Single-crystalline transition metal phosphide superconductor WP studied by Raman spectroscopy and first-principles calculations

We present a study of phonon modes and electron-phonon coupling in superconductor WP using angle-resolved polarized Raman spectroscopy. Seven out of twelve Raman-active modes are detected, with frequencies in good accordance by the first-principles calculations. We analyze parallel- and cross-polarized configurations of ${\mathrm{A}}_{\mathrm{g}}$ and ${\mathrm{B}}_{3\mathrm{g}}$ phonon modes with polarization- and angle-resolved Raman measurements. For the observed seven modes, the total electron-phonon coupling parameter $\lambda$ is only about 0.07, which is too small to achieve superconductivity. Our results elucidate that the low-frequency phonon modes play a key role in the superconductivity within the Bardeen-Cooper-Schrieffer theory framework.

Figure 1

(a) Crystal structure of WP in $yz$ plane. We defined $x, y$, and $z$ as the directions along the unit cell $a, b$, and $c$ axes, respectively. (b) Schematic of the ARPRS. The electric field of the linearly polarized incident and the scattered light are ${\mathit{E}}_{\mathrm{i}}$ and ${\mathit{E}}_{\mathrm{s}}$, respectively. The bottom-right inset depicts the optical image of the WP single crystal measured by ARPRS. The angle $\theta$ was defined as the angle between the light polarization direction and the $b$ axis. (c) Raman spectra of the WP. The colored curves are the resulting fitted spectra with Lorentz functions. The upper-right inset shows Raman spectra of the WP which were measured in the ($yy$) parallel-polarized (i.e., ${\mathit{E}}_{\mathrm{i}} \parallel {\mathit{E}}_{\mathrm{s}} \parallel b$-axis) and ($yz$) cross-polarized (i.e., ${\mathit{E}}_{\mathrm{i}} \perp {\mathit{E}}_{\mathrm{s}}, {\mathit{E}}_{\mathrm{i}} \parallel b$-axis, and ${\mathit{E}}_{\mathrm{s}} \parallel c$-axis) configurations, respectively.

Figure 2

Angle-dependent Raman spectra of the measured WP single crystal. Contour map of the measured Raman phonon intensities as a function of angle and energy under parallel- and cross-polarized configurations [(a), (b)]. Angle dependence of the Raman-active phonon peak intensities measured in parallel-polarized [(c), (e), (g), (i), (k), (m)] and cross-polarized [(d), (f), (h), (j), (l), (n)] configurations. The blue dots show the measured data. The black solid curves display the fit results based on Eqs. (2, 3, 4, 5).

Figure 3

(a) Raman spectra for ${\mathrm{A}}_{\mathrm{g}}$-1 and ${\mathrm{A}}_{\mathrm{g}}$-4 modes of WP at $\theta$ = ${20}^{\circ }, {10}^{\circ }, 0^{\circ }$, and ${350}^{\circ }$ under parallel-polarized configuration. (b) Waterfall plot of temperature-dependent Raman spectra for ${\mathrm{A}}_{\mathrm{g}}$-4 mode of WP. The vertical dashed lines indicate the ${\mathrm{A}}_{\mathrm{g}}$-4 mode. The colored dots with different shaped curves are the experimental data at the corresponding temperature. The colored curves are the resulting fitted spectra with Fano functions. (c) Temperature evolution of the phonon energies ${\omega }_0$. (d) Temperature evolution of the linewidth $\gamma$. (e) Fano symmetry parameter $q$ of the ${\mathrm{A}}_{\mathrm{g}}$-4 mode from the fit in (b).

Figure 4

Phonon vibration patterns for the acoustic and optical phonon modes in WP, which are determined from the DFT-based lattice dynamics calculations. The W and P atoms in the unit cell are denoted by blue and black spheres, respectively. The atomic displacements of the W and P atoms are indicated by the colored arrows.

Figure 5

(a) Calculated electronic band structure of WP along the high-symmetry directions of $q$ points ($\mathrm{\Gamma } \to$ X $\to$ S $\to$ Y $\to \mathrm{\Gamma } \to$ Z $\to$ U $\to$ R $\to$ T $\to$ Z). We plot in the energy range from $-3$ to $+3$ eV. (b) The projection of the P $p$-orbital and W $d$-orbital. The Fermi level is set at zero energy. (c) The phonon spectra of WP resolved in terms of the vibration directions of W and P atoms. (d) PhDOS for WP. (e) The magnitude of the EPC ${\lambda }_{\mathbf{q}\nu }$. (f) The Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ and the cumulative frequency-dependent EPC $\lambda \left(\omega \right)$.