Penetration Depth & Reflectivity For w<wp (“low” frequency regime) The skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a micrometer: Thus at microwave frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver 3 mm thick evaporated on a piece of glass is thus an excellent conductor at such frequencies. In copper, the skin depth can be seen to fall according to the square root of frequency: Conductor Skin depth (μm) Aluminum 0.80 Copper 0.65 Gold 0.79 Silver 0.64 Frequency Skin depth (μm) 60 Hz 8470 10 kHz 660 100 kHz 210 1 MHz 66 10 MHz 21 100 MHz 6.6 At low frequency, penetration decreases with frequency!

Penetration Depth & Reflectivity For w>wp (“High” frequency regime) Real dielectric function  no attenuation  transparency No attenuation independently of the frequency n is close to 1–  R=0 and T=1 Subtle point here! It should also be noted that we are only specifying the decay of the field which may be due to absorption of the electromagnetic energy in a dissipative “lossy” medium or may simply describe the penetration of the field in a medium where no loss occurs (or a combination of the two). For instance, a hypothetical substance may have a complex index of refraction  A wave will enter that medium without significant reflection and will be totally absorbed in the medium with a penetration depth (in field strength) of , where  l  is the vacuum wavelength. A different hypothetical material with a complex index of refraction will also have a penetration depth of 16 wavelengths, however, since n=0, R=1 hence in this case the wave will be perfectly reflected from the material! No actual absorption of the radiation takes place, however the electric and magnetic fields extend well into the substance. In either case the penetration depth is found directly from the imaginary part of the material's refractive index as is detailed above.

Surface Reflectivity and surface plasmons Novotny Ch. 12 Lüth Ch. 5 Metals look grey until intra-band transitions are considered. If not: Inter-band transitions Gold absorbs green/blue photons (2.75 eV), And reflects yellow ones. Copper absorbs green photons (2.1 eV), and reflects orange/red ones. Gold (Fig. 12.2): incident light at l=450 nm (blue) has a d.f.: reflected blu light transmitted (and absorbed) blu light These are bulk properties…

Bulk phonon polariton: ions oscillate in an elastic regime under the action of an external electromagnetic field “coupling” between phonons & e.m.-field oscillating dipole momentum phonon susceptibility reset of the dielectric constant plotted in next slide DC field dielectric constant longitudinal oscillation

Bulk phonon polaritons within the optical gap e is real and negative! Reflectivity

VACUUM-DIELECTRIC INTERFACE Note: The so called “evanescent fields” are due to surface polaritons. These fields are also “used” in scanning near-field optical microscopy (“SNOM”) to scan and determine surface morphology by analysis of the changes induced by surface fields on the “enhanced” fields of a tip.

Total internal reflection from inside dielectric VACUUM-DIELECTRIC INTERFACE No solutions These solutions are partly reflected waves Evanescent waves inside the dielectric (<0, high reflectivity) w Ibach Ch. 7 Total internal reflection from inside dielectric

Quasi-particle (phonon/polariton) creation/annihilation Figure 4.27, curve 1 illustrates the micro-Raman spectrum from a bulk n-GaP crystal. Curves 2 and 3 were obtained from the samples anodized at current densities of 5 and 15 mA/cm2. The as-grown crystal exhibits only one Raman band centered at 404 cm-1, which corresponds to the Brillouin-zone-center LO phonon. The q = 0 TO phonon is forbidden in (100) geometry and therefore it is absent in the bulk GaP spectrum. The anodization process leads to a small downward frequency shift and broadening by 15 to 20 % of the LO-phonon band accompanied by a significant Raman signal intensification (up to 5–6 times). A complete breakdown of the polarization selection rules is reflected in Figure 4.27 by the appearance of a strong TO-phonon band. Like the LO phonon, the TO phonon also exhibits a downward frequency shift with increasing the anodization current. wq q Incident light (ki, wi) Scattered light (ks, ws) Quasi-particle (phonon/polariton) creation/annihilation wq≡Dw=wi-ws, q=ki-ks In technical context, usually frequency is expressed in m-1 or more often cm-1, i.e. through the corresponding reduced wavevector k=k/2p unit of measurement: Raman: inelastic scattering of photons on surface: the reflected photons have different energy and k, and the shift is due to creation/annihilation of excitations with energy the energy shift, and wavevector the wavevector shift

(SEMI)METAL-DIELECTRIC INTERFACE Plasmonic dielectric function Metal/Dielectric interface (surface) dispersion relation Note: w+, in both limits, would give a dispersion of the kind: w=vk//, with v>c (physical nonsense) wp is at 1.0 in ordinate! (silver) Lüth: case of metal/vacuum interface (e1=1) Free electron

(SEMI)METAL-DIELECTRIC INTERFACE Dissipation included Surface Plasmon Polariton electric field

Surface polariton dispersion for free electrons + inter-band transitions

Electron energy loss spectra see below for basics on the technique. Inelastic scattering of low energy e- (20÷200 eV) on crystal surface: the scattered e- energy loss is due to creation of a quantum of excitation ω either phonon or plasmon polaritons Scattering cross section (event probability): i.e. large peaks (“structure”) in the spectrum Fuchs-Kliewer SPhP Red: surface-plasmon-polariton (DE=40 meV) extremely sensitive to doping concentration and exposure Blue: surface-phonon-polariton (DE=37 meV) not sensitve to doping concentration and exposure

Loss peak position Phonon-like With varying doping with the GaAs dielectric function this relation becomes a quadratic equation, which provides two “hybrid” solutions plotted in the graph dashed line in the graph is: Plasmon-like With varying doping or adsorbate density we have that: nphonon is constant nelectron varies Phonon-like ideal phonon-like curve, independent of electron concentration n Phonon-like Plasmon-like (semi-Log scale!)

EELS and HREELS Lüth Panel IX (Ch. 4) (Ch. 4 Dielectric Theory EELS: primary energy of the electron beam: 20÷500 eV (but <200 eV for surf. analysis) HREELS: (High Resolution EELS): energy below 20 eV

Scattering on adsorbed atoms In order to have momentum conservation K=Dk = kmode The activated/destroyed mode of the adsorbate must possess a wavevector parallel to Dk. If e.g. bending oscillations are involved, the scattering is off-specular:

Surface plasmon polariton excitation Novotny Ch. 12

Sensitivity to adsorbates SPP reflectivity curves Sensitivity to adsorbates

Confinement effects in nanogold particles (size dlA) Localized plasmons Confinement effects in nanogold particles (size dlA) Novotny Ch. 12 Electric field distribution for a nanogold/dielectric system The e.m. field can penetrate the nanogold particle and make it “resonating” as an antenna (at w=wSPP) re-irradiating a scattered field only thanks to free electron “oscillations” i.e. plasmon polaritons, (even if these oscillations are quasi-static, and the field is uniform inside the particle). Uniform (!) internal field Scattered field

Estinction(a)=Absorption-a3+Scattering-a6 “a” being the particle radius Lycurgus’cup

DIATHRETE CUP