Mechanisms of electron-phonon coupling unraveled in momentum and time: The case of soft phonons in TiSe2

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Science Advances  12 May 2021:
Vol. 7, no. 20, eabf2810
DOI: 10.1126/sciadv.abf2810
  • Fig. 1 TiSe2 properties and experimental schematic.

    (A) Illustration of valence and conduction bands in TiSe2. Photons (1.55 eV) drive electronic transitions into a partially occupied conduction band minima at the M point of the Brillouin zone (BZ). The valence (conduction) band at Γ (M/L) is formed by Ti-3d (Se-4p) orbitals. (B) Fermi surface contours showing the bands in (A) in the TiSe2 BZ. (C) Dispersion of the transverse phonon in TiSe2 illustrating softening of the frequency at the M and L points yielding a Kohn anomaly. (D) BZ of TiSe2 in the high-temperature (P3m1) and low-temperature (P3c1) phases. Dashed black (solid blue) line illustrates the BZs of the normal (CDW) phase. The M points of the high-temperature phase are Γ points in the CDW phase. (E) Experimental configuration for the ultrafast electron scattering experiments, with the electron beam oriented along the [001] zone axis of TiSe2.

  • Fig. 2 Ultrafast electron scattering of TiSe2.

    (A) Equilibrium electron scattering pattern with various Bragg peaks (Γ points) and high-symmetry points (M and K) of the BZ identified. Inset: Intensity linecuts through M points along the a* (gray) and b*-a* (green) directions in reciprocal space. The green linecut intersects a thermal diffuse peak because of a populated transverse soft phonon mode. The peak is not present in the gray linecut in the other direction because of the magnitude of the one-phonon structure factor in Eq. 3. (B) Normalized intensity change at a pump-probe time delay of 400 fs (4 mJ/cm2). Pattern has been threefold symmetrized for improved signal-to-noise ratio. Regions of decreasing intensity are found not only at the Γ points but also at particular M points where strong TDS intensity from the transverse soft mode appears. Some of these regions are indicated in multiple BZs by the black circles. These results are in strong agreement with scattering intensity simulations using density function theory results in Materials and Methods. Inset: Intensity change of green linecut shown in (A) for various time delays. The noise level of the measurement is indicated by the error bar of 0.2%.

  • Fig. 3 Ultrafast electron diffuse intensity dynamics at various points of the BZ along with the Γ110 (Bragg peak) Debye-Waller dynamics.

    The M trace is shifted for clarity. The Γ110 trace is scaled by a factor of 1/4. The error bars are determined from the statistics of the intensity before photoexcitation (t = 0).

  • Fig. 4 Time-resolved soft-mode scattering in TiSe2.

    (A) Diffuse scattering from the transverse soft mode at q=(32,32,0), (M) shown under equilibrium conditions (gray) and at a pump-probe delay of 400 fs (green), illustrating a suppression of the peak amplitude due to phonon stiffening. (B) Scattering intensity after 5 ps, also with the equilibrium data from (A), where the dominant effect is the increased diffuse background due to lattice heating. The peaks are fit at all pump-probe time delays to extract the time-dependent amplitudes and diffuse background offsets. (C) Fit results for M amplitude (absolute value shown) and diffuse background versus time (note that not all time-delay points are fit at later times). The background amplitude is scaled by a factor of 10 for presentation (the actual background rise at late times in roughly 1% consistent with Figs. 2 and 3). Photocarrier density at M determined from time- and angle-resolved photoelectron spectroscopy (tr-ARPES) (34) is shown as the gray curve. The error bars in both (A) and (B) are determined by intensity counting statistics, and the SE in (C) is determined from the fitting routine covariance matrix.

  • Fig. 5 Magnitude of frequency renormalization for the soft transverse mode in TiSe2 determined by transient scattering intensities at M points (see section S5 for details).

    The gray Gaussian curve depicts instrumental temporal response function. The inset is a schematic representation of the change in phonon band for the soft mode.

  • Fig. 6 Comparison of experimental diffraction data with density functional theory calculations.

    (A) Computed phonon dispersion curve of TiSe2 in the CDW phase and normal phase at two different electronic temperatures. Low = 270 K and High = 1000 K. The stiffening of the zone-boundary transverse mode at M and L points is evident with increasing electron temperature, as is the flat dispersion along the M-L direction. (B) Experimental differential intensity at a pump-probe time delay of 400-fs (from Fig. 2, B and C) computation of the difference in diffuse scattering around the (210) peak for the high and low electron temperature phonon band structures shown in (A) at the same value of kBT = 25 meV (Eq. 3). The pattern of intensity suppression observed in (B) is well reproduced by the computational results shown in (C), which primarily reflect the stiffening of the zone-boundary soft modes at elevated electron temperature.

  • Table 1 Electron-lattice equilibration rates.
    Momentum qTime constant (fs)Rate (THz)
    Γ1070 ± 300.93 ± 0.27
    K1270 ± 1900.79 ± 0.12
    M980 ± 2901.03 ± 0.31
    M1170 ± 3000.88 ± 0.21

Supplementary Materials

  • Supplementary Materials

    Mechanisms of electron-phonon coupling unraveled in momentum and time: The case of soft phonons in TiSe2

    Martin R. Otto, Jan-Hendrik Pöhls, Laurent P. René de Cotret, Mark J. Stern, Mark Sutton, Bradley J. Siwick

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    • Supplementary Text
    • Figs. S1 to S5
    • Table S1
    • References

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