Historical research guide · 0704 · 2017
Thermal effect in the Casimir force for graphene and graphene-coated substrates: Impact of nonzero mass gap and chemical potential
G. Bimonte · G. L. Klimchitskaya · V. M. Mostepanenko
In one minute
The Casimir force is the vacuum’s signature you can weigh: bring two surfaces close and the field between them pushes them together. Graphene makes it stranger. Because a sheet of graphene is one atom thick and its electrons behave like massless particles moving at a three-hundredth of the speed of light, its Casimir force picks up a temperature contribution that becomes dominant at separations where ordinary materials show almost none. Giuseppe Bimonte in Naples with Galina Klimchitskaya and Vladimir Mostepanenko in Saint Petersburg compute that giant thermal effect for graphene as it really comes, not as an idealisation — with a small energy gap in its spectrum and with doping, described exactly by the finite-temperature polarization tensor of quantum electrodynamics. Their finding is that the two imperfections pull in opposite directions and partly cancel, so a real, doped, slightly gapped sheet behaves much closer to the idealised one than you would fear — and the differential experiment already proposed to see the effect survives both of them with room to spare.
Connection
Chapter 2’s case that the vacuum is a real, structured medium rests on the Casimir force being calculable to the last decimal and then measured, and this is that calculation carried into a material where the vacuum’s temperature signature becomes large enough to dominate. It also names the bench test: a differential setup whose predicted signal sits far above an error bar already achieved.
From the source
Using the rigorous finite-temperature quantum electrodynamics of the polarization tensor, the authors show that the mass gap and the chemical potential of real graphene act on the Casimir force in opposite directions — the force falls as the gap is opened and rises as the doping is raised — so for a real sample the two effects partly cancel.Abstract; Section III, Eq. 32 and Figure 1(a)
The reason is conductivity. Raising the chemical potential raises graphene’s conductivity and so raises the force, while opening a mass gap lowers the carriers’ mobility and therefore the conductivity, and brings the force down. The relative influence of both parameters shrinks as the sphere is moved further from the sheet.Section III, the paragraph following Figure 1(a)
For graphene deposited on silica or silicon the Casimir force is much stronger than for a free-standing sheet, yet the thermal correction and its fractional weight in the total force are smaller, and the influence of the graphene coating fades as the substrate’s static dielectric permittivity rises. Since most applications use coated substrates, the authors judge that controlling the Casimir force by doping alone is problematic there.Section IV; Section VI, paragraphs 3 and 4
The thermal correction is not a monotonic function of the mass gap, and its derivative with respect to the gap energy is discontinuous where that energy equals the chemical potential — because below that point the zero-temperature polarization tensor does not depend on the chemical potential at all.Section III, Figures 2(b) and 2(c) and the paragraph discussing them; Section VI, paragraph 4
The bench test is a differential measurement: a gold-coated sphere over a silicon plate whose two halves are, respectively, bare and graphene-coated, with the difference of the two Casimir forces recorded as a function of separation. Taking the doping measured in real samples, 0.02 electronvolts, together with the conservative upper bound of 0.2 electronvolts on the gap energy, the smallest possible thermal signal still runs far above the 1 femtonewton error already achieved in a differential Casimir measurement, across the whole range from 220 nanometres to 1.5 micrometres.Section V, Eqs. 34 to 36, Figure 8(c) and the final paragraph
The authors name what a direct observation would buy: fitting the measured giant thermal effect against computations at different gap energies would determine the mass gap of a graphene sheet, and the same handle would open the modification and control of the Casimir force in graphene-based micromechanical systems.Section VI, final paragraph
The way in
https://doi.org/10.1103/PhysRevB.96.115430How to cite it
G. Bimonte, G. L. Klimchitskaya, V. M. Mostepanenko (2017) Thermal effect in the Casimir force for graphene and graphene-coated substrates: Impact of nonzero mass gap and chemical potential doi:10.1103/PhysRevB.96.115430