First and second-order Raman process

Resonance Raman scattering In typical experiments, the laser excitation energy is much higher than the phonon energy. Although the exchange in energy between light and the medium is transferred to the atomic vibrations, it is the electron system that mediates the light-matter interaction. If the energy of the photon does not match a real transition in the electronic band structure, the electron that absorbs the energy is said to be in a virtual state. From there, it can couple to the lattice. If, however, the laser photon energy matches a real electronic transition from an occoupied initial state to an unoccupied final state Eif, the probability of this process can exceed the former by orders of magnitude

(≈103, [51]). This process is called resonance Raman effect. The same is valid if the scattered light (ELaser±Eq) is equal to an electronic transitionEif. Therefore,

the Raman intensity should increase when ELaser → Eif, being resonant with

incident light and ELaser → Eif ±Eq, being resonant with scattered light. The

Raman intensity profile is given by [51]:

I(ELaser) = A

(ELaser−Eif −iγr)(ELaser−(Eif ±Eq)−iγr)

2 (2.61)

The full width half maximum (FWHM) of each peak in the Raman intensity profile is the resonance window width γr, which is related to the lifetime of the

excited states in the Raman scattering process. It gives the time delay between absorption of a photon and emission of a phonon. The resonance effect is impor- tant in nano-scale structures, as the Raman signal is in general very weak due to the small size of the samples.

Higher-order Raman processes A detailed description of the Raman spec- trum of graphite and graphene will be given in chapter A. For the discussion, higher-order Raman processes will be important. The order of the Raman pro- cess is given by the number of scattering events that contribute to the Raman process. In the first-order Stokes Raman scattering event, the photon exchange energy creates one phonon in the crystal with a very small momentum (q ≈ 0

[51]). If the scattering involves two, three or more phonons the Raman process is called second-order, third-order, .. respectively. First-order processes give the basic quantum of vibration, higher-order processes can give very important information about overtones (n·Eq) and combination of several phonon modes

(Eq1+Eq2). One interesting point is that for higher-order processes the restriction

q ≈ 0 in the first-order process is relaxed. A photoexcited electron at ki can be

scattered toki+q and with a second phonon with wavevector −q backscattered

to its original position, which allows the recombination with their corresponding holes. The propability of such processes is usually very small and not imortant for solids, but as will be shown in chapter 6, in the case ofsp2 _{systems there are} Raman signals from q6= 0 scattering events.

The Raman spectrum of 1-layer graphene The Raman spectrum of 1- layer graphene exhibits different features. The two most prominent peaks are the G-peak appearing at 1582 cm−1 and the 2D-peak appearing at about 2700 cm−1, when using 2.41 eV as excitation laser energy. Besides these two modes, different peaks appear in the spectra, as shown in Fig. 2.8(a) [52]. There is an additional peak with low intensity near 2400 cm−1_{, called G*-band. Fig. 2.8(a)(ii) shows the} Raman spectrum for 1-layer graphene with some disorder or defects in the sample, where an additional peak, labeled D, at about 1350 cm−1 _{appears. Close to} 1620 cm−1 another disorder-induced mode appears, called D’. Fig.2.8(b) depicts the relevant physical processes responsible for the peaks in the Raman spectrum of graphene [52]. This is the common picture used in the literature to explain the Raman scattering in graphene. We start with (i): The G peak is associated with the doubly degenerated iTO and LO (see Fig.2.6 from the last section) phonon mode with E2g symmetry at the Γ-point in the Brillouin zone (BZ). A

2.4. Raman scattering

Figure 2.8.:(a) The (i) Raman spectrum of a defect free monolayer graphene showing the main Raman features, G, G* and the 2D-mode with a laser excitation energy of 2.41 eV. And (ii) the Raman spectrum of a defective monolayer, in which two disorder-induced D and D’ Raman modes appear additionally. (b) The (i) first-order Raman process which gives rise to the G-mode. Second-order Raman process showing the (ii) inter-valley D-mode and (iii) intra-valley D’-mode. (iv Second-order Raman process involving two phonons and (v) a possible triple resonance Raman process. All figures are taken from Ref.[52].

The latter is a first-order process. Scattering within the K (K’)-point is called intra-valley scattering. (ii): The photo-excited electron scatters elastically by a defect and a iTO phonon from K to K’ and back. Scattering from K to K’ points in the hexagonal BZ is called inter-valley scattering, involving q 6= 0 phonons. The latter is a second-order process. It can generally be distinguished between first- and second-order process by the frequency shift of the second-order Raman peaks with the laser wavelength used. For example, the disorder-induced D-mode shifts with laser excitation energy about 53 cm−1_{/eV [53]. (iii): Disorder-induced} intra-valley process involving a q6= 0 iLO-phonon. (iv): In the double-resonance process the photo-excited electron is inelastically scattered by a iTO phonon with wavevectorqfrom K to K’ point and then back scattered by a iTO phonon with opposite momentum. In a double-resonance Raman process with three scattering events, two resonance conditions have to be fullfilled. The intermediate state is always a real electronic state and either the initial or the final state is a real electronic state. Because of the special band structure of graphene, with an

almost symmetric conduction- and valence-band, this can lead to case (v), the triple resonance. The electron and hole are scattered from K to K’ point, where electron-hole generation is a resonant process, both electron and hole scattering are resonant and electron-hole recombination at the opposite side will be resonant. This triple resonant condition might explain why the 2D-band in 1-layer graphene is more intense than the G-band [52].