tunnel effect
Tunneling effect
tunnel effect (tunneling) - the passage of a particle (or system) through a region of space in which it is forbidden to stay classical mechanics. Most famous example such a process is the passage of a particle through a potential barrier when its energy E is less than the height of the barrier U 0 . In classical physics, a particle cannot be in the area of such a barrier, much less pass through it, since this violates the law of conservation of energy. However, in quantum physics the situation is fundamentally different. A quantum particle does not move along any particular trajectory. Therefore, we can only talk about the probability of finding a particle in a certain region of space ΔрΔх > ћ. At the same time, neither potential nor kinetic energies have definite values in accordance with the uncertainty principle. It is allowed to deviate from the classical energy Е by the value ΔЕ during the time intervals t given by the uncertainty relation ΔЕΔt > ћ (ћ = h/2π, where h is Planck's constant).
The possibility of a particle passing through a potential barrier is due to the requirement of a continuous wave function on the walls of the potential barrier. The probability of detecting a particle on the right and on the left is related by a relation that depends on the difference E - U(x) in the region of the potential barrier and on the width of the barrier x 1 - x 2 at a given energy.
As the height and width of the barrier increase, the probability of the tunneling effect decreases exponentially. The probability of the tunnel effect also decreases rapidly with increasing particle mass.
Penetration through the barrier is probabilistic. Particle with E< U 0 , натолкнувшись на барьер, может либо пройти сквозь него,
либо отразиться. Суммарная вероятность этих двух возможностей равна 1. Если
на барьер падает поток частиц с Е < U 0 , то часть этого
потока будет просачиваться сквозь барьер, а часть – отражаться. Туннельное
прохождение частицы через потенциальный барьер лежит в основе многих явлений
ядерной и atomic physics: alpha decay, cold emission of electrons from metals, phenomena in the contact layer of two semiconductors, etc.
There is a possibility that a quantum particle will penetrate the barrier, which is insurmountable for a classical elementary particle.
Imagine a ball rolling inside a spherical hole dug in the ground. At any moment in 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 edge of the hole, two scenarios are possible. If its total energy exceeds potential energy gravitational field, determined by the height of the point where the ball is located, 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 the laws of Newtonian mechanics, the ball will never leave the hole without giving it additional momentum if it does not have enough energy of its own 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 pass over the raised side of such a pit, it will roll further. It is important to remember that in the Newtonian world of the ball and the hole, the very fact that, having gone over the side of the hole, the ball will roll further, it does not make sense if the ball does not have enough kinetic energy to reach the top. If he does not reach the edge, he simply will not get out of the pit and, accordingly, under no circumstances, at any speed, will he roll anywhere further, no matter at what height above the surface the edge of the side is outside.
In the world of quantum mechanics, things are different. Imagine that there is a quantum particle in something like such a well. In this case we are talking is no longer about a real physical well, but about a conditional situation when a particle needs a certain amount 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 tension energy field lower than the energy possessed by the particle, it has a chance to be "overboard", even if the real kinetic energy of this particle is not enough to "pass" over the edge of the board in the Newtonian sense. This mechanism of passage of a particle through a potential barrier is called the quantum tunneling effect.
It works like this: in quantum mechanics, a particle is described in terms of a wave function, which is related to the probability of the particle's location in this place in this moment time. If a particle collides with a potential barrier, the Schrödinger equation allows us to calculate the probability of the particle penetrating through it, since the wave function is not only 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, hinders the motion of the particle, but is not a solid, impenetrable boundary, as is the case in Newton's classical mechanics.
If the barrier is low enough, or if the total energy of the particle is close to the threshold, the wave function, although decreasing 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 found on the other side of the potential barrier - in the world of Newtonian mechanics, this would be impossible. And since the particle has passed over the edge of the barrier (let it have the form lunar crater), she will roll freely down its outer slope away from the hole from which she got out.
A quantum tunneling transition can be viewed as a kind of "leakage" or "leakage" of a particle through a potential barrier, after which the particle moves away from the barrier. There are enough examples of such phenomena in nature, as well as in modern technologies. Let's take a typical radioactive decay: heavy core emits an alpha particle consisting of two protons and two neutrons. On the one hand, this process can be imagined in such a way that a heavy nucleus holds an alpha particle inside itself by means of intranuclear binding forces, just as the ball was held in a hole in our example. However, even if the alpha particle does not have enough free energy to overcome the barrier of intranuclear bonds, there is still a possibility of its detachment from the nucleus. And by observing spontaneous alpha emission, we get experimental confirmation reality of the tunnel effect.
Another important example tunnel effect - the process of thermonuclear fusion that feeds energy to stars (see Evolution of stars). One of the stages of thermonuclear fusion is the collision of two deuterium nuclei (one proton and one neutron each), as a result of which a helium-3 nucleus (two protons and one neutron) is formed and one neutron is emitted. According to Coulomb's law, between two particles with the same charge (in this case protons that make up the nuclei of deuterium) 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 get close enough to synthesize a helium nucleus. However, in the interiors 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- and the stars shine.
Finally, the tunnel effect is already being used in practice in the technology of electron microscopes. The action of this tool is based on the fact that the metal tip of the probe approaches the surface under examination at an ultra-small distance. In this case, the potential barrier does not allow electrons from the metal atoms to flow to the surface under study. When moving the probe to the limit close range along the surface under study, he, as it were, goes over atom by atom. When the probe is in close proximity to the atoms, the barrier is lower than when the probe passes between them. Accordingly, when the device "gropes" an atom, the current increases due to the increase in the leakage of electrons as a result of the tunneling effect, and in the gaps between the atoms, the current decreases. This allows in the most detailed way explore the atomic structures of surfaces, literally "mapping" them. By the way, electron microscopes just give the final confirmation of the atomic theory of the structure of matter.
1.7. The concept of the tunnel effect.
The tunnel effect is the passage of particles through a potential barrier due to wave properties particles.
Let a particle moving from left to right encounter a potential barrier with a height U 0 and width l. According to classical concepts, a particle passes unhindered over a barrier if its energy E greater than the barrier height ( E> U 0 ). If the particle energy is less than the barrier height ( E< U 0 ), then the particle is reflected from the barrier and starts moving in the opposite direction, the particle cannot penetrate through the barrier.
Quantum mechanics takes into account the wave properties of particles. For a wave, the left wall of the barrier is the boundary of two media, on which the wave is divided into two waves - reflected and refracted. Therefore, even with E> U 0 it is possible (although with a low probability) that the particle is reflected from the barrier, and when E< U 0 there is a non-zero probability that the particle will be on the other side of the potential barrier. In this case, the particle, as it were, "passed through the tunnel".
We will decide the problem of the passage of a particle through a potential barrier for the simplest case of a one-dimensional rectangular barrier shown in Fig. 1.6. The shape of the barrier is given by the function
. (1.7.1)
We write the Schrödinger equation for each of the regions: 1( x<0 ), 2(0< x< l) and 3( x> l):
;
(1.7.2)
; (1.7.3)
. (1.7.4)
Denote
(1.7.5)
. (1.7.6)
General solutions of equations (1), (2), (3) for each of the regions have the form:
Solution of the form 
corresponds to a wave propagating in the direction of the axis x, a 
a wave propagating in the opposite direction. In region 1, the term 
describes the wave incident on the barrier, and the term 
the wave reflected from the barrier. In region 3 (to the right of the barrier) there is only a wave propagating in the x direction, so 
.
The wave function must satisfy the continuity condition, so the solutions (6), (7), (8) at the boundaries of the potential barrier must be "sewn". To do this, we equate the wave functions and their derivatives at x=0 and x = l:
;
;
;
. (1.7.10)
Using (1.7.7) - (1.7.10), we obtain four equations to determine five coefficients BUT 1 , BUT 2 , BUT 3 ,AT 1 and AT 2 :
BUT 1 +V 1 =A 2 +V 2 ;
BUT 2 exp( l) + B 2 exp(- l)= A 3 exp(ikl) ;
ik(BUT 1 - AT 1 ) = (BUT 2 -AT 2 ) ; (1.7.11)
(BUT 2 exp( l)-AT 2 exp(- l) = ikBUT 3 exp(ikl) .
To obtain the fifth relation, we introduce the concepts of reflection coefficients and barrier transparency.
Reflection coefficient let's call the relation
, (1.7.12)
which defines probability particle reflections from the barrier.
transparency ratio
(1.7.13)
gives the probability that the particle will pass through the barrier. Since the particle will either be reflected or pass through the barrier, the sum of these probabilities is equal to one. Then
R+ D =1; (1.7.14)
. (1.7.15)
That's what it is fifth relation that closes the system (1.7.11), from which all five coefficients.
Of greatest interest is transparency ratioD. After transformations, we get
,
(7.1.16)
where D 0 is a value close to unity.
It can be seen from (1.7.16) that the transparency of the barrier strongly depends on its width l, on how much the height of the barrier U 0 exceeds particle energy E, as well as on the mass of the particle m.
With 
classical point of view, the passage of a particle through a potential barrier at E<
U 0
contradicts the law of conservation of energy. The fact is that if a classical particle were at some point in the barrier region (region 2 in Fig. 1.7), then its total energy would be less than the potential energy (and its kinetic energy would be negative!?). With quantum dot there is no such contradiction. If a particle moves towards a barrier, then it has a well-defined energy before it collides with it. Let the interaction with the barrier last for a while
t, then, according to the uncertainty relation, the energy of the particle will no longer be determined; energy uncertainty 
. When this uncertainty turns out to be of the order of the height of the barrier, it ceases to be an insurmountable obstacle for the particle, and the particle will pass through it.
The transparency of the barrier decreases sharply with its width (see Table 1.1.). Therefore, particles can only pass through very narrow potential barriers due to the tunneling mechanism.
Table 1.1
The values of the transparency coefficient for an electron at ( U 0 – E ) = 5 eV = const
|
l, nm | |||||
We considered a rectangular barrier. In the case of a potential barrier of arbitrary shape, for example, as shown in Fig. 1.7, the transparency coefficient has the form
. (1.7.17)
The tunnel effect manifests itself in a number of physical phenomena and has important practical applications. Let's give some examples.
1. Autoelectronic (cold) electron emission.
AT 
In 1922, the phenomenon of cold electron emission from metals under the action of a strong external electric field was discovered. Potential Energy Graph U electron from the coordinate x shown in fig. At x
<
0 is the region of the metal in which electrons can move almost freely. Here the potential energy can be considered constant. A potential wall appears at the metal boundary, which does not allow the electron to leave the metal, it can do this only by acquiring additional energy, equal to work exit A.
Outside the metal (at x >
0) the energy of free electrons does not change, therefore, for x> 0, the graph U(x)
goes horizontally. Let us now create a strong electric field near the metal. To do this, take a metal sample in the form of a sharp needle and connect it to the negative pole of the source. Rice. 1.9 How the tunneling microscope works
ka voltage, (it will be the cathode); we will place another electrode (anode) nearby, to which we will attach the positive pole of the source. With a sufficiently large potential difference between the anode and cathode, an electric field with a strength of about 10 8 V/m can be created near the cathode. The potential barrier at the metal-vacuum boundary becomes narrow, the electrons seep through it and leave the metal.
Field emission was used to create electronic tubes with cold cathodes (now they are practically out of use), at present it has found application in tunneling microscopes, invented in 1985 by J. Binning, G. Rohrer and E. Ruska.
In a tunnel microscope, a probe, a thin needle, moves along the surface under study. The needle scans the surface under study, being so close to it that electrons from the electron shells (electron clouds) of surface atoms due to wave properties can get to the needle. To do this, we apply “plus” from the source to the needle, and “minus” to the test sample. The tunneling current is proportional to the coefficient of transparency of the potential barrier between the needle and the surface, which, according to formula (1.7.16), depends on the width of the barrier l. When scanning the sample surface with a needle, the tunneling current changes depending on the distance l, repeating the profile of the surface. Precise movement of the needle over short distances is carried out using the piezoelectric effect, for this purpose the needle is fixed on a quartz plate, which expands or contracts when an electrical voltage is applied to it. Modern technology makes it possible to make a needle so thin that a single atom is located at its end.
And 
the image is formed on the computer display screen. The resolution of a tunneling microscope is so high that it allows you to "see" the arrangement of individual atoms. Figure 1.10 shows an example of an atomic surface of silicon.
2. Alpha radioactivity ( - decay). In this phenomenon, a spontaneous transformation of radioactive nuclei occurs, as a result of which one nucleus (it is called the parent) emits a -particle and turns into a new (daughter) nucleus with a charge less than 2 units. Recall that the particle (the nucleus of the helium atom) consists of two protons and two neutrons.
E 
If we assume that the -particle exists as a single formation inside the nucleus, then the plot of its potential energy versus coordinate in the field of the radioactive nucleus has the form shown in Fig. 1.11. It is determined by the energy of the strong (nuclear) interaction, due to the attraction of nucleons to each other, and the energy of the Coulomb interaction (electrostatic repulsion of protons).
As a result, is a particle in the nucleus, which has the energy E is behind the potential barrier. Due to its wave properties, there is some probability that the -particle will be outside the nucleus.
3. Tunnel effect inp- n- transition used in two classes of semiconductor devices: tunnel and inverted diodes. A feature of tunnel diodes is the presence of a falling section on the straight branch of the current-voltage characteristic - a section with a negative differential resistance. In reversed diodes, the most interesting thing is that when turned back on, the resistance is less than when it is turned back on. See section 5.6 for details on tunnel and reverse diodes.
tunnel effect - amazing phenomenon, which is completely impossible from the point of view of classical physics. But in the mysterious and mysterious quantum world, there are somewhat different laws of the interaction of matter and energy. The tunnel effect is a process of overcoming a certain potential barrier, provided that its energy is less than the height of the barrier. This phenomenon has an exclusively quantum nature and completely contradicts all laws and dogmas. classical mechanics. Tem more amazing world in which we live.
To understand what the quantum tunnel effect is, the best way is to use the example of a golf ball launched with some force into the hole. At any unit of time, the total energy of the ball is in opposition to potential power 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 his way, he will definitely need an additional kinetic momentum. So the great Newton spoke.
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 deepening in the earth, but about what physicists conventionally call a "potential hole". This value also has an analogue of the physical board - an energy barrier. This is where the situation changes dramatically. For the so-called quantum transition and the particle is outside the barrier, another condition is necessary.
If the intensity of the external energy field smaller particle then she has 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. description of any particle is characteristic not with the help of some physical quantities, but through the wave function associated with the probability of the location of the particle at a certain point in space in each specific unit of time.
When a particle collides with a certain barrier, using the Schrödinger equation, one 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 terms, at which the particle can be outside these barriers. Various obstacles, of course, interfere with the movement of particles, but by no means are solid impenetrable boundaries. Conditionally speaking, this is a kind of borderline between two worlds - physical and energy.
The tunnel effect has its analogue in nuclear physics - the autoionization of an atom in a powerful electric field. Physics also abounds with examples of the manifestation of tunneling. solid body. These include field emission, migration, as well as effects that arise at the contact of two superconductors separated by a thin dielectric film. Tunneling plays an exceptional role in the implementation of numerous chemical processes at low and cryogenic temperatures.
