Horst Störmer, Dan Tsui & Bob Laughlin - 1998 Nobel Laureates in Physics at MIT Symposium 10/8/1999
KASTNER: Experiments were done up the street. And if you walk across through Building 9 into Building 13, you can find the offices where Bob Laughlin yelled at me and his thesis advisor, John Joannopoulos, for many years. We really feel as though this is a family affair. And we're really delighted that Bob and Dan and Horst all agreed to come and speak today.
I want to express thanks on behalf of the Physics Department to Dave Litster, the Vice President for research and Dean for research for helping to support this. And for the Francis Bitter Magnet Lab. Bob Griffin also helped support it.
And I especially want to mention the David Harris lecture fund. David Harris was a student at MIT. And before his death, he discussed with his family a memorial that he would like. And he wanted a memorial which would be a living memorial as opposed to hardware or bricks and mortar. And so his family endowed a fund which provides resources for us to bring distinguished lectures to the physics department each year.
And this year we're especially delighted that that's being used to help support this symposium. So now, I'd like to call upon Dave Litster, who would like to say a few words. Dave.
LITSTER: Thank you, Marc. It says here another welcome. I'm not sure you need another welcome. But I think the reason I was asked to do this is to indicate that this lecture series is an important part of the intellectual life of the Institute as a whole and that what's happening this afternoon is also important to the rest of the Institute and not just the Physics Department.
So I welcome you. I was thinking about this prize a little bit. And the thing that-- the idea that popped into my head is that, very often, those of us who do basic science, experimental basic science, especially owe an enormous debt of gratitude to technology. But we seldom acknowledge it publicly. But it's very important.
And this Nobel Prize would never have happened without some very important pieces of technology. Sometimes, the basic scientists have to develop the technology themselves. And here, I see there were three important technological ingredients. The first was these very special materials with very high electron mobility. And those were developed by Horst and Dan together with Art Gossard at Bell Laboratories. Without those, the experiments could not have been done.
Secondly, the experiments required very high magnetic fields. And those happened to be available at the Francis Bitter Laboratory and no place else in the country. And so the experiments were done here. I think Peter Wolff will be telling us more later this evening for those who were there about it.
And the third bit of technology is something that people refer to as wet technology. It was the unusual brain of Bob Laughlin who happens to be able to think about things in a way different from most of us and, therefore, was able to explain this. So I think it's going to be an interesting afternoon. In fact, it's going to be a real intellectual treat. And I will sit down so that Bob Griffin can proceed and we can get it started. Thank you.
GRIFFIN: Well, thank you, Marc and Dave. I'm here because I'm Director, presently, the Director of the Francis Bitter Magnet Laboratory. And as many of you know, being Director of the Magnet Lab has been like being captain of a-- the last few years, at least-- being captain of a ship sailing on a really pretty rough sea. Some people-- one of my colleagues described us like being captain of the Titanic even.
But we've had some very good experiences, nevertheless during the last few years. And many of those were associated with visits from Horst Stormer and Dan Tsui. I mean, we're very closely connected, of course, with the experimental aspects of the fractional quantum Hall effect.
As Dave Litster just mentioned, the fractional quantum Hall effect was actually discovered at the Francis Bitter National Magnet Laboratory at that time. And it was actually approximately 18 years ago today, yesterday rather. I don't know if you realize that, October 7, 1981. And we're converging to the opinion that the experiment may have been done in cell four of the Magnet Lab. Maybe Horst and Dan can comment on that later on.
But we owe Horst and Dan a very great thank you, a very big thank you. They have been an enormous help to the lab. And I think everybody at the lab, particularly myself, really appreciates what they have done for the lab. And perhaps more than whatever we did for them.
And so I would like to express the gratitude of the laboratory and its members to both of those individuals for all of the nice interactions we've had. And also to add our congratulations to everyone here on the award of the 1998 Nobel Prize.
KASTNER: Thanks, Bob. So now I'm going to call on Ray Ashoori, who was post-doc with Horst Stormer at Bell Labs before he came to the MIT faculty to introduce Horst.
ASHOORI: Well, it's a pleasure and an honor for me to introduce Horst. Horst has been at Bell Labs since completing his PhD at the University of Stuttgart in 1977. And he started at Bell as a post-doc and a year later became a regular member of technical staff. And following sort of a meteoric rise, three or four years later became a department head, head of the famous semiconductor physics department which is where the transistor was invented.
And he became, after that, director of physics research in 1991. He's now a professor at Columbia. And he still holds an adjunct director position at Bell.
By the mid-1980s, when I was in grad school, Horst was already quite famous. In fact, I remember one of the older students in my lab having boasted, I saw Horst Stormer. And a lot of other adjectives came along with that. But you can imagine how thrilled I was when he chose me as a post-doc back in 1990.
So the first thing I noticed coming into the lab was just an amazing amount of inventiveness and a little bit of outrageousness that I think led Horst to do a lot of great things. The first or second day in the lab, he asked me, can you solder? I said, sure, I can solder. I thought it was just soldering wires together.
What he meant was, can you sit up on this table on top of a stool and solder a couple of wires-- solder, under a microscope, on the end of a long probe that we used, a low temperature probe, and solder little microscopic wires on this sort of flimsy set up. And the idea was-- we had an idea-- he had an idea for a very novel experiment.
And I was a little bit reticent about sitting up on the stool. So he showed me first that it could, indeed, be done. So it's this kind of spirit that made it just a tremendous amount of fun to be there.
Way back when he was a post-doc, he invented a technique for growing ultra pure semiconductor materials called modulation doping. And this discovery-- or this invention was one of the key items, I think, in finally giving rise to the discovery of fractional quantum Hall effect that he and Dan Tsui did here, as you've all heard.
And this kind of structure, actually-- it has had some technological relevance, in that, if you have a cell phone, you've got transistors made out of this kind of ultra pure material that Horst invented. And some of these ideas that he's had, in retrospect they seem so simple and maybe even obvious. But of course, before you make the discovery, before you have the idea, it's not at all obvious. And Horst is known, I think, for just seeing with amazing clarity what needs to be done.
Here's another sort of outrageous thing that he did. He invented a technique called cleaved edged overgrowth, which you take a wafer that you grow. It's very thin wafer. You cleave it inside of a machine that's used for growing these wafers. And you actually grow on this thin edge of a wafer. OK.
This allows you to make, it turns out, perfect one-dimensional structures. And the technique itself is difficult enough, but on top of this, when I came to his lab he's actually doing lithography on this half-millimeter edge. So imagine doing lithography-- the kind that's normally done on big semiconductor wafers-- on a very tiny edge.
And I was just amazed at the outrageousness of this quest. He actually made it work. And he has done beautiful 1D experiments-- experiments on one-dimensional wires since.
I also-- here, I'm blessed with being near Peter Wolff who was Director of the Magnet Lab at the time Horst and Dan made the discovery. And I just love listening to his stories about Horst and Dan during that time and how they-- when they discovered the fractional quantum Hall effect.
First of all, they're working inside of a very noisy big magnet where there's giant currents running through the spinner magnet, water rushing through it, making the whole system vibrate. This bore is about two centimeters wide.
Now, imagine you have room temperature outside this bore. Inside the bore, you want it to be 50 millikelvin. So you somehow have to develop a refrigerator that cools your samples down to 50 millikelvin. Horst and Dan did it, and that, of course, is what got them the prize.
So you can see how this kind of inventiveness is just critical to this kind of discovery. Here's another story about the first day on the job with Horst. He has tremendous energy. And you see this in the way he walks, because you just can't keep up with him.
So what I would notice is that he walks so fast that I would-- I couldn't walk as fast. So I'd end up running a few steps, then walking a few steps, and running a few steps. And this is how we progressed to lunch every day during my two years of Bell Labs.
After Horst became director, and that was after my first year at Bell, after a year, he would still come down to the lab, and he would do things like help me change the pump oil. You know, what a completely mundane job for Nobel Prize-winning director.
He's getting up because he's getting sick of this. But you could see you could see in him that there was a real love of just being in the lab. All right. Let me end with this.
Horst has received every physics award imaginable, the Buckley Prize, the Nobel Prize, the Franklin Medal, he's in the National Academy of Sciences. He's a truly, I think, extraordinary and rare person. And it's just a thrill to have them here just for the day. Here's Horst.
STORMER: Is this on? Is this on?
STORMER: Ray, you haven't gotten any briefer. But thanks very much. I'm-- it's just too much honor. Thanks. Thanks very-- I must say-- you hear more and more of these things, and you start wondering about yourself. But, all I can say, I'm a very lucky guy that has worked with tremendous people, and not just collaborators but also then in the form of post-docs, and Ray is one of those.
And there are several others around. And they give me all this credit. And it's just not true. It's these guys that did it. And that Ray says help changing pump oil was very easy. It's the only thing I can do.
It's a pleasure to be back at what was, for many years, our part-time home, not far down here. Sort of a place you really wouldn't want to call home when you walk into it. It's a loud and noisy environment. But Dan Tsui and I came there for many years doing our experiments, eating cold pizza and drinking warm Coke, looking at data on chart recorder coming out, and singing together with our grad students funny songs from Monty Python.
It was a really good time for a few years that-- well, actually, it must be about a decade that we came up here. So it's a pleasure to be back at MIT. It doesn't mean that I haven't been back in the meantime visiting old friends, and many of them are in the audience. There's actually too many to go through one by one. If I do it, I know I'm going to forget one of them, so I better don't.
So we have three hours. I'm not sure the audience how long you're going to hold out for three hours. But I have the advantage of having the first hour, and Bob has the last hour. [LAUGHS] So let's see what we're going to do.
That's my first-- can I take this off? No. Almost not. You can imagine we're doing a lot of these recently. Actually, the two of us, Bob and I, have been on a tour through Scandinavia. And we started feeling very much like these gentlemen. It's just a good one that--
And it hasn't it hasn't stopped much since then. I'm looking very much forward to Wednesday, next week. I hope that some of the burden is going to be taken off from us. But I'm afraid it's not going to be condensed metaphysics, so it still will be rough going.
In fact, when I told Bob that-- I did this when we went to Stockholm, and I had only the pieces with me. I told Bob that I would be doing this. And he said, yeah, and I knew it who's going to be Pavarotti.
I told him, of course, the best. So I'm actually-- along in this vein, I thought, this is, of course, all this it's a great honor and it is a wonderful, wonderful week, and we had a very good time. And how good a time we had is perhaps shown in this one.
You come in in the morning, and you are sleepy and not shaved and washed, because you get off the plane. And so they want you to take a photograph in front of the King. This is the King. Yeah.
I remember, I swear, Lou told a Viagra joke.
No. I tell you why we're laughing. And you know why we're laughing. The Viagra jokes were old by then.
So they get you in front of the camera. And what all photographers do is-- you're never close enough, right? Photographers always do that. And because they have one hand on the camera they do it like this.
So they wanted us to stand closer together. And, you know, Bob and I say, I can't. He's, after all, in the middle there, and it's not so easy to get-- so Bob started out complaining. He said, what do you want? And I started laughing.
So the two of us just blurted out. And you see this is Louis Ignarro, one of the physiology prize winners. And he looks around. He's just totally flabbergasted at how poorly-- how badly physicists can behave in front of the kings. Well, of course, I mean, it was that one that ended up in the newspaper the next day.
My fellow countrymen have had good fun with me. Because, in fact, there was always this-- it raises sort of the question, is he German, or is he not? Can we can we count him as one of ours or not?
And this question actually goes on, and particularly because my umlaut-- I have an umlaut in my name, and I dropped it when I came to this country. Because a good friend of mine, Michael Schluter, who is no more-- but he told me drop you umlaut. Because A, the typewriter doesn't have it. They don't know what to do with it.
And then the smart guys put an E afterwards. So half of them drop the umlaut. The other half puts an E after the O. Then you have appear twice in the alphabet, and it's not next to each other. So I dropped my umlaut. But, now, of course, they have found the umlaut again. And you wouldn't believe, in the last APS meeting, I was in twice, once with and once without umlaut.
But this is what, actually, somebody from [INAUDIBLE] sent me the next day in the mail from Germany. This was-- this is sort of the equivalent of the, I don't know, daily news probably. This was the front page of it.
And when I got this in the mail, my first reaction is to fly home and kill my mother. Because that's me--
--as a five-year-old playing Indians and Cowboys. In Germany, then, of course, no concept of the whole thing. But I thought my mother had given him that picture. But indeed, they had shot it off the wall, because it's hanging together with my little brother who's here that they cut out. It's hanging on the wall in my life my parents' living room.
In Germany-- this is, here, referring to the picture our Nobel laureate. And then it says under here, already as a boy, he built a toy train with a rocket propulsion, which is actually-- which is true. What it doesn't say there.
So in very small letters here-- and, by the way, it blew up in his hand and ripped off his thumb, which is also true. It was a long time ago. I have no problem with this anymore. But it certainly doesn't show me as a very good experimentalist. So it was a little bit off.
Well, enough of the jokes. Let me try to talk a bit about the fractional quantum Hall effect and see how far I get in. And Dan said that however I get, he takes the second half.
So I want to talk about fractional charges and fractional quantum numbers. And I want to give you sort of an impression what this whole thing about. The way I judge the audience is actually that most of you actually know everything about it, and I might as well sit down. But, after all, you paid my ticket up here. So why don't I deliver.
I don't think I have to say much to this audience about condensed metaphysics. But I think-- let me just, perhaps, pick up at one point along this thought. And that is one of the big problems nowadays in condense metaphysics is electron-electron correlations, meaning that electrons are strongly interacting, of course, because they are charged.
And taking into account the correlation, the motion between electrons as it's forced upon them by the other electrons around them is a very difficult problem. Most of through our life, we get away with very crude approximations. Number one, forget it. Just don't take it into account. Twiddle a little bit around with the math and with that dispersion relation, and you're done.
But, in some cases, electron-electron interaction is everything. For example, in superconductivity, it's everything, mediated by phonons there. And what I want to show is another case where electron-electron interaction is everything.
And it's only electrons this time around, electrons in 2D that really show very strange behavior, indeed. In fact, the strange behavior that it shows-- and it's just the pool of two-dimension electrons, nothing else, in a magnetic field.
You get fractional charges out, which was, of course, a big surprise. The new electron quantum liquids, fractional quantum numbers, flux-- magnetic flux can be attached to electrons. At least, it looks as if it were attached to electrons. New particles are popping up that can sometimes be fermions and bosons, depending on what magnetic fields you have.
Huge magnetic fields, as big as the ones that you get next door-- or got next door, and you still get a good fraction of them next door-- can apparently disappear, 10 Tesla, or something like that, a million times the Earth's magnetic field, disappears. And the carriers that you have seem to be moving along as if there were no magnetic field. There's a mass popping up, which has nothing to do with the mass of the electron. And they may even superconduct in some sense. Let me see whether I get to this, give you a bit of an impression.
We're working on two-dimensional electron systems. And you can create two-dimensional electron system, and three-dimensional worlds are always at an interface between-- can you see-- always at an interface between two things. So you could do it on the surface of helium, and it is being done on the surface of helium, on the surface of anything. Typically, things are dirty. This is no comment about your floor. It's just dirty, in the sense microscopically, that there are dangling bonds where the electrons get stuck.
So it turns out, actually, the best way to make a two-dimensional electron system is that the interface between two things, between two semiconductors inside a semiconductor. The technique to do this was actually developed, invented, in the '70s by Al Cho at Bell Labs.
And it's a high vacuum evaporation technique. We basically evaporate two semiconductors on top of each other. And so this is a high vacuum chamber. And you evaporate gallium and arsenic, gives you one semiconductor. Gallium and aluminum and arsenic gives you another semiconductor. That's all you need to know.
And you grow it on the substrate, which provides a template so that the atoms go into the right place. Otherwise, it's just like ball bearings, in the sense, otherwise, it would be random. So you provided a nice underlying lattice for it to grow.
And, as you know, in order for semiconductors to conduct, you have to throw some dirt into it. You have to doping them. And, in this case, actually silicon becomes the doping. So silicon is the stuff that gives off an additional electron that then can move around in the material.
The way this looks is like this. This is about two-inch diameter. What is dark here-- disregard these things up there. They actually do have something to do with this what Ray brought up. And that is this cleave that you overgrow. This is not import.
It's just this part down here. This is the wafer. So they're shiny semiconductors, look like a silicon wafer, basically. And the reason that the black here is because the background was black. But this is a shiny surface of about 2-inch diameter, which then has been grown by MBE.
And the important aspect of this is that you have incredibly controlled in this E direction. You can grow these layers one layer after the other. So what you see here in the cross-section is four layers of gallium arsenide, four layers of aluminum arsenide, four layers of gallium arsenide. This is the black and white stripes.
You have a control. This was in one atomic layer. And this is-- so you form these layers that fluctuate by not more than one atomic layer, so about an angstrom. So this is the flattest surfaces that you can create with a control of, really, one layer. You know you can stop at five, or six, or seven layers. And this technique that was developed and invented in the 1970s we had then used in order to make very high mobility two-dimension electron system.
In three dimensions, there's sort of a catch-22 when you dope something. You throw in the silicon, which gives off the electron that then moves through the semiconductor. But your left with the charged the impurities of which the electron then, very often, bounces off.
So, yes, you introduce carriers. But you also introduce a scattering for it. In 3D, you hardly can get around it. In 2D, you can. Because in 2D, you can put the electrons in one layer and the dirt in the other layer. And therefore, they hardly see each other.
And this is the method to do it. It's called modulation doping, which is-- so the green is one semiconductor. The blue is the other semiconductor. You put the impurities into the green semiconductor, and they wander over into the blue semiconductor, because there they find lower energy.
This is a quantum well of about 100 angstrom thickness. And the electrons now can move in this plane. But they cannot move in this E direction because they are tightly confined.
In fact, they are so tightly confined that quantum mechanics plays an important role. They're confined that you see the first bound state and the second bound state, and they're separated by about, let's say, 10 millielectron volts. So quantum mechanics is an important aspect of this.
If you only have the lowest state occupied, and not the second one, the electron in the Z direction has no degree of freedom. It's just tight. It cannot do anything. But it can move in the x-y plane easily, in fact, very readily because it's such a nice plane, and then the impurities are away from it. So you get a very high mobility two-dimensional electron system in which the electron can run for very long distances before they scatter as anything.
The electron density is about 10 to the 11, which makes about an electron every 300 angstrom, something like this. Typically 10 to the 11. You can vary, but that's a typical density that you will find.
Mobility is nowadays as high as 10 million. In fact, last week-- no, it was the week before. Two weeks ago, we had a record of 23 million mobility, which is unheard of. I mean, I'll show you a little bit more about this.
You may not understand what mobility means. But it's a mean-free pass of about 100 microns. In fact, 20 million is about 200 microns. So the electron travels in this plane for about 200 microns without bouncing into anything.
And 200 microns is really big. It's about the thickness of your fingernail, which doesn't look so big. But on a microscopic scale, and on a little graphic scale what we can do nowadays, it is a huge mean-free pass. And it's these two-dimensional electron system, which then we can regard as a two-dimensional metal. They're basically nothing else but that with a very long mean-free pass. So it means that the lattice behind it hardly plays any role. It's like a two-dimension electron system just in free space.
Now, this technique was invented around 1978, something like this, by these four guys. And I show it to you for two reasons. The one is sort of a historical picture in terms of MBE machines. This is how MBE machines looked then.
In fact, what you did is this is a very high vacuum. These days, you still open it up, let all the air in, stuck your sample in, pumped it out over night. And the next morning, you get your growth. You don't do this anymore, much cleaner than that.
The people standing around here is Willy Wiegmann, Art Gossard, Ray Dingle, and this is me in an earlier incarnation, standing around the MBE machine for the photographer. And the important part-- the other important part why I'm showing to you is Art Gossard.
Art Gossard was the person that grew the sample on which we then discovered the fractional quantum Hall effect. And this was the first time mobilities went as high as 90,000. And that made it all possible. I mean, it brought it actually about.
So the sample is the most important ingredient to all of this. And Art Gossard and his assistant, Billy Wiegmann, they grew the sample. And it's really unfortunate that they didn't share with us in that prize. They shared with us in some previous prizes, but not in this one.
And I say it's unfortunate, because I think for us condensed metaphysicists, materials is just the most important aspect of our work. The people that have first hand on materials are the ones that very often are ahead in the science that they're doing because it's really the materials that get them to this initial step from where you then can start playing. So Art and MBE technology is a very important aspect of all of this.
So what you end up with then is a two-dimension electron system where by the separation of electrons from the doping impurities and then the suppression of scattering from those-- is there any kind of a pointer? No? All right.
You get very high mobilities. At very low temperatures, they get a very long mean-free pass. So it looks a bit like a billiard table in two dimensions. Electrons can only move into two--
Oh, there is one. Right. I should-- oh, this is this little guy there. Great. Thanks, Marc.
Let me show you how an MBE machine looks nowadays and the people that are running it. This is the crew nowadays that is making this 25 million-- 23 million mobility sample. Actually, the most important guy, Loren Pfeiffer, and his assistant Ken West. This is Kirk Baldwin, who's an assistant with me. And this is Amir Yacoby, who was a post-doc at the time.
So now, the MBE machine is about that thick. And you use much cleaner methods in order to get such high mobility. But again, these are the important guys to which we owe everything.
I'm very often being asked, why at Bell Labs? Why at fraction of charges? Is it ever going to be in the telephone system? And my only answer to this is this is only going to make our vacuum cleaners run only 1/3 speed, and that's about all. And it's a joke. So fractional charges will not have-- I don't see fractional charges having any kind of application, even in the long run.
But there's another aspect of all of this work, which has an application, and that's this one. Out of exactly the same material-- or exactly the same material in which we discovered, Dan and I discovered the fractional quantum Hall effect-- out of the same material, you can make field effect transistors.
It just looks like it. This is what is in every PC. It's a MOSFET. Never mind what it is. This thing just looks the same. But it's another material, and this specific technique of modulation doping creates mobilities that are much higher than here. And contact resistance are lower in all of this.
So these transistors, or transistors made out of this materials are the fastest and lowest noise transistors in the world still today. This is one built by Fujitsu, who actually was one of the first that made a microscopic transistor out of this. Never mind what the structure is. But this is the fastest, lowest noise transistor that you can get.
And it's in any of those around the world. Each one has one of those, radio telescopes. This is not a very big market. There's just about probably 100 or 200 around the world. So you cannot make money with that.
But you can make quite a lot of money with these things. So a lot of these high frequency 2 gigahertz telephones that-- cellular phones-- have in the front end, which is-- well, the microphone is a front-end maybe up there-- have one of these HEM transistors for very good performance. And the base stations have it in there.
So there's another part of this modulation doping, which is not basic science, but, really, technology. And I feel very fortunate that on one side this is modulation doping that we invented led to very basic science. And on the other side, it also led to an application.
And it's not so that the one came after the other. I think it's an important lesson. And we're not the first to teach this lesson, in a sense. Or the material is not the first to teach this lesson, that is that it's not always an invention, fundamental physics, and then there is application. Very often, these things go in parallel, and very often, actually, they go backwards, because the progress in technology is so important for us to do the next step in science too. OK.
Let's get a little bit closer to the fraction quantum Hall effect. If you have such a system, particularly if it's semiconductors, the easiest thing you can do apart from measuring its resistance is to measure the Hall resistance. All it is it's like a resistance measurement, but in addition, you put a magnetic field up.
So this is a two-dimension electron system. You run a current down it, and you put a magnetic field onto it. Then you measure two characteristic voltages, the one is along the current path. The other one is across the current path. The one is nothing else but the resistance, and the other one should be 0, as long as there's no magnetic field because the thing is symmetric.
But when you put a magnetic field on it, then you push the carriers to one side because of the Lorentz force. So you build up a voltage. Eventually, you'll build up a voltage from one side to the other. That's what you measure.
And these-- the voltage that you measure-- divided by the current, because it's proportional to it. So if you double the current, you get double the voltage. So you normalize it to the current. You get a resistance, of characteristic resistance, which is called the Hall resistance.
And as Edwin Hall has already figured out in the last century, I'll show this next the next few graph, Rxy, this resistance is proportional to E, to the magnetic field, and inversely proportional to the carrier density in the material. So it's a wonderful way of measuring carrier densities in material, particularly in semiconductors. Because, you see, the smaller end, the easier it is to measure. The smaller end, the bigger this number is. And typically, carrier concentrations are small in semiconductor as compared to metals.
So this is a wonderful way of measuring carrier concentration. And it's an industry standard. So this is the first thing you do.
And we've done this, of course, but Edwin Hall had done this already in the last century. This is why it's called Hall effect. He was a graduate student, by the way, at Johns Hopkins in 1879, when he discovered that in a very thin foil of gold.
He just did this experiment that I showed you. And this is the result here. In fact, there's another aspect to all of this, which I always enjoy, is for all of us that are writing papers that have to be 3 and 1/2 page and not any longer. It's interesting to read this introduction here. We have to fight with every word, take it word out, and make an acronym out of it, so that you save another five words. This starts out with, "Sometimes during the last university year, while I was reading Maxwell's Electricity and Magnetism--" I mean, by that time, you had already-- we're already over your page limit nowadays.
The other aspect is, just like it is expensive nowadays to publish color figures, it must have been expensive in those days to publish any kind of a figure. So his results are here only in form of a table. So I had to sit down and, for the first time, plot up Edwin Hall's data. And here is Edwin Hall's data now on a graph. He has plotted.
This is what we call nowadays the Hall resistance. It's actually a current through the gold leaf divided by the current through a magnetometer versus magnetic field in units of the Earth's magnetism. One does not know what these units are, but one knows that it's proportional to the magnetic field. So that's fine.
So this is magnetic field. This is Hall resistance. And you see, those are his data. It's a very nice straight line, just as we know. And, of course, since then, this experiment has been done in many ways by many people, and it's a perfectly nice straight line. And the slope of this line here gives you the carrier density of the material. The steeper it is, the less carriers are in it.
So this was a status until 1980 when Klaus von Klitzing came along and told us that this is not so at all. In fact, in two dimensions, it's not so at all. This is the original result, Klitzing, Pepper, and Dorda in 1980.
And it's plotted in a different way Edwin Hall would have plotted it. So I show you the way-- similar data. It's not the same data, but similar data locks if you plot it like Edwin Hall plotted it.
Magnetic field Hall resistance. This is the Hall resistance. I'll talk about the other one in a second.
So Edwin Hall would have told us this is a straight line. And so Klaus von Klitzing found-- of course, this is a very nice example. The steps were much weaker when he saw him. But there are lots of steps in there.
This is not all a straight line. These steps are in there. And not only are these steps very flat and sometimes very wide, but they are also quantized to units of H over E squared to an accuracy, nowadays, of a few parts and 10 to the 8. So a few parts in a billion, almost, a few 10 billionths accurately quantized to this ratio of natural constant.
This came as a big surprise. In fact, there are some inklings in the theoretical literature that sort of foresaw this in some sense, but not really pointed it out. You experimentalists should-- and it was obscure. But Klaus von Klitzing discovered this, actually, at the Magnet Lab in Grenoble, the sort of sister institution in Europe.
So this is not all a straight line. But it's plateaus and it's quantized. And I show you also what the resistance does as a function of magnetic field. That's the other property.
It looks like the derivative of this. So whenever this is flat, it's actually a deep minimum. And then when it makes a transition, you get a spike out of it. So the magnetic resistance looks like the derivative of the Hall resistance.
I show you this because, later I've sometimes show you these data, because they're sort of more dramatic than even the Hall resistance. But it's the same-- the origin is it has the same origin. Here comes my two view graphs explanation of the integer quantum Hall effect.
If you put a magnetic field on a two-dimensional electron gas, than the electrons classically perform rotary motion around the magnetic fields. So quantum mechanically, this is just a harmonic oscillator, so this is my quantum mechanical oscillator. And then there are bound states in there.
And there's only a certain number of bound-- well, there's an infinite number actually. But they are discrete, and there's nothing in between. So the electron can be in this state, in this state, or in this state, and nothing in between.
So in a three-dimensional electron gas, there is actually something in between. But in 2D, there's nothing in between. And therefore, there's only allowed states at this energy, at this energy, and this energy, and nothing in between. I forget the spin, or leave the spin out for most of my talk. It's just complicates matters here, but it's very easy to take into account.
The other aspect of this quantization is that the number of electrons you can fit into a squared per one of these Landau levels, as they are called, is only a ratio of a natural constant in proportion to the magnetic field. So you take-- if you take this wave function, it makes sort of a size out of it. You take a cookie cutter and just cut out, see how many fit in there. And you see that the number of electrons per level is just e times B over h.
And it's independent of the mass. It's independent of epsilon of the materials, independent of everything. It's eB, only proportional to the magnetic field. So the integral quantum Hall effect actually happens when I fill up an exact number of these states, of these Landau levels.
And you can see that very easily. You take the number of electrons per level, which is eB over h. Then if i, 1, 2, 3, 4 of these levels is filled, then it's just i times d. You put this into the equation that I showed you earlier that Edwin Hall derived already in the last century, and you get the result, h divided by ie squared. So it's really true.
There's a big but here. And this took ingenious people to figure out. Because what you assume in there is that the number of electrons, in a certain sense, is quantized, and that the system is ideal. You do it here. But it's never like this, the number of electrons is fixed once and for all. And you never have an exact number of Landau levels occupied except for singular points.
So all this says is that the long-- right, this is my Hall line here along the magnetic field line. There's a few discrete points. But when I go up, this resistance is exactly this number. But it doesn't say that there's any plateaus. I wouldn't even know where they are.
So this is a bit of a fake doing this way. But I cannot go much deeper into it. What it tells you is that this connection between the quantization of the Hall resistance and the two-dimensional system, the two-dimension electron system has something to do with filling up exact numbers of Landau levels.
And this big but here has to do with localization in two dimensions. There's actually an important aspect. If you had an ideal two-dimension system, you would not see the quantum effect. It requires dirt in order to see it.
I think it's this sort of totally counterintuitive. You need dirt. You need impurities. You need localization of electrons in order to see it.
In fact, the guy that figured it out is going to give a talk sometime today. This is the-- I will not go through this argument, of course. But the bottom line off of Bob's argument, one way at least of looking at it, is that Rxy, this-- you can actually regard it-- which is this number-- you can actually regard it as a ratio of flux, of magnetic flux, quantum to the electronic charge. So it's the ratio of flux quantum to electronic charge.
And therefore, you can look at this, the quantum Hall effect, if you turn this argument around, as being a measure of e. So that is a very accurate measurement of the electronic charge, which is then the flux quantum divided by whatever you measure. And then this is an integer. And the integer you always know whether it's 1, 2, 3, or 4. You cannot be that far off. So it is one way of looking at the integer quantum Hall effect that is a very accurate measurement of the electronic charge.
The other aspect that I always found perplexing is that you measure something to an accuracy of a few parts 10 to the 8 in a sample that looks as terrible as that one. This is about three millimeters wide, about 15 millimeter long.
This is indium here that is put on and diffused in order to make contact. The black part is actually the surface of the semiconductor, which is really shiny. These parts have been scratched away with an ultrasonic scratcher, really a very intrusive method. And those are gold wires running off here.
This is a sample that was made by Dan Tsui. I would have done a much nicer job, but I'm sure that my contacts wouldn't have worked. Dan's worked. And, in fact, that's the sample in which we discovered the fractional quantum Hall effect.
So in terms of the integral quantum Hall effect already, it is always so amazing that, independent of geometry, independent of how you put your contacts on, you measure something, a quantity to a few parts 10 to the 8. And, of course, this is the essence of a standard, that it does not depend on, in this case, on the geometry at all. You can even make a round sample with all the contacts on the edges. As long as you do that, you also measure it.
This is a picture that's probably going to bring some emotions up in some of the people that are sitting here in the audience, certainly, Larry, I would think. This is-- I don't know which cell it is, but Larry can tell us. Which cell is it, Larry? It was downstairs someplace, yeah, I think so.
This is 23 Tesla, I suppose. Picture that was shot much later, dilution refrigerator-- that's the fridge. This is the can for around it. This is the pumping system. A person is about as big as this pumping system.
And we went with samples through the Magnet Lab and did experiments in high magnetic fields, and later on then with a dilution refrigerator. This is about 10 years later, I would think, but still when the Francis Bitter National Magnet Lab was in-- up here. Well, it still is up here, only without the National.
Dan and I were coming up here for several years to do experiments. And we were typically getting up on Sunday morning, driving up here, having a long four-hour chat. Depending on who was driving, it was either three or four hours-- four or five, I should say, I think. We're not that fast.
And we had this brand new sample form Art Gossard. We were doing experiments down in whatever cell. I don't know which cell it was. We were working on silicon MOSFETs, actually.
Dan had invited me to come up with him. He was there already since quite a while doing experiments on silicon. And then with this new sample, we started working together on this modulation dopes systems.
So here's the result of October, 6, 1981. This is the Hall resistance. This is the magnetic field, just what Edwin Hall would have done. And it's a very nice straight line.
This is an offset here, so they get it still on the chart recorder paper. It's an old technology to take data. You take a pen and pull it over a piece of paper. And then you look at the trace. And that's what your data is like.
There is-- actually, this is my handwriting-- this says laser upstairs. You see these spikes here? There was-- I don't know which laser. There was a powerful laser upstairs, and whenever they yanked it, we got a big signal down there. So this is from the laser upstairs. We just had to make sure that this was not physics, but just a third effect.
But otherwise, it's just a nice straight line, just as Edwin Hall would have said. But they're already deviations coming up here. And these deviations are from the integral quantum Hall effect. We knew exactly where they were, where they had to be.
So the top line is basically a repetition of this, taken at 4 degrees. And when we pumped lower, to lower temperatures, indeed, these structures became nice flat Hall plateaus. And that was the last Hall plateau, on the second last Hall plateau. Everything, all the world was in order.
But it got out of order up here. What you see here is that the magnetic field is about three times as high as this one. Remember, this is the last Hall plateau. This is the last time the Fermi energy can be between two Landau levels. From then on, it's always in the lowest Landau level.
So there was nothing expected out here. Actually, I should say, the reason we looked at these things was because Dan Tsui at the time wanted to look at the Wigner lattice. He had done that already in silicon. And we thought, well, let's look at these materials and see where the Wigner lattice.
The Wigner lattice is this when the electron condense into a regular array of electrons, a hexagonal array, or triangular lattice-- triangular, I should say. It's not hexagonal-- triangular lattice of electrons that become immobilized and they form a nice two-dimensional lattice.
This was a prediction back then. There are now some indications over the years that something like that indeed exists. And that's what we were looking for.
And what we saw is this plateau there. Well, it's not a plateau yet. But it's sort of a inflection point. We got all excited about it.
But, of course, it could be all sort of things. I mean, samples, resistances very often shot up or shot down. But this one looked sort of clean up here, but then it developed this thing, which just looks like the integral quantum Hall effect at a higher temperature, but at a different point.
And we were standing there down in whatever cell floor. And in about 15 seconds, Dan did the following. And I always tell this, because I think it's just wonderful and tells you a lot about Dan Tsui.
He took this into his finger, the distance from here to here. And he counted off how often he could count it off until it got to this magnetic field. So it was 1, 2, 3. And he said 1, 2, 3 quarks.
And we both had a very good chuckle about it, because it was so humorous that, of course, quarks, of course, are elementary particles with which a charge of 1/3 is associated. And, of course, we are not doing elementary particle physics. The highest voltage we have applied in our system is about 10 millielectron volts.
So this is a small m. This is not a capital M. And it's certainly not a G. So it was a good joke. But, at the same time, he was dead right. Because that it had something to do with the charge of 1/3 really turned out to be correct. And that just shows you tremendous insight of Dan.
So this is then-- this is the argument written out. If you take Bob Laughlin's argument at the time, which is that the electronic charge, or whatever you measure, is just the flux quantum divided by this. And you take it seriously, that's what you get out. Just put the numbers in, and this is just this 1, 2, 3. It's 1/3 of an electronic charge somehow involved if you take it literally.
So, of course, at the time, it wasn't clear whether it was really a step or not. After all, it was only a slight inflection. It was only an inflection here. So the question was whether it was a step or not.
And we had to look closely whether it actually was a step. You see this says here, this is not a step.
I often show-- I often show this view graph. And I do it nowadays in different continents. This is the only one that can laugh about it. You go to Europe, they just-- there's just dead silence. They don't get this joke. And you know why not. This country is run by lawyers.
So going to lower temperatures, what was an inflection point really became a pretty nice plateau. And I always say is a few parts and 10 to the 4 that this was at 1/3, and it was quantized to about 3 times H over e-- so H over 1/3 e squared, I should probably put it. And simultaneously, the resistance, the magnetoresistance, actually developed a deep minimum, which wasn't going yet all the way down. But it looked like it. As a function of temperature, it was going down.
Another picture of the time a little bit later, down that was now around the hybrid magnet is this one. This was taken in February 16, 1984, about three years later. But it's the only one that I got.
And those were people that were working with us. This is Albert Chang. This is Peter Berglund, Greg Boebinger, Dan Tsui, and myself here holding up a sign. 85 millikelvin, 80 kilogauss. Peter Berglund was the guy that developed this and built this dilution refrigerator that's sitting down here.
And there was a post-doc on a grad student, actually an MIT grad student, that Peter Wolff told us about. Peter said, I got a good grad student. You want one? And we said, sure. And Greg just became a terrific collaborator.
He's nowadays director of the Magnet Lab at Los Alamos. There's one section that's in Los Alamos, and Greg is running this now. So he's still sticking to magnets, like most of us. We fell in love with magnets. Cannot get away from them. Must have something to do with the attractive force.
So it very quickly became clear that different from the integral quantum Hall effect-- and you'll have seen all this, so I'm not going to go through it. There was the argument that the fraction quantum Hall effect is not a single particle effect. It does not come about from the interaction between-- sorry-- from the quantization of individual electrons in the presence of an electric magnetic field in a semiconductor. But it had to do with interaction between electrons.
It must come from this, because all of the steps that come about from individual-- quantization of individual electrons were all expressed in the integer quantum Hall effect already. There shouldn't be anything else.
And there were, I think, the smartest and brightest of the theorists who were trying to figure it out for quite a while. And this here is Bob. And he says, "Nothing yet. How about you, Newton?" With an apple from hell about to drop onto him.
And, of course, you're going to hear more about it. But this is what he came up with. This is his wave function. And I'm always-- I always express that I find this an immense achievement that the behavior of some 10 to the 11 electrons you can describe everything with an equation that has just about 15 digits in them, 15 letters in them.
This is Bob Laughlin's wave function. He's going to tell us more about it. But this is the solution. This is, within quantum mechanics, everything we can know about it, including all the interactions between the electrons. Bob will modify that, I suppose.
Let me tell you about the fraction quantum Hall effect and its origin from a slightly different point of view that we have developed over the-- when I say we, the theorists have. And I'm just a few-- a poor interpretation of the theorists' work.
One way of looking at the fraction of quantum Hall effect nowadays is from the point of view of the attachment of magnetic flux to the electrons, which is driven by electron-electron interaction. The electrons try to be at the position where there's no other electron. They have to do that for Pauli principle reasons. But the more you get the electrons away from you, the better it is for Coulomb reason because of the Coulomb interaction with the other electrons.
So since the wave function have 0s, many particles may function as also 0s. They try to attract as many 0s of the wave function onto them. It is advantageous in terms of energy.
And one way of describing that is in terms of what's called [INAUDIBLE] flux. It's similar as if you had attached to the electrons a magnetic flux. I won't go any further into this. It takes theorists to do it, and certainly much more time.
But then this particle statistics is actually changing. If you take two electrons, you change them around. And their fermions, the wave function gets a minus 1 in front. So they're fermions.
If you take two electrons with a vortex attached-- well, with a [INAUDIBLE] flux-- let's just call it magnetic flux-- attached to them, you exchange them, you get also this minus sign. But because there's a magnetic field and you move the electrons around the magnetic field that's stuck to the other electron, it gets an additional phase change. And therefore, the minus 1 is multiplied by another minus 1, so it's a plus 1. If you stick 2 on it, it becomes a minus 1 again. If you take 3, it becomes a plus 1.
So these behave like fermions. These behave like bosons. These behave like fermions. These behave-- in fact, whenever m is odd, it looks like a boson. And whenever m is even, it looks like a fermion. And there's a long list of people-- I couldn't list everybody on here-- a long list of people that, over time, have developed this picture.
So there's a way, which I call here another way-- because I don't have the time to show the-- and Bob is going to speak later anyway. So one way of looking at the fractional quantum Hall effect at 1/3 then is to take electrons. Imagine there are 3 flux quanta attached to them by virtue of the electron-electron interaction, which forces them to have 0s on the wave function. But just look at that as being magnetic flux attached to them.
Then I look at these particles as new particles, which are no longer fermions but bosons now. And these new particles now are green little spheres rather than red little spheres with three arrows on them. So I have now a system of green little balls. And the red balls plus flux quanta have been transformed into these green balls.
And they are now bosons rather than the electrons. In the magnetic field, actually, they become bosons in no magnetic field, because all of the magnetic field has been sucked in by the electrons. They're attached to the electrons. They don't see the magnetic fields anymore.
This is sort of a jump that I'm making. You have to really think more deeply about it, particularly since I'm seeing it's [INAUDIBLE] flux, not real flux. There are deep issues down there. But just for the purpose of this talk, we take the electron with three flux quanta attached to them and make out of them what's now called composite bosons in the absence of magnetic field.
Then you have a system, a new system, of composite bosons where the magnetic flux has disappeared. And this system then is at 0 magnetic field. And bosons condense into the bose condensate superficially, certainly. And, indeed, that's one of the way you can look at the 1/3 state, that you do this, take the electrons, put flux quanta, make new particles, and bosons in 0 magnetic field, and they condense into a bose condensate, and that's what the 1/3 state is.
So you can look at it as a bose compensation of composite bosons. This is, actually, historically backwards but. It's pedagogically a little bit easier to do it this way.
Now, that's the 1/3 state. How about all these other states? And there's a slew of them. The 1/3 was just the first. And then there are tons of other states that came about.
And here, I show you what is the magnetoresistance, which looks like the derivative because it's more impressive. And I apologize, actually, for the picture because it's old by now. In the meantime, there's probably 5, 6 more oscillations here.
So there's 1/3, 2/5 3/7, 4/9, 5/11, 6/13, 7-- whatever-- up to 21. So you see all these other-- the integer quantum Hall effect are these deep minima here. All the rest is fractions. It's not just the 1/3. It's just a slew of things, of fractions.
This is a list of fractions that have been seen. We can argue about to what degree we trust each of these fractions. But this is just my way of writing it down. I say this one is OK. And, I think, actually, recently, we just have seen something that looks like 6/23.
So it's up here. See, it's a slew of fractions, not just one third. It was just the first one to be discovered. So what's the origin of this? OK, so it makes it a little bit busy that suddenly all of these fractions--
A little bit like these chaps here. There's a guy reading the music down here. And you say, gee, look at all these little black dots. It's totally confusing what's happening there.
The first thing when I was looking into-- and it's this state at 1/2. See, there's something at 1/2. It doesn't look at all like these other fraction that have this nice minima. The 1/2 state is a broad little thing. It doesn't do much as a function of temperature.
This is a low temperature of 100 millikelvin, 123 millikelvin, perhaps. It doesn't do much. It just sits there. And the 1/2 state was for long a long time a miracle, sort of, what this thing is all about.
But then when I heard the description of it-- and the people that are most responsible for it-- actually, these few here. But then, also, Jainendra Jain, who actually-- again, this is historically backwards. I'll get to it.
So in the same vein of what I told you about electrons flux quanta onto it, you can then look at the 1/2 state where there's only two flux per electron. You attach those to it. And as I told you earlier, now, they look like fermions again. But they look like fermions without a magnetic field, or the magnetic fluxes on the electrons.
And therefore, you get now pink particles, pink particles that behave like fermions, but in the absence of a magnetic field. And therefore, the magnetic field has been eaten up. The electrons are moving around just like in 0 magnetic field. But they're fermions.
Not but they're fermions. They remain fermions in spite of the fact that two flux have attached to them, or just because actually. We know what a system that is fermions in 0 magnetic field does. It fills up a Fermi surface.
That is exactly, actually, what was found and was predicted first, but then was found experimentally in various experiments by Bob Laughlin-- sorry, I wanted to say Bob Willard. Bob Laughlin has just become an experimentalist, what he always wanted to be in life. So I made you one.
This is just a caricature, just a cartoon of it. So you find it in k space, a circle like that, the Fermi edge. And you can measure this.
Never mind the fact of 2. In fact, this fact-- this difference of fact of 2, actually, square root of two was an important aspect of that, yes, this really looks like it. So this was found that the electrons at 1/2 form a new Fermi surface, just like in 0 magnetic field.
And then you-- OK, so we have only 1/2. So what of all the other fractions? In all the other fractions then, one can look at in this way. And that's really, historically, sort of the starting point of it. But for this talk, better to do this way.
You remember this view graph. That was one of the very first view graphs I showed you. Just integral quantum Hall effect, quantization. This is the 2D surface, quantized Hall-- Landau levels, et cetera.
And then one thinks of all these other states in a way that is analogous to the fractional quantum Hall state-- sorry, to the integer quantum Hall state. See, the integer quantum Hall state is down here, and you get these oscillations from the integer Hall effect.
Now we have another 0 of magnetic field, so to say, at 1/2, where all the magnetic field has been eaten up. And if I slightly change the magnetic field, then I get a residual magnetic field. And these particles here that see 0 magnetic field is actually exactly 1/2, see a finite magnetic field if you get a little bit off.
And then they develop Landau levels. Just like the electrons develop Landau levels down here, these new particles, composite fermions, develop Landau levels by themselves. And these Landau levels of the composite fermions become the fractional quantum Hall states.
So you can look at the fractional quantum Hall effect also as the integral quantum effects, but not of regular electrons, but of composite fermions. And there's a very nice symmetry. This is just the same plot only, one's in red and one's in black.
And if I pull this over here, where I put the 1/2 on top of the 0, then you see that the black minima, which are from the fractional quantum Hall effect, and the red minima, which were from the integral quantum Hall effect, are just on top of each other. So we should have seen this experimentally. But it was really the theorists that were pushing down our throat.
I suppose I have about two minutes. Is this correct? OK.
So this picture is now confirmed in many ways. There have been experiments done on these fractional states that then let you extrapolate back to some kind of a mess around 1/2. There's a lot of questions still associated what's happening exactly at 1/2.
What is the mass? What does it really mean? What kind of phenomena do you see there exactly? What else could you see? Is it a real Fermi system or not?
These are all things that are up in the air and still experiments being done on it. But the concept of a composite fermion, composite boson is really something that has become very important in the fractional quantum Hall effect.
The spin of the electron also has back in and became a very important tool to measure some of the energy gaps. I don't have time to get into it. We can actually associate a mass with these composite fermions. We can associate a G factor with the composite fermions. All this has been done over the past few years.
Let me show you just one other state that is of great interest nowadays. And that's a state out here, at 5/2. Never mind how I get to this. This is the 1/2. This is the 3/2. This guy is called 5/2.
It basically means, for every five flux quanta, you get 2 electrons. And this state looks sort of like this one. But when you cool-- this state looks sort of like this one. But when you cool it down, it doesn't.
And this is the newest experiments on that. When you cool the 5/2 state down-- sorry, it's a bit busy-- it goes to 0, and it gets a plateau. Now, this is a 1/2 state, so it has an even denominator. So it should be a fermion, and fermions do not show a fractional quantum Hall effect. Composite fermions do not show a fractional quantum Hall effect, but they show a Fermi surface.
This one shows a fractional quantum Hall effect. It was discovered in '87, I think, Bob Willard, et al. And that was a big surprise, an even denominator became a fractional quantum Hall effect. How come?
We still don't know. Well, let me actually point out. This is the lowest temperature ever have cooled down in an electron system. These data were just taken at the Magnet Lab in-- Talla-- sorry, not Tallahassee-- Gainesville-- where they have very low temperatures, milli-- microkelvin, for that matter.
We use it only down to millikelvin in high magnetic fields. This is not that high, actually. This is only 3.7 Tesla. But we were able to cool the electron system down to 4 millikelvin, which is a real achievement-- the elections, not the lattice-- this is the electron temperature-- by various trick that the people at the Gainesville, [INAUDIBLE] and then together with our student down there, Wei Pan, developed that really let us do this.
Now, the speculation-- and there still is a speculation, although the theorists tell us more and more that they're very convinced about this. And they even tell us that our data is supported. I don't understand that argument yet.
Is that what's happening is just like at 1/2, the electrons actually do form fermions. They take an electron, put 2 flux quanta onto it, and they are fermions. So now we have b equal to 0 with new fermions called composite fermions.
But then they discover that they can actually mutually attract pairwise. This is an interaction in real space. But, in fact, the wave function that you can write down for it-- it's called a pfaffian. I'm way out of my league here, but it's called a pfaffian. But it looks similar to BCS wave function with an l equal to one in there.
But there is an interaction in there, and they form pairs. And these pairs-- sort of in far analogy with Cooper pairs, if you think of it-- they can condense. They form a new quantum liquid now.
So you form first fermions. Then you have an interaction which is just all Coulomb interaction. And they form pairs, and these pairs then open up a new gap, which is something like a superconducting gap. So in a sense, they form a new kind of superconductor made out of composite fermions, and that's what you see in the experiment.
Let me leave it at that. There are various other experiments that are very-- other parts of the magnetic field scale. There are, nowadays, anisotropies that are being discovered in high magnetic fields. They're called stripe phases.
There is the question, for example, how many electrons are required in order to see the fractional quantum Hall effect? Obviously, one is not good enough. I say obviously, but this is an important part. One is not good enough. It's a many particle effect. And some people like Ray and others are looking whether 3 is enough or 5 is enough, or do you need 10 in order to see it.
There's a Wigner solid out there where electrons are supposed to condense into a solid of electrons. And that one, there are indications for it. There are things called skyrmions, which are many particle spin systems that are playing a role.
So this field is alive and well, to say the least. I believe the 5/2 state, if we really can pin it down, that at this kind of a state, it's really opening up a whole new universe of experiments that can be done. Let me show you my last view graph, and that is sort of where are we going.
This is the data, or the result that Edwin Hall gave us about 100 years ago. Magnetic field, Hall effect, straight line. So this is just the standard Hall effect.
Then Klaus von Klitzing came along and told us that this is not so. There's actually-- you see this is a used view graph. I better fix it up. There's these big steps in here. This is the integral quantum Hall effect.
And then Dan and I were so lucky to discover the 1/3 state. So suddenly, there were more steps here. And then, the 5 something came along, and the 7, and 11, and the 13, and the 17, and the 23. And as we keep going, we're going to end up with that, which is a perfectly straight line. Only what you don't see is that there are infinitely many steps along here.
It's called the Devil's staircase, Devil for obvious reasons. And it's going to be fractions all over the place. So eventually, we're going to end up just like Edwin Hall, and he's going to tell us, I told you so.
So let me just show you this graph here, which is probably the most important. All of this-- I only change pump oil. Of course, Dan, who's going to talk, long friend and collaborator and, really, my mentor in many ways. Lots of collaborators, post-docs, students at Bell Labs that have worked on the fractional quantum Hall effect.
And then there is particularly the people that are the crystal growers that are here pointed out in-- all important ones here in red, Art Gossard and Willy Wiegmann, and then also Loren Pfeiffer and Ken West who are growing the samples nowadays.
So it was these people working on the fractional quantum Hall effect that really brought it all about. I always feel that Dan and I, we were so lucky to discover-- it's a little bit like archeology. We discovered the first tile sticking out from a roof out of the sand, and then it was all these terrific people that continued the work and really were cleaning up the whole-- not cleaning up, actually discovering the whole building, which finally made us to receive the Nobel Prize for it.
I feel like the luckiest person in the world having been able to work with such terrific people and being part of this endeavor. So, in a sense, I think I owe a lot of-- not in a sense, I do owe a lot of gratitude to all of these people that have brought it about. I feel that I was only one of the lucky ones that was just there at the right time at the right place. Thanks very much.
And, of course, the right place at the time was sort of one street down the road.
ASHOORI: Thanks for a beautiful overview. You don't get away so soon. Are there questions?
STORMER: I don't want to cut into Dan's--
KASTNER: First, couldn't it have been done without the Magnet Lab?
STORMER: Yes, of course. No, no. Look, no doubt. We see the fractional quantum Hall effect now at 0.8 kilogauss, something like this. So this is exaggerated. But it's a few kilogauss.
So, yes, of course today. But at this time, no. This sample is required. This sample, which was 90,000 mobility, in the field of whatever it was, 19 Tesla or something like that were this is where the inflection was to see it.
At any lower than that, it wouldn't have happened. Nowadays, we have superconducting magnets that go that high. At that point superconducting magnets were typically 8 Tesla. So one would not have seen it.
Yes, at that point, it required the sample and this magnetic field. A few years later, if actually it had been pursued any further, maybe people would have said, well, forget modulation doping. It doesn't look interesting. But a few years later, you could do it at home, meaning in a superconducting magnet, but not at the time.
ASHOORI: - Bob--
AUDIENCE: Of course, there are two prototypes I know about for the 1/2. And they're distinguished, so you allegedly can find the edge properties.
STORMER: I what?
STORMER: Edge state?
AUDIENCE: Are you guys currently working on--
STORMER: No. This is-- edge states are always the next generation. I mean, you first have to get the bulk. Let me not call it bulk. You first have to get whatever we are doing, the trivial stuff, right before you look into the edge states. Tunneling into the edge state is a difficult problem.
You know, I mean, it's still-- there's lots of question about what you really see when you chiral systems, et cetera. So I don't have anything. Maybe Dan is going to tell you something. But I think-- no, we don't have anything on the edge.
You know, the other thing to differentiate the two states is whether it's been polarized or unpolarized. And that's already a difficult one. That's the experiment that we wanted to do since a year. But it's so hard to turn a sample at 4 millikelvin electron temperature down in a high magnetic field and see whether one sees the lifting of the spin degeneracy.
That, I think, is a trivial experiment. Although, it's very hard. And then comes the edge.
ASHOORI: Why do they call the 5/2 state a superconductor, when the longitudinal conductance doesn't-- it goes to 0? So it's that--
STORMER: You know why. I mean, it's-- the analogy with superconductivity is a long one. There's lots of things that are different. But since it looks like-- it has sort of the character of BCS wave function in there, so you can look at it as being superconductivity. But then, of course, everything else is different. Don't catch me at my weak points.
AUDIENCE: [INAUDIBLE] Is there anybody else you can think of who's sitting here?
STORMER: I don't think so. I don't think so. I mean, there is a description of these states here, a theoretical description. I think all the connections are not yet made. In fact, it was theoretically predicted that there was stripe phases.
Let's also keep in mind there's a prediction of stripe phases-- something. And then there are experiments that are shown anisotropy. So far, the connection that the anisotropy is due to stripe phases had not really been made.
We just think this is the area where there is anisotropy. This is the area where there should be stripe phase. Probably, that's what it is. One doesn't really know that. What has been discovered is anisotropy. But it's probably stripe phases. From point of theory, I think we got the experts in here. What?
AUDIENCE: Almost certainly yes.
STORMER: OK. This is one-- Patrick?
ASHOORI: So we sit back at this time in the Magnet Lab when you came up with Horst first law that I teach my students all the time.
STORMER: Horst first law? I haven't heard about this.
ASHOORI: That crummiest looking samples always work best.
STORMER: Oh. Yeah, I guess I learned this from Dan in many ways.
ASHOORI: OK. If there are no more questions, let's thank Horst again.
LEE: Well, it is a real pleasure to welcome Dan Tsui to MIT. Dan, soon after he got his PhD from Chicago then went to Bell Labs, where he spent about 15 years before he went to Princeton University as a professor in the electrical engineering department.
I got to know Dan in the '70s when we were both in Bell Labs. And, as some of you might know, especially in those days, Bell Labs was known for having a bunch of very aggressive people around. Well, there are a few exceptions to this rule, and Dan is one of them.
But Dan may not be the one who walks the fastest or talks the fastest, but when it comes to matters that count, it doesn't take long for one to figure out that he's usually the one that gets there first. I had the experience, for example, in the late '70s when a lot of us were doing work on the localization, weak localization effects, and disorder effects, and so on.
Dan was really very quick. He instantly understood the importance of what's going on, understood the theory. And he realized immediately that the MOSFET system is the ideal system to study these new theories, because you can control the electron density and everything else.
So within a matter of a month or two, he and David Bishop came up with the experiment, a definitive experiment that showed all these weak localization effects, and so on. And, of course, the next great achievement was the fractional quantum Hall effect that we'll hear about.
Since that time, I came to MIT in '82. And one of the great pleasures is to have periodic visits from Dan when he comes to do experiments at the Magnet Lab. And every time he would come and he would show me all these new data that he found that he's excited about that he wants to understand.
But every time I talk to Dan, I always come away with the impression that he's someone who is really fascinated by nature, by what he's doing. He loves being at the frontier and exploring this new world. And he is someone who really wants to understand what he's studying.
And to him, getting credit, or being the first, or getting honor seems to be secondary. These are things that would come naturally for him. Science is really what drives them. And so it's wonderful that this great honor has come to someone who really does not seek it. And so I present to you Dan.
TSUI: Thank you, Patrick. It is an honor and always a pleasure to be back on MIT campus. I think Horst probably would say-- well, I enjoyed it so much, I even spent my first sabbatical in the Magnet Lab, in the Francis Bitter Magnet Lab, which-- which I wanted to, for the talk, is to take this opportunity to have a closer look of some of the longstanding problems as characterized by some of the questions-- some of the questions and the comments we used to hear very often.
So more specifically, what I'm going to do is I'm going to give you a very brief superficial overview, basically relation the overall, the other even-denominator fractions, which Horst already touched upon. But before I go into any detail, into those experiments, I'm going to spend some time to, how you say, tunneling experiment into the sample edges.
And I don't think I'll-- or Richard and I thought I'll also go through some of the microwave absorption experiments in the very small fractional limit, where the system, the electrons are insulating, are true insulators.
And if you like, it's a belief-- well, it is a punitive linear crystal regime. But I probably will not have time to go into that. So this is what I would like to do.
Let me-- I have this view graph here just to remind me to say that in our modern day electronics world, and mostly, device functions are performed by these two-dimensional electrons, which confine to move along the interface of two different semiconductors, or a semiconductor and an insulator.
But the only point I want to make here is under normal ambience, these electrons behaves as just ordinary GaAs of two-dimensional particles. The strange, or the extraordinary physics, which Horst has shown us becomes dominant, or very apparent, when you take a system at low temperatures exposed to high magnetic field, and when the system actually is clean. It's a clean two-dimensional system.
That's the point. But the other point I want to make is, really, the only reason I want to show that view graph before-- actually, let me put it back. Because in the magnetic field, the energy, or the so-called Landau quantization quantizes the energy spectrum into discrete levels. If you like, they're massively degenerate discrete levels. And these levels are called Landau levels.
And the point I want to make here is mainly the terminology I want to keep straight. So the Landau level index, this big N, is usually integers 0, 1, 2, 3, large numbers, you know, whatever numbers. So when you take the electron spin into account, each Landau level composed-- or has two spin levels. That's the only point I want to make, because what we need to keep in mind to look at some of the data in some detail. OK.
We have already heard from Horst's talk that in the so-called fractional quantum Hall effect regime, correlation is everything. And the-- for the odd denominator fractions-- which we know the systems go into so-called Hall liquids. At the odd-denominator quantum Hall liquids, we have these fractional states, you have seen.
But also, we have seen there is the even-denominator fractions. Correlation favors normal metal-like Fermi sea of composite particles, or the composite Fermions. And this is the case for the lowest Landau level, if you like, the n equal to 0 Landau level.
And this is the data, actually from very recent sample, Loren Pfeiffer and Ken West grow out in September. And the data was taken by Wei Pan, who is graduate student Horst and I supervised. And what you notice here, the filling factor 2 is somewhere here, and down.
So this is for the n equal to 0 Landau level. So there will be two spin-- if a spin split, it would spin up down here somewhere, and have the spin down here. OK. So in this level, indeed, the composite fermion is what the ground state for even denominator fractions is indicated here.
And, for example, the 1/2, you have already heard from-- Horst already showed to you, except in this sample, because the perfection of the 2D electron system, this [INAUDIBLE], I think Horst said is, denominator 21, or 20 whatever. And here is similarly for 1/4.
And the main thing is for the even fraction where the composite fermion is the ground state, the structure is essentially featureless and is flanked by two series. In this case, it's the p over for p plus minus series of fractional quantum Hall states. And here is a p over 2p plus minus 1 series of fractional Hall states. And similarly for the 3/4 and 3/2. So for the n equal to 0 Landau level, indeed, the odd-denominator fractions are the fractional quantum Hall liquids. The even-denominator fractions are these composite fermion states.
For the higher Landau level-- that's what I want to show here-- and that, apparently, is not the case. The data looks very different. Again, here is the lowest Landau level n equal to 0. You see that the even-denominator fraction is essentially featureless. Just the fractional quantum Hall states converges unto this. The fractional quantum Hall state is this deep minimum and just convergence into this even composite fermion, the ground state, the composite fermion gas state, essentially featureless.
But when you go to the next Landau level, and you go to one level-- that's for the 7/2 and 5/2-- and you have deep minimum, which is very similar to the fractional quantum states. And also, these structures, you know, on the flanks, on the sides, those are indeed the fractional quantum Hall states. OK.
Now, when you go to the next higher level, the index is-- and you could do 2. And you actually see are large spikes at the 9/2, 11/2. And similar for the next level, and you could do 3, 13/2, 15/2. These actually have large peaks.
But more importantly, whether they are a peak or a minimum, a very deep minimum, depends on how you do the measurement, depend on the measurement. The current is measured in what direction? The black's measured along this direction, defined as x direction, and the other one is the y direction.
So this is, basically, what-- well, I'm going to do is I'll come back-- I'll come back to show you more data, look at these higher Landau level. And you go to 1, and you go to 2. And pick-- in particular, just pick out the 5/2 and 9/2 to go through the experiment in some detail.
But before I do that, I want to address, basically, the other comments you have seen about edge states. Halperin and Wen-- so we got one probably sitting somewhere in the audience, I know-- were the first told us how to think about the sample edges, how to think about the edge states.
In case of the integer quantum Hall effect is quite easy to visualize or to picture them. And that's what is illustrated on this view graph. If this is the bulk material, in a strong magnetic field, we look at the 1 electron energy levels, they are just these Landau levels separated by an energy gap, which is just the cyclotomic energy.
Now, when I go to the surface, approaching the surface, sample surface, then when the electron orbit in the magnet field, the electron feels the surface potential, or the surface feel. Then it changes. It increases the electron energy. So why would my Landau level would have to be drawn in this way?
So now if I am at the integer quantum Hall states-- so my Fermi level is somewhere between these two levels, below are all filled. Top, my n, below n is all filled. My n plus 1 is all empty. Then, of course, inside the sample in the bulk is an insulator gapped.
But when come to the surface, there is a state right at the Fermi level. And these states, or if you like, the electrons in these states are one one-dimensional electron system. And this becomes more clear in the next view graph.
In the real space, this is how it would look. This, of course, you would recognize this. This is probably some graduate student's thesis. Who else would do this neat think? This is taken from Matt Grayson's thesis. As a matter of fact, the data going to show are all from him.
So in the bulk in quantum Hall effect, the levels in all these side for orbits are all occupied. And the next higher energy level ones are empty. It's not shown here.
But come to the edges, the edge states are just the so-called, in metaphysics in the '60s, we called them the skipping orbits. These are the skipping orbits. And because of the magnetic field, they all skip in one direction.
And the language, the proper language to describe, a fancier language, they're a motion of chiral. They're moving in one direction. And this is for the integer case. And there has been a lot of experiment evidence to this.
For the fractional case, it's-- for the fractional case, of course, again, the bulk-- in the bulk, it will be gapped. The charge of excitation will have a gap.
And the edges-- the electrons and edges will count to, again, a one-dimensional electronic system. But it's an interacting system. And it's well known that for one-dimensional interacting electron system, the ground state is a liquid. And usually it's called a Luttinger liquid, again, because the magnetic field is called the Chiral Luttinger liquid.
And the main characteristic of a Luttinger liquid is it has phonon like excitation spectrum. So the density states, the other Fermi lab, if you like, vanishes. But edge, given the finite energy excitation, the density of states, because it's over some power law dependence, as indicated here.
And this would have direct consequences on the current due to tunneling either into the edge or tunneling between the edges. So if you look at the current, with a tunneling current into the edge, fractional edge, or the tunneling current between the fractional edges, this will have direct influence, direct consequences. OK.
The earlier experiment on this is the Milliken, Umbach, Webb experiment where they look at the temperature dependence of the tunneling current between the edges using a point contact geometry. But what I'll do, basically showing the data Matt Grayson took, Matt Grayson has, is following the tunneling, which is tunneling from, essentially, a normal metal into the edge, which in some sense-- many of you-- I'll show a view where it will become very clear. If you just remember the normal metal superconductor geometry in superconductivity.
This is just repeat what I said already. So what I'm going to show is the experiment Matt Grayson-- Matt Grayson's data, and using the sample Loren Pfeiffer and Ken West grown. This is the edge cleaved overgrowth Ray Ashoori mentioned earlier.
The structure at the end is you have the 2 dimensional electron system on this side. And the 2D electron gas is confined in a quantum well indicated here. Then from the edge cleave overgrowth a tunnel barrier is grown, which is in this case, be the aluminum gallium arsenide tunnel barrier.
Then very heavily doped semiconductors electron and plus get it out, which is just a normal metal electron. So the experiment is to inject our tunnel electrons into the fractional quantum Hall system. That's the experiment.
And if you remember from the normal metal superconductor tunnel junction, the classical [INAUDIBLE] tunneling geometry, which your measurements are in a tunnel inject electron into this interacting system, the one-dimensional Luttinger liquid system. And if you look at the so-called tunnel conductance, derivative of the curve, you measure directly the tunneling conductance.
So the experiment is just to measure the IV characteristics, the IV curve. And this is just another view what you have. This is for the 1/3 stage, the bulk. Charging stations gapped. And the edge, or the actions on the edge, that's the one-dimensional Luttinger liquid. And the experiment is basically just measuring the IV characteristic. You expect a power law dependence. And the exponent alpha is basically, well, if you see the right-- that's basically characterized the edge Luttinger liquid.
Well, I think I have another picture. This is a more realistic picture from Matt Grayson. This is a 2D electron layer. So you can put contacts so you can characterize the fractional quantum Hall characteristics used to characterize your sample. And here is the tunneling the normal electrode injecting into that. OK.
So you can get look at the-- red curve is the IV characteristic, the Hall resistance characteristic of the 2D electron. And this is the 1/3 plateau. This the 2/5. This is the 2/3. This is the plateau 1.
And this is the tunneling IV. And if you look at the edge, add the fractional 1/3 state, you would expect the Chiral Luttinger liquid-- expect to have a Chiral Luttinger liquid. And the IV, you should have-- you expect the power law dependence with an exponent. It's expected actually to be for 3.
This is the first piece of data from Albert Chang's early experiment. And this is right at filling factor 1/3. We know the bulk is gapped. And the edge, we expect to have the Luttinger liquid. And there is more than two decades as the power law, clear power law dependence indicator. And the power is 2.7 is sufficiently close to the expected value of 3.
Which Matt Grayson as he looked at a large number of samples. On each sample, not only at the 1/3 or 2/3, also away from those at a large-- you know, just continued a various magnetic field, very close grid look at it. How do they look?
And surprisingly, when even you're not at 1/3, I think it's great. When you're away from the 1/3, you also have the same power law dependence. And in this wide range from 15 Tesla to 7 Tesla, you all have this power or dependence. It's a matter of fact.
This power law, this is semi-local. This is [INAUDIBLE] plot. This power law dependence would cover, basically, about four decades. I think in a range of certainly unambiguous.
And in the next few view graphs, I just flip them very quickly, it just the fit we made. The fit is to a theory-- well, you can even just-- you can use just a resistor network the same way. This fits with [INAUDIBLE] theory.
This is right on 1/3. His fit gets 2.8, where Albert Chang had earlier, 2.7. And this is the higher field. And the fit, again, this is about four or five decades. It's about 4.2, that fit.
And again, you're away from the-- I'll just very quickly flip to give you some sense, the fit. This is probably about three decades or so. And that give a power law about 2.5. And this one probably narrower range, but still-- no, it's not. It's still about four decades, about four.
So I just want to mention-- what I want to show is, if you take all his data from the power law dependence, the exponent is alpha. And all the detail plotted up, this is from these four samples. These circles are Albert Chang's early data. They essentially all fall roughly in a straight line.
I mean, the immediate first surprise-- it's surprising as-- when you're away from the plateau region-- away from the fractional quantum Hall states, the bulk is not kept. The bulk has conduction. But the edge state, the Luttinger liquid picture, the edge mote, which are Luttinger liquids are still just proper description of it.
And the other one, of course, that dashed curve is a theory from Shytov, Levitov, and Halperin. And this discrepancy between experiment and theory I think still remains a problem.
The other point I want to just mention that for the filling factor one and higher in this region is expected-- the edge mold, channels are Fermi liquid. And the exponent is just your normal IV curve, normal ohmic, and it is expected to be 1.
More recently, Michael Hilke, who was a post-doc at Princeton, looking into between basically between one filling factor-- well, one somewhere here just exploded out more data. And in data, this is flatten out 1 occurs of earlier, but indeed continues.
But his sample, or the sample he looked at, the data to date is basically is not good enough to say anything about what about the composite fermion ground state at 3/2 or the 5/2 at those particular fittings. That basically is all I want to say in relation to tunneling into the edges of the fractional quantum Hall effects.
Now, I would like to come back to the even denominator fractions in higher Landau levels, Landau levels for the Landau index N equal to 1 and higher. This is Rui Du's data on samples actually Loren Pfeiffer and Ken West grown in, I think, it's in 1991 this data, or early '92, or so.
But the point, I already mentioned, if I look at the 5/2, there's clear minimum. It's very different from the 5/2, very different from what you've seen, 1/2 and 1/4 for the composite fermion signature in the data.
But more strikingly is when you go to 9/2 and 11/2, which is for the Landau level index n equal to 2 for the 9/2, 11/2, and 13/2. And they give a sharp, very large resistance peak, very strong resistance peak. And it depends on the current orientation which direction you measure the resistance.
In the 90 degree off resistance direction, it's a deep minimum. And in order to be able to go through the data later more, it's kind of confusing once told the magnetic field. I want you to keep in mind two things. One is these lines, these vertical lines indicates the easy conduction direction.
If you measure the current that direction, in this case, the resistance does not have resistance small. Perpendicular to that direction, it's a hard conduction direction. So this is measured at perpendicular.
And the hard direction is x direction. And the easy direction is the y direction. So that's one point I wanted you to keep in mind.
The other point is that in addition to this sharp peak there are also other structures. And these structures, they don't fit the fractional quantum Hall scheme. They don't fit this 1/2 or 2/3 or 2/5.
And when the temperature goes-- you know, when you lower the temperature as indicated here, and everything disappear, just this spike, for this current direction, this spike dominates. And in addition to that, because I want to mention, this is a sample in almost a decade ago.
And the samples grown in those days are not rotated. So you can't have some very density gradient, some very small. In this sample, the density gradient is sort of about or less than 1% in this direction.
But I think if we look at it very carefully, it's very convincing this anisotropy difference cannot be attributed to this slight density grating. But I am going to show you the data on the next view graph from Jim Eisenstein in Caltech. Those are sample were very recently grown. They're rotated, and there is no uniformity. The results is essentially the same.
But before I do that, let me just also say another point. For this sample, the plan, the sample plan is 100. Roy made an x-ray diffraction very carefully. The orientation is a better than 0.5 degree.
And the edges are 110. They're cleavage edges. You cleave them. They're just essentially right out. These are a few things I want to mention.
Keeping that in mind-- well, first let me just show you the data from Lilly et al, which are more recent samples in terms of 2D electron quality, 2D ground system quality. But the data is essentially the same. In one direction, it's large resistance, and the other direction is minimum. Simple.
And they also look at the different orientations. That data you have seen before is, essentially, the current measure along the two different 110 directions. Whereas here, the also edge 1 rotate the sample by 45 degree. And you still see some anisotropy. But it's much smaller. So this experiment actually pinned it down, located what the anisotropy direction is.
Now, with this in mind, now I want you to just stare at a few foils. We'll go through the overall. First, this one is a temperature characteristic.
Anisotropy is basically, well, 115 millikelvin it's isotropic. There's hardly any difference. The anisotropy started basically about 100 millikelvin, and it becomes stronger and stronger. That's one point.
And the other structures I've already mentioned that they don't fit any fractional Hall states. And this data curve here is the Hall resistance. The Hall resistance has structure in this region. Basically, there is some-- at the lowest temperature, I think is the solid line. They're clearer structures, but so far, we haven't been able to identify the structures for any-- clearly identify the physics.
The next view graph is a summary of this one. Just plot the temperature dependence. This is for the 9/2 maximum and minimum. They are same isotropic, about 100 millikelvin. By about 60 millikelvin, they essentially saturate the growth.
And this is just plot the structure on the side. It does not really fit to any fractional state on the side, which is for these two different orientation. For two different current orientations. That's basically the data and the futures I want to show you.
The application of a in-play magnetic field has a very large effect on this anisotropy, thus indicated here. I probably should-- this is one. Let me look at it first. It's indicated here.
I apply a magnetic field. This is implant field. I apply a field in the plane of the sample. And this is the implant field.
If the implant field is applied in the difficult-- this is in the easy conduction direction. That's the y direction. If I apply it in the easy conduction direction, it suppresses the resistance in the hard conduction direction. In the hard direction, there is a 9/2 large become small, small, very small.
At the same time, the easy conduction direction, the resistance getting larger, larger, then getting very large. If you like this, put a field, put an implant field in the easy now changes the conduction direction to the hard direction. And the hard conduction direction change to the easy one.
I know it's confusing as well. What basically I'm trying to say-- I am trying to say what the data tells you tells you this. An implant field makes the transport along it hard. It makes the transport for along the implant field direction hard.
If you put a large implant field, at the end of the day, the easy conduction direction is perpendicular to the implant field. That's really all I'm going to say. Yeah?
AUDIENCE: I was wondering, does it go back? When you take it off-- the in-plane field off-- does it go back to the way it was before?
TSUI: Right. There are no histories. It doesn't go back. And this is consistent. In some sense, once you look at it this one, which would make that conclusion even more clear. See, this of course you put a few already in the hard conduction direction.
So in some sense the things already align in the right direction. Actually, you don't change anything. But in between, it changes things. So at the end, when you put a field in the plant, the implant field orients the system in such a way at the end the easy conduction direction is perpendicular to the implant field.
That's the message I want you to take away. That's really it. That's what the experiment is. The gross part of the experiment, that's what it is. OK.
Now, the experiment, of course, has much more detail. All I'm saying at the end, so at the end, if you start out with a line in the right direction, the implant field is perpendicular to the easy one. At the end, nothing changes, just integer at some point.
And if it's the wrong direction, they switch. Easy becomes hard. Hard becomes easy. In between, they're definitely more than that. And later, in some sense, you can feel, well, the properties are reasonable.
You put an implant field. Not only you do some kind of anisotropy related thing, but you also change the Zeeman energy, all kinds of things. But the main thing at this stage I want it us to keep in mind is that this is just the same thing, this plot in anisotropy, the difference of the resistance the two direction normalized by the sum.
But the message, really, is that you put an implant field. And that field forces the system into such a condition the easy conduction direction is perpendicular to the magnetic field.
Our current understanding of it or at least that's what I hear people telling me is the so-called stripe phase. The understanding of it has to go back to Shklovskii and his collaborators earlier his paper and also Moessner and Chalker on the Hartree-Fock calculations for the ground state of 2D electron system in a magnetic field.
What they find is they find for Landau index bigger and equal than 2 and the charge unidirectional. This u stands for unidirectional. Charge of density waves state is favored over the fractional quantum Hall state.
I have sort of a cartoon of a picture here just try to give you some flavor of what all those big words mean. In some sense what it means essentially is that a possibility when you go to when you're at 9/2 filling, the system face separates.
It separates filling factor 4 whole states and for that in fact are five Hall liquids, four five for these stripes. So if I look at the 2D density. And basically this was a density wave with this as my periodicity. That's really the only point I tried to convey with this
Very recently, I think must be too late. I got lazy. I didn't write. It's Jungwirth. I think it's MacDonald, Girvin, and Smrcka. They have calculated the anisotropy energy when you put it in plain and they find this they implemented a few tends to align the stripes perpendicular to it.
What they're finding with say if the stress this way they playing field you put a few at the end of the day in prime you will tend to align these and anisotropy. And it's just such a line that drives perpendicular to the plane. In that sense in a sense well we just since we don't have a transport theory.
We don't really know how to go trends for us in these phase separation. I would say in this direction, I try to indicate these are the edges there's four. Four that would be a lot of four edge states. And the edge states can't there are compressible, bulk are incompressible.
So this certainly is easy interaction. It's a long stripe. So the magnetic field tends to make a stripe. For the stripe model, stripe face model with a line that perpendicular to the magnet. In that sense, it's consistent. This was the experimental observation.
So qualitative are the main features of the experiment or the experimental data is consistent with this stripe phase. For this picture, of a cross to relate to more qualitatively to the experiment and also the more detailed structures. You'll need a data transport storage.
In particular, how do you transform a charge in this kind of model across the stripes in these in the artery FARC model. There is also, in addition to the stripes, is there a source of bubble phase. So you know just a while can one attribute those additional structures in that region.
Other than that peak, we've run them on half. Can one attribute that or assigned to be the bubble phase? I think we have to wait and see.
Let me see. OK. The next part let me mention, go with it and you could wind up with a 5/2. Or as Horst has already pointed out in his talk, let me go back to this earlier data. You see that it's Horst's data also. Basically I want to remind you as for why you have same story half earlier with 1/4 essentially.
The campus of fermions signature essentially is just a very broad minimum. This is the data in 1987 which I mentioned. But if you look at a 5/2, this structure the minimum is much stronger than was expected.
The more detailed result, the more detail expressed that it was. On this view graph, when you change the temperature from hundreds Kelvin to the lowest about 25 millikelvin. This middle amount becomes very strong. It looks like a-- it certainly is not very different from the natural 1/2 or 3/2.
And when you look at the Hall resistance, that's this data here. It clearly shows clear deviation from the classical Hall. And this certainly strongly suggests the 5/2. Or the 5/2 could be or maybe a fractional quantum Hall.
The more recent data you have seen already Horst has shown. Those are the dirty that are using the microkelvin in Gainesville that you can fit. It's already mentioned somebody pine and shot and Dwight Adams microkelvin lab.
And the experiment here is 5/2 clearly shows you are indeed cooling the electron temperature. So the electron temperature is very, very clear. That's one point but you are not at all on that.
Let me just mention to you once you could blow the time to run at the lowest temperature indicator here. And there is a very clear plateau quantized to the quantum number 5/2. Just essentially using the sort of calibration.
And as you calibrate against something else, this is quantized actually to better than 2 ppm. The limitation is, as you know, really didn't it just to install know instrumentation was designed for that or quantizations.
So this clearly says the 5/2 stage is indeed a fractional quantum Hall state. Indeed, at incompressible fluids that are pressing, there are two candidates. Or at the present, or I should say, through all the years, two candidates for the ground state.
What is the ground state? One is the single state of what usually is called Haldane-Rezayi hollow core type. It's basically the interaction. Potential has this particular hollow core model. One is this not.
And the other one is the Pfaffian state. Whenever you have so many of you. But what was already mentioned in the Pfaffian, which is the p-wave pairing of the composite Fermi compounds, or is one to some detail of the BCS type of candidates.
The early support for the experimental support evidence or-- the support from the experiment in the early days for this singular state is the total magnetic field experiment. See, one way to view the total magnetic field is since the electron system is two dimensional indicators here you can increase the total magnetic field without changing. Or put it away, keep the perpendicular field constant and still increase the total field.
Of course, the fractional quantum Hall correlation. The correlation give you rise to the fractional quantum Hall state in the polarized. In the simplest case, it's just so that this is the given by that, or the simplest, it's the perpendicular magnetic field. Whereas the Zeeman energy is related to the total magnetic field.
So by totaling the magnetic field, essentially, you keep the perpendicular field constant. You can-- one of the simplest ways to view this, the only thing you do nothing to the system, except you're changing Zeeman energy. That would be the simplest way to view the total field.
So if you do that, view that way and do the total magnetic field experiment, this is the earlier data from Jim Eisenstein and Bob Willard. This is a 0 magnetic field, very strong about 5/2. When you total magnetic field, that meant a decrease.
As you increase, when you total, you increase the total field. Basically, this happened a total field earlier. Or so for some test like now is five some Tesla. But the perpendicular view is the at this point. It's 4.8 Tesla.
And by the time the pearl field, or Zeeman energy increased about 5.7, it's very weak. By that time, more than 7 Tesla for the total field are the amount of energy, then this 5/2 structure disappear. Or the 5/2 status quenched.
This in some sense, was the strongest support for the single stage. But very quickly there, the more recent experiment makes this. It certainly shows there is a problem with this.
Now, this was the simplest. One is shown by this data here the state at the 5/2 is still equally strong and appeared to bow through. And I have tested almost 13 Tesla. That's very large, say, about an inch.
So that's one problem. I think it are more definite problems of this sort of suggests maybe the singular state there is something not right. And so the other problem came in some sense became more clear, became clearer, more recently. That's the total few experiment you have already seen the total few related to that 9/2 earlier.
Now, it's essentially the same experiment, the same type of experiment. I just looked at the data at 5/2. What do you do if you look at the 5/2? In this case, the in plain field is put along the hard conduction direction.
The hard conduction direction is for the 9/2, for 5/2 here. This is isotropic anyway. So that but no you put it in for you once you put it in plain view. Then you see that become an isotropic. Then depending on what direction you've eventually in the plane field is sufficiently large.
So you have one direction becomes hard to conduction. The other direction is the easy conduction. This is what the implant field in the hard conduction direction.
Put an implant field in the other direction, again, it's the same thing. At the end when the field is sufficiently large that you get because the system becomes an isotropic 5/2. In the easy conduction, for the 9/2 easy conduction direction becomes larger. And it is difficult to come down the direct direction it's easier.
So the main message here on this is you put it an implant field. Or do a total experiment. But keep on tilt toward or even higher than you'll find now of an isotrope.
So what you have seen earlier just the suppression of the minimum. That's only part of the story. So after you keep on totaling, it's not only suppress. The orient disappear. But if you in the right direction are actually in them, to measure it in the right direction you get a maximum.
That's really the main message. The other is just put these data since I have all the data. So plot it up. The really the main message.
And somewhere here, or this say like in this-- let me just not even bother. That's just the plot. But what I want to show you is this is the data from Lilly and Eisenstein from Caltech.
It's the same result as it put it here. You produce a strong and isotropic transport. And you find what you put in making it a feat. So what the experiment says, well for this experiment tell us this that dominate a third of the total experiment, the dominant effect. It's not the same as the dominant effect is in plane field is induce anisotropy.
It's stated this anisotropy energy we see in something like this, the hard to find calculations. So the dominant effect is in plane view induced anisotropy. Not as a Zeeman energy. So the imprint field of what it does destroys the-- it destroys the 5/2 fraction of whole states.
And in its place, it creates a-- if we use the 9/2 state, or if we believe the 9/2 state, is that a rational charge density wave state grants that. Then it creates a unidirectional charge of density. We've stated that's the stripe phase.
And the experiment you see the total few experiment which really is a phase transition, phase transition from this Pfaffian state or p-wave paired state of the BCS type to stripe phase. And this transition is driven by when you apply this in plain magnetic field.
There is a pre-print. I think I have the-- there are two pre-prints. One has three names. One, two I think is the one that Rezzayi and Haldane. Their calculation is a finite number, a certain number electron calculation. Their calculations sort of kind of showed that for the spin polarized system and, essentially, the far field state energy.
And stripe phase energy are very close together. So there we see if you can change the interaction potential. Some more it's very easy to change it from one to the other. That's right.
I think my time is up. I have already said hard to see anything of what. Let me just read this these few words, then I'll stop. OK. Which basically I was going to-- I was going to say this. The picture that has emerged from a more recent experiment is that the states of a clean-- I sort of emphasize a clean two-dimension electron system is magnetic field.
At fractional fillings for the Landau levels 1 and 0, is the first four if you count the spin levels. Landau for filling factor is bigger than-- roughly bigger than 1/6. You have a fractional quantum Hall liquids at the odd denominator fractions.
And the even denominator fractions, you have is composite fermion guess. If a composite fermion liquid in the lowest Landau case. And you go to 1 for 5/2. Which you have very likely, I think. Sort of it's p-wave paired.
Then for n bigger than 2, the higher levels it appears I think the current data sort of indicates Hartree-Fock ground states has a lower energy than the fractional quantum Hall states. I will not have time to say anything about the winner crystal.
But it's believed in the clean system for the filling factor, less than 1/6 there's no crystal. But in real life, of course, the physical phenomena we see are always in play. The results of it interplay between this order. And electron-electron interaction.
It turns out this crystal case for that regime in the water, it becomes very important. And this order at the present certainly is one of the most difficult problems. And we really don't know how to characterize it. I better stop here. Thank you.
LEE: Thank you, Dan, for the very current update on this exciting story. Obviously, it's still ongoing. There may be time for just quick questions from the audience. Since we are running late, maybe we should just move on. Let's thank Dan again.
KASTNER: So now I'm going to give John Joannopolous the joy of introducing Bob Laughlin.
JOANNOPOULOS: Thank you, Marc. It is, indeed, with particular pleasure and pride that I introduce our next speaker, Bob Laughlin. As I'm sure many of you know, Bob's talents his brilliance and creativity as a theorist have by now become legendary.
But what you may not know is that some of these talents actually overflow into other areas, such as music, and art. It turns out that Bob is an accomplished pianist. He is also a composer, a composer of classical music. He's written for piano concertos.
He's also a skillful sketch artist and a very clever cartoonist. You can see-- you can find some of his work on the web on his home page. Actually, some of my favorite cartoons are jibes at the National Science Foundation, which Bob calls the National Seance Foundation.
And in the first two talks, we saw some very beautiful and pioneering experiments led to the discovery of a new two-dimensional electron-electron system with some very unusual properties. And, of course, now the question was, how can one account for these experiments? How can one understand what the nature of this system is?
Now, interestingly, historically, fascination with two-dimensional fluids actually dates back to antiquity. The first written record that we know of appears in Babylonian cuneiform about 4,000 years ago, about the times of Hammurabi, and describes a kind of divination known as lecanomancy where the diviner, or the baru, as he was called, would drop little droplets of oil into a bowl of water and then would very, very carefully study the nature and intricacies of this expanding film across the surface.
And in doing so, there by prophecy the fortunes of kings. And so it came to pass that it would take a modern day baru, Bob Laughlin, to decipher the nature of the intricacies of this novel two-dimensional electron system and lead to, really, what is a truly beautiful theory of a new state of matter, a novel quantum fluid with the fractional charge quasi particles. And so please join me in giving Bob a very hearty welcome back home.
LAUGHLIN: It's always a pleasure to be back here at the place where they number the buildings. I was many times in this wonderful lecture hall not listening to physics, listening to music mostly. MIT's music program is, obviously, very special.
I remember vividly hearing Murray Gell-Mann here confuse us all about quarks. It was at MIT that I first heard Ray Dingle talk about heterostructures and all the amazing things they could do, demonstrating elementary quantum mechanics. I later heard some great stories about Ray from Horst, and I hope any of you that put enough liquor in him will get those stories out.
Now, I apologize for this lengthy wordy thing. But I have a point to make this afternoon that is summarized succinctly in it. So it is about my story of why the fractional quantum Hall effect is important.
It's fascinating for lots of reasons, but it's important in my view because it shows experimentally-- this is the key word experimentally-- that both particles carrying a simple elementary fraction of the electric charge of the electron and interacting by gauge forces not postulated in the underlying equations of motion can arise in an emergent manner as collective effects from ordinary particles obeying the ordinary laws of quantum mechanics.
Now, both particles carrying fractional quantum numbers and powerful gauge fields between them are key postulates of the standard model of elementary particles, something that is allegedly fundamental in our universe. And as I'm going to claim this afternoon the importance of these experiments is that they impugn this idea. They suggest that maybe such things in the universe itself could be emergent, which is to say. Could not necessarily be fundamental at all but just collective phenomenon.
Frank Wilczek has a wonderful name for this. He calls it solid state imperialism. And I am the worst of the solid state imperialists. At least, I hope so.
Now, on that note, let me start off with a thing that's been a lot in the news lately, which is the theory of everything. Now, this is not the theory of everything, because there is no light in here. And there is no nuclear physics in here. And there is no gravity in here. And there are no weak interactions in here and so forth.
But for chemistry and the world of physics at people scales, this might as well be the theory of everything. And what I mean by everything is shown down here. Now, look, everybody, I apologize. I made this view graph in Stockholm the day after we just had a press conference that was completely monopolized by Professor Sin who won the conops prize.
And he had, of course, done fundamental work in welfare policy. And you can imagine that nothing sells better in Sweden than talks about welfare policy. And so I was kind of-- I was feeling my oats here when I finished out the things that the theory of everything could describe. Notice what I meant by Ebola virus.
Now, one of the things that these are-- by the way, those of you who are not physicists, what is this? This is the elementary Schrodinger equation for nonrelativistic matter. And H is the equation of motion for this matter. It consists of kinetic energy, of electrons, kinetic energy, of ions, coulombic attraction between them, and also coulombic repulsions between pairs of light particles. And that's it.
Now, one of the things that this theory of everything describes, it is alleged, is this experiment. This is not von Klitzing's original quantum Hall experiment, but it's conceptually the same. So let's talk about it as though it were.
So let me remind everybody that the way the experiments work is electric current is forced along a sample that's typically about a millimeter, or even several millimeters in size. In fact, this distance here was one millimeter. In the presence of a big magnetic field pointing out of the board, then one measures two voltages this one between this pad and the this one between those two pads. That's the wiggly line here. And also between this pad and that one, and that's the line with the steps.
And those voltages are measured as a function of field strength, magnetic field strength. And what the effect is that at certain values of the magnetic field are more precisely certain ranges of it. The potential drop in the direction of the current is going dies away to nothing.
And at the same time, the whole conductance which is across the current path becomes constant and what Klaus discovered was that the value of this constant, which you can measure accurately because it's constant, is a related to the current being forced by a hollow resistance that's multiples of elementary constants. In this case, the electric charge of the electron and plugs constant.
Now, as of this moment, the accuracy of this experiment is about apart in 10 to the 8, slightly less. And I ask you all to ponder how accuracy that big could come out of these equations. You see, everybody who's solid studying quantum physics knows that computing things like air, water, and fire from this underlying equation of everything is a ludicrous undertaking. It's too hard. The equations here cannot really be solved accurately when the number of atoms exceeds about 10.
Accurately means apart in 10 to the 4, maybe. So beyond that, it isn't possible to solve first principles a solution. So when we get elementary constants, Planck's constant and e coming out of a measurement with 10 to the 20th particles in it, we have to pause and ask, why.
Why is it so accurate? Certainly, you can't deduce this starting from the underlying equations. That's too hard. Now, you may or may not know accuracy of this kind occurs all the time-- all the time in physics.
The first examples actually come from chemistry. The ones I like to cite, electrochemistry turns out to be a very accurate way of measuring the electric charge of the electron. However, the sexy physics examples mainly come from superconductivity and the Josephson effect. The Josephson quantum of magnetic flux is the most accurate quantity known in mensuration aside from the speed of light itself.
And so there is plenty of precedent in the physics literature for many, many particles giving rise to accurate measurements of elementary constants. And then all those cases we know the reason for this accuracy is higher organizing principles the matter itself is acting in some collective way that you cannot deduce from these equations. You can't go deductively from these equations to this behavior, because the accuracy isn't there.
So the intellectual problem that's posed by questions like this is, what is the principal? Why are these experiments so accurate? Now, a part and 10 to the 8, you begin to think about this a little bit. And you see that, well, the theory of everything that I showed you actually has corrections.
The biggest one is here, as all practical purposes, that spin orbit coupling and things of that sort in this material would be parts in 10 the the 4 effect. The radiation field actually is bigger, but it's subsumed by normalization effects in QED.
However, once we begin to worry about these things, then we begin to worry about all the other possible corrections to the experiment. How accurate can it get? What is limiting its accuracy, if anything?
Now, there is an idea that is well known to people who worry a lot about the quantum mechanics of empty space. And that is that empty space isn't empty. Empty space is full of stuff. That is, the sum total of the accelerator experiments that have been done since the Second World War have told us that space is a lot more like a piece of glass then it is ideal Newtonian vacuum.
It's filled with stuff. And so when I talk by analogy between issues we have in the vacuum of space, and issues that happen in solids-- in fact, this analogy is apt. Empty space is not empty in exactly the same way that a solid is not empty.
So all these corrections to the laws of physics in ordinary space matter, at least in principle, they matter. And one of the questions we have to ask is, why don't they show up in mensuration? Why do all these-- why are all these little corrections to the underlying equations irrelevant?
Now, the answer, of course, is there's a principle involved. And in order to talk about this principle, I have to remind you all of something you know, or teach you it if you don't. And that is something we call the principle of adiabatic continuity.
So here's the problem. Suppose we've got something we want to understand, and it's very complicated like, for example, the Hall conductance, but we know there's a principle involved that makes it exact. So we can imagine a thought experiment where we perturb the equations of motion slowly and interrelate between lambda equals 0, which is the thing we're studying and lambda equals 1, which is some idealized limit to the problem.
Then, if it turns out the thing we're measuring is constant, as I adiabatically deform the equations of motion, then I can understand what's going on over here by studying what's going on over here. In solid state physics, we do this all the time without calling it this fancy name. We call it modeling.
But I have to be a little careful now, because the thing we're trying to understand is an accuracy. And so we can't be cavalier about modeling and have to really think hard about what continuation principle allows us to understand something exact in terms of a model. Now, the model that we-- like, I would say, let's say the model would be, I guess, this end of the line-- that we like to think about this problem in terms of is the one shown here.
So what this is is ordinary nonrelativistic electrons feeling a magnetic field B and also moving in what we call a dirt potential, or random potential. Now, at the risk of telling a joke that will bomb, the dirt potential is like pornography. I can't define it, but I know it when I see it.
In this case, the random potential is something that we could sketch but the truth is that, in physics, we don't know how to define around a potential. We only know how to define the behavior that you see in one.
And the behavior that you see in random potentials has a name. It's called localization. And localization is an emergent principle that we know to be true, because it's been seen in experiments many, many times.
Now, before we get to that, let's look at this random potential that turns out is crucial to having Hall plateaus. It's very easy to show that if it's missing, then you don't have them.
So suppose-- let's do that-- suppose the sample is system is translation invariant. So we have some density of particles in there and a magnetic field. Well, then you all know you can run past it, make a Lorentz transformation, then by the relativity principle, there will be in the moving frame an electric field across the sample, a current going along the sample.
And the ratio of the two after some algebra turns out to be the number of particles per square centimeter divided by the field strength times some elementary constants. So measurements like this Hall effect, measurements usually measure the particle density. And, in fact, this is a standard and rather boring experiment that one does in semiconductors to tell how many extra carriers you have.
A corollary of that idea, then, is that quantum Hall effect can't occur in samples that are perfect. Because if the sample were perfect, it would be measuring the number of carriers in there. And the number of carriers in the sample is not quantized.
Now, let me now tell you all about localization. This, unfortunately, is confused by dimensionality issues. So let me tell you the old lore back in the old days when people were thinking about three-dimensional system semiconductors.
In particular, the idea of localization was born-- it was born in the mind of Phil Anderson. The basic idea of it is that eigenstates of one electron quantum mechanics, which is what we're talking about here, come in two varieties. They are extended states and localized ones.
Extended states are contiguous from one side of the sample to the other. And they are capable of carrying electric current. Localized states are the opposite. They are not contiguous from one side of the sample to the other. And they are not capable of carrying electric current.
Now, in an actual sample, it is found experimentally that not only is this true, but there is an energy scale called the mobility edge below which all the states are localized and above which all the states are extended. And the way the system approaches these this edge is part and parcel of the metal insulator transition in disordered one electron problems.
Now, there was, in the old days lore about this, that you couldn't have localized unextended states at the same energy. Because if you did, then the tiniest amount of perturbation would mix then, and you'd have two extended states. But nobody knew whether to believe that law or not. So the way it came down eventually was that this was adopted as true, because the experiments told us it was true. Localization is a phenomenon that happens in nature that you cannot deduce from microscopics.
You can get part of the way there if you cheat and make some assumptions about what a random potential is. But the truth is, we don't do that. So the right way to look at localization is not as a theoretical ideas so much as an established experimental fact.
Now, what happens in a magnetic field? Well, at the time of this experiment, nobody knew. Nobody knew what had happened, and there were some things we didn't know, which was the random potential should have taken Landau levels and broadened them.
Landau levels are highly degenerate manifolds of states. And the first thing a perturbation does to degenerate states is it breaks the degeneracy. So everybody expected the density of states of this one electron problem to look something like this.
And, actually, I think it's fair to say, in the beginning, everybody expected everything to be localized. That it should have been localized in the tail region was completely obvious from anybody's point of view. That there should be localization in the center was very non-obvious.
The Klitzing experiments told us that it was not so, that there were extended states in the centers of these Landau bands. And the reason is because it conducted electricity. So we knew that it must do so. There were extended state bands there. But we didn't know much more about that in those days.
It was one year ago, I guess, more like two years after the publication of the famous gang of four paper predicting or asserting that localization must be complete in two dimensions. And we are, in fact, in two dimensions in this problem. So many people felt that localization should have been complete and something else must have been going on.
Now, as I said, the experiments told us otherwise. And now, in the light of hindsight, we understand that localization physics is different in magnetic fields than it is in their absence. In particular, this problem has a density of states, which actually looks like I've drawn it. It's tails of extent of localized states separated by distinct bands of extended ones.
Now, how do you get perfection out of imperfection? This is relevant to the fractional quantum Hall effect. So let me lead you down to this idea historically.
Here is this now infamous argument to me which accounts for why localization makes exact quantization of all conductance. And as Horst said, the essence of it is that the system can be made by a thought experiment into a charge meter. The thought experiment which does that is illustrated here.
So we imagine wrapping the quantum Hall system into a loop. There's a magnetic field coming out of the surface of this loop. I don't know how you do that. Just do it.
And we will imagine filling up these one electron states up to some chemical potential that is in the local state band. Now, rather than computing currents, which is hard, let us compute energies and then take the adiabatic derivative of the total energy of the system with respect to a little bit of magnetic flux that's poked through the center. This is not hard to understand. It's just Faraday's laws of induction.
When you change the flux to the center, there is some EMF that winds around the loop. And if there is current going, then it will do work on it. Why is this allowed? Well, it's allowed by the fact that the system doesn't dissipate.
So that if we are in a region of localized stage, then the conductance, which is to say the dissipative part of the response function, has to be identically 0, because localized things are insulators. And so that means that this flux can be put into the loop and out adiabatically. So it's because there's no dissipation that adiabatic derivative makes sense.
Now, while one is doing this adiabatic derivative, the localized states do nothing. And the reason for that is that since they go down, they don't go around the loop. They can't tell the difference between addition of flux and a gauge transformation. So it's exactly true that the localized states contribute nothing to the adiabatic derivative. And for that reason, I can just erase them, which I've done here.
Now, we've got a much nicer problem, which is little bands of states separated by finite gaps in which the chemical potential lies. Now, as I do this adiabatic derivative, it is not possible for a state to cross the gap adiabatically. The reason for that is that if it did, it would be coincident in energy with a localized state and, therefore, would have to be localized. So that's out.
So this little localized state region here is like a brick wall. It prevents adiabatic motion of states. And so that means when I wind the flux, the only state motion that's allowed is from one edge to the other.
Now, the trick to making this calculation simple is to make the sample big enough that the adiabatic derivative can be replaced by a differential where the denominator is a flux quantum. Doing this means that adiabatically changing slowly, slowly, slowly, stop, maps the Hamiltonian back to itself up to a gauge transformation. And this means that the system can have changed its energy only by repopulating the original states.
But that is only possible at edges. Because only at the edges are there states at the fermion energy. So the current carried is 1 over C times the change in energy, which is the number of electrons, the number of states, which are transferred from one side of the sample to the other times the chemical potential difference between the two sides divided by a flux quantum, which is the answer.
So in this view of accurate quantization, the proof, if you will, that localization physics will not change the quantum of Hall conductance, even though the number of states, is changing, as is their occupation, is that the experiment measures the charge of the electron. The electron-- the quantum of Hall conductance is quantized because the electric charge of the electron is quantized. And the quanta can be viewed in this context as the same.
Now, that's historical perspective. You can imagine my surprise, great joy, actually, when I got this pre-print from Horst and Dan announcing their discovery of the fractional quantum Hall effect. So what you see here are-- this experiment is just the same as I showed you. We're sending a current along the axis measuring voltages. The voltage drop in the direction of current flow as, shown here.
The Hall voltage between that pad and that one is shown here. And down on the left, you see Klitzing's effect. And then up here where there should have been nothing, you see a new feature developing, and a dip in the parallel inductance corresponding to it failing to get deeper as you lower the temperature.
No, no. We checked all that. We checked for nonlinearities. It looks very good. You'll see it hasn't gotten to 0 yet.
In this experiment, it's on its way, obviously, but hasn't gotten there. So I made a guess based on my knowledge of these guys and also the way the experiment was going, namely that localization physics was also happening here and that it would be complete when the temperature was lowered. Also, if you look at this line with your eye, you'll center it with your eye, you'll see it's already 1% accurate.
And that was already too much. There should have not-- there should have been no reason for it to be 1% accurate, even that accurate. So there was every reason to believe that the effect was the same as Klitzing's effect, except at the wrong fraction. The filling factor, rather than being integer, was 1/3.
Now, Horst-- here is-- the experiments got better. Here is a later one. So the guess was-- right here is a later one showing the 1/3 plateau, very well-developed. It's very, very deep minimum here. And you can notice also subsidiary fractions here at 2/5 and 3/7.
Now, as the samples improved, fractions of all sorts began popping out. And Horst likes to say, well, what a neat thing it was that good old Bob invented this equation to explain it all. But the truth is, I didn't.
And even worse than that, I was really lucky-- in fact, the world was lucky-- that samples were so lousy when the original experiment was done. Because had we known about all these other fractions, we would have been very confused. So anyway, it was a lucky break for everybody that this one just happened to be the biggest one and, therefore, the most obvious.
Now, how accurate is this experiment? Well, as I told you, you can eyeball it. The original ones were a percent. The modern error bar is about a part in 10 to the 5.
And most of us see the only impediment to getting it better being the temperatures to which you can cool the 2D electron gas. That means we have the same problem with the accuracy of this fractional plateau that we had with the integer ones. That it's already too accurate to be explained by modeling. There really must be principal involved to make it so accurate.
Unfortunately, the localization principle that we knew won't work here because any problem that can be adiabatic continued to noninteractive electrons must have integer plateaus. That's the essence of the thought experiment that I showed you. That means that this quantum state, whatever it is, is not continual to noninteracting electrons but is rather some sort of many body effect that involves the coulomb interactions.
Now, by far, the easiest theory to make right now is that this effect is the same as this one, except the thing being localized is not an electron but a piece of an electron, and elementary excitation of the problem that has fractional charge. And please note that I'm talking in the sense of the principle of adiabatic continuability.
So I'm imagining a limit of the problem where the objects, if you will, elementary excitations, become noninteractive and free. In that case, we have localization physics, just as before, but not localization of electrons. But instead, localizations of the elementary excitations in this state, whatever it is.
Now, I can't emphasize too strongly that the reason to believe this is the experiments, not the theory that I'm about to show you. The experiments tell you that there must be fractionally charged excitations in the problem because there is no other way out of accounting for the accuracy. The accuracy is the key thing. There has to be principle. The principle you have available is localization.
Localization of things deformable to electrons, free electrons, won't work as a matter of principle. So, by far, the easiest thing left is just to have a new vacuum where the excitation is a fraction. Now, all that was left now was to invent a prototype that everybody could understand.
Now, you folks probably many of you are too young to remember that in those days, there was already a big literature on fractionally charged excitations that goes way, way back to papers on solitons by Jackiw and Rebbi-- done here, actually. And also, work done by Bob Schrieffer at Santa Barbara in systems which have-- will be called discrete broken symmetries in them. So the idea that there should be vacuo with fractionally charged excitations is not originally to me. It's due to people who went before me, chiefly particle physics and Bob Schrieffer.
So one already knew that the idea was not nuts. The only thing left was to figure out what it might be. Now, I should tell you when-- I knew about the soliton work, and so I made a theory of this effect based on an analogy with fractional quantization in solitons. And I sent it into Phys Rev Letters. And I like to say, thank god the referee rejected it.
I know later that this referee was Steve Kivelson. And Steve pointed out, well, that your discrete symmetry breaking that you've made in your theory is really continuous image rating. It's like the stripe idea, and that means it's got a pin. And, therefore, it wouldn't conduct electricity, and so the whole idea is nonsense.
So I got this referee report. And I said, well, unfortunately, he's right. So I need to go think about it more. So I thought, well, this thing-- experiments are telling us it has a great big energy gap.
So when things have energy gaps, you can get to first base in modeling them just by brute force. So I thought, OK, let's try to solve the problem by computer, starting with one electron. That I can do. Then let's do two. I did two. That turns out to be easier than doing it on paper.
What does that mean? Nothing. Hello? Then 3, then 4. So the model Hamiltonian one couldn't study is just a few electrons that are repelling each other coulombically. And for balance, let's put a little pressure in here to keep them in these old calculations.
There are two energy scales. There's a so-called cyclotron frequency and coulomb energy, which is e squared over the length associated with that harmonic oscillator energy. And the limit we usually prefer as theorists is this one, where the cyclotron energy is much larger than the coulomb energy, even though in real samples, we're not always deeply in that limit.
But it doesn't matter because of the adiabatic continuability. This is the right limit to be in. So I studied, looked at these guys, and solved them, and got the wave functions out, and showed them here. There they are.
These wave functions I'm showing you are completely meaningless. I just like to show them. Then I found those, sure enough. The little cluster is kind of doing what the experiments do. As you increase the pressure going this way, the mean square error changes and jumps, and the angular momentum also changes and jumps as though it's contracting. It has preferred densities, and it's contracting discretely.
So I thought, that's good. Then I said, OK, well that's what's happening then. Maybe something like this thing down here is the ground state. So I knew this function is exact. When the number of particles is 2, it's pretty close when the number of particles was 3, 4, or 5 and 6.
So I spent several weeks trying to prove that it was the exact ground state of the relevant Hamiltonian. Relevant Hamiltonian, in this case, is noninteracting electrons plus a little bit of coulomb balance potential in the background, jellium basically, plus pairwise coulomb forces. And it wouldn't work, and it wouldn't work, and it wouldn't work.
And, finally, I said, well, it's not going to work. This doesn't work. It's a great idea, but it failed.
So I went in the physics library at Livermore and began reading books on helium. I thought, oh, well, helium is like this, and I could learn from helium. And the very first book I pulled down was Gene Feenberg's book on many body theory.
And I opened it at random, and there on the very page I had opened was this wave function. It turns out that this functional form for many body problems is used very commonly. It's called Jastrow form, and it's variational.
So I thought, oh, my god, that's it. The function is an exact. Wave functions in many body physics never are. So the thing is variational.
So I quickly went to find out how good it was. Now, just so you folks know, what's written here, z 1 through z n are the locations of the electrons in the plane expressed as complex numbers. So this is a pairwise difference raised to a power. The power has to be an odd integer. And then because of the background magnetic field, it has to be canceled by an exponential one for each electron, and the length scale in this experiment is the magnetic length which has been set to 1.
So, as luck would have it, I was in a laboratory that specializes in plasma physics. And the nature of the calculations that I had to do were all the same as those done in dense plasmas. So I went to my plasma buddies Forrest Rogers and Hugh DeWitt and asked them how to calculate. And they showed me, and then one by one, things began to fall into place.
So we squared the wave function to get the probability to find the electrons at various positions. Integrate out all but one of those coordinates, and we get the density, the charge density of the state. And, sure enough, it was exactly uniform out to the boundary, and the density is exactly what we want.
It's one of the ideal density of a filled Landau level. The first nontrivial value of n, by the way, is 3, which is the 1/3 effect. Also, one can integrate out all but two of the coordinates and get what's called the correlation function.
And that's important for-- well, here's what it looks like. So what this is is this is the density of the fluid given that as a function of distance, given that one of the particles is at the origin. So in the electron gas lingo, this is the exchange correlation hole of the gas.
And what you see is that the electrons avoid each other. So if one electron is at the origin than the other electrons avoid it. Although, in the m equals 3 state, which is the interesting one, they pile up a little bit at the near neighbor position. So once you have this function, then you can weight it by 1 over R to get the energy.
Now, just to show you there was, at the time, a strawman-- which was Hide Fukuyama's-- Hide and Phil Platzman, I guess-- their Hartree-Fock calculations for the Wigner crystal. And so it was possible to weight this pair distribution function by 1 over R and find that the wave function I made was a little better energetically because it keeps the particles a little bit farther apart than the crystal.
And so here is total energy as a function of filling factor. Here is the strawman Wigner crystal energy. Down at the bottom is the 1/3 state I constructed. 1/5 looks like it's better, and then 1/7 is a little hard to tell.
So on the basis of this calculation, I predicted that there should be one more of these fractions, namely 1 at 1/5. When I finished this calculation, I phoned up Horst. I said, Horst, Horst, I think there's another fraction. He said, really, what? I said, 1/5. He said, well, it's close.
What had happened in the meantime was that these guys had discovered the famous 2/5 state, which, unfortunately, I did not predict. And then they began to find all sorts of fractions that I didn't predict and didn't find 1/5. And that was, as my father used to say, cheap thrills.
And then they didn't find it, and they didn't find it, and they didn't find it. And then, finally, I believe it was Emilio Mendez found it in this experiment.
Now, there are better-- wait, wait.
Wait-- there are better experiments. I just wanted to show this one to demonstrate the lengths to which theorists will go to declare victory of their theories. In fact, modern samples have a nice plateau here, although it is one of the weakest ones ever found. Well, let's just get that right off.
Now, what about the fractionally charts excitations? Turns out, you get those for free. Once you have a uniform state with an energy gap, then you can imagine a little thought experiment where you poke the sample with an infinitely thin solenoid, adiabatically wind a little flux-- a magnetic flux through it, a whole flux quantum.
And then once you've done that, again, the Hamiltonian has returned to itself just like it is in the loop problem. And so you can remove the solenoid by gauge transformation. And this process gives the next iteration of charge plus or minus 1/3, depending on which way you wind.
The basic reason for this is the following. That the winding process does something complicated near the solenoid you don't understand. But far away from the solenoid, the effect is very simple. It's exactly the same as in the integral quantum Hall effect. It just takes the state and moves it over by one orbital.
And so the net charge transported through an imaginary Gauss's law box here is the average charge per orbital of the ground state, which, in this case, was 1/3. One thing led to another, and I cooked up these variational wave functions for the positive and negative quasi particle.
Z0 here is not a variable. It's a place locating the center of this particle. So it's a quantum number. These wave functions came out of the blue. They were just the simplest approximations I could think of that would look something like this thought experiment.
Later, Duncan Haldane and Ed Rezayi did numerical calculations on the problem and found out these wave functions were vastly better than they had any right to be. So these are actually pretty good representations of the problem with coulomb interactions.
What these particles look like is this. This is what they look like. This is a plot of charge density versus how far you are from the center of the particle. And we want to look at the solid line here. That's the positively charged one.
What you see is near the center of it, there is a place where the charge density of the fluid is missing. And when you add up the deficit, it's exactly 1/3. The quasi hole wave function is this dashed line, and it's the opposite. The charge density of the vacuum near the center is in excess. And when you add it up, the amount in excess is exactly 1/3.
Now, with these wave functions, then it's possible to compute the coulomb repulsion of the piled up charge with itself. And then compare it with measurements of the energy to make such a particle. These measurements-- measurements which do that, or which we believe do that-- are shown here.
This is a plot of the parallel resistance versus 1 over temperature down in the dead center in a plateau. And what you see is, as you cool it off, which goes to the right, there's a region where you get Arrhenius behavior, transitioning to what we call variable range hopping behavior at lower temperatures.
Both of these behaviors have lore associated with them. This is a strong localization effect, and this is what you expect to get very generally with thermal activation of particles from a chemical potential up to a mobility edge. The mobility edge distance ought to be something like the gap of the state.
So one can take this slope and convert it into an energy gap and compare it with what one gets in theory. Now, when this was first done, it didn't work so well. This is a plot of measured activation gap versus field strength.
Here, the measurements down here-- and the theory is this dash line. However, shortly thereafter, Alan MacDonald realized that the model is too crude. There's a big effect missing, which is the thickness of the electron gas in the vertical direction. So it has a finite thickness.
And when one takes into account, a good hefty fraction of the deficit goes away. And the rest of it-- at least, I just described too well being good enough for government work-- because these this relation of the mobility edge energy to the actual gap is a little vague in the first place. Being comparable in size is enough, especially when the trend is right.
Notice, everybody, please, that the energy scale we're talking about here is quite low-- quite low compared with the natural coulomb energy scale. So the fact that has this trend with magnetic field and is also the right size is very significant. So we got that right.
Now, the last thing I want to talk about is gauge forces. It somehow came to pass in the development of composite fermions, which are perfectly right in the way that the following amazing property of these quasi particles got overlooked. I think this has always struck me as one of the most important and most amazing properties of these objects. And that is that they exert forces on each other that are velocity dependent.
This was first figured out by Frank Wilczek and Dan Arovas and Bob Schrieffer who took the trouble of adiabatically exchanging two of these quasi particles theoretically and discovering that the so-called berry phase for adiabatic exchange was a fraction.
In this experiment, you get an odd multiple of pi when the particles are fermions, an even multiple of pi when they're bosons. This particular multiple was either one. Between, it's a fraction, and a fraction is associated with the charge of the particle.
What that means physically is that when one of these particles moves in the presence of another, it feels a fictitious vector potential as though there were a solenoid being carried by the stationary particle carrying a fraction of the elementary flux quantum where the fraction is associated with the charge. Now, the really neat thing about fractional statistics was that it was figured out about the same time-- and I think, actually, before-- empirically by Bert Halperin through graphs like this.
Bert figured out that if the fractional statistics were present, then the quasi particles themselves could recondense into new fractions. And those fractions would come out to be the observed ones, namely 2/5, 3/7, and so forth, If? They had these solenoidal forces on them.
So this is a total energy calculation pulled out of one of Bert's papers. And we don't have to worry about the details of it. Just notice that it has some interesting features at the fractions at which the so-called hierarchical states occur.
Now, my interpretation of this state of affairs has really not changed over the many years, namely that the observation of these fractions other than 1/3 amounts to experimental proof that the gauge forces are there. I mean, the gauge forces actually are emergent in this problem, even though there is more than one way to describe it. The physics of emergent velocity dependent forces is definitely present in the problem and has physical measurable effect.
Now, we go back to the premise. Now, after all is said and done, what have we learned? Well, like most wonderful things in physics, this particular discovery has implications. It was not expected. It was completely out of the blue. It has very many interesting aspects, particularly those relevant to small devices in solid state physics, semiconductor engineering, and so forth.
But the thing that matters to me mostly as a theorist is the bigger issue of where the strange rules of the vacuum might come from. And always, when you have an experiment, god is on your side. In this case, we do have experiments. We have accurate quantization, which is telling us what that there are, indeed, fractionally charged objects in the world that are collective in nature.
They carry a quantum of charge that is as exact as the electric charge itself, of the electron itself. And they also have very powerful forces between them, which are not postulated in the underlying equations of motion but simply emerge a new vacuum and a new kind-- fundamentally new kind of quantum emergence.
At least for me, and I hope for you too eventually, this has been an exceedingly thought-provoking thing. And I believe and hope that it will have major implications for physics of all kinds as we move into the next century and millennium. All right. That's all I have to say. Thank you very much.
KASTNER: Thank you, Bob. That was a very beautiful talk. And we have time, I think, for a couple of questions. Yes? Patrick?
LAUGHLIN: Yeah. Maybe I lied. Sorry, Patrick. We have a parody problem. Wait, wait. Yeah, here we go. Here we go.
KASTNER: The picture that had the loop?
AUDIENCE: So I also try to teach this in my class, but I've never been able to move on because I don't understand how you can go from the derivative to the discreet--
LAUGHLIN: Excellent question. Let me answer that now so that your students will understand this henceforth. OK, let me answer this now, Patrick. Because you know-- you all should know that Patrick and I have been arguing about this for 10 years.
LAUGHLIN: Well, yeah, right. My hair is grayer than yours, so I can't admit it. The answer, of course, is that when the loop is small so that there's a Bohm-Aharonov effect, then the Hall conductance is not quantized.
AUDIENCE: That true?
AUDIENCE: Are you predicting [INAUDIBLE]?
LAUGHLIN: Yes, but that's a no-brainer, Patrick.
AUDIENCE: No, but then it tries to [INAUDIBLE].
LAUGHLIN: Wait, wait, wait. Patrick, this is already known. If you have something that's one or two atoms, and one or two magnetic lengths big, there isn't any quantization of anything.
AUDIENCE: So the question is, how does it depend on help?
LAUGHLIN: Ah, interesting. Now, interesting question. Now, nobody knows. Patrick, you know when you-- I'm sorry. I'm preaching here. Look, I don't know how it depends on l. What I know is that when the sample is too small, you can see it. It's there, and therefore, the quantization argument is false.
So one has to be in the thermodynamic limit for the argument to work. And that's because localization is a thermodynamic concept. Now, you're absolutely right that the disappearance of the Bohm-Aharonov effect in the thermodynamic limit is part of the argument.
And I guess the best answer I can give you is, like localizations-- see, I can't prove that either. But the scaling hypothesis leads me to plus experimental facts, lead me to believe very strongly in localization. And so by the same token, I'd say that if we have a loop that's a centimeter in size, I assure you that you will never see the Bohm-Aharonov effect. I assure you.
LAUGHLIN: Yes. Now, Ray-- now, Ray-- I made Ray a bet that an effect that he saw was due to-- what did I say? I said it was time-dependent delays and transport.
And Ray made me this bet in public at a semiconductor meeting. And then, well, should I tell the rest of the story, Ray? Well, I don't know. I think probably we ought to pay each other $50 and then go have a beer.
What actually happened was the experiment was-- the voltage in the experiment was driven a little too hard. And the effect was an artifact of the measurement. So who is culpable there?
LAUGHLIN: True. It's a different artifact. But I say if you demand the $100 under those circumstances, you are a scoundrel. I think we should buy each other a beer and call it a day, Ray.
KASTNER: I think on that note, let's thank Bob for a beautiful report.