Coherent dynamics of the charge density wave gap in tritellurides

The dynamics of the transient electronic structure in the charge density wave (CDW) system RTe3 (R 1⁄4 rare-earth element) is studied using timeand angle-resolved photoemission spectroscopy (trARPES). Employing a three-pulse pump–probe scheme we investigate the effect of the amplitude mode oscillations on the electronic band structure and, in particular, on the CDW energy gap. We observe coherent oscillations in both lower and upper CDW band with opposite phases, whereby two dominating frequencies are modulating the CDW order parameter. This demonstrates the existence of more than one collective amplitude mode, in contrast to a simple Peierls model. Coherent control experiments of the two amplitude modes, which are strongly coupled in equilibrium, demonstrate independent control of the modes suggesting a decoupling of both modes in the transient photoexcited state.


Introduction
Cooperative effects in low-dimensional materials represent a fascinating topic of condensed matter research. Various interactions of the electronic, orbital, spin and lattice degrees of freedom can lead to instabilities and broken-symmetry ground states leading to new emergent properties, which are of both fundamental and technological interest. Examples for such low-temperature broken-symmetry phases are superconducting, charge density wave (CDW) or magnetically ordered states, which are connected to other typically lower symmetry states by phase transitions as function of temperature, pressure, external elds, or are induced by optical excitation. As a model system for emergent order, the formation of a CDW ground state found in many materials of reduced dimensionality is one of the well-established and intensely studied examples at the heart of quantum many-body physics. 1 CDW formation refers to a periodic modulation of the charge density (with wave vector q) which is accompanied by a periodic lattice distortion. A predominant mechanism for CDW formation is the Peierls scenario, 2 which is most pronounced in a one-dimensional (1D) system or in systems where nesting (q ¼ 2k F ) across the Fermi surface occurs. In this mechanism a key ingredient is a high density of states at the Fermi level together with strong, anisotropic electronphonon coupling giving rise to the opening of an energy gap D(k) at the Fermi surface along the nesting direction. 1 The energy gap D(k) is a direct measure of the order parameter of the CDW phase and can be determined by appropriate spectroscopic tools from the occupied and unoccupied single-particle band structure.
Fluctuations or impulsive (e.g. optical) excitations can lead to collective excitations of the amplitude or phase of the complex order parameter D ¼ |D|exp(if). While for the phase mode the magnitude of D remains constant, the so-called amplitude mode directly modulates the magnitude of the gap leading to oscillations of | D | (t) in the free energy potential with a frequency associated to the phonon mode driving the Fermi surface nesting 1 ( Fig. 1(a)). A sketch of a simple linear band model shown in Fig. 1(b) illustrates the effect of the amplitude mode on the electronic band structure: The linear dispersing band (dashed black line) is modied by the opening of a CDW gap of magnitude D CDW at E F , which is  proportional to the CDW order parameter, and is split into a lower and upper band (solid blue and red lines). The effect of the amplitude mode results in a modication of the magnitude of the gap, leading to a shi of both lower and upper CDW bands towards E F (dashed lines) and to a transient modulation of the gap size D(t).
As the formation of broken symmetry ground states and emergent properties in general depends on a complex interplay of various elementary interactions, which occur on intrinsically different energy-and time scales, studies of the nonequilibrium dynamics of complex materials can reveal new insights into the underlying mechanisms and strengths of interaction between the various degrees of freedom. 3,4 Ultrafast time-resolved spectroscopy provides direct access to these dynamics because the relevant elementary scattering and relaxation processes in solids occur typically on femto-to picosecond time scales. In the case of CDW formation the most direct measurement would be the complete mapping of the band structure to extract the order parameter D and to study the interaction of the electronic system with other (phonon) degrees of freedom. In the experiment this is achieved by time-and angle-resolved photoemission spectroscopy (trARPES). Thereby, an ultrashort optical pump pulse excites the system inducing e.g. the melting and subsequent recovery of the CDW phase and time evolution of the electronic structure changes are probed as snapshots by ARPES using a timedelayed ultrashort UV probe pulse (see e.g. ref. [4][5][6][7][8]. The material family of Tritellurides, RTe 3 (R ¼ rare-earth element), represents a quasi-1D model system, well suited to study the Fermi surface nesting driven CDW formation. 9-13 Angle-resolved photoemission spectroscopy (ARPES) has revealed a great deal of information about the electronic structure in RTe 3 . The Fermi surface is modied by large energy gaps of several 100 meV in the CDW phase, making RTe 3 an ideal candidate to study the dynamics intrinsically linked to the CDW state. Substitution of various lanthanides allows direct tuning of the CDW transition temperature T CDW by chemical pressure, whereby for the heavier elements (e.g. DyTe 3 , HoTe 3 ), a second, perpendicular CDW transition at a lower transition temperature occurs. 13,14 In previous trARPES studies on TbTe 3 a transition into a transient metallic state together with a coherent oscillation at $2.3 THz of the CDW band was observed aer ultrafast optical excitation. 6,15 The analysis was based on the changes of the transient dispersion of the gapped CDW band close to E F and the coherent oscillation was attributed to the CDW amplitude mode due to its exclusive occurrence close to k F and in the gapped region of the Brillouin zone of the CDW phase below T CDW . In particular, the observed oscillation modulates the peak position of the occupied CDW band, as well as its amplitude and spectral width. In addition to the dynamics of the occupied band, a strong downshi of the unoccupied band was observed in the rst $300 fs, leading to an initial suppression of the gap size. 15 However, this previous work was lacking a direct measurement of the order parameter D(t) and its temporal evolution over an extended range of time delays to extract detailed information about the coupling of various phonon modes to the band structure and, in particular, to the CDW gap. Therefore, a novel approach is necessary to investigate the coherent dynamics of the full CDW energy gap, as will be presented below.
In this work we study the complete dynamics of the CDW order parameter and investigate the occupied and unoccupied band structure in the gapped region as well as the inuence of the amplitude mode oscillations on the position and size of the CDW gap in full detail. We employ a novel three-pulse excitation scheme to repopulate and probe the population in the upper (normally unoccupied) CDW band over extended periods of time (see Fig. 1(d)). From a detailed analysis of the frequencies of the modes modulating the gap size we obtain two dominating contributions at 2.2 THz and 1.75 THz, which demonstrates the existence of more than one amplitude mode, in contrast to a simple Peierls picture. Furthermore, by employing coherent control of the amplitude modes by a two pulse excitation scheme, we demonstrate a transient decoupling of the amplitude modes, which show strong coupling in thermal equilibrium. 16

Experimental methods
The experimental setup for trARPES has been described in detail elsewhere. 15,17 Single crystals of RTe 3 (R ¼ Tb, Dy) grown by slow cooling of a binary melt 14 were mounted on a 45 slanted sample holder. The samples where cleaved in ultrahigh vacuum (p < 5E À 11 mbar) at T ¼ 30 K, where also the trARPES measurements were performed. For the three-pulse experiments, the output of a Ti:Sa regenerative amplier (Coherent RegA) operating at a repetition rate of 300 kHz was frequency quadrupled using two successive b-bariumborate (BBO) crystals to yield hn probe ¼ 6.0 eV used as probe pulses. The rest of the fundamental beam at hn pump ¼ 1.5 eV was split into two parts and delayed separately using two individual optical delay stages in a Mach-Zehnder interferometer conguration, used as the two pump pulses. For the momentum-and energy-resolved detection of photoelectrons emitted by the probe pulses, the sample was rotated in front of a time-of-ight (TOF) spectrometer with an effective opening angle of AE3.5 . The overall time and momentum resolution was 100 fs and 50 meV, respectively.

Results
The occupied and unoccupied band structure of TbTe 3 in the gapped region probed 50 fs aer excitation is shown in Fig. 2 in a false color representation, along the Brillouin zone (BZ) cut indicated in Fig. 1(c). Here, a single weak pump pulse with a uence of F Pop $ 90 mJ cm À2 (population pulse), set at t Pop ¼ 50 fs before the probe pulse was used to transiently populate the normally unoccupied states at E > E F . Both lower and upper CDW bands are visible below and above the Fermi level, respectively, separated by the CDW energy gap of D CDW $ 400 meV. The delay t Pop was carefully chosen such that the transient population of the unoccupied states was maximal.
In order to investigate the inuence of the amplitude mode oscillation on the size and position of the gap, the three-pulse excitation scheme in Fig. 1(d) was employed. A rst pump pulse (hn Pump ) with a uence of F Pump ¼ 240 mJ cm À2 arriving at a variable time delay t with respect to the probe pulse is used to excite the coherent amplitude mode oscillations. The weaker population pulse is kept at the xed delay t Pop with respect to the probe pulse to transiently repopulate the unoccupied states. The subsequent probe pulse hn Probe then monitors both occupied and unoccupied bands. The much weaker uence F Pop ( F Pump ensures minimizing the inuence of the population pulse on the amplitude mode oscillation.   Fig. 2). For better visibility, the intensity scale for energies above E F is enhanced by 30 times. Aer excitation at t ¼ 0, we nd the characteristic oscillations of the amplitude mode in position and intensity of the occupied peak, 6,15 where the dashed lines are guides to the eye to highlight the oscillation period. In the unoccupied band, also a weak intensity modulation by the amplitude mode is observed, which is highlighted by the contour lines. The transient trARPES spectra shown in Fig. 3(b) at a maximum (blue) and a minimum (green) of the oscillations in the position of the lower band reveal a shi in the band position also in the unoccupied band, which exhibits an opposite sign with respect to the occupied band. For further analysis, the peak positions of the occupied and unoccupied band are determined by Lorentzian line ts and the results are depicted in the upper panel of Fig. 4. We clearly nd oscillations of the same magnitude in the position of both bands with a frequency of f $ 2.2 THz, which, however, show a beating pattern with a period of t $ 3 ps, indicating the presence of more than one mode. Most remarkably, the oscillations are clearly anti-correlated, where a maximum in the lower band corresponds to a minimum in the upper band and vice versa. This behavior corresponds to a transient oscillation of the gap size D CDW ¼ E upper À E lower , as depicted in the lower panel (green). In contrast, the gap center E center ¼ (E upper + E lower )/2 (yellow) shows virtually no oscillations.  Fig. 1(c) in a false color representation. The upper color scale is enhanced by a factor of 70. Red and blue lines are guides to the eye to highlight the dispersion of the upper and lower CDW band, respectively, which are separated by the CDW energy gap D CDW (see Fig. 1(b)).

Paper Faraday Discussions
This This is indeed the expected behavior for the inuence of the amplitude mode of the CDW order parameter, which modulates the amplitude of the complex order parameter and hence the CDW gap size (see Fig. 1(b)). The observation of the coherent oscillations of the CDW gap size as demonstrated by our experiments thus provides an unambiguous and direct proof that the oscillations previously only observed in the lower CDW band 6,15 indeed correspond to amplitude mode excitations of the system.
The main frequencies of the oscillations are analyzed by tting the transient gap size with two damped oscillators and an exponential background function, depicted as dark green line in Fig. 4. The t, which shows a good agreement with the oscillations and in particular with the beating pattern, yields central frequencies of f 1 ¼ 2.230(6) THz (2.2 THz mode) and f 2 ¼ 1.77(2) THz (1.75 THz mode). The initial phases of both modes are found to be cosine-like, in agreement with a displacive excitation of the modes. A fast Fourier transformation (FFT) of the residual data show indications of additional frequencies around 2.6 THz and 3.5 THz, which are however too weak in amplitude to be unambiguously tted. Both the observed frequencies and the ratio of amplitudes, A 1 /A 2 $ 6 agree very well with the coherent phonon modes observed for TbTe 3 with transient optical reectivity. 18 The observation of multiple frequencies that couple to the magnitude of the order parameter is remarkable, as in the mean-eld description of the Peierls transition, only one specic phonon mode at the CDW wave vector q couples to the electronic degrees of freedom and shows a critical soening at  the phase transition. 1 In TbTe 3 , this mode has been identied with the 2.2 THz mode, that shows sizable soening close to the phase transition. 18,19 However, this previous work also demonstrated that there exists a complex interplay between the 1.75 THz mode and 2.2 THz mode, suggesting a strong coupling between both modes §. Our results show that the amplitude of the CDW order parameter D CDW is modulated by both modes, demonstrating the existence of more than one amplitude mode. This can be understood from the quasi-1D nature of RTe 3 , which exhibits more than one phonon branch at the CDW wave vector. These additional phonon branches can also linearly couple to the modulated electronic charge density and exhibit the characteristic features of amplitude modes, as explained in the time-dependent Ginzburg-Landau model. 21,22

Coherent control of the amplitude mode
The observation of both the 2.2 THz and the 1.75 THz mode in the transient oscillations of the order parameter raises the question, how far these modes are mutually coupled and inuence each other or whether they independently modulate the CDW gap. The mode crossing of the two modes observed as a function of temperature in transient reectivity measurements 18 and Raman scattering 19 indeed suggests a substantial coupling, especially near the crossing point of the modes. Additional insight into this coupling can be gained from time- resolved coherent control experiments. Here, a second time-delayed pump pulse is employed to control the oscillation of a coherent phonon launched by the rst pump pulse. Such control of coherent phonon oscillations in the time domain has been demonstrated in a number of materials, [23][24][25][26][27] including selective excitation of certain vibrational modes 28,29 and asymmetric control schemes exploring collective dynamics in correlated material systems. 30,31 For the coherent control experiments, two excitation pulses hn Pump1 and hn Pump2 of equal uence F $ 250 mJ cm À2 separated by the pump-pump delay t 12 were used employing the scheme in Fig. 1(e). Here, the rst pump pulse triggers the amplitude mode oscillation at t 0 , and the second pump pulse is used for coherent control of the oscillations, by coherently enhancing or suppressing the oscillations depending on t 12 . The resulting oscillations are observed by the probe pulse aer the pump1-probe delay t. For each scan of t, t 12 was held x, and multiple scans for various t 12 spanning a whole period of the amplitude mode were performed.
The resulting trARPES intensities of DyTe 3 for a pump-pump delay of t 12 ¼ 0.67 ps and t 12 ¼ 0.93 ps are shown in Fig. 5(a) and (b), corresponding to a coherent suppression or enhancement of the amplitude mode oscillation, respectively. The cross-correlation (XC) of the two pump pulses with the probe pulse are depicted in the lower panels, showing the arrival times and equal excitation amplitudes of both pump pulses. Clearly observed is the efficient suppression of the coherent oscillation for out-of-phase excitation in Fig. 5(a), whereas the oscillation is enhanced for in-phase excitation in Fig. 5(b).
The transient peak position of the lower CDW band determined by line ts is shown for various pump-pump delays t 12 in Fig. 6(a), where the arrival of the control pulse is indicated by the red markers. Aer the initial shi of the peak position associated with the arrival of each pump pulse, the coherent oscillations  are well resolved in all cases. Traces of out-of-phase and in-phase excitation, corresponding to Fig. 5(a) and (b) are highlighted by thick orange and green lines, respectively. In order to quantify the coherent control of the amplitude mode, fast FFT traces obtained from the transient peak position aer background subtraction for t > 2 ps are shown in Fig. 6(b). The position of the two modes at f ¼ 2.2 THz and f ¼ 1.75 THz are marked by red and black dashed lines, respectively. While both peaks are present at t 12 ¼ 0.49 ps (bottommost curve), at t 12 ¼ 0.68 ps (orange curve), the intensity of the 2.2 THz mode is almost completely suppressed, and only the weak mode at 1.75 THz remains. Similarly, at t 12 ¼ 0.93 ps, only the enhanced oscillation of the 2.2 THz mode is observed and the 1.75 THz mode is absent. A t of two Lorentzian line shapes centered at f ¼ 1.75 THz and f ¼ 2.2 THz to the FFT spectra is used to determine the oscillation amplitudes, which are shown in Fig. 6(c) as a function of pump-pump delay t 12 as black and red symbols, respectively. From the response of the spectral weight of the two modes to the control pulse, information on the coupling of the two modes can be gained. Neglecting the decay of the coherent oscillations between the two pump pulses, the coherent oscillation amplitude A(t 12 ) of independent modes can be described by 25 where f is the frequency of the oscillation and A 0 is the maximal amplitude for resonant excitation with both pump pulses. Fits to eqn (1) are shown as solid black and red lines in Fig. 6(c) for the 1.75 THz and 2.2 THz mode, respectively. The data are described nicely by a model with two independent modes.

Discussion
The simple model captures the general behavior of the oscillation amplitudes as a function of t 12 of both modes remarkably well and especially perfectly reproduces their different periods with t 12 . Slight deviations in the extracted amplitudes from the model could be explained due to the phase shi occurring to the oscillations when crossing the resonance condition, which leads to a deviation of the FFT spectra from Lorentzian line shapes, or nonlinearities in the uence dependence of the oscillation amplitudes, which could result in a saturation of the oscillation amplitude. The good agreement of the coherent oscillation amplitude control to the simple model of independent oscillators indicates weak coupling of the different amplitude modes. Especially the fact that we can selectively control the amplitude of each mode, in particular that we can completely suppress the 2.2 THz mode with almost maximal amplitude of the 1.75 THz mode at t 12 ¼ 0.68 ps, and the absence of the 1.75 THz mode at t 12 z 0.8 ps, underlines the independent nature of the two amplitude modes in the coherent control experiment. The observation of weak coupling of the amplitude modes in the photoexcited phase is surprising, as both modes couple to the same electronic order parameter and modulate similar coordinates of the same atoms. 16 Our observation is in contrast to temperature-dependent studies of the amplitude modes in RTe 3 observed by time-resolved optical spectroscopy 18 and Raman spectroscopy. 19 These works demonstrated a mixing and avoided crossing of the two modes at 1.75 THz and 2.2 THz due to strong coupling during the soening of the modes with increasing temperature. In the time-dependent Ginzburg-Landau model this behavior arises due to the coupling of the lattice degrees of freedoms, described as structural order parameters, to the electronic part of the order parameter, which leads to an indirect coupling of the different phonon modes in the system. 22 Thus, the observation of weakly coupled amplitude modes in the coherent control experiments suggests a transient decoupling of the lattice degrees of freedom in the photoexcited state, which is not observed in the static case. A more complete description of the transient order parameter dynamics would require a proper description in the framework of the time-dependent Ginzburg-Landau model, 22 which is however beyond the scope of the current manuscript. In summary, using a three-pulse pump-probe scheme in time-and angle-resolved photoemission experiments we demonstrated the effect of the amplitude mode oscillations on the electronic band structure in the prototypical charge-density wave compounds RT3 3 . We observe coherent oscillations in both lower and upper CDW band with opposite phases, which result in a modulation of the CDW energy gap. The observation of two dominating frequencies modulating the CDW order parameter demonstrates the existence of more than one collective amplitude mode, which is in contrast to the simple Peierls model. This can be understood from the quasi-1D nature of RTe 3 , which allows more than one phonon branch at the CDW wave vector q to couple to the order parameter. Coherent control experiments of the two amplitude modes, which are strongly coupled in equilibrium, allow independent control of the modes and demonstrate a transient decoupling of the amplitude modes in the photoexcited state.