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 electron potential energy versus position. 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 tunnel 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, a quantum effect consisting in the penetration of a quantum particle through a region of space, into which, according to the laws of classical physics, finding a particle is prohibited. Classic a particle with total energy E and in potential. field can only reside in those regions of space in which its total energy does not exceed the potential. energy U of interaction with the field. Since the wave function of a quantum particle is nonzero throughout space and the probability of finding a particle in a certain region of space is given by the square of the modulus of the wave function, then in forbidden (from the point of view of classical mechanics) regions the wave function is nonzero.

T It is convenient to illustrate the tunnel effect using a model problem of a one-dimensional particle in a potential field U(x) (x is the coordinate of the particle). In the case of a symmetrical double-well potential (Fig. a), the wave function must “fit” inside the wells, i.e., it is a standing wave. Discrete energy sources levels that are located below the barrier separating the minima of the potential form closely spaced (almost degenerate) levels. Energy difference levels, components, called. tunnel splitting, this difference is due to the fact that the exact solution of the problem (wave function) for each of the cases is localized in both minima of the potential and all exact solutions correspond to non-degenerate levels (see). The probability of the tunnel effect is determined by the coefficient of transmission of a wave packet through the barrier, which describes the non-stationary state of a particle localized in one of the potential minima.

Potential curves energy U (x) of a particle in the case when it is acted upon by an attractive force (a - two potential wells, b - one potential well), and in the case when a repulsive force acts on the particle (repulsive potential, c). E is the total energy of the particle, x is the coordinate. Thin lines depict wave functions.

In potential field with one local minimum (Fig. b) for a particle with energy E greater than the interaction potential at c =, discrete energy. there are no states, but there is a set of quasi-stationary states, in which the great relates. the probability of finding a particle near the minimum. Wave packets corresponding to such quasi-stationary states describe metastable ones; wave packets spread out and disappear due to the tunnel effect. These states are characterized by their lifetime (probability of decay) and energy width. level.

For a particle in a repulsive potential (Fig. c), a wave packet describing a non-stationary state on one side of the potential. barrier, even if the energy of a particle in this state is less than the height of the barrier, it can, with a certain probability (called the probability of penetration or the probability of tunneling), pass on the other side of the barrier.

Naib. important for the manifestation of the tunnel effect: 1) tunnel splitting of discrete oscillations, rotation. and electronic-co-lebat. levels. Splitting of oscillations. levels in with several. equivalent equilibrium nuclear configurations is inversion doubling (in type), splitting of levels in with inhibited internal. rotation ( , ) or in , for which intra-mol. rearrangements leading to equivalent equilibrium configurations (eg PF 5). If different equivalent minima are not separated by potential. barriers (for example, equilibrium configurations for right- and left-handed complexes), then an adequate description of real piers. systems is achieved using localized wave packets. In this case, stationary states localized in two minima are unstable: under the influence of very small perturbations, the formation of two states localized in one or another minimum is possible.

The splitting of quasi-degenerate groups rotates. states (the so-called rotational clusters) is also due to the tunneling of the mol. systems between several neighborhoods. equivalent stationary axes of rotation. Splitting of electron vibrations. (vibronic) states occurs in the case of strong Jahn-Teller effects. Tunnel splitting is also associated with the existence of bands formed by electronic states of individual or molecular states. fragments in periodic structure.

2) Phenomena of particle transfer and elementary excitations. This set of phenomena includes non-stationary processes that describe transitions between discrete states and the decay of quasi-stationary states. Transitions between discrete states with wave functions localized in different states. minimums of one adiabatic. potential, correspond to a variety of chemicals. r-tions. The tunnel effect always makes a certain contribution to the rate of transformation, but this contribution is significant only at low temperatures, when the above-barrier transition from the initial state to the final state is unlikely due to the low population of the corresponding energy levels. The tunnel effect manifests itself in the non-Arrhenius behavior of the r-tion velocity; A typical example is the growth of a chain during radiation-initiated solids. The speed of this process at temperature is approx. 140 K is satisfactorily described by the Arrhenius law with

1.7. The concept of the tunnel effect.

The tunnel effect is the passage of particles through a potential barrier due to the wave properties of the particles.

Let a particle moving from left to right encounter a potential barrier of 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 begins to move 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, at which the wave is divided into two waves - reflected and refracted. Therefore, even with E> U 0 it is possible (albeit with a small probability) that a particle is reflected from the barrier, and when E< U 0 there is a nonzero probability that the particle will be on the other side of the potential barrier. In this case, the particle seemed to “pass through a tunnel.”

Let's decide the problem of a particle passing 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 specified by the function

. (1.7.1)

Let us 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)

Let's denote

(1.7.5)

. (1.7.6)

General solutions of equations (1), (2), (3) for each of the areas 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 term
describes a wave incident on a barrier, and the term
- 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, therefore solutions (6), (7), (8) at the boundaries of the potential barrier must be “stitched”. 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 A 1 , A 2 , A 3 ,IN 1 And IN 2 :

A 1 +B 1 =A 2 +B 2 ;

A 2 exp( l) + B 2 exp(-  l)= A 3 exp(ikl) ;

ik(A 1 - IN 1 ) =  (A 2 -IN 2 ) ; (1.7.11)

 (A 2 exp( l)-IN 2 exp(-  l) = ikA 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 reflection of a particle from a barrier.

Transparency factor

(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)

This is it fifth relationship that closes the system (1.7.11), from which all five coefficients

Of greatest interest is transparency coefficientD. After transformations we get

, (7.1.16)

Where D 0 – value close to unity.

From (1.7.16) it is clear that the transparency of the barrier strongly depends on its width l, on how high the barrier is U 0 exceeds the particle energy E, and also on the mass of the particle m.

WITH from the 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 the kinetic energy would be negative!?). From a quantum point of view, there is no such contradiction. If a particle moves towards a barrier, then before colliding with it it has a very specific energy. 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 definite; energy uncertainty
. When this uncertainty turns out to be on 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 pass through only very narrow potential barriers due to the tunneling mechanism.

Table 1.1

Values ​​of the transparency coefficient for an electron at ( U 0 – E ) = 5 eV = const

We considered a rectangular-shaped 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. Field electron (cold) emission of electrons.

IN In 1922, the phenomenon of cold electron emission from metals under the influence of a strong external electric field was discovered. Potential Energy Graph U electron from 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, preventing the electron from leaving the metal; it can do this only by acquiring additional energy equal to the work function A. Outside the metal (at x > 0) the energy of free electrons does not change, so when 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 shape of a sharp needle and connect it to the negative pole of the source. Rice. 1.9 Operating principle of a tunnel microscope

ka voltage, (it will be the cathode); We will place another electrode (anode) nearby, to which we will connect the positive pole of the source. If the potential difference between the anode and the cathode is large enough, it is possible to create an electric field with a strength of about 10 8 V/m near the cathode. The potential barrier at the metal-vacuum interface becomes narrow, electrons leak through it and leave the metal.

Field emission was used to create vacuum tubes with cold cathodes (they are now practically out of use); it has now found application in tunnel 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 reach the needle. To do this, we apply a “plus” from the source to the needle, and a “minus” to the sample under study. The tunnel current is proportional to the transparency coefficient of the potential barrier between the needle and the surface, which, according to formula (1.7.16), depends on the barrier width l. When scanning the surface of a sample with a needle, the tunneling current varies depending on the distance l, repeating the surface profile. Precision movements of the needle over short distances are carried out using the piezoelectric effect; for this, the needle is fixed on a quartz plate, which expands or contracts when an electrical voltage is applied to it. Modern technologies make it possible to produce a needle so thin that there is only one atom 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 image of the 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 mother nucleus) emits an  particle and turns into a new (daughter) nucleus with a charge less than 2 units. Let us recall that the  particle (the nucleus of a helium atom) consists of two protons and two neutrons.

E If we assume that the α-particle exists as a single formation inside the nucleus, then the graph of the dependence of its potential energy on the 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, caused by 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 with energy E is located behind the potential barrier. Due to its wave properties, there is some probability that the  particle will end up outside the nucleus.

3. Tunnel effect inp- n- transition used in two classes of semiconductor devices: tunnel And reversed diodes. A feature of tunnel diodes is the presence of a falling section on the direct branch of the current-voltage characteristic - a section with a negative differential resistance. The most interesting thing about reverse diodes is that when connected in reverse, the resistance is less than when connected in reverse. For more information on tunnel and reverse diodes, see section 5.6.

• Translation

I'll start with two simple questions with fairly intuitive answers. Let's take a bowl and a ball (Fig. 1). If I need to:

The ball remained motionless after I placed it in the bowl, and
it remained in approximately the same position when moving the bowl,

So where should I put it?

Rice. 1

Of course, I need to put it in the center, at the very bottom. Why? Intuitively, if I put it somewhere else, it will roll to the bottom and flop back and forth. As a result, friction will reduce the height of the dangling and slow it down below.

In principle, you can try to balance the ball on the edge of the bowl. But if I shake it a little, the ball will lose its balance and fall. So this place doesn't meet the second criterion in my question.

Let us call the position in which the ball remains motionless, and from which it does not deviate much with small movements of the bowl or ball, “stable position of the ball.” The bottom of the bowl is such a stable position.

Another question. If I have two bowls like in fig. 2, where will be the stable positions for the ball? This is also simple: there are two such places, namely, at the bottom of each of the bowls.

Rice. 2

Finally, another question with an intuitive answer. If I place a ball at the bottom of bowl 1, and then leave the room, close it, ensure that no one goes in there, check that there have been no earthquakes or other shocks in this place, then what are the chances that in ten years when I If I open the room again, I will find a ball at the bottom of bowl 2? Of course, zero. In order for the ball to move from the bottom of bowl 1 to the bottom of bowl 2, someone or something must take the ball and move it from place to place, over the edge of bowl 1, towards bowl 2 and then over the edge of bowl 2. Obviously, the ball will remain at the bottom of the bowl 1.

Obviously and essentially true. And yet, in the quantum world in which we live, no object remains truly motionless, and its position is not known with certainty. So none of these answers are 100% correct.

Tunneling

If I place an elementary particle like an electron in a magnetic trap (Fig. 3) that works like a bowl, tending to push the electron towards the center in the same way that gravity and the walls of the bowl push the ball towards the center of the bowl in Fig. 1, then what will be the stable position of the electron? As one would intuitively expect, the average position of the electron will be stationary only if it is placed at the center of the trap.

But quantum mechanics adds one nuance. The electron cannot remain stationary; its position is subject to "quantum jitter". Because of this, its position and movement are constantly changing, or even have a certain amount of uncertainty (this is the famous “uncertainty principle”). Only the average position of the electron is at the center of the trap; if you look at the electron, it will be somewhere else in the trap, close to the center, but not quite there. An electron is stationary only in this sense: it usually moves, but its movement is random, and since it is trapped, on average it does not move anywhere.

This is a little strange, but it just reflects the fact that an electron is not what you think it is and does not behave like any object you have seen.

This, by the way, also ensures that the electron cannot be balanced at the edge of the trap, unlike the ball at the edge of the bowl (as below in Fig. 1). The electron's position is not precisely defined, so it cannot be precisely balanced; therefore, even without shaking the trap, the electron will lose its balance and fall off almost immediately.

But what's weirder is the case where I'll have two traps separated from each other, and I'll place an electron in one of them. Yes, the center of one of the traps is a good, stable position for the electron. This is true in the sense that the electron can remain there and will not escape if the trap is shaken.

However, if I place an electron in trap No. 1 and leave, close the room, etc., there is a certain probability (Fig. 4) that when I return the electron will be in trap No. 2.

Rice. 4

How did he do it? If you imagine electrons as balls, you won't understand this. But electrons are not like marbles (or at least not like your intuitive idea of ​​marbles), and their quantum jitter gives them an extremely small but non-zero chance of "walking through walls" - the seemingly impossible possibility of moving to the other side. This is called tunneling - but don't think of the electron as digging a hole in the wall. And you will never be able to catch him in the wall - red-handed, so to speak. It's just that the wall isn't completely impenetrable to things like electrons; electrons cannot be trapped so easily.

In fact, it's even crazier: since it's true for an electron, it's also true for a ball in a vase. The ball may end up in vase 2 if you wait long enough. But the likelihood of this is extremely low. So small that even if you wait a billion years, or even billions of billions of billions of years, it won’t be enough. From a practical point of view, this will “never” happen.

Our world is quantum, and all objects are made of elementary particles and obey the rules of quantum physics. Quantum jitter is always present. But most objects whose mass is large compared to the mass of elementary particles - a ball, for example, or even a speck of dust - this quantum jitter is too small to be detected, except in specially designed experiments. And the resulting possibility of tunneling through walls is also not observed in ordinary life.

In other words: any object can tunnel through a wall, but the likelihood of this usually decreases sharply if:

The object has a large mass,
the wall is thick (large distance between two sides),
the wall is difficult to overcome (it takes a lot of energy to break through a wall).

In principle the ball can get over the edge of the bowl, but in practice this may not be possible. It can be easy for an electron to escape from a trap if the traps are close and not very deep, but it can be very difficult if they are far away and very deep.

Is tunneling really happening?

Or maybe this tunneling is just a theory? Definitely not. It is fundamental to chemistry, occurs in many materials, plays a role in biology, and is the principle used in our most sophisticated and powerful microscopes.

For the sake of brevity, let me focus on the microscope. In Fig. Figure 5 shows an image of atoms taken using a scanning tunneling microscope. Such a microscope has a narrow needle, the tip of which moves in close proximity to the material being studied (see Fig. 6). The material and the needle are, of course, made of atoms; and at the back of the atoms are electrons. Roughly speaking, electrons are trapped inside the material being studied or at the tip of the microscope. But the closer the tip is to the surface, the more likely the tunneling transition of electrons between them is. A simple device (a potential difference is maintained between the material and the needle) ensures that electrons prefer to jump from the surface to the needle, and this flow is a measurable electrical current. The needle moves over the surface, and the surface appears closer or further from the tip, and the current changes - it becomes stronger as the distance decreases and weaker as it increases. By tracking the current (or, alternatively, moving the needle up and down to maintain a constant current) as it scans a surface, the microscope infers the shape of that surface, often with enough detail to see individual atoms.

Rice. 6

Tunneling plays many other roles in nature and modern technology.

Tunneling between traps of different depths

Now let us assume that one electron trap in Fig. 4 deeper than the other - exactly the same as if one bowl in fig. 2 was deeper than the other (see Fig. 7). Although an electron can tunnel in any direction, it will be much easier for it to tunnel from a shallower to a deeper trap than vice versa. Accordingly, if we wait long enough for the electron to have enough time to tunnel in either direction and return, and then start taking measurements to determine its location, we will most often find it deeply trapped. (In fact, there are some nuances here too; everything also depends on the shape of the trap). Moreover, the difference in depth does not have to be large for tunneling from a deeper to a shallower trap to become extremely rare.

In short, tunneling will generally occur in both directions, but the probability of going from a shallow to a deep trap is much greater.

Rice. 7

It is this feature that a scanning tunneling microscope uses to ensure that electrons only travel in one direction. Essentially, the tip of the microscope needle is trapped deeper than the surface being studied, so electrons prefer to tunnel from the surface to the needle rather than vice versa. But the microscope will work in the opposite case. The traps are made deeper or shallower by using a power source that creates a potential difference between the tip and the surface, which creates a difference in energy between the electrons on the tip and the electrons on the surface. Since it is quite easy to make electrons tunnel more often in one direction than another, this tunneling becomes practically useful for use in electronics.

(solving the problems of the PHYSICS block, as well as other blocks, will allow you to select THREE people for the full-time round who scored the highest number of points when solving the problems of THIS block. Additionally, based on the results of the head-to-head round, these candidates will compete for a special nomination “ Physics of nanosystems" Another 5 people with the highest scores will also be selected for the full-time round. absolute number of points, so after solving problems in your specialty, it makes complete sense to solve problems from other blocks. )

One of the main differences between nanostructures and macroscopic bodies is the dependence of their chemical and physical properties on size. A clear example of this is the tunnel effect, which consists in the penetration of light particles (electrons, protons) into areas that are energetically inaccessible to them. This effect plays an important role in processes such as charge transfer in the photosynthetic devices of living organisms (it is worth noting that biological reaction centers are among the most efficient nanostructures).

The tunnel effect can be explained by the wave nature of light particles and the uncertainty principle. Due to the fact that small particles do not have a specific position in space, there is no concept of trajectory for them. Consequently, to move from one point to another, a particle does not have to pass along the line connecting them, and thus can “bypass” energy-forbidden regions. Due to the absence of an exact coordinate for an electron, its state is described using a wave function that characterizes the probability distribution along the coordinate. The figure shows a typical wave function when tunneling under an energy barrier.

Probability p penetration of an electron through a potential barrier depends on the height U and the width of the latter l ( formula 1, left), Where m– electron mass, E– electron energy, h – Planck’s constant with a bar.

1. Determine the probability that an electron tunnels to a distance of 0.1 nm if the energy differenceU –E = 1 eV ( 2 points). Calculate the energy difference (in eV and kJ/mol) at which an electron can tunnel a distance of 1 nm with a probability of 1% ( 2 points).

One of the most noticeable consequences of the tunnel effect is the unusual dependence of the rate constant of a chemical reaction on temperature. As the temperature decreases, the rate constant does not tend to 0 (as can be expected from the Arrhenius equation), but to a constant value, which is determined by the probability of nuclear tunneling p( f formula 2, left), where A– pre-exponential factor, E A – activation energy. This can be explained by the fact that at high temperatures only those particles whose energy is higher than the barrier energy enter into the reaction, and at low temperatures the reaction occurs exclusively due to the tunnel effect.

2. From the experimental data below, determine the activation energy and tunneling probability ( 3 points).

Modern quantum electronic devices use the resonant tunneling effect. This effect occurs if an electron encounters two barriers separated by a potential well. If the electron energy coincides with one of the energy levels in the well (this is a resonance condition), then the overall probability of tunneling is determined by passing through two thin barriers, but if not, then a wide barrier stands in the way of the electron, which includes a potential well, and the overall probability of tunneling tends to 0.

3. Compare the probabilities of resonant and non-resonant tunneling of an electron with the following parameters: the width of each barrier is 0.5 nm, the width of the well between the barriers is 2 nm, the height of all potential barriers relative to the electron energy is 0.5 eV ( 3 points). Which devices use the tunneling principle ( 3 points)?