3 - Quantum transport via evanescent waves

Summary

Zitterbewegung as an intrinsic disorder

The Berry phase, the existence of a topologically protected zero-energy level and the anomalous quantum Hall effect are striking manifestations of the peculiar, ‘ultrarelativistic’ character of charge carriers in graphene.

Another amazing property of graphene is the finite minimal conductivity, which is of the order of the conductance quantum e²/h per valley per spin (Novoselov et al., 2005a; Zhang et al., 2005). Numerous considerations of the conductivity of a two-dimensional massless Dirac fermion gas do give us this value of the minimal conductivity with an accuracy of some factor of the order of one (Fradkin, 1986; Lee, 1993; Ludwig et al., 1994; Nersesyan, Tsvelik & Wenger, 1994; Ziegler, 1998; Shon & Ando, 1998; Gorbar et al., 2002; Yang & Nayak, 2002; Katsnelson, 2006a; Tworzydlo et al., 2006; Ryu et al., 2007).

It is really surprising that in the case of massless two-dimentional Dirac fermions there is a finite conductivity for an ideal crystal, that is, in the absence of any scattering processes (Ludwig et al., 1994; Katsnelson, 2006a; Tworzydlo et al., 2006; Ryu et al., 2007). This was first noticed by Ludwig et al. (1994) using a quite complicated formalism of conformal field theory (see also a more detailed and complete discussion in Ryu et al., 2007). After the discovery of the minimal conductivity in graphene (Novoselov et al., 2005a; Zhang et al., 2005) I was pushed by my experimentalist colleagues to give a more transparent physical explanation of this fact, which has been done in Katsnelson (2006a) on the basis of the concept of Zitterbewegung (Schrödinger, 1930) and the Landauer formula (Beenakker & van Houten, 1991; Blanter & Büttiker, 2000).