The tunnel effect is an amazing phenomenon, completely impossible from the standpoint of classical physics. But in the mysterious and mysterious quantum world, slightly different laws of interaction between matter and energy operate. The tunnel effect is the process of overcoming a certain potential barrier, provided that its energy is less than the height of the barrier. This phenomenon is exclusively quantum in nature and completely contradicts all the laws and dogmas of classical mechanics. The more amazing is the world in which we live.
The best way to understand what the quantum tunneling effect is is to use the example of a golf ball thrown into a hole with some force. At any given unit of time, the total energy of the ball is in opposition to the potential force of gravity. If we assume that it is inferior to the force of gravity, then the specified object will not be able to leave the hole on its own. But this is in accordance with the laws of classical physics. To overcome the edge of the hole and continue on its way, it will definitely need additional kinetic impulse. This is what the great Newton said.
In the quantum world, things are somewhat different. Now let’s assume that there is a quantum particle in the hole. In this case, we will no longer be talking about a real physical depression in the ground, but about what physicists conventionally call a “potential hole.” Such a value also has an analogue of the physical side - an energy barrier. Here the situation changes most radically. In order for the so-called quantum transition to take place and the particle to appear outside the barrier, another condition is necessary.
If the strength of the external energy field is less than the particle, then it has a real chance regardless of its height. Even if it does not have enough kinetic energy in the understanding of Newtonian physics. This is the same tunnel effect. It works as follows. It is typical to describe any particle not using any physical quantities, but through a wave function associated with the probability of the particle being located at a certain point in space at each specific unit of time.
When a particle collides with a certain barrier, using the Schrödinger equation, you can calculate the probability of overcoming this barrier. Since the barrier not only absorbs energy but also extinguishes it exponentially. In other words, in the quantum world there are no insurmountable barriers, but only additional conditions under which a particle can find itself beyond these barriers. Various obstacles, of course, interfere with the movement of particles, but are by no means solid, impenetrable boundaries. Conventionally speaking, this is a kind of borderland between two worlds - physical and energetic.
The tunnel effect has its analogue in nuclear physics - autoionization of an atom in a powerful electric field. Solid state physics also abounds in examples of tunneling manifestations. This includes field emission, migration, as well as effects that occur at the contact of two superconductors separated by a thin dielectric film. Tunneling plays an exceptional role in the implementation of numerous chemical processes under conditions of low and cryogenic temperatures.
Can a ball fly through a wall, so that the wall remains in place undamaged, and the energy of the ball does not change? Of course not, the answer suggests itself, this doesn’t happen in life. In order to fly through a wall, the ball must have sufficient energy to break through it. In the same way, if you want a ball in a hollow to roll over a hill, you need to provide it with a supply of energy sufficient to overcome the potential barrier - the difference in the potential energies of the ball at the top and in the hollow. Bodies whose motion is described by the laws of classical mechanics overcome the potential barrier only when they have a total energy greater than the maximum potential energy.
How is it going in the microcosm? Microparticles obey the laws of quantum mechanics. They do not move along certain trajectories, but are “smeared” in space, like a wave. These wave properties of microparticles lead to unexpected phenomena, and among them perhaps the most surprising is the tunnel effect.
It turns out that in the microcosm the “wall” can remain in place, and the electron flies through it as if nothing had happened.
Microparticles overcome the potential barrier, even if their energy is less than its height.
A potential barrier in the microcosm is often created by electrical forces, and this phenomenon was first encountered when atomic nuclei were irradiated with charged particles. It is unfavorable for a positively charged particle, such as a proton, to approach the nucleus, since, according to the law, repulsive forces act between the proton and the nucleus. Therefore, in order to bring a proton closer to the nucleus, work must be done; The potential energy graph looks like that shown in Fig. 1. True, it is enough for a proton to come close to the nucleus (at a distance of cm), and powerful nuclear forces of attraction (strong interaction) immediately come into play and it is captured by the nucleus. But you must first approach, overcome the potential barrier.
And it turned out that the proton can do this, even when its energy E is less than the barrier height. As always in quantum mechanics, it is impossible to say with certainty that the proton will penetrate the nucleus. But there is a certain probability of such a tunnel passage of a potential barrier. This probability is greater, the smaller the energy difference and the smaller the particle mass (and the dependence of the probability on the magnitude is very sharp - exponential).
Based on the idea of tunneling, D. Cockcroft and E. Walton discovered artificial fission of nuclei in 1932 at the Cavendish Laboratory. They built the first accelerator, and although the energy of the accelerated protons was not enough to overcome the potential barrier, the protons, thanks to the tunnel effect, penetrated into the nucleus and caused a nuclear reaction. The tunnel effect also explained the phenomenon of alpha decay.
The tunnel effect has found important applications in solid state physics and electronics.
Imagine that a metal film is applied to a glass plate (substrate) (usually it is obtained by depositing metal in a vacuum). Then it was oxidized, creating on the surface a layer of dielectric (oxide) only a few tens of angstroms thick. And again they covered it with a film of metal. The result will be a so-called “sandwich” (literally, this English word refers to two pieces of bread, for example, with cheese between them), or, in other words, a tunnel contact.
Can electrons move from one metal film to another? It would seem not - the dielectric layer interferes with them. In Fig. Figure 2 shows a graph of the dependence of the electron potential energy on the coordinate. In a metal, an electron moves freely and its potential energy is zero. To enter the dielectric, it is necessary to perform a work function, which is greater than the kinetic (and therefore total) energy of the electron.
Therefore, electrons in metal films are separated by a potential barrier, the height of which is equal to .
If electrons obeyed the laws of classical mechanics, then such a barrier would be insurmountable for them. But due to the tunneling effect, with some probability, electrons can penetrate through the dielectric from one metal film to another. Therefore, a thin dielectric film turns out to be permeable to electrons - a so-called tunnel current can flow through it. However, the total tunnel current is zero: the number of electrons that move from the lower metal film to the upper one, the same number on average moves, on the contrary, from the upper film to the lower one.
How can we make the tunnel current different from zero? To do this, it is necessary to break the symmetry, for example, connect metal films to a source with voltage U. Then the films will play the role of capacitor plates, and an electric field will arise in the dielectric layer. In this case, it is easier for electrons from the upper film to overcome the barrier than for electrons from the lower film. As a result, a tunnel current occurs even at low source voltages. Tunnel contacts make it possible to study the properties of electrons in metals and are also used in electronics.
TUNNEL EFFECT(tunneling) - quantum transition of a system through a region of motion prohibited by classical mechanics. A typical example of such a process is the passage of a particle through potential barrier when her energy
less than the height of the barrier. Particle momentum R in this case, determined from the relation 
Where U(x)- potential particle energy ( T- mass), would be in the region inside the barrier, an imaginary quantity. IN quantum mechanics thanks to uncertainty relationship Between the impulse and the coordinate, subbarrier motion becomes possible. The wave function of a particle in this region decays exponentially, and in the quasiclassical case (see Semiclassical approximation)its amplitude at the point of exit from under the barrier is small.
One of the formulations of problems about the passage of potential. barrier corresponds to the case when a stationary flow of particles falls on the barrier and it is necessary to find the value of the transmitted flow. For such problems, a coefficient is introduced. barrier transparency (tunnel transition coefficient) D, equal to the ratio of the intensities of the transmitted and incident flows. From the time reversibility it follows that the coefficient. Transparencies for transitions in the "forward" and reverse directions are the same. In the one-dimensional case, coefficient. transparency can be written as
integration is carried out over a classically inaccessible region, X 1,2 - turning points determined from the condition At turning points in the classical limit. mechanics, the momentum of the particle becomes zero. Coef. D 0 requires for its definition an exact solution of quantum mechanics. tasks.
If the condition of quasiclassicality is satisfied
along the entire length of the barrier, with the exception of the immediate neighborhoods of turning points x 1.2 coefficient D 0 is slightly different from one. Creatures difference D 0 from unity can be, for example, in cases where the potential curve. energy from one side of the barrier goes so steeply that the quasi-classical the approximation is not applicable there, or when the energy is close to the barrier height (i.e., the exponent expression is small). For a rectangular barrier height U o and width A coefficient transparency is determined by the file
Where 
The base of the barrier corresponds to zero energy. In quasiclassical case D small compared to unity.
Dr. The formulation of the problem of the passage of a particle through a barrier is as follows. Let the particle in the beginning moment in time is in a state close to the so-called. stationary state, which would happen with an impenetrable barrier (for example, with a barrier raised away from potential well to a height greater than the energy of the emitted particle). This state is called quasi-stationary. Similar to stationary states, the dependence of the wave function of a particle on time is given in this case by the factor 
The complex quantity appears here as energy E, the imaginary part determines the probability of decay of a quasi-stationary state per unit time due to T. e.:
In quasi-classical When approaching, the probability given by f-loy (3) contains an exponential. factor of the same type as in-f-le (1). In the case of a spherically symmetric potential. barrier is the probability of decay of a quasi-stationary state from orbits. l determined by f-loy
Here r 1,2 are radial turning points, the integrand in which is equal to zero. Factor w 0 depends on the nature of the movement in the classically allowed part of the potential, for example. he is proportional. classic frequency of the particle between the barrier walls.
T. e. allows us to understand the mechanism of a-decay of heavy nuclei. Between the particle and the daughter nucleus there is an electrostatic force. repulsion determined by f-loy At small distances of the order of size A the nuclei are such that eff. potential can be considered negative: 
As a result, the probability A-decay is given by the relation
Here is the energy of the emitted a-particle.
T. e. determines the possibility of thermonuclear reactions occurring in the Sun and stars at temperatures of tens and hundreds of millions of degrees (see. Evolution of stars), as well as in terrestrial conditions in the form of thermonuclear explosions or CTS.
In a symmetric potential, consisting of two identical wells separated by a weakly permeable barrier, i.e. leads to states in wells, which leads to weak double splitting of discrete energy levels (so-called inversion splitting; see Molecular spectra). For an infinitely periodic set of holes in space, each level turns into a zone of energies. This is the mechanism for the formation of narrow electron energies. zones in crystals with strong coupling of electrons to lattice sites.
If an electric current is applied to a semiconductor crystal. field, then the zones of allowed electron energies become inclined in space. Thus the post level. electron energy crosses all zones. Under these conditions, the transition of an electron from one energy level becomes possible. zones to another due to T. e. The classically inaccessible area is the zone of forbidden energies. This phenomenon is called. Zener breakdown. Quasiclassical the approximation corresponds here to a small value of electrical intensity. fields. In this limit, the probability of a Zener breakdown is determined basically. exponential, in the cut indicator there is a large negative. a value proportional to the ratio of the width of the forbidden energy. zone to the energy gained by an electron in an applied field at a distance equal to the size of the unit cell.
A similar effect appears in tunnel diodes, in which the zones are inclined due to semiconductors R- And n-type on both sides of the border of their contact. Tunneling occurs due to the fact that in the zone where the carrier goes there is a finite density of unoccupied states.
Thanks to T. e. electric possible current between two metals separated by a thin dielectric. partition. These metals can be in both normal and superconducting states. In the latter case there may be Josephson effect.
T. e. Such phenomena occurring in strong electric currents are due. fields, such as autoionization of atoms (see Field ionization)And auto-electronic emissions from metals. In both cases, electric the field forms a barrier of finite transparency. The stronger the electric field, the more transparent the barrier and the stronger the electron current from the metal. Based on this principle scanning tunneling microscope- a device that measures the tunneling current from different points of the surface under study and provides information about the nature of its heterogeneity.
T. e. is possible not only in quantum systems consisting of a single particle. Thus, for example, low-temperature motion in crystals can be associated with tunneling of the final part of a dislocation, consisting of many particles. In problems of this kind, a linear dislocation can be represented as an elastic string, initially lying along the axis at in one of the local minima of the potential V(x, y). This potential does not depend on at, and its relief along the axis X is a sequence of local minima, each of which is lower than the other by an amount depending on the mechanical force applied to the crystal. . The movement of a dislocation under the influence of this stress is reduced to tunneling into an adjacent minimum defined. segment of a dislocation with subsequent pulling of its remaining part there. The same kind of tunnel mechanism may be responsible for the movement charge density waves in Peierls (see Peierls transition).
To calculate the tunneling effects of such multidimensional quantum systems, it is convenient to use semiclassical methods. representation of the wave function in the form 
Where S- classic system action. For T. e. the imaginary part is significant S, which determines the attenuation of the wave function in a classically inaccessible region. To calculate it, the method of complex trajectories is used.
Quantum particle overcoming potential. barrier may be connected to the thermostat. In classic Mechanically, this corresponds to motion with friction. Thus, to describe tunneling it is necessary to use a theory called dissipative. Considerations of this kind must be used to explain the finite lifetime of current states of Josephson contacts. In this case, tunneling occurs. quantum particle through the barrier, and the role of a thermostat is played by normal electrons.
Lit.: Landau L. D., Lifshits E. M., Quantum Mechanics, 4th ed., M., 1989; Ziman J., Principles of Solid State Theory, trans. from English, 2nd ed., M., 1974; Baz A. I., Zeldovich Ya. B., Perelomov A. M., Scattering, reactions and decays in nonrelativistic quantum mechanics, 2nd ed., M., 1971; Tunnel phenomena in solids, trans. from English, M., 1973; Likharev K.K., Introduction to the dynamics of Josephson junctions, M., 1985. B. I. Ivlev.
There is a possibility that a quantum particle will penetrate a barrier that is insurmountable for a classical elementary particle.
Imagine a ball rolling inside a spherical hole dug in the ground. At any moment of time, the energy of the ball is distributed between its kinetic energy and the potential energy of gravity in a proportion depending on how high the ball is relative to the bottom of the hole (according to the first law of thermodynamics) . When the ball reaches the side of the hole, two scenarios are possible. If its total energy exceeds the potential energy of the gravitational field, determined by the height of the ball's location, it will jump out of the hole. If the total energy of the ball is less than the potential energy of gravity at the level of the side of the hole, the ball will roll down, back into the hole, towards the opposite side; at the moment when the potential energy is equal to the total energy of the ball, it will stop and roll back. In the second case, the ball will never roll out of the hole unless additional kinetic energy is given to it - for example, by pushing it. According to Newton's laws of mechanics , the ball will never leave the hole without giving it additional momentum if it does not have enough of its own energy to roll overboard.
Now imagine that the sides of the pit rise above the surface of the earth (like lunar craters). If the ball manages to fall over the raised side of such a hole, it will roll further. It is important to remember that in the Newtonian world of the ball and the hole, the fact that the ball will roll further over the side of the hole has no meaning if the ball does not have enough kinetic energy to reach the top edge. If it does not reach the edge, it simply will not get out of the hole and, accordingly, under no conditions, at any speed and will not roll anywhere further, no matter what height above the surface outside the edge of the side is.
In the world of quantum mechanics, things are different. Let's imagine that there is a quantum particle in something like such a hole. In this case, we are no longer talking about a real physical hole, but about a conditional situation when a particle requires a certain supply of energy necessary to overcome the barrier that prevents it from breaking out of what physicists have agreed to call "potential hole". This pit also has an energy analogue of the side - the so-called "potential barrier". So, if outside the potential barrier the level of energy field intensity is lower , than the energy that a particle possesses, it has a chance to be “overboard”, even if the real kinetic energy of this particle is not enough to “go over” the edge of the board in the Newtonian sense. This mechanism of a particle passing through a potential barrier is called the quantum tunneling effect.
It works like this: in quantum mechanics, a particle is described through a wave function, which is related to the probability of the particle being located in a given place at a given moment in time. If a particle collides with a potential barrier, Schrödinger's equation allows one to calculate the probability of a particle penetrating through it, since the wave function is not just energetically absorbed by the barrier, but is extinguished very quickly - exponentially. In other words, the potential barrier in the world of quantum mechanics is blurred. It, of course, prevents the particle from moving, but is not a solid, impenetrable boundary, as is the case in classical Newtonian mechanics.
If the barrier is low enough or if the total energy of the particle is close to the threshold, the wave function, although it decreases rapidly as the particle approaches the edge of the barrier, leaves it a chance to overcome it. That is, there is a certain probability that the particle will be detected on the other side of the potential barrier - in the world of Newtonian mechanics this would be impossible. And once the particle has crossed the edge of the barrier (let it have the shape of a lunar crater), it will freely roll down its outer slope away from the hole from which it emerged.
A quantum tunnel junction can be thought of as a kind of "leakage" or "percolation" of a particle through a potential barrier, after which the particle moves away from the barrier. There are plenty of examples of this kind of phenomena in nature, as well as in modern technologies. Take a typical radioactive decay: a heavy nucleus emits an alpha particle consisting of two protons and two neutrons. On the one hand, one can imagine this process in such a way that a heavy nucleus holds an alpha particle inside itself through intranuclear binding forces, just as the ball was held in the hole in our example. However, even if an alpha particle does not have enough free energy to overcome the barrier of intranuclear bonds, there is still a possibility of its separation from the nucleus. And by observing spontaneous alpha emission, we receive experimental confirmation of the reality of the tunnel effect.
Another important example of the tunnel effect is the process of thermonuclear fusion, which supplies energy to stars ( cm. Evolution of stars). One of the stages of thermonuclear fusion is the collision of two deuterium nuclei (one proton and one neutron each), resulting in the formation of a helium-3 nucleus (two protons and one neutron) and the emission of one neutron. According to Coulomb's law, between two particles with the same charge (in this case, protons that are part of deuterium nuclei) there is a powerful force of mutual repulsion - that is, there is a powerful potential barrier. In Newton's world, deuterium nuclei simply could not come close enough to synthesize a helium nucleus. However, in the depths of stars, the temperature and pressure are so high that the energy of the nuclei approaches the threshold of their fusion (in our sense, the nuclei are almost at the edge of the barrier), as a result of which the tunnel effect begins to operate, thermonuclear fusion occurs - and the stars shine.
Finally, the tunnel effect is already used in practice in electron microscope technology. The action of this tool is based on the fact that the metal tip of the probe approaches the surface under study at an extremely short distance. In this case, the potential barrier prevents electrons from metal atoms from flowing to the surface under study. When moving the probe at an extremely close distance along the surface being examined, he sorts it out atom by atom. When the probe is in close proximity to atoms, the barrier is lower , than when the probe passes in the spaces between them. Accordingly, when the device “gropes” for an atom, the current increases due to increased electron leakage as a result of the tunneling effect, and in the spaces between the atoms the current decreases. This allows for a detailed study of the atomic structures of surfaces, literally “mapping” them. By the way, electron microscopes provide the final confirmation of the atomic theory of the structure of matter.
The most striking representative of quantum size effects is the tunnel effect - a purely quantum phenomenon that played an important role in the development of modern electronics and instrument making. The phenomenon of tunneling was discovered in 1927 by our compatriot G. A. Gamow, who was the first to obtain solutions to the Schrödinger equation, which describe the possibility of a particle overcoming a potential barrier, even if its energy is less than the height of the barrier. The solutions found helped to understand many experimental data that could not be understood within the framework of the concepts of classical physics.
For the first time in physics, the tunnel effect was used to explain radioactive decay of atomic nuclei, for example:
The fact is that the particle - the nucleus of a helium atom - does not have sufficient energy to leave the unstable nucleus. On this path, the particle needs to overcome a huge (28 MeV), but rather narrow (10 -12 cm - radius of the nucleus) potential barrier. The Soviet scientist G. Gamow (1927) showed that the disintegration of the atomic nucleus in this case becomes possible precisely due to the tunneling of the particle transfer. Thanks to the tunnel effect, cold emission of electrons from metals and many other phenomena also occur. Many believe that for the grandeur of the results of his work, which became fundamental for many sciences, G.A. Gamow was to be awarded several Nobel Prizes. Only thirty years after the discovery of G. A. Gamow, the first devices based on the tunnel effect appeared - tunnel diodes, transistors, sensors, thermometers for measuring ultra-low temperatures, and, finally, scanning tunnel microscopes, which laid the foundation for modern research on nanostructures. The tunneling effect is the process of a microparticle overcoming a potential barrier in the case when its total energy (remaining unchanged during tunneling) is less than the height of the barrier. The tunnel effect is a phenomenon of exclusively quantum nature, which could not be explained within the framework of classical concepts. An analogue of the tunnel effect in wave optics can be the penetration of a light wave into a reflecting medium (at distances on the order of the light wavelength) under conditions where, from the point of view of geometric optics, total internal reflection occurs. In general, the tunneling effect is the process of a microparticle overcoming a potential barrier in the case when its total energy (which remains unchanged during tunneling) is less than the height of the barrier. In classical mechanics, motion occurs under the condition that the total energy of the particle is greater than its potential energy, i.e. there is an inequality:
Since the total energy is equal to the sum of the kinetic and potential energies:
and kinetic energy is greater than zero, then, accordingly, the difference between the total and potential energies will also be greater than zero:
and thus the following condition will be satisfied:
It should be noted that the problem of particle motion in a potential box satisfies this condition, since inside the box the potential energy is zero. However, in quantum mechanics, motion is also possible under the condition that the total energy is less than the potential energy. Such tasks are united by a common name - potential barriers. Consider a potential barrier of a rectangular shape. Let the potential value in region I be zero, . In region II, the value of the potential energy is equally determined by the height of the barrier and thus . In region III, the potential energy value is zero, . Let us denote the wave functions for the regions: for region I, for region II and for region III. In this problem we will be interested in the case when the total energy of the particle is less than the height of the potential barrier, i.e. provided that .
Fig.8. Passage of a particle through a potential barrier
For each of the three regions, we write down the Schrödinger equation, bring it to standard form and describe its general solutions. Let us consider the motion of a particle in region I. Let us denote the wave function of the particle in this case . As in the case of free particle motion, the corresponding Schrödinger equation will be written as:
from which it follows that:
the general solution of the Schrödinger equation for region I can be written as:
the first part of the function can be interpreted as a wave incident on the potential barrier (particle movement from left to right in region I). The coefficients and are called the amplitudes of the incident and reflected waves, respectively. They determine the probability of a wave passing through a potential barrier, as well as the probability of its reflection from the barrier. Since the expansion coefficients in the expression for the wave function are related to the intensity of the beam of particles moving towards the barrier or reflected from it, then, accordingly, taking the amplitude of the incident wave , we will have:
Let us now consider the motion of a particle in region II. In the conditions of this problem, the case of physical interest for us will be when the total energy of the particle is less than the height of the potential barrier, which corresponds to the fulfillment of a condition of the form:
since for area II:
those. The value of the potential energy of a particle is determined by the height of the barrier - the size of the region:
then the Schrödinger equation for region II will have the form:
from which it follows that:
