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arXiv:2609.27864

Temperature dependence of the Casimir pressure on two graphene sheets placed on a substrate

Michael V. Davidovich

quant-phSubmitted Aug 20, 2026• Announced Aug 21, 2026
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Abstract

The dispersion interaction between two infinite sheets of graphene located on a substrate is considered based on the Lifshitz formula, taking temperature into account. The result is obtained in the form of a series that is easy to calculate. The influence of the interband conductivity of graphene, which turned out to be small, is examined.

Comments
32 pages, 4 figures

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The dispersion interaction between two infinite sheets of graphene located on a substrate is considered based on the Lifshitz formula, taking temperature into account. The result is obtained in the form of a series that is easy to calculate. The influence of the interband conductivity of graphene, which turned out to be small, is examined.
Based on the van Kampen-Schram method, a formula for the Casimir force between two plane-layered structures is obtained, which includes the sum of the Matsubara frequencies and a contour integral. It is shown that the contour integral is not taken into account in the Lifshitz formula. Taking it into account gives a small contribution at high temperatures when the Lifshitz formula is approximately correct. However, at low temperatures, its contribution is significant, and the Lifshitz formula for the finite temperature does not transform into the Lifshitz formula at zero temperature. An example of interaction of metal plates is considered.1
We investigate the Casimir force on thin suspended dielectric layers covered with graphene, both theoretically and experimentally. Thermal effects of the Casimir force on graphene occur at separations significantly smaller compared to ordinary metals and dielectrics. At such small separations, the Casimir force is sufficiently strong to be measured experimentally. However, the presence of substrates to support the graphene significantly reduces the enhancement. We show theoretically that replacing the solid substrate beneath graphene with a thin dielectric membrane partially restores this strong thermal dependence. One of the interacting objects is a solid gold plate. The other object is a suspended dielectric membrane made of either 15 nm of silicon oxide or 13 nm silicon nitride, covered with graphene. At a separation of $\sim$500 nm, theoretical predictions yield changes in the Casimir pressure of up to 54 % and 40 % for silicon oxide and silicon nitride, respectively, when the temperature is increased from 10 K to 294 K. Measurements were performed for graphene on a suspended silicon nitride membrane at room temperature, using a gold-coated sphere attached to a cantilever as the force sensor. The measured force gradient is consistent with the theoretical prediction, although the thermal dependence could not be resolved, in part because of uncertainties in the optical properties of silicon nitride. Replacing a bulk substrate with a thin dielectric membrane covered with graphene provides a substantial reduction of the Casimir force, which may facilitate the stable operation of neutral nanomechanical elements at small separations.
The corrections to Casimir pressure (force) on two plates of real metal described by the dielectric constant of the Drude, Drude-Lorentz, Drude-Smith and plasma models at the finite temperature and conductivity are considered. We consider the contour integral lost in the Lifshitz formula and discuss inconsistencies for the corrections to the Casimir force given in the literature. Numerical and analytical results in the near and far zones are presented. The two first-order correction obtained due to the actual dielectric constant of the metal coincided with the literature data.
Casimir forces, arising from quantum and thermal fluctuations, depend strongly on the electromagnetic susceptibilities of the interacting materials. Here, we theoretically investigate the Casimir force between two graphene sheets separated by a nanometer-thin layer of liquid water. We find that deviations from the bulk case can be as large as 24 \%, mostly due to the static susceptibility components. This effect does not decrease monotonically as a function of the layer thickness: it increases up to a maximum at 70 nm and it slowly declines to the bulk limit only at 1 micron. This is due to two competing effects: on one hand, the anisotropy decreases with the layer thickness; on the other hand, the relative contribution of the most anisotropic, static susceptibility components increases.
We investigate the fluctuation-induced Casimir interactions between two parallel graphene sheets carrying steady-state drift currents. The graphene properties are modeled based on the shifted Fermi disk model to capture the non-equilibrium optical response of the system. We find that the drift current introduces a repulsive correction to the perpendicular to the layers Casimir interaction, thereby reducing the overall attractive force. Although the correction is repulsive, it does not overcome the underlying attraction between the layers. It also generates a lateral force that opposes the carrier flow direction. Both contributions are studied in terms of distance and drift velocity functionalities showing pathways for Casimir force control.