When doing a photoluminescence experiment, an electron from the valence band is excited (by a laser) to the conduction band, leaving a hole in the valence band. There will be a Coulomb interaction between this electron and hole, creating an electron-hole pair called an exciton. When the electron and hole recombine, and thus the exciton annihilates, a photon is created and emitted, which can then be detected to provide information about the quantum dots band structure. In the neutral exciton X0, consisting of one electron-hole pair, the electron has spin sz = ±12, while the hole has pseudospin Jz = ±32. This results in four possible spin configurations, denoted as |X, Xz , where X = |sz+ Jz| and Xz its projection along the z-axis, which are degenerate when the spin-Hamiltonian is neglected.
In a quantum dot strain causes a splitting between the light hole and heavy hole states of several tens of meV. As the fine-structure interaction energies are in the order of several meV, only the heavy hole has to be taken into account and the light hole-heavy hole mixing can be neglected. When we choose | ↑ and | ↓ to represent the electron spin and | ⇑ and | ⇓ to represent the heavy hole pseudospin, these four spin configurations are
| ↓ | ⇑ = |1, 1
| ↑ | ⇓ = |1, −1
| ↑ | ⇑ = |2, 2
| ↓ | ⇓ = |2, −2
(2.17)
Within the dipole approximation only the X = 1 are optically active. As photons are bosons with spin ±1, the momentum of the X = 1 states can be transferred to a photon.
Therefore these states are referred to as bright states, while the X = 2 states are referred to as dark states. It can also be derived that the photons emitted by bright excitons are either right handed (σ+) or left handed (σ−) circularly polarized. Using this description, it is possible to demonstrate how the exchange interaction affects the exciton states. The exchange Hamiltonian can be written in terms of spin. When not assuming rotational invariance, which will be the case for most quantum dots, and neglecting heavy hole-light hole mixing the exchange Hamiltonian becomes [16]
Hexchange= − X
Here Jiis the hole spin, sithe electron spin and ai, bimaterial parameters. Representing this in a matrix with the possible spin configurations as basis we obtain [17]
Hexchange= 1 has a block diagonal form, the bright and dark excitons do not mix with each other. The two dark states however do mix with each other, while the two bright states will only mix in the absence of rotational symmetry (bx 6= by). The mixing between the excitons will cause an energy splitting, which is generally referred to as the finestructure splitting, but which we will neglect here as we generally do not observe it in our experiments. This is true when measuring in the Faraday geometry, where the growth direction of the quantum dots is the same as the direction of the magnetic field. In the Voigt geometry the magnetic field is perpendicular to the growth direction, which results in a mixing between the bright and dark states. This allows for detection of both the bright and dark states. Applying a magnetic field will result in a diamagnetic shift αex and a Zeeman splitting gX. When we assume the diamagnetic shift to be the same for dark and bright states, the excitonic energies are related to them by
E(H) = EX=2±1
2gX=2µBH + αexH2 (2.20) E(H) = EX=1±1
2gX=1µBH + αexH2 (2.21) The Coulomb and exchange interactions will result in small corrections on 2.20 and 2.21, but these will be neglected here. Simple expressions for the diamagnetic shift and g-factors can be then obtained;
In our experiment these quantities will be determined by measuring the exciton energy as a function of magnetic field. Using the expressions in 2.24, the electron and hole g-factors can be obtained from measuring the exciton g-g-factors. The experiments discussed in this thesis are performed in the Faraday geometry. Because of this only the exciton g-factor can be determined. To determine the hole and electron g-g-factors separately, the Voigt geometry would have to be used. The main results of the theory discussed in this section are summarized in figure 2.2.
Figure 2.2: Excitons at zero magnetic field (top) as opposed to non-zero magnetic field (bottom) for the single particle picture (left) and the many particle picture (right). In the single particle picture both the hole and electron are confined to the quantum dot, where recombination of the two results in a photon with either left-handed or right-handed circular polarization. When applying a magnetic field, the electron and hole each experience a different diamagnetic shift and Zeeman splitting. In the many particle picture, the electron and hole form an exciton through their Coulomb interaction. At zero magnetic field the fourfold degenerate ground state already experiences mixing. As a function of magnetic field, these mixed states shift and split further. Only the excitons with X = 1 (the bright excitons) are luminescent and are able to annihilate upon emission of a photon.
Chapter 3
The effects of strain on quantum dot properties
In the previous chapter we introduced a simple model based on the harmonic oscillator potential, which was already capable of describing some fundamental properties of quantum dots. However, investigating the effects of strain on these properties requires a more sophis-ticated model. A very popular approach to do so is the so-called k · p-theory, which will be introduced in this chapter. To get a better understanding of the physics behind strained quantum dots, first the concepts of strain and band structures have to be introduced.
3.1 Strain
This section will discuss the origin of strain and how it can be defined. To explain what strain is, first consider a one dimensional rod with length L. When elongating this rod to length l, the strain is given by
=l − L
L . (3.1)
When considering a three dimensional body, and dividing it into small elements, each element may experience both a normal strain and a shear strain. Normal strain acts per-pendicular to the face of the element, while shear strain acts along the face. This builds up a three-dimensional strain tensor, where the diagonal components describe the normal strain and the off-diagonal components the shear strain. Each element can be described by a displacement vector u = ux+ uy+ uz[18]. When choosing infinitesimally small elements, the components of the strain tensor consist of derivatives of this displacement vector via
ij =1
2(ui,j+ uj,i) (3.2)
where ui,j represents the derivative of displacement ui with respect to j. The complete strain tensor is then given by
=
For symmetry reasons xy = yx will be true. In a semiconductor crystal, the elements are generally chosen as Bravais cells containing one atom each. The strain tensor is then discretized by using the next-neighbours for each atom. In this thesis only the normal strain is treated and a few terms regarding strain are used. First there is hydrostatic strain, where the normal strain components all equal to each other (xx = yy = zz ≡ ). The strain component is then equal to the relative change in volume of the system, ∆VV . In uniaxial strain, one of the diagonal components will be non-zero, while in biaxial strain there will be two non-zero components on the diagonal. As we choose z to be the growth direction of our quantum dots, xxand yy are the in-plane strain components k, while zz represents the out-of-plane strain component ⊥. In our experiments, an in-plane biaxial strain will be added to the quantum dots. This strain will be assumed to be isotropic, meaning xx= yy. Let us now consider a thin epilayer of one semiconductor material grown on a substrate of another. If the layer is sufficiently thin, all of the strain will be incorporated in the layer, and not in the substrate. The difference in lattice constants of the epilayer and substrate, denoted by respectively ae and as, will result in an in-plane strain in the epilayer given by [19]
k= xx= yy= as− ae
ae
. (3.4)
The in-plane strain leads to a strain component in the growth direction, which is of opposite sign and given by [20] [21]
⊥ = − 2σ
1 − σk. (3.5)
Here σ represents the Poisson ratio, which for the type of semiconductor studied in this report is roughly 13. Using this value yields ⊥ ≈ −k. In general the radius of a quantum dot is much larger that its height. Because of this, the analysis for the epilayer is also to some extent applicable to the inner region of a quantum dot. The edges of the dot will show some transitional behaviour.