## Dynamics of collective modes in an unconventional charge density wave system BaNi2As2

We studied the *T*– and *F*-dependence of the photoinduced near-infrared reflectivity dynamics in undoped BaNi_{2}As_{2} using an optical pump-probe technique. The single crystals were cleaved along the a-b plane with the pump and the probe beams at near-normal incidence (they were cross-polarized for higher signal-to-noise ratio). We performed also pump- and probe-polarization dependence of the photoinduced reflectivity, with no significant variation being observed (see Supplementary Note 4). The reported temperature dependence measurements were performed upon warming. Continuous laser heating was experimentally determined to be about 3 K for *F* = 0.4 mJ cm^{−2}, and has been taken into account (see Supplementary Note 6).

### Photoinduced reflectivity dynamics in the near-infrared

Figure 1a presents the *T*-dependence of photoinduced reflectivity transients, Δ*R*/*R*(*t*), recorded upon increasing the temperature from 10 K, with *F* = 0.4 mJ cm^{−2}. This fluence was chosen such that the response is still linear, yet it enables high enough dynamic range to study collective dynamics (see also section on excitation density dependence).

Clear oscillatory response is observed up to ≈ 150 K, with the magnitude displaying a strong decrease near and above *T*_{S}. Similarly to the oscillatory signal, the overdamped response is also strongly *T*-dependent. As shown in Fig. 1b the response can be decomposed into an overdamped and oscillatory response. To analyze the dependence of the oscillatory response on *T*, we first subtract the overdamped components. These can be fit by

$$\frac{{{\Delta }}R}{R}=H\left(\sigma ,t\right)\left[{A}_{1}{e}^{-t/{\tau }_{1}}+B+{A}_{2}\left(1-{e}^{-t/{\tau }_{2}}\right)\right],$$

(1)

where *H*(*σ*, *t*) presents the Heaviside step function with an effective rise time *σ*. The terms in brackets represent the fast decaying process with *A*_{1}, *τ*_{1} and the resulting quasi-equilibrium value *B*, together with the slower buildup process with *A*_{2} and *τ*_{2}, taking place on a 10 ps timescale—see inset to Fig. 1b. Figure 1c presents the oscillatory part of the signal subtracted from the overdamped response at selected temperatures together with the fit (black dashed lines) using sum of four damped oscillators (discussed below).

### Collective modes in BaNi_{2}As_{2}

Figure 2 presents the results of the analysis of the oscillatory response. Figure 2a shows the *T**–*dependence of the Fast Fourier Transformation spectra in the contour plot, where several modes up to ≈ 6 THz can be resolved, with the low-T mode frequencies depicted by red arrows. To analyze the temperature dependence of the modes’ parameters we fit the oscillatory response to a sum of damped oscillators, \({\sum }_{i}{S}_{i}\cos \left(2\pi {\widetilde{\nu }}_{i}t+{\phi }_{i}\right){e}^{-{{{\Gamma }}}_{i}t}\).

Figure 2b–d presents *T*-dependences of the extracted mode frequencies *ν*_{i} (here \({\nu }_{i}^{2}={\widetilde{\nu }}_{i}^{2}+{({{{\Gamma }}}_{i}/2\pi )}^{2}\)—see ref. ^{30}), dampings *Γ*_{i}, and spectral weights (*S*_{i}) of the four dominant modes (see also Supplementary Notes 2 and 3). Noteworthy, all these low-frequency modes are observed up to ≈ 150 K, well above *T*_{S} = 138 K and \({T}_{{{{{{{{\rm{S}}}}}}}}^{\prime} }=142\) K. While their spectral weights are dramatically reduced upon increasing the temperature through *T*_{S}, their frequencies and linewidths remain nearly constant through *T*_{S} and \({T}_{{{{{{{{\rm{S}}}}}}}}^{\prime} }\).

In Fig. 3 we present the result of the phonon dispersion calculations for the high-temperature tetragonal structure. None of the experimentally observed low-frequency modes matches the calculated **q** = 0 mode frequencies. Therefore, and based on their *T*– and *F*-dependence, discussed below, we attribute these modes to collective amplitude modes of the CDW order^{19,21,27,28,31}. In particular, we argue that these low-temperature **q** = 0 amplitude modes are a result of linear (or higher order^{19,21}) coupling of the underlying electronic modulation with phonons at the wavevector **q**_{CDW} (or *n* ⋅ **q**_{CDW} for the *n*-th order coupling^{19,21}) of the high-*T* phase. Within this scenario,^{19,21,27,28,31,32} the low-*T* frequencies of amplitude modes should be comparable to frequencies of normal state phonons at **q**_{CDW} (or *n*⋅**q**_{CDW} for the higher-order coupling), with renormalizations depending on the coupling strengths. Moreover, *T*-dependences of modes’ parameters *ν*_{i}, Γ_{i}, and *S*_{i} should reflect the temperature variation of the underlying electronic order parameter^{19,21,27,28,31,32}.

The first support for the assignment of these modes to amplitude modes follows from calculations of the phonon dispersion, presented in Fig. 3. Note that, since these modes appear already above *T*_{S}, their frequencies must be compared to phonon dispersion calculations in the high-temperature tetragonal phase. Figure 3 presents the calculated phonon dispersion in the [100] and [101] directions, along which the modulation of the I-CDW and C-CDW, respectively, is observed. In Figure 3, frequencies of the experimentally observed modes are denoted by the dashed horizontal lines (the line thicknesses reflect the modes’ strengths).

Indeed, the frequencies of strong 1.45 THz and 1.9 THz modes match surprisingly well with the calculated phonon frequencies at the I-CDW modulation wavevector (given by the vertical dashed line in Fig. 3a), supporting the linear-coupling scenario. The corresponding (calculated) frequencies of phonons at the C-CDW wavevector, shown in Fig. 3b, are quite similar. As shown in Fig. 2, both modes display a pronounced softening upon increasing temperature, much as the dominant amplitude modes in the prototype quasi-1D CDW system K_{0.3}MoO_{3},^{19,21} as well as dramatic drop in their spectral weights at high temperatures^{19}. Finally, the particular *T*-dependence of Γ for the 1.45 THz mode clearly cannot be described by an anharmonic phonon decay model, given by \({{\Gamma }}(\omega ,T)={{{\Gamma }}}_{0}+{{{\Gamma }}}_{1}(1+2/{e}^{h\nu /2{k}_{B}T}-1)\)^{33}. Instead, the behavior is similar to prototype CDW systems, where damping is roughly inversely proportional to the order parameter^{19,21}.

Given the fact that the structural transition at *T*_{S} is of the first order, such a strong T-dependence of frequencies and dampings at *T* < *T*_{S} may sound surprising. However, as amplitude modes are a result of coupling between the electronic order and phonons at the CDW wavevector,^{19,21,28} the *T*-dependence of the mode frequencies and dampings reflect the *T*-dependence of the electronic order parameter^{19,21}. Indeed, the *T*-dependence of PLD^{10} as well as of the charge/orbital order^{15} do display a pronounced *T*-dependence within the C-CDW phase.

A strongly damped mode at 0.6 THz also matches the frequency of the calculated high-temperature optical phonon at **q**_{I−CDW}. We note, however, that the calculations imply this phonon to have an instability near **q**_{I−CDW}, thus the matching frequencies should be taken with a grain of salt. The extracted mode frequency does show a pronounced softening (Fig. 2b), though large damping and rapidly decreasing spectral weight result in a large scatter of the extracted parameters at high temperatures. We further note the anomalous reduction in damping of the 0.6 THz mode upon increasing the temperature (Fig. 2c). Such a behavior has not been observed in conventional Peierls CDW systems,^{19,21} and may reflect the unconventional nature of the CDW order in this system. We note, that phonon broadening upon cooling was observed for selected modes in Fe_{1+y} Te_{1−x}Se_{x}^{34,35} and NaFe_{1−x}Co_{x}As^{36} above and/or below the respective structural phase transitions. Several interpretation have been put forward for these anomalous anharmonic behaviors, that can have distinct origins^{34,35,36}.

A weak narrow mode at 1.65 THz is also observed, which does not seem to have a high temperature phonon counterpart at the **q**_{I−CDW}. Its low spectral weight may reflect the higher-order coupling nature of this mode.

Finally, several much weaker modes are also observed (see Fig. 2a). Comparison with phonon calculations suggest 3.3 THz and 5.4 THz modes are likely regular **q** = 0 phonons, the 5.9 THz mode could also be the amplitude collective mode, while the nature of 0.17 THz mode is unclear (see Supplementary Note 3 for further discussion and Supplementary Note 5 for complementary data obtained by simultaneous Raman spectroscopy). We note that, as the pump-probe technique is mostly sensitive to A_{g} symmetry modes that couple directly to carrier density^{22,30}, the stronger the coupling to the electronic system, the larger the spectral weight of the mode. Correspondingly, in time-resolved experiments the spectral weights of amplitude modes are much higher than regular **q** = 0 phonons.

### Overdamped modes in BaNi_{2}As_{2}

Further support for the CDW order in BaNi_{2}As_{2}^{10,15} is provided by the *T*-dependence of overdamped components. Figure 4a presents the *T*-dependence of signal amplitudes *A*_{1} + *B*, which corresponds to the peak value, and *A*_{2} extracted by fitting the transient reflectivity data using Eq. (1). In CDW systems the fast decay process with *τ*_{1} has been attributed to an overdamped (collective) response of the CDW condensate,^{19,21} while the slower process (*A*_{2}, *τ*_{2}) has been associated to incoherently excited collective modes^{21}. As both are related to the CDW order, their amplitudes should reflect this. Indeed, both components are strongly reduced at high temperatures, with a pronounced change in slope in the vicinity of *T*_{S}—see Fig. 4a. Component *A*_{2} displays a maximum well below *T*_{S}, similar to the observation in K_{0.3}MoO_{3}^{37}. Above ≈ 150 K the reflectivity transient shows a characteristic metallic response, with fast decay on the 100 fs timescale.

The evolution of timescales *τ*_{1} and *τ*_{2} is shown in Fig. 4b. In the C-CDW phase, up to ≈110−120 K, the two timescales show qualitatively similar dependence as in prototype 1D CDWs:^{19,20,21}*τ*_{1} increases with increasing temperature while *τ*_{2} decreases^{19,20,21}. As *τ*_{1} is inversely proportional to the CDW strength,^{19,21} its *T*-dependence is consistent with the observed softening of the amplitude modes. Its increase with increasing temperature is, however, not as pronounced as in CDW systems with continuous phase transitions, where timescales can change by an order of magnitude when gap is closing in a mean-field fashion^{18,19,20,21}. From about 130 K *τ*_{1} remains nearly constant up to ≈150 K. On the other hand, for *T* ≳ 120 K *τ*_{2} displays a pronounced increase, though the uncertainties of the extracted parameters start to diverge as signals start to faint. Importantly, all of the observables seem to evolve continuously through *T*_{S}, despite the pronounced changes in the electronic and structural properties that are observed, e.g., in the c-axis transport^{38} or the optical conductivity^{16,17}.

### Excitation density dependence

Valuable information about the nature of CDW order can be obtained from studies of dynamics as a function of excitation fluence, *F*. In conventional Peierls CDW systems a saturation of the amplitude of the overdamped response is commonly observed at excitation fluences of the order of 0.1–1 mJ cm^{−2}^{20,24,25,26}. The corresponding absorbed energy density, at which saturation is reached, is comparable to the electronic part of the CDW condensation energy^{24,26}. Similarly, the spectral weights of amplitude modes saturate at this saturation fluence. The modes are still observed up to excitation densities at which the absorbed energy density reaches the energy density required to heat up the excited volume up to the CDW transition temperature^{24}. The reason for this is an ultrafast recovery of the electronic order on a timescale *τ*_{1}, which is faster than the collective modes’ periods^{24}.

We performed *F*-dependence study at 10 K base temperature, with *F* varied between 0.4 and 5.6 mJ cm^{−2}. The reflectivity transients are presented in Fig. 5a. Unlike in prototype CDWs, no saturation of the fast overdamped response is observed up to the highest *F* (inset to Fig. 5b). The absence of spectroscopic signature of the CDW induced gap in BaNi_{2}As_{2}^{17} suggest that most of the Fermi surface remains unaffected by the CDW order. Thus, the photoexcited carriers can effectively transfer their energy to the lattice,^{39} just as in the high-*T* metallic phase. Nevertheless, the fact that the excitation densities used here do exceed saturation densities in conventional CDW systems by over an order of magnitude suggests an unconventional mechanism driving the CDW in BaNi_{2}As_{2}. We note that signal *A*_{2} displays a super-linear dependence for *F* > 2 mJ cm^{−2}.

Figure 5b presents *τ*_{1}(*F*) and *τ*_{2}(*F*) for the data recorded at 10 K. Qualitatively, the *F*-dependence of the two timescales resembles their temperature dependence, similar to observations in Peierls CDW systems^{24}. Since *τ*_{1} reflects the recovery of the electronic part of the order parameter, Δ, and follows *τ*_{1} ∝ 1/Δ,^{19,21} this observation supports a continuous suppression of the electronic order with increasing *F*. However, in Ni-122 no discontinuous drop in *τ*_{1}(*F*) is observed up to the highest fluences. In K_{0.3}MoO_{3}^{24} such a drop in *τ*_{1}(*F*) is observed at the fluence corresponding to the full suppression of the electronic order.

Figure 5c–e presents the *F*-dependence of the extracted amplitude mode parameters. A softening upon increasing the fluence is observed for all four modes (Fig. 5c). However, above ≈ 3 mJ cm^{−2} the values reach a plateau. Such an unusual behavior is not observed in Peierls CDWs^{19,21} and may hold clues to the interplay between the periodic lattice distortion and the underlying electronic instability. An indication of suppression of the underlying electronic order is observed also as saturation of spectral weights of some of the amplitude modes near *F* ≈ 3 mJ cm^{−2}, see Fig. 5e. On the other hand, the mode at 1.45 THz, which is the most similar to main modes in K_{0.3}MoO_{3}, shows no such saturation up to the highest fluences. While the observed anomalies seen near *F* ≈ 3 mJ cm^{−2} may be linked to the underlying microscopic mechanism of CDW order in Ni-122, one could also speculate the anomalies may be related to the photoinduced suppression of commensurability.

To put the observed robustness of the CDW against optical excitation into perspective, we note that *F* = 1 mJ cm^{−2} corresponds to the absorbed energy density of about 180 J cm^{−3} (110 meV per formula unit). Assuming rapid thermalization between electrons and the lattice, and no other energy decay channels, the resulting temperature of the excited sample volume would reach ≈ 160 K (see also Supplementary Notes 6 and 7). However, with high conductivity also along the c-axis^{38} and the estimated electronic mean free path on 7 nm^{40}, transport of hot carriers into the bulk on the (sub)picosecond timescale cannot be excluded. Nevertheless, the fact that even at 5.6 mJ cm^{−2} (0.6 eV per formula unit) the CDW order has not collapsed, underscores an unconventional CDW order in in BaNi_{2}As_{2}^{10,15}.