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Multi-Band Effective Mass Approximations pp 129–154Cite as

Finite Elements for k⋅p Multiband Envelope Equations

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 94))

Abstract

This chapter applies the finite element method to the k ⋅ p equations describing electronic states in semiconductor nanostructures. It highlights advantages over other discretization methods and discusses the crucial ingredients in order to obtain accurate results. One particular issue, the appearance of unphysical or spurious solutions, is demonstrated to emerge from an inconsistency of the continuum equation system, not the discretization, and two causes are identified whose correct treatment leads to the elimination of such solutions.

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References

  1. D.J. BenDaniel, C.B. Duke, Space-charge effects on electron tunneling. Phys. Rev. 152, 683–692 (1966)

    Article  Google Scholar 

  2. S. Ben Radhia, K. Boujdaria, S. Ridene, H. Bouchriha, G. Fishman, Band structures of GaAs, InAs, and Ge: A 24-k ⋅p model. J. Appl. Phys. 94, 5726–5731 (2003)

    Article  Google Scholar 

  3. G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (John Wiley & Sons, 1974)

    Google Scholar 

  4. D. Braess, Finite Elemente (Springer-Verlag Berlin, Heidelberg, New York, 1997)

    Book  MATH  Google Scholar 

  5. M.G. Burt, A new effective-mass equation for microstructures. Semicond. Sci. Technol. 3, 1224–1226 (1988)

    Article  Google Scholar 

  6. M.G. Burt, The justification for applying the effective-mass approximation to microstructures. J. Phys.: Condens. Matter 4, 6651–6690 (1992)

    Google Scholar 

  7. M.G. Burt, Direct derivation of effective-mass equations for microstructures with atomically abrupt boundaries. Phys. Rev. B 50, 7518 (1994)

    Article  Google Scholar 

  8. M.G. Burt, Fundamentals of envelope function theory for electronic states and photonic modes in nanostructures. J. Phys.: Condens. Matter 11, R53–R83 (1999)

    Google Scholar 

  9. M. Cardona, N.E. Christensen, G. Fasol, Relativistic band structure and spin-orbit splitting of zinc-blende-type semiconductors. Phys. Rev. B 38, 1806–1827 (1988)

    Article  Google Scholar 

  10. X. Cartoxia, D.Z.Y. Ting, T.C. McGill, Numerical spurious solutions in the effective mass approximation. J. Appl. Phys. 93, 3974 (2003)

    Article  Google Scholar 

  11. S.L. Chuang, Physics of Optoelectronic Devices (John Wiley & Sons, 1995)

    Google Scholar 

  12. S.L. Chuang, C.S. Chang, k ⋅p method for strained wurtzite semiconductors. Phys. Rev. B 54, 2491 (1996)

    Google Scholar 

  13. P. Enders, A. Barwolff, M. Woerner, D. Suisky, k ⋅p theory of energy bands, wave functions, and optical selection rules in strained tetrahedral semiconductors. Phys. Rev. B 51, 16695 (1995)

    Google Scholar 

  14. B.A. Foreman, Elimination of spurious solutions from eight-band k ⋅p theory. Phys. Rev. B 56, R12748 (1997)

    Article  Google Scholar 

  15. B.A. Foreman, Effective-mass hamiltonian and boundary conditions for the valence bands of semiconductor microstructures. Phys. Rev. B 48, 4964 (1993)

    Article  Google Scholar 

  16. B.A. Foreman, First-principles envelope-function theory for lattice-matched semiconductor heterostructures. Phys. Rev. B 72, 165345 (2005)

    Article  Google Scholar 

  17. B.A. Foreman, Choosing a basis that eliminates spurious solutions in k ⋅p theory. Phys. Rev. B 75, 235331 (2007)

    Article  Google Scholar 

  18. N. Fraj, I. Saidi, R. Ben, K. Boujdaria, Band structures of AlAs, GaP, and SiGe alloys: A 30 k ⋅p model. J. Appl. Phys. 21, 8979 (2007)

    Google Scholar 

  19. M.J. Godfrey, A.M. Malik, Boundary conditions and spurious solutions in envelope function theory. Phys. Rev. B 53, 16504 (1996)

    Article  Google Scholar 

  20. M. Grundmann, O. Stier, D. Bimberg, InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons and electronic structure. Phys. Rev. B 52, 11969 (1995)

    Article  Google Scholar 

  21. J. Hader, J.V. Moloney, A. Thranhardt, S.W. Koch, Interband Transitions in InGaN Quantum Wells. In: Nitride Semiconductor Devices: Principles and Simulation. NUSOD Institute Newark, DE 19714-7204, USA (2007), 145–167

    Google Scholar 

  22. M. Holm, M.E. Pistol, C. Pryor, Calculations of the electronic structure of strained inas quantum dots in InP. J. Appl. Phys. 92, 932 (2002)

    Article  Google Scholar 

  23. H.T. Johnson, L.B. Freund, The influence of strain on confined electronic states in semiconductor quantum structures. Int. J. Sol. Struct. 38, 1045 (2001)

    Article  MATH  Google Scholar 

  24. E.O. Kane, Energy Band Theory In: Handbook on Semiconductors, Vol. 1, W. Paul (ed.) (1982), 194–217

    Google Scholar 

  25. K.I. Kolokolov, J. Li, C.Z. Ning, k ⋅p Hamiltonian without spurious-state solutions Phys. Rev. B 68, 161308(R) (2003)

    Google Scholar 

  26. R.B. Lehoucq, D.C. Sørensen, C. Yang, Arpack users guide: Solution of large scale eigenvalue problems by implicitly restarted arnoldi methods (SIAM, Philadelphia, 1998)

    Book  Google Scholar 

  27. A.T. Meney, B. Gonul, E.P. O’Reilly, Evaluation of various approximations used in the envelope-function method. Phys. Rev. B 50, 10893 (1994)

    Article  Google Scholar 

  28. F. Mireles, S.E. Ulloa, Ordered hamiltonian and matching conditions for heterojunctions with wurtzite symmetry: GaN/Al x Ga1−x N quantum wells. Phys. Rev. B. 60, 13659 (1999)

    Article  Google Scholar 

  29. R.A. Morrow, K.R. Brownstein, Model effective-mass hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions Phys. Rev. B 30, 678–680 (1984)

    Article  Google Scholar 

  30. S.H. Park, D. Ahn and Y.T. Lee, Finite element analysis of valence band structures in quantum wires. J. Appl. Phys. 96 2055 (2004)

    Article  Google Scholar 

  31. C.R. Pidgeon, R.N. Brown, Interband magneto-absorption and Faraday rotation in insb. Phys. Rev. 146, 146 (1966)

    Article  Google Scholar 

  32. M. Povolotskyi, A.D. Carlo, Elasticity theory of pseudomorphic heterostructures grown on substrates of arbitrary thickness. J. Appl. Phys. 100, 063514 (2006)

    Article  Google Scholar 

  33. L.R. Ram-Mohan, Finite Element and Boundary Element Applications in Quantum Mechanics (Oxford University Press, 2003)

    Google Scholar 

  34. L.R. Ram-Mohan, K.H. Yoo, Wavefunction engineering of layered semiconductors: theoretical foundations. J. Phys.: Condens. Matter 18, R901–R917 (2006)

    Google Scholar 

  35. A.V. Rodina, A.Y. Alekseev, A.L. Efros, M. Rosen, B.K. Meyer, General boundary conditions for the enevelope function in the multiband k ⋅p model Phys. Rev. B. 65, 125302 (2002)

    Article  Google Scholar 

  36. U. Rössler, Nonparabolicity and warping in the conduction band of GaAs. Solid State Commun. 49, 943–947 (1984)

    Article  Google Scholar 

  37. M.F.H Schuurmans, G.W. t’Hooft, Simple calculations of confinemenet states in a quantum well. Phys. Rev. B. 31, 8041 (1985)

    Google Scholar 

  38. C. Schwab, p- and hp- Finite Element Methods (Clarendon Press Oxford, 1998).

    Google Scholar 

  39. T. Sandu, Optical matrix elements in tight-binding models with overlap. Phys. Rev. B 72, 125105 (2005)

    Article  Google Scholar 

  40. D.C. Sørensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations. Technical Report TR-96-40 (1996)

    Google Scholar 

  41. S. Steiger, R.G. Veprek, B. Witzigmann, Unified simulation of transport and luminescence in optoelectronic nanostructures. J. Comput. Electron. 7, 509–520 (2008)

    Article  Google Scholar 

  42. F. Szmulowicz, Envelope-function theory without spurious real solutions Europhys. Lett. 69, 249 (2005)

    Google Scholar 

  43. F. Szmulowicz, Solution to spurious bands and spurious real solutions in the envelope-function approximation. Phys. Rev. B 71, 245117 (2005)

    Article  Google Scholar 

  44. R.G. Veprek, S. Steiger, B. Witzigmann, Ellipticity and the spurious solution problem of k ⋅p envelope equations. Phys. Rev. B 76, 165320 (2007)

    Article  Google Scholar 

  45. R.G. Veprek, S. Steiger, B. Witzigmann, Reliable k ⋅p band structure calculation for nanostructures using finite elements. J. Comput. Electron. 7, 521–529 (2008)

    Article  Google Scholar 

  46. R.G. Veprek, Computational modeling of semiconductor nanostructures for optoelectronics, Ph.D. dissertation, ETH Zürich (2009)

    Google Scholar 

  47. R.G. Veprek, S. Steiger, B. Witzigmann, Operator ordering, ellipticity and spurious solutions in k ⋅p calculations of iii-nitride nanostructures. Opt. Quant. Electr. (2009)

    Google Scholar 

  48. I. Vurgaftman, J.R. Meyer, Band parameters for iii-v compound semiconductors and their alloys. Appl. Phys. Rev. 89, 5815 (2001)

    Article  Google Scholar 

  49. L.W. Wang, Real and spurious solutions of the 8 × 8 k ⋅p model for nanostructures. Phys. Rev. B 61, 7241 (2000)

    Article  Google Scholar 

  50. S.R. White, L.J. Sham, Electronic properties of flat-band semiconductor heterostructures. Phys. Rev. Lett. 47, 879 (1981)

    Article  Google Scholar 

  51. O.C. Zienkiewicz, R.L. Taylor, Finite Element Method Volume 1 – The Basis (Elsevier, 5th edition, 2000)

    Google Scholar 

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Acknowledgements

This work was funded by a Swiss National Science Foundation (SNSF) fellowship for prospective researchers. The chapter is based on results the authors obtained whilst working on the SNF-GAIN project, funded through SNSF grant 200021-107932.

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Authors and Affiliations

  1. Condensed Matter Theory Group, School of Physics, University of Sydney, Sydney, NSW, 2006, Australia

    Ratko G. Veprek

  2. Network for Computational Nanotechnology, Purdue University, West Lafayette, IN, 47907, USA

    Sebastian Steiger

Authors
  1. Ratko G. Veprek
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  2. Sebastian Steiger
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Correspondence to Ratko G. Veprek .

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  1. & Numerische Mathematik, Bergische Universität Wuppertal Lehrstuhl für Angewandte Mathematik, Wuppertal, Germany

    Matthias Ehrhardt

  2. Forschungsgruppe “Partielle Differentialgleichungen", Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Thomas Koprucki

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Veprek, R.G., Steiger, S. (2014). Finite Elements for k⋅p Multiband Envelope Equations. In: Ehrhardt, M., Koprucki, T. (eds) Multi-Band Effective Mass Approximations. Lecture Notes in Computational Science and Engineering, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-01427-2_4

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