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[Kunie Ishioka](https://orcid.org/0000-0002-2285-8839), Oleg V. Misochko

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[Suppression of shear ionic motions in bismuth by coupling with large-amplitude internal displacement](https://mdr.nims.go.jp/datasets/b281b80a-44aa-4dac-9b26-d9c6ea434ece)

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Suppression of shear ionic motions in bismuthby coupling with large-amplitude internal displacementKunie Ishioka1, ∗ and Oleg V. Misochko21National Institute for Materials Science, Tsukuba, 305-0047 Japan2Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow region, Russia(Dated: September 3, 2024)Bismuth, with its rhombohedral crystalline structure and two Raman active phonon modes corre-sponding to the internal displacement (A1g) and shear (Eg) ionic motions, offers an ideal target forthe investigation of the non-equilibrium phonon-phonon and electron-phonon couplings. We inves-tigate the Eg phonon dynamics under intense photoexcitation by performing anisotropic transientreflectivity (TR) measurements on a 1-mm thick bismuth single crystal at 11 K. The amplitude ofcoherent Eg phonon is found to increase with incident pump fluence up to 10mJ/cm2 and then turnsto an apparent decrease. This behavior is in stark contrast to the amplitude of the A1g phonon instandard TR measurements, which increases monotonically up to 20mJ/cm2 and then saturates.The contrasted behaviors of the two phonon modes can be interpreted in terms of the strong cou-pling of the Eg oscillation with large-amplitude A1g displacement on a highly excited electronicstate, where dynamic fluctuation of the vibrational potential would lead to a quick loss in the Egvibrational coherence. Unlike the previous studies on thin Bi films on substrates we observe no signof a transition to a high-symmetry phase but instead a sign of partial damage of the crystallinesurface at 28mJ/cm2, possibly due to less efficient cooling at the surface of the bulk crystal.I. INTRODUCTIONBismuth (Bi) is recently attracting renewed attentionbecause of its surface accommodating topological insu-lating electronic states. As for the bulk Bi electronicstates, it is established that Bi is a semimetal with itsFermi surface consisting of three electron pockets at theL points of the Brillouin zone and one hole pocket atthe T point [1], though their topological classification isstill under debate [2–6]. The crystalline structure of Bicrystal is rhombohedral with A7 symmetry, as illustratedin Fig. 1(a), with the ground-state internal displacementalong the trigonal (z) axis u ≡ c1/2(c1+c2) = 0.2357 andthe trigonal shear angle θA7 = 57.23◦ [3, 7, 8]. This struc-ture would be transformed to simple cubic by a slightdeformation to u = 0.25 and θA7 = 60◦ [1, 9]. The-oretical simulations predicted that the electronic bandstructure of Bi could undergo phase transitions from asemimetal to a semiconductor and to a metal by tweak-ing u and/or θA7 [1, 3, 7, 8]. Correspondingly, a pressure-induced semimetal-semiconductor phase transition of Biwas reported experimentally [10, 11].The rhombohedral Bi crystal features two Raman-active modes of A1g- and Eg-symmetries, which are as-sociated with the variations in u and θA7, respectively,as shown in Fig. 1(a). It has been theoretically demon-strated that introduction of photocarriers [12, 13] trans-forms the potential energy surface (PES) along the A1gcoordinate from a double well into a flattened single well,as schematically shown in Fig. 1(b). The susceptibilityof the PES to the electronic excitation leads to a sig-nificant enhancement of coherent A1g phonons via “dis-∗ ishioka.kunie@nims.go.jpFIG. 1. (a) Crystalline structure of Bi together with thedirections of the A1g and Eg displacements. (b, c) Schematicillustrations of potential energy surfaces (PESs) along the A1g(b) and Eg (c) coordinates for the ground and photoexcitedstates.placive excitation of coherent phonons (DECP)” mecha-nism [14, 15] and has thereby made the A1g mode a modeltarget for optical studies on non-equilibrium electron-phonon coupling [14, 16–29]. In the DECP mechanism aphotoinduced shift in the equilibrium coordinate gives adriving force that is dependent on excited carrier densityN [15, 30]. If the carriers are created sufficiently fast andlive sufficiently long (displacive limit), the coherent ionicdisplacement Qz can be described by a shifted dampedharmonic function:Qz(t) = QEz −Q0z exp(−Γzt) cos(2πνzt), (1)mailto:ishioka.kunie@nims.go.jp2with the initial amplitude defined by the equilibriumshift between the ground and excited states: Q0z(N) ≡QEz (N)−Qminz . The DECP enhancement of the A1g modeof Bi was experimentally confirmed by means of time-resolved x ray diffraction (trXRD) [31–36] and was sup-ported by density functional theory (DFT) calculations[12, 13, 32, 37]. One of the interests of recent ultrafastspectroscopic experiments on Bi lies on the optical con-trol of its structural and electronic phases. A single-shottransient reflectivity (TR) study on thin Bi films [38] ob-served the disappearance of the coherent A1g oscillationsignal under intense photoexcitation and attribute it tothe theoretically predicted transition to a high-symmetryphase with a single-well PES. The threshold fluence forthe phase transition was found to depend critically onthe Bi film thickness, suggesting that the phase transi-tion competes with the electronic transport in the depthdirection on sub-picosecond time scale [26, 39].The Eg-symmetry mode of Bi has been much less ex-plored by time-resolved experiments, despite of the theo-retical predictions that it could lead to a semiconductor-semimetal phase transition [1, 3, 7]. This is partly be-cause of its much smaller amplitude compared to the A1gmode [34, 40, 41] as a result of the insusceptibility ofits equilibrium position to photoexcitation, as schemat-ically illustrated in Fig. 1(c). The anisotropic opticalpolarization-dependence confirmed the generation of theEg phonon to be dominated by impulsive stimulated Ra-man scattering (ISRS) mechanism at low excitation den-sities [19]. With a sufficiently short light field and ashort-lived intermediate state (impulsive limit) the ionicdisplacement can be expressed by [30, 42, 43]:Qy(t) = Q0y exp(−Γyt) sin(2πνyt). (2)A density functional theory (DFT) study [13] predicteda strong coupling between the Eg and A1g modes underintense photoexcitation. The coupling was experimen-tally demonstrated as the emergence of a combinationmode at the difference frequency in a prior TR study[44]. The literature also reported coherent Eg amplitudeas a function of pump fluence based on its fast Fourier-transform (FFT) peak height obtained from the stan-dard (isotropic) TR detection. The FFT spectra also fea-tured much larger A1g peak, however; at high fluences theasymmetrically broadened A1g peak overlapped and ob-scured the Eg peak, which left considerable uncertaintyin the quantitative evaluation of the latter mode.In the present study we experimentally investigate co-herent Eg phonons in Bi under intense photoexcitationby employing an anisotropic detection scheme, in whichthe isotropic A1g contribution to the TR signals could inprinciple be cancelled. To maximize the Eg phonon sig-nal we choose a bulk single crystal Bi whose crystallineaxes are well specified and keep it at a low temperatureof 11 K [19, 44]. For comparison we also examine thefluence-dependences of the A1g mode and photoexcitedcarriers in the standard (isotropic) detection scheme. Wefind that the amplitudes of the two phonon modes exhibitstrikingly contrasted fluence-dependences, and interpretthe results in the context of dynamic coupling of the Egphonons with the A1g mode and with non-equilibriumphotocarriers. On the other hand, the initial phases ofthe two coherent phonons shift in parallel with increasingfluence, suggesting a fluence-dependent transition timefrom the ground state to the excited state.II. EXPERIMENTALThe sample studied is a 1-mm thick bulk single crystalBi with a (0001)-oriented polished surface in hexagonalnotation (or (111) in cubic notation), which was pur-chased from MaTeck and is used without further treat-ment. The crystal is mounted in a closed-cycle cryostatwith its [1120] axis in vertical direction and is kept at 11K.Single-color pump-probe reflectivity measurements areperformed on the Bi crystal with an output of a regener-ative amplifier with 120 fs duration, 810 nm center wave-length (1.53 eV photon energy), and 100 kHz repetitionrate as the light source. A f = 100-mm plano-convexlens focuses the linearly polarized pump and probe beamsto the ∼80 and 40µm spots on the sample with inci-dent angles of < 5◦ and 15◦ from the surface normal,respectively. Incident pump fluence is adjusted betweenFinc = 0.40 and 28mJ/cm2 by rotating a half-wave-platebefore a fixed plate-type polarizer located in front of thefocusing lens. Pump beam is modulated with an opticalchopper for lock-in detection.To examine the Eg phonons we measure anisotropictransient reflectivity, ∆Reo ≡ ∆RH −∆RV , by employ-ing the incident probe polarized at ∼ 45◦ from horizon-tal (H) and by detecting the H and vertical (V ) polar-ization components of the reflected probe light with apair of matched photodiode detectors. For comparisonwe also measure electronic and A1g responses in stan-dard (isotropic) TR scheme, in which the pump-inducedchange in reflectivity ∆R is measured by detecting theprobe light before and after reflection at the sample sur-face with a pair of matched photodetectors. In both de-tections configurations, the signal from the detector pairis amplified with a current pre-amplifier and a lock-inamplifier. Time delay t between the pump and probepulses is scanned step by step with a translational stage(slow scan).For a Bi crystal with moderate coherent ionic displace-ments Qi, photoexcited carrier density N , and latticetemperature Tl, the standard TR signal can be approxi-mately expressed by the sum of their linear combinations;∆R(t)R=1R[∑i(∂R∂Qi)Qi(t) +(∂R∂N)N(t)+(∂R∂Tl)∆Tl(t)], (3)with i = y and z denoting the displacements along the3FIG. 2. (a) Anisotropic TR signals of Bi(0001) surface obtained at 11 K with different pump fluences. Pump light ispolarized parallel to the [1120] axis. (b) Fast Fourier transform (FFT) spectra of the oscillatory part of (a). (c-g) Pump fluencedepehdences of the amplitude (c), dephasing rate (d), initial frequency (e), frequency chirp (f), and initial phase (g) of thecoherent Eg phonon obtained by fitting the reflectivity signals in (a) to Eq. (5). Dotted line in (c) indicates a linear fit in thelow fluence regime.Eg and A1g coordinates. In the anisotropic detectionscheme the last two terms of Eq. (3) as well as the A1gcontribution to the first term are to be cancelled, sincethey are isotropic within the surface plane. The schemetherefore enables us to monitor the Eg phonon responseexclusively:∆Reo(t)R=1R(∂R∂Qy)Qy(t). (4)For a crystal far from the equilibrium under extremely in-tense photoexcitation, however, Eqs. (3,4) may no longerbe adequate because of the contributions of the higher-order terms and/or because the crystal may be approach-ing photo-induced phase transition [45, 46].III. RESULTSA. Coherent Eg PhononWe first examine the Eg phonon dynamics in theanisotropic detection scheme. Figure 2(a) shows theanisotropic TR signals ∆Reo at selected incident fluencesFinc. Here the pump polarization is set to be parallel tothe [1120] crystalline axis to maximize the Eg phononcontribution, and the probe is polarized at ∼ 45◦ to it forthe anisotropic detection [19]. At the minimum fluenceof Finc = 0.40mJ/cm2 the TR signal shows a negativespike at t = 0, followed by a periodic modulation predom-inantly due to the Eg phonon at ∼2THz. Fast Fouriertransform (FFT) spectrum in Fig. 2(b) shows a smallA1g peak at 3THz as well, however, due to the imperfectoptical polarization in the experiments. As the fluenceis increased, the Eg peak is broadened and redshifted, inagreement with the prior TR studies [40, 44]. Meanwhilethe A1g peak grows with fluence and eventually becomeshigher than the Eg peak. At Finc = 16mJ/cm2 the base-line of the signal starts to exhibit large fluctuation duringthe slow scan of the time delay, and the as shown in Fig.S1(a) in Supplemental Materials (SM) [30]. We checkthe reversibility of the fluence-dependence by measuringthe same spot again at a low fluence, whose results aresummarized in Fig. S1(b,c) in SM [30]. We find that thesignal after exposure to the maximum fluence is as noise-less as that from a fresh spot. The Eg amplitude beforeand after the exposure is comparable, and so is the de-phasing rate (the linewidth). These results indicate thatno significant irreversible damage such as melting was4induced by the exposure.To quantitatively analyze the fluence-dependence ofthe coherent Eg phonons we fit the oscillatory compo-nent of the time-domain TR signals to the sum of twodamped harmonic functions:∆RoscR= Ae exp(−Γet) sin[2πνet+ ϕe]+Aa exp(−Γat) sin[2πνat+ ϕa], (5)where the subscripts e and a denote the Eg and A1gmodes. For simplicity we assume linear chirps in thefrequencies:νi = ν0i + βit, (6)with i denoting e or a. This function can give excellentfits to the TR signals for t > 0.3 ps, as demonstratedin Fig. S3 in SM [30], whereas for t < 0.3 ps the fittingis somewhat poorer because of the large negative spikeoverlapping around t = 0. Figure 2 (c-g) summarizesthe Eg phonon parameters obtained by the fitting as afunction of incident pump fluence. The initial frequencyν0e redshifts from 2.1 to 1.8THz and the linear chirpβe increases from < 10−4 to 0.06THz/ps, whereas thedephasing rate Γe increases from 0.08 to 0.8 ps−1, withincreasing fluence from 0.40 to 12mJ/cm2.Surprisingly, the Eg amplitude is found to increase al-most linearly only up to Finc ≃ 8mJ/cm2 and then toturn to an apparent decrease [Fig. 2(c)]. The decreaseis not due to an irreversible photoinduced damage, sincethe fluence-dependence is reversible, as we have alreadyseen in Fig. S1 in SM [30]. We also confirm that the be-havior is independent of analysis method; the area underthe Eg peak in the FFT spectra similarly increases andthen decreases with fluence, as shown in Fig. S4 in SM[30]. We note that a previous TR study [44] reported afluence-dependence of the Eg amplitude that appearedto be contradict the present result. This is because theliterature estimated the Eg amplitude as the FFT peakheight from the standard (isotropic) TR signals. In thiscase it is likely that the Eg peak height was increasedconsiderably by the overlapping A1g peak and that thetwo peaks were hardly separable, as we have also checkedourselves in standard TR experiments shown in Fig. S5of SM [30].The initial phase of the Eg phonon [Fig. 2(g)] is foundto be close to a sine function of time (ϕe = 0), as is ex-pected for an excitation in the impulsive limit [Eq. (2)]with a δ-function-like driving force. With increasing flu-ence the phase first steeply decreases to ϕe ∼ −π/4 andthen recovers partially to ϕe ∼ −π/6. The deviation ofthe initial phase from zero indicates that the duration ofthe driving force can no longer be negligible comparedto the phonon period. We note that in the low fluenceregime (Finc < 6mJ/cm2), where we observe most promi-nent phase shift the fitting results are excellent [Fig. S3in SM [30]], which ensures the reliable determination ofthe phase. In the high fluence regime (Finc > 6mJ/cm2)FIG. 3. (a,b) Standard TR signals of Bi (solid curves) ob-tained at 11 K at different pump fluences. Broken curvesrepresent the non-oscillatory response obtained by fittingto Eq. (7). (c) Height of the non-oscillatory response att = 0.2 ps (AC , plotted to the left axis) and the baseline ata long time delay (Ath , to the right axis) as a function ofpump fluence. Dotted curve represents the extrapolation ofthe fitting to a power function (AC ∝ Fninc) in the low fluenceregime (Finc < 6mJ/cm2).the fit leaves somewhat larger uncertainty, however, dueto the faster dephasing of the Eg oscillation as well asthe larger contributions from the negative spike at t = 0.The poorer fitting may be the cause for the larger scat-tering in the initial phase in the high fluence regime. Wealso perform similar anisotropic TR measurements usinga longer pump pulse, whose results are summarized inFig. S6 in SM [30]. We find ϕe to exhibit a qualitativelysimilar, if less pronounced, shift with fluence for a longerpulse duration, confirming that the phase shift is no ar-tifact of the experiments.B. Electronic and Thermal ResponsesTo examine the origin of the unconventional fluence-dependence of the Eg phonon we also examine the carrierand A1g phonon dynamics of the same Bi crystal sam-ple excited under the same condition in the standard TRdetection scheme. The pump and probe light polariza-tions are set at 45◦ and 0◦ to the [1120] axis, respectively,to minimize the Eg contribution to the signals [19]. The5standard TR signals, shown in Fig. 3(a), feature a sizablenon-oscillatory response of photoexcited carriers in addi-tion to the oscillatory coherent phonon response. Theformer can be fitted to a multi-exponential function ontop of a baseline,∆RnonR= ΣjAj exp(−t/τj) +Ath, (7)whose results are indicated with broken curves inFig. 3(a,b).At the lowest fluence examined (Finc = 0.40mJ/cm2)the non-oscillatory component can be fitted reasonablyto the sum of an exponential function for the rise and twofor the decay. The obtained rise time, τrise = 0.6 ps, is inagreement with that reported for the intervalley electron-phonon scatterings [47, 48]. The decay time constants areobtained to be τfast = 0.85 ps and τslow =13ps; the latteris a few times slower than that of the electron-hole recom-bination reported in previous two-photon photoemissionstudies [47, 49], possibly because of the thicker Bi crys-tal and the lower temperature employed in the presentstudy. At long time delays (t > 30 ps) the non-oscillatorycomponent approaches a negative baseline Ath, as shownin Fig. 3(b). The value of Ath can be used to estimate thelattice temperature rise ∆Tl K at the long time delays byadopting the temperature-dependence of the reflectivity[50]:∂(∆R/R)/∂T = −8× 10−5K−1. (8)We would obtain ∆Tl < 1K at the minimum fluence.With increasing fluence the initial rise of the TR signalbecomes faster in time, plausibly due to the larger con-tribution from the intraband scatterings among highlyexcited electrons and holes. At Finc > 4mJ/cm2 therise is complete before the first maximum of the coher-ent oscillation (τrise ≪ 0.2 ps) and can no longer be fit-ted uniquely to an exponential function. We thereforefit only the decay (t > 0.1 ps) to two exponentials. Inthe following analyses we assume that the electrons andholes come to follow the Fermi-Dirac distribution beforethe first maximum of the oscillation at t ≃ 0.2 ps, andregard the non-oscillatory amplitude:AC ≡ ∆Rnon(t = 0.2 ps)R(9)as a semi-quantitative measure for photoexcitated carrierdensity N , though the assumption of the ultrafast ther-malization may not hold at extremely high densities. Wefind that AC grows first superlinearly (AC ∝ Fninc withn = 1.5) up to Finc ≃ 6mJ/cm2 (dotted curve in thefigure), then turns to a linear increase until AC reaches asaturation at ∼ 20mJ/cm2, as shown with filled symbolsin Fig. 3(d). Further increase in the fluence eventuallyleads to an emergence of large noise in the TR signal atFinc = 28mJ/cm2, as shown in Fig. S2 in SM [30]. Whenthe fluence is lowered after the exposure to the maximumfluence, we obtain a noiseless TR signal with distinct co-herent phonon oscillation again, which excludes photoin-duced melting and amorphization of the crystal by theexposure.The saturation of AC at Finc ≥ 20mJ/cm2 may be in-terpreted as the saturation of linear optical absorption,which is mainly responsible to the early time response.To assess this interpretation we estimate the carrier den-sity as follows. At t = 0 the photoexcited carrier densityN should have a depth distribution described by an ex-ponential function of the distance from the surface z:N(t = 0, z) = N0 exp(−αz). A recent TR study onthin (≪ 1µm) Bi films on Si substrates reported thatthe depth distribution becomes homogenized within 150fs and its density at the surface reduces to 1/3 - 1/5 de-pending on the film thickness [26]. Such ultrafast homog-enization of photocarriers would be less likely in our bulkcrystal, and for simplicity we assume the incident pumplight with photon energy hω=1.53 eV to be absorbeduniformly and completely within the optical penetrationdepth 1/α = 14.7 nm:N(t ≃ 0) ={(1−R)Fincα/(hω) for 0 < z < 1/α,0 for z > 1/α,(10)which would give an upper limit of the carrier densityat the surface. As for the reflectivity at 11K we ten-tatively estimate R = 0.95 by extrapolating the reflec-tivity reported for higher temperatures [36, 51], as dis-cussed in Fig. S7 in SM [30]. The absorbed fluenceFabs ≡ (1 − R)Finc obtained by using this reflectivityvalue is plotted on the top axes of Figs. 2(c-g), 3(c) and4(b-f). This choice is justified by the quantitative agree-ment of the fluence-dependence of the A1g frequency,shown in Fig. S9 in SM [30], with that reported fora 197-nm thick Bi film at room temperature [26]. AtFinc = 20mJ/cm2, at which AC reaches a saturation, wewould obtain N = 2.8 × 1021 cm−3. This density wouldcorrespond to 2% of the valence electrons of Bi:Nv = 5NAw/M = 1.4× 1023cm−3, (11)with 5 being the number of valence electrons per Bi atom,NA, the Avogadro constant, w = 9.747 g/cm3, the den-sity of Bi, and M = 209, the atomic number. It wouldbe reasonable to expect the linear absorption to be sat-urated at such a high carrier density.At long time delays (t ≫ 10 ps) we obtain the baselineAth by fitting the TR signals to Eq. (7) and estimatethe equilibrium lattice temperature using Eq. (8). Athincreases almost linearly up to Finc = 20mJ/cm2 andthen turns to a steeper increase with fluence, as plottedwith open symbols in Fig. 3(c). At the maximum fluenceexamined, the lattice temperature rise is estimated to be∆Tl ≃ 35K.6FIG. 4. (a) FFT spectra of the oscillatory part of TR sig-nals shown in Fig. 3(a). (b-f) Fluence-dependences of theamplitude (b), dephasing rate (c), initial frequency (d), fre-quency chirp (e), and Initial phase (f) of the coherent A1gphonon, obtained by fitting the oscillatory reflectivity signalsto the second term of Eq. (4). Dotted line in (b) representsthe extrapolation of a linear fitting in the low fluence regime(Finc < 3mJ/cm2).C. Coherent A1g PhononThe A1g phonon response can be extracted from theTR signals in Fig. 3(a) by subtracting the non-oscillatorycomponent (broken curves). The FFT spectra shown inFig. 4(a) features only the A1g peak at 3THz, confirmingthat the Eg contribution is minimized with the selectedpump and probe light polarizations. We accordingly fitthe oscillatory oscillatory signals to only the second termof Eq. (5), either over the entire time range or over thefirst three cycles of the oscillations as was done in someof the previous studies. The results of the two fittings arecompared in Figs. S8 and S9 of SM [30]. Although neitherreproduces the experimental oscillations perfectly at allthe pump fluences, the A1g phonon parameters obtainedfrom the two fittings are in reasonable agreement. Wetherefore discuss on only the phonon parameters obtainedfrom the fitting over the entire time window, which aresummarized in Fig. 4(b-f).The A1g amplitude increases linearly in the low fluenceregime (Finc < 3mJ/cm2), as indicated with a dottedline in Fig. 4(a), but grows superlinearly (Aa ∝ F 1.5inc ) inthe intermediate fluence regime (Finc = 3− 15mJ/cm2).Further increase in the fluence leads to a saturation of theamplitude in the high fluence regime (Finc > 20mJ/cm2)and eventually to an emergence of large noise in the TRsignal during the time delay scan, as we have already seenin Fig. S2(a) in SM [30]. When the fluence is loweredafter the exposure to the maximum fluence, we obtaincoherent A1g oscillation whose initial amplitude is almostas large as that before the exposure, but the dephasingis much faster, as shown in Fig. S2(b,c). The comparisonindicates a small irreversible change in the crystal, suchas a slight damage at the surface, but no sign of completemelting or other phase transition.The observed fluence-dependence of the A1g amplitudeis in quantitative agreement with a prior TR study per-formed in a similar condition (on a 1-mm thick Bi sin-gle crystal at 5K) [52]. The behavior is also in rough,though not perfect, agreement with that of the electronicresponse AC [filled symbols in Fig. 3(c)], suggesting thatthe saturation of A1g amplitude has the same origin asthat ofAC . On the other hand, it is in striking contrast tothe fluence-dependences of the Eg amplitude [Fig. 2(c)],which turns to decrease already at Finc ≃ 10mJ/cm2while the A1g amplitude is still growing superlinearly.The contrast excludes the possibility that the Eg am-plitude decreases because the entire crystalline lattice isbecoming unstable, by approaching the high-symmetryphase for example.The initial phase ϕa is close to −π/2 at the mini-mum fluence of Finc = 0.4mJ/cm2, as expected for co-herent phonons excited in the displacive limit (Eq. (1))with a Heaviside-step-function-like driving force. Withincreasing fluence the initial phase first shifts steeplydown to ϕa ≃ −π and then gradually recovers towardϕa = −2π/3. We note that fitting results are excellentthroughout the entire fluence range examined, as shownin Fig. S8 of SM [30], ensuring the reliable determina-tion of the initial phase. We also observe a consistentphase shift with fluence in the standard TR measure-ments using a longer pump pulse, as shown inFig. S6(b)in SM [30]. On one hand, in the DECP model the initialphase can deviate from ±π if the driving force is not aHeaviside-step-function but has a finite decay time [15].On the other hand, we notice that the initial phase of theEg phonon [Fig. 2(g)] also shifts with fluence, almost inparallel to that of the A1g. The parallel trends of the twomodes could be better explained by a finite rise time inthe driving forces, which could affect both the Heaviside-step-function-like and δ-function-like driving forces forthe A1g and Eg modes.The rise of carrier-phonon coupling after photoexci-tation may be visualized by performing time-windowedFourier transform on the whole TR signals contain-ing both the electronic and phononic responses [colored7FIG. 5. False-color plots of time-windowed FFT amplitude as a function of time window position and frequency obtainedat different fluences. A Gaussian time window of 330-fs half width at half maximum is used. Solid curves represent the peakpositions within the frequency range of 2 − 3.5 THz. Broken lines reproduce the linearly chirped A1g frequency νa = νa0 + βatreproduced with parameters plotted in Fig. 4(d,e). Chained lines represent the intrinsic A1g frequency.curves in Fig. 3(a)]. The results obtained with a Gaus-sian time window of half width ∆t = 0.33 ps, whichallows us the best balance between the temporal andfrequency resolutions (∆ν ≃ 0.5THz), are presented asfalse-color plots in Fig. 5. At the lowest fluence [Fig. 5(a)]the phononic response at ∼3THz and the electronic re-sponse at ≲ 1THz are well separated from each other;they both emerge within the time window width used inthe analysis. With increasing fluence the electronic re-sponse acquires a higher-frequency component at earlytime delays, which interconnects it with the phononic re-sponse in the first fraction of ps. At later time delays thecoupled electron-phonon response [solid curves in Fig. 5]approaches the intrinsic A1g phonon frequency [chainedcurves in the same figure]. The time scale for this tran-sient blueshift is comparable to the time window width atthe minimum fluence (Finc = 0.4mJ/cm2) but becomesas slow as ≳1 ps at the maximum fluence. This observa-tion may be interpreted in terms of the fluence-dependenttransition time from the ground-state PES to the excited-state PES, as was predicted for the A1 phonon of tel-lurium, which is also associate with the Peierls stability,by recent time-dependent DFT simulations [53]. We notethat there is a notable discrepancy between the tran-sient blue shift obtained from the time-windowed anal-yses [solid lines in Fig. 5] and that estimated from thetime-domain fitting to Eq. (5) [dotted lines]. The discrep-ancy can be regarded as the manifest of the excited-statePES still shifting and deforming towards a new equilib-rium. The two frequencies eventually coincide after thesystem reaches the equilibrium in the low to medium flu-ence regime. In high fluence regime (Finc > 15mJ/cm2),however, the two frequencies no longer coincide on rea-sonable time scale, implying the failure of assuming alinear frequency chirp in this regime.IV. DISCUSSIONWe now discuss the origin of the unconventionalfluence-dependence of the Eg phonon amplitude [Fig. 2c],which is in stark contrast to the behavior of the A1g am-plitude [Fig. 4b].A prior DFT study [13] calculated excited-state PESsas functions of Eg and A1g coordinates. Figure 6(a,b)reconstructs partially the reported two-dimensional (2D)PESs around the ground-state equilibrium (z = Qminz =0.234 and y = 0) for two selected densities of photoex-cited electron-hole pairs Ne−h. The 2D PESs over theentire calculated displacements, showing the double-wellpotential along the A1g coordinate, are reconstructed inFig. S10(a) in SM [30] for all the four densities reportedin the literature. Upon photoexcitation the equilibriumalong the z coordinate shift from Qminz to QEz (Ne−h).This gives a driving force to Bi ions to oscillate betweenthe ground-state equilibrium Qminz and the maximumQmaxz = 2QEz −Qminz via the DECP mechanism. In the ycoordinate the ions oscillate around y = 0 via ISRS mech-anism, whose amplitude is not obtained from the calcu-lations but was estimated to be an-order-of-magnitudesmaller than the A1g by a previous trXRD study[34].At a low excitation density, where the displacementsare small and the PES can be approximated to be har-monic, one could assume that the Eg and A1g oscilla-tions are independent of each other. Even in the case ofNe−h = 0.5% of valence electrons, however, the PES slicealong the y coordinate, shown in Fig. 6(c), is not perfectlyindependent of the A1g displacement but receives a smalldisturbance as a function of z. With increasing Ne−h themaximum Qmaxz , indicated with a solid line in Fig. 6(a,b),approaches toward the central barrier at z = 0.25, whilethe barrier becomes lower. This introduces a significantdeformation in the PES slice along the y coordinate as zvaries from Qminz to Qmaxz , i.e., within a half cycle of theA1g oscillation. At Ne−h = 1.42%, the PES along the8FIG. 6. (a,b) Excited state PESs at photoexcitation of 0.5%(a) and 1.42% (b) of valence electrons, as reproduced by usingthe parameters obtained by DFT simulations in Ref. 13. yand z represents the Eg and A1g coordinates in unit of thehexagonal lattice constant a and in the form of the internaldisplacement parameter u, respectively. Crosses (X) representthe ground-state minimum at z = Qminz = 0.234 and y = 0.Broken and solid lines denote the equilibrium on the excitedstate QEz and the maximum displacement Qmaxz in the A1gcoordinate. (c,d) Slices of the two-dimensional PES along theEg coordinate at different values of u at photoexcitation of0.5% (c) and 1.42% (d) of valence electrons.y coordinate suffers so significant deformation that thecurvature of the PES slice becomes negative when theion reaches Qmaxz , as shown in Fig. 6(d). We infer thatthis deformation of the PES would lead to a quick loss ofthe vibrational coherence of the Eg mode within a singlecycle of the A1g oscillation, and thereby to an effectivesuppression of coherent Eg phonons at high fluences asobserved in Fig. 2.Further increase in the excitation density above 2%would transform the double-well PES in the z coordinateto a single well, as illustrated in Fig. 1(c), according toanother DFT simulations [12]. A single-shot TR study[38] reported a gradual decrease in the A1g oscillationamplitude under intense photoexcitation and completedisappearance at Finc > 10mJ/cm2 (Fabs > 3mJ/cm2)for a 275-nm thick Bi film at room temperature. The dis-appearance was interpreted as a result of photoinducedtransition to the theoretically predicted high-symmetryphase. A trXRD study on a 50-nm thick Bi film at roomtemperature reported a similar disappearance of the os-cillation at Fabs > 3mJ/cm2 [54]. In the present study,which we believe stayed below the reported threshold flu-ence, we did not observe such complete disappearance ofthe A1g oscillation. Instead we observed large noise inthe TR signal starting to appear during the scan of timedelay [Figs. S1 and S2 in SM [30]], which is indicative ofpartial damage of the crystal surface as a result of contin-uous heating during the repetitive excitation at 100 kHzat a high fluence. We speculate that the lattice coolingat the surface of our 1-mm thick Bi crystal could be lessefficient than in a sub-µm-thick film on a substrate dueto the low thermal conductivity of Bi [55] than that ofthe substrate.V. CONCLUSIONUltrafast dynamics of coherent Eg phonons of bulk sin-gle crystal Bi was investigated under intense photoexcita-tion at low temperature. With increasing pump fluencethe Eg amplitude reached its maximum and turned toa decrease at a significantly lower fluence than the A1gamplitude became saturated. 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