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).

The carbon atom

Carbon is the sixth element in the Periodic Table. It has two stable isotopes, 12C (98.9% of natural carbon) with nuclear spin I = 0 and, thus, nuclear magnetic moment μn = 0, and 13C (1.1% of natural carbon) with I = ½ and μn = 0.7024μN (μN is the nuclear magneton), see Radzig & Smirnov (1985). Like most of the chemical elements, it originates from nucleosynthesis in stars (for a review, see the Nobel lecture by Fowler (1984)). Actually, it plays a crucial role in the chemical evolution of the Universe.

The stars of the first generation produced energy only by proton–proton chain reaction, which results in the synthesis of one α-particle (nucleus 4He) from four protons, p. Further nuclear fusion reactions might lead to the formation of either of the isotopes 5He and 5Li (p + α collisions) or of 8Be (α + α collisions); however, all these nuclei are very unstable. As was first realized by F. Hoyle, the chemical evolution does not stop at helium only due to a lucky coincidence – the nucleus 12C has an energy level close enough to the energy of three α-particles, thus, the triple fusion reaction 3α → 12C, being resonant, has a high enough probability. This opens up a way to overcome the mass gap (the absence of stable isotopes with masses 5 and 8) and provides the prerequisites for nucleosynthesis up to the most stable nucleus, 56Fe; heavier elements are synthesized in supernova explosions.

Preface

I do not think that I need to explain, in the preface to a book that is all about graphene, what graphene is and why it is important. After the Nobel Prize for physics in 2010, everybody should have heard something about graphene. I do need, however, to explain why I wrote this book and what is special about it.

I hope it will not be considered a disclosure of insider information if I tell you that Andre Geim is a bit sarcastic (especially with theoreticians). Every time I mentioned that I was somewhat busy writing a book on graphene, he always replied ‘Go to amazon.com and search for “graphene”.’ Indeed, there are many books on graphene, many more reviews and infinitely many collections of papers and conference proceedings (well, not really infinitely many . . . in the main text I will use the mathematical terminology in a more rigorous way, I promise). Why, nevertheless, has this book been written and why may it be worthwhile for you to read it?

Of course, this is a personal view of the field. I do love it, and it has been my main scientific activity during the last seven years, from 2004 when graphene started to be the subject of intensive and systematic investigations. Luckily, I was involved in this development almost from the very beginning. It was a fantastic experience to watch a whole new world coming into being and to participate in the development of a new language for this new world. I would like to try to share this experience with the readers of this book.