Casimir interactions between two parallel graphene sheets carrying steady-state drift currents

Technical Summary: Casimir Interactions Between Parallel Graphene Sheets Carrying Steady-State Drift Currents

Problem Statement
The paper investigates fluctuation-induced Casimir interactions between two parallel graphene sheets when one or both sheets carry steady-state drift currents. While the Casimir force between neutral objects in equilibrium is well-established as an attractive force dependent on geometry and material properties, the behavior of this interaction under non-equilibrium conditions—specifically when charge carriers are in motion—remains an area of active theoretical inquiry. Previous studies have explored drift currents in semiconducting plates or treated the effect in graphene simply as a Doppler shift of the electromagnetic response. However, these approaches often rely on semiclassical Drude-Boltzmann transport or assume a moving medium, which fails to capture the specific microscopic physics of graphene, where the lattice remains static while only the carriers drift. The authors aim to resolve the boundary conditions for current-carrying graphene sheets using a more rigorous model to determine how drift currents modify both the perpendicular (vertical) and lateral components of the Casimir force.

Methodology
The authors model the system using two parallel graphene sheets separated by a distance $d$, with drift currents driven by external voltage biases. To accurately describe the non-equilibrium optical response of the current-carrying graphene, they employ the Shifted Fermi Disk (SFD) model. Unlike simpler Doppler-shift approximations valid only for parabolic two-dimensional electron gases, the SFD model accounts for the intrinsic redistribution of carriers in momentum space caused by steady-state drift, preserving graphene's linear Dirac dispersion and resulting in an anisotropic conductivity tensor.

The calculation of forces is performed using fluctuational electrodynamics within the Rytov framework. The authors:

1. Derive the modified conductivity tensor components ($\sigma_{xx}, \sigma_{yy}$) incorporating the drift velocity $v_d$ relative to the Fermi velocity $v_F$.
2. Solve the electromagnetic boundary conditions for the planar geometry to obtain reflection coefficients for both $s$ (TE) and $p$ (TM) polarizations, accounting for Doppler-shifted frequencies ($\omega' = \omega - \mathbf{q} \cdot \mathbf{v}_d$) in the moving carrier frame.
3. Calculate the fluctuation-induced forces by evaluating the Maxwell stress tensor at the boundaries. This involves integrating over frequency and wave vectors, separating contributions from propagating ($q < \omega/c$) and evanescent ($q > \omega/c$) modes.
4. Distinguish between the vertical force component ($F_z$) and the lateral force component ($F_x$), the latter arising due to the breaking of time-reversal symmetry by the drift current.

Key Contributions and Results
The study yields several specific findings regarding the modification of Casimir interactions by drift currents:

• Vertical Force Modification: The drift current introduces a repulsive correction ($\Delta F_z$) to the inherently attractive equilibrium Casimir force. While the total force remains attractive, the repulsive correction reduces the overall magnitude. This correction scales approximately as $\beta_d^2$ (where $\beta_d = v_d/v_F$) and exhibits a distance dependence of $\Delta F_z \sim 1/d^{5.8}$, indicating a shorter-range interaction compared to the equilibrium $1/d^4$ scaling. The effect is most pronounced at small separations and high drift velocities.
• Lateral Force Generation: The system generates a lateral Casimir-like force ($F_x$) that opposes the direction of the carrier flow, analogous to non-contact quantum friction. For small drift velocities ($\beta_d < 1$), this force scales linearly with $\beta_d$. At larger velocities, the dependence becomes nonlinear. The magnitude of $F_x$ is significantly smaller than the equilibrium vertical force but scales as $1/d^{4.2}$.
• Configuration Dependence:
  • Single vs. Double Current: When currents flow in both sheets in opposite directions, the magnitude of the corrections to both $F_z$ and $F_x$ is enhanced compared to the single-current case.
  • Co-directional Currents: When currents flow in the same direction with equal velocity, the lateral force vanishes ($F_x = 0$) because there is no relative motion between the electron populations of the two layers. The correction to the vertical force in this scenario is also minimal, as the Doppler shift effects cancel out, leaving only the secondary modification from the SFD conductivity.
• Parameter Dependencies: The magnitude of the current-induced corrections depends on temperature ($T$) and Fermi energy ($E_F$). The corrections show an approximately linear dependence on temperature (driven by the thermal Bose factor in the low-frequency regime) and a near-linear correlation with the Fermi energy, as higher $E_F$ enhances the metallic character of the graphene.

Significance and Claims
The paper claims to provide a physically consistent framework for treating non-equilibrium optical responses in current-carrying graphene by moving beyond simple Doppler-shift approximations. By utilizing the SFD model, the authors demonstrate that the microscopic description of carrier redistribution is crucial for accurately capturing the dielectric response of graphene under drift.

The primary significance of the work lies in establishing that current flow provides an effective route to tune Casimir interactions in graphene-based systems. The results suggest that the attractive Casimir force can be modulated (reduced) and a lateral frictional force can be induced by controlling the drift velocity and direction of carriers. This offers a pathway for engineering fluctuation-induced forces at the nanoscale in two-dimensional materials and van der Waals heterostructures, contributing to the broader understanding of non-equilibrium fluctuation phenomena. The authors emphasize that while the repulsive correction does not overcome the underlying attraction, the ability to control the force magnitude and direction via electrical bias represents a significant step in manipulating quantum vacuum forces in non-equilibrium systems.