1.2 Scattering by Step Potential and Klein Para-dox

As for an application, you are suggested to solve Dirac equation for step-potential barrier. You consider the scattering of an electron of energy E and momentum p=pzby the following electrostatic step potential

which is illustrated in Fig.1.5.

In this case, the Dirac equation reads

where =E-V0θ(z). You can certainly assume the incident wave ψL, +and reflected wave ψL, - in the left region as follows

Substituting it into Eq. (1.37), you can obtain a solution, such as

and

with pc=. Likewise, you can get the transmitted wave in the right region

with p′c=.

You are suggested to consider the case of strong field V0＞E+mc2 so that p′is a real number. The continuity boundary condition ψL, +(0)+ψL, -(0)=ψR, +(0)requires that

where

unless V0→∞or m=0.

According to (1.13), you can easily calculate the probability flow j=j(z)ez, i.e., j(z)=cψ†(z)αzψ(z), consequently

where ezis the unit vector along z-axis. The subscripts in (1.42) specify the incident, reflected and transmitted flows respectively. Furthermore, you obtain the reflection and transmission ratios of those flows

Since r＞ 1, you perceive that the reflection ratio |jre/jin| is larger than one. These extraordinary“phenomena”were first realized by Klein and used to be called Klein paradox.

The Klein paradox can be interpreted with the help of the concept of Dirac sea. See Fig. 1.6, if the“sea level”of Dirac sea (or the top edge of valence band of a semiconductor)in the right region is lifted to a certain height, some particles in the Dirac sea in the right region become easier to escape into the left region as long as they can attain sufficient energy to overcome the threshold energy Eth. You perceive that, once this process occurs, there will be more electrons emerging in the left region and leave holes in the right region. The aforementioned case V0＞E+mc2means that the“sea level”of Dirac sea is above the energy of the incident particle. Making a tiny Gauss box across the barrier of the two regions, you can verify that the conservation of probability still holds, (-ez)·jre+ez·jtr=ez·jin. Note that the transmitted flow jtr has a minus sign. This means the transmitted particle flows along -z direction, which is equivalent to that the antiparticle (hole) flows along z direction. There are several experiments of electron transport in graphene on the Klein paradox for massless particles.