Tomographic electron flow in confined geometries: Beyond the dual-relaxation time approximation

Hydrodynamic-like electron flows are typically modeled using the Stokes-Ohm equation or a kinetic description that is based on a dual-relaxation time approximation. Such models assume a short intrinsic mean free path $\ell_e$ due to momentum-conserving electronic scattering and a large extrinsic mean free path $\ell_\text{MR}$ due to momentum-relaxing impurity scattering. This assumption, however, is overly simplistic and falls short at low temperatures, where it is known from exact diagonalization studies of the electronic collision integral that another large electronic mean free path $\ell_o$ emerges, which describes long-lived odd electron modes -- this is sometimes known as the tomographic effect. Here, using a matched asymptotic expansion of the Fermi liquid kinetic equation that includes different electron relaxation times, we derive a general asymptotic theory for tomographic flows in arbitrary smooth boundary geometries. Our key results are a set of governing equations for the electron density and electron current, their slip boundary conditions and boundary layer corrections near diffuse edges. We find that the tomographic effect strongly modifies previous hydrodynamic theories for electron flows: In particular, we find that (i) an equilibrium is established in the bulk, where the flow is governed by Stokes-Ohm like equations with significant finite-wavelength corrections, (ii) the velocity slip conditions for these equations are strongly modified from the widely-used hydrodynamic slip-length condition (iii) a large kinetic boundary layer arises near diffuse boundaries of width $\sim\sqrt{\ell_e \ell_o}$, and (iv) all these effects are strongly suppressed by an external magnetic field. We illustrate our findings for electron flow in a channel. The equations derived here represent the fundamental governing equations for tomographic electron flow in arbitrary smooth geometries.