Band model of the graphene bilayer

213.1 Simplified modelAsimplifiedmodelthatonlyconsiderstheinterplanehoppingtermbetweenA atoms employsa matrix of the form⎛⎞0 v F pe iφ(p) t ⊥ 0v F pe −iφ(p) 0 0 0H 0 (p) =. (3.3)⎜ t ⊥ 0 0 v F pe −iφ(p)⎟⎝⎠0 0 v F pe iφ(p) 0From now on in this section, we use units such that v F =1asdiscussedinSection2.1. ThisHamiltonian has the advantage that it allows for relatively simple calculations. Some of thefine details of the physics might not be accurate but it will work as a minimal model andcapture most of the important physics. It is important to know thequalitativenatureoftheterms that are neglected in this approximation, this will be discussed later in this Chapter.It is also an interesting toy model as it allows for (approximately) “chiral” particles withmass (i.e., a parabolic spectrum) at low energies as we will discuss in the next Section. Foralargepartofthisthesiswewillstudythepropertiesofthesystem with this simplifiedHamiltonian.3.2 Approximate effective two-band modelsThere are two main reasons for constructing approximate two-band models:First, onphysical grounds the high-energy bands (far away from the Dirac point) should not be veryimportant for the low-energy properties of the system. Second, it is often easier to workwith 2 × 2matricesinsteadof4× 4matrices. Inthissection,wederivethelow-energyeffective model by doing degenerate second order perturbation theory. The quality of theexpansion is good as long as v F p ≪ t ⊥ ≈ 0.35 eV. We first present the general expressionfor the second-order 2 × 2effectiveHamiltonian,thereaftervarioussimplifiedformswillbeintroduced. Analyses similar to the one presented here were presented in (McCann andFal’ko, 2006) and (Nilsson et al., 2006c).