1. Introduction

Knowledge of the fundamental bandgaps of group IV alloys, both direct and indirect, is important for photonic and electronic applications. However, those gaps are not known at present for the majority of alloys. One way to find the gaps is to determine the shapes of valence and conduction energy-bands in momentum-space. However, this first-principles band-structure calculation is complicated: the numerical simulations required are both sophisticated and time consuming. So the question arises as to whether there is a simple (but not simplistic) way to estimate the direct and indirect gaps of the various alloys. The answer given here is that a rudimentary study of six binary alloys can shed considerable light upon the properties of the ternary and quaternary alloys.

The quaternary alloy CSiGeSn is the ultimate group-IV alloy and the four ternary alloys CSiGe, CSiSn, CGeSn, SiGeSn, together with the six binary alloys CSi, CGe, CSn, SiGe, SiSn, GeSn, are subsets of the quaternary. Epitaxial growth of semiconductor heterostructure devices constructed from these alloys is motivated by their considerable potential in photonic and electronic technologies. The goal is to integrate the group IV components on a silicon substrate; thereby to manufacture cost-effective group-IV opto-electronic integrated circuits in a high-volume silicon foundry. Although strained-layer devices are indeed feasible (and desireable in some cases), another objective is to grow unstrained group-IV alloy heterostructures that are lattice matched to a virtual substrate (VS) upon silicon. This VS would consist of one or more strain-relaxed buffer layers on Si, such as Ge and/or GeSn layers.

In the above-mentioned device-alloy heterostructures, the active layers should preferably have a “truly direct” bandgap in order to maximize the device performance, especially in band-to-band laser diodes, LEDs, amplifiers, photodetectors and modulators. There is ample experimental evidence in the literature that this directness is achievable in GeSn and SiGeSn materials, albeit at mid-infrared (MIR) wavelengths that are longer than those in the 1.55 μm telecoms band.

2. Approach

This paper goes beyond direct SiGeSn [1] to investigate all of the other possible IV alloys over their full composition range. In other words, the undecided question is whether there are additional alloy compositions–binary, ternary and quaternary–for which the fundamental bandgap is direct. This paper attempts to answer that question by estimating the direct and indirect gaps of CSi, CGe, CSn, SiGe, SiSn and GeSn using a simple linear-interpolation method. Applying those results to quaternary space, it is found that directness is available in principle over −0.41 eV ≤ E_g ≤ 1.5 eV (setting aside the real world issues of solubility, segregation, and stability in the actual epitaxy of these crystalline alloys).

In the binary AB comprised of elements A and B, its bandgap is taken as a weighted average of the A and B gaps, where the weights are the A and B fractions. However, experiments on GeSn show that this linear-interpolation procedure is only a rough estimate because the observed bandgaps exhibit a nonlinear composition dependence. The direct gap has a “negative bowing” in which the gap energy is lower than that given by the linear approximation. Nevertheless, the linear method gives a guideline on what to expect in practice. It shows in an approximate way the composition region for which the alloy is truly direct.

A framework for the quaternary is illustrated in Fig. 1 where we have plotted the three lowest-energy indirect bandgaps and the lowest-energy direct bandgap of the four group-IV elements, C, Si, Ge and Sn.

 

Fig. 1 Fundamental bandgaps of group IV elements as a function of lattice size for unstrained, bulk crystal.

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Here the cubic phase of diamond was used along with the alpha-phase of tin. This bandgap-versus-lattice parameter diagram is for the bulk, unstrained, crystalline elements. The top of the degenerate LH, HH valence band is taken as a reference, and the energy of the k-space conduction-band minima at Γ, L, Δ, and X are shown. The gap energies E_g and lattice constants a cover wide ranges. Going into more detail about Fig. 1: for C the gaps are E_g(Γ) = 7.35 eV [2], E_g(L) = 5.6 eV [2], E_g(Δ) ≅ 6.2 eV [3, Fig. 3], E_g(X) = 6.4 eV [2]; for Si, E_g(Γ) = 4.1 eV [2], E_g(L) = 2.0 eV [2], E_g(Δ) = 1.12 eV [2], E_g(X) ≅ 1.17 eV [2]; for Ge, E_g(Γ) = 0.8 eV [2], E_g(L) = 0.66 eV [2], E_g(Δ) = 0.85 eV [2], E_g(X) = 1.2 eV [2]; and for grey tin the s-like Γ₇^- CB edge is situated in energy below the p-like Γ₈⁺ VB edge; namely E_g(Γ_7c) = −0.41 eV [4], together with E_g(Δ_5c) ≅ 0.85 eV [5], E_g(X_5c) = 0.90 eV [4] and E_g(L_6c) = 0.14 eV [4]. It is necessary to include both the Δ and X properties for the following reasons. In Si, the fundamental gap is at Δ, not at X. Similarly, in Ge and Sn, the indirect gap at Δ is lower in energy than the X gap. In Fig. 1, the 4.1 eV direct gap of Si has been utilized because it correlates with the 0.8 eV direct gap of Ge, whereas the 3.4 eV direct gap of Si (where two bands cross at k = 0) is believed to correspond to the second direct gap of Ge at 3.2 eV.

It is not sufficient in Fig. 1 to connect the Γ-points with line segments to determine quaternary directness. What is needed is the composition space wherein the Γ-gaps are narrower than the L, Δ and X gaps. To find that regime, we examined E_g-versus-composition for all six binaries. After examining the E_g-vs-a plots of CSi, CGe, CSn, SiGe, GeSn and SiSn, looking there for regions in which the Γ-line was lower in energy than the L, Δ and X lines, the result was that the alloys CSi, CGe and SiGe were never direct and only the Sn-containing alloys CSn, SiSn and GeSn had direct compositions. Such directness is shown in Fig. 2, Fig. 3 and Fig. 4. In these Figures, Vegard’s law is used to find the composition of the alloy that corresponds to a particular lattice size on the horizontal axis.

 

Fig. 2 Fundamental E_g-versus-a for C_1-xSn_x using the linear-theory method. Dashed line shows bowing at Γ.

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Fig. 3 Fundamental E_g-versus-a for Si_1-ySn_y using the linear-theory method. Dashed line shows bowing at Γ.

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Fig. 4 Fundamental E_g-versus-a for Ge_1-zSn_z using the linear-theory method. Dashed line shows bowing at Γ.

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3. Results and Discussion

By examining the line crossings in Figs. 2, 3 and 4, respectively, we can determine the energy E_c and the lattice constant a_c at which the indirect-to-direct bandgap crossover occurs, as follows: E_c = 1.5 eV, a_c = 0.576 nm for the crossover composition C(25)Sn(75); E_c = 0.64 eV, a_c = 0.622 nm for Si(25.5)Sn(74.5); and E_c = 0.55 eV, a_c = 0.582 nm for Ge(81.5)Sn(19.5). In all three of these binary cases, directness continues as the Sn content is raised above the crossover Sn content and is increased towards 100%. First-principles theory for CSn [6] and SiSn [6] and experimental evidence for GeSn [7,8] indicate that there is in fact bowing in the E_g(Γ)-vs-a curves. We have plotted in Figs. 2, 3, and 4 the E_g(Γ) bowing taken from the literature cited.

The E_g-bowing theory does give a more accurate prediction of E_c for CSn, SiSn and GeSn. However, that theory requires knowledge of the E_g(L) and E_g(X) bowing curves. The virtual-crystal approximation and the mixed-atom theory disagree on the extent of E_g(L) and E_g(X) bowing in SiSn [6], and those bowings are not known for CSn. In the absence of trustworthy information on L and X bowing in CSn and SiSn, L-X bowing has not been drawn in Figs. 2-4. Nevertheless, its qualitative effects are clear: the above-listed a_c will shift to smaller lattice values and the fraction of Sn in each E_c alloy will be smaller; whereas the E_c found from bowing will be similar to those in the linear case.

Returning now to the linear cases, we take the “directness ranges” for CSn, SiSn and GeSn in Figs. 2-4 and plot those on the same E_g-vs-a diagram in Fig. 5 where the E_c are three of the four “vertices.” The lines joining the vertices in Fig. 5 determine a four-sidedcomposition zone that indicates (approximately) where the quaternary alloy CSiGeSn will be truly direct. The upper two boundaries show where the L valley gives way to the Γ valley. For C_1-x-y-zSi_xGe_ySn_z, this diagram locates directness in the region (shaded blue) where 0 ≤ x ≤ 0.255 and 0 ≤ y ≤ 0.815 and 0.195 ≤ z ≤ 1.0 with x + y + z ≤ 1. Figure 5 suggests that CSiGeSn directness is available over 1.50 ≤ E_g ≤ −0.41 eV with lattice parameters in the 0.576 nm ≤ a ≤ 0.649 nm range.

 

Fig. 5 “Truly direct” bandgap regimes of three binary group-IV alloys as a function of lattice size. The four-sided region defined by the outer boundaries determines the fundamentally direct region of CSiGeSn.

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4. Conclusion

A search was carried out here to find those alloy compositions for which the fundamental bandgap of CSiGeSn is direct because few of its bandgaps are currently known over its wide composition space. The present search is motivated by a host of potential silicon-based opto-electronic integrated “circuit” applications. When plotted on a E_g-vs-a diagram, the direct and indirect gaps of the elements C, Si, Ge and Sn are data points on the CSiGeSn “space.” On that diagram, the direct-bandgap regions of binaries can offer guidelines about quaternary directness.

The simplest possible analysis of a binary alloy uses linear interpolation between the properties of its constituent elements. With such interpolation, the direct and indirect gaps of six binaries were examined. This simple analysis led to a simple conclusion: that the quaternary, the ternaries and the binaries must contain tin in their composition in order to possess a fundamental direct gap; the alloys must be CSn, SiSn, GeSn, CSiSn, CGeSn, SiGeSn, and CSiGeSn. A corollary is that CSi, CGe, SiGe and CSiGe are always indirect. The linear theory predicts that the CSiGeSn direct gap will be in the range from 1.50 eV down to −0.41 eV while its lattice parameter is in the 0.576 nm to 0.649 nm range.

Experiments on GeSn as well as first-principles theory for CSn and SiSn indicate that E_g-vs-alloy composition exhibits negative-bowing rather than being linear, and the known E_g(Γ) bowing was plotted here. Because information on the E_g(L) and E_g(X) bowing of CSn and SiSn was not available, the bowed direct CSiGeSn space was not determined here, even though the bowing theory is more accurate. However, it is clear that the bowed space extends to smaller lattice parameters than the linear-theory space. Specifically, with negative bowing, the upper three vertices in Fig. 5 shift to the left. This means that bowing theory implies an Sn content of less than 75%, 74.5%, and 19.5% in the crossover alloys of CSn, SiSn and GeSn, respectively, where the bowed E_g(Γ) curve crosses the bowed E_g(L) curve. With bowing, the shaded region in Fig. 5 would distort and shift slightly leftward.

Although Vegard’s law is assumed in Fig. 5, there is deviation from the law according to Eq. (2) of [9] which predicts a small positive bowing of the lattice parameter in the Sn-rich region of SiSn and GeSn. If such bowing is also assumed for CSn, then the three upper vertices in Fig. 5 would be shifted to slightly larger lattice parameters. There would be a small rightward shift of the distorted, shaded region.