Ultrafast electron calorimetry uncovers a new long-lived metastable state in 1T-TaSe2 mediated by mode-selective electron-phonon coupling

Ultrafast laser pulses uncover a new metastable state in 1T-TaSe2 by exciting electrons and then tracking their temperature.


The PDF file includes:
Section S1. Data analysis of trARPES spectra Section S2. DFT calculations of electronic structure Section S3. Evolution of the electron temperature Section S4. The electronic band shift and the new metastable states Section S5. Relationship between the band shift and the CDW order Section S6. Caption of the supplementary movie Fig. S1. Fit of the trARPES spectra. Fig. S2. Band structure for 1T-TaSe 2 in the metallic state (1 × 1) with spin-orbit coupling. Fig. S3. Partial density of states projected onto three kinds of Ta atoms in the CDW state ( 13 13  ) with spin-orbit coupling. Fig. S4. Analysis of the electron temperature. Fig. S5. Analysis of the band shift. Fig. S6. Schematic of the new long-lived metastable state mediated by mode-selective electronphonon coupling. Fig. S7. The long-lasting metastable state. Fig. S8. ARPES spectra at selected time delays for the laser fluence of 0.86 mJ/cm 2 . Fig. S9. ARPES spectra at two time delays as a function of laser fluence. Legend for movie S1 References (46-51)

Other Supplementary Material for this manuscript includes the following:
(available at advances.sciencemag.org/cgi/content/full/5/3/eaav4449/DC1) Movie S1 (.mp4 format). Transforming a material into a new state after heating the electrons with an ultrafast laser.

Section S1. Data analysis of trARPES spectra
We take the data with laser fluence of 0.86 mJ/cm 2 as an example to demonstrate the extraction of the electron temperature and band shift dynamics. Figure S1A displays the temporal evolution of the energy distribution curves (EDC) at the momentum point indicated in Fig. 2A. They are fitted with a Fermi-Dirac function multiplied by a Lorentzian-shape density of states (DOS) where B is the Boltzmann constant, F is the Fermi level, is the temperature, is an amplitude, b is the band position, Γ is the width of the Lorentzian function, 0 is a constant background.
Since we need to fit the EDCs at each delay with the same model and some fitting parameters should share the same value for all data, we implement a global fitting process in the data analysis. Here we discuss three methods. In the first one, we link F , Γ in each EDC, take , b , , 0 as independent, and fit all the EDCs simultaneously. From the fitted and b at each delay time, we extract the electron temperature and band shift dynamics. Note that since is influenced by the energy resolution (effectively as ), we derive electron temperature as = √ 2 − 2 . is determined by the fact that = 300 K before laser excitation.
The results are shown as red circles in fig. S1, B and C. In the second method, we take the width of the Lorentzian Γ as independent for each EDC, considering the possible variation of the imaginary part of the self energy. The results (black circles in fig. S1, B and C) are consistent with that from the first method.
In the third method, we take the DOS as a more generalized form taking the asymmetry of the line shape into account (46) where is a factor indicating the asymmetry of the DOS. When = 0 , there is no asymmetry, Equation (S2) reduces to Equation (S1). The global fitting to the data with this function yields an value of 0.02, which is very small as seen in fig. S1D. The extracted electron temperature and band shift show good agreements with those from the first two methods.
Based on the above discussions, the first method with a minimal number of fitting parameters provides already good and reliable fits to the EDCs. We apply this method in the data analysis for all laser fluences and the results are discussed in the main text.
We note that the resolution of this calorimetry technique is mainly limited by the energy resolution of the whole instrument, and also depends on the stability of the laser -so that the fluence can be accurately recorded. Based on the data and the statistics presented in Fig. 2, fig. S1 and fig. S4, we estimate the current resolving power of this technique in temperature is about 10, e.g., the resolution is about 100 K when measuring a temperature of 1000 K.   Figure S2 shows the calculated band structure in the metallic state (1 × 1 cell). In the CDW state, there are three kinds of non-equivalent Ta atoms. We plot the partial density of states projected onto these three Ta atoms in fig. S3. It clearly shows that the electrons tend to "localize" around the center of star-of-David on the occupied side (below EF), which is consistent with charge density modulation in the CDW state. While on the unoccupied side, there are more states around the outer Ta atoms.

Section S3. Evolution of the electron temperature
We fit the electron temperature dynamics by a two-exponential function where 1 and 2 are the exponential rise and decay constants. The temporal resolution is with the left term representing the laser energy density at the sample surface (note that ARPES is surface sensitive at this probe photon energy), where and are the reflectance (~40%) and penetration depth (22 nm), respectively (6, 47).
is the sample temperature (300 K) in the experiment. The result is shown as the solid red curve in Fig. 2E and fig. S4F, it captures the fluence dependence of Te at 4 ps only at F < Fc. Note that there is a change of slope (i.e., a kink) in the maximum Te as a function of fluence ( fig. S4D), indicating that the electronic heat capacity at F < Fc is effectively larger than that at F > Fc. The enlarged heat capacity at low fluences originates from the energy required to change the state (e.g., the electronic condensation energy or energy for band shift) of 1T-TaSe2, which saturates at Fc (orange curve in Fig. 3D). This part of energy corresponds to the latent heat in the first-order transition under thermal equilibrium. Taking this into account, we roughly estimate that the fluence required for the phase transition in an equilibrium way would be ~ 2.2 mJ/cm 2 . However, as shown in Fig. 3D and supplementary S5, a significantly lower fluence of 1.35 mJ/cm 2 is already enough for 1T-TaSe2 to go through and complete the transition in a quasi-equilibrium and metastable way.

Section S4. The electronic band shift and the new metastable states
We fit the EDCs at different momentum cuts around 0.3-0.4 Å -1 and extract the band shift, as shown in fig. S5A. The amplitude and dynamics look pretty similar for the results at each k//, so it is fair to average them to improve the signal-to-noise ratio. It is worth mentioning that the band shift observed here is clearly different from the chemical potential shift reported in BaFe2As2 (48). Figure S5B shows the temporal evolution of the EDC at k// ~ 0.5 Å -1 . We can observe both the dynamics of Ta 5d and Se 4p bands by optimizing the polarization of the probe beam. The position of Ta 5d band (which is the one we discussed in the text unless specified otherwise) exhibits a pronounced shift, while the Se 4p band remains nearly unchanged. The temporal evolution of the band shift can be modelled by Equations (1) and (2)  Upon laser excitation, the evolution of the sample is determined first by the electron temperature and then by electron-phonon coupling, which are fluence dependent. For strong laser excitation, the electron-phonon coupling switches from nearly homogeneous to mode-selective. The resulting inhomogeneity within the phonon bath drives the material into a new long-lived metastable CDW state. The blue shading represents the electron density in the real-space, the grey circles represent Ta atoms, both amplitudes are exaggerated for better visualization. Te, Tp and Tl refer to the temperatures of the electron, strongly-coupled phonons, and the rest of the phonon bath, respectively.
This new state lasts for a long time as evidenced by the mild changes of both the band shift and the electron temperature at long time delays. We take the results at the fluence of 0.92 mJ/cm 2 as an example, as shown in fig. S7. For the band shift, the oscillation part is subtracted since it is insignificant in the long-time dynamics. Then we fit the data with a three-exponential function: initial rise, decay to the metastable state (selective electronphonon coupling) and decay to the quasi-equilibrium state (anharmonic decay of the strongly coupled phonons). The fit yields a timescale of 553 ps for the metastable state to recover. We note that this value is not quantitatively reliable since we only have data before 5 ps, however, it provides the qualitative magnitude of the long lasting time of the new metastable states. This is further confirmed by the nearly flat part around 4 ps in the electron temperature dynamics (since 1T-TaSe2 is a CDW material rather than a simple metal, we avoid using Ntemperature model here). It is worth mentioning that this long-lived metastable state and the related timescale are consistent with the results of previous ultrafast electron diffraction experiments (27, 49-51), some of which reported a "second-order character" of the optically induced phase transition but without a clarification of its nature (27,49). Moreover, we confirm that the observed new metastable state at F > Fc is indeed an CDW ordered state, with the same wave vector as the ground state but with smaller CDW amplitude. As shown in the bottom panels in fig. S9, the spectra in such states show band folding at the same momentum, but are less pronounced than the normal CDW state. These new states are not reachable under thermal equilibrium conditions, and bridge the first order transition to a continuous one, as illustrated in Figs. 1 and 4.

Section S6. Caption of the supplementary movie
Movie S1. Transforming a material into a new state after heating the electrons with an ultrafast laser. Top panel: schematic of the Ta atoms in the 1T-TaSe2 crystal lattice (black dots) and electrons (blue shading) in the two phases that can be reached by simply heating the material under thermal equilibrium conditions, e.g., with a hot plate. At low temperatures, the electrons localize (charge order) and the lattice is highly distorted, while at high temperatures, the electrons are smeared and the atomic spacing is uniform. Bottom panel: Movie of the electron energy in the material after rapid heating with a femtosecond laser, i.e., the band structure. The charge order quickly melts, before the material evolves into a new metastable state that lasts for hundreds of picoseconds, characterized by a state with less charge order and lattice distortion. Another remarkable property of this new state is that the electrons are much less coupled to most of the lattice, so that the heat capacity is ≈1/3 that of either equilibrium phases. As a result, significantly less energy is required to melt the charge order and transform the state of the material than under thermal equilibrium conditions.