Shear viscosity at finite magnetic field for graphene, non-relativistic and ultra-relativistic cases

TL;DR

This work addresses the problem of calculating the shear viscosity of the electron fluid in graphene under a finite magnetic field, and compares it with non-relativistic and ultra-relativistic regimes. It employs a kinetic theory approach in the relaxation-time approximation, deriving five anisotropic shear-viscosity coefficients $\eta_n$ that couple to five velocity-gradient tensors, and expresses them in terms of the cyclotron time $\tau_B$ and the momentum-relaxation time $\tau_c$, with Fermi-Dirac integrals $f_\nu(A)$ encoding the thermodynamic phase space. The key contributions are the explicit microscopic expressions for $\eta_n$, the mapping to three observable components $\eta_\perp$, $\eta_\parallel$, $\eta_\times$, and the identification of field thresholds for observable anisotropy and maximal Hall viscosity across graphene (GHD), non-relativistic (NRHD), and ultra-relativistic (URHD) systems. The findings have practical significance for experiments on graphene Dirac fluids and for magneto-viscous phenomena in high-energy plasmas, offering guidance on the required magnetic-field strengths and doping levels to observe anisotropic and Hall-viscosity effects.

Abstract

Present article has addressed finite magnetic field extension of previous work by Cho et al. (Phys. Rev. B 108, 235172, 2023) on microscopic calculation of shear viscosity for electron fluid in graphene system. Our calculation is based on the kinetic theory approach in the relaxation time approximation. In the absence of magnetic field, transport is governed by a single shear viscosity coefficient, whereas the application of a finite magnetic field induces anisotropy, give rise to the five independent shear viscosity coefficients associated with distinct velocity gradient tensors. These coefficient can be physically categorized into perpendicular, parallel and Hall components relative to the magnetic field direction. When the scattering time equals to the cyclotron time, the perpendicular component is suppressed by 80% and parallel component by 50%, and Hall effect can reach maximum. Corresponding magnetic field strength for electron fluid in graphene is around 0.01-0.1 Tesla and the same for non-relativistic electron fluid and ultra-relativistic quark fluid are around 10 Tesla and 10^14 Tesla respectively. They may be considered as required magnetic field strength in three different fluid systems to observe noticeable magnetic field response in their shear viscosity coefficients.

Figures (4)

• Figure 1: Shear viscosity to entropy density ratio for GHD, URHD and NRHD cases for their corresponding temperatures.
• Figure 2: Magnetic field dependent of perpendicular shear viscosity ratio in graphene, non-relativistic and ultra-relativistic systems.
• Figure 3: Magnetic field dependent of parallel shear viscosity ratio in graphene, non-relativistic and ultra-relativistic systems.
• Figure 4: Magnetic field dependent of Hall viscosity ratio in graphene, non-relativistic and ultra-relativistic hydrodynamic regimes. GHD and NRHD results are shown as at $T=60K$ and $\mu=0.2~meV$ while URHD case corresponds to $10^{12}K$ and $\mu=0.3~GeV$.