The research presented in this thesis can be continued in many ways. It is a very promising first step to use strain to control the exciton g-factor over a wide range, opening the door for many new research possibilities, both numerically and experimentally.
Numerically the next steps are quite trivial; in order to get a better understanding of the experiment, the calculations presented in this work should be repeated for different dot sizes and shapes. Even the influence of the exact quantum dot composition could be investigated.
A way to improve the calculations is to include the Coulomb interactions in the model. This will hopefully also lead to more insight in the behaviour of the valence band, as this has been a big question mark for years now. Experimentally our work can be continued in many ways. A list:
• We have only investigated the exciton and biexciton in this thesis. It would be interest-ing to see how the behaviour of charged excitons compares to what we have observed here. As the difference between excitons and biexcitons was found to be small, it is expected that the charged excitons will be not much different either.
• By performing experiments on more quantum dots, clearer correlations between the emission energy, g-factors and their changes can be observed.
• Here only biaxial strain was used. The use of uniaxial strain or an asymmetric biaxial strain may cause interesting effects, especially regarding the finestructure splitting, as it breaks the symmetry in the plane of the quantum dot. Tensile strain is another regime that was untouched during our work; tensile strain can cause interesting effects as it will bring the light hole and heavy hole bands closer to each other. At some point, the mixing between the two will be significant, influencing the behaviour of the g-factor.
• The electron and hole g-factors can be determined separately by measuring in the Voigt geometry. This will provide more insight in the behaviour of the electron and hole.
• In our work, the observed change in the exciton g-factor was a few %. This effect could greatly be enhanced by making use of state of the art samples. We anticipate that enhancements of at least a factor 10 are possible. The maximum achievable strain of 0.4% was shown already several years ago. It is not likely that the membrane-technology has developed since then and still is developing, making even higher strains a possibility for the future.
• By experimenting with different quantum dot sizes, shapes and compositions, it will be possible to find exciton g-factors relatively close to zero. Numerical calculations may be of assistance in this search. By using a sample exhibiting large strains and g-factors close to zero, it should eventually be possible to tune the g-factor around zero.
This is the ultimate goal, as accomplishing this will lead to real possible applications of this technology.
• Finally, strain can be used in combination with other effects. One combination that has already proved to be very promising, as energy coincidence between the exciton and biexciton was achieved by making use of it, is strain with external electric fields.
The addition of the electric field could be used in two ways. One way is to gain even more control over the exciton g-factor, as both strain and electric field are capable of changing it. It is not unlikely that the electric field affects the exciton emission energy and g-factor in a different way than strain does. Therefore it might be possible to use a combination of the two to independently control the exciton energy and g-factor.
Whether this is indeed the case and to what extent it can be used is a very interesting research possibility for the future.
Appendix A
A.1 Eight band model
The strain independent interaction Hamiltonian was derived by Bahder and given by
where the equations in A.2 define all relevant parameters.
A = Ec+ (F + 2m~2
In A.2 the parameter B represents the symmetry-breaking in the crystal, which we as-sume to be 0, causing W and T to be 0 as well. The parameter F represents the contribution
of the kinetic term in the conduction band. The terms γ1, γ2and γ3are modified Luttinger parameters, which are related to the Luttinger parameters, γ1L, γL2 and γ3L by
γ1= γ1L−3EEP valence and conduction bands. The valence band wave functions are usually denoted by |Xi,
|Y i and |Zi. It can be shown that the only non-zero optical matrix element is hX|px|Γti = hY |py|Γti = hZ|pz|Γti = ıP0. The optical matrix element P0 is related to EP by
EP = 2m0
~2
P02. (A.4)
The strain part of the Hamiltonian is given by
where the equations in A.6 define all relevant parameters.
w = ı√13b0xy,
Analogous to the interaction Hamiltonian, b0 represents the symmetry breaking in the crystal, which will be zero in our case. Because of this, the terms w and q will also be zero.
Bibliography
[1] Epitaxy of nanostructures. V.A. Shchukin, N.N. Ledentsov, D. Bimberg, Springer 2004 [2] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57(1), 120-126 (1998), Qauntum computation
with quantum dots.
[3] Quantum dots. L. Jacak, A. W´ojs, P. Hawrylak, Springer 1998
[4] R. Trotta, P. Atkinson, J.D. Plumhof, E. Zallo, R.O. Rezaev, S. Kumar, S. Baunack, J.R. Schr¨oter, A. Rastelli, O.G. Schmidt, Adv. Mater. 24, 2668-2672 (2012), Nanomem-brane Quantum-Light-Emitting Diodes Integrated onto Piezoelectric Actuators.
[5] T. Nakaoka, T. Kakitsuka, T. Saito, S. Kako, S. Ishida, M. Nishioka, Y. Yoshikuni, Y.
Arakawa, Journal of App. Phys. 94, 6812 (2003), Strain-induced modifications of the electronic states of InGaAs quantum dots embedded in bowed airbridge structures.
[6] R. Trotta, E. Zallo, E. Magerl, O.G. Schmidt, A. Rastelli, Phys. Rev. B 88, 155312 (2013), Indepedent control of exciton and biexciton energies in single quantum dots via electroelastic fields.
[7] J.E. Avron, G. Bisker, D. Gershoni, N.H. Linder, E.A. Meirom, Phys. Rev. Letters 100, 120501 (2008), Entanglement on Demand through Time Reordering
[8] F. Klotz, V. Jovanov, J. Kierig, E.C. Clark, D. Rudolph, D. Heiss, M. Bichler, G.
Abstreiter, M.S. Brandt, J.J. Finley App. Phys. letters 96, 053113 (2010), Observation of an electrically tunable exciton g factor in InGaAs/GaAs quantum dots.
[9] R.J. Warburton, B.T. Miller, C.S. D¨urr, C. B¨odefeld, K. Karrai, J.P. Kotthaus, G.
Medeiros-Ribeiro, P.M. Petroff, S. Huant, Phys. Rev. B 58(24), 16221-16231 (1998), Coulomb interactions in small charge-tunable quantum dots.
[10] Classical electrodynamics., J.D. Jackson, Wiley 1999
[11] Introduction to Quantum Mechanics. D.J. Griffiths, Pearson Educations 2005 [12] Quantum mechanics. C. Cohen-Tannoudji, B. Diu, F. Laloe, Wiley 1977
[13] V. Fock Zeitschrift f¨ur Physik A Hadrons and Nuclei 47, 446-448 (1929), Bemerkung zur quantelung des harmonischen oszillators im magnetfeld.
[14] The Feynman lectures on physics. R.P. Feynman, Cambridge 1995
[15] P.A.M. Dirac Proceedings of the Royal society of London A 123, 714-744 (1929), Quan-tum mechanics of many-electron systems.
[16] H.W. van Kesteren, E.C. Cosman, W.A.J.A. van der Poel, C.T. Foxon, Phys. Rev. B 41, 5283-5292 (1990), Fine structure of exciton in type-II GaAs/AlAs quantum wells.
[17] M. Bayer, G. Ortner, O. Stern, A. Kuther, A.A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T.L. Reinecke, S.N. Walck, J.P. Reithmaier, F. Klopf, F. Sch¨afer, Phys. Rev. B 65(19), 195315 (2002), Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots.
[18] Introduction to Solid State Physics, Charles Kittel, Wiley 2005
[19] D.J. Dunstan, Journal of Mat. Science 8, 337-375 (1997), Review: Strain and strain relaxation in semiconductors.
[20] M. Grundmann, O. Stier, D. Bimberg, Phys. Rev. B 52, 11969 (1995), InAs/GaAs pyramidal quantom dots: Strain distribution, optical phonons, and electronic structure.
[21] E.P. O’Reilly, Semicond. Sci. Technol. 4, 121-137 (1989), Valence band engineering in strained-layer structures
[22] Fundamentals of Semiconductors: Physics and Materials Properties, Peter Y. Yu, Manuel Cardona, Springer 2010
[23] E.O. Kane, J. Phys. Chem. Solids, 1, 249 (1957), Band structure of indium antimonide.
[24] J.M. Luttinger, W.Kohn, Phys. Rev. 97, 4 (1955), Motion of Electrons and Holes in Perturbed Periodic Fields.
[25] J. Bardeen, W. Shockley Phys. Rev. 80(1) 72-80 (1950), Deformation Potentials and mobilities in Non-Polar Crystals
[26] Symmetry and Strain Induced Effects in Semiconductors, G.L. Bir, G.E. Pikus, Wiley 1974
[27] T.B. Bahder, Phys. Rev. B 41, 17 (1990), Eight-band k · p-model of strained zinc-blende crystals.
[28] Group Theory, Application to the Physics of Condensed Matter, M.S. Dresselhaus, G.
Dresselhaus and A. Jorio, Springer-Verlag 2008
[29] T.B. Bahder, Phys. Rev. B 45, 4 (1992), Analytic dispersion relations near the Γ-point in strained zinc-blende crystals.
[30] I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, Journal of Applied Physics, 89(11) (2001), Band parameters for IIIV compound semiconductors and their alloys.
[31] L.M. Roth, B. Lax, S.Zwerdling, Phys. Rev. 114(1) 90-104 (1959), Theory of Optical Magneto-Absorption Effects in Semiconductors
[32] L.M. Roth, Phys. Rev. 118(6), 1534-1540 (1960), g-factor and donor spin-lattice relax-ation for electrons in germanium and silicon.
[33] Spin-orbit coupling effects in two-dimensional electron and hole systems. R. Winkler, Springer 2003
[34] C.E. Pryor, M.E. Flatt´e, Phys. Rev. Letters 96(2) 026804 (2006), Land´e g-factors and orbital momentum quenching in semiconductor quantum dots.
[35] C.E. Pryor, M.E. Flatt´e, Phys. Rev. Letters 99(17) 179901 (2007), Erratum: Land´e g-factors and orbital momentum quenching in semiconductor quantum dots.
[36] N.A.J.M. Kleemans, J. van Bree, M. Bozkurt, P.J. Van Veldhoven, P.A.M. Nouwens, R. N¨otzel, A.Y. Silov, P.M. Koenraad, M.E. Flatt´e, Phys. Rev. B 79 045311 (2009), Size-dependent exciton g-factor in self-assembled InAs/InP quantum dots.
[37] J. van Bree, A.Y. Silov, P.M. Koenraad, M.E. Flatt´e, C.E. Pryor, Phys. Rev. B 65 165323 (2012), g-factors and diamagnetic coefficients of electrons, holes, and excitons in InAs/InP quantum dots
[38] Introduction to optics. F.L. Pedrotti, L.S. Pedrotti, Pearson Education 1993
[39] Magneto-optical properties of self-assembled III-IV semiconductor nanostructure.
N.A.J.M. Kleemans, Technische Universiteit Eindhoven 2010
[40] F. Ding, R. Singh, J.D. Plumhof, T. Zander, V. Kˇr´apek, Y.H. Chen, M. Benyoucef, V.
Zwiller, K. D¨orr, G. Beste, A. Rastelli, O.G. Schmidt, Phys. Rev. Letters 104, 067405 (2010), Tuning the Exciton Binding Energies in Single Self-Assembled InGaAs/GaAs Quantum Dots by Piezoelectric-Induced Biaxial Stess.
[41] Spin and circulating currents in nanostructures - PhD thesis. J. van Bree, Technische Universiteit Eindhoven 2014