Electron-Phonon Coupling and the Soft Phonon Mode in ${\mathrm{TiSe}}_2$

We report high-resolution inelastic x-ray measurements of the soft phonon mode in the charge-density-wave compound ${\mathrm{TiSe}}_2$. We observe a complete softening of a transverse optic phonon at the $L$ point, i.e., $\mathbf{q}=(0.5,0,0.5)$, at $T\approx T_{\mathrm{CDW}}$. Detailed ab initio calculations for the electronic and lattice dynamical properties of ${\mathrm{TiSe}}_2$ are in quantitative agreement with experimental frequencies for the soft phonon mode. The observed broad range of renormalized phonon frequencies, $(0.3,0,0.5)\le \mathbf{q}\le (0.5,0,0.5)$, is directly related to a broad peak in the electronic susceptibility stabilizing the charge-density-wave ordered state. Our analysis demonstrates that a conventional electron-phonon coupling mechanism can explain a structural instability and the charge-density-wave order in ${\mathrm{TiSe}}_2$ although other mechanisms might further boost the transition temperature.

Figure 1

Temperature dependence of soft phonon mode at $\mathbf{Q}=(-0.5,1,0.5)$ ($L$ point). (a)–(c) Energy scans for temperatures $204\le T\le 300\mathrm{K}$. Solid (red) lines are fits consisting of a damped harmonic oscillator (inelastic) and a pseudo-Voigt function (elastic) (blue dashed lines). (d) Soft-mode energy ${\omega }_q$ observed via IXS as function of temperature. The dashed line in (d) is the temperature dependence deduced from thermal diffuse scattering reproduced from Ref. 23.

Figure 2

Energy scans at $\mathbf{Q}=(-h,1,0.5)$, $h=0.35-0.5$, for temperature $T=190\mathrm{K}$ ($\approx T_{\mathrm{CDW}}$). Solid lines represent the total fit result consisting of a damped harmonic oscillator for the inelastic and a pseudo-Voigt function for the elastic scattering (blue dashed lines). The inset of (d) shows the scan at $h=0.5$, which can be well described just by the pseudo-Voigt function as used in (a)–-(d).

Figure 3

(a) Dispersion of the soft phonon mode at $T=190\mathrm{K}$ ($\approx T_{\mathrm{CDW}}$, filled symbols) and $T=30\mathrm{K}$ (open circles) along $\mathbf{Q}=(-h,1,0.5)$, $h=0-0.475$. Dashed lines are results of DFPT calculations using the experimental crystal structure and $\sigma =0.15\mathrm{eV}$. The solid (blue) line indicates the dispersion of the soft-mode character across the anticrossing at $h\approx 0.33$. Frequencies of the upward dispersing branch [e.g., see Fig. 2a] agree well with the calculation but are not shown for the sake of clarity. (b) Damping ratio of the DHO at $T=190\mathrm{K}$ (filled symbols) and $T=30\mathrm{K}$ (open circles).

Figure 4

(a) Results of the calculation are shown as function of $h$ along the $A\mathrm{\text{−}}L$ line, i.e., $\mathbf{q}=(h,0,0.5)$ with $h=0-0.5$: (1) the calculated electronic contribution to the phonon linewidth $2\gamma$ of the soft phonon mode (FWHM, solid line, left-hand scale), (2) the total electronic JDOS (open symbols, right-hand scale), and (3) interband contributions to the JDOS between the Se $4p$ and Ti $3d$ bands near $\Gamma$ and $L$ (filled symbols, right-hand scale). (b) Experimental linewidth $\Gamma$ (dots) compared to scaled $2\gamma$ ($\times 2.5+\mathrm{\text{offset}}$, solid line).