Casimir-Lifshitz force with graphene: theory versus experiment, role of spatial non-locality and of losses

Abstract

We analyze the impact of spatial non-locality and losses in the electromagnetic response of graphene on the Casimir-Lifshitz interaction. To this purpose, we calculate the Casimir-Lifshitz force (CLF) between a gold sphere and a graphene-coated SiO$_2$ plane and compare our finding with the recent experiment in PRL {\bf 126}, 206802 (2021) and PRB {\bf 104}, 085436 (2021). We calculated the CLF using three different models for the electromagnetic response of graphene: electric conductivity using a non-local and lossy Kubo model, electric conductivity using the local and lossy Kubo model, and the non-local and lossless polarization operator model. The relation between these three models has been recently explored in PRB {\bf 111}, 115428 (2025). We show that, for the parameters of the available experiments, the theoretical predictions for the Casimir-Lifshitz force using the three models are practically identical (having a relative differences smaller than $10^{-3}$). This implies that for those given experiments, both non-local and lossy effects in the graphene response are completely negligible. We also find that this experiment cannot distinguish between the Drude and Plasma prescriptions for the involved materials (gold and graphene). Our findings are relevant for present and future comparisons with experimental measurement of the Casimir-Lifshitz force involving graphene structures. Indeed, we show that an extremely simple local Kubo model for the electric conductivity, explicitly depending on Dirac mass, chemical potential, losses and temperature, is largely enough for a totally comprehensive comparison with typical experimental configurations. We also show how the Polarization tensor must be used and modified in general, for phenomena needing a more fine response function, i.e. requiring both the spatial non-locality and losses.