Phonon-limited transport and Fermi arc lifetime in Weyl semimetals

Weyl semimetals harbor topological Fermi arc surface states that determine the nontrivial charge current response to external fields. We study here the quasiparticle decay rate of Fermi arc states arising from their coupling to acoustic phonons, as well as the phonon-limited conductivity tensor for a clean Weyl semimetal slab. Using the phonon modes for an isotropic elastic continuum with a deformation potential coupling to electrons, we determine the temperature dependence of the quasiparticle decay rate, both near and far away from the arc termination points. By solving the coupled Boltzmann equations for the bulk and arc state distribution functions in the slab geometry, we show how the linear-response conductivity depends on key parameters such as the temperature, the chemical potential, the geometric shape of the Fermi arcs, or the slab width. The chiral nature of Fermi arc states causes an enhancement of the longitudinal conductivity along the chiral direction at low temperatures, together with a $1/T^2$ scaling regime at intermediate temperatures without counterpart for the conductivity along the perpendicular direction.

Figure 1

WSM slab geometry. (a) The sample is infinitely extended along the $(y,z)$ directions, with conserved momentum ${\mathbf{k}}_{\parallel }=(k_y,k_z)$, and it has the transverse width $L$ along the $x$-direction. Fermi-arc surface states are sketched in the surface Brillouin zone for $\alpha >0$ and constant energy $\varepsilon >0$. (b) Arc shapes in the $(k_y,k_z)$ plane, see Eq. (2.19), for the surface state at $x=-L/2$ with constant energy $\varepsilon =0.2k_{\mathrm{W}}v$ and different angles $\alpha =0,0.4,0.8,1.2$ (from left to right). The shaded disks indicate phase space regions where bulk states are present. They appear slightly elongated in the $z$-direction due to the anisotropy of the bulk dispersion relation.

Figure 2

Arc-arc decay rate for $\alpha =0.8$. In all figures, we use $c_l/v=0.0170$ and $c_t/c_l=0.571$. Panel (a) shows the temperature dependence of ${\mathrm{\Gamma }}^{\left(ss\right)}(\varepsilon ,k_z)$, see Eq. (4.2), on double-logarithmic scales for $\varepsilon =0.1v{k}_{\mathrm{W}}$ and two values of $k_z$, with ${\mathrm{\Gamma }}_0$ in Eq. (3.25) and $T_{\mathrm{BG}}=c_l{k}_{\mathrm{W}}$, see Eq. (2.43). We distinguish contributions from the Rayleigh ($R$) mode and from bulk ($B$) phonon modes ($\lambda =l,t)$. The dashed lines represent the $T^3$ and $T^2$ scaling near the arc center, see Eq. (4.8), and close to the arc edge, see Eq. (4.10), respectively. Panel (b) shows (for two different energies) the $k_z$-dependence of the rate for $T\gg T_{\mathrm{BG}}$ as encoded by the coefficient ${\mathrm{\Xi }}^{(ss,1)}\left(k_z\right)$ in Eq. (4.5).

Figure 3

Arc-bulk decay rate ${\mathrm{\Gamma }}^{\left(bs\right)}(\varepsilon ,k_z)$ for energy $\varepsilon =0.1v{k}_{\mathrm{W}}$ and boundary angle $\alpha =0.8$ as obtained by numerical integration of Eq. (4.13). Panel (a) shows the temperature dependence of ${\mathrm{\Gamma }}^{\left(bs\right)}$ on double-logarithmic scales for two values of $k_z$. Equation (4.19) gives ${\mathrm{\Omega }}_{\min }/T_{\mathrm{BG}}\simeq 4.6\times {10}^{-6}$ for $k_z=0.999{\overline{k}}_{\mathrm{W}}$. As in Fig. 2, we separately show contributions from Rayleigh ($R$) and bulk ($B$) phonon modes. The dashed lines indicate power-law scaling with the respective exponents. Panel (b) shows the $k_z$-dependence of the coefficient ${\mathrm{\Xi }}^{(bs,1)}$ determining the arc-bulk rate for $T\gg T_{\mathrm{BG}}$, see Eq. (4.16), for two energies.

Figure 4

Phonon threshold energy for arc-bulk scattering, ${\mathrm{\Omega }}_{\min }(\varepsilon ,k_z)$, vs position along the arc, $k_z/{\overline{k}}_{\mathrm{W}}\left(\varepsilon \right)$, for $\alpha =0.5$ and several energies.

Figure 5

Temperature dependence of the conductivities ${\sigma }_{yy}$ and ${\sigma }_{zz}$ for different values of the boundary angle $\alpha$. We take $k_{\mathrm{W}}L=1000$ and $\mu =0.08v{k}_{\mathrm{W}}$, where $T_{\mathrm{BG}}^{\left(b\right)}/T_{\mathrm{BG}}=0.16$ and $T_{\mathrm{BG}}^{\prime }/T_{\mathrm{BG}}\approx 0.29$. The data points have been obtained by numerical solution of the Boltzmann equation; curves are guides to the eye only. Panel (a) shows ${\sigma }_{yy}$ in units of ${\sigma }_0$, see Eq. (5.13), vs $T/T_{\mathrm{BG}}$ on a double-logarithmic scale. Straight dashed lines indicate the quoted power laws. Panel (b) shows ${\sigma }_{yy}/{\sigma }_{zz}$ vs $T/T_{\mathrm{BG}}$ for the same parameters on a semilogarithmic scale.

Figure 6

Temperature dependence of ${\sigma }_{yy}$ and ${\sigma }_{zz}$ as in Fig. 5 but for fixed boundary angle $\alpha =0$ and with $\mu /v{k}_{\mathrm{W}}\in (0.05,0.08)$ and $k_{\mathrm{W}}L\in (1000,10000)$. Panel (a) shows ${\sigma }_{yy}/{\sigma }_0$ vs $T/T_{\mathrm{BG}}$ on double-logarithmic scales. Panel (b) shows ${\sigma }_{yy}/{\sigma }_{zz}$ vs $T/T_{\mathrm{BG}}$ for the same parameters on a semilogarithmic scale.