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

TL;DR

This work addresses how spatial non-locality and losses in graphene affect the Casimir-Lifshitz force in a Au sphere–graphene–SiO$_2$ system. It systematically compares three EM response models—non-local lossy Kubo, local lossy Kubo, and non-local lossless QFT polarization—within Lifshitz theory and the Proximity Force Approximation, against recent experiments. The key finding is that, for the experimental parameters, all models predict CLF gradients that differ by less than $10^{-3}$, indicating non-locality and losses are negligible in these measurements and that the Drude vs Plasma prescriptions for the metals cannot be distinguished. Practically, a simple local Kubo model, with parameters such as Dirac mass $\Delta$, chemical potential $\mu$, and losses $\Gamma$, suffices for comprehensive comparison, while the polarization tensor framework can be invoked for more refined non-local or magnetic-response phenomena.

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.

Figures (16)

• Figure 1: Scheme of the systems, as in the experiment PRL_MohideenPRB_Mohideen. An SiO$_{2}$ plate covered with a single sheet of graphene placed at a distance $d$ of a gold covered sphere of radius $R$.
• Figure 2: Log-log plot of the imaginary part of the dielectric susceptibility of gold(in black) on real frequencies take from Palik_Handbook. The extrapolations at low and high frequencies are plotted in red, the Drude model for gold $\epsilon = 1 - {\omega_{P}^{2}}/{[\omega(\omega + \mathrm{i} \Gamma_{\rm{Au}})]}$ is plotted in green. In the insert, the dielectric susceptibility of gold for imaginary frequencies ($\epsilon(\mathrm{i}\hbar\xi) - 1$) obtained from the application of the Kramers-Krönig formula to real data following PhysRevB.77.035439 is plotted in black, while the Drude model of gold is plotted in green.
• Figure 3: Log-Plot of the dielectric susceptibility of SiO$_{2}$ for imaginary frequencies, taken from Palik_Handbook. In the insert, the dielectric susceptibility of SiO$_{2}$ ($\epsilon(\mathrm{i}\hbar\xi) - 1$) for imaginary frequencies is represented in a double logarithmic plot.
• Figure 4: Plots of the CLF gradient $G_{\rm{r}}(d)$ of (\ref{['Grough']}) and as a function of the distance $d$. The experimental data are represented with the black (results of the PRL PRL_Mohideen) and cyan (results of the PRB PRB_Mohideen) points respectively, the numerical results at $T=294\text{ K}$ are in purple ($\Delta = 0\text{ eV}$ and $\mu=0.25\text{ eV}$) and in red ($\Delta = 0.1\text{ eV}$ and $\mu=0.23\text{ eV}$), while the results at $T=0\text{ K}$ are in blue ($\Delta = 0\text{ eV}$ and $\mu=0.25\text{ eV}$) and green ($\Delta = 0.1\text{ eV}$ and $\mu=0.23\text{ eV}$). For the numerical evaluation we used here the general non-local Kubo model $\sigma^{\rm{K}}$ for graphene non-local_Graphene_Lilia_PabloPabloMauroComparisonKuboQFT2024 with losses $\hbar\Gamma_{\rm{Gr}} = 10^{-3}\text{ eV}$. For the $n=0$ Matsubara term we used the Drude prescription for gold (\ref{['DrudeGoldxi']}), with $\hbar\omega_{P} = 9\text{ eV}$ and $\hbar\Gamma_{\rm{Au}} = 35\times 10^{-3}\text{ eV}$.
• Figure 5: Logarithmic Plot of the relative difference $\mathcal{D}(Exp,\sigma^{\rm{K}})$ (Eq. \ref{['def_rel_diff']}) of the theoretical predictions using the non-local Kubo model $\sigma^{\rm{K}}$ with the experimental result of PRL_Mohideen. The numerical results at $T=294\text{ K}$ are in purple ($\Delta = 0\text{ eV}$) and in red ($\Delta = 0.1\text{ eV}$), while the results at $T=0\text{ K}$ are in blue ($\Delta = 0\text{ eV}$) and green ($\Delta = 0.1\text{ eV}$), the same as in Fig. \ref{['Fig_Comparison_NOLocal_Model']}. The black curve is the error bar compared with the experimental value of each experimental point $\mathcal{D}(Exp,Exp\text{ lower bar})$. Theoretical values below this black curve show cases when the theoretical results are inside the experimental error-bars. We can observe a small discrepancy at short distances.