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Description
Attenuated total reflection (ATR) is a convenient way to probe thin films of wide-bandgap semiconductors in the far infrared. For the ZnO/6H-SiC system, whose reststrahlen bands lie at about 378–590 and 788–970 $\mathrm{cm}^{-1}$, the prism, incidence angle and air gap determine which bands can be excited. We simulate the Otto configuration (prism / air gap / ZnO film / 6H-SiC substrate), where the evanescent wave couples to the film-substrate structure.
Both crystals are treated as uniaxial media with the optic axis normal to the surface. The ordinary ($\varepsilon_{\perp}$) and extraordinary ($\varepsilon_{\parallel}$) permittivities are described separately by a Lorentz phonon oscillator plus a Drude free-carrier term. Reflectance for s and p polarization is computed using one multilayer characteristic-matrix (Abelès) algorithm covering external reflection and ATR; for a single film it reproduces the Airy formula to about $10^{-15}$. Calculations use a 0.8 $\mu\mathrm{m}$ film (carrier density $10^{18}$ $\mathrm{cm}^{-3}$, mobility 50 $\mathrm{cm}^{2}/(\mathrm{V}\,\mathrm{s})$) on a substrate ($10^{17}$ $\mathrm{cm}^{-3}$, 100 $\mathrm{cm}^{2}/(\mathrm{V}\,\mathrm{s})$), a uniform plane-parallel gap and a frequency-independent prism index. Unless stated otherwise, the prism is KRS-5 ($n=2.37$) and the internal incidence angle is $\theta=35^{\circ}$.
The critical angle $\theta_c=\arcsin(1/n)$ is $25.0^{\circ}$ for KRS-5, $17.1^{\circ}$ for Si and $14.5^{\circ}$ for Ge. Ge transmits only above about 780 $\mathrm{cm}^{-1}$, making the ZnO band inaccessible, whereas KRS-5 (cutoff about 250 $\mathrm{cm}^{-1}$) covers both bands. For p polarization, clear reflectance minima appear at about 510–555 $\mathrm{cm}^{-1}$ (ZnO) and 830–880 $\mathrm{cm}^{-1}$ (6H-SiC). For s polarization, ZnO-band reflectance stays above 43% for all gaps considered, so p polarization is required.
The ZnO-band minimum depth depends non-monotonically on gap thickness $d_{\mathrm{gap}}$: $R_{\min}$ is about 19% at 0.5 $\mu\mathrm{m}$, 5.7% at 1 $\mu\mathrm{m}$, 0.1% at 1.5 $\mu\mathrm{m}$, 3.5% at 2 $\mu\mathrm{m}$, 74% at 5 $\mu\mathrm{m}$ and 99% at 10 $\mu\mathrm{m}$. The minimum shifts from 511 to 554 $\mathrm{cm}^{-1}$ as the gap grows from 0.5 to 5 $\mu\mathrm{m}$. For a 2 $\mu\mathrm{m}$ gap, $R_{\min}$ is about 8.9% at $28^{\circ}$, 2.8% at $30^{\circ}$, 3.5% at $35^{\circ}$, 23% at $40^{\circ}$ and 74% at $55^{\circ}$. Useful angles thus extend from a few degrees to about $10^{\circ}$ above $\theta_c$.
The minima are associated with surface and guided phonon-polariton modes. Practical starting conditions are a KRS-5 prism, p polarization, a 1.5–2 $\mu\mathrm{m}$ gap and a $30$–$35^{\circ}$ angle. Calculated spectra will be compared with measured ATR spectra, and the inverse problem for film thickness, carrier density and mobility will be solved using the same model.
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