Quantum oscillations in acoustic phonons in Weyl semimetals

We theoretically study the modification of the energy spectrum of long-wavelength acoustic phonons due to the electron-phonon interaction in a three-dimensional topological Weyl semimetal under the influence of quantizing magnetic fields. We find that the dispersion and attenuation of phonons show striking oscillatory behaviors when varying the magnetic field at low temperatures. These oscillations are distinct when the Fermi energy is in different energy regimes. Moreover, the van Hove singularity of the Weyl spectrum can manifest as a transition between different oscillation patterns when increasing the magnetic field. These phonon behaviors could provide testable fingerprints of the Fermi-surface morphology and relativistic feature of the Weyl semimetal.

Figure 1

(a) Energy spectrum of the model (1) with respect to $k_z$ in the absence of the magnetic field. The dotted lines mark the boundaries between three energy regimes. (b) Landau bands with respect to $k_z$ in the presence of an 8-T magnetic field in the $z$ direction. The pink thick curve stands for the chiral Landau band. Other parameters are $k_c=0.1{\mathring{\text{A}}}^{-1}$, $m=10$ $\mathrm{eV}{\mathring{\text{A}}}^2$, and $v=2.5\mathrm{eV}\mathring{\text{A}}$.

Figure 2

Modification of the phonon velocity $\mathrm{\Delta }{v}_p$ (or equivalently modification of phonon dispersion $\mathrm{\Delta }{\mathrm{\Omega }}_q=\mathrm{\Delta }{v}_{p}q$) as a function of ${\ell }_B^2$ at $T=0$ and $\mu =0.1\mathrm{eV}$ (a), 0.5 eV (b), and 1.2 eV (c), respectively. $\mathrm{\Delta }{v}_p$ is in units of ${10}^{-4}\mathrm{eV}\mathring{\text{A}}=15\hslash \mathrm{m}/\mathrm{s}$. The pink dots are calculated numerically from Eq. (11) with $T=0.1\mathrm{mK}$ and $\delta ={10}^{-6}$ eV; the blue dots and yellow solid lines are plots of Eqs. (29) (a), (30) (b), and (31) (c) with the first ${10}^4$ harmonics and only the first harmonic, respectively. The black dashed lines plot the background $\mathrm{\Delta }{\mathrm{\Omega }}_q^b/q$ of oscillations. The insets in each panel are the corresponding Fermi surfaces with the extremal cross sections in gray color. Other representative parameters are $v=6\mathrm{eV}\mathring{\text{A}}$, $k_c=0.15{\mathring{\text{A}}}^{-1}$, $m=15\mathrm{eV}{\mathring{\text{A}}}^2$, $\rho =7\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$, $D=20\mathrm{eV}$, $v_p=0.015\mathrm{eV}\mathring{\text{A}}$, and $q={10}^{-4}{\mathring{\text{A}}}^{-1}$.

Figure 3

The periods ${\mathcal{P}}_{1,2}$ (in units of ${10}^3{\mathring{\text{A}}}^2$) (a), phase shifts ${\gamma }_{1,2}$ (in units of $\pi$) (b), and amplitudes ${\mathrm{\Lambda }}_{1,2}$ (in units of ${\text{eV}}^{-1}$) (c) of the two harmonics in $\mathrm{\Delta }{\mathrm{\Omega }}_q$ as functions of $\mu$ in the moderate-energy regime. Other parameters are the same as those in Fig. 2.

Figure 4

(a) Band structure and (b) density of states of the topological Weyl semimetal in the case with $v_z>\sqrt{2}v$. A van Hove singularity happens at $E_{\text{v}}=v\sqrt{v_z^2-v^2}/2m$. The inset of (b) shows the Fermi surface in the energy regime $E_{\text{v}}<\mu <\text{min}(E_{\text{L}},v{k}_c)$. The parameters are $k_c=0.23{\mathring{\text{A}}}^{-1}$, $m=10\mathrm{eV}{\mathring{\text{A}}}^2$ and $v=2.5\mathrm{eV}\mathring{\text{A}}$.

Figure 5

Modification of the phonon velocity $\mathrm{\Delta }{v}_p$ (or equivalently $\mathrm{\Delta }{\mathrm{\Omega }}_q=\mathrm{\Delta }{v}_{p}q$) as a function of ${\ell }_B^2$ for $v<m{k}_c$. $\mathrm{\Delta }{v}_p$ is in units of ${10}^{-4}\mathrm{eV}\mathring{\text{A}}=15\hslash \mathrm{m}/\mathrm{s}$. The beating oscillations have two periods ${\mathcal{P}}_2$ and ${\mathcal{P}}_3$, different from that in Fig. 2 the periods of which are ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$. $\mu =0.8\mathrm{eV}$, $k_c=0.3{\mathring{\text{A}}}^{-1}$, and other parameters are the same as those in Fig. 6.

Figure 6

Transition between different patterns of oscillations. $\mathrm{\Delta }{v}_p$ is in units of ${10}^{-4}\mathrm{eV}\mathring{\text{A}}=15\hslash \mathrm{m}/\mathrm{s}$. The blue dots are numerical results from Eq. (11). (a) $\mu =0.5\mathrm{eV}$. In the large field (i.e., small ${\ell }_B$) region, ${\mathrm{\Omega }}_q$ shows single-periodic oscillations, whereas in the small field (i.e., large ${\ell }_B$) region, it shows beating oscillations with three periods. (b) $\mu =-0.475\mathrm{eV}$. In the small field region, ${\mathrm{\Omega }}_q$ shows single-periodic oscillations, where in the large field region it shows beating oscillations with three periods. The black dashed lines are the plots of $\mathrm{\Delta }{\mathrm{\Omega }}_q^{\text{b}}$ stated in Eq. (34). The dashed red vertical line denotes the transition of the patterns. Other parameters are $k_c=0.23{\mathring{\text{A}}}^{-1}$, $m=10\mathrm{eV}{\mathring{\text{A}}}^2$, $v=2.5\mathrm{eV}\mathring{\text{A}}$, $D=10\mathrm{eV}$, $v_p=0.015\mathrm{eV}\mathring{\text{A}}$, $\rho =7000\mathrm{kg}/{\mathrm{m}}^3$, and $q={10}^{-4}{\mathring{\text{A}}}^{-1}$.

Figure 7

Phonon attenuation as a function of ${\ell }_B^2$ for $\mu =0.1\mathrm{eV}$ (a), 0.5 eV (b), and 1.2 eV (c), respectively. The results are calculated numerically from Eq. (12). The red, blue, and black lines correspond to the temperatures of $3\mu /E_L$, $5\mu /E_L$, and $8\mu /E_L$ (in units of K), respectively. In (a) and (c), the dots are the plots of the peak lines in Eqs. (43) and (48), respectively. Other parameters are the same as those in Fig. 2.

Figure 8

The solid blue lines are the same as Fig. 2 but only for $T=3\mu /E_L$. The phonon attenuation can be decomposed as two different sets of harmonics, as distinguished by the dashed red and blue lines. The two dotted lines are the plots of the peak lines in Eqs. (43) and (48), respectively. Other parameters are the same as those in Fig. 2.