Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2012 | 1 | 58-79

Tytuł artykułu

Mathematical modeling of semiconductor quantum dots based on the nonparabolic effective-mass approximation


Treść / Zawartość

Warianty tytułu

Języki publikacji



Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version are then presented for calculating the wanted eigenvalues that are clustered in the interior of the spectrum and may have small gaps and degeneracy. These are challenging issues arising from modeling a great variety of semiconductor nanostructures fabricated by advanced technology in semiconductor industry and science. Generic algorithms for many-electron simulations under this framework are also provided. Numerical results obtained within this framework are summarized to three eminent aspects, namely, accuracy of models, physical novelty, and effectivity of nonlinear eigensolvers. Concerning numerical accuracy, important details related to experimental data are also addressed.







Opis fizyczny




  • Department of Applied Mathematics, National Hsinchu University of
    Education, Hsinchu 300, Taiwan


  • Y. Alhassid, The statistical theory of quantum dots, Rev. Mod. Phys. 72, 895 (2000).
  • R. C. Ashoori, Electrons in artificial atoms, Nature 379, 413 .
  • Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia (2000).
  • J. A. Barker, R. J. Warburton, and E. P. O’Reilly, Electron and hole wave functions in self-assembled quantum rings, Phys. Rev. B 69, 035327 (2004).
  • G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Edition de Physique, Les Ulis (1990).
  • M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, Coupling and entangling of quantum states in quantum dot molecules, Science 291, 451 (2001).
  • L. Bergamaschi, G. Pini, and F. Sartoretto, Computational experience with sequential and parallel, preconditioned Jacobi–Davidson for large, sparse symmetric matrices, J. Comp. Phys. 188, 318 (2003).
  • D. Bimberg and N. Ledentsov, Quantum dots: lasers and amplifiers, J. Phys.: Condens. Matter 15, R1063, (2003). [Crossref]
  • S. Birner, T. Zibold, T. Andlauer, T. Kubis, M. Sabathil, A. Trellakis, and P. Vogl, Nextnano: General purpose 3-D simulations, IEEE Electron. Trans. Electron Devices 54, 2137 (2007) . [Crossref]
  • F. Boxberg and J. Tulkki, Theory of the electronic structure and carrier dynamics of strain-induced (Ga, In)As quantum dots, Rep. Prog. Phys. 70, 1425 (2007).
  • P. K. Chakraborty, L. J. Singh, and K. P. Ghatak, Simple theory of the optical absorption coefficient in nonparabolic semiconductors, J. Appl. Phys. 95, 5311 (2004) .
  • T. Chakraborty and P. Pietiläinen, Optical signatures of spin-orbit interaction effects in a parabolic quantum dot, Phys. Rev. Lett. 95, 136603 (2005).
  • J.-H. Chen and J.-L. Liu, A numerical method for exact diagonalization of semiconductor quantum dot model, Comput. Phys. Commun. 181, 937 (2010).
  • H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comp. Phys. 155, 468 (1999).
  • J. I. Climente, J. Planelles, and J. L. Movilla, Magnetization of nanoscopic quantum rings and dots, Phys. Rev. B 70, 081301 (2004).
  • M. Crouzeix, B. Philippe, and M. Sadkane, The Davidson method, SIAM J. Sci. Comput. 15, 62 (1994).
  • E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani, Spin-split subbands and magneto-oscillations in III-V asymmetric heterostructures, Phys. Rev. B 50, 8523 (1994).
  • J. G. Díaz, J. Planelles, G. W. Bryant, and J. Aizpurua, Tight-binding method and multiband effective mass theory applied to CdS nanocrystals: Single-particle effects and optical spectra fine structure, J. Phys. Chem. B 108, 17800 (2004).
  • D. P. DiVincenzo, Quantum computation, Science 270, 255 (1995).
  • H.-A. Engel and D. Loss, Fermionic Bell-state analyzer for spin qubits, Science 309, 586 (2005).
  • T. Ezaki, N. Mori, and C. Hamaguchi, Electronic structures in circular, elliptic, and triangular quantum dots, Phys. Rev. B 56, 6428 (1997).
  • I. Filikhin, E. Deyneka, and B. Vlahovic, Non-parabolic model for InAs/GaAs quantum dot capacitance spectroscopy, Solid State Comm. 140, 483 (2006).
  • I. Filikhin, V. M. Suslov, and B. Vlahovic, Modeling of InAs/GaAs quantum ring capacitance spectroscopy in the nonparabolic approximation, Phys. Rev. B 73, 205332, (2006).
  • C. Filippi, C.J. Umrigar, and M. Taut, Comparison of exact and approximate density functionals for an exactly soluble model, J. Chem. Phys. 100, 1290 (1994).
  • R. W. Freund, Band Lanczos method, in Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., SIAM, Philadelphia, 80 (2000).
  • S. Giblin, One electron makes current flow, Science 316, 1130 (2007).
  • P. Giannozzi et al, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  • S. Goedecker, Linear scaling electronic structure methods, Rev. Mod. Phys. 71, 1085 (1999).
  • T. Gokmen, M. Padmanabhan, K. Vakili, E. Tutuc, and M. Shayegan, Effective mass suppression upon complete spin-polarization in an isotropic two-dimensional electron system, Phys. Rev. B 79, 195311 (2009).
  • G. H. Golub, F. T. Luk, and M. L. Overton, A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix, ACM Trans. Math. Software 7, 149 (1981).
  • L. Greengard, Fast algorithms for classical physics, Science 265, 909 (1994).
  • R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007).
  • P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  • Y.-C. Hsieh, J.-H. Chen, S.-C. Tseng, and J.-L. Liu, The effect of band nonparabolicity on modeling few-electron ground states of charge-tunable InAs/GaAs quantum dot, Physica E 41, 403 (2009). [Crossref]
  • C.-J. Huang and C.J. Umrigar, Local correlation energies of two-electron atoms and model systems, Phys. Rev. A 56, 290 (1997).
  • F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang, A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation, J. Comp. Phys. 229, 2932 (2010).
  • T.-M. Hwang, W.-W. Lin, J.-L. Liu, and W. Wang, Fixed point methods for a semiconductor quantum dot model, Math. Comp. Modelling. 40 519 (2010).
  • T.-M. Hwang, W.-W. Lin, J.-L. Liu, and W. Wang, Jacobi-Davidson methods for cubic eigenvalue problems, Num. Lin. Alg. Appl. 12, 605 (2005).
  • K. Jacobi, Atomic structure of InAs quantum dots on GaAs, Prog. Surf. Sci. 71, 185 (2003).
  • T. Jamieson, R. Bakhshi, D. Petrova, R. Pocock, M. Imani, and A. M. Seifalian, Biological applications of quantum dots, Biomaterials 28, 4717 (2007).
  • H. Jiang, D. Ullmo, W. Yang, and H. U. Baranger, Electron-electron interactions in isolated and realistic quantum dots: A density functional theory study, Phys. Rev. B 69, 235326 (2004).
  • B. E. Kane, A silicon-based nuclear spin quantum computer, Nature 393, 133 (1998).
  • E. O. Kane, Band structure of indium antimonide, J. Phys. Chem. Sol. 1, 249 (1957).
  • M. A. Kastner, Artificial atoms, Physics Today 46, 24 (1993). [Crossref]
  • M. V. Kisin, B. L. Gelmont, and S. Luryi, Boundary-condition problem in the Kane model, Phys. Rev. B 58, 4605 (1998).
  • G. Klimeck, S. S. Ahmed, H. Bae, N. Kharche, R. Rahman, S. Clark, B. Haley, S. Lee, M. Naumov, H. Ryu, F. Saied, M. Prada, M. Korkusinski, and T. B. Boykin, Atomistic simulation of realistically sized nanodevices using NEMO 3-D-Part I: Models and benchmarks, IEEE Electron. Trans. Electron Devices 54, 2079 (2007). [Crossref]
  • G. Klimeck, S. S. Ahmed, N. Kharche, M. Korkusinski, M. Usman, M. Prada, and T. B. Boykin, Atomistic simulation of realistically sized nanodevices using NEMO 3-D-Part II: Applications, IEEE Electron. Trans. Electron Devices 54, 2090 (2007). [Crossref]
  • A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput. 23, 517 (2001).
  • W. Kohn and L. J. Sham, Self consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  • A. Kongkanand, K. Tvrdy, K. Takechi, M. Kuno, and P. V. Kamat, Quantum dot solar cells. Tuning photoresponse through size and shape control of CdSe-TiO2 architecture, J. Am. Chem. Soc. 130, 4007 (2008).
  • N. Kotera, H. Arimoto, N. Miura, K. Shibata, Y. Ueki, K. Tanaka, H. Nakamura, T. Mishima, K. Aiki, and M. Washima, Electron effective mass and nonparabolicity in InGaAs/InAlAs quantum wells lattice-matched to InP, Physica E 11, 219 (2001). [Crossref]
  • L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Few-electron quantum dots, Rep. Prog. Phys. 64, 701 (2001).
  • M. L. Leininger , C. D. Sherrill , W. D. Allen, and H. F. Schaefer, Systematic study of selected diagonalization methods for configuration interaction matrices, J. Comp. Chem. 22, 1574 (2001). [Crossref]
  • Y. Li, J.-L. Liu, O. Voskoboynikov, C. P. Lee, and S. M. Sze, Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot, Comput. Phys. Commun. 140, 399 (2001).
  • J.-L. Liu, J.-H. Chen, and O. Voskoboynikov, A model for semiconductor quantum dot molecule based on the current spin density functional theory, Comput. Phys. Commun. 175, 575 (2006).
  • D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998).
  • F. Malet, M. Barranco, E. Lipparini, R. Mayol, M. Pi, J. I. Climente, and J. Planelles, Vertically coupled double quantum rings at zero magnetic field, Phys. Rev. B 73, 245324 (2006).
  • R. V. N. Melnik, Coupled effects in low-dimensional nanostructures and multiphysics modeling, Encyclopedia of Nanoscience and Nanotechnology, Ed. by H. S. Nawla, American Scientific Publishers 12, 517 (2011).
  • D. V. Melnikov, J.-P. Leburton, A. Taha, and N. Sobh, Coulomb localization and exchange modulation in two-electron coupled quantum dots, Phys. Rev. B 74, 041309(R) (2006). [Crossref]
  • X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, Quantum dots for live cells, in vivo imaging, and diagnostics, Science 307 538 (2005).
  • B. T. Miller, W. Hansen, S. Manus, R. J. Luyken, A. Lorke, and J. P. Kotthaus, Few-electron ground states of charge-tunable self-assembled quantum dots, Phys. Rev. B 56, 6764 (1997).
  • R. M. Nieminen, From atomistic simulation towards multiscale modelling of materials, J. Phys. Condens. Matter 14, 2859 (2002). [Crossref]
  • G. Ortner, I. Yugova, G. von Högersthal, A. Larionov, H. Kurtze, D. Yakovlev, M. Bayer, S. Fafard, Z. Wasilewski, P. Hawrylak, Y. Lyanda-Geller, T. Reinecke, A. Babinski, M. Potemski, V. Timofeev, and A. Forchel, Fine structure in the excitonic emission of InAs/GaAs quantum dot molecules, Phys. Rev. B 71, 125335 (2005).
  • S. R. Patil and R. V. N. Melnik, Thermoelectromechanical effects in quantum dots, Nanotechnology 20, 125402 (2009).
  • J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B 45, 13244 (1992).
  • E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity, Phys. Rev. L. 95, 067401 (2005).
  • P. Pietiläinen and T. Chakraborty, Energy levels and magneto-optical transitions in parabolic quantum dots with spin-orbit coupling, Phys. Rev. B 73, 155315 (2006).
  • E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin, and J. T. Devreese, Development of an eight-band theory for quantum dot heterostructures, Phys. Rev. B 64, 245328 (2001).
  • M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Observation of discrete electronic states in a zero-dimensional semiconductor nanostructure, Phys. Rev. Lett. 60, 535 (1988).
  • S. M. Reimann and M. Manninen, Electronic structure of quantum dots, Rev. Mod. Phys. 74, 1283 (2002). [Crossref]
  • V. Rokhlin, Rapid solution of integral equations of classic potential theory, J. Comp. Phys. 60, 187 (1985).
  • M. Rontani, F. Rossi, F. Manghi, and E. Molinari, Coulomb correlation effects in semiconductor quantum dots: The role of dimensionality, Phys. Rev. B 59, 10165 (1999).
  • B. Rössner, H. von Känel, D. Chrastina, G. Isella, and B. Batlogg, Effective mass measurement: the influence of hole band nonparabolicity in SiGe/Ge quantum wells, Semicond. Sci. Technol. 22, S191 (2007). [Crossref]
  • M. Roy and P. A. Maksym, Efficient method for calculating electronic states in self-assembled quantum dots, Phys. Rev. B 68, 235308 (2003).
  • H. Saarikoski, E. Räsänen, S. Siljämaki, A. Harju, M. J. Puska, and R. M. Nieminen, Testing of two-dimensional local approximations in the current-spin and spin-density-functional theories, Phys. Rev. B 67, 205327 (2003).
  • N. Schildermans, M. Hayne, and V. V. Moshchalkov, Nonparabolic band effects in GaAs/AlxGa1-xAs quantum dots and ultrathin quantum wells, Phys. Rev. B 72, 115312 (2005).
  • G. L. G. Sleijpen, A. G. Booten, D. R Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36, 595 (1996). [Crossref]
  • G. L. G. Sleijpen and H. A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Rev. 42, 267 (2000).
  • O. Stier, M. Grundmann, and D. Bimberg, Electronic and optical properties of strained quantum dots modeled by 8-band k · p theory, Phys. Rev. B 59, 5688 (1999). [Crossref]
  • E. A. Stinaff, M. Scheibner, A. S. Bracker, I. V. Ponomarev, V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, and D. Gammon, Optical signatures of coupled quantum dots, Science 311, 636 (2006).
  • Y. Tanaka and H. Akera, Many-body effects in transport through a quantum dot, Phys. Rev. B 53, 3901 (1996).
  • M. B. Tavernier, E. Anisimovas, F. M. Peeters, B. Szafran, J. Adamowski, and S. Bednarek, Four-electron quantum dot in a magnetic field, Phys. Rev. B 68, 205305 (2003).
  • F. Tisseur, Backward error analysis of polynomial eigenvalue problems, Lin. Alg. Appl. 309, 339 (2000) .
  • C. A. Ullrich and M. E. Flatté, Intersubband spin-density excitations in quantum wells with Rashba spin splitting, Phys. Rev. B 66, 205305 (2002).
  • W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Electron transport through double quantum dots, Rev. Mod. Phys. 75, 1 (2003).
  • G. Vignale and M. Rasolt, Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields, Phys. Rev. B 37, 10685 (1988).
  • C. Vömel, S. Z. Tomov, O. A. Marques, A. Canning, L.-W. Wang, and J. J. Dongarra, State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems, J. Comp. Phys. 227, 7113 (2008).
  • O. Voskoboynikov, C. M. J. Wijers, J.-L. Liu, and C. P. Lee, The magneto-optical response of layers of semiconductor quantum dots and nano-rings, Phys. Rev. B 71, 245332 (2005).
  • O. Voskoboynikov, C. M. J. Wijers, J.-L. Liu, and C. P. Lee, Interband magneto-optical transitions in a layer of semiconductor nano-rings, Europhys. Lett. 70, 656 (2005).
  • H. Voss, Iterative projection methods for computing relevant energy states of a quantum dot, J. Comp. Phys. 217, 824 (2006).
  • D. D. Vvedensky, Multiscale modelling of nanostructures, J. Phys.: Condens. Matter 16, R1537 (2004). [Crossref]
  • W. Wang, T.-M. Hwang, W.-W. Lin, and J.-L. Liu, Numerical methods for semiconductor heterostructures with band nonparabolicity, J. Comp. Phys. 189, 579, (2003).
  • H.-G. Weikert, H.-D. Meyer, and L. S. Cederbaum, Block Lanczos and many-body theory: Application to the oneparticle Green’s function, J. Chem. Phys. 104, 7122 (1996).
  • C. Wetzel, et al., Electron effective mass and nonparabolicity in Ga0.47In0.53As/InP quantum wells, Phys. Rev. B 53, 1038 (1996).
  • C. M. J. Wijers, O. Voskoboynikov, and J.-L. Liu, A hybrid model for the magneto-optics of embedded nano-objects, Phys. Stat. Sol. (C) 3, 3782 (2006). [Crossref]
  • A. D. Yoffe, Semiconductor quantum dots and related systems: electronic, optical, luminescence and related properties of low dimensional systems, Advances in Physics 50, 1 (2001).
  • Y. Zhang and S. D. Sarma, Spin polarization dependence of carrier effective mass in semiconductor structures: Spintronic effective mass, Phys. Rev. Lett. 95, 256603 (2005).
  • Y. Zhou and Y. Saad, Block Krylov-Schur method for large symmetric eigenvalue problems, Numer. Alg. 47, 341 (2008).
  • Y. Zhou, Y. Saad, M. L. Tiago, and J. R. Chelikowsky, Self-consistent-field calculations using Chebyshev-filtered subspace iteration, J. Comp. Phys. 219, 172 (2006) .

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.