**3. Results**

Figure 1 shows reflectance spectra for all samples measured at 300 K for two orthogonal linear polarizations of the incident light. A distinct absorption dip can clearly be seen for TM polarization (red curves), which corresponds to a case where there is an electric field component perpendicular to the sample surface. Black curves denote spectra obtained for s polarized light (TE) where there is no corresponding minimum observed, which is consistent with theory. Also visible in the red curves of the TM polarization is how the Berreman minima shift towards lower wavenumber (longer wavelength) with decreasing doping and finally even disappear for sample D due to the cut-o ff wavelength of our setup (detector limit at 600 cm<sup>−</sup>1).

**Figure 1.** The reflectance spectra for two orthogonal polarizations of the probing light denoted TE (red curves) and TM (black curves): (**a**) 1.9 × 10<sup>19</sup> cm<sup>−</sup>3; (**b**) 9.9 × 10<sup>18</sup> cm<sup>−</sup>3; (**c**) 5.2 × 10<sup>18</sup> cm<sup>−</sup>3; (**d**) 1.1 × 10<sup>18</sup> cm<sup>−</sup>3.

Figure 2 shows intensity normalized reflectance spectra for the p-polarized light of three samples A, B and C. The absorption minima can be ascribed to plasma frequency <sup>ω</sup>*p*, of free electrons in the layers. We can see that <sup>ω</sup>*p* shifts to lower energies with carrier concentration decreasing according to Equation (1).

$$
\omega\_p^2 = \frac{m^2}{\varepsilon\_0 \varepsilon\_\infty m^\*} \tag{1}
$$

where ε∞ is the material dielectric constant equal to 11.64 and ε0 is the permittivity of free space. The value of *m*\* is 0.0453 of electron rest mass.

**Figure 2.** Normalized p-polarized reflectance spectra for Sample A (red curve), Sample B (black curve), and Sample C (blue curve) together with given characteristic reflectance minima at plasma frequencies and indication of carrier nominal concentration.

Figure 3 shows a comparison of the dependence of the obtained nominal carrier concentration versus plasma frequencies (black squares) obtained for InGaAs alloy with similar composition by Charache et al. [22]. This allowed us to ge<sup>t</sup> a useful function after exponential fitting (red line). A similar procedure has been proven to be successful when applied for the analysis of Berreman minima in the case of carrier concentration determination in InAs layers [23], using data from Hinkey et al. [24]. By open squares we marked nominal concentration values on the fitted line at the respective wavenumbers. Open solid points denote the plasmon frequencies measured by reflectivity measurements (given in wavenumbers). By placing these frequencies on the fitted curve, we were able to determine the actual electron concentrations of 1.3 × 10<sup>19</sup> cm<sup>−</sup>3, 7.0 × 10<sup>18</sup> cm<sup>−</sup>3, and 4.0 × 10<sup>18</sup> cm<sup>−</sup><sup>3</sup> for samples A, B, and C, respectively. Furthermore, we can see that the revealed differences between the nominal and the measured concentrations, despite not being large, show for all cases the measured values to be smaller than the nominal ones. This is just the first approximation approach, which can be used as an attempt for growth parameters verification in order to establish a better match between nominal and achieved concentrations in the layers.

A more sophisticated and precise method requires including the effective mass dependence on the carrier concentration in the calculation of the plasma frequencies as a function of concentration, since it cannot be neglected for concentrations above 10<sup>18</sup> cm<sup>−</sup>3, as in ref. [25]. Due to non-parabolic behavior of energy dispersion far from Γ point of the Brillouin zone, a correction of carrier effective mass must be taken into account, which can be expressed by Equation (2) taken from ref. [26].

$$m^\*(n) = m\_\varepsilon \left[ 1 + \frac{4P^2}{3E\_\mathcal{S}} \left( 1 + \frac{8P^2 \hbar^2 \left( 3\pi^2 n \right)^{\frac{2}{3}}}{3m\_\varepsilon E\_\mathcal{S}^2} \right)^{\frac{2}{3}} \right]^{-1},\tag{2}$$

where *me* is the electron rest mass and *P*<sup>2</sup> is the momentum matrix element of coupling between valence and conduction bands, calculated to be 15.4 eV.

**Figure 3.** Carrier concentration in a function of Plasma frequency. Black squares denotes points after ref. [22] together with respective fit (red line). Triangular points represents nominal concentrations and open circles those established.

Figure 4a shows the calculated *m\**/*me* ratio as a function of electron concentration. As we can see, the role of correction is more important for larger concentrations in the range of nominal concentrations considered within this paper (indicated by the red dotted square), where the relative changes of the effective mass up to around 30% can occur. Figure 4b shows calculated plasma frequencies assuming the mass correction (black curve) together with the nominal values (black squares) and those determined here (red circles). The obtained values are still slightly lower than nominal, however closer to the nominal values when compared to those obtained in the first, simpler approach. The respective summary is shown in Table 2. The resolution of our experimental setup was 2 cm<sup>−</sup>1, while the full widths at half maximum (FWHM) of measured spectra were ~50 cm<sup>−</sup>1. This allowed us to estimate uncertainty of the carrier concentration at ±2%, which is not enough to explain the difference between nominal concentration.

**Figure 4.** (**a**) Electron mass changes as a function of their concentration after Equation (2); (**b**) Plasma frequencies as a function of electron concentration. Black curve represents calculated Plasma frequencies assuming mass correction shown in panel (**a**). Open triangles depicts nominal values and open circles those measured.


**Table 2.** The samples' description and growth protocol.
