STM-D-0836Paper2017Published and peer-reviewed

Reflection and transmission in nonlocal susceptibility models with multiple resonances

R. J. Churchill · T. G. Philbin

Abstract and summary · read the original at the source

In one page

Every material answers a light wave slightly sideways: what happens at one point depends on the field in the neighbourhood around that point, not only on the field at the point itself. Physicists call this spatial dispersion, and it makes a surface awkward, because Maxwell’s usual boundary conditions no longer pin down all the waves that get into the material and an extra rule has to be supplied by hand. Robert Churchill and Thomas Philbin solve the awkward case — a material carrying several such resonances at once, as real semiconductors do. They derive exact reflection and transmission coefficients, leave the surface rule as a free parameter so that every choice in the literature falls out of one formula, and then compute what it all means for the zero-point and thermal radiation just outside the surface. That last part is the payoff. In the simplified local picture the energy density blows up as you approach the wall; treated properly it climbs and levels off at a finite value.

Why it matters hereChapter 2 holds that the vacuum near a surface is a structured medium with a real, calculable energy density, and this paper does that calculation honestly for materials with the messy multi-band structure real crystals have. The divergence that made the textbook version look unphysical disappears once the material’s own spatial response is kept — which matters to anyone designing a device that depends on knowing the field right at a wall.

What it claims

01The authors derive exact expressions for the electromagnetic reflection and transmission coefficients at the planar boundary of a semi-infinite dielectric whose susceptibility carries multiple spatially dispersive resonances, extending the single-resonance treatment of Halevi and Fuchs. The surface enters through an arbitrary phenomenological reflection coefficient for the polarization waves, so that the various additional boundary conditions proposed in the literature — Pekar, Ting and coworkers, Fuchs-Kliewer, Rimbey-Mahan, Agarwal and coworkers — are recovered as particular values of one parameter.Abstract; Section II C, Reflection and Transmission Coefficients; Conclusions, Section VIII
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02Keeping the material’s spatial dispersion removes the unphysical divergence that the purely local model predicts for the spectral energy density of zero-point and thermal radiation near a surface. Instead of growing as one over the cube of the distance to the boundary, the computed energy density climbs and then saturates to a finite value as the distance goes to zero, confirming for multi-resonance media the result the authors found for one resonance.Section VII, Spectral energy density, the local divergent result Equation 64 and Figure 8 for ZnO at 3.44 electronvolts
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03Every resonance contributes two peaks to the spectral energy density outside the medium — one near the transverse frequency from the s-polarization part of the integral and one near the longitudinal frequency from the p-polarization part — so the three exciton bands of ZnO give six peaks in the spectrum measured 8 nanometres from the boundary. The size of every peak depends strongly on which additional boundary condition is assumed, largest for the Ting condition and smallest for the Pekar condition.Section VII, Figure 9 and the accompanying discussion
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04Multiple spatially dispersive resonances cannot be treated one at a time. Closely spaced bands suppress each other’s peaks — the ZnO peak at the first longitudinal frequency is cut down by the nearby second one until it is no larger than the s-polarization peaks — and for the heavy and light exciton bands of GaAs both single-exciton approximations underestimate the peak energy density, because at short distances the result is dominated by the evanescent-wave contribution where the two models differ most.Section VII, Figures 7, 10 and 11 and the closing paragraph of the section
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05For heavy and light exciton bands the authors give an improved single-band approximation, taking the mean of the square roots of the two band parameters rather than an effective single value; and they extend the model to exciton dispersion relations carrying a linear term in the wave vector, as uniaxial crystals have, where the splitting adds a further peak and turns the CdS spectral energy density into an overall three-peak structure with the crystal axis normal to the surface.Section V B, Heavy and Light Excitons; Section VI and Figure 12; Conclusions, Section VIII
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06The authors name the next extensions of the model: separate transverse and longitudinal susceptibilities, higher-order nonlocal terms, using the multi-band results to identify which additional boundary condition is appropriate for a medium with a complex exciton band structure, and applying the coefficients to the calculation of Casimir self-forces.Conclusions, Section VIII, final paragraph
What to watch

Read it · abstract

Abstract

We consider a semi-infinite dielectric with multiple spatially dispersive resonances in the susceptibility. The effect of the boundary is described by an arbitrary reflection coefficient for polarization waves in the material at the surface, with specific values corresponding to various additional boundary conditions (ABCs) for Maxwell’s equations. We derive exact expressions for the electromagnetic reflection and transmission coefficients and present the results for a variety of materials with multiple exciton bands. We find an improved single-band approximation for heavy/light exciton bands and extend our model to exciton dispersion relations with linear k terms which occur in uniaxial crystals. Finally, we calculate the spectral energy density of thermal and zero-point radiation for a variety of multi-resonance models and ABCs.

R. J. Churchill and T. G. Philbin, Physics and Astronomy Department, University of Exeter. Physical Review B 95, 205406 (2017); preprint arXiv:1702.05058.

(Abstract only. The complete paper — the infinite-medium theory, the surface impedance and boundary treatment, the worked results for ZnO, GaAs and CdS, and the spectral energy density calculation — is at the source; see the rights note above for why the full text is not reproduced here.)

The way in

https://doi.org/10.1103/PhysRevB.95.205406Published as Physical Review B 95, 205406 (2017) under the APS default licence, and posted to arXiv as 1702.05058 under the arXiv non-exclusive distribution licence 1.0 — neither is an open licence and no Creative Commons statement appears in the text or on the arXiv record, so this page carries the summary, the claims and the authors’ own abstract and sends the reader to the source. The abstract below is transcribed from the arXiv preprint.

How to cite it

R. J. Churchill, T. G. Philbin (2017) Reflection and transmission in nonlocal susceptibility models with multiple resonances. doi:10.1103/PhysRevB.95.205406

Where it sits in the curriculum

What the vacuum is