By Mikhail I. Katsnelson
Graphene is the thinnest recognized fabric, a sheet of carbon atoms prepared in hexagonal cells a unmarried atom thick, and but more advantageous than diamond. It has very likely major purposes in nanotechnology, 'beyond-silicon' electronics, solid-state cognizance of high-energy phenomena and as a prototype membrane which may revolutionise delicate subject and 2nd physics. during this publication, top graphene examine theorist Mikhail Katsnelson offers the fundamental suggestions of graphene physics. subject matters lined comprise Berry part, topologically safe 0 modes, Klein tunneling, vacuum reconstruction close to supercritical fees, and deformation-induced gauge fields. The e-book additionally introduces the speculation of versatile membranes correct to graphene physics and discusses digital shipping, optical homes, magnetism and spintronics. regular undergraduate-level wisdom of quantum and statistical physics and reliable kingdom concept is believed. this can be an incredible textbook for graduate scholars in nanoscience and nanotechnology and a very good creation for physicists and fabrics technological know-how researchers operating in comparable parts.
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Additional resources for Graphene: Carbon in Two Dimensions
Example text
One can demonstrate by a straightforward calculation (Mikitik & Sharlai, 2008) that each of the three side conical points contributes p to the Berry phase and the central one contributes p, so the total Berry phase is 3p p ¼ 2p, in agreement with Eq. 111). One can also see straightforwardly that the winding number of the transformation À Á À Á À Á2 ð2:123Þ Rx þ iRy $ kx þ iky þ a kx iky is the same (two) as for Eq. 112). ~ k~ ¼ i n~ rk~n The distribution of the Berry ‘vector potential’ O demonstrating singularities at four conical points is shown in Fig.
25), which gives us immediately ^ ¼ e2 c b^þ bc 2 2 ð2:28Þ with the well-known eigenvalues e2n ¼ n ¼ 0; 1; 2; . . ð2:29Þ 28 Electron states in a magnetic field Thus, the eigenenergies of massless Dirac electrons in a uniform magnetic field are given by p ð2:30Þ EnðÆÞ ¼ Æhoc n; where the quantity hoc ¼ p 2hv ¼ lB r 2hjejBv2 c ð2:31Þ will be called the ‘cyclotron quantum’. In the context of condensed-matter physics, this spectrum was first derived by McClure (1956), in his theory of the diamagnetism of graphite.
However, this is not the case for a degenerate spectrum and, in particular, for the case in which conical points exist, like in graphene. Using Stokes’ theorem, Eq. 81) can be written in terms of the surface integral over the area, restricted by the contour C: ð D ð E E D ~ ~ ~ ~ ~ ~ ~ ~ gn ðCÞ ¼ Im dS Árk~  n; k rk~ n; k ¼ Im dS rk~ n  rk~ n ð2:82Þ E E rk~n; k~ . , ~ rk~ n ¼ ~ To demonstrate explicitly the role of crossing points of the energy spectrum (such as the conical points in graphene), we introduce, following Berry (1984), the summation over a complete set of eigenstates jmi: E D E E XD D ~ ~ ~ ð2:83Þ rk~ n : rk~ n m  m ~ rk~n  rk~ n ¼ m The term with m ¼ n in Eq.