Electron--phonon cooling noise in monolayer and bilayer graphene

We study the electron--phonon heat-current fluctuations in monolayer and Bernal bilayer graphene coupled to longitudinal acoustic phonons. To do this, we construct a symmetric heat-current operator and evaluate the non-equilibrium current--current correlator at finite frequency for unequal electron and phonon temperatures. The heat-current vertices then involve the probe energy combined with twice the phonon energy, and the nonequilibrium spectral density must be calculated directly rather than inferred from a dynamic thermal fluctuation--dissipation relation. Further, we obtain full analytical expressions in the Bloch--Grüneisen (BG) and Equipartition (EP) regimes, derive their low-frequency limits, and compare them with a complete doped-intraband on-shell calculations for monolayer graphene. The numerical results connect the two temperature regimes and show how finite-frequency noise changes out of equilibrium. For representative parameters, the corrections become percent-level in the upper-gigahertz to sub-terahertz range. It is worth mentioning that in equilibrium, the results satisfy the zero-frequency thermal fluctuation--dissipation relation(FDT) and reproduce the well established cooling-power and thermal-conductance laws.