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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19957–19965
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Design of a Si-based lattice-matched room-temperature GeSn/GeSiSn multi-quantum-well mid-infrared laser diode

G. Sun, R. A. Soref, and H. H. Cheng  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19957-19965 (2010)
http://dx.doi.org/10.1364/OE.18.019957


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Abstract

This paper presents modeling and simulation of a silicon-based group IV semiconductor injection laser diode in which the active region has a multiple quantum well structure formed with Ge0.9Sn0.1 quantum wells separated by Ge0.75Si0.1Sn0.15 barriers. These alloy compositions were chosen to satisfy three conditions simultaneously: a direct band gap for Ge0.9Sn0.1, type-I band alignment between Ge0.9Sn0.1 and Ge0.75Si0.1Sn0.15, and a lattice match between wells and barriers. This match ensures that the entire structure can be grown strain free upon a relaxed Ge0.75Si0.1Sn0.15 buffer on a silicon substrate – a CMOS compatible process. Detailed analysis is performed for the type I band offsets, carrier lifetime, optical confinement, and modal gain. The carrier lifetime is found to be dominated by the spontaneous radiative process rather than the Auger process. The modal gain has a rather sensitive dependence on the number of quantum wells in the active region. The proposed laser is predicted to operate at 2.3 μm in the mid infrared at room temperature.

© 2010 OSA

1. Introduction

2. Band structure of GeSn/GeSiSn quantum wells

Energy-band theory [8

8. R. A. Soref and C. H. Perry, “Predicted band gap of the new semiconductor SiGeSn,” J. Appl. Phys. 69(1), 539 (1991). [CrossRef]

] and FTIR absorption experiments [9

9. H. P. L. de Guevara, A. G. Rodriguez, H. Navarro-Contreras, and M. A. Vidal, “Nonlinear behavior of the energy gap in Ge1-xSnx alloys at 4 K,” Appl. Phys. Lett. 91(16), 161909 (2007). [CrossRef]

] have indicated that the bandgap of unstrained crystalline GeSn makes a transition from indirect to direct as the percent of α-Sn is increased. Since the band offsets between ternary Sn-containing alloys and Si or Ge are not known experimentally, we follow the assumptions made in Ref [1

1. V. R. D'Costa, Y.-Y. Fang, J. Tolle, J. Kouvetakis, and J. Menéndez, “Ternary GeSiSn alloys: New opportunities for strain and band gap engineering using group-IV semiconductors,” Thin Solid Films 518(9), 2531–2537 (2010). [CrossRef]

]. and calculate the conduction-band minima for the lattice-matched heterostructure consisting of Ge1-zSnz and a ternary Ge1- x - ySixSny. We used Jaros' band offset theory [10

10. M. Jaros, “Simple analytic model for heterojunction band offsets,” Phys. Rev. B Condens. Matter 37(12), 7112–7114 (1988). [CrossRef] [PubMed]

] which gives results in good agreement with experiment for many heterojunction systems. For example, this theory predicts an average valence band offset, ΔEv,av=0.48eV for a Ge/Si hetero-interface (higher energy on the Ge side), close to the accepted value of ΔEv,av=0.50eV. The basic ingredients of our band-alignment calculation are the average valence-band offset between the two materials (an average between heavy, light, and split-off hole bands) and the compositional dependence of the ternary-alloy’s band structure. For the Ge/-Sn interface. Jaros’ theory predicts ΔEv,av=0.69eV (higher energy on the α-Sn side). Thus, relative to the average valence band of Ge, the average valence band position for Ge1- x - ySixSny is simply a linear interpolation

Ev,av(Ge1xySixSny)=0.48x+0.69y.
(1)

Similarly, with these spin-orbit splitting values ΔEv,av=0.69eV ΔSO(Si)=0.043 eV, ΔSO(Sn)=0.800eV [12

12. V. R. D'Costa, C. S. Cook, A. G. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, and J. Menendez, “Optical critical points of thin-film Ge1−ySny alloys: A comparative Ge1−ySny/Ge1−xSix study,” Phys. Rev. B 73(12), 125207 (2006). [CrossRef]

], the spin-orbit splitting for Ge1- x - ySixSny is

ΔSO(Ge1xySixSny)=0.295(1xy)+0.043x+0.800y.
(2)

The top of the valence band for Ge1- x - ySixSny can then be determined as

Ev(Ge1xySixSny)=Ev,av(Ge1xySixSny)+ΔSO(Ge1xySixSny)3.
(3)

The minima of the conduction band at points L and Γ can then be calculated by evaluating the compositional dependence of the band gaps of the ternary alloy as
E(Ge1xySixSny)=EGe(1xy)+ESix+ESnybGeSi(1xy)x               bGeSn(1xy)ybSiSnxy
(4)
where EGe, ESi, and ESn are the bandgaps of Ge, Si, and α-Sn, respectively, at those points, and the bowing parameters bGeSi,bGeSn,bSiSn have been discussed in Refs [13

13. V. R. D'Costa, C. S. Cook, J. Menendez, J. Tolle, J. Kouvetakis, and S. Zollner, “Transferability of optical bowing parameters between binary and ternary group-IV alloys,” Solid State Commun. 138(6), 309–313 (2006). [CrossRef]

,14

14. J. Weber and M. I. Alonso, “Near-band-gap photoluminescence of Si-Ge alloys,” Phys. Rev. B Condens. Matter 40(8), 5683–5693 (1989). [CrossRef] [PubMed]

]. These values at L and Γ points have been given in Table 1

Table 1. Band parameters at various valleys used in the band alignment calculation [1214]

table-icon
View This Table
.

Finally, for the indirect conduction band minimum near the X-point, Weber and Alonso find
EX(Ge1xSix)=0.931+0.018x+0.206x2
(5)
(in eV) for Ge1- xSix alloys [14

14. J. Weber and M. I. Alonso, “Near-band-gap photoluminescence of Si-Ge alloys,” Phys. Rev. B Condens. Matter 40(8), 5683–5693 (1989). [CrossRef] [PubMed]

]. On the other hand, the empirical pseudopotential calculations of Chelikovsky and Cohen place this minimum at 0.90 eV in α-Sn, virtually the same as its value in pure Ge [15

15. M. L. Cohen and T. K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev. 141(2), 789–796 (1966). [CrossRef]

]. We thus assume that the position of this minimum in ternary Ge1- x - ySixSny alloys is independent of the Sn concentration y, and thus is also given by Eq. (5). Obviously, the calculation of band structures outlined above is an approximation that is subject to experimental corrections as more measurements become available. This implies that the compositions of Ge1-zSnz and Ge1- x - ySixSny are necessarily adjusted in the QW structure to arrive at the band structure that is being proposed here. But it should be pointed out that the laser behavior depends only on the band structure, and that the results obtained in this design should be valid albeit at slightly different binary and ternary compositions, and possibly at slightly different lasing wavelengths.

The α-Sn composition dependence of the conduction band gaps for Ge1-zSnz at the three valleys , Γ, and X is first calculated using Eqs. (4) and (5) to establish the crossing point where the Γ-point band gap drops below that of the L-point. Figure 1(a)
Fig. 1 (a) Band gaps of Ge1-zSnz at L, Γ, and X vs. Sn composition z, and (b) Γ-point band alignment between lattice matched Ge1- x - ySixSny and Ge0.9Sn0.1 vs. the Sn composition y.
shows that for α-Sn composition greater than z6%, the Ge1-zSnz gap becomes direct. We thus choose Ge0.9Sn0.1 to be the QW layer with a direct band gap of Eg=0.505eV. Fixing at this Ge0.9Sn0.1 composition, we then looked for a lattice matched Ge1- x - ySixSny that can be used as barriers that form type-I band alignment with Ge0.9Sn0.1. Such a simultaneous requirement for lattice parameter and band alignment can be satisfied by the additional degree of freedom in the Ge1- x - ySixSny where both and can be tuned. Using Vegard’s law for the lattice constant of Ge1- x - ySixSny, the lattice constant of Ge1- x - ySixSny is
a(Ge1xySixSny)=aGe(1xy)+aSix+aSny
(6)
where the lattice constants are aGe=5.64613Å, aSi=5.43095Å, andaSn=6.48920Å for Ge, Si, and α-Sn [11

11. S. Adachi, Properties of Group-IV, III–V, and II–VI Semiconductors, (John Wiley and Sons, England, 2005)

], respectively, and we can vary the Si and α-Sn compositions in GeSiSn simultaneously to yield exactly the lattice constant of Ge0.9Sn0.1. Adding the band gaps to the top of the valence band Eq. (3), we obtain the band alignment between at the Γ-point as shown in Fig. 1(b).

There is a wide range of Sn composition 0.10<y<0.19 over which the ternary Ge1- x - ySixSny forms type-I confinement with Ge0.9Sn0.1, i.e., both elections and holes are confined in the Ge0.9Sn0.1 QWs by the Ge1- x - ySixSny barriers. In particular, we choose barriers with the composition Ge0.75Si0.1Sn0.15 that gives the largest conduction band offset of ΔEc,Γ=88meV, and offset of the valence band ΔEv=68meV as shown in Fig. 1(b). The laser device shall be grown on a relaxed Ge0.75Si0.1Sn0.15 buffer on a Si substrate to ensure that the entire structure is strain free as illustrated in Fig. 2
Fig. 2 Illustration of the GeSn/GeSiSn QW laser device on a lattice matched SiGeSn relaxed buffer upon a Si or SOI substrate. The device’s band alignment is also shown. This elongated mesa forms a strip channel waveguide.
.

3. Carrier lifetime

The proposed laser device has MQWs in its active region. Since the compositions of Sn and Si are relatively small in comparison with that of Ge for either QWs or barrier layers, we shall use Γ-point Ge parameters in the following calculations. The quantum confinement leads to energy subbands in both conduction and valence bands. The energy levels of these subbands can be calculated by solving the one-dimensional Schrödinger equation following the envelope function approximation [16

16. G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).

]. The laser device under consideration has a forward-biased PIN structure where the active MQW region is undoped. The band-to-band lasing transitions occur as stimulated emissions triggered by recombination of electron-hole pairs that are injected into this region. The analysis proceeds as follows. For a given carrier density, we can derive quasi Fermi levels at a specific temperature (T) for electrons in the conduction band (Efc) and for light holes (LHs) and heavy holes (HHs) in the valence band (Efv). However, only the electron-HH pair recombination contributes to lasing transitions since the ground-state HH subband lies lower-in-valence energy than that of the LH subband. The structure that we have calculated was chosen to have 20nm Ge0.9Sn0.1 QWs that are separated by 20nm Ge0.75Si0.1Sn0.15 barriers. The energy separation between the ground-state electron and HH subbands has been determined to be Eq=0.541eV, which is 36meV larger than the Ge0.9Sn0.1 band gap (0.505eV) due to the quantum confinement.

In order to estimate the carrier lifetime, it is necessary to calculate the radiative as well as the nonradiative Auger recombination rate in the MQW active region. The radiative process is spontaneous consisting of electron-heavy hole (e-hh) as well as electron-light hole (e-lh) recombination. The spontaneous emission rate per unit area in the energy interval EE+dE due to the e-hh process can be calculated as [16

16. G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).

],
Rsp,ehh=n¯e2mr|Mb|2π2m02ε04c3dEqEfc(Ee)fv(Ehh)dE
(7)
where e is the electron charge, m0 the free electron mass, ε0 the permittivity of vacuum, the Planck constant, c the speed of light in vacuum, n¯ the index of refraction, d the QW width, and |Mb|2the average matrix element for the Bloch states [16

16. G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).

]. The occupation probabilities at the states that are separated by a photon energy E with the same k in the reciprocal space are
fc(Ee)=[1+exp(EeEfckBT)]1,fv(Ehh)=[1+exp(EfvEhhkBT)]1,
(8)
respectively, where kB is the Boltzmann constant, and the electron and hole energies in the conduction and valence subbands
Ee=mrme(EEq),Ehh=mrmhh(EEq)
(9)
are computed through the reduced effective mass
mr=memhhme+mhh
(10)
in which meand mhh are the electron and HH effective mass, respectively. Obviously, the spontaneous emission rate for the radiative e-lh process can be obtained similarly. The total spontaneous emission rate per unit area can be obtained as

Rrad=Rsp,ehh+Rsp,elh
(11)

The radiative lifetime can then be obtained by τrad=n/Rrad where n is the area carrier density. The result for the area carrier density of n=2×1012/cm2, corresponding to a carrier concentration of 1018/cm2 in QW layers for the well thickness of d = 20nm, is shown in Fig. 3
Fig. 3 Radiative and Auger recombination lifetime as a function of temperature for the carrier density of 2×1012/cm2 in the Ge0.9Sn0.1 /Ge0.75Si0.1Sn0.15 MQW active region.
for a range of temperature. It can be seen that the radiative lifetime for a fixed carrier density is rather insensitive to temperature change, showing a slight increase with the temperature.

This radiative carrier lifetime can be shown to be much shorter than that of the Auger process where the recombination of an electron-hole pair takes place by transferring energy and momentum to a third particle which could be either an electron or a hole. For comparison, we have also estimated the Auger lifetime by following the calculation procedure outlined in Ref [16

16. G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).

]. The result for the same area carrier density of n=2×1012/cm2 determined by the Auger process is shown in Fig. 3. Clearly the Auger lifetime decreases rapidly with the increase of temperature, but even at T = 300K, it remains longer than that of the radiative process. We therefore conclude that the spontaneous radiative recombination is the dominant process in determining the carrier lifetime. In comparison with the DH laser that we have simulated earlier [3

3. G. Sun, R. A. Soref, and H. H. Cheng, “Design of an electrically pumped SiGeSn/GeSn/SiGeSn double-heterostructure mid-infrared laser,” J. Appl. Phys. 108(3), 033107 (2010). [CrossRef]

], this represents a significant improvement as a result of reduced density of states in QWs relative to that of bulk material, which leads to its potential room temperature operation.

4. Optical gain

In contrast to their advantage of having longer carrier lifetime, single-QW lasers typically have a very small optical confinement factor compared to that of DH lasers because their active regions are too thin relative to the lasing wavelength. Fortunately, the MQW structure offers a practical solution to the mode-overlap problem by increasing the effective thickness of the active region. We calculate the confinement factor by treating the MQW active region as having an index of refraction that is averaged between the QW index and the barrier layers index. Figure 4
Fig. 4 Optical confinement factor for the fundamental TE mode as a function of the number of QWs in the active region of a Ge0.9Sn0.1 /Ge0.75Si0.1Sn0.15 QW laser having Ge0.75Si0.1Sn0.15 cladding layers.
shows the optical confinement factor for the fundamental TE mode of the Ge0.9Sn0.1 /Ge0.75Si0.1Sn0.15 QW laser for a range of QW numbers whose active regions consist of 20nm Ge0.9Sn0.1 QWs that are separated by 20nm Ge0.75Si0.1Sn0.15 barriers, and where the active region is cladded above and below by thick Ge0.75Si0.1Sn0.15 layers. The confinement factor Γ, defined as the spatial overlap integral of the TEo mode profile with the gain profile, increases from 0.003 for a single QW to 0.90 for 35 QWs.

It may be desired to increase the laser’s emission wavelength into the 3 to 5 μm band (the atmospheric transmission window). Then it is necessary to increase the Sn content of the QWs beyond 10% and to change the barrier composition to lattice-match the new QWs. Having done this for λ = 3.5 μm, we found that the well/barrier conduction-band offset decreased to about 20 meV, a value not sufficient for good confinement of electrons. However, this offset problem may be solvable by employing SiGeSn QWs along with ternary barriers.

5. Conclusion

Acknowledgement

This work was supported in part by the Air Force Office of Scientific Research, Dr. Gernot Pomrenke, Program Manager.

References and links

1.

V. R. D'Costa, Y.-Y. Fang, J. Tolle, J. Kouvetakis, and J. Menéndez, “Ternary GeSiSn alloys: New opportunities for strain and band gap engineering using group-IV semiconductors,” Thin Solid Films 518(9), 2531–2537 (2010). [CrossRef]

2.

R. A. Soref, J. Kouvetakis and J. Menendez, “Advances in SiGeSn/Ge technology,” Mater. Res. Soc. Symp. Proc., 958, 0958–L01–08 (2007).

3.

G. Sun, R. A. Soref, and H. H. Cheng, “Design of an electrically pumped SiGeSn/GeSn/SiGeSn double-heterostructure mid-infrared laser,” J. Appl. Phys. 108(3), 033107 (2010). [CrossRef]

4.

J. Tolle, R. Roucka, V. D'Costa, J. Menendez, A. Chizmeshya, and J. Kouvetakis “Sn-based group-IV semiconductors on Si: new infrared materials and new templates for mismatched epitaxy,” Mater. Res. Soc. Symp. Proc.891, 0891–EE12–08 (2006).

5.

G.-E. Chang, S.-W. Chang, and S. L. Chuang, “Theory for n-type doped, tensile-strained Ge-Si(x)Ge(y)Sn1-x-y quantum-well lasers at telecom wavelength,” Opt. Express 17(14), 11246–11258 (2009). [CrossRef] [PubMed]

6.

Y.-H. Zhu, Q. Xu, W.-J. Fan, and J.-W. Wang, “Theoretical gain of strained GeSn0.2/Ge1-x-y’SixSny’ quantum well laser,” J. Appl. Phys. 107(7), 073108 (2010). [CrossRef]

7.

G. Sun, H. H. Cheng, J. Menendez, J. B. Khurgin, and R. A. Soref, “Strain-free Ge/GeSiSn quantum cascade lasers based on L-valley intersubband transitions,” Appl. Phys. Lett. 90(25), 251105 (2007). [CrossRef]

8.

R. A. Soref and C. H. Perry, “Predicted band gap of the new semiconductor SiGeSn,” J. Appl. Phys. 69(1), 539 (1991). [CrossRef]

9.

H. P. L. de Guevara, A. G. Rodriguez, H. Navarro-Contreras, and M. A. Vidal, “Nonlinear behavior of the energy gap in Ge1-xSnx alloys at 4 K,” Appl. Phys. Lett. 91(16), 161909 (2007). [CrossRef]

10.

M. Jaros, “Simple analytic model for heterojunction band offsets,” Phys. Rev. B Condens. Matter 37(12), 7112–7114 (1988). [CrossRef] [PubMed]

11.

S. Adachi, Properties of Group-IV, III–V, and II–VI Semiconductors, (John Wiley and Sons, England, 2005)

12.

V. R. D'Costa, C. S. Cook, A. G. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, and J. Menendez, “Optical critical points of thin-film Ge1−ySny alloys: A comparative Ge1−ySny/Ge1−xSix study,” Phys. Rev. B 73(12), 125207 (2006). [CrossRef]

13.

V. R. D'Costa, C. S. Cook, J. Menendez, J. Tolle, J. Kouvetakis, and S. Zollner, “Transferability of optical bowing parameters between binary and ternary group-IV alloys,” Solid State Commun. 138(6), 309–313 (2006). [CrossRef]

14.

J. Weber and M. I. Alonso, “Near-band-gap photoluminescence of Si-Ge alloys,” Phys. Rev. B Condens. Matter 40(8), 5683–5693 (1989). [CrossRef] [PubMed]

15.

M. L. Cohen and T. K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev. 141(2), 789–796 (1966). [CrossRef]

16.

G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).

17.

Y.-Y. Fang, J. Xie, J. Tolle, R. Roucka, V. R. D’Costa, A. V. G. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular-based synthetic approach to new group IV materials for high-efficiency, low-cost solar cells and Si-based optoelectronics,” J. Am. Chem. Soc. 130(47), 16095–16102 (2008). [CrossRef] [PubMed]

OCIS Codes
(140.5960) Lasers and laser optics : Semiconductor lasers
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: June 25, 2010
Revised Manuscript: August 11, 2010
Manuscript Accepted: August 12, 2010
Published: September 3, 2010

Citation
G. Sun, R. A. Soref, and H. H. Cheng, "Design of a Si-based lattice-matched room-temperature GeSn/GeSiSn multi-quantum-well mid-infrared laser diode," Opt. Express 18, 19957-19965 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19957


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References

  1. V. R. D'Costa, Y.-Y. Fang, J. Tolle, J. Kouvetakis, and J. Menéndez, “Ternary GeSiSn alloys: New opportunities for strain and band gap engineering using group-IV semiconductors,” Thin Solid Films 518(9), 2531–2537 (2010). [CrossRef]
  2. R. A. Soref, J. Kouvetakis and J. Menendez, “Advances in SiGeSn/Ge technology,” Mater. Res. Soc. Symp. Proc., 958, 0958–L01–08 (2007).
  3. G. Sun, R. A. Soref, and H. H. Cheng, “Design of an electrically pumped SiGeSn/GeSn/SiGeSn double-heterostructure mid-infrared laser,” J. Appl. Phys. 108(3), 033107 (2010). [CrossRef]
  4. J. Tolle, R. Roucka, V. D'Costa, J. Menendez, A. Chizmeshya, and J. Kouvetakis “Sn-based group-IV semiconductors on Si: new infrared materials and new templates for mismatched epitaxy,” Mater. Res. Soc. Symp. Proc.891, 0891–EE12–08 (2006).
  5. G.-E. Chang, S.-W. Chang, and S. L. Chuang, “Theory for n-type doped, tensile-strained Ge-Si(x)Ge(y)Sn1-x-y quantum-well lasers at telecom wavelength,” Opt. Express 17(14), 11246–11258 (2009). [CrossRef] [PubMed]
  6. Y.-H. Zhu, Q. Xu, W.-J. Fan, and J.-W. Wang, “Theoretical gain of strained GeSn0.2/Ge1-x-y’SixSny’ quantum well laser,” J. Appl. Phys. 107(7), 073108 (2010). [CrossRef]
  7. G. Sun, H. H. Cheng, J. Menendez, J. B. Khurgin, and R. A. Soref, “Strain-free Ge/GeSiSn quantum cascade lasers based on L-valley intersubband transitions,” Appl. Phys. Lett. 90(25), 251105 (2007). [CrossRef]
  8. R. A. Soref and C. H. Perry, “Predicted band gap of the new semiconductor SiGeSn,” J. Appl. Phys. 69(1), 539 (1991). [CrossRef]
  9. H. P. L. de Guevara, A. G. Rodriguez, H. Navarro-Contreras, and M. A. Vidal, “Nonlinear behavior of the energy gap in Ge1-xSnx alloys at 4 K,” Appl. Phys. Lett. 91(16), 161909 (2007). [CrossRef]
  10. M. Jaros, “Simple analytic model for heterojunction band offsets,” Phys. Rev. B Condens. Matter 37(12), 7112–7114 (1988). [CrossRef] [PubMed]
  11. S. Adachi, Properties of Group-IV, III–V, and II–VI Semiconductors, (John Wiley and Sons, England, 2005)
  12. V. R. D'Costa, C. S. Cook, A. G. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, and J. Menendez, “Optical critical points of thin-film Ge1−ySny alloys: A comparative Ge1−ySny/Ge1−xSix study,” Phys. Rev. B 73(12), 125207 (2006). [CrossRef]
  13. V. R. D'Costa, C. S. Cook, J. Menendez, J. Tolle, J. Kouvetakis, and S. Zollner, “Transferability of optical bowing parameters between binary and ternary group-IV alloys,” Solid State Commun. 138(6), 309–313 (2006). [CrossRef]
  14. J. Weber and M. I. Alonso, “Near-band-gap photoluminescence of Si-Ge alloys,” Phys. Rev. B Condens. Matter 40(8), 5683–5693 (1989). [CrossRef] [PubMed]
  15. M. L. Cohen and T. K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev. 141(2), 789–796 (1966). [CrossRef]
  16. G. P. Agrawal, and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Company Inc. New York, 1986).
  17. Y.-Y. Fang, J. Xie, J. Tolle, R. Roucka, V. R. D’Costa, A. V. G. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular-based synthetic approach to new group IV materials for high-efficiency, low-cost solar cells and Si-based optoelectronics,” J. Am. Chem. Soc. 130(47), 16095–16102 (2008). [CrossRef] [PubMed]

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