1. Introduction

Metals are the standard materials for plasmonics. Because of their possibility to sustain surface waves on flat interfaces or nanoparticles, metals are the building blocks to design original waveguides [1], sub-wavelength imaging [2], metasurfaces [3] and metamaterials [4] such as hyperbolic media [5]. Hyperbolic metamaterials are very promising because their large number of electromagnetic states leads to a peak in the photonic density of states allowing broadband control over light-matter interaction at room temperature. Hyperbolic media can be made of stacks of dielectric and metal layers [6] or of doped and un-doped semiconductor layers [7]. The main property of these systems is their plasma frequency, ω_p, which characterizes the collective oscillation of free carriers in the metal, the well-known volume plasmon. For metals, the plasma frequency is in the near UV range. In this paper we consider highly doped semiconductors with typical plasma frequencies in the mid infra-red (IR). These materials display plasmons with adjustable plasma frequency and lower ohmic losses than metal plasmons. It is essential to know accurately ω_p for the design of hyperbolic media or semiconductor devices such as, e.g., quantum cascade lasers [8].

It is possible to measure ω_p using electron energy loss spectroscopy (EELS) to excite directly volume plasmon [9], or to observe a radiation near ω_p in the case of thin metal foil [10–14] but the technique is complex and cumbersome to implement. More simply, transmission measurements in oblique incidence and under p-polarized light performed on thin metal layer have demonstrated absorption peak near ω_p [15]. This absorption peak has been recently observed on doped semiconductors [16] and has been ascribed to a surface effect, the Berreman effect [17]. In this article, we demonstrate that Fresnel coefficients which involve only transverse waves can account for the observation of a resonant dip in the reflectivity near ω_p in highly doped semiconductors thin layers. We attribute this reflectivity dip to the resonant excitation of the so-called Brewster “mode” [18] which is a transverse leaky mode. This provides an easy way to measure the plasma wavelength and gives an efficient tool to measure the carrier concentration. In situ measurement during epitaxial growth is possible.

2. Samples

The samples consist in a Silicon-doped InAs_0.91Sb_0.09 layer grown lattice-matched onto n-doped GaSb substrates (~10¹⁸ cm⁻³) with varying thickness. The InAsSb layer thickness is either 100 nm or 1 µm. The nominal value of the incorporated Si atoms lies between 2x10¹⁹ cm⁻³ and 1x10²⁰ cm⁻³. We used secondary ion mass spectroscopy (SIMS) to evaluate the density of Si atoms incorporated in the InAsSb layers and Hall-effect measurements on substrate-free samples to control the carrier density. Details on the substrate removal technique are available in ref [19]. Hall-effect measurements have been done with an ACCENT HL5500 equipment. Reflectance measurements in normal incidence in the case of thick layer allow estimating the plasma frequency by adjusting a model based on transfer matrix and using a Drude model, ε_Drude, to model the optical properties of the InAsSb layer.

All parameters are displayed in Table 1. We can see differences between nominal, SIMS and Hall-effect values notably at the highest doping level. They are due to the limited efficiency of activation of the Si donors [20]. The incorporation of Sb to lattice match the InAs layer to the GaSb substrate seems to perturb the efficiency of the Si doping.

Table 1. Characteristics of the three samples. Columns from the left to the right correspond to the thickness, the plasma wavelength, the plasma wavenumber, the broadening, the carrier mobility, the carrier density obtained from Hall-effect, SIMS measurements and nominal values expected from the growth parameters.

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3. The Brewster mode: the theoretical point of view

The typical system that we model is a thin Si-doped InAs_0.91Sb_0.09 layer deposited on a GaSb substrate (see inset Fig. 1). Air and GaSb substrate are modeled by a dielectric constant respectively equal to ε₁ = 1 and ε₃ = 14.4 [21]. We chose to not consider phonon effects in GaSb because their spectral signatures are out of our range of interest. So we only consider phonon effect into the InAsSb layer. We assume phonon characteristics of InAs to model InAsSb because of the weak Sb concentration. Indeed, we want to know what happens for a highly doped semiconductor layer with ω_p far from the Transverse-Optical phonon frequency, ω_TO. The 1 µm thick InAsSb layer is modeled using the following expression:

 

Fig. 1 Dispersion relation of sample A (1 µm thick InAsSb layer doped at 5.8 x10¹⁹ cm⁻³ deposited on a GaSb substrate). The magenta dashed lines correspond to the four horizontal asymptotes corresponding to the plasma frequency ω_p, the SPP frequency at the interface air/semiconductor ω_spp1, the SPP frequency at the interface semiconductor/substrate ωs_pp2 and the TO-phonon frequency ω_TO. The magenta dotted lines correspond to the two oblique asymptotes for the light line in air (most vertical) or in substrate (most horizontal). Inset schematizes the structure.

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They are due to surface-plasmon polaritons (SPP) at each interface, to the Brewster mode and to TO-phonons. If the InAsSb thickness d is large enough (Im(kz)d >> 1). Then, both interfaces are independent so the Brewster angle corresponds to the bulk plasmon polariton dispersion branch. Below ω_p, we obtain the two dispersion relations of the SPP at each interface. Their asymptotical values for large q are represented by the dashed magenta lines at ω_spp1 for the SPP1 at the Air/InAsSb interface and at ω_spp2 for the SPP2 at the InAsSb/GaSb interface. By linearizing tan(kzd) Eq. (3) gives:

If d decreases, the SPP at each interface couples to give mixed modes of even (SPP1 E_x(z) even, H_y(z) and E_z(z) are odd functions, green open up-triangles in Fig. 1) and odd (SPP2 E_x(z) odd, H_y(z) and E_z(z) are even functions, red open circles in Fig. 1) modes. These two modes repel when d decreases. It follows that the SPP1 modes blue shift towards ω_p. If d is small enough, SPP1 approaches ω_p, that is, the permittivity approaches 0. We obtain an “ε-near-zero” (ENZ) mode below the light line [22]. In parallel, above the light line the Brewster mode flattens close to ω_p. Also the permittivity approaches zero, impacting the electric field profile (see Fig. 2). The leaky Brewster mode is represented by the blue open down-triangles in Fig. 1.

 

Fig. 2 Field profile |H_y|² and |E_z|² (arbitrary units) for the Brewster mode (point I in Fig. 1), the SPP mode at the upper interface (point II in Fig. 1) and the SPP mode at the lower interface (point III in Fig. 1). Coefficient in the right bottom part of the electric field intensity profile is calculated relative to the magnetic field intensity profile.

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At this stage, it is interesting to investigate the structure of the modes. Figure 2 shows the profile of the magnetic field and of the y-component of the electric field taken at the points labelled I, II and III on the dispersion relation in Fig. 1. Colors in each plot of Fig. 2 delimit the GaSb substrate (gray), the InAsSb layer (cyan) and the air (white) as in the inset of Fig. 1. Coefficient in the right bottom part of the electric field intensity profile is calculated relative to the magnetic field intensity profile. Point III corresponds to the SPP2 mode. Both |H_y|² and |E_z|² field intensity profiles are preferentially pinned at the InAsSb/GaSb interface and extend deeply into the dielectric. This is the typical characteristic of a SPP. Point II corresponds to the SPP1 mode. |H_y|² and |E_z|² are preferentially pinned at the Air/InAsSb interface and extend deeply into the air. The main difference compared to the previous mode is the fact that the electric field increases in InAsSb due to the permittivity decrease. Indeed, this mode blue shifts towards ω_p, i.e. towards ε₂(ω_p) = 0, when the InAsSb thickness decreases. The boundary condition on the x component of the electric displacement, leads to ${\left|E_{z2}\right|}^2={\left|{\epsilon }_{zz1}/{\epsilon }_{zz2}\right|}^2{\left|E_{z1}\right|}^2$, where E_i and ε_ij are the electric field and the dielectric tensor, respectively. When ε_yy2 approaches 0, the ratio |ε_zz1/ε_zz2|² approaches infinity. This drastically enhances E_y2. Because of the high losses due to free carriers in this spectral range the transmitted light is absorbed. This is the ENZ mode [22].

Point I corresponds to the Brewster mode. This is a surface leaky mode as the Berreman mode [22] which can be observed in a thin dielectric layer deposited on a metal. This is clearly seen in the |H_y|² profile which diverges out of the InAsSb layer. Contrariwise, the |E_z|² profile is enhanced in the InAsSb layer compared to the air or the GaSb substrate. This enhancement is similar to that of the ENZ mode and is due to the decrease of the permittivity value when the mode reaches ω_p. The behavior of this mode with the InAsSb thickness is the same as that of the Berreman mode. The Brewster mode approaches ω_p when d decreases. So Berreman and Brewster modes are similar. They both exist because of the zeros of ε(ω) but their microscopic origin is not exactly the same. In both cases the absorption peak is due to the enhancement of the electric field associated with the ENZ effect. The difference stems from the fact that Berreman effect was introduced in the context of absorption peaks by ultrathin dielectric layers deposited on metals. The absorption was due to losses Γ of the phonons. In contrast, the Brewster mode has been introduced in the context of metals [18]. The absorption is due to the free carriers of the surface plasmon polariton mode. This is the fundamental difference between both modes.

4. Reflectance experiments

Samples are investigated by angular dependent reflectance measurements. All spectra have been taken with a Nicolet Nexus 870 FTIR spectrometer equipped with a KBr beam splitter, a glow-bar IR source and a DTGS detector. The angular dependent reflectance setup consists of gold mirrors mounted on two arms rotating around the optical axis giving the possibility to access to angles from 0° up to 85°. A KRS holographic wire grid polarizer has been used to polarize the incident light. Finally, a diaphragm allows controlling the angular aperture of the setup at ± 0.5°. Dispersion relations are obtained by gathering spectra taken each 5°. Each reflectance spectrum represents 2000 accumulations of sample spectra normalized by the reference sample spectrum (which is a 200 nm thick gold film deposited on a Si substrate). Numerical arrangement and smoothing of the spectra have been used to obtain the dispersion relation.

Figures 3-(a) and 3-(b) show the reflectance dispersion under s and p polarized light, respectively, taken by angular resolved reflectance from sample A which is 1 µm thick. The black dashed line is the light line in air. The light gray and dark gray colors correspond respectively to low and high intensity reflectance. Reflected light is accessible above the light line only. Below this line (white part) experiments are not possible because of the wavevector mismatch between the incident light and the guided modes. The oblique lines are experimental artifacts due to weak intensity fluctuations between spectra. For both polarizations we can see that the reflectance is maximum below a wavenumber ν_p ~0.175 µm⁻¹ (dark gray part). In this range of wavenumber the InAsSb layer is metallic because of the high doping level. ν_p corresponds to the plasma wavenumber associated to the plasma frequency, ω_p, or the critical wavelength, λ_p = 5.6 µm.

 

Fig. 3 Reflectance dispersion of sample A, a) under TE or s-polarized light, b) under TM or p-polarized light obtained by angular resolved reflectance. The black dashed line is the light line in air. The black solid line corresponds to the Brewster angle of the highly doped InAsSb (72°).

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Beyond ν_p, the semiconductor becomes “transparent” even if the band gap of InAsSb is around 0.335 eV, i.e 0.270 µm⁻¹. At the high-doping levels investigated here, the Moss-Burstein effect [23] blue-shifts the Fermi edge and increases artificially the effective optical band gap of the material. The InAsSb layer can then be considered as a dielectric up to 0.8 µm⁻¹ (~1 eV). In this spectral range one can clearly see under both s and p polarized light, interferences due to the multi-layer structure. We do not observe any significant difference between both polarizations except i) for large wavevectors or grazing incidence and ii) near the plasma wavenumber, ν_p. In the former case, under p-polarized light the signal decreases up to a specific angle θ_B (corresponding to the black line in Fig. 3-(b)) whereas under s-polarized light the signal increases monotonously. θ_B is the Brewster angle, corresponding to a near-zero value of the reflected light. The Brewster angle is 72° ± 1° which corresponds to refractive index of 3.2 ± 0.2. This value is consistent with the free carrier contribution to the refractive index [24], typically between −0.2 and −0.6 in this range of wavenumber for this doping level. Beyond the Brewster angle, because of the sign inversion of the Fresnel coefficients we observe a phase inversion. The interference minima (dark gray) become maxima (light gray) and vice-versa. In the latter case, near ν_p, under p-polarized light a dip appears in the reflectance front at ν_p (white circled in Fig. 3-(b)). This resonance is observed near ν_p and corresponds to the range of wavenumber where the permittivity is between 0 and 1.

We now move to sample C which consists in a much thinner (100 nm) layer of doped semiconductor on GaSb. Figure 4 shows the reflectance dispersion under p polarized light. We can see at 0.187 µm⁻¹ a dip in the reflectance signal more pronounced than in Fig. 3 (Sample A). The spectral signature of this mode is stronger for sample C than for sample A which shows that this resonance is highly sensitive to the layer thickness. This spectral signature is due to what we will name the “Brewster mode” [18] in the reminder of the article. It is important to notice the quasi q independence of the Brewster mode in Fig. 4 which is not exactly the case of the Fig. 3-(b).

 

Fig. 4 Reflectance dispersion under TM or p polarization of sample C obtained by angular resolved reflectance experiment. The dark dashed line is the light line in air.

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To summarize this experimental part, we have identified under oblique p-polarized light a dip in the reflectance spectra of highly doped semiconductor layers regardless of the thickness. Reducing the thickness of the InAsSb layer seems even to flatten the dispersion relation of this Brewster mode provoking its q independence and strengthening its spectral signature.

5. Discussion

Figures 5-(a) and 5-(b) show respectively the simulated and measured reflectance dispersion relations of the Brewster mode of sample B. This sample has a lower doping level and thus a weaker broadening than sample A. Simulations (Fig. 5-(a)) are done using transfer matrix method and Fresnel coefficients. Figure 5-(a) and 5-(b) show sharper resonances in the reflectance spectra below the first minimum of reflection due to the fact that the refractive index is close to 1, so the reflection coefficient is close to 0. Simulation and experiment are in excellent agreement. The dip in reflection is due to the coupling of the incident beam to the lossy leaky mode. This is similar to the dip observed when coupling an incident beam to a surface plasmon through a grating or a prism. The mode is only visible under p polarization. The mode appears when the wavevector increases between 0.3 and 0.9 µm⁻¹. The calculated dispersion relation of the Brewster mode from Eq. (3) is represented by black circles. They are into the light gray part corresponding exactly to the mode. This is the spectral range of the Brewster mode.

 

Fig. 5 Reflectance dispersion under TM or p polarization of sample B obtained by angular resolved reflectance a) simulation or b) experiment. The black dashed line is the light line in air. The black circles correspond to wavenumbers associated to the dispersion relation calculated with Eq. (3) with sample B parameters. Insets correspond to the spectra for q = 0 (white) or 0.800 µm⁻¹ (black) corresponding to the black dashed line of the 2D image.

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The inset of Fig. 5-(b) shows the spectra obtained for q = 0 µm⁻¹ (white line) and 0.800 µm⁻¹ (black line). The latter spectrum corresponds to the vertical black dashed line in the 2D image. We can clearly see the spectral signature labeled Brewster mode. This is the dip in the reflectance signal near the wavenumber 0.142 µm⁻¹. Similar spectrum is obtained in the inset of Fig. 5-(a) (black line) at q = 0.800 µm⁻¹ (dashed black line in the 2D image). The dip near ν = 0.142 µm⁻¹ is the Brewster mode. The Brewster mode follows the expected behavior of the metal modeled by the Drude function in the case of InAs (black circles).

Our theoretical point of view explains all the experimental results obtained on highly doped semiconductors. Because of the enhancement of the electric field it is possible to observe this Brewster mode especially for thin layer of InAsSb. In Fig. 4 for a 100 nm thick layer of InAsSb we can see a dip in the reflectance signal at 0.1786 µm⁻¹ which is the Brewster mode. Thanks to the electric field enhancement, the spectral signature of the Brewster mode is stronger. Finally for this small thickness the Brewster mode is q independent. This is fully coherent with the theory.

Finally, it is noticeable that the spectral position of the Brewster mode can be used to accurately measure the doping level of doped semiconductors, e.g. by in situ measurements during epitaxial growth. Indeed, in the case of thin layers the Brewster mode is exactly at ν_p, or λ_p, which is proportional to the carrier density. Carrier density and plasma frequency are linked by the following expression:

Figure 6 shows the variation of λ_p as a function of the carrier density in the case of InAs [25]. The red dashed line represents Eq. (6) for InAs taking into account the non parabolicity effect. It corresponds to the power law: λ_p = 1.058 N^-0.37, where λ_p and N are respectively in µm and cm⁻³ units. Note that we do not take into account the strong coupling between plasmon and phonon, when λ_p reaches the phonons frequencies delimiting the Reststrahlen band (grey stripe in Fig. 6). Indeed, because of this strong coupling the power law will fail near the Reststrahlen band.

 

Fig. 6 Critical wavelength, λ_p, versus the carrier density in the case of InAs (dark square). The red dashed line corresponds to the power law extracted from Fig. 6. The grey part of the figure corresponds to the Reststrahlen band due to optical phonon. The open symbols correspond to our experimental values.

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The power law depends directly on the non-parabolicity of the effective mass and it is thus material dependent. Realizing series of samples of different materials (GaAs, GaSb, …) with different carrier concentrations will allow establishing abacus linking the carrier density in the layer to the wavelength of the p-polarized light in reflectance experiments, similarly to Fig. 6 established for InAs. We plot the plasma wavelength of sample A, B and C in the Fig. 6 (open circles). They are in good agreement with the power law that we obtained for InAs. The differences between experiments and the empirical law are due to incertitude on the carrier concentration (notably for Sample C because of the impossibility to make Hall measurement) and the Sb incorporation in InAs that reduce the maximum doping elements incorporation [20]. Series of samples will be necessary to obtain abacus. These abacus can then be used to determine the carrier concentration needed to achieve a given plasma frequency. Conversely, they will be useful for semiconductor hetero-structures growth and notably in the case of in situ doping level monitoring, in particular when the dopant activation is far from unity.

6. Conclusion

We have studied by angle-resolved reflectance highly-doped InAsSb layers lattice-matched onto GaSb substrates. We have observed a reflectivity dip near the plasma frequency. This absorption peak is analogous to the Berreman absorption peak observed on thin dielectric films deposited on metals. In our case however, we have shown that this reflectivity dip can be associated to the excitation of a Brewster mode. In both cases, the physical mechanism responsible for absorption can be traced back to the strong enhancement of the field in the thin film due to the epsilon near zero effect. For thin films, the reflectivity dip is observed for a broad range of angles of incidence in agreement with the dispersion relation of the Brewster mode. It is thus an effect which is easy to observe. Finally, measuring p-polarized reflectance spectrum at oblique incidence provides a tool for in situ measurement of the carrier concentration in doped semiconductors.