Vladan Mlinar Ph.D. defense (2007)

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Ph.D. defense at the University of Antwerp, Belgium.

The thesis can be downloaded from:
http://www.cmt.ua.ac.be/thesis/mlinar.pdf

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Vladan Mlinar Ph.D. defense (2007)

  1. 1. Universiteit Antwerpen Electronic Structure Calculation of Single and Coupled Self – Assembled Quantum Dots Kandidaat: Promotor: Vladan Mlinar Prof. Dr. François Peeters Theorie van de Gecondenseerde Materie, Departement Fysica vladan.mlinar@ua.ac.be 5. July, 2007
  2. 2. QUANTUM NANOSTRUCTURES: material 1 material 2 material 1
  3. 3. QUANTUM NANOSTRUCTURES: material 1 material 2 material 1 CB G6 kx Eg G8 kz HH D G7 LH SO
  4. 4. QUANTUM NANOSTRUCTURES: material 1 material 2 material 1
  5. 5. QUANTUM NANOSTRUCTURES: material 1 material 2 material 1 • Delta-function atomic like density of states • Self-assembly • strain field • piezoelectricity • 3D confinement • band-mixing • inter-valley mixing • Coulomb interaction • External magnetic field
  6. 6. APPLICATIONS: • Quantum dot laser (Kirstaedter et al. (1994)) • (In,Ga)As/GaAs QDs used as an active medium • Low threshold current density • High characteristic temp. • greater size uniformity • higher QD densities • Quantum dot infrared photodetector • high responsivity • high temperature operation • high light coupling to normal incidence light • Single polarized photon sources • Emits one and only one photon in each pulse • Usually InAs/GaAs QDs were investigated for usage as single photon sources Seguin et al. Appl. Phys. Lett. 95, 263109 (2006)
  7. 7. OUR TASKS: • Deeper understanding of the quantum dot electronic structure not only on qualitative but also on quantitative level. • Modeling of the electronic and optical properties of quantum dots (different theoretical models) • Comparison with experiment (identification of the results of optical spectroscopy performed on QD systems, exciton complexes etc.)
  8. 8. OUTLINE: • Modeling of semiconductor nanostructures • Electronic and optical properties of QDs • Unstrained QDs in an external magnetic field • Influence of the substrate orientation
  9. 9. OUTLINE: • Modeling of semiconductor nanostructures • What do we know from experiment ? • How do we approach the problem? • k.p model for nanostructures • Electronic and optical properties of QDs
  10. 10. WHAT DO WE KNOW FROM EXPERIMENT: • Growth conditions determine electronic and optical properties of QDs I. Drouzas, J. Ulloa, D. J. Mowbray Group V – sensitive scan Group III – sensitive scan • Measurements: PL Intensity (arb. units) 9000 63593-GaAs -QDs 4K X 6000 ? 2X 3000 4X ? 3X 5x ? 0 B. Urbaszek, et al., PRL 90, 247403 (2003) S. Godefroo, et al. JAP 96, 2535 (2004) 754 755 756 757 758 759 760 Wavelength (nm) A. Rastelli et al.
  11. 11. STEPS IN THE MODELING:
  12. 12. ENVELOPE FUNCTION APPROACH: k.p theory?    Envelope function : (r )  U n r Fn r  n Hamiltonian: CB H = Hk + Hstrain (Pikus-Bir Hamiltonian) G6 kx Eg G8 kz Magnetic field: HH H = Hk + Hstrain + HZeeman D G7    eA LH k k   SO
  13. 13. FROM BULK TO NANOSTRUCTURES: Model is valid at the abrupt interface ? 1  Bulk -> nanostructures kj  i x j Conventional approach: Burt-Foreman approach: Mk x2  k x Mkk Mk x2  k x Mk x Nk x k y  1 k x Nk y  k y Nk x  N XY  k x N  k y  k y N  k x ' ' 2 M. Burt, J. Phys. Condens. Matter, 6651 (1992). G. Bastard, PRB 24, 5963 (1981). B. A. Foreman, PRB 56, R12748 (1998). material 1 material 2 material 1 (M, N – effective mass parameters)
  14. 14. FROM BULK TO NANOSTRUCTURES: Model is valid at the abrupt interface ? Operator ordering (nanostructure): Mk x2  k x Mk x N XY  k x N  k y  k y N  k x ' ' In the presence of a magnetic field: ˆ k  C{k , k }  1 K [k , k ] Ck xi ˆx j ˆ ˆ ˆ ˆ xi xj xi xj 2 Analogy: The k operators fail to commute with the effective-mass parameters, whereas in the “bulk” Hamiltonian when a magnetic field is included, the k operators fail to commute with each other. Vladan Mlinar et al., PRB 71, 205305 (2005).
  15. 15. k.p MODEL FOR NANOSTRUCTURES: GaAs/Al0.3Ga0.7As 0  10meV , h  6nm 0  15meV , h  4nm Hole energy Vladan Mlinar et al., PRB 71, 205305 (2005). levels
  16. 16. k.p MODEL FOR NANOSTRUCTURES: 0  10meV Hole energy levels InAs/GaAs Vladan Mlinar et al., PRB 71, 205305 (2005).
  17. 17. k.p MODEL FOR NANOSTRUCTURES: B = 40T 0  10meV Hole energy levels InAs/GaAs Vladan Mlinar et al., PRB 71, 205305 (2005).
  18. 18. k.p MODEL FOR NANOSTRUCTURES: InAs/GaAs system Quantum dot: Quantum well: E(B) dependence E(kt) dependence Hole energy levels Vladan Mlinar et al., PRB 71, 205305 (2005).
  19. 19. k.p MODEL FOR NANOSTRUCTURES: InAs/GaAs system Quantum dot: Quantum well: E(B) dependence E(kt) dependence ˆ 'ˆ ˆ ˆ k xi N  k x j  k x j N  k xi   3 ˆ  3 ˆ    3   3 ˆ ˆ  2 ˆ ˆ 3 (i kx j  i k xi   3 k xi , k x j  [k xi , k x j ]) 2m xi x j 2 Hole energy levels Vladan Mlinar et al., PRB 71, 205305 (2005).
  20. 20. NUMERICAL IMPLEMENTATION:
  21. 21. SUMMARY (First part): • Experiment versus theory • k.p model for nanostructures • “Correct” boundary conditions at the interface • Existance of non-physical solutions in the conventional k.p model applied to nanostructures • 3D model for nanostructures (numerical problems)
  22. 22. OUTLINE: • Modeling of semiconductor nanostructures • Electronic and optical properties of QDs: • Unstrained QDs in an external magnetic field • Influence of the substrate orientation • Type II QDs
  23. 23. UNSTRAINED QDs: MOTIVATION GaAs/AlGaAs QD: (1) XSTM image of GaAs/AlGaAs QD: (2) Experimental data (KU Leuven): E1 E2 1,655 E3 1,650 1,645 energy (eV) 1,640 1,635 1,630 1,625 1,620 0 10 20 30 40 50 magnetic field
  24. 24. UNSTRAINED GaAs/AlxGa1-xAs QDs: Collaboration with TU Berlin N=9 Intensity (arb. units) N=7 N=5 N=3 N=0 1600 1620 1640 1660 1680 Energy (meV) Position of the measured PL peak Vladan Mlinar et al., PRB 75, 205308 (2007).
  25. 25. UNSTRAINED GaAs/AlxGa1-xAs QDs: The wave function isosurfaces Electron and hole energy level plotted for 65% probability (with respect to the GaAs density conduction band) as a function of a magnetic field
  26. 26. COMPARISON WITH EXPERIMENT: GaAs/AlxGa1-xAs quantum dot:
  27. 27. SUMMARY (second part): • Interface roughness was observed to sensitively affect the transition energies, but hardly intraband energies. • For a magnetic field applied in the growth direction and in the direction perpendicular to the growth direction (where B ≤50T), we find good agreement between the exciton diamagnetic shift obtained from our calculations and the experimental data of N. Schildermas et al. (KU Leuven)
  28. 28. GROWTH ON [11k] MOTIVATION: P. Caroff et al., APL 87, 243107 (2005) M. Schmidbauer et al., PRL 96, 66108 (2006)
  29. 29. INFLUENCE OF SUBSTRATE ORIENTATION:  cos  cos  sin  cos   sin     xi  U ij x j U    sin   cos  sin  cos  0   sin  sin  cos    z y x
  30. 30. INFLUENCE OF SUBSTRATE ORIENTATION:  cos  cos  sin  cos   sin     xi  U ij x j U    sin   cos  sin  cos  0   sin  sin  cos    z - For QDs grown on [hkl] substrates: z´ θ k h2  k 2 tg  , tg  y h l x´ x Φ - For QDs grown on [11k] substrates: y´    / 4 h  1, k  1, l  2 / tg
  31. 31. PROBLEM: • How are the electronic structure and transition energies influenced by the substrate orientation? • What is new as compared to [001] grown QDs? Model QD: lens and truncated pyramidal InAs/GaAs QDs grown on [11k] substrates, where k=1,2,3. L1 P1 L2 P2 L3 P3
  32. 32. [11k] GROWN QDs – strain distribution L3 P1 • Isotropic strain is increased in [11k] grown flat dots. • The isotropic strain is almost constant in the growth direction of the larger dots.
  33. 33. [11k] GROWN QDs – strain distribution L3 P1 • Isotropic strain is increased in [11k] grown flat dots. • The isotropic strain is almost constant in the growth direction of the larger dots. Biaxial component of the strain is decreased regardless of the dot size!
  34. 34. [11k] GROWN QDs – strain distribution Simplified picture: Unstrained
  35. 35. [11k] GROWN QDs – strain distribution Simplified picture: Unstrained Electron & hole energy levels + isotropic of [11k] grown flat dots will lie energetically higher as compared to [001] grown QDs
  36. 36. [11k] GROWN QDs – strain distribution Simplified picture: Unstrained Electron & hole energy levels + isotropic of [11k] grown flat dots will + biaxial lie energetically higher as compared to [001] grown QDs Increased hole band mixing!
  37. 37. ROLE OF PIEZOELECTRICITY: •Piezoelectric effect: • Shear strain leads to piezoelectric polarization P P = eijkεjk • The polarization induces a fixed charge: ρP = -divP • Piezoelectric potential VP is obtained from the Poisson equation ρP = ε0εrΔVP
  38. 38. ROLE OF PIEZOELECTRICITY: The asymmetric piezoelectric potential influences the distribution of the electron & hole wavefunction.
  39. 39. [11k] GROWN QDs – single particle states L1 L2 L3 • Increased hole band mixing! • The maximum effective-mass occurs for (111) surfaces (JAP 79, 15 (1996)) P1 P2 P3
  40. 40. [11k] GROWN QDs: (i) hydrostatic component of the strain tensor (ii) biaxial component of the strain tensor influencing the degree of the valence band mixing, (iii) variation of the hole effective mass with the substrate orientation, since it can significantly alter the effects of the size quantization in QD. QD size in the growth direction determines the degree of the influence of the substrate orientation on the electronic and optical properties of [11k] grown QDs, whereas the influence of the shape is of secondary importance. Vladan Mlinar and Francois M. Peeters., Appl. Phys. Lett. 89, 261910 (2006); Vladan Mlinar and Francois M. Peeters, Appl. Phys. Lett 91 (2007).
  41. 41. COMPARISON WITH EXPERIMENT: • InAs/GaAs QDs in an external magnetic field • Experimental data taken from S. Godefroo et al., J. Appl. Phys. 96, 2535 (2004).
  42. 42. [11k] GROWN QDMs: Isotropic (hydrostatic) part of strain tensor for [11k] grown QDM: InAs/GaAs QDM Piezoelectric potential of QDM with isosurfaces at ±32meV (blue –32meV, Model QDM: red +32meV) -Eight identical lens shaped InAs/GaAs QDs with R = 7.91nm, h = 4.52nm Vladan Mlinar and Francois M. Peeters., J. Mater. Chem (2007).
  43. 43. [11k] GROWN QDMs: For [111] grown QDMs, changing the interdot Distance varies the transition energies up to 50meV V. Mlinar and F.M. Peeters., J. Mater. Chem (2007).
  44. 44. SUMMARY (third part): • QDs grown on high index surfaces • Continuum elastic model for strain calculation • k·p model for single-particle energy levels • QD size dependent influence of substrate orientation on the electronic and optical properties of QDs • the flatter the dot the larger the difference from the reference [001] case • Influence of the shape is of secondary importance
  45. 45. TYPE II QDs: InP/(In,Ga)P QDs InP/InGaP double quantum dot molecule: InP/InGaP triple QDM Comparison with experiment Vladan Mlinar et al., PRB 73, 235336 (2006).
  46. 46. CONCLUSIONS: Modeling: Substrate orientation: Unstrained QDs: • Experiment versus theory • QDs grown on high index • Interface roughness was observed surfaces to sensitively affect the transition • k.p model for nanostructures -CM model for strain calc energies, but hardly intraband -“Correct” boundary conditions -k.p model for single-particle energies. at the interface energy levels - Existance of non-physical • For a magnetic field applied in solutions in the conventional • QD size dependent influence of the growth direction and in the k.p model applied to nanostr. substrate orientation on the direction perpendicular to the electronic and optical properties growth direction (where B ≤50T), • 3D model for nanostructures of QDs (the flatter the dot the we find good agreement between (numerical problems) larger the difference from the the exciton diamagnetic shift reference [001] case) obtained from our calculations and the experimental data of N. Schildermas et al. (KU Leuven) Dank u voor uw aandacht!

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