Semiclassical theory of potential scattering for massless Dirac fermions
Section snippets
Preliminary considerations: three regimes of scattering
The wave function of a massless Dirac fermion obeys the effective Dirac equation where is the Fermi velocity, is the two-dimensional vector of Pauli matrices, is the momentum operator and is the characteristic scale of change of the potential. In this paper we consider a potential that depends on only. Then the separation of variables gives , and we obtain Denoting the
Semiclassical scattering states
Before we can solve the scattering problem for the different regimes outlined in the previous section, we first have to define the asymptotic scattering states. From now on we assume that is an analytic function in the complex plane. Therefore we can consider Eqs. (10), (11) for in the complex plane: and The semiclassical solution for Eq. (12) has the form where is a power series in
Stokes diagrams and the WKB approximation in the complex plane
In the previous section we established that the functions , given by (21), solve the reduced Eq. (10) for up to order . Let us now choose the constant to be a turning point, i.e. a point where vanishes; . There are lines, called anti-Stokes lines [18], [19], [23], emanating from the point along which the imaginary part of the function , given by Eq. (20), vanishes, that is .1
Tunneling through a barrier supporting hole states
In this section we solve the scattering problem for the first case of Section 1, that is, tunneling through a barrier supporting hole states. The classically allowed region for this case has been extensively studied by the authors in [14] with the canonical operator method [34], and particular emphasis was placed there on its geometric interpretation. In this section we are mainly interested in the transition through the classically forbidden region.
In Fig. 6 we show the potential of the
Semiclassical treatment of above-barrier transmission
In this section we consider the second regime from Section 1, namely above-barrier scattering. This regime can be split in two cases, (i) scattering above a potential hump, and (ii) scattering above a monotonous finite range potential.
The first case describes for instance finite-range gating in graphene. The Stokes diagram for this potential is shown in Fig. 3(b). One sees that there are four turning points, two in the upper half-plane and two in the lower half-plane. In what follows we will
Tunneling through a barrier without hole states
Now let us consider the third regime from Section 1, the conventional tunneling regime. We consider a short-range potential, for which the Stokes diagram is shown in Fig. 2(c). Two of the four turning points are real, and the other two are imaginary. In the previous section, we saw that imaginary turning points give rise to exponentially small reflections, so we will start by neglecting their influence.
To relate the transmission coefficient to the reflection coefficient, we use the contour
The exactly solvable model of the monotonous finite range potential
Up to now we have considered scattering of massless Dirac fermions by a potential hump. Implicitly, we assumed that we only have to keep four turning points in the Stokes diagram to find the major contribution to the scattering. This assumption naturally led to Stokes diagrams topologically equivalent to those for a parabolic potential, . In contrast to a potential hump, a monotonous finite range potential (finite increase) cannot be modeled by a polynomial function. This leads to a
Comparison with numerical results
In this section we compare our semiclassical predictions from the previous sections to numerical results. These are obtained by approximating the potential by a series of small steps. Since the potential is constant between each of them, one can use the exact solution for a constant potential [10]. Matching the coefficients at each interface with the help of a computer, we obtain the reflection and transmission coefficients.
Let us start by considering a finite increase of the potential, which
Conclusion
In this paper we have studied potential scattering of massless Dirac fermions in semiclassical approximation. We have shown that, depending on the energy of the incoming particle and its angle of incidence, there are three different regimes. These are (i) the regime of the Klein tunneling, i.e. the regime when the scattering of electrons is mediated by hole states supported by the barrier, (ii) the above-barrier scattering regime, and (iii) the conventional tunneling regime. For each of these
Acknowledgments
We are grateful to Sergey Dobrokhotov and Andrey Shytov for helpful discussions.
We acknowledge financial support from the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). This work is supported by the Dutch Science Foundation NWO/FOM and the EU-India FP-7 collaboration under MONAMI.
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