Excitons in self-assembled type- II quantum dots and coupled dots CMT Condensed Matter Theory Karen Janssens Milan Tadic Bart Partoens François Peeters Universiteit Antwerpen
Self-assembled quantum dots Necessary ingredients: 2 semiconductor materials with a substantially different lattice parameter, e.g. InP : a ~ 5.869 Å and GaInP : a ~ 5.653 Å (mismatch ~ 3.8%) MBE growth P 2 In InP Result GaInP lattice mismatch strain fields formation of islands self-assembled quantum dots
Stack of InAs Quantum Dots self assembly & self organization 5 nm 18 nm Bruls et al. TU/e Appl. Phys. Lett. 81, 178 (22)
Different type-ii systems electron in the dot, hole outside e.g. InP/GaInP dots CB InP InGaP InGaP e hole in the dot, electron outside e.g. GaSb/GaAs dots CB GaAs e GaSb GaAs e VB h h InAs/Si dots CB (Γ) Si InAs Si VB h CB (X) e e VB h
Theoretical approach Strain: approach of J.R. Downes et al., JAP 81, 67 (1997); J.H. Davies, JAP 84, 1358 (1998): ε r ε 1+ ν ' o ' i i ij () r = εoδij dsj 3 4π 1 ν r ur ' ' S r r Band structure: effective anisotropic mass (following the ideas of L.R. Wilson et al., Phys. Rev. B 57, R273 (1998) which was successfully applied to InAs/GaAs SAQD s). Exciton energy:hartree-fock approximation advantages: much faster numerical program + magnetic field can easily be included r r
No strain Only Coulomb interaction
Single dots the hole can sit: - at the radial boundary of the dot - above/below the disk R d z h (nm) d (nm) 2 15 1.4 5-5.6-1.2-15 -2 5 1 15 2 r h (nm) 14 12 1 8 6 4 1 2 1.14.36.18 5 1 15 2 r h (nm) 2 75% 5% 25%.7.28.18.35 5 1 15 2 r h (nm) 3 3 Study of the influence of the disk parameters d and R (at B = T) ( ) 2 P 2 π dz r ψ r, z dr = side h h h h h h R Distinguish between 2 regimes: disk-like regime: d << 2R pillar-like regime: d >> 2R ( 1) ( 3) 2 2 4 6 8 1 12 14 R (nm)
Pillar-like system exciton energy (mev) 85 8 75 7 65 6 z (nm) 1 5-5 electron wavefunction at B = T -1 2 4 6 8 1 ρ (nm) = = 1 = 2 = 3 55 1 2 3 4 5 6 7 8 9 B (T) the hole is sitting at the radial boundary of the quantum disk appearance of angular momentum transitions z (nm) z (nm) 15 1 5-5 -1-15 1 5-5 -1.4.2.84.6.12.14.56.8.2.1.11 d =12 nm, R = 4nm.4.28 B = T = B = 5T = 2.72.48.12.96.24 B = 25T = 1 B = 9T = 3-15 2 4 6 8 1 12 14 2 4 6 8 1 12 14 r (nm) r (nm)
Vertically coupled quantum dots exciton energy (mev) 64 62 6 58 56 54 52 5 48 Two vertically coupled dots Result for: R = 6nm, d = 6nm and d d = 3.6nm z (nm) 1-1 1.2.88.18 (a) 2 4 6 8 1 r (nm) = = 1 = 2 = 3 1 2 3 4 5 B (T) z (nm) 2 1-1.22.22.79.11 (b) -2 5 1 15 2 r (nm) d d d extra parameter to vary: interdot-distance d d z R ρ B easier realization of the pillar-like system
Spontaneous symmetry breaking enhancement of the Coulomb attraction magnetic field induces a permanent dipole moment 2.23 1 1.37.11 z (nm) z (nm) -1-2 2 1.18.12 = B = 15T 2..33.45 = B = 3T < z h > - < z e > (nm) 8 6 4 2 = = 1-1 -2 = B = 5T.22 = 3 B = 5T 2 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 2 r (nm) r (nm) 1 2 3 4 5 B (T)
Stark effect Single and coupled type-ii dots exciton energy (mev) 7.2 7. 6.8 6.6 6.4 6.2 density ψ(,z) 2 1..5. Single type-ii disk hole at:. kv / cm.5 kv / cm electron at F = 1 kv / cm.5 kv / cm -2-1 1 2 z (nm) dipole moment (e nm) 12 1 8 6 4 2..2.4.6.8 1. F (kv / cm) -.2..2.4.6.8 1. F (kv / cm) R = 1nm, d = 8nm non-parabolic Stark shift (cfr. coupled type I disks) creation of a strong dipole moment for F linear contribution is more important EF ( ) = EF ( ) pf ( F) β ( F F ) 2
Two coupled type II-disks R = 12nm, d = 3nm, d d = 3nm exciton energy (mev) 75. 74.5 74. 73.5 73. (a) z (nm) 2 1.35.1 hole.5.35.88-1 electron (b) F = kv / cm -2 5 1 15 r (nm) dipole moment (e nm) 1 5-5 -1 (c) -.5..5 F (kv / cm) hysteresis due to spontaneous symmetry breaking system is trapped in the energy minimum permanent dipole moment -.5..5 1. 1.5 F (kv / cm)
Effects due to strain Hole band engineering
Pseudomorphic quantum wells: case of compressive strain Hydrostatic strain Hydrostatic strain HH LH Tetrahedral strain splits the HH and LH bands
The effective potentials: for different SAQD s height InP/In.49 Ga.51 P Conduction band Heavy hole band Light hole band
Effective potentials 1 5 heavy hole z potential (mev) -5-1 unstrained VB offset light hole R = 8nm d = 4nm B R d r -15-1 -5 1 2 3 z (nm) r (nm)
Heavy hole: type I type II potential (mev) 4 2-2 -4-6 -8-1 d = 2nm d = 8nm R = 8nm -12-15 -1-5 1 2 3 z (nm) r (nm) d (nm) 8 7 6 5 4 3 z (nm) 15 1 5-5 -1 R = 8nm d = 8nm -15 5 1 15 2 25 3 r (nm) 5% Heavy hole type II - like z (nm) type I - like 3 4 5 6 7 8 9 1 11 12 13 14 6 4 2-2 -4-6 R (nm) R = 8nm d = 2nm 5 1 15 r (nm)
Heavy light hole transition disk thickness d (nm) 5.4 5.2 5. 4.8 4.6 4.4 4.2 4. 3.8 3.6 3.4 3.2 3. Light hole groundstate B = T B = 5T Heavy hole groundstate 6. 6.5 7. 7.5 8. 8.5 9. 9.5 1. disk radius R (nm) exciton energy (mev) 375 37 365 36 355 z (nm) 5-5 Mass (m o ) B = T hh lh 5 1 15 r (nm) B = 5T 35 1 2 3 4 5 B (T) z (nm).12 15 1 5-5 -1 z.61.15.3-15 5 1 r (nm)
Comparison with experiment 1 Diamagnetic shift: E = E( B) E( B= T) 15 experimental result heavy hole exciton light hole exciton R = 8nm influence of radius? influence of masses? E (mev) 1 5 d = 2.5nm R = 6nm with fitted masses Fitted masses: m =.15m.77m e m =.5m.1515m hh, m = 1.5m.269m lh, 1 2 3 4 5 B (T) 1 M. Hayne, R. Provoost, M.K. Zundel, Y.M. Manz, K. Eberl, V.V. Moshchalkov, Phys. Rev. B 62, 1324 (2).
Start with a positive unstrained VB offset 1 Still discrepancy between theory and experiment increase of the masses needed to obtain good fit m =.15 m ; m =.5 m ; m = 1.5m e hh, lh, E (mev) 15 1 5 R = 8nm d = 2.5nm experimental result heavy hole exciton light hole exciton V h = -45meV V h = 55meV with fitted masses 1 2 3 4 5 B (T) CB Negative offset e VB h h Positive offset CB e h VB 1 S.-H. Wei and A. Zunger, Appl. Phys. Lett. 72, 211 (1998)
Coupled cylindrical SAQD s Two-dot stack Three-dot stack
Energy levels in vertically coupled quantum dots R=8nm h=2nm Electron Heavy hole Light hole
Effective potentials in two vertically coupled quantum dots Conduction band Heavy hole band Light hole band
Hole states F z =J z +L z =fħ: total angular momentum J z =jħ : Bloch part L z =lħ : envelope part f =3/2 : Heavy hole (j=3/2), l=,-3 Light hole (j=1/2), l=-1, -2 f =1/2 : Heavy hole (j=3/2), l=-1,-2 Light hole (j=1/2), l=,-1
Probability density of the odd S 3/2 state in the two-dot stack (ground state for d>2 nm)
Ground + S 1/2 Second + S 1/2 Ground - S 1/2
Energy levels in a three-dot stack
Probability density of the even S 3/2 state in the three-dot stack (ground state for d>2 nm)
Excitons in the two-dot stack Dependence of the ground states for F exc =-1 and F exc = on the spacer thickness (h=2nm; r=8nm). F exc =-1 (heavy-hole-like exciton) F exc = (light-hole-like exciton). Three-dot stack There appears substantial overshoots in the exciton energies.
Conclusions (1/2) Exciton in a type II quantum dot: pillar disk systems Magnetic field effect: disk-like system: parabolic -> linear increase of energy with B (Cfr. Type I) pillar-like system: angular momentum transitions vertically coupled dots with small interdot distance spontaneous symmetry breaking -> magnetic field induced dipole moment Electric field effect: Stark shift Parabolic field dependence only for single type-i disk Strongly linear dependence for coupled type-i and single and coupled type II disks -> creation of dipole moment
Conclusions (2/2) Strain effect: hole band engineering (light heavy hole). The total angular momentum of the ground state of holes changes with the thickness of the quantum dot. In coupled quantum dots: strain acts opposite to quantum mechanical coupling. It increases the electron ground state above the single quantum dot value. - Electron coupling is effective only for spacers thinner than the coupling length. - Holes exhibit electronic coupling only for very thin spacers. Strain predominantly influences the holes. Therefore, our simple model calculation will be more appropriate for e.g. GaSb/GaAs and InAs/Si dots.
The end
For details see: http://cmt.uia.ac.be Magnetoexcitons in planar type-ii quantum dots in a perpendicular magnetic field K.L. Janssens, B. Partoens, and F.M. Peeters, Phys. Rev. B 64, 155324 (21). Single and vertically coupled type-ii quantum dots in a perpendicular magnetic field: exciton grounstate properties K.L. Janssens, B. Partoens, and F.M. Peeters, Phys. Rev. B 66, 75314 (22). Stark shift in single and vertically coupled type-i and type-ii quantum dots K.L. Janssens, B. Partoens, and F.M. Peeters, Phys. Rev. B 65, 23331 (22). Effect of isotropic versus anisotropic elasticity on the electronic structure of cylindrical InP/In.49 Ga.51 P self-assembled quantum dots M. Tadić, F.M. Peeters, and K.L. Janssens, Phys. Rev. B 65, 165333 (22). Strain and band edges in single and coupled cylindrical InAs/GaAs and InP/InGaP self-assembled quantum dots M. Tadić, F.M. Peeters, K.L. Janssens, M. Korkusiński, and P. Hawrylak, J. Appl. Phys. 92, 5819 (22). Electronic structure of the valence band in cylindrical strained InP/InGaP quantum dots in an external magnetic field M. Tadić and F.M. Peeters, Physica E 12, 88 (22). Electron and hole localization in coupled InP/InGaP self-assembled quantum dots M. Tadić, F.M. Peeters, B. Partoens, and K.L. Janssens, Physica E 13, 237 (22).