Generalized design principles for hydrodynamic electron transport in anisotropic metals

Interactions of charge carriers with lattice vibrations, or phonons, play a critical role in unconventional electronic transport of metals and semimetals. Recent observations of phonon-mediated collective electron flow in bulk semimetals, termed electron hydrodynamics, present new opportunities in the search for strong electron-electron interactions in high carrier density materials. Here we present the general transport signatures of such a second-order scattering mechanism, along with analytical limits at the Eliashberg level of theory. We study electronic transport, using ab initio calculations, in finite-size channels of semimetallic ZrSiS and ${\mathrm{TaAs}}_2$ with and without topological band crossings, respectively. The order of magnitude separation between momentum-relaxing and momentum-conserving scattering length scales across a wide temperature range make both of them promising candidates for further experimental observation of electron hydrodynamics. More generally, our calculations suggest that the hydrodynamic transport regime does not, to first order, rely on the topological nature of the bands. Finally, we discuss general design principles guiding future search for hydrodynamic candidates, based on the analytical formulation and our ab initio predictions. We find that systems with strong electron-phonon interactions, reduced electronic phase space, and suppressed phonon-phonon scattering at temperatures of interest are likely to feature hydrodynamic electron transport. We predict that layered and/or anisotropic semimetals composed of half-filled $d$ shells and light group V/VI elements with lower crystal symmetry are promising candidates to observe hydrodynamic phenomena in the future.

Figure 1

(a) Schematic energy dispersion diagram along a high symmetry direction for a semimetal, with highly dispersive $s/p$ electronlike (blue) and “heavy” $d/f$ holelike (red) bands crossing the Fermi energy. Due to the separation of the Fermi pockets, lattice excitations with a large wave number are needed for transition between $k_s$ and $k_d$. (b) Fermi surface diagram of the semimetal shown in (a) across the first Brillouin zone, showing intrapocket (top left) and interpocket (bottom) scattering events. The square lattice Brillouin zone, with reciprocal lattice vector $G$, is shown in solid lines. Adapted from Ref. [22]. (c) Electron-phonon momentum relaxing scattering event, where initial electronic state $\mathit{k}$ is scattered into final state ${\mathit{k}}^{\prime }$, exchanging both energy and momentum to the phonon mode $\mathit{q}={\mathit{k}}^{\prime }-\mathit{k}$. The thermally accessible states with energy $\sim k_{B}T$ around the Fermi energy are indicated by the shaded area. (d) Electron-electron momentum conserving scattering event, where two initial electronic states ${\mathit{k}}_{\mathit{1}}$ and ${\mathit{k}}_{\mathit{2}}$ are scattered into final states ${\mathit{k}}_1^{\prime }$ and ${\mathit{k}}_2^{\prime }$. The scattering process conserves both energy and momentum. (e) A faceted Fermi surface, on which a scattering event between wave vectors from $\mathit{k}$ to ${\mathit{k}}^{\prime }$ may lead to negligible change in the electron velocity such that the total momentum stays “quasi” conserved.

Figure 2

(a) Crystal lattice of ZrSiS, highlighting the layered structure. (b),(c) Electron-phonon lifetimes (${\tau }_{\mathrm{eph}}$) at (b) 298 K and (c) 4 K projected on the Fermi surface of ZrSiS. Similarly, while both electron and hole pockets have long lived carriers, lowering temperature quenches the scattering events on the hole pockets much more significantly.

Figure 3

(a) Temperature dependence of first-order electron-phonon ($l_{\mathrm{eph}}$) and second-order phonon mediated electron-electron ($l_{\mathrm{ee}\left(\mathrm{ph}\right)}$) mean free paths in ZrSiS calculated from ab initio. The impurity mean free path is taken as 1 $\mu \mathrm{m}$ from Uykur et al. [39]. Assuming that, at low temperatures, impurity scattering is the dominant electron resistivity mechanism, we estimate the impurity concentration to be $1.5\times {10}^{15}{\mathrm{cm}}^{-3}$ according to ${\rho }_{\mathrm{unitary}}^{3\text{D}}=\frac{4\pi \hslash n_d}{n{e}^2{k}_F}$ [40, 41, 42]. (b) Normalized $j_x$ curvature phase diagram. Overlaid lines show the trajectories with the decreasing temperature with different channel widths.

Figure 4

(a) Crystal lattice of ${\mathrm{TaAs}}_2$, highlighting the monoclinic structure. (b),(c) Electron-phonon lifetimes (${\tau }_{\mathrm{eph}}$) at 298 K and (d),(e) 4 K projected on the Fermi surface of ${\mathrm{TaAs}}_2$. Different views are presented due to the low symmetry Brillouin zone and the complexity of the electron and hole pockets. The features are highlighted that hole pockets at low temperatures feature much longer momentum relaxing lifetimes.

Figure 5

(a) Temperature dependence of first-order electron-phonon ($l_{\mathrm{eph}}$) and second-order phonon mediated electron-electron ($l_{\mathrm{ee}\left(\mathrm{ph}\right)}$) mean free paths in ${\mathrm{TaAs}}_2$ calculated from ab initio. The impurity mean free path is estimated as 10 $\mu \mathrm{m}$ from earlier works [48, 49]. (b) Normalized $j_x$ curvature phase diagram. Overlaid lines show the trajectories with the decreasing temperature with different channel width.