Optical phonons and the soft mode in 2$H$-NbSe${}_2$

We present an investigation of the lattice dynamics of the charge density wave (CDW) compound 2$H$-NbSe${}_2$. We analyze the precise nature of the wave vector-dependent electron-phonon coupling (EPC) and derive the bare dispersion of the CDW soft phonon mode using inelastic x-ray scattering combined with ab initio calculations. Experimentally, phonon modes along the Γ−$M$ line, i.e., q $=$ ($h$,0,0), with 0 ≤ $h$ ≤ 0.5 and the same longitudinal symmetry (${\Sigma }_1$) as the CDW soft mode, were investigated up to 32 meV. In agreement with our calculations, we observe significant EPC in the optic modes at $h$ ≤ 0.2. We analyze the EPC in the optic, as well as acoustic, mode and show that the q dependences stem from scattering processes between two bands at the Fermi surface that both have a Nb 4$d$ character. Finally, we demonstrate that the soft mode dispersion at $T$ $=$ 33 K (=$T_{\mathrm{CDW}}$) can be well described on the basis of a strongly q-dependent EPC matrix element and an acousticlike bare phonon dispersion in agreement with observations near room temperature.

Figure 1

(a) Calculated phonon dispersion along the (100) direction in NbSe${}_2$. Different line types (colors) denote branches with different symmetries (see legend). (b) Branches with longitudinal symmetry (${\Sigma }_1$) [solid lines in (a)]. Labels correspond to the discussion. Vertical bars denote the calculated electronic contribution to the phonon line width 2γ, scaled by a factor of 10 for visibility.

Figure 2

(a)–(c) The q dependence of the calculated absolute atomic displacements for the acoustic (AC), lower optic (LO1), and highest optic (LO2) branches. Three displacements are shown: (a) Nb ∥ $a$, (b) Se ∥ $a$, and (c) Se ∥ $c$. Other components are zero. (d)–(f) Calculated electronic contribution to the phonon line width γ for the (d) LO2, (e) LO1, and (f) AC branches. Each panel shows the total line width (solid line) and the dominant contributions related to specific electronic scattering processes.

Figure 3

Raw inelastic x-ray scattering data obtained at $T$ $=$ 50 K and (a) Q $=$ (2.7,0,1) and (b) Q $=$ (2.7,0,0). Solid (red) lines are fits consisting of an elastic line, DHOs for the inelastic peaks, and a linear background shown as dashed (blue) lines. The inset in (b) shows the temperature-dependent phonon energy of the second-lowest longitudinal mode at q $=$ (0.3,0,0). The dashed line is a visual guide.

Figure 4

(a) Raw data showing two optic phonon modes at Q $=$ (2.2,0,0) at $T$ $=$ 33 K. The solid (red) line is a fit consisting of two DHOs for the inelastic peaks convoluted with the experimental resolution and a linear background shown as dashed (blue) lines. Measured phonon dispersion is at (b) $T$ $=$ 33 K and (c) $T$ $=$ 250 K. Solid lines denote the dispersion associated with the soft phonon mode (see text), as published in Ref. 13. Dashed lines are the calculated energies of the LO1 and LO2 optic branches (see Fig. 1).

Figure 5

(a) Observed phonon energies at $T$ $=$ 33 K along a different high symmetry direction (different from [100]), starting from the CDW vector ${\mathbf{q}}_{\mathrm{CDW}}$. Energies are shown as function of the distance from ${\mathbf{q}}_{\mathrm{CDW}}$ in absolute units. The solid line is a linear fit of the data. The dashed lines indicate the corresponding phonon energies along the (100) direction, as published in (Ref. 4) (two lines for q < ${\mathbf{q}}_{\mathrm{CDW}}$ and q > ${\mathbf{q}}_{\mathrm{CDW}}$). (b) Dispersion of the LO1 branch at $T$ $=$ 33 K for different values of $l$ along the [001] direction. The line is a visual guide.

Figure 6

(a) Wave vector-dependent line widths (FWHM) at $T$ $=$ 33 K (filled symbols) and 250 K (open symbols) of the LO1 and LO2 branches. (b) Calculated electronic contribution to the phonon line width 2γ for the LO1 and LO2 branches with two numerical smearing parameters: σ $=$ 0.1 and 1.0 eV.

Figure 7

(a) Fit (solid line) of the observed square soft mode energies at $T$ $=$ $T_{\mathrm{CDW}}$ (dots) using Eq. (3) and $Re{\chi }_{q,\mathrm{FS}}\left(h\right)$ and $g_q\left(h\right)$ as extracted from DFPT. ${\omega }_{\mathrm{bare}}$ (dashed line) was approximated by a Brillouin function with two fit parameters.