In the previous chapter the origin of the g-factor was explained; an external magnetic field couples to the spin of a particle (hole or electron) which results in a splitting of the energy levels called the Zeeman splitting. This splitting is proportional to the g-factor of the particle. For a free electron this value is 2. For electrons in a semiconductor crystal the effective g-factor will deviate from the free electron value. Using k · p-theory an analytical expression for electron g-factors in bulk can be derived. The deviations from the free electron
g-factor arise from the coupling of the conduction bands to other bands in the system. When taking into account only the influence of the valence band on the conduction band it can be shown that
ge= 2 − 2EP∆
3Eg(Eg+ ∆) (3.21)
where EP denotes a coupling between the conduction and valence band. The equation in 3.21 was derived by Roth [31] [32] and is therefore called the Roth formula. This simple formula shows the necessity of the spin-orbit coupling; without spin-orbit coupling, ∆ = 0, the effective electron g-factor is equal to the free electron g-factor. Also, without a coupling between the conduction and valence band, the deviations from the free electron g-factor vanish. This is due to the absence of an orbital momentum (L = 0) in the conduction band;
only a coupling to the valence band will induce an effective orbital momentum. The Roth formula is capable of reproducing bulk electron g-factors quite well. For example, for InAs the electron g-factor can be calculated to be −14.4, while experiments show −14.9 [33]. For the hole g-factor it is not possible to derive such an analytical formula. This is due to its degeneracy and the fact that the band itself carries an orbital momentum. For free holes the g-factor is given by the Land´e-factor [11]
gJ,L,S= 1 +J (J + 1) − L(L + 1) + S(S + 1)
2J (J + 1) (3.22)
Applying this formula to the J = 32 and J +12 valence bands yields respectively gh=43 and gh = 23. Again, in a semiconductor crystal coupling to the other bands will cause deviations from these values.
It was shown by Pryor and Flatt´e that for confined electrons the Roth formula no longer holds [34] [35]. To do so, they performed numerical calculations on spherical InAs quantum dots in vacuum. They used high energy barriers to prevent the wave function from exceeding the quantum dot radius. Due to their approach, effects like strain were not present in their calculations. They found for very large quantum dots values of the effective electron g-factor as predicted by the Roth formula. By decreasing the quantum dot size, thus increasing the confinement and the band gap, both their calculations and the Roth formula start to approach the free electron g-factor value of 2. However, their calculations reached this value much quicker. As the only thing that is taken into account by the calculations and not by the Roth formula is the confinement, this must be the cause of this. They introduced the concept of orbital momentum quenching in order to explain their observation. We already mentioned that usually particles in the conduction band carry no angular momentum, but that the coupling to the valence band induces an effective orbital momentum. This effective orbital momentum causes the effective electron g-factor to deviate from the free electron value. In the framework of the eight-band model, due to coupling between the bands, the ground state is always a linear combination of the eight band states. The conduction band ground state will logically be dominated by the conduction band states. Increasing the confinement leads to an increased band gap and therefore a decreased coupling between the conduction and valence band. Therefore the conduction band ground state will be more heavily dominated by the conduction band states, leading to a decrease of the contribution of the valence band states. This will decrease the effective orbital momentum induced by the valence band; the orbital momentum is said to be quenched. This will prove to be an important concept for qualitatively explaining the results of both our numerical model and our experiments later in this report. For the valence band the concept of orbital momentum
quenching is not easily applicable. It is likely that the confinement does have an effect on the effective orbital momentum in the valence band, however to our best knowledge it has not yet been investigated how exactly this is happening. Analyzing this problem would be far from straightforward, as the valence band itself already carries a non-zero orbital momentum. We can therefore not intuitively predict what will happen to the effective hole g-factor under the application of strain. In other research it has been shown both numerically and experimentally that the hole and exciton g-factor decrease with increasing confinement energy [34] [36] [37]. As compressive strain leads to an increase of the confinement energy, we hypothesize that this will have the same effect on the g-factor. Whether this hypothesis is correct will be discussed in the following chapters.
Chapter 4
Photoluminescence and confocal microscopy
In this chapter the experimental techniques used in this research are discussed. First the working principle of our main investigation method, the confocal microscope, is discussed, after which the used setup will be described. To finish the chapter, a description of the sample structure is given.
4.1 Photoluminescence
When focussing a laser on a quantum dot, an electron can be excited from the valence band to the conduction band, creating an electron-hole pair (exciton). When the electron and hole recombine, a photon is emitted. The energy of this photon depends on the characteristics of the quantum dot. In a photoluminescence experiment, these photons are collected, providing valuable information about the properties of the quantum dot. One possibility is to do photoluminescence experiments on an ensemble of quantum dots (macro-PL). This will yield information of the basic characteristics of the sample, among which are the quantum dot density and the average size. Using sophisticated optical techniques, it is also possible to measure PL of single quantum dots (micro-PL). This way valuable information of each individual quantum dot can be determined, like the emission energy and the g-factor, which is of great importance in this report. Only micro-PL measurements are performed in the work discussed in this thesis.