Indirectly pumped 3.7 THz InGaAs/InAlAs quantum-cascade lasers grown by metal-organic vapor-phase epitaxy

Kazuue Fujita; Masamichi Yamanishi; Shinichi Furuta; Kazunori Tanaka; Tadataka Edamura; Tillmann Kubis; Gerhard Klimeck

doi:10.1364/OE.20.020647

Indirectly pumped 3.7 THz InGaAs/InAlAs quantum-cascade lasers grown by metal-organic vapor-phase epitaxy

Kazuue Fujita, Masamichi Yamanishi, Shinichi Furuta, Kazunori Tanaka, Tadataka Edamura, Tillmann Kubis, and Gerhard Klimeck

Optics Express
Vol. 20,
Issue 18,
pp. 20647-20658
(2012)
•https://doi.org/10.1364/OE.20.020647

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Kazuue Fujita, Masamichi Yamanishi, Shinichi Furuta, Kazunori Tanaka, Tadataka Edamura, Tillmann Kubis, and Gerhard Klimeck, "Indirectly pumped 3.7 THz InGaAs/InAlAs quantum-cascade lasers grown by metal-organic vapor-phase epitaxy," Opt. Express 20, 20647-20658 (2012)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Infrared and far-infrared lasers (140.3070)
- Semiconductor lasers, quantum cascade (140.5965)
- Quantum-well, -wire and -dot devices (230.5590)

About this Article
History
- Original Manuscript: July 18, 2012
- Revised Manuscript: August 10, 2012
- Manuscript Accepted: August 20, 2012
- Published: August 23, 2012

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Open Access

Abstract

Device-performances of 3.7 THz indirect-pumping quantum-cascade lasers are demonstrated in an InGaAs/InAlAs material system grown by metal-organic vapor-phase epitaxy. The lasers show a low threshold-current-density of ~420 A/cm² and a peak output power of ~8 mW at 7 K, no sign of parasitic currents with recourse to well-designed coupled-well injectors in the indirect pump scheme, and a maximum operating temperature of T_max~100 K. The observed roll-over of output intensities in current ranges below maximum currents and limitation of T_max are discussed with a model for electron-gas heating in injectors. Possible ways toward elevation of T_max are suggested.

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References

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Publication

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[Crossref] [PubMed]
B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007) (and references cited therein).
[Crossref]
M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16(5), 3242–3248 (2008).
[Crossref] [PubMed]
S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]
S. Fathololoumi, E. Dupont, C. W. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~ 200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express 20(4), 3866–3876 (2012).
[Crossref] [PubMed]
M. Fischer, G. Scalari, K. Celebi, M. Amanti, C. Walther, M. Beck, and J. Faist, “Scattering processes interahertz InGaAs/InAlAs quantum cascade lasers,” Appl. Phys. Lett. 97(22), 221114 (2010).
[Crossref]
M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]
M. Yamanishi, K. Fujita, T. Edamura, and H. Kan, “Indirect pump scheme for quantum cascade lasers: dynamics of electron-transport and very high T0-values,” Opt. Express 16(25), 20748–20758 (2008).
[Crossref] [PubMed]
K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (~450 K) of long-wavelength (~15 μm), low-threshold-current density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97(20), 201109 (2010).
[Crossref]
H. Yasuda, T. Kubis, P. Vogl, N. Sekine, I. Hosako, and K. Hirakawa, “Nonequilibrium Green’s function calculation for four-level scheme terahertz quantum cascade lasers,” Appl. Phys. Lett. 94(15), 151109 (2009).
[Crossref]
A. Wacker, “Extraction-controlled quantum cascade lasers,” Appl. Phys. Lett. 97(8), 081105 (2010).
[Crossref]
T. Kubis, S. R. Mehrotra, and G. Klimeck, “Design concepts of terahertz quantum cascade lasers: Proposal for terahertz laser efficiency improvements,” Appl. Phys. Lett. 97(26), 261106 (2010).
[Crossref]
S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]
M. Yamanishi, K. Fujita, T. Kubis, N. Yu, T. Edamura, K. Tanaka, G. Klimeck, and F. Capasso, “Indirect pumping operation of THz InGaAs/InAlAs quantum-cascade-lasers,” paper presented at Eleventh International Conference on Intersubband Transitions in Quantum Wells, Badesi, Italy, 11–17, September 2011.
E. Dupont, S. Fathololoumi, Z. R. Wasilewski, G. Aers, S. R. Laframboise, M. Lindskog, S. G. Razavipour, A. Wacker, D. Ban, and H. C. Liu, “A phonon scattering assisted injection and extraction based terahertz quantum cascade laser,” J. Appl. Phys. 111(7), 073111 (2012).
[Crossref]
S. Fathololoumi, E. Dupont, Z. R. Wasilewski, G. Aers, S. R. Laframboise, S. G. Razavipour, M. Lindskog, A. Wacker, D. Ban, and H. C. Liu, “Terahertz quantum cascade lasers based on phonon scattering assisted injection and extraction,” paper presented at Conference on Lasers and Electro-Optics (CLEO 2012), CTh4N.4, San Jose, CA, USA, 6–11, May 2012.
M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Quantum Electron. 14(2), 431–435 (2008).
[Crossref]
S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[Crossref]
T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B 79(19), 195323 (2009).
[Crossref]
T. C. Kubis, “Quantum Transport in semiconductor nanostructures,” in Selected Topics of Semiconductor Physics and Technology (Munich, Germany, 2009) vol. 114.
The energy-diffusion model has been recently proposed by one (MY) of the authors; M. Yamanishi, unpublished note (2012).
P. Harrison and R. W. Kelsall, “The relative importance of electron-electron and electron-phonon scattering in terahertz quantum cascade lasers,” Solid-State Electron. 42(7-8), 1449–1451 (1998).
[Crossref]
M. S. Vitiello, R. C. Iotti, F. Rossi, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, Q. Hu, and G. Scamarcio, “Non-equilibrium longitudinal and transverse optical phonons in terahertz quantum cascade lasers,” Appl. Phys. Lett. 100(9), 091101 (2012).
[Crossref]
M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett. 86(11), 111115 (2005).
[Crossref]
T. Liu, T. Kubis, Q. Jie Wang, and G. Klimeck, “Design of three-well indirect pumping terahertz quantum cascade lasers for high optical gain based on noequilibrium Green’s function analysis,” Appl. Phys. Lett. 100(12), 122110 (2012).
[Crossref]

2012 (4)

E. Dupont, S. Fathololoumi, Z. R. Wasilewski, G. Aers, S. R. Laframboise, M. Lindskog, S. G. Razavipour, A. Wacker, D. Ban, and H. C. Liu, “A phonon scattering assisted injection and extraction based terahertz quantum cascade laser,” J. Appl. Phys. 111(7), 073111 (2012).
[Crossref]

M. S. Vitiello, R. C. Iotti, F. Rossi, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, Q. Hu, and G. Scamarcio, “Non-equilibrium longitudinal and transverse optical phonons in terahertz quantum cascade lasers,” Appl. Phys. Lett. 100(9), 091101 (2012).
[Crossref]

T. Liu, T. Kubis, Q. Jie Wang, and G. Klimeck, “Design of three-well indirect pumping terahertz quantum cascade lasers for high optical gain based on noequilibrium Green’s function analysis,” Appl. Phys. Lett. 100(12), 122110 (2012).
[Crossref]

S. Fathololoumi, E. Dupont, C. W. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~ 200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express 20(4), 3866–3876 (2012).
[Crossref] [PubMed]

2011 (1)

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

2010 (4)

M. Fischer, G. Scalari, K. Celebi, M. Amanti, C. Walther, M. Beck, and J. Faist, “Scattering processes interahertz InGaAs/InAlAs quantum cascade lasers,” Appl. Phys. Lett. 97(22), 221114 (2010).
[Crossref]

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (~450 K) of long-wavelength (~15 μm), low-threshold-current density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97(20), 201109 (2010).
[Crossref]

A. Wacker, “Extraction-controlled quantum cascade lasers,” Appl. Phys. Lett. 97(8), 081105 (2010).
[Crossref]

T. Kubis, S. R. Mehrotra, and G. Klimeck, “Design concepts of terahertz quantum cascade lasers: Proposal for terahertz laser efficiency improvements,” Appl. Phys. Lett. 97(26), 261106 (2010).
[Crossref]

2009 (4)

H. Yasuda, T. Kubis, P. Vogl, N. Sekine, I. Hosako, and K. Hirakawa, “Nonequilibrium Green’s function calculation for four-level scheme terahertz quantum cascade lasers,” Appl. Phys. Lett. 94(15), 151109 (2009).
[Crossref]

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[Crossref]

T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B 79(19), 195323 (2009).
[Crossref]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]

2008 (3)

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16(5), 3242–3248 (2008).
[Crossref] [PubMed]

M. Yamanishi, K. Fujita, T. Edamura, and H. Kan, “Indirect pump scheme for quantum cascade lasers: dynamics of electron-transport and very high T0-values,” Opt. Express 16(25), 20748–20758 (2008).
[Crossref] [PubMed]

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Quantum Electron. 14(2), 431–435 (2008).
[Crossref]

2007 (2)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007) (and references cited therein).
[Crossref]

2005 (1)

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett. 86(11), 111115 (2005).
[Crossref]

2002 (1)

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[Crossref] [PubMed]

1998 (1)

P. Harrison and R. W. Kelsall, “The relative importance of electron-electron and electron-phonon scattering in terahertz quantum cascade lasers,” Solid-State Electron. 42(7-8), 1449–1451 (1998).
[Crossref]

Aers, G.

Amanti, M.

Ban, D.

Beck, M.

Beere, H. E.

Belkin, M. A.

Beltram, F.

Benz, A.

Capasso, F.

Celebi, K.

Chan, C. W.

Chan, C. W. I.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

Davies, A. G.

Deutsch, C.

Dupont, E.

Edamura, T.

Faist, J.

Fan, J. A.

Fasching, G.

Fathololoumi, S.

Fischer, M.

Fujita, K.

Furuta, S.

Harrison, P.

Hirakawa, K.

Hormoz, S.

Hosako, I.

Hu, Q.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[Crossref]

Iotti, R. C.

Jie Wang, Q.

Jirauschek, C.

Kan, H.

Kelsall, R. W.

Khanna, S. P.

Klimeck, G.

Köhler, R.

Kubis, T.

Kumar, S.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[Crossref]

Lachab, M.

Laframboise, S. R.

Lindskog, M.

Linfield, E. H.

Liu, H. C.

Liu, T.

Mahler, L.

Mátyás, A.

Mehrotra, S. R.

Razavipour, S. G.

Reno, J. L.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]

Ritchie, D. A.

Rossi, F.

Scalari, G.

Scamarcio, G.

Sekine, N.

Spagnolo, V.

Sugiyama, A.

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Tredicucci, A.

Vitiello, M. S.

Vogl, P.

Wacker, A.

A. Wacker, “Extraction-controlled quantum cascade lasers,” Appl. Phys. Lett. 97(8), 081105 (2010).
[Crossref]

Walther, C.

Wasilewski, Z. R.

Williams, B. S.

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007) (and references cited therein).
[Crossref]

Yamanishi, M.

Yasuda, H.

Yeh, C.

Appl. Phys. Lett. (9)

A. Wacker, “Extraction-controlled quantum cascade lasers,” Appl. Phys. Lett. 97(8), 081105 (2010).
[Crossref]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[Crossref]

IEEE J. Quantum Electron. (1)

J. Appl. Phys. (1)

Nat. Photonics (2)

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007) (and references cited therein).
[Crossref]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Nat. Phys. (1)

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quntum cascade laser operating significantly above the temperature of ħω/kB,” Nat. Phys. 7(2), 166–171 (2011).
[Crossref]

Nature (1)

Opt. Express (3)

Phys. Rev. B (2)

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[Crossref]

Solid-State Electron. (1)

Other (4)

S. Fathololoumi, E. Dupont, Z. R. Wasilewski, G. Aers, S. R. Laframboise, S. G. Razavipour, M. Lindskog, A. Wacker, D. Ban, and H. C. Liu, “Terahertz quantum cascade lasers based on phonon scattering assisted injection and extraction,” paper presented at Conference on Lasers and Electro-Optics (CLEO 2012), CTh4N.4, San Jose, CA, USA, 6–11, May 2012.

T. C. Kubis, “Quantum Transport in semiconductor nanostructures,” in Selected Topics of Semiconductor Physics and Technology (Munich, Germany, 2009) vol. 114.

The energy-diffusion model has been recently proposed by one (MY) of the authors; M. Yamanishi, unpublished note (2012).

M. Yamanishi, K. Fujita, T. Kubis, N. Yu, T. Edamura, K. Tanaka, G. Klimeck, and F. Capasso, “Indirect pumping operation of THz InGaAs/InAlAs quantum-cascade-lasers,” paper presented at Eleventh International Conference on Intersubband Transitions in Quantum Wells, Badesi, Italy, 11–17, September 2011.

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Fig. 1 Conduction band diagrams and moduli squared of the relevant wavefunctions in the designed active/injector regions of type-A and -B THz IDP QCLs. The lattice-matched In_0.53Ga_0.47As/In_0.52Al_0.48As layer sequences of one period of the active/injector layers, in angstroms, starting from the injection barrier (toward the right side) are as follows: (a) 20/89/7/121/10/119/18/210/14/118 and (b) 22/93/7/137/9/133/18/225/14/127 where In_0.52Al_0.48As barrier layers are in bold and In_0.53Ga_0.47As QW layers in roman, and Si-donors are quasi-δ-doped in the underlined layers. The bias fields are assumed to be high enough, (a) 13.1 kV/cm and (b) 11.5 kV/cm, to almost align the ground state 1′ of the injector to the IDP state, level 4.

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Fig. 2 Light intensity-voltage-current (current density) characteristics of a type-A IDP QCL at different temperatures, 7 K−83 K.

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Fig. 3 Light intensity-voltage-current (current density) characteristics of a type-B IDP QCL at different temperatures, 7 K-100 K. The insertion shows the band diagram and moduli squared of the relevant wavefunctions in the active/injector regions of the type-B QCL biased by a low electric field of ~3.5 kV/cm.

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Fig. 4 THz spectra emitted from (a) the type-A and (b) type-B QCLs for different biases at temperatures, 14 K, and 7 K and 80 K, respectively.

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Fig. 5 Differential resistance R_d in logarithmic scale as functions of current in a direct pump InGaAs/InAlAs QCL at 10 K, and the type-A and –B IDP QCLs at 7 K.

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Fig. 6 Computed spectral functions (contour lines) at k_// = 0 and energy-resolved current densities (colored areas) in the first module of the type-A QCL at 40 K. The simulation was carried out, based on the NEGF formalism, assuming the mean-field approximation for electron-electron interaction. In the simulation, we used an effective interface roughness height, Λ = 2.4 Å and a roughness correlation length, λ_c = 90 Å, and an alloy-scattering potential, δV_alloy = 0.6 eV, and, also, included phonon and impurity scattering. The bias field was assumed to be ~12 kV/cm.

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Fig. 7 Light intensity, estimated excess energy, and tunneling time as functions of normalized current J/J_{max, A or B} for the type-A (dashed curves) and –B (solid curves) IDP QCLs. J_{max, A or B} is the maximum current density at 7 K in the type-A or –B QCL.

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Fig. 8 Computed normalized energy-resolved electron populations, n(E)/g₀τ_tun (blue curves) for normalized generation spectra, g(E)/g₀ (dash-dotted green curves), where g₀ is the peak value of g(E) in an injector subband of the first module for the bias conditions I (τ_tun = 2.1 ps and E_excess = 20 meV), II (τ_tun = 1.1 ps and E_excess = 25 meV), and III (τ_tun = 0.5 ps and E_excess = 30 meV). The calculated population distribution taking only into account the electron-electron scatterings is shown by the red dash-curve and is labeled with “e-e only”. For reference, the initial kinetic energy, E_LO−E₃₂, required for thermally-activated-electron-LO-phonon relaxation in the upper laser subband in the next module is shown by the horizontal dashed lines. The computation was based on the energy-diffusion model. Hereby the parameters were D_e = 1.16 x 10¹⁴ (meV)²/s and τ_LO = 0.2 ps.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

τ_{tun} / τ_{tun, 0} = J_{max} / J + [J_{max} / J - 1] (τ_{relax} / τ_{tun, 0}),

V {(q)}^{2} \propto {| \iint d z d z^{'} Ψ {(z)}^{2} Ψ {(z^{'})}^{2} exp [- Q | z - z^{'} |] |}^{2} / Q^{2},

g (E) + D_{e} d^{2} n (E) / d E^{2} - n (E) [N_{phonon} / τ_{LO}] + n (E + E_{LO}) [(1 + N_{phonon}) / τ_{LO}] - n (E) / τ_{tun} = 0,

\begin{array}{l} g (E) + D_{e} d^{2} n (E) / d E^{2} - [n (E) - n (E - E_{LO})] [N_{phonon} / τ_{LO}] \\ + [n (E + E_{LO}) - n (E)] [(1 + N_{phonon}) / τ_{LO}] - n (E) / τ_{tun} = 0, \end{array}