Tunneling

History

Applications

Resources

Tunneling, also known as the tunnel effect, is a quantum mechanical phenomenon by which a tiny particle can penetrate a barrier that it could not, by any classical or obvious means, pass. Though seemingly miraculous, the effect does have some intuitive characteristics. For instance, thin barriers allow more particles to tunnel than do thick ones, and low barriers permit more tunneling than do high ones.

Tunneling does not generally show itself in the macroscopic world. It only starts to become a factor for microscopic items. Atoms can tunnel, as can electrons, but things such as tennis balls and grapes, easily seen with the naked eye, will not. For microscopic particles, the barrier heights are described in terms of energy instead of distance, but for conceptual purposes there is little difference (Figure 1).

It is important to note that the effect can only be understood with the aid of quantum mechanics. Classical mechanics, the system pioneered by Isaac Newton that stood unchallenged until the early twentieth century, has no way of explaining tunneling. In the Newtonian philosophy, all particles, even the tiniest of microscopic particles, can be located precisely. Any uncertainty is

seen as the result of an imperfect measuring device or a sloppy scientist. In addition, each microscopic particle is considered to be like a tiny pebble and there can be nothing wave like about it.

The quantum view of the universe is fundamentally different from the Newtonian view. Each particle is said to have both corpuscular (pebble like) and wave-like properties. Furthermore, a quantum particle cannot, in general, be located precisely. It has a built-in uncertainty that cannot be taken away by the best of measuring instruments. For these reasons, a particle in quantum mechanics is often treated as a wave function. This is another way of saying that the particle is akin to a small bundle of waves.

Representing a quantum particle as such a bundle has two advantages. For one, it reveals that the particle is, in some sense, blurry and can never be exactly pinned down. It exists over a range of space, not a specific point. For another, the wave function format allows particles to exhibit wave like properties (see interference and diffraction). Tunneling is a one of these wave like properties. By the more general name of “barrier penetration,” it is a well-established characteristic of waves. Light waves, for instance, have long been observed to overcome daunting optical barriers.

In a fundamental sense, the quantum mechanical explanation of tunneling can be illustrated by an analogy. If we roll a ball very slowly toward, say, a cement speed bump, we confidently say that it will not surpass the barrier and predict that it will roll back toward us. However, if a very modest ocean swell approaches an offshore sandbar, we are not as sure of the results and rightly so. The character of the wave, even if diminished in size, often makes its way past the sandbar. Similarly, treating a microscopic particle with the mathematical model of a ball clearly tells us tunneling is impossible, while using the mathematical model of a wave just as clearly states that particles will always have a chance to tunnel.

History

some scattering experiments of the early twentieth century paved the way for tunneling. Ernest Rutherford, the pioneer of the scattering method, came across a paradox in a series of uranium experiments in 1910. Scattering involves the probing of a microscopic object by bombarding it with other particles and then keenly observing how the particles are, literally, scattered by the object in question. Specifically, Rutherford tried to pinpoint where the bombarding particles were scattered and how fast they were moving.

An example using tennis balls gives one a simple and accurate picture of a scattering experiment. Say a statue exists in a dark courtyard and we want information about the statue without being able to see it.

KEY TERMS

Electron —One of the fundamental particles of the universe; carries a negative charge and has very little mass.

Energy barrier —An obstacle analogous to a physical wall, where any object that passes must either possess an energy greater than the barrier’s or tunnel through the barrier.

Macroscopic —Not needing a microscope to be seen, readily observed by the human eye.

Potential —Closely related to potential energy, which is known as the “energy of position” or the energy that a body possesses due to its circumstances as opposed to its motion.

Quantum mechanics —The theory that has been developed from Max Planck’s quantum principle to describe the physics of the very small. The quantum principle basically states that energy only comes in certain indivisible amounts designated as quanta. Any physical interaction in which energy is exchanged can only exchange integral numbers of quanta.

Scattering —An experimental technique by which an item of interest is studied by inducing other particles to collide with it.

Wave function —A useful mathematical construct commonly employed in quantum mechanics to represent both a particle’s wavelike characteristics and its uncertainty in location.

Mainly, we want to know what its shape is (e.g., a horse or a person) and from what materials it is made (e.g., granite or clay). To learn more about the statue, we stand at the edge of the courtyard and toss tennis balls into the dark area, hoping to hit the statue. After many thousands of tennis balls, we’ve learned a lot. If only a few of the tennis balls have hit anything, we know the statue is probably quite small. Also, the directions that tennis balls were scattered are important. If most of them come straight back to us, we can ascertain that the statue has a large, flat front like a wall. Moreover, by observing the speed of the scattered tennis balls, we can get an idea of how hard the statue is. Atomic scattering set a similar scene for Rutherford, but he was interested in the nuclei of atoms and he preferred alpha particles to tennis balls.

Through a series of remarkable scattering experiments, Rutherford, working in conjunction with his students Hans Geiger and Ernest Marsden, had accurately mapped the insides, or nucleus, of the uranium atom. Uranium was of great interest at the time because of its radioactive properties. The summation of Rutherford’s results for uranium came in the form of a potential diagram. Potential refers to electric potential energy, so his diagram mapped out an energy barrier (Figure 2). Any particle that escaped the nucleus (i.e., any particle of nuclear radiation) would have to overcome this energy obstacle before it left. Rutherford estimated the top of this barrier to be at least about nine units of energy high. However, when observing the occasional radiated alpha particle (see radiation), he found that it had only four units of energy. The paradox was born. How could a particle with so little energy overcome a barrier of such height? Such a problem can be compared to a man walking next to a huge baseball stadium and suddenly seeing a baseball floating toward him, as if gently tossed. Surely any ball hit out of the stadium would have been moving fast enough to at least sting his palm.

The question lingered for 18 years until George Gamow, assisted by his colleagues Edward Uhler Condon and Ronald Gurney, proposed a solution. In 1928, quantum mechanics was gaining credibility, and the three physicists performed a relatively simple calculation, treating the alpha particle as a quantum mechanical wave function. In essence, they analyzed the problem from the viewpoint that an alpha particle was not located precisely at any given spot, but rather that its existence was spread out, like a wave. Their explanation proposed that the alpha particles tunneled out of uranium’s energy barrier, and it fit Rutherford’s observations perfectly. The acceptance and practical application of tunneling theory had begun.

Applications

one of the first applications of tunneling was an atomic clock based on the tunneling frequency of the nitrogen atom in an ammonia (chemical formula NH₃) molecule. The rate at which the nitrogen atom tunnels back and forth across the energy barrier presented by the hydrogen atoms is so reliable and so easily measured that it was used as the timing mechanism in one of the earliest atomic clocks.

A current and quickly advancing application of tunneling is Scanning Tunneling Microscopy (abbreviated STM). This technique can render high-resolution images, including individual atoms, that accurately map the surface of a material. As with many high-tech tools, its operation is fairly simple in principle, while its actual construction is quite challenging.

The working part of a tunneling microscope is an incredibly sharp metal tip. This tip is electrically charged and held near the surface of an object (known as the sample) that is to be imaged. The energy barrier in this case is the gap between the tip and the sample. When the tip gets sufficiently close to the sample surface, the energy barrier becomes thin enough that a noticeable number of electrons begin to tunnel from the tip to the object. Classically, the technique could never work because the electrons would not pass from the tip to the sample until the two actually touched. The number of tunneling electrons, measured by incredibly sensitive equipment, can eventually yield enough information to create a picture of the sample surface.

Another application of tunneling has resulted in the tunnel diode. The tunnel diode is a small electronic switch and, by incorporating electron tunneling, it can process electronic signals much faster than any ordinary physical switch. At peak performance, it can switch on and then off again ten billion times in a single second.

Resources

BOOKS

Hewitt, Paul. Conceptual Physics. New York: Prentice Hall,
2001.

Meriam, J.L., and L.G. Kraige. Engineering Mechanics, Dynamics. 5th ed. New York: John Wiley & Sons, 2002.

Brandon Brown

Tunneling

Tunneling, also known as the tunnel effect, is a quantum mechanical phenomenon by which a tiny particle can penetrate a barrier that it could not, by any classical or obvious means, pass. Though seemingly miraculous, the effect does have some intuitive characteristics. For instance, thin barriers allow more particles to tunnel than do thick ones, and low barriers permit more tunneling than do high ones.

Tunneling does not generally show itself in the macroscopic world. It only starts to become a factor for microscopic items. Atoms can tunnel, as can electrons, but things such as tennis balls and grapes , easily seen with the naked eye , will not. For microscopic particles, the barrier heights are described in terms of energy instead of distance, but for conceptual purposes there is little difference.

It is important to note that the effect can only be understood with the aid of quantum mechanics . Classical mechanics, the system pioneered by Isaac Newton that stood unchallenged until the early twentieth century, has no way of explaining tunneling. In the Newtonian philosophy, all particles, even the tiniest of microscopic particles, can be located precisely. Any uncertainty is seen as the result of an imperfect measuring device or a sloppy scientist. In addition, each microscopic particle is considered to be like a tiny pebble and there can be nothing wave-like about it.

The quantum view of the universe is fundamentally different from the Newtonian view. Each particle is said to have both corpuscular (pebble-like) and wave-like properties. Furthermore, a quantum particle cannot, in general, be located precisely. It has a built-in uncertainty that cannot be taken away by the best of measuring instruments. For these reasons, a particle in quantum mechanics is often treated as a wave function. This is another way of saying that the particle is akin to a small bundle of waves.

Representing a quantum particle as such a bundle has two advantages. For one, it reveals that the particle is, in some sense, blurry and can never be exactly pinned down. It exists over a range of space, not a specific point. For another, the wave function format allows particles to exhibit wave-like properties (see interference and diffraction ). Tunneling is a one of these wave-like properties. By the more general name of "barrier penetration," it is a well-established characteristic of waves. Light waves, for instance, have long been observed to overcome daunting optical barriers.

In a fundamental sense, the quantum mechanical explanation of tunneling can be illustrated by an analogy. If we roll a ball very slowly toward, say, a cement speed bump, we confidently say that it will not surpass the barrier and predict that it will roll back toward us. However, if a very modest ocean swell approaches an offshore sandbar, we are not as sure of the results and rightly so. The character of the wave, even if diminished in size, often makes its way past the sandbar. Similarly, treating a microscopic particle with the mathematical model of a ball clearly tells us tunneling is impossible, while using the mathematical model of a wave just as clearly states that particles will always have a chance to tunnel.

History

Some scattering experiments of the early twentieth century paved the way for tunneling. Ernest Rutherford, the pioneer of the scattering method, came across a paradox in a series of uranium experiments in 1910. Scattering involves the probing of a microscopic object by bombarding it with other particles and then keenly observing how the particles are, literally, scattered by the object in question. Specifically, Rutherford tried to pinpoint where the bombarding particles were scattered and how fast they were moving.

An example using tennis balls gives one a simple and accurate picture of a scattering experiment. Say a statue exists in a dark courtyard and we want information about the statue without being able to see it. Mainly, we want to know what its shape is (e.g., a horse or a person) and from what materials it is made (e.g., granite or clay). To learn more about the statue, we stand at the edge of the courtyard and toss tennis balls into the dark area, hoping to hit the statue. After many thousands of tennis balls, we've learned a lot. If only a few of the tennis balls have hit anything, we know the statue is probably quite small. Also, the directions that tennis balls were scattered are important. If most of them come straight back to us, we can ascertain that the statue has a large, flat front like a wall. Moreover, by observing the speed of the scattered tennis balls, we can get an idea of how hard the statue is. Atomic scattering set a similar scene for Rutherford, but he was interested in the nuclei of atoms and he preferred alpha particles to tennis balls.

Through a series of remarkable scattering experiments, Rutherford, working in conjunction with his students Hans Geiger and Ernest Marsden, had accurately mapped the insides, or nucleus, of the uranium atom. Uranium was of great interest at the time because of its radioactive properties. The summation of Rutherford's results for Uranium came in the form of a potential diagram. Potential refers to electric potential energy, so his diagram mapped out an energy barrier. Any particle that escaped the nucleus (i.e., any particle of nuclear radiation ) would have to overcome this energy obstacle before it left. Rutherford estimated the top of this barrier to be at least about nine units of energy high. However, when observing the occasional radiated alpha particle (see radiation), he found that it had only four units of energy. The paradox was born. How could a particle with so little energy overcome a barrier of such height? Such a problem can be compared to a man walking next to a huge baseball stadium and suddenly seeing a baseball floating toward him, as if gently tossed. Surely any ball hit out of the stadium would have been moving fast enough to at least sting his palm.

The question lingered for 18 years until George Gamow, assisted by his colleagues Edward Uhler Condon and Ronald Gurney, proposed a solution. In 1928, quantum mechanics was gaining credibility, and the three physicists performed a relatively simple calculation, treating the alpha particle as a quantum mechanical wave function. In essence, they analyzed the problem from the viewpoint that an alpha particle was not located precisely at any given spot, but rather that its existence was spread out, like a wave. Their explanation proposed that the alpha particles tunneled out of uranium's energy barrier, and it fit Rutherford's observations perfectly. The acceptance and practical application of tunneling theory had begun.

Applications

One of the first applications of tunneling was an atomic clock based on the tunneling frequency of the nitrogen atom in an ammonia (chemical formula NH3) molecule . The rate at which the nitrogen atom tunnels back and forth across the energy barrier presented by the hydrogen atoms is so reliable and so easily measured that it was used as the timing mechanism in one of the earliest atomic clocks.

A current and quickly advancing application of tunneling is Scanning Tunneling Microscopy (abbreviated STM). This technique can render high-resolution images, including individual atoms, that accurately map the surface of a material. As with many high-tech tools, its operation is fairly simple in principle, while its actual construction is quite challenging.

The working part of a tunneling microscope is an incredibly sharp metal tip. This tip is electrically charged and held near the surface of an object (known as the sample) that is to be imaged. The energy barrier in this case is the gap between the tip and the sample. When the tip gets sufficiently close to the sample surface, the energy barrier becomes thin enough that a noticeable number of electrons begin to tunnel from the tip to the object. Classically, the technique could never work because the electrons would not pass from the tip to the sample until the two actually touched. The number of tunneling electrons, measured by incredibly sensitive equipment, can eventually yield enough information to create a picture of the sample surface.

Another application of tunneling has resulted in the tunnel diode . The tunnel diode is a small electronic switch and, by incorporating electron tunneling, it can process electronic signals much faster than any ordinary physical switch. At peak performance, it can switch on and then off again ten billion times in a single second.

Resources

books

Hewitt, Paul. Conceptual Physics. New York: Prentice Hall, 2001.

Meriam, J.L., and L.G. Kraige. Engineering Mechanics, Dynamics. 5th ed. New York: John Wiley & Sons, 2002.

Wiesendanger, Roland, and Hans-Joachim Goentherodt, eds. Scanning Tunneling Microscopy I. New York: Springer-Verlag, 1993.

Brandon Brown

KEY TERMS

Electron

—One of the fundamental particles of the universe; carries a negative charge and has very little mass.

Energy barrier

—An obstacle analogous to a physical wall, where any object that passes must either possess an energy greater than the barrier's or tunnel through the barrier.

Macroscopic

—Not needing a microscope to be seen, readily observed by the human eye.

Potential

—Closely related to potential energy, which is known as the "energy of position" or the energy that a body possesses due to its circumstances as opposed to its motion.

Quantum mechanics

—The theory that has been developed from Max Planck's quantum principle to describe the physics of the very small. The quantum principle basically states that energy only comes in certain indivisible amounts designated as quanta. Any physical interaction in which energy is exchanged can only exchange integral numbers of quanta.

Scattering

—An experimental technique by which an item of interest is studied by inducing other particles to collide with it.

Wave function

—A useful mathematical construct commonly employed in quantum mechanics to represent both a particle's wavelike characteristics and its uncertainty in location.

Tunneling

Tunneling is a phenomenon in which a tiny particle penetrates an energy barrier that it could not, according to the classical laws of science, pass across. One way of describing this process, also known as the tunnel effect, is shown in Figure 1. Notice that the y-axis in this graph represents energy, while the x-axis represents position. The graph shows that in order for the particle to move from left to right, it must surmount an "energy barrier." In other words, the particle must absorb enough energy to climb over the barrier.

An everyday example of this phenomenon is rolling a ball over a small hill. Suppose you stand on one side of the hill and want to roll the ball to a friend on the other side of the hill. You can do so only if you roll the ball hard enough for it to climb up your side of the hill. If you

don't push the ball with enough force, it goes only part way up the hill and then rolls back down to you.

The tunnel effect would mean in this example that you might give the ball only a slight nudge—not enough to get it over the hill. But after the slight nudge, the ball would suddenly appear on the other side of the hill, right in front of your friend. You might be tempted to say that the ball had "tunneled through" the hill rather that going over it. In fact, that analogy explains the way in which the tunneling phenomenon got its name.

In fact, you would never observe an effect like this with a ball, a friend, and a hill. Tunneling occurs only with particles the size of atoms and smaller. The physical laws that describe very small particles such as these are somewhat different from the laws we use to describe large-scale everyday events and objects. The physical laws of very small particles are included in the field of science known as quantum mechanics.

One of the interesting discoveries resulting from quantum mechanics is that tunneling can occur with very tiny particles. The chance that the particle can get from position A to position B in Figure 1 is not zero. That probability may be very small (one chance out of one million, for example), but it is something greater than zero.

Words to Know

Energy barrier: An obstacle somewhat similar to a physical wall, such that any object must either possess an energy greater than that of the barrier or be able to tunnel through the barrier in order to pass the barrier.

Macroscopic: Not needing a microscope to be seen; readily observed by any one of the human senses.

Quantum mechanics: A system of physical principles that arose in the early twentieth century to improve upon those developed earlier by Isaac Newton, specifically with respect to submicroscopic phenomena.

The interesting point is that once the probability of tunneling is greater than zero, than we know that it probably will occur from time to time. When that happens, we observe physical phenomena that do not and cannot be observed at the macroscopic level.

Applications

One of the first applications of tunneling was an atomic clock based on the tunneling frequency of the nitrogen atom in an ammonia (NH₃) molecule. In this molecule, the nitrogen atom tunnels back and forth across the energy barrier presented by the hydrogen atoms in a pattern that is reliable and easily measured. This characteristic made it ideal for use as one of the earliest atomic clocks.

A current and rapidly growing application of tunneling is Scanning Tunneling Microscopy (abbreviated STM). This technique can produce high-resolution images that accurately map the surface of a material. Some of the best STM pictures may actually show us what individual atoms look like. As with many high-tech tools, the operation of a scanning tunneling microscope is fairly simple in principle, while its actual construction is quite challenging.

The working part of a tunneling microscope is an incredibly sharp metal tip. This tip is electrically charged and held near the surface of an object that is to be imaged. The energy barrier in this case is the gap between the tip and the sample. When the tip gets sufficiently close to the sample surface, the energy barrier becomes thin enough that a noticeable number of electrons begin to tunnel from the tip to the object. Classically, the technique could never work because the electrons would not pass from the tip to the sample until the two actually touched. The number of tunneling electrons, measured by highly sensitive equipment, can eventually yield enough information to create a picture of the sample surface.

Another application of tunneling has resulted in the tunnel diode. The tunnel diode is a small electronic switch that can process electronic signals much faster than any ordinary physical switch. At peak performance, it can switch on and then off again ten billion times in a single second.

[See also Quantum mechanics ]