Raman Spectroscopy of Graphene

TL;DR

This paper reviews the theoretical foundations of Raman spectroscopy in graphene, explaining how it reveals information about the material's structure, defects, and optical properties through peak analysis.

Contribution

It provides a comprehensive theoretical framework for understanding Raman spectra of graphene, linking spectral features to electronic and phononic properties.

Findings

Raman spectra of graphene mainly show two peaks.

Peak variations reveal information about defects, doping, and strain.

Raman spectroscopy is a versatile tool for graphene analysis.

Abstract

Raman spectroscopy of graphene is reviewed from a theoretical perspective. After an introduction of 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 peak positions 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 is used routinely to analyze layer number, defects, doping and strain of graphene samples. At the same time, it is an intriguing playground to study the optical properties of graphene.

Tables

Table 1. Table 1: Feynman rules for the basic building blocks for Raman scattering. Feynman diagrams and associated factors appearing in the perturbation series for the Raman scattering amplitudes. Each factor is a 2 × \times 2-matrix in the basis of π 𝜋 \pi and π ∗ superscript 𝜋 \pi^{*} states, i.e., each element of the matrix is a matrix element between π 𝜋 \pi and π ∗ superscript 𝜋 \pi^{*} states.

	Diagram	Feynman rule
\pbox5cmPropagation
of electron		$i{𝐆}_{𝐤}\left(\omega \right)$
\pbox5cmElectron-light interaction		$\left(-i\right){𝜸}_{𝐤}^{\nu }$
		$\left(-i\right){\widetilde{𝜸}}_{𝐤}^{\mu \nu }$
\pbox5cmElectron-phonon interaction		$\left(-i\right){𝐠}_{𝐤-𝐪}^{\lambda }$
		$\left(-i\right){\widetilde{𝐠}}_{𝐤-{𝐪}_1-{𝐪}_2}^{{\lambda }_1{\lambda }_2}$

Equations

$$H_0=\frac{{\mathbf{p}}^2}{2m_e}+V(\mathbf{r};\{{\mathbf{R}}_{n\alpha ,0}\}),$$

$$\begin{array}{rl}V(\mathbf{r};\{{\mathbf{R}}_{n\alpha }\})& =V(\mathbf{r};\{{\mathbf{R}}_{n\alpha ,0}\})+\sum _{n,\alpha }\frac{\partial V}{\partial {\mathbf{R}}_{n\alpha }}(\mathbf{r};\{{\mathbf{R}}_{n\alpha ,0}\})\cdot \mathbf{u}({\mathbf{R}}_{n\alpha ,0})\\ & +\frac{1}{2}\sum _{n,m,\alpha ,\beta }\sum _{i,j}\frac{{\partial }^{2}V}{\partial R_{n\alpha ,i}\partial R_{m\beta ,j}}(\mathbf{r};\{{\mathbf{R}}_{n\alpha ,0}\})u_i({\mathbf{R}}_{n\alpha ,0})u_j({\mathbf{R}}_{m\beta ,0})+\mathcal{O}(u^3).\end{array}$$

$$H_0\to H_0+H_{\mathrm{el}-\mathrm{ph}}^{(1)}+H_{\mathrm{el}-\mathrm{ph}}^{(2)},$$

$$\mathbf{u}({\mathbf{R}}_{n\alpha ,0})=\sum _{\mathbf{q},\lambda }\sqrt{\frac{\hslash }{2M_C{\omega }_{\mathbf{q},\lambda }}}\left[b_{\lambda ,\mathbf{q}}{\mathbf{v}}_{\lambda ,\mathbf{q}}^{\alpha }+b_{\lambda ,-\mathbf{q}}^{†}{\left({\mathbf{v}}_{\lambda ,-\mathbf{q}}^{\alpha }\right)}^{\ast }\right]{\mathrm{e}}^{i\mathbf{q}\cdot {\mathbf{R}}_n},$$

$$\begin{split}g^{\lambda}_{s,\mathbf{k}-\mathbf{q};s^{\prime},\mathbf{k}}&=\langle s,\mathbf{k}-\mathbf{q};\lambda,\mathbf{q}|H^{(1)}_{\mathrm{el-ph}}|s^{\prime},\mathbf{k}\rangle\\ &=\sqrt{\frac{\hbar}{2M_{C}\omega_{\mathbf{q},\lambda}}}\sum_{n,\alpha}\mathrm{e}^{i\mathbf{q}\cdot\mathbf{R}_{n}}\left(\mathbf{v}^{\alpha}_{\lambda,\mathbf{q}}\right)^{*}\\ &\quad\quad\cdot\bigg{\langle}s,\mathbf{k}-\mathbf{q}\bigg{|}\frac{\partial V}{\partial\mathbf{R}_{n\alpha}}(\mathbf{r};\{\mathbf{R}_{n\alpha,0}\})\bigg{|}s^{\prime},\mathbf{k}\bigg{\rangle},\end{split}$$

$${\left|g_{\pi ,\mathbf{K};{\pi }^{\ast },2\mathbf{K}}^{\mathrm{TO}}\right|}^2\propto \underset{d\to 0}{\lim }{\left(\frac{\left[{\epsilon }_{\mathbf{K}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{K}}^{\pi }\right](d)}{d}\right)}^2,$$

$$\begin{array}{rl}H& =\frac{{\left[\mathbf{p}+\frac{e}{c}\mathbf{A}(\mathbf{r})\right]}^2}{2m_e}+V(\mathbf{r};\{{\mathbf{R}}_0\})\\ & =H_0+\frac{e}{c{m}_e}\mathbf{A}(\mathbf{r})\cdot \mathbf{p}+\frac{e^2}{2c^2{m}_e}[\mathbf{A}(\mathbf{r}){]}^2\\ & =H_0+H_{\mathrm{el}-\mathrm{R}}^{(1)}+H_{\mathrm{el}-\mathrm{R}}^{(2)}.\end{array}$$

$$\mathbf{A}(\mathbf{r})=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}\sum _{\mu }\sqrt{\frac{4\pi c^2\hslash }{2{\omega }_k}}\left[{\mathbf{ϵ}}_{\mu ,\mathbf{k}}{a}_{\mu ,\mathbf{k}}+{\left({\mathbf{ϵ}}_{\mu ,-\mathbf{k}}\right)}^{\ast }{a}_{\mu ,-\mathbf{k}}^{†}\right]{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{r}},$$

$$\begin{split}\gamma^{\mu}_{s,\mathbf{k}-\mathbf{k}^{\prime};s^{\prime},\mathbf{k}}&=\langle s,\mathbf{k}-\mathbf{k}^{\prime};\mu,\mathbf{k}^{\prime}|H^{(1)}_{\mathrm{el-R}}|s^{\prime},\mathbf{k}\rangle\\ &=\frac{e}{cm_{e}}\sqrt{\frac{4\pi c^{2}\hbar}{2\omega_{k^{\prime}}}}\boldsymbol{\epsilon}^{*}_{\mu,\mathbf{k}^{\prime}}\cdot\Big{\langle}s,\mathbf{k}-\mathbf{k}^{\prime}\Big{|}\mathrm{e}^{i\mathbf{k}^{\prime}\cdot\mathbf{r}}\mathbf{p}\Big{|}s^{\prime},\mathbf{k}\Big{\rangle},\end{split}$$

$$G_{s,\mathbf{k}}^R(t)=\{\begin{array}{ll}{\mathrm{e}}^{-i{\epsilon }_{\mathbf{k}}^{s}t/\hslash }\left(1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^s)\right),& t>0\\ -{\mathrm{e}}^{-i{\epsilon }_{\mathbf{k}}^{s}t/\hslash }{n}_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^s),& t<0,\end{array}$$

$$G_{s,\mathbf{k}}(\omega )=\frac{n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^s)}{\hslash \omega -{\epsilon }_{\mathbf{k}}^s-\frac{i}{2}{\gamma }_{\mathbf{k}}^s}+\frac{1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^s)}{\hslash \omega -{\epsilon }_{\mathbf{k}}^s+\frac{i}{2}{\gamma }_{\mathbf{k}}^s}.$$

$${\omega }_{\mathrm{in}}-{\omega }_{\mathrm{out}}=\sum _{i=1}^n{\omega }_{{\mathbf{q}}_i,{\lambda }_i}$$

$$\sum _{i=1}^n{\mathbf{q}}_i=0\text{(no defects)}.$$

$${\mathcal{M}}_1^G$$

$${\mathcal{M}}_2^G$$

$${\mathcal{M}}_3^G$$

$$\begin{split}\mathcal{M}^{G}_{2}=\sum_{\mathrm{spin}}\sum_{\mathbf{k}}\left(-\frac{1}{\hbar}\right)\sum_{s=\pi,\pi^{*}}n_{\mathrm{F}}(\varepsilon^{s}_{\mathbf{k}})\Bigg{\{}&\left.\left[\boldsymbol{\gamma}^{\nu}_{\mathbf{k}}\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{out}})\mathbf{g}^{\lambda}_{\mathbf{k},\mathbf{k}}\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{in}})\left(\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}\right)^{\dagger}\right]_{ss}\right|_{\hbar\omega=\varepsilon^{s}_{\mathbf{k}}+\frac{i}{2}\gamma^{s}_{\mathbf{k}}}\\ +&\left.\left[\mathbf{g}^{\lambda}_{\mathbf{k},\mathbf{k}}\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{in}})\left(\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}\right)^{\dagger}\mathbf{G}_{\mathbf{k}}(\omega)\boldsymbol{\gamma}^{\nu}_{\mathbf{k}}\right]_{ss}\right|_{\hbar\omega=-\hbar\omega_{\mathrm{out}}+\varepsilon^{s}_{\mathbf{k}}+\frac{i}{2}\gamma^{s}_{\mathbf{k}}}\\ +&\left.\left[\left(\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}\right)^{\dagger}\mathbf{G}_{\mathbf{k}}(\omega)\boldsymbol{\gamma}^{\nu}_{\mathbf{k}}\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{out}})\mathbf{g}^{\lambda}_{\mathbf{k},\mathbf{k}}\right]_{ss}\right|_{\hbar\omega=-\hbar\omega_{\mathrm{in}}+\varepsilon^{s}_{\mathbf{k}}+\frac{i}{2}\gamma^{s}_{\mathbf{k}}}\Bigg{\}}\end{split}$$

$${\mathcal{M}}_{G\text{ peak}}^{\mu \nu \lambda }({\omega }_{\mathrm{in}};{\gamma }_{\mathrm{el}};{\omega }_{\mathrm{ph}},{\gamma }_{\mathrm{ph}})=\sum _{d=1}^3{\mathcal{M}}_d,$$

$$I_G({\omega }_{\mathrm{out}})\propto \frac{\mathrm{d}{P}_{G\text{ peak}}}{\mathrm{d}t}=2\pi \sum _{\lambda =\text{LO,TO}}{\left|{\mathcal{M}}_{G\text{ peak}}^{\mu \nu \lambda }({\omega }_{\mathrm{in}};{\gamma }_{\mathrm{el}};{\omega }_{\mathrm{ph}},{\gamma }_{\mathrm{ph}})\right|}^2\times \delta ({\omega }_{\mathrm{in}}-{\omega }_{\mathrm{out}}-{\omega }_{\mathrm{ph}})$$

$$\delta ({\omega }_{\mathrm{in}}-{\omega }_{\mathrm{out}}-{\omega }_{\mathrm{ph}})\to \frac{1}{\pi }\frac{{\gamma }_{\mathrm{ph}}/2}{({\omega }_{\mathrm{in}}-{\omega }_{\mathrm{out}}-{\omega }_{\mathrm{ph}}{)}^2+({\gamma }_{\mathrm{ph}}/2{)}^2}$$

$$\begin{array}{rl}{\mathcal{M}}_2^G\supset & \sum _{\mathrm{spin}}\sum _{\mathbf{k}}\left(-\frac{1}{\hslash }\right)n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^{\pi })\left(1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }})\right)\left(1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }})\right)\\ & \times \frac{⟨{\pi }_{\mathbf{k}};{\omega }_{\mathrm{out}}^{\nu }|H_{\mathrm{el}-\mathrm{R}}^{(1)}|{\pi }_{\mathbf{k}}^{\ast }⟩⟨{\pi }_{\mathbf{k}}^{\ast };{\omega }_{\mathrm{ph}}^{\lambda }|H_{\mathrm{el}-\mathrm{ph}}^{(1)}|{\pi }_{\mathbf{k}}^{\ast }⟩⟨{\pi }_{\mathbf{k}}^{\ast }|H_{\mathrm{el}-\mathrm{R}}^{(1)}|{\pi }_{\mathbf{k}};{\omega }_{\mathrm{in}}^{\mu }⟩}{\left[\hslash {\omega }_{\mathrm{in}}-({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{k}}^{\pi })+\frac{i}{2}({\gamma }_{\mathbf{k}}^{\pi }+{\gamma }_{\mathbf{k}}^{{\pi }^{\ast }})\right]\left[\hslash ({\omega }_{\mathrm{in}}-{\omega }_{\mathrm{ph}})-({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{k}}^{\pi })+\frac{i}{2}({\gamma }_{\mathbf{k}}^{\pi }+{\gamma }_{\mathbf{k}}^{{\pi }^{\ast }}+{\gamma }_{\mathrm{ph}})\right]},\end{array}$$

$${\mathcal{M}}_1^{\text{2 ph.}}$$

$$\displaystyle\times\mathbf{G}_{\mathbf{k}+\mathbf{q}}(\omega+\omega_{\mathrm{out}}+\omega_{\mathbf{q},\lambda_{1}})\mathbf{g}^{\lambda_{2}}_{\mathbf{k}+\mathbf{q},\mathbf{k}}\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{in}})\left(\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}\right)^{\dagger}\Bigg{]}$$

$${\mathcal{M}}_2^{\text{2 ph.}}$$

$$\displaystyle\times\mathbf{G}_{\mathbf{k}+\mathbf{q}}(\omega+\omega_{\mathbf{q},\lambda_{1}}+\omega_{\mathbf{q},\lambda_{2}})\boldsymbol{\tilde{\gamma}}^{\mu\nu}_{\mathbf{k}}\Bigg{]}$$

$${\mathcal{M}}_3^{\text{2 ph.}}$$

$$\displaystyle\times\mathbf{G}_{\mathbf{k}}(\omega+\omega_{\mathrm{in}})\left(\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}\right)^{\dagger}\Bigg{]}$$

$${\mathcal{M}}_4^{\text{2 ph.}}$$

$${\mathcal{M}}_{\text{2 ph.}}^{\mu \nu {\lambda }_1{\lambda }_2}(\mathbf{q};{\omega }_{\mathrm{in}};{\gamma }_{\mathrm{el}};{\omega }_{\mathrm{ph}},{\gamma }_{\mathrm{ph}})=\sum _{d=1}^{11}{\mathcal{M}}_d,$$

$$I_{\text{2 ph.}}({\omega }_{\mathrm{out}})\propto \frac{\mathrm{d}{P}_{\text{2 ph.}}}{\mathrm{d}t}=2\pi \sum _{{\lambda }_1,{\lambda }_2}\sum _{\mathbf{q}}{\left|{\mathcal{M}}_{\text{2 ph.}}^{\mu \nu {\lambda }_1{\lambda }_2}(\mathbf{q};{\omega }_{\mathrm{in}};{\gamma }_{\mathrm{el}};{\omega }_{\mathrm{ph}},{\gamma }_{\mathrm{ph}})\right|}^2\times \delta ({\omega }_{\mathrm{in}}-{\omega }_{\mathrm{out}}-{\omega }_{\mathbf{q},{\lambda }_1}-{\omega }_{\mathbf{q},{\lambda }_2}).$$

$$\begin{array}{rl}{\mathcal{M}}_1^{\text{2 ph.}}\supset & \sum _{\mathrm{spin}}\sum _{\mathbf{k}}\left(-\frac{1}{\hslash }\right)n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^{\pi })n_{\mathrm{F}}({\epsilon }_{\mathbf{k}+\mathbf{q}}^{\pi })\left(1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }})\right)\left(1-n_{\mathrm{F}}({\epsilon }_{\mathbf{k}+\mathbf{q}}^{{\pi }^{\ast }})\right)\\ & \times \frac{⟨{\pi }_{\mathbf{k}+\mathbf{q}};{\omega }_{\mathrm{out}}^{\nu }|H_{\mathrm{el}-\mathrm{R}}^{(1)}|{\pi }_{\mathbf{k}+\mathbf{q}}^{\ast }⟩⟨{\pi }_{\mathbf{k}};{\omega }_{\mathbf{q}}^{{\lambda }_1}|H_{\mathrm{el}-\mathrm{ph}}^{(1)}|{\pi }_{\mathbf{k}+\mathbf{q}}⟩⟨{\pi }_{\mathbf{k}+\mathbf{q}}^{\ast };{\omega }_{-\mathbf{q}}^{{\lambda }_2}|H_{\mathrm{el}-\mathrm{ph}}^{(1)}|{\pi }_{\mathbf{k}}^{\ast }⟩⟨{\pi }_{\mathbf{k}}^{\ast }|H_{\mathrm{el}-\mathrm{R}}^{(1)}|{\pi }_{\mathbf{k}};{\omega }_{\mathrm{in}}^{\mu }⟩}{\left[\hslash {\omega }_1-({\epsilon }_{\mathbf{k}+\mathbf{q}}^{\pi }-{\epsilon }_{\mathbf{k}}^{\pi })+\frac{i}{2}(-{\gamma }_{\mathbf{k}+\mathbf{q}}^{\pi }+{\gamma }_{\mathbf{k}}^{\pi }-{\gamma }_1)\right]\left[\hslash ({\omega }_{\mathrm{out}}+{\omega }_1)-({\epsilon }_{\mathbf{k}+\mathbf{q}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{k}}^{\pi })+\frac{i}{2}({\gamma }_{\mathbf{k}}^{\pi }+{\gamma }_{\mathbf{k}+\mathbf{q}}^{{\pi }^{\ast }}-{\gamma }_1)\right]}\\ & \times \frac{1}{\left[\hslash {\omega }_{\mathrm{in}}-({\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{k}}^{\pi })+\frac{i}{2}({\gamma }_{\mathbf{k}}^{\pi }+{\gamma }_{\mathbf{k}}^{{\pi }^{\ast }})\right]}.\end{array}$$

$$\hslash {\omega }_{\mathrm{in}}={\epsilon }_{\mathbf{k}}^{{\pi }^{\ast }}-{\epsilon }_{\mathbf{k}}^{\pi }\Rightarrow \mathbf{k}={\mathbf{k}}_{\mathrm{res}}({\omega }_{\mathrm{in}}).$$

$$\hslash {\omega }_{\mathbf{q},{\lambda }_1}={\epsilon }_{{\mathbf{k}}_{\mathrm{res}}({\omega }_{\mathrm{in}})+\mathbf{q}}^{\pi }-{\epsilon }_{{\mathbf{k}}_{\mathrm{res}}({\omega }_{\mathrm{in}})}^{\pi }\Rightarrow \mathbf{q}={\mathbf{q}}_{\mathrm{res}}({\omega }_{\mathrm{in}}).$$

$$\begin{array}{rl}& \hslash ({\omega }_{\mathrm{out}}+{\omega }_{{\mathbf{q}}_{\mathrm{res}},{\lambda }_1})-({\epsilon }_{{\mathbf{k}}_{\mathrm{res}}+{\mathbf{q}}_{\mathrm{res}}}^{{\pi }^{\ast }}-{\epsilon }_{{\mathbf{k}}_{\mathrm{res}}}^{\pi })\\ =& -\hslash {\omega }_{{\mathbf{q}}_{\mathrm{res}},{\lambda }_2}+{\epsilon }_{{\mathbf{k}}_{\mathrm{res}}}^{{\pi }^{\ast }}-{\epsilon }_{{\mathbf{k}}_{\mathrm{res}}+{\mathbf{q}}_{\mathrm{res}}}^{{\pi }^{\ast }}.\end{array}$$

Full text

1. I Introduction

2. II Theoretical description of the Raman spectrum of graphene

3. II.1 Basic building blocks

4. II.1.1 Electronic band structure

5. II.1.2 Phonon dispersion

6. II.1.3 Electron-phonon coupling

7. II.1.4 Electron-light coupling

8. II.1.5 Electron-defect scattering

9. II.1.6 Summary and Feynman rules

10. II.2 General kinematic considerations and possible Raman processes

11. II.3 Diagrammatic calculation of the Raman amplitudes

12. II.3.1 One-phonon, defect-free processes ($G$ peak)

13. II.3.2 Two-phonon, defect-free processes (e.g., $2D$, $2D^{\prime}$, and $D+D^{\prime\prime}$ peaks)

14. II.3.3 Defect-assisted peaks (e.g., the $D$, $D^{\prime}$, and $D+D^{\prime}$ peaks)

15. III Influence of internal and external parameters on the Raman spectrum

16. III.1 Layer number

17. III.2 Defect concentration

18. III.3 Doping

19. III.4 Strain

20. IV Conclusions

I Introduction

Raman scattering is the inelastic scattering of light. The frequency of a photon can change by transferring energy to and/or receiving energy from the lattice vibrations of the material. In quantum mechanics, the “allowed” energies of an oscillation are quantized and - in the case of a harmonic oscillation - equidistant. The vibrations of the lattice can thus be described in the language of quasi-particles and the term phonon is usually used. The incoming photon changes its energy by effectively exciting or absorbing one or several phonons. If a phonon is excited, the photon looses energy and the process is called Stokes scattering. In the case of the absorption of a phonon, the photon gains energy and the process is referred to as Anti-Stokes scattering. The spectrum of the inelastically scattered light therefore features discrete peaks, whose positions can be directly associated with certain vibrational modes of the crystal.

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 $2D$ line, is due to second order Raman scattering, where two phonons are excited at the same time. The overall Raman spectrum of graphene thus consists 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, and the 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 of the principal characterization tools for graphene and related graphitic materials.

Raman spectroscopy is often categorized as “vibrational spectroscopy” because it probes primarily the phonon frequencies. However, a quantitative description of the Raman intensities is deeply connected to the optical (absorption and 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 crossing of the optically active $\pi$ and $\pi^{*}$ bands, the Raman effect is resonant for all laser frequencies in the infrared, visible, and near UV ranges. Different “quantum pathways” Chen et al. (2011) (corresponding to contributions of different microscopic processes in perturbation theory) contribute to the intensity of a given peak. This gives rise to intriguing quantum interference effects with sometimes surprising consequences for the Raman intensities (e.g., as a function of laser energy and of sample doping). Furthermore, the intensities and the widths of the Raman peak are strongly influenced by the lifetimes of intermediate electronic excitations. Raman scattering thus also bears information on the complex dynamics of electron-hole pairs in graphene. All this makes a chapter on Raman spectroscopy an integral part of a book about the optical properties of graphene.

A number of books/book chapters Saito et al. (1998); Reich et al. (2004) and review articles Ferrari (2007); Malard et al. (2009); Ferrari and Basko (2013) have already been devoted to the topic of Raman scattering in graphitic materials. This book chapter puts particular emphasis on the theoretical foundations of resonant Raman scattering in graphene. It explains the basic theory for Raman scattering in graphene using perturbation theory. Together with the study of the groundbreaking theoretical research articles (in particular Basko (2008, 2009); Venezuela et al. (2011); Herziger et al. (2014)), it should enable the newcomer to the field to learn how an exact quantitative calculation can be achieved in principle. For the computationally less ambitious reader, the chapter gives detailed qualitative understanding how the different terms in perturbation theory depend on external stimuli (such as doping, defects, or strain) and what conclusions one can draw from the Raman spectra.

The chapter is structured as follows: In section II, we first introduce the basic building blocks for the understanding of resonant Raman spectra in graphene: The electronic band structure, the phonon dispersion, the coupling between electrons and phonons, and the coupling of the electronic states to light. Afterwards, the formalism of time-dependent perturbation theory is applied to give the reader an intuitive understanding and enable her/him to study the relevant research literature. In section III, the theory is then applied to understand qualitatively and quantitatively, how Raman spectroscopy can be used to study the quality of graphene flakes as well as to understand the effects of environment, 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 particular the behavior, shape, and size of its various peaks. To this end, we will use time-dependent perturbation theory to calculate expressions for the Raman scattering amplitudes. This approach has the advantage that the individual terms of the perturbation series can be represented by so-called Feynman diagrams. The graphical representation of the calculation allows for both easier bookkeeping of the terms in the perturbation series and at the same time gives a 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 theory calculation. We briefly discuss the electronic band structure of graphene before having a more detailed look at the phonon dispersion of graphene. Thereafter, we will discuss the coupling between electrons and phonons and their treatment in perturbation theory and show how to include the effects of an external electromagnetic field, i.e. of incoming and outgoing photons. This first part will be concluded by briefly touching upon electron-defect scattering and a summary in which we give a graphical representation of the mathematical expressions for the various building blocks in terms of Feynman diagrams.

The second part contains an overview over the kinematics of Raman scattering. We will derive what kind of peaks are 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-dependent perturbation theory and the language of Feynman diagrams, we will demonstrate how the Raman spectrum of graphene can be calculated. in this section we will mainly focus on the two most prominent Raman peaks and study their behavior from the scattering amplitude obtained in perturbation theory.

II.1 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 will have a look at each of these building blocks before we conclude this section with a summary.

II.1.1 Electronic band structure

Graphene is a monolayer of carbon atoms arranged in a two-dimensional hexagonal lattice. Each carbon atom possesses four valence electrons, three of which occupy hybridized $sp^{2}$ orbitals, which form strong, in-plane covalent $\sigma-$bonds. The forth valence electron occupies a $p_{z}$-orbital, which form $\pi$-bonds, in which the electrons are delocalized. The electronic band structure, obtained from an ab initio calculation using density functional theory (DFT) is shown in Fig. 1.

Since we will be interested in describing and calculating the Raman spectrum for excitation energies in or near the visible spectral range ($\sim$1.5-4 eV), it is sufficient to focus on the bands near the Fermi energy, which are the $\pi$ and $\pi^{*}$ bands (full lines in Fig. 1). Note that for a quantitatively accurate calculation of the Raman spectrum, the DFT band structure needs to be corrected to properly include electron-electron-interaction effects. These corrections can be computed, for example, within the $GW$ approximation Hedin (1965); Onida et al. (2002) and their main effect is a steepening of the conic $\pi$ bands around the $K$ point with their slope increasing by roughly 18% Grüneis et al. (2008); Venezuela et al. (2011).

II.1.2 Phonon dispersion

Besides the electronic band structure, the phonon dispersion of graphene also plays a very important role for the understanding of the Raman spectrum. Since the lattice structure of graphene can be described by a hexagonal lattice with two atoms per unit cell, the phonon band structure consists of 2$\times$3=6 branches: Three acoustic ones, which can be further divided into an in-plane longitudinal (LA), an in-plane transverse (TA) and an out-of-plane transverse (ZA) branch, and three optical branches, which can also be separated into in-plane longitudinal (LO), in-plane transverse (TO), and out-of-plane transverse (ZO). The terms “longitudinal” and “transverse” refer to displacements in directions parallel and perpendicular to the quasimomentum $\mathbf{q}$ of the phonon, respectively. Note that these notions only make sense for small phonon wave vectors, i.e for $\mathbf{q}$ near $\Gamma$. Away from $\Gamma$, 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 $\Gamma$ by following the specific branch along the high-symmetry lines. The phonon dispersion of graphene as obtained from a density functional perturbation theory (DFPT) calculation is shown in Fig. 2a.

The most prominent features of the phonon dispersion are the kink in the TO branch at $K$ and the positive slope of the LO branch at $\Gamma$. As we will see in a later section, these are precisely the parts of the phonon dispersion that play the most important role in the description of the Raman spectrum as the phonons that contribute the most to it are the two degenerate in-plane optical phonons at $\Gamma$ and the phonons of the TO branch around the $K$ point, whose lattice vibration patterns are shown in Fig. 3. For a description of the Raman spectrum it is therefore of utmost importance to accurately describe these parts of the phonon dispersion.

The pronounced kinks arise due to so-called Kohn anomalies Kohn (1959). When the lattice ions vibrate according to a phonon mode with wave vector $\mathbf{q}$, they induce an electronic charge density of the same periodicity, whose magnitude becomes large when $\mathbf{q}$ connects two points on the Fermi surface. Since the Fermi surface of graphene consists of the two points $K$ and $K^{\prime}$ at the corners of the Brillouin zone (which are separated by a vector $\mathbf{K}$), phonons with $\mathbf{q}=\mathbf{0}$ or $\mathbf{q}=\mathbf{K}$ lead to these large, periodic electronic charge densities. These electronic charge densities in turn lead to descreased screening of the ionic charges, leading to a great enhancement of electron-phonon interaction. This large electron-phonon coupling for $\mathbf{q}$ near $\Gamma$ 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 $\Gamma$ and $K$, the slope of the phonon dispersion is directly proportional to the squared electron-phonon coupling between the corresponding phonon and electrons in the $\pi$ and $\pi^{*}$ bands averaged over a small circle of electronic $\mathbf{k}$ vectors around the Fermi points $K$ and $K^{\prime}$ Piscanec et al. (2004).

The increased electron-phonon coupling at $\Gamma$ and especially at $K$ is further strongly enhanced by electronic correlation effects Lazzeri et al. (2008). 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 by exchange-correlation functionals in the commonly used local density or generalized gradient approximations. As will be commented upon in the next subsection, it is possible to obtain a more accurate description of the electron-phonon coupling for phonons at $\Gamma$ 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 important result is a steepening of the kink near $K$ and a decrease of the phonon frequency of the TO branch at $K$ by more than 100 cm*-1* Lazzeri et al. (2008), which matches the experimental data well (compare Fig. 2b).

II.1.3 Electron-phonon coupling

Besides a description of the electron and phonon band structures, one needs a description of the coupling between the two. For this, we consider the electronic Hamiltonian (discarding electron-electron interaction for the moment), which has the form

[TABLE]

where $\mathbf{r}$ and $\mathbf{p}$ denotes the position and momentum operator, respectively, and $V$ is the lattice potential which depends parametrically on the position of the nuclei, which we collectively denote with $\{\mathbf{R}_{n\alpha}\}$, where $n$ labels the unit cell and $\alpha$ 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 $\mathbf{R}_{n\alpha,0}$ changes by a small displacement: $\mathbf{R}_{n\alpha,0}\to\mathbf{R}_{n\alpha}=\mathbf{R}_{n\alpha,0}+\mathbf{u}(\mathbf{R}_{n\alpha,0})$, where $\mathbf{u}(\mathbf{R}_{n\alpha,0})$ denotes the displacement of the nucleus at equilibrium position $\mathbf{R}_{n\alpha,0}$. As a consequence, the lattice potential changes as well and for small displacements $\mathbf{u}(\mathbf{R}_{n\alpha,0})$ the potential can be expanded into a Taylor series:

[TABLE]

Here, the sums run over all unit cells $n$, all atoms $\alpha$ within the unit cell, and all cartesian coordinates $i$. The electronic Hamiltonian thus acquires two more terms (up to quadratic order in the displacement), describing the coupling of electrons to one and two phonons, respectively:

[TABLE]

where $H^{(i)}_{\mathrm{el-ph}}$ 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 $\mathbf{R}_{n\alpha,0}$ can be written as a superposition of the vibrational eigenmodes:

[TABLE]

where $\mathbf{q}$ and $\lambda$ denote the quasimomentum and phonon branch (e.g. $\lambda$=TO for the transverse optical branch) of the eigenmode, respectively, $\mathbf{R}_{n}$ is the position, i.e., lattice site, of the $n$th unit cell, and $b^{(\dagger)}_{\lambda,\mathbf{q}}$ is an operator that destroys (creates) one phonon of branch $\lambda$ with quasimomentum $\mathbf{q}$. The vector $\mathbf{v}^{\alpha}_{\lambda,\mathbf{q}}$ denotes one of the two three-dimensional parts of the six-dimensional eigenvector of the dynamical matrix describing the displacement of atom $\alpha$ of the unit cell.

To describe the Raman scattering amplitude within perturbation theory, one needs the matrix elements of the electron-phonon Hamiltonian for the $\pi$ and $\pi^{*}$ states and for one or two specific phonons. For example, the matrix elements of the electron-one-phonon Hamiltonian for a phonon of branch $\lambda$ and with quasimomentum $\mathbf{q}$ in the final state are given by

[TABLE]

where $s$ and $s^{\prime}$ specify the electronic band and $|s,\mathbf{k}-\mathbf{q};\lambda,\mathbf{q}\rangle$ is a state consisting of an unperturbed electronic state specified by the band index $s$ and Bloch wave vector $\mathbf{k}-\mathbf{q}$ and a phonon from branch $\lambda$ with quasimomentum $\mathbf{q}$. For the Raman spectrum of graphene, only the $\pi$ and $\pi^{*}$ bands are important, i.e. $s,s^{\prime}=\pi,\pi^{*}$ and for every $\mathbf{k}$, $\mathbf{q}$, and $\lambda$ the electron-phonon coupling matrix elements can be written as a 2$\times$2-matrix $\mathbf{g}^{\lambda}_{\mathbf{k}-\mathbf{q},\mathbf{q}}=\left(g^{\lambda}_{s,\mathbf{k}-\mathbf{q};s^{\prime},\mathbf{k}}\right)_{s,s^{\prime}=\pi,\pi^{*}}$ in the space of the $\pi$ and $\pi^{*}$ bands.

By the same procedure, one can obtain a 2$\times$2-matrix containing the matrix elements of the electron-two-phonon interaction Hamiltonian $H^{(2)}_{\mathrm{el-ph}}$. For the purpose of this chapter, only those matrix elements are important that contain the two phonons in the final state. Again, these can conveniently written as a 2$\times$2 basis in the space of $\pi$ and $\pi^{*}$ bands, which we will denote by $\mathbf{\tilde{g}}^{\lambda_{1},\lambda_{2}}_{\mathbf{k}-\mathbf{q}_{1}-\mathbf{q}_{2},\mathbf{k}}$, where $\lambda_{1(2)}$ and $\mathbf{q}_{1(2)}$ are the quantum numbers of phonon 1(2).

Numerical values for the electron-phonon matrix elements can be obtained from density functional perturbation theory. It should be noted, however, that the underestimation of electron-electron interaction effects in DFT leads to an underestimation of the electron-phonon matrix elements as well, especially for $\mathbf{k}$ and $\mathbf{q}$ near $K$, where long-range electronic correlation effects play an important role. For specific $\mathbf{k}$ and $\mathbf{q}$, such as $\mathbf{k}=\mathbf{q}=K$, it is possible to incorporate these effects by relating the electron-phonon matrix elements at these high-symmetry points to changes of the band energies when the lattice ions are displaced statically according to the phonon displacement pattern shown in Fig. 3b. For instance, for a phonon from the TO branch with quasimomentum $\mathbf{q}=\mathbf{K}$, the square of the electron-phonon coupling for a $\pi$ and $\pi^{*}$ state with $\mathbf{k}=2\mathbf{K}=\mathbf{K}^{\prime}$ obeys Piscanec et al. (2004); Lazzeri et al. (2008):

[TABLE]

where $d$ is the amplitude of the displacement and $\varepsilon^{\pi(\pi^{*})}_{\mathbf{K}}(d)$ is the energy of the $\pi$ ($\pi^{*}$) band at $K$ when the atoms are displaced.

When incorporating electron-electron interaction effects on the level of the $GW$ approximation into the band energies, this procedure allows one to calculate the electron-phonon matrix elements at these high-symmetry points while taking into account correlation effects more accurately. This procedure, however, is limited to high-symmetry points for $\mathbf{k}$ and $\mathbf{q}$ as only there the electron-phonon matrix elements can be related to changes of the band energies.

II.1.4 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 $\mathbf{A}(\mathbf{r})$ and a scalar potential $\phi(\mathbf{r})$. In Coulomb gauge $\mathbf{\nabla}\cdot\mathbf{A}(\mathbf{r})=0$, the scalar potential must be zero for fields vanishing at infinity. The external electromagnetic field can therefore be described entirely in terms of a vector potential $\mathbf{A}(\mathbf{r})$. The Coulomb gauge imposes one constrain on $\mathbf{A}(\mathbf{r})$, leaving two degrees of freedom for the external electromagnetic field, which can be identified with the two possible linearly independent polarization directions of light.

The coupling of electrons to an external vector potential $\mathbf{A}(\mathbf{r})$ can be introduced via the minimal coupling prescription 111We work with Gaussian (or cgs) units. In SI units, the factor $e/c$ is to be replaced by $e$. $\mathbf{p}\to\mathbf{p}+\frac{e}{c}\mathbf{A}(\mathbf{r})$. The electronic Hamiltonian then becomes:

[TABLE]

The coupling to the external electromagnetic field generates two more terms in the Hamiltonian that can be treated in perturbation theory.

To obtain the matrix elements for the coupling of electrons to individual photons, the electromagnetic field has to be quantized. For this, one expands the vector potential in the eigenmodes of a free electromagnetic field, i.e. into plane waves, and promotes it to an operator:

[TABLE]

where $\mu$ labels the two possible polarizations of a photon, $\omega_{k}=c|\mathbf{k}|$ and $\boldsymbol{\epsilon}_{\mu,\mathbf{k}}$ are the frequency and polarization vector of a photon with polarization $\mu$ and wave vector $\mathbf{k}$, respectively, and $a^{(\dagger)}_{\mu,\mathbf{k}}$ is an operator that destroys (creates) a photon with polarization $\mu$ and wave vector $\mathbf{k}$.

The matrix elements of the electron-one-photon Hamiltonian between two unperturbed electronic states and with one photon with polarization $\mu$ and wave vector $\mathbf{k}^{\prime}$ in the initial state then reads:

[TABLE]

where $|s,\mathbf{k}-\mathbf{k}^{\prime};\mu,\mathbf{k}^{\prime}\rangle$ denotes a state consisting of an unperturbed electronic state specified by the band index $s$ and Bloch wave vector $\mathbf{k}-\mathbf{k}^{\prime}$ and a photon with polarization $\mu$ with wave vector $\mathbf{k^{\prime}}$. This expression is often simplified further by the dipole approximation $\exp(i\mathbf{k^{\prime}}\cdot\mathbf{r})\approx 1$, which is justified by the fact that the wave length of the light ($\sim$300-800 nm for light in the visible spectrum) is much larger than the typical crystal length scale given by the lattice constant ($\sim$2.46 Å for graphene), i.e. $\mathbf{k}^{\prime}\cdot\mathbf{r}\approx 0$ if $\mathbf{r}$ varies on the crystal scale only as it does in the matrix elements between Bloch states. In other words, for the description of Raman spectra, we can treat the external photons as having zero wave vector (i.e. we set $\mathbf{k}^{\prime}=0$), yet finite frequency. Again, we will collect the matrix elements between the possible combinations of $\pi$ and $\pi^{*}$ states for fixed $\mathbf{k}$ and light polarization $\mu$ in a 2$\times$2-matrix $\boldsymbol{\gamma}^{\mu}_{\mathbf{k}}=\left(\gamma^{\mu}_{s,\mathbf{k};s^{\prime},\mathbf{k}}\right)_{s,s^{\prime}=\pi,\pi^{*}}$.

Finally, one can follow a similar procedure to obtain the matrix elements between $\pi$ and $\pi^{*}$ states for the electron-two-photon Hamiltonian $H^{(2)}_{\mathrm{el-R}}$. For a calculation of the Raman spectrum only those matrix elements are important which involve one photon with polarization $\mu$ in the initial state and one photon with polarization $\nu$ in the final state. We will denote the corresponding 2$\times$2-matrix of matrix elements in the basis of $\pi$ and $\pi^{*}$ states by $\boldsymbol{\tilde{\gamma}}^{\mu\nu}_{\mathbf{k}}$.

II.1.5 Electron-defect scattering

The final building block needed for a complete description of the Raman spectrum within the framework of perturbation theory is a description of electron-defect scattering. If the graphene sheet contains defects, electrons may scatter from them elastically, leading to additional observable peaks in the Raman spectrum. Since a first, qualitative understanding of these defect-assisted peaks does not require a detailed model for electron-defect scattering, we will not have a closer look at this issue but merely refer the reader to the relevant literature Hwang and Das Sarma (2008); Venezuela et al. (2011). While we will discuss defect-assisted peaks in the following sections, we will not have a detailed look at its treatment in perturbation theory and hence we will not discuss the modeling of electron-defect scattering here.

II.1.6 Summary and Feynman rules

To conclude the section, we summarize the elemental building blocks that are needed for the calculation of the Raman 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 represent electrons (and holes) by full lines with arrows, phonons by dashed lines, and photons by wavy lines. The elemental Feynman diagrams for the various building blocks are shown in Table II.1.6.