Efficient method for the calculation of dissipative quantum transport in quantum cascade lasers Peter Greck,1 Stefan Birner,1,2,3,∗ Bernhard Huber,1,3 and Peter Vogl1 1 Walter
Schottky Institut, Physik-Department, Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany 2 Institute for Nanoelectronics, Technische Universit¨ at M¨unchen, Arcisstr. 21, 80333 Munich, Germany 3 nextnano GmbH, S¨ udm¨ahrenstr. 21, 85586 Poing, Germany ∗ [email protected]
Abstract: We present a novel and very efficient method for calculating quantum transport in quantum cascade lasers (QCLs). It follows the nonequilibrium Green’s function (NEGF) framework but sidesteps the calculation of lesser self-energies by replacing them by a quasi-equilibrium expression. This method generalizes the phenomenological B¨uttiker probe model by taking into account individual scattering mechanisms. It is orders of magnitude more efficient than a fully self-consistent NEGF calculation for realistic devices. We apply this method to a new THz QCL design which works up to 250 K – according to our calculations. © 2015 Optical Society of America OCIS codes: (140.5965) Semiconductor lasers, quantum cascade; (250.5590) Quantum-well, -wire and -dot devices.
References and links 1. S. Datta, “Nanoscale device modeling: the Green’s function method,” Superlattices Microstruct. 28 (4), 253–278 (2000). 2. A. Wacker, “Semiconductor superlattices: a model system for nonlinear transport,” Phys. Rep. 357 (1), 1–111 (2002). 3. T. Grange, “Nanowire terahertz quantum cascade lasers,” Appl. Phys. Lett. 105 (14), 141105 (2014). 4. T. Kubis and P. Vogl, “Assessment of approximations in nonequilibrium Green’s function theory,” Phys. Rev. B 83, 195304 (2011). 5. P. Greck, Efficient calculation of dissipative quantum transport properties in semiconductor nanostructures, Selected Topics of Semiconductor Physics and Technology (G. Abstreiter, M.-C. Amann, M. Stutzmann, and P. Vogl, eds.), vol. 105 (Verein zur F¨orderung des Walter Schottky Instituts der Technischen Universit¨at M¨unchen e.V., M¨unchen, 2012). 6. M. B¨uttiker, “Role of quantum coherence in series resistors,” Phys. Rev. B 33 (5), 3020–3026 (1986). 7. M. B¨uttiker, “Symmetry of electrical conduction,” IBM J. Res. Develop. 32 (3), 317–334 (1988). 8. A. Trellakis, A. T. Galick, A. Pacelli, and U. Ravaioli, “Iteration scheme for the solution of the two-dimensional Schr¨odinger-Poisson equations in quantum structures,” J. Appl. Phys. 81 (12), 7880–7884 (1997). 9. S. Li and E. Darve, “Extension and optimization of the FIND algorithm: Computing Green’s and less-than Green’s functions,” J. Comput. Phys. 231 (4), 1121–1139 (2012). 10. P. Harrison, “The nature of the electron distribution functions in quantum cascade lasers,” Appl. Phys. Lett. 75 (18), 2800–2802 (1999). 11. M. Troccoli, G. Scamarcio, V. Spagnolo, A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, and M. Striccoli, “Electronic distribution in superlattice quantum cascade lasers,” Appl. Phys. Lett. 77 (8), 1088–1090 (2000). 12. 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).
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6587
13. M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett. 88 (24), 241109 (2006). 14. M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alton, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett. 89 (13), 131114 (2006). 15. M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91 (16), 161111 (2007). 16. N. Jukam, S. S. Dhillon, D. Oustinov, J. Mad´eo, J. Tignon, R. Colombelli, P. Dean, M. Salih, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Terahertz time domain spectroscopy of phonon-depopulation based quantum cascade lasers,” Appl. Phys. Lett. 94 (25), 251108 (2009). 17. A. Benz, G. Fasching, A. M. Andrews, M. Martl, K. Unterrainer, T. Roch, W. Schrenk, S. Golka, and G. Strasser, “Influence of doping on the performance of terahertz quantum-cascade lasers,” Appl. Phys. Lett. 90 (10), 101107 (2007). 18. A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147 (2–3), 152–155 (2008). 19. 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). 20. 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). 21. A. Vizbaras, S. Katz, G. Boehm, and M.-C. Amann, “Short-injector quantum cascade laser emitting at 8-µm wavelength with high slope efficiency,” IEEE Photon. Technol. Lett. 21 (19), 1384–1386 (2009). 22. A. Friedrich, G. Boehm, M. C. Amann, and G. Scarpa, “Quantum-cascade lasers without injector regions operating above room temperature,” Appl. Phys. Lett. 86 (16), 161114 (2005). 23. J. S. Yu, S. Slivken, and M. Razeghi, “Injector doping level-dependent continuous-wave operation of InP-based QCLs at λ ∼ 7.3 µm above room temperature,” Semicond. Sci. Technol. 25 (12), 125015 (2010). 24. E. Benveniste, A. Vasanelli, A. Delteil, J. Devenson, R. Teissier, A. Baranov, A. M. Andrews, G. Strasser, I. Sagnes, and C. Sirtori, “Influence of the material parameters on quantum cascade devices,” Appl. Phys. Lett. 93 (13), 131108 (2008). 25. C. Sirtori, H. Page, and C. Becker, “GaAs-based quantum cascade lasers,” Phil. Trans. R. Soc. Lond. A 359 (1780), 505–522 (2001). 26. M. Garcia, E. Normand, C. R. Stanley, C. N. Ironside, C. D. Farmer, G. Duxbury, and N. Langford, “An AlGaAs– GaAs quantum cascade laser operating with a thermoelectric cooler for spectroscopy of NH3 ,” Opt. Commun. 226 (1–6), 39–43 (2003). 27. S. Katz, A. Vizbaras, G. Boehm, and M.-C. Amann, “Nonlinear gain behavior in injectorless quantum cascade lasers,” Opt. Eng. 49 (11), 111107 (2010). 28. G. Scalari, M. I. Amanti, C. Walther, R. Terazzi, M. Beck, and J. Faist, “Broadband THz lasing from a photonphonon quantum cascade structure,” Opt. Express 18 (8), 8043–8052 (2010). 29. A. Matyas, R. Chashmahcharagh, I. Kovacs, P. Lugli, K. Vijayraghavan, M. A. Belkin, and C. Jirauschek, “Improved terahertz quantum cascade laser with variable height barriers,” J. Appl. Phys. 111 (10), 103106 (2012). 30. 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). 31. T. Kubis and P. Vogl, “Predictive quantum theory of current and optical emission in quantum cascade lasers,” Proc. SPIE 7230, 723019 (2009). 32. A. Wacker, “Gain in quantum cascade lasers and superlattices: A quantum transport theory,” Phys. Rev. B 66 (8), 085326 (2002). 33. M. A. Belkin, Q. J. Wang, C. Pfl¨ugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE J. Sel. Topics Quantum Electron. 15 (3), 952–967 (2009). 34. T. Aellen, S. Blaser, M. Beck, D. Hofstetter, J. Faist, and E. Gini, “Continuous-wave distributed-feedback quantum-cascade lasers on a Peltier cooler,” Appl. Phys. Lett. 83 (10), 1929–1931 (2003). 35. S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. M´aty´as, 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).
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6588
1.
Introduction
A detailed understanding of carrier dynamics is crucial for the design and improvement of modern semiconductor nanodevices such as quantum cascade lasers (QCLs). However, neither a classical or semiclassical nor a strictly coherent, ballistic quantum mechanical theory can capture the tight interplay between incoherent relaxation processes and quantum interference effects. A general and rigorous framework to capture all of these effects is the nonequilibrium Green’s function (NEGF) theory. Today, it is well established that this theory is among the most general schemes for the prediction of quantum transport properties [1–3]. Unfortunately, the basic NEGF equations are complex, mathematically demanding, and a quantitative implementation is still a highly challenging task, even with the recent advances of modern computer hardware. Furthermore, it is very difficult to develop approximations within the NEGF formalism which maintain charge and current conservation and obey the Pauli principle [4]. In this work, we present a novel method to calculate stationary quantum transport properties that we termed multi-scattering B¨uttiker probe (MSB) model [5]. It generalizes the so-called B¨uttiker probe model [1, 6, 7] and accounts for individual scattering mechanisms. If the semiconductor device equations are discretized on a spatial grid, then the standard B¨uttiker probe scattering model associates a single momentum and energy sink with each device grid point. Unfortunately, it does not capture effects of different scattering mechanisms. In contrast, our new MSB model accounts for individual scattering mechanisms by considering multiple scattering potentials at each grid position [Fig. 1]. In this article we first outline the concepts of semiconductor heterostructures most readers are familiar with. Then we briefly summarize the NEGF scheme and the original B¨uttiker probe model before we explain the details of our novel method. Finally, we apply our MSB model to a new quantum cascade laser structure and present results which is the main motivation for this work. 2. 2.1.
Method The one-particle Hamiltonian
A quantum cascade laser is a heterostructure consisting of different layers that are laterally homogeneous. Therefore we can separate the growth coordinate (z axis) and the homogeneous layers (xy plane). We describe the electronic structure by a single-band effective mass Hamiltonian H(z, kk ) = HL (z) + HT (z, kk ), (1) with the longitudinal and transverse parts given as 1 d h¯ 2 d + Ec,0 (z) − eφ (z), HL (z) = − 2 dz m⊥ (z) dz HT (z, kk ) =
h¯ 2 kk2 2m∗k (z)
,
(2)
(3)
where h¯ is the reduced Planck constant, Ec,0 (z) is the conduction band edge energy profile of the heterostructure and kk = |kk | is the absolute value of the two-dimensional wave vector in the xy plane. Throughout this work we assume the electron effective mass to be isotropic, m∗ = m∗⊥ = m∗k . Additionally, we assume that the effective mass is independent of energy thereby neglecting any effects due to nonparabolicity. This assumption is reasonable for THz quantum cascade lasers where all relevant energy levels are close to the conduction band edge.
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6589
2.2.
The Poisson equation
The electrostatic potential φ is obtained by solving the nonlinear Poisson equation ∇ · (ε0 εs (z)∇φ (z)) = −e ND+ (z) − n(z, φ ) .
(4)
The right hand side of this equation is the charge density which consists of the electron concentration n and the ionized donor concentration ND+ . Here, e is the positive elementary charge, ε0 is the electric vacuum permittivity and εs is the static relative permittivity. The Poisson equation includes electron–electron scattering within the Hartree approximation. We solve this equation by a Newton–Raphson method. In order to solve it very efficiently, we apply a predictor– corrector approach to ‘predict’ the quantum density n(z) upon small changes dφ (z) of the electrostatic potential [8]. We thus avoid the computationally expensive calculation of the Green’s functions [Eq. (5)–(8)] within each Poisson iteration cycle. 2.3.
The NEGF scheme
The quantum transport problem of a QCL can be regarded as a carrier scattering problem between reservoirs. Hence, the system of interest must be treated as an open quantum system. A common approach is to divide the quantum system into a device region and two lead regions (source and drain). The leads are assumed to be in equilibrium and consequently, all relevant scattering processes occur inside the device. The effect of the coupling of the device to the source and drain leads is described via a so-called contact self-energy Σ RC , which incorporates transitions between the device and the leads. A full implementation of the NEGF method requires the self-consistent solution of the retarded and lesser Green’s functions, GR and G< . The first one characterizes the width and energy of the scattering states and is related to the local density of states. The second quantity characterizes the state occupancy and determines the charge and current densities, and the optical gain. Under stationary conditions, the retarded and lesser Green’s functions are sufficient to completely describe a nonequilibrium system. The self-consistent cycle involves the solution of four coupled integro-differential equations. In operator form, they read for a given energy E, GR R
Σ Σ< G
Schottky Institut, Physik-Department, Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany 2 Institute for Nanoelectronics, Technische Universit¨ at M¨unchen, Arcisstr. 21, 80333 Munich, Germany 3 nextnano GmbH, S¨ udm¨ahrenstr. 21, 85586 Poing, Germany ∗ [email protected]
Abstract: We present a novel and very efficient method for calculating quantum transport in quantum cascade lasers (QCLs). It follows the nonequilibrium Green’s function (NEGF) framework but sidesteps the calculation of lesser self-energies by replacing them by a quasi-equilibrium expression. This method generalizes the phenomenological B¨uttiker probe model by taking into account individual scattering mechanisms. It is orders of magnitude more efficient than a fully self-consistent NEGF calculation for realistic devices. We apply this method to a new THz QCL design which works up to 250 K – according to our calculations. © 2015 Optical Society of America OCIS codes: (140.5965) Semiconductor lasers, quantum cascade; (250.5590) Quantum-well, -wire and -dot devices.
References and links 1. S. Datta, “Nanoscale device modeling: the Green’s function method,” Superlattices Microstruct. 28 (4), 253–278 (2000). 2. A. Wacker, “Semiconductor superlattices: a model system for nonlinear transport,” Phys. Rep. 357 (1), 1–111 (2002). 3. T. Grange, “Nanowire terahertz quantum cascade lasers,” Appl. Phys. Lett. 105 (14), 141105 (2014). 4. T. Kubis and P. Vogl, “Assessment of approximations in nonequilibrium Green’s function theory,” Phys. Rev. B 83, 195304 (2011). 5. P. Greck, Efficient calculation of dissipative quantum transport properties in semiconductor nanostructures, Selected Topics of Semiconductor Physics and Technology (G. Abstreiter, M.-C. Amann, M. Stutzmann, and P. Vogl, eds.), vol. 105 (Verein zur F¨orderung des Walter Schottky Instituts der Technischen Universit¨at M¨unchen e.V., M¨unchen, 2012). 6. M. B¨uttiker, “Role of quantum coherence in series resistors,” Phys. Rev. B 33 (5), 3020–3026 (1986). 7. M. B¨uttiker, “Symmetry of electrical conduction,” IBM J. Res. Develop. 32 (3), 317–334 (1988). 8. A. Trellakis, A. T. Galick, A. Pacelli, and U. Ravaioli, “Iteration scheme for the solution of the two-dimensional Schr¨odinger-Poisson equations in quantum structures,” J. Appl. Phys. 81 (12), 7880–7884 (1997). 9. S. Li and E. Darve, “Extension and optimization of the FIND algorithm: Computing Green’s and less-than Green’s functions,” J. Comput. Phys. 231 (4), 1121–1139 (2012). 10. P. Harrison, “The nature of the electron distribution functions in quantum cascade lasers,” Appl. Phys. Lett. 75 (18), 2800–2802 (1999). 11. M. Troccoli, G. Scamarcio, V. Spagnolo, A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, and M. Striccoli, “Electronic distribution in superlattice quantum cascade lasers,” Appl. Phys. Lett. 77 (8), 1088–1090 (2000). 12. 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).
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6587
13. M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett. 88 (24), 241109 (2006). 14. M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alton, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett. 89 (13), 131114 (2006). 15. M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91 (16), 161111 (2007). 16. N. Jukam, S. S. Dhillon, D. Oustinov, J. Mad´eo, J. Tignon, R. Colombelli, P. Dean, M. Salih, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Terahertz time domain spectroscopy of phonon-depopulation based quantum cascade lasers,” Appl. Phys. Lett. 94 (25), 251108 (2009). 17. A. Benz, G. Fasching, A. M. Andrews, M. Martl, K. Unterrainer, T. Roch, W. Schrenk, S. Golka, and G. Strasser, “Influence of doping on the performance of terahertz quantum-cascade lasers,” Appl. Phys. Lett. 90 (10), 101107 (2007). 18. A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147 (2–3), 152–155 (2008). 19. 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). 20. 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). 21. A. Vizbaras, S. Katz, G. Boehm, and M.-C. Amann, “Short-injector quantum cascade laser emitting at 8-µm wavelength with high slope efficiency,” IEEE Photon. Technol. Lett. 21 (19), 1384–1386 (2009). 22. A. Friedrich, G. Boehm, M. C. Amann, and G. Scarpa, “Quantum-cascade lasers without injector regions operating above room temperature,” Appl. Phys. Lett. 86 (16), 161114 (2005). 23. J. S. Yu, S. Slivken, and M. Razeghi, “Injector doping level-dependent continuous-wave operation of InP-based QCLs at λ ∼ 7.3 µm above room temperature,” Semicond. Sci. Technol. 25 (12), 125015 (2010). 24. E. Benveniste, A. Vasanelli, A. Delteil, J. Devenson, R. Teissier, A. Baranov, A. M. Andrews, G. Strasser, I. Sagnes, and C. Sirtori, “Influence of the material parameters on quantum cascade devices,” Appl. Phys. Lett. 93 (13), 131108 (2008). 25. C. Sirtori, H. Page, and C. Becker, “GaAs-based quantum cascade lasers,” Phil. Trans. R. Soc. Lond. A 359 (1780), 505–522 (2001). 26. M. Garcia, E. Normand, C. R. Stanley, C. N. Ironside, C. D. Farmer, G. Duxbury, and N. Langford, “An AlGaAs– GaAs quantum cascade laser operating with a thermoelectric cooler for spectroscopy of NH3 ,” Opt. Commun. 226 (1–6), 39–43 (2003). 27. S. Katz, A. Vizbaras, G. Boehm, and M.-C. Amann, “Nonlinear gain behavior in injectorless quantum cascade lasers,” Opt. Eng. 49 (11), 111107 (2010). 28. G. Scalari, M. I. Amanti, C. Walther, R. Terazzi, M. Beck, and J. Faist, “Broadband THz lasing from a photonphonon quantum cascade structure,” Opt. Express 18 (8), 8043–8052 (2010). 29. A. Matyas, R. Chashmahcharagh, I. Kovacs, P. Lugli, K. Vijayraghavan, M. A. Belkin, and C. Jirauschek, “Improved terahertz quantum cascade laser with variable height barriers,” J. Appl. Phys. 111 (10), 103106 (2012). 30. 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). 31. T. Kubis and P. Vogl, “Predictive quantum theory of current and optical emission in quantum cascade lasers,” Proc. SPIE 7230, 723019 (2009). 32. A. Wacker, “Gain in quantum cascade lasers and superlattices: A quantum transport theory,” Phys. Rev. B 66 (8), 085326 (2002). 33. M. A. Belkin, Q. J. Wang, C. Pfl¨ugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE J. Sel. Topics Quantum Electron. 15 (3), 952–967 (2009). 34. T. Aellen, S. Blaser, M. Beck, D. Hofstetter, J. Faist, and E. Gini, “Continuous-wave distributed-feedback quantum-cascade lasers on a Peltier cooler,” Appl. Phys. Lett. 83 (10), 1929–1931 (2003). 35. S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. M´aty´as, 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).
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6588
1.
Introduction
A detailed understanding of carrier dynamics is crucial for the design and improvement of modern semiconductor nanodevices such as quantum cascade lasers (QCLs). However, neither a classical or semiclassical nor a strictly coherent, ballistic quantum mechanical theory can capture the tight interplay between incoherent relaxation processes and quantum interference effects. A general and rigorous framework to capture all of these effects is the nonequilibrium Green’s function (NEGF) theory. Today, it is well established that this theory is among the most general schemes for the prediction of quantum transport properties [1–3]. Unfortunately, the basic NEGF equations are complex, mathematically demanding, and a quantitative implementation is still a highly challenging task, even with the recent advances of modern computer hardware. Furthermore, it is very difficult to develop approximations within the NEGF formalism which maintain charge and current conservation and obey the Pauli principle [4]. In this work, we present a novel method to calculate stationary quantum transport properties that we termed multi-scattering B¨uttiker probe (MSB) model [5]. It generalizes the so-called B¨uttiker probe model [1, 6, 7] and accounts for individual scattering mechanisms. If the semiconductor device equations are discretized on a spatial grid, then the standard B¨uttiker probe scattering model associates a single momentum and energy sink with each device grid point. Unfortunately, it does not capture effects of different scattering mechanisms. In contrast, our new MSB model accounts for individual scattering mechanisms by considering multiple scattering potentials at each grid position [Fig. 1]. In this article we first outline the concepts of semiconductor heterostructures most readers are familiar with. Then we briefly summarize the NEGF scheme and the original B¨uttiker probe model before we explain the details of our novel method. Finally, we apply our MSB model to a new quantum cascade laser structure and present results which is the main motivation for this work. 2. 2.1.
Method The one-particle Hamiltonian
A quantum cascade laser is a heterostructure consisting of different layers that are laterally homogeneous. Therefore we can separate the growth coordinate (z axis) and the homogeneous layers (xy plane). We describe the electronic structure by a single-band effective mass Hamiltonian H(z, kk ) = HL (z) + HT (z, kk ), (1) with the longitudinal and transverse parts given as 1 d h¯ 2 d + Ec,0 (z) − eφ (z), HL (z) = − 2 dz m⊥ (z) dz HT (z, kk ) =
h¯ 2 kk2 2m∗k (z)
,
(2)
(3)
where h¯ is the reduced Planck constant, Ec,0 (z) is the conduction band edge energy profile of the heterostructure and kk = |kk | is the absolute value of the two-dimensional wave vector in the xy plane. Throughout this work we assume the electron effective mass to be isotropic, m∗ = m∗⊥ = m∗k . Additionally, we assume that the effective mass is independent of energy thereby neglecting any effects due to nonparabolicity. This assumption is reasonable for THz quantum cascade lasers where all relevant energy levels are close to the conduction band edge.
#229178 - $15.00 USD © 2015 OSA
Received 4 Dec 2014; revised 12 Jan 2015; accepted 14 Jan 2015; published 3 Mar 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006587 | OPTICS EXPRESS 6589
2.2.
The Poisson equation
The electrostatic potential φ is obtained by solving the nonlinear Poisson equation ∇ · (ε0 εs (z)∇φ (z)) = −e ND+ (z) − n(z, φ ) .
(4)
The right hand side of this equation is the charge density which consists of the electron concentration n and the ionized donor concentration ND+ . Here, e is the positive elementary charge, ε0 is the electric vacuum permittivity and εs is the static relative permittivity. The Poisson equation includes electron–electron scattering within the Hartree approximation. We solve this equation by a Newton–Raphson method. In order to solve it very efficiently, we apply a predictor– corrector approach to ‘predict’ the quantum density n(z) upon small changes dφ (z) of the electrostatic potential [8]. We thus avoid the computationally expensive calculation of the Green’s functions [Eq. (5)–(8)] within each Poisson iteration cycle. 2.3.
The NEGF scheme
The quantum transport problem of a QCL can be regarded as a carrier scattering problem between reservoirs. Hence, the system of interest must be treated as an open quantum system. A common approach is to divide the quantum system into a device region and two lead regions (source and drain). The leads are assumed to be in equilibrium and consequently, all relevant scattering processes occur inside the device. The effect of the coupling of the device to the source and drain leads is described via a so-called contact self-energy Σ RC , which incorporates transitions between the device and the leads. A full implementation of the NEGF method requires the self-consistent solution of the retarded and lesser Green’s functions, GR and G< . The first one characterizes the width and energy of the scattering states and is related to the local density of states. The second quantity characterizes the state occupancy and determines the charge and current densities, and the optical gain. Under stationary conditions, the retarded and lesser Green’s functions are sufficient to completely describe a nonequilibrium system. The self-consistent cycle involves the solution of four coupled integro-differential equations. In operator form, they read for a given energy E, GR R
Σ Σ< G
