Abstract

Silicon-based lasers have long been sought, since they will permit monolithic integration of photonics with high-speed silicon electronics and thereby significantly broaden the reach of silicon technology. Among the various approaches that are currently being pursued to overcome the intrinsic limitations of Si as an efficient light source, intersubband transitions in Si-based quantum well structures offer a rather feasible alternative that conveniently circumvents the indirect-band nature of Si. Various approaches for achieving lasing action based on intersubband transitions within the group-IV materials are reviewed. Relevant theories are presented in detail. Challenges facing the valence band approach, which has not so far been successful, are analyzed, and proposals that bring the intersubband process to the conduction band are discussed.

© 2010 Optical Society of America

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2008 (5)

Y. Bai, S. Slivken, S. R. Darvish, M. Razeghi, “Room temperature continuous wave operation of quantum cascade lasers with 12.5% wall plug efficiency,” Appl. Phys. Lett. 93, 021103 (2008).
[Crossref]

A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanbun-Ek, R. Maulini, A. Tsekoun, R. Go, C. Kumar, N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature quantum cascade laser emitting at 4.6 μm,” Appl. Phys. Lett. 92, 111110 (2008).
[Crossref]

D. J. Paul, G. Matman, L. Lever, Z. Ikonić, R. W. Kelsall, D. Chrastina, G. Isella, H. von Känel, E. Müller, A. Neels, “Si∕SiGe bound-to-continuum terahertz quantum cascade emitters,” ECS Trans. 16, 865–874 (2008).
[Crossref]

D. J. Paul, G. Matmon, L. Lever, Z. Ikonić, R. W. Kelsall, D. Chrastina, G. Isella, H. von Känel, “Si∕SiGe bound-to-continuum quantum cascade terahertz emitters,” Proc. SPIE 6996, 69961C (2008).
[Crossref]

L. Lever, A. Valavanis, Z. Ikonić, R. W. Kelsall, “Simulated [111] Si–SiGe THz quantum cascade laser,” Appl. Phys. Lett. 92, 021124 (2008).
[Crossref]

2007 (5)

B. S. Williams, “Tereahertz quantum-cascade lasers,” Nat. Photonics 1, 517–525 (2007).
[Crossref]

D. G. Revin, J. W. Cockburn, M. J. Steer, R. J. Airey, M. Hopkinson, A. B. Krysa, L. R. Wilson, S. Menzel, “InGaAs∕AlAsSb∕InP quantum cascade lasers operating at wavelengths close to 3 μm,” Appl. Phys. Lett. 90, 021108 (2007).
[Crossref]

M. P. Semtsiv, M. Wienold, S. Dressler, W. T. Masselink, “Short-wavelength (λ≈3.05 μm) InP-based strain-compensated quantum-cascade laser,” Appl. Phys. Lett. 90, 051111 (2007).
[Crossref]

M. A. Wistey, Y.-Y. Fang, J. Tolle, A. V. G. Chizmeshya, J. Kouvetakis, “Chemical routes to Ge∕Si(100) structures for low temperature Si-based semiconductor applications,” Phys. Rev. Lett. 90, 082108 (2007).

G. Sun, H. H. Cheng, J. Menéndez, J. B. Khurgin, R. A. Soref, “Strain-free Ge∕GeSiSn quantum cascade lasers based on L-valley intersubband transitions,” Phys. Rev. Lett. 90, 251105 (2007).

2006 (4)

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

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

B. Jalali, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006).
[Crossref]

K. Driscoll, R. Paiella, “Silicon-based injection lasers using electrical intersubband transitions in the L valleys,” Appl. Phys. Lett. 89, 191110 (2006).
[Crossref]

2005 (4)

J. H. Park, A. J. Steckl, “Demonstration of a visible laser on silicon using Eu-doped GaN thin films,” J. Appl. Phys. 98, 056108 (2005).
[Crossref]

Z. Mi, P. Bhattacharya, J. Yang, K. P. Pipe, “Room-temperature self-organized In0.5Ga0.5As quantum dot laser on silicon,” Electron. Lett. 41, 742–744 (2005).
[Crossref]

H. Park, A. W. Fang, S. Kodama, J. E. Bowers, “Hybrid silicon evanescent laser fabricated with a silicon waveguide and III–V offset quantum wells,” Opt. Express 13, 9460–9464 (2005).
[Crossref] [PubMed]

R. Roucka, J. Tolle, C. Cook, A. V. G. Chizmeshya, J. Kouvetakis, V. D’Costa, J. Menendez, Zhihao D. Chen, S. Zollner, “Versatile buffer layer architectures based on Ge1−xSnx alloys,” Phys. Rev. Lett. 86, 191912 (2005).

2004 (6)

P. Aella, C. Cook, J. Tolle, S. Zollner, A. V. G. Chizmeshya, J. Kouvetakis, “Optical and structural properties of SixSnyGe1−x−y alloys,” Phys. Rev. Lett. 84, 888–890 (2004).

J. Menéndez, J. Kouvetakis, “Type-I Ge∕Ge1−x−ySixSny strained-layer heterostructures with a direct Ge bandgap,” Phys. Rev. Lett. 85, 1175–1177 (2004).

A. Liu, H. Rong, M. Paniccia, O. Cohen, D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004).
[Crossref] [PubMed]

O. Boyraz, B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[Crossref] [PubMed]

M. J. Chen, J. L. Yen, J. Y. Li, J. F. Chang, S. C. Tsai, C. S. Tsai, “Stimulated emission in a nanostructured silicon pn junction diode using current injection,” Appl. Phys. Lett. 84, 2163–2165 (2004).
[Crossref]

H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, J. L. Reno, “Importance of electron impurity scattering for electron transport in terahertz quantum-cascade lasers,” Appl. Phys. Lett. 84, 645–647 (2004).
[Crossref]

2003 (6)

R. Bates, S. A. Lynch, D. J. Paul, Z. Ikonic, R. W. Kelsall, P. Harrison, S. L. Liew, D. J. Norris, A. G. Cullis, W. R. Tribe, D. D. Arnone, “Interwell intersubband electroluminescence from Si∕SiGe quantum cascade emitters,” Appl. Phys. Lett. 83, 4092–4094 (2003).
[Crossref]

J. Ruan, P. M. Fauchet, L. Dal Negro, C. Mazzoleni, L. Pavesi, “Stimulated emission in nanocrystalline silicon superlattice,” Appl. Phys. Lett. 83, 5479–5481 (2003).
[Crossref]

R. Calps, D. Dimitropoulos, V. Raghunathan, Y. Han, B. Jalali, “Observation of stimulated Raman scattering in silicon waveguides,” Opt. Express 11, 1731–1739 (2003).
[Crossref]

M. Bauer, C. Ritter, P. A. Crozier, J. Ren, J. Menéndez, G. Wolf, J. Kouvetakis, “Synthesis of ternary SiGeSn semiconductors on Si(100) via SnxGe1−x buffer layers,” Phys. Rev. Lett. 83, 2163–2165 (2003).

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, G. Davies, “Far infrared (λ≃87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett. 82, 3165–3167 (2003).
[Crossref]

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, J. L. Reno, “Terahertz quantum cascade laser at λ∼100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83, 2124–2126 (2003).
[Crossref]

2002 (4)

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002).
[Crossref]

L. Diehl, S. Mentese, E. Müller, D. Grützmacher, H. Sigg, U. Gennser, I. Sagnes, Y. Campiedelli, O. Kermarrec, D. Bensahel, J. Faist, “Electroluminescence from strain-compensated Si0.2Ge0.8∕Si quantum cascade structures based on a bound-to-continuum transitions,” Appl. Phys. Lett. 81, 4700–4702 (2002).
[Crossref]

I. Bormann, K. Brunner, S. Hackenbuchner, G. Zindler, G. Abstreiter, S. Schmult, W. Wegscheider, “Midinfrared intersubband electroluminescence of Si∕SiGe quantum cascade structures,” Appl. Phys. Lett. 80, 2260–2262 (2002).
[Crossref]

S. A. Lynch, R. Bates, D. J. Paul, D. J. Norris, A. G. Cullis, Z. Ikonić, R. W. Kelsall, P. Harrison, D. D. Arnone, C. R. Pidgeon, “Intersubband electroluminescence from Si∕SiGe cascade emitters at terahertz frequencies,” Appl. Phys. Lett. 81, 1543–1545 (2002).
[Crossref]

2001 (4)

J. Faist, M. Beck, T. Aellen, E. Gini, “Quantum-cascade lasers based on a bound-to-continuum transition,” Appl. Phys. Lett. 78, 147–149 (2001).
[Crossref]

L. Friedman, G. Sun, R. A. Soref, “SiGe∕Si THz laser based on transitions between inverted mass light-hole and heavy-hole subbands,” Appl. Phys. Lett. 78, 401–403 (2001).
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R. A. Soref, G. Sun, “Terahertz gain in SiGe∕Si quantum staircase utilizing the heavy-hole inverted effective mass,” Appl. Phys. Lett. 79, 3639–3641 (2001).
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R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. 78, 2620–2622 (2001).
[Crossref]

2000 (2)

G. Dehlinger, L. Diehl, U. Gennser, H. Sigg, J. Faist, K. Ensslin, D. Grützmacher, E. Müller, “Intersubband electroluminescence from silicon-based quantum cascade structures,” Science 290, 2277 (2000).
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L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzo, F. Priolo, “Optical gain in silicon nanocrystals,” Nature 408, 440–444 (2000).
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1999 (1)

M. Hartig, J. D. Ganière, P. E. Selbmann, B. Deveaud, L. Rota, “Density dependence of carrier-carrier induced intersubband scattering in GaAs∕AlGaAs quantum wells,” Phys. Rev. B 60, 1500–1503 (1999).
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1998 (4)

L. Friedman, R. A. Soref, G. Sun, Y. Lu, “Theory of the strain-symmetrized silicon-based Ge-Si superlattice laser,” IEEE J. Sel. Top. Quantum Electron. 4, 1029–1034 (1998).
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L. Friedman, R. A. Soref, G. Sun, Y. Lu, “Asymmetric strain-symmetrized Ge–Si interminiband laser,” IEEE Photon. Technol. Lett. 10, 1715–1717 (1998).
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G. Sun, Y. Lu, J. B. Khurgin, “Valence intersubband lasers with inverted light-hole effective mass,” Appl. Phys. Lett. 72, 1481–1483 (1998).
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F. Priolo, G. Franzò, S. Coffa, A. Carnera, “Excitation and nonradiative deexcitation processes of Er3+ in crystalline Si,” Phys. Rev. B 57, 4443–4455 (1998).
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1997 (2)

G. Franzò, S. Coffa, “Mechanism and performance of forward and reverse bias electroluminescence at 1.54 μm from Er-doped Si diodes,” J. Appl. Phys. 81, 2784–2793 (1997).
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G. He, H. A. Atwater, “Interband transitions in SnxGe1−x alloys,” Phys. Rev. Lett. 79, 1937–1940 (1997).
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1996 (1)

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, M. S. Hybertson, A. Y. Cho, “Quantum cascade lasers without intersubband population inversion,” Phys. Rev. Lett. 76, 411–414 (1996).
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1995 (1)

G. Sun, L. Friedman, R. A. Soref, “Intersubband lasing lifetimes of SiGe∕Si and GaAs∕AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425–3427 (1995).
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1994 (2)

J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, C. Sirtori, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
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1993 (3)

R. A. Soref, “Silicon-based optoelectronics,” Proc. IEEE 81, 1687–1706 (1993).
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G. SunKhurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. QE-29, 1104–1111 (1993).
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1991 (1)

R. A. Soref, C. H. Perry, “Predicted bandgap of the new semiconductor SiGeSn,” J. Appl. Phys. 69, 539–541 (1991).
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1989 (4)

A. Fasolino, E. Molinari, J. C. Maan, “Resonant quasiconfined optical phonons in semiconductor superlattices,” Phys. Rev. B 39, 3923–3926 (1989).
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R. Ferreira, G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single-and multiple-quantum-well structures,” Phys. Rev. B 40, 1074–1086 (1989).
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C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Appl. Phys. Lett. 39, 1871–1883 (1989).

J. Weber, M. I. Alonso, “Near-band-gap photoluminescence of Si–Ge alloys,” Phys. Rev. B 40, 5683–5693 (1989).
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1988 (1)

M. Jaros, “Simple analytic model for heterojunction band offsets,” Appl. Phys. Lett. 37, 7112–7114 (1988).

1987 (1)

D. W. Jenkins, J. D. Dow, “Electronic properties of metastable GexSn1−x alloys,” Phys. Rev. B 36, 7994–8000 (1987).
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1986 (1)

R. A. Soref, J. P. Lorenzo, “All-silicon active and passive guided-wave components for λ=1.3 and 1.6 μm,” IEEE J. Quantum Electron. QE-22, 873–879 (1986).
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1985 (1)

L. C. West, S. J. Eglash, “First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well,” Appl. Phys. Lett. 46, 1156–1158 (1985).
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1982 (1)

B. K. Ridley, “The electron–phonon interaction in quasi-two-dimensional semiconductor quantum-well structures,” J. Phys. C 15, 5899–5917 (1982).
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1981 (2)

P. J. Price, “Two-dimensional electron transport in semiconductor layers. I. Phonon scattering,” Ann. Phys. 133, 217–239 (1981).
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S. R. White, L. J. Sham, “Electronic properties of flat-band semiconductor heterostructures,” Phys. Rev. Lett. 47, 879–882 (1981).
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1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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1971 (1)

R. F. Kazarinov, R. A. Suris, “Possibility of the amplification of electromagnetic waves in a semiconductor with a superlattice,” Sov. Phys. Semicond. 5, 707–709 (1971).

1970 (1)

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1966 (1)

M. L. Cohen, T. K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev. 141, 789–796 (1966).
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1957 (1)

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1, 249–261 (1957).
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1956 (1)

J. M. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: general theory,” Phys. Rev. 102, 1030–1041 (1956).
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1955 (1)

J. M. Luttinger, W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
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Abstreiter, G.

I. Bormann, K. Brunner, S. Hackenbuchner, G. Zindler, G. Abstreiter, S. Schmult, W. Wegscheider, “Midinfrared intersubband electroluminescence of Si∕SiGe quantum cascade structures,” Appl. Phys. Lett. 80, 2260–2262 (2002).
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Aella, P.

P. Aella, C. Cook, J. Tolle, S. Zollner, A. V. G. Chizmeshya, J. Kouvetakis, “Optical and structural properties of SixSnyGe1−x−y alloys,” Phys. Rev. Lett. 84, 888–890 (2004).

Aellen, T.

J. Faist, M. Beck, T. Aellen, E. Gini, “Quantum-cascade lasers based on a bound-to-continuum transition,” Appl. Phys. Lett. 78, 147–149 (2001).
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Airey, R. J.

D. G. Revin, J. W. Cockburn, M. J. Steer, R. J. Airey, M. Hopkinson, A. B. Krysa, L. R. Wilson, S. Menzel, “InGaAs∕AlAsSb∕InP quantum cascade lasers operating at wavelengths close to 3 μm,” Appl. Phys. Lett. 90, 021108 (2007).
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Ajili, L.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, G. Davies, “Far infrared (λ≃87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett. 82, 3165–3167 (2003).
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Alonso, M. I.

J. Weber, M. I. Alonso, “Near-band-gap photoluminescence of Si–Ge alloys,” Phys. Rev. B 40, 5683–5693 (1989).
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Arnone, D. D.

R. Bates, S. A. Lynch, D. J. Paul, Z. Ikonic, R. W. Kelsall, P. Harrison, S. L. Liew, D. J. Norris, A. G. Cullis, W. R. Tribe, D. D. Arnone, “Interwell intersubband electroluminescence from Si∕SiGe quantum cascade emitters,” Appl. Phys. Lett. 83, 4092–4094 (2003).
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S. A. Lynch, R. Bates, D. J. Paul, D. J. Norris, A. G. Cullis, Z. Ikonić, R. W. Kelsall, P. Harrison, D. D. Arnone, C. R. Pidgeon, “Intersubband electroluminescence from Si∕SiGe cascade emitters at terahertz frequencies,” Appl. Phys. Lett. 81, 1543–1545 (2002).
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Atwater, H. A.

G. He, H. A. Atwater, “Interband transitions in SnxGe1−x alloys,” Phys. Rev. Lett. 79, 1937–1940 (1997).
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Y. Bai, S. Slivken, S. R. Darvish, M. Razeghi, “Room temperature continuous wave operation of quantum cascade lasers with 12.5% wall plug efficiency,” Appl. Phys. Lett. 93, 021103 (2008).
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R. Ferreira, G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single-and multiple-quantum-well structures,” Phys. Rev. B 40, 1074–1086 (1989).
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G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Editions de Physique, 1998), Chap. 3.

Bates, R.

R. Bates, S. A. Lynch, D. J. Paul, Z. Ikonic, R. W. Kelsall, P. Harrison, S. L. Liew, D. J. Norris, A. G. Cullis, W. R. Tribe, D. D. Arnone, “Interwell intersubband electroluminescence from Si∕SiGe quantum cascade emitters,” Appl. Phys. Lett. 83, 4092–4094 (2003).
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S. A. Lynch, R. Bates, D. J. Paul, D. J. Norris, A. G. Cullis, Z. Ikonić, R. W. Kelsall, P. Harrison, D. D. Arnone, C. R. Pidgeon, “Intersubband electroluminescence from Si∕SiGe cascade emitters at terahertz frequencies,” Appl. Phys. Lett. 81, 1543–1545 (2002).
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Bauer, M.

M. Bauer, C. Ritter, P. A. Crozier, J. Ren, J. Menéndez, G. Wolf, J. Kouvetakis, “Synthesis of ternary SiGeSn semiconductors on Si(100) via SnxGe1−x buffer layers,” Phys. Rev. Lett. 83, 2163–2165 (2003).

Beck, M.

J. Faist, M. Beck, T. Aellen, E. Gini, “Quantum-cascade lasers based on a bound-to-continuum transition,” Appl. Phys. Lett. 78, 147–149 (2001).
[Crossref]

Beere, H.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, G. Davies, “Far infrared (λ≃87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett. 82, 3165–3167 (2003).
[Crossref]

Bensahel, D.

L. Diehl, S. Mentese, E. Müller, D. Grützmacher, H. Sigg, U. Gennser, I. Sagnes, Y. Campiedelli, O. Kermarrec, D. Bensahel, J. Faist, “Electroluminescence from strain-compensated Si0.2Ge0.8∕Si quantum cascade structures based on a bound-to-continuum transitions,” Appl. Phys. Lett. 81, 4700–4702 (2002).
[Crossref]

Bergstresser, T. K.

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

Z. Mi, P. Bhattacharya, J. Yang, K. P. Pipe, “Room-temperature self-organized In0.5Ga0.5As quantum dot laser on silicon,” Electron. Lett. 41, 742–744 (2005).
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Bir, G. L.

G. L. Bir, G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).

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V. R. D’Costa, C. S. Cook, A. G. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, J. Menéndez, “Optical critical points of thin-film Ge1−ySny alloys: a comparative Ge1−ySny∕Ge1−xSix study,” Phys. Rev. B 73, 125207 (2006).
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Bormann, I.

I. Bormann, K. Brunner, S. Hackenbuchner, G. Zindler, G. Abstreiter, S. Schmult, W. Wegscheider, “Midinfrared intersubband electroluminescence of Si∕SiGe quantum cascade structures,” Appl. Phys. Lett. 80, 2260–2262 (2002).
[Crossref]

Bowers, J. E.

Boyraz, O.

Brunner, K.

I. Bormann, K. Brunner, S. Hackenbuchner, G. Zindler, G. Abstreiter, S. Schmult, W. Wegscheider, “Midinfrared intersubband electroluminescence of Si∕SiGe quantum cascade structures,” Appl. Phys. Lett. 80, 2260–2262 (2002).
[Crossref]

Buongiorno, C.

M. E. Castagna, S. Coffa, L. carestia, A. Messian, C. Buongiorno, “Quantum dot materials and devices for light emission in silicon,” Proceedings of the 32nd European Solid-State Device Research Conference, 2002, held at Fireze, Italy (2002), paperD21.3.

Callebaut, H.

H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, J. L. Reno, “Importance of electron impurity scattering for electron transport in terahertz quantum-cascade lasers,” Appl. Phys. Lett. 84, 645–647 (2004).
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B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, J. L. Reno, “Terahertz quantum cascade laser at λ∼100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83, 2124–2126 (2003).
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Calps, R.

Campiedelli, Y.

L. Diehl, S. Mentese, E. Müller, D. Grützmacher, H. Sigg, U. Gennser, I. Sagnes, Y. Campiedelli, O. Kermarrec, D. Bensahel, J. Faist, “Electroluminescence from strain-compensated Si0.2Ge0.8∕Si quantum cascade structures based on a bound-to-continuum transitions,” Appl. Phys. Lett. 81, 4700–4702 (2002).
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Canonico, M.

V. R. D’Costa, C. S. Cook, A. G. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, J. Menéndez, “Optical critical points of thin-film Ge1−ySny alloys: a comparative Ge1−ySny∕Ge1−xSix study,” Phys. Rev. B 73, 125207 (2006).
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Capasso, F.

A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanbun-Ek, R. Maulini, A. Tsekoun, R. Go, C. Kumar, N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature quantum cascade laser emitting at 4.6 μm,” Appl. Phys. Lett. 92, 111110 (2008).
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K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002).
[Crossref]

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. 78, 2620–2622 (2001).
[Crossref]

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, M. S. Hybertson, A. Y. Cho, “Quantum cascade lasers without intersubband population inversion,” Phys. Rev. Lett. 76, 411–414 (1996).
[Crossref] [PubMed]

J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, C. Sirtori, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

carestia, L.

M. E. Castagna, S. Coffa, L. carestia, A. Messian, C. Buongiorno, “Quantum dot materials and devices for light emission in silicon,” Proceedings of the 32nd European Solid-State Device Research Conference, 2002, held at Fireze, Italy (2002), paperD21.3.

Carnera, A.

F. Priolo, G. Franzò, S. Coffa, A. Carnera, “Excitation and nonradiative deexcitation processes of Er3+ in crystalline Si,” Phys. Rev. B 57, 4443–4455 (1998).
[Crossref]

Castagna, M. E.

M. E. Castagna, S. Coffa, L. carestia, A. Messian, C. Buongiorno, “Quantum dot materials and devices for light emission in silicon,” Proceedings of the 32nd European Solid-State Device Research Conference, 2002, held at Fireze, Italy (2002), paperD21.3.

Chang, J. F.

M. J. Chen, J. L. Yen, J. Y. Li, J. F. Chang, S. C. Tsai, C. S. Tsai, “Stimulated emission in a nanostructured silicon pn junction diode using current injection,” Appl. Phys. Lett. 84, 2163–2165 (2004).
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Chen, M. J.

M. J. Chen, J. L. Yen, J. Y. Li, J. F. Chang, S. C. Tsai, C. S. Tsai, “Stimulated emission in a nanostructured silicon pn junction diode using current injection,” Appl. Phys. Lett. 84, 2163–2165 (2004).
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Chen, Zhihao D.

R. Roucka, J. Tolle, C. Cook, A. V. G. Chizmeshya, J. Kouvetakis, V. D’Costa, J. Menendez, Zhihao D. Chen, S. Zollner, “Versatile buffer layer architectures based on Ge1−xSnx alloys,” Phys. Rev. Lett. 86, 191912 (2005).

Cheng, H. H.

G. Sun, H. H. Cheng, J. Menéndez, J. B. Khurgin, R. A. Soref, “Strain-free Ge∕GeSiSn quantum cascade lasers based on L-valley intersubband transitions,” Phys. Rev. Lett. 90, 251105 (2007).

Chizmeshya, A. V. G.

M. A. Wistey, Y.-Y. Fang, J. Tolle, A. V. G. Chizmeshya, J. Kouvetakis, “Chemical routes to Ge∕Si(100) structures for low temperature Si-based semiconductor applications,” Phys. Rev. Lett. 90, 082108 (2007).

R. Roucka, J. Tolle, C. Cook, A. V. G. Chizmeshya, J. Kouvetakis, V. D’Costa, J. Menendez, Zhihao D. Chen, S. Zollner, “Versatile buffer layer architectures based on Ge1−xSnx alloys,” Phys. Rev. Lett. 86, 191912 (2005).

P. Aella, C. Cook, J. Tolle, S. Zollner, A. V. G. Chizmeshya, J. Kouvetakis, “Optical and structural properties of SixSnyGe1−x−y alloys,” Phys. Rev. Lett. 84, 888–890 (2004).

Cho, A. Y.

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002).
[Crossref]

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. 78, 2620–2622 (2001).
[Crossref]

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, M. S. Hybertson, A. Y. Cho, “Quantum cascade lasers without intersubband population inversion,” Phys. Rev. Lett. 76, 411–414 (1996).
[Crossref] [PubMed]

J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, C. Sirtori, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

Chrastina, D.

D. J. Paul, G. Matman, L. Lever, Z. Ikonić, R. W. Kelsall, D. Chrastina, G. Isella, H. von Känel, E. Müller, A. Neels, “Si∕SiGe bound-to-continuum terahertz quantum cascade emitters,” ECS Trans. 16, 865–874 (2008).
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D. J. Paul, G. Matmon, L. Lever, Z. Ikonić, R. W. Kelsall, D. Chrastina, G. Isella, H. von Känel, “Si∕SiGe bound-to-continuum quantum cascade terahertz emitters,” Proc. SPIE 6996, 69961C (2008).
[Crossref]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cockburn, J. W.

D. G. Revin, J. W. Cockburn, M. J. Steer, R. J. Airey, M. Hopkinson, A. B. Krysa, L. R. Wilson, S. Menzel, “InGaAs∕AlAsSb∕InP quantum cascade lasers operating at wavelengths close to 3 μm,” Appl. Phys. Lett. 90, 021108 (2007).
[Crossref]

Coffa, S.

F. Priolo, G. Franzò, S. Coffa, A. Carnera, “Excitation and nonradiative deexcitation processes of Er3+ in crystalline Si,” Phys. Rev. B 57, 4443–4455 (1998).
[Crossref]

G. Franzò, S. Coffa, “Mechanism and performance of forward and reverse bias electroluminescence at 1.54 μm from Er-doped Si diodes,” J. Appl. Phys. 81, 2784–2793 (1997).
[Crossref]

M. E. Castagna, S. Coffa, L. carestia, A. Messian, C. Buongiorno, “Quantum dot materials and devices for light emission in silicon,” Proceedings of the 32nd European Solid-State Device Research Conference, 2002, held at Fireze, Italy (2002), paperD21.3.

Cohen, M. L.

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

Cohen, O.

Colombelli, R.

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002).
[Crossref]

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. 78, 2620–2622 (2001).
[Crossref]

Cook, C.

R. Roucka, J. Tolle, C. Cook, A. V. G. Chizmeshya, J. Kouvetakis, V. D’Costa, J. Menendez, Zhihao D. Chen, S. Zollner, “Versatile buffer layer architectures based on Ge1−xSnx alloys,” Phys. Rev. Lett. 86, 191912 (2005).

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K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002).
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G. Sun, Y. Lu, J. B. Khurgin, “Valence intersubband lasers with inverted light-hole effective mass,” Appl. Phys. Lett. 72, 1481–1483 (1998).
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L. Friedman, G. Sun, R. A. Soref, “SiGe∕Si THz laser based on transitions between inverted mass light-hole and heavy-hole subbands,” Appl. Phys. Lett. 78, 401–403 (2001).
[Crossref]

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. 78, 2620–2622 (2001).
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Y. Bai, S. Slivken, S. R. Darvish, M. Razeghi, “Room temperature continuous wave operation of quantum cascade lasers with 12.5% wall plug efficiency,” Appl. Phys. Lett. 93, 021103 (2008).
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A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanbun-Ek, R. Maulini, A. Tsekoun, R. Go, C. Kumar, N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature quantum cascade laser emitting at 4.6 μm,” Appl. Phys. Lett. 92, 111110 (2008).
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D. G. Revin, J. W. Cockburn, M. J. Steer, R. J. Airey, M. Hopkinson, A. B. Krysa, L. R. Wilson, S. Menzel, “InGaAs∕AlAsSb∕InP quantum cascade lasers operating at wavelengths close to 3 μm,” Appl. Phys. Lett. 90, 021108 (2007).
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R. A. Soref, G. Sun, “Terahertz gain in SiGe∕Si quantum staircase utilizing the heavy-hole inverted effective mass,” Appl. Phys. Lett. 79, 3639–3641 (2001).
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H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, J. L. Reno, “Importance of electron impurity scattering for electron transport in terahertz quantum-cascade lasers,” Appl. Phys. Lett. 84, 645–647 (2004).
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D. J. Paul, G. Matman, L. Lever, Z. Ikonić, R. W. Kelsall, D. Chrastina, G. Isella, H. von Känel, E. Müller, A. Neels, “Si∕SiGe bound-to-continuum terahertz quantum cascade emitters,” ECS Trans. 16, 865–874 (2008).
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Figures (15)

Fig. 1
Fig. 1

Photon emission process in (a) the direct and (b) the indirect bandgap semiconductors.

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Fig. 2
Fig. 2

(a) Conduction and valence subband formations in a semiconductor QW; (b) in-plane subband dispersions with optical transitions between conduction and valence subbands.

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Fig. 3
Fig. 3

Two subbands formed within the conduction band confined in a QW and their election envelope functions; (b) in-plane energy dispersions of the two subbands. The radiative IST between the two subbands is marked with a green arrow.

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Fig. 4
Fig. 4

Schematic band diagram of two periods of a QCL structure with each period consisting of an active and an injector region. Lasing transitions are between the states 3 and 2 in the active regions, with rapid depopulation of lower state 2 into state 1, which couples strongly with the minibands formed in injector regions that transport carriers to state 3 in the next period. The magnitude-squared wavefunctions for the three subbands in active regions are illustrated.

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Fig. 5
Fig. 5

In-plane dispersions of subbands HH1, LH1, and HH2 for a 70   Å ∕ 50   Å Ga As ∕ Al Ga As SL [35].

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Fig. 6
Fig. 6

Intersubband and intrasubband transitions due to electron–phonon scattering. (b) 22 → 11 transition induced by the electron–electron scattering.

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Fig. 7
Fig. 7

HH valence band diagram of one period of a Si Ge ∕ Si SL with a hole energy increase in the upward direction. (b) Comparison of lifetime difference ( τ 3 − τ 2 ) between the Si Ge ∕ Si and Ga As ∕ Al Ga As SL (in a similar three-level scheme) as a function of the transition energy ( E 3 − E 2 ) [38].

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Fig. 8
Fig. 8

Dispersions of subbands HH1, LH1, and HH2 in a 90   Å ∕ 50   Å Si 0.7 Ge 0.3 ∕ Si SL strained balanced on a Si 0.81 Ge 0.19 buffer, obtained with a 6 × 6 valence band matrix, taking into account HH, LH, and SO interactions and strain effect [50].

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Fig. 9
Fig. 9

Band diagram of the Si 0.8 Ge 0.2 ∕ Si SL under an electric bias of 30   kV ∕ cm . The labels ( n − 1 , n , n + 1 , … ) represent the QWs in which the wave functions are localized [51]. (b) Dispersions of the four levels (two doublets) in a QW.

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Fig. 10
Fig. 10

Illustration of two periods of a bound-to-continuum QCL. Lasing transition occurs between an isolated bound upper state 2 (formed in the minigap) and a delocalized lower state 1 (sitting on top of a miniband).

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Fig. 11
Fig. 11

Conduction band minima at L, Γ, X points of Ge 1 − x − y Si x Sn y that is lattice matched to Ge [71].

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Fig. 12
Fig. 12

L-valley conduction-band profile and squared envelope functions under an electric field of 10   kV ∕ cm . Layer thicknesses in ångströms are marked with bold numbers for Ge QWs and regular numbers for GeSiSn barriers. The array marks the injection barrier [71].

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Fig. 13
Fig. 13

Upper-state lifetime τ 3 , lower-state lifetime τ 2 , and scattering time τ 32 between them as a function of temperature.

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Fig. 14
Fig. 14

Schematic of a ridge plasmon waveguide with the Ge ∕ Ge Si Sn QCL sandwiched between two metal layers.

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Fig. 15
Fig. 15

Simulated threshold current density of the Ge ∕ Ge Si Sn QCL as a function of temperature.

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Equations (53)

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E i , k = E i + ℏ 2 k 2 2 m e * ,
[ − ℏ 2 2 d d z 1 m e * ( z ) d d z + V c ( z ) ] φ i ( z ) = E i φ i ( z )
Φ i ( r , z ) = φ i ( z ) u e ( R ) e j k ⋅ r ,
φ i ( z ) = A l e j k z ( z − d l ) + B l e − j k z ( z − d l ) ,
φ i ( z ) and 1 m e * ( z ) d φ i ( z ) d z continous ,
E i = ℏ 2 k z 2 2 m e * ( z ) + V c ( z ) .
V c ( z ) = V c , l − e E z − e ϕ ( z ) ,
∂ 2 ∂ z 2 ϕ ( z ) = − e ε 0 ε ( z ) [ ∑ i n i | φ i ( z ) | 2 − N d ( z ) ]
H = − | 3 2 , 3 2 ⟩ | 3 2 , 1 2 ⟩ | 3 2 , − 1 2 ⟩ | 3 2 , − 3 2 ⟩ | 1 2 , 1 2 ⟩ | 1 2 , − 1 2 ⟩ [ P + Q − S R 0 − 1 2 S 2 R − S † P − Q 0 R − 2 Q 3 2 S R † 0 P − Q S 3 2 S † 2 Q 0 R † S † P + Q − 2 R † − 1 2 S † − 1 2 S † − 2 Q 3 2 S − 2 R P + Δ 0 2 R † 3 2 S † 2 Q − 1 2 S 0 P + Δ ] + V v ( z )
P = ℏ 2 2 m 0 γ 1 ( k x 2 + k y 2 + k z 2 ) − a v ( ϵ x x + ϵ y y + ϵ z z ) ,
Q = ℏ 2 2 m 0 γ 2 ( k x 2 + k y 2 − 2 k z 2 ) − b 2 ( ϵ x x + ϵ y y − 2 ϵ z z ) ,
S = ℏ 2 2 m 0 2 3 γ 3 ( k x − j k y ) k z ,
R = ℏ 2 2 m 0 3 [ − γ 2 ( k x 2 − k y 2 ) + 2 j γ 3 k x k y ] .
ϵ x x = ϵ y y = a 0 − a a , ϵ y y = − 2 C 12 C 11 ϵ x x
Ψ i ( r , z ) = e j k ⋅ r [ χ 1 ( z ) | 3 2 , 3 2 ⟩ + χ 2 ( z ) | 3 2 , 1 2 ⟩ + χ 3 ( z ) | 3 2 , − 1 2 ⟩ + χ 4 ( z ) | 3 2 , − 3 2 ⟩ + χ 5 ( z ) | 1 2 , − 1 2 ⟩ + χ 6 ( z ) | 1 2 , − 1 2 ⟩ ] ,
χ ( z ) and [ p + q − s 0 0 − 1 2 s 0 − s † p − q 0 0 − 2 q 3 2 s 0 0 p − q s 3 2 s † 2 q 0 0 s † p + q 0 − 1 2 s † − 1 2 s † − 2 q 3 2 s 0 p 0 0 3 2 s † 2 q − 1 2 s 0 p ] χ ( z ) continuous .
p = γ 1 ∂ ∂ z ,
q = − 2 γ 2 ∂ ∂ z ,
s = − 3 j γ 3 ( k x − j k y ) .
g 12 = 2 π ℏ | H m | 2 δ ( E 2 − E 1 − ℏ ω ) ,
L ( E ) = Γ ∕ 2 π ( E − E 0 ) 2 + Γ 2 ∕ 4 ,
g 12 = 2 π ℏ ∫ | H m | 2 δ ( E − ℏ ω ) L ( E ) d E = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 0 ) + Γ 2 ∕ 4 .
H ex = e A ⋅ P m e * ,
H m = e A m e * P 12 ,
P 12 = ⟨ φ 1 | − j ℏ ∂ ∂ z | φ 2 ⟩
z 12 = ⟨ φ 1 | z | φ 2 ⟩ = i ℏ m 0 E 12 P 12
f 12 = 2 m 0 m e * 2 E 12 | P 12 | 2 .
g net = ∫ g 12 { f 2 ( E 2 , k ) [ 1 − f 1 ( E 1 , k ) ] − f 1 ( E 1 , k ) [ 1 − f 2 ( E 2 , k ) ] } ρ r ( E 2 , k − E 1 , k ) d ( E 2 , k − E 1 , k ) ,
g net = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 [ ∫ f 2 ( E 2 , k ) ρ 2 d E 2 , k − ∫ f 1 ( E 1 , k ) ρ 1 d E 1 , k ] = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,
γ ( ω ) = g net ℏ ω I L p ,
A = A 0 cos ( β ⋅ r − ω t ) z ̂ = 1 2 A 0 z ̂ [ e j ( β ⋅ r − ω t ) + e − j ( β ⋅ r − ω t ) ] ,
γ ( ω ) = e 2 | P 12 | 2 2 ε 0 c n eff m e * 2 ω L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) = e 2 m 0 2 ω z 12 2 2 ε 0 c n eff m e * 2 L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,
γ ( ω 0 ) = 2 e 2 m 0 2 ω z 12 2 ε 0 c n eff m e * 2 Γ L p ( N 2 − N 1 ) .
g net ( v ) = 2 π ℏ | H m ( v ) | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − F HH ( E h ) ] | E l − E h = ℏ ω .
H m ( v ) = ∑ n = 1 6 e A m n * ⟨ χ n ( l ) | − j ℏ ∂ ∂ z | χ n ( h ) ⟩ ,
γ ( v ) = π e 2 ε 0 c n eff ω L p | ∑ n = 1 6 P n ( l h ) m n * | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − f HH ( E h ) ] | E l − E h = ℏ ω = π e 2 m 0 2 ω ε 0 c n eff L p | ∑ n = 1 6 z n ( l h ) m n * | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − f HH ( E h ) ] | E l − E h = ℏ ω .
1 τ 12 = 2 π ℏ ∫ | H e p | 2 δ ( E 2 , k − E 1 , k ′ ∓ ℏ ω Q ) d N f ,
| H e p | 2 = { Ξ 2 K B T 2 c L Ω δ q , ± ( k ′ − k ) | G 12 ( q z ) | 2 acoustic phonon ℏ D 2 2 ρ ω 0 Ω δ q , ± ( k ′ − k ) [ n ( ω 0 ) + 1 2 ∓ 1 2 ] | G 12 ( q z ) | 2 nonpolar optical phonon } ,
n ( ω 0 ) = 1 exp ( ℏ ω 0 ∕ K B T ) − 1 .
G 12 ( q z ) = ⟨ φ 1 | e j q z z | φ 2 ⟩
G 12 ( q z ) = ∑ n = 1 6 ⟨ χ n ( 1 ) | e j q z z | χ n ( 2 ) ⟩ .
d N f = Ω ( 2 π ) 3 q d q d θ d q z ,
1 τ 12 = { Ξ 2 K B T m e * 4 π c L ℏ 3 ∫ | G 12 ( q z ) | 2 d q z acoustic D 2 m e * [ n ( ω 0 ) + 1 2 ∓ 1 2 ] 4 π ρ ℏ 2 ω 0 ∫ | G 12 ( q z ) | 2 d q z nonpolar optical } .
Δ E v , av ( x , y ) = E v , av ( Ge Si Sn ) − E v , av ( Ge ) = 0.48 x + 0.69 y .
E Ge Si Sn ( x , y ) = E Ge Si Sn ( 1 − x − y ) + E Si x + E Sn y − b Ge Si ( 1 − x − y ) x − b Ge Sn ( 1 − x − y ) y − b Si Sn x y .
E x = 0.931 + 0.018 x + 0.206 x 2
∂ N 3 ∂ t = η J e − N 3 − N ¯ 3 τ 3 ,
∂ N 2 ∂ t = N 3 − N ¯ 3 τ 32 − N 2 − N ¯ 2 τ 2 ,
N 3 − N 2 = τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ,
γ ( ω 0 ) = 2 e 2 m 0 2 ω 0 z 23 2 ε 0 c n eff m z * 2 Γ L p [ τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ] .
ε M = 1 − ω p 2 ω 2 + j γ m ω ,
E = { E 0 ε cosh ( k d 2 ) ( j β z ̂ + q x ̂ ) e − q ( z − d ∕ 2 ) e j ( β x − ω t ) z > d 2 E 0 [ j β cosh ( k z ) z ̂ − k sinh ( k z ) x ̂ ] e j ( β x − ω t ) | z | < d 2 E 0 ε cosh ( k d 2 ) ( j β z ̂ − q x ̂ ) e q ( z + d ∕ 2 ) e j ( β x − ω t ) z < − d 2 } ,
k 2 [ ε 2 tanh 2 ( k d 2 ) − 1 ] = k D 2 ( 1 − ε ) ,

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