# Spectral features due to inter-Landau-level transitions in the Raman spectrum of bilayer graphene 

Marcin Mucha-Kruczy´ nski, 1 Oleksiy Kashuba, 1, 2 and Vladimir I. Fal’ko 1 

$^{1}$Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom 

$^{2}$Institute for Theoretical Physics A, RWTH Aachen, D-52074 Aachen, Germany 

We investigate the contribution of the low-energy electronic excitations towards the Raman spec- trum of bilayer graphene for the incoming photon energy Ω ≳ 1eV. Starting with the four-band tight-binding model, we derive an effective scattering amplitude that can be incorporated into the commonly used two-band approximation. Due to the influence of the high-energy bands, this ef- fective scattering amplitude is different from the contact interaction amplitude obtained within the two-band model alone. We then calculate the spectral density of the inelastic light scattering ac- companied by the excitation of electron-hole pairs in bilayer graphene. In the absence of a magnetic field, due to the parabolic dispersion of the low-energy bands in a bilayer crystal, this contribution is constant and in doped structures has a threshold at twice the Fermi energy. In an external mag- netic field, the dominant Raman-active modes are the n − → n + inter-Landau-level transitions with crossed polarisation of in/out photons. We estimate the quantum efficiency of a single n − → n + transition in the magnetic field of 10T as I$_{n}$ −$_{→}$$_{n}$ + ∼ 10 − $^{12}$. 

Bilayer graphene 1,2 is a representative of the re- cently discovered family of new carbon allotropes. 2 It attracted attention by the observation of an unusual se- quencing of plateaus in the quantum Hall effect 1,2 and the possibility to modify its spectrum by opening a small gap and induce an insulating state with an exter- nal electric field. 2–4 The electronic properties of bilayer graphene have been characterised using angle-resolved photoemission spectroscopy 5,6 and optical absorption in the visible 7,8 and infrared 9–14 spectral range. Bilayer graphene has also been studied using inelastic scattering of light, subject to the detection of Raman-active lattice vibrations in this two-dimensional crystal. 15–23 However, no theoretical or experimental study has yet been per- formed, on the contribution of electronic excitations to- wards Raman spectra of bilayer graphene. This paper presents a theory of such a contribution. 

The electronic Raman spectroscopy can provide infor- mation about various single particle and collective elec- tron excitations in the system studied. In semiconduc- tors, it has been, for example, employed to investigate donor and acceptor states, plasmons and spin-density fluctuations involving electron spin-flip due to the spin- orbit interaction. 24,25 The inelastic scattering of photons on electrons in semiconductor placed in an external mag- netic field was first discussed in Ref. 26, where it was pointed out that nonparabolicity of the electronic bands is crucial for the electron-photon interaction matrix ele- ments not to vanish. The features corresponding to the electronic contribution to the Raman scattering in an external magnetic field were observed in many semicon- ductors, for example, InSb (Ref. 27) and GaAs. 28 

Recently, the Raman spectroscopy of electronic ex- citations in monolayer graphene has been investigated theoretically. 29 It has been shown that at high mag- 

PACS numbers: 73.22.Pr, 71.70.Di, 78.67.Wj, 81.05.ue 

## I. INTRODUCTION 

netic fields the inelastic light scattering accompanied by the excitation of the electronic mode with the highest quantum efficiency involves the generation of inter-band electron-hole pairs. At high (quantizing) magnetic fields this leads to the electron excitations from the Landau level (LL) n − at the energy − √$_{2}$$_{n}$ ℏ v/λ$_{B}$ in the valence band to the Landau level n + at the energy √$_{2}$$_{n}$ ℏ v/λ$_{B}$ in the conduction band with energies ω$_{n}$ $_{= 2}$√$_{2}$$_{n}$ ℏ v/λ$_{B}$ and crossed polarisation of in/out photons, in contrast to the Δ n = ± 1 transitions between Landau levels which are dominant in the absorption of left and right- handed circularly polarised infrared photons. 30 Raman spectroscopy, therefore, provides data supplementary to that obtained in optical absorption. This fact could be of interest in particular for bilayer graphene placed in an external magnetic field. Recent measurement of the in- frared absorption in fields up to 20T showed 31 deviations from the tight-binding model for an ideal bilayer. Here, we offer a theory for an alternative experimental probe, with different selection rules for the inter-Landau-level excitations, which could bring some new insight into the properties of the bilayer graphene. 

We study the Raman spectroscopy of electronic exci- tations in bilayer graphene both with and without an external magnetic field using the tight-binding approach. First, we describe processes in which after the inelastic scattering of the optical photon, an electron-hole pair is created in the low-energy bands of the bilayer. Then, we evaluate the scattering amplitude corresponding to such a process within the framework of the four-band tight-binding model and extract an effective scaterring amplitude that is incorporated into the two-band low- energy Hamiltonian for bilayer graphene. 1 This effective scattering amplitude contains the influence of the high- energy bands and is different from the scattering ampli- tude found using only the approximate low-energy Hamil- tonian. Next, we find the low-energy electronic contribu- 

