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The rigorous finite-temperature QED formalism of the polarization tensor isused to study the combined effect of nonzero mass gap $m$ and chemicalpotential $\mu$ on the Casimir force and its thermal correction in theexperimentally relevant configuration of a Au sphere interacting with a realgraphene sheet or with graphene-coated dielectric substrates made of differentmaterials. It is shown that for both a free-standing graphene sheet and forgraphene-coated substrates the magnitude of the Casimir force decreases as $m$is increased, while it increases as $\mu$ is increased, indicating that theseparameters act in opposite directions. According to our results, the impact of$m$ and/or $\mu$ on the Casimir force for graphene-coated plates is muchsmaller than for a free-standing graphene sheet. Furthermore, computations showthat the Casimir force is much stronger for graphene-coated substrates than fora free-standing graphene sample, but the thermal correction and its fractionalweight in the total force are smaller in the former case. These results areapplied to a differential setup that was recently proposed to observe the giantthermal effect in the Casimir force for graphene. We show that this experimentremains feasible even after taking into account the influence of the nonzeromass-gap and chemical potential of real graphene samples. Possible furtherapplications of the obtained results are discussed.

Thermal effect in the Casimir force for graphene and graphene-coated substrates: Impact of nonzero mass gap and chemical potential