### 5 Experiment and Results

As the starting point of the project I had planned to recreate a graph that clearly shows quantum cutting. This graph I didn’t make. Early on I detected an anomaly which I pursued further, after consulting with my supervisors. As described earlier, a sample used in these experiments is an ensemble of NCs. Each size has a speciﬁc bandgap energy, and so a Gaussian distribution of crystals will show a Gaussian luminescence curve (a gaussian) in the frequency domain. The measurements showed a blue shift in the luminescence curve as the excitation wavelength was lowered, which is a new ﬁnd. Also, because of time and problems with the sample mounting piece of the integrating sphere, I did most of my measurements without the sphere. I selected three samples to measure, each having a diﬀerent average NC size.

#### 5.1 Calibration Curves and Software

First thing to do is to calibrate the setup. As we’ve seen in the previous section there are quite a few parts to consider. All of these introduce their own imperfections and errors to the measurement, and they have to be dealt with. The calibration measurement in this case is done with a dedicated calibration lamp, which has a known spectrum. This lamp is then put in place of the sample, and this is measured. The output of the CCD or PMT is then compared with the theoretical curve, from which a correction curve is obtained. This should correct for all the imperfections introduced by material and devices from the sphere or ﬁber to the CCD or PMT. This curve can be used to correct any other measurements.

The CCD and PMT are connected to the computer, with proprietary software that outputs data as a simple ascii text ﬁle, so that data can be easily accessed. The measurement window is from around 500nm to around 1100nm. The region of interest is for these samples is well within that windows, typically the gaussian distribution peak between 700 and 900nm. I’ve used the grating (from the monochromator) with the widest window, but that window is still only 150nm wide. This means the window must be changed a few times to cover the entire spectrum we’re interested in. Each window also has its own ﬁlter, to suppress any photons scattering inside the monochromator8. Each measurement therefore consists of a number of ﬁles, which require ample post-processing to produce a set of data points that can be easily plotted. I decided early on that post-processing manually was too much ’dumb’ work and that I should write software that would do it for me. I did, successfully. The software ﬁrst reads all the relevant data into memory, and organizes this data based on parameters provided by the user. Then the data is converted into a single table, on which certain operations can be performed. Basic operations like calibrating based on correction data, but also more advanced operations such a noise removal. At the end the table is output as a .csv ﬁle with which many applications can import. Plotting curves and ﬁtting curves is instantly possible, no further preprocessing is required. I’m not sure whether I’ve saved much time considering I’m a slow writer, but in the later stages it was satisfying to be able to ’develop’ the raw data in a matter of seconds into a graph to see immediately if I should do another measurement or to establish that I’d found what I was looking for. Manual development is not quite that fast. Plotting and ﬁtting was done with Origin and QtiPlot.

#### 5.2 Photoluminescence

Photoluminescence (PL) curves are measurements of the spectrum that the nanocrystals emit after optical excitation, done with the CCD and very roughly gaussian in shape. I started with the sample ’Si6t14T1150’, the sample with which I was to built a graph that shows quantum cutting. I excited this sample at diﬀerent wavelengths, as can be seen in ﬁgure 8.

Fitting the data turned out to succeed best by adding up two gaussians, and a third one for the small bump around 635nm which becomes more pronounced as the excitation wavelength is lowered. Figure 9 shows the curve of excitation at 211nm, ﬁtted with three gaussians. It ﬁts perfectly. These results correspond with Sa’ar 16.

Naturally, the question is where these diﬀerent components come from and why the PL spectrum blueshifts as excitation wavelength is decreased. My supervisor mentioned that the bump around 635nm could be due to impurities in the quartz, because he’s seen it in blank samples, consisting of only the substrate.

The two main peaks could have a diﬀerent explanation. This sample has a NC size of 4.1nm. The next step was to see whether diﬀerent NC sizes would exhibit the same behavior. Diﬀerent sizes will have a gaussian centered at diﬀerent wavelengths, so looking at other samples could give some more information. Wolkin et al. 22 discusses a model in which the oxygen of the matrix introduces oxygen bonds (Si=O) at the edge of the nanocrystals, as displayed in ﬁgure 10.

This bond localizes a hole and an electron. He argues that this gives rise to additional energy levels, inside the bandgap. These levels become apparent in NCs with diameters from about 2nm and down, according to his model. The bad news is that the smallest sample we have is 2.6nm, which is not small enough to exhibit this splitting. Now, Wolkin’s paper is a theoretical treatise, so it’s not unreasonable to suspect that his model may be oﬀ a bit. Also, the samples have a Gaussian distribution of NC sizes, so there will be a certain fraction of NCs small enough to exhibit the eﬀect anyway. Figure 9 clearly shows something shifting, so maybe Wolkin’s model is oﬀ by a certain margin. I decided to measure two other samples as well, one with smaller NCs (Si4t7T1150, 2.6nm) and one with larger NCs (Si10t14T1150, 4.6nm).

##### 5.2.1 Fitting

As can be seen in ﬁgure 11, the sample with the larger NC size doesn’t show a clear blue shift. The 2.6nm sample does show something however, but very much less pronounced than the ﬁrst sample (Si6t14T1150). I’ve tried to ﬁt these two samples in the same manner as the ﬁrst sample. For the 2.6nm sample this could be done, but the 4.6 sample was so close to a single gaussian already; adding more peaks didn’t make any sense. What is nonetheless interesting is that the spacing between the two large peaks (for any ﬁt that would ﬁt well) are around 132meV apart, which is close to the energy associated with the Si=O bond, as discussed by Wolkin. Table 1 shows all the ﬁts and their peak separation.

 Delta Peaks (meV) at Sample NC-size 322nm 251nm 211nm 405nm Si4t7T1150 2.6nm 141.9 76.0 138.0 – Si6t14T1150 4.1nm 152.9 222.1 207.9 168.0 Si10t14T1150 4.6nm – – – –

Table 1: Table of the diﬀerence in meV of the peaks for each sample and excitation wavelength. Missing values indicate no accurate ﬁts could be produced.

My supervisor has determined the NC sizes with a formula that relates the center of a gauss ﬁt, the peak energy, to a certain NC size. It’s an experimental formula, but it can be motivated by the idea discussed earlier: decreasing the nanocrystal size decreases the width of the potential well in which the free electrons move about. This causes the energy levels of those electrons to separate further apart. Mistrustful as I am, I decided to verify this formula. I compared it to a collection of diﬀerent measurements by diﬀerent groups19. Figure 12 shows that the formula ﬁts pretty well to multiple datasets. Any discrepancy between the actual and calculated NC sizes cannot be determined with the data at hand. The slightly strange behavior in above measurements may be explained by this possible discrepancy.

Concluding, it seems that frequency response is, for certain samples, dependent on excitation wavelength. Lowering excitation wavelengths caused a slight blue shift of the light emitted by the crystals. Also, it introduces a third gaussian (centered around 635nm) at lower excitation wavelengths. Regretfully I wasn’t able to suﬃciently determine whether the model of Wolkin holds any merit. Another obvious explanation is that there are multiple distinct distributions on the sample, considering the gaussian distribution of sizes a certain set of crystals tend to have. The annealing process is yet not very well understood. It is assumed that the NC concentration is constant throughout the sample, resulting in the laser beam ﬂux to drop oﬀ exponentially.

A possible explanation is that the NCs get saturated with excitons. Timmerman et al. 21 shows saturation for excitation diﬀerent wavelengths. As the photon energy is increased, the absorption of the Si NCs is increased as well (ﬁgure 17). Although the power was kept constant through all the excitation wavelengths, the increased number of absorbed photons at shorter wavelengths could compensate. The number of absorbed photons through the crystal falls oﬀ exponentially with penetration depth: the closer the NC’s are to the top edge, the more intense the laser beam is and the more likely they are to absorb a photon. The diﬀerence in absorption between 400nm and 200nm is about a factor 100, much larger than the decrease in photon count (halving the wavelength at constant power means halving the photon count). The NCs in the top layer could still be saturated. The shift would then be explained by the Si=O bonds not saturating and thus increasing relatively more.

#### 5.3 Time Resolved Measurements

Another way to ﬁnd out where the multiple gaussians come from is to have a look at decay times, or kinetics. The CCD measures once with a long time window, in my case typically ﬁve seconds. With the PMT, with its resolution of 250ps, it is possible to create a graph that shows the decay of the signal. By taking PMT measurements of a wide range of wavelengths, speciﬁcally wavelengths that have a PL strongly associated with one of two gaussians, we can put a number on the decays of each of two gaussians and perhaps ﬁnd something in literature.

Unfortunately the PMT only provided accurate measurements up to 800nm, precisely in the middle of our region of interest and therefore precisely useless to distinguish the two major peaks seen in the previous section. On the short wavelength end of the spectrum the PMT was good down to 400nm. I chose to measure on two diﬀerent timescales, 1$\mu$s (ﬁgure 14) and 500$\mu$s (ﬁgure 13), because these are the typical timescales on which the Si-O and Si=O decay are visible on, respectively. From Wolkin et al. 22 we know that the main (750nm) component here is the band-to-band NC PL (ﬁgure 13) and the Si=O eﬀect (132meV shift). The 425nm peak is Si-O4,11,14. Both are eﬀects at the edge of the NC. Here the NCs are bound in some way to the matrix, which means that some kind of oxygen bonds can form (ﬁgure 10).

##### 5.3.1 Fitting

Fitting the data would hopefully show that there are indeed multiple eﬀects at work. If a ﬁt built up from a number of components, it stands to reason that each of these components is associated with a separate physical process. Fitting the data with multiple exponentials resulted in nothing however. Some argue13 that ﬁtting the kinetics of NCs should be done with a stretched exponential (3). This diﬀers from the regular exponential (2) by only a an extra power in the exponent. This is motivated by the idea that at a single wavelength there’s a collection of contributions from diﬀerently sized NCs shining, not only the ones with the speciﬁc corresponding bandgap. This will happen according to a Gaussian distribution (which the ensemble of NCs is assumed to be), which results in the stretched exponential. After trying many combinations of (stretched) exponentials, ﬁtting the long timescale with a single stretched exponential, and the short timescale with a single exponential, made the most physical sense. For the some wavelengths no ﬁts made sense, because there was only noise.

 $\phi \left(t\right)=\alpha {e}^{-\left(t∕\tau \right)}$ (2)
 $\phi \left(t\right)=\alpha {e}^{-{\left(t∕\tau \right)}^{\beta }}$ (3)

Figure 15 shows the two ﬁtting parameters $\alpha$ and $\tau$ for the exponential curve (2) on the 1$\mu$s measurement. Figure 16 shows the three ﬁtting parameters $\alpha$, $\beta$ and $\tau$ for the exponential curve (3) on the 1$\mu$s measurement. From the plots of these parameters it’s at least clear that on the short timescale there are two diﬀerent processes visible, each with a diﬀerent timescale. On the long timescale only the longer process is still visible. This agrees nicely with the assumption that Si-O decay is fast luminescent compared to the normal NC decay. The second interesting ﬁnd is that the $\beta$ is fairly constant over the whole range of wavelengths, which is to be expected since it’s a single sample with a single normal distribution. Linnros et al. 13 shows a similar $\beta$.

#### 5.4 Higher Energies or More Power?

I’ve ﬁnalized my project by investigating whether increasing the power showed the same eﬀect as lowering the excitation wavelength. Because silicon has diﬀerent absorption coeﬃcient at diﬀerent photon energies, I’ve used the graph in ﬁgure 17 to determine with what range of laser power I’d be dumping enough energy in the sample to match the energy deposited by the high energy laser. It can be argued that NCs in quartz could have a diﬀerent absorption curve from bulk silicon times the volume percentage of Si NCs. I’ve compared an experimental curve made by the group with the one in ﬁgure 17, and they agree pretty well up to the high energies, exactly the region we’re interested in. Because of the way it is measured, the bulk curve is probably more correct in, which is why I’ll be using it.

Exciting the sample with wavelengths of 400nm and longer showed no shifting. That’s why I chose to do the variable power measurement at 416nm. From ﬁgure 17 I determined that I should increase laser power with a factor 1000 to match the energy absorbed from the laser at 211nm.

To make it possible to distinguish a trend, if there should be one, I measured the spectrum at a few diﬀerent laser powers. Figure 18 shows a blueshifting similar to what we saw in the wavelength dependent measurement. Also, at ﬁrst glance the primary peak seems to be at around 900nm, with the center of the power dependent region at about 800nm. This diﬀerence is about 170meV, perhaps corresponding to the 132meV of the Si=O bond. Table 2 shows the ﬁtting parameters on the data in ﬁgure 18. As is also slightly visible from the ﬁgure, the NC PL peak shifts towards the red, albeit very slowly. The supposed Si=O peak grows, but the energy diﬀerence seems too large (132meV versus 220+meV). Considering Wolkin et al. 22, it might be possible the Si=O band is still visible when it is inside the conduction band. A second explanation is that the defect is introduced by the amorphous SiO${}_{2}$. One thing is certain, the schematic in ﬁgure 10 doesn’t correspond to the observations. Figure 20 introduces a new proposal for the location of the energy level.

 Laser Power Parameter 15$\mu$J 45$\mu$J 110$\mu$J 460$\mu$J 1000$\mu$J 1550$\mu$J Right center (nm) 918.7 913.5 909.4 901.8 898.2 893.5 Right amplitude (a.u.) 164.6 161.6 165.6 171.2 167.8 168.6 Left center (nm) 770.9 769.8 770.7 763.7 770.6 768.8 Left amplitude (a.u.) 9.9 11.3 14.7 16.7 24.1 25.7 Energy delta (meV) 258.9 253.4 245.4 248.7 228.7 225.0

Table 2: Table containing the centers of the peaks and the diﬀerence in meV. Fits done on the data in ﬁgure 18, sample Si4t7T1150 excited on 416nm with diﬀerent laser powers (NC size: 2.6nm).

Inside the monochromator light is reﬂected multiple times between gratings and mirrors, through ﬁlters and slits. To rule out the possibility that any eﬀect observed could be due to misaligned components inside the monochromator, I’ve done some measurements to double check that no weird reﬂections were present. Using ﬁlters right after the grating you can block certain frequency ranges. By blocking the frequency range you’re actually trying to measure, nothing should be visible. Any pattern still visible will be due to scattered light. Luckily, there was only noise, indicating the monochromator is ﬁne and our measurements correct as far as the monochromator goes.