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Carbon Wonderland by 2010 Nobel Laureate in Physics Andre Geim and Philip Kim

Graphene, a new isolated form of carbon, is a rich source of innovative fundamental physics and practical applications

A clean graphene surface as scanned with an electron microscope. Image: Vanderbilt University
A clean graphene surface as scanned with an electron microscope. Image: Vanderbilt University

Think for a moment about the humble pencil. You may not be surprised if we tell you that this writing instrument, which is so common today, once stood at the top of the list of desirable high-tech toys. In fact, the simple pencil was even once banned from being exported due to it being a strategic military asset. But the not-so-expected news about him is that every time someone draws a line with a pencil, the resulting mark contains bits of the hottest new material in physics and nanotechnology: graphene.

Graphene is made from graphite, the black filling inside the pencil: a type of pure carbon built from flat layers of atoms stacked on top of each other. The layered structure of graphite was discovered hundreds of years ago. Naturally, the physicists and materials scientists tried to split this mineral into the sheets that make it up - if only for the opportunity to study a material that might turn out to be endowed with a geometry of such elegant simplicity. Graphene is the name given to this type of single sheet. It is made entirely of carbon atoms bound together in a periodic network of hexagons arranged in a plane only one atom thick.

But for many years all attempts to create graphene have failed miserably. The most popular early approach was to insert molecules between the atomic planes of the graphite to act as a wedge to separate the planes from each other, that is, to apply a technique called chemical exfoliation. Although there is almost no doubt that graphene layers did detach from the graphite at an intermediate stage of the process, they were never identified as such. Instead the resulting final product was usually a pulp of black graphitic particles resembling wet soot. Early interest in chemical peels waned and faded.

A short time later the experimenters took a more direct approach. They split graphite crystals into increasingly fine slices by scraping or rubbing them against another surface. Despite the crudeness of the method, known as micromechanical fragmentation, surprisingly, its performance was impressive. Researchers have succeeded in peeling off thin layers of graphite that are less than 100 atomic planes thick. Already in 1990, for example, German physicists from the RWTH University in the city of Aachen isolated layers of graphite that were so thin that they were transparent to light.

After ten years one of us (Kim) included the micromechanical fragmentation in joint work with Yuanbo Zhang, who at the time was a research student at Columbia University. With the help of the improved method, they created a high-tech version of the pencil: a "nano pencil", of course. "Writing" with the nano-pencil produced slices of graphite that were only a few tens of atomic layers thick. But the resulting material was thin graphite and not graphene. No one believed anymore that matter could exist in nature.

The end of this pessimistic assumption came in 2004. One of us (Gaim), in collaboration with Kostya S. Novoslov, who was then a postdoctoral fellow, and his partners at the University of Manchester in England, was engaged in researching a variety of approaches to create even thinner graphite samples. At this time, most laboratories started such experiments with soot, but Geim and his colleagues randomly started with graphite chips left after mechanically breaking graphite, and this way turned out to be incredibly profitable. They simply attached such a chip to plastic adhesive tape, folded the sticky side of the tape over the flake and then pulled the two parts apart, thus splitting the flake in half. The experimenters repeated the operation over and over, and the resulting shards became thinner and thinner. When the researchers had many thin shards in their hands, they examined the pieces very carefully - and were amazed to find that the nucleus of some of them had only one atom. The surprise was even greater when they saw that the newly identified pieces of graphene had high crystalline quality and chemical stability even at room temperature.

The actual discovery of graphene generated a flurry of international research interest. Not only is it the thinnest material possible, it is also incredibly strong and hard. Furthermore, in its pure form it conducts electrons at room temperature faster than any other material. Today, engineers in laboratories around the world are busy investigating this material to determine if it is possible to process it into products such as super-hard structures, smart displays, super-speed transistors and computers that operate using tiny structures called quantum dots.

Meanwhile, the strange nature of graphene at the atomic scale allows physicists to study phenomena that must be described using relativistic quantum physics. Investigating such phenomena, which are among the most exotic in nature, has until now been the exclusive domain of astrophysicists and physicists who specialized in high-energy particles, and for that they used telescopes that cost tens of millions of dollars or particle accelerators that cost tens of billions of dollars. Graphene allows experimenters to test the predictions of relativistic quantum mechanics with the help of devices standing on the laboratory table.

Please meet - the Grafen family

In view of the widespread distribution of the pencil nowadays, it is quite interesting to note that the material known as graphite did not play a role in the ancient educated cultures such as China and Greece. It was only in the 16th century that the English discovered a large deposit of pure graphite, then called plumbago ("lead ore" in Latin. This is the origin of the incorrect expression "the lead in the pencil" from which the Hebrew word pencil derives - the editors). Its effectiveness as a marking material was immediately apparent, and the English did not hesitate and created a convenient substitute for quill and ink. Soon the pencil became the "last cry" among the educated in Europe.

However, it was not until 1779 that the Swedish chemist Carl Scheel showed that plumbago is carbon and not lead. Ten years later, the German geologist Avraham Gottlob Werner said that perhaps the material should be called graphite, which means "to write" in Greek. Meanwhile, gun manufacturers discovered another use for this crumbly ore: they found it to be a perfect candidate for the role of inner coating of casting molds for cannonballs. This use became a closely guarded military secret. During the Napoleonic Wars, for example, the British Crown embargoed the sale of graphite and pencils to France.

In the last decades, graphite regained a little of its advanced technological status, when scientists investigated the properties and possible applications of some molecular structures of carbon that were not known until then and appear in ordinary graphitic materials. The first of these, a football-shaped molecule called a bucky ball, was discovered in 1985 by the American chemists Robert Carle and Richard A. Smalley in collaboration with their English colleague Harry Croteau. Six years later, Sumio Ijima, a Japanese physicist, identified the honeycomb-like cylindrical carbon structures called carbon nanotubes. Many researchers reported on nanotubes in the decades before that, but then no one recognized their importance. The two new molecular structures were given the name fullerenes (fullerenes - this name, like the name "bucky balls", was determined in honor of the visionary architect and engineer, Buckminster Fuller, who explored these shapes in the structures he designed even before the carbon structures themselves were discovered).

Molecular hexagonal network

The atoms that make up graphite and the fullerenes and graphene are all arranged in the same basic structural array: a ring of six carbon atoms linked together in the form of an elaborate hexagon - a structure that chemists call a benzene ring.

At the second level of organization, which forms the graphene itself, many benzene rings connect and form a sheet of hexagons similar to a lattice of chicken coops. The other forms of carbon in the family are built from sheets of graphene. Bucky spheres and many other non-tubular fullerenes are actually sheets of graphene folded on an atomic scale into spheres, elongated spherical shapes (spheroids), and the like. Carbon nanotubes are essentially sheets of graphene rolled into tiny cylinders. And as we mentioned here, graphite is a thick, three-dimensional stack of graphene sheets. The sheets are attached to each other with the help of weak intermolecular forces of attraction called van der Waals forces. It is the weak bond between adjacent sheets of graphene that allows the graphite to easily break into tiny, thin slices that make up the mark left by the pencil on the paper.

Looking back, it is clear that the fullerenes were around us all along, even though they were hidden from our eyes until recently. They appear, for example, in the soot that covers every barbecue, although in tiny amounts. In exactly the same way, pieces of graphene can be found in every pencil mark - although they too remained hidden from our eyes for a long time. But since they were discovered, the scientific community has devoted a lot of attention to all these molecules.

The main importance of Bucky balls is as an example of a fundamentally new type of molecule, although it is possible that they also have important applications, mainly in drug delivery techniques. Carbon nanotubes are endowed with a cluster of unusual properties - chemical, electrical, mechanical, optical and thermal - that have inspired a wide range of innovative potential applications, such as materials that could possibly replace silicon in micro-chips and fibers that could be used to weave incredibly lightweight and strong cables. Although graphene itself - the father of all graphitic structures - was associated with such visions only a few years ago, it is conceivable that the insights into basic physics and the intriguing technological applications that this material will bring in its wings will be many more than those of its carbon cousins.

exceptional exceptional

Two properties of graphene make it an extraordinary material. First, despite the relatively crude ways in which it is produced, graphene exhibits an incredibly high quality - resulting from a combination of the purity of its carbon content and the systematic arrangement of its atoms in the lattice. Until now, the researchers have not been able to find even one single atomic defect in graphene - for example, a place in the lattice that is missing an atom or an atom that has moved out of place. Apparently, this perfect crystalline order is due to the incredibly strong and flexible interatomic bonds that make a material harder than diamond, but allow the planes to bend when a mechanical force is applied to them. The flexibility allows the structure to tolerate large deformation before the atoms have to rearrange themselves to accommodate the stretch.

The quality of graphene's crystalline structure is also responsible for its high electrical conductivity. The electrons can move in the lattice without defects and foreign atoms diverting them from the direction of the current. Even the surrounding carbon atom holes, which the electrons in graphene must contend with at room temperature, are relatively small because of the great strength of the interatomic bonds.

Graphene's second unusual property is that its conduction electrons, in addition to moving around the lattice almost unimpeded, also move much faster than the free electrons in ordinary metals and semiconductors. Indeed, the electrons in graphene - or perhaps the "electric charge carriers" is a more appropriate term - are strange creatures living in a strange world, where laws parallel to the laws of relativistic quantum mechanics play an important role. This type of interaction within a solid is unique to graphene, as far as scientists know. Thanks to this innovative material, produced from a pencil, relativistic quantum mechanics is no longer limited to cosmology or high-energy physics. Now she enters the laboratory.

Big Bang in Carboniferous Stochlandia

To appreciate the strange behavior of the electric charge carriers in graphene, it might be worth comparing it to the way normal electrons move in a normal conductor. The "free" electrons that make up an electric current in a metal, for example, are not really free because they do not behave exactly like electrons moving in a vacuum. Electrons naturally carry a negative electrical charge, so when they move through a metal they leave a positive charge on their parent atoms in the metal. Therefore, as electrons move through the lattice, they are affected by the electrostatic fields it creates, which pull and push them in all directions in a complicated way. The end result is that the moving electrons behave as if their mass is different from that of normal electrons - in other words, they have an "effective mass". Physicists call such charge carriers quasi-particles. These charged particles, like electrons, move much slower than the speed of light through the conducting metal. There is therefore no need to apply Einstein's relativity corrections to their movements, since the theory only becomes important at speeds approaching the speed of light. The interactions of the quasi-particles in the conductor can be described with the help of Newton's well-known classical physics or with the help of "normal" (that is, non-relativistic) quantum mechanics.

The electrons moving in the hexagonal network of the carbon atoms in graphene also behave as quasi-particles, but surprisingly their behavior is completely different from that of electrons. In fact, the most similar thing is another elementary particle: the nearly massless neutrino. Of course, a neutrino, as its name implies, is electrically neutral (the meaning of the word neutrino in Italian is "the little neutral"), while the quasi-particles in graphene carry the same electric charge as the electron charge. But the neutrino moves almost at the speed of light, so it must be described in terms of relativity. Similarly, a quasi-particle in graphene always moves at a constant and high speed, albeit about 300 times slower than the speed of light. Despite its reduced speed, its behavior is largely similar to the relativistic behavior of the neutrino.

The relativistic nature of the quasi-particles in graphene renders ordinary, non-relativistic quantum mechanics useless when it comes to describing how they work. Physicists must choose a more complex framework from their arsenal of theories: relativistic quantum mechanics, now known as quantum electrodynamics. This theory has its own language, and a cornerstone of this language is the probabilistic equation named after the English physicist Paul A. M. Dirac, who first wrote it in the 20s. That's why theorists sometimes describe the electrons moving in graphene as massless Dirac quasi-particles.

Particles "come out of nowhere"

Unfortunately, an interpretation of quantum electrodynamics always requires a vigorous struggle against ordinary intuition. It is necessary to become familiar with phenomena that seem paradoxical, despite the discomfort involved. The paradoxes of quantum electrodynamics arise due to the fact that a relativistic particle is always accompanied by its alter ego from the opposite world: the antiparticle. The electron, for example, is accompanied by an antiparticle known as a positron, whose mass is exactly the same as the electron's mass, but whose electric charge is positive. A pair of a particle and an antiparticle can appear under relativistic conditions, because due to its extremely high energy and speed of movement, it does not need much energy to create a pair of "virtual particles". Strange as it sounds, the couple emerges straight out of nothing - out of the void.

The reason for such an occurrence is the result of one of the many versions of Heisenberg's uncertainty principle in quantum mechanics: in a comprehensive formulation, the more precisely the time of an event is defined, the less precise the amount of energy associated with this event. Thus on very small scales of time, the energy can have almost any value. Energy equivalent to mass, according to Einstein's famous formula, E = mc2, so the energy equivalent to mass of a particle and an antiparticle can appear out of nowhere. For example, a virtual electron and a virtual positron can suddenly emerge and exist through "borrowing energy" from the void, provided that the lifetime of the virtual particles is so short that the energy deficit is repaid before it can be detected.

The intriguing vitality of the vacuum in quantum electrodynamics generates many strange effects. Klein's paradox is a good example of this. The paradox describes conditions under which a relative object can pass through any potential barrier, however high and wide. A familiar type of potential barrier is plain hills surrounding a valley. A truck leaving the valley accumulates potential energy as it climbs up the hill, at the expense of the energy released from the fuel burned in its engine. However, the truck can slide from the top of the hill to the other side with the engine off and idling. The potential energy gained by the truck as it climbed up is converted back into kinetic energy as it rolls down.

An examination of the twilight zone

Particles can also move very easily by themselves "downhill" from relatively high areas of potential energy to relatively low areas of it. But if a "hill ridge" of high potential energy surrounds a particle in an energy "valley", the particle is no less stuck than a truck without fuel in a real valley. There is one major caveat to this conclusion, expressed in ordinary, non-relativistic quantum mechanics. According to Heisenberg's uncertainty principle, in another version of it, it is impossible to know the exact position of a particle. Physicists therefore describe the position of a particle in probabilistic terms. One of the strange results is that although a low-energy particle appears to be "trapped" by a high barrier, there is some probability that after some time the particle will be found outside the barrier. If this indeed happens, the passage of his ghost through the energy barrier is called quantum tunneling.

In non-relativistic quantum tunneling there is a finite probability that a low energy particle will tunnel through a high potential barrier. This probability can have different values, but never 100%. The probability of quantum tunneling decreases as the barrier becomes higher and thicker. However, the Klein paradox completely changes the nature of quantum tunneling. It states that the probability of relativistic particles tunneling through high-energy barrier regions spanning a wide area is 100%. Near the barrier the particles simply connect with their antiparticle twins, who experience the world in reverse, so the hills in the real world look like valleys to the antiparticles. After gliding through the strange valley of the barrier in the antiworld, the antiparticles are converted back into particles and emerge undisturbed on the other side. This prediction seems to be deeply counterintuitive even to many physicists.

Such a strange prediction begs to be tested experimentally, but it has long been unclear whether Klein's paradox can stand up to the test, even theoretically. The massless Dirac quasiparticles in graphene now came to the rescue. In graphene, Klein's paradox is made into a routine effect with easily predictable effects. When Dirac's quasi-particles, which are massless and carry the charge, move inside a graphene crystal along which obstacles such as an electric voltage or a potential energy difference have been placed, the experimenters can test the material's electrical conductivity. Perfect tunneling (100%) will prevent the increase in electrical resistance that could be expected due to the added barriers and boundaries. Researchers are now measuring the flux of such particles that are able to tunnel through potential barriers at different heights. Physicists expect graphene to help demonstrate many more unusual effects predicted by quantum electrodynamics.

Love from a second dimension

It is still too early to assess the technological applications of graphene, but more than ten years of research on carbon nanotubes - rolled graphene - gives graphene a major head start. It is likely that almost every useful role that scientists predict for the nanotubes is also open to their flat cousin. The hi-tech industries have blueprints for some commercial applications, and some companies are already betting on their promise. To meet the demand for such applications will require commercial production of graphene, and many technology research teams are currently working on developing improved production techniques. Although it is already possible to produce graphene powder in commercial quantities, it is still difficult to produce graphene sheets, and they are probably considered the most expensive material on the planet. A graphene crystal created by micromechanical splitting and smaller than the thickness of a human hair can currently cost more than $1,000. Groups in Europe and several institutes in the United States - which include, among others, the Georgia Institute of Technology, the University of California at Berkeley and Northwestern University - have grown graphene layers on thin slices of silicon carbide, similar to the slices common in the semiconductor industry.

Meanwhile, engineers around the world are trying to take advantage of the incredibly desirable electrical and physical properties that make graphene unique [see box on previous page and left]. The large ratio of its volume to its surface area, for example, would be useful for the production of rigid composites. Graphene's extreme minuteness can improve the efficiency of field emitters - needle-like devices that release electrons in the presence of strong electric fields.

Graphene's properties can be carefully tuned by applying electric fields, making it possible to build improved superconducting transistors, known as spin valve transistors, as well as incredibly sensitive chemical detectors. In addition to this, thin layers, produced from graphene patches that cover each other, have great potential to be used as conductive and transparent coatings for LCD screens and solar cells. The list is far from comprehensive, but we expect some niche applications to hit the market within a few years.

Amnesty for Moore's Law?

One engineering direction deserves a special mention: graphene-based electronics. We emphasized that the charge carriers in graphene move at high speed and lose relatively little energy in scattering or in collisions with atoms in the lattice. This feature should enable the construction of components known as ballistic transistors, very high frequency devices that will respond much faster than existing transistors.

There is an even more interesting possibility: graphene could help the microelectronics industries extend the life of Moore's Law. Gordon Moore, a pioneer of the electronics industry, determined about 40 years ago that the number of transistors that can be crammed into a given area doubles approximately every 18 months. The inevitable end of this ongoing minimization has been prematurely announced many times already. Graphene's stability and electrical conductivity, which are remarkable even at nanoscale scales, could enable the production of single transistors considerably shorter than tens of nanometers, and perhaps even as small as a single benzene ring. In the long run it is possible to imagine a complete system of integrated circuits carved from a single sheet of graphene.

Whatever the future brings, there is almost no doubt that the single-atom-thick wonderland will remain in the spotlight for decades to come. Engineers will continue to work to bring the innovative byproducts of this wonderland to market, and physicists will continue to test its exotic quantum properties. But the truly amazing thing is that all this richness and complexity has been hidden for centuries in almost every simple mark drawn by a pencil.

key concepts

Graphene is a sheet of carbon that is a single atom thick. A stack of such sheets creates graphite, the filling material for pencils. Only recently have physicists isolated the substance.

The pure, flawless crystal conducts electricity faster than any other material at room temperature.

In the vision of the engineers there is a wide range of products made of graphene, such as super-speed transistors. Physicists are finding that the material allows them to theorize exotic phenomena previously thought to be observable only in black holes and high-energy particle accelerators.

molecular structures

The head of the graphite family

Graphene, a plane of carbon atoms that resembles a grid of chicken coops, is the basic building block of all the "graphitic" materials pictured below. Graphite, the main component of the pencil filling, is a crumbling material that resembles a layered cake made of graphene sheets connected to each other in a weak bond. When graphene folds into round shapes, fullerenes are formed, which include cylinders called carbon nanotubes and football-like molecules known as buckyballs, as well as a host of shapes that combine the two structures.

About the authors

Andre K. Geim and Philip Kim are condensed matter physicists, who in recent years have studied the nanoscale properties of "two-dimensional" crystalline materials that are a single atom thick. Geim is a member of the British Royal Society and holds the Langworthy Chair in Physics at the University of Manchester in England. He also directs the Center for Mesoscopic Sciences and Nanotechnology. Geim received his PhD from the Institute of Solid State Physics in Chernogolovka, Russia. Kim is a member of the American Physical Society and an associate professor of physics at Columbia University. He received his doctorate from Harvard University. His research focuses on quantum, electrical and thermal transition processes in materials on a nanoscopic scale.

Quantum electrodynamics enters the laboratory

Electrons move essentially undisturbed through the incredibly elaborate atomic structure of graphene, reaching such high speeds that their behavior cannot be described by "normal" quantum mechanics. The alternative theory used is known as relativistic quantum mechanics or quantum electrodynamics (QED). Until now, it was thought that the distinct (and strange) predictions of this theory could only be observed in black holes or high-energy particle accelerators, but with the help of graphene, physicists can test one of the strangest predictions of QED in the laboratory: "perfect quantum tunneling".

In classical, or Newtonian, physics, a low-energy electron behaves like an ordinary particle. If it does not have enough energy to carry it over the potential barrier, it is quite clear that it will remain trapped on one side of the barrier, just as a truck without fuel will remain abandoned on one side of the hill.

In the picture of the usual quantum mechanics the electron behaves in certain contexts like a wave propagating in space. In a simplistic formulation, the wave represents the probability of finding the electron at a certain point in space and time. When this "slow" wave approaches a potential barrier, it penetrates through the barrier, and there is a certain probability, which is neither 0% nor 100%, that the electron will be on the other side of the barrier. In practice the electron passes through the barrier via tunneling.

When a high-speed electron wave in graphene reaches a potential barrier, the QED theory makes an even more dizzying prediction: the probability that the electron wave will be found on the other side of the barrier is 100%. The observation that graphene conducts electricity well seems to confirm this prediction.

DIY: Graphene

1. Work in a clean environment; Stray hairs and dirt wreak havoc on graphene samples.

2. Prepare a silicon oxide (quartz) plate, which will help you see graphene layers through the microscope. Smooth the surface of the surface so that the graphene adheres to it and clean it thoroughly using a mixture of hydrochloric acid (HCl) and hydrogen peroxide.

3. Attach a flake of graphite to a plastic adhesive tape about 15 cm long using tweezers. Fold the tape near the flake at a 45 degree angle so that it is trapped between the sticky sides. Press it gently, slowly separate the two sides of the tape, and you will see the graphite easily split in two.

4. Repeat the third step about ten times. This procedure will become more and more difficult as you make more folds.

5. Carefully place on the quartz plate the split graphite sample that remained stuck to the film. Using plastic tongs, remove all the air between the sample and the film by gently pressing. Move the forceps lightly but firmly over the sample for about ten minutes. Hold the slice on the surface with the tongs while you slowly peel off the film. For this step, 30 to 60 seconds are needed to reduce as much as possible tearing of the graphene you created.

6. Place the graphene under a microscope to which you have fitted aperture lenses that magnify 50x or 100x. You will see many remains of graphite: large and shiny pieces in all kinds of shapes and colors (top picture), and if you are lucky you will also see graphene: crystalline and very transparent shapes with color little relative to the rest of the slice. The top sample is magnified 115 times; The lower sample is magnified 200 times.


Graphene based technology

Graphene has not been in the hands of engineers long enough to develop products that use it, but the list of future technologies based on graphene is long. Here are two examples:

A single electron transistor

A single electron transistor, also known as a quantum dot transistor, can be designed from a graphene plane on a nanometer scale. The diagram schematically shows how two electrodes, "source" and "sink", are connected by an "island" of conductive material or a quantum dot that is only 100 nanometers long. The island – shown in an electron microscope image of such a device and magnified here 40,000 times – is too small to contain more than one new electron at a time. The rest of the electrons are left out due to electrostatic repulsion. An electron from the source moves to the island via quantum tunneling, then leaves to the sink via tunneling. The voltage applied to a third electrode called the gate checks whether a single electron can enter or leave the island, and indicates a 0 or 1.

composite materials

Often two or more complementary materials can be combined to achieve the desired properties of both. It is common to use a skeleton made of one material that strengthens another material: see, for example, the body of a fiberglass boat made of plastic with strong glass fibers running through it. The researchers are examining the physical properties of composite materials made from polymers reinforced with graphene-based materials such as graphene oxide, which is a stiffer and stronger material than graphene. Unlike graphene, graphene oxide "paper" is relatively easy to make, and it may soon find its own useful applications in multilayer composites. The scale is one micron.

And more on the subject

Electrons in Atomically Thin Carbon Sheets Behave Like Massless Particles. Mark Wilson in Physics Today, Vol. 59, pages 21-23; January 2006.

Drawing Conclusions from Graphene. Antonio Castro Neto, Francisco Guinea and Nuno Miguel Peres in Physics World, Vol. 19, pages 33-37; November 2006.

Graphene: Exploring the Carbon Flatland. AK Geim and AH MacDonald in Physics Today, Vol. 60, pages 35-41; August 2007.

The Rise of Graphene. AK Geim and KS Novoselov in Nature Materi als, Vol. 6, pages 183-191; 2007.

Andre K. Geim's Mesoscopic Physics Team at the University of Manchester

Philip Kim's research group at Columbia University


  1. Quantum tunneling has been known for about 50 years. At the time, they developed diodes based on this, which produces negative resistance. Negative resistance is imaginary and not measurable. Connecting this diode in resonance circuits, produces super high frequencies, with extremely low noise. The frequency limit is determined only by the length of the wires. When measuring with a curve tracer you see an empty section where the negative resistance is. Since physically there is no such resistance. There is a classified use for these diodes to this day.

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