Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ ternary alloys are a candidate material system for use in solar cells and other optoelectronic devices. We report on the direct transition energies and structural properties of Ge-rich Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ alloys with six different compositions (up to 10% Si and 3% Sn), lattice-matched to Ge or GaAs substrates. The direct interband transitions occurring at energies between 0.9 and 5.0 eV were investigated using spectroscopic ellipsometry, and the resulting data were used to obtain the dielectric functions of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer by fitting a multilayer model. Values for the $ E 0 $, $ E 1 $, $ \Delta 1 $, $ E 0 \u2032 $, and $ E 2 $ transition energies were then found by identifying critical points in the dielectric functions. Structurally, the composition of the samples was measured using energy-dispersive x-ray measurements. The lattice constants predicted from these compositions are in good agreement with reciprocal space maps obtained through x-ray diffraction. The results confirm that a 1 eV absorption edge due to direct interband transitions can be achieved using relatively low Si and Sn fractions ( $<10$% and $<3$%, respectively), although the bandgap remains indirect and at lower energies. The higher-energy critical points show smaller shifts relative to Ge and match results previously observed or predicted in the literature.

## I. INTRODUCTION

Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ ternary alloys have been a topic of research interest for a variety of applications, including group IV lasers,^{1,2} photodiodes,^{3} light-emitting diodes (LEDs),^{4,5} and photovoltaics.^{4,6–9} By controlling the alloy composition, it is possible to tune the bandgap, higher-energy interband transitions, and the lattice constant. For photovoltaic applications, semiconductors with an absorption edge around 1 eV are an area of current research interest for use in multijunction solar cells with well-established material systems such as Ge, GaAs, and In $ 0.5 $Ga $ 0.5 $P. However, identifying materials that can be grown with sufficiently high material quality at the required bandgap and lattice constant is a challenge. In the industry-standard two-terminal three-junction solar cell using In $ 0.5 $Ga $ 0.5 $P, (In)GaAs, and Ge (in order of highest to lowest bandgap), the Ge junction produces significantly more current than the top two junctions; this excess current cannot be extracted, thus resulting in a loss. Despite this, Ge remains the standard choice for the bottom junction due to a good lattice-match with (In)GaAs and In $ 0.5 $Ga $ 0.5 $P and a lack of lattice-matched higher-bandgap materials which can be grown with the required material quality. Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ can be engineered to have the same lattice constant as Ge while having a higher bandgap and absorption edge and is, therefore, a suitable candidate material for these applications.

Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ alloys with a direct absorption edge around 1 eV have a high atomic fraction of Ge ( $x>$ 85%) and an atomic Si:Sn ratio of around 3.7:1, necessary to maintain the desired lattice constant according to Vegard’s law.^{10} The addition of Si to Ge increases the energy of both the fundamental bandgap (0.67 eV in pure Ge^{11}) and the lowest-energy direct transition (0.80 eV in pure Ge^{11}) but will reduce the lattice constant as Si has a smaller atomic radius than Ge. Conversely, the addition of Sn increases the lattice constant and reduces the fundamental bandgap. By balancing the relative amounts of Si and Sn in the alloy, it is possible to engineer a material, which has a higher bandgap and absorption edge than Ge but the same lattice constant, although this material is expected to retain an indirect bandgap lower in energy than the 1 eV transition; it is not possible to achieve a 1 eV direct fundamental bandgap.^{12–14} We have previously reported on the near-bandgap behavior of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ alloys, investigating both the indirect fundamental bandgap and the circa-1 eV direct transition through spectroscopic measurements and theoretical calculations.^{14} Here, we confirm the circa-1 eV direct absorption edge and investigate the higher-energy interband transitions for the same set of samples.

High-quality Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ alloys have been grown mainly, and most successfully, through chemical vapor deposition (CVD) and molecular beam epitaxy (MBE).^{15} Ultrahigh vacuum CVD (UHV-CVD) growth of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ was first reported by Bauer *et al.*,^{16} followed by the first reports of photoluminescence signal from Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ by Soref *et al.*^{17} These samples were grown using deuterated tin (SnD $ 4 $); this precursor was also used in the fabrication of the first demonstration of a multijunction cell with a Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ subcell.^{8} Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ has also been grown through both reduced-pressure CVD, using the commercially available precursor SnCl $ 4 $,^{18–20} including for multijunction solar cell applications,^{9} and MBE.^{21–23} This development of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ epitaxy has happened in conjunction with the epitaxy of direct-bandgap GeSn alloys for laser applications, mostly in heterostructures containing (Si)GeSn. For research applications, custom-built epitaxy reactors or highly customized commercial reactors are often used, in combination with different precursors for Si and Ge depending on the composition being grown.^{24} In addition to SnD $ 4 $, silicon and more complex germanium hydrides (Ge $ n $H $ 2 n + 2 $ and Si $ n $H $ 2 n + 2 $) with $n>1$ are commonly used, which are more expensive than the common commercial precursors. The samples investigated here were grown using a commercial CVD reactor and precursor materials, with the aim of developing a Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ growth process which reduces the complexity and cost associated with multijunction cell growth. To study the direct interband transitions that dominate the dielectric function of these materials, spectroscopic ellipsometry (SE) was performed over the wavelength range 250–1800 nm (0.69–4.96 eV). The composition of the samples was measured through energy-dispersive x-ray (EDX) spectroscopy using a scanning electron microscope (SEM) while structural features were studied using x-ray diffraction (XRD) and optical microscopy. These results confirm that with these growth methods, it is possible to fabricate material with a 1 eV absorption edge, significantly blueshifted from pure Ge, on Ge or GaAs substrates, providing an important step toward successfully fabricating a multijunction solar cell incorporating Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $.

## II. METHODS

### A. Sample growth

Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ samples were grown through low-pressure CVD (chemical vapor deposition) in an ASM Epsilon 2000 CVD epitaxy reactor. Germane (GeH $ 4 $), disilane (Si $ 2 $H $ 6 $), and tin chloride (SnCl $ 4 $) precursors were used, with H $ 2 $ as the carrier gas. The growth temperatures for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ were in the range 330–480 $ \xb0 $C and were kept constant throughout the growth of each layer. The Si and Sn source flows were adjusted to obtain lattice-matched growth at different compositions. Growth rates were in the range of 200–300 nm/min. The relatively low growth temperature lowers the cracking efficiency of the germane precursor, resulting in a high flow rate; this has a significant cost implication, and thus while this growth method was used as a proof of concept, it is likely that for commercial applications, different growth strategies would need to be considered.

Two sets of samples across the composition range are considered here: one set grown on GaAs substrates with a thin ( $\u224860$ nm) Ge seed layer, which will be referred to as set A, and one set grown on Ge substrates which were overgrown with approximately 500 nm of lattice-matched In $ 0.012 $Ga $ 0.988 $As grown through metal-organic vapor-phase epitaxy (MOVPE), referred to as set B. The layer structure of the samples is shown in Fig. 1. Samples within both of these sets were grown at three different compositions, all aiming for an approximately 3.7:1 ratio of Si:Sn to achieve the same lattice constant as Ge, assuming the lattice constant of the alloy obeys Vegard’s law.^{10,23,25} For set B, two different Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ thicknesses were grown for each composition, giving a total of three samples in set A and six samples in set B. The resulting sample structures, compositions, layer thicknesses, and the labels used for the samples are summarized in Table I and Fig. 2.

Sample . | d_{SiGeSn} (nm)
. | Nonuniformity (%) . | Si (atomic %) . | Sn (atomic %) . | a_{SiGeSn} (Å)
. | a_{SiGeSn,XRD} (Å)
. |
---|---|---|---|---|---|---|

A1 | 1779.6 (0.2) | 1.1 | 4.6 (0.2) | 0.9 (0.1) | 5.65(4) | 5.6537 |

A2 | 1850.0 (0.3) | 1.2 | 6.8 (0.5) | 1.6 (0.1) | 5.65(5) | 5.6553 |

A3 | 1886.5 (0.4) | 1.7 | 9.6 (0.4) | 2.2 (0.1) | 5.65(4) | 5.6533 |

B1 (thin) | 334.3 (0.2) | 2.5 | n.m. | n.m. | 5.65(8) | n.m. |

B2 (thin) | 365.4 (0.2) | 2.4 | n.m. | n.m. | 5.65(7) | n.m. |

B3 (thin) | 364.1 (0.2) | 2.6 | n.m. | n.m. | 5.65(7) | n.m. |

B1 (thick) | 2040.4 (0.6) | 1.6 | 2.6 (0.3) | 0.8 (0.2) | 5.65(8) | n.m. |

B2 (thick) | 2112.3 (0.6) | 1.6 | 5.9 (0.2) | 1.5 (0.1) | 5.65(7) | n.m. |

B3 (thick) | 2073.9 (0.8) | 1.5 | 8.4 (0.2) | 2.5 (0.1) | 5.65(7) | n.m. |

Sample . | d_{SiGeSn} (nm)
. | Nonuniformity (%) . | Si (atomic %) . | Sn (atomic %) . | a_{SiGeSn} (Å)
. | a_{SiGeSn,XRD} (Å)
. |
---|---|---|---|---|---|---|

A1 | 1779.6 (0.2) | 1.1 | 4.6 (0.2) | 0.9 (0.1) | 5.65(4) | 5.6537 |

A2 | 1850.0 (0.3) | 1.2 | 6.8 (0.5) | 1.6 (0.1) | 5.65(5) | 5.6553 |

A3 | 1886.5 (0.4) | 1.7 | 9.6 (0.4) | 2.2 (0.1) | 5.65(4) | 5.6533 |

B1 (thin) | 334.3 (0.2) | 2.5 | n.m. | n.m. | 5.65(8) | n.m. |

B2 (thin) | 365.4 (0.2) | 2.4 | n.m. | n.m. | 5.65(7) | n.m. |

B3 (thin) | 364.1 (0.2) | 2.6 | n.m. | n.m. | 5.65(7) | n.m. |

B1 (thick) | 2040.4 (0.6) | 1.6 | 2.6 (0.3) | 0.8 (0.2) | 5.65(8) | n.m. |

B2 (thick) | 2112.3 (0.6) | 1.6 | 5.9 (0.2) | 1.5 (0.1) | 5.65(7) | n.m. |

B3 (thick) | 2073.9 (0.8) | 1.5 | 8.4 (0.2) | 2.5 (0.1) | 5.65(7) | n.m. |

### B. Structural characterization

The samples’ surface morphology was examined by taking optical images using an Olympus BX51 Microscope equipped with polarizer, rotatable analyzer, and Nomarski prism taken at 100 $\xd7$ magnification. Compositions were obtained from EDX measurements performed using a JEOL JSM-6700F Field-emission Scanning Electron microscope equipped with Oxford instruments X-Max 150 mm $ 2 $ taken at an accelerating voltage of 15 kV. The sample composition was measured at ten points on the surface of each sample (averages and standard deviations are reported in Table I); measurements of the same samples during different measurement sessions showed these results to be repeatable. The crystalline quality and lattice constant of the samples were investigated using a Malvern-Panalytical Empyrean x-ray diffractometer by measuring the symmetrical (004) and asymmetrical (224) rocking curves and reciprocal space maps (RSMs).

### C. Spectroscopic ellipsometry

$\Psi $ quantifies the magnitude of the ratio and lies in the range 0 $ \xb0 $–90 $ \xb0 $, while $\Delta $ is the phase and lies in the range 0 $ \xb0 $– $ 180 \xb0 $.

For a sample with multiple layers, it is generally necessary to fit a multilayer model to the data which accounts for reflection at the front surface and each interface, and the resulting interference at each wavelength. A parametric Herzinger–Johs^{26} model is used to fit the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ optical constants. Such a parametric model has the advantages of reducing the number of fitting parameters (compared to fitting the $ \epsilon 1 $ and $ \epsilon 2 $ values point-by-point at each measured wavelength) and enforcing Kramers–Kronig consistency,^{26,27} while increasing the flexibility of the peak shapes compared to a more physically constrained model such as a Critical Point Parabolic Band (CPPB) model.^{28,29} The drawback compared to a CPPB model is that due to the flexible peak shape; the parameters drawn from fitting a parametric model are not necessarily physically meaningful; e.g., the center energy of a peak may not correspond directly to a specific transition energy in the material’s band structure. For this reason, rather than obtaining the transition energies directly from the parametric model, the dielectric function of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ is differentiated in order to obtain the relevant center energies of the higher-energy (2–5 eV) transitions, as described below. The lowest-energy direct absorption edge (0.8 eV for Ge, up to $\u22481$ eV for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ samples studied) was represented using the Tanguy model for the Hulthén potential,^{30} which was previously used to fit the absorption edge of Ge;^{31} the $ E 0 $ values can thus be obtained directly from the fit to the ellipsometry data.

The structure of the different sample types, used as a starting point for these models, is shown in Fig. 1, comprising the GaAs (set A) or Ge (set B) substrate, the Ge (set A) or In $ 0.012 $Ga $ 0.988 $As (set B) intermediate layer, and the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer of interest. A surface roughness layer (modeled in the WVASE using the Bruggeman effective medium approximation,^{27} with 50% air and 50% Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $) was also included. This surface roughness layer (found to be 1–2 nm thick for all the samples) is necessary to achieve a good fit to the data, which indicates the presence of a lower-index surface layer. A similar method to the fitting procedure for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ optical constants outlined in Ref. 32 is used. The starting points for the layer thicknesses are the nominal thicknesses targeted during CVD growth. Initially, the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer is assumed to be pure Ge; since we are considering Ge-heavy ( $ > 85$%) samples of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $, using the well-known optical constants of Ge as a placeholder for Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ gives a good starting point for finding the layer thicknesses. For the set B samples, the InGaAs buffer layer thickness is also allowed to vary at this stage; in each case, the deviation from the nominal thickness of 500 nm was found to be $\u226410$ nm. The starting point for the Tanguy/Herzinger–Johs model fits for Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ is the built-in Herzinger–Johs model for Ge^{27} in the WVASE software, with the lowest-energy absorption edge instead described using a Tanguy oscillator rather. The Herzinger–Johs model allows for very flexible peak shapes; each feature is described by seven variable parameters that control the peak’s strength, center energy, and broadening, with further parameters controlling the shape and asymmetry of the peak. Here, only the peak strength ( $ A i $), center energy ( $ E i $), and broadening ( $ B i $) are allowed to vary, with the other parameters kept fixed at the Ge values. This is to avoid unreasonable peak shapes, including some with discontinuities in the dielectric function, especially in the low-energy region where thin-film interference and noise due to signal depolarization affect the SE measurements.^{33} It is found that excellent fits to the data could be achieved without varying the other peak shape parameters. The parameters for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ optical constants are fitted starting at high energies, initially allowing only the center energies of the peaks to vary, followed by the peak strength and broadening. The layer thickness of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ is allowed to continue to vary during the optical constant fits. The data measured at all the incidence angles (75 $ \xb0 $, 77 $ \xb0 $, and 79 $ \xb0 $) are used simultaneously in the fitting procedure. For the set A samples, where the depolarization at long wavelengths is high ( $ > 10$% at some wavelengths) due to the GaAs substrate becoming transparent, the effect of backside reflections is included in the model. For set B, the depolarization was low over the whole wavelength range, and depolarization was not included. For both sample sets, thickness nonuniformity of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer was included; this has the effect of smoothing out the interference fringes and is necessary to obtain a good fit to the data. The optical constants for the Ge, InGaAs, and GaAs came from the WVASE software database and are shown in the supplementary material.^{34}

After the layer thicknesses have been determined exactly, and a parametric model for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ optical constants has been obtained as described above, a point-by-point fit where $ \epsilon 1 $ and $ \epsilon 2 $ are allowed to vary freely at each measurement wavelength is also performed, keeping the layer thicknesses fixed. The two methods agree very well for energies above $\u22481$ eV. Below these energies (i.e., below the lowest direct transition in the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $, where Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ is only very weakly absorbing), the presence of thin-film interference fringes in the data significantly affects the point-by-point fits, as shown in the supplementary material.^{34}

The fundamental indirect gap (0.67 eV for Ge at room temperature and blueshifted to 0.7–0.8 eV for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ samples, as determined by photoluminescence measurements^{14}) is not included in the Herzinger–Johs model. Even considering the expected blueshift in this bandgap relative to Ge for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ samples, this transition occurs very close to or beyond the longest wavelength which can be measured using the V-VASE ellipsometer and is expected to contribute so weakly to the dielectric function that it would be extremely difficult to observe through ellipsometry. This transition was investigated through photoluminescence measurements of the same set of samples as reported in Ref. 14.

### D. Critical point fits

^{11}attributes the $ E 1 $ and $ E 1 + \Delta 1 $ features to a 3D M1-type CP, this feature has been treated as a 2D CP by others

^{28,35}and it was found that using the expression for a 2D CP resulted in better fits. What is treated here as a single $ E 0 \u2032 $ feature around 3 eV is likely made up of contributions from $ E 0 \u2032 $, $ E 0 \u2032 + \Delta 0 \u2032 $ and $ E 0 \u2032 + \Delta 0 \u2032 + \Delta 0 $, but these are not clearly resolved and are thus treated as a single peak.

^{28}Note that these expressions for the behavior of $ d 2 ( E 2 \epsilon ) d E 2 $ around the critical point assume parabolic bands. The second derivative of the real and imaginary parts of the dielectric function was calculated from the results of the point-by-point fit. The choice of this differential (specifically the choice to differentiate the quantity $ E 2 \epsilon $ rather than the more common choice of simply differentiating $\epsilon $) and how the differentiation is performed computationally are discussed in the supplementary material.

^{34}

For the lowest-energy direct transition, $ E 0 $, no fit to the second derivative was performed; the point-by-point fit at low energies contains artifacts from the interference fringes; additionally, the set A samples have high measurement error at low energies due to backside reflections from the substrate. Because the $ E 0 $ transition was modeled using a Tanguy oscillator in the ellipsometry data fits, as discussed above, we instead report the bandgap obtained from these fits directly.

Due to the relative weakness of this transition, and the aforementioned issues with interference fringes and measurement error, we did not evaluate $ E 0 + \Delta 0 $, although it is visible in the second derivatives of the dielectric function shown in Fig. 7. It was evaluated from photoreflectance measurements of the same samples in Ref. 14.

## III. RESULTS

### A. Structural measurements

The composition and lattice constant of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ samples were characterized using SEM-EDX and XRD (rocking curve and reciprocal space mapping) measurements, clearly showing the increasing Si and Sn compositions within the sample sets. Table I gives the composition obtained through EDX measurements, and the predicted lattice constant through Vegard’s law and from the XRD measurements (for set A only). Figure 2 shows the Si and Sn fractions of each sample; all samples lie between a 4.6:1 and 3.2:1 Si:Sn ratio, with the set B samples having a lower ratio than the set A samples, meaning the set B samples are more closely lattice-matched to their Ge substrates, while the set A samples are more closely lattice-matched to their GaAs substrates. For all the samples, the thickness of the epitaxial Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer is expected to be below the critical thickness,^{43} as shown in the supplementary material.^{34}

Surface imaging shows the presence of defects, but no cross-hatching or features that indicate relaxation of the epitaxial layer. For set B, only the thicker samples ( $\u22482000$ nm) were measured in these structural measurements, and it was assumed that the corresponding thinner ( $\u2248400$ nm) Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layers have very similar compositions; this is corroborated by the consistency of the dielectric function across the different thicknesses, as discussed in Sec. III B. The surfaces of both sets of samples show “bubble” and “pyramid”-like defects, as shown in Figs. 4(a)–4(c). Given the known difficulties of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ growth, especially regarding Sn incorporation, it is reasonable to assume these defects may be related to segregation of Sn during growth; however, additional EDX measurements comparing flat areas of the surface with the defect features do not indicate a measurably different composition within either of these feature types. The pyramid-type defects penetrate into the bulk of the material significantly and appear to be the results of growth along a different crystal axis to the bulk film, although further investigation is required.

Rocking curves and RSMs were measured around the (004) and (224) lattice points. Figures 4(d) and 4(e) show the RSM scans for sample A1; RSM data for the other set A samples around both points are given in the supplementary material.^{34} Only the out-of-plane lattice constant can be observed in the (004) scans, while the asymmetric (224) scan will contain contributions from both the in-plane and out-of-plane lattice constants. For samples A1 and A3, some spread around a single RSM peak is observed in both the (004) and (224) RSMs, but only a single clearly defined peak is visible, indicating that the epi-layer is very closely lattice-matched to the substrate. The peak broadening in reciprocal space at constant $ Q x $ in the (224) RSMs also indicates pseudomorphic (constant lattice constant) growth, with the in-plane lattice parameter strained to the substrate (GaAs) lattice constant. For sample A2, a clear split in the peak in the (004) RSM is observed (see supplementary material^{34}), showing that the epi-layer has a different out-of-plane lattice constant than the substrate. The XRD peaks for sample A2 correspond to lattice constants of 5.6537 and 5.6567 Å, with the former value being the GaAs lattice constant. This indicates that the relaxed lattice constant of sample A2 is larger than that of GaAs, causing in-plane compressive strain in the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ and expansion of the out-of-plane lattice constant. Assuming the in-plane lattice constant is pinned to that of GaAs, as indicated by the RSM in the (224) direction, and taking the Poisson ratio^{36} of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ to be 0.3 (a typical value used for Ge^{37}), the relaxed lattice constant is calculated to be 5.6553 Å, in excellent agreement with the value of 5.655 Å calculated through Vegard’s law using the composition measurements from SEM-EDX (see Table I). Sample A2 was the set A sample with the largest predicted deviation from the GaAs lattice constant through Vegard’s law, and the only sample for which a clear second peak in the RSM can be observed. This indicates that the samples are closely lattice-matched to the substrates, with some strain, and the SEM-EDX composition measurements and XRD give consistent results.

### B. Ellipsometry

The ellipsometry data for the set B samples, and the result of the model fit, are shown in Fig. 5. The Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ thickness and the thickness nonuniformity percentage determined from the multilayer model fit are given in Table I. In all cases, the model converged well to a single value of the thickness; an example of a uniqueness fit for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ thickness is shown in the supplementary material,^{34} in addition to all the measured data and fit results for the set A samples, and Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ model parameters.^{34} Tabulated SE data and resulting fits of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ optical constants are also available in Ref. 38. The ellipsometry data for all samples show clear effects due to thin-film interference in the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer; while the higher-energy transitions (2 eV and above) are clearly visible in the raw data, the near-bandgap data are dominated by interference fringes that depend on the thickness of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer. This can be seen clearly by comparing the data in Fig. 5 from the samples with the same composition but different thicknesses, with the thinner samples showing much wider interference fringes.

For the set B samples, the optical constants of the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ are obtained only through fitting to the data from the thicker samples. To fit the SE data of the thinner samples, only the layer thicknesses are allowed to vary. This allowed for a validation of the optical constant fits and a check that the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layer has the same composition for the different thicknesses; the models for the thinner set B samples show excellent agreement with the data despite only the layer thicknesses being allowed to change as shown in Figs. 5(d)–5(f). Figure 6 shows the dielectric function for each Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ composition in set A and set B, showing how the absorption edge and lower-energy critical points ( $<2.5$ eV) blueshift with increasing Si/Sn composition while the higher-energy critical points redshift. The dielectric functions shown in Fig. 6 are for the parametric model fit for Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $; the point-by-point fit is extremely similar above $\u22481.5$ eV but is more noisy at low energies. A comparison between the two fits is shown in the supplemental material.^{34} Figure 7 shows the real and imaginary parts of the dielectric function for the Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ layers of the set A samples and their first and second derivatives. Through fitting Eq. (2), the exact location of the transition energies can be extracted from these derivatives; these sharp features are clearly visible in the second derivative shown in Fig. 7.

### C. Critical point fitting

^{34}). The results for the critical point and Tanguy model fits are summarized in Table II and shown in Fig. 9. Figure 9 also shows the predicted values of $ E 0 $, $ E 1 $, and $ E 2 $ at the measured compositions, calculated through equations of the form

^{44}

^{,}$ E Ge $ was fixed in each case at the values obtained in our measurements, with values for $ E Si $ and $ E Sn $ from the literature.

^{39,40}The bowing parameters used were from Ref. 13 for $ E 0 $ and Ref. 35 for $ E 1 $. For $ E 2 $, a linear interpolation was used as no previous reports of the bowing parameter were found. All the end point energies and bowing parameters are listed in the supplementary material.

^{34}For $ E 0 $, we see excellent agreement between the calculated and measured values, with deviations of $<3$% from the calculated theoretical values across the composition range. The theoretical values are slightly lower than the measured values, but follow the same trend. For $ E 1 $, the agreement is also excellent, and the deviation from calculated values is $<1$%. Note that we did not make a theoretical comparison for $ E 0 \u2032 $; as discussed above, this does not correspond to a single clearly defined critical point, and thus values reported across the literature vary significantly.

Comp. . | E_{0}
. | E_{1}
. | E_{1} + Δ_{1}
. | E_{0}′
. | E_{2}
. |
---|---|---|---|---|---|

Ge | 0.801 | 2.093 | 2.300 | 3.044 | 4.349 |

A1 | 0.925 | 2.123 | 2.331 | 3.041 | 4.340 |

A2 | 0.966 | 2.132 | 2.338 | 3.031 | 4.328 |

A3 | 1.015 | 2.145 | 2.347 | 3.025 | 4.322 |

B1 | 0.870 | 2.106 | 2.311 | 3.047 | 4.338 |

B2 | 0.933 | 2.125 | 2.327 | 3.033 | 4.331 |

B3 | 0.980 | 2.137 | 2.338 | 3.030 | 4.321 |

Comp. . | E_{0}
. | E_{1}
. | E_{1} + Δ_{1}
. | E_{0}′
. | E_{2}
. |
---|---|---|---|---|---|

Ge | 0.801 | 2.093 | 2.300 | 3.044 | 4.349 |

A1 | 0.925 | 2.123 | 2.331 | 3.041 | 4.340 |

A2 | 0.966 | 2.132 | 2.338 | 3.031 | 4.328 |

A3 | 1.015 | 2.145 | 2.347 | 3.025 | 4.322 |

B1 | 0.870 | 2.106 | 2.311 | 3.047 | 4.338 |

B2 | 0.933 | 2.125 | 2.327 | 3.033 | 4.331 |

B3 | 0.980 | 2.137 | 2.338 | 3.030 | 4.321 |

As expected, the lowest direct interband transition energy $ E 0 $ of Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ shifts to higher energies as the Si and Sn composition increases. The values obtained here from fitting SE data are in good agreement with the results of photoreflectance measurements of the same samples reported in Ref. 14; the deviation in the $ E 0 $ values between the two measurements is $<4$% for all samples. A 1 eV $ E 0 $ transition can be achieved with $<10$% Si and $<3$% Sn with a Si:Sn ratio of around 4:1, while maintaining a good lattice match to Ge/In $ 0.012 $Ga $ 0.988 $As. The trend in $ E 0 $ shows an effect from higher relative Sn composition in the set B samples: these samples show less blueshift than the set A samples, as can be seen in Fig. 9. The $ E 1 $ and $ E 1 + \Delta 1 $ energies, around 2.2 and 2.3 eV respectively, also blueshift with increased Si/Sn composition as expected.^{24,41} Meanwhile, $ E 0 \u2032 $ and $ E 2 $ redshift very slightly (Fig. 9), each by less than 30 meV between Ge and the highest-composition sample (A3), as does the split-off energy $ \Delta 1 $ though by only 4 meV. The reduction in $ E 0 \u2032 $ matches previously reported results.^{24} Although different sources report different values for the $ E 0 \u2032 $ energies of Ge and Si,^{11} the values for Si are generally slightly higher by around 0.2 eV, yet we observe a slight redshift here; this could be due to the presence of $\alpha $-Sn (which has a lower $ E 0 $ by $\u22481$ eV) outweighing the effect of the Si, or because what is treated here as a single transition is, in fact, made up of at least two contributing critical points. This can be seen in Figs. 7(e) and 7(f), where there appears to be a second weaker feature visible around 3.4–3.5 eV. The reduction in $ E 2 $ is also consistent with previous literature results,^{24,42} and expected from the lower $ E 2 $ energy of both Si and $\alpha $-Sn compared to Ge. Note that we did not extract values of $ E 0 + \Delta 0 $ from fits to the ellipsometry data; this contribution is relatively much weaker than that of $ E 0 $, which is already much weaker than the higher-energy transitions. It was not possible to obtain a reliable, unique fit to this critical point, so it was excluded from the second-derivative fits. Photoreflectance measurements of the same samples indicate that $ \Delta 0 $ stays almost constant.^{14}

## IV. CONCLUSIONS

Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ is a promising candidate material for solar cells and other optoelectronic applications, with previous experimental and theoretical results indicating that a direct transition energy at 1 eV can be achieved at atomic compositions $<10$% Si and $<3$% Sn, with the resulting alloy being lattice-matched to Ge or GaAs. The results of ellipsometry measurements presented here confirm this, showing $ E 0 $ energies between 0.98 and 1.02 eV for alloys with compositions of 6.8–9.6% Si and 1.6–2.5% Sn. The higher-energy (up to 5 eV) direct transitions at the $\Gamma $, $X$, and $L$ points in the band structure were also investigated, showing a blueshift in $ E 1 $ and small redshifts in $ \Delta 1 $, $ E 0 \u2032 $, and $ E 2 $, in good agreement with previously published results.

At these compositions, Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ retains a fundamentally indirect “Ge-like” bandgap. This indirect transition was studied for this set of samples using photoluminescence, showing bandgaps in the range 0.7–0.75 eV.^{14} While this lower-lying indirect transition is expected to affect the voltage of a solar cell adversely, Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ will have a higher fundamental bandgap and thus voltage than Ge [if used instead of Ge in, e.g., a triple-junction device with InGaP and (In)GaAs]. In a four-junction InGaP/(In)GaAs/SiGeSn/Ge device, where current-matching is important, absorption across a Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ thin film will be dominated by the lowest-energy direct interband transitions, rather than the indirect bandgap, and the additional Si $ x $Ge $ 1 \u2212 x \u2212 y $Sn $ y $ junction will still provide a voltage boost compared to the standard triple-junction architecture. Thus, the 1 eV direct absorption edge is still expected to provide benefits in terms of current-matching, and the higher indirect gap compared to Ge can provide an increase in voltage.

## ACKNOWLEDGMENTS

This work was supported by the Engineering and Physical Sciences Research Council, UK (EPSRC; via a CASE Studentship, held by P.P. and sponsored by IQE plc.) and by the Royal Society (via an Industry Fellowship, held by N.J.E.-D.). We would like to thank Aled Morgan (IQE plc.) for contributing additional information on sample growth.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Phoebe M. Pearce:** Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Sheau Wei Ong:** Data curation (supporting); Formal analysis (supporting); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Andrew D. Johnson:** Conceptualization (equal); Funding acquisition (equal); Methodology (supporting); Project administration (supporting); Resources (equal); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Eng Soon Tok:** Funding acquisition (supporting); Methodology (equal); Resources (supporting); Writing – original draft (supporting). **Nicholas J. Ekins-Daukes:** Conceptualization (equal); Data curation (supporting); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The spectroscopic ellipsometry data that support the findings of this study are openly available in Zenodo, Ref. 38 and other data are available from the corresponding author upon reasonable request.

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