Resonant tunnel effect

The Tunnel effect indicates the capacity for a particle of energy lower than a threshold to cross this threshold. Whereas in traditional mechanics this phenomenon is unexplainable, in quantum mechanics a particle of energy lower than the height of the barrier has a Probabilité very low, but nonzero, to pass to through.

A spectacular phenomenon appears if the barrier has, in its part tunnel, localized states whose energy corresponds to that of the incidental particle. Such a state localized can be a atomic state of an impurity, or on the contrary a discrete level of a quantum well created.

To suppose that by an unspecified process the discrete state is occupied at a given moment, the probability so that it is depopulated towards the accessible states outside the barrier will cause the depopulation by tunnel effect through the barriers which border it (the level is thus only quasi-dependant). It is the source of the lifespan, and the width of the level of the well. On the level of description, it is an effect similar to that which is responsible for the natural width of the lines of emission of the atoms.

The resonant tunnel effect appears when such a quantum system is approached outside with a close energy, or equal to that of the quasi-dependant level. The transition probability through each barrier of entry or exit alone is very weak but resonance with the level of the well will trap the quantum particle, during a relatively long time, of about size of the lifespan of the quasi-dependant level, but this trapping will make it possible the particle to cross the unit. The Transmittivité of the barrier can approach the unit, even for thick barriers.

One can note that several resounding barriers can be followed, revealing bands of important transmittivity.

Example

Let us take as simple model two distant rectangular barriers of potential of some Nm. If the height of the barrier would be infinite, the calculation of the discrete levels would be very simple; finitude in height, and the extension finished on the right, and on the left here cause a displacement of the levels, to the bottom, and the appearance of a level width (or in other words, one lifespan).

The figure opposite watch transmittivity of the barrier, obtained by a sweeping in energy. The curve marries the form characteristic of Lorentzian, specific to resonances. The description of the temporal aspects can be observed if one follows a Paquet of wave, associated with an incidental particle, coming from the left, whose spectrum in energy recovers the curve of resonance.

It is observed that the Fonction of wave is mainly transmitted, that it is deformed considerably compared to the entering package, and that it is late on a package which would not have had to cross the barrier. At the moment when the function of wave is calculated the central well is still populated: it is emptied little by little towards the two free half spaces. The delay between the top of the “free” package and the transmitted part corresponds to the lifespan $\ Delta t$ of the quasi-free level, that is to say:

$\ Delta E \ Delta t= \ hbar$ .

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