given in Fig. 5（c）. Thus， among the temperature interval between 0.77 T_C （200 K） and T_C （260 K）， the competition between ferromagnetic metal and paramagnetic insulator dominates the terahertz conductivity.

According to Drude-Lorentz model in Eq. 2， some temperature dependent fitting parameters are plotted in Fig. 5. Figure 5（a） shows the temperature dependent plasma frequency （ω_p， ωP=Ne2/m*ε∞ε01/2） of LCMO film between 40 K and 290 K. As the plasma frequency is proportional to the square root of carrier density， it is seen from Fig. 5（a） that the carrier density in LCMO film decreases with temperature from 40 K to 180 K. By further increasing the temperature， the carrier density is seen to increase rapidly from 180 K to 200 K， and reach a maximum at 200 K. As the temperature is higher than 200 K， the carrier density of the film decreases gradually with increasing temperature to 290 K. The fitting scattering time （τ） presented in Fig. 5（b） demonstrates a non-monotonous change with temperature in the temperature range from 40 K to 200 K： the magnitude of τ is seen to increase from 40 K to 90 K， then decrease from 90 K to 200 K. It is noted from Fig. 2（b） that the magnetization of LCMO film remains almost a constant from 40 K to 90 K， but decreases slowly from 90 K to 200 K， and decreases sharply in the temperature range from 200 K to 260 K. Because the magnetization is strongly correlated with spin order of Mn³⁺ and Mn⁴⁺ ions， non-temperature dependence of magnetization in the temperature interval between 40 K and 90 K suggests that electron spin order does not exhibit obvious change in this temperature region. On the other hand， the plasma frequency （or say carrier density） is seen to decrease， but the carrier scattering time is seen to increase from 40 K to 90 K. Therefore， the decrease in conductivity from 40 K to 90 K is not derived from spin disorder， but due to the decrease of the electron's density with temperature in the film. In addition， the film magnetization decreases slowly with temperature from 90 K to 200 K as shown in Fig. 2（b）， which indicates the electrons spin tending to lose order slowly. The carrier density as a function of temperature shows a non-monotonous change （Fig. 5（a））， and the carrier scattering time shows a monotonous change （decrease） with temperature （Fig. 5（b））. The decrease in scattering time with temperature is mainly due to the temperature induced electrons spin disorder from 90 K to 200 K. Above 200 K， the LCMO film magnetization decreases rapidly with temperature， which is indicative of the phase transition occurring from FM to PM.

While the carrier scattering time remains at very small value at this high temperature region. Finally， as shown in Fig. 5（c）， the fitting Lorentz resonant frequency is seen to increase with temperature from 210 K to 270 K， and the decrease with temperature from 270 K to 290 K. In the temperature range from 210 K to 270 K， the magnetic phase transition occurs as shown in Fig. 2（b） and （d）， it is noted that the magnetic phase transition is accompanied with the metal-insulator phase transition， and both kinds of phase transitions take place in the same temperature range from 210 K to 270 K. On the other hand， metal-to-insulator phase transition means Drude-like carriers in metallic phase could be trapped by lattice to form polarons with Lorentz-like conductivity. The strong coupling between spin and polaron leads to the magnetic phase transition taking place at the same temperature as that of metal-to-insulator phase transition ^［36］.

In summary， we have investigated the temperature-dependent carrier dynamic of a colossal magnetoresistance LCMO film by terahertz transient spectroscopy. The FM-to-PM phase transition temperature T_C is almost same with the metal-to-insulator phase transition temperature， i.e.， T_MI=T_C≈260 K. Thus， the FM and PM states are closely related with the metallic and insulating states around T_C≈260 K. At FM phase， the electron spin disorder induces the decrease of the carrier scattering time via weakening the double change energy of the film （90∼200 K）. Near the phase transition temperature between FM and PM， the rapid decrease of spin order could lead to the decrease in the double exchange energy and density of free carriers in the film. The rapid decrease in double exchange energy leads to the coupling between spin and polaron in the temperature region from 210 K to 270 K. Our work could pave the way to clarify the thermal dynamics of double exchange electrons in La_0.7Ca_0.3MnO₃ film around the Curie temperature T_C. It also improves the analysis of the microscopic mechanism and theory of the film and affords a crucial reference for manganese oxide to be more widely utilized in thermoelectric device， contactless reluctance switch and other devices.