Strong anisotropy of the electron-phonon interaction in NbP probed by magnetoacoustic quantum oscillations

In this study, we report on the observation of de Haas–van Alphen–type quantum oscillations (QOs) in the ultrasound velocity of NbP as well as “giant QOs” in the ultrasound attenuation in pulsed magnetic fields. The difference in the QO amplitude for different acoustic modes reveals a strong anisotropy of the effective deformation potential, which we estimate to be as high as $9\mathrm{eV}$ for certain parts of the Fermi surface. Furthermore, the natural filtering of QO frequencies and the tracing of the individual Landau levels to the quantum limit allows for a more detailed investigation of the Fermi surface of NbP, as was previously achieved by means of analyzing QOs observed in magnetization or electrical resistivity.

Figure 1

Magnetoacoustic quantum oscillations in NbP for pulsed magnetic fields $\mathit{H}\parallel c$ at $T=1.35\mathrm{K}$ for different acoustic modes. (a) Change in the relative ultrasound velocity $-\mathrm{\Delta }v/v$ versus magnetic field. (b) Change in ultrasound attenuation $\mathrm{\Delta }\alpha$ versus magnetic field. The curves are shifted with respect to each other for better visibility.

Figure 2

Frequency analysis of the quantum oscillations in ultrasound velocity for different modes at $T=1.35\mathrm{K}$. (a) Projections of the electron pockets, E1 and E2, and the hole pockets, H1 and H2, of NbP parallel to the $k_x-k_z$ plane (or, similarly, to the $k_y-k_z$ plane due to the fourfold rotational symmetry). Extremal orbits for $\mathit{H}\parallel c$ are shown in red. For an illustration of the full Fermi surface in the first Brillouin zone see, for instance, Refs. [23, 34]. (b) Top: Shubnikov–de Haas oscillations subtracted from the magnetoelectrical resistivity ${\rho }_{xx}$ at $T=2\mathrm{K}$ for comparison. Bottom: Landau level peaks assigned to the maximum orbit ${\alpha }_1$. (c) Landau level peaks assigned to the minimum orbit ${\beta }_1$. (d) Low-frequency oscillation visible in the $C_{44}$ mode assigned to the minimum orbit ${\beta }_2$. (e) Assignment of the remaining peaks in the high-field range to the second maximum orbit of E1, ${\gamma }_1$, and possibly the maximum orbit ${\delta }_1$. (f) Oscillation assigned to the maximum orbit ${\alpha }_2$ visible in the $C_{11}$ mode, emphasized by applying a low-pass Fourier filter. (g) Assigned Landau levels plotted versus inverse magnetic field. Solid lines represent linear fits. The inset enlarges the high-field range.

Figure 3

Temperature evolution of the quantum oscillations in the ultrasound velocity for the (a) and (c) $C_{44}$ and (e) $C_{33}$ modes and Lifshitz-Kosevich fit for frequencies (b) ${\beta }_1$, (d) ${\beta }_2$, and (f) ${\alpha }_1$ dominant in $C_{44}$ and $C_{33}$, respectively.

Figure 4

Extraction of the oscillation amplitude for superimposed peaks in the high-field range by fitting two Lorentzian functions (green) at fixed inverse-field values. Extraction of the height for ${\gamma }_1^1$ and ${\delta }_1^1$ from the ultrasonic quantum oscillations in the (a) $C_{11}$ and (c) $C_{33}$ modes. Extraction of the height for ${\beta }_1^2$ for the (b) $C_{11}$ and (d) $C_{33}$ modes.

Figure 5

Frequency analysis of the giant quantum oscillations in ultrasound attenuation for different modes at $T=1.35\mathrm{K}$. (a) Landau level peaks assigned to the resonant orbits $F_1$ and $F_2$. (b) Assigned Landau levels plotted vs inverse magnetic field. Solid lines represent linear fits.