Historical research guide · 0470 · 2008
Contribution of Drifting Carriers to the Casimir-Lifshitz and Casimir-Polder Interactions With Semiconductor Materials
Diego A. R. Dalvit · Steve K. Lamoreaux
In one minute
Diego Dalvit of Los Alamos and Steve Lamoreaux of Yale — the physicist whose 1997 experiment first measured the Casimir force to a few per cent — ask a practical question: what does a semiconductor plate look like to the fluctuating vacuum field? Metals are easy and ideal insulators are easy, but semiconductors sit awkwardly between them, because the few free charges they carry drift and pile up in response to a field. The authors couple Maxwell’s equations to the classical Boltzmann equation that governs that drift, and derive the reflection amplitudes a semiconductor surface actually presents, frequency by frequency. Two results matter. The common shortcut — bolting a conductivity term onto the material’s permittivity — gives the wrong force at separations shorter than the material’s Debye screening length, because thermal diffusion counterbalances the current the field drives. And their theory satisfies the Nernst heat theorem, the Casimir entropy going to zero at zero temperature. They aimed the calculation at a germanium-plate measurement then under way.
Connection
Chapter 2 rests on the Casimir force being a measured property of real materials rather than an idealisation, and this is the calculation that tells an experimenter what the plates in front of them actually reflect. Anyone designing a Casimir device out of semiconductors needs exactly this: the force as a function of the carrier density and the screening length you can engineer.
From the source
A theory of the Casimir-Lifshitz and Casimir-Polder interactions with semiconductor or insulator surfaces can be built by coupling Maxwell’s equations to the classical Boltzmann transport equation for the drifting carriers, yielding frequency-dependent TM and TE reflection amplitudes for the surface.Abstract; Field equations section; Equations 6 and 7
The standard shortcut of adding a term four-pi-i-sigma-zero over omega to a bare dielectric permittivity is not correct at separations smaller than the Debye-Hückel screening length, because the current driven by the field is counterbalanced by thermal diffusion.Conclusions, first paragraph
In the quasi-static limit the derived amplitudes reduce to the zero-frequency values Pitaevskii proposed — the transverse electric reflection vanishing and the transverse magnetic reflection taking the Debye-screened form — so the theory recovers the thermal Lifshitz force between an atom and a surface of small carrier density, and the crossover between good conductors and ideal dielectrics.Limiting cases, case a; Ideal dielectric limit, case b
For intrinsic semiconductors with typical parameters, only the zero-order transverse magnetic Matsubara mode is significantly modified, so to very high accuracy the whole effect of drifting carriers can be modelled by the Debye-Hückel screening length alone.Free energy and entropy section, the germanium paragraph accompanying Figure 1
The theory is compatible with the Nernst theorem: the Casimir-Lifshitz entropy vanishes at zero temperature, and so does the Casimir-Polder entropy when the plate is a semiconductor, because the intrinsic carrier density falls exponentially as the temperature goes to zero.Free energy and entropy section, closing paragraph
The two conductivity models diverge measurably: with drifting carriers the plates look like perfect conductors for the zero-order transverse magnetic mode once the gap exceeds the Debye length — 0.68 micrometres for germanium and 24 micrometres for silicon — while the additive-conductivity model puts that crossover at the thermal wavelength of about 7 micrometres regardless of the material. The authors set out to test this on an ongoing measurement between pure germanium plates.Figure 2 and its caption; Conclusions, final paragraph
The way in
https://doi.org/10.1103/PhysRevLett.101.163203How to cite it
Diego A. R. Dalvit, Steve K. Lamoreaux (2008) Contribution of Drifting Carriers to the Casimir-Lifshitz and Casimir-Polder Interactions With Semiconductor Materials doi:10.1103/PhysRevLett.101.163203