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Theory of Optical Nonlocality in Polar Dielectrics

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Pith's one-line read The paper demonstrates that the standard local dielectric approximation fails for phonon polaritons in nanometric polar dielectrics, and that a nonlocal continuum theory with quadratic phonon dispersion quantitatively reproduces…

desk verdict Solid nonlocal theory for phonon polaritons with independent sphere predictions, but the 'first experimental verification' claim overreaches because the key nonlocal parameter is fit to the same data and the model misses a dip in the few-atomic-lattice sample. read the letter →

arxiv 1908.03806 v2 pith:UC3OF6CR submitted 2019-08-10 cond-mat.mes-hall physics.optics

Christopher R. Gubbin , Simone De Liberato This is my paper · ORCID
classification cond-mat.mes-hallphysics.optics
keywords phononpolaritonsnonlocalopticspolardielectricsspatialdispersionReststrahlenbandepsilon-near-zeromodesAlN/GaNsuperlatticesMietheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the usual local dielectric model, which assumes the material response at a point depends only on the field at that point, breaks down for phonon polaritons in nanometric polar crystals. It develops a macroscopic nonlocal theory in which optical phonons have spatial dispersion, and shows that this changes the predicted optical response of nanospheres, thin films, and superlattices. The theory explains previously unexplained infrared reflectance features in AlN/GaN atomic-scale superlattices that local theory cannot capture. If correct, local models are inadequate for phonon-polariton structures below roughly ten nanometers, and a lightweight continuum theory can serve as a design tool for mid-infrared nanophotonics.

What carries the argument

The carrying object is a continuum model of the polar crystal as an ionic displacement field $\mathbf{X}$ coupled to Maxwell's equations, with the equation of motion $\left[\omega_T^2 - \omega(\omega+i\gamma) + \beta_L^2\nabla(\nabla\cdot) - \beta_T^2\nabla\times\nabla\times\right]\mathbf{X} = (\mu/\rho)\mathbf{E}$. Fourier transforming this equation yields the longitudinal and transverse nonlocal dielectric functions $\varepsilon_L(\omega,k)$ and $\varepsilon_T(\omega,k)$, whose zeros define dispersive LO phonon modes. The system is closed by an additional boundary condition at interfaces, taken here as the vanishing of the ionic displacement at a dielectric-vacuum boundary, equivalent to specular reflection. This machinery lets longitudinal modes be excited at boundaries and propagate inside the nanostructure, coupling to transverse photonic modes and producing the predicted spectral features.

What would settle it

A direct check would be to measure the infrared reflectance of an AlN/GaN superlattice with layer thicknesses around 1 nm and compare the position and shape of the dip near 795 cm$^{-1}$, where the paper itself states the quadratic approximation fails; a corrected nonlocal theory must reproduce that feature. Alternatively, measure the extinction spectrum of 5 nm radius 3C-SiC nanospheres and look for the discrete longitudinal peaks below the LO frequency that the nonlocal theory predicts: if no such peaks appear, the central mechanism is wrong.

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Extended reading notes

Core claim

The central claim is that LO phonons in polar dielectrics acquire spatial dispersion that becomes optically relevant at the nanoscale, so the local dielectric function $\varepsilon_{\mathrm{LRA}}(\omega)$ must be replaced by wavevector-dependent functions $\varepsilon_L(\omega,k)$ and $\varepsilon_T(\omega,k)$ of the form given in Eqs. (2) and (3), with quadratic terms $\beta_L^2 k^2$ and $\beta_T^2 k^2$. Because optical phonons have negative dispersion, propagative longitudinal phonon modes coexist with the negative-dielectric Reststrahlen region, unlike in metals where longitudinal plasma waves are evanescent. This leads to discrete longitudinal resonances that couple to the Fröhlich mode, Fano-like interference, a small redshift of the main resonance, and size-dependent damping. Applied to AlN/GaN superlattices, the nonlocal model reproduces reflectance peaks and redshifts that local theory misses, with fitted parameters $\beta^{\mathrm{AlN}}_L = 5.1\times10^5\ \mathrm{cm\,s^{-1}}$ and $\gamma_{\mathrm{AlN}} = 10.3\ \mathrm{cm^{-1}}$, giving a nonlocal skin depth near 1.5 nm and indicating validity down to a few atomic layers.

Load-bearing premise

The load-bearing premise is that the true optical-phonon dispersion in layers only one or two nanometers thick is well approximated by a single quadratic term with a constant velocity, and that the chosen interface boundary condition is the correct one; if the real dispersion bends away from that quadratic curve at large wavevectors, the predicted spectra will not generalize.

Editorial extensions

If this is right

  • Below about 10 nm, local-response simulations of phonon-polariton structures will systematically miss extra resonances and mispredict mode frequencies; nonlocal terms must be included.
  • The extra peaks in small SiC nanospheres and in AlN/GaN superlattices are discrete longitudinal optical phonon modes confined by the particle or film boundaries, hybridizing with the Fröhlich or epsilon-near-zero resonances.
  • Nonlocal damping gives a size-dependent broadening of the form $\gamma_{\mathrm{NL}} = \gamma + A\beta_L/R$ with $A \approx 0.03$ for 3C-SiC, meaning smaller particles lose more energy to LO phonon emission even at radii where the local model appears sufficient.
  • In thin films, confined LO phonons shift the effective edge of the Reststrahlen band, so epsilon-near-zero resonances follow the quantized mode frequency $\omega_n$ rather than the zone-centre LO frequency for films thinner than about 10 nm.
  • A continuum nonlocal theory can reproduce experimentally observed infrared spectra of atomic-scale superlattices with only one additional fitted parameter, the longitudinal phonon velocity $\beta_L$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same continuum nonlocal formalism could be applied to other polar dielectrics with negative LO dispersion, such as quartz or hexagonal boron nitride, predicting analogous discrete longitudinal resonances and nonlocal redshifts in their Reststrahlen bands.
  • The fitted value of $\beta^{\mathrm{AlN}}_L$ could be checked independently against measured AlN phonon dispersion; a significant discrepancy would indicate that the single-quadratic-term approximation absorbs other physics and may not extrapolate to untested geometries.
  • Replacing the constant $\beta_L$ with a wavevector-dependent piecewise-constant velocity, which the paper mentions as possible, would extend the model across the full Brillouin zone and should also capture the 795 cm$^{-1}$ dip where the quadratic approximation currently fails.
  • Because the nonlocal effects rely on boundary-induced coupling between longitudinal and transverse modes, patterned or roughened surfaces could be used to tune the strength of these features without changing material composition.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a macroscopic continuum nonlocal dielectric theory for polar dielectrics, in which the LO and TO phonon dispersions are treated as quadratic in wavevector (Eqs. 2 and 3) and an additional boundary condition (Fuchs-Kliewer) closes the interface problem. The theory is applied to three systems: SiC nanospheres, where it predicts extra extinction peaks and size-dependent damping; thin AlN films, where it predicts a redshift of the ENZ mode with decreasing thickness; and AlN/GaN atomic-scale superlattices, where it aims to reproduce unexplained reflectance features. The central claim is that the local dielectric description fails at nanometric scales and that the nonlocal theory provides the first experimental verification of quantitatively correct results down to a few atomic lattices. The paper includes detailed appendices deriving the continuum model, the additional boundary conditions, the nonlocal Mie theory, and the scattering-matrix treatment of layered systems.

Significance. If the central claim were fully supported, the paper would establish that local effective-medium descriptions of phonon polaritons break down below roughly 10 nm and would provide a computationally light continuum tool for nanophotonic design. The SiC nanosphere predictions are the strongest part: the parameters β_SiC_L and β_SiC_T are taken from independent ab initio phonon dispersion calculations, so those predictions are not circular, and the predicted extra peaks and Fano-like features are concrete and falsifiable. The appendix derivations are careful and the numerical methods are clearly described. However, the superlattice validation, which is the basis of the 'first experimental verification' claim, is substantially weakened by the fact that β_AlN_L, γ_AlN, and a thickness shift are all fitted to the same reflectance data, and by the paper's own admission that the quadratic dispersion fails at the 795 cm⁻¹ dip in Fig. 5b.

major comments (3)
  1. [Sec. IV.B and Fig. 5b] The authors explicitly acknowledge that the quadratic dispersion approximation fails to reproduce the dip at 795 cm⁻¹ in Fig. 5b, because the 1.2/1.4 nm layers of heterostructure B probe wavevectors where the constant β_L underestimates the LO frequency. Since heterostructure B is precisely the 'few atomic lattices' system used to support the conclusion that the theory provides quantitatively correct results down to few atomic lattices in size, this admission undercuts that claim. The authors should either restrict the quantitative claim to the features that are actually reproduced, or improve the dispersion model (e.g., using piecewise constant velocities) and refit the data.
  2. [Appendix F and Fig. 5] The nonlocal superlattice fit uses three parameters (β_AlN_L, γ_AlN, and the thickness shift δ) fitted to the same experimental reflectance data, whereas the local theory uses two parameters, and no uncertainties, confidence intervals, or sensitivity analysis are reported. Consequently, the improved agreement at 835/865 cm⁻¹ in heterostructure A and the redshifted ENZ mode in heterostructure B is partly a result of parameter adjustment rather than an independent confirmation that a single constant β_AlN_L captures the nonlocal response at the relevant wavevectors. The authors should provide an independent determination of β_AlN_L from phonon dispersion data, or at least quantify how the fit quality varies with β_AlN_L and show that the qualitative conclusions are robust.
  3. [Abstract and Sec. V] The abstract and the concluding section state that the paper provides 'the first experimental verification that our theory provides quantitatively correct results for systems down to few atomic lattices in size.' Given the acknowledged 795 cm⁻¹ mismatch and the fitted nature of β_AlN_L, this wording overstates what the comparison demonstrates. The authors should soften the claim to something like 'first experimental evidence consistent with the nonlocal theory' and explicitly delineate which spectral features are and are not reproduced.
minor comments (5)
  1. [Sec. V] The paragraph beginning 'By modelling recently published experimental data...' is a sentence fragment without a main verb; it should be rewritten as a complete sentence.
  2. [Appendix B] The phrase 'through it's surface' should read 'through its surface'.
  3. [Appendix C] The name 'Debarnardi' should be 'Debernardi' to match the cited reference [55] (Debernardi et al., 1999).
  4. [Fig. 4 caption and text] The caption refers to 'The dots' while the main text describes the n = 1 longitudinal mode as green circles; please make the notation consistent.
  5. [Appendix F] The fitting procedure would benefit from a statement of the uncertainty on δ, γ_AlN, and β_AlN_L, and from a discussion of the correlation between δ and β_AlN_L, since both affect the effective confinement length.

Circularity Check

1 steps flagged · score 4.0 of 10

Superlattice 'verification' partly reduces to parameters fitted to the same data; the core nonlocal formalism for spheres is independent.

  1. fitted input called prediction [Appendix F (fitting procedure), used in Sec. IV.B and Sec. V]
    "In Appendix F: 'Our fitting procedure used two unknown parameters for the local theory and three in the nonlocal one: the shift in layer thickness δ, the damping rate in the AlN γAlN , and the longitudinal velocity in the AlN layer βAlN L , the latter being unique to the nonlocal theory.' In Sec. V: '...providing a first experimental verification that our theory provides quantitatively correct results for systems down to few atomic lattices in size.'"

    β_AlN_L is one of three parameters fitted to the same experimental reflectance curves that are then displayed as agreement in Fig. 5. Because β_L directly sets the longitudinal-mode frequencies through Eq. (11), the extra peaks near 835/865 cm⁻¹ and the redshifted ENZ feature are partly produced by the fitted parameter rather than independently predicted. The skin-depth estimate l_ph = 1.5 nm and the ENZ film shifts in Fig. 4 additionally inherit the fitted β_AlN_L and γ_AlN. The 'first experimental verification' claim is therefore statistically forced to some degree; the honest scope is that the nonlocal model captures qualitative features the local model cannot, not that the fitted parameter set is independently confirmed.

full rationale

The formal derivation of the nonlocal dielectric response is self-contained: Eqs. (2)-(3) follow from the continuum equation of motion in Appendix A, and the additional boundary condition choice is justified by symmetry and by the stated practical equivalence of Fuchs-Kliewer and Pekar-Ridley conditions. No self-citation chain carries the mathematical content, and the SiC sphere calculations use β_SiC values taken from independent ab initio phonon dispersion work, so those predictions are not circular. The main circularity concern is confined to the superlattice validation: the paper fits β_AlN_L, γ_AlN, and a thickness shift δ to the same experimental reflectance data it then cites as 'first experimental verification.' Agreement after such fitting is not an independent test of the quadratic-dispersion assumption, especially since the paper itself acknowledges that the quadratic approximation fails at the 795 cm⁻¹ dip in Fig. 5b. That failure is a correctness risk rather than a circularity, but it compounds the fitted-input issue. Accordingly, the paper is not wholly circular: the central nonlocal theory has independent content and external benchmarks for SiC, but the headline superlattice verification is partly a fitted-input-called-prediction, giving a moderate circularity score of 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new particles or forces are introduced. The longitudinal optical phonon modes are standard crystal excitations; the nonlocal modes are the same LO phonons with a wavevector-dependent frequency. The ledger records four fitted parameters and five modeling assumptions that the central claim depends on, with the AlN fit parameters carrying the main circularity burden.

free parameters (4)
  • beta_AlN_L = 5.1e5 cm/s
    Longitudinal phonon velocity in AlN, fitted to superlattice reflectance data in Appendix F and used in Figs. 4 and 5 and in Eq. (7).
  • gamma_AlN = 10.3 cm^-1
    AlN damping rate fitted to the same experimental reflectance data in Appendix F.
  • delta = 0.94 nm
    Layer thickness reduction per interface, fitted to account for atomic intercalation or screening; used in both local and nonlocal fits (Appendix F).
  • A = 0.03
    Dimensionless nonlocal broadening constant in Eq. (9), fitted to the numerical nonlocal Mie data for SiC spheres.
assumptions (5)
  • domain assumption Polar crystal modeled as an isotropic continuum with equation of motion Eq. (A1), including only quadratic spatial dispersion through beta_L and beta_T.
    This is the starting model from which the longitudinal and transverse dielectric functions are derived; it neglects higher-order dispersion and anisotropy.
  • domain assumption A single damping rate gamma is used for both LO and TO phonon responses, independent of frequency and polarization.
    Stated in Appendix A as a simplification; the paper does not test polarization-dependent damping.
  • ad hoc to paper Fuchs-Kliewer additional boundary condition (X_z = 0 at interfaces) is applied, equivalent to specular reflection.
    Chosen by symmetry arguments (Appendix B); the paper notes Pekar-Ridley gives practically indistinguishable observables, so the ABC ambiguity remains unresolved.
  • domain assumption All materials are treated as isotropic, including wurtzite AlN and GaN.
    Acknowledged in the Discussion; anisotropic extension would require numerical modal dispersion and is left for future work.
  • domain assumption Quadratic phonon dispersion is valid for the wavevectors probed in these structures.
    The paper explicitly states this approximation breaks down for narrower layers, causing the failure to reproduce the 795 cm^-1 dip in Fig. 5b.

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Cite this review

Pith. "Pith review of Theory of Optical Nonlocality in Polar Dielectrics." pith.science (2026). https://pith.science/paper/UC3OF6CR

@misc{pith2026190803806,
  author       = {Pith},
  title        = {Pith review of: Theory of Optical Nonlocality in Polar Dielectrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UC3OF6CR}},
  note         = {Machine review of arXiv:1908.03806}
}
read the original abstract

Sub-wavelength confinement of mid-infrared light can be achieved exploiting the metal-like optical response of polar dielectric crystals in their Reststrahlen spectral region, where they support evanescent modes termed surface phonon polaritons. In the past few years the investigation of phonon polaritons localised in nanoresonators and layered heterostructures has enjoyed remarkable success, highlighting them as a promising platform for mid-infrared nanophotonic applications. Here we prove that the standard local dielectric description of phonon polaritons in nanometric objects fails due to the nonlocal nature of the phonon response and we develop the corresponding nonlocal theory. Application of our general theory to both dielectric nanospheres and thin films demonstrates that polar dielectrics exhibit a rich nonlocal phenomenology, qualitatively different from the one of plasmonic systems, due to the negative dispersion of phononic optical modes.

Figures

Figures reproduced from arXiv: 1908.03806 by the authors.

Figure 1
Figure 1. FIG. 1. a) Illustration of nonlocal effects in gold and SiC nan [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Panels a) and b) show a comparison of nonlocal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. a) Absolute nonlocal frequency shift of the main [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the local (blue solid) and nonlocal (re [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of experimental reflectance data [31] [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    & De Liberato, S

    Ballarini, D. & De Liberato, S. Polaritonics: from micro - cavities to sub-wavelength confinement. Nanophotonics 8, 641–654 (2019)

    work page 2019

  2. [2]

    Relative merits of phononics vs

    Khurgin, J. Relative merits of phononics vs. plasmonics : the energy balance approach. Nanophotonics 7, 2305–316 (2017)

    work page 2017

  3. [3]

    Schuller, J. A. et al. Plasmonics for extreme light concen- tration and manipulation. Nature Materials 9, 193–204 (2010)

    work page 2010

  4. [4]

    L., Halas, N

    Brongersma, M. L., Halas, N. J. & Nordlander, P. Plasmon-induced hot carrier science and technology. Na- ture Nanotechnology 10, 25–34 (2015)

    work page 2015

  5. [5]

    Greffet, J.-J. et al. Coherent emission of light by thermal sources. Nature 416, 61–64 (2002)

    work page 2002

  6. [6]

    & Keilmann, F

    Hillenbrand, R., Taubner, T. & Keilmann, F. Phonon- enhanced lightmatter interaction at the nanometre scale. 12 Nature 418, 159–162 (2002)

    work page 2002

  7. [7]

    Caldwell, J. D. et al. Low-loss, infrared and tera- hertz nanophotonics using surface phonon polaritons. Nanophotonics 4, 44–68 (2015)

    work page 2015

  8. [8]

    Caldwell, J. D. et al. Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators. Nano letters 13, 3690–3697 (2013)

    work page 2013

Show all 58 references
  1. [9]

    Ellis, C. T. et al. Aspect-ratio driven evolution of high- order resonant modes and near-field distributions in lo- calized surface phonon polariton nanostructures. Scien- tific Reports 6, 32959 (2016)

    2016

  2. [10]

    R., Maier, S

    Gubbin, C. R., Maier, S. A. & De Liberato, S. Theoreti- cal investigation of phonon polaritons in SiC micropillar resonators. Physical Review B 95, 035313 (2017)

    2017

  3. [11]

    Spann, B. T. et al. Photoinduced tunability of the rest- strahlen band in 4H − SiC. Physical Review B 93, 085205 (2016)

    2016

  4. [12]

    R., Martini, F., Politi, A., Maier, S

    Gubbin, C. R., Martini, F., Politi, A., Maier, S. A. & De Liberato, S. Strong and coherent coupling between localized and propagating phonon polaritons. Physical review letters 116, 246402 (2016)

    2016

  5. [13]

    Dunkelberger, A. D. et al. Active tuning of surface phonon polariton resonances via carrier photoinjection. Nature Photonics 12, 50–56 (2018)

    2018

  6. [14]

    Passler, N. C. et al. Strong coupling of epsilon-near- zero phonon polaritons in polar dielectric heterostruc- tures. Nano letters 18, 4285–4292 (2018)

    2018

  7. [15]

    Berte, R. et al. Sub-nanometer thin oxide film sensing with localized surface phonon polaritons. ACS Photonics 5, 2807–2815 (2018)

    2018

  8. [16]

    Gubbin, C. R. & De Liberato, S. Theory of nonlinear polaritonics: χ(2) scattering on a β − SiC surface. ACS Photonics 4, 1381–1388 (2017)

    2017

  9. [17]

    Razdolski, I. et al. Second harmonic generation from strongly coupled localized and propagating phonon- polariton modes. Physical Review B 98, 125425 (2018)

    2018

  10. [18]

    Li, P. et al. Reversible optical switching of highly con- fined phonon-polaritons with an ultrathin phase-change material. Nature Materials 15, 870 (2016)

    2016

  11. [19]

    Li, P. et al. Infrared hyperbolic metasurface based on nanostructured van der waals materials. Science 359, 892–896 (2018)

    2018

  12. [20]

    Chaudhary, K. et al. Polariton nanophotonics using phase-change materials. Nature Communications 10, 4487 (2019)

    2019

  13. [21]

    & Mortensen, N

    Raza, S., Toscano, G., Jauho, A.-P., Wubs, M. & Mortensen, N. A. Unusual resonances in nanoplasmonic structures due to nonlocal response. Physical Review B 84, 121412 (2011)

    2011

  14. [22]

    & Pendry, J

    Fern´ andez-Dom ´ ınguez, A., Wiener, A., Garc ´ ıa-Vidal, F., Maier, S. & Pendry, J. Transformation-optics description of nonlocal effects in plasmonic nanostructures. Physical review letters 108, 106802 (2012)

    2012

  15. [23]

    Probing the ultimate limits of plasmonic enhancement

    Cirac ` ı, C.et al. Probing the ultimate limits of plasmonic enhancement. Science 337, 1072–1074 (2012)

    2012

  16. [24]

    A., Raza, S., Wubs, M., Sndergaard, T

    Mortensen, N. A., Raza, S., Wubs, M., Sndergaard, T. & Bozhevolnyi, S. I. A generalized non-local optical response theory for plasmonic nanostructures. Nature Communications 5, 3809 (2014)

    2014

  17. [25]

    & Mortensen, N

    Wubs, M. & Mortensen, N. A. Nonlocal response in plasmonic nanostructures. In Bozhevolnyi, S. I., Martin- Moreno, L. & Garcia-Vidal, F. (eds.) Quantum Plasmon- ics, 279–302 (Springer International, 2017)

    2017

  18. [26]

    R., Mortensen, N

    Maack, J. R., Mortensen, N. A. & Wubs, M. Size- dependent nonlocal effects in plasmonic semiconductor particles. EPL (Europhysics Letters) 119, 17003 (2017)

    2017

  19. [27]

    Gubbin, C. R. et al. Hybrid longitudinal-transverse phonon polaritons. Nature Communications 10, 1682 (2019)

    2019

  20. [28]

    & Wolf, M

    Paarmann, A., Razdolski, I., Gewinner, S., Sch¨ ollkop f, W. & Wolf, M. Effects of crystal anisotropy on opti- cal phonon resonances in midinfrared second harmonic response of sic. Phys. Rev. B 94, 134312 (2016)

    2016

  21. [29]

    & Greffet, J.-J

    Chalopin, Y., Dammak, H., Hayoun, M., Besbes, M. & Greffet, J.-J. Size-dependent infrared properties of mgo nanoparticles with evidence of screening effect. Applied Physics Letters 100, 241904 (2012)

    2012

  22. [30]

    Paudel, T. R. & Lambrecht, W. R. L. Computational study of phonon modes in short-period aln/gan superlat- tices. Phys. Rev. B 80, 104202 (2009)

    2009

  23. [31]

    Ratchford, D. C. et al. Controlling the infrared dielec- tric function through atomic-scale heterostructures. ACS Nano 13, 6730–6741 (2019)

    2019

  24. [32]

    R., Maier, S

    Gubbin, C. R., Maier, S. A. & De Liberato, S. Real-space hopfield diagonalization of inhomogeneous dispersive me- dia. Phys. Rev. B 94, 205301 (2016)

    2016

  25. [33]

    & Arakawa, Y

    Li, X.-Q. & Arakawa, Y. Dielectric function associated with dispersive optic-vibrations. Solid State Communi- cations 108, 211 – 213 (1998)

    1998

  26. [34]

    Trallero-Giner, C., Garc ´ ıa-Moliner, F., Velasco, V. R. & Cardona, M. Analysis of the phenomenological models for long-wavelength polar optical modes in semiconductor layered systems. Phys. Rev. B 45, 11944–11948 (1992)

    1992

  27. [35]

    & Kliewer, K

    Fuchs, R. & Kliewer, K. L. Surface plasmon in a semi- infinite free-electron gas. Phys. Rev. B 3, 2270–2278 (1971)

    1971

  28. [36]

    Cirac, C., Pendry, J. B. & Smith, D. R. Hydrody- namic model for plasmonics: A macroscopic approach to a microscopic problem. ChemPhysChem 14, 1109–1116 (2013)

    2013

  29. [37]

    Schnitzer, O., Giannini, V., Craster, R. V. & Maier, S. A. Asymptotics of surface-plasmon redshift saturation at subnanometric separations. Phys. Rev. B 93, 041409 (2016)

    2016

  30. [38]

    Garcia de Abajo, F. J. Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveg- uides. The Journal of Physical Chemistry C 112, 17983– 17987 (2008)

    2008

  31. [39]

    M., Gray, S

    McMahon, J. M., Gray, S. K. & Schatz, G. C. Nonlocal optical response of metal nanostructures with arbitrary shape. Physical review letters 103, 097403 (2009)

    2009

  32. [40]

    Raza, S. et al. Blueshift of the surface plasmon resonance in silver nanoparticles studied with eels. Nanophotonics 2, 131–138 (2013)

    2013

  33. [41]

    & Fuchs, R

    Rojas, R., Claro, F. & Fuchs, R. Nonlocal response of a small coated sphere. Phys. Rev. B 37, 6799–6807 (1988)

    1988

  34. [42]

    Giannini, V., Francescato, Y., Amrania, H., Phillips, C. C. & Maier, S. A. Fano resonances in nanoscale plas- monic systems: A parameter-free modeling approach. Nano Letters 11, 2835–2840 (2011)

    2011

  35. [43]

    Gennaro, S. D. et al. Spectral interferometric microscopy reveals absorption by individual optical nanoantennas from extinction phase. Nature Communications 5, 3748 (2014)

    2014

  36. [44]

    & Maier, S

    Simoncelli, S., Li, Y., Cort´ es, E. & Maier, S. Imag- ing plasmon hybridization of fano resonances via hot- 13 electron-mediated absorption mapping. Nano Letters 18, 3400–3406 (2018)

    2018

  37. [45]

    & Fragstein, C

    Kreibig, U. & Fragstein, C. v. The limitation of electro n mean free path in small silver particles. Zeitschrift f¨ ur Physik 224, 307–323 (1969)

    1969

  38. [46]

    Current-dependent potential for nonlocal ab- sorption in quantum hydrodynamic theory

    Cirac ` ı, C. Current-dependent potential for nonlocal ab- sorption in quantum hydrodynamic theory. Phys. Rev. B 95, 245434 (2017)

    2017

  39. [47]

    & Marquier, F

    Campione, S., Brener, I. & Marquier, F. Theory of epsilon-near-zero modes in ultrathin films. Phys. Rev. B 91, 121408 (2015)

    2015

  40. [48]

    de Ceglia, D. et al. Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms. Scientific Reports 8 (2018)

    2018

  41. [49]

    & Strauch, D

    Karch, K., Pavone, P., Windl, W., Sch¨ utt, O. & Strauch, D. Ab initio calculation of structural and lattice- dynamical properties of silicon carbide. Phys. Rev. B 50, 17054–17063 (1994)

    1994

  42. [50]

    I., Wiener, A., Maier , S

    Luo, Y., Fernandez-Dominguez, A. I., Wiener, A., Maier , S. A. & Pendry, J. B. Surface plasmons and nonlocality: A simple model. Phys. Rev. Lett. 111, 093901 (2013)

    2013

  43. [51]

    & Cardona, M

    Trallero-Giner, C., Garcia-Moliner, F., Velasco, V. & Cardona, M. Analysis of the phenomenological models for long-wavelength polar optical modes in semiconduc- tor layered systems. Physical Review B 45, 11944 (1992)

    1992

  44. [52]

    The propagation of electromagnetic energy through an absorbing dielectric

    Loudon, R. The propagation of electromagnetic energy through an absorbing dielectric. Journal of Physics A: General Physics 3, 233–245 (1970)

    1970

  45. [53]

    Pekar, S. I. Theory of electromagnetic waves in a crysta l with excitons. Journal of Physics and Chemistry of Solids 5, 11–22 (1958)

    1958

  46. [54]

    & Englman, R

    Ruppin, R. & Englman, R. Optical properties of nonlo- cal dielectrics independent of additional boundary condi- tions. Phys. Rev. Lett. 53, 1688–1691 (1984)

    1984

  47. [55]

    & Cardona, M

    Debernardi, A., Ulrich, C., Syassen, K. & Cardona, M. Raman linewidths of optical phonons in 3 c − SiC under pressure: First-principles calculations and experimenta l results. Phys. Rev. B 59, 6774–6783 (1999)

    1999

  48. [56]

    & Demchenko, I

    Derkachova, A., Kolwas, K. & Demchenko, I. Dielec- tric function for gold in plasmonics applications: Size de- pendence of plasmon resonance frequencies and damping rates for nanospheres. Plasmonics 11, 941–951 (2015)

    2015

  49. [57]

    G., Estevez, G

    Barrera, R. G., Estevez, G. A. & Giraldo, J. Vector spherical harmonics and their application to magneto- statics. European Journal of Physics 287–294 (1985)

    1985

  50. [58]

    & Bernholc, J

    Bungaro, C., Rapcewicz, K. & Bernholc, J. Ab initio phonon dispersions of wurtzite aln, gan, and inn. Phys. Rev. B 61, 6720–6725 (2000)

    2000

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