Raman Spectroscopy of Graphene
RRaman Spectroscopy of Graphene
Sven Reichardt
1, 2 and Ludger Wirtz Physics and Materials Science Research Unit, Universit´e du Luxembourg, 1511 Luxembourg, Luxembourg JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany
Raman spectroscopy of graphene is reviewed from a theoretical perspective. After an introductionof the building blocks (electronic band structure, phonon dispersion, electron-phonon interaction,electron-light coupling), Raman intensities are calculated using time-dependent perturbation theory.The analysis of the contributing terms allows for an intuitive understanding of the Raman peakpositions and intensities. The Raman spectrum of pure graphene only displays two principle peaks.Yet, their variation as a function of internal and external parameters and the occurrence of secondary,defect-related peaks, conveys a lot of information about the system. Thus, Raman spectroscopy isused routinely to analyze layer number, defects, doping and strain of graphene samples. At thesame time, it is an intriguing playground to study the optical properties of graphene.
CONTENTS
I. Introduction 1II. Theoretical description of the Raman spectrum of graphene 2A. Basic building blocks 31. Electronic band structure 32. Phonon dispersion 33. Electron-phonon coupling 54. Electron-light coupling 65. Electron-defect scattering 76. Summary and Feynman rules 8B. General kinematic considerations and possible Raman processes 9C. Diagrammatic calculation of the Raman amplitudes 101. One-phonon, defect-free processes ( G peak) 112. Two-phonon, defect-free processes (e.g., 2 D , 2 D (cid:48) , and D + D (cid:48)(cid:48) peaks) 133. Defect-assisted peaks (e.g., the D , D (cid:48) , and D + D (cid:48) peaks) 17III. Influence of internal and external parameters on the Raman spectrum 18A. Layer number 18B. Defect concentration 18C. Doping 20D. Strain 22IV. Conclusions 24Acknowledgments 24References 24 I. INTRODUCTION
Raman scattering is the inelastic scattering of light. The frequency of a photon can change by transferring energyto and/or receiving energy from the lattice vibrations of the material. In quantum mechanics, the “allowed” energiesof an oscillation are quantized and - in the case of a harmonic oscillation - equidistant. The vibrations of the latticecan thus be described in the language of quasi-particles and the term phonon is usually used. The incoming photonchanges its energy by effectively exciting or absorbing one or several phonons. If a phonon is excited, the photon loosesenergy and the process is called
Stokes scattering. In the case of the absorption of a phonon, the photon gains energyand the process is referred to as
Anti-Stokes scattering. The spectrum of the inelastically scattered light thereforefeatures discrete peaks, whose positions can be directly associated with certain vibrational modes of the crystal. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r In first order Raman scattering, only optical phonons at the Brillouin zone center can be excited. Furthermore,selection rules only allow for vibrations with special symmetry properties. For graphene, only one peak, the so-called G peak, is due to first-order Raman scattering. An additional line, the so-called 2 D line, is due to second orderRaman scattering, where two phonons are excited at the same time. The overall Raman spectrum of graphene thusconsists mainly of two peaks (see Fig. 4) and is quite similar to the spectrum of graphite and of carbon nanotubes.Nevertheless, via the exact peak positions, their width, their dispersion as a function of the exciting laser energy, andthe occurrence of side peaks, Raman spectroscopy yields a surprising amount of information about graphene samples:For example, it gives information on the number of layers, their stacking order, the underlying substrate, defects,impurities, doping, and strain. Since Raman scattering is a fast and non-invasive method, it has evolved into one ofthe principal characterization tools for graphene and related graphitic materials.Raman spectroscopy is often categorized as “vibrational spectroscopy” because it probes primarily the phononfrequencies. However, a quantitative description of the Raman intensities is deeply connected to the optical (absorptionand emission) properties of the concerned material. This is particularly the case for resonant Raman spectroscopy,where the emitted/absorbed light is in resonance with electronic transitions. In graphene, due to the linear crossingof the optically active π and π ∗ bands, the Raman effect is resonant for all laser frequencies in the infrared, visible,and near UV ranges. Different “quantum pathways” [1] (corresponding to contributions of different microscopicprocesses in perturbation theory) contribute to the intensity of a given peak. This gives rise to intriguing quantuminterference effects with sometimes surprising consequences for the Raman intensities (e.g., as a function of laserenergy and of sample doping). Furthermore, the intensities and the widths of the Raman peak are strongly influencedby the lifetimes of intermediate electronic excitations. Raman scattering thus also bears information on the complexdynamics of electron-hole pairs in graphene. All this makes a chapter on Raman spectroscopy an integral part of abook about the optical properties of graphene.A number of books/book chapters [2, 3] and review articles [4–6] have already been devoted to the topic of Ramanscattering in graphitic materials. This book chapter puts particular emphasis on the theoretical foundations of resonantRaman scattering in graphene. It explains the basic theory for Raman scattering in graphene using perturbationtheory. Together with the study of the groundbreaking theoretical research articles (in particular [7–10]), it shouldenable the newcomer to the field to learn how an exact quantitative calculation can be achieved in principle. For thecomputationally less ambitious reader, the chapter gives detailed qualitative understanding how the different termsin perturbation theory depend on external stimuli (such as doping, defects, or strain) and what conclusions one candraw from the Raman spectra.The chapter is structured as follows: In section II, we first introduce the basic building blocks for the understandingof resonant Raman spectra in graphene: The electronic band structure, the phonon dispersion, the coupling betweenelectrons and phonons, and the coupling of the electronic states to light. Afterwards, the formalism of time-dependentperturbation theory is applied to give the reader an intuitive understanding and enable her/him to study the relevantresearch literature. In section III, the theory is then applied to understand qualitatively and quantitatively, howRaman spectroscopy can be used to study the quality of graphene flakes as well as to understand the effects ofenvironment, doping and strain. II. THEORETICAL DESCRIPTION OF THE RAMAN SPECTRUM OF GRAPHENE
In this first section, we will have a look at the microscopic origin of the Raman spectrum of graphene, in particularthe behavior, shape, and size of its various peaks. To this end, we will use time-dependent perturbation theory tocalculate expressions for the Raman scattering amplitudes. This approach has the advantage that the individualterms of the perturbation series can be represented by so-called
Feynman diagrams . The graphical representation ofthe calculation allows for both easier bookkeeping of the terms in the perturbation series and at the same time givesa simple physical picture for the microscopic processes governing Raman scattering.In the first part of this section, we will introduce the basic building blocks needed for the perturbation theorycalculation. We briefly discuss the electronic band structure of graphene before having a more detailed look at thephonon dispersion of graphene. Thereafter, we will discuss the coupling between electrons and phonons and theirtreatment in perturbation theory and show how to include the effects of an external electromagnetic field, i.e. ofincoming and outgoing photons. This first part will be concluded by briefly touching upon electron-defect scatteringand a summary in which we give a graphical representation of the mathematical expressions for the various buildingblocks in terms of Feynman diagrams.The second part contains an overview over the kinematics of Raman scattering. We will derive what kind of peaksare expected in the Raman spectrum of graphene based on kinematical considerations alone.In the third subsection, we will focus on the actual calculation of the Raman spectrum. Using time-dependentperturbation theory and the language of Feynman diagrams, we will demonstrate how the Raman spectrum of graphenecan be calculated. in this section we will mainly focus on the two most prominent Raman peaks and study theirbehavior from the scattering amplitude obtained in perturbation theory.
A. Basic building blocks
We start by introducing the basic ingredients needed to theoretically describe the Raman spectrum of graphene.These ingredients are the electronic band structure, the phonon dispersion, the electron-phonon coupling, the electron-light coupling, and (if the sample is not perfect) electron-defect scattering. With the exception of the latter, we willhave a look at each of these building blocks before we conclude this section with a summary.
1. Electronic band structure
Graphene is a monolayer of carbon atoms arranged in a two-dimensional hexagonal lattice. Each carbon atompossesses four valence electrons, three of which occupy hybridized sp orbitals, which form strong, in-plane covalent σ − bonds. The forth valence electron occupies a p z -orbital, which form π -bonds, in which the electrons are delocalized.The electronic band structure, obtained from an ab initio calculation using density functional theory (DFT) is shownin Fig. 1. Γ K M Γ B a n d e n e r g y ( e V ) FIG. 1:
Electronic band structure of monolayer graphene.
Ab initio electronic band structure of monolayergraphene along the high symmetry line Γ − K − M − Γ obtained from density functional theory (DFT) in thegeneralized gradient approximation (GGA). The full lines are the π and π ∗ bands that are by far the mostimportant bands for the Raman scattering process.Since we will be interested in describing and calculating the Raman spectrum for excitation energies in or near thevisible spectral range ( ∼ π and π ∗ bands (full lines in Fig. 1). Note that for a quantitatively accurate calculation of the Raman spectrum, the DFTband structure needs to be corrected to properly include electron-electron-interaction effects. These corrections canbe computed, for example, within the GW approximation [11, 12] and their main effect is a steepening of the conic π bands around the K point with their slope increasing by roughly 18% [9, 13].
2. Phonon dispersion
Besides the electronic band structure, the phonon dispersion of graphene also plays a very important role for theunderstanding of the Raman spectrum. Since the lattice structure of graphene can be described by a hexagonal latticewith two atoms per unit cell, the phonon band structure consists of 2 × q of the phonon, respectively. Note that these notions only makesense for small phonon wave vectors, i.e for q near Γ. Away from Γ, the character of the modes is a mixture of acoustic(in-phase oscillations of neighboring atoms) and optical (opposite phase oscillations of neighboring atoms) behavior.However, for simplicity the labels LA etc. are usually also used to designate the phonon branches away from Γ byfollowing the specific branch along the high-symmetry lines. The phonon dispersion of graphene as obtained from adensity functional perturbation theory (DFPT) calculation is shown in Fig. 2a. (a) Γ K M Γ P h o n o n f r e q u e n c y ( c m − ) LOTOZOLA TAZA (b)
FIG. 2:
Phonon dispersion of monolayer graphene. (a)
Ab initio phonon dispersion of monolayer graphenealong the high symmetry line Γ − K − M − Γ obtained from density functional perturbation theory (DFPT) in thelocal density approximation (LDA). (b) Zoom-in into the phonon dispersion for phonon wave vectors around K .Symbols denote experimental data obtained from inelastic X-ray scattering (IXS) and Raman scattering. Full linesrepresent the TO branch obtained from density functional theory within the generalized gradient approximation(GGA). Dashed lines represents TO phonon frequencies corrected within perturbation theory, where theelectron-phonon coupling was obtained on the level of the GW approximation. (Figure taken from Ref. 14.)The most prominent features of the phonon dispersion are the kink in the TO branch at K and the positive slopeof the LO branch at Γ. As we will see in a later section, these are precisely the parts of the phonon dispersion thatplay the most important role in the description of the Raman spectrum as the phonons that contribute the most to itare the two degenerate in-plane optical phonons at Γ and the phonons of the TO branch around the K point, whoselattice vibration patterns are shown in Fig. 3. For a description of the Raman spectrum it is therefore of utmostimportance to accurately describe these parts of the phonon dispersion. (a) c) b) K -A’ a) Γ -E (b) c) b) K -A’ a) Γ -E FIG. 3:
Lattice vibration patterns. (a) Vibration pattern corresponding to one of the two degenerate, opticalin-plane phonons at Γ. In the commonly used
Mulliken notation , this mode is called an E g mode, as it transformsin the E g representation of the point group of Γ. The dotted line marks the boundaries of the Wigner-Seitz cell ofthe graphene lattice. (b) Vibration pattern corresponding to the TO branch at K . In the Mulliken notation, thismode is denoted as A (cid:48) , as it transforms in the A (cid:48) representation of the point group of K . The dashed lines denotethe boundaries of a supercell of six atoms. (Figure taken from Ref. 14.)The pronounced kinks arise due to so-called Kohn anomalies [15]. When the lattice ions vibrate according to aphonon mode with wave vector q , they induce an electronic charge density of the same periodicity, whose magnitudebecomes large when q connects two points on the Fermi surface. Since the Fermi surface of graphene consists of thetwo points K and K (cid:48) at the corners of the Brillouin zone (which are separated by a vector K ), phonons with q = or q = K lead to these large, periodic electronic charge densities. These electronic charge densities in turn lead todescreased screening of the ionic charges, leading to a great enhancement of electron-phonon interaction. This largeelectron-phonon coupling for q near Γ or K is responsible for the comparatively steep slope of the phonon dispersion(i.e. the kinks) in these parts of the Brillouin zone, since it can be shown that at Γ and K , the slope of the phonondispersion is directly proportional to the squared electron-phonon coupling between the corresponding phonon andelectrons in the π and π ∗ bands averaged over a small circle of electronic k vectors around the Fermi points K and K (cid:48) [16].The increased electron-phonon coupling at Γ and especially at K is further strongly enhanced by electronic corre-lation effects [14]. This manifests itself in the inaccurate quantitative description of the phonon dispersion near K in density functional theory, where correlation effects due to electron-electron interaction are not well-described byexchange-correlation functionals in the commonly used local density or generalized gradient approximations. As willbe commented upon in the next subsection, it is possible to obtain a more accurate description of the electron-phononcoupling for phonons at Γ and K on the level of the GW approximation. These corrected values for the electron-phonon coupling lead to a correction of the phonon frequencies within perturbation theory. The most importantresult is a steepening of the kink near K and a decrease of the phonon frequency of the TO branch at K by morethan 100 cm − [14], which matches the experimental data well (compare Fig. 2b).
3. Electron-phonon coupling
Besides a description of the electron and phonon band structures, one needs a description of the coupling betweenthe two. For this, we consider the electronic Hamiltonian (discarding electron-electron interaction for the moment),which has the form H = p m e + V ( r ; { R nα, } ) , (1)where r and p denotes the position and momentum operator, respectively, and V is the lattice potential which dependsparametrically on the position of the nuclei, which we collectively denote with { R nα } , where n labels the unit celland α specifies the atom within the unit cell. The index “0” refers to the equilibrium positions of the nuclei.When the lattice atoms are oscillating, the position of the nucleus at position R nα, changes by a small displacement: R nα, → R nα = R nα, + u ( R nα, ), where u ( R nα, ) denotes the displacement of the nucleus at equilibrium position R nα, . As a consequence, the lattice potential changes as well and for small displacements u ( R nα, ) the potentialcan be expanded into a Taylor series: V ( r ; { R nα } ) = V ( r ; { R nα, } ) + (cid:88) n,α ∂V∂ R nα ( r ; { R nα, } ) · u ( R nα, )+ 12 (cid:88) n,m,α,β (cid:88) i,j ∂ V∂R nα,i ∂R mβ,j ( r ; { R nα, } ) u i ( R nα, ) u j ( R mβ, ) + O ( u ) . (2)Here, the sums run over all unit cells n , all atoms α within the unit cell, and all cartesian coordinates i . The electronicHamiltonian thus acquires two more terms (up to quadratic order in the displacement), describing the coupling ofelectrons to one and two phonons, respectively: H → H + H (1)el − ph + H (2)el − ph , (3)where H ( i )el − ph is given by the i th order term in the expansion of the potential given above.After quantizing the phonon field, the operator corresponding to a displacement of the atom at equilibrium position R nα, can be written as a superposition of the vibrational eigenmodes: u ( R nα, ) = (cid:88) q ,λ (cid:115) ¯ h M C ω q ,λ (cid:104) b λ, q v αλ, q + b † λ, − q (cid:0) v αλ, − q (cid:1) ∗ (cid:105) e i q · R n , (4)where q and λ denote the quasimomentum and phonon branch (e.g. λ =TO for the transverse optical branch) of theeigenmode, respectively, R n is the position, i.e., lattice site, of the n th unit cell, and b ( † ) λ, q is an operator that destroys(creates) one phonon of branch λ with quasimomentum q . The vector v αλ, q denotes one of the two three-dimensionalparts of the six-dimensional eigenvector of the dynamical matrix describing the displacement of atom α of the unitcell.To describe the Raman scattering amplitude within perturbation theory, one needs the matrix elements of theelectron-phonon Hamiltonian for the π and π ∗ states and for one or two specific phonons. For example, the matrixelements of the electron-one-phonon Hamiltonian for a phonon of branch λ and with quasimomentum q in the finalstate are given by g λs, k − q ; s (cid:48) , k = (cid:104) s, k − q ; λ, q | H (1)el − ph | s (cid:48) , k (cid:105) = (cid:115) ¯ h M C ω q ,λ (cid:88) n,α e i q · R n (cid:0) v αλ, q (cid:1) ∗ · (cid:28) s, k − q (cid:12)(cid:12)(cid:12)(cid:12) ∂V∂ R nα ( r ; { R nα, } ) (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) , k (cid:29) , (5)where s and s (cid:48) specify the electronic band and | s, k − q ; λ, q (cid:105) is a state consisting of an unperturbed electronic statespecified by the band index s and Bloch wave vector k − q and a phonon from branch λ with quasimomentum q . Forthe Raman spectrum of graphene, only the π and π ∗ bands are important, i.e. s, s (cid:48) = π, π ∗ and for every k , q , and λ the electron-phonon coupling matrix elements can be written as a 2 × g λ k − q , q = (cid:16) g λs, k − q ; s (cid:48) , k (cid:17) s,s (cid:48) = π,π ∗ in thespace of the π and π ∗ bands.By the same procedure, one can obtain a 2 × H (2)el − ph . For the purpose of this chapter, only those matrix elements are important thatcontain the two phonons in the final state. Again, these can conveniently written as a 2 × π and π ∗ bands, which we will denote by ˜g λ ,λ k − q − q , k , where λ and q are the quantum numbers of phonon 1(2).Numerical values for the electron-phonon matrix elements can be obtained from density functional perturbationtheory. It should be noted, however, that the underestimation of electron-electron interaction effects in DFT leads toan underestimation of the electron-phonon matrix elements as well, especially for k and q near K , where long-rangeelectronic correlation effects play an important role. For specific k and q , such as k = q = K , it is possible toincorporate these effects by relating the electron-phonon matrix elements at these high-symmetry points to changesof the band energies when the lattice ions are displaced statically according to the phonon displacement patternshown in Fig. 3b. For instance, for a phonon from the TO branch with quasimomentum q = K , the square of theelectron-phonon coupling for a π and π ∗ state with k = 2 K = K (cid:48) obeys [14, 16]: (cid:12)(cid:12) g TO π, K ; π ∗ , K (cid:12)(cid:12) ∝ lim d → (cid:32) (cid:2) ε π ∗ K − ε π K (cid:3) ( d ) d (cid:33) , (6)where d is the amplitude of the displacement and ε π ( π ∗ ) K ( d ) is the energy of the π ( π ∗ ) band at K when the atoms aredisplaced.When incorporating electron-electron interaction effects on the level of the GW approximation into the bandenergies, this procedure allows one to calculate the electron-phonon matrix elements at these high-symmetry pointswhile taking into account correlation effects more accurately. This procedure, however, is limited to high-symmetrypoints for k and q as only there the electron-phonon matrix elements can be related to changes of the band energies.
4. Electron-light coupling
The final ingredient needed for a theoretical description of Raman spectra is the coupling of electrons to light, i.e.an external electromagnetic field. An external electromagnetic field can be described in terms of a vector potential A ( r ) and a scalar potential φ ( r ). In Coulomb gauge ∇ · A ( r ) = 0, the scalar potential must be zero for fields vanishingat infinity. The external electromagnetic field can therefore be described entirely in terms of a vector potential A ( r ).The Coulomb gauge imposes one constrain on A ( r ), leaving two degrees of freedom for the external electromagneticfield, which can be identified with the two possible linearly independent polarization directions of light.The coupling of electrons to an external vector potential A ( r ) can be introduced via the minimal coupling prescrip-tion p → p + ec A ( r ). The electronic Hamiltonian then becomes: H = (cid:2) p + ec A ( r ) (cid:3) m e + V ( r ; { R } )= H + ecm e A ( r ) · p + e c m e [ A ( r )] = H + H (1)el − R + H (2)el − R . (7)The coupling to the external electromagnetic field generates two more terms in the Hamiltonian that can be treatedin perturbation theory.To obtain the matrix elements for the coupling of electrons to individual photons, the electromagnetic field has tobe quantized. For this, one expands the vector potential in the eigenmodes of a free electromagnetic field, i.e. intoplane waves, and promotes it to an operator: A ( r ) = (cid:90) d k (2 π ) (cid:88) µ (cid:115) πc ¯ h ω k (cid:104) (cid:15) µ, k a µ, k + ( (cid:15) µ, − k ) ∗ a † µ, − k (cid:105) e i k · r , (8)where µ labels the two possible polarizations of a photon, ω k = c | k | and (cid:15) µ, k are the frequency and polarization vectorof a photon with polarization µ and wave vector k , respectively, and a ( † ) µ, k is an operator that destroys (creates) aphoton with polarization µ and wave vector k .The matrix elements of the electron-one-photon Hamiltonian between two unperturbed electronic states and withone photon with polarization µ and wave vector k (cid:48) in the initial state then reads: γ µs, k − k (cid:48) ; s (cid:48) , k = (cid:104) s, k − k (cid:48) ; µ, k (cid:48) | H (1)el − R | s (cid:48) , k (cid:105) = ecm e (cid:115) πc ¯ h ω k (cid:48) (cid:15) ∗ µ, k (cid:48) · (cid:68) s, k − k (cid:48) (cid:12)(cid:12)(cid:12) e i k (cid:48) · r p (cid:12)(cid:12)(cid:12) s (cid:48) , k (cid:69) , (9)where | s, k − k (cid:48) ; µ, k (cid:48) (cid:105) denotes a state consisting of an unperturbed electronic state specified by the band index s andBloch wave vector k − k (cid:48) and a photon with polarization µ with wave vector k (cid:48) . This expression is often simplifiedfurther by the dipole approximation exp( i k (cid:48) · r ) ≈
1, which is justified by the fact that the wave length of the light( ∼ ∼ k (cid:48) · r ≈ r varies on the crystal scale only as it does in the matrixelements between Bloch states. In other words, for the description of Raman spectra, we can treat the externalphotons as having zero wave vector (i.e. we set k (cid:48) = 0), yet finite frequency. Again, we will collect the matrixelements between the possible combinations of π and π ∗ states for fixed k and light polarization µ in a 2 × γ µ k = (cid:16) γ µs, k ; s (cid:48) , k (cid:17) s,s (cid:48) = π,π ∗ .Finally, one can follow a similar procedure to obtain the matrix elements between π and π ∗ states for the electron-two-photon Hamiltonian H (2)el − R . For a calculation of the Raman spectrum only those matrix elements are importantwhich involve one photon with polarization µ in the initial state and one photon with polarization ν in the final state.We will denote the corresponding 2 × π and π ∗ states by ˜ γ µν k .
5. Electron-defect scattering
The final building block needed for a complete description of the Raman spectrum within the framework of per-turbation theory is a description of electron-defect scattering. If the graphene sheet contains defects, electrons mayscatter from them elastically, leading to additional observable peaks in the Raman spectrum. Since a first, qualitativeunderstanding of these defect-assisted peaks does not require a detailed model for electron-defect scattering, we willnot have a closer look at this issue but merely refer the reader to the relevant literature [9, 17]. While we will discussdefect-assisted peaks in the following sections, we will not have a detailed look at its treatment in perturbation theoryand hence we will not discuss the modeling of electron-defect scattering here. We work with
Gaussian (or cgs ) units. In SI units, the factor e/c is to be replaced by e .
6. Summary and Feynman rules
To conclude the section, we summarize the elemental building blocks that are needed for the calculation of theRaman spectrum of graphene. Each of these elemental processes can be represented graphically by a so-called
Feynman diagram . In these diagrams each involved (quasi-)particle is represented by a specific line. We representelectrons (and holes) by full lines with arrows, phonons by dashed lines, and photons by wavy lines. The elementalFeynman diagrams for the various building blocks are shown in Table I.
Diagram Feynman rulePropagationof electron ω ; k i G k ( ω )Electron-lightinteraction k k µ ( − i ) γ ν k k k µ ν ( − i ) ˜ γ µν k Electron-phononinteraction k k − q λ, q ( − i ) g λ k − q k k − q − q λ , q λ , q ( − i ) ˜g λ λ k − q − q TABLE I:
Feynman rules for the basic building blocks for Raman scattering.
Feynman diagrams andassociated factors appearing in the perturbation series for the Raman scattering amplitudes. Each factor is a2 × π and π ∗ states, i.e., each element of the matrix is a matrix element between π and π ∗ states.In addition to the matrix elements for scattering processes between particles, there also appears a diagram for thepropagation of an unperturbed electron state between two scattering events. This amplitude is given by the retardedGreen’s function , defined as: G Rs, k ( t ) = (cid:40) e − iε s k t/ ¯ h (1 − n F ( ε s k )) , t > − e − iε s k t/ ¯ h n F ( ε s k ) , t < , (10)where ε s k is the energy of the state | s = π, π ∗ , k (cid:105) and n F ( ε ) denotes the Fermi-Dirac distribution. The retarded Green’sfunction describes the propagation of an unperturbed electron state for positive times ( t >
0) and of an unperturbedhole state for negative times t <
0. Since the perturbation theory calculation will be done in frequency space, we alsogive the Fourier transform of the retarded Green’s function: G s, k ( ω ) = n F ( ε s k )¯ hω − ε s k − i γ s k + 1 − n F ( ε s k )¯ hω − ε s k + i γ s k . (11)Here, we accounted for the fact that an electronic state | s, k (cid:105) has a finite lifetime γ s k , due to electron-phonon, electron-defect, and electron-electron scattering. As in the sections before, it is convenient to write the Fourier-transformedGreen’s function as a 2 × π and π ∗ states at fixed k : G k ( ω ) = ( G s, k ( ω ) δ s,s (cid:48) ) s,s (cid:48) = π,π ∗ , where δ s,s (cid:48) is the Kronecker delta. B. General kinematic considerations and possible Raman processes
Before we use the elementary processes and their associated matrix elements to compute the quantum mechanicalamplitude for the various possible Raman processes, we will first discuss some kinematic aspects of Raman processesin general and discuss which Raman processes are expected on kinematic grounds. In a general Raman process, theinitial state consists of one photon of energy ε in = ¯ hω in and polarization µ and the final state contains one photon ofenergy ε out = ¯ hω out and polarization ν plus n phonons with wave vectors q , . . . , q n from branches λ , . . . , λ n . Theenergy conservation law determines the observed Raman shift ω in − ω out = n (cid:88) i =1 ω q i ,λ i (12)as the sum of the frequencies of the involved phonons. Traditionally, Raman shifts (and phonon frequencies) areconverted to shifts of the inverse wavelength, ∆ λ − = λ − − λ − , and given in units of cm − , where the inversewavelength can be expressed in terms of the frequency ω via the relation λ − = ω/ (2 πc ). For reference, an energyof ¯ hω = 1 meV corresponds to a Raman shift of approximately 8 cm − . The wave vectors of the latter are furtherrestricted by momentum conservation. In the case that there are no defects in the sample, the sum of the wave vectorsof the phonons must be zero, since the wave vectors of photons are negligible and hence don’t need to be consideredin the momentum conservation law: n (cid:88) i =1 q i = 0 (no defects) . (13)In the presence of defects, the sum of the momenta of the phonons need not be equal to zero anymore since electronicscattering at defects can make up for it. If we limit ourselves to a maximum of two phonons, these kinematicconsiderations lead to the following categorization of Raman processes and their corresponding designations: a) One-phonon processes without defects ( G peak)Momentum conservation forces the phonon momentum to be equal to zero, i.e. the phonon must be a phonon fromthe Brillouin zone center, i.e. the Γ point. Non-zero Raman shifts further require the phonon to be an optical onesince the frequencies of acoustic phonons vanish at Γ. Furthermore, horizontal mirror symmetry forbids processeswith one out-of-plane phonon. Thus the phonons that contribute to the one-phonon processes in the absence ofdefects are the two degenerate, optical in-plane phonons at Γ. These are doubly degenerate due to rotation symmetry,i.e. they possess the same energy leading to the same Raman shift for both. The corresponding peak in the Ramanspectrum, located at around 1580 cm − in pristine graphene, is universally called the G peak as it was first observedin graphite. Note that, as will be discussed in a later section, the breaking of rotation symmetry, for example bystrain, leads to the two optical phonons having different frequencies and a splitting of the G peak. b) Two-phonon processes without defects (most prominently the 2 D , 2 D (cid:48) , and D + D (cid:48)(cid:48) peaks)Here momentum conservation requires the wave vectors of the involved phonons to be opposite, i.e. q = q = − q .The vector q , however, is not constrained by any further condition and can in principle correspond to any pointin the Brillouin zone. This explains why the corresponding Raman peaks are much broader than the G peak, as acontinuum of phonon frequencies contributes to them. The fact that there are still clear, observable peaks is relatedto the fact that certain phonon wave vectors and frequencies lead to a resonance in the Raman amplitudes as will bediscussed in the next section. The peaks that belong to this category are the 2 D peak, whose dominant contributioncomes from two phonons of the TO branch with q near K as is located at around 2700 cm − , the 2 D (cid:48) peak at around3200 cm − , whose main contribution stems from two phonons of the LO branch near Γ, and the D + D (cid:48)(cid:48) peak, whichis dominated by one phonon of the TO branch with q near K and one phonon from the LA branch with oppositewave vector and appears at around 2450 cm − . c) One-phonon processes with defects (most prominently the D and D (cid:48) peaks)In contrast to the G peak, the presence of defects allows phonons away from the Γ point to participate in a one-phononRaman process as the momentum of the phonon can be compensated by electron-defect scattering (see next sectionfor an example). Thus, in principle, any phonon can contribute to the Raman spectrum. However, just as for the0peaks in b) , the spectrum is dominated by a few selected phonons that lead to a resonance in the Raman amplitude.The two most prominent peaks that fall into this category are the D peak, which is mostly due to a TO phonon near K and is located at around 1350 cm − , and the D (cid:48) peak, which is mostly caused by a phonon from the LO branchnear Γ and occurs at around 1600 cm − . As their names suggest, these phonons are the same as the ones observedin the two-phonon overtones 2 D and 2 D (cid:48) from the previous paragraph. d) Two-phonon processes with defects (most prominently the D + D (cid:48) peak)Finally, there are additional two-phonon Raman processes that are made possible by the presence of defects. In thesesprocesses the two phonons do not need to have opposite wave vectors and can thus stem from different points in theBrillouin zone. The most prominent peak in this category is the D + D (cid:48) peak at around 2950 cm − , in which onephonon from the TO branch with a wave vector from near K and an LA phonon with a wave vector close to Γ makeup the resonant contribution.A typical, experimentally obtained Raman spectrum for a sample with defects is shown in Fig. 4, where the mostprominent peaks are labeled and the most dominant phonons are identified. I n t en s i t y ( A . U . ) Raman S hift (cm -1 ) D GD' D+D'' 2D D+D' 2D' P honon fr e qu e n c y ( c m - ) Γ K M Γ LOTOZOLA SH ZA
D'DD''D D D FIG. 4:
Example of a Raman spectrum and contributing phonons.
Two example of a Raman spectrum fora sample without (upper spectrum) and with defects (lower spectrum). The most prominent peaks as described inthe text have been labeled. Inset: Phonon dispersion of graphene, including the corrected TO branch. The phononswhich most prominently contribute to the various Raman peaks are marked by crosses. (Figure assembled fromfigures taken from Refs. 6 and 9.)Having given an overview over the kinematic constraints that govern Raman spectrum, we will now turn to thedynamical aspects in the next section and show how to obtain expressions for the Raman amplitudes and how thesean be used to determine which phonons play the most important role in the various processes.
C. Diagrammatic calculation of the Raman amplitudes
In this section we will illustrate the technique of diagrammatic perturbation theory to obtain mathematical expres-sions for the quantum mechanical amplitudes for the various Raman processes. The reader can find a more detailedaccount of this technique in Ref. 18–20. We will mainly focus on two examples: One-phonon and two-phonon processeswithout defects. Afterward, we will briefly comment on the defect-assisted peaks.1
1. One-phonon, defect-free processes ( G peak) For the first example, consider a defect-free process with one photon with polarization µ and energy ¯ hω in in theinitial state and one photon with polarization ν and energy ¯ hω out plus a phonon of branch λ and with momentum q in the final state. This is the process that gives rise to the G peak. The three leading-order, topologically inequivalentFeynman diagrams that can be assembled from the elementary diagrams of Table I are shown in Fig. 5. (a) (b) (c) FIG. 5:
Leading-order Feynman diagrams for the G peak. The three leading-order Feynman diagrams thatcan be constructed from the basic building blocks of the theory.Note that momentum conservation dictates q = 0 since the momentum of photons is negligible as discussed earlier.The wave vectors of the involved virtual electronic states are therefore all the same. The condition q = 0 also fixesthe type of the phonon to be optical as acoustic phonons with zero momentum do not couple to π and π ∗ states [21].The phonon branch index λ is thus restricted to the two branches TO and LO which are degenerate at the Γ point.Energy conservation fixes the frequency of the outgoing photon to ω out = ω in − ω q =0 ,λ . The finite lifetime, i.e., decaywidth, of the phonon can be included via the substitution ω q =0 ,λ → ω q =0 ,λ − i/ γ q =0 ,λ , where the relation betweenlifetime and decay width is given by γ = ¯ h/τ . With these kinematic constraints, application of the Feynman rulesyields the following mathematical expressions for the amplitudes corresponding to the three diagrams: M G = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:104) G k ( ω ) ( γ µ k ) † G k ( ω − ω in ) g λ k , k G k ( ω − ω out ) γ ν k (cid:105) (14) M G = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:104) G k ( ω ) γ ν k G k ( ω + ω out ) g λ k , k G k ( ω + ω in ) ( γ µ k ) † (cid:105) (15) M G = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:2) G k ( ω ) g λ k , k G k ( ω − ω q =0 ,λ ) ˜ γ µν k (cid:3) , (16)where it is understood that ω out is expressed through ω in and ω q =0 ,λ and the latter is a complex number that includesthe finite lifetime of the phonon, as discussed before. Notice that, in our convention, where the dimensions of theelectron-phonon and electron-light matrix elements are those of an energy and the dimension of the Green’s functionis inverse energy, the dimension of each matrix element is that of a frequency, i.e. a rate.To obtain the amplitude for each diagram from the elementary Feynman rules, one goes against the fermion flow(i.e. against the direction of the arrows) and writes down, from left to right the factors for each piece one encounters,imposing momentum and energy (frequency) conservation at each vertex. Since all elementary pieces are 2 × −
1) that accompanies a fermion loop, one sums over allinternal degrees of freedom, which includes a sum over spin, and takes the trace of the matrix product and alsoincludes summing over any undetermined, internal momenta and frequencies.The integral over the loop frequency ω can be evaluated with the residue theorem of complex analysis. Note thatin the leading order diagrams for Raman scattering, only the electronic Green’s functions have poles that contributeto the contour integral. The factor exp( iω + ) prescribes that the contour be closed in the upper half of the complexplane and therefore only those electronic states contribute that feature a positive imaginary part, which are thosestates that are occupied. Thus the integration over ω picks out the occupied states. After the integral has been carriedout, the expression for each Feynman diagram becomes a sum of terms. For example, the amplitude M G becomes a2sum of three terms: M G = (cid:88) spin (cid:88) k (cid:18) − h (cid:19) (cid:88) s = π,π ∗ n F ( ε s k ) (cid:40) (cid:104) γ ν k G k ( ω + ω out ) g λ k , k G k ( ω + ω in ) ( γ µ k ) † (cid:105) ss (cid:12)(cid:12)(cid:12) ¯ hω = ε s k + i γ s k + (cid:104) g λ k , k G k ( ω + ω in ) ( γ µ k ) † G k ( ω ) γ ν k (cid:105) ss (cid:12)(cid:12)(cid:12) ¯ hω = − ¯ hω out + ε s k + i γ s k + (cid:104) ( γ µ k ) † G k ( ω ) γ ν k G k ( ω + ω out ) g λ k , k (cid:105) ss (cid:12)(cid:12)(cid:12) ¯ hω = − ¯ hω in + ε s k + i γ s k (cid:41) (17)The number of terms generated in this way is equal to the number of internal propagators (i.e. the lines that make upthe loop). For a simple loop, as it is encountered in the diagrams for the leading order terms for Raman scattering,the number of internal lines is equal to the number of vertices. This means that a diagram with three external linesthat do not meet in a common vertex will lead to a sum of three terms after the loop frequency has been integratedover. Some books and articles, e.g. the commonly cited Ref. 22, prefer to work directly with these terms and writedown a diagram for each one of them. These kinds of diagrams are known as Goldstone diagrams and should not beconfused with the Feynman diagrams used in this text. In fact, every Feynman diagram can be viewed as a short-handnotation for the sum of several Goldstone diagrams with the sum being written as an integral over the loop frequency.Unfortunately, the distinction between Goldstone and Feynman diagrams is note always made in the literature onesometimes encounters Goldstone diagrams being labeled as Feynman diagrams and vice versa . As pointed out, bothare essentially equivalent as long as one uses the corresponding set of rules to translate a diagram into its correspond-ing amplitude and as long as one makes sure to write down all possible topologically non-equivalent diagrams of thechosen approach. In the Goldstone diagram formulation this entails several different time-ordering of the externallines, while in the Feynman diagram formalism the loop integral takes care of this technicality. We thus prefer towork with the Feynman diagram formalism as it greatly reduces the number of diagrams that need to be drawn. Inour case, the three Feynman diagrams for the process giving rise to the G peak correspond to eight Goldstone diagrams.Once one has obtained the quantum mechanical amplitudes for the individual diagrams, the total amplitude forthe process can be obtained by adding the individual amplitudes: M µνλG peak ( ω in ; γ el ; ω ph , γ ph ) = (cid:88) d =1 M d , (18)where we explicitly noted all quantities which the total matrix element depends on. These are the frequency of theincoming photon, the lifetimes of the involved electronic states (summarily denoted as γ el ), and the frequency anddecay width of the phonon, abbreviated as ω ph and γ ph , respectively. Note that the phonon frequency of an opticalphonon at Γ is independent of the branch λ , as noted before. Finally, Fermi’s golden rule yields the total rate of theRaman process, i.e., the probability per unit time for the process to happen, which is directly proportional to therecorded Raman intensity: I G ( ω out ) ∝ d P G peak d t = 2 π (cid:88) λ =LO,TO (cid:12)(cid:12)(cid:12) M µνλG peak ( ω in ; γ el ; ω ph , γ ph ) (cid:12)(cid:12)(cid:12) × δ ( ω in − ω out − ω ph ) (19)For a phonon with finite lifetime, the δ -function has to be replaced by a Lorentzian: δ ( ω in − ω out − ω ph ) → π γ ph / ω in − ω out − ω ph ) + ( γ ph / (20)From this expression we can immediately see that the position and width of the peak are entirely determined bythe frequency (approx. 1580 cm − in pristine graphene) and width (5-15 cm − depending on the sample quality) ofthe phonon. The intensity of the G peak, however, depends predominantly on the excitation frequency ω in as wellas on the lifetime of the involved electronic states, while the frequency and lifetime of the phonon only have someminor influence on the intensity of the peak. Regarding the latter, it should be noted that due to the absence of aband gap in graphene, there are always electronic transitions that are in resonance with the energy of the incominglight. The condition for resonance is that the energy of the in- or outgoing light matches the energy of an electronictransition. To give an example of a resonant term appearing in the Raman amplitude, we take a look at one of the3terms contained in the matrix product of the first term of M G : M G ⊃ (cid:88) spin (cid:88) k (cid:18) − h (cid:19) n F ( ε π k ) (cid:16) − n F ( ε π ∗ k ) (cid:17) (cid:16) − n F ( ε π ∗ k ) (cid:17) × (cid:104) π k ; ω ν out | H (1)el − R | π ∗ k (cid:105)(cid:104) π ∗ k ; ω λ ph | H (1)el − ph | π ∗ k (cid:105)(cid:104) π ∗ k | H (1)el − R | π k ; ω µ in (cid:105) (cid:2) ¯ hω in − ( ε π ∗ k − ε π k ) + i ( γ π k + γ π ∗ k ) (cid:3) (cid:2) ¯ h ( ω in − ω ph ) − ( ε π ∗ k − ε π k ) + i ( γ π k + γ π ∗ k + γ ph ) (cid:3) , (21)where we replaced the frequency of the outgoing light by the difference of the frequency of the incoming light andthe phonon frequency and accounted for the finite lifetime of the phonon by adding a negative imaginary part tothe phonon frequency. This term corresponds to a process where an electron in the π band is resonantly excited tothe π ∗ band by the incoming photon from which it drops to a virtual state in the π ∗ band by emitting a phononand then recombines with the hole by emitting the photon found in the final state. When the resonance condition∆ ε ( k ) = ¯ hω in or ¯ hω out is met, the denominator becomes minimal, leading to the resonant behavior. Note that, sincethe band structure is monotonically increasing away from the Fermi surface (i.e. the K point), the density of statesincrease monotonically with energy as well and hence the number of states that are in resonance with the energy ofthe incoming light increases with ω in , leading to the observed strong behavior of I G as a function of ω in [8, 23]. Bycontrast, Raman scattering in semi-conductors with sizable band gaps is mostly non-resonant, unless the excitationenergy matches the energy of the band gap or an excitonic state within the gap, whereupon a strong resonance isrecorded.In the limit of small ω in , the matrix element vanishes, which can easily be explained by the following argument:For small ω in , the electronic states that are in resonance with the excitation energy are located very close to the K point. However, very close to the K point, the band structure of graphene can be considered to be conic (the conicband structure is often called Dirac cone due to the Hamiltonian reducing to the Dirac Hamiltonian in this limit). Aconic band structure has a continuous in-plane rotation symmetry, which implies angular momentum conservation.If we consider circularly polarized light and phonons at Γ, each of them has an angular momentum of l z = ± ¯ h .The initial state thus has angular momentum ± ¯ h , while the final state can have an angular momentum of +2¯ h , 0¯ h ,or − h , which would violate angular momentum conservation. Therefore, in the case of a conic band structure, the G peak process would not be possible in leading order in perturbation theory as it would violate angular momentumconservation. If one goes one order beyond the linear, conic approximation for the band structure, however, the termsthat break the continuous rotation symmetry down to the 120 ◦ rotation symmetry of the lattice cause a trigonalwarping of the band structure. The reduced 120 ◦ symmetry means that angular momentum is only conserved modulo ± h , in other words, states with angular momenta that differ by an integer multiple of 3¯ h get mixed. This in turnleads to a non-vanishing Raman amplitude at the leading order in perturbation theory, since now the total angularmomenta of the initial and final states can differ by an integer multiple of 3¯ h and hence an incoming photon withangular momentum +¯ h can generate a phonon and photon both with angular momentum − ¯ h . This argument alsoillustrates that it is important to go beyond the Dirac cone approximation of the band structure when consideringthe G peak and that it is necessary to include the trigonal warping terms, which make the G peak process possiblein the first place.
2. Two-phonon, defect-free processes (e.g., D , D (cid:48) , and D + D (cid:48)(cid:48) peaks) As a second example for the diagrammatic calculation of the Raman spectrum, consider again a defect-free process,this time with two phonons in the final state. This kind of process gives rise to the 2 D , 2 D (cid:48) , and D + D (cid:48)(cid:48) peaks, amongothers. With the available building blocks, the amplitude in lowest order in perturbation theory can be constructedfrom eleven Feynman diagrams (corresponding to 38 Goldstone diagrams), four arbitrarily chosen ones of which areshown in Fig. 6. The case of linearly polarized photons and phonons can be constructed by taking suitable linear combinations of the circular polarizedmodes. (a) (b)(c) (d) FIG. 6:
Four examples of leading-order Feynman diagrams for the two-phonon, defect-free peaks.
Fourof the eleven Feynman diagrams that appear in leading order of perturbation theory. Of special importance is thetype of diagram shown in a) as it can become double resonant (see text).Again, application of the Feynman rules leads to the following expressions for the four example diagrams: M = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:34) G k ( ω ) g λ k , k + q G k + q ( ω + ω q ,λ ) γ ν k + q × G k + q ( ω + ω out + ω q ,λ ) g λ k + q , k G k ( ω + ω in ) ( γ µ k ) † (cid:35) (22) M = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:34) G k ( ω ) g λ k , k + q G k + q ( ω + ω q ,λ ) g λ k + q , k × G k + q ( ω + ω q ,λ + ω q ,λ ) ˜ γ µν k (cid:35) (23) M = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:34) G k ( ω ) ˜g λ λ k , k G k ( ω + ω q ,λ + ω q ,λ ) γ ν k × G k ( ω + ω in ) ( γ µ k ) † (cid:35) (24) M = (cid:88) spin (cid:88) k (cid:90) d ω π e + iω + ( − − i ) i × tr (cid:34) G k ( ω ) ˜g λ λ k , k G k ( ω + ω q ,λ + ω q ,λ ) ˜ γ µν k (cid:35) , (25)The total amplitude for this process can again be constructed by summing all (in the leading order eleven) amplitudes: M µνλ λ ( q ; ω in ; γ el ; ω ph , γ ph ) = (cid:88) d =1 M d , (26)The Intensity is then calculated form Fermi’s golden rule after summing over all possible phonon branches λ , λ andwave vectors q : I ( ω out ) ∝ d P d t = 2 π (cid:88) λ ,λ (cid:88) q (cid:12)(cid:12)(cid:12) M µνλ λ ( q ; ω in ; γ el ; ω ph , γ ph ) (cid:12)(cid:12)(cid:12) × δ ( ω in − ω out − ω q ,λ − ω q ,λ ) . (27)Again, for phonons with finite lifetimes, the δ -function is to be replaced by a Lorentzian. From the above expression,we can see that the two-phonon, defect-free process will result in a Raman spectrum that consists of the sum ofLorentzians, one for each combination of phonon branches λ , λ and for each phonon wave vector q . The posi-tion of each of these Lorentzians in the Raman spectrum (i.e., the recorded Raman shift) is given by the sum of5the frequencies of the two involved phonons. Since the phonon frequencies for the different wave vectors q forma continuum, one does not record individual peaks, however, but rather a continuous distribution in the Ramanspectrum. Note that the weight of each Lorentzian, i.e. the weight of the contribution of each wave vector q and ofeach of the branches λ and λ , is given by the corresponding Raman matrix element, which depends again, on theexcitation frequency ω in , the electronic lifetimes γ el and the phonon frequencies and lifetimes ω ph and γ ph , respectively.As will be discussed in detail in the next paragraph, for each pair ( λ , λ ) of phonon branches considered, theRaman amplitude will have a dominating maximum due to a double resonance for specific phonon wave vectors q λ ,λ res ( ω in ) that depend on the excitation frequency. These will be the phonons that contribute the most to theRaman amplitude and hence the corresponding Lorerentzians will have a much bigger weight than those of the otherphonons. This in turn leads to the observation of a clear peak for the two-phonon processes. This peak will have itsmaximum at the center of the Lorentzian corresponding to the q λ ,λ res , i.e. at ω q res ,λ + ω q res ,λ . While the Ramanmatrix element for these specific wave vectors might be maximal, this does not mean, however, that other “nearby”phonons do not sizably contribute to the observed Raman amplitude. In fact, the condition that fixes q λ ,λ res dependson the excitation energy, which in case of a resonance in the Raman amplitude is equal to some specific electronicexcitation energy. Since the electronic levels are smeared out by a finite decay width γ el , so is the resonance conditionfor q λ ,λ res . Thus, the range of the phonon wave vectors that give a sizable contribution to the Raman spectrum isdetermined by the electronic lifetime γ el and since the range of q determines the range of phonon frequencies that con-tribute to the two-phonon peaks, the electronic lifetime has a major impact on the width of two-phonon peaks. Notethat γ el = O (100 meV) (cid:29) γ ph = O (15 meV) and hence the phononic lifetimes hardly play any role for the width of thetwo-phonon peaks. This should be contrasted with the width of the G peak which is entirely determined by the widthsof the degenerate phonon involved and thus is much narrower than the two-phonon-induced peaks. For example,typical widths of the 2 D peak range from 20-40 cm − compared to the typical widths of 5-15 cm − for the G peak. [24]To understand the mentioned double-resonant behavior of the Raman amplitude for the two-phonon processes,we take a look at one of the amplitudes corresponding to one of the Feynman diagrams. In particular, we will lookat one of the terms generated by the contour integration over ω that are contained in the matrix product in M : M ⊃ (cid:88) spin (cid:88) k (cid:18) − h (cid:19) n F ( ε π k ) n F ( ε π k + q ) (cid:16) − n F ( ε π ∗ k ) (cid:17) (cid:16) − n F ( ε π ∗ k + q ) (cid:17) × (cid:104) π k + q ; ω ν out | H (1)el − R | π ∗ k + q (cid:105)(cid:104) π k ; ω λ q | H (1)el − ph | π k + q (cid:105)(cid:104) π ∗ k + q ; ω λ − q | H (1)el − ph | π ∗ k (cid:105)(cid:104) π ∗ k | H (1)el − R | π k ; ω µ in (cid:105) (cid:104) ¯ hω − ( ε π k + q − ε π k ) + i ( − γ π k + q + γ π k − γ ) (cid:105) (cid:104) ¯ h ( ω out + ω ) − ( ε π ∗ k + q − ε π k ) + i ( γ π k + γ π ∗ k + q − γ ) (cid:105) × (cid:2) ¯ hω in − ( ε π ∗ k − ε π k ) + i ( γ π k + γ π ∗ k ) (cid:3) . (28)It can be interpreted as describing the excitation of an electron with wave vector q from the π to the π ∗ band due tothe absorption of a photon with frequency ω in and polarization µ , followed by scattering of the excited electron to astate with wave vector k + q in the π ∗ band by emitting a phonon with frequency ω ≡ ω − q ,λ . In the meantime theempty state at wave vector k in the π band has been filled by an electron that was scattered from a state with wavevector k + q in the π band by emitting a phonon with frequency ω ≡ ω q ,λ , leaving behind a hole. Finally, this holerecombines with the scattered excited electron in the π ∗ band by via emission of a photon with frequency ω out andpolarization ν . This process is illustrated in Fig. 7a for two phonons of the same branch (TO) with wave vectors ± q near K and electronic states with wave vectors k near K and k + q near K (cid:48) , respectively.The amplitude for this process becomes maximal when one or several of the denominators become minimal, i.e.when their real part vanishes. While the amplitude for the G peak could only become singly-resonant since bothdenominators could not vanish simultaneously, here the presence of electronic states with different wave vectors allowsfor two of the denominators to vanish simultaneously if the phonon wave vector obeys some resonance condition. Notethat irrespective of the phonon wave vector q , single resonance is always possible, as already discussed for the G peak.To derive the condition for double resonance for the phonon wave vector q , we look at the amplitude for fixed q andfixed phonon branches λ and λ and require two of the three factors in the denominator of Equation 28 to vanishfor some electronic wave vector k . We will see that this is only possible for a specific choice of q , which we alreadydenoted as q λ ,λ res in the previous paragraph. In principle, any two combinations of the three denominators can vanishat once. We thus restrict ourselves to one of the three possible pairs of vanishing terms to illustrate how to find theresonance condition for the phonon wave vector q . The two terms we seek to minimize are the first and third of thefactors in the denominator. By setting the real part of the latter to zero, we immediately find a condition for the6 (a) (b) FIG. 7:
Illustration of one possible double-resonant Raman process and dispersion of the D line. (a)Illustration of the exemplary microscopic Raman process described in the text, which involves two phonons withwave vectors ± q near K and electronic states with wave vectors k near K and k + q near K’, respectively. Full linesrepresent the conic electronic band structure around K and K (cid:48) and the shaded regions represent the filled π band.The vertical, full arrows represent the excitation and subsequent recombination of an electron-hole pair, while thediagonal, dashed and dotted arrows, represent the scattering of electron and hole, respectively, from a state withwave vector near K to a state with wave vector near K (cid:48) . (Figure taken from Ref. 6.) (b) Observed dependence of theposition of the 2 D peak on the energy of the incoming photon. Dots and squares refer to data taken on freestandingand supported graphene, respectively. Full lines represent fits to the experimental data. (Figure taken from Ref. 25.)wave vector of the electronic states that are in resonance with the light:¯ hω in = ε π ∗ k − ε π k ⇒ k = k res ( ω in ) . (29)Note that since k is a two-dimensional vector, this condition describes an entire resonance surface (a closed line intwo dimensions) of k -vectors in the Brillouin zone. For example, in the limit of a conic band structure, the set of k res describes a circle in the Brillouin zone. By also requiring that the first factor in the denominator is minimal, wefinally arrive at a condition for q :¯ hω q ,λ = ε π k res ( ω in )+ q − ε π k res ( ω in ) ⇒ q = q res ( ω in ) . (30)Again, q is a two-dimensional vector and thus this equations defines a resonance surface for q q , which, for example,is a circle for conic band structure and a conic phonon dispersion as found for q near K or Γ. Due to the shape of thegraphene bands and the shape of the optical phonon dispersion near K , solutions for these equations always exist.By plugging the resonant wave vectors k res and q res and the constraints into the remaining factor that we did notexplicitly minimize, we find by using ω in = ω out + ω + ω :¯ h ( ω out + ω q res ,λ ) − ( ε π ∗ k res + q res − ε π k res )= − ¯ hω q res ,λ + ε π ∗ k res − ε π ∗ k res + q res . (31)In case the two phonons stem from them same branch, i.e., λ = λ , this equation can be further simplified by pluggingin the constraint for ω q ,λ : − ¯ hω q res ,λ + ε π ∗ k res − ε π ∗ k res + q res =( ε π k res + ε π ∗ k res ) − ( ε π k res + q res + ε π ∗ k res + q res ) . (32)Notice that the terms in parenthesis are a measure for the electron-hole asymmetry at k res and k res + q res . Since k res and k res + q res have to be near K for the resonance conditions to be fulfilled [9, 26], the third factor will be small aswell since the electron-hole asymmetry is small near K .On the other hand, for two phonons of different branches, i.e. λ (cid:54) = λ , the third factor will in general not be smalland this explains why the Raman peaks due to phonons of the same branch are much more visible than the Ramanbands with two phonons of different branches.It should be noted that one can construct similar conditions for q λ ,λ res by demanding that the two other possiblepairs of terms be minimal. In general, this will result, however, in slightly different wave vectors q λ ,λ res . Also notethat the resonant conditions resulting from the other diagrams have q depend on λ or even on both λ and λ , whichjustifies the notation q λ ,λ res . By explicitly evaluating the double-resonance conditions, one finds [9] that the phononsthat give the biggest contributions to the various Raman peaks are a phonon from the TO branch with q near K forthe D peak, and phonon from the LO branch with q near Γ for the D (cid:48) peak and a phonon from the LA branch with7 q near K for the D (cid:48)(cid:48) peak.Finally, we want to point out that q λ ,λ res depends on the excitation frequency ω in as mentioned before. Sincethe phonon wave vector determines the resonant contribution and hence the position of the maximum of the peak,the two-phonon peaks will shift when the excitation frequency ω in is varied, which leads to the these peaks beingcalled dispersive . In the approximation of a linear band structure (i.e. a Dirac cone) and a linear phonon dispersionnear K and Γ and for the resonant electron and phonon wave vectors being parallel along high-symmetry lines, thetwo conditions for double resonance become linear equations for | k | and | q | , which are also linear in ω in . Hence thesolutions to these equations | k res | and | q res | are also linear in ω in and because of the approximately linear dispersionof the phonons near K and Γ, the resonant phonon frequency and with it the peak position are linear in ω in as well.This linear dispersion of the two-phonon frequencies has been observed experimentally, e.g., in Refs. 25–27 and isshown in Fig. 7b. Note that contrary to the process for the G peak, the Raman amplitude does not vanish for a conicband structure, since the final state contains three particles (one photon, two phonons), so that angular momentumconservation as implied by a conic band structure can always be obeyed.So far we have focused on the double-resonant contribution to the Raman amplitude and determined which phononwave vectors q maximize it, which in turn determines the Raman shift at which the two-phonon Raman peaks areobserved. However, we want to stress that while the double-resonant terms are undoubtedly of high importance for acorrect description of the Raman peaks, quantum interference effects also play a crucial role in the correct descriptionof the Raman spectrum [9]. While the contributions of different phonons are added after squaring the matrix element(different phonons mean different final states), the sum over electronic wave vectors k has to be done before squaringthe matrix element. As pointed out in the literature [9], the doubly-resonant electronic states k res can constructivelyor destructively interfere with one another, which can significantly influence the peak shape and height.
3. Defect-assisted peaks (e.g., the D , D (cid:48) , and D + D (cid:48) peaks) After the detailed discussion of defect-free processes in the last two subsections, we conclude this section by brieflyaddressing the defect-assisted processes. In the presence of defects, electron-phonon scattering events can be comple-mented or replaced by elastic electron-defect scattering events. For example, if one replaces one of the two electron-phonon scattering events in the defect-free two-phonon process by an electron-defect scattering event, one obtains aprocess that has one instead of two phonons in the final state. This phonon can in principle have an arbitrary wavevector q , as already discussed in the section on kinematic constraints. A simple way to obtain the amplitude forsuch a process is to replace the electron-phonon(-light) matrix element that involves the second phonon by one of thedefect-scattering amplitudes and set the corresponding phonon frequency in the denominator to zero.While the double resonance conditions that involve the second phonon can thus no longer be fulfilled, the doubleresonance conditions that involve the first phonon, however, are unaffected by this. Therefore, the correspondingRaman process will still be doubly resonant for the same phonon wave vectors q res , we found for the two-phonon,defect-free case above. Accordingly, the maximum of the corresponding Raman peak will be located at the position ofthe frequency of the resonant wave vectors. Compared to the defect-free two-phonon process involving two phononsof the same branch, where the Raman shift of the peak maximum is given by the sum of the frequencies of the twophonons, the Raman shift for the corresponding one-phonon, defect-assisted peak will thus be halved. This explainsthe designation 2 D and 2 D (cid:48) for two of the two-(same-)phonon peaks as their corresponding defect-assisted peaks aredenoted D (for resonant phonons of the TO branch with q near K ) and D (cid:48) (for resonant phonons of the LA branchnear Γ), respectively. Since the double resonance conditions are the same for the two-phonon process and one-phononprocess with defects, the position of the Raman peaks of the latter also depend on the excitation frequency ω in .In addition to the matrix element squared, the intensity of the defect-assisted peaks will also contain a factor ofthe defect concentration [9]. This direct linear dependence of the height of the peak on the defect concentration iseasily understandable since the height of the peak is proportional to the probability per unit time of the correspondingRaman process. Since the latter requires a defect to be non-zero, the probability per unit time for the process tohappen is proportional to the defect concentration and hence the peak height is as well. A more detailed discussionof the dependence of the intensity of theses peaks can be found in the final section of this chapter.Finally, there are also Raman processes which involve two phonons and additional electron-defect scattering events.This allows for more and different conditions for double resonance, leading to several Raman sidebands and peaks,the most prominent of which is the D + D (cid:48) peak.8 III. INFLUENCE OF INTERNAL AND EXTERNAL PARAMETERS ON THE RAMAN SPECTRUM
In the final section of this chapter, we discuss how various internal properties (e.g. defects or layer number) of agraphene sample or externally controllable parameters (e.g. doping or strain) influence the Raman spectrum. We willrestrict ourselves to the discussion of the arguably most common and arguably best understood external influences,namely the number of layers, the defect concentration, doping, and strain. Other influences not covered here includedisorder and edge effects [4], substrate influences [24, 25, 28], and temperature [29], for which the interested reader isreferred to the literature.
A. Layer number
In the previous parts of this chapter, we were only concerned with monolayer graphene. While it is still by farthe most often studied graphene system in contemporary research and allows one to illustrate the general theoreticalapproach to the calculation of Raman spectra, multi-layer graphene also has received some attention in the literature.Indeed, one important application of Raman spectroscopy is the determination of the number of layers during sampleexfoliation or growth. We therefore briefly summarize the most important changes to the Raman spectrum if thesample consists not of one but of several layers of graphene.By increasing the number of layers, the number of atoms in the unit cell changes by 2 for each additional layer.The number of atoms in the unit cell determines the number of (optical) phonon branches. For N atoms in the unitcell the number of optical branches is given by 3 N −
3, two thirds of which are in-plane phonons. However, almostacross the whole Brillouin zone, the new phonon branches that are relevant for Raman scattering are degenerate withthe ones already existing in monolayer graphene. Only near the K point do the elsewhere degenerate TO branchessplit slightly [10].By contrast, the electronic band structure, undergoes a more severe change. Both the π and π ∗ bands split into N bands each. As the different layers of multilayer graphene are bound mostly by van der Waals forces, which are muchweaker than the intralayer covalent forces, the splitting of the additional bands is small compared to the energy scaleof the bands. Thus all of these bands now play a role in the Raman processes.By taking both the changes to the phonon dispersion and the electronic band structure into account, we cansummarize the changes to the various Raman processes and peaks as follows:For the one-phonon, defect-free process, i.e. the G peak, very little change is observed as the position and de-cay width of the degenerate phonon at Γ are as good as unchanged. The separation in energy of the additional bandsfrom the monolayer ones is so large that the additional possible electronic transitions have transition energies out ofrange of the phonon. Hence they couple only weakly to it, which means that the frequency and decay width of thephonon at Γ are not measurably different from the monolayer ones.For the two-phonon and/or defect-assisted processes, however, the situation is different. Recall that the existenceof a clear peak in monolayer graphene is attributed to the fact that for certain electron wave vectors k res and phononwave vectors q res the Raman amplitude becomes double resonant. The additional electronic bands in multilayergraphene allow for additional double-resonance conditions with slightly different solutions k res and q res (compareFig. 8a for an example of a double-resonant process in the case of bilayer graphene). In addition, the slight splitting ofthe TO branch of the phonon dispersion for q near K further allows for different q res . Altogether, this leads to morephonon wave vectors q that satisfy a double resonance condition and hence several close-by phonon frequencies nowcontribute significantly to the observed Raman shift. This leads to a measurable broadening of the two-phonon anddefect-assisted peaks, in particular to a splitting/broadening of the most prominent peak, the 2 D peak. As expectedfrom the above considerations, the broadening of these peaks increases with every additional number of layers [30, 31](see Fig. 8b) and converges towards the peak structure of bulk graphite.The broadening of the 2 D peak in particular is nowadays routinely used to reliably estimate the number of layersduring sample exfoliation or growth. B. Defect concentration
As already mentioned before, the presence of defects in a graphene sample changes the Raman spectrum in twoways: On the one hand, it leads to more peaks in the spectrum as the defect-assisted Raman processes becomepossible. On the other hand, it modifies the height and widths of the non-defect-assisted peaks. For both groups ofRaman peaks, we will briefly discuss the dependence of their height and width on the defect concentration in thissubsection.9 (a) (b) (c)
FIG. 8:
Change of the D peak width with layer number. (a,b) Two examples of double-resonant processesfor bilayer graphene. Shown are the two π and π ∗ bands of bilayer graphene and examples of electronic transitionsthat can go double-resonant. (Figure taken from Ref. 10.) (c) Evolution of the FWHM of the 2 D peak with thenumber of layers, from monolayer graphene to highly oriented pyrolithic graphite (HOPG). (Figure taken fromRef. 30.)For the defect-assisted peaks, e.g. the D , D (cid:48) , and D + D (cid:48) peaks, the peak height and width depends non-monotonicallyon the defect concentration n D .Firstly, there is the direct dependence of the intensity on n D , which appears as a scaling factor in the intensity.This factor represents the fact that the probability for a defect-assisted Raman process is proportional to the numberof defects.Secondly, there is the indirect dependence of the quantum mechanical amplitude on n D through the decay widths,i.e., the lifetimes, of the intermediate electronic states, γ el = γ el ( n D ), which appear in the denominator of the matrixelement of the Raman process and thus higher widths lead to smaller quantum mechanical amplitudes. In general,the decay widths of electronic states contains several contributions related to the different ways in which an electroncan scatter. The three most prominent ones are electron-electron interaction, i.e., Coulomb scattering with anotherelectron, electron-phonon interaction, i.e., activation of a lattice vibration, and scattering from a defect. The lattercontribution to the total decay width of an electronic state will be proportional to n D , as more defects leads to ahigher probability for an electron to scatter from a defect.Depending on the defect concentration, the first or the second aspect will dominate [9].For high defect concentrations, the electronic decay widths will be dominated by electron-defect scattering and hencewill be proportional to n D . Since the electronic decay widths appear in each of the multiple factors in the denominatorof the Raman matrix element, this effect will eventually outweigh the overall linear factor of n D appearing in theexpression for the Raman intensity. Therefore, for large enough densities, the heights of the defect-assisted peaks willdecrease.While the peak heights thus behave non-monotonically as a function of n D , the peak widths monotonically increasewith n D . Here, the only relevant effect is the increase of the electronic decay widths with n D . As discussed before, thebroadening of the electronic states leads to a smearing of the resonance condition and hence a wider range of phononfrequencies will non-negligibly contribute to the Raman peaks. Thus, the widths of the peaks increase monotonicallywith defect concentration.The properties of the non-defect-assisted peaks on the other hand, depend on the defect concentration in a purelymonotonic fashion.Their intensities monotonically decrease as a function of the defect concentration, since they only depend on n D through the decay widths of the electronic states, which enter the quantum mechanical matrix element throughfactors in the denominator. Since for low defect concentrations, the dependence of the electronic decay widths on n D is negligible, the height of the non-defect-assisted peaks is independent of the defect concentration as well for low n D .The different behavior of the intensity of the defect-assisted and non-defect-assisted peaks as a function of n D has0been used to estimate the amount of defects and disorder in a sample by measuring the ratio of the intensity of the D and G peaks [32] (see Fig. 9).FIG. 9: Dependence of the intensity of the Raman peaks on the defect concentration.
Ratio of theintensity of the D and G peaks as a function of the average distance between defects L D , which is a measure for thedefect concentration n D . (Figure taken from Ref. 32.)The behavior of the widths of the defect-free peaks under changes of n D depends on the specific peak. The widthof the G peak, as shown in the previous section, does not depend on the electronic decay widths as it is purelydetermined by the lifetime of the created phonon. The widths of the multi-phonon peaks, such as the 2 D , 2 D (cid:48) , or D + D (cid:48)(cid:48) peaks, by contrast, increases with n D , for the same reason as for the defect-assisted peaks. C. Doping
Doping, i.e., changes of the charge carrier density, leads to a shift of the Fermi level. Hole ( p -type) doping leads tostates in the π band not being occupied, while electron ( n -type) doping causes states in the π ∗ band to be occupied.This will influence the Raman spectrum in three ways: (a) (b) (c) FIG. 10:
Effects of doping on electrons, phonons, and Raman processes. (a,b) Change of the frequency (a)and the decay width (b) of the in-plane optical phonons at Γ with doping, which determine the frequency (position)and width of the Raman G peak. The dots represent experimental data, while the full lines represent the results of atheoretical calculation. (Figures taken from Ref. 33.) (c) Intensity of the Raman G peak as a function of Fermienergy, i.e., doping. The dots represent experimental data, while the full line represent the result of a theoreticalcalculation using a simplified version of the theory presented in the previous section. (Figure taken from Ref. 1.)1Firstly, the electronic band structure is slightly affected as the screening of the electron-electron interaction isincreased by the presence of additional charge carriers. The most important effect of this is decrease of the slope ofthe π and π ∗ bands near the K point [34]. This effects the two-phonon and/or defect-assisted Raman processes bychanging the condition for double resonance.An additional effect is the increase of the decay width of excited electronic states (or equivalently a decrease oftheir lifetime) γ el , as the scattering rate of an electron due to electron-electron interaction increases with additionalcharge carriers in the system. This will affect the height of the Raman peaks and correspondingly their area, sincethey are proportional to the matrix element squared of the corresponding Raman process, which in turn decreases forhigher decay widths of the electronic states as they appear in the denominator of the amplitude. Since the amplitudefor the two-phonon processes (e.g., the 2 D peak process) contains one more electronic Green’s functions than theamplitude for the one-phonon, defect-free process (leading to the G peak), it contains one more factor of γ el , andhence the height and area of the two-phonon peaks reduces much more with doping than the height and area of the G peak [35]. In addition to the height, the width of the two-phonon and/or defect-assisted peaks changes as well asexperimentally observed for the 2 D peak [25], since it is also related to the electronic decay width, as mentioned inthe last section.Secondly, the phonon frequencies and widths change. This is most noticeable for those phonons which are af-fected by Kohn anomalies: the degenerate optical in-plane phonons at Γ and the TO branch at and around K .The phonon at Γ is affected by doping through a change of its self-energy , i.e., its change of frequency due toelectron-phonon interaction. The commonly associated picture behind this is the phonon’s temporary decay into anelectron-hole pair which recombines to produce the same phonon again. This process is only relevant if the electronor the hole have not been scattered in the meantime, i.e., if they are long-lived enough or, equivalently, if their decaywidth is low enough. Otherwise this virtual process does not play a role as recombination to the original phononis not possible. The decay width is ’low enough’ if it is much lower than the transition energy of the electron-holepair which is equal to the energy of the phonon. To put it the other way around, for this process to play a role, thelifetime of the excited electron or the hole must be much longer than than the period of the phonon oscillation. If thisis the case, the nuclei are oscillating while the electron stays in its exciting state without ’instantly’ dropping backto the ground state and without adiabatically following the nuclei, as assumed in the adiabatic Born-Oppenheimerapproximation. In graphene, the decay widths of the exited electron and the hole are indeed much lower comparedto the phonon energy, leading to the statement that the adiabatic Born-Oppenheimer approximation breaks down ingraphene [33]. With doping, the occupation of the excited states and holes is changed and this in turn changes theself-energy of the phonon, resulting in an increase of the phonon frequencies of the degenerate optical branches atΓ [33, 36], which means that the position of the G peak shifts towards higher frequencies, as shown in Fig. 10a. Notethat in other materials, the ratio of electronic decay widths and phonon frequencies is usually much larger than ingraphene and hence the exited electrons and holes have time to relax to the ground state and adiabatically follow theoscillating nuclei and hence the self-energy correction in other materials is negligible.At the same time, doping also blocks a possible decay channel of the phonons, by preventing the decay into anelectron-hole pair due to the Pauli principle. If the Fermi level exceeds half of the phonon energy, than the decayinto an electron and a hole with each carrying away half of the phonon energy becomes impossible due to the Pauliexclusion principle since the electronic state that would be occupied after the decay is already occupied due to dop-ing. Thus, if the Fermi level exceeds half of the phonon energy, the phonon decay widths drops noticeably (compareFig. 10b), as the decay channel into electron-hole pairs makes up a sizable part of the total decay width of thephonons. In the case of finite temperature, this sharp drop of the decay width is smeared out with the Fermi-Diracdistribution. The strong decrease of the decay width of the phonon at Γ with doping directly amounts to a strongdecrease of the width of the Raman G peak, since the latter is equal to the decay width of the phonon, as explainedin the theoretical section. By monitoring the G peak width, one thus has a qualitative measure to check if the Fermilevel exceeds half of the phonon energy.For the two-phonon and/or defect-assisted peaks, the physical picture is roughly the same as for the G peak.As mentioned earlier, the phonons involved in the processes leading to the D , D (cid:48)(cid:48) , D + D (cid:48)(cid:48) and, most importantly,the 2 D peak have wave vectors near K . In the vicinity of K , effects of the Kohn anomaly play an important rolesince it leads to steepening of the slope of the phonon dispersion, which plays an important role in determining whichphonons obey the double-resonance conditions and hence where the center of the Raman peaks is situated. TheKohn anomaly is very sensitive to the Fermi surface and thus a change of the Fermi level through doping means thatit has a large influence on the phonon dispersion around K and consequently on the position of the Raman peaks.Indeed, it is found experimentally that the two-phonon and/or defect-assisted peaks [37] shift with doping, albeit notas strongly as the G peak.The decay width of the phonons that fulfill the double-resonance condition, on the other hand, are not affected for2low doping levels. The finite momentum of these phonons does not allow the phonon to decay into an electron-holepair due to energy and momentum conservation, as the necessary hole states would have to have an energy of aroundhalf of the laser energy for which the phonon fulfills the double-resonance condition. For typical experimental laserenergies of 2-3 eV, it would require a Fermi level shift of 1-1.5 eV to block the decay width of the phonons involvedin the doubly resonant process. The influence of doping on the decay widths of the phonons can thus be neglectedfor typical doping levels achieved in experiment.Finally, doping influences the Raman amplitudes directly. As seen in the previous section, the Raman ampli-tudes contain factors of the Fermi-Dirac distribution of the involved electronic states. By changing the Fermi leveland thus varying the occupation of the electronic states, some of the Raman amplitudes are set to zero. For the G peak amplitude, which does not rely primarily on the resonant terms to exist, doping in general has only minoreffects. It has been demonstrated experimentally however [1], that if the Fermi level comes close to half of thefrequency of the outgoing light, the total amplitude for the G peak process increases dramatically. The underlyingreason for this increase is the blocking of terms in the amplitude that destructively interfere with the resonant terms,leading to a large increase of the amplitude, as depicted in Fig. 10c.For the two-phonon and/or defect-assisted processes, the situation is different. Here, the interference between thedifferent terms in the amplitude is believed to be constructive [1] and therefore raising the Fermi level to half of thefrequency of the light only leads to a sharp decrease of the peak height and ultimately to its disappearance, as hasbeen demonstrated for the 2 D peak.We can summarize the effects of doping on the Raman spectrum as follows: The G peak is expected to shift toa higher frequency. In addition, it will narrow once the Fermi level is bigger than ¯ hω ph / − ¯ hω ph / − [24]. The two-phonon and/or defect-assisted peaks, inparticular the 2 D peak, are expected to shift as well, albeit not as strong as the G peak. Furthermore the ratio ofthe areas of the 2 D and G peaks decreases as well with doping. D. Strain
Mechanical strain has a large influence on the Raman spectrum. This is hardly surprising as it changes the bondlengths of the atoms, which leads to both a change in the phonon frequencies as the effective ’springs’ between theatoms are softened or hardened and to a change in the electronic band structure since the unit cell becomes distortedand hence the Brillouin zone does as well.The G peak is less affected by these changes than the two-phonon and/or defect-assisted peaks as the electronicstructure only directly influences the intensity, i.e., the peak height. The peak position and width, by contrast, are onlydetermined by the frequency and width of the degenerate optical phonons at Γ. The width of these phonons remainsalmost completely unaffected by strain since it is mostly determined by the decay of the phonon into electron-holepairs. The position, however, is rather sensitive to strain since it equal to the frequency of the phonon. As mentionedbefore, strain leads to a change of the bond strength between the atoms, which in turn modifies the effective springconstant between the atoms and hence the phonon frequency. In general, compressive strain leads to a hardening ofthe bonds as the charge density becomes higher between the atoms thus binding the atoms stronger together, whichresults in a higher phonon frequency and therefore a larger Raman shift. Tensile strain, on the other hand, leadsto the opposite effect and reduces the phonon frequency and the Raman shift. For experimentally accessible strainvalues, the relation between Raman shift of the G peak and strain is a linear one, with the constant of proportionalitygiven by the Gr¨uneisen parameter for the in-plane optical phonon at Γ.The two-phonon and/or defect-assisted peaks such as the 2 D peak, however, undergo a larger change with strain.Here, both the change in the phonon frequencies and the distortion of the electronic band structure play a role. Whenthe electronic band structure is distorted, the conditions for double resonance are changed. When combined withthe change of the phonon frequencies this leads to much larger shift of the two-phonon and/or defect-assisted peakpositions with strain compared to that of the G peak.Additionally, one has to distinguish between uniaxial strain and biaxial strain: Uniaxial strain means that strain is applied along a specific direction. The fact that a specific direction is nowsingled out has a big influence on the various Raman peaks. For the G peak, the most important effect is the liftingof the degeneracy of the in-plane optical phonon frequencies at Γ. In unstrained graphene, the two in-plane opticalphonon branches are degenerate at Γ because of rotation symmetry. By singling out a specific direction, uniaxial strainbreaks this rotation symmetry and as a result the two phonon branches split. Since the Raman peak corresponding3to one-phonon, defect-free processes is due to the emission of an in-plane optical phonon at Γ, the G peak splitsinto two peaks corresponding to the two now non-degenerate phonon branches. The difference between these twophonon frequencies is proportional to the applied strain and for low strain ( < (a) (b) (c) FIG. 11:
Effects of strain on the Raman G and D peaks. (a,b) Influence of uniaxial strain on the Raman G (a) and 2 D (b) peaks. Shown are experimentally recorded Raman spectra for increasing amounts of uniaxial strain,leading to a splitting of the G and 2 D peaks. (Figures taken from Ref. 38 and Ref. 39, respectively.) (c) Illustrationof the separation of doping and strain-induced shifts of the Raman G and 2 D peak positions as presented in Ref. 40.The two clusters of points represent measurements of the positions of the G and 2 D peaks on various spots of twodifferent samples. The dashed lines represent axes of slopes 0.7 and 2.2, corresponding to the linear relation betweenshifts of the 2 D and G peaks due to doping or strain, respectively. By projecting a point in the ( ω G , ω D )-planeonto these to axes, one can gain estimates for the amount of charge carrier doping and strain in the sample. Thescaling of the axes is valid for hole doping and uniaxial strain, respectively. (Figure taken from Ref. 40.)For the two-phonon and/or defect-assisted peaks, uniaxial strain can also cause observable peak splitting. Themechanism, however, is a different one than that for the G peak. The observed peak splitting of, for example, the2 D peak has its roots in a breaking of the threefold rotation symmetry of the resonance surface of the electronand phonon wave vectors. Recall that the phonons mostly contributing to the peak are determined from resonanceconditions derived in the previous section. These resonance conditions can be fulfilled or approximately fulfilled bya whole set of wave vectors for the electrons and phonons, which however obey the lattice symmetries. By breakingthe threefold rotation symmetry of the lattice by applying uniaxial strain, the shape of the resonance surfaces in theBrillouin zone is distorted and this leads to different phonon wave vectors that satisfy them that correspond to phononfrequencies that are a bit separated from each other. This leads to the observation of a splitting in the two-phononand/or defect-assisted bands, as has been measured, for example, for the 2 D peak [39] (compare Fig. 11b) As in thecase of the G peak, the applied uniaxial strain has to be large enough to observe such a splitting with the thresholdvalue being roughly in the same range as for the G peak.It should be noted that, for a low-quality graphene sheet and/or an unfortunate choice of substrate, the intrinsicwidth of the Raman peaks can be too large to allow the peak splitting to be observed. Instead one will only measure afurther broadening of the peaks (compare for instance Ref. 38, where no 2 D peak splitting was observed, and Ref. 39,where it was observed). Biaxial strain, on the other hand, does not lead to the splitting of the Raman peaks. Due to the hexagonal latticestructure, biaxial strain leads mostly only to a uniform stretching or contraction of the unit cell and does not breakthe rotation symmetry in a major way. Therefore, both mechanisms described in the previous paragraph do not applyin the case of biaxial strain and the Raman bands do not split. However, the mechanism that leads to a shift of thepeaks remains applicable. As one would very naively expect, the change of the peak positions for a given amount ofstrain is indeed much larger for biaxial strain than for uniaxial strain [40].Finally, we want to point out that while strain and doping both lead to a shift of the positions of the variouspeaks, it has recently been demonstrated that it is possible to disentangle these two sources of peak shifts [40]. Byfocusing on the positions of G and 2 D peaks, one can exploit the fact that the two peak shifts are linearly correlated,both for doping and strain as the source of the peak shifts. The ratio of the two shifts, however, depends on the4source of the shifts: For hole doping, the ratio of the 2 D and G peak shifts is roughly ∆ ω D / ∆ ω G | hole dop. ∼ .
7. Forelectron doping, the slope is much slower and deviates soon from the linear trend. For strain, the 2 D peak, as beingrelated to a two-phonon Raman process, is subject to much larger changes of the position compared to the G peak,with the experimentally extracted ratio of the peak shifts given by ∆ ω D / ∆ ω G | strain ∼ .
2. Under the assumptionthat the doping- and strain-induced shift are simply additive and not correlated, one can project the coordinates ofa point in the ω G - ω D plane onto the axes of slopes 0.7 and 2.2 and separate the doping- and strain-induced shifts(compare Fig. 11c), whereupon both can be converted to a value for the charge carrier concentration or the strain,respectively, using theoretical predictions. IV. CONCLUSIONS
This chapter has given a glimpse at the rich variety of phenomena that occur in Raman spectroscopy of graphene.Due to the linear band structure of graphene, there are always electron-hole pairs in resonance with the incom-ing/outgoing laser light. This leads to the relatively large Raman cross section and the strong sensitivity of theRaman spectrum on the sample parameters. A full quantitative explanation of Raman peak positions, peak widths,and intensities requires the somewhat lengthy formalism of time-dependent perturbation theory. However, we haveseen that kinematic considerations and a careful look at the matrix elements and energy denominators allow in manycases an intuitive understanding of the Raman spectrum without explicit calculations.The G line position is given by the E g phonon frequency at Γ and thus it does not disperse as a function of thelaser energy. Its position and width are sensitive measures for doping and strain. At the same time it remains largelyunaffected by defects and by the number of layers. The 2 D line, by contrast, disperses as a function of the laserwave length. Its splitting into subpeaks (sometimes only seen as an increased broadening of the line) is, in particular,used as a convenient measure for the number of layers in the sample. Its position is also affected by strain, but onlymoderately by doping. The intensity of defect related peaks is a direct measure of the defect concentration in thesample.One of the main challenges for quantitative calculations is still the absolute position and the width of the 2 D line(and thus also of the D line). The difficulties are twofold: (i) a fully ab initio calculation of the highest opticalphonon branch around K is still missing (only the linear slope at K has been determined quantitatively using the GW approximation). (ii) The width is a result of both the effect of k-space integration and also depends on thevarious broadenings of intermediate electron and hole states as well as on the phonon frequency.For the time being, ab initio calculations of the Raman spectrum of graphene (such as, for example, the calculationpresented in Ref. 10) still rely on the input of phonon and electron lifetimes as semi-empirical parameters. All thelifetimes (due to electron-electron, electron-phonon, and phonon-phonon interaction) can, in principle, be calculated,but calculations remain quite expensive in terms of computing time. With the increasing available computing powerand recent progress in code development for lifetime calculations, we are optimistic that a fully ab initio calculationof the Raman spectrum of graphene can be achieved in the near future. ACKNOWLEDGMENTS
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