Electron-phonon coupling and Kohn anomaly due to floating two-dimensional electronic bands on the surface of ZrSiS

$\mathrm{ZrSiS}$, an intriguing candidate of topological nodal line semimetals, was discovered to have exotic surface floating two-dimensional (2D) electrons [Phys. Rev. X 7, 041073 (2017)], which are likely to interact with surface phonons. Here, we reveal a prominent Kohn anomaly in a surface phonon branch by mapping out the surface phonon dispersions of $\mathrm{ZrSiS}$ using high-resolution electron energy loss spectroscopy. Theoretical analysis via an electron-phonon coupling (EPC) model attributes the strong renormalization of the surface phonon branch to the interactions with the surface floating 2D electrons. With the random phase approximation, we calculate the phonon self-energy and evaluate the mode-specific EPC constant by fitting the experimental data. The EPC picture provided here may be important for potential applications of topological nodal line semimetals.

Figure 1

Sample characteristic and the surface floating 2D states. (a) Crystal structure of ZrSiS. (b) Illustration of the bulk BZ and the SBZ. (c) The optical image of a sheeny surface of cleaved ZrSiS crystal pasted on a Mo sample holder. (d) Room temperature LEED pattern of a freshly cleaved (001) surface with an incident electron energy of 120 eV. The white triangle shows the least irreducible SBZ. (e) The Fermi surfaces from the surface electronic band calculations. The light blue lines are the projection from the bulk bands, while the dark blue ellipses denote the surface floating 2D states. (f) The conelike surface floating 2D band around $\overline{X}$ point. The orange arch is the elliptical Fermi surface with major axis $2k_a$ and minor axis $2k_b$.

Figure 2

HREELS measurement results. (a), (b) 2D momentum-energy mappings at RT along $\overline{\mathrm{\Gamma }}\overline{X}$ and $\overline{\mathrm{\Gamma }}\overline{M}$, respectively. The negative energy range corresponds to anti-Stokes peaks (phonon annihilation), while the positive range corresponds to Stokes peaks (phonon creation). The white dashed lines denote the SBZ boundaries. The brightness, representing the signal intensity, is plotted in logarithmical scale. (c), (d) Energy distributed curves along the red dash lines at $q=0.3{\AA }^{-1}$ and $q=0.1{\AA }^{-1}$ in (a), respectively. The orange background is the elastic peak. Phonon peaks are fitted by Gaussian functions and labeled by AP (acoustic phonon) or OP (optical phonon) according to their dispersions.

Figure 3

Calculated and experimental phonon dispersions. (a) Calculated bulk phonon dispersions along high symmetry directions of the bulk BZ. The green dots correspond to the Raman-active modes [64, 66]. (b) Calculated LDOS for phonons of the (001) surface along $\overline{X}-\overline{\mathrm{\Gamma }}-\overline{M}-\overline{X}$, with the intensity plotted in logarithmical scale. The yellow shades are the projection of bulk phonon dispersions onto the surface and red sharp lines represent surface phonons with the LDOS larger than 2.5. The dash-dot lines denote the Rayleigh modes splitting from the bulk acoustic branch. (c) The second derivative image along $\overline{M}-\overline{\mathrm{\Gamma }}-\overline{X}$ of the SBZ at RT obtained from the 2D HREELS mapping shown in Fig. 2. The superimposed red lines are the calculated surface phonon dispersions extracted from (b). The white dashed block designates the zone where the anomalous phonon softening occurs. The experimentally observed OP1 mode is compared with the theoretical $\alpha$ mode (white line). (d) The same comparison as (c) with the experimental data obtained at 35 $\mathrm{K}$.

Figure 4

Illustrations of the electron-phonon scattering process. (a) The allowed $(\mathrm{\Omega },Q)$ space for phonons defined by the constraint conditions (Cst. 1), (Cst. 2), and (Cst. 3). Regions (3), (2), and (1) correspond to (b), (c), and (d), respectively. Panels (b), (c), and (d): Schematic drawings of the initial and final Fermi ellipses before and after interacting with phonons in three cases of different momentum and energy transfer. The light blue zones denote the allowed area of initial states for electron-phonon interactions, while the dark blue zones are the banned area. The royal blue hyperbolas are the lines that satisfy the conservation of energy.

Figure 5

Renormalized dispersion and fitting results from the EPC model. (a) The dispersions of the OP1 mode. The black line is the calculated bare surface phonon dispersion from first principle calculation, while the blue dots with error bars (using the instrument resolution here) are the extracted experimental data. The yellow line is the renormalized dispersion from the EPC model. (b) Calculated real (light blue) and imaginary (green) parts of the phonon self-energy and the EPC constant (magenta) for the OP1 mode.

Figure 6

Schematics of the scattering momentum. A simple illustration of the relation between momentums of electrons before $\left(\mathit{k}\right)$ and after $(\mathit{k}+\mathit{q})$ interacting with phonons $\mathit{q}$.