Strain engineering of transverse electric and transverse magnetic mode of material gain in GeSn/SiGeSn quantum wells

8-band k · p Hamiltonian together with envelope function approximation and planewave expansion method are applied to calculate the electronic band structure and material gain for Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw quantum wells (QWs) grown on virtual Ge1-zSnz substrates integrated with Si platform. It is clearly shown how both the emission wavelength in this material system can be controlled by the content of virtual substrate and the polarization of emitted light can be controlled via the built-in strain. In order to systematically demonstrate these possibilities, the transverse electric (TE) and transverse magnetic (TM) modes of material gain, and hence the polarization degree, are calculated for Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw (QWs) with the strain varying from tensile (ε = +1.5%) to compressive (ε = −0.9%). It has been predicted that the polarization can be changed from 100% TE to 80% TM. In addition, it has been shown that SiyGe1−x−ySnx barriers, lattice matched to the virtual Ge1-zSnz substrate (condition: y = 3.66(x-z)), may ensure a respectable quantum confinement for electrons and holes in this system. With such material features Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw QW structure unified with Ge1-zSnz/Si platform may be considered as a very prospective one for light polarization engineering.

concerning the operating temperature (T 0 parameter), emission wavelength, and light polarization still exist for this material system.
In our recent work we have presented that the compressively stained GeSn/Ge quantum well (QW) is a very gifted gain medium for lasers integrated with Si platform 34 . The material gain for such QWs is stronger than for bulk GeSn and can be tuned in a broad spectral range, which is motivating for gas sensing 34 . Since Ge 1−w Sn w QWs can be integrated with Si platform via Ge 1-z Sn z virtual substrates, it is also probable to grow tensile strained GeSn QWs when w < z. In this case the built-in strain enhances the direct character of bandgap and due to this, is very satisfactory for laser applications. Alternatively, the material gain spectrum for tensile strained QWs can be dominated by the transverse magnetic (TM) mode since the ground state transition in such QWs is between the electron and the light-hole subband in place of the heavy-hole subband. It suggests that the light polarization engineering via stain should be achievable in GeSn QWs, which is very important for real laser applications and their functionality. However, a careful experimental studies of polarization of light emitted by GaSn QWs have not been carried out up to now. A few reports on gain calculations in GeSn-based QWs can be found in the literature [34][35][36][37][38][39][40][41] but there are no studies which focus on light polarization engineering in this material system.
In order to show that GeSn QW is a very promising gain medium, which allows to control the light polarization, and to stimulate experimental works in this field, we have calculated the electronic band structure and determined the transverse electric (TE) and transverse magnetic (TM) modes of material gain for the active region composed of Ge 1−w Sn w QW with the strain tuned from tensile to compressive and Si y Ge 1−x−y Sn x barriers lattice matched to Ge 1-z Sn z virtual substrate which can be integrated with Si platform. The obtained results are discussed in this paper in the context of recent progress in the growth and characterization of GeSn QWs and possibilities of engineering the electronic band structure and the built-in strain in SiGeSn material system. Figure 1 schematically shows Ge 1−w Sn w QW with Si y Ge 1−x−y Sn x barriers grown on Ge 1-z Sn z virtual substrate which is integrated with Si platform. Since the lattice constant in such heterostructures is determined by the virtual substrate, the built-in strain in Ge 1−w Sn w QW can be tuned from tensile (w < z) to compressive (w > z). In order to ensure the quantum confinement for electrons and holes in Ge 1−w Sn w QW, barriers with larger bandgap and type I bandgap alignment are needed. In this case Si y Ge 1−x−y Sn x barriers are a very natural candidate, since the incorporation of Si into GeSn opens up the bandgap and allows full strain engineering in this material system. Such barriers can be lattice matched to the virtual substrate and thereby Ge 1−w Sn w QW can be surrounded by barriers which create the waveguide for the proper wavelength.

Material parameters
The proposed QW structures can be useful in lasers if two conditions are fulfilled. The first condition is determined by the nature of the material, it is a positive material gain for the Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QW. The second condition is a good quality of (Si)GeSn materials which is determined by the technology applied and is always a big challenge. In this paper, the first condition is examined, since positive results can strongly motivate to work on the growth of lasers with the active Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QW region on Si platform.
In order to calculate gain spectra for Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs, a credible model and material parameters are required. As shown in our previous papers 42, 43 , the 8-band k · p model together with envelope function approximation and planewave expansion method are advanced enough to be used to calculate electronic band structures for QWs near the center of the Brillouin zone and determine gain spectra from the calculated dispersion of electron and hole subbands. This model can also be applied to calculate material gain spectra for GeSn QWs if the L valley is properly treated 34 . Thus this model has been applied to study Ge 1−w Sn w /Si y Ge 1−x−y Sn x / Ge 1−w Sn w QWs in this work. Details on 8-band k · p Hamiltonian, its elements, and formulas used for calculations of TE and TM mode of the material gain are given in the Methods section. Figure 2 shows the positions of conduction (Γ − 7 , − L 6 and Δ 5 states) and valence (Γ + 8 and Γ + 7 states) bands in Si, Ge, and α-Sn together with interpolations of these positions for the three alloys (SiGe, GeSn, and SiGe). Nonzero bowing parameters used to calculate the conduction and valence band positions in these alloys are shown in Fig. 2 and the mathematical formula for these calculations is given below In Eq. (1) Q A and Q B are the values of the quantity (conduction or valence band position) in the parent material A and B, respectively. The first two terms on right side in Eq. (1) represent the linear interpolation, and the third term is a parabolic deviation from the linearity characterized by the bowing parameter b Q . Bowing parameters for GaSn used in our calculate ons are taken from the recent publication on a comprehensive and detailed study of the composition dependence of lattice constants, bandgaps and band offsets for bulk Ge 1−x Sn x alloy in the full composition range by using state-of-the-art density functional theory (DFT) methods 16 . These bowing parameters are consistent with the bowing parameters reported in refs 2,[11][12][13][44][45][46][47][48][49] , where the authors experimentally determined the bandgap bowing for the direct gap to be 2.42 eV 44 , 1.97-2.61 eV 11 , 2.30-2.84 eV 45 , 2.46 eV 46 , and 2.8 eV 2 . In this case it is worth noting that the Burstein-Moss effect plays an important role in GeSn 16 and this effect can be responsible for some scattering of experimental data. In addition the compressive strain, which is present in GeSn layers grown on Ge substrates (or templates), opens the bandgap and thereby the bowing parameter for the bandgap determined experimentally can be adjusted correctly and lower than the one expected for unstrained GeSn 16 . From first principles calculations the bandgap bowing parameter for the bandgap at the Γ point was reported to be 2.49 eV 48 , 2.55 eV (2.75 eV for Sn 12.5%) 12 , ~2.4-3.0 eV for Sn < 12% 13 , and 3.1 eV 14 . Moreover many authors suggest that the bowing parameter varies with Sn concentration 12,13,47 . It can be another reason for some scattering of bandgap bowing parameters reported in the literature. Taking into account the cited literature data the bowing parameter for the direct gap taken to our calculations (i.e., 3.02 eV 16 ) is fully acceptable. The bowing parameter for the indirect gap in GeSn was reported to be 0.89 eV 12 , 0.9 eV 49 , and 1.03 46 , 1.23 eV 11 , 2.28 eV 48 . To our calculations we took the value of 1.23 eV 16 , which is very consistent with the bandgap bowing determined experimentally 11,46 . In this case it is worth noting that the bowing parameters in refs 2,[11][12][13][14][44][45][46][47][48][49] are determined for the bandgap while in ref. 16 the bowing parameters are determined also for both the conduction and the valence band and thereby they are more useful for our calculations. In addition the band offset between Ge and Sn is determined in ref. 16 and therefore it is preferred to adopt the band offset and bowing parameters from one reference where they are determined within one consistent approach. In this case it is also worth noting that the band offset reported in ref. 16 is quite close to the band offset reported in ref. 12 (0.53 vs. 0.91 eV). For SiGe the bowing parameter for the direct gap has been assumed to be zero in order to be consistent with the recent article on this issue 50 . For SiSn the bowing parameter for the direct gap is assumed to be 4.0 eV according to ref. 51 . In this case it is worth noting that larger bowing parameters are expected for SiSn than for GeSn since Sn differ more significantly from Si than from Ge in terms of atom size and electronegativity. This difference also suggests that the empirical formula with the constant bowing parameter (i.e., Eq.(1)) can be invalid for this alloy in full content range since the bowing parameter can vary with Sn concentration. So far this issue is not explored very carefully but larger bowing parameter for the direct gap in SiSn (13.2 eV) was suggested by D'Costa et al. 52 from the analysis of results for SiGeSn with low Sn concentration. The lower value of bowing parameter selected to our calculations is fully justified since we are considering SiGeSn alloys with larger Sn concentration which are close to those which are reported in ref. 51 . In case of SiSn the whole bandgap bowing is incorporated into the conduction band since its distribution between the conduction and the valence band is unknown at this moment (note that it is a typical first approximation in semiconductor alloys in such a situation).
Remaining material parameters for (Si)GeSn alloys, which are needed for the electronic band structure and the material gain calculations, are obtained from the linear interpolation of material parameters for the parent materials (Si, Ge, and Sn) taken from refs 16,48,[53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69] and are given in the Table 1. Currently it is experimentally documented that the lattice constant of GeSn is described by the linear interpolation 70 . This conclusion is also very consistent with the recent DFT calculations of the lattice constant for GeSn 16 . For remaining parameters (elastic constants (C 11 and C 12 ), deformation potentials, etc.) it is expected that the deviation from linearity of interpolation will be negligible like in other semiconductor alloys 71 . Even if there is significant nonlinearity for given parameter, but its value is unknown, the best solution is to take the linear interpolation as the first approximation and such solution is applied in our case.

Materials selection
The diagram shown in Fig. 2 is a starting point for the selection of materials for building a type I QW. From this diagram it is clearly visible that Ge 1-x Sn x with around 0.1 < x < 0.3 is a direct gap material which is a good candidate for QWs dedicated for light emitters. Direct gap materials are not necessary for barriers and SiGeSn is the most optimal choice in this case due to very flexible strain engineering in this alloy. Since the conduction and valence band positions move due to the built-in strain, different diagrams are necessary to evaluate the selection of materials for QW and barriers. Such diagrams are shown in Fig. 3. In this case the conduction (Γ and L point) band minima and the valence (HH, LH, and SO) band maxima are shown by solid lines for Ga 1−w Sn w QW material strained to virtual Ga 1-z Sn z substrate with z = 0.05, 0.10, and 0.15. In addition, the conduction and valence band positions for Si y Ge 1−x−y Sn x barrier lattice matched to the virtual substrate are shown in this diagram, see dashed lines. Si concentration (y) in Si y Ge 1−x−y Sn x barrier is changing according to following formula: y = 3.66(x-z), which fulfills the condition of lattice matching to virtual Ga 1-z Sn z substrate. When analyzing solid lines in Fig. 3 for the same bands (CB, HH, LH, or SO) it is clearly visible that the built-in strain strongly modifies the position of a given band and thereby can be used to engineer the bandgap in this QW system (for a better comparison all curves are plotted on each panel and curves corresponding to a given panel are thicker while the remaining curves are plotted by thin grey lines). In this case the strain-related shifts for CB, HH, LH, and SO band are calculated using the Bir-Pikus theory 72 , see proper formulas in the Methods section. So far it has been shown that this theory works very well in this material system in the range of low built-in strains 15,73 , i.e., the range which is considered in this work.
With the diagram shown in Fig. 3 it is relatively easy to identify compositions promising for type I Ge 1−w Sn w / Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs through identifying proper quantum confinement trends of the conduction and the valence band with the QW and/or the barrier content change. Thus, in the next part of the paper, the electronic band structure and gain spectra are discussed for sets of QWs built on the conclusions drawn from the diagrams shown in Fig. 3.  www.nature.com/scientificreports www.nature.com/scientificreports/ conduction bands which is very favorable for laser applications (in general, a larger T 0 temperature is expected for deeper QWs). However, this increase is connected with the increase in Sn concentration because of the lattice matching condition for this alloy. It is possible that the optical quality of lattice matched Si y Ge 1−x−y Sn x deteriorates with the increase in Si concentration and therefore Si y Ge 1−x−y Sn x barriers with large Si concentration are promising from the band structure point of view but can be difficult to manufacture from the technological point of view. Moreover it is worth noting that an electron leakage to the L point of the Brillouin zone in Si y Ge 1−x−y Sn x barriers can determine T 0 in these QWs. The energy separation between the electron ground state and the L point is not very favorable in this case even for Si y Ge 1−x−y Sn x barriers with large Si concentration since the energy position of L point does not change significantly in this case, see Fig. 4. A significant electron confinement improvement is observed with the Sn concentration increase in the well layer, see the electronic band structure in Figs 4, 5 and 6. Thereby, an energy separation between the ground electron state energy in Ge 1−w Sn w QW and the L point energy in Si y Ge 1−x−y Sn x barrier significantly improves (and thereby a higher T 0 is expected) for lasers containing Ge 1−w Sn w QWs with larger Sn concentration, but large T 0 will be a challenge for this material system.

Unstrained system
With the increase in Sn concentration in Ge 1−w Sn w QW, the gain spectra shift to red. For lower Sn concentration their intensities increase, which is connected with improving the confinement mentioned before, and decreases for higher Sn concentration, which seems to be connected with confined electrons effective mass decrease visible in conduction QW bands, see gain spectra in Figs 4, 5 and 6. It is also visible that, in case of the unstrained QWs, intensities of TE and TM modes are comparable and the every TM mode peak position is slightly blue-shifted in comparison to the corresponding TE mode peak position as it should be due to properties of heavy and light holes states. Moreover, the intensity of both modes of material gain decrease with the increase in Si concentration in Si y Ge 1−x−y Sn x barriers, which is hardly visible for lower and distinctly visible for cases with higher Sn concentration in QW.
Additional carrier current related information has been attached to Figs 4, 5 and 6. One may ask about values of quasi-Fermi levels energy (QFLE) of electrons and holes in presented unstrained QWs as an important parameter for designing real devices. In these QWs, all confined holes always seem to be close enough to valence band edge to not worry about them and leakage phenomena. Different situation appears in case of electrons which, for carrier concentrations chosen in calculations, seems to easily be able to fill almost all accessible confined states with energies below L valley energy. So some information, about how deep below L valley energy QFLE for electrons is set in every case, is marked by color of numbers which describes carrier concentrations, see Figs 4, 5 and 6. Red means that QFLE is higher than L valley energy. Blue means that QFLE is below L valley energy but not deeper than k b T, where T = 300 K. Black means that QFLE is more than k b T below L valley energy, where T = 300 K. To show QFLE value ranges in calculations for carrier concentrations from 2 · 10 18 cm −3 to 5 · 10 18 cm −3 there are grey regions put in charts with proper confinements.
Summarizing the results for unstrained Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs we conclude that the change of the content of virtual Ge 1-z Sn z substrate and the growth of unstrained Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs on such substrate (w = z) is a good strategy for tuning the emission wavelength of radiation with two comparable polarization components (TE and TM mode) possible to be emitted from this material system. In the next step, strain engineering in the QW region can be applied for tuning the contribution of TE and TM modes to lasing from the QW.   Figure 7 shows spectra of TE (solid lines) and TM (dashed lines) modes of material gain for Ge 1−w Sn w /Si 0.1 Ge 0.9-x Sn x /Ge 1−w Sn w QWs of various Sn concentrations (w) grown on virtual Ge 0.90 Sn 0.10 , Ge 0.875 Sn 0.125 , and Ge 0.85 Sn 0.15 substrates. By changing the Sn concentration in Ge 1−w Sn w QW the built-in strain can be tuned from tensile to compressive. For tensile strained QWs the gain spectrum is dominated by the TM mode while for compressively strained QWs the TM mode of material gain is very weak and significantly blue-shifted in comparison to the TE mode. Therefore, for compressively strained Ge 1−w Sn w /Si 0.1 Ge 0.9-x Sn x /Ge 1−w Sn w QWs, emission of TE polarized light is expected, while a mixed light polarization (TE mode mixed with TM mode) is expected for tensile strained QWs. In addition, it is worth noting, that high dominating TE material optical gain is easier to achieve with virtual substrates containing less Sn (set along line III in Fig. 7) than in case when one may want to have dominating TM material optical gain (set along line I in Fig. 7).
Because there is less energetic separation between TE and TM modes of material gain spectra when the system is tensely strained, an analysis of optical gain peak values and positions for Ge 1−w Sn w /Si 0.1 Ge 0.723 Sn 0.177 /Ge 1−w Sn w QWs grown on virtual Ge 0.85 Sn 0.15 substrate (spectra along line I in Fig. 7) is presented below. In Fig. 8 TE and TM mode material gain peak positions are plotted in the left panel, values together with a TM polarization degree, both at TM gain peak energy are plotted in the right panel, where the polarization degree is defined as the ratio of the values of TM mode to the sum of the intensities of TE and TM mode at the given energy ((polarization(E) = g TM (E)/[g TE (E) + g TM (E)]). It is visible that TE peak changes its energy faster than TM peak with the less than 3% Sn concentration in the Ge 1−w Sn w QW changing (incorporating strain to QW layer). Situation is different for higher tensile strain, where both peaks www.nature.com/scientificreports www.nature.com/scientificreports/ go together and even have similar energy. This change in the TE mode material gain behavior is connected with less confined hole states number existing in the QWs with lower Sn concentrations and therefore also shallower confinements and higher tensile strain induced effects. In the right plot (Fig. 8) it is presented that calculated TM polarization degree increases from 50% to 80% with the tensile strain increase in the Ge 1−w Sn w QWs. In this case un-polarized light (50% polarization degree) is observed for unstrained QW and TM polarized light is observed for tensile strained QWs (75-80% polarization degree is observed for strains larger than 0.6%). The increase in polarization degree is mostly associated with the stronger TM mode and drop of TE mode material gain values for tensile strained QWs. The relative gain peaks spectral shift appearing due to the built-in strain is important in cases closer to unstrained situation only. In general, the last factor (the shift of TM and TE gain peaks) seems not to be controllable too much by the QW width when higher tensile strain situation is achieved. The reason is there is only one degenerated confined electron band and one degenerated confined hole band for these structures. However the less strain is present in the QW layer the more control of this factor should be apparent, since reducing strain in this system is connected with deepening its confinements in gamma point which increases number of bounded states. In order to study this phenomenon for the tensely strained QWs, gain spectra have been calculated with concerning various widths. Figure 9 shows spectra of TE (solid lines) and TM (dashed lines) mode of material gain for Ge 1−w Sn w / Si 0.1 Ge 0.9-x Sn x /Ge 1−w Sn w QWs of various widths grown on virtual Ge 0.85 Sn 0.15 substrate. For all QW contents it is clearly visible that gain peaks shifts to blue by ~60 meV with the reduction of QW width from 14 to 8 nm, but differences between shifts of TE and TM peaks are negligibly small except case with low strain. With the QWs width, while the intensity of TE and TM modes almost does not change for QWs with larger Sn concentration, it decreases quite significantly for QWs with lower Sn concentration. It is visible, that calculated ratio between the values of TE and TM mode does not change significantly. Therefore the polarization degree should not vary with www.nature.com/scientificreports www.nature.com/scientificreports/ the QW width even if modes are changing their relative positions, see green lines for a given Sn concentration in Fig. 9. It means that the QW width does not significantly influence the mixing of TE and TM modes in this QW system, and the built-in strain together with number of confined states are the main factors which control the polarization degree of light emitted from Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs grown on virtual Ge 1-z Sn z substrates integrated with Si platform.

scattering of Material parameters
Since some scattering of material parameters (i.e., bandgap bowing parameters) is present in the literature for (Si)GeSn alloys, see the previous discussion, it is interesting to analyze and discuss the influence of the accuracy of these parameters on our conclusions on light polarization engineering in this QW system. Such analysis has been performed changing the bowing parameter for the direct and indirect bandgap in GeSn by 20% (from 3.02 to 2.43 eV for the direct gap and from 1.23 eV to 0.98 eV for the indirect gap) and tuning the valence band offset by 20% (from 0.53 eV to 0.64 eV). It has been observed that these changes shift to blue the TE and TM peak positions but do not change their relative positions and intensities. Therefore the polarization degree of light does not change in this case. It allows us to claim that our conclusions on polarization engineering are valid for this material system even if the spectral position of the gain peak is predicted less accurately. It has been observed that the blueshift depends on QW width and content but in general this shift is smaller than 70 meV. Moreover it has been concluded that these changes do not influence trends in variation of the gain peak intensity with changes in Sn concentration in QW but slightly influence the range of QW contents for which a positive material gain is observed. Regarding changes in bowing parameters for SiSn, which were used for calculations of the bandgap for Si y Ge 1−x−y Sn x barriers, it has been observed that their influence on the spectral position and the intensity of TE and TM mode of material gain is negligible for deep QWs and starts to be important for shallow QWs. It suggests that results for Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1−w Sn w QWs with Si rich barriers are more reliable.  www.nature.com/scientificreports www.nature.com/scientificreports/ Since the alloy disorder strongly influences the electronic band structure of highly mismatched alloys (HMAs), see for example ref. 74 , and GeSn is on the border of regular alloys and HMAs, it is interesting to know and comment how the alloy disorder is important for QWs containing GeSn. So far the influence of alloy disorder on the electronic band structure of GeSn was studied in ref. 16 . It was observed that there is no variation in the lattice constant but there is a significant difference in energy gap values when passing from the clustered to the uniform distribution. It is worth commenting that it can be one of the reasons for scattering the bowing parameter reported in the literature for the GeSn band gap. As mentioned before, bowing parameters used to our calculations are taken from ref. 16 and correspond to uniform distribution of Sn atoms. In general such atom distribution is expected for the non-equilibrium growth process with the properly low growth temperature. An epitaxial growth at higher temperatures can lead to an atom segregation which is a well-known phenomenon even for regular semiconductor alloys like InGaAs 75 . In means that the regular atom distribution can be achieved in this alloy by lowering the growth temperature. On the other hand too low growth temperature leads to spontaneous formation of native point defects which are unfavorable for light emitters. According to motivation of this work the proper optimization of the growth temperature and/or other growth conditions/parameters is another challenge for this material system, but this issue is not out of the scope of this work.

Conclusions
We have presented the first systematic study of light polarization engineering in Ge 1−w Sn w /Si y Ge 1−x−y Sn x /Ge 1− w Sn w QWs integrated with Ge 1-z Sn z /Si platform. It has been shown how the emission wavelength in this system can be tuned by the content of virtual Ge 1-z Sn z substrate and the light polarization can be engineered via the built-in strain in Ge 1−w Sn w QW. Using virtual Ge 1-z Sn z substrate the built-in strain in Ge 1−w Sn w QW can be tuned from tensile (w < z) to compressive (w > z), changing the polarization degree from 80% TM to 100% TE polarization. Good quantum confinement for electrons and holes in this QW seems to be ensured by Si y Ge 1−x−y Sn x barriers lattice matched to Ge 1-z Sn z substrate. In addition, it is shown that the QW width should influence the degree of TM polarization for tensile strained QWs very weakly. We believe that the reported results will strongly motivate experimental studies of light polarization in GeSn-based QWs.

band k · p Hamiltonian.
For unstrained GeSn the 8 band k · p Hamiltonian is given below   where the subscripts CB, HH, LH, and SO stand for the conduction (Γ point), heavy-hole, light-hole, and spin-orbit split off bands, correspondingly. ℏ is the Planck constant distributed by 2π, m 0 is the electron mass, and = + k k k where Δ SO and Γ E g are, respectively, the spin-orbit splitting and the energy gap for Ge.
Built-in strains in (Si)GeSn. The strain-related shifts for CB and VB are considered using the Bir-Pikus Hamiltonian 72 . These strains change the quantum confinement potential for electrons and holes as shown in our earlier papers 42,43 . In the QW region the matrix elements of the 8-band k · p Hamiltonian are improved as below and η ax = −b ax (1 + 2c 12 /c 11 )ε xx describe influence of the hydrostatic and shear strain components on the band structure. a C and a V are the hydrostatic deformation potentials for CB and VB, respectively; c 11 and c 12 are elastic constants, b ax is the axial deformation potential and ε xx is the in-plane strain in the GeSn layer. For GeSn QWs grown on virtual Ge 1-z Sn z substrate this strain is defined by the lattice parameters of Ge 1−w Sn w (QW material) and Ge 1-z Sn z substrate: (2019) 9 . Material parameters for (Si)GeSn are determined consuming linear interpolation between parameters of relevant materials (Si, Ge, and Sn).
Material gain. The optical gain for GeSn/SiGeSn QWs is designed for a given carrier density, which defines the quasi-Fermi levels E c F and E v F for the conduction and valence bands, correspondingly, and vice versa. The carrier density in a band in the QW is set by integration of result of the density of states, ρ(k), and the occupation probability of carriers (i.e. the Fermi-Dirac distribution) over the entire band. The Fermi-Dirac distribution for electrons (f c ) and holes (f v ) in the QW is given by Eqs (6) and (7) where k B is the Boltzmann's constant and T is the temperature. The number of carrier in conduction (N) and valence (P) band is obtained by First relations on the right hand sides of equations (8) and (9) are related to the 2-D Fermi gases defined by electronic band structure of the QW obtained in the planewave expansion of the 8 k · p model, while the ones following are related to 3-D Fermi gases, which are defined by the electronic band structure of unstrained bulk Ge crystal designed using the 8 k · p model. V well and S well are correspondingly the QW volume and surface. The third part in integral (7) profits account of 3-D electron Fermi gas in L valley, where E k ( ) Ge L c is parabolic approximation of this band in bulk crystal. It is shifted to the center of Brillouin zone in order to make simpler the quasi-Fermi level calculations and hence the factor 4 appears at this term (this explanation is equivalent to eight L valleys at the border of the Brillouin zone). Transitions related to this valley are omitted in the gain calculations. Further, for non-parabolic bands, integration is carried out in a k space with the density of states determined from the k · p calculations. k max is the integration limit determined by convergence of the integrals (7) and (8).
A conventional method based on the relaxation time approximation convoluted with a Lorentzian function ( 2 , where ℏ is Plank's constant and Δ is the proper energy difference) with a proper broadening time (τ b = 0.1 ps) has been used to calculate the gain spectra 42,43 . In this approximation the transverse electric (TE) and transverse magnetic (TM) gain is given by Eq.