{'fraction_non_alphanumeric': 0.02724875476120715, 'fraction_numerical': 0.017579841781423966, 'mean_word_length': 5.034653465346534, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Ultrafast Relaxation Dynamics of Photoexcited Dirac Fermion in The Three Dimensional Dirac Semimetal Cadmium Arsenide

Three dimensional (3D) Dirac semimetals which can be seen as 3D analogues of graphene have attracted enormous interests in research recently. In order to apply these ultrahigh-mobility materials in future electronic/optoelectronic devices, it is crucial to understand the relaxation dynamics of photoexcited carriers and their coupling with lattice. In this work, we report ultrafast transient reflection measurements of the photoexcited carrier dynamics in cadmium arsenide (Cd3As2), which is one of the most stable Dirac semimetals that have been confirmed experimentally. By using low energy probe photon of 0.3 eV, we probed the dynamics of the photoexcited carriers that are Dirac-Fermi-like approaching the Dirac point. We systematically studied the transient reflection on bulk and nanoplate samples that have different doping intensities by tuning the probe wavelength, pump power and lattice temperature, and find that the dynamical evolution of carrier distributions can be retrieved qualitatively by using a two-temperature model. This result is very similar to that of graphene, but the carrier cooling through the optical phonon couplings is slower and lasts over larger electron temperature range because the optical phonon energies in Cd3As2 are much lower than those in graphene.

Three dimensional (3D) Dirac semimetals which can be seen as 3D analogues of graphene have attracted enormous interests in research recently. In order to apply these ultrahigh-mobility materials in future electronic/optoelectronic devices, it is crucial to understand the relaxation dynamics of photoexcited carriers and their coupling with lattice.
In this work, we report ultrafast transient reflection measurements of the photoexcited carrier dynamics in cadmium arsenide (Cd3As2), which is one of the most stable Dirac semimetals that have been confirmed experimentally. By using low energy probe photon of 0.3 eV, we probed the dynamics of the photoexcited carriers that are Dirac-Fermi-like approaching the Dirac point. We systematically studied the transient reflection on bulk and nanoplate samples that have different doping intensities by tuning the probe wavelength, pump power and lattice temperature, and find that the dynamical evolution of carrier distributions can be retrieved qualitatively by using a two-temperature model. This result is very similar to that of graphene, but the carrier cooling through the optical phonon couplings is slower and lasts over larger electron temperature range because the optical phonon energies in Cd3As2 are much lower than those in graphene.
Steady state transport and angle-resolved photoemission spectroscopy (ARPES) measurements mainly confirm the linear band dispersion of the Dirac Fermion and study the behavior of electron near the Fermi level. On the other hand, the transport behaviors in high speed devices, as promised by ultrahigh mobility of Dirac semimetals, are determined by the dynamic conductivity of excited carriers due to the presence of high/dynamic fields in these devices. Therefore, it is crucial to understand electron-electron (e-e) scattering, the cooling of the electrons due to the coupling to lattice phonons and the response of quasiparticle plasma to dynamical fields, if we want to apply the Dirac semimetals in high speed devices. Although the photoexcited Dirac Fermion dynamics are well studied in 2D systems especially that of graphene [44][45][46][47][48][49][50][51][52][53][54][55], very few experiments have been performed so far on photoexcited Dirac Fermion dynamics in 3D Dirac semimetals. Time resolved ARPES were studies on ZrTe5 [56] and SrMnBi2 [57] respectively. A transient-grating measurement was performed on Cd3As2 with photon energy of 1.5 eV which is far above the Dirac point and relates to the transitions of multiple bands [58]. In this work, we performed ultrafast transient reflection measurements of photoexcited carrier dynamics close to the Dirac point on both bulk and nanoplate forms of Cd3As2 and retrieved the dynamical evolution of carrier distributions qualitatively by using a two-temperature model (TTM). The photoexcited carrier dynamics of Cd3As2 is found to be very similar to that of graphene: after the photoexcitation with 1.5-eV pump photons, photoexcited carriers experience fast thermalization through carrier-carrier scattering and initial relaxation to low energy state by emitting high energy optical phonons with a several ps time constant.
The cooling thereafter is dominated by relatively slow, low energy optical phonon and acoustic phonon coupling processes. However, the optical phonon couplings are slower and last over larger electron temperature range, because the optical phonon energies in Cd3As2 are much lower compared to those in graphene.

II. METHODS
The bulk Cd3As2 single crystals were grown out of a Cd-rich melt with the ratio of Cd:As = 85:15 by flux techniques following the same recipe that is described in previous works [42,59]. Cd3As2 nanoplates were synthesized from Cd3As2 powders (Alfa Aesar 99.99% purity) by chemical vapor deposition with a silicon substrate placed at downstream to collect the products [38,[60][61][62]. Before the growth, the system was flushed several times with Argon gas to thoroughly remove the oxygen in the tube. Then the system was elevated to 650 °C in 20 min under an Ar pressure of 0.8 atm, and held at this condition for 20 min for growth with 20 sccm Ar flow.
The crystalline properties of bulk Cd3As2 were determined by X-ray diffraction (XRD) measurement (Rigaku MiniFlex 600 diffractometer, with a Cu-Kα1 radiation source). The thickness of Cd3As2 nanoplate was measured by SPI3800N atomic force microscope (AFM) system with contact mode. Transmission electron microscopy (TEM) and selected area electron diffraction (SAED) characterizations were performed by a FEI Tecnai F20 TEM equipment at 200 keV. For the hall transport measurement, individual nanoplates were transferred onto a silicon substrate with an oxide layer of 285 nm to fabricate devices, and then the measurement was performed in an Oxford cryostat system with magnetic field from 0 T to 14 T and temperature from 10 K to 300 K.
where R represents the reflection, ΔR is differential reflection, t is the pump-probe delay; amplitudes Ai and decay time constants τ or τ1 (2) are the fitting parameters. The standard error of τ and τ1(2) calculated by Origin Pro 8 software is used for the error bar of fitting.

B. Ultrafast Transient Spectroscopy Scheme
Ultrafast transient spectroscopy is one of the standard experimental tools to characterize photoexcited carrier dynamics and high field transport behaviors of condensed matters. As illustrated in the schematic diagram ( Fig. 1a and 1b) This is especially true after the initial relaxation from highly nonequilibrium states when dynamical evolutions of carrier distributions are mostly around the Fermi level that is close to the probe transitions. Moreover, as described later, the experimentally observed features with 4-μm probe can be easily interpreted with a two-temperature model [66].  . 2c). From the spectra plot that is normalized with negative peak at time zero (Fig. 2d), we notice the initial decay is faster when it is probed by shorter wavelength. Figure 3 shows the effect of pump power on the transient dynamic of bulk at 10 K. As the pump power increases from 1 mW to 10 mW, the magnitude of negative peak around time zero clearly increases, then ΔR/R decays and crosses zero at different delays (tc) as plotted in Fig. 3c. After that, the ΔR/R reaches a positive maximum and then gradually relaxes to zero within 40 ps. It is noteworthy that the maximum amplitude of positive ΔR/R almost stays as a constant regardless of pump power as shown in Fig. 3c and the inset of Fig. 3a. A normalized plot according to ΔR/R|t = 0 (Fig. 3b) reveals that it takes longer for photoexcited carriers to relax to zero crossing point with higher pump excitation power.

C. Ultrafast Transient Dyamics Features of Photoexcited Carriers
Further analysis indicates the transient signal from negative peak to zero crossing can be well fitted with a mono-exponential decay function (Fig. 3b) with the decay time constant increasing with pump power (Fig. 3d). These tendencies persist at 77 K as shown in  Further analyses with bi-exponential fittings of the transient responses (Fig. 4b) indicate that the slow time constant τ2 increases with temperature. The fast time constant τ1 also increases with temperature below 150K, but it decreases slightly above 150 K. Since the two exponential decay time constants from bi-exponential fittings are fairly close to each other as shown in Fig. 4b, semi-log scale plots are given in Fig. S4 of Supplemental Materials [65] for judgement of the reliability of bi-exponential fittings.

D. Interpretation of Transient Dyamics
Although there is no general consensus on quantitative model of the optically excited non-equilibrated carriers' behavior in the field on similar questions and a quantitative understanding from first principle is still lacking, we notice that a simple two temperature model can qualitatively interpret the transient measurement results of our bulk sample, especially for the 4-μm probe results which only involve two bands. The TTM involves two characteristic temperatures for electron Te and lattice TL has been successfully applied to metal film and graphene for the similar issues [47,67]. As shown in Fig. 5, immediately after the 1.5-eV pump photon excitation and subsequent initial thermalization through e-e scattering, the carriers will reach a quasi-equilibrium distribution that can be characterized by an electron temperature (Te) using Fermi-distribution function as shown in Fig. 5b. This leaves a smeared Fermi surface that is characterized by Te shortly after pump excitations (~ps). The TTM is different from the Rothwarf-Taylor model that is used to describe a gapped semiconductor [68], where the separated quasi Fermi-distributions of electrons and holes have to be considered before the electron-hole recombination in a nanosecond scale [69,70]. Indeed, there is some evidences showing that during the first picosecond, electrons and holes each maintain a distinct transient chemical potential in graphene [71][72][73][74]. The TTM is valid in Cd3As2 because the electron scattering time is usually much faster than the pulse width (~100 fs) in solids and the photoexcited holes can be instantaneously filled with electrons through efficient e-e scattering due to the gapless structure which is similar to that in graphene and other semimetallic materials [57,[72][73][74].
After the rapid thermalization among electrons, the electron temperature cools down through the e-p scattering processes (Fig. 5c), which include a fast cooling process by emitting optical phonons with relatively large energy and a slow cooling process through scattering with low energy acoustic phonons. The energies of optical phonon branches of Cd3As2 are around 25 meV (~288K) [75,76], much lower than the optical G phonon of 195 meV in graphene [77]. For this reason, the optical phonon cooling is slower in Cd3As2 (~2-5 ps) compared to that in graphene (<1 ps). On the other hand, because the optical phonon coupling survives until much lower electron temperature, the electron temperature of Cd3As2 exhibits faster cooling within the low electron temperature range due to the existence of optical phonon coupling [75]. The theoretically predicted decay time constant through acoustic phonon coupling is on the order of microsecond around the Dirac point and decreases to nanosecond for heavily doped samples. In the presence of short-range disorder, the cooling power is enhanced by a factor of 250, which makes the decay time constant falling into few picosecond region [78]. However, considering the good crystal quality revealed by high-resolution TEM of the nanoplate ( Compared to the carrier distributions before pump excitation (Fig. 5a), the change after pump excitation and subsequent thermalization process is mainly around the Fermi surface as illustrated in Fig. 5d. There are more carrier-occupy states above the Fermi level and less carrier-occupy states below the Fermi level. This carrier distribution change mainly accounts for the transient reflection signal probed by the corresponding probe photon transitions. From this diagram, if the probe transition is closer to the Fermi level of unexcited distributions, larger |ΔR/R| signal should be observed. This immediately explains why the magnitude of ΔR/R is larger with 4-μm probe on the bulk sample comparing to that with larger probe photon energies (Fig. 2c) and that on the nanoplate sample with lower Fermi level (Fig. 2a and 2b). Although many factors such as pump photon flux and morphology may contribute to the different transient responses of nanoplate and bulk observed in Fig. 2a and 2b,  As shown in Fig. 5e, and can be directly extracted from the transient reflection curves, which helps to fix the characteristic electron temperature range for the relaxation dynamics fittings of Te. If it takes 5-7 ps to relax from to , and ≤ 77 K, we speculate > 300 K, thus the relaxation from to should be dominated by optical phonon couplings. The mono-exponential fitting in Fig. 3b indicates the decay time constant increases with pump power when the electron temperature cools from to .
As is determined by the pump power in pump power dependent measurements, longer relaxation time from higher with higher pump excitation to the fixed is expected.
That explains the power dependence of tc observed in Fig. 3c. The increase of the decay time constant with increasing pump power (Fig. 3d) is mainly due to the elevated phonon temperature because both the e-p coupling strength and lattice heat capacity are temperature dependent. Although the heat capacity of lattice is orders larger than that of electrons, the phonon temperature still increases slightly as the electrons dissipate heat to the phonons through e-p couplings. We note the slightly elevated phonon temperature difference at different pump power also accounts for the very weak dependence of ΔR/R ( ) on pump power shown in Fig. 3c.
In temperature dependent measurements, is mainly determined by the pump excitation with only weak dependence on the initial lattice temperatures when the pump photon flux is large ( Fig. S5 and S6 in the Supplemental Materials [65]). As the lattice temperature increases, the relaxation gets slower (Fig. 4b), which has to correlate to the temperature dependence of electron and phonon heat capacities and the e-p coupling strength. To fit the results at 10 K, a mono-exponential decay can fit the range from negative peak to with a decay time constant 3.17 ps, similar to that is used in the power dependent measurement. However, because lower energy optical phonon and acoustic phonon coupling certainly play a role during the relaxation [78], the mono-exponential fitting is not sufficient for the cooling after , while a bi-exponential fitting is used to fit the full transient reflection dynamics of temperature dependent measurement results as shown in Fig. 4. Furthermore, we find that although the bi-exponential function can fit the 77 K, 150 K and 300 K data, the 10 K data in both Fig. 3 and 4 require tri-exponential rather than bi-exponential to be fit well as shown in Fig. S2 of Supplemental Materials [65].
The two exponential components from the bi-exponential fitting do not necessarily correspond to the optical phonon coupling and acoustic phonon decay processes respectively. The acoustic phonon cooling process may not be resolved in our measurement due to the limited signal to noise ratio, while the decay time constant should be far longer if the theoretical prediction is correct [78]. Thus the two decay constants may be dominated by the optical phonon coupling with different branches, considering the rich optical phonon modes available in Cd3As2 [75]. Alternatively, it is simply a magnification of lattice heat capacity variation over different temperature ranges.

VI. CONCLUSIONS
In conclusion, the photoexcited carrier dynamics of Cd3As2 are investigated using ultrafast transient reflection measurements with probe photon transitions close to the Dirac point and Fermi level. We found that the carrier dynamics of Cd3As2 can be qualitatively interpreted with a two-temperature model similar to that of graphene, but the optical phonon couplings are slower and last over larger electron temperature range. The fast transient time of the photoexcited carriers observed in this work promises Cd3As2 as an excellent candidate for ultrafast optoelectronics and photonics applications such as ultrafast photodetectors [79], optical switches [80] and saturate absorbers, especially for those working in the challenging middle/far IR and THz wavelength range.