Valley-Controlled Viscosity of Two-Dimensional Dirac Fluids

Technical Summary: Valley-Controlled Viscosity of Two-Dimensional Dirac Fluids

Problem Statement
The hydrodynamic regime of electronic transport, where electron-electron collisions dominate over momentum-relaxing processes, is characterized by shear viscosity governing momentum diffusion. While recent experiments in small-angle twisted bilayer graphene (SA-TBG) have demonstrated that weak interlayer hybridization combined with displacement fields can lift valley degeneracy, the impact of this valley imbalance on the macroscopic viscous properties of the electron fluid remains unexplored. Specifically, it is unclear how shifting the energy of two low-energy Dirac cones relative to one another (valley splitting, $\Delta$) alters the shear and kinematic viscosities, particularly given the complex interplay between intraband scattering, electron-hole scattering, and screening.

Methodology
The authors employ a linearized semiclassical Boltzmann approach to calculate the shear viscosity ($\eta$) and kinematic viscosity ($\nu = \eta/\rho_{\text{eff}}$) of electron fluids.

• Framework: The study utilizes the Boltzmann equation retaining only momentum-conserving electron-electron scattering in the collision integral, assuming Umklapp processes, acoustic phonons, and static disorder are subdominant.
• Models:
  1. Valley-Polarized Two-Cone Dirac System: An effective model for SA-TBG where two Dirac cones are separated by an energy offset $\Delta$. The model assumes no intervalley scattering (valley index conservation) but allows intravalley and inter-cone scattering within the same valley index.
  2. Monolayer Graphene (MLG): A benchmark single-cone Dirac system with valley degeneracy.
  3. Parabolic-Band 2DEG: A conventional Fermi-liquid control case.
• Calculation: The viscosity transport time ($\tau_v$) is derived by projecting the linearized collision integral onto the shear channel using a traceless stress vertex. The authors explicitly account for dynamically screened Coulomb interactions and, crucially, include static screening to regularize divergences in the shear channel.
• Parameters: Numerical results are presented for a fixed carrier density $n_0 = 10^{11} \text{ cm}^{-2}$ across a range of temperatures ($T$) and valley splittings ($\Delta$).

Key Contributions and Results

1. Nonmonotonic Viscosity in Valley-Imbalanced Dirac Fluids:
   The primary finding is that the kinematic viscosity of the two-cone Dirac system is a strongly nonmonotonic function of the valley splitting $\Delta$. As $\Delta$ increases from zero:

   • The viscosity initially rises.
   • It reaches a local maximum ($\Delta_{\text{max}}$) when the shifted valley is driven close to charge neutrality.
   • It decreases to a secondary minimum ($\Delta_{\text{min}}$) as the shifted valley becomes hole-doped, opening new electron-hole scattering channels.
   • It rises again at large $\Delta$ when both valleys are deeply degenerate and Pauli blocking suppresses scattering.
   • The position of the maximum $\Delta_{\text{max}}(T)$ shifts toward $\Delta=0$ as temperature increases, eventually collapsing into a broad peak at high $T$ ($T > T_c \approx 260$ K).

2. Analytic Estimates for Extrema:
   The authors derive analytic expressions for the locations of the viscosity extrema in the degenerate regime:

   • $\Delta_{\text{max}}$ occurs when the shifted valley is neutral ($\mu_1 \approx 0$), leading to an estimate $\Delta_{\text{max}}(T) \simeq \sqrt{\Delta_0^2 - \frac{\pi^2}{3}(k_B T)^2}$.
   • $\Delta_{\text{min}}$ is associated with the onset of significant hole doping in the shifted valley, estimated via a phenomenological fit involving the thermal energy scale.

3. Distinct Behavior in Monolayer Graphene (MLG):
   In contrast to the two-cone system, MLG exhibits a strictly monotonic decrease in kinematic viscosity with temperature.

   • While the shear viscosity ($\eta$) in MLG shows a minimum at the crossover between the degenerate Fermi-liquid and non-degenerate Fermi-gas regimes, the kinematic viscosity ($\nu$) does not.
   • This is attributed to the strong temperature dependence of the inertial mass density ($\rho_{\text{eff}}$) in Dirac fluids, which scales with the enthalpy density. As $T$ increases, $\rho_{\text{eff}}$ increases sufficiently to prevent a minimum in $\nu$, causing it to decrease monotonically.

4. Role of Screening:
   The paper highlights that static screening is not merely a quantitative correction but a strict physical requirement for obtaining correct low-temperature asymptotics in the shear channel.

   • In MLG, the projection onto shear modes removes collinear divergences, but the bare Coulomb interaction leaves a residual logarithmic divergence ($\sim 1/[T^2 \ln T]$).
   • Including static screening restores the conventional Fermi-liquid-like $T^{-2}$ asymptotic behavior for the transport time.
   • Similarly, in the parabolic-band 2DEG, static screening is essential to regularize the transport integral and yield a finite viscosity.

5. Comparison with 2DEG:
   In the parabolic 2DEG, where mass density is fixed, both shear and kinematic viscosities exhibit a minimum as functions of temperature, resembling the behavior of simple fluids but with algebraic ($T^{-2}$) rather than exponential growth at low temperatures due to Pauli blocking.

Significance and Claims
The paper claims to identify valley control as a direct "knob" for tuning hydrodynamic transport in Dirac materials. By demonstrating that valley splitting induces a pronounced nonmonotonic response in viscosity, the work clarifies the interplay between band structure, scattering phase space, and screening. The authors assert that these viscosity changes should manifest directly in tunable nonlocal resistance profiles, current flow patterns, and the critical thresholds for current-vortex formation in experimental settings. Furthermore, the study establishes that the unique temperature dependence of the inertial mass density in Dirac fluids leads to distinct hydrodynamic signatures (such as the monotonic kinematic viscosity in MLG) that differentiate them from conventional Fermi liquids and simple fluids.