2 tion to the Raman spectra and obtain an analytic descrip- tion for the selection rules and intensity of such Raman scattering in a bilayer placed in a strong (quantizing) magnetic field. Bilayer graphene consists of two coupled sheets of graphene with AB (Bernal) stacking characteristic of bulk graphite, 32 Fig. 1(a). The unit cell contains four in- equivalent atoms A 1, B 1, A 2 and B 2 where letters A and B denote two sublattices in the same layer and 1 / 2 stands for the bottom/top layer. The Fermi level in graphene lies in the vicinities of the corners of the hexagonal Bril- louin zone (also called valleys) known as K$_{+}$ and K$_{−}$ (Fig. 1(b)). The conventional tight-binding Hamiltonian based on π -orbitals of carbon atoms (one per atom, four in the unit cell) and expanded in momentum around the valleys reads Here, σ = ( σ$_{x}$, σ$_{y}$ ) and σ$_{x}$ , σ$_{y}$ , σ$_{z}$ are the Pauli matrices, electron momentum p is measured from the center of the valley, v ∼ 10 $^{6}$m/s (Refs. 11,12,18,23) is a parameter related to the nearest neighbour intralayer coupling γ$_{0}$ ∼ 3eV, $^{11,18,23}$, γ$_{1}$ ∼ 0 . 4eV (Refs. 5,9–12,18,19,21,23) is the direct interlayer coupling, v$_{3}$ is related to the weak direct A 1 ↔ B 2 interlayer hops [ v$_{3}$/v ∼ 0 . 1 (Refs. 1,18,23)] and ξ = ± is the valley index. The basis is constructed using components corresponding to atomic sites A 1 , B 2 , A 2 , B 1 in the valley K$_{+}$ and B 2 , A 1 , B 1 , A 2 in K$_{−}$ . One can also take into account terms quadratic in the electron momentum p : The part of the resulting electronic dispersion relevant for the Raman scattering of photons with energies Ω < 2 . 5eV, is illustrated in Fig. 1(c) for the valley K$_{+}$ . Two bands, later referred to as low-energy ones, touch each other at the neutrality point - the position of the Fermi energy in the neutral structure. Two other, referred to as high-energy bands, are split by the interlayer coupling, γ$_{1}$ , from the neutrality point. The v$_{3}$ parameter leads to the trigonal warping of the electronic dispersion. Its influence is most important for very low energies, ϵ < 5meV. To describe the process of inelastic scattering of light on electrons, we consider an experimental setup in which incoming laser light of energy Ω ≫ γ$_{1}$ , in-plane momen- tum q (out-of-plane component of momentum equal to q$_{z}$ = √ Ω $^{2}$/c2 − q $^{2}$) and polarisation l is shined onto to the sample. Scattered photon has polarisation ˜ l , in-plane momentum ˜ q and energy ˜ Ω = Ω − ω , where ω is the Raman shift. We also assume the temperature T to be smaller than the Raman shift, k$_{B}$T < ω ( k$_{B}$ is the Boltz- mann’s constant). In our case, the inelastic light scatter- ing may occur via a one-step process (so called contact interaction) or a two-step process involving an interme- diate state. The two-step process, such as shown in Fig. 1(c), involves: the absorption (or emission) of a photon with energy Ω ( ˜ Ω) transferring an electron with momen- tum p from an occupied state in the valence band into a virtual intermediate state (energy is not conserved at this stage), followed by another electron emission (or ab- sorption) of the second photon with energy ˜ Ω (Ω). The one-step process is the usual inelastic scattering of an incoming photon on an electron with transfer of energy to the latter. As a result of both one and two-step pro- cesses, an electron-hole pair in the low-energy bands is created with the electron and the hole having almost the same momentum ( p + q − ˜ q and p , respectively), since q , ˜ q ≪ p and the momentum transfer from light is neg- ligible ( v/c ∼ 3 · 10 − $^{3}$). Therefore, p + q − ˜ q ≈ p and due to the approximately electron-hole symmetric band FIG. 1: (a) Schematic of the bilayer graphene crystal lattice. (b) The Brillouin zone of bilayer graphene with two inequiv- alent valleys denoted as K + and K − . (c) The band structure of bilayer graphene in the vicinity of the K + point along the p x axis. Also shown are some of the two-step processes lead- ing to the creation of an electron-hole pair in the low-energy bands accompanied by the absorption of a photon followed by emission. Gray solid (dashed) lines indicate the first (second) step of the process. The black (white) circle denotes the hole (electron) in the final electron-hole pair, while the hatched circle represents the intermediate virtual state. Note that for any intermediate state | ν 〉 with energy ϵ ν , Ω , ˜ Ω ≫ ϵ ν . (d) Diagrammatic representation of the scattering amplitudes R , discussed in Sec. II, and δ R (discussed in Appendix). ˆ H$_{0}$ = ( ξv$_{3}$ ( σ$_{x}$p$_{x}$ − σ$_{y}$p$_{y}$ ) ξv σ · p ξv σ · p γ$_{1}$σ$_{x}$ ) . (1) δ ˆ H = µ ( v$_{3}$ $_{v}$[ σ$_{x}$ ( p 2 x − p 2 $_{y}$)+2 σ$_{y}$p$_{x}$p$_{y}$ ] σ$_{x}$ ( p 2 x − p 2 $_{y}$) − 2 σ$_{y}$p$_{x}$p$_{y}$ σ$_{x}$ ( p 2 x p 2 $_{y}$) − 2 σ$_{y}$p$_{x}$p$_{y}$ 0 ) , where µ = − v 2 6 $_{γ$_{0}$}$. However, the influence of the δ ˆ H term on the results of the Raman spectra analysis is negligibly small, as shown in the Appendix. II. THEORY OF THE INELASTIC LIGHT SCATTERING IN BILAYER GRAPHENE GLYPH<G67> GLYPH<G65> GLYPH<G20> GLYPH<G67> GLYPH<G57> GLYPH<G20> GLYPH<G67> GLYPH<G57> GLYPH<G64>