Evanescent-mode Casimir-Josephson force and gate-controlled resonances in ballistic graphene Josephson junctions
We develop a microscopic scattering theory of the phase-dependent equilibrium mechanical response of a ballistic superconductor-graphene-superconductor Josephson junction. The calculation is based on an energy-dependent Dirac scattering matrix embedded in a normalized Matsubara determinant, so the reported force correction contains the complete Bogoliubov-de Gennes spectrum of the stated ideal model. At charge neutrality the full determinant approaches a closed-form evanescent-mode result proportional to $ΔW/L^2$, reaching $2\ln 2\,ΔW/(πL^2)$ at phase difference $π$ in the zero-temperature short-junction limit. Gate doping produces propagating channels and Fabry-Pérot structure, causing large oscillations and sign reversals of the phase-dependent force correction $δF=F(ϕ)-F(0)$. We distinguish the interband-to-intraband Andreev crossover, controlled by $|μ|/Δ$, from the evanescent-to-propagating crossover, controlled by $|μ|L/(\hbar v_F)$. Exact real-energy subgap poles obtained from the same energy-dependent scattering matrix are used to diagnose specular/interband and retro/intraband character; these labels are not treated as separately measurable thermodynamic forces in the mixed regime. We also quantify the difference between the complete determinant result and the frozen-scattering short-junction approximation without identifying that difference with a pure continuum force. The resulting gate- and phase-dependent mechanical signal provides a Dirac-material extension of earlier superconductivity-induced mechanical-force proposals.