Lecture 18: The Perfect Solid - Crystals

Lecture 18: The Perfect Solid: Crystals Summary Crystalline solids possess long-range order: the arrangement of the bonds plays an important role in determining the properties of crystals. Each crystal has a unit cell, a repeating unit like a stamp that can be tiled into a pattern. Often, there are many unit cells that could be chosen for a given crystal; it’s conventional to select the smallest unit cell that contains all of the necessary information. In the figure, (a), (b), and (c) all show valid unit cells and how they tile, but (d) isn’t valid because when tiled, it doesn’t cover all the space: there are gaps. In 3D space, unit cells comprised of various atomic arrangements give information about crystalline solids. The unit cell also gives information about how atoms pack together. The packing fraction is a measure of how much of the 2D area of the unit cell is filled up by atoms: Packing fraction = (area occupied by atoms)/(total area available) In 3D space, there are only 7 distinct unit cells that can fill all of space with no voids. Bravais figured out how to arrange atoms into unit cells; it turns out there are only 14 unique Bravais lattices. In 3.091, we’ll focus on cubic unit cells, which have all sides the same length and all angles = 90 degrees. There are three distinct cubic unit cells. The top row indicates the position of atom centers for each of the cubic lattices. Next, the atoms are shown as filling up the maximum amount of volume: this helps to visualize how many atoms are in each unit cell. Finally, each of the unit cells are shown in the context of a bigger section of a crystal. Several key metrics help determine the macroscopic properties of a crystal, including the atomic packing fraction (APF) and the coordination number. The atomic packing fraction is calculated in 3D space: APF=( atoms in unit cell)*(volume of atom)/(volume of unit cell) The volume of an atom can be approximated as the volume of a hard sphere: V = 4/3 ∗ π ∗ r3 . For each of the cubic lattices, the radius of the appropriate hard sphere can be related to the lattice constant (width of the unit cell). Finally, the coordination number, or the number of nearest neighbors, gives information about how many atoms are available to form bonds. The number of 47 Unit cells figure (upper right) © source unknown. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse Cubic lattices diagram(lower right) from Chem LibreTexts. License: CC BY-NC-SA. atoms in the unit cell is the sum of the parts of atoms shared between neighboring unit cells. For example, the body-centered cubic (BCC) unit cell has a full atom at the center and also an atom at each corner that is shared equally between 8 cells. Therefore, the total number of atoms in a BCC unit cell is 8*1/8+1=2 atoms/unit cell. Why this matters Let’s go back to an application of semiconductors that I mentioned a few lectures ago but didn’t delve into. It’s about using light energy to excite electrons above the bandgap, and the fact that this means you can harness light as a form of electricity with the right materials. That’s called a solar cell, or a photovoltaic device. Here’s the energy we receive on planet earth from the sun. We saw this plot when we gave the example of the ozone molecule as a key reason we don’t receive much UV on the planet (since ozone absorbs it high up in the atmosphere). This time, I’m using the plot to show that not only does the intensity strongly depend on the wavelength of light, but much of the energy we receive comes either in the visible or the infrared parts of the spectrum. Naively, you may think the best solar cell would simply absorb everything; in the language of our bandgap and semiconductor lectures, that means it would have a very small bandgap, since the smaller the gap, the lower the energy of the photon can be to excite an electron above the gap. In a solar cell, once an electron is excited above the gap, that negative charge then gets extracted out of the device from the conduction band, while the positive hole gets extracted from the valence band. Those extracted charges can then do work (like charge your phone). Ah, but were it only so easy! The challenge is that once the electron is excited it very quickly “thermalizes” down all the way to the bottom of the conduction band. It’s a process that happens so quickly that there’s very little we can do about it (although there is an active area of research to extract “hot electrons” out of materials, before they can thermalize, but no working devices at this point). This thermalization means that the amount of energy the electron can have when it is ready to do work, no matter how high in energy the photon was that excited it, is equal to the energy of the band gap. That’s shown here in this energy diagram as the blue squiggly line. A photon of this blue energy would be absorbed by the material, and the excited electron wouldn’t lose any energy since it didn’t go higher than the bandgap. But if a higher frequency light is absorbed, like that purple squiggle, then the photon still creates an excited electron in the material, although any excess energy above the bandgap is lost, since the electron quickly drops down to the conduction band minimum before it can be extracted. 48 Figures above © sources unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. On the other hand, if the photon that comes in has an energy lower than the bandgap, say that green squiggle in the diagram, then it won’t be absorbed at all (no states are present to be excited into). And here is why understanding the chemistry of solar cells is so important, for we have competing drivers: the lower the gap, the more of that spectrum can be absorbed, but the less energy the electron has to do work and the more energy lost to thermalization, while the higher the gap the more energy the electron has in that CBM, but fewer photons can be absorbed. This is because, as we learned, the semiconductor can only absorb light equal or greater in energy to its band gap. This trade-off means there’s an optimal bandgap for a solar cell made from one single material, and that optimal gap is right around 1.3 eV. That’s quite close to the bandgap of silicon, which is 1.1eV. Yes, but that is only if we have a perfect crystal! In fact, silicon could have a much lower bandgap, depending on how good of a crystal it is. If it has defects (these are interruptions in the lattice and stay tuned, that topic is coming soon!), then the bandgap can go way down, with the limit being that silicon could even become metallic if the crystallinity is messed up too much. This explains part of the reason why silicon has been expensive and also has to go onto glass as opposed to plastic substrates: namely, because a silicon solar cell has to be made very thick (100’s of microns). The truth of the matter is that silicon is a terrible light absorber. That’s right, the solar cell material that dominates the world market at > 85% is horrible at doing one of the most important things it needs to do: absorb sunlight. Silicon is still is capable of absorbing, of course, but it’s just inefficient so it just has to be made very thick in order to capture all of the light. Compare that to other materials that can absorb light efficiently like a dye molecule, which could absorb all the sun’s energy in 1000x thinner layers. The thickness of silicon is a double-whammy: first, it means the solar cell is brittle, cannot bend, and must go on glass which makes the whole device heavy, and 2) it means the excited charges have a much further distance to travel in the material before they can reach an electrode and be extracted for work. And that means high cost, since it’s a lot of material that needs to be made as perfectly crystalline as possible. It can be (almost) done, but it’s difficult, expensive, and requires huge fabrication plants. But getting that quality, that crystallinity over large distances (yes, microns are very large distances for an electron or an atom!), is worth it because crystallinity holds the key to the semiconductor properties. In this case, it’s the electron transport we’re talking about, and the more ordered and crystalline the lattice is, the better for the charges to move around. Why this employs Since this lecture is all about crystallinity, let’s take a look at the employment opportunities in making crystals. It is true that you can dig up rocks from the ground and they may already have crystalline order in them, like the inside of this rock shown here. But most of the time these crystals are useless for technology since they contain all sorts of orientations all jumbled up together, they often have a lot of defects in them, and also they’re usually not pure elements but rather minerals containing the element of interest along with impurities (like oxy- 49 gen). So the question is: how does one make as pure of a crystal as possible out of a single element? Let’s take silicon as an example. Here’s a figure of all the steps that it takes to turn sand into crystalline silicon. As you can see, there are a lot of them, and many involve high temperatures or chemical mixing. A lot of companies operate at different stages of this chart. For example, getting from sand to “metallurgical grade silicon” is something a lot of smaller companies do, like Mississippi Silicon, Elkem, or Silicon Materials, Inc., to name only a few. These are often companies with a few hundred employees so they may not have a large number of job postings, but there are many companies so if you do a deep dive into the full landscape of silicon producers you’ll likely find some good opportunities. Once a company has made high enough purity silicon, they can then mold it into an ingot. This also requires further purification and also again high temperatures. Most of the companies that make high-purity ingots are currently based in China, although there are others scattered around the world including in places like Japan, Germany, and Korea. And once there’s an ingot, it can be sliced into wafers that can then be made into devices. Here we have a lot of large companies specializing in this process, like Applied Materials or Lam Research, among many others. Example problems 1. Calculate the atomic packing fraction (APF) of a simple cubic unit cell. 2. Which of the three cubic structures do you think is more likely to form at high pressure? Why? 50 Diagram of Si refinement process © source unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. 3. CsCl forms an interpenetrating SC lattice, which looks like a BCC lattice but with 2 different types of atoms. For this problem, we’ll use a BCC lattice and assume that the Cl- ions are at the corners. a) How many of each type of atom are in the unit cell? b) What is the lattice constant, a (the length of one side of the unit cell), in terms of the atomic radii rCs and rCl ? What is the value of a in Å? BCC unit cell © source unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. 51 Solutions 1. Volume of unit cell = a3 = 8r3 (because lattice constant, a = 2r) Volume of one atom = r3 of atoms in simple cubic unit cell = 1 Lattice constant, a = 2r APF =(volume of one atom)/(volume of the unit cell)x(number of atoms in a cell) 4 3 3 πr 8r3 ×1= π = 52.4% 6 2. FCC is more likely than BCC which is more likely than SC. Higher packing fraction means atoms are closer together 3. a) 1 Cl- ion and 1 Cs+ ion b) Close packed direction = body diagonal √ 3a = 2rCs + 2rCl √ 3a = 2(2.38A) + 2(1.00A) = 3.90A 52