Band gap and strain engineering of pseudomorphic Ge_1−x−ySi_xSn_y alloys on Ge and GaAs for photonic applications

Germanium-tin alloys are attractive as an electronic and optoelectronic material, because they are compatible with silicon and form a direct band gap at sufficiently large tin concentrations between 6% and 10%.¹ Adding silicon to this alloy to form the ternary compound Ge_1−x−ySi_xSn_y has the added advantage that the lattice constant and band gap are tunable independently.² Such alloys find applications as photodetectors³ and semiconductor lasers.⁴ 

Most Ge_1−x−ySi_xSn_y alloys for device applications have been grown as relaxed epilayers, usually on Si substrates.^1,3,4 In such relaxed epilayers, the strain is relieved at the substrate/epilayer interface by the formation of misfit dislocations. These dislocations also form a cross-hatched network leading to rough surfaces and tilted crystal mosaics.¹ The dislocations are expected to form nonradiative recombination centers reducing the carrier lifetimes, increasing dark currents, and reducing light emission efficiency.

It is therefore interesting to ask, if similar beneficial electronic properties can also be achieved using pseudomorphic Ge_1−x−ySi_xSn_y alloys. A few such studies have been performed on epitaxial layers grown using molecular beam epitaxy (MBE)⁵ and chemical vapor deposition.⁶ Unfortunately, band structure calculations based on deformation potentials indicate that such pseudomorphic alloys cannot become direct.^7,8

In this manuscript, we explore the Si-Ge-Sn parameter space with deformation potential calculations to determine if a suitable alloy composition and substrate can be found to yield a direct band gap for a pseudomorphic Ge_1−x−ySi_xSn_y. Using detailed band structure charts, we show that this is not the case. Even for very high Sn content (>20%), the direct gap is never significantly lower than the indirect gap. We confirm our results for binary Ge_1−ySn_y alloys with spectroscopic ellipsometry measurements to determine the critical-point energies of E₀, E₁, and E₁ + Δ₁ transitions. We expand on prior work by others⁵ to discuss the impact of alloy disorder on the excitonic effects contributing to the E₁ transitions.

Several groups have reported the compositional dependence of the various band gaps of relaxed Ge_1−x−ySi_xSn_y alloys at room temperature.^2,9–14 $E0Γ$ is the direct band gap at the Γ-point. $EiL$ and $EiΔ$ are the indirect band gaps between the valence band (VB) maximum at Γ and the conduction band (CB) minima at the L-point and along Δ. E₁ and E₁ + Δ₁ are the direct band gaps at or near the L-point split by spin–orbit interactions of the VB. The compositional dependence of the various room temperature band gaps of the relaxed (unstrained) Ge_1−x−ySi_xSn_y alloys can be expressed as

where E^j are the corresponding band gaps of the elements and b^jk (j, k = Si, Ge, Sn) the bowing parameters defining the deviation of the band gap from the linear interpolation. Gaps and bowing parameters are given in Table I for the various gaps.

By convention, our figures always use the VB maximum as our zero energy level. Under hydrostatic strain, this choice implies that the absolute deformation potential for the VB a_VB is zero. Theory shows that a_VB is small and its sign is not known.¹⁵ Under shear strain, when the VB splits, our figures still use the highest VB maximum as our energy reference level, which leads to an unphysical value of a_VB. The actual value of a_VB does not matter for our results, since we only report band gaps and splittings.

The compositional dependence of the CBs can then be obtained from the dependence of the corresponding band gaps of the alloys. To take into account how these bands and band gaps are changed in pseudomorphic alloys, we must first calculate the strain tensor caused by the lattice mismatch using continuum elasticity theory. Then, we calculate band gap shifts and splittings using deformation potentials.

The lattice parameters of relaxed Ge_1−x−ySi_xSn_y alloys were calculated using a linear interpolation (Vegard's law) from the lattice parameters a_j of the elements

with small quadratic deviations from linearity described by the bowing parameters θ_jk. Although extensive work has been done to characterize the compositional dependence of the lattice parameters of Ge_1−ySn_y alloys, a discrepancy exists in the literature over the value of θ_GeSn.^3,16–18 According to most recent findings,^16,17,19,20 the bowing parameter θ_GeSn is very small and Vegard's Law (θ_GeSn = 0) can be used to accurately calculate the lattice parameters of Ge_1−ySn_y alloys. The bowing parameter for the lattice constant of Ge_1−xSi_x alloys θ_GeSi = –0.026 (see Ref. 21); and θ_SiSn was assumed to be zero, justified by the low Si and Sn content.

Depending on composition, the lattice parameter of relaxed Ge_1−x−ySi_xSn_y will be larger, equal to (lattice matched to Ge), or smaller than the lattice parameter of the Ge substrate. Under pseudomorphic growth conditions, Ge_1−x−ySi_xSn_y on Ge experiences a biaxial stress along the interface due to the lattice mismatch between the alloy layer and the Ge substrate, creating an in-plane strain given by

Within continuum elasticity theory and for (001) surface orientation, the pseudomorphic alloy layer is tetragonally deformed on the Ge substrate and the out-of-plane strain

is related to $ϵ∥(x,y)$ through the ratio of the elastic constants C₁₂/C₁₁. For the elements, the elastic constants C_mn were taken from the literature,^22–24 see Table I, and scaled linearly with composition

The strain resulting from biaxial stress in the (001) surface has two components: The hydrostatic strain ϵ_H shifts the CB positions and the [001] shear strain ϵ_S lifts the degeneracy of some electronic bands. They are related to the in-plane and out-of-plane strain through²² 

In the absence of shear strain, the heavy (v₁) and light hole (v₂) bands are degenerate at the Γ point. Spin-orbit interactions separate them from the split-off band (v₃). The [001] shear strain lifts the degeneracy of the VB edge, leading to an additional splitting. The energy positions of the top three VBs at the Γ point can be expressed as^22,32–34

The sign of the shear strain ϵ_S determines whether v₁ or v₂ is the highest VB. Note that the [001] shear deformation potential b for the VB has a negative value. Δ₀ is the spin-orbit splitting of the VB at the Γ point. Its compositional dependence was obtained by linearly interpolating the elemental end values, see Table I 

The energy shifts of the CB at the Γ and L points due to the hydrostatic deformation can be calculated using^25,26,35,36

where a and $[Ξd+(1/3)Ξu]L$ are the hydrostatic deformation potentials for the CB at the Γ and L points, respectively. The [001] shear strain does not split the CB at Γ (because it is nondegenerate) or at the L-point.

At the Δ-minimum, however, we have to consider both hydrostatic and shear contributions. The [001] shear strain splits the CB at the six Δ minima into a doublet and a quadruplet, in addition to the hydrostatic strain shift. The energies can be expressed as^25,26,35,36

where $ΞuΔ$ is the [001] shear deformation potential and $[Ξd+(1/3)Ξu]Δ$ the hydrostatic deformation potential of the CB at the Δ minima.

Combining the compositional dependence of the band gaps of the relaxed (unstrained) ternary alloys, given by Eq. (1), with the energy shifts and splittings of the CBs and VBs under strain using deformation potential theory, the compositional dependence of the band gaps of pseudomorphic Ge_1−x−ySi_xSn_y on Ge at the Γ, L, and Δ CB minima can be obtained. All parameters used for the calculation are listed in Table I. Results are shown in Fig. 1.

The band gaps at the Γ, L, and Δ minima as a function of Si and Sn compositions show a similar qualitative behavior: The band gaps are widening with increasing Si and narrowing with increasing Sn content, as expected. In Fig. 1, the thick solid line, y = 0.3x, indicates the lattice matched compositions of Ge_1−x−ySi_xSn_y on Ge, reasonably consistent with the previously reported value y = 0.27x (see Ref. 2). The small discrepancy is caused by slightly different bowing parameters used in the calculation of the lattice parameters. This line separates a positively strained (tensile) region from a negatively strained (compressive) region. A Ge-like band gap between 0.66 and 0.82 eV can be obtained for unstrained Ge_1−x−ySi_xSn_y by varying the Si and Sn content.

A first-order singularity (kink) of the band gap at L is observed in Fig. 1(b) as the sign of the in-plane strain changes from tensile to compressive. As mentioned earlier, the sign of the in-plane strain determines whether v₁ or v₂ is the highest VB. The light hole v₂, given by Eq. (9), is the highest VB for compressively strained pseudomorphic Ge_1−x−ySi_xSn_y alloys on Ge, while the heavy hole v₁, given by Eq. (8), becomes the highest VB under tensile in-plane strain. Also, the Δ4 quadruplet is lower in energy than the Δ2 doublet in the compressively strained region, while tensile strain makes the Δ2 energy lower. Therefore, the band gap $EiΔ$ at the Δ minimum is associated with the Δ4 quadruplet under compressive strain, while the Δ2 doublet determines $EiΔ$ in tensile pseudomorphic Ge_1−x−ySi_xSn_y on Ge. Hence, the behaviors of the highest VB and the lowest CB at the Δ minimum with respect to strain explain the singularity in the predicted band gaps when the sign of the strain changes.

The smallest band gap, either direct or indirect, of pseudomorphic Ge_1−x−ySi_xSn_y alloys on Ge is shown in Fig. 2. The dashed-dotted curve between 15% and 20% tin represents an indirect to direct band gap crossover: Above this line, pseudomorphic Ge_1−x−ySi_xSn_y on Ge becomes a direct band gap material. This condition requires a very large Sn content, which has not yet been achieved experimentally due to the low solubility of Sn in Ge and Si. The dashed line, y = 0.71x – 0.12, shows a transition of the smallest band gap from the L valley (Ge like) to the Δ valley of the CB (Si like). The smallest band gap is associated with the L-valley of the CB (Ge like) in the region bounded by the dashed-dotted and dashed lines.

As a special case, we consider pseudomorphic Ge_1−ySn_y alloys grown on Ge (x = 0), see Fig. 3. This figure shows the position of the top three VB maxima calculated from Eqs. (8)–(10) and the various band gaps $E0Γ, EiL, EiΔ2$⁠, and $EiΔ4$ from Eqs. (1) and (12)–(14) versus Sn content We clearly see that the indirect gap $EiL$ is lower than the direct gap $E0Γ$ for y < 25%, indicating that pseudomorphic Ge_1−ySn_y alloys on Ge will not become a direct band gap material for practically achievable Sn compositions, which is consistent with the models recently reported.^31,37

There is strong interest in growing Ge_1−x−ySi_xSn_y alloys on Ge buffered Si substrates instead of directly on a Ge substrate.³⁸ The Ge buffer on Si is under tensile biaxial stress because of the different thermal expansion coefficients of Si and Ge,³⁹ which can be accurately modeled, assuming full relaxation of the Ge buffer at the growth temperature T_g (of the buffer) and no additional relaxation while cooling down.⁴⁰ The thermal strain increases the in-plane lattice parameter of the Ge buffer, which in turn affects the strain of the pseudomorphic Ge_1−x−ySi_xSn_y alloy layer grown on top of the Ge buffer. This reduces the compressive strain in the pseudomorphic Ge_1−x−ySi_xSn_y on the Ge buffer and thus lowers the critical Si and Sn contents needed to achieve a direct band gap material. Increasing the growth temperature of the Ge buffer increases its tensile strain, which reduces the critical Sn content more, but even the highest possible growth temperature of the Ge buffer (melting point of Ge, T_g= 1200 K) reduces the critical Sn content for the indirect to direct transition by only 1%. In Figs. S1 and S2 of the supplementary material,⁶⁰ we show the band gaps of Ge_1−x−ySi_xSn_y alloys grown pseudomorphically on Ge buffered Si substrates, where the Ge buffer was grown at T_g = 770 K. For such alloys, the strain-free condition for the alloy composition becomes y = 0.3x + 0.01.

For a sample with 19% Si and 6% Sn grown on a Ge virtual substrate, our calculations predict a direct gap E₀ of about 1.02 eV, somewhat larger than photoluminescence results by Wendav et al.,⁴¹ shown by the symbol in Fig. S1. Effects of residual strain, errors in composition (obtained by Rutherford backscattering), and nonuniform composition are probably too small to explain this discrepancy. However, we note that the photoluminescence peak in Ref. 41 is quite broad. Due to microscopic statistical alloy fluctuations, one expects a redshift of the photoluminescence peak relative to the onset of absorption measured by ellipsometry. Indeed, our calculations agree well with the high-energy shoulder of the photoluminescence peak, see Figs. 3 and 4 in Ref. 41. We therefore believe that our calculations are not in conflict with the results of Ref. 41.

Similarly, one can also grow Ge_1−x−ySi_xSn_y alloys pseudomorphically on a bulk GaAs substrate.⁴² Since the GaAs lattice constant is slightly smaller than that of Ge, growth on GaAs reduces the tensile strain or increases the compressive strain of Ge_1−x−ySi_xSn_y alloys compared to growth on bulk Ge. For such alloys on GaAs, the strain-free condition for the alloy composition becomes y = 0.3x – 0.01. In Figs. S3 and S4 of the supplementary material, we show the band gaps of Ge_1−x−ySi_xSn_y alloys grown pseudomorphically on bulk GaAs. To achieve a direct band gap in pseudomorphic Ge_1−x−ySi_xSn_y alloys on GaAs, a slightly higher tin content is required than for similar pseudomorphic alloys grown on bulk Ge.

Finally, we also discuss the compositional dependence of the E₁ and E₁ + Δ₁ interband transitions in pseudomorphic Ge_1−ySn_y alloys (x = 0) on Ge. These transitions occur at the L-point of the Brillouin zone and along the Λ direction. They are easily observed in the optical constants of Ge and related materials using spectroscopic ellipsometry measurements.⁴⁰ The strain dependence of these transitions is given by^5,6,11,22

where the superscripts s and 0 denote the band gaps of the strained and relaxed alloys, respectively. The spin-orbit splitting Δ₁ of the VB at the L point is taken as the difference between the E₁ and E₁ + Δ₁ energies with parameters in Table I. Note the negative bowing for the spin-orbit splitting Δ₁, which is common for semiconductor alloys.^9,43 ΔE_H and ΔE_S are the energy shifts due to hydrostatic and shear strain, respectively, calculated using

where $D11=−5.4 eV$ and $D33=−3.8 eV$ are the hydrostatic and shear deformation potentials for Ge_1−ySn_y alloys taken from Ref. 11 (significantly lower than for bulk Ge). The sign of $D33$ affects the intensities of the two critical points (CPs)²² (as discussed below) and therefore we follow the sign convention from Ref. 22. The band gaps of the relaxed alloys and other parameters were taken from the literature and are listed in Table I. As explained below, our own experimental values for these deformation potentials are slightly different, but agree within the errors, and therefore we used the parameters established in Ref. 11 for our calculations.

To avoid the complexity of the ternary Ge_1−x−ySi_xSn_y system with too many compositional parameters to be determined precisely, pseudomorphic Ge_1−ySn_y binary alloy layers on Ge were prepared by MBE to validate the theory described in Sec. II. These epilayers were grown using a modified EPI 620 MBE system with a base pressure of 1.3 × 10⁻⁸Pa and utilizing Knudsen thermal effusion cells with pyrolytic BN crucibles for both Ge and Sn. Intrinsic (001) Ge substrates were prepared by wet chemical cleaning⁴⁴ before quickly being loaded into the MBE introduction chamber and taken to ultrahigh vacuum. The substrates were slowly heated to 450 °C overnight, then taken to 650 °C for one hour before individually being transferred into the main MBE growth chamber. Each wafer was flash heated to 850 °C for 10 min, then cooled to the growth temperature between 150 and 250 °C (as measured via thermocouple) prior to opening the Ge and Sn cell shutters for growth. The Sn composition was varied by changing the Sn effusion cell temperature while keeping the Ge cell temperature constant across all growths, achieving a growth rate of 0.6 to 0.7 nm/min.

A PANalytical Empyrean diffractometer, operated at 45 kV and 40 mA, was used for the high resolution x-ray diffraction analysis of Ge_1−ySn_y epitaxial layers. Our high resolution configuration consists of a two-bounce Ge(220) hybrid monochromator, which offers a high-intensity well-collimated beam of monochromatic Cu Kα₁ radiation (1.5406 Å) and a three-bounce Ge (220) analyzer in front of the Xe proportional detector. We acquired symmetric (004) ω – 2θ diffraction curves and 2θ/ω – 2θ reciprocal space maps (RSMs) for symmetric (004) and asymmetric $(2¯2¯4)$ grazing exit reflections at room temperature to investigate the pseudomorphic nature, crystalline quality, lattice parameters, and thicknesses of the layers.

The optical properties of the Ge_1−ySn_y alloys were characterized using spectroscopic ellipsometry with two ellipsometers for different spectral ranges. The ellipsometric angles ψ and Δ were acquired from 0.5 to 6.6 eV with 0.01 eV steps at four angles of incidence (60°, 65°, 70°, 75°) on a J.A. Woollam vertical variable angle-of-incidence rotating-analyzer ellipsometer⁴⁵ with a computer-controlled Berek waveplate compensator, as described elsewhere.⁴⁶ To reduce experimental errors, all data were obtained by averaging two-zone measurements with equal and opposite polarizer angles. In the infrared spectral range from 250 to 7000 cm⁻¹, we measured on a Woollam FTIR-VASE ellipsometer, which is based on a fixed analyzer (0° and 180°) and polarizer (±45°) and a rotating compensator, at the same four angles of incidence. In the FTIR experiments, 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 scans per spectrum) to improve the signal to noise ratio. The ellipsometric angles (ψ and Δ) and the Fresnel reflectance ratio $ρ=eiΔ tan ψ$ are related to the pseudorefractive index $n̂$ and the pseudodielectric function $ϵ̂=n̂2$ of the sample through

where ϕ₀ is the angle of incidence and ϕ₁ the angle of refraction.

The dielectric functions $ϵ(ℏω)$ of pseudomorphic Ge_1−ySn_y alloys on Ge were obtained by analyzing the ellipsometric angles with a multilayer model (GeO₂/Ge_1−ySn_y/Ge). $ϵ(ℏω)$ for GeO₂ and Ge was used in tabulated form.⁴⁷ The dielectric functions of Ge_1−ySn_y were described with a parametric oscillator model,⁴⁸ which imposes Kramers–Kronig consistency between the real and imaginary parts of ϵ. In the first step of the fit, the oxide and Ge_1−ySn_y layer thicknesses and all parameters in the parametric oscillator model for Ge_1−ySn_y were adjusted. In the second step, all parameters were kept fixed at the values obtained in the first step, and the data were fitted again at each measured photon energy by taking the values of ϵ₁ and ϵ₂ for Ge_1−ySn_y as adjustable parameters (known as point-by-point fit) to obtain the final tabulated dielectric function of Ge_1−ySn_y. We confirmed that the results from the second step, shown in Fig. 4, were still Kramers-Kronig consistent. Redshifting of the dielectric function and broadened CPs with respect to bulk Ge with increasing Sn percentage indicates the alloying and strain effects of Sn on the Ge band structure.

Our main focus here was to investigate the compositional dependence of the CPs associated with the E₀, E₁, and E₁ + Δ₁ optical transitions. The E₁ and E₁ + Δ₁ CPs can be described using an expression for a mixture of a two-dimensional minimum and a saddle point^49,50

where $ℏω$ is the photon energy, A the amplitude, E_g the CP energy, Γ the broadening parameter, and ϕ the excitonic phase angle, which describes the amount of mixing. The contribution of the CPs to the dielectric function can be enhanced by analyzing the second derivatives of ϵ. The real and imaginary parts of the tabulated dielectric function obtained by point-by-point fitting were numerically differentiated and smoothed using ten Savitzky–Golay coefficients for second-order derivatives with a polynomial degree of three to obtain a good signal to noise ratio without distorting the line shape.⁵¹ The second derivative spectra of the dielectric function for the Ge_1−ySn_y were fitted using Eq. (19). Both E₁ and E₁ + Δ₁ were fitted simultaneously and the excitonic phase angle ϕ was forced to take the same value for both CPs.

The poor signal to noise ratio of the ellipsometric angles ψ and Δ obtained from FTIR ellipsometry between 0.1 and 0.9 eV makes the second derivative analysis of the tabulated dielectric function impractical. A different approach has to be followed to extract the compositional dependence of the E₀ band gap: We calculated the second derivatives of the parametric dielectric function as D'Costa et al.⁹ For a given sample, two data sets from 0.5 to 3 eV and from 0.1 to 0.9 eV, taken on two instruments under slightly different conditions (Fig. 5), were fitted simultaneously using a single model to obtain the parametric dielectric function of the Ge_1−ySn_y alloy layer, assuming two slightly different surface oxide thicknesses. This parametric dielectric function was then used for the second derivative line shape analysis and fitted with a three dimensional critical point⁹ 

High resolution x-ray diffraction was used to characterize the crystalline quality, composition, strain, and thickness of the Ge_1−ySn_y epilayers on Ge. ω – 2θ scans of symmetric (004) reflections showed two peaks, see the inset in Fig. 6(a). The sharp peak near 33.0° arises from the Ge substrate. The broader peak corresponding to the Ge_1−ySn_y epilayer is shifted to lower angles (away from the Ge peak) as the Sn composition increases, as a result of the increased out-of-plane lattice constant a_⊥ of the epilayer due to composition and biaxial compressive stress. The peak separation δθ is related to the ratio a_Ge/a_⊥. Assuming that the epilayer is not tilted relative to the Ge substrate, which we confirmed by (004) symmetric RSMs, a_⊥ can be calculated from Ref. 52: $aGe/a⊥=1+δθcotθ,$ where θ is the Bragg angle of bulk Ge. From the out-of-plane lattice constant a_⊥ and the in-plane lattice constant $a∥=aGe$ (pseudomorphic condition), we calculated the in-plane strain

and the relaxed lattice constant

from which the tin content y could be determined using Vegard's law, see Eq. (2) and the solid line in Fig. 7. The resulting value y is plotted along the horizontal axis in our figures showing experimental results and also listed in Table II.

All samples showed (004) Pendellösung fringes around the epilayer peak, indicating a uniform thickness of a high quality (coherent) Ge_1−ySn_y epilayer. The spacing between the fringes can be used to extract the thickness of the epilayer,^17,52,53 in good agreement with spectroscopic ellipsometry, see Table II.

Asymmetric $(2¯2¯4)$ grazing exit ω – 2θ/2θ RSMs, shown in Fig. 6(a) for a Ge_1−ySn_y sample with y = 0.096, were also acquired to extract the reciprocal lattice coordinates $Q∥$ and Q_⊥ along the [110] and [001] directions of the Ge_1−ySn_y epilayers and to investigate the degree of relaxation of the epilayer on Ge.⁵⁴ The Ge substrate and the epilayer peaks have the same $Q∥$⁠, indicating that the Ge_1−ySn_y epilayer is fully strained to the Ge substrate. Our layers are near the critical thickness reported by Wang et al.,⁵⁵ but nevertheless fully strained. The AFM characterization of the surface roughness for the same sample is shown in Fig. 6(b). The sample has an RMS roughness of 1.6 nm, indicating the reliability of the optical characterization techniques used in this paper.⁵⁶ 

The $Q∥$ and Q_⊥ maxima in the $(2¯2¯4)$ RSMs are related to the out-of-plane

and in-plane lattice parameters⁵⁷ 

which are shown by symbols in Fig. 7. The pseudomorphic condition $(a∥=aGe)$ is satisfied very well. Overall, the $(2¯2¯4)$ XRD results are consistent with the (004) XRD results, which we used to determine composition. Therefore, the relaxed lattice constants calculated from Eqs. (21) and (22) shown by symbols agree well with Vegard's law (solid line).

The sign and magnitude of the in-plane, out-of-plane, hydrostatic, and shear strain calculated from Eqs. (2)–(4), (6), and (7) are shown by lines in Fig. 8. Strains determined from the in-plane and out-of-plane lattice parameters measured by $(2¯2¯4)$ XRD are also shown (symbols). These pseudomorphic Ge_1−ySn_y alloys are described well by continuum elasticity theory with the elastic constants given by Eq. (5).

The measured Sn composition and thickness from (004) XRD and the in-plane strain $ϵ∥$ and out-of-plane strain ϵ_⊥ from $(2¯2¯4)$ RSMs for pseudomorphic Ge_1−ySn_y alloys on Ge samples are summarized in Table II.

The compositional dependence of the direct and indirect band gaps derived from the bands (in Fig. 3) is shown in Fig. 9 (lines). The E₀ band gaps measured by FTIR ellipsometry (•, ◻) are in excellent agreement with the theoretical predictions. The slight discrepancy with the values reported in Ref. 6 (+), where Ge_1−ySn_y is grown on a Ge buffer layer on Si, may be due in part to the tensile stress caused by the thermal expansion mismatch between the Ge buffer and the Si substrate.⁴⁰ The resulting strain affects the lattice parameter of the Ge buffer and reduces the compressive strain of the pseudomorphic Ge_1−ySn_y epilayer (dotted). This buffer also is very defective and cannot be described as a single layer with uniform optical constants.⁶ Finally, while our work used second derivatives to calculate critical point energies, Ref. 6 used a third derivative analysis, which is expected to yield a small systematic shift compared to our second-derivative results.⁵⁸ 

The compositional dependence of the E₁ and E₁ + Δ₁ energies of pseudomorphic Ge_1−ySn_y alloys on Ge, predicted from Eqs. (15) and (16), is shown in Fig. 10 (dashed). Both are blue shifted compared to relaxed Ge_1−ySn_y alloys (solid); the blue shift is larger for the E₁ + Δ₁ critical point (CP). Our experimental results from ellipsometry are shown by symbols and also listed in Table II. The agreement is good if the deformation potentials from Ref. 11 are used in the calculation (dashed), but it can be improved slightly with revised values of $D11=−5.1±0.3 eV$ and $D33=−4.5±0.1 eV$ (dotted lines in Fig. 10). Within the errors, our deformation potentials are the same as those in Ref. 11.

The small deviation of the E₁ CP of the 11% Sn sample from the prediction may be caused by a thickness interference fringe in the pseudodielectric function, which could be removed only partially by our data analysis techniques. It therefore influences the second derivative spectrum of the dielectric function used to determine the CP energies.

Our data analysis also determines the other CP parameters (amplitude, broadening, and phase angle), which are shown in Figs. 11, S6, and S7. The broadenings (see Fig. S6) increase nearly linearly with composition (with small quadratic corrections) due to alloy scattering and statistical or macroscopic compositional fluctuations, which is common in semiconductor alloys.^9,43,59 The broadenings for E₁ are only slightly smaller than for E₁ + Δ₁. The phase angle ϕ (see Fig. S7) decreases slightly with increasing Sn content, similar to Al_xGa_1−xAs alloys,⁴³ where a phase angle below 90 ° indicates a decreasing contribution of excitonic effects to the E₁ and E₁ + Δ₁ transitions due to alloy disorder.⁴³ 

Finally, the amplitudes of the critical points (shown in Fig. 11) show interesting trends. While the E₁ + Δ₁ amplitude decreases monotonically, the E₁ amplitude appears to first increase, and then decrease for y > 0.07. The same trends were also seen (but not discussed) in Ref. 5. The decrease of the E₁ amplitudes seen for larger Sn contents is attributed to the decreased enhancement of the transition by excitonic effects, as mentioned above. For low Sn contents, the relative change in amplitudes is driven by strain and given by²² $ΔA(ϵS)/A0=±6D33ϵS/Δ1$ from $k→·p→$ theory, where the – (+) sign is for the E₁ (E₁ + Δ₁) amplitude. For ϵ_S = 0.005 (compare Fig. 8) and $D33=−3.8 eV$ (note the sign), we indeed find a 25% increase in the E₁ amplitude for y = 0.07, in agreement with our data. The inline equation also predicts a more rapid initial decrease of the E₁ + Δ₁ amplitudes for small y, consistent with our data.

In summary, the compositional dependence of the band gaps of pseudomorphic Ge_1−x−ySi_xSn_y alloys on Ge and GaAs was calculated using deformation potential theory. We predict an indirect to direct band gap transition only for very large Sn compositions. Specifically, no indirect to direct cross-over can be achieved for pseudomorphic Ge_1−ySn_y on Ge or GaAs with practically approachable Sn (y < 25%) compositions.

Our predictions were confirmed for Ge_1−ySn_y alloys (x = 0) using spectroscopic ellipsometry. The complex pseudodielectric functions of pseudomorphic Ge_1−ySn_y alloys grown on Ge by MBE were measured using FTIR and UV-visible ellipsometry in the 0.1–6.6 eV energy range for Sn contents up to 11%, to investigate the compositional dependence of the band gaps. Critical point energies and related parameters were obtained by analyzing the second derivative spectrum of the dielectric function. The $E0Γ$⁠, E₁, and E₁ + Δ₁ band gaps of pseudomorphic Ge_1−ySn_y alloys measured from ellipsometry were in good agreement with the theoretical predictions from continuum-elasticity theory and deformation potentials. The broadenings and phase angles versus composition indicate a decrease in the importance of excitonic contributions to E₁ for larger tin contents. The critical-point amplitudes are dominated by strain effects for lower tin content and by the decrease of excitonic effects for y > 0.07.

This work was supported by the Air Force Office of Scientific Research (Nos. FA9550-13-1-00222 and FA9550-14-1-0207) and by the Army Research Office (Nos. W911NF-14-1-0072 and W911NF-12-1-0380). Support since 2016 has been provided by the National Science Foundation (No. DMR-1505172).