Lecture 15: Electronic Bands in Solids

Lecture 15: Electronic bands in solids Summary When two atoms come together, their atomic orbitals combine to form a set of bonding and antibonding molecular orbitals (MOs). When 1024 atoms come together to form a solid, their atomic orbitals combine to form a continuous band of electronic states. The bands are filled up with electrons: how full they get depends on how many electrons each constituent contributes and how closely spaced in energy the bands are. For example, here is the band energy diagram for Na, Mg, and Al. All of these elements have full 1s, 2s, and 2p bands: these “core bands” are therefore completely filled with electrons. Na has one additional 3s1 valence electron, which fills up 1/8 of the combined 3s/3p band. Similarly, Mg has valence electron occupation of 3s2 and Al has 3s2 3p1 , which fill up the 3s/3p band even more. Here, the 3s and 3p bands are shown as one combined band: when bands are very close in energy, they can overlap, and essentially combine to form one continuous set of states. When there is an energy gap between the most energetic electrons that live in one band and the least energetic electrons that live in the next band, a band gap forms, like those shown here between 1s and 2s, 2s and 2p, and 2p and 3s/3p bands. Note that the term band gap is reserved to describe only one of these band gaps, as discussed below. Knowing the band filling tells us a lot about the properties of the solid. If the highest energy band is partially filled, the material is a metal. If the highest filled band is completely full of electrons, then there is a gap between it and the next-highest band: this is the band gap of the material, and these solids are called semiconductors. In semiconductors, the highest energy band that is completely filled with electrons is called the valence band, and the next highest unoccupied band is called the conduction band. If the band gap is very large, say >4 or 5 eV, then the material is electrically insulating. The band gap has a property similar to that in the antibonding MO or nodes in higher-energy AOs: namely, all of the energies within the band gap are forbidden, so all of the electrons must be in a band, not in-between. Within a band, electrons can move around: if there is no gap and the material is a metal, it is easy for electrons to freely move around, and conductivity is high. It’s harder for electrons to move around in a semiconductor, but it’s possible for an electron to get an energy boost and transition from one band to another; often, these boosts come from light or heat. Of course, it takes less energy to cross a small gap between two states than a large one: it is 33 Figure above by B.A. Averill and P. Eldridge in General Chemistry. License: CC BY-NC-SA. Source: Open Textbook Library. much more likely for thermal energy (about 0.025 eV/atom at room temperature, 300K) to excite an electron into the conduction band of a semiconductor like Si than an insulator like diamond (C). Electrons can also be excited into the conduction band via light: semiconductors with energy in the range from about 1.5-3eV can be excited by visible light (about 400-800nm). If a photon’s wavelength corresponds to just the right energy to match the bandgap, that electron can jump across the forbidden energy region. In this case, it jumps to the conduction band and becomes free to move around! With this in mind, it is possible to control the flow of electrons across the gap—and the conductivity of the semiconductor—with light. If the band gap is larger than the lowest wavelength visible light, the material can’t absorb higher-energy photons, and the material is transparent in the visible light range. Why this matters The world we live in uses a lot of light. And in case you haven’t noticed, more and more of that light comes from LEDs, or light emitting diodes. Whether it’s your phone or laptop or television or car or fridge or house or office or street, odds are the way it lights up is with LEDs. And those work because they are semiconductors. As discussed in lecture, a semiconductor forms when the bonding in a material together with its structural arrangement leads to the formation of a band gap of just the right size (a few eV). From this we immediately have two well-known devices. First, electrons in the valence band maximum (VBM) can be excited by light into the conduction band minimum (CBM) where they conduct, a device otherwise known as a photo-detector. Second, we can have the opposite: put electrons into the CBM and as they cascade down to the VBM, photons are emitted, a device otherwise known as an LED. In lecture 3, I used television as the topic of Why This Matters and mentioned that the reason color TV took so long to develop was that stable, inexpensive, red phosphors were a challenge. For LEDs it turns out it was also a particular color that delayed their ubiquity, but in this case it was blue, on the higher end of the gap. For phosphors it was chemistry that held the key to inventing new molecules that could absorb electrons and shine red. Once again, 50 years later, it was chemistry that held the key to unlocking a new material, in this case a semiconductor that could take a current of electrons and convert them into blue light. The trick comes in being able to engineer the band gap of the solid. Check out this plot of band gap vs. the lattice constant, which is a measure of the spacing between atoms in a periodically repeating crystal. We’ll be digging into crystals in a few weeks, but for now you can consider the x-axis to be simply any structural feature of the solid. The point is that there is a strong dependence, and once this dependence was known and understood, then new alloys could be developed with just the right chemistry and therefore structure, to give just the right bandgap and therefore color. Take a look at GaN on the plot and notice that its bandgap is just a tad too high if we want 34 blue to be emitted as electrons cascade from its CBM to VBM. On the other hand, other materials like InN have bandgaps that are far lower in energy than what is needed to emit blue. It turns out that alloying different materials together is one of the most effective ways to “tune the bandgap,” and that’s exactly how scientists got to blue. They already had red and green, so once they got blue they were able to combine them all together to get white LEDs, which have completely taken over the market since. The idea of purposefully modifying the bandgap of a material is called bandgap engineering, and it is the centerpiece of the semiconductor revolution. It reaches far beyond LEDs, as the bandgap is a crucial property in lasers, transistors, detectors, and so much more of our electronic world. Why this employs How can knowledge of the band structure of a solid lead to a job? This may, in fact, be the easiest Why This Employs section to write of the whole semester! Semiconductors are so ubiquitous in our world that it’s nearly impossible to move without interacting with them. They’re in almost any and every electronic device. They’re so important some would call this the age of the semiconductor. Moore’s law depends on making tinier devices out of them each year. They make electricity from the sun. They provide light from electricity. They communicate for us. They compute for us. I could go on. The semiconductor industry is massive and there are many jobs in many companies, and many MIT students go on to work in this industry. There are so many jobs in the semiconductor industry that there’s a website called semiconductorjobs.com. And many other similar websites. Even more interestingly, the industry is no longer just about making faster chips or specialized chips like for wireless communication. It’s gotten so much bigger because of all of the various needs for these materials and devices, from consumer electronics to smart cars to Hollywood to space exploration to artificial intelligence. There are jobs related to all of these areas, both in the traditional semiconductor industry (think, Intel, AMD, Semtech, TI, and dozens more), but also in the companies working on these applications (think, Bosch, Toyota, Dreamworks, Microsoft, Google, and hundreds, maybe thousands more), as well as companies doing both, like Samsung. Example Problems 1. Suppose your LED is made of silicon and you want to make it absorb higher wavelength light. Should you alloy it with germanium or carbon? Explain your answer. 35 Bandgap vs. lattice constant © Light Emitting Diodes website. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. 2. Below is a plot of sunlight intensity at Earth’s surface as a function of wavelength. Solar radiation spectrum vs. wavelength diagram removed due to copyright restrictions. Calculate the band gap wavelength of each of these well-known semiconductors, and mark the range of wavelengths of light that each semiconductor could absorb. Which do you think would make the worst solar cell? Choose one: A. Si (band gap 1.14eV) B. Ge (band gap 0.67eV) C. GaAs (band gap 1.39eV) D. InSb (band gap 0.16eV) 3. You have three different materials: AlP (band gap 2.45 eV), SiC (band gap 3.0 eV), and CdSe (band gap 1.74 eV). Which of the following three geometric arrangements is likely to be the most efficient at converting sunlight to electricity? Explain your answer. 36 Solutions 1. Absorb higher wavelength = smaller bandgap. Thus alloy with Ge since larger atom = smaller bandgap. 2. Use the good old energy to wavelength conversion trick: A. Si (band gap 1.14eV) = 1088nm = 1.088 microns B. Ge (band gap 0.67eV) = 1851nm = 1.851 microns C. GaAs (band gap 1.39eV) = 892nm = 0.892 microns D. InSb (band gap 0.16eV) = 7750nm = 7.750 microns Each material will absorb light with wavelengths shorter than this band gap wavelength (so everything to the left of that wavelength on the graph). The InSb absorption spectrum will extend well off the side of the graph (it absorbs EVERYTHING) - but because the bandgap of InSb is so small, the electrons it excites don’t carry very much energy (they lose the excess to heat) so it would be a horrible solar cell. 3. Top left: all wavelengths absorbed and least thermal energy released. 37