Detecting electron-phonon couplings during photo-induced phase transition
Takeshi Suzuki, Yasushi Shinohara, Yangfan Lu, Mari Watanabe, Jiadi Xu, Kenichi L. Ishikawa, Hide Takagi, Minoru Nohara, Naoyuki Katayama, Hiroshi Sawa, Masami Fujisawa, Teruto Kanai, Jiro Itatani, Takashi Mizokawa, Shik Shin, Kozo Okazaki
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Detecting electron-phonon couplings during photo-induced phase transition
Takeshi Suzuki , ∗ , Yasushi Shinohara , , Yangfan Lu , Mari Watanabe , Jiadi Xu , Kenichi L.Ishikawa , , , Hide Takagi , , Minoru Nohara , Naoyuki Katayama , Hiroshi Sawa , Masami Fujisawa ,Teruto Kanai , Jiro Itatani , Takashi Mizokawa , Shik Shin , , ∗ , and Kozo Okazaki , , , ∗ Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan Photon Science Center, Graduate School of Engineering,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Department of Nuclear Engineering and Management,Graduate School of Engineering, The University of Tokyo,7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Department of Physics, University of Tokyo, Hongo, Tokyo, 113-0033, Japan Research Institute for Photon Science and Laser Technology,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany Research Institute for Interdisciplinary Science,Okayama University, Okayama, 700-8530, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan School of Advanced Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan Office of University Professor, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan Material Innovation Research Center, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan (Dated: August 13, 2020)Photo-induced phase transitions have been intensively studied owing to the ability to control amaterial of interest in the ultrafast manner, which can induce exotic phases unable to be attainedat equilibrium. However, the key mechanisms are still under debate, and it has currently been acentral issue how the couplings between the electron, lattice, and spin degrees of freedom are evolvingduring photo-induced phase transitions. Here, we develop a new analysis method, frequency-domainangle-resolved photoemission spectroscopy, to gain precise insight into electron-phonon couplingsduring photo-induced insulator-to-metal transitions for Ta NiSe . We demonstrate that multiplecoherent phonons generated by displacive excitations show band-selective coupling to the electrons.Furthermore, we find that the lattice modulation corresponding to the 2 THz phonon mode, whereTa lattice is sheared along the a-axis, is the most relevant for the photo-induced semimetallic state. Strongly-correlated electron systems display very richphases owing to intertwined couplings between multipledegrees of freedom including the charge, orbital, spin,and lattice [1]. Moreover, external fields, such as elec-tronic and magnetic fields or physical pressure, can in-duce phase transitions in these systems by breaking theirsubtle balances between multiple competing phases [2, 3].In this respect, photo-excitation is a very promising wayto control the physical properties because it can instanta-neously change physical properties of a targeting materialin various manners by exploiting many degrees of free-dom such as polarization or wavelength [4]. For study-ing photo-excited nonequilibrium states, time- and angle-resolved photoemission spectroscopy (TARPES) has astrong advantage because it can track nonequilibriumelectronic band structures after photoexcitations [5–8].For photo-induced phase transitions, although manystrongly-correlated electron systems have been inten-sively studied [9–13], the precise mechanisms are stillunder debate [14–16]. Recently, we revealed thephoto-induced insulator-to-metal transitions (IMTs) inTa NiSe [17], where we also showed strong evidence indynamical behaviors as an excitonic insulator. Moreover,other interesting photo-excited phenomena in Ta NiSe have been reported previously, which include photo-induced enhancement of the excitonic insulator [18, 19]or emergence of collective modes [20, 21]. In most re-ports, the key roles for such phenomena are played bysignificant electron-phonon couplings.In this Letter, we report an analysis method extendedfrom TARPES, namely, frequency-domain angle-resolvedphotoemission spectroscopy (FDARPES), to reveal howelectron-phonon couplings play roles in the photo-excitednonequilibrium states for Ta NiSe . We observe that thelattice modulation corresponding to the phonon mode,where Ta lattice is sheared along the a-axis, is the mostrelevant for the photo-induced semimetallic state. Fur-thermore, this method can be generally applicable to de-tect the temporal modulations of photoemission intensityinduced by any coupling beyond coherent phonons.TARPES, as illustrated in Fig. 1(a), allows us to di-rectly observe the temporal evolution of the electronicband structure. We used an extremely stable commer-cial Ti:sapphire regenerative amplifier system (Spectra-Physics, Solstice Ace) with a center wavelength of 800nm and pulse width of ∼
35 fs for the pump pulse. Sec-ond harmonic pulses generated in a 0.2-mm-thick crystalof β -BaB2O4 were focused into a static gas cell filled e - k ( (cid:973) (cid:16)(cid:20) ) E - E F ( e V ) D t = -500 fs D t = 150 fs D t = 1000 fs D t = 2000 fs k ( (cid:973) (cid:16)(cid:20) ) E - E F ( e V ) MaxMin Inc.Dec. (b)(a) (f) abc Ta Ni Se
XUV IR (c) (d) (e)(g) (h)
Analyzer
FIG. 1. (a) Schematic illustration of time- and angle-resolved photoemission spectroscopy (TARPES), as applied to Ta NiSe .The pump pulse is infrared light whereas the probe pulse is extreme ultraviolet light produced by high-harmonic generation.Photoelectrons are detected by a hemisphere analyzer. (b)-(e) TARPES spectra of Ta NiSe . The delay time between thepump and probe is indicated in each panel. (f)-(h) Difference images of TARPES. Red and blue points represent increasingand decreasing photoemission intensity, respectively. with Ar to generate higher harmonics. By using a setof SiC/Mg multilayer mirrors, we selected the seventhharmonic of the second harmonic ( hν = 21.7 eV) for theprobe pulse. The temporal resolution was determinedto be ∼
70 fs from the TARPES intensity far above theFermi level, which corresponded to the cross correlationbetween the pump and probe pulses. The hemisphericalelectron analyzer (Omicron-Scienta R4000) is used to de-tect photoelectrons. All the measurements in this workwere performed at the temperature of 100 K.The sample is a high-quality single crystals ofTa Ni(Se . S . ) grown by chemical vapour trans-port method using I as transport agent, as was re-ported in the refs [22, 23]. Whereas a relatively largecleaved surface is necessary for TARPES measurementscompared with static ARPES, because Ta NiSe has aone-dimensional crystal structure, a large cleaved sur-face of the pristine Ta NiSe that was sufficient forTARPES measurements was difficult to obtain. However,sufficiently-large cleaved surface of 3 % S-substitutedTa NiSe could be obtained. This is why we used 3 %S-substituted Ta NiSe rather than pristine Ta NiSe inthis study. Clean surfaces were obtained by cleaving insitu.Figures 1(b)-1(e) show the TARPES snapshots ofTa NiSe at various delays shown as a function of mo-mentum and energy. The pump fluence is set to be 2.27mJ/cm . The delay between the pump and probe is indi-cated in each panel. To enhance the temporal variations,the difference images between the before and after pho-toexcitation are shown in Figs. 1(f)-1(h), where red andblue represent an increase and decrease in photoemis- sion intensity, respectively. After strong photoexcitation,new semimetallic electron- and hole-dispersions appear,as has previously been reported [17]. This is the directsignature of photo-induced IMT. To highlight the changeof the electronic band structure, we show the peak posi-tions of the TARPES spectra before and after photoex-citation in Figs. 2(a) and 2(b). One can notice that thehole band is shifted upward and crosses the Fermi level, E F , while the electron band appears and crosses E F atthe same Fermi wavevector, k F , as the hole band.To more specifically reveal the photo-induced profilein Ta NiSe , we investigated the TARPES images interms of electron-phonon couplings. Figure 2(c) showsthe time-dependent intensities for representative regionsin the energy and momentum space indicated as I-IV inFigs. 2(a) and 2(b). As a background, carrier dynamicscorresponding to overall rise-and-decay or decay-and-risebehaviors were observed. Additionally, oscillatory behav-iors were clearly seen superimposed onto background car-rier dynamics, which indicated strong electron-phononcouplings as a result of excitations of coherent phonons.To extract the oscillatory components, we first fit thecarrier dynamics to a double-exponential function con-voluted with a Gaussian function, shown as the black-solid lines in Fig. 2(c), and then subtracted the fit-ting curves from the data. Fourier transformations wereperformed for the subtracted data and the intensitiesfor each frequency component are shown in Fig. 2(d).One can clearly see that distinctively different frequency-dependent peak structures appeared depending on theregions in the energy and momentum space denoted asI-IV in Figs. 2(a) and 2(b).Considering that the frequencies for the peak positionsobserved in this work matched the A g phonon modes re-ported in previous work [24, 25] and that we dominantlyexcited the electron system under our experimental con-dition, the observed coherent phonons were likely to arisefrom displacive excitation of coherent phonons (DECPs)[26]. According to the DECP theory, photoexcitationsuddenly changes the minimum energy position in thelattice coordinates of the potential energy surface (PES),and lattice coordinates oscillate around the new energyminimum position with its own frequency determinedby the curvature of the PES. Figure 3(a) schematicallyshows this situation. The ground and excited states aredenoted as | G i and | E i , respectively, while Q and Q are the lattice coordinates corresponding to the k ( (cid:973) (cid:16)(cid:20) ) E - E F ( e V ) (a) k ( (cid:973) (cid:16)(cid:20) ) (b)(a) (b) +- I II IIIIV I II IIIIV (c) (d) I n t en s i t y ( C oun t s ) -4-2010501.00.50.0 3.02.01.00.0 Delay Time (ps) P o w e r S pe c t r a Frequency (THz)
IIIIIIIV IIIIIIIV
FIG. 2. TARPES spectra (a)before and (b)after pump exci-tation. The peak positions in the TARPES spectra are indi-cated as circles. (c) Time-dependent TARPES intensities atdifferent energy and momentum regions. I-IV corresponds tothe regions indicated in (a) and (b). Data are shown as redcircles whereas the fitting results by double-exponential decayfunctions convoluted with a Gaussian function are shown asblack solid lines. (d) Amplitude of Fourier transforms of theoscillation components in (c)I-(c)IV obtained by subtractingthe fitting curve from the data. Q Q Energy|G>|E> (a) (b)(c) -1.00.01.0 D a t a - f i t ( C oun t s ) Delay Time (ps) -0.20.00.2
FIG. 3. (a) Schematic illustration of the free energy curvesfor the ground and photo-excited states as a function of latticecoordinates corresponding to the directions of the 2-THz and3-THz phonon modes. (b), (c) Oscillation components for Iand II corresponding to the data subtracted by the fits. Thefitting curve using a single cosine function are shown as dottedand solid black curves. We use data for the fits shown by solidblack curves. ± π and 2.97 ± ± π and 2.07 ± π .Thus, the modulation of photoemission intensity in re-gion I along the 3-THz phonon mode was triggered im-mediately after photoexcitation. On the other hand, therelatively positive phase shift in region II compared withregion I indicated the modulation of the photoemissionintensity along the 2-THz phonon mode occurred with adelay of 120 fs.We will now discuss the electron-phonon couplings inmore detail. Since we observed that the amplitude ofeach oscillation significantly changed depending on theregions in the energy and momentum space, we furthermapped out the frequency-dependent intensity of theFourier component in the energy and momentum space,which we call FDARPES. Figures 4(a)-4(e) show theFDARPES spectra corresponding to the frequencies of1, 2, 3, 3.75, and 4 THz, respectively. The correspondingphonon modes were calculated by ab initio calculationand their modes are shown in Figs. 4(f)-4(j). The de-tails of calculations are found in Supplemental Material[27]. We assigned all the phonon modes as A g modesin the monoclinic phase. In the previous results of Ra-man measurements at different polarization settings [21], E - E F ( e V ) k ( (cid:973) (cid:16)(cid:20) )1 THz 2 THz 3 THz 3.75 THz 4 THz MaxMin k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) (e)(d)(c)(b)(a)(f) (g)(h) (i) bca TaNiSe b c a (h) (j)
FIG. 4. (a)-(e) Frequency-domain angle-resolved photoemission spectroscopy (FDARPES) spectra shown as frequency-dependent intensities of the oscillation components as a function of energy and momentum. The peak positions in the TARPESspectra after and before photoexcitation are plotted as blue and green circles in Fig. 4(a) and Fig. 4(c), respectively. (f)-(j)Calculated phonon modes corresponding to (a)-(e). eleven Raman-active phonon modes are observed in theY(ZZ)Y setting while the three of them show strongerpeak intensities in the Y(ZX)Y than in the Y(ZZ)Y. Thisis considered to be due to the fact that all of the observedmodes have A g character in the monoclinic phase andthree of them turn to be B g character in the orthorhom-bic phase. Precise procedures of phonon assignments arefound in Supplemental Material [27].Noticeably, the FDARPES spectra exhibit distinc-tively different behaviors depending on the frequency,which demonstrates that each phonon mode is selectivelycoupled to the specific electronic bands. Particularly, the2-THz phonon mode has the strongest signal around E F , where it consists of a mixture of Ta 5 d and Se 3 p orbitals[41], and this signature is responsible for the collapse ofthe excitonic insulator. Because the IMT of Ta NiSe occurs by melting the excitonic insulating flat band, thisstrongest signal suggests that the photo-induced metallicstates drive the lattice distortions corresponding to the2-THz phonon motion.In order to see spectral features of FDARPES inmore detail, we compare the FDARPES spectra withthe band dispersions before and after photoexcitation.Full results are found in Supplemental Material [27].We find that FDARPES spectrum at 1 THz matchesthe band dispersions after photoexcitation better thanthat before photoexcitation shown in Fig. 4(a) while theFDARPE spectrum at 3 THz is closer to the band dis-persions before photoexcitation shown in Fig. 4(c). Re-cent theoretical investigation reported that the intensityof FDARPES spectra reflects the strength of electron-phonon coupling matrix elements [42]. Our results sug-gest the strong electron-phonon couplings for 1- and 3-THz phonon modes are associated with semimetalic andsemiconducting bands, respectively.As clearly seen in this work, our developed analy-sis method, FDARPES, can offer new routes to inves-tigate the electron-phonon couplings through the non-equilibrium states. Our work using this method pro-vides direct evidence for the DECP mechanisms respon-sible for the photo-induced IMTs in Ta NiSe . Thus,FDARPES can be used to study many other photo-induced phase transitions by observing how the electronband structure is influenced by the specific phonon-mode.We also emphasize the versatility of FDARPES. By us-ing the multiple degrees of freedom in the excitationpulses, we can drive different quasiparticles; for example,circularly-polarized pulses can promote a specific spinpopulation or appropriate mid- and far-infrared wave-length can resonantly excite IR-active phonons. Further-more, FDARPES can detect couplings of electrons toany quasiparticles or collective modes as long as theircouplings manifest as oscillations of intensities in theTARPES spectra.We would like to acknowledge Y. Ohta, for valuablediscussions and comments. This work was supportedby Grants-in-Aid for Scientific Research (KAKENHI)(Grant No. JP18K13498, JP19H00659, JP19H01818,JP19H00651 JP18K14145, and JP19H02623) from theJapan Society for the Promotion of Science (JSPS), byJSPS KAKENHI on Innovative Areas Quantum LiquidCrystals (Grant No. JP19H05826), by the Center of In-novation Program from the Japan Science and Technol-ogy Agency, JST, the Research and Education Consor-tium for Innovation of Advanced Integrated Science byJST, and by MEXT Quantum Leap Flagship Program(MEXT Q-LEAP) (Grant No. JPMXS0118067246, JP-MXS0118068681), Japan. 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Most coherent phonons excited by displacive excitation of coherent phonon (DECP) mechanisms [S1] show asimilar phonon frequency to ground state frequency evaluated with Raman scattering measurements [S2-S5]. Thefrequency-domain ARPES (FDARPES) spectra shown in Figs. 4(a)-4(e) are attributed to the Fourier transformof ARPES modulation owing to the coherent phonon excited by the pump pulse. To identify the phonon modes,we extracted electronic structure modulation due to phonon modes from theoretical simulations. We employed anab-initio theoretical framework based on density-functional theory (DFT) [S6, S7]. We evaluated the phonon modesand take all-symmetric modes, A g irreducible representation, from the framework. Details of theoretical simulation
Our theoretical simulation relied on the PerdewBurke-Ernzerhof (PBE) functional [S8] within density-functionaltheory (DFT). Whole calculations were performed by the ABINIT code [S9]. We used the fhi98PP pseudopotential[S10] for abinit [S11] with plane-wave basis set whose energy cut-off was chosen as 50 Hartree, 1361 eV. To enforcethe proper crystal spatial symmetry, C2/c (No. 15), we had to use the Bravais lattice as the simulation cell ratherthan the primitive cell. The supercell contains two primitive cells. We employ 24 × × Atomic position optimization
First, we performed atomic position optimization based on PBE-DFT with the same cell shape as an experimentalvalue, a = 3.496 ˚A, b = 12.829 ˚A, c = 15.641 ˚A, α = γ = 90 ◦ , β = 90 . ◦ [S12]. The tolerance of the optimizationwas 5 . × − a.u. = 0.0257 eV nm − . The DFT-optimal atomic and experimental positions are shown in Table S1and Table S2. The input file for SCF within ABINIT is attached as supplemental material (input SCF.in). TABLE S1. Atomic positions of the optimal crystal structurefrom DFT x y z
Ta -0.004440 0.220835 0.109287Ni 0.0 0.701355 1/4Se(1) 0.5046 0.077651 0.138568Se(2) -0.003119 0.142849 0.950279Se(3) 0.0 0.32910 1/4 TABLE S2. Atomic positions of the experimental crystalstructure x y z
Ta -0.007930 0.221349 0.110442Ni 0.0 0.701130 1/4Se(1) 0.5053 0.080385 0.137979Se(2) -0.005130 0.145648 0.950866Se(3) 0.0 0.32714 1/4
Phonon mode calculation and its assignment
Second, we obtained phonon modes based on density-functional perturbation theory (DFPT) [S13]. The normalmode U qa,α for atom a was obtained as an eigenfunction of the dynamical matrix through DFPT as X bβ D aα ; bβ U qb,β = ω ( q ) U qa,α , α, β = x, y, z, X aα (cid:12)(cid:12) U qa,α (cid:12)(cid:12) = 1 (S1)where q is an index for the eigenmode. The input file for DFPT within ABINIT is attached as supplemental material(input DFPT.in). The actual atomic displacement was obtained by δR qa,α = U qa,α / √ M a , using mass M a of an atom a .Because the Bravais lattice is a double size supercell, the phonon modes obtained by DFPT contain Γ-point phononsand finite wavenumber phonons. Since the equivalent atomic displacements for the primitive cell of Γ-point modes isin-phase, the relative motions of the equivalent atoms are the reference to extract the relevant phonon modes. Werejected modes that showed counter displacements between two-equivalent atoms for the primitive cell.We identified an irreducible representation for each mode by performing symmetry operations for the atomicdisplacement and evaluating characters by symmetry operation for the general points of the Bravais lattice, R x x x = x x x , R x x x = − x x − x + 1 / , R x x x = − x − x − x , R x x x = x − x x + 1 / (S2)and character table, Table S3. TABLE S3. Characters table of the C2/c space group R R R R A g A u B g B u Numerically evaluated characters for the displacement were evaluated as C qi = X a (cid:16) δr qb, , δr qb, , δr qb, (cid:17) ˜ R i δr qa, δr qa, δr qa, (cid:30) X a [( δr qa, (cid:1) + (cid:0) δr qa, (cid:1) + (cid:0) δr qa, (cid:1) (cid:3) (S3)where δr qa is the coordinate for the lattice vectors rather than Cartesian coordinate, atom b is that transformed by R i from atom a , and ˜ R i is a translation-free symmetry operation as˜ R x x x = x x x , ˜ R x x x = − x x − x , ˜ R x x x = − x − x − x , ˜ R x x x = x − x x (S4)because the displacement itself is a translation-free object. Errors of the numerical character were smaller than1 . × − compared with the expected unity.We focused on the A g mode because the DECP mechanism informs us that the all-symmetric A g mode is excited.The frequencies are shown in Table S4. Actual modes are attached as supplemental material (PBE phonon.csv). TABLE S4. Frequencies of A g mode, quasicharacters and assigned experimental modesFrequency(THz) Quasicharacter for Assigned to C2 C4 C6 C8orthorhombic phase experimental mode [S5]0.99 A g B g B g A g A g B g A g A g A g A g A g We assigned theoretical modes to an experimental frequency by the approximated symmetry of the oscillation. InTable S5, quasicharacters for the orthorhombic phase are shown as well. By increasing the temperature, the Ta NiSe crystal adopted an orthorhombic crystal structure, Cmcm (No. 63), whose operation and character table are givenin Eq. (S5) and Table S5, respectively. We numerically evaluated the characters based on the Cmcm symmetry andused its value to assign the phonon modes to whether A g or B g modes. This analysis was not rigorously well definedand is an approximated treatment. We called the numerically evaluated character for the higher symmetry spacegroup as the quasicharacter here. C x x x = x x x , C x x x = − x − x x + 1 / , C x x x = − x x − x + 1 / , C x x x = x − x − x ,C x x x = − x − x − x , C x x x = x x − x + 1 / , C x x x = x − x x + 1 / , C x x x = − x x x . (S5) TABLE S5. Characters table of the Cmcm space group C C C C C C C C A g A u B g B u B g B u B g B u To obtain the quasicharacter, we introduced an approximated identification of the atomic position related to Eq.(S5). We allowed error value ǫ =0.1 Bohr = 0.00592 nm for the identification as vuut"X pα A α,p ( C i r a,α − r b,α ) < ǫ (S6)where r a,α is the reduced coordinate of atom a and A is a matrix to convert the reduced coordinate to a Cartesiancoordinate. The error value was chosen such that a one-to-one correspondence of atoms was realized for the wholeoperation given by Eq. (S5). Although the fourth A g mode, 2.88 THz, had an almost B g -like character, we assignedit as an A g mode. In the orthorhombic phase, the number of B g mode is three. We assign three modes, 1.96 THz,2.42 THz, and 4.46 THz, that had the closest quasicharacter value compared to minus unity as the B g mode and donot 2.88 THz mode. Follow-up
To check robustness of our conclusion with this specific framework based on PBE-DFT, we performed calculationsin different conditions, 1) Perdew-Wang (PW) functional [S14] was used rather than PBE, 2) PBE-DFT frameworkbut with experimental atomic positions, and 3) PW-DFT framework with experimental atomic positions. We use LDApseudopotential [S15] for the PW-DFT calculation. The atomic position optimized with the PW functional is shownin Table S6. To obtain the identification of approximately equivalent atoms for the orthorhombic phase, ǫ = 0.2 Bohrwas employed in Eq. (S6) for the experimental crystal structure. The phonon frequencies and the quasicharacters forthese calculation conditions are summarized in Table S7. TABLE S6. Atomic position of DFT with the PW functional x y z
Ta -0.007407 0.223358 0.112221Ni 0.0 0.7049391 1/4Se(1) 0.5047 0.077647 0.138656Se(2) -0.004972 0.149029 0.952662Se(3) 0.0 0.32730 1/4 TABLE S7. Frequencies of A g mode, quasicharacters and assigned experimental modesFrequency(THz) Quasicharacter Frequency (THz) Quasicharacter Frequency (THz) QuasicharacterPW DFT PW DFT PBE Exp. PBE Exp. PW Exp. PW Exp.0.97 A g A g A g B g B g B g A g A g B g B g B g A g A g A g A g B g B g B g A g A g A g A g A g A g A g A g A g A g A g A g A g A g A g Here, we comment on a trade-off between the computational cost and convergence accuracy of the results of thissimulation. The most computationally demanding part of this procedure was the DFPT part because many and largelinear systems must be solved. When 384 cores of the system-B (Sekirei), supercomputer at the Institute of SolidState Physics, the University of Tokyo, were used within the flat-MPI parallelization, the DFPT calculation took65.13 hours. Roughly speaking, the DFPT cost with the GGA-functional obeys a linear scaling for the number of k -points. An estimation for 36 × × N K = 36 × ×
9. The residual force, evaluated with the optimal atomic position of sparser Brillouin zone sampling,was maximally 1 . × − = 0.068 eV nm − , which was fairly close to our criteria to identify the atomic equilibriumposition. The optimized atomic positions with denser sampling are shown in Table S8. By converting these to aCartesian coordinate, maximal error is 6 . × − (a.u.) = 0.36 fm. These errors were small compared with thefollowing case for the energy cut off.We also checked the energy convergence error by using energy cut off 60 a.u. = 1633 eV with the same Brillouinzone sampling, N K = 24 × ×
6. The residual force, evaluated with ecut = 50 a.u., is maximally 2 . × − a.u. =0.11 eV nm − . The optimized atomic positions are shown in Table S9. The maximal error of the atomic positionin the Cartesian coordinate was 1 . × − a.u. = 0.69 fm. Performing DFPT calculation for the energy cut off 60a.u., we obtained the phonon frequencies and quasicharacter shown in Table S10. The error for the frequency of thephonon modes was maximally 0.02 THz. Thus, the error in the Brillouin zone sampling is expected to be smallerthan this error. TABLE S8. Atomic positions of optimal crystal structurefrom PBE-DFT with 36 × × x y z Ta -0.004442 0.220801 0.109289Ni 0.0 0.7013654 1/4Se(1) 0.5046 0.077653 0.138552Se(2) -0.003147 0.142847 0.950286Se(3) 0.0 0.32910 1/4 TABLE S9. Atomic positions of optimal crystal structurefrom PBE-DFT with energy cut off 60 a.u. x y z
Ta -0.004440 0.220819 0.109280Ni 0.0 0.7014205 1/4Se(1) 0.5046 0.077647 0.138557Se(2) -0.003124 0.142859 0.950281Se(3) 0.0 0.32910 1/4 TABLE S10. Frequencies of optimal crystal structure from PBE-DFT energy cut off 60 a.u.Frequency(THz) Quasicharacter for C2 C4 C6 C8orthorhombic phase0.97 A g B g -1.000 -1.000 -1.000 -1.0002.42 B g -0.979 -0.979 -0.979 -0.9792.87 A g -0.713 -0.713 -0.713 -0.7133.56 A g B g -0.998 -0.998 -0.998 -0.9985.16 A g A g A g A g A g Correlation between the orthorhombic and monoclinic phases
By reducing the symmetry from the orthorhombic to monoclinic phase, the number of irreducible representationsis also reduced. As a result, Raman-active phonon modes of both A g and B g characters in the orthorhombic phaseturn to be A g character in the monoclinic phase. Tables S11 and S12 show Wyckoff positions and Raman-activephonon modes for each atom in the orthorhombic (Cmcm) and monoclinic (C2/c) phases, respectively. Table S13shows correlation for each irreducible representation between the orthorhombic (Cmcm) and monoclinic (C2/c) phase,and Table S14 shows the number of total Raman-active phonon modes and its correlation relations. TABLE S11. Wyckoff positions and Raman-active phononmodes for each atom in the orhorhombic (Cmcm) phaseWyckoff position Raman-active phonon modesTa 8f 2 A g + B g + B g + 2 B g Ni 4c A g + B g + B g Se(1) 8f 2 A g + B g + B g + 2 B g Se(2) 8f 2 A g + B g + B g + 2 B g Se(3) 4c A g + B g + B g TABLE S12. Wyckoff positions and Raman-active phononmodes for each atom in the monoclinic (C2/c) phaseWyckoff position Raman-active phonon modesTa 8f 3 A g + 3 B g Ni 4e A g + 2 B g Se(1) 8f 3 A g + 3 B g Se(2) 8f 3 A g + 3 B g Se(3) 4e A g + 2 B g TABLE S13. Correlation table between the orhorhombic(Cmcm) and monoclinic (C2/c) phaseCmcm C2/c A g A g A u A u B g B g B u B u B g B g B u B u B g A g B u A u TABLE S14. The number of total Raman-active phonon mon-des and its correlation relationsCmcm C2/c8 A g + 3 B g A g B g + 8 B g B g Comparison between band dispersions and FDARPES spectra
We compare band dispersions before and after photoexcitation with FDARPES spectra for each frequency as shownin Fig. S1.2 E - E F ( e V ) k ( (cid:973) (cid:16)(cid:20) )1 THz 2 THz 3 THz 3.75 THz 4 THz MaxMin k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) (e)(d)(c)(b)(a) E - E F ( e V ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) k ( (cid:973) (cid:16)(cid:20) ) MaxMin (j)(i)(h)(g)(f)
FIG. S1. Comparison between FDARPES spectra and band dispersions before (Figs. S1(a)-S1(e)) and after photoexcitation(Figs. S1(f)-S1(j)). Band dispersions are shown as green (Figs. S1(a)-S1(e)) and blue circles (Figs. S1(f)-S1(j)).
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