Klein Tunneling
PHYS 503 Physics Colloquium Fall 2008
9/11
Deepak Rajput
Graduate Research Assistant
Center for Laser Applications
University of Tennessee Space Institute
Email: drajput@utsi.edu
Web: http://drajput.com

Outline
* Classical picture
* Tunneling
* Klein Tunneling
* Bipolar junctions with graphene
* Applications

Classical Picture
Tunneling

* Transmission of a particle through a potential barrier higher than its kinetic energy (V>E).

* It violates the principles of classical mechanics.

* It is a quantum effect.
Quantum tunneling effect

* On the quantum scale, objects exhibit wave-like characteristics.

* Quanta moving against a potential hill can be described by their wave function.

* The wave function represents the probability amplitude of finding the object in a particular location.
Quantum tunneling effect

* If this wave-function describes the object as being on the other side of the potential hill, then there is a probability that the object has moved through the potential hill.

* This transmission of the object through the potential hill is termed as tunneling.
Tunneling
Klein Tunneling

> In quantum mechanics, an electron can tunnel from the conduction into the valence band.

> Such tunneling from an electron-like to hole-like state is called as interband tunneling or Klein tunneling.

> Here, electron avoids backscattering

Tunneling in Graphene
> In graphene, the massless carriers behave differently than ordinary massive carriers in the presence of an electric field.

> Here, electrons avoid backscattering because the carrier velocity is independent of the energy.

> The absence of backscattering is responsible for the high conductivity in carbon nanotubes (Ando et al, 1998).

Absence of backscattering

Let?s consider a linear electrostatic potential

Electron trajectories will be like:

Absence of backscattering

* For py = 0, no backscattering.

* The electron is able to propagate through an infinitely high potential barrier because it makes a transition from the conduction band to the valence band.

Band structure
Absence of backscattering

* In this transition from conduction band to valence band, its dynamics changes from electron-like to hole-like.

* The equation of motion is thus,

at energy E with

* It shows that in the conduction band (U < E) and in the valence band (U > E).

Klein tunneling

Klein tunneling

Transmission resonance

Bipolar junctions

* Electrical conductance through the interface between p-doped and n-doped graphene: Klein tunneling.

Bipolar junctions
* Top gate: Electrostatic potential barrier
* Fermi level lies
Bipolar junctions
* Carrier density ncarrier is the same in the n and p regions when the Fermi energy is half the barrier height U0.
Bipolar junctions

* The Fermi wave vector for typical carrier densities of is > 10-1 nm-1.

* Under these conditions kFd >1, p-n and n-p junctions are smooth on the Fermi wavelength.

* The tunneling probability expression can be used.

Bipolar junctions
* The conductance Gp-n of a p-n interface can be solved by integration of tunneling probability over the transverse momenta

* The result of integration ?:

where W is the transverse dimension of the interface.

Applications of tunneling

* Atomic clock

* Scanning Tunneling Microscope

* Tunneling diode

* Tunneling transistor

Questions ?

Who got the Nobel prize (1973) in Physics for his pioneering work on electron tunneling in solids?