Excitonic and lattice contributions to the charge density wave in 1T-TiSe$_2$ revealed by a phonon bottleneck

Understanding collective electronic states such as superconductivity and charge density waves is pivotal for fundamental science and applications. The layered transition metal dichalcogenide 1T-TiSe2 hosts a unique charge density wave (CDW) phase transition whose origins are still not fully understood. Here, we present ultrafast time- and angle-resolved photoemission spectroscopy (TR-ARPES) measurements complemented by time-resolved reflectivity (TRR) which allows us to establish the contribution of excitonic and electron-phonon interactions to the CDW. We monitor the energy shift of the valence band (VB) and coupling to coherent phonons as a function of laser fluence. The VB shift, directly related to the CDW gap closure, exhibits a markedly slower recovery dynamics at fluences above Fth = 60 microJ cm-2. This observation coincides with a shift in the relative weight of coherently coupled phonons to higher frequency modes in time-resolved reflectivity (TRR), suggesting a phonon bottleneck. Using a rate equation model, the emergence of a high-fluence bottleneck is attributed to an abrupt reduction in coupled phonon damping and an increase in exciton dissociation rate. Thus, our work establishes the important role of both excitonic and phononic interactions in the CDW phase transition, as well as the Bose-Einstein condensation of excitons in 1T-TiSe2.

Charge density waves (CDWs) are an important component in phase diagrams of many correlated electron systems 1,2 . Typically observed in low-dimensional materials, the signatures of a CDW phase have been reported in two-dimensional transition metal dichalcogenides (TMDs) 3 , cuprate superconductors 4 , -conjugated polymers 5 and metal oxides 6 . The central importance of CDW states arises from the relationship between fluctuations in their order parameter and superconductivity, Mott insulating states, and spin density waves 1,7 . In the TMD 1T-TiSe2 superconductivity appears in proximity to CDW incommensurability 8 , which can be achieved by pressure 9 , copper doping 10 , or electrostatic gating 11 . Thus, understanding of the CDW transition mechanism for this material has attracted considerable scientific interest, especially concerning its driving mechanism 12,13,14 .
The CDW phase in 1T-TiSe2 is achieved upon cooling below TCDW = 202 K, where the hexagonal lattice of the normal phase found at room temperature undergoes a reconstruction forming a 2a  2a  2c superlattice. This structural fingerprint, denoted here as periodic lattice distortion (PLD), occurs together with the opening of an electronic gap ( = 130 meV at 80 K), which is large compared to other TMDs exhibiting a CDW 2,15 . Following early experiments on TiSe2, it was suggested that the CDW state is in fact an excitonic insulator stabilised by Coulomb interactions 16,17 , owing to the semi-metallic character of the band structure, featuring holes in the Se-4p valence band (VB) at the Γ ̅ point and electrons in the Ti-3d conduction band (CB) at the M ̅ -point of the first Brillouin Zone (BZ) (Fig. 1a). Thus, the presence of a PLD and excitons motivated several experimental and theoretical studies aimed at identifying the role of phonons compared to Coulomb interactions 12,15,18,19 . These efforts have highlighted 1T-TiSe2 as a model system for studying many-body electron and phonon interactions in condensed matter physics and recently culminated with the report of Bose-Einstein condensation of excitons in this material 20 .
When compared to other CDW materials of the TMD family, 1T-TiSe2 has small atomic displacements in going from the normal phase structure to the PLD, only ~0.02 Å 21 , in stark contrast to changes of up to 0.1 Å observed in 1T-TaS2 for example 18 . The small PLD has been argued to indicate the limited importance of electron-phonon coupling in driving the CDW, thus supporting a purely excitonic mechanism 2,22 . The typical and consistently reported signatures of the CDW transition in 1T-TiSe2 are a downwards shift of the VB energy 23 , as a consequence of the CDW gap () opening, and the presence of backfolded VB appearing in momentum space at M ̅ , consistent with the superlattice structure as sketched in Fig. 1a (the in plane 2a  2a reconstruction results in M ̅ being at the centre of the reconstructed BZ) 24 .
However, these important observations, reported by steady-state ARPES, have failed to conclusively identify the excitonic or lattice contribution to the CDW.
Ultrafast spectroscopy is the experimental tool of choice to probe out-of-equilibrium phenomena in correlated electron systems 25,26,27,28 . One of the central themes in this research field is bottleneck dynamics, where out-of-equilibrium phonons impede excited carriers from re-joining the CDW or superconducting condensate 25,28,29 . Previous ultrafast studies on TiSe2 have not reported such bottleneck effects or used it to disentangle the excitonic and electronphonon contributions to the CDW 30,31,32 . Nevertheless, recent experimental evidence based on optical-pump THz-probe has shown how excitonic order can be transiently suppressed at any sample temperature below TCDW, but with the PLD remaining robust only up to 150 K 13 . Also, signatures of phonon driven oscillations in the CDW recovery have been seen by TR-ARPES clearly suggesting a role for phonons 32  Here, we use TR-ARPES and time resolved optical reflectivity (TRR) to clarify how phonon dynamics in 1T-TiSe2 influence the CDW recovery following transient perturbation by 30 fs, 1.82 eV light pulses. In TR-ARPES, thanks to a purposely designed combination of time resolution (< 70 fs), energy resolution (~ 53 meV), and sensitivity at low laser fluence (enabled by the 80 kHz laser repetition rate), we are able to probe the VB dynamics in unexplored conditions. We find three distinct out-of-equilibrium regimes as a function of excitation fluence. At fluences below Fth = 60 J cm -2 the weakly perturbed CDW recovers within a short timescale of 2 ps. Above this fluence the CDW is still partially present, but its recovery exhibits a bottleneck concomitant with a change in the coherently coupled phonons seen in TRR. With the help of a rate equation model we describe how phonons contribute to the recovery dynamics. Finally, for a fluence above FCDW = 200 J cm -2 , we enter a third regime with a transient complete suppression of the CDW.
In TR-ARPES, the infrared pump pulse first promotes electrons from the occupied to the unoccupied states with the same momenta, while the subsequent UV probe pulse is used to photoemit electrons and the transient energy dispersion is mapped in momentum space (see methods section for details). Figures 1b,c,d show the evolution of the TR-ARPES maps at three different time delays between the pump pulse and the probe (6.05 eV) for a TiSe2 single crystal at 80 K. The sample temperature is ideal since it is below TCDW = 202 K, but sufficiently high to allow perturbations to the PLD. Figure 1b shows the TR-ARPES map at -225 fs delay (i.e. before the pump photoexcitation), which reflects the VB dispersion in the vicinity of the Γ ̅ point ( ∥ = 0) in the CDW phase along the K ̅ Γ ̅ K ̅ direction. The effect of the pump is apparent in Fig. 1c at time delay +25 fs where electrons from the VB have been promoted into a high energy CB for k-states at the edges of our detection window. As a high energy CB becomes transiently populated, photoelectron signal is observed above the Fermi level. The effect of the pump and the TR-ARPES maps exhibit features similar to what has been reported in other TR-ARPES studies 31,33,34 . Like in other semimetals, 35,36 carrier relaxation from high energy states occurs within a few hundred fs (see Fig. S10).
The key signature of the CDW is the gap, Fig 1d shows that the VB is shifted upwards in energy (i.e.  is reduced) even at +425 fs delay. This VB shift lasting longer than the pump laser duration (30 fs) is a signature of laser-induced perturbation of . Changes in the VB binding energy are extensively documented in steady-state ARPES, when heating the 1T-TiSe2 lattice from T < TCDW to the normal phase 15,23 . In order to accurately study the VB energy shift, we have performed an analysis at the fixed detection angle -14º ( ∥ ≈ -0.1 Å -1 for the VB) (dotted white line in Fig.1 Figure 2b reports the VB shift as a function of fluence at specific time delays corresponding to the maximum shift (dots) and at t = 3 ps (triangles), the latter will be discussed below. To gauge the VB maximum shift with respect to the level of CDW perturbation, we have also plotted the shift in the equilibrium VB binding energy as horizontal lines in Fig. 2b, taken from high resolution steady-state ARPES 23 in going from 70 K to 300 K. Two key observations are apparent from the trend of the VB maximum. First, the maximum shift is initially linear for low fluence before reaching a plateau for F > 93 J cm -2 equivalent to a shift in the equilibrium VB binding energy observed at temperatures between 180 K and 200 K.
Second, this plateau persists until a critical fluence of FCDW = 200 J cm -2 , beyond which the VB transiently shifts above the 200 K line of equilibrium data and is consistent with complete suppression of  and disappearance of CDW order. These nonlinear trends are not due to saturation of absorption from TiSe2 or average laser heating, as shown in Supplementary, but are instead an intrinsic characteristic of the CDW dynamics.
Returning to the VB dynamics in Fig. 2a for the low fluences of 31 and 62 J cm -2 we find a fast recovery, described by a mono-exponential decay with equal time constants of ~ 770 fs.
This leads to a complete VB recovery, i.e. 0 eV shift, at time delays > 2 ps. For higher fluences, multi-exponential decays with lifetimes longer than 1100 fs are found. The inset in Fig. 2a gives a clearer comparison of the same data and plotted on a logarithmic normalised intensity scale. The dynamics can be grouped into two well-defined categories and points to a threshold fluence, Fth, between these regimes of fast and slow VB recovery. The residual VB shift at 3 ps (triangles), shown in Fig. 2b, clearly identifies Fth > 62 J cm -2 . Note that Fth  FCDW/3.3, and so does not coincide with the complete suppression of  occurring at FCDW.
Following these observations, we look at the spectral intensity dynamics from the TR-ARPES.
The left panel in Fig. 2c shows the spectral weight obtained by integrating the PES up to the Fermi level for F < Fth, normalised as described in caption. For both fluences the spectral weight is depleted upon photoexcitation and re-established within 500 fs. However, for the 62 J cm -2 data (orange curve) at a delay > 500 fs the spectral weight shows a small and short-lived intensity gain. Similar behaviour but more pronounced and longer lasting, is observed for the two higher fluences reported in the right panel of Fig. 2c. Photoexcitation with F ≥ Fth, therefore increases spectral weight in the VB. To confirm the origin of this gain we compare the traces at 80 K and 300 K in Fig. 2d. At negative delay, the total intensity measures electrons in the VB up to the Fermi level. This is diminished when going from 300 K to 80 K because CDW formation transfers spectral weight from Γ ̅ to the VB folded at M ̅ (Fig. 1a). Crucially, the 300 K data in the normal phase do not show any increase in intensity above the initial value.
Thus, the observed gain (blue arrow) is caused by the photo-induced unfolding of the VB from M ̅ to Γ ̅ points and indicates breaking of exciton pairs in the CDW condensate 33 , and/or a disturbance of the PLD.
The results of Fig. 2 report the VB dynamics probed in a fluence regime so far unexplored by TR-ARPES in 1T-TiSe2. It is important to point out that previous time resolved all-optical experiments in the low fluence regime F < Fth have clarified how the initial perturbation of the CDW by fs laser pulses is non-thermal, i.e. it concerns mainly the electronic order in the system represented by the exciton condensate and to a negligible extent the lattice degrees of freedom 13,14 . This is consistent with electron-electron and electron-exciton scattering times on the order of hundreds of femtoseconds 13 . When discussing possible scenarios for the CDW in TiSe2, we consider that the CDW formation is due to both excitonic and lattice interactions, where the relative contributions and relationship are currently unknown. For fluences below Fth, the rapid VB recovery is consistent with electronic dynamics and suggests that mainly the excitonic part of the CDW is perturbed. Above Fth there is an additional contribution with a longer recovery time, indicating a bottleneck in re-establishing the CDW ground state.
Interestingly, Fth identified from the 3 ps data in Fig. 2b corresponds to a VB position at 150 K from steady-state measurements. This is the sample temperature at which recent femtosecond THz experiments reported the disappearance of the phonon fingerprint of the PLD 13 . All together we suggests that the < 200 fs non-thermal shift of ~0.05 eV at Fth can be interpreted as the initial excitonic contribution to the CDW. Above Fth a second weakening process for  sets in, with a maximum contribution of ~0.03 eV, estimated from the 3 ps VB shift at FCDW.
A simple picture in which excitonic and phononic contributions can be identified and summed to obtain the full  does not apply. Rather we now discuss how phonons can influence the dynamics of  Understanding of the CDW bottleneck dynamics benefits from monitoring lattice degrees of freedom. While steady-state ARPES experiments can provide indirect information on lattice structure from changes in the electronic band structure, time-resolved experiments allows the dynamics of a subset of phonons to be probed in real time 37 . In TR-ARPES, we observe that below Fth the VB dynamics are modulated by periodic oscillations, very likely connected to coherent phonons, whereas above Fth their amplitude weakens or is undetectable ( Supplementary Information section 4). Oscillations of the VB in TR-ARPES signify the presence of phonons connected with the order parameter,  thus it is conceivable that such oscillations will weaken as the CDW is perturbed above Fth. Further information on phonon dynamics can be obtained from optical TRR experiments which offer a slightly higher time resolution and probe the change in refractive index of our crystal modulated by phonons.  Fig. S7 that the A1g* mode is selectively coupled to the CDW. At F  132 J cm -2 the oscillations are instead dominated by the higher-frequency mode, similar in frequency to the A1g phonon of the normal phase, or two zone-edge modes triggered by a second order process seen in other solids with phase transitions to broken symmetry states 39,40 . We argue that both assignments for the high frequency 6.03 THz oscillations are consistent with a perturbed PLD and excitation of phonon modes of the normal phase structure (Section 4 in Supplementary information).
The progressive change of amplitude between the selectively coupled phonons (SCP) as the laser intensity increases is symptomatic of two phenomena: (i) A1g* phonons that do not couple coherently to the CDW recovery, (ii) a loss of PLD (disappearance of A1g*) and thus rearrangement of the lattice towards the normal phase, as also supported by the unfolding results (Fig. 2c). Information on phonons is relevant for the bottleneck, since a substantial population of excited vibrational modes (hot phonons) can transfer energy back into the already perturbed exciton condensate and suppress its re-establishment. Most important is the observation that phonons linked to the normal phase structure modulate the dynamics when the bottleneck in the recovery appears, as shown by the comparison of FT amplitudes and VB shift in Fig. 3d.
We have performed a series of simulations capable of describing the VB dynamics as a function of laser fluence. These are inspired by the Rothwarf-Taylor model which was initially used to describe the equilibration of Cooper pairs in superconductors with hot electrons and phonons 25,40 and has also been applied to CDW materials 41 where 80K = 130 meV is the CDW gap at 80 K(ref. 23  They impinged on the sample surface at about 45° with crossed polarisations in order to avoid interference artefacts. All the experiments were performed at a sample temperature ranging from 80 K to 300 K as specified in the text.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author.  lines are linked to the right y-axis and are the shift in VB energy determined by high resolution steady state ARPES as the sample temperature is increased from 70 K, adapted from reference [23]. The VB position at 70 K from steady state ARPES has been set to coincide with the VB energy at negative delays in our experiments. Energy error bars on VB shift data points are < 2 meV. c, Intensity of the VB at 80 K and for different laser fluences as indicated, normalised to ARPES intensity at negative delays and at room temperature (shown in d). d, Comparison between the normalised VB intensity in the CDW phase (80 K) and normal phase (300 K). A gain in intensity is observed in the 80 K data indicated by the blue shaded region.