Optical response of thin amorphous films to infrared radiation

We briefly review the electrical-optical response of materials to radiative forcing within the formalism of the Kramers-Kronig relations. A commensurate set of criteria is described that must be met by any frequency-domain model representing the time-domain response of a real (i.e., physically possible) material. The criteria are applied to the Brendel-Bormann (BB) oscillator, a model that was originally introduced for its fidelity at reproducing the non-Lorentzian peak broadening experimentally observed in the infrared absorption by thin amorphous films but has since been used for many other common materials. We show that the BB model fails to satisfy the established physical criteria. Taking an alternative approach to the model derivation, a physically consistent model is proposed. This model provides the appropriate line-shape broadening for modeling the infrared optical response of thin amorphous films while adhering strictly to the Kramers-Kronig criteria. Experimental data for amorphous alumina (${\mathrm{Al}}_2{\mathrm{O}}_3$) and amorphous quartz silica (${\mathrm{SiO}}_2$) are used to obtain model parametrizations for both the noncausal BB model and the proposed causal model. The proposed model satisfies consistency criteria required by the underlying physics and reproduces the experimental data with better fidelity (and often with fewer parameters) than previously proposed permittivity models.

Figure 1

BB oscillator. The real (thick blue line) and imaginary (thin orange line) parts of the BB oscillator with ${\nu }_p=310.83$, $\gamma =96.49$, ${\nu }_0=816.20$, and $\sigma =219.9$. The units are ${\mathrm{cm}}^{-1}$. The parameters are from a multioscillator model for ${\mathrm{SiO}}_2$ (given in full in the results section). Here $\sigma$ was exaggerated by a factor of 10 to keep the response stable for $\nu <0$ and to more clearly demonstrate the divergence resulting from the singularity at $\nu =0$. Note the lack of symmetry (antisymmetry) in the real (imaginary) signal, which indicates the model is not Hermitian. The divergence at $\nu =0$ indicates the model is not causal.

Figure 2

Parity correction. The same as Fig. 1, except with the addition of the real (thick dashed line) and imaginary (thin dashed line) parts of the model using the corrected expressions in (15), given as a function of wave number $({\mathrm{cm}}^{-1}$). The models are identical for $\nu >0$, but the corrected model retains symmetry (antisymmetry) in the real (imaginary) signals for $\nu <0$. Near $\nu =0$, both models diverge, but the corrected model does so in a manner consistent with the even (odd) character of the real (imaginary) signal. This demonstrates that the divergence and parity-breaking issues are independent. In other words, the Fourier transform of the parity-corrected model will be real valued, but it will not generally be causal.

Figure 3

Singularity removal by approximation. The model parameters are the same as used in Fig. 1. In each plot the thick blue line is the real part, and the thin orange line is the imaginary part, given as a function of wave number $({\mathrm{cm}}^{-1}$). The top row gives model shapes (a) before and (b) after singularity removal, with the shapes given by ${\mathcal{S}}_1\triangleq (1/\alpha )[f_{\mathit{w}}\left(z_{+}\right)-f_{\mathit{w}}\left(z_{+}\right)]$ and ${\mathcal{S}}_2\triangleq f_{\mathit{w}}\left(z_{+}\right)+f_{\mathit{w}}\left(z_{+}\right)$, respectively. The shape functions have been normalized to the imaginary part for ease of comparison. The bottom row gives the component functions (c) $1/\alpha$, (d) $f_{\mathit{w}}\left(z_{+}\right)$, and (e) $f_{\mathit{w}}\left(z_{-}\right)$. The insets of (d) and (e) provide a closer look at the behavior of the imaginary part near the origin. With a sensible recomposition of the component functions, one obtains a continuous (Hermitian) shape where the singularity has been removed. One notes that the maxima of (a) and (b) do not occur at the same wave number. This is a desirable feature that is discussed in detail later.

Figure 4

Comparison with the CDHO profile. (a) Imaginary and (b) real parts of the BB model (thin solid line), the proposed model (thin dashed line), and the CDHO (thick solid line) as a function of wave number $({\mathrm{cm}}^{-1}$). The imaginary parts have been normalized by their respective right half-plane maxima for ease of comparison. The model parameters are the same as those used in Fig. 1. The proposed model is asymptotically equivalent to the CDHO (i.e., at $\nu =0$ and $\nu =\pm \infty$). The frequency of the energy absorption maximum is in nearly perfect agreement with the CDHO. Both have an energy absorption maximum occurring at approximately ${\nu }_0$, which is indicated in (a) by the vertical dotted line. The proposed model clearly represents a continuous complex function satisfying the KKRs and preserving the relevant characteristics of the underlying classical Lorentz harmonic oscillator theory.

Figure 5

${\mathrm{SiO}}_2$ permittivity as a function of wave number $({\mathrm{cm}}^{-1}$). A six-oscillator model is shown, which has been fit to the data of Popov et al. [38]. In the model plots, the BB model is indicated by $+$ symbols and the proposed model is indicated by $\times$ symbols, so that the $✳$ symbol implies consensus. The solid black line indicates the data. In the error plots, the BB model relative error is shown by the solid red line, and the proposed model relative error is shown by the dots. Also shown in the error plots is the relative consensus error (dashed gray line) between the BB model and the proposed model, which is effectively zero everywhere. In each case, the relative errors have been obtained by scaling the true error pointwise by the magnitude of the corresponding experimental data. The parameters for each model are given in Table 1.

Figure 6

${\mathrm{Al}}_2{\mathrm{O}}_3$ permittivity as a function of wave number $({\mathrm{cm}}^{-1}$). A three-oscillator model is shown, which has been fit to the data of Eriksson et al. [39, 40]. In the model plots, the BB model is indicated by $+$ symbols, and the proposed model is indicated by $\times$ symbols, so that the $✳$ symbol implies consensus. The solid black line indicates the data. In the error plots, the BB model error is shown by the solid red line, and the proposed model error is shown by the dots. Also shown in the error plots is the consensus error (dashed gray line) between the BB model and the proposed model, which is effectively zero everywhere. In each case, the relative errors have been obtained by scaling the true error pointwise by the magnitude of the corresponding experimental data. The parameters for each model are given in Table 2.