Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting

Marta M. Betcke; Heinrich Voss

Applications of Mathematics (2007)

  • Volume: 52, Issue: 3, page 267-284
  • ISSN: 0862-7940

Abstract

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In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25 % of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.

How to cite

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Betcke, Marta M., and Voss, Heinrich. "Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting." Applications of Mathematics 52.3 (2007): 267-284. <http://eudml.org/doc/33288>.

@article{Betcke2007,
abstract = {In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25 % of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.},
author = {Betcke, Marta M., Voss, Heinrich},
journal = {Applications of Mathematics},
keywords = {quantum dot; nonlinear eigenvalue problem; minmax characterization; iterative projection method; electronic state; spin orbit interaction; quantum dot; nonlinear eigenvalue problem; minmax characterization; iterative projection method; Hamiltonian; nonlinear Arnoldi method},
language = {eng},
number = {3},
pages = {267-284},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting},
url = {http://eudml.org/doc/33288},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Betcke, Marta M.
AU - Voss, Heinrich
TI - Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 267
EP - 284
AB - In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25 % of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.
LA - eng
KW - quantum dot; nonlinear eigenvalue problem; minmax characterization; iterative projection method; electronic state; spin orbit interaction; quantum dot; nonlinear eigenvalue problem; minmax characterization; iterative projection method; Hamiltonian; nonlinear Arnoldi method
UR - http://eudml.org/doc/33288
ER -

References

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  1. Eigenvalue problems, In: Handbook of Numerical Analysis, Vol. II, P. G.  Ciarlet, J.-L.  Lions (eds.), North Holland, Amsterdam, 1991, pp. 641–787. (1991) MR1115240
  2. Wave Mechanics Applied to Semiconductor Heterostructures, Les editions de physique, Les Ulis Cedex, 1988. (1988) 
  3. Iterative projection methods for symmetric nonlinear eigenvalue problems with applications, In preparation. 
  4. 10.1016/j.future.2003.07.003, Future Generation Computer Systems 20 (2004), 363–372. (2004) MR2213179DOI10.1016/j.future.2003.07.003
  5. Physics of Optoelectronic Devices, John Wiley & Sons, New York, 1995. (1995) 
  6. 10.1103/PhysRevB.50.8523, Phys. Rev.  B 50 (1994), 8523–8533. (1994) DOI10.1103/PhysRevB.50.8523
  7. FEMLAB, Version  3.1, COMSOL, Inc., Burlington, 2004. (2004) 
  8. The k · p method, In: Semiconductors and Semimetals. Physics of III–V  Compounds, Vol 1, R. K.  Willardson, A. C.  Beer (eds.), Academic Press, , 1966, pp. 75–100. (1966) 
  9. 10.1140/epjb/e2002-00250-6, Eur. Phys.  J. B 28 (2002), 475–481. (2002) DOI10.1140/epjb/e2002-00250-6
  10. Spin dependent boundary conditions and spin splitting in cylindrical quantum dots, In: Techn. Proc. of Internat. Conference on Modeling and Simulation of Microsystems, 2001, pp. 562–565. (2001) 
  11. 10.1103/PhysRev.97.869, Phys. Rev. 97 (1954), 869–883. (1954) DOI10.1103/PhysRev.97.869
  12. 10.1137/0722055, SIAM J.  Numer. Anal. 22 (1985), 914–923. (1985) Zbl0594.65026MR0799120DOI10.1137/0722055
  13. Hardy-Type Inequalities. Pitman Research Notes in Mathematics Vol.  219, Longman Scientific & Technical, Harlow, 1990. (1990) MR1069756
  14. Effect of the heterointerface on the spin splitting in the modulation doped In x Ga 1 - x As/InP quantum wells for B → 0 , J. Appl. Phys. 83 (1998), 4324–4333. (1998) 
  15. 10.1063/1.1614426, J.  Appl. Phys. 94 (2003), 5891–5895. (2003) DOI10.1063/1.1614426
  16. 10.1002/1521-3951(200107)226:1<175::AID-PSSB175>3.0.CO;2-I, Phys. Stat. Sol. 226 (2001), 175–184. (2001) DOI10.1002/1521-3951(200107)226:1<175::AID-PSSB175>3.0.CO;2-I
  17. 10.1103/PhysRevB.63.165306, Phys. Rev.  B 63 (2001), 165306-1–165306-6. (2001) DOI10.1103/PhysRevB.63.165306
  18. Initializing iterative projection methods for rational symmetric eigenproblems, In: Online Proceedings of the Dagstuhl Seminar Theoretical and Computational Aspects of Matrix Algorithms, Schloss Dagstuhl, 2003ftp://ftp.dagstuhl.de/pub/Proceedings/03/03421/03421.VoszHeinrich.Other.pdf, 2003. 
  19. 10.1023/B:BITN.0000039424.56697.8b, BIT 44 (2004), 387–401. (2004) Zbl1066.65059MR2093512DOI10.1023/B:BITN.0000039424.56697.8b
  20. A rational eigenvalue problem governing relevant energy states of a quantum dots, J.  Comput. Phys. 217 (2006), 824–833. (2006) MR2260626
  21. 10.1002/mma.1670040126, Math. Methods Appl. Sci. 4 (1982), 415–424. (1982) MR0669135DOI10.1002/mma.1670040126

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