Persistent order due to transiently enhanced nesting in an electronically excited charge density wave

Non-equilibrium conditions may lead to novel properties of materials with broken symmetry ground states not accessible in equilibrium as vividly demonstrated by non-linearly driven mid-infrared active phonon excitation. Potential energy surfaces of electronically excited states also allow to direct nuclear motion, but relaxation of the excess energy typically excites fluctuations leading to a reduced or even vanishing order parameter as characterized by an electronic energy gap. Here, using femtosecond time- and angle-resolved photoemission spectroscopy, we demonstrate a tendency towards transient stabilization of a charge density wave after near-infrared excitation, counteracting the suppression of order in the non-equilibrium state. Analysis of the dynamic electronic structure reveals a remaining energy gap in a highly excited transient state. Our observation can be explained by a competition between fluctuations in the electronically excited state, which tend to reduce order, and transiently enhanced Fermi surface nesting stabilizing the order.

The quasi two-dimensional electronic band structure close to the Fermi level in RTe 3 can be well approximated by a simplified tight binding (TB) model of the in-plane Te 5p x /z orbitals, that exhibit coupling along and perpendicular to the chains, t and t ⊥ , see Fig. 1a of the main manuscript. This leads to the following band dispersion [1]: with E F = −2t sin(π/8) fixed by the band filling, and the TB coupling parameters t ≈ −1.9 eV and t ⊥ ≈ 0.35 eV chosen to match the experimental Fermi surface. The bands of this noninteracting TB model, shown in Supplementary Fig. 1(a) as black solid lines, gain a small curvature proportional to the ratio t ⊥ /t leading to the diamond-shaped Fermi surface (compare to Fig. 1 of the main manuscript).
Nesting of this metallic FS along the nesting vector q CDW along c * transfers spectral weight into shadow bands translated by ±q CDW (green dashed lines). The coupling between bare bands |k and shadow bands |k ± q of strength ∆ is taken into account by considering the following wave function which allows for the coupling of states |k and |k ± q , which was termed interacting TB model in the literature [2,1,3]: where the coefficients are determined as the solutions of the coupling matrix: This leads to the opening of gaps 2∆ around the crossing of main and shadow bands, and splitting of the bands into two branches [4]: see Supplementary Fig. 1(b). Due to the slight mismatch in curvature of main and shadow band, the band crossing and hence the position of the gap relative to E F shifts with momentum, leading to residual metallic pockets on the FS where the lower band shifts above E F , see FS in Supplementary Fig. 1(a) and blue curve in Supplementary Fig. 1(b). To model the data in the main manuscript we included in addition a coupling of p x and p z bands and 3-dimensional folding of the bands [1,3], leading to a slight shift of the gap position with respect to the band crossing.
The gain in electronic energy due to the opening of the gaps in the electron dispersion can be estimated from the density of states (DOS) ρ( ). Supplementary Fig. 1(c) shows the DOS of interacting and non-interacting TB model, as red and black lines, respectively. The opening of the gap leads to a transfer of states from the Fermi level to lower energies, shown by red (increase) and green (decrease) colors. The electronic energy in the valence band is given by where F ≈ 6 eV from the band bottom. Supplementary Fig. 1(d) shows the electronic energy for the interacting and non-interacting TB model, yielding a gain in electronic energy δE = E non−interacting − E interacting ≈ 250 meVu.c. −1 . For these estimates, q CDW has been adjusted accordingly to keep the integrated DOS at E F constant.

Supplementary Note 2 Determination of the transient CDW band positions
For the determination of the position of the CDW bands, transient trARPES spectra are fitted, as shown for exemplary pump-probe delays in Supplementary Fig. 2. To account for the highly non-thermal distribution of electrons within the first picosecond, a fitting model composed of two independent parts for the occupied and unoccupied band structure was considered, shown as green and yellow areas, respectively. Each part consists of a linear background and a Lorentzian peak function, multiplied with a Fermi-Dirac distribution: Here, i = 1, 2 are the fit functions of occupied and unoccupied bands, A i and B i the coefficients of a constant and linear backgrounds, C i , E i and W i the amplitudes, positions and widths of the Lorentzians, and µ i and T i the chemical potential and electronic Temperature of the Fermi-Dirac distributions. The position of the Fermi-Dirac distribution of the unoccupied bands was used as an auxiliary parameter to mimic the cut-off at high energies due to the dispersion of the acceptance angle of the pTOF (see the image plots in Supplementary Fig. 2). We find an excellent description of the data by the fits for all pump-probe delays. Due to the highly non-thermal electronic system after excitation, even the Fermi-Dirac distribution used for the formally occupied part of the spectrum needs to be considered as an auxiliary parameter, as no Fermi level can be defined any more in the spectrum e.g. at t = 200 fs, and is held fixed with a high value of T 1 to produce a smooth curve. In order to crosscheck the results of the fitting procedure, peak positions are compared to the extrema in the second derivative of the data with respect to energy, as commonly used in ARPES [5], confirming the results of the fitting procedure (black markers in Supplementary Fig. 2).

Supplementary Note 3 Determination of the CDW gap size
Due to the progressing thermalization of the electronic system, the position of the upper CDW band becomes hard to determine on top of the background of the hot thermalized electron distribution for t > 500 fs and is indistinguishable from the background for delays t > 1000 fs. In order to determine the transient gap size for all pump-probe delays, we compare the occupied band position inverted and scaled by a factor of ×1.8, shown as green markers in Supplementary  Fig. 3(a), to the unoccupied band position. We find a close agreement for 0 fs < t < 500 fs (black arrow in Supplementary Fig. 3), indicating a constant asymmetry of the gap closing with pumpprobe delay. The deviations towards larger band shift of the unoccupied band found for t > 500 fs correspond to a systematic underestimation of the peak energy, which can be explained by the asymmetric peak shape due to the exponential background of the hot electron distribution and manifests in the large error bars (see fits in Supplementary Fig. 2). Thus, for delays at t < 0 fs and t > 500 fs, where no reliable determination of the unoccupied peak position is possible, we extrapolate the unoccupied peak position by the scaled occupied peak position, assuming a constant proportion of occupied and unoccupied peak shift. This procedure yields the transient CDW gap size, shown in Supplementary Fig. 3(b) and Fig. 3 of the main text.

Supplementary Note 4
Determination of the pump energy density The absorbed pump fluence per pulse F abs in the limit of a Gaussian pump beam profile much larger than the probe beam is given by [4]: where E hν is the pump energy per pulse, R the reflectivity of the sample, α the angle of incidence with respect to the surface normal, and FWHM x/y the full-width at half maximum of the horizontal and vertical pump profiles. The absorbed pump energy density per unit cell uc (z) as a function of depth z from the surface decays with the optical penetration depth d hν ≈ 24 nm as where 1st is the energy density in the 1st unit cell with volume V uc ≈ 480Å 3 [6] at the surface For a photoemission probe depth d probe much smaller than d hν , the effective pump energy density is just given by 1st .

Supplementary Note 5 Transient CDW band dispersion parallel to the Fermi surface
The electronic band dispersion along k 2 parallel to the Fermi surface (see Fig. 1c and Fig. 4 of the main manuscript) were determined separately for occupied and unoccupied states, as a function of momentum. Peak positions were determined by fits of a Lorentzian line shape, plus an exponential background function: Here, A is a constant offset, B and the amplitude and characteristic energy of the exponential background, and C, E 0 and W the amplitude, position and width of the Lorentzian. Solid lines in Supplementary Fig. 4(c) show the fits to the data for two momentum positions, and for selected pump-probe delays. For these fits, no Fermi function was considered, and fits restricted in energy and momentum accordingly. The transient change of the gap dispersion discussed in the main manuscript manifests in the slightly larger shift with pump probe delay of the occupied peak at k x = 0.183Å −1 , and of the unoccupied peak at k x = 0.211Å −1

Position of the gap center in the tight binding model
The energetic position of the CDW gap in the band structure is determined in the tight binding model by the crossing of the main p x /p z bands and the CDW shadow bands that are translated by ±q CDW (compare Supplementary Fig. 1(b)). In order to estimate the effect of the various parameters of the TB model, the position of this crossing of main and shadow band in the non-interacting TB model has been determined for a variation of t ⊥ , t and q CDW , shown in Supplementary Fig. 5(a-c) as a function of k x coordinate, respectively 1 . While we find virtually no change in the gap position upon changing t , a change of t ⊥ nicely reproduces the change in slope of the gap dispersion observed in the transient excited state (see Supplementary Fig. 6 and Fig. 4 of the main text). We find the slope to be proportional to t ⊥ over a large range of values. In contrast, a change in q CDW mostly leads to a shift of the whole gap dispersion, while affecting the slope only very little ( Supplementary Fig. 5(c)).

Supplementary Note 7
Fluence dependent shift of the gap center In order to investigate the fluence dependence of the transient change of the gap dispersion discussed in the main manuscript, the gap position along the Fermi surface (direction k 2 in Fig. 1 (c) of the main manuscript) has been determined from peak fits to the upper and lower CDW peak for various fluences. Supplementary Fig. 6(a) shows the gap center for various fluences as a function of momentum along the Fermi surface at t = 0 fs (circles) and t = 200 fs (triangles), where the gap is minimal. For all fluences, we observe a momentum-dependent shift of the gap center towards the Fermi level during the collapse of the gap, corresponding to a change of the band structure towards a more square-like Fermi surface. In order to quantify this change of the band structure, the slope of the gap center along the Fermi surface has been determined by linear fits (lines in Supplementary Fig. 6(a)) and the relative change of the gap dispersion is shown in Supplementary Fig. 6(b) as a function of absorbed fluence (also shown in the inset of Fig. 4(d) of the main manuscript). Despite the large error bars, the change in slope is distinctly different from zero for all fluences and a clear trend towards larger change in the band dispersion for higher excitation is observed.