Phonon Self-Energy Corrections to Nonzero Wave-Vector Phonon Modes in Single-Layer Graphene

P. T. Araujo, D. L. Mafra, K. Sato, R. Saito, J. Kong, and M. S. Dresselhaus

Phys. Rev. Lett. 109, 046801 – Published 24 July 2012

Abstract

Phonon self-energy corrections have mostly been studied theoretically and experimentally for phonon modes with zone-center ( $q = 0$ ) wave vectors. Here, gate-modulated Raman scattering is used to study phonons of a single layer of graphene originating from a double-resonant Raman process with $q \neq 0$ . The observed phonon renormalization effects are different from what is observed for the zone-center $q = 0$ case. To explain our experimental findings, we explored the phonon self-energy for the phonons with nonzero wave vectors ( $q \neq 0$ ) in single-layer graphene in which the frequencies and decay widths are expected to behave oppositely to the behavior observed in the corresponding zone-center $q = 0$ processes. Within this framework, we resolve the identification of the phonon modes contributing to the $G^{⋆}$ Raman feature at $2450 {cm}^{- 1}$ to include the $iTO + LA$ combination modes with $q \neq 0$ and also the 2iTO overtone modes with $q = 0$ , showing both to be associated with wave vectors near the high symmetry point $K$ in the Brillouin zone.

Received 27 October 2011

DOI:https://doi.org/10.1103/PhysRevLett.109.046801

Authors & Affiliations

P. T. Araujo^1,*, D. L. Mafra^1,2, K. Sato³, R. Saito³, J. Kong¹, and M. S. Dresselhaus^1,4

¹Department of Electrical Engineering and Computer Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
²Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30123-970 Brazil
³Department of Physics, Tohoku University, Sendai 980-8578, Japan
⁴Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

^*ptaraujo@mit.edu

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Vol. 109, Iss. 4 — 27 July 2012

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Images

Figure 1
(a) Possible ( $E_{F} = 0$ ) and not possible ( $E_{F} \neq 0$ ) AV $q = 0$ processes for $e − h$ pair creation (annihilation) due to phonon (with energy $ℏ ω_{q}$ ) absorption (emission). $E^{eh}$ stands for the $e − h$ pair energy. (b) and (c) show, respectively, the frequency $ω_{G}$ hardening and decay width $γ_{G}$ narrowing for the $G$ -band Raman feature as a function of gate voltage $V_{G}$ . The insets in (b) and (c) are theoretical predictions based on Ref. 2 of the $E_{F}$ dependence of $ℏ ω_{q} - ℏ ω_{q}^{0}$ and $γ_{q}$ for an AV $q = 0$ process. The $E_{F}$ values on the upper scales are obtained with $E_{F} = ℏ | v_{F} | \sqrt{π C_{g} V_{G} / e}$ , where $C_{g} = 115 aF / μ m^{2}$ is the gate capacitance per unit of area, $e$ is the electron charge, and $| v_{F} | = 1.1 \times 10^{6} m / s$ is the carrier Fermi velocity.Reuse & Permissions
Figure 2
(a) Possible ( $E_{F} = 0$ and $E_{F} \neq 0$ ) $q = 0$ (measured from the $K$ point) EV processes, (b) not possible ( $E_{F} = 0$ ) and possible ( $E_{F} \neq 0$ ) AV processes, and (c) possible EV processes for electron-hole pair creation (annihilation) due to phonon (with energy $ℏ ω_{q}$ ) absorption (emission) when the phonon wave vector is not zero ( $q \neq 0$ ). (d) shows illustrative predictions for the $V_{G}$ dependence of the phonon frequency correction $ω_{q} - ω_{q}^{0}$ (black solid line) and the corresponding decay width $γ_{q} = | γ_{q} - γ_{q}^{0} |$ (gray line) when $q \neq 0$ , both as a function of $E_{F}$ . The $ω_{q} - ω_{q}^{0}$ and $γ_{q}$ values in (d) were normalized to illustrate the concept of $ω_{q}$ softening and $γ_{q}$ broadening. $E^{eh}$ is the $e − h$ pair energy and $E_{K \to K^{'}}$ is the energy required to translate an electron from $K$ to $K^{'}$ [26].Reuse & Permissions
Figure 3
(a) The experimental $G^{⋆}$ and the $G^{'}$ bands as they appear in the resonant Raman spectrum. The asymmetric $G^{⋆}$ feature is a combination of the $iTO + LA$ ( $q = 2 k$ read from the $K$ point) modes and the 2iTO ( $q = 0$ read from the $K$ point) mode. The $G^{'}$ mode is an overtone of the iTO mode ( $q = 2 k$ ). For illustrative purposes, the signal of the $G^{⋆}$ feature was multiplied by a factor of 10 and the Lorentzian profiles used to fit the spectrum are shown in constructing (a). (b) The frequency dispersion of the $G^{⋆}$ peaks as a function of laser energy ( $E_{Laser}$ ), showing that the $iTO + LA$ ( $q = 2 k$ ) is a dispersive mode, while the 2iTO ( $q = 0$ ) is nondispersive [19, 20]. (c) The gate voltage $V_{G}$ dependence of the 2iTO ( $q = 2 k$ ) $ω_{G^{'}}$ and $γ_{G^{'}}$ [inset in (c)]. (d) and (e) show, respectively, the $ω_{q}$ and $γ_{q}$ dependencies on $| E_{F} |$ seen for the $iTO + LA$ and 2iTO modes. In (c),(d), and (e), the $E_{F}$ values were calculated as explained in the caption of Fig. 1.Reuse & Permissions

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