Controlling the Infrared Dielectric Function through Atomic-Scale Heterostructures

Surface phonon polaritons (SPhPs), the surface-bound electromagnetic modes of a polar material resulting from the coupling of light with optic phonons, offer immense technological opportunities for nanophotonics in the infrared (IR) spectral region. However, once a particular material is chosen, the SPhP characteristics are fixed by the spectral positions of the optic phonon frequencies. Here, we provide a demonstration of how the frequency of these optic phonons can be altered by employing atomic-scale superlattices (SLs) of polar semiconductors using AlN/GaN SLs as an example. Using second harmonic generation (SHG) spectroscopy, we show that the optic phonon frequencies of the SLs exhibit a strong dependence on the layer thicknesses of the constituent materials. Furthermore, new vibrational modes emerge that are confined to the layers, while others are centered at the AlN/GaN interfaces. As the IR dielectric function is governed by the optic phonon behavior in polar materials, controlling the optic phonons provides a means to induce and potentially design a dielectric function distinct from the constituent materials and from the effective-medium approximation of the SL. We show that atomic-scale AlN/GaN SLs instead have multiple Reststrahlen bands featuring spectral regions that exhibit either normal or extreme hyperbolic dispersion with both positive and negative permittivities dispersing rapidly with frequency. Apart from the ability to engineer the SPhP properties, SL structures may also lead to multifunctional devices that combine the mechanical, electrical, thermal, or optoelectronic functionality of the constituent layers. We propose that this effort is another step toward realizing user-defined, actively tunable IR optics and sources.


S1. Polar material dielectric function and calculated reflectance
In order to calculate the reflectance of bulk AlN, GaN, and SiC as shown in Fig. 1b of the paper, we used an analytical expression for the dielectric function of each material. Each material belongs to the hexagonal crystal system, and is therefore birefringent, having an out-of-plane dielectric function for light polarized along the c-axis of the lattice (extraordinary) and another in-plane ∥ dielectric function for light polarized perpendicular to the c-axis (ordinary). In the mid-infrared ⊥ (IR), the dielectric function for a uniaxial polar, undoped semiconductor is given by the following formula: where or for the permittivity parallel or perpendicular to the c-axis.  We calculated the reflectance in Fig. 1 using Fresnel equations for normal incidence, with the dielectric function of each material given by Eq. (S1) with the appropriate values from Table S1.
For the calculation, we let = = 5 cm -1 for each material. To calculate the reflectance of the ⊥ ∥ SL structures shown in Fig. 1e

S2. AlN/GaN SL vs. AlGaN IR responses
We briefly note an important point that the IR response of a SL of two materials is, in general, distinct from an alloy of the same two materials. In Fig. S1, we overlay the reflectivity of AlGaN thin film (500 nm thick Al 0.42 Ga 0.58 N on SiC wafer) and Sample B for comparison. The difference in IR response is because the two materials have distinct sets of phonon modes. AlGaN displays two-mode and one-mode behavior for zone-center TO and LO phonons, respectively, with the phonon frequencies depending on the AlGaN composition. In contrast, the AlN/GaN SL displays multiple TO and LO phonon modes. The SL phonon density of states depends on the structural geometry, and the phonon modes correspond to confined and interface-like modes.

Figure S1
Comparison of the reflectivity of an Al 0.42 Ga 0.58 N alloy thin film (500 nm on SiC wafer) and AlN/GaN SL Sample B from manuscript.

S3. Chemical composition and structure analysis of Sample B
We used electron energy loss spectroscopy (EELS) in order to characterize the chemical structure of the SL. Figure S2a and b show a high-angle annular dark-field imaging (HAADF) image of Sample B and EELS data acquired along the red dashed line in the STEM image. From the oscillating Z-contrast observed in the HAADF image and line profile, there is obvious chemical segregation between the layers. The EELS results show that there is a relatively high concentration of Ga everywhere on the sample, which likely results from residual Ga used in the milling process to prepare the cross-sectional sample. Even so, there appears to be a significant amount of Al in the Ga layers, which suggests some intermixing between the layers. The effect is seen more clearly in Figure S2c where we compare the EELS spectra from the Ga-and Al-rich layers. Here, it can be seen that both Al and Ga are present in each layer, but that that the Al almost disappears in the Ga rich layer, while the Ga peak only reduces slightly in the Al layer indicating that much of the Ga signal is due to Ga implantation from the FIB and not from layer intermixing. In addition to the EELS results, intermixing between the layers is also suggested by the sinusoidallike HAADF intensity along the SL growth direction. This is shown in Fig. S3, in which the HAADF intensity is vertically binned along the red rectangle in the STEM image ( that such intermixing is not the origin of the changing dielectric function.

S4. Determination of optic phonon modes using IR second harmonic generation
In addition to the SHG spectra shown in the main text of this work, here we provide the complete data set measured on Sample A (see Fig. S4a). This sample exhibits a spatial gradient in both the frequency/amplitude appears non-trivial, the full data set further confirms the previous observation in demonstrating the tunability of the phonon modes in the XH.
The same behavior is observed in the corresponding reflectivity spectra (see Fig. S4b), even though it is much harder to assess as the resonances appear as edges rather than peaks as they are observed in the SHG data. The latter have been analyzed using least-squares fitting of a Lorentzian line shape function. The resulting peak shifts and amplitudes are shown in Fig. S4c-d.

Figure S4
Full data set of the SHG and reflectivity spectra. a) and b) show, respectively, the SHG and reflectivity spectra in false colors at each of the labelled positions on the SL sample corresponding to different GaN and AlN layer thicknesses. c) and d) show, respectively, the spectral positions and peak amplitudes of the two hybrid TO phonon modes, H1 (red) and H2 (green), also marked in a.

S5. XH dielectric function
We used ellipsometry to determine the mid-IR permittivity of the SL structures. Ellipsometry measures the change in the polarization of reflected light from the surface of a sample and is generally reported with the parameters  and . These parameters are related to the ratio of the Fresnel reflection coefficients for p-and s-polarized light, r p and r s , respectively, by The SL permittivity was determined by modeling the permittivity with an analytic formula and then fitting the ellipsometric data by using the permittivity model to compute the Fresnel reflection coefficients. Since ellipsometric measurements are determined from the ratio of two measured quantities, the derived permittivity values are very precise. We used the ellipsometric software WVASE to fit the ellipsometric data and extract the permittivity. To account for the additional observed phonon modes, we instead model the entire AlN-GaN SL as a single material with multiple phonon modes using Eq. 2. The best fit for each sample is shown in blue in Fig. S5. For Sample A, a good fit was achieved for k = 2 and k = 3 for the in-plane and out-of-plane dielectric function, respectively. The resulting dielectric function is shown in Fig. S6.
We found that the same number of phonon modes could be used to achieve a reasonable fit for Sample B, however, including an additional phonon mode for the out-of-plane dielectric function significantly improved the fit. For the Sample B fit shown in Fig. 3a-b, we used k = 2 and k = 4 for the in-plane and out-of-plane dielectric function, respectively. The derived dielectric function from the fit is plotted in Fig. 3 of the manuscript. Table S2 summarizes the best-fit parameters for both samples.

S7. Computational results of phonon modes for AlN 4 /GaN 5 SL
Density-functional perturbation theory (DFPT) was used to compute the phonon modes of an infinite AlN/GaN SL in the [0001] direction. The calculation was performed for 4 monolayers and 5 monolayers of AlN and GaN, respectively, as informed by the layer thicknesses for Sample B.
The SL was fully relaxed before calculation of the phonon modes. An illustration of the simulation unit cell is shown in Fig. S8. Because the different constituent layers have opposite parity for the number of monolayers, it was necessary for the simulation unit cell to contain two layers of each material to achieve appropriate bonding at the AlN-GaN interface. Therefore, the unit cell contained N = 36 atoms, which resulted in phonon modes. Of these modes, 35 are 3 -3 = 105 double degenerate E-symmetry modes and 35 are A-symmetry modes.  and are not optically distinct modes. As a result, these modes with overlapping TO and LO frequencies generate only a single pole and zero crossing.

Ga Al
As discussed in the main text, exhibits two Reststrahlen band regions: one narrow region around ⊥ 560 cm -1 and another broad region from about 630 cm -1 to 800 cm -1 . While in Fig. 3c-d was ⊥ calculated from many E-symmetry modes, only two of the modes (mode # 27 and 34 in Table S3) have significant TO-LO splitting, and these modes largely define the Reststrahlen regions. From the atomic displacements associated with these two modes (Fig. S9), it is immediately clear that mode # 27 resembles a GaN-like confined mode, while mode # 34 appears to be mostly an interface phonon in the AlN layers. In contrast to , a number of A-symmetry modes have significant TO-LO splitting and contribute ⊥ to . Figure S10 shows the vibrational pattern of the 4 A-symmetry modes with the largest TO-∥ LO splitting. Overall, these modes are not tightly confined to one material. The out-of-plane dielectric function in Fig. 3c-d has a broad Reststrahlen region from around 540 -700 cm -1 . From   Fig. S10, mode #18, which contributes to the permittivity in this region does not appear to be confined to either layer. Within the 740 -830 cm -1 , exhibits somewhat complex behavior which ∥ appears to arise from a number of phonon modes. These modes are weakly localized to the AlN layer.

S8. XH polaritonic response
The theoretical reflectivity spectra in Fig. 4 of the main text are calculated using a 4 x 4 transfer matrix formalism 5 . The incident medium is set to be KRS5 (n=2.4), thus providing the same optical response as obtained experimentally in the Otto geometry. The formalism allows for the calculation of the reflection and transmission coefficients at arbitrary incidence angles for any number of material layers, each described by either an isotropic or anisotropic dielectric tensor.
For the simulation labeled 'XH' in Fig. 4 of the main text, the formalism was applied on a system of 4 layers (KRS5 / air (2.8 μm) / XH (585 nm) / 6H-SiC), giving rise to an excellent agreement with the experimental data. Here, the dielectric function extracted from ellipsometry shown in Fig.   3 of the main text was used in its parametrized form (Eq. 2) using the values reported in Table S2 to describe the XH layer. The spectra labeled 'Macroscopic' that fail to reproduce the experimental results were calculated accounting for each AlN and GaN layer individually, i.e. for a system of 503 layers (KRS5 / air (2. 8 μm) / 250x (AlN (1.2 nm) / GaN (1.4 nm)) / 6H-SiC). Here, the AlN and GaN layers were described by their bulk dielectric functions using the parameters given in Table S1. All calculations explicitly account for the anisotropy of the dielectric response for the polar materials, i.e., SiC, AlN, GaN, XH.
To understand the nature of the different polariton modes observed as dips in the XH Otto geometry reflectivity data shown in Fig. 4 and as peaks in the Im(r p ) plots in Fig. 5  (ii) Thin film: In the experiment, we use a 585 nm thick film of the XH material, which is more than 10 times smaller than the wavelengths employed and also on the length scale of the absorption length on the strongest phonon resonances. Therefore, optical thin film effects are heavily affecting the optical response. Also, volume-confined hyperbolic modes only emerge in thin films. 6 (iii) Substrate: Since SiC can also support a surface polariton, and its dielectric function is strongly dispersive in the frequency range of interest, the substrate also strongly affects the polariton response of the XH thin film. Notably, the film thickness employed here is well beyond the strong coupling regime observed in previous work with AlN ultrathin films. 7 Qualitative understanding of these effects is gained by inspecting Fig. S11, where we show theoretical polariton dispersions of four different samples: a) a thick semi-infinite XH crystal/air interface, b) a thin slab of the XH interfaced with air on both sides and c) a thin slab of the XH on a SiC substrate interface to air, and d) a thick semi-infinite SiC/air interface. As in Fig. 5 a-b) of the main text, these plots show the imaginary part of the reflection coefficient as a function of inplane momentum, which conveniently reveals all polariton modes. 6 Calculations were performed with the same transfer matrix formalism as the reflectance calculations in Fig. 4 of the main text 5 , but without explicitly treating the prism. Additionally, we also extracted the dispersion of all modes by tracing the peaks in similar calculations for the XH with reduced damping, i.e. by using Eq. 2 with parameters in Table S2, but with all phonon dampings γ divided by 5.
The mode assignment given in the main text is supported by analyzing the XH slab-thickness dependence of the polariton dispersion shown in Fig. S12, where the mode progression of the volume-confined hyperbolic modes 6, 8 is directly visible.