Separating electron and hole g-factor

To separately determine the electron and hole g-factor, it is necessary to access the dark states discussed in section 2.3. In general this is achieved by measuring in the Voigt geometry, meaning the magnetic field is perpendicular to the detection direction. This will split the exciton line into four lines; two bright states and two dark states. As in this research we have only measured in the Faraday geometry, with the magnetic field parallel to the detection direction, this splitting is not usually observed. However in one of the examined dots this splitting into four peaks is observed, however it cannot conclusively be assigned to the dark state splitting. An analysis of the results is given in this section.

Figure 6.12 shows the spectra as a function of magnetic field for this dot. The right panel is zoomed in on the concerning lines, where the splitting of the left line can clearly be seen. The splitting of the right line is a lot less visible. To emphasize this, figure 6.13 shows the spectrum of this dot at 10T .

The fourth peak in the spectrum is very small and therefore leads to large uncertainties.

Figure 6.13: Quantum dot spectrum showing the four split peaks. These peaks can be assigned to the bright states (inner two) and the dark states (outer two). One of the dark states is only a very weak peak.

The separation into four peaks is visible from 3T , but can only be accurately distinguished at magnetic fields of 6T and higher. The observed splitting which is unusual for the Faraday geometry can have several explanations. First there is the possibility of the magnetic field not being perfectly parallel to the detection direction. This is unlikely however, as the splitting would then also be observable in other dots. Another possibility is that the sample is locally not exactly horizontal due to a small defect in the membrane, or the quantum dot not being horizontal in the sample due to a growth defect. The inhomogeneity of the wetting layer that was observed may make the latter a plausible explanation; an inhomogeneous wetting layer signal indicates that the surface of this wetting layer might be very rough on an atomic level. This roughness could cause some quantum dots to grow under an angle, resulting in a configuration where the growth direction of the dot is no longer parallel to our applied magnetic field. As the angle between the base of the quantum dot growth direction and the direction of the magnetic field must already be significant we can not prove that this is indeed the cause. Therefore we cannot give a satisfying explanation for observing this effect. What is another source of concern, is the difference in intensity between the bright and dark state peaks. As the dark-bright state mixing should be small in this case, it is expected to observe bright state peaks of similar intensity as before and dark state peaks of low intensity. This contradicts the spectrum in figure 6.13, where there are two possible configurations. Either the two outer lines are the bright states and the two inner lines the dark states, or vice versa. Our analysis further in this report will prove the latter to be the case. One problematic thing in this configuration is that one of the dark state peaks is surprisingly intense, even more intense than the bright states. This may again be caused by the polarization-dependence of the collection fiber. Additionally, similar splitting behaviour has been observed before, where at a critical magnetic field an additional line splits off from each Zeeman-split line. This behaviour has never been explained and its origin remains unknown. As in our experiment the splitting of the peaks is observable already at 3T , one tends to think the observed splitting into four lines is in this case indeed the bright-dark state mixing, but conclusive evidence cannot be given. For future experiments on this it would help to use smaller intervals of magnetic field to perform the measurements. One

Figure 6.14: Diamagnetic shift of the outer peaks (blue) and the inner peaks (red) as a function of applied voltage. As the coefficients are very similar, this indicates that the observed splitting is indeed caused by the bright-dark state mixing.

Figure 6.15: G-factor for the bright exciton (left) and the dark exciton (right) as a function of applied voltage. The dark exciton g-factor barely increases with applied voltage, while the bright exciton g-factor decreases strongly. This means the change in electron g-factor must approximately cancel out the change in hole g-factor.

additional aspect that indicates this observation is indeed the bright-dark mixing can be found in the diamagnetic shift, which should be very similar for the bright and dark states.

To investigate this, figure 6.14 shows the diamagnetic coefficient of both bright and dark states as a function of applied voltage. For visibility the errorbars are omitted.

As expected the diamagnetic shifts turn out to be approximately the same for both bright and dark states. When assuming this splitting to be caused by the mixing of bright and dark states, this can be used to separately determine the hole and electron g-factor. We have previously shown that the dark exciton g-factor is equal to the sum of both separate g-factors, while the bright exciton g-factor is given by the difference between the two. From calculations it is known that the electron g-factor is relatively small. Therefore the bright and dark exciton g-factors must be equal in sign. The results for the bright and dark exciton g-factors are shown in figure 6.15.

As our previous results showed that the bright exciton g-factor decrease with applied

Figure 6.16: Hole (left) and electron (right) g-factor as a function of applied voltage. A linear dependence is clearly visible. The change in both g-factors is roughly the same, but of opposite signs.

voltage for all investigated dots, we conclude that the left panel, which shows the g-factor of the inner lines of the spectrum, must correspond to the bright exciton. Knowing that

g_bright= g_h− ge

gdark= gh+ ge

(6.6)

we derive that the electron and hole g-factors must be carry the same sign. As the dark exciton g-factor barely changes with applied voltage, the change in electron g-factor must cancel out the change in hole g-factor. Using

|g_h| =gbright+ gdark

2

|g_e| = gbright− gdark

2

(6.7)

the electron and hole g-factors can be determined separately. Results are shown in figure 6.16.

The hole g-factor shows a decent agreement with our calculations. The electron g-factor is a different matter; the effect on this is much greater than predicted by our calculations.

However, we have stated there are still some unresolved issues with the electronic calcula-tion of the conduccalcula-tion band which may be the cause of these differences. Moreover, we only performed calculations on one particular size and composition of quantum dot, which coinci-dentally showed an unusually small electron g-factor. Dots of other sizes will probably have larger electron g-factors and may therefore also show a larger response to external strain.

In the previous chapter we have already discussed the origin of the change in electron g-factor; it is caused by a change in orbital momentum. The degeneracy and non-zero angular momentum of the valence band makes the behaviour of the hole g-factor more complicated.

However, a hole g-factor decreasing with increasing confinement energy has previously been shown [41] for similar nanostructures, which agrees with our observations presented here.

Moreover, the behaviour agrees with our numerical calculations performed in the previous chapter.