Transient three-dimensional structural dynamics in 1 T -TaSe 2

We report on thermal and optically driven transitions between the commensurate (C) and incommensurate (IC) charge-density wave (CDW) phases of 1 T -TaSe 2 . Optical excitation results in suppression of the C-CDW on a subpicosecond timescale. The optically driven C to IC transition involves a short-lived ( ∼ 1 ps) unreconstructed phase. Nucleation of an IC phase stacking order is observed already at ∼ 4 ps following photoexcitation. The short timescales involved in establishing the stacking order implies that the nucleation of the IC phase is inﬂuenced by the local geometry of the adjacent layers such that the stacking direction of the C phase determines the stacking direction of the IC phase. From this follows that the nucleation of the IC-CDW is inherently three dimensional (3D). We observe the activation of a coherent shear mode in the optically driven transitions to the transiently stabilized unreconstructed phase. The activation mechanism starts with a rapid lifting of the periodic lattice distortions (PLD) of the Ta sublattice which results in formation of local transient velocity disparities in the Se sublattice. The local differences in Se-phonon amplitudes result in noncompensated shear forces between the layers. This is an example of a multistep coherent launching mechanism. The energy of the optically excited electronic state dissipates energy into modes of the PLD through strong electron-phonon coupling. The rapid suppression of the PLD launches the third step, a coherent vibrational shear mode with low dissipation. The results highlight the importance in considering the 3D nature of the CDWs in the analysis of both structure and dynamics in transition-metal dichalcogenides.


I. INTRODUCTION
Two-dimensional (2D) materials are formed from extended covalently bonded crystalline sheets that may serve as building blocks for three-dimensional (3D) structures through stacking of the 2D sheets. The properties of the resulting 3D material are influenced by the weak interplane van der Waals bonds. A particularly exciting group of 2D materials are the transition-metal dichalcogenides (TMDs) that exhibit intriguing electronic and structural properties [1,2]. The formation of commensurate electronic charge-density waves (C-CDW) or incommensurate electronic charge-density waves (IC-CDW) and their relation with periodic lattice distortions (PLD) have been the subject of particular interest [3][4][5][6]. The strong electron-phonon coupling makes these systems especially well suited for studies of quasiparticle interactions [7] and photo-driven phase transitions [8]. Under the assumption of a weak interplane coupling in TMDs, most previous studies have restricted the analysis of the PLD to the * jonas@kth.se two-dimensional superstructure within the layers. However, the ground states of several 2D CDW materials exhibit a complex three-dimensional structure with a preferential stacking order in the out of plane direction [9]. An increasing number of theoretical and experimental studies have demonstrated the importance in including the interplane van der Waals interactions in the description of the physical properties of layered materials [10][11][12][13]. In a time-resolved diffraction study by Erasmus et al. [14], it was shown that strong out of plane coupling facilitates CDW formation and that 3D properties should be considered in analysis of the formation of periodic lattice distortions in TMDs. Other studies have predicted how a change in stacking order may result in metal to semiconductor transition or interlayer exciton formation [10,11].
The combination of a CDW and long-range stacking order in 1T -TaSe 2 makes it a prime candidate for studying the coupling of the in-plane PLD with the interlayer van der Waals interaction [9,15]. The formation of a long-range stacking order indicates the significance of the interlayer interaction on the 3D properties of the material. The C-CDW phase of 1T -TaSe 2 hosts 13 in-plane Ta atoms displaced from their unreconstructed lattice positions to form a so-called Star of David cluster with a hexagonal √ 13a 0 × √ 13a 0 superstructure, as shown in Fig. 1 [3,16]. The C in-plane superlattice exhibits a 13.9°rotation relative to the unreconstructed lattice. Results from diffraction and nuclear magnetic resonance have determined that the C-CDW of 1T -TaSe 2 exhibits a T i = 2a 0 translation vector between subsequent layers, with a threefold rotational symmetry [17][18][19][20]. The three symmetric translation directions are not randomly populated, instead 1T -TaSe 2 exhibits a long-range stacking order indicating interactions extending even to next-nearest layers [15]. After a c 0 -axis period of 13 layers the in-plane atomic positions will reside in equivalent positions to the first layer of the stack.
1T -TaSe 2 exhibit a commensurate to incommensurate CDW phase transition at T c = 473 K [16]. The superlattice of the IC-CDW phase rotates 13.9°relative to the C-CDW phase to align with the unit-cell vectors of the unreconstructed lattice and forms a fcc-like out of plane stacking [9]. The fcc-like stacking minimizes the electrostatic interaction between the in-plane triangular charge-density lattices [21,22]. A detailed structural description of the change in stacking at the C-IC phase transition may contribute to the understanding of the correlation between the intralayer structure and the interlayer van der Waals interaction in TMDs. An understanding of this correlation may shed light on the processes that underpin the many intriguing fundamental properties of this class of materials such as the long-lived metastable state in 1T -TaSe 2 [23], hidden states in 1T -TaS 2 [24], and the relation between CDW and superconductivity [25].
Ultrafast electron diffraction has successfully been employed in studies of photo-driven intraplane structural dynamics in TMDs [8,14,[26][27][28]. Recently, Le Guyader et al. [29] extended the scope of analysis to include dynamics in stacking order of 1T -TaS 2 . The large Se 4p-Ta 5d transfer interaction in 1T -TaSe 2 results in a well-established 3D Fermi surface. From the 3D electronic structure follows a stronger interlayer interaction and higher degree of long-range order compared to 1T -TaS 2 [30,31]. Furthermore, the phase diagram of 1T -TaSe 2 has an additional advantage over 1T -TaS 2 , in that it does not include the rather complicated nearly commensurate phase. All taken together makes 1T -TaSe 2 a well-suited model system for studies of the influence of interlayer interaction on transient structure dynamics. Here, we investigate the 3D structural dynamics in 1T -TaSe 2 during a photo-driven CDW transition and subsequent recovery by means of ultrafast electron diffraction. Information of the stacking order is accessed by application of precise sample 094303-2 tilt angles to selected zone axes. The dynamics of the C-IC CDW phase transition and the change in out of plane stacking order in conjunction with the phase transition is described. A coherent shear phonon is observed following a photo-driven suppression of the C-CDW. The excitation of the shear phonon is explained through a mechanism including residual interlayer forces originating from Se sites at different corrugation amplitude (out of plane displacement) in combination with the break of symmetry resulting from the alternating rotational directions in stacking order.

A. Experiment
Single crystals of 1T -TaSe 2 were grown from high-purity elements by a chemical vapor transport method using iodine as transport agent [32]. Transmission electron microscopy (TEM) samples were carefully microtomed at the preferential cleave plane using a diamond knife and were placed on a single-layer graphene TEM grid (Ted Pella). The sample thickness was measured, on the grid, to be 20 nm using a JPK NanoWizard 3 atomic force microscope (Supplemental Material, Fig. S1 [33]). Electron-energy-loss spectroscopy confirmed the thickness as 20 nm (Supplemental Material, Fig. S1 [33]). Variable-temperature electron diffraction experiment was performed using a Gatan double-tilt heating holder (model 652). Time-resolved diffraction experiments were performed in an ultafast electron microscope described in detail in Ref. [34]. Briefly, the instrument is a modified JEOL JEM 2100 TEM with a conventional thermionic electron gun operating at 200 keV. All the time-resolved experiments were performed at room temperature. The electron bunches were generated through photoemission by a UV (258 nm, Amplitude Systemes) laser pulse. The full width at half maximum (FWHM) of the electron bunches was approximately 1.2 ps as characterized by photo-induced near-field electron microscopy ( Fig. S2 [33]) [35]. Each electron bunch contains approximately 1000 electrons. The sample was excited by a coupled 300-fs (Amplitude Systemes) 515-nm laser pulse (hv = 2.4 eV) focused to a spot on the sample with a FWHM of 120 μm. The sample was excited at a repetition rate of no more than 70 kHz to allow for complete relaxation between excitation pulses. No sample degradation or twin domain formation was observed in the analysis.
Diffraction patterns were simulated using CRYSTAL MAKER SINGLECRYSTAL 3 with intensities estimated from the structure factor under a kinematic assumption. The analysis used atomic structure information of the commensurate phase from Brouwer and Jellinek [19]. A 39-layer cell was constructed in order to consider the coexistence of threefold rotationaloriented domains in out of plane direction (Fig. S3). The atomic structure models were drawn by VESTA [36].

Total energies and phonons
The total energy calculations used to assess the stacking order were performed with the VASP software and similar plane-wave and energy cutoffs as used for the phonon calculations. The Brillouin zone was sampled by a -centered 3 × 3 × 6 mesh of k points. The supercell was set up in the C-CDW phase with a translation vector 2a 0 , in accordance with Fig. 1(a).

Band structure and optical absorption
The band-structure calculations and the calculation of the joint density of states to assess the momentum-resolved optical absorption across the Brillouin zone were performed using the ELK code [37]. The cutoff for the plane-wave basis functions was determined by setting the maximum | G + k| · R MT to 9, where R MT is the average radius of the augmentation muffin tins, G is a reciprocal lattice vector, and k a crystal momentum within the first Brillouin zone. We used a -centered 3 × 3 × 3 mesh of k points to sample the Brillouin zone for the self-consistent density, followed by a band-structure calculation along the high-symmetry lines as specified in Fig. S13.

Shear phonon
The first-principles calculation of the phonon modes used to identify the shear-mode excitation was performed using density-functional theory implemented using the projectoraugmented waves method in the VASP code [38,39]. The exchange and correlation was treated with the density functional by Hamada, with explicit inclusion of van der Waals interactions [40]. A cutoff of 500 eV was used for the plane-wave basis, and an energy convergence criterion of 1 × 10 −9 eV in order to capture the weak energy dispersion of the interlayer shear modes in a finite-displacement framework. The Brillouin zone of a 12-atom supercell consisting of 4 vertically stacked unit cells of 1T -TaSe 2 in the unreconstructed phase was sampled by a 12 × 12 × 6 mesh. The time-dependent density-functional theory (TD-DFT) coupled to Ehrenfest molecular-dynamics simulations was performed with the TDAP package implemented on top of the SIESTA software [41,42]. Norm-conserving pseudopotentials were used together with a double-zeta+polarization basis set for computational efficiency. The exchange-correlation potential was treated within the adiabatic approximation using the functional by Perdew, Burke, and Ernzerhof [43]. A supercell composed of two unit cells of 1T -TaSe 2 in the C-CDW phase, consisting of 156 atoms, was used in order to allow for sufficient lattice dynamics. Additional details are provided in the Supplemental Material [33].

III. RESULTS AND DISCUSSION
This section begins with a description of the static structures of the commensurate and incommensurate CDW phases of 1T -TaSe 2 , with an emphasis on the structural properties central to establishing the out of plane stacking order. It is followed by results from a thermally driven phase transition demonstrating the strong influence of a change in PLD and stacking order on the experimental diffraction patterns. These results lay the framework for the interpretation of the processes included in the photo-driven transition, especially the fast change of stacking order and the excitation of a shear phonon. Figure 1(a) illustrates the in-plane atomic structure of the commensurate CDW phase of 1T -TaSe 2 projected along the [001] direction with the Se atoms color coded according to their relative coordinate along the c axis of the C-CDW phase (in fractions of c) [19]. For clarity of presentation of the PLD-induced buckling of the Se atoms only the top-layer Se atoms are included in the figure. Red-shaded circular regions indicate the most protruding, convex, sites of the layer. Conversely, blue-shaded circular regions indicate concave sites. In the stacking of layers it is energetically favorable for convex protruding positions to reside above concave positions in adjacent layers as revealed in Fig. 1(b), resulting in a stacking translation vector of T i = 2a 0 [18]. The transition to the IC phase results in a rotation of the unit cell of the superstructure from (a,b) to a direction parallel to the fundamental lattice (a',b') [ Fig. 1(a)]. In the IC phase the layer stacking order is fcc-like; thus the Star of David cluster will form an ABC stacking in the out of plane direction.  Figure 2(g) shows the subtracted diffraction intensity for the two phases. The rotation of spots is evident from the results, as is the elimination of some spots. At the C-CDW to IC-CDW transition temperature the q 1 C satellite spots in Fig. 2(e) rotate 13.9°to the q 1 IC positions in Fig. 2(f), while q 2 C disappear without a corresponding spot appearing at the q 2 IC position. This implies that q 1 IC is placed in bright Bragg diffraction condition at the [102] zone axis while q 2 IC is out of Bragg condition. This difference in diffraction condition for the q 1 IC and q 2 IC satellite spots can be traced from the three-dimensional superstructure of the IC-CDW phase. Figure 2(c) shows the intensity of the q 1 C and q 1 IC satellite 094303-4 spots as a function of temperature, across the C-CDW to IC-CDW phase transition. The intensity of q 1 C decreases steadily with temperature below the phase transition. In the same temperature range, there is no intensity at the q 1 IC position. The q 1 C spot disappears immediately above the transition temperature. Simultaneously we observe a dramatic increase in the intensity of q 1 IC , to levels similar to what was observed for q 1 C at temperatures below the transition. Detailed traces of the intensity of q 1 IC and q 2 IC as a function of temperature are shown in Fig. 2(d). At the transition temperature of approximately 498 K the intensity of q 1 IC increases dramatically compared to q 2 IC . This implies that both the in plane superstructure and the out of plane stacking order change simultaneously at the transition, suggesting that the three-dimensional character of the IC-CDW phase is immediately established above the transition. This will be further investigated in the time-resolved photoinduced phase transition studies below. Note that the temperature in Fig. 2 is measured at the sample holder position which may be slightly higher than the actual temperature of the sample. The continued increase in intensity of the q 1 IC with temperature above the transition may be a result of a domain growth process. This interpretation is corroborated by a slight decrease in satellite spot width at increasing temperature.

C. Photo-driven commensurate to incommensurate transition
Next, we will discuss the photoinduced transient structural changes in 1T -TaSe 2 . In the C-CDW phase, electronicstructure calculations indicate that absorption occurs predominantly by transitions from p states on Se to d states on Ta [44]. The electronic bands in the C-CDW phase do not cross the Fermi level along any of the high-symmetry directions in the in plane of the layers. However, there are band crossings between -A and L-M that in the hexagonal Brillouin zone correspond to the out of plane direction. This is indicating that charge transport along the stacking direction is more favorable than in the planes (Supplemental Material, Fig. S13 [33]). This band structure is similar to TaS 2 that also shows no band crossings in the planes, but in the out of plane direction. [10] The suppression of the PLD starts already within the duration of the optical pump pulse; this fast initiation is warranted by the strong electron-phonon coupling between the photoexcited electrons, as can be deduced from the strong dependence of the band energies to the amplitude of the mode responsible for the PLD as seen in Fig. S13. During the melting process of the PLD, additional intraplanar electronic band crossings of the Fermi level appear, allowing for a more efficient recombination of electrons and holes. In order to estimate where in the Brillouin zone the optical pump will introduce the majority of electron-hole pairs, we calculate the absorption probability as a function of crystal momentum. To model the absorption, we calculate the joint density of states for a range of PLD amplitudes. For a photon energy of approximately 2.4 eV we deduce that the absorption in the C-CDW phase allows for absorption in a large part of the Brillouin zone for the pump photon energy under consideration. The increased band dispersion resulting from the melting of the PLD leads to absorption in a wider energy range. The initial stages of the melting of the PLD are likely different for optical and thermal melting. This can be inferred from the computed momentum-resolved optical absorption probability, as seen in Fig. S13. For an energy range of about 2.4 eV we can see that, in the initial phase, we have optical absorption throughout the Brillouin zone. As the PLD is suppressed and the unreconstructed structure is restored to the lattice, the electronic bandwidth increases. An increased degeneracy in specific parts of the Brillouin zone also indicates the increase in symmetry. Similarly, we can, by looking at the evolution of the band structure, see that upon thermal melting we will have the initial thermal excitations of primarily intraband character in a very limited portion of the Brillouin zone, between and A Please note that it is the first Brillouin zone of the 1T structure in the C-CDW phase that is depicted; for clarity the same representation is also used in the unreconstructed phase. We expect that the differences along the course of the melting have an impact on transient properties whereas the final stage is similar. Figures 3(a) and 3(b) show the transient intensity of the superstructure satellite spots along the [102] zone axis for the C and IC phases following excitation by a 515-nm laser pulse. The corresponding results for the [001] zone axis are shown in Figs. S7 and S8 [33]. In Fig. 3(b) the pump fluence (2.5 mJ/cm 2 ) was sufficient to drive the sample across the C-IC CDW phase transition as observed from the increase in intensity of the satellite spots associated with the IC phase, while in Fig. 3 the electronic relaxation process due to the strong electronphonon coupling between the photoexcited states and the PLD mode. This process tentatively happens within a single oscillation period of the PLD amplitude mode, or less than 300 fs [8,26]. No apparent change in intensity is observed at positions corresponding to q 1 IC and q 2 IC for a laser excitation fluence of 1.4 mJ/cm 2 [ Fig. 3(a)]. Instead a partial recovery from the transient unreconstructed phase can be observed at the q 1 C and q 2 C positions. The transient results allow for extraction of a 0.7-ps recovery time constant for the C-CDW and the starting point for the recovery is estimated to ∼1.3 ps (Supplemental Material, section 2.1 [33]). The recovery time constant is marginally shorter than that observed in 1T -TaS 2 (∼4 ps) [8] but significantly shorter than in the weakly interlayer interacting 4Hb-TaSe 2 (∼150 ps) [14]. These results indicate that stronger interlayer interaction results in shorter recovery times of the C-CDW. At 2.5-mJ/cm 2 pump fluence, the onset for formation of the diffraction spots associated with the nucleation of the IC phase is observed approximately ∼1 ps after time zero [ Fig. 3(b)]. The 1-ps delay between the extinction of the C phase and the emergence of the IC phase directly implies that the transition involves a transient phase with a completely suppressed PLD, with a structure similar to an unreconstructed 1T phase. Figure 3 traces the change in q 1 IC and q 2 IC intensities individually as filled and empty circles. The observed onset time is similar for both satellite spots and the same timescale is observed for establishing the IC-CDW phase along the [001] zone axis (Supplemental Material, Figs. S7 and S8). A 1-ps delay time before nucleation of the IC-phase is approximately twice the period of the CDW amplitude mode reported at 2.4 THz for 1T -TaS 2 [45,46], and implies that a phononic thermalization is involved in the decay of this highly perturbed photoinduced state before the onset of incommensurate phase formation. A tentative thermalization pathway may involve the longitudinal acoustic mode that in NbSe 2 and TaS 2 has been associated with the thermally driven CDW transition [47][48][49]. A softening of this mode to ∼1 THz has been reported close to the CDW transition in TaS 2 [49], a period similar to the ∼1-ps delay observed for the onset of IC-CDW formation in the photo-driven experiments here.
At times immediately following the delayed onset of the IC phase we note that the change in intensity of the q 1 IC and q 2 IC satellite spots do not exhibit the same transient behavior. q 2 IC show a minor increase in intensity up until 4 ps while the temporal trace of q 1 IC shows a continued increase in intensity also after this point in time. After 4 ps the temporal traces for q 1 IC and q 2 IC qualitatively resemble the temperature traces from Fig. 2(d) except that the intensity of the q 1 IC at 8 ps is only 20% of the intensity of q 1 C before time zero. The relative diffraction intensities deviate from the thermal-driven phase transition where the intensity of q 1 IC above and q 1 C below the transition temperature are similar, as shown in Fig. 2(c). The deviation of the transient diffraction intensity from the steady state implies that the atomic structure has yet to reach its equilibrium positions and may be explained by a process of domain growth that is expected to persist on the nanosecond timescale [50]. The development of individual CDW satellite diffraction spots at selected time delays are shown in Fig. 4. Already at 5 ps we observe significantly higher diffraction intensity at the position of q 1 IC compared to the position of q 2 IC and at 8 ps the diffraction pattern is qualitatively rather similar to the steady-state pattern shown in Fig. 2(f). A weak spot at the center-inverted position to q 2 IC is observed in both Fig. 2(f) and Fig. 4 that probably stems from a slight misalignment of the sample from the [102] zone axis. The divergence of the q 1 IC and q 2 IC intensities at around 4 ps allows us to reach the conclusion that the stacking order of the IC phase has been initiated already at this point, a manifestation of the strong coupling between the layers of 1T -TaSe 2 . The rapid formation of IC-phase stacking order indicates that local nucleation of IC-phase PLD within the TaSe 2 TaSe 2 layers is influenced by the positions of the Star of David clusters in adjacent layers such that the stacking direction of the IC-CDW phase is decided by the previous C-CDW phase stacking direction of the domain. From this follows that the nucleation of the IC-CDW is three dimensional. Our DFT modeling did not find an energy difference of the three symmetric stacking directions in the commensurate phase that may explain a preferred stacking direction (Supplemental Material, sec. 3.1). Therefore, an explanation for preservation of stacking direction throughout the complete C-IC-C excitation and relaxation cycle is interpreted to stem from kinetical restrictions. Under our experimental conditions we did not observe any change in C-CDW phase rotation chirality (clockwise or anticlockwise rotation of the in-plane angle of the C-PLD relative to the unreconstructed phase) or laser-induced formation of mirror domain walls as has been reported for high pump fluence in the nearly commensurate phase of 1T -TaS 2 [51]. Wiegers et al. [18] have previously reported a correlation between the rotation chirality and stacking translation direction in the C phase of 1T -TaSe 2 . This was explained by the strong interlayer interaction in 1T -TaSe 2 in combination with small distortions in the Star of David cluster that minimize the interlayer strain for a single stacking direction within a domain.

D. Excitation of shear phonon
Immediately following excitation, the Bragg spots of the main lattice increase in intensity simultaneously with the suppression of the C-CDW. This is in agreement with what has been reported for similar CDW materials [8]. After 1-2 ps the main lattice Bragg peaks will again decrease in intensity due to reestablishment of the PLD (either C or IC) combined with heating of lattice (the Debye-Waller effect). More interestingly, in 1T -TaSe 2 , we also observed a slow intensity oscillation of the main Bragg peaks following photoexcitation. The individual diffraction spots show an in-phase change as demonstrated in Fig. S6. The average intensity oscillation at different pump power is shown in Fig. 5. The procedure for fitting of the experimental data is described in the Supplemental Material, sec. 2.2. From the fitting procedure, which includes the temporal resolution, we may extract the oscillation frequencies as 0.43, 0.36, and 0.30 THz for 1.3, 2.0, and 2.4 mJ/cm 2 , respectively (Fig. 5). The oscillation is interpreted as excitation of a phonon and the softening observed at increased fluence is consistent with an anharmonic response at large excitation amplitudes [52]. A redshift can also be expected with increasing sample temperature. pump fluence reveal a temporal broadening of the first peak after time zero. The broadening is however not symmetric, instead the peak is observed to extend to longer temporal delays. The broadening may stem from slower relaxation processes of photoexcited electronic states at high pump power [53] or stabilization of the transient unreconstructed state as discussed in connection with Fig. 3(b). The starting point of the oscillation is also delayed at high fluence according to the parameters extracted from the fitting process (see Supplemental Material, sec. S2 for more details [33]). In order to assess the intensity oscillations of the Bragg spots, we performed first-principles calculations of phonon frequencies in the C-CDW phase. The outcome shows that in principle only shear modes appear in the corresponding frequency range, 0.46 THz according to calculations. The frequency is very close to the observed value at low pump power as shown in Fig. 6(a). An assessment using a sound velocity of ∼2000 m/s [54] and the sample-thickness (∼20 nm, Supplemental Material, Fig. S1) rules out breathing modes (with resulting periods of T = 20 ps). Through modulation of the lattice with the shear mode in the stacking direction we may calculate the corresponding structure factor change as a function of shear displacement. The simulated diffraction intensities reproduce the experimental intensity 094303-7 oscillations (Fig. S10). As indicated by the analysis, this mode is almost undamped during the timescales probed here. This indicates a coherent motion, albeit with a delayed onset from the pump pulse. In the following, we analyze the underlying mechanism for the coherence and onset.
Upon melting of the PLD, the Ta sublattice is believed to quickly attain unreconstructed phase positions. However, as indicated by our TD-DFT simulations, the dynamics carry remnant features of the PLD, namely the amplitude of the vibrational modes, as visualized in Fig. 6. We investigate the effect of these remnant features using first-principles calculations for a two-layer model of the material.
We perform a simulation of 1T -TaSe 2 (see Supplemental Material for details) using time-dependent density functional theory coupled to molecular dynamics, as implemented in the TDAP code [41]. The simulation is started from a completely relaxed structure in the C-CDW phase. The external field is introduced as a vector potential in the Hamiltonian, exciting the electrons primarily from Se-p to Ta-d states (see Fig. S14 for details). The first 130 fs of the trajectory indicates a large velocity disparity between Se atoms (and Ta atoms), indicating that the Se atom with the largest deviation from the ideal bond lengths have a higher velocity. Note that only one layer of material is simulated using the TD-DFT framework due to the computational complexity, hence the stacking vector is T i = 0 . The use of a single layer introduces additional symmetries that are in the real material broken by the stacking order. The result of the TD-DFT simulation ties the bond-length disparity of Se atoms to the velocity as a function of time.
We then proceed with static modeling of a two-layer shear displacement in a hexagonal stacking mimicking the stacking vector T i = 2a 0 of the trigonal phase. The stacking order is such that the center of the Star of David cluster in the adjacent layers will be situated directly on top of a local region with high Se velocities (Fig. 6). The center Ta atom itself has in fact a Ta-Se bond length similar to those in the unreconstructed 1T phase. The shear-mode displacement vector is represented by Q S The bond-length disparity of Se atoms that leads to different velocities for different Se atoms is represented as a collective mode with displacement vector Q C indicating the bond distortions inherent to the C-CDW phase. Q C corresponds to a different local kinetic energy related to the Se-atoms' position in relation to the stacking order and the intralayer PLD.
Assume a potential-energy surface V of the form where ω i represents frequencies and γ represents the linear coupling between the modes; H.c. represents higher-order corrections. The collective Se amplitude mode Q C is not an eigenmode of the system, hence the vectorial form of the frequency. By modulating the amplitude of the Se atoms, we uncover a coupling to the shear mode indicating that | dV dQ s | > 0 when |Q C | > 0, with the resulting force in the direction of the stacking order [ Fig. 6(b)]. Note that for a perfect trigonal stacking the force will be compensated. However, our diffraction experiments indicate that the three equivalent stacking vectors are all present; hence, there will always remain uncompensated restoring forces from end points and layers in the vicinity of a transition in stacking-order direction. This will launch a coherent shear motion, that due to the weak coupling between the layers remains a coherent oscillation throughout the recovery of the CDW and PLD. Bear in mind that the coherent activation of the shear mode must occur within a half period, roughly 1 ps of time. Hence, the initial large velocity differences between different Se sites will launch the shear mode. The subsequent thermalization of the Se layer to a homogeneous temperature distribution will result in formation of a transient unreconstructed phase with reduced interlayer interaction that may contribute to a reduced decoherence rate of the shear excitation. This explains the initiation of the shear modes uncovered by the Bragg spot oscillations discussed in the previous section.
A thermally driven transition excites all phononic and lowlying electronic degrees of freedom in an incoherent manner. We do not see any signature of a homogeneous transition to the unreconstructed phase in our thermal excitation data, although it is not experimentally feasible to study the thermally driven transition on the picosecond timescale. An incoherent transition directly from commensurate to incommensurate phase will not result in a net force between the layers since local atomic displacements will be incommensurate, averaging out any resulting interlayer forces. This shows an important difference with respect to the optically induced transition, where the ultrafast coherently driven transition passes through the transient unreconstructed structure, resulting in an uncompensated net force between layers. However, the two processes reach similar end points on very long timescales.

IV. CONCLUSIONS
In summary, we report a detailed study of the transition between the C-CDW and IC-CDW phases of 1T -TaSe 2 using both thermal and optical activation. Our results indicate that the C-CDW in 1T -TaSe 2 is a highly three-dimensional entity due to strong interlayer interactions that establish long-range out of plane stacking order. Optical excitation results in suppression of the C-CDW on a subpicosecond timescale. A recovery time-constant of ∼0.7 ps is observed for the commensurate PLD at an optical pump power insufficient to drive a transition into the IC phase. The recovery time constant is shorter than what has been reported for the related compounds 1T -TaS 2 and 4Hb-TaSe 2 . This may be rationalized as a result of the stronger interlayer coupling in 1T -TaSe 2 . At a laser pump fluence sufficient to optically drive nucleation of the IC phase is an ∼1-ps delay observed between the extinction of the C-CDW phase and the onset for formation of the IC-CDW phase, a manifestation of that the optically driven phase transition involves a transient unreconstructed state. The ∼1-ps delay time for the nucleation of the IC-CDW phase implies that a phononic thermalization is involved in the decay of this highly perturbed photoinduced transient state. Nucleation of an fcc-like stacking of the IC phase is observed already ∼4 ps after photoexcitation. Such rapid stacking order formation implies that the nucleation of the IC phase within the layers is influenced by the local geometry of the adjacent layers such that the stacking direction of the C phase determines the 094303-8 direction of stacking in the IC phase. From this follows that the nucleation of the IC-CDW is inherently 3D. In the optically driven transitions to the transiently stabilized unreconstructed phase, we observe the activation of a coherent shear mode along the stacking direction. We analyze the launching mechanism in terms of locations with high and low velocities on the Se sublattice that result from the rapid melting of the PLD. The bond lengths of Se-Ta bonds in the C-CDW phase are related to the Se-atoms' position in the reconstructed lattice. During the melting, this results in differences in Sephonon amplitudes, which results in noncompensated shear forces between the layers. This is an example of a multistep coherent launching mechanism. The energy of the optically excited electronic state dissipates energy into the PLD through the strong electron-phonon coupling. This rapid suppression of the PLD launches the third step, a coherent vibrational shear mode with low dissipation. Our results highlight the importance in considering the three-dimensional nature of the CDW in both structure and dynamics in 1T -TaSe 2 .