# Tunnel effect

The tunnel effect indicates the property which has a quantum object to cross a Barrière of potential, impossible crossing according to the traditional Mécanique. Generally, the Fonction of wave of a particle, whose square of the module represents the Amplitude its Probabilité of presence, is not cancelled on the level of the barrier, but attenuates inside the barrier, practically exponentially for a rather broad barrier. If, on the outlet side of the barrier of potential, the particle has a probability of nonnull presence, it can cross this barrier. This probability depends on both sides on the states accessible from the barrier as well as its space extension.

## Analyzes

At the theoretical level the behavior tunnel is not basically different from the traditional behavior of the quantum particle vis-a-vis the barrier of potential; it satisfies the equation of Schrödinger, differential equation implying the continuity of the function of wave and its derivative first in all space. Just as the equation of the electromagnetic waves leads to the phenomenon of the waves évanescentes, in the same way the function of wave encounters cases where the amplitude of probability of presence is nonnull in places where the potential energy is higher than total energy.

If, at the mathematical level the evaluation of the tunnel effect can sometimes be simple, the interpretation which one seeks to give to the solutions reveals the ditch which separates traditional mechanics, field of the material point following a trajectory defined in the space time, of quantum mechanics where the concept of simple trajectory disappears with the profit from a whole unit from possible trajectories, of which trajectories where the time appears complex or imaginary pure… where speeds become imaginary.

One will note on this subject that the lasted of crossing tunnel of a particle through a quantum barrier was, and is still, the subject of rough discussions. Rather many studies in the electromagnetic or photonic field revealed the appearance of what one can interpret like supraluminic speeds, respecting however restricted relativity: it is about the phenomenon known under the name of Effet Hartmann.

## Applications

The tunnel effect is with work in:
• molecules: , for example,
• modelings of disintegrations (Fission, Radioactivity alpha),
• the transistors,
• some diodes,
• different types of microscopes,
• the Effect Josephson.

Particular case: the resonant Tunnel effect.

## Examples

A wave planes correspondent with a particle of a effective Masse of 0,067 times the mass of the electron, of energy 0,08 eV is incidental on a barrier of simple rectangular potential, of 0,1 eV. The diagram reveals the density of probability of presence associated in this stationary state . The left side reveals the phenomenon of Interférence between the incidental wave and the considered wave. The tunnel part (in the barrier) comes from the combination of two exponential, respectively decreasing from left to right, and from right to left. On the right, the transmitted plane wave appears by a density of probability of constant presence.

Function of wave of an electron, representing the Density of probability of its position. The greatest probability is that the electron " rebondisse". There exists a weak probability that the electron crosses the barrier of potential.

## Mathematical analyzes

### Introduction to the concept of transmittivity

The quantum barrier separates the space into three, of which the parts left and right-hand side are regarded as having constant potentials until the infinite one ($V_G$ on the left, $V_D$ on the right). The intermediate part constitutes the barrier, which can be complicated, revealing a soft profile, or on the contrary formed of rectangular barriers, or others possibly in series.

One often is interested in research of the stationary states for such geometries, states whose energy can be higher than the height of potential, or on the contrary lower. The first case corresponds to a situation named sometimes like traditional , although the answer reveals a typically quantum behavior; the second corresponds if the energy of the state is lower than the height of the potential. The particle to which the state corresponds crosses the barrier by tunnel effect then, or, in other words, if the energy diagram is considered, by leapfrog effect.

Considering an incidental particle since the left, the stationary state takes the following simple form:

$\ varphi \left(X\right) = \ exp \left(ik_Gx\right) +r \ exp \left(- ik_Gx\right)$ for

$\ varphi \left(X\right) = \ varphi_ \left\{int\right\} \left(X\right)$ for $a \ Leq X \ Leq b$;
$\ varphi \left(X\right) = T \ exp \left(ik_Dx\right)$ for $b \ Leq x$;

where R and T is respectively the coefficients of reflection and transmission in amplitude for the incidental plane wave $\ varphi \left(X\right) = \ exp \left(ik_Gx\right)$. $\ varphi_ \left\{int\right\} \left(X\right)$ is the function of wave inside the barrier, whose calculation can be rather complicated; it is related to the expressions of the function of wave in the right and left half spaces by the relation of continuity of the function of wave and its derivative first.

Enough often one is interested in the probability of transmission (giving place to the current tunnel, for example), and thus one privileges the study of the coefficient of transmission T , more precisely the value in amplitude and phase of the coefficient $t_ \left\{ab\right\} = \ mathcal \left\{T\right\}$, characterizing the relations between the incidental plane wave, taken at the entry has and the plane wave of exit taken at the point B . The probability of transmission is named the transmittivity $T= |\ mathcal \left\{T\right\}|^2$.

These are the transmittivities which is presented in some particular cases, limited below (makes some for certain formulas only) to the case tunnel.

### Examples of transmittivities tunnel

#### simple rectangular barrier, simple associations of barriers

The majority of the characteristics of the tunnel effect appear at the time of the consideration of the simplest barrier of the potential, a symmetrical rectangular barrier, for which the potential is constant (equal to U ) between the points has and B , and no one on the right and on the left. In this case, the vectors of wave incident (considered) and transmitted have even module, noted $k= \ sqrt \left\{2nd\right\}/\ hbar$ while the interior part of the function of wave is a linear combination of the exponential functions decreasing and increasing ($\ exp$ and $\ exp$), $\ Kappa= \ sqrt \left\{2m \left(U-E\right)\right\}/\ hbar$.

The conditions of continuity to the interfaces then make it possible easily to evaluate the four complexes, R , has , B and T . From this last term, one deduces the transmittivity:

$T= \ frac \left\{4 \ Kappa^2k^2\right\} \left\{\left(\ Kappa^2+k^2\right) ^2 \ sinh^2 \left(\ Kappa d\right)+4 \ Kappa^2k^2\right\}$,
D being the thickness of the barrier.

In the case of thick barrier ($\ large Kappa d$), one obtains the formula simple to retain:

$T=16 \ frac \left\{\ Kappa^2k^2\right\} \left\{\left(\ Kappa^2+k^2\right) ^2\right\} \; \ exp \left(- 2 \ Kappa d\right)$.
In this case there, one can regard the transmittivity as produces it obtained by approach BKW (cf will infra the exponential term) by a prefactor who is only the product of the square modules of the coefficients of transmission specific to the interfaces of entry and exit.

This structure is a form simplified of that which in the case of appears a barrier of an unspecified form broken up like a series of rectangular barriers. The structure of calculation rests then on the taking into account of a matric writing of the equations, connecting the components progressive and regressive in each layer, allowing the establishment of the matrix of transfer of the stationary mode between the space of entry and the space of exit.

This method is illustrated on the case of a structure met in electronics or optics, the resounding barrier tunnel, consisted of a barrier of entry of an internal part of low potential (well of potential, of width L ) and of a spar gate (cf diagram). It is shown that, if the potential in the well is constant (defining a real vector of wave $k_3= \ sqrt \left\{2m \left(U_b-E\right)\right\}/\ hbar$), the transmittivity of the barrier can be written:

$T=\frac\left\{T_E\;T_S\right\}$

#### trapezoidal barrier

The trapezoidal barrier is obtained by the application of a potential difference between the two extremities of the simple rectangular barrier. What gives the following diagram, which offers the advantage of admitting exact analytical solutions; indeed, for this barrier the expression of the function of wave, inside is a linear combination of functions of Airy, Have and Bi, which one can connect to the plane waves solutions in the left and right parts.

A particular case appears within the framework of this description. If the potential difference is sufficiently important so that the barrier shows the existence of a traditional point of return (passage of a part tunnel with a traditional part , at the point $x_2$), one then obtains the effect of emission of field, usually used in electronic microscopy. The particle, located in the band of conduction on the left, crosses by tunnel effect and is accelerated towards outside, on the right.

Possibly, according to the values of energy and the shape of the barrier, of resonances of transmittivity can appear, due to the jump of potential on the walk of right-hand side. This resonance refers certain common with those of the Effet Ramsauer. The diagram opposite corresponds to an accumulation the instantaneous ones of the density of presence associated with an incidental package of wave since bottom on the left. The effect of resonance appears here by the appearance of the three maximum ones in the traditional part of the barrier. At the end of the crossing the considered and transmitted parts move away to the top of the figure, on the left and on the right respectively.

#### approximation BKW

If the barrier of potential presents a soft profile, it is possible to show, starting from the equation of Schrödinger, or a fine discretization of the potential in a series of small successive rectangular barriers, that the function of wave, in a point of coordinate X in the barrier can be written:

$\ varphi_ \left\{BKW\right\} \left(X\right) = \ frac \left\{1\right\} \left\{\ sqrt \left\{K \left(X\right)\right\}\right\} \; \ int^x \, of the \, K \left(U\right)\right).$

This approximation, studied by Brillouin, Kramers and Wentzel, is obviously nonvalid for the traditional points of return, $x_1 \; , \; x_2$ (cf diagram), where the potential V (X) is equal to energy E of the state ( K (X) is then null), it is necessary to proceed with some care for the connection on both sides of these points.

Within the framework of the study of the transmittivity this expression is especially useful in the case tunnel, where K (X) becoming imaginary pure, the two exponential ones appearing in the expression above corresponds in the terms decreasing of the left towards the line (factor term of constant a) and decreasing of the right-hand side towards the left (factor term of B). In the case of an incidental wave coming from left, and for the sufficiently broad barriers, the source of the regressive part (expression B) is tiny. The transmittivity due to this tunnel part is then obtained by the consideration of the reduction in the amplitude of the wave between the traditional points of return of entry and of exit, that is to say:

$T_ \left\{BKW\right\} = \ exp \ quad; \ quad \ Kappa \left(U\right) = \ sqrt \left\{\ frac \left\{2m\right\} \left\{\ hbar^2\right\} \left(V \left(U\right) - E\right)\right\}\;.$

It is this expression which it is then necessary to calculate, by the method of the reversed potential, for example. It should be noted that this approximation must be corrected by préfacteurs, characteristic of the steeply sloping potentials (jump of potential), which one meets with the interface between two materials, and which are currencies in the current electronics components (quantum wells).

### Semi-traditional approach and use of the turned over potential

Before with the development of the means of fast and powerful calculations, which allow precise evaluations of the transmittivities, approximate methods developed which allowed, in an effective way, to discover characteristics of some transmittivities tunnel of certain barriers of theoretical and practical importance: barrier of the Coulomb type (model of Radioactivity alpha) or triangular barrier associated with the field effect.

It is a question of evaluating the argument of exponential appearing in approximation BKW. It is easy to calculate the integrals for the hyperbolic or linear potentials, but it is interesting to note the possible approach by the method of the turned over potential for which the evaluation of $\ exp \ int^x p \left(U\right) of the \ hbar$ is obtained via that of $\ exp- S \left(E\right) /2 \ hbar$ in which $S \left(E\right)$ is the action calculated on the traditional orbit that a of the same particle energy in the turned over potential would follow, obtained within the framework of the use of the Symétrie of Corinne.

The interest rests then on the fact that for the sufficiently thick barriers, corresponding to broad wells, the action is, in the semi-traditional approximation, prone to the quantification $S \left(E\right) = N \, h$.

Transmittivity BKW of such a barrier is written then:

$T \left(E\right) = \ exp N \left(E\right) \,$

where the quantum number N ( E ) is the reciprocal function of the energy E postulated like discrete energy level of the well of potential corresponding to the turned over barrier.

#### Application to the radioactivity alpha

The barrier of potential which the particle alpha must cross, of energy E , after its random appearance within the core of atomic number Z , is transformed into a Coulomb well, whose energy levels are those of a Hydrogénoïde. This allows the calculation of the number N ( E ) directly starting from well-known formulas:

$E = - \ frac \left\{m e^4 \left(Z-2\right) 2\right\} \left\{2 \ hbar^2\right\} \ times \ frac \left\{1\right\} \left\{n^2 \left(E\right)\right\}\,$

where the reduced mass, and they appear loads of the particle alpha and the core wire (atomic number Z -1).

The carryforward of the number N ( E ) in the expression of the transmittivity then reveals the behavior observed of the Demi-vie (proportional contrary to the transmittivity) of the transmitters alpha according to $\ sqrt \left\{E\right\}$, energy of the particle meeting the barrier.

#### Application to the Fowler-Nordheim effect

Under the action of an electric field F , one can make leave the electrons of a metal (load Q , mass m , energy E compared to the bottom of the band of conduction), in particular of an alkaline metal of Output $\ Phi$. The electron is then subjected to a triangular potential which can, at first approximation, being treated by method BKW: the transmittivity which deduced (taking into account the traditional points of return $x_1=0$ and $x_2= \left(\ Phi - E\right) /qF\right)$) is

$T_ \left\{BKW\right\} = \ exp \; = \; \ exp qF\right\} \ int_ \left\{0\right\} ^ \left\{\ Phi-E\right\} \; dX \; \ sqrt \left\{X\right\}$

$\ qquad= \ exp qF\right\} \left(\ Phi-E\right) ^ \left\{3/2\right\} \;.$

Obtaining the current tunnel must of course take account of the distribution in energy and direction of the whole of the electrons of the band, for the temperature of the driver.

Here also the transmittivity could have been obtained by using a turned over potential. It is then about the '' half-well of Torricelli '', whose energy level can be calculated and allow obtaining the number N ( E ).

## See too

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