3 Physics

First, let’s deﬁne more precisely the topic at hand: Si NCs. What is silicon, what are nanocrystals, and why would one study this at all in the context of solar power generation?

3.1 Electrons and Photons

When studying solar power generation, you’re studying electron-photon interactions. After all, the point is to convert the energy of photons emitted by the sun into electrons moving around an electrical circuit. The reason that semiconductors are studied in the context of solar power generation is that these materials are an excellent playground to ﬁne-tune electron-photon interactions to the whatever is required. Band Theory explains why semiconductors exhibit this behavior.

In short, a semiconductor has electrons below a certain energy level localized to atomic electron shells or covalent bonds. Above that energy level the electrons are free to move throughout the material, like in metals. In bulk materials it’s useful to describe the electrons with their energy and the collective electronic behavior as an energy spectrum. There can be gaps in the spectrum of allowed electron energies, which can be bridged by either absorbing or emitting a photon. When the material is in it’s ground state, the spectrum is ﬁlled up to a certain energy level. For semiconductors, this maximum energy level, the Fermi energy level, lies inside a gap. This means that when the material is in its ground state, all the electrons are in the valence band. If the material is perturbed with a high enough energy, for example by heat (phonons) or light (photons), an electron can bridge the gap to the conduction band, where it’s free to move through the material and can be captured to power a device. Conversely, an electron in the conduction band can transfer back to the valence band by emitting photons or phonons. Thus, a semiconductor is like a machine that can convert light into electricity and vice versa.

Silicon is a popular semiconductor material, because it’s abundant and relatively chemically inert, in other words: cheap and easy to work with. Silicon as a bulk material is well understood, and it’s used on huge industrial scales in the solar power sector and the computer chip industry. While in computer chips the most important aspects are its electrical properties, in solar panels the electron-photon interaction is of most importance. After all, the point is to catch photons and use their energy to pump electrons around.

3.2 The Indirect Bandgap and Solar Panels

We’ve seen that semiconductors are good materials for interactions between electrons and photons, and that’s why they are the primary material used for devices such as solar panels and photosensitive surfaces. The reason that silicon is a popular material, in industry and research, is almost a vicious circle: these devices need to be mass produced at a low cost, which means there’s great interest in characterizing silicon in all it forms and functions, which leads to further use in commercial applications. Not only the conversion of light to electrons is of interest, the other way around too. Commercial lasers are an important ingredient in modern day electronics, which is why there is great interest in a silicon laser9. The reason we don’t have a silicon laser yet is a result of the band structure of silicon (ﬁgure 2). Silicon has a large bandgap at the $\Gamma$ point, but the smallest possible energygap between the conduction band and the valence band is between the $\Gamma$ and close to the X point. This indirect bandgap requires a phonon to be part of the electron-photon interaction, but requires less energy, an advantage when the solar spectrum is considered. In other words, electrons have to change momentum for this transition to happen, as can be seen in ﬁgure 3.

Because the indirect bandgap excitation is less likely than a direct bandgap excitation, the indirect bandgap excited state is long lived: recombination is equally less likely. In solar panels this is an advantage, because electrons in the conduction band have to be harvested somehow. In solar panels these electrons cause a charge diﬀerence over the front and backside, which are harvested with electrodes. On the back, this means you can cover the entire surface with a thin metallic ﬁlm, but on the front this would mean you block all the incoming sunlight. The solution is to cover only a small portion with a metal, which are the thin metallic lines you can easily discern when looking a the photosensitive area of a traditional panel. As you can also see, these lines are fairly widely spaced and cover only a very small portion of the panel. The primary reason is that electrons in bulk silicon have a long mean free path, due to the purity of the silicon and the indirect bandgap causing the long lived state. The downside is that silicon panels have a typical thickness of 300-500$\mu$m to compensate for the reduced absorption coeﬃcient: an unlikely transition means a lower absorption coeﬃcient. A higher crystalline quality means a better panel, and a higher price. Materials with a direct bandgap can be only a few micrometers thick. Other tricks to increase panel eﬃciency for example are having some sort of transparent conductor, wavy bulk silicon surfaces to increase scattering of photons, or silicon nanowires instead of nanospheres12.

3.3 Silicon Nanocrystals

Nanoparticles, quantumdots, ﬁne particles, nanodots, ultraﬁne particles. These are all buzzwords sometimes used interchangeably to refer to particles on the nanometer scale. The kind that’s researched in the context of solar panels is the Si NC. Recall the indirect bandgap of bulk silicon and remember band theory. As the crystalline structure gets smaller, the band model reverts to the single atom model of many separate energy levels. Typical Si NCs are a few nanometers in diameter, and have a few thousand Si atoms. These NCs are therefore on the limits where band theory is applicable. They are nonetheless small enough for the Heisenberg uncertainty principle to come into play5. The smaller the nanocrystal is, the more conﬁned the free electrons in it are. As the position of the electrons are more precisely deﬁned, their momenta are less precisely deﬁned. Thus, the problem of the indirect bandgap, the wide gap in momentum between the $\Gamma$ and X points, is mitigated. Also, as the NC size decreases, the potential well in which the free electrons are gets smaller. As can be remembered from basic quantum mechanics, this causes the energy levels of these electrons to separate further apart. In summary, the likelihoods of excitation and decay are increased with nanocrystals. Figure 3 shows a schematic view of this discussion.

Solar panels have to convert solar photons into electricity, which means that these panels should use the energy in these solar photons as optimally as possible. The energy that’s usable per photon is the energy of the bandgap: excess energy is wasted as phonons, also known as heat. Nanocrystals may provide a way to reclaim this energy, increasing the eﬃciency of the conversion.

3.4 Decay and Quantum Cutting

When a photon is captured by an electron in a NC, it transfers all of its energy to the electron. This means for high energy photons, a photon is excited pretty high into the conduction band. UV photons for example have an energy of around 3-5eV, and the bandgap of nanocrystals of 4nm is in the order of 1.5eV19. In semiconductors it is well understood that this extra energy is usually quickly dissipated as phonons in the crystal, called cooling. Note that the electron cools down, but the crystal heats up, which in turn dissipates its energy to the surrounding environment, SiO${}_{2}$.

It is this excess energy, that’s usually wasted, that NCs can potentially put to good use. They can do this through the process of Quantum Cutting (QC), also known as Multiple Electron Generation (MEG) or Carrier Multiplication (CM). How does it work? Provided the electron absorbs a photon with at least twice the bandgap of the NC, the excess energy can be transferred to another electron of the valence band, thereby kicking it into the conduction band. Voila, with a single high energy photon, the nanocrystal has created two excited electrons, instead of just one5,7,20. You can see where the process names come from.

3.5 The Mechanisms for Quantum Cutting

First it’s important to note the electronic characteristics of nanocrystals. The Si NCs are often embedded in a SiO${}_{2}$ substrate. The substrate clearly is an isolator, so, from the point of view of electrons, the NCs are potential wells with steep edges. It is clear how electrons that enter the conduction band are conﬁned to the NC: they’d have to cross a high potential wall to move to another NC. Of course, it’s important to consider that for each excited electron, there’s also a hole created in the valence band. Because of the small dimensions of the NCs, it is easy to imagine that the electron-hole pair, also known as an exciton, can be seen as a dipole. The scientiﬁc community currently doesn’t agree on a mechanism which would explain quantum cutting. Generally, there are thought to be three serious contenders20:

3.5.1 Impact ionization by hot carrier.

The case of impact ionization, also known as Coulomb interaction or inverse Auger recombination, is the easiest to imagine. The electrons being charges will repel each other. Because of that it is possible that a hot carrier (an electron high in the conduction band) can transfer part of its kinetic energy to a second electron, simply by bumping into it. In this way the excitation of the second electron can happen in the same NC. But, because the electronic wave functions are large compared to the size of the NC (the Bohr exciton radius in bulk Si is 4.9nm3), this state isn’t energetically favorable. The charges will repel each other, but when conﬁned to the small space of the NC, Auger recombination will happen fast and generally results in only one exciton surviving in a NC. In some samples the average distance between NCs isn’t much larger than the average diameter of the NCs, which means the substrate is a high, but not so wide barrier. Because the electronic wave functions are so large, they can overlap with the next NC. This means the second electron can be excited in the next NC.

3.5.2 Coherent superposition of single and multiexciton states.

In this case it’s important to consider both the electron and the hole17. The exciton can be described as a dipole. The idea is that a hot dipole can be seen as a superposition of two cool dipoles. These two dipoles are coupled to one another and this way a single hot exciton is cut up in two cool excitons. Basically this mechanism is Coulomb coupling (as opposed to interaction). The story about interactions crossing the potential between NCs is true here also: excitons can be induced in neighboring crystals.

3.5.3 Multiexciton formation through a virtual state.

The use of a virtual state makes this explanation rather exotic15. Equation (1), colloquially referred to as Fermi’s golden rule, is an expression for the probability of a transition from state i to state f.

 ${T}_{i\to f}=\frac{2\pi }{\hslash }{\left|⟨f|{H}^{\prime }|i⟩\right|}^{2}\rho$ (1)

${H}^{\prime }$ is the process that governs the transition and $\rho$ the density of states. In the case of excitons in NCs, ${H}^{\prime }$ is a matrix given by a sum over energies around the virtual state. What is summed is the product of two matrices, divided by a the energy diﬀerence with the actual virtual state. The ﬁrst matrix is a measure of the probability that a particle in the virtual state is excited and the second matrix is a measure for decay and therefore a measure for the creation of another exciton. With these matrices, Fermi’s golden rule is a measure for the creation of other excitons.

3.6 Measuring Quantum Cutting

Whatever the exact mechanism may be, it is well established that the eﬀect exists. The creation of the second exciton can be either in the same NC or in the neighboring NC. The creation of a second exciton in the same NC will be short-lived because of Auger recombination, so the quantum cutting in our samples will be generally spatially separated, hence the abbreviation SSQC.

Remember that we’ve talked about cooling. Excitons with high energies still can lose energy to phonons, instead of quantum cutting. The investigation of quantum cutting is motivated by its possible end use: increased eﬃciency of solar panels. Measuring this eﬃciency in this early stage is a bit of a problem: there’s no easy way to harvest the electrons from the NCs yet. That’s a ﬁeld of research of its own. For the time being, it’s important to think of ways to measure data about the NCs themselves. How many excitons are there? How does that depend on photon energy? How does that depend on photon ﬂux? How long do the electrons take to decay?

To precisely control the power and wavelength of the photons hitting the sample lasers are a favorite instrument. Counting the amount of excited NCs for example, can be done by measuring the absorption, basically subtracting the ﬂux with sample from the ﬂux without sample. Doing such a measurement is however complicated, as it’s very easy to lose photons anisotropically. In section 4 a setup that covers all these diﬃculties will be discussed.

Another way of counting the number of electrons in the conduction band is through induced absorption (IA). IA is a method in which the sample is ﬁrst ’pumped’ and then ’probed’. With Si NCs, a wavelength with enough energy to push the electron up into the conduction band is shot at the sample. The result is that almost all NCs are in an excited state: the sample is pumped. Secondly, a second pulse is shot onto the sample. In these experiments, this pulse has a lower photon energy, less than the bandgap. Only electrons already in the conduction band can absorb these photons, because the electrons in the valence band can only absorb photons with at least the energy of one bandgap. So, the amount of absorbed light is a measure for the amount of excited NCs: the sample is probed.

An alternative way to make the competition between cooling and multiple exciton generation more visible, is by decreasing the nanocrystal size. As the size reaches a certain point, the band starts to break up into distinctly separate energy levels (ﬁgure 5). As the levels separate, the creation of phonons gets more diﬃcult: there are not many energies left. The idea is that multiple exciton generation gets more and more visible as NC size goes down.

3.7 Eﬃciency

As mentioned previously, eﬃciency is key. Because there are many diﬀerent kinds of eﬃciency, it’s a good idea to distinguish between them by deﬁning them properly.

3.7.1 Energy Conversion Eﬃciency

The Energy Conversion Eﬃciency (ECE) is the metric that is deﬁned as the ratio between power going into the solar panel in the form of photons, and the power coming out in the form of electrons. For commercial purposes, this is the only metric that matters, but to understand and improve this ratio, it’s useful to brake this factor down into the parts that make up a solar panel.

3.7.2 Quantum Eﬃciency

Wikipedia2 deﬁnes it as the "percentage of photons hitting the photo reactive surface that will produce an electron-hole pair". The ECE is a very broad deﬁnition that is a combination of a lot of independent factors, of which the Quantum Eﬃciency (QE) is one. Every solar panel component has an eﬃciency worthy of investigation, but in this research group, and in this project, only the QE of the material that actually converts the photons to electrons is studied.

3.7.3 External and Internal Quantum Eﬃciency

With respect to solar panels the QE is also known as the External Quantum Eﬃciency (EQE). The Internal Quantum Eﬃciency (IQE) is deﬁned as ratio between the incoming photons that are not reﬂected by and don’t penetrate through the panel. When researching speciﬁc parts that make up the solar panel, the photo reactive surface in our case, it’s useful to exclude eﬀects of for example the antireﬂective coating.

In the case of this project, the IQE refers to the eﬃciency of the photoluminescence. Light of a certain frequency and intensity illuminates the sample and light of another certain frequency and intensity is emitted by the sample. Light passing through the sample is discarded in data post-processing. In the nanocrystals, some energy ’leaks’ away through processes not necessarily known and some NCs don’t shine at all. Whatever the multiple causes and their respective eﬃciencies may be, the IQE measured in these experiments is ratio between light in and light out these processes cause.

3.7.4 Absolute and Relative Quantum Eﬃciency

It’s not always feasible to build an experimental setup so that you can actually measure any of these eﬃciencies, as the setup always introduces complications (how do you make sure you collect all emitted photons?). Measuring the QE means measuring the fraction of photon power illuminating the sample and the photon power emitted by the sample (usually at a completely diﬀerent wavelength). Samples can emit and reﬂect anisotropically and it’s not always possible to work around this. If you don’t, and you measure the QE, it will be relative, because it’s unknown how the anisotropy aﬀects that particular measurement. Solving the problem of anisotropy means that you can actually put numbers on your axes, because now it’s known that the fraction measured pertains to each and every angle the sample might reﬂect and emit at.

3.8 From Theory to Experiment

The group I have performed my Bachelor’s Project at has a new setup that allows for absolute QE measurements. The point is to obtain more precise and simply more results on carrier multiplication. Once measuring CM/QC/MEG is routine, we can see if we can go a step further and disprove some of the explanations. Also, we can begin to tune the samples to see what conﬁgurations provide the maximum eﬀect of CM, and therefore may be the most interesting for the solar power industry. All in all, an interesting experiment!