Strain

The first step in our calculation is to determine the strain field in the quantum dot. This is done by creating the previously discussed grid and minimizing its elastic energy. The results of this calculation are shown in figure 5.1. The top row shows intersections of the grid with the InGaAs (red) quantum dot, the GaAs (green) host material and the GaAs (blue) substrate. In this case the strain was unchanged, meaning the GaAs substrate had the same lattice constant as the GaAs host material. In the left panel the intersection goes through the base of the quantum dot and is perpendicular to the z-axis. The right plane is perpendicular to the x-axis and crosses the center of the quantum dot. The strain field is then computed by the methods described in the previous section. From this the different components of the strain tensor can be separately extracted. The second row shows the in plane strain component _xx (due to the dots radial symmetry _yy is the same), while the bottom row shows the z-component of the strain _zzin the same planes. The in-plane strain in the center of the quantum dot is around −2.5%. This arises from the lattice mismatch between the quantum dot and host material. The difference in lattice constant between InAs and GaAs is about 7%. Assuming this decreases linearly with In-concentration in the quantum dot, one would expect for 40% In-concentration a lattice mismatch of 2.8%. This is in rough agreement with the calculations. At the radial edge of the quantum dot, the host material will experience a small tensile strain to compensate for this lattice mismatch.

This strain relaxes rather quickly. A compression in one direction causes an expansion in the other directions through the Poisson ratio. As the size of the quantum dot in the x- and y-directions is much larger than in the z-direction, the lattice in the z-direction will experience more expansion than compression, resulting in a positive (tensile) strain as can be seen in the third row. In chapter 3 we predicted this to be equal to the in-plane compressive strain. As the tensile strain is found to be 1.7%, this is apparently not the case. Our predictions were made using an infinite slab of semiconductor material. The deviations from our numerical calculations can easily be understood by considering the effect of the GaAs-host that surrounds the quantum dot also in the z-direction. This will slightly compress the dot in the z-direction and therefore also expand it in the xy-direction.

As the lateral size of the quantum dot is much larger than its height, this effect will be small compared to the in-plane compression. This leads to a both a smaller in-plane compressive strain and growth-direction tensile strain than expected. The trace of the strain tensor is

−3.3%.

To illustrate how strain affects the bandedges of the quantum dot, these are shown in figure 5.2. The intersection is in the xy-plane, through the center of the base of the quantum dot. The figure shows the bandedges on this intersection after computing the

Figure 5.1: a) and b): intersections of the grid on which the quantum dots are calculated.

Red represents the InGaAs quantum dot, while green and blue represent respectively the GaAs host and substrate. c) and d): in-plane strain. e) and f): out-of-plane strain.

Figure 5.2: Band structure of the system on an intersection through the base and center of the quantum dot and a comparison with the unstrained In₄₀Ga₆₀As and GaAs bandedges (inset). The strain induces a splitting of the light and heavy hole levels and increases the band gap.

strain. In the inset, the bandedges in the center of this intersection are compared to the unstrained bandedges of bulk In₄₀Ga₆₀As and GaAs. Without strain, the heavy hole and light hole bands are degenerate. This changes after the strain computation; the degeneracy between heavy hole and light hole is lifted.

At the radial edges of the quantum dot the bandedges of the host material are also influenced by the relaxing strain. This vanishes when going further away from the dot and it can be verified that the bandedges eventually return to the unstrained GaAs values.

Additional strain increases the bandgap of the quantum dot material. This is logical, as the strain is compressive and therefore decreases the lattice constant and increases the confinement energy. It is therefore also expected that if we include additional compressive strain later in our calculations, this will lead to a further increase of the bandgap and with it also the emission energy. We can compare the results from our numerical calculations to the values predicted in bulk in section 3.4.2. Table 5.1 compares this to our numerical quantum dot calculations. It is important to note that there we assumed the strain to be biaxial, with a z-component of equal size and opposite sign with respect to the in-plane component. Our calculations have shown that this is not entirely true. Therefore we also include the corrected bulk values for the strains that have been determined by our numerical strain calculation.

In our numerical quantum dot case the heavy hole and light hole are split by 120meV . This is significantly different from the bulk value which shows a splitting of roughly 200meV , but already agrees much better to our corrected bulk value (160meV ). The conduction band

∆ECB(meV ) ∆EHH(meV ) ∆ELH(meV ) ∆ELH(meV )

Bulk 160 134 −77 −28

Bulk (corrected) 188 46 −112 −33

QD 227 46 −74 −82

Table 5.1: Shifts of the conduction and valence band edges when including strain. A compar-ison is made between the bulk values calculated in the previous chapter and the numerical quantum dot calculations presented in this one.

Figure 5.3: Bandedges of the conduction band (left) and valence band (right) on an inter-section through the base and center of the quantum dot. Indicated are the electron and hole ground states in absence of magnetic field. The electron ground state shows to be very delocalized.

and band gap changes also agree quite well. Deviations can be explained by the influence of confinement, and the inhomogeneity of the strain inside the dot, especially near the edges.

5.2.2 Magnetic field

With the strain field computed, the electron and hole state energies at zero magnetic field can be determined by numerically diagonalizing the Hamiltonian. This will yield as many eigenvalues as requested, which in our case will be two; one for the electron ground state and one for the hole ground state. Figure 5.3 shows where the electron and hole ground states are located in there respective bands.

The electron ground state is only separated from the GaAs bandedge by slightly more than 20meV . This means that the electron wavefunction is very delocalized. One of the consequences of this is that this is most likely the only possible electron level in this system.

It also affects the calculations itself, as the computational box has to be chosen large enough for the wavefunction to have reached zero at the edge. This significantly increases the computation time. The hole state is much better confined, with ∼ 100meV between the ground state and the valence band edge. The next step is then to include a magnetic field and investigate the behaviour of the electron and hole ground states. This is done by redoing the electronic calculation with magnetic fields ranging from 0T to 10T in steps of 2T . In order to reduce computation time, this is done only once for the case of no additional strain.

In future calculations, only one magnetic field will be used to determine the g-factor and diamagnetic shifts. At non-zero magnetic field this will yield two different states for each the electron and hole. The Zeeman energy and diamagnetic shift can be calculated from

Energy(eV ) Γ6 Γ8 Γ7

0.695886 0.962 0.000 0.002 0.000 0.007 0.019 0.006 0.003 0.695882 0.000 0.962 0.019 0.007 0.000 0.002 0.003 0.006

−0.671565 0.000 0.004 0.034 0.942 0.000 0.016 0.003 0.002

−0.671713 0.03 0.000 0.015 0.000 0.944 0.033 0.002 0.003

Table 5.2: Contributions of the eight Bloch basis functions in the wavefunctions of the four ground states. The electron ground states is dominated by the conduction band, while the hole ground states are dominated by the heavy hole.

these energy levels using

E_Zeeman^e,h = E_↑,⇑− E_↓,⇓ (5.2)

E_dia^e,h= 1

2(E_↑,⇑+ E_↓,⇓) (5.3)

where ↑, ↓ represent the electron spin and ⇑, ⇓ the heavy hole pseudo-spin. To determine the sign of the electron and hole g-factors it is necessary to identify which state corresponds to which (pseudo-)spin. As the wavefunction of each state is a linear combination of the eight base wave function of the k.p-theory, investigating the contribution of each base state to the final states will reveal the sign of the g-factors. For the four (two electron, two hole) states this is shown in table 5.2.

From this can be derived that the electron g-factor is negative while the hole g-factor is positive. Table 5.2 also confirms our previous remark that the hole state is dominated by the heavy hole. Using

ge,h= EZeeman

µBB (5.4)

αe,h= E_dia

B² (5.5)

the g-factors and diamagnetic coefficients can be extracted from the calculations.Figure 5.4 shows the evolution of the electron (right) and hole (left) levels with magnetic field. In the left panel the levels mainly split, while experiencing little shift. In the right panel the opposite case is present; there is only a very small splitting while the shift is significant. This indicates that the g-factor for the electron (hole) will be small (large) and the diamagnetic shift will be large (small).

To emphasize this figure 5.5 shows E_Zeeman (E_dia) as a function of magnetic field (squared). For the Zeeman energy of both the electron and hole a clear linear behaviour is found. The hole g-factor in this case is indeed found to be quite large (2.57), while the electron g-factor is very small (−0.06). Therefore the exciton g-factor is mainly dominated by the hole contribution. For different sizes of quantum dots the absolute values may vary, however generally the hole factor is an order of magnitude larger than the electron g-factor. This is quite intuitive, as the hole has a larger spin than the electron, which will therefore experience a larger coupling to the magnetic field.

For the diamagnetic shift a linear dependence is expected and found, aside from some small discrepancies for especially the electron diamagnetic shift. These are most likely caused by higher order terms in the diamagnetic shift. The electron diamagnetic shift is in

Figure 5.4: Evolution of the hole (left) and electron (right) ground state with magnetic field.

The hole shows a large Zeeman splitting but small diamagnetic shift. For the electron the opposite is true.

Figure 5.5: Zeeman (left) and diamagnetic (right) energies for the electron, hole and com-bined exciton. The Zeeman energy shows a clear linear behaviour, while the diamagnetic shift shows some deviations from the anticipated quadratic dependency. By fitting the calculated data, the exciton g-factor can be determined to be 2.51, while the diamagnetic coefficient is 7.38.

absolute value larger than the hole diamagnetic shift. This is as expected as according to equation 2.13 the diamagnetic shift depends linearly on the square of the lateral extension length. We have already shown that the electron ground state is less strongly confined than the hole ground state and therefore will also have a larger lateral extension length, thus diamagnetic coefficient.

It is to be noted that Coulomb interactions have not been taken into account in these calculations. This will change the absolute values of the diamagnetic shifts as including Coulomb interactions will change the confinement, however the behaviour will be the same.