强磁场下GaInAs合金量子阱中非平衡载流子弛豫机制
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摘要
在最近的一些实验中,沿着量子级联激光器的生长方向加一个磁场B作为可调控手段,人们可以调节激光器内的无辐射跃迁渠道,例如合金无序散射以及光学声子散射,并引起辐射功率的1/B振荡。理论上,对于激光器的合金材料量子阱中载流子弛豫的问题,若使用费米黄金规则计算声子散射速率,我们必须假定电子—声子相互作用对合金系统能级只是微扰。在弱电子—声子耦合情况下,我们采用费米黄金规则对GaInAs量子阱中合金无序态的声学声子散射给出合理的估计,但是对于光学声子散射的计算结果却存在疑问。因此在强LO声子散射的情况下,我们采用数值对角化办法来研究这个问题。在电子—LO声子强耦合情况下,我们采用磁极化子形式来计算上能级的电子停留几率,结果也证明在光学声子共振散射的磁场下电子的快速弛豫过程,同时表明合金散射在其中起到重要作用。出于对这种非平衡电子系统中磁极化子态的兴趣,我们对理想材料加入一个平面内电场,研究了磁极化子的解离过程。我们计算了理想量子阱的中红外线性光吸收系数,并且预言与磁极化子态有关的子带间跃迁吸收峰结构。对于合金材料,我们也计算了在这种交叉的磁场和电场下电子的态密度。研究表明合金散射造成态密度边缘的特殊结构。
In a few recent experiments, a magnetic field B applied parallel to the growthdirection of a quantum cascade laser (QCL) as an external parameter modulates thenon-radiative relaxation channels in the device, such as alloy disorder scattering andoptical phonon scattering, and results in a 1/B oscillation of the output. In the theoryof carrier relaxation in the alloy material quantum wells as units of the QCL, any com-putation using a regular Fermi’s golden rule to estimate phonon scattering rates has toassume the electron-phonon interaction only results in perturbations to the alloy states.In the weak electron-phonon coupling regime, we use the Fermi’s golden rule andhave a reasonable estimate of the acoustic phonon scattering from the alloy disorderedstates in a GaInAs well, while the result of fast scattering by optical phonons is ratherquestionable. Therefore,in the case of strong LO phonon scattering, a numerical di-agonalization computation is carried out. In the strong electron-LO phonon couplingregime, we use a magneto-polaron formalism and compute the electron upper levelsurvival probabilities, which also demonstrates rapid relaxation processes around a op-tical phonon scattering resonance magnetic field, where the alloy disorder in the wellplays a key role. With the interests in the magneto-polarons as high energy states in thenonequilibrium electron system, we take into accounts of an additional in-plane electricfield in a perfect material system to investigate the dissociation of the magneto-polaronstates. We compute a mid-infrared linear optical absorption by an ideal well, and pre-dict fine structures of intersubband absorption peaks related to the polaron states. Thedensity of states (DOS) of an alloy material in the crossed magnetic and electric fieldsare also computed, and structures at the DOS tails due to alloy scattering are found.
In a few recent experiments, a magnetic field B applied parallel to the growthdirection of a quantum cascade laser (QCL) as an external parameter modulates thenon-radiative relaxation channels in the device, such as alloy disorder scattering andoptical phonon scattering, and results in a 1/B oscillation of the output. In the theoryof carrier relaxation in the alloy material quantum wells as units of the QCL, any com-putation using a regular Fermi’s golden rule to estimate phonon scattering rates has toassume the electron-phonon interaction only results in perturbations to the alloy states.In the weak electron-phonon coupling regime, we use the Fermi’s golden rule andhave a reasonable estimate of the acoustic phonon scattering from the alloy disorderedstates in a GaInAs well, while the result of fast scattering by optical phonons is ratherquestionable. Therefore,in the case of strong LO phonon scattering, a numerical di-agonalization computation is carried out. In the strong electron-LO phonon couplingregime, we use a magneto-polaron formalism and compute the electron upper levelsurvival probabilities, which also demonstrates rapid relaxation processes around a op-tical phonon scattering resonance magnetic field, where the alloy disorder in the wellplays a key role. With the interests in the magneto-polarons as high energy states in thenonequilibrium electron system, we take into accounts of an additional in-plane electricfield in a perfect material system to investigate the dissociation of the magneto-polaronstates. We compute a mid-infrared linear optical absorption by an ideal well, and pre-dict fine structures of intersubband absorption peaks related to the polaron states. Thedensity of states (DOS) of an alloy material in the crossed magnetic and electric fieldsare also computed, and structures at the DOS tails due to alloy scattering are found.
引文
[1] Paiella R, (eds.). Intersubband transitions in quantum structures. New York: McGraw-Hill,2006.
[2] Tonouchi M. Cutting-edge terahertz technology. Nature Photon., 2007, 1:97.
[3] Williams B S. Terahertz quantum-cascade lasers. Nature Photon., 2007, 1:517.
[4] Kazarinov R F, Suris R A. Possibility of amplification of electromagnetic waves in a semi-conductor with a supperlattice. Fiz. Tekh. Poluprovodn., 1971, 5:797.
[5] Faist J, Capasso F, Sivco D L, et al. Quantum Cascade Laser. Science, 1994, 264:553.
[6] Faist J, Capasso F, Sirtori C, et al. High power mid-infraread (λ～5μm) quantum cascadelasers operating above room temperature. Appl. Phys. Lett., 1996, 68:3680.
[7] Sirtori C, Kruck P, Barbieri S, et al. GaAs/AlxGa1?xAs quantum cascade lasers. Appl. Phys.Lett., 1998, 73:3486.
[8] Ko¨hler R, Tredicucci A, Beltram F, et al. Terahertz semiconductor-heterostructure laser.Nature, 2002, 417:156.
[9] Amanti M I, Fischer M, Scalari G, et al. Low-divergence single-mode terahertz quantumcascade laser. Nature Phtoton., 2009, 3:586.
[10] Kumar S, Chan C W I, Hu Q, et al. Two-well terahertz quantum-cascade laser with directintrawell-phonon depopulation. Appl. Phys. Lett., 2009, 95:141110.
[11] Liu P Q, Ho?man A J, Escarra M D, et al. Highly power-e?cient quantum cascade lasers.Nature Phtoton., 2010, 4:95.
[12] Yang Q, Manz C, Bronner W, et al. GaInAs/AlAsSb quantum-cascade lasers operating upto 400 K. Appl. Phys. Lett. 86, 2005, 86:131107.
[13] Diehl I, Mentese S, Mu¨ller E, et al. Electroluminescence from strain-compensatedSi0.2Ge0.8/Si quantum-cascade structures based on a bound-to-continuum transition. Appl.Phys. Lett., 2002, 81:4700.
[14] Jovanovic′V D, Indjin D, Ikonic′Z, et al. Simulation and design of GaN/AlGaN far-infrared(λ～34μm) quantum-cascade laser. Appl. Phys. Lett., 2004, 84:2995.
[15] Bellotti E, Driscoll K, Moustakas T D, et al. Monte Carlo study of GaN versus GaAsterahertz quantum cascade structures. Appl. Phys. Lett., 2008, 92:101112.
[16] Nevou L, Tchernycheva M, Julien F H, et al. Short wavelength (λ= 2.13μm) intersubbandluminescence from GaN/AlN quantum wells at room temperature. Appl. Phys. Lett., 2007,90:121106.
[17] Driscoll K, Liao Y, Bhattacharyya A, et al. Optically pumped intersubband emission ofshort-wave infrared radiation with GaN/AlN quantum wells. Appl. Phys. Lett., 2009,94:081120.
[18] Beck M, Hofstetter D, Aellen T, et al. Continuous Wave Operation of a Mid-Infrared Semi-conductor Laser at Room Temperature. Science, 2002, 295:301.
[19] Franz K J, Menzel S, Ho?man A J, et al. High k-space lasing in a dual-wavelength quantumcascade laser. Nature Photon., 2008, 3:50.
[20] Williams B S, Callebaut H, Kumar S, et al. 3.4-THz quantum cascade laser based onlongitudinal-optical-phonon scattering for depopulation. Appl. Phys. Lett., 2003, 82:1015.
[21] Williams B S, Kumar S, Callebaut H, et al. Terahertz quantum-cascade laser operating upto 137 K. Appl. Phys. Lett., 2003, 83:5142.
[22] Sirtori C, Kruck P, Barbieri S, et al. Low-loss Al-free waveguides for unipolar semiconduc-tor lasers. Appl. Phys. Lett., 1999, 75:3911.
[23] Capasso F, Gmachl C, Paiella R, et al. New frontiers in quantum cascade lasers and appli-cations. Selected Topics in Quantum Electronics, IEEE Journal of, 2000, 6(6):931–947.
[24] Ulrich J, Zobl R, Unterrainer K, et al. Magnetic-field-enhanced quantum-cascade emission.Appl. Phys. Lett., 2000, 76:19.
[25] Smirnov D, Becker C, Drachenko O, et al. Control of electron–optical-phonon scatteringrates in quantum box cascade lasers. Phys. Rev. B, 2002, 66(12):121305.
[26] Becker C, Vasanelli A, Sirtori C, et al. Electron–longitudinal optical phonon interaction be-tween Landau levels in semiconductor heterostructures. Phys. Rev. B, 2004, 69(11):115328.
[27] Leuliet A, Vasanelli A, Wade A, et al. Electron scattering spectroscopy by a high magneticfield in quantum cascade lasers. Phys. Rev. B, 2006, 73(8):085311.
[28] Alton J, Barbieri S, Fowler J, et al. Magnetic field in-plane quantization and tuning ofpopulation inversion in a THz superlattice quantum cascade laser. Phys. Rev. B, 2003,68(8):081303.
[29] Scalari G, Blaser S, Faist J, et al. Terahertz Emission from Quantum Cascade Lasers inthe Quantum Hall Regime: Evidence for Many Body Resonances and Localization E?ects.Phys. Rev. Lett., 2004, 93(23):237403.
[30] Wade A, Fedorov G, Smirnov D, et al. Terahertz optics Magnetic-field-assisted terahertzquantum cascade laser operating up to 225 K. Nature Photon., 2008, 3:41.
[31] Regnault N, Ferreira R, Bastard G. Broadening e?ects due to alloy scattering in a quantumcascade laser. Phys. Rev. B, 2007, 76(16):165121.
[32] Vasanelli A, Leuliet A, Sirtori C, et al. Role of elastic scattering mechanisms inGaInAs/AlInAs quantum cascade lasers. Appl. Phys. Lett., 2006, 89:172120.
[33] Chen Y, Regnault N, Ferreira R, et al. Optical-phonon scattering and theory of magneto-polarons in a quantum well in a strong magnetic field. Phys. Rev. B, 2009, 79(23):235314.
[34] Heinonen O, Taylor P L, Girvin S M. Electron-phonon interactions and the breakdown ofthe dissipationless quantum Hall e?ect. Phys. Rev. B, 1984, 30(6):3016–3019.
[35] Chaubet C, Raymond A, Dur D. Heating of two-dimensional electrons by a high electricfield in a quantizing magnetic field: Consequences in Landau emission and in the quantumHall e?ect. Phys. Rev. B, 1995, 52(15):11178–11192.
[36] Chow E, Wei H P, Girvin S M, et al. Phonon Emission from a 2D Electron Gas: Evidenceof Transition to the Hydrodynamic Regime. Phys. Rev. Lett., 1996, 77(6):1143–1146.
[37] Chaubet C, Geniet F. Nonequilibrium occupation of Landau levels and universal criticalfield in the quantum-Hall-e?ect breakdown. Phys. Rev. B, 1998, 58(19):13015–13027.
[38] Kawano Y, Okamoto T. Imaging of intra- and inter-Landau-level scattering in quantum Hallsystems. Phys. Rev. B, 2004, 70(8):081308.
[39] Ikushima K, Sakuma H, Komiyama S, et al. Visualization of quantum Hall edge channelsthrough imaging of terahertz emission. Phys. Rev. B, 2007, 76(16):165323.
[40] Zhang J q, Vitkalov S, Bykov A A, et al. E?ect of a dc electric field on the longitudinal resis-tance of two-dimensional electrons in a magnetic field. Phys. Rev. B, 2007, 75(8):081305.
[41] Zhang J Q, Vitkalov S, Bykov A A. Nonlinear resistance of two-dimensional electrons incrossed electric and magnetic fields. Phys. Rev. B, 2009, 80(4):045310.
[42] Helm M, Gornik E. Landau-level population inversion in crossed electric and quantizingmagnetic fields. Phys. Rev. B, 1986, 34(10):7459–7462.
[43] Helm M, Unterrainer K, Gornik E. Hot-carrier quantum distribution function in crossedelectric and magnetic fields. Phys. Rev. B, 1989, 39(9):6212–6215.
[44] Kim K W, Stroscio M A. Novel Landau level laser in the quantum Hall regime. Appl. Phys.Lett., 1986, 48:559.
[45] Ashcroft N W, Mermin N D. Solid states physics. New York: HoltRinehart and Winston,1976.
[46] Bastard G. Wave mechanics applied to semiconductor heterostructures. New York: Wiley-Interscience, 1991.
[47] Soven P. Coherent-Potential Model of Substitutional Disordered Alloys. Phys. Rev., 1967,156(3):809–813.
[48] Velicky′B, Kirkpatrick S, Ehrenreich H. Single-Site Approximations in the Electronic The-ory of Simple Binary Alloys. Phys. Rev., 1968, 175(3):747–766.
[49] Bastard G. Energy levels and alloy scattering in InP-In(Ga)As heterojunctions. Appl. Phys.Lett., 1983, 43:591.
[50] Fromhold T M, Wilkinson P B, Sheard F W, et al. Manifestations of Classical Chaos in theEnergy Level Spectrum of a Quantum Well. Phys. Rev. Lett., 1995, 75(6):1142–1145.
[51] Gutzwiller M C. Chaos in classical and quantum mechanics. New York: Springer-Verlag,1990.
[52] Ando T, Uemura Y. Theory of Quantum Transport in a Two-Dimensional Electron Systemunder Magnetic Fields. : I. Characteristics of Level Broadening and Transpot under StrongFields. Journal of the Physical Society of Japan, 1974, 36:959.
[53] Ando T, Fowler A B, Stern F. Electronic properties of two-dimensional systems. Rev. Mod.Phys., 1982, 54(2):437–672.
[54] Trugman S A. Localization, percolation, and the quantum Hall e?ect. Phys. Rev. B, 1983,27(12):7539–7546.
[55] Hashimoto K, Sohrmann C, Wiebe J, et al. Quantum Hall Transition in Real Space: FromLocalized to Extended States. Phys. Rev. Lett., 2008, 101(25):256802.
[56] Fro¨hlich H. Polarons and Excitons. Edinburgh: Oliver and Boyd, 1963: 1.
[57] Lyddane R H, Sachs R G, Teller E. On the Polar Vibrations of Alkali Halides. Phys. Rev.,1941, 59(8):673–676.
[58] Herring C, Vogt E. Transport and Deformation-Potential Theory for Many-Valley Semicon-ductors with Anisotropic Scattering. Phys. Rev., 1956, 101(3):944–961.
[59] Chang I F, Mitra S S. Long wavelength optical phonons in mixed crystals. Advances inPhysics, 1971, 20:359.
[60] Elliott R J, Krumhansl J A, Leath P L. The theory and properties of randomly disorderedcrystals and related physical systems. Rev. Mod. Phys., 1974, 46(3):465–543.
[61] Barker A S, Sievers A J. Optical studies of the vibrational properties of disordered solids.Rev. Mod. Phys., 1975, 47(S2):S1–S179.
[62] Cochran W. Z. Kristallogr., 1959, 112:465.
[63] Toyozawa Y. Polarons in Ionic Crystals and Polar Semiconductors. North-Holland, Ams-terdam, 1972: 1.
[64] Nash K J, Skolnick M S, Bass S J. Electron-phonon interactions in indium gallium arsenide.Semicond. Sci. Technol., 1987, 2:329.
[65] Ridley B K. Quantum processes in semiconductors. 4th ed., Oxford: Clarendon Press, 1999.
[66] Vurgaftman I, Meyer J R, Ram-Mohan L R. Band parameters for III–V compound semi-conductors and their alloys. Journal of Applied Physics, 2001, 89:5815.
[67] Landau L D. U¨ber die Bewegung der Elektronen in Kristalgitter. Phys. Z. Sowjetunion,1933, 3:644.
[68] Fro¨hlich H. Electrons in lattice fields. Adv. Phys., 1954, 3:325.
[69] Devreese J T. Fro¨hlich polarons from 0D to 3D: concepts and recent developments. J. Phys.:Condens. Matter, 2007, 19:255201.
[70] Holstein T. Studies of polaron motion : Part I. The molecular-crystal model. Annals ofPhysics, 1959, 8:325.
[71] Salje E K H, Alexandrov A S, Liang W Y, (eds.). Polarons and Bipolarons in High-TcSuperconductors and Related Materials. New York: Cambridge University Press, 1995.
[72] Alexandrov A S, (eds.). Polarons in Advanced Materials. Berlin: Springer, 2007.
[73] Fomin V M, Gladilin V N, Devreese J T, et al. Photoluminescence of spherical quantumdots. Phys. Rev. B, 1998, 57(4):2415–2425.
[74] Rossi F, (eds.). Semiconductor macroatoms : basic physics and quantum-device applica-tions. London: Imperial College Press, 2005: 183–219.
[75] Grange T, Ferreira R, Bastard G. Polaron relaxation in self-assembled quantum dots: Break-down of the semiclassical model. Phys. Rev. B, 2007, 76(24):241304.
[76] Benisty H, Sotomayor-Torre`s C M, Weisbuch C. Intrinsic mechanism for the poor lumines-cence properties of quantum-box systems. Phys. Rev. B, 1991, 44(19):10945–10948.
[77] Heitz R, Veit M, Ledentsov N N, et al. Energy relaxation by multiphonon processes inInAs/GaAs quantum dots. Phys. Rev. B, 1997, 56(16):10435–10445.
[78] Li X Q, Nakayama H, Arakawa Y. Phonon bottleneck in quantum dots: Role of lifetime ofthe confined optical phonons. Phys. Rev. B, 1999, 59(7):5069–5073.
[79] Verzelen O, Ferreira R, Bastard G. Polaron lifetime and energy relaxation in semiconductorquantum dots. Phys. Rev. B, 2000, 62(8):R4809–R4812.
[80] Cohen-Tannoudji C, Dupont-Roc J, Grynberg G. Photons and atoms : introduction to quan-tum electrodynamics. New York: Wiley, 1989.
[81] Stroscio M, (eds.). Phonons in nanostructures. Cambridge; New York: Cambridge Univer-sity Press, 2001.
[82] Kim K W, Stroscio M A. Electron-optical-phonon interaction in binary/ternary heterostruc-tures. J. Appl. Phys., 1990, 68:6289.
[83] Zibik E A, Wilson L R, Green R P, et al. Intraband relaxation via polaron decay in InAsself-assembled quantum dots. Phys. Rev. B, 2004, 70(16):161305.
[84] Fano U. E?ects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev.,1961, 124(6):1866–1878.
[85] Merkulov I A, Efros A L, Rosen M. Electron spin relaxation by nuclei in semiconductorquantum dots. Phys. Rev. B, 2002, 65(20):205309.
[86] Khaetskii A, Loss D, Glazman L. Electron spin evolution induced by interaction with nucleiin a quantum dot. Phys. Rev. B, 2003, 67(19):195329.
[87] Shah J, (eds.). Ultrafast spectroscopy of semiconductors and semiconductor nanostructures.Berlin: Springer, 1999.
[2] Tonouchi M. Cutting-edge terahertz technology. Nature Photon., 2007, 1:97.
[3] Williams B S. Terahertz quantum-cascade lasers. Nature Photon., 2007, 1:517.
[4] Kazarinov R F, Suris R A. Possibility of amplification of electromagnetic waves in a semi-conductor with a supperlattice. Fiz. Tekh. Poluprovodn., 1971, 5:797.
[5] Faist J, Capasso F, Sivco D L, et al. Quantum Cascade Laser. Science, 1994, 264:553.
[6] Faist J, Capasso F, Sirtori C, et al. High power mid-infraread (λ～5μm) quantum cascadelasers operating above room temperature. Appl. Phys. Lett., 1996, 68:3680.
[7] Sirtori C, Kruck P, Barbieri S, et al. GaAs/AlxGa1?xAs quantum cascade lasers. Appl. Phys.Lett., 1998, 73:3486.
[8] Ko¨hler R, Tredicucci A, Beltram F, et al. Terahertz semiconductor-heterostructure laser.Nature, 2002, 417:156.
[9] Amanti M I, Fischer M, Scalari G, et al. Low-divergence single-mode terahertz quantumcascade laser. Nature Phtoton., 2009, 3:586.
[10] Kumar S, Chan C W I, Hu Q, et al. Two-well terahertz quantum-cascade laser with directintrawell-phonon depopulation. Appl. Phys. Lett., 2009, 95:141110.
[11] Liu P Q, Ho?man A J, Escarra M D, et al. Highly power-e?cient quantum cascade lasers.Nature Phtoton., 2010, 4:95.
[12] Yang Q, Manz C, Bronner W, et al. GaInAs/AlAsSb quantum-cascade lasers operating upto 400 K. Appl. Phys. Lett. 86, 2005, 86:131107.
[13] Diehl I, Mentese S, Mu¨ller E, et al. Electroluminescence from strain-compensatedSi0.2Ge0.8/Si quantum-cascade structures based on a bound-to-continuum transition. Appl.Phys. Lett., 2002, 81:4700.
[14] Jovanovic′V D, Indjin D, Ikonic′Z, et al. Simulation and design of GaN/AlGaN far-infrared(λ～34μm) quantum-cascade laser. Appl. Phys. Lett., 2004, 84:2995.
[15] Bellotti E, Driscoll K, Moustakas T D, et al. Monte Carlo study of GaN versus GaAsterahertz quantum cascade structures. Appl. Phys. Lett., 2008, 92:101112.
[16] Nevou L, Tchernycheva M, Julien F H, et al. Short wavelength (λ= 2.13μm) intersubbandluminescence from GaN/AlN quantum wells at room temperature. Appl. Phys. Lett., 2007,90:121106.
[17] Driscoll K, Liao Y, Bhattacharyya A, et al. Optically pumped intersubband emission ofshort-wave infrared radiation with GaN/AlN quantum wells. Appl. Phys. Lett., 2009,94:081120.
[18] Beck M, Hofstetter D, Aellen T, et al. Continuous Wave Operation of a Mid-Infrared Semi-conductor Laser at Room Temperature. Science, 2002, 295:301.
[19] Franz K J, Menzel S, Ho?man A J, et al. High k-space lasing in a dual-wavelength quantumcascade laser. Nature Photon., 2008, 3:50.
[20] Williams B S, Callebaut H, Kumar S, et al. 3.4-THz quantum cascade laser based onlongitudinal-optical-phonon scattering for depopulation. Appl. Phys. Lett., 2003, 82:1015.
[21] Williams B S, Kumar S, Callebaut H, et al. Terahertz quantum-cascade laser operating upto 137 K. Appl. Phys. Lett., 2003, 83:5142.
[22] Sirtori C, Kruck P, Barbieri S, et al. Low-loss Al-free waveguides for unipolar semiconduc-tor lasers. Appl. Phys. Lett., 1999, 75:3911.
[23] Capasso F, Gmachl C, Paiella R, et al. New frontiers in quantum cascade lasers and appli-cations. Selected Topics in Quantum Electronics, IEEE Journal of, 2000, 6(6):931–947.
[24] Ulrich J, Zobl R, Unterrainer K, et al. Magnetic-field-enhanced quantum-cascade emission.Appl. Phys. Lett., 2000, 76:19.
[25] Smirnov D, Becker C, Drachenko O, et al. Control of electron–optical-phonon scatteringrates in quantum box cascade lasers. Phys. Rev. B, 2002, 66(12):121305.
[26] Becker C, Vasanelli A, Sirtori C, et al. Electron–longitudinal optical phonon interaction be-tween Landau levels in semiconductor heterostructures. Phys. Rev. B, 2004, 69(11):115328.
[27] Leuliet A, Vasanelli A, Wade A, et al. Electron scattering spectroscopy by a high magneticfield in quantum cascade lasers. Phys. Rev. B, 2006, 73(8):085311.
[28] Alton J, Barbieri S, Fowler J, et al. Magnetic field in-plane quantization and tuning ofpopulation inversion in a THz superlattice quantum cascade laser. Phys. Rev. B, 2003,68(8):081303.
[29] Scalari G, Blaser S, Faist J, et al. Terahertz Emission from Quantum Cascade Lasers inthe Quantum Hall Regime: Evidence for Many Body Resonances and Localization E?ects.Phys. Rev. Lett., 2004, 93(23):237403.
[30] Wade A, Fedorov G, Smirnov D, et al. Terahertz optics Magnetic-field-assisted terahertzquantum cascade laser operating up to 225 K. Nature Photon., 2008, 3:41.
[31] Regnault N, Ferreira R, Bastard G. Broadening e?ects due to alloy scattering in a quantumcascade laser. Phys. Rev. B, 2007, 76(16):165121.
[32] Vasanelli A, Leuliet A, Sirtori C, et al. Role of elastic scattering mechanisms inGaInAs/AlInAs quantum cascade lasers. Appl. Phys. Lett., 2006, 89:172120.
[33] Chen Y, Regnault N, Ferreira R, et al. Optical-phonon scattering and theory of magneto-polarons in a quantum well in a strong magnetic field. Phys. Rev. B, 2009, 79(23):235314.
[34] Heinonen O, Taylor P L, Girvin S M. Electron-phonon interactions and the breakdown ofthe dissipationless quantum Hall e?ect. Phys. Rev. B, 1984, 30(6):3016–3019.
[35] Chaubet C, Raymond A, Dur D. Heating of two-dimensional electrons by a high electricfield in a quantizing magnetic field: Consequences in Landau emission and in the quantumHall e?ect. Phys. Rev. B, 1995, 52(15):11178–11192.
[36] Chow E, Wei H P, Girvin S M, et al. Phonon Emission from a 2D Electron Gas: Evidenceof Transition to the Hydrodynamic Regime. Phys. Rev. Lett., 1996, 77(6):1143–1146.
[37] Chaubet C, Geniet F. Nonequilibrium occupation of Landau levels and universal criticalfield in the quantum-Hall-e?ect breakdown. Phys. Rev. B, 1998, 58(19):13015–13027.
[38] Kawano Y, Okamoto T. Imaging of intra- and inter-Landau-level scattering in quantum Hallsystems. Phys. Rev. B, 2004, 70(8):081308.
[39] Ikushima K, Sakuma H, Komiyama S, et al. Visualization of quantum Hall edge channelsthrough imaging of terahertz emission. Phys. Rev. B, 2007, 76(16):165323.
[40] Zhang J q, Vitkalov S, Bykov A A, et al. E?ect of a dc electric field on the longitudinal resis-tance of two-dimensional electrons in a magnetic field. Phys. Rev. B, 2007, 75(8):081305.
[41] Zhang J Q, Vitkalov S, Bykov A A. Nonlinear resistance of two-dimensional electrons incrossed electric and magnetic fields. Phys. Rev. B, 2009, 80(4):045310.
[42] Helm M, Gornik E. Landau-level population inversion in crossed electric and quantizingmagnetic fields. Phys. Rev. B, 1986, 34(10):7459–7462.
[43] Helm M, Unterrainer K, Gornik E. Hot-carrier quantum distribution function in crossedelectric and magnetic fields. Phys. Rev. B, 1989, 39(9):6212–6215.
[44] Kim K W, Stroscio M A. Novel Landau level laser in the quantum Hall regime. Appl. Phys.Lett., 1986, 48:559.
[45] Ashcroft N W, Mermin N D. Solid states physics. New York: HoltRinehart and Winston,1976.
[46] Bastard G. Wave mechanics applied to semiconductor heterostructures. New York: Wiley-Interscience, 1991.
[47] Soven P. Coherent-Potential Model of Substitutional Disordered Alloys. Phys. Rev., 1967,156(3):809–813.
[48] Velicky′B, Kirkpatrick S, Ehrenreich H. Single-Site Approximations in the Electronic The-ory of Simple Binary Alloys. Phys. Rev., 1968, 175(3):747–766.
[49] Bastard G. Energy levels and alloy scattering in InP-In(Ga)As heterojunctions. Appl. Phys.Lett., 1983, 43:591.
[50] Fromhold T M, Wilkinson P B, Sheard F W, et al. Manifestations of Classical Chaos in theEnergy Level Spectrum of a Quantum Well. Phys. Rev. Lett., 1995, 75(6):1142–1145.
[51] Gutzwiller M C. Chaos in classical and quantum mechanics. New York: Springer-Verlag,1990.
[52] Ando T, Uemura Y. Theory of Quantum Transport in a Two-Dimensional Electron Systemunder Magnetic Fields. : I. Characteristics of Level Broadening and Transpot under StrongFields. Journal of the Physical Society of Japan, 1974, 36:959.
[53] Ando T, Fowler A B, Stern F. Electronic properties of two-dimensional systems. Rev. Mod.Phys., 1982, 54(2):437–672.
[54] Trugman S A. Localization, percolation, and the quantum Hall e?ect. Phys. Rev. B, 1983,27(12):7539–7546.
[55] Hashimoto K, Sohrmann C, Wiebe J, et al. Quantum Hall Transition in Real Space: FromLocalized to Extended States. Phys. Rev. Lett., 2008, 101(25):256802.
[56] Fro¨hlich H. Polarons and Excitons. Edinburgh: Oliver and Boyd, 1963: 1.
[57] Lyddane R H, Sachs R G, Teller E. On the Polar Vibrations of Alkali Halides. Phys. Rev.,1941, 59(8):673–676.
[58] Herring C, Vogt E. Transport and Deformation-Potential Theory for Many-Valley Semicon-ductors with Anisotropic Scattering. Phys. Rev., 1956, 101(3):944–961.
[59] Chang I F, Mitra S S. Long wavelength optical phonons in mixed crystals. Advances inPhysics, 1971, 20:359.
[60] Elliott R J, Krumhansl J A, Leath P L. The theory and properties of randomly disorderedcrystals and related physical systems. Rev. Mod. Phys., 1974, 46(3):465–543.
[61] Barker A S, Sievers A J. Optical studies of the vibrational properties of disordered solids.Rev. Mod. Phys., 1975, 47(S2):S1–S179.
[62] Cochran W. Z. Kristallogr., 1959, 112:465.
[63] Toyozawa Y. Polarons in Ionic Crystals and Polar Semiconductors. North-Holland, Ams-terdam, 1972: 1.
[64] Nash K J, Skolnick M S, Bass S J. Electron-phonon interactions in indium gallium arsenide.Semicond. Sci. Technol., 1987, 2:329.
[65] Ridley B K. Quantum processes in semiconductors. 4th ed., Oxford: Clarendon Press, 1999.
[66] Vurgaftman I, Meyer J R, Ram-Mohan L R. Band parameters for III–V compound semi-conductors and their alloys. Journal of Applied Physics, 2001, 89:5815.
[67] Landau L D. U¨ber die Bewegung der Elektronen in Kristalgitter. Phys. Z. Sowjetunion,1933, 3:644.
[68] Fro¨hlich H. Electrons in lattice fields. Adv. Phys., 1954, 3:325.
[69] Devreese J T. Fro¨hlich polarons from 0D to 3D: concepts and recent developments. J. Phys.:Condens. Matter, 2007, 19:255201.
[70] Holstein T. Studies of polaron motion : Part I. The molecular-crystal model. Annals ofPhysics, 1959, 8:325.
[71] Salje E K H, Alexandrov A S, Liang W Y, (eds.). Polarons and Bipolarons in High-TcSuperconductors and Related Materials. New York: Cambridge University Press, 1995.
[72] Alexandrov A S, (eds.). Polarons in Advanced Materials. Berlin: Springer, 2007.
[73] Fomin V M, Gladilin V N, Devreese J T, et al. Photoluminescence of spherical quantumdots. Phys. Rev. B, 1998, 57(4):2415–2425.
[74] Rossi F, (eds.). Semiconductor macroatoms : basic physics and quantum-device applica-tions. London: Imperial College Press, 2005: 183–219.
[75] Grange T, Ferreira R, Bastard G. Polaron relaxation in self-assembled quantum dots: Break-down of the semiclassical model. Phys. Rev. B, 2007, 76(24):241304.
[76] Benisty H, Sotomayor-Torre`s C M, Weisbuch C. Intrinsic mechanism for the poor lumines-cence properties of quantum-box systems. Phys. Rev. B, 1991, 44(19):10945–10948.
[77] Heitz R, Veit M, Ledentsov N N, et al. Energy relaxation by multiphonon processes inInAs/GaAs quantum dots. Phys. Rev. B, 1997, 56(16):10435–10445.
[78] Li X Q, Nakayama H, Arakawa Y. Phonon bottleneck in quantum dots: Role of lifetime ofthe confined optical phonons. Phys. Rev. B, 1999, 59(7):5069–5073.
[79] Verzelen O, Ferreira R, Bastard G. Polaron lifetime and energy relaxation in semiconductorquantum dots. Phys. Rev. B, 2000, 62(8):R4809–R4812.
[80] Cohen-Tannoudji C, Dupont-Roc J, Grynberg G. Photons and atoms : introduction to quan-tum electrodynamics. New York: Wiley, 1989.
[81] Stroscio M, (eds.). Phonons in nanostructures. Cambridge; New York: Cambridge Univer-sity Press, 2001.
[82] Kim K W, Stroscio M A. Electron-optical-phonon interaction in binary/ternary heterostruc-tures. J. Appl. Phys., 1990, 68:6289.
[83] Zibik E A, Wilson L R, Green R P, et al. Intraband relaxation via polaron decay in InAsself-assembled quantum dots. Phys. Rev. B, 2004, 70(16):161305.
[84] Fano U. E?ects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev.,1961, 124(6):1866–1878.
[85] Merkulov I A, Efros A L, Rosen M. Electron spin relaxation by nuclei in semiconductorquantum dots. Phys. Rev. B, 2002, 65(20):205309.
[86] Khaetskii A, Loss D, Glazman L. Electron spin evolution induced by interaction with nucleiin a quantum dot. Phys. Rev. B, 2003, 67(19):195329.
[87] Shah J, (eds.). Ultrafast spectroscopy of semiconductors and semiconductor nanostructures.Berlin: Springer, 1999.