Zitterbewegung as an intrinsic disorder

The Berry phase, the existence of a topologically protected zero-energy level andthe 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 isof the order of the conductance quantum e2/h per valley per spin (Novoselov etal., 2005a; Zhang et al., 2005). Numerousconsiderations of the conductivity of a two-dimensional massless Dirac fermiongas do give us this value of the minimal conductivity with an accuracy of somefactor of the order of one (Fradkin, 1986; Lee, 1993; Ludwig etal., 1994; Nersesyan, Tsvelik & Wenger, 1994; Ziegler, 1998;Shon & Ando, 1998; Gorbar et al., 2002; Yang &Nayak, 2002; Katsnelson, 2006a; Tworzydlo et al., 2006; Ryuet al., 2007).

It is really surprising that in the case of massless two-dimentional Diracfermions there is a finite conductivity for an ideal crystal,that is, in the absence of any scattering processes (Ludwig etal., 1994; Katsnelson, 2006a; Tworzydlo et al.,2006; Ryu et al., 2007). This was first noticed by Ludwiget al. (1994) using a quite complicated formalism ofconformal field theory (see also a more detailed and complete discussion in Ryuet al., 2007). After the discovery of the minimalconductivity in graphene (Novoselov et al., 2005a; Zhanget al., 2005) I was pushed by my experimentalist colleaguesto give a more transparent physical explanation of this fact, which has beendone in Katsnelson (2006a) on the basis of the concept ofZitterbewegung (Schrödinger, 1930) and the Landauerformula (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 momentμn = 0, and 13C (1.1% ofnatural carbon) with I = ½ andμn =0.7024μN(μN is the nuclear magneton), see Radzig& Smirnov (1985). Like most of the chemical elements, it originates fromnucleosynthesis 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–protonchain reaction, which results in the synthesis of one α-particle (nucleus4He) from four protons, p. Further nuclear fusion reactions mightlead to the formation of either of the isotopes 5He and5Li (p + α collisions) or of 8Be (α +α collisions); however, all these nuclei are very unstable. As was firstrealized by F. Hoyle, the chemical evolution does not stop at helium only due toa lucky coincidence – the nucleus 12C has an energy levelclose enough to the energy of three α-particles, thus, thetriple fusion reaction 3α → 12C,being resonant, has a high enough probability. This opens up a way to overcomethe mass gap (the absence of stable isotopes with masses 5 and 8) and providesthe prerequisites for nucleosynthesis up to the most stable nucleus,56Fe; heavier elements are synthesized in supernovaexplosions.

Preface

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

I hope it will not be considered a disclosure of insider information if I tellyou that Andre Geim is a bit sarcastic (especially with theoreticians). Everytime I mentioned that I was somewhat busy writing a book on graphene, he alwaysreplied ‘Go to amazon.com and search for“graphene”.’ Indeed, there are many books on graphene, manymore reviews and infinitely many collections of papers and conferenceproceedings (well, not really infinitely many . . . in the maintext 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 foryou to read it?

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