Open Access
Add to BibSonomy
Abstract
We propose a lattice-matched Ge/GeSiSn quantum cascade detector (QCD) capable of operating in the longwave infrared. The optical absorption and carrier transport based on intersubband transitions all occur within the L-valley of the conduction band of the group-IV material system using N-doped quantum wells (QWs). The waveguided lattice matched structure can be deposited strain free on top of a Ge buffer grown on Si substrate, and is end-coupled to low-loss on-chip Ge waveguides. We optimized the QCD structure through the analysis of the photoresponsivity and detectivity D*. The QCD operates in photovoltaic mode with narrow spectral response that is peaked anywhere in the 9 to 16 µm range, tunable by design. This work aims to push the optical response of the photodetectors made from the SiGeSn material system to longer wavelengths. The study suggests the QCD response can indeed significantly extend the spectral range beyond that of the photodiodes and photoconductors made from the same group-IV system for a wide variety of applications in imaging, sensing, lidar, and space-and-fiber communications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Silicon, as the most successful semiconductor material for the electronics industry, has severe limitations when it comes to photonics applications owing to its indirect bandgap. Among the complete set of photonic devices needed for Si-based optoelectronic integrated circuits (OEICs), light emitting devices such as LEDs and lasers, as well as photodetectors (PDs), are the most challenging components to develop because the large momentum mismatch between the conduction-band electrons and valence-band holes across the indirect bandgap makes them highly inefficient to interact with photons which have practically zero momentum. Given the promise of tremendous economic payoff in developing Si-based CMOS-compatible photonics, a concerted effort in material engineering is being directed towards a fully monolithic approach, named group-IV photonics, that integrates photonic devices made of group-IV elements such as Si, Ge, Sn and their binary and ternary alloys on Si substrates [1,2]. Despite the material challenges, exciting breakthroughs have recently been reported on the device front [3–19]. Although improvement in device performance is still needed for commercial applications, the successful demonstration of these devices has established their operating feasibility in the short-wave infrared (SWIR) (1-3 µm) to the mid infrared (MIR) (3-6 µm) range. At the heart of material engineering is the incorporation of α-Sn, which is a semi-metal, into the group-IV semiconductors such as Si, Ge or SiGe alloy, altering the material’s energy band structure. Gradual increase of the α-Sn composition in Ge has resulted in reduction of its bandgap, pushing its spectral response to longer IR wavelengths. Indeed, PDs made of GeSn alloys have exhibited optical response with cut-off of 4.6 µm [3–19]. Further increase of the Sn composition beyond a certain critical value, has shown that GeSn turns into a direct bandgap alloy. LEDs [13–16], optically pumped lasers [17–19] and laser diodes [20] have been demonstrated in the MIR range. While in principle the operating wavelengths can be pushed out further into the long-wave infrared (LWIR) (6-15 µm) region by incorporating even more Sn, it is a given reality that the higher Sn composition is becoming harder and harder to achieve in material growth without sacrificing material quality because of the large lattice mismatch between Si, Ge and Sn. Furthermore, even if we can continue to push the limit and are able to further reduce the material bandgap, the device performance based on band-to-band transitions will deteriorate near room temperature due to thermal issues and will be limited to operation at cryogenic temperatures.
One way to overcome the challenge is to explore intersubband transitions in quantum wells (QWs) where optical transitions occur between quantum confined subbands within the conduction band (CB). Both quantum cascade lasers (QCLs) [21] and quantum cascade detectors (QCDs) [22] are such devices that have been demonstrated in III-V and II-VI materials and the devices operate in the wide spectral range of SWIR through MIR and LWIR to THz. These devices are unipolar and their operating wavelength no longer depends upon the value of the material bandgap. The devices can be tuned by designing QW structures with proper barrier and well thicknesses. Such devices can also be developed in the group-IV system to expand the group-IV photonics into the LWIR and potentially THz range. This approach alleviates the material growth challenge in pushing for higher Sn incorporation in the group-IV system in order to reach longer wavelength spectral regimes. In other words, one no longer requires higher Sn compositions to reduce the bandgap; instead only enough Sn in the mix for a “longwave” CB offset between the QW and barrier. These intersubband devices are expected to have much improved temperature performance in comparison to their band-to-band counterparts operating in the same wavelength range. A THz QCL based on lattice- matched Ge/GeSiSn with only 5% Sn was proposed and predicted to operate at room temperature [23]. In this paper, we show that such a Ge/Ge0.76Si0.19Sn0.05 QW structure can also be designed to work as a QCD with LWIR response. What is different from the existing QCDs that are constructed in III-V or II-VI material systems is that the electrons responsible for optical absorption and photocurrent in the Ge/Ge0.76Si0.19Sn0.05 QCD reside in the L-valleys of the CB instead of in the Γ-valley. We present a QCD design that employs the scheme of diagonal optical transition which occurs between two subbands that are localized in two neighboring Ge QWs separated by a lattice-matched Ge0.76Si0.19Sn0.05 barrier. Such a design has been shown in III-V QCDs to hold advantages over the conventional QCD design in which the optical absorption takes places vertically between the ground-state and excited-state subbands confined in the same QW [24]. By eliminating the need for resonant tunneling to extract excited electrons (which is highly sensitive to the subband alignment), the diagonal transition scheme offers improvement on the extraction efficiency and an increase of the device resistance, resulting in higher optical responsivity and specific detectivity. The current design uses nonpolar LO-phonon scattering to divert the excited electrons to the next period, physically displacing electrons in a cascading manner that is eventually collected as photocurrent. The performance of the proposed Ge/Ge0.76Si0.19Sn0.05 QCD is modeled by analyzing the underlying physical processes such as optical absorption, electron-phonon scattering, population dynamics, and thermally induced resistance. The optimization of the QCD design is carried out as a trade-off between the optical transition strength and suppression of the excited electrons being scattered back to their originating ground-state subband. This work brings forth the feasibility of extending the detecting functionality of the group-IV material system far beyond the demonstrated MIR operation to the LWIR range.
2. QCD design
It is well known that alloys of GeSiSn have their CB valleys at L, Γ, and X points of the Brillouin zone that are close to one another in energy, potentially complicating the QCD design. It is desirable to isolate the intersubband transitions due to either optical absorption or phonon scatterings in the QCD within one particular CB valley to simplify the design. It has been established in a previous study that the CB offset in the lattice-matched Ge/Ge0.76Si0.19Sn0.05 QW has the value of around 150 meV at the L-valley based on the Jaros’ band offset theory that has been shown to yield good agreement with experiment for many heterojunction systems [25] as shown in the inset of Fig. 1(a) where the other two CB valleys ($\varGamma ,\;X$) are all sitting above the L-valley of Ge0.76Si0.19Sn0.05 barriers that confine the Ge well. Thus the QCD can be designed without the complexity arising from $\varGamma $ and $X$-valleys.
Fig. 1. Illustration of (a) the L-valley CB with its three subbands and associated envelop functions and intersubband transitions including optical absorption and phonon scattering processes and (b) the lattice matched Ge/GeSiSn QCD designed for waveguide operation.
Since the photocurrent generated by the optical absorption in QCD relies on the phonon scattering for electrons to move from one QW to the next, the energy separation of the subbands in the Ge/Ge0.76Si0.19Sn0.05 QW structure should be slightly higher than the optical phonon energy to facilitate the emission of optical phonons – a process known to be efficient in intersubband scattering. If we use the optical phonon energy of 36 meV in Ge, it is not difficult to see that the band offset of 150 meV allows for three subbands that can be supported by three QWs in one period of the Ge/Ge0.76Si0.19Sn0.05 QCD as shown in Fig. 1(a) where two cascading periods are illustrated. This QCD design employs the scheme of diagonal transition between the ground state 1 and excited state 3 for the optical absorption which has been shown to yield better performance for QCDs made of InGaAs/InAlAs QWs on InP substrate because of the higher extraction efficiency [24]. Under the illumination of an LWIR signal, the electron initially sitting in subband 1 (ground state) can absorb a photon and makes the transition to subband 3. This electron can be scattered by either an acoustic or optical phonon towards subband 2 which is confined in the QW on the right at the scattering rate of ${\gamma _{32}}$, followed by another scattering process to subband 1’ at the rate of ${\gamma _{21^{\prime}}}$, which is the ground-state subband in the next period, resulting in the displacement of the electron from one period to another, which is the origin of the photocurrent. It is not difficult to see the QCD is a photovoltaic device that requires no bias to operate. The intersubband optical absorption requires that the optical fields in the LWIR signal be polarized perpendicular to the QW plane in the QCD. While it can operate under normal incidence with gratings or wedge to couple signals in, the QCD is most efficient working as a waveguided detector as shown in Fig. 1(b), in which the IR signals are TM-polarized. We shall analyze and optimize the lattice matched Ge/Ge0.76Si0.19Sn0.05 QCD operating in the waveguided mode.
In order to obtain strong photocurrent, the QCD should be designed in such a way that the backscattering rate ${\gamma _{31}}$ from subband 3 to subband 1 is minimized to prevent photogenerated elections from falling back, which obviously requires a thicker barrier ${b_{31}}$ to reduce the electron envelop function overlap between subbands 1 and 3. On the other hand, the optical absorption responsible for exciting electrons from subband 1 to 3 favors a thinner barrier ${b_{31}}$ with larger overlapping of the same envelop functions. An optimization is therefore necessary. We have considered a lattice-matched Ge/Ge0.76Si0.19Sn0.05 QCD structure with the following layer thicknesses in a period as 2.0/5.5/${b_{13}}$/2.0/2.0/2.5 all in nm. The three QWs indicated in bold of 5.5, 2.0 and 2.5 nm in thickness are chosen to yield three subbands with energy differences between them are slightly above the optical phonon energy of 36 meV when separated by 2.0-nm barriers except ${b_{13}}$ which needs to be optimized by considering scattering rates and the optical absorption between the involved subbands. Our calculation indicates that the subband energies remain roughly constant as we vary ${b_{13}}$ and the subbands 1 and 3 are separated by about 90 meV, corresponding to $\lambda \sim $13.8 µm at which the QCD is designed to operate. This wavelength can be tuned with variation of the QW thicknesses.
The scattering rate from subband i to j should take into account both acoustic and optical phonon scattering which can be written respectively as [26]
The scattering time from subband i to j is related to the acoustic and optical phone scattering rate as,
The absorption rate per QCD period, on the other hand, depending on the dipole matrix element $|\left\langle 1\textrm{|}z\textrm{|}3\right\rangle{|^2}$, can be written as [27]
The barrier thickness ${b_{13}}$ in the QCD should be optimized between the absorption rate from subband 1 and 3, ${\gamma _a}$, and the scattering branching ratio ${\tau _{31}}/{\tau _{32}}$ which is the ratio of the scattering rate from subband 3 to 2, $\tau _{32}^{ - 1}$, to that of 3 to 1, $\tau _{31}^{ - 1}$. The ${b_{13}}$-dependence of the scattering time in Fig. 2(a) calculated at 300 K shows that the scattering time ${\tau _{31}}$ increases rapidly with ${b_{13}}$ which is expected as the spatial separation between the envelop functions of subbands 1 and 3 increases, resulting in the reduced phonon scattering matrix ${G_{ij}}({{q_z}} )$ in Eqs. (1) and (2) which in turn reduces the scattering rate from subband 3 to 1, $\tau _{31}^{ - 1}$. The scattering time ${\tau _{32}}$, on the other hand, shows little dependence on ${b_{13}}$ since the spatial separation between subbands 3 and 2 are largely unaffected. As a result, the branching ratio ${\tau _{31}}/{\tau _{32}}$ clearly favors larger barrier thickness ${b_{13}}$ as shown in Fig. 2(b). While the thicker ${b_{13}}$ lowers the scattering rate $\tau _{31}^{ - 1}$, it also reduces the dipole matrix element $|\left\langle{1\textrm{|}z\textrm{|}3}\right\rangle |$ in Eq. (4), resulting in smaller optical absorption rate ${\gamma _a}$. The optimization of ${b_{13}}$ is a combination of these factors that ultimately determine the performance of the QCD.
Fig. 2. (a) Phonon scattering times ${\tau _{31}}$, ${\tau _{32}}$ and (b) dipole matrix $|{\left\langle 1\textrm{|}z\textrm{|}3\right\rangle} |$ and branching ratio ${\tau _{31}}/{\tau _{32}}$ as a function of barrier thickness ${b_{13}}$ calculated at 300 K.
3. Optimization and performance analysis
The photoresponsivity of QCD can be calculated through the following population analysis. Consider that the QCD structure is doped to produce total areal density of electrons, ${{\boldsymbol N}_{\boldsymbol d}} = {\bar{{\boldsymbol N}}_1} + {\bar{{\boldsymbol N}}_2} + $ ${\bar{{\boldsymbol N}}_3} + $ ${\bar{{\boldsymbol N}}_{\boldsymbol C}}$, that is distributed among subband ${\boldsymbol i}$ (${\bar{{\boldsymbol N}}_{\boldsymbol i}},{\boldsymbol i} = 1,2,3$), and in the CB continuum sitting above the barriers (${\bar{{\boldsymbol N}}_{\boldsymbol C}})$ at thermal equilibrium under dark conditions. With the incidence of an IR signal in close resonance to the subband energy separation, ${{\boldsymbol E}_3} - {{\boldsymbol E}_1}$, population dynamics can be analyzed with the optical absorption and scattering processes as,
Under the condition of weak signals, we should have $\frac{1}{{{\gamma _a}}}\left( {1 + \frac{{{\tau_{32}}}}{{{\tau_{31}}}}} \right) \gg {\tau _{{{21}^{\prime}}}} + 2{\tau _{32}}$, the photocurrent can be approximated as ${J_{ph}} \approx e({{{\bar{N}}_1} - {{\bar{N}}_3}} ){\gamma _a}/\left( {1 + \frac{{{\tau_{32}}}}{{{\tau_{31}}}}} \right)$, and using Eq. (4), we obtain the photoresponsivity for one QCD period,
Fig. 3. (a) Photoresponsivity as a function of the as a function of barrier thickness ${b_{13}}$ calculated at 77 and 300 K and (b) its temperature dependence for the optimized QCD with ${b_{13}}$=3.0 nm.
An important figure of merit for detectors is the detectivity, which is normally defined as $D = 1/NEP$ where $NEP$ is a noise equivalent power. In the absence of bias as is the case for QCD, the dark current noise is of thermal (Johnson) nature and can be found as
where $\Delta f$ is the signal bandwidth (determined by the electrical filter) and R is the resistance that can be calculated as where $\rho $ is the resistivity of the QCD, ${l_{QCD}}$, ${l_{abs}}$ and w are the dimensions of the QCD as shown in Fig. 1(b). The thickness of QCD is simply, ${l_{QCD}} = {N_{QCD}}L$, where ${N_{QCD}}$ is the number of periods and L the period of the QCD. The smallest dark current noise is achieved with QCD resistance of smallest resistance area, $w{l_{abs}}$, where the optical absorption length can be calculated through the optical absorption coefficient, ${l_{abs}} = 1/\alpha $, which is related to the absorption rate ${\gamma _a}$ per period through $\alpha = {N_{QCD}}{\alpha _0}$ in which the absorption coefficient per QCD period of L, ${\alpha _0} = \hbar \omega ({{N_1} - {N_3}} ){\gamma _a}/LI$. With Eq. (4), we thus obtainThus the detectivity is
where the photoresponsivity is related to that calculated for one period in Eq. (7) as ${R_{ph}} = {R_{p0}}/{N_{QCD}}$. It follows that the detectivity D does not depend on the number of periods ${N_{QCD}}$ because ${l_{QCD}} = {N_{QCD}}L$ and ${l_{abs}} = 1/\alpha = 1/{N_{QCD}}{\alpha _0}$.Unlike the normal detector where the optical area can be reduced in order to reduce the resistance area and hence the Johnson noise, the optical area remains fixed in the waveguide. Determined by the optical confinement, both the width w and the thickness ${l_{QCD}}$ [Fig. 1(b)] are on the scale of $\lambda /2n$, hence it only makes sense to normalize detectivity to the bandwidth and introduce
The resistivity $\rho = L/{\bar{N}_c}e{\mu _{QCD}}$ depends on the electron areal density ${\bar{N}_c}$ per period in the CB continuum which is responsible for the Johnson noise. We have calculated electron densities thermally distributed in various subbands ${\bar{N}_i}$ ($i = 1,2,3$) and continuum ${\bar{N}_c}$, and the results are shown in Fig. 4(a) for a fixed doping of 1011/cm2 per QCD period. As the temperature increases, electron densities in the excited subbands (${\bar{N}_2}$ and ${\bar{N}_3}$) and continuum (${\bar{N}_c}$) all go up, but the increase of ${\bar{N}_c}$ outpaces ${\bar{N}_2}$ and ${\bar{N}_3}$ because the 3D density of states (DOS) in the continuum is much higher than the 2D DOS of the confined subbands in the QCD. Indeed, ${\bar{N}_c}$ surpasses ${\bar{N}_3}$ and ${\bar{N}_2}$ at the temperature of 125 K and 300 K, respectively.
Fig. 4. Temperature dependence of (a) electron densities in subbands ${\bar{N}_i}$ ($i = 1,2,3$) and continuum ${\bar{N}_c}$, and (b) detectivity $D\mathrm{^{\prime}}$.
Another parameter that $\rho $ depends upon is the mobility of QCD, ${\mu _{QCD}}$. Since the electron transport in the QCD following the optical excitation relies on phonon scattering not the electrical conductance, it is interesting to see that the detectivity of the QCD actually favors smaller electron mobility. For QCDs constructed with large number of coupled QWs with multiple interfaces, its actual mobility is expected to be much lower than those of constituent materials of Ge and Ge0.76Si0.19Sn0.05. For lack of reliable experimental value of the QCD, we have used the electron mobility of Ge in the calculation which should lead to a conservative estimate of the detectivity. The result is shown in Fig. 4(b) where the strong temperature dependence is due to the combination of temperature dependence of electron density in the continuum ${\bar{N}_c}$ as well as the photoresponsivity ${R_{p0}}$.
4. Conclusion
In summary, we proposed a Ge/Ge0.76Si0.19Sn0.05 QCD that can operate in the 9 to 16 um longwave-infrared range. The QCD operates within the L-valley CB with a band offset of 150 meV between the Ge QW and the lattice matched Ge0.76Si0.19Sn0.05 barrier. The whole structure can be grown strain free over a Ge buffer on a Si substrate. Through the analysis of the optical absorption and phonon scattering processes as well as the electron population dynamics, we have optimized the Ge/Ge0.76Si0.19Sn0.05 QCD and determined its photoresponsivity and detectivity. The results suggest that optical response of the waveguided Ge/Ge0.76Si0.19Sn0.05 QCD can be extended deep into the LWIR far beyond the reach of the photodiodes or photoconductors made from the same group-IV material system that rely on band-to-band transitions. By exploring other alloy combinations, it is feasible to expand the operating wavelength spectrum.
Funding
Air Force Office of Scientific Research (FA9550-19-1-0341, FA9550-21-1-0347); Ministry of Science and Technology, Taiwan (MOST 110-2636-E-194-002, MOST 111-2636-E-194-002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]
2. R. Soref, D. Buca, and S.-Q. Yu, “Group IV photonics: driving integrated optoelectronics,” Opt. Photonics News 27(1), 32–39 (2016). [CrossRef]
3. T. Pham, W. Du, B. Conley, J. Margetis, G. Sun, R. Soref, J. Tolle, B. Li, and S. Q. Yu, “Si-based Ge0. 9Sn0. 1 photodetector with peak responsivity of 2.85 A/W and longwave cutoff at 2.4 µm,” Electron. Lett. 51(11), 854–856 (2015). [CrossRef]
4. H. Tran, T. Pham, J. Margetis, Y. Zhou, W. Dou, P. C. Grant, J. M. Grant, S. Al-Kabi, G. Sun, and R. A. Soref, “Si-based GeSn photodetectors toward mid-infrared imaging applications,” ACS Photonics 6(11), 2807–2815 (2019). [CrossRef]
5. S. Su, B. Cheng, C. Xue, W. Wang, Q. Cao, H. Xue, W. Hu, G. Zhang, Y. Zuo, and Q. Wang, “GeSn pin photodetector for all telecommunication bands detection,” Opt. Express 19(7), 6400–6405 (2011). [CrossRef]
6. M. Oehme, M. Schmid, M. Kaschel, M. Gollhofer, D. Widmann, E. Kasper, and J. Schulze, “GeSn pin detectors integrated on Si with up to 4% Sn,” Appl. Phys. Lett. 101(14), 141110 (2012). [CrossRef]
7. H. Tseng, H. Li, V. Mashanov, Y. Yang, H. Cheng, G. Chang, R. Soref, and G. Sun, “GeSn-based pin photodiodes with strained active layer on a Si wafer,” Appl. Phys. Lett. 103(23), 231907 (2013). [CrossRef]
8. H. Tran, T. Pham, W. Du, Y. Zhang, P. C. Grant, J. M. Grant, G. Sun, R. A. Soref, J. Margetis, and J. Tolle, “High performance Ge0. 89Sn0. 11 photodiodes for low-cost shortwave infrared imaging,” J. Appl. Phys. 124(1), 013101 (2018). [CrossRef]
9. Y. Dong, W. Wang, S. Y. Lee, D. Lei, X. Gong, W. K. Loke, S.-F. Yoon, G. Liang, and Y.-C. Yeo, “Germanium-tin multiple quantum well on silicon avalanche photodiode for photodetection at two micron wavelength,” Semicond. Sci. Technol. 31(9), 095001 (2016). [CrossRef]
10. C.-H. Tsai, B.-J. Huang, R. A. Soref, G. Sun, H. Cheng, and G.-E. Chang, “GeSn resonant-cavity-enhanced photodetectors for efficient photodetection at the 2 µm wavelength band,” Opt. Lett. 45(6), 1463–1466 (2020). [CrossRef]
11. K.-C. Lee, M.-X. Lin, H. Li, H.-H. Cheng, G. Sun, R. Soref, J. R. Hendrickson, K.-M. Hung, P. Scajev, and A. Medvids, “Planar GeSn photodiode for high-detectivity photodetection at 1550 nm,” Appl. Phys. Lett. 117(1), 012102 (2020). [CrossRef]
12. M. R. Atalla, S. Assali, A. Attiaoui, C. Lemieux-Leduc, A. Kumar, S. Abdi, and O. Moutanabbir, “All-Group IV Transferable Membrane Mid-Infrared Photodetectors,” Adv. Funct. Mater. 31(3), 2006329 (2021). [CrossRef]
13. H. Tseng, K. Wu, H. Li, V. Mashanov, H. Cheng, G. Sun, and R. Soref, “Mid-infrared electroluminescence from a Ge/Ge0. 922Sn0. 078/Ge double heterostructure pin diode on a Si substrate,” Appl. Phys. Lett. 102(18), 182106 (2013). [CrossRef]
14. B. Schwartz, M. Oehme, K. Kostecki, D. Widmann, M. Gollhofer, R. Koerner, S. Bechler, I. A. Fischer, T. Wendav, and E. Kasper, “Electroluminescence of GeSn/Ge MQW LEDs on Si substrate,” Opt. Lett. 40(13), 3209–3212 (2015). [CrossRef]
15. W. Du, Y. Zhou, S. A. Ghetmiri, A. Mosleh, B. R. Conley, A. Nazzal, R. A. Soref, G. Sun, J. Tolle, and J. Margetis, “Room-temperature electroluminescence from Ge/Ge1-xSnx/Ge diodes on Si substrates,” Appl. Phys. Lett. 104(24), 241110 (2014). [CrossRef]
16. B.-J. Huang, C.-Y. Chang, Y.-D. Hsieh, R. A. Soref, G. Sun, H.-H. Cheng, and G.-E. Chang, “Electrically injected GeSn vertical-cavity surface emitters on silicon-on-insulator platforms,” ACS Photonics 6(8), 1931–1938 (2019). [CrossRef]
17. S. Wirths, R. Geiger, N. Von Den Driesch, G. Mussler, T. Stoica, S. Mantl, Z. Ikonic, M. Luysberg, S. Chiussi, and J.-M. Hartmann, “Lasing in direct-bandgap GeSn alloy grown on Si,” Nat. Photonics 9(2), 88–92 (2015). [CrossRef]
18. V. Reboud, A. Gassenq, N. Pauc, J. Aubin, L. Milord, Q. Thai, M. Bertrand, K. Guilloy, D. Rouchon, and J. Rothman, “Optically pumped GeSn micro-disks with 16% Sn lasing at 3.1 µ m up to 180 K,” Appl. Phys. Lett. 111(9), 092101 (2017). [CrossRef]
19. Y. Zhou, W. Dou, W. Du, S. Ojo, H. Tran, S. A. Ghetmiri, J. Liu, G. Sun, R. Soref, and J. Margetis, “Optically pumped GeSn lasers operating at 270 K with broad waveguide structures on Si,” ACS Photonics 6(6), 1434–1441 (2019). [CrossRef]
20. Y. Zhou, Y. Miao, S. Ojo, H. Tran, G. Abernathy, J. M. Grant, S. Amoah, G. Salamo, W. Du, and J. Liu, “Electrically injected GeSn lasers on Si operating up to 100 K,” Optica 7(8), 924–928 (2020). [CrossRef]
21. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef]
22. L. Gendron, M. Carras, A. Huynh, V. Ortiz, C. Koeniguer, and V. Berger, “Quantum cascade photodetector,” Appl. Phys. Lett. 85(14), 2824–2826 (2004). [CrossRef]
23. R. Soret, G. Sun, H. Cheng, J. Menendez, and J. Khurgin, “Strain-free Ge/GeSiSn Quantum Cascade Lasers Based on L-Valley Intersubband Transitions,” Appl. Phys. Lett. 90(25), 251105 (2007). [CrossRef]
24. P. Reininger, B. Schwarz, H. Detz, D. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, O. Baumgartner, H. Kosina, and G. Strasser, “Diagonal-transition quantum cascade detector,” Appl. Phys. Lett. 105(9), 091108 (2014). [CrossRef]
25. M. Jaros, “Simple analytic model for heterojunction band offsets,” Phys. Rev. B 37(12), 7112–7114 (1988). [CrossRef]
26. B. K. Ridley, Electrons and phonons in semiconductor multilayers (Cambridge University Press, 2009).
27. G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29(4), 1104–1111 (1993). [CrossRef]
