Abstract
A recently developed Fourier-transform-based formulation of the longitudinal optical phonon scattering rates in heterostructures allows us to separate the wave functions from multidimensional integrals, which depend on the intersubband transition energy, the chemical potential, and the electron temperature. Here, we discuss an efficient determination of these integrals and an automatic fitting procedure in order to provide a compact table of pre-calculated integrals. As a result, computation times on a scale of minutes for the scattering rates are achieved for any reasonable set of parameters.
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Faist, J., Capasso, F., Sivco, D.L., Sirtori, C., Hutchinson, A.L., Cho, A.Y.: Quantum cascade laser. Science 264, 553–556 (1994). doi:10.1126/science.264.5158.553
Köhler, R., Tredicucci, A., Beltram, F., Beere, H.E., Linfield, E.H., Davies, A.G., Ritchie, D.A., Iotti, R.C., Rossi, F.: Terahertz semiconductor-heterostructure laser. Nature 417, 156–159 (2002). doi:10.1038/417156a
Harrison, P.: Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures. Wiley, Chichester (2005)
Williams, B.S.: Terahertz quantum-cascade lasers. Nat. Photon. 1, 517–525 (2007). doi:10.1038/nphoton.2007.166
Jirauschek, C., Kubis, T.: Modeling techniques for quantum cascade lasers. Appl. Phys. Rev. 1, 011307 (2014). doi:10.1063/1.4863665
Lü, X., Schrottke, L., Luna, E., Grahn, H.T.: Efficient simulation of the impact of interface grading on the transport and optical properties of semiconductor heterostructures. Appl. Phys. Lett. 104, 232106 (2014). doi:10.1063/1.4882653
Schrottke, L., Lü, X., Grahn, H.T.: Fourier transform-based scattering-rate method for self-consistent simulations of carrier transport in semiconductor heterostructures. J. Appl. Phys. 117, 154309 (2015). doi:10.1063/1.4918671
Lü, X., Schrottke, L., Grahn, H.T.: Fourier-transform-based model for carrier transport in semiconductor heterostructures: Longitudinal optical phonon scattering. J. Appl. Phys. 119, 214302 (2016). doi:10.1063/1.4952741
Schrottke, L., Giehler, M., Wienold, M., Hey, R., Grahn, H.T.: Compact model for the efficient simulation of the optical gain and transport properties in THz quantum-cascade lasers. Semicond. Sci. Technol. 25, 045025 (2010). doi:10.1088/0268-1242/25/4/045025
Taj, D., Iotti, R.C., Rossi, F.: Microscopic modeling of energy relaxation and decoherence in quantum optoelectronic devices at the nanoscale. Eur. Phys. J. B 72, 305–322 (2009). doi:10.1140/epjb/e2009-00363-4
Arumugam, K., Godunov, A., Ranjan, D., Terzić, B., Zubair, M.: A memory efficient algorithm for adaptive multidimensional integration with multiple GPUs. In: 20th Annual International Conference on High Performance Computing, pp. 169–175 (2013). doi:10.1109/HiPC.2013.6799120
Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Software 17, 437–451 (1991). doi:10.1145/210232.210233
Hahn, T.: Cuba-a library for multidimensional numerical integration. Comput. Phys. Commun. 168, 78–95 (2005). doi:10.1016/j.cpc.2005.01.010
Berntsen, J., Espelid, T.O., Genz, A.: Algorithm 698: Dcuhre: an adaptive multidimensional integration routine for a vector of integrals. ACM Trans. Math. Software 17, 452–456 (1991). doi:10.1145/210232.210234
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran 90: The art of Parallel Scientific Computing. Press Syndicate of the University of Cambridge, Cambridge (2002)
Vurgaftman, I., Meyer, J.R., Ram-Mohan, L.R.: Band parameters for iiiv compound semiconductors and their alloys. J. Appl. Phys. 89, 5815–5875 (2001). doi:10.1063/1.1368156
Schrottke, L., Lü, X., Rozas, G., Biermann, K., Grahn, H.T.: Terahertz GaAs/AlAs quantum-cascade lasers. Appl. Phys. Lett. 108, 102102 (2016). doi:10.1063/1.4943657
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The authors would like to thank O. Marquardt for a careful reading of the manuscript and G. Rozas for helpful discussions.
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Lü, X., Schrottke, L. & Grahn, H.T. Efficient numerical procedure for the determination of the wave function-independent terms in longitudinal optical phonon scattering rates formulated in the Fourier domain. J Comput Electron 15, 1505–1510 (2016). https://doi.org/10.1007/s10825-016-0902-6
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DOI: https://doi.org/10.1007/s10825-016-0902-6