Tracking quasiparticle energies in graphene with near-field optics

Phillip E. C. Ashby and J. P. Carbotte

Phys. Rev. B 86, 165405 – Published 2 October 2012

Abstract

Advances in infrared nanoscopy have enabled access to the finite momentum optical conductivity $σ (q, ω)$ . The finite momentum optical conductivity in graphene has a peak at the Dirac fermion quasiparticle energy $ε (k_{F} - q)$ , i.e., at the Fermi momentum minus the incident photon momentum. We find that the peak remains robust even at finite temperature as well as with residual scattering. It can be used to trace out the fermion dispersion curves. However, this effect depends strongly on the linearity of the Dirac dispersion. Should the Dirac fermions acquire a mass, the peak in $σ (q, w)$ shifts to lower energies and broadens as optical spectral weight is redistributed over an energy range of the order of the mass gap energy. Even in this case structures remain in the conductivity, which can be used to describe the excitation spectrum. By contrast, in graphene strained along the armchair direction, the peak remains intact but shifts to a lower value of $q$ determined by the anisotropy induced by the deformation.

Received 13 August 2012

DOI:https://doi.org/10.1103/PhysRevB.86.165405

Authors & Affiliations

Phillip E. C. Ashby^*

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1

J. P. Carbotte^†

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 and The Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

^*ashbype@mcmaster.ca
^†carbotte@mcmaster.ca

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Vol. 86, Iss. 16 — 15 October 2012

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Images

Figure 1
(Top) The scattering geometry in $k$ space for the optical conductivity and polarizability. Both the optical conductivity and polarizability contain coherence factors $F_{s s^{'}} (φ) = \frac{1}{2} (1 + s s^{'} \cos φ)$ . The important difference is that $φ = θ_{k} + θ_{k + q}$ in the optical conductivity, while $φ = θ_{k} - θ_{k + q}$ in the polarizability. We orient our axes so that $x$ is along the so-called zigzag direction of graphene, while the $y$ axis is along the armchair direction. (Bottom) Energy dispersions for bare, strained, and gapped graphene. The effect of strain distorts the Dirac cones from a circular to an elliptical cross section. Gapped graphene retains its circular shape, but the low-energy dispersion is now quadratic, and the Dirac point is no longer accessible.Reuse & Permissions
Figure 2
The real part of the longitudinal and transverse conductivity, $σ^{L}$ and $σ^{T}$ , for $q = 0.4$ scaled by $q / π$ as a function of $ω / μ$ . JDOS and the polarizability $Π$ as a function of $ω / μ$ . The prefactors have been chosen to make the longitudinal conductivity and JDOS agree at $ω = q$ .Reuse & Permissions
Figure 3
(Top panel) The real part of the finite momentum optical conductivity $σ (q, ω)$ as a function of $ω / μ$ for $q / k_{F} = 0.4$ and $T / μ$ $=$ 0, 0.03, 0.07, 0.15, 0.3 for bare bands. There is a strong quasiparticle peak at $ω = q$ , which is unaffected by the finite temperature. The finite temperature smears the interband contribution and begins to fill in the Pauli-blocked region for large enough temperature. (Bottom panel) The real part of the finite momentum optical conductivity $σ (q, ω)$ as a function of $ω / μ$ for $q / k_{F}$ $=$ 1.0 and $T / μ$ $=$ 0, 0.03, 0.3 for bare bands. Included for comparison is the $T = 0$ result with a residual scattering rate $γ / μ$ $=$ 0.005. (Insets): The insets show the optical spectral weight for $T / μ$ $=$ 0, 0.3. The quasiparticle peak holds less spectral weight at larger momentum transfer ( $q$ ). For the $q / k_{F} = 0.4$ case, we can see that the finite temperature has transferred spectral weight to the previously Pauli-blocked region.Reuse & Permissions
Figure 4
The real part of the longitudinal optical conductivity $σ (q, ω)$ for $q / k_{f} = 0.4$ as a function of $ω / μ$ . Included are the result for bare bands at $T = 0$ in black, a numerical evaluation of Eq. (1) including impurity scattering with $γ / μ = 0.005$ in dashed green, and our analytic expression for the quasiparticle peak, Eq. (25), in light green circles.Reuse & Permissions
Figure 5
The quasiparticle peak in the real part of the optical conductivity for momentum transfer $q = 0.4$ (main panel) and $q = 1.0$ (inset) as a function of $ω / μ$ . We show impurity concentrations of $γ / μ$ $=$ 0, 0.05, and 0.005. The position of the peak stays robust even with large impurity content and is broadened as the impurity content increases.Reuse & Permissions
Figure 6
The quasiparticle peak in the real part of the optical conductivity as a function of $ω / μ$ for $q / k_{F} = 0.4$ for graphene with mass gap $Δ$ $=$ 0, 0.05, 0.1, 0.15, 0.2, and 0.25. The peak is pulled to smaller values of $q$ and slowly broadened as the mass gap increases. The broadening happens over an energy scale approximately given by the mass gap and is physically caused by changes in the joint density of states (shown in the inset).Reuse & Permissions

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