# Tunnel effect

The probability of an electron hitting a potential barrier. There is little probability that it will pass through the barrier, which would not be possible according to classical physics.

In physics, the tunnel effect is an illustrative term for the fact that a particle can overcome a potential barrier of finite height even if its energy is less than the "height" of the barrier. According to the ideas of classical physics this would be impossible, but according to quantum mechanics it is possible. With the help of the tunnel effect, among other things, the alpha decay of atomic nuclei is explained. Technical applications are, for example, the scanning tunneling microscope and flash memory .

## discovery

In 1897 Robert Williams Wood observed the effect in an experiment with the field emission of electrons in a vacuum, without being able to interpret it.

In 1926/1927 Friedrich Hund first discovered and described what was later known as the tunnel effect (the discovery of which is often attributed to George Gamow ) in isomeric molecules.

In 1926 Gregor Wentzel , Hendrik Anthony Kramers and Léon Brillouin laid the foundation stone for the quantum mechanical explanation of tunnel processes with the WKB method named after them . Using this method, George Gamow was able to explain the alpha decay during his stay with Max Born in Göttingen as well as Ronald W. Gurney and Edward U. Condon in 1928 . At the same time, Ralph Howard Fowler and Lothar Wolfgang Nordheim succeeded in explaining the field emission of electrons.

The Swedish physicist Oskar Klein gave in 1929 a refined theory of the tunneling through barriers by very fast particles.

## Quantum mechanical explanation

Schematic representation of the tunnel effect:
A particle coming from the left hits a potential barrier. The energy of the tunneled particle remains the same, only the amplitude of the wave function becomes smaller and thus the probability of finding the particle.

The quantum mechanical approach is based on the (non-relativistic) Schrödinger equation , a differential equation for the wave function that indicates where a particle can be. This wave function is nowhere equal to zero in the “forbidden” area, i.e. inside or beyond the barrier, but decays exponentially there with increasing penetration depth . Even at the end of the forbidden area, their value is not zero. Since the square of the magnitude of the wave function is interpreted as the probability density for the location of the particle, there is a non-zero probability for the particle to appear on the other side of the barrier. ${\ displaystyle \ Psi}$${\ displaystyle \ left | \ Psi \ right | ^ {2}}$

Like many effects in quantum theory, the tunnel effect only plays a role for extremely short distances and very short periods of time or high energies.

The naming of the tunnel effect takes into account the fact that the particles cannot conventionally cross the barrier, and the effect, if at all, should rather be imagined as a kind of " tunneling through " the barrier.

Quantum mechanical phenomena lead to considerations that are correct, but not realistic: Because a curiosity in quantum mechanics is that the attempt to go through a house wall does not necessarily fail. There is a non-zero probability that every single particle in the human body will overcome the potential barriers of the wall and then be on the other side of the wall. However, this probability is extremely low.

## Appearance and applications

### Nuclear fusion in stars

From an energetic point of view, the pressure and temperature in the sun and other stars would not be sufficient for a thermonuclear fusion of atomic nuclei, which is the source of the emitted radiation. Due to the tunnel effect, however , the Coulomb potential is overcome quantum mechanically with a certain probability. The tunnel effect is therefore one of the decisive factors for life on earth.

### Chemical reaction

The tunnel effect of atoms in chemical reactions means that they can take place faster and at lower temperatures than through classic movement via the activation energy . At room temperature it plays a role primarily in hydrogen transfer reactions. At low temperatures, however, many astrochemical syntheses of molecules in interstellar dark clouds can be explained by including the tunnel effect , including the synthesis of molecular hydrogen , water ( ice ) and the prebiotically important formaldehyde .

### Quantum biology

The tunnel effect is one of the central effects in quantum biology . The genetic code , for example, is not completely stable due to the occurrence of proton tunnels in the DNA . As a result, the tunnel effect is jointly responsible for the occurrence of spontaneous mutations . In contrast, electron tunneling plays an important role in many biochemical redox and catalytic reactions.

### Alpha decay

Among other things, the alpha decay of atomic nuclei is based on the tunnel effect . According to classical physics , the nucleus should not disintegrate because of the strong attractive interaction . However, due to the tunnel effect, there is a non-zero probability per unit of time ( decay probability ) that the alpha particle will leave the nucleus, because the quantum mechanical probability of the alpha particle being located beyond the energy barrier is not zero; once the positively charged alpha particle is outside the barrier, it finally leaves the nucleus by repelling it from the rest of the nucleus, which is also positively charged. The decay probability results in a half-life for this stochastic process .

### Two-electrode tunnel

In 1933, Hans Bethe and Arnold Sommerfeld approximately calculated the tunnel current density between two electrodes with a small potential difference and a trapezoidal potential barrier. A somewhat better approximation could then be given by R. Holm and B. Kirschstein in 1935, who approximated the shape of the potential barrier with a parabola . Holm refined his theory in 1951 to the effect that he could enter the tunnel current density for potential differences in the magnitude of the work function is of conventional electrode materials. It was not until 1963 that J. Simmons was able to provide a generalized formula with which the tunnel current density can be calculated for all potential differences between two electrodes, whereby the field emission is then also included.

### Field electron and field ion microscope

The tunnel effect found an important application in the high-resolution microscopes developed by Erwin Wilhelm Müller in Berlin. In 1936 he described the field electron microscope and then in 1951 the field ion microscope , which was the first instrument to enable atomic resolution.

### Tunnel diode

In 1957 Leo Esaki developed the first tunnel diode , an electronic high-frequency semiconductor component with negative differential resistance . For this he received the Nobel Prize in Physics in 1973 .

### Superconductivity

In 1960 Ivar Giaever and JC Fisher discovered one-electron tunneling between two superconductors . In 1962, Brian D. Josephson discovered that Cooper pairs can also tunnel ( Josephson effect ). This was demonstrated experimentally in 1963 by Philip Warren Anderson , JM Rowell and DE Thomas for the direct current case and by Sidney Shapiro for the alternating current case . Josephson received the Nobel Prize in Physics for this in 1973 .

### Scanning tunnel microscope

Gerd Binnig and Heinrich Rohrer developed a process that made controlled two-electrode tunneling in a vacuum possible for the first time, which ultimately led to the invention of the scanning tunneling microscope . The patent for this technology was applied for in 1979. In 1986 you and Ernst Ruska were awarded the Nobel Prize in Physics.

### Magnetic tunnel resistance

The magnetic tunnel resistance makes use of the fact that the tunnel current between two ferromagnetic materials separated by a thin insulator at a magnetic tunnel contact changes due to an external magnetic field. This effect is used, for example, when reading out data in modern hard drives ( TMR effect ).

### Flash memory

Flash memory media such as USB sticks and memory cards use floating gate (MOS) FETs and are therefore also based on the tunnel effect.

## Tunnel effect using the example of the box potential

For the mathematical description of the tunnel effect, we consider the potential

Potential diagram
${\ displaystyle V (x) = {\ begin {cases} V_ {0} & {\ text {if}} x \ in D: = [- a, a] \\ 0 & {\ text {if}} x \ notin D \ end {cases}} \ quad}$

and divide the room into three areas (I) (left of the barrier), (II) (in the barrier) and (III) (right of the barrier). The particle falling from the left (area I) has the energy E mit . Classically, a particle falling from the left would be reflected at the barrier . ${\ displaystyle 0 ${\ displaystyle x = -a}$

The stationary Schrödinger equation for the wave function of a particle of mass and energy in this potential is: ${\ displaystyle \ Phi (x)}$${\ displaystyle m}$${\ displaystyle E}$

${\ displaystyle - {\ frac {\ hbar ^ {2}} {2 \, m}} {\ frac {\ mathrm {d} ^ {2}} {\ mathrm {d} x ^ {2}}} \ Phi (x) + V (x) \ Phi (x) = E \ Phi (x),}$

where is the reduced Planck constant . To solve the equation, we choose the approach for the wave function in areas (I) and (III) : ${\ displaystyle \ hbar}$

${\ displaystyle \! \, \ Phi (x) = A \, \ mathrm {e} ^ {ikx} + B \, \ mathrm {e} ^ {- ikx}}$

This is a superposition of a left to right and from right to left running plane wave with yet to be determined and . The wave vector is through ${\ displaystyle (\ mathrm {e} ^ {ikx})}$${\ displaystyle (\ mathrm {e} ^ {- ikx})}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle k}$

${\ displaystyle E = {\ frac {\ hbar ^ {2} k ^ {2}} {2 \, m}}}$

certainly.

In area (I) it is clearly clear that and must be. It is the complex reflection coefficient , which describes the portion of the wave entering from the left that is reflected by the potential. If we are at the classical limit and the incoming particles is totally reflected. So we have ${\ displaystyle A = 1}$${\ displaystyle B = R}$${\ displaystyle R}$ ${\ displaystyle | R | ^ {2} = 1}$

${\ displaystyle \! \, \ Phi _ {\ text {I}} (x) = \ mathrm {e} ^ {ikx} + R \, \ mathrm {e} ^ {- ikx}}$

In area (III), since there is no particle coming from the right, we only have a part of the incident wave that may have passed through and we start with:

${\ displaystyle \! \, \ Phi _ {\ text {III}} (x) = T \, \ mathrm {e} ^ {ikx}}$

Here is the complex-valued transmission coefficient . Since the probability current density must be preserved, it follows from the continuity equation (without proof): ${\ displaystyle T}$

${\ displaystyle \! \, | R | ^ {2} + | T | ^ {2} = 1}$

This is vividly clear, since the particle cannot disappear.

In area (II) we choose the general approach

${\ displaystyle \! \, \ Phi _ {\ text {II}} (x) = \ alpha \ mathrm {\,} {e} ^ {\ kappa x} + \ beta \ mathrm {e} ^ {- \ kappa x}}$

is there and real, there is. ${\ displaystyle \ kappa = {\ sqrt {{\ frac {2 \, m} {\ hbar ^ {2}}} \ left (V_ {0} -E \ right)}}}$${\ displaystyle \! \, V_ {0} -E> 0}$

This concludes the physical considerations and mathematical manual work remains. Due to the continuity condition of the wave function and its derivative at the points (x = -a) and (x = a) is obtained four equations for the four unknowns , , and . The solutions then apply to all energies > 0 and one obtains, for example, the transmission coefficient at : ${\ displaystyle R}$${\ displaystyle T}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle E}$${\ displaystyle E

${\ displaystyle T (E) = \ mathrm {e} ^ {- 2ika} {\ frac {2 \, k \ kappa} {2 \, k \ kappa \ cosh (2 \, \ kappa a) -i (k ^ {2} \, - \ kappa ^ {2}) \ sinh (2 \, \ kappa a)}}}$
Course of the transmission coefficient as a function of the ratio of the height of the potential barrier to the energy of the tunneling particle. The different colored curves differ in the parameter . Here is the width of the barrier and the amount of the wave vector of the free particle.${\ displaystyle | T | ^ {2}}$${\ displaystyle V_ {0}}$${\ displaystyle E}$${\ displaystyle k \ cdot b}$${\ displaystyle b = 2a}$${\ displaystyle k = \ left | {\ vec {k}} \ right | = {\ sqrt {2mE}} / \ hbar}$

The probability for a transmission is then the square of the absolute value of and reads: ${\ displaystyle T}$

${\ displaystyle P_ {T} (E) = {\ frac {1} {1 + {\ frac {V_ {0} ^ {2}} {4E \ left (V_ {0} -E \ right)}} \ sinh ^ {2} (2 \ kappa a)}}}$

The function sequence can be seen in the graphic opposite. You can see that the transmission probability is also not zero, i.e. that there is a finite probability of finding the particle on the classically forbidden side. This is the tunnel effect. It is interesting that the transmission probability for is not necessarily 1, i.e. H. the particle can also be reflected if it always came over the barrier in the classic way. ${\ displaystyle E ${\ displaystyle E> V_ {0}}$

To make the above formula a little clearer, consider the borderline case ( ) , for example . Here the transmission probability approaches 1, which is also clearly evident: no barrier, no reflection. ${\ displaystyle V_ {0} \ rightarrow 0}$

## Measurement of the time required

The above equations do not provide any information on how long the particle needs to get from one end of the tunnel to the other. The estimates for electrons were between zero and about 500 · 10 −18  seconds. Current experiments at the ETH Zurich (2008) have shown a maximum time requirement of 34 · 10 −18  s, which is the measurement accuracy of the arrangement. In the experiment, a circularly polarized laser pulse of only 5 · 10 −15  s duration (during this time the electric field vector rotates once through 360 °) was shot at an electron that was bound to a helium atom “behind” a potential wall of 24.6 eV . The probability of passage of the electron at this wall height is so low that no spontaneous ionization of the He atom is observed.

As a result of the short laser pulse, the height of the potential wall was reduced for a defined time so that one of the two electrons could leave the atom. Then it was accelerated by the electric field of the light pulse and removed from the He + ion. The time course could be calculated from the departure direction. According to the researchers, the electron reappeared on the outside immediately after its "disappearance" on the inside of the potential wall. The experiment was not a photoionization because photon energy in the UV range would have been necessary. The femtosecond laser used does not have an exactly definable wavelength, but the focus of its broadband range is clearly in the IR range. Here the photon energy is not sufficient to ionize helium.

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