Transient Electronic Structure and Melting of a Charge Density Wave in TbTe3

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Science  19 Sep 2008:
Vol. 321, Issue 5896, pp. 1649-1652
DOI: 10.1126/science.1160778


Obtaining insight into microscopic cooperative effects is a fascinating topic in condensed matter research because, through self-coordination and collectivity, they can lead to instabilities with macroscopic impacts like phase transitions. We used femtosecond time- and angle-resolved photoelectron spectroscopy (trARPES) to optically pump and probe TbTe3, an excellent model system with which to study these effects. We drove a transient charge density wave melting, excited collective vibrations in TbTe3, and observed them through their time-, frequency-, and momentum-dependent influence on the electronic structure. We were able to identify the role of the observed collective vibration in the transition and to document the transition in real time. The information that we demonstrate as being accessible with trARPES will greatly enhance the understanding of all materials exhibiting collective phenomena.

In quantum matter, the emergence of order and collective modes associated with order are key areas to gain knowledge on electronic correlations and collective behavior. Charge density wave (CDW) materials are among the well-established examples that have had a major impact on our understanding of quantum many-body problems (1). Recent efforts on RTe3 (where R = rare earth element) have identified it to be a model system to study Fermi surface (FS) nesting-driven CDW formation (28). The study of the electronic structure and FS of RTe3 by means of angle-resolved photoemission spectroscopy (ARPES) has played a major role in developing our current knowledge (2, 3, 5).

However, conventional ARPES can only provide very limited information on the collective excitations that are important to understand the nature of the many-body state. Important many-body collective modes, such as the CDW amplitude mode (9), can only be detected, for example, by Raman spectroscopy (10), which cannot be directly linked to the CDW gap modulation (the defining signature of the mode) (1). Time- and angle-resolved photoemission spectroscopy (trARPES) offers the capability to simultaneously capture the single-particle (frequency domain) and collective (time domain) information, thus making it possible to directly probe the link between the collective modes and single-particle states that form the charge density wave.

We report on femtosecond time- and angle-resolved photoemission of TbTe3, where we identify the amplitude mode of the CDW state through an inspection of time- and Math-dependent modulations of the single-particle spectral function A(ω,Math,t) in the CDW state (here ω is the electron energy, Math is the electron momentum, and t is the time between excitation and probing). For sufficiently high excitation densities, melting of the charge-ordered state is observed by closing of the CDW band gap through transient recovery of the ungapped electronic dispersion that otherwise can only be observed above the CDW transition temperature.

The details of the ARPES and trARPES experiments, as well as a short introduction to the CDW physics in TbTe3, are presented in (11). First, we used ARPES to characterize the charge dynamics of the CDW system TbTe3 in thermal equilibrium. TbTe3 is a member of the RTe3 family of compounds that exhibit a FS nesting-driven CDW formation (4, 7, 8). The diamond-shaped normal-state FS of RTe3 can be described well by a tight binding (TB) model (2, 3, 5, 11, 12) (Fig. 1A). When the transition temperature Tc for TbTe3 is 335 K (8), the CDW gap at 300 K is almost closed (Fig. 1A). At a Tc of 100 K, the gap extends further around the FS (Fig. 1A′). We analyzed the Brillouin zone (BZ) near the diamond tips through the region in which the CDW gap appears (indicated by red arcs in Fig. 1, A and A′) as a function of energy EEF (where E is the electron energy and EF is the Fermi energy). The intensity maps in Fig. 1, B and B′, show a p-like (11) Te band at ∼0.5-eV binding energy that disperses weakly and a p-like (11) Te conduction band (CB) that disperses through EF above Tc and exhibits the CDW gap well below (Fig. 1, C and C′).

Fig. 1.

Measurements taken at 300 K (upper panels) and 100 K (lower panels). Throughout the figures, the photoelectron intensity is encoded in a false-color scale [inset in (A)]. (A and A′) FS maps from ARPES of TbTe3 at 300 K (A) and 100 K (A′). TB model (black dotted lines) and CDW nesting vector QCDW (white arrow) are indicated. The position of the cuts shown in (B) and (B′) and (C) and (C′) is marked in (A) and (A′) (red curve). (B and B′) Cuts extracted by interpolation from the data taken by conventional ARPES at a photon energy of 23 eV. (C and C′) Respective cuts done using the trARPES system at a probe photon energy of 6 eV. See (11) for experimental details.

We leveraged the marked progress in pump-probe spectroscopy at ultrafast time scales for trARPES (1320). We pumped the CDW system optically with an infrared (1.5-eV) laser pulse of 50-fs width and subsequently probed it after a variable delay by photoemitting electrons with an ultraviolet (6-eV) laser pulse of 90-fs width, resulting in a total experimental time resolution of 0.1 ps (11). Thus, the time-dependent evolution of the electron occupation (1315) and the transient single-particle spectral function (16, 17) was probed, and with that, the fingerprint of collective excitations (21) and transitions (22, 23) were also probed. Figure 1, C and C′, shows the same cuts across the BZ as the ones seen in Fig. 1, B and B′, but the former were taken with our trARPES setup using 6-eV femtosecond laser pulses (but without optical excitation). The good agreement ensures that we probe the same BZ position in conventional ARPES as well as with trARPES in what follows.

The electron spectral intensity is shown (Fig. 2, A and B) at the FS well below TCDW (where TCDW is the CDW phase-transition temperature) as a function of pump-probe delay for two excitation densities F, representing incident pump fluence. Starting with F = 0.3 mJ/cm2, Fig. 2C depicts spectra before, at, and after the optical excitation. Optical excitation populates states above EF by hot electrons and has minor influence on the occupied states at 0 fs.

Fig. 2.

(A and B) Photoelectron intensity as a function of energy and delay taken at kF and 100 K. (A), (C), and (E) were measured at weak fluences (0.3 mJ/cm2); (B), (D), and (F) were measured at strong fluences (2 mJ/cm2). (C and D) Spectra from (A) and (B) at selected delays on a logarithmic (linear) intensity scale. (E and F) Time-dependent binding energy of the Te band (blue lines) and the CB (black lines) for weak and strong fluences, respectively. Analysis was done by fitting Lorentzians to the Te [blue lines in (E) and (F)] and CB [bold black line in (E)] states. Far from equilibrium at small delays and high fluences, the center of mass was determined for the CB [thin black line in (E), bold black line in (F)] [see (11) for method details]. The dotted black section in (E) indicates the region in which Lorentz fitting is not applicable (11), whereas the dotted black curve in (F) is a magnification of the center of mass (thick black curve). (G) The determined frequencies of the collective modes are summarized for 300 and 100 K as a function of incident pump fluence. Circles, Te band; squares, CB; solid symbols, energy variation; symbols with a center dot, amplitude; symbols with a center cross, linewidth variation. Diamonds indicate the frequency of the first moment from (F).

The continuous population of states above EF occurs through electron-electron scattering within the experimental time resolution because, under the excitation conditions used, inelastic electron-electron scattering is known to be the dominant relaxation process within the first few 100 fs (1315, 24). This efficient scattering ensures that effects of coherent polarization are negligible and that we observe a transient electronic population. Within 400 fs, the electron distribution has thermalized, and the broadening of the Fermi-Dirac distribution indicates a significantly enhanced electron temperature Tel. In addition, the CB state shows a considerable shift of its spectral weight toward EF after pumping, maximized at 400 fs (Fig. 2A). During this time, the excess energy resides in the electronic system. As Tel is enhanced with respect to the lattice temperature, we attribute these effects to the hot electron distribution (16, 17). Electrons and lattice then equilibrate through electron-phonon scattering within ∼1 ps. Beyond that time, the analysis of the transient binding energy and center of mass reveals an oscillatory behavior (Fig. 2E). Oscillatory changes of the CB state are well resolved to persist for several picoseconds, and we determined a frequency ΩCB ≈ 2.3 THz. For the Te band, the changes are much weaker, and a Fourier transformation (11) yields a periodic contribution at ΩTe ≈ 3.6 THz. Thus, the CB is clearly more susceptible to the excitation.

Photo-doping (electron-hole pair excitations generated by optical pumping) causes the electronic part of the CDW to decrease almost instantaneously. This starts a movement of the ionic part of the CDW because of the instantaneous change in electronic screening. Because the pump pulse duration is shorter than the oscillation period, a phase relation with respect to time zero is established (25). When the CDW amplitude is above its equilibrium value, the ions will be driven back by a steep rise in the lattice strain energy. When the CDW amplitude is below equilibrium, the steep loss of the energy gained from electron-phonon interaction will drive them back. The ions thus start to oscillate (1). This amplitude mode would then be excitable and directly observable with our technique. Based on the data presented so far and those to follow, we attribute the 2.3-THz oscillation to the amplitude mode. Our data directly reveal that the amplitude mode oscillation drives the CB and thus the CDW gap modulation.

We increased F to 2 mJ/cm2 to show further evidence that is consistent with this assignment. Respective data are shown in Fig. 2B. Clearly, the impact of oscillations on the electronic states has increased. The spectral function of the CB state depends on the oscillation phase, as seen from spectra at 160 and 300 fs given in Fig. 2D. However, these spectra do not suggest a simple shift and broadening of the line, but rather a more complex transient state with this strong perturbation of the CDW state. Variations of the CB line are analyzed by center-of-mass determination, accounting for the transfer of spectral weight and avoiding assumptions of particular spectral profiles. The time evolution appears to be separated into two regimes: (i) Two pronounced initial oscillations display a different periodicity of 280 and 380 fs, respectively. (ii) After 1 ps, weak oscillations are found at 3.6 THz (Fig. 2F, black line).

Within the first ps, no phase correlation between the impact of modes on the CB and Te bands is encountered (Fig. 2F). After 1 ps, maxima in the CB oscillations tend to fall onto minima in the Te transient. The Te band oscillations are now well resolved, and we find that ΩTe = 3.6 THz (Fig. 2F). Recent Raman spectroscopy on RTe3 shows that the ΩTe mode cannot be assigned to one of the four Raman-active A1g modes predicted for the undistorted lattice and that it probably results from the CDW (26). This oscillation is robustly present in all of our measurements, regardless of BZ position, temperature, or pump power. However, Lavagnini et al. (26) show the oscillation to decrease and eventually vanish with increasing temperature or pressure above TCDW (26).

To strengthen the evidence leading to our previous assignment, we include fluence variations at 100 and 300 K. Figure 2G summarizes these results and displays the observed frequencies for both modes as a function of fluence. At 300 K, only the Te mode (3.6 THz) is observed, whereas at 100 K, two modes are identified for F ≤ 1 mJ/cm2. As for 300 K, thermodynamic fluctuations are expected to hamper the excitation of the amplitude mode, which again suggests that the 2.3-THz mode is the CDW amplitude mode. We also note that this frequency regime is typical of the CDW systems (27). Recent Raman (26) and time-dependent reflectivity (28) studies also come to the conclusion that the 2.3-THz mode is connected to the formation of the CDW.

The trARPES experiment provides an explicit link between the amplitude mode and CDW gap modulation via a momentum-dependent analysis: In a nesting-driven CDW, the electrons are most susceptible to scattering by the CDW phonon wave vector QCDW at the FS as the Lindhardt response function increases rapidly upon approaching kF (here, kF is the electron momentum at the Fermi energy) (1). We therefore expect that a modulation in the CDW amplitude and thus a modulation in CDW order parameter (i.e., the gap magnitude) influences the electronic band structure the most at or near kF. Following this simple argument, this k-dependence should be present independent of perturbation strength. Figure 3A defines the BZ position where the data of Fig. 3, B to D, have been recorded. Figure 3, B to D, establishes a pronounced momentum-dependent excitation of the electronic structure. While away from the FS (below EF), the CB energy does not vary; the largest change indeed occurs on the FS.

Fig. 3.

(A) Detail of the FS plot in Fig. 1A′ with indicated positions (white circles) of time-resolved data shown in (B) to (D) for fixed k as a function of time delay. Indicated cut position (red line) of photoelectron intensity is shown as a function of energy, and position [(E) to (I)] for a momentum scan is shown as a function of time delays. All data were collected at 100 K and F = 2 mJ/cm2. kF is marked in (E) to (I) (red dot). Error bars indicate the distance to the neighboring sample points, which is a good estimate for the error of kF.

Our finding also has implications well beyond the specific materials we study. Conventional ARPES offers superior energy and angle resolution and is able to detect the effect of collective modes in the form of dispersion kinks or spectral dips (29). The interpretation of their assignment and the impact on the low-lying electronic structure, however, requires sophisticated theoretical and often model-dependent calculations. Current technology limits the accessible BZ region, energy, and angle resolution of trARPES. Nevertheless, trARPES is able to directly probe the transient interplay between collective modes and single-particle states in theory-independent ways and thus complements conventional ARPES.

We now turn to a momentum-specific analysis of the electronic excitation and the response of the lattice to reveal the process of the ultrafast melting of the CDW state. Figure 3, E to I, shows the momentum-dependent data for selected delays; the full data set is available as a supplemental movie online. At 0 fs, the CDW gap is preserved, and the dispersion of the CB represents a localized state as prior to pumping. However, in the vicinity of kF (red dot in Fig. 3, E to I), an intensity increase at EEF = 0.15 eV coinciding with an unoccupied state as observed in Fig. 2C is encountered and represents instantaneous photo-doping of the system. At 100 fs (i.e., delayed with respect to the excitation), the gap is closed and the dispersion of the band resembles the quasi-free electron dispersion known from spectra taken at 300 K (Fig. 1). We term this excitation regime “strongly perturbative,” as the excitation results in an ultrafast melting of the charge-ordered state. Notably, the conduction electrons recover their wavelike nature, as they are still strongly k-dependent, despite the intense excitation and resulting intense scattering.

Thus, we have identified a delay in the ultrafast CDW melting with respect to the photo-doping. This is related to the mechanism of CDW formation. Because of coupling between lattice vibrations (phonons) to the electrons, electronic energy is gained if a phonon with a wave vector Q, which nests two regions of the FS, freezes in the crystal, leading to a charge density modulation. Upon cooling, this energy gain becomes larger than the energy loss from the lattice deformation, and the system undergoes a phase transition into the CDW state. Because formation of the CDW requires freezing of a phonon (i.e., nuclear motion), the inverse process of melting cannot proceed faster than the respective motion. Considering that photo-doping modifies the screened ion potential such that it sustains a delocalized state, the ion cores still have to propagate to the potential minima in the presence of screening carriers, which explains the observed delay in the CDW melting. This result provides a direct and vivid demonstration of the electron-phonon interaction being the origin of the CDW formation.

This result also raises the question of whether the transition is complete after 100 fs. We return to the strong perturbation case with F = 2 mJ/cm2 (Fig. 2 B, D, and F), where we have indicated earlier that the physics can be divided into two regimes according to the time delays. Between 1 and 3 ps, when the melting of the CDW is complete and the amplitude mode no longer exists, only the 3.6-THz Te phonon oscillation is observed. This is most clearly seen in the black dotted curve in Fig. 2F, which has the same frequency. Below 1 ps, the situation is complex. Whereas the gap is closed at 100 fs, a transient “equilibrium” outside the CDW state has not been established yet. The ions cannot stop once excited and will continue to oscillate as previously discussed. Thus, we expect the system to oscillate between a localized CDW state and a delocalized molten CDW state en route to the transient equilibrium beyond 1 ps. The mode associated with this oscillation is related to the amplitude mode but is not quite the same.

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