| Title:
|
Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting (English) |
| Author:
|
Betcke, Marta M. |
| Author:
|
Voss, Heinrich |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
267-284 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quantum dot |
| Keyword:
|
nonlinear eigenvalue problem |
| Keyword:
|
minmax characterization |
| Keyword:
|
iterative projection method |
| Keyword:
|
electronic state |
| Keyword:
|
spin orbit interaction |
| MSC:
|
65F15 |
| MSC:
|
65F50 |
| MSC:
|
65H17 |
| MSC:
|
81Q10 |
| idZBL:
|
Zbl 1164.65412 |
| idMR:
|
MR2316156 |
| DOI:
|
10.1007/s10492-007-0014-5 |
| . |
| Date available:
|
2009-09-22T18:29:44Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134675 |
| . |
| Reference:
|
[1] I. Babuška, J. Osborn: Eigenvalue problems.In: Handbook of Numerical Analysis, Vol. II, P. G. Ciarlet, J.-L. Lions (eds.), North Holland, Amsterdam, 1991, pp. 641–787. MR 1115240 |
| Reference:
|
[2] G. Bastard: Wave Mechanics Applied to Semiconductor Heterostructures.Les editions de physique, Les Ulis Cedex, 1988. |
| Reference:
|
[3] M. Betcke: Iterative projection methods for symmetric nonlinear eigenvalue problems with applications.In preparation. |
| Reference:
|
[4] T. Betcke, H. Voss: A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems.Future Generation Computer Systems 20 (2004), 363–372. MR 2213179, 10.1016/j.future.2003.07.003 |
| Reference:
|
[5] S. L. Chuang: Physics of Optoelectronic Devices.John Wiley & Sons, New York, 1995. |
| Reference:
|
[6] 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 (1994), 8523–8533. 10.1103/PhysRevB.50.8523 |
| Reference:
|
[7] : FEMLAB, Version 3.1.COMSOL, Inc., Burlington, 2004. |
| Reference:
|
[8] E. O. Kane: The $k \cdot p$ 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. |
| Reference:
|
[9] Y. Li, O. Voskoboynikov, C. P. Lee, S. M. Sze, and O. Tretyak: Electron energy state spin-splitting in 3d cylindrical semiconductor quantum dots.Eur. Phys. J. B 28 (2002), 475–481. 10.1140/epjb/e2002-00250-6 |
| Reference:
|
[10] Y. Li, O. Voskoboynikov, J.-L. Liu, C. P. Lee, and S. M. Sze: 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. |
| Reference:
|
[11] J. M. Luttinger, W. Kohn: Motion of electrons and holes in perturbed periodic fields.Phys. Rev. 97 (1954), 869–883. 10.1103/PhysRev.97.869 |
| Reference:
|
[12] A. Neumaier: Residual inverse iteration for the nonlinear eigenvalue problem.SIAM J. Numer. Anal. 22 (1985), 914–923. Zbl 0594.65026, MR 0799120, 10.1137/0722055 |
| Reference:
|
[13] B. Opic, A. Kufner: Hardy-Type Inequalities. Pitman Research Notes in Mathematics Vol. 219.Longman Scientific & Technical, Harlow, 1990. MR 1069756 |
| Reference:
|
[14] T. Schäpers, G. Engels, J. Lange, T. Klocke, M. Hollfelder, and H. Lüth: Effect of the heterointerface on the spin splitting in the modulation doped In$_{x}$Ga$_{1-x}$As/InP quantum wells for $B\rightarrow 0$.J. Appl. Phys. 83 (1998), 4324–4333. |
| Reference:
|
[15] O. Voskoboynikov, O. Bauga, C. P. Lee, and E. Tretyak: Magnetic properties of parabolic quantum dots in the presence of the spin-orbit interaction.J. Appl. Phys. 94 (2003), 5891–5895. 10.1063/1.1614426 |
| Reference:
|
[16] O. Voskoboynikov, C. P. Lee, and E. Tretyak: Spin-orbit energy state splitting in semiconductor cylindrical and spherical quantum dots.Phys. Stat. Sol. 226 (2001), 175–184. 10.1002/1521-3951(200107)226:1<175::AID-PSSB175>3.0.CO;2-I |
| Reference:
|
[17] O. Voskoboynikov, C. P. Lee, and E. Tretyak: Spin-orbit splitting in semiconductor quantum dots with a parabolic confinement potential.Phys. Rev. B 63 (2001), 165306-1–165306-6. 10.1103/PhysRevB.63.165306 |
| Reference:
|
[18] H. Voss: 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. |
| Reference:
|
[19] H. Voss: An Arnoldi method for nonlinear eigenvalue problems.BIT 44 (2004), 387–401. Zbl 1066.65059, MR 2093512, 10.1023/B:BITN.0000039424.56697.8b |
| Reference:
|
[20] H. Voss: A rational eigenvalue problem governing relevant energy states of a quantum dots.J. Comput. Phys. 217 (2006), 824–833. MR 2260626 |
| Reference:
|
[21] H. Voss, B. Werner: A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems.Math. Methods Appl. Sci. 4 (1982), 415–424. MR 0669135, 10.1002/mma.1670040126 |
| . |
