The sample used in this research, which will be referred to as the O717 sample, consists of quantum dots mounted on top of piezo-electric material. This allows us to strain the quantum dots in order to control their electronic properties. Applying strain to the system will change the band structure in the quantum dots as explained in chapter 2. This will influence fundamental properties like the emission energy and effective masses. In this thesis especially the g-factor is of interest.
Significant strain is accomplished by a combination of two things. First there is the use of [P b(M g1
3N b2
3)O3]0.72[P bT iO3]0.28 (P M N − P T ) piezo-electric material. This material exhibits a very large piezo-electric constant ∼ 2000pN/C (as compared to the most com-monly used PZT, which typically has a piezo-electric constant in the order of ∼ 300pN/C), allowing for large in-plane strains. Secondly, the use of a thin nanomembrane containing the quantum dots allows for a very good transfer of strain from the P M N − P T to the quantum dot layer. Changes of strain in the quantum dot layer up to ∆k = 0.4% have been reported.
A GaAs nanomembrane containing self-assembled InGaAs quantum dots is constructed using molecular beam epitaxy (MBE). A 148nm-thick layer of intrinsic GaAs is grown on top of a 100nm-thick Al0.75Ga0.25As sacrificial layer. On top of this layer, InGaAs quantum dots are grown with an In-concentration of 50%. By flushing the system with Ga afterwards, the In-concentration is reduced to 20 − 40%. The quantum dots are then capped with a 149nm-thick layer of intrinsic GaAs, which results in a maximum height of the quantum dots of 2.5nm. The nanomembrane is then released from the substrate by complete etching of the sacrificial layer.
A 120nm-thick layer of gold is deposited onto a P M N − P T piezo-electric substrate.
The nanomembrane is then integrated onto this through thermo-compression bonding. In thermo-compression bonding, two surfaces are brought into atomic contact, while applying both force and heat. Due to this heat treatment, atoms from one crystal lattice migrate to the other and vice versa. This connects both interfaces. The final sample structure is shown in figure 4.3. Figure 4.4 shows an image of the actual sample. The sample is placed on a gold surface, which will be connected to the ground of our power supply. The positive voltage of the supply is connected to another gold island, which in turn is connected to the top side of the sample by three thin bonded wires. In the bottom corner of the sample, the quantum dot membranes are located. The absence of these membranes on the rest of the sample already indicates that the deposition of the membranes onto the P M N −P T was not very successful. This will later be verified by the fact that the possible shifts in quantum dot emission energy are much lower than those reported in literature. The membranes generally have a size of 50µm by 100µm. In this report, investigation was focussed on the four membranes in the corner. Figure 4.5 illustrates how the strain is transferred from the piezo-electric PMN-PT to the quantum dot layer. A positive voltage applied in the z-direction causes an elongation in that direction. Due to the Poisson ratio this induces a biaxial compression in the xy-plane. This strain is then transferred to the QD membranes, where this also leads to an elongation in the z-direction.
The sample is able to sustain voltages up to 1000V . Getting this voltage down through
Figure 4.3: Schematic representation of the final sample structure. The InGaAs quantum dot layer (orange) is located in between two layers of intrinsic GaAs. Through thermo-compression bonding the quantum dot structure is integrated onto the piezoelectric material with a layer of gold in between.
Figure 4.4: Images of the O717 sample, containing a small number of quantum dot mem-branes in the bottom corner.
Figure 4.5: Schematic representation of the way the strain in the PMN-PT is transferred to the QD layer. Solid lines show the unstrained system, while the dotted lines show the strained system. Voltage applied in the z-direction of the PMN-PT induce an elongation (green arrows) in that direction, and therefore a compression (red arrows) in the xy-plane.
This compression is translated to the QD layer, where it will also induce an elongation in the z-direction.
the microscope stick to the sample requires well isolated wiring, as the entire environment consists out of metal. This isolation needs to be able to withstand the high magnetic fields and low temperatures our setup is operated at. This was eventually solved by using regular copper wires and isolating them with a teflon sleeving, which remains somewhat flexible at these low temperatures as opposed to other isolating materials. However, it was still not flexible enough for the final connection to the sample, as the sample moves around. To solve this, the teflon sleeving was cut up into small segments. By doing this the wiring was ensured to be flexible enough, however, this leaves open some gaps in the isolation.
During testing, there was a breakdown at 800V . Therefore in all experiments the voltage was restricted to 500V to be safe.
Chapter 5
Tuning electronic properties using strain: a numerical
approach
The simple phenomenological model discussed in the previous chapter is capable of making qualitative predictions of the features that are observed in magnetoluminescence exper-iments. However, to also make more quantitative predictions of quantum dots, a more complicated, numerical model is required. This model has to take into account the real semiconductor bandstructure and determine how confinement influences this. We make use of a numerical model based on the eight band k · p-theory discussed in the previous chap-ter, after which the numerical model with which our calculations were performed will be described. We do this to test the predictions made in the previous chapter and to later compare this to our experiments.
5.1 Numerical model
A reliable model needs to take into account enough bands to obtain confined hole and electron states. The k · p-approximation can be used for this. Due to a lattice mismatch between the quantum dot material and the surrounding host material, a non-uniform strain will be present in the system and influence the electronic states of the quantum dot. This strain will therefore also have to be included in the model. The wavefunction of an exciton in a quantum dot generally extends into the surrounding host material. As the bulk electronic properties of both materials are different, it is not sufficient to only include the quantum dot material into the model. The host material will also have to be incorporated. This is done by creating a grid, where each element of the grid is allowed to have different parameters. In the end the g-factor will be calculated from the magnetic field-dependency of the quantum dot state energies. The magnetic field will couple to both the orbital and the spin part of the wave function. Using the gauge invariance of the Schr¨odinger equation, a vector potential can be introduced into the Hamiltonian which accounts for the orbital part. To include also the coupling to the spin part of the wave function, a Zeeman Hamiltonian HZeemanis added to the Hamiltonian.
These aspects have been integrated into a model that is able to calculate any arbitrary nanostructure. The steps performed in these calculations are as followed:
1. To start, a uniform cubic grid is created. A quantum dot is defined by its size and shape. To each point of the grid, the material parameters of the corresponding semi-conductor material (either host or dot) are assigned.
2. The strain of the system is calculated by minimizing the elastic energy of the grid.
This is done by varying the displacements at each grid point, calculate the elastic energy of the system and then use a conjugate gradient algorithm to minimize this energy.
3. Due to the wave functions decaying exponentially in the host material, only a small area around the quantum dot is required to do the electronic calculations. The grid is truncated to a more suitable size to speed up the electronic calculations.
4. Next for each grid point the material dependent Hamiltonian is determined. The total interaction matrix of the grid then consists out of all the Hamiltonians at each grid point. For the grid size used in our calculations, 100x100x100, this leads to a square matrix with a size of eight million, as each Hamiltonian is eight by eight.
5. The derivatives in the created matrix are discretized using only the nearest and next-nearest neighbouring grid points. Magnetic field is included by modifying these deriva-tives with a phase factor. All this leads to a sparse matrix, which can be diagonalized using the Lancsoz algorithm. This will yield the eigenvalues (energies) of the system, and if requested, also the eigenfunctions (wavefunctions).