Breit-Wigner-Fano line shapes in Raman spectra of graphene

Eddwi H. Hasdeo, Ahmad R. T. Nugraha, Mildred S. Dresselhaus, and Riichiro Saito

Phys. Rev. B 90, 245140 – Published 23 December 2014

Abstract

Excitation of electron-hole pairs in the vicinity of the Dirac cone by the Coulomb interaction gives rise to an asymmetric Breit-Wigner-Fano line shape in the phonon Raman spectra in graphene. This asymmetric line shape appears due to the interference effect between the phonon spectra and the electron-hole pair excitation spectra. The calculated Breit-Wigner-Fano asymmetric factor $1 / q_{BWF}$ as a function of the Fermi energy shows a V-shaped curve with a minimum value at the charge neutrality point and gives good agreement with the experimental results.

Received 16 June 2014
Revised 12 December 2014

DOI:https://doi.org/10.1103/PhysRevB.90.245140

Authors & Affiliations

Eddwi H. Hasdeo^1,*, Ahmad R. T. Nugraha¹, Mildred S. Dresselhaus^2,3, and Riichiro Saito¹

¹Department of Physics, Tohoku University, Sendai 980-8578, Japan
²Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4037, USA
³Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

^*hasdeo@flex.phys.tohoku.ac.jp

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Vol. 90, Iss. 24 — 15 December 2014

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Figure 1
All possible Raman scattering processes considered in Eq. (2): (a) the G Raman scattering with the Raman shift $ω_{s} = ω_{G}$ , (b)–(f) the electronic Raman scattering (ERS) processes. The ERS processes include the Coulomb interactions between a photoexcited electron (PE) and electrons on the Dirac cone (DEs): (b) a first-order intravalley interaction and intravalley scattering (Aa), (c) a first-order intervalley interaction and intravalley scattering (Ea), (d) a second-order intravalley interaction and intravalley scattering (Aa), (e) a second-order intervalley interaction and intravalley scattering (Ea), and (f) a second-order intervalley interaction and intervalley scattering (Ee). Capital (small) letters A (a) and E (e) label the intravalley and the intervalley interactions (scatterings). Here $E_{L}$ and $E_{S}$ are the laser excitation energy and scattered photon energy, respectively. The Raman shift for the first- (second-) order processes is $ω_{s} = ω_{e} (ω_{s} = ω_{1} + ω_{2})$ .
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Figure 2
(a) Illustration of the direct Coulomb intravalley (A) interaction and intervalley (E) interaction involving states around $K$ and $K^{'}$ Dirac cones. (b) The averaged absolute value of the direct Coulomb interaction matrix element $K^{d}$ as a function of momentum transfer $q$ for the intravalley (A) interaction and for the intervalley (E) interaction. The intervalley scattering is not shown in this figure.
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Figure 3
(a) Calculated Raman intensity according to Eq. (2) for $E_{F} = 0.00 eV$ (solid line) compared with the Lorentzian $G$ mode intensity by taking the square of $A_{G}$ in Eq. (3) (dashed line). The $G$ -mode constituents, i.e., iTO and LO, are indicated by a dotted line and a dot-dashed line, respectively. An asymmetric line shape (solid line) appears due to the interference effect between the $G$ mode with the ERS. The inset shows calculated results of the first-order (dashed line) and the second-order (solid line) ERS spectra, indicating that the second-order processes have an intensity value six orders of magnitude greater than that of the first-order processes. (b) Calculated Raman intensity for $E_{F} = 0.00 eV$ (solid line) and $E_{F} = 0.20 eV$ (dashed line). The BWF asymmetric factor $1 / q_{BWF}$ decreases by increasing the absolute value of $| E_{F} |$ away from the Dirac point because the ERS intensity also decreases by increasing $| E_{F} |$ (inset).
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Figure 4
Comparison between (a) the calculated results (this work) and (b) the experimental results taken from Ref. [9] for the $G$ -band Raman intensity as a function of the Raman shift, which is plotted for various values of $E_{F}$ in the range $- 0.20 \leq E_{F} \leq 0.40$ eV. The values of $1 / q_{BWF}$ obtained from the calculation and the experiment are also given on each plot. (c) Comparison of the BWF asymmetric factor $1 / q_{BWF}$ as a function of $E_{F}$ and gate voltage $V_{G}$ between theory (squares) and experiment (circles). Both the linewidth and the phonon peak frequency shift due to the Kohn anomaly effect are fitted from the experimental results in Ref. [9].
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Figure 5
(a) The G-band intensity as a function of electron initial states in the first Brillouin zone, (b) the G-band intensity as a function of electron energy, (c) the G-band phase $cos φ (ω_{s})$ , where $φ (ω_{s}) = {tan}^{- 1} [Re A_{G} (ω_{s}) / Im A_{g} (ω_{s})]$ , (d) the G-band peak intensity as a function of cutoff energy measured from $E_{L}$ . In this plot we use $E_{L} = 2.4 eV, E_{F} = 0 eV$ , and resonance window $γ = 0.1 eV$ .
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Figure 6
(a) Schematic of the zeroth-order ERS process, which is equivalent to the ERS Feynman diagram given by Kashuba and Fal'ko in Ref. [22]. Panels (b) and (c) show the calculated Raman intensity for $E_{F} = 0$ eV and $E_{F} = 0.1$ eV, respectively. Insets show contribution from the zeroth-, first-, and second-order ERS processes to the Raman intensity.
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