Quantum Dots

Nanoha online simulation and more for nanotechnology so the topic of today's lecture or overview is

quantum

dots

nano 101 toin and

dots

and for some of you you might just say

quantum

dot which means today I'm going to spend about the half my time talking about actually where it is. Quantum mechanics arises and why these quantum dots are interesting from that perspective and from the applications that we want to get from these quantum dots, but to really understand where they come from and what we can do with them, we really have to understand where the mechanics comes from quantum and what are some of its fundamentals, so really the overview of the presentation is such that I will talk about classical systems first just to give you reminders, so you will be more or less drinking from a fire hose in terms of information, particles that propagate waves, standing waves, and chromatography, and then I want to highlight some strange experimental results that came out in the early 1900s that really led to the advent of quantum mechanics, so they were discrete photoelectric optical spectra. particle wave effect and duality and then that brings us to the point that electrons are waves and electrons are particles and we can think of quantum dots, so I'm going to show you what a quantum dot is, some experimental examples sometimes and, last but not least.

I'm going to show you some of the research my group is doing on quantum dots with multi-million dollar atom simulations on parallel machines, so it's kind of a firehose approach and just as a start and a reminder to draw such classic contrasts. and macroscopic particles the way we normally think of this iPod or anything we normally put in our hands has a finite extension, it has a finite weight and it can be counted with integers, we can take an element and we can deal with it, so that is how we think of a particle instead of a fluid, for example, so the laws of motion, our classical Newtonian mechanics, so F is equal to M multiplied by the right, is something very classical that you learned a long time ago, when interactions with other particles are based on continuity of energy and continuity of momentum, so those are the things that you generally derive classical particle interactions from, so if one billiard ball bounces off another, you apply conservation of energy and conservation of momentum and you figure out where that second ball goes, something you've done in mechanics.

So the example is a billiard ball. The key as I showed this slide is in classical, the extension is continuous if the weight continues and the particles are discrete, we can count them. What I mean by this is if you want to make it a little bit bigger, just expand it a little bit or if you want to make it a little bit smaller, you shave the thing a little bit, but in our thinking it's continuous, there's nothing discrete about this thing. in terms of weight. and extension, then there are propagating plane waves that you probably dealt with in math class.

They have an infinite extent, they have a finite wavelength and a finite frequency, so you could have a sine wave that propagates like this and the law of motion is typically a wave equation which is a second order differential equation in T, time and space and a solution that you for this equation would be a sine wave, so if you plug this expression here into this equation you will find that it will satisfy this equation and you will have to satisfy some dispersion relationship between frequency and momentum which is determined in this or that is used in this equation here this way interactions with other waves with each other you know they can interfere with each other so you could take one wave in another wave and if they have a slightly different frequency you would see a double type pattern oscillation so that you know that they can be coherently added to each other.

I mean, that's something you've seen in math. in the signal processing clauses so that they can have constructive and destructive interference and can even cancel each other so that there is no amplitude right there, then you may have heard of Huygens' principle that any plane wave away from that can be reconstructed as a start of an individual set of circular waves, so if I start from here you draw a little circle here and here and here in each of these sources you will find a new wave front that is constructed so that you can actually make up a plane wave of a lot of little circular waves and that shows you intuitively that waves can go around corners, if you inject something here with a plane wave, it can bend around corners and that's why light can go around corners, you have diffraction, so The bad thing is that I had all kinds of animations on my Mac, so I'm frantically trying to bring this to life.

I hope to show you this animation later. It was an animation that showed this Huygens principle, so now one thing is that in the late 1800s, people did an experiment, the so-called double slit experiment, where the light came from a small source and then through of a filter, it had two slits or two slits here and the light was coming from one source but it was split into two and what was predicted if light was a wave there should be interference patterns like this, but if light was a particle then there would be no interference pattern there is no reason why there should be interference, but experimentally an interference pattern was in fact observed.

So that was kind of accepted proof that light is an electromagnetic wave, that light can interfere, therefore it can and does have wave properties, so it is an electromagnetic form that consists of an electric and magnetic field that They oscillate in space and time, so it was somewhat accepted and again I want to show here what the characteristics are. Well, the wave has an infinite extension, they are not and they are not countable, right, you can't really measure the extension this way, there is a finite extension. wavelength but it is continuous, you can choose any frequency in principle, it is something continuous if you have a certain source, you should be able to choose any frequency and the frequency is finite, it is not something infinite, so there is a concept of standing waves and that is a sort of a mix between a propagating wave and a particle, so you've seen like a string on a guitar as a classic standing wave, here you have a fixed boundary condition and you might have a wave that just sits in the middle of the wavelength of the oscillator length L or you could put two full lobes here, that would be the wavelength would then be a full L, so these properties are standing waves, they have a finite extent, they have discrete wavelengths and discrete frequencies. because they have discrete wavelengths, the law of motion is again a wave equation, but the solution you choose is finite in space, so it is a sine function in this space, but it is zero outside this space, so it is confined, not infinitely extended and In the interaction with the other wave state, there can be coherent superpositions again, for example, sounds can be added in an instrument.

You can think of a standing wave as a resonator, for example your guitar string is a resonator and the case around the guitar is also a resonator and they talk to each other and exchange energy and exchange vibrational energy, so you can think of this resonator of one resonator coupled to another like a guitar string and there is energy transfer and energy conservation again, it's a concept you remember. of classical mechanics, but it also holds for these wave properties. The key element to this is that these resonators must be in tune with each other. If the box under the guitar string is simply not in tune with the string, there will simply not be an efficient coupling, meaning that if you move the string too far from the box, they are no longer in tune with each other, the coupling is weak or if you drill a bunch of holes in this resonator, it won't either. can no longer be strongly coupled, but the key element here is that this momentum, this K, is quantized now is a function that can only take finite values ​​because these wavelengths have a very discrete spectrum, so the extent of these Standing waves can be counted in 1/2 wavelengths.

They are discrete, they come in integer multiples and frequency comes in integer fractions of integers. Well, this is in terms of countable integers, but it is infinitely extended. In principle, you could put many lobes here, so there are infinitely many, but you're discrete, okay, from continuous to discrete, that's the step you want to get off the slide, so earlier we said that light is an electromagnetic wave and have you seen a prison, it can be used to split white light into a spectrum of light, so the colors of the rainbow come out, so white light consists of a wide spectrum of colors and each individual color is associated with a frequency particular to the form.

Now, on that note, this prison dissects white light into its frequency components, okay? Look, it's not an analyzer and that, in a sense, is also called a chromatograph, it looks at the colors, the chroma, this is the Greek word for color and it analyzes it, so why do I put these things together now? Well, I showed you that classical particles have a finite extension, the waves that propagate are infinite in their extension, standing waves are waves but of finite extension, so these are waves of finite extension and then I talked a little about how these chromatographs can actually dissect light that looks white and can dissect it into components, so the next part of my talk is going to be talking a little bit about strange experimental results and what really led to quantum mechanics, so people will like to burn things properly, so what they did was they burned properties and they shined the light that came out of it through a gift, what they discovered is that if you take clean elemental material and burn it, you don't get really white light and you don't get a continuous spectrum, there are not all the colors of the rainbow. in that spectrum they heated elemental materials and discovered that it is actually a discrete spectrum, so instead of having all the colors of the rainbow they found discrete lines very sharp lines yes yes yes, let's talk, let's think in terms of elements of the periodic table it also works for compounds, but then it gets more complicated, so here's a spectrum of hydrogen. here is a spectrum of what I earn.

It turns out that these spectra are actually fingerprints of material. Each elemental material has its own spectrum, which was an important idea. about the chemistry and an important idea about what the material is made of. You saw this yellow light almost every night. It is a bright sodium vapor lamp. If you go out to parking lots at night, bright orange lights are highly efficient lights. because they convert all this thermal energy into a single line and it turns out that we like to see that single line. I mean, you can't see ultraviolet light, you can't see infrared light, but all this energy is actually converted into something and it's the light that we see.

So it's a very high efficiency light because there's not a lot of loss in the spectra that we don't see, so you're kind of exposed to that through experience, but this actually led to the development of the first atomic models. that people discovered that there is something special in these spectral lines and they explain them with a model in which there is an atomic nucleus and the electrons move in orbitals and add energy to them, heating them basically excites the electrons from an orbital lower than an orbital superior. and eventually they fall again and when they fall again that energy is put into light and we see that light and within that model there is discretion that there is a discrete jump from one orbital to the next and that orbital is a discrete energy, so That was the advent of early quantum mechanics, then their more sophisticated models were developed, which are not actually circular orbits, but there are probability density functions like here is an S orbital, here is a p orbital, here is a D orbital , so you may have seen it in chemistry, but this.

It really all originally came from these experimental observations of spectra, so in a sense that was the first idea that electrons are standing waves that are attached to a certain nucleus. Well, these electrons are treated like a wave and they are bound to a certain box and the box is basically held together by a nucleus and there are discrete transitions in this spectrum of levels and they lead to discrete optical lines and each atom is different, it has different sets of orbitals, different sets of electrons and that's why each material has a different spectrum, that's why it's a fingerprint of the material, so there was another fundamental experimental evidence that people scratched their headscolloidal particles that are introduced into the quantum dots of the tissues for quantum computing, where quantum computing processing is performed in which it is not calculated with a charge, but with a quantum mechanical state. and from a very new term application, people are talking about quantum dots being used in detectors and I'll talk a little bit about this in the next one, so quantum wells are currently used for infrared detection and other optical detections, but the problem is a quantum well is a one dimensional structure like that, so it's a sandwich that grows from the bottom up, but it doesn't absorb the light coming in directly somehow, you have to rotate the axis of the light, it's a problem These quantum wells are blind to light coming directly in, but what people do is put something cool on top, an electromagnetic classification that effectively turns on the light, changes the angle of the light and then the light can be absorbed and that is It's done through electromagnetic modeling and we've done it at JPL, but these quantum dots have the property of being three-dimensional objects, they don't just have one-dimensional symmetry, so they can actually absorb light. that is orthogonal directly to the surface, it is not necessary to have this classification that loses a lot of light and a lot of sensitivity, so there is one more application here that stood out for lighting, if you can do as I pointed out earlier. these sodium lamps are very high efficiency lamps, if you can create systems that don't get as hot and you can make them in small screens, you are a winner and people dream of using these quantum dots in lighting applications, so last but not least not less important.

I want to talk a little bit about Moore.law, so Lou Moore's law is vaguely formulated: overall device performance doubles every 18 months and I'm talking about transistor devices in the computer and that's been historically true for more than 20 years, that's why these computers are getting faster and faster. and more powerful, so technically it was achieved by making these transistors that grow in a 2D plane, making them in two dimensions, making them smaller and smaller and people usually plot them on the feature size, so this feature size Lateral is a function of years of development. on a logarithmic scale that actually decreases the mania, which shows that there is an exponential decrease in this in size, but we are getting to this area where quantum effects are creeping in and certainly if you are doing this side feature size up to 210 nanometers of 0.02 microns, then certainly the quantum effects are going to be very important.

What this graph generally ignores is that there are layers in these devices that are already extremely thin and quantum mechanics is already governing this in this dimension and the quantum device or device simulators that are used to design current devices have to take care of the features of quantum mechanics that are already in this dimension, they usually neglect them in this dimension. Another feature they generally neglect is the number of electrons underneath a CMOS. gate, so in typical devices produced today there are between 10,000 and 100,000 electrons that operate this valve well. If you make things quite small, you'll only have a few tens of electrons, a few hundred electrons in there, now each electron has a certain capacitance associated with it. with it, a capacitive energy from sitting on this door or not now if you have it, if the device is large, the kaepa capacitance is very large and sorry, yes, and really the room temperature KT, the energy in the room temperature or in the environment is much greater than the energy associated with a single electron jumping or not jumping into the device, so an electron jumping in or out doesn't make a big difference in the electrostatics of the device, but if you're getting into the realm where The devices are very small, the capacitances are very small, so this charge energy for an electron to jump into the system makes a big difference, and suddenly you're in the realm where adding an electron can dramatically change the performance of the device.

It's called Coulomb blocking, where a second electron that wants to come in sees the other electron in the device and is being repelled, it can just keep going, so soon commercial devices will be in the realm where they fall between these two gray areas of stability, this area is very stable because adding one electron to the device or not, it does not change the electrostatics here, adding an electron is impossible unless it has enough energy, but that is very stable and this is very stable in the medium, it is quite unstable and it has a noisy system where if you add some electrons they significantly change all the characteristics of the device, so that's why quantum dots come into the technology and what they are, and now I would like to talk a little bit at the end about this tool that we have been building at JPL and now here it is called electronic nano modeling or Nemo 3d, it allows multi-million dollar atom simulations and admires these artificial atoms and molecules, so the modeling agenda starts with the desire to model physical structures that contain multi-million atoms, okay, so these are crystal geometries like this or a nanotube, but each atom must be represented in this structure and each atom has individual orbitals associated with them and we need these orbitals to describe the electronic structure in these atoms, so We describe each one. atom with a set of orbitals we map them into a real structure and then in this big structure we calculate these nanoscale quantum states like here is an S orbital and then we use these quantum states to map them in sensors or map them in computing now this is a little bit more For experts, the idea now is that if you want to do quantum mechanics, you have to choose a base system.

The fitness, if you do signal processing, you have to choose a set of foundations for your system. Usually physicists. I like to use simple e waves for the ikx type simple wave because they are very easy to handle and you can do Fourier transforms of infinitely periodic systems. They work wonderfully and are effectively called pseudopotentials for systems like this. As an electrical engineer my devices are not infinitely periodic, they are finite in size and I like to put a contact on them and I like to put electrons in one side and take them out the other side, that is not an infinitely periodic system so I would like to have a framework of modeling that is based on local orbitals where my basis set is not infinitely extended, but I actually know where my device ends and where my contact begins and couple them, so Lou, you will reuse the local orbitals and we. we use s, P and D orbitals for each atom and then we end up with a huge matrix that we need to solve and I'll talk about this a little later.

Here's another example C, so here you see a quantum dot crystal and the dot. It was spinning and depending on how you look at the crystal you would get different views, but for each atom that described it, here is basically the wave function that we want to build is a sum of individual orbitals and they have a coefficient in front of them and the individual orbitals are orbitals. P s P and D and in total there are ten of these to spin up and down so you map this to a physical structure and this is a zinc blende structure like this so this is a typical gallium arsenide structure It is a silicon structure each atom has four nearest neighbors and there are coupling elements whose 1s orbital here will communicate with a p orbital here and a p orbital here will communicate with another p orbital here so basically there are matrix elements So , and you write this like this, each atom has its own coefficients and then adds another sum of the four neighboring atoms.

Well, effectively, what you have is a Hamiltonian matrix where each atom has a ten by ten matrix. and then you talk to four other atoms that are sitting somewhere like this and then there's another array of them, another atom and you have other blocks that you can talk to, maybe they're neighbors or something, so you're starting to build a sparse matrix and this matrix is ​​of the order of if you have a million atoms you would have a million of these blocks well in terms of size of the overall matrix but the matrix only fills in one, two, three, four, five blocks each one because each sub block will only communicate with four other sub blocks it will not be an atom this atom will not communicate with this atom because there is no link with it okay so this array has a certain structure and what we do is to first model a subsystem, a small system like this adequately represents the material property of, for example, gallium arsenide and indium arsenide and then we map these orbital interactions into a huge structure that gives us a system where we can now handle a system of twenty-one million atoms to be able to have a system. i.e. where this array is 21 million block sizes, that's a pretty big array.

I don't think your little laptop can hold this properly to give it a size, the matrix sorts 4 by 10 to 8, okay, so the key is us. I want to get the values ​​from this array. The way we do it is we divide this matrix into slices and put each slice on a different processor and what we do is in the letter we need to do a matrix vector multiplication where Now these huge blocks are located on different computers, so what each portion on different computers and we process in a parallel system, we have different algorithms available for this and I show you this slide to give you a kind of overview of where supercomputing has gone and when.

I started my job at JPL. I told my boss I wouldn't recommend anyone do this. I told him I hate supercomputers. He was the leader of the supercomputing group and he said to me, "Well, you could change your attitude." And I did it because in 1997 he built number two of a Beowulf cluster. Tom Stirling, who invented Beowulf computing, was in our group, so we started with a bunch of PCs in four or five years. Now you can buy commercial products, PC racks where this is widely adopted for you. guys, 100,000 dollars seems like a lot of money, I understand that, but compared to five million dollars, a hundred thousand dollars is a lot cheaper and these hundred thousand dollar computers that can work can compete with five million dollar computers at that time, so small research groups.

They can afford to have their own individual supercomputers to themselves and that is why parallel computing has become more and more popular because people can actually afford to have their own supercomputer here is a graph of the performance of this Nemo 3d code from this matrix vector multiplied and what you want to achieve in parallel computing, if you double the number of CPUs, you would like to divide your computing time by two. It's true, it makes an investment, it gives you more CPU. You would like to get the computing time divided by two, so in a log-log. scale, if you plot time versus number of CPUs, you would like to have a graph like this as an ideal graph, okay, it looks linear on a logarithmic scale or if you do it on a linear scale, it's a hyperbola, right? two and the two again, so here is the 3d Nemo plot for different system sizes, so for an eighth of a million atoms 1/4 1/2 1 2 4 and 8 million atoms and it scales very well and linearly in this register. logarithmic scale, so the performance of this code is pretty good and we can calculate several eigenstates of a system like this, so here's the first one, the ground state here is some kind of px, py and a PZ orbital and then things get more complicated in the pyramidal structure they are no longer normal quantum numbers the wave functions look very strange there may be three lobes and a small lobe up there there may be overlapping lobes or things that really look like spaceships and you can do all kinds of things ordered things because these are their own atoms, they have their own wave functions, they have their own characteristics, so what we do within a network for computational nanotechnology is we try to employ high-performance computing and visualization, so We created a team that started working on this.

Nemo 3d tool, there are a group of people from engineering, computer science and iTap IT at Purdue who are working on this, we started. I transferred this code last year when I came to Purdue from JPL to here, so I'm laughing, basically, last year we went. From one hundred thousand atoms for which we were able to calculate eigenvectors we went to 21 million atoms, which is a tremendous advance and here is the result of an end-to-end calculation time based on the number of atoms for it to be executed. The NSF Terra network is a large accumulation of supercomputers and this is the end-to-end computing time, so 21 millionatoms took about 11 hours to compute on 64 CPUs, so we parked a job on 64 high-performance computing CPUs for 12 hours. and we got a result if we run this here there is a kind of scale mark if we run it on 64 CPUs it will cost us about half an hour per million albums if we run only half the CPUs it will cost us about double the time which shows that the code scales very well, you don't have to pay much to add more CPU, they are not wasted, they are useful, so to give you an idea, 21 million atoms correspond to a semiconductor cube of approximately 80 by 80 by 80 nanometers, no one most in the world can do this.

This is a capability that we have to do that no one else can do to prove that this really works. So here's a quantum dot and now I'm showing a conduction band. and the edge of the valence band and we are trying to extract these eigenstates from an energy spectrum that is about 40 electrons around the world, but we are trying to get interior states from an algorithmic sense which is actually a bit difficult and they are separated by about ten thousand involved or approximately or maybe a relationship involved, so we have to analyze in terms of resolution very small objects resolved from a wide energy spectrum of a medium of a spectrum, so if we take a small system of only three hundred thousand atoms and we calculate these systems we get this result, so it would be approximately 22 nanometers 22 nanometers times 11 nanometers if we take the same point and simply enlarge the buffer area, we know that the wave functions will be confined even within So if we make the system now six point seven million, we get the same states and if we make it twenty one million, we still get the same states, so this is kind of a proof of concept that we can in fact extract these eigen. states of a huge matrix and it basically means that we can get unique and specific eigenstates of correct symmetry from this huge matrix, so why would you care now that we can deal with really large systems that we can take for example quantum dots in those who like to grow? one on top of the other and we can consider a system where two quantum dots on top of each other and that is something like a quantum dot molecule, and it happens that these quantum dots on the top are slightly larger than the ones on the bottom. bottom and so the ground state will want to be located at the top point if these points are far apart, they are almost like independent points, then the next state will be at the bottom point.

Now the question is do you bring these points closer together so that they can have In some quantum mechanical interaction they will form a molecular state where now the ground state has parts at the top point but also parts at the bottom point and the so-called antibonding state is in a sense the opposites where the electrons are mostly sitting at the bottom point and have some component at the top point, that's something that's well studied in quantum mechanics, where if you put two atoms together, here's this distance between points where if you put these states together, They start to repel each other, so this is like doing quantum mechanics on a scale of anonymity.

So it's very exciting because it's a laboratory at the nanometer scale where we can study fundamental quantum mechanics. People have noted that, for example, with hydrogen, if you bring hydrogen. the atoms together eventually bond and there is a bonding state and an antibonding state, okay and they are separated from each other by more energy than just separating the individual states, so we can make tremendously large systems now where we can, people are interested in lasers, so they have huge stacks of quantum dots, so if you have seven identical dots, we can calculate the states in this and it's interesting that if you make it perfectly symmetrical and don't include the voltage, the top state looks very similar. to the lower state in its symmetry between these groups of states, now you can do this analytically if you had given me a single point wave function, I could have constructed them for a perfectly symmetric system, but nature is not so typical, typically these systems These dots are getting bigger and bigger at the top and these mini bands that formed there are broken and very unintuitive?

Actually, the ground state of this general system is such that the electrons like to reside at the bottom point and not at the top. dot and yet you couldn't do it analytically, you would have to have a simulation tool like this and if you want to design a laser based on the system, you better know the distribution of the states like this. The modeling capability is unique. I'll show you two more slides and then I'll finish. We are also developing visualization tools for this, where we want to be able to drill down into these wave functions and that is done with a collaboration with an IT department. where we can visualize these wave functions like this, but we can also take slices through them and then we can also do a statistical analysis of the components of these wave functions, mostly they are like P, like D or like a star, and We found something really cool, if you just look at the S orbitals in this wave function we see a symmetry, if you look at the P and D orbitals we see a very different symmetry and if we add the two together we see a composite wave function, but I was really very surprised. that the individual contributions of these general wave functions were different for the s and P orbitals, so it is really a new result obtained by visualization and we are starting to investigate it, and this project, interestingly, was started last year. year with a group from Sri, so some of the Sri students participated in that and a graduate student continued in it, so that's pretty much the end of my talk.

I wanted to remind you that we started with the classical system where I try to Remember that the classical system is typically discrete, you can count them. Classical waves are generally not countable, they have infinite extent, but standing waves are in the middle, between the particle and them in a continuous wave, and that leads to this particle wave duality. that eventually led to quantum mechanics and these quantum dots are basically little laboratories that we can build together to study quantum mechanics or use quantum mechanical effects for real applications, so that's kind of a quantum cue, do you have questions or concerns? , okay, no questions.

Should I be completely clear or very confused, yes, with the fondant molecules that have realized that it has actually been done correctly. Yeah, how my others are people who maybe do student Peters with quantum dots or I haven't seen Alec do the electronics applications. I've seen our people propose making memory cells, it's similar to a flash memory where a flash memory works where you basically have a capacitor and you pour charge into it and in today's flash memory you have thousands of electrons that are sitting on that capacitor and Now people are trying to replace this huge capacitor with a small capacitor, so that is the short-term electronic application that I see has a chance of surviving; the longer term ones would have logic arrays made from quantum dots, but that is much further down the road. pipe, that's still a pipe dream I would say, and that's why I wouldn't say that they're building supercomputers out of that in the new pipe, okay, yeah, so if my Mac comes to life, I can show you the animation, it's the animation of the Huygens principle that you did not see well