Theory of double-resonant Raman spectra in graphene: Intensity and line shape of defect-induced and two-phonon bands

Pedro Venezuela, Michele Lazzeri, and Francesco Mauri

Phys. Rev. B 84, 035433 – Published 25 July 2011

Abstract

We calculate the double-resonant (DR) Raman spectrum of graphene, and determine the lines associated to both phonon-defect processes (such as in the $D$ line at $\sim$ 1350 cm $^{- 1}$ , $D^{'}$ at $\sim$ 1600 cm $^{- 1}$ , and $D^{''}$ at $\sim$ 1100 cm $^{- 1}$ ), and two-phonon ones (such as in the $2 D$ , $2 D^{'}$ , or $D + D^{''}$ lines). Phonon and electronic dispersions reproduce calculations based on density-functional theory corrected with GW. Electron-light, -phonon, and -defect scattering matrix elements and the electronic linewidth are explicitly calculated. Defect-induced processes are simulated by considering different kinds of idealized defects. For an excitation energy of $ε_{L} = 2.4$ eV, the agreement with measurements is very good and calculations reproduce the relative intensities among phonon-defect or among two-phonon lines; the measured small widths of the $D$ , $D^{'}$ , $2 D$ , and $2 D^{'}$ lines; the line shapes; the presence of small intensity lines in the 1800–2000-cm $^{- 1}$ range. We determine how the spectra depend on the excitation energy, on the light polarization, on the electronic linewidth, on the kind of defects, and on their concentration. According to the present findings, the intensity ratio between the $2 D^{'}$ and $2 D$ lines can be used to determine experimentally the electronic linewidth. The intensity ratio between the $D$ and $D^{'}$ lines depends on the kind of model defect, suggesting that this ratio could possibly be used to identify the kind of defects present in actual samples. Charged impurities outside the graphene plane provide an almost undetectable contribution to the Raman signal. The present analysis reveals that, for both $D$ and $2 D$ lines, the dominant DR processes are those in which electrons and holes are both involved in the scattering, because of a destructive quantum interference that kills processes involving only electrons or only holes. The most important phonons belong to the $K \to Γ$ direction ( $i n n e r$ phonons) and not to the K $\to$ M one ( $o u t e r$ phonons), as usually assumed. The small $2 D$ linewidth at $ε_{L} = 2.4$ eV is a consequence of the interplay between the opposite trigonal warpings of the electron and phonon dispersions. At higher excitation, e.g., $ε_{L} = 3.8$ eV, the $2 D$ line becomes broader and evolves in an asymmetric double peak structure.

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Received 23 March 2011

DOI:https://doi.org/10.1103/PhysRevB.84.035433

Authors & Affiliations

Pedro Venezuela^1,2, Michele Lazzeri¹, and Francesco Mauri¹

¹IMPMC, Université Pierre et Marie Curie, CNRS, 4 place Jussieu, F-75252 Paris, France
²Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, RJ, Brazil

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Vol. 84, Iss. 3 — 15 July 2011

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Figure 1
Goldstone diagrams for the double-resonant Raman processes considered in this work. In this paper, the term “ $a b$ processes” refers to the processes highlighted by the gray area ( $e h 1$ , $e h 2$ , $h e 1$ , and $h e 2$ ). The other processes are referred to as “ $a a$ processes.” The largest part of the Raman intensity is due to the $a b$ processes. The reader might be familiar with an alternative representation of the processes, reported in Fig. 2.Reuse & Permissions
Figure 2
An alternative representation (customary for graphene and graphite) of the processes associated to the diagrams of Fig. 1. The crosses represent the electronic dispersion near the conic region. The vertical arrows represent the electron/hole creation and recombination. The horizontal arrows represent the scattering with a defect or with a phonon. For simplicity we show only the processes involving a phonon with momentum along the K-M line. In this paper, the term “ $a b$ processes” refers to the processes highlighted by the gray area ( $e h 1$ , $e h 2$ , $h e 1$ , and $h e 2$ ). The other processes are referred to as “ $a a$ processes.”Reuse & Permissions
Figure 3
Graphene electronic dispersion obtained with the five-neighbor tight binding described in the text (solid line). For a comparison we also show the dispersion obtained with the first-neighbor TB having the same Fermi velocity at K (dashed line). The electron/hole ( $e / h$ ) asymmetry is defined as $ε_{k}^{π^{*}} + ε_{k}^{π}$ and is constant for the first-neighbor TB.Reuse & Permissions
Figure 4
Calculated graphene phonon dispersion from DFT (lines) vs IXS measurements on graphite from Refs. 30 (filled dots), 31 (triangles), and28 (open dots). The highest optical branch near K is “corrected” to include GW effects following Refs. 27 and 28, and is plotted with a thicker gray (red) line. The dashed line is the same branch calculated from standard DFT, without GW correction. The cross at $Γ$ is the measured Raman $G$ line frequency in graphite (1582 cm $^{- 1}$ ).Reuse & Permissions
Figure 5
Electronic linewidth as a function of energy. (a) Contribution of electron-phonon scattering to the electronic linewidth, $γ^{(e p)}$ , compared to conical bands results [Eq. (10)]. (b) Contribution of on-site and hopping impurity scattering to the electronic linewidth. $γ^{(on)}$ is proportional to $α_{on} = n_{d} {(δ V_{0})}^{2}$ and $γ^{(hopp)}$ is proportional to $α_{hopp} = n_{d} {(δ t_{1})}^{2}$ (Sec. 2e). We thus plot $γ^{(i)} / α_{i}$ , where the label “i” refers to “on” (on-site defect) or to “hopp” (hopping defect).Reuse & Permissions
Figure 6
Intensity vs Raman shift for $ɛ_{L} = 2.4$ eV. Comparison of the present calculations with the measurements from Ref. 11. Notice that our model includes only double-resonant processes and thus the $G$ line is not present. Measurements correspond to a defect concentration $n_{d} =$ $10^{12}$ cm $^{- 2}$ . Calculations are done using $γ^{tot} = 96$ meV, and hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . All the intensities are normalized to the maximum value of the $2 D$ line.Reuse & Permissions
Figure 7
Intensity vs Raman shift for $ɛ_{L} = 2.4$ eV. Comparison of the present calculations with the measurements from Ref. 1. The figure reports only two-phonon processes. Calculations are done using $γ^{tot} = 84$ meV. All the intensities are normalized to the maximum value of the $2 D$ line. The inset shows the $D$ + $D^{''}$ band in a different scale.Reuse & Permissions
Figure 8
Calculated Raman spectra for $ɛ_{L} = 1.2$ eV and $γ^{tot} = 32$ meV, $ɛ_{L} = 2.4$ eV and $γ^{tot} = 84$ meV, $ɛ_{L} = 3.8$ eV and $γ^{tot} = 170$ meV. Calculations are done using hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . All the intensities are normalized to the corresponding $2 D$ line maxima.Reuse & Permissions
Figure 9
Phonon dispersion of graphene along high-symmetry lines. Bold crosses indicate the phonons that mostly contribute to the $D$ , $D^{'}$ , $D^{''}$ , $D^{3}$ , $D^{4}$ , and $D^{5}$ Raman bands, for $ε_{L} = 2.4$ eV. Dotted crosses indicate phonons that also contribute to the $D$ , $D^{'}$ , $D^{3}$ , and $D^{4}$ bands but with smaller intensity. The crosses are determined from the maximum of $I_{q}$ as defined in Sec. 3b2.Reuse & Permissions
Figure 10
Raman shift as a function of excitation energy. Upper panel: two-phonon bands. Lower panel: disorder-induced bands. Our results compared to experimental data from Refs. 9 (circles) and 35 (triangles).Reuse & Permissions
Figure 11
Calculated Raman spectra for small intensity bands. Calculations are done using $ε_{L} = 2.0$ eV and $γ^{tot} = 65$ meV (upper), $ε_{L} = 2.4$ eV and $γ^{tot} = 84$ meV (middle), $ε_{L} = 2.8$ eV and $γ^{tot} = 106$ meV (lower). We consider hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . All the intensities are normalized to the corresponding $2 D$ line maxima.Reuse & Permissions
Figure 12
Raman shift vs excitation energy for the small intensity bands of Fig. 11. Upper and lower panels display results for two-phonon and defect-induced bands, respectively. Upper panel calculations are compared with measurements from Refs. 36 (dots) and 37 (diamonds).Reuse & Permissions
Figure 13
Comparison of calculated Raman spectra done with different light polarizations. Calculations are done using $ε_{L}$ = 2.4 eV and $γ^{tot}$ = 84 meV (upper plot), or $ε_{L}$ = 3.8 eV and $γ^{tot}$ = 170 meV (lower plot). We used hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . The intensities are normalized to the corresponding $2 D$ line maxima calculated with unpolarized light. “Parallel” and “transverse” refer to $I_{∥}$ and $I_{⊥}$ as defined in Sec. 2d.Reuse & Permissions
Figure 14
Integrated areas under the $2 D$ , $2 D^{'}$ , and $D + D^{''}$ lines [ $A$ $(2 D)$ , $A$ $(2 D^{'})$ , and $A$ $(D + D^{''})$ ] as a function of the electron $=$ hole linewidth ( $γ^{tot}$ ), for $ɛ_{L}$ = 2.4 eV. The areas are normalized to $A$ $(2 D)$ calculated with $γ^{tot} = {\tilde{γ}}^{(e p)}$ = 84 meV. For clarity, $A$ $(2 D^{'})$ and $A$ $(D + D^{''})$ are multiplied by 20. Symbols are calculations, lines are the fit from Eq. (15). Inset: $A$ $(2 D) /$ $A$ $(2 D^{'})$ ratio.Reuse & Permissions
Figure 15
Integrated area under the $2 D$ line as a function of the excitation energy $ε_{L}$ . The defect concentration is zero. The full line is obtained by including the dependence of the broadening on $ε_{L}$ , $γ^{tot} = {\tilde{γ}}^{(e p)} (ε_{L})$ (see Sec. 2f). The dashed line is from an unrealistic simulation in which $γ^{tot}$ has been kept fixed to a constant value $γ^{tot} = {\tilde{γ}}^{(e p)} (2.4 eV) = 84$ meV, independent from $ε_{L}$ .Reuse & Permissions
Figure 16
Ratio of the integrated areas under Raman bands as a function of excitation energy. (a) Two-phonon bands: our results compared to experimental data from Ref. 1. (b) Disorder-induced bands from hopping impurities, with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ .Reuse & Permissions
Figure 17
Intensity of the $D$ and $2 D$ Raman lines as a function of the defect concentration for $ε_{L} = 2.4$ eV. Calculations are done using hopping defects and are reported as a function of the parameter $α_{hopp} = n_{d} {(δ t_{1})}^{2}$ ( $n_{d}$ is the defect concentration and $δ t_{1}$ is the hopping parameter), in the upper horizontal scale. The lower horizontal scale is obtained by considering $δ t_{1} = 8.0$ eV. (a) $I_{D}$ is the maximum of the intensity of the $D$ line; symbols are experimental data from Ref. 7. The dashed line is a linear fit of the $I_{D}$ calculated values for $n_{d} < 5 \times 10^{11} {cm}^{- 2}$ . Theoretical and experimental intensities have been normalized by their maximum values. (b) Integrated areas under $D$ and $2 D$ bands, $A (D)$ and $A (2 D)$ . Experimental data are from Ref. 11. Theoretical and experimental areas are normalized by $A (2 D)$ at minimum defect concentration. The vertical line indicates the defect concentration of $7 \times 10^{12}$ cm $^{- 2}$ ( $α_{hopp} = 4.5 \times 10^{14}$ cm $^{- 2}$ eV $^{2}$ ) for which the two contributions to the electronic broadening are equal: ${\tilde{γ}}^{(D)} = {\tilde{γ}}^{(e p)}$ .Reuse & Permissions
Figure 18
Calculated Raman spectra obtained for three different kinds of defects (hopping, on-site, and Coulomb), compared with the measurements of Ref. 11 done at $ε_{L} = 2.4$ eV. The Raman $G$ line is not described by the present model. Calculations are done using $γ^{tot}$ = 96 meV. Other relevant parameters are given in the text. All intensities are normalized by the corresponding $2 D$ maximum. The intensity of the Coulomb impurity spectrum is enhanced by $10^{2}$ for clarity.Reuse & Permissions
Figure 19
The upper panels compare the calculated Raman spectrum $I_{tot}$ with spectra determined considering only $a a$ processes, $I_{a a}$ , or $a b$ processes, $I_{a b}$ . More precisely, $I_{tot}$ is determined considering all the processes shown in Fig. 1; $I_{a a}$ is computed by considering only $e e 1$ , $e e 2$ , $h h 1$ , and $h h 2$ processes; $I_{a b}$ is computed by considering only $e h 1$ , $e h 2$ , $h e 1$ , and $h e 2$ processes (see the text). The lower panels display fictitious Raman intensities $\tilde{I}$ obtained by substituting to the DR scattering amplitudes $K$ in Eqs. (3) their modulus $| K |$ (see the text). The two lines ${\tilde{I}}_{a a}$ and ${\tilde{I}}_{a b}$ are obtained by considering only $a a$ and $a b$ processes, as before. Calculations are done using $ε_{L} = 2.4$ eV, $γ^{tot} = 84$ meV, and hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . All the intensities are normalized to the $2 D$ line maximum of $I_{tot}$ .Reuse & Permissions
Figure 20
Decomposition the intensity of the most important Raman bands into their components associated to phonons with a given wave vector q, $I_{q}$ . The rhombi are the graphene first Brillouin zone. For each band, we consider the contribution to the Raman intensity in a window of frequencies corresponding to that particular band (Ref. 46). The intensities are normalized to the maximum of each band. Calculations are done using $ε_{L} = 2.4$ eV, $γ^{tot} = 84$ meV, and hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ .Reuse & Permissions
Figure 21
Electron and phonon states relevant for the $2 D$ line. The rhombi are the graphene Brillouin zone. (a) The triangularly distorted contour around K is obtained from $ε_{k}^{π^{*}} - ε_{k}^{π} = 2.4$ eV and represents the electronic states near K that are excited by a laser with energy $ε_{L} = 2.4$ eV. The contour around K $^{'} = 2 K$ is obtained from $ε_{k}^{π^{*}} - ε_{k}^{π} = 2.06$ eV and represents the electronic states near K $^{'}$ that are deexcited by the emission of a quantum of light with energy $ε_{L} - 2 ω_{ph}$ eV, with $ω_{ph} = 1354$ cm $^{- 1}$ (half the energy of the $2 D$ line for $ε_{L} = 2.4$ eV). (b) q $_{n}$ is one of the vectors such that the contour near K translated by q $_{n}$ is tangent to the contour near K $^{'}$ . (c) $I_{q}$ decomposition of the $2 D$ intensity (same figure as the $2 D$ panel in Fig. 20). The dashed closed line is defined by the ensemble of the q $_{n}$ vectors. (d) The dashed line is the same as in (c). The thick gray (red) line is the phonon isoenergy contour obtained from $ω_{q}^{ν} = 1354$ cm $^{- 1}$ . The relevant phonon branch, thick gray line in Fig. 4, is disentangled from the other branches as in Fig. 2 of Ref. 28. Notice that the isoenergy contours of electron states [panels (a) and (b)] and phonons [panels (c) and (d)] have opposite trigonal warpings. Notice also that phonon isoenergy contours in Fig. 2 of Ref. 28 are plotted with respect to the K $^{'}$ of the present notation.Reuse & Permissions
Figure 22
Scheme of the double-resonant process associated to the $2 D$ line. The momenta of the phonons mostly involved are indicated as “inner” and “outer.”Reuse & Permissions
Figure 23
Lower panel: momenta of the inner and outer high-symmetry phonons which mostly contribute to the $2 D$ band. The lines are obtained from the vectors connecting the isoenergy electronic contours corresponding to that excitation energy. For example, the values for $ε_{L} = 2.4$ eV are the moduli of the inner and outer vectors reported in Figs. 21 and 22. The symbols are obtained from the maximum intensity in the $I_{q}$ plots (as those in Fig. 20 or in the left panels of Fig. 26) corresponding to that excitation energy. Upper panel: frequency of the inner and outer phonons reported in the lower panel.Reuse & Permissions
Figure 24
Electron and phonon states relevant for the $2 D^{'}$ line. (a) The triangularly distorted contours around K are obtained from $ε_{k}^{π^{*}} - ε_{k}^{π} = 2.4$ eV and $ε_{k}^{π^{*}} - ε_{k}^{π} = 2.0$ eV. They represent the electronic states that are excited by a laser with energy $ε_{L} = 2.4$ eV and those that are deexcited by the emission of a quantum of light with energy $ε_{L} - 2 ω_{ph}$ eV, with $ω_{ph} = 1602$ cm $^{- 1}$ (half the energy of the $2 D^{'}$ line for $ε_{L} = 2.4$ eV). q $_{n}$ is one of the vectors such that the excited-states’ contour translated by q $_{n}$ is tangent to the second contour. The analogous construction arond $K^{'}$ is also shown. (b) $I_{q}$ decomposition of the $2 D^{'}$ intensity (same figure as the $2 D^{'}$ panel in Fig. 20). The two dashed (green) closed lines (almost indistinguishable in the scale of the figure) are defined by the ensemble of the nesting q $_{n}$ vectors among K states or among $K^{'}$ states.Reuse & Permissions
Figure 25
Angular dependence of the intensity (lower panels) and of the weighted average Raman shift (upper panels) for the $D$ , $2 D$ , and $2 D^{'}$ bands. The angles are measured taking the horizontal direction in Fig. 20 as reference. Thus for the $D$ and $2 D$ bands, $0^{\circ}$ is the K $\to$ $Γ$ direction in the BZ, while $\pm$ 60 $^{\circ}$ are the K $\to$ M one. For the $2 D^{'}$ band, zero degrees is the $Γ$ $\to$ K direction, while, $\pm$ 30 $^{\circ}$ are the $Γ$ $\to$ M direction. In the lower panels, the solid lines correspond to our most precise calculation. Dashed lines correspond to an approximated simulations in which the electron-light, electron-phonon, and electron-defect scattering matrix elements are kept constant (see the text). Calculations are done using $ε_{L} = 2.4$ eV, $γ^{tot} = 84$ meV, and hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ .Reuse & Permissions
Figure 26
Calculated $2 D$ line for the excitation $ε_{L} = 3.8$ eV and $γ^{tot} = 170$ meV. Top right panel: intensity vs Raman shift. The line appears as a broad band with two maxima near 2790 cm $^{- 1}$ ( $2 D^{-}$ ) and 2840 cm $^{- 1}$ ( $2 D^{+}$ ). Left panels: mapping of the Raman intensity in the Brillouin zone (as in Fig. 20) of the two components $2 D^{-}$ $2 D^{+}$ obtained by integrating in the corresponding frequency windows (Ref. 46). Central panels: angular dependence of the weighted average Raman shift and of the intensity, as in Fig. 25Reuse & Permissions
Figure 27
Comparison of a typical Raman spectrum (full calculation) with a test calculation in which the phonon energies in the denominators of the DR scattering amplitudes $K$ are considered zero. Calculations are done using $ε_{L} = 2.4$ eV, $γ^{tot} = 96$ meV, and hopping defects with $α_{hopp} = 6.4 \times 10^{13}$ eV $^{2}$ cm $^{- 2}$ . All the intensities are normalized to the $2 D$ line maximum value of the full calculation.Reuse & Permissions
Figure 28
Numerical solution of Eqs. (D1) using $ε_{L} = 2.4$ eV, $γ = 84$ meV, and $ℏ v_{F} = 6.49$ eV Å. $\bar{q} = 2 q ℏ v_{F} / ε_{L}$ is an adimensional momentum and $\bar{q} = 2$ corresponds to the double-resonance condition. $I_{a a}$ is magnified by 10 $^{2}$ for clarity. ${\tilde{I}}_{a a}$ and ${\tilde{I}}_{a b}$ are intensities in which quantum interference has been artificially suppressed (see the text).Reuse & Permissions
Figure 29
DR scattering amplitudes $\bar{K^{r}}$ as defined in Eq. (D2) for the $a a$ and $a b$ processes, as a function of the adimensional momentum $\bar{k} = 2 k ℏ v_{F} / ε_{L}$ . The real and imaginary part of the complex number $\bar{K^{r}}$ are plotted as two different lines. Calculations are done using $ε_{L} = 2.4$ eV, $γ = 84$ meV, and $ℏ v_{F} = 6.49$ eV Å.Reuse & Permissions

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