Infrared dielectric response, index of refraction, and absorption of germanium-tin alloys with tin contents up to 27% deposited by molecular beam epitaxy

Germanium-Tin (Ge_1-xSn_x) alloys have received considerable attention as a possible direct bandgap group IV semiconductor optoelectronic material compatible with Si and Ge fabrication processes.^1,2 The introduction of Sn into the Ge lattice alters the electronic band structure and causes the Γ-valley to decrease faster than the L-valley in relaxed films, allowing the bandgap to become direct for Sn contents greater than around 6%–12%.² Photodetectors,^3,4 photoemitters,^5–8 photoconductors,⁹ and optically pumped lasers fabricated¹⁰ from Ge_1-xSn_x have been recently demonstrated. Group IV photonic devices that operate in the 2–5 μm wavelength mid-infrared (IR) regime are particularly attractive for sensing applications and can integrate directly with Si to overcome the challenges associated with the integration of group III-V materials.¹¹ To get the maximum utility from devices made from such materials, it is important to understand the relationship between the alloy composition and the effect on the optical properties such as the index of refraction and absorption coefficient. For example, the ability to tune the index of refraction is desirable for IR anti-reflective coatings and as layers for photonic devices.¹² Yet in Ge_1-xSn_x alloys, the relationship between the alloy composition and the effect on the index of refraction in the mid-IR is still not well understood for Sn contents above 15%, which is addressed in this report. As a consequence of lowering the bandgap by increasing the Sn concentration in Ge_1-xSn_x alloys, the sensitivity to photon absorption persists further into the IR. For example, a relaxed Ge_1-xSn_x alloy with x = 0.15 is predicted to have a direct bandgap of 0.3 eV, which corresponds to a wavelength value of 4.1 μm, whereas for a relaxed alloy with x = 0.18, the direct bandgap is predicted to be 0.22 eV which corresponds to a wavelength value of 5.6 μm.^2,13,14 There have been few publications, however, of the optical properties for such high Sn contents, which is addressed in this report.

The low thermal stability, low solid solubility (<1%),¹⁵ and large lattice mismatch between Ge and Sn (14.7%)¹⁶ require non-equilibrium deposition techniques. Because of these challenges, the availability of Ge_1-xSn_x films of appreciable thickness (>20 nm)¹⁷ has been restricted to atomic Sn contents of less than 22%,¹⁸ until now. To enable larger Sn percentages (>18%), molecular beam epitaxy (MBE) was used with an electron beam (e-beam) evaporation source for elemental Ge to reduce the substrate temperatures below about 150 °C so that Sn does not segregate on the layer surface.

There have been many investigations into the dielectric function and associated optical critical points of Ge_1-xSn_x alloys into the mid-IR, but with lower Sn fractions x <0.20,^14,19–23 as well as studies of the index of refraction and absorption with x ≤0.10.¹² In this report, we used visible and IR variable angle spectroscopic ellipsometry (VASE), to characterize the optical properties of Ge_1-xSn_x thin films with x = 0.15 to x = 0.27, including the index of refraction and absorption coefficient, for wavelengths from 0.190 μm to 6 μm.

Ge_1-xSn_x thin film alloys with up to 27 at. % of Sn were deposited on 76.2 mm diameter undoped (001) oriented Ge substrates (u-Ge) with a resistivity of 40 Ω-cm by MBE as described previously.²⁴ Using thermal (Knudsen) effusion cell sources, the relatively low vapor pressure of Ge necessitated high cell temperatures upwards of 1240 °C, which approached the thermal limit of the pBN crucible, and radiated a substantial amount of heat to the substrate. In the current configuration of our MBE system, the growth of Ge_1-xSn_x alloys with Sn concentrations above 19% required an e-beam evaporation source (MBE Komponenten) for Ge, which had a smaller heated volume than the Knudsen source, allowing for less heat radiated to the substrate so that substrate temperatures below 150 °C could be achieved, as well as much higher growth rates and higher Sn contents. The native oxide of germanium was difficult to remove from the wafer surface, and care was taken when cleaning the substrates, including heating in the growth chamber to remove the surface germanium oxide, as described elsewhere.²⁴ 

The film quality, crystallinity, strain state, surface quality, thickness, and Sn concentrations were confirmed by high-resolution X-ray diffraction (HRXRD), reciprocal space mapping (RSM), atomic force microscopy (AFM), X-ray reflectivity (XRR), and channeling Rutherford backscattering spectrometry (RBS) as described elsewhere.²⁴ RSM plots of diffracted X-ray intensity versus reciprocal lattice vectors were used to determine the crystalline state, strain state, and the Sn composition of the films. As shown in Fig. 1, the broadened shape of the Ge_1-xSn_x film peaks was attributed to relaxation²⁵ and to possible disorder at the substrate interface but where the film regains high-quality crystallinity as it grows thicker. The cause for interface disorder, which was previously reported in channeling RBS measurements, was likely the lattice mismatch between the Ge_1-xSn_x and the Ge substrate.²⁴ Channeling RBS was performed on other samples in this series which verified alloy composition uniformity and the accuracy of these composition calculations.¹⁶ RBS backscattered yield ratios showed that over 95% of the Sn was substitutional in the lattice.²⁴ The films in Fig. 1 are typical of the films in this study. The Sn concentration and film relaxation along with other film characterization parameters are given in Table I.

To determine the optical constants of the Ge_1-xSn_x thin films, VASE ellipsometry measurements were performed at room temperature in the ambient atmosphere. All samples were cleaved into 2 cm × 2 cm pieces and then sonicated for 15 min in deionized water, 15 min in isopropyl alcohol, then for again 15 min in deionized water, and finally dried with nitrogen gas to remove any surface contaminants. Measurements of the ratio of the Fresnel reflection coefficients for parallel and perpendicular components of linearly polarized light at each selected wavelength were used to obtain the ellipsometric angles ψ and Δ, given by

The films were characterized with a combination of two ellipsometers to cover different spectral ranges. In the wavelength range from 0.190 to 1.690 μm, measurements were performed at multiple angles of incidence (60°, 65°, and 70°) with a spectral resolution of 4 nm, using a Berek waveplate compensator (J.A. Woollam, V-VASE). In the infrared wavelength range from 1.532 to 6.173 μm (833–1620 cm⁻¹ wavenumbers), measurements were performed on a J.A. Woollam FTIR-VASE ellipsometer, which is based on a fixed analyzer (0° and 180°) and polarizer (±45°) with a rotating compensator, at multiple angles of incidence (60°, 65°, and 70°). Above about 4 μm, the signal-to-noise ratio decreased; to improve the signal-to-noise ratio, the spectral resolution was set to 16 cm⁻¹ with long signal averaging (three measurement cycles, each with 15 spectra per compensator revolution and 20 revolution scans per spectrum).

To extract the optical constants from the ellipsometric angles, we assumed a model comprising four layers including a vacuum ambient, a surface layer incorporating both surface roughness and any native oxide layer that may be present,²⁶ the Ge_1-xSn_x film layer, and finally a semi-infinite bulk Ge substrate. The dielectric function of the surface layer was obtained using an effective medium approximation layer that consisted of 50% film and 50% ambient. The dielectric functions of the Ge_1-xSn_x films and Ge substrate were described using a Johs-Herzinger parametric oscillator model, which imposes Kramers-Kronig (K-K) consistency between the real and imaginary parts to ensure causality. The parametric oscillator model consists of 8 oscillators with multiple fit parameters (about 40 total) each representing, in principle, critical point energy transitions for Ge and Ge_1-xSn_x. Although this model has been successfully used to determine the optical properties of many semiconductors, one must be careful not to assign too much physical meaning to the fit parameters. Because of strong correlations between parameters, two solutions with different fit parameters could yield the same dispersion results, whereas the usefulness of the model is strictly in the optical constants it provides not in the values of the fitting parameters. The Ge_1-xSn_x film thicknesses from XRR measurements were used as an initial starting point for the parametric model thicknesses which, along with the parametric oscillator parameters, were treated as adjustable parameters during the model fitting iterations. A dielectric function dispersion model fit was considered satisfactory when a good agreement between the calculated and measured ellipsometric angles was achieved, as determined by regression analysis using the Levenberg-Marquardt algorithm until the weighted mean squared error (MSE) between the calculated and experimental data was minimized. All MSE values were less than 1.5. The largest differences between the calculated and measured ellipsometric angles occurred below 0.3 eV (about 0.4° for ψ and 3.5° for Δ, corresponding to 1 in units of the pseudo-dielectric function).

The spectral dependence of the complex index of refraction, $ñ=n+iκ$⁠, was determined from the parametric dielectric function dispersion model, $ε̃=ñ2$⁠, and the absorption coefficient from the relation

where κ is the imaginary part of the complex index of refraction, also known as the extinction coefficient, and λ the wavelength of light. The complex dielectric function from the parametric model is given in Fig. 2. As the Sn concentration increased, there was a red-shifting and broadening of the complex dielectric function. The 25% and 27% Sn concentration samples have a broadened peak in ε₂, near 0.9 eV, which was not observed in the other samples.

To verify whether these peaks were an artifact of the fittings or due to a property of these samples, two additional fittings to the measured ellipsometry data were performed on the 25% and 27% Sn concentration films. In the first additional fitting, we modeled the 25% and 27% Sn concentration films with a K-K consistent model that consisted of a combination of 11 Gaussian and Lorentz oscillators, rather than parametric oscillators. In the second additional fitting, we performed an uncorrelated all-wavelength inversion of the ellipsometric angles with the thickness fixed to the values obtained in the parametric oscillator fit, where K-K consistency is not enforced (this fitting is known as a point-by-point fit). The MSE of the two additional fittings changed only slightly from the initial fitting and the broadened peak in ε₂ near 0.9 eV was still present in the optical constants obtained from these new fits, indicating that this peak is a result of a property of these samples rather than an artifact of the parametric oscillator fitting. The broadened peak is possibly caused by an intrinsic feature such as from the band structure, an extrinsic one such as from a defect, or an interference fringe from the substrate.

Since Ge and Sn are group-IV elements, they both have four electrons per atom. Therefore, ∫ωε₂(ω)dω should be approximately independent of the Sn content, if the integral is taken over the measured spectral range from 0.2 to 6.5 eV. This is known as the sum rule for the oscillator strength²⁷ and can be used to test the accuracy of optical constants for new materials. In our case, the integral deviates no more than 7% from its bulk Ge value for all Sn contents, if we assume a surface layer thickness of about between 4 and 8 nm. The sum rule analysis therefore provides a strong argument for the accuracy of the optical constants presented in this work.

At wavelengths above about 1.5 μm, the index of refraction, n, increased with the Sn concentration (Fig. 3). The peak at the lower wavelengths in each curve is due to the E₁ energy critical point.¹³ It is known that alloying with Sn causes a red-shifting and broadening of the electronic critical points²⁴ due to lowering of the bandgap, and this is observed more easily in the inset to Fig. 3, where the energy level for the E₁ critical point is red-shifted and broadened with the increasing Sn concentration. The inflection in the E₁ critical point peak for the 25% Sn concentration sample near 0.8 μm that is not present in the same peak for the 27% sample was also confirmed with the three fitting methods described above and attributed to spin-orbit splitting of the valence band. The dispersion characteristics of Ge_1-xSn_x are not well known, and thus far, there has been little quantitative treatment of a relation to determine their value at mid-IR wavelengths.¹² At a wavelength of 5 μm, an empirical linear fitting of the index of refraction was found to be $n=mx+b$⁠, for Sn% x, where slope $m=0.01373±6.12489× 10−4$ and the intercept $b=4.01707±0.01195$⁠. A wavelength of 5 μm was chosen because for small deviations, the index was slowly changing. The empirical linear fitting was found useful for the Sn compositions from bulk Ge to 27% measured here but was not evaluated for other Sn contents.

The absorption coefficients determined from the experimentally measured data for different Ge_1-xSn_x compositions are given in Fig. 4. With the increasing Sn content, the absorption persisted to longer wavelengths out to the far IR and also increased in strength at a given wavelength, which is expected as the bandgap at the Γ-point is predicted to decrease with the increasing Sn concentration.^2,14 The 25% and 27% Sn concentration samples continued absorbing past our measurement spectral range of 6 μm, while there is a decrease in the absorption coefficient for the 15% and 18% Sn concentration samples because the bandgaps for these concentrations are predicted to lie within the measured spectral range. The bandgap at the Γ-point of Ge_1-xSn_x alloys depends on both the composition and the film strain state. For purposes of comparison, the theoretically predicted bandgap at the Γ-point, determined from compositional dependence and deformation potential theory as described elsewhere,¹³ is given in Table I. As seen in Table I, at high Sn concentrations, the bandgap at the Γ-point is predicted to correspond to wavelengths in the far-IR regime beyond about 8 μm, raising the possibility that Ge_1-xSn_x may be used for devices in far-IR applications beyond the mid-IR. It should be noted that defects in a Ge_1-xSn_x film could alter its optical properties, as with any material. Although we believe that the films in this study are of high quality as evidenced by XRD, we cannot rule out the presence of defects nor their effects on film properties.

Using ellipsometry, we have measured the dielectric function of Ge_1-xSn_x thin film alloys with Sn concentrations of 15, 18, 25, and 27%, in the spectral range of 0.190–6 μm. From the dielectric function, we have determined that the index of refraction and absorption coefficient of these Ge_1-xSn_x thin film alloys can be tuned by varying the Sn concentration which alters these optical properties. As the Sn concentration increased, both the index of refraction and the absorption coefficient increased to larger values at longer wavelengths. For instance, for the 27% Sn alloy, the absorption at a wavelength of 4.4 μm was comparable to that of unalloyed Ge at about 1.5 μm. An empirical linear fit to the index of refraction at 5 μm was obtained, which can be used to estimate the Sn concentration needed for an application-specific refractive index, such as for use in mid-IR optical devices or coatings. We have shown that for high Sn concentrations, the absorption persists past 6 μm, which corresponds to a higher photodetector sensitivity, responsivity, and specific detectivity at the mid-IR wavelengths. The characterization of these parameters will be critical in the design of future Ge_1-xSn_x optoelectronic devices that utilize high Sn concentrations.

This work was funded in part by grants from the Air Force (AFOSR Award Nos. FA9550-14-1-0207, FA9550-13-1-0022, and FA9550-17-1-0134), the Army Research Office (ARO Award No. W911NF-12-1-0380), and by gifts from Thorlabs, Inc. The work at NMSU was supported by the National Science Foundation (No. DMR-1505172). Special thanks to D. Beatson, C. Pinzone, J. Wei, J. Kouvetakis, J. Menendez, and S.-Q. (Fisher) Yu for valuable discussions. The work reported here was partially carried out in the Nanofabrication Facility at the University of Delaware.