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In 1929, Klein<ref> |
In 1929, Klein<ref>O. Klein "Die reflexion von elektronen an einem potentialsprung nach der relativischen dynamik von Dirac" Z. Phys 53, 157-165, 1929</ref> obtained a surprising result by applying the [[Dirac equation]] to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. However, Klein’s result showed that if the potential is on the order of the electron mass, V ~ mc², the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted. |
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Revision as of 14:51, 22 December 2008
In 1929, Klein[1] obtained a surprising result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. However, Klein’s result showed that if the potential is on the order of the electron mass, V ~ mc², the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.
The Klein Paradox for 1D potential step
Massless Particles
Consider a massless relativistic particle approaching a potential step of height with energy and momentum .
The particle's wave function, follows the time-independent Dirac equation:
And is the pauli matrix:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Dispersion1.png/400px-Dispersion1.png)
Assuming the particle is propagating from the left, we obtain two solutions - one before the step ,in region (1) and one under the potential, in region (2):
Where the coefficients A, A' and B are complex numbers. Both the incoming and transmitted wave functions are associated with positive group velocity (Blue lines in Fig.1), whereas the reflected wave function is associated with negative group velocity. (Green lines in Fig.1)
We now want to calculate the transmission and reflection coefficients, They are derived from the probability amplitude currents.
The definition of the probability current associated with the Dirac equation is:
In this case:
The transmission and reflection coefficients are:
Continuity of the wavefunction at , yields:
And so the transmission coefficient is 1 and there is no reflection.
One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle. This explanation best suits the single particle solution cited above. Other, more complex interpretations are suggested in literature, in the context of quantum field theory.
Other Cases
For the massive case, the calculations are similar to the above. The results are as surprising as in the massless case. The transmission coefficient is always larger than zero, and approaches 1 as the potential step goes to infinity.
These results were expanded to higher dimensions, and to other types of potentials, such as a linear step, a square barrier, etc. Many experiments in electron transport in graphene rely on the Klein Paradox for massless particles.[2][3]
Links
References
- ^ O. Klein "Die reflexion von elektronen an einem potentialsprung nach der relativischen dynamik von Dirac" Z. Phys 53, 157-165, 1929
- ^ M. Katsnelson, K. Novoselov, A. Geim, "Chiral Tunneling and the Klein paradox in graphene" Nat. Phys. 2, 620, 2006
- ^ V. Cheianov, V. Fal'ko B. Altshuler "The focusing of Electron Flow and a Veselago Lens in graphene p-n junctions" Science, v. 315, 1252, 2007