Drift-induced modifications to the dynamical polarization of graphene

Mohsen Sabbaghi, Hyun-Woo Lee, Tobias Stauber, and Kwang S. Kim

Phys. Rev. B 92, 195429 – Published 25 November 2015

Abstract

The response function of graphene is calculated in the presence of a constant current across the sample. For small drift velocities and finite chemical potential, analytic expressions are obtained and consequences on the plasmonic excitations are discussed. For general drift velocities and zero chemical potential, numerical results are presented and a plasmon gain region is identified that is related to interband transitions.

Received 31 August 2015
Revised 23 October 2015

DOI:https://doi.org/10.1103/PhysRevB.92.195429

Authors & Affiliations

Mohsen Sabbaghi^1,2,3, Hyun-Woo Lee^2,*, Tobias Stauber^3,†, and Kwang S. Kim^1,‡

¹Center for Superfunctional Materials, Department of Chemistry and Department of Physics, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, Korea
²Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
³Instituto de Ciencia de Materiales de Madrid, CSIC, E-28049 Madrid, Spain

^*hwl@postech.ac.kr
^†tobias.stauber@csic.es
^‡kimks@unist.ac.kr

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Issue

Vol. 92, Iss. 19 — 15 November 2015

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Images

Figure 1
Color-mapped values of the (a) real and (b) imaginary parts of $Δ Π (q, ω)$ obtained from Eq. (8) and presented in units of $(k_{shift} / k_{F}) D (E_{F})$ . The positive and negative $q_{x}$ axes, respectively, correspond to the cases in which $q$ is antiparallel ( $θ_{q} = 0^{\circ}$ ) and parallel ( $θ_{q} = 180^{\circ}$ ) to the drift velocity. The computed $Δ Π (q, ω)$ values are corrected according to Mermin's approach [40] for a phenomenological scattering rate of $ℏ / τ = 5 meV$ (see Appendix pp2).
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Figure 2
The numeric-analytic comparison of $Im [Δ Π (q, ω)]$ for Dirac fermions indicates that the analytic expression given by Eq. (8) becomes more accurate for smaller amounts of $(k_{shift} / k_{F})$ . The numerical evaluation of $Δ Π$ is performed based on $Δ n_{F}$ values computed from Eq. (6), and the results are normalized by $(k_{shift} / k_{F}) D (E_{F})$ .
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Figure 3
The computed (a) decay rate, (b) $ω_{pl} / γ_{pl}$ ratio, and (c) frequency of the TM-SPP modes of a suspended graphene channel, propagating parallel (red solid curves) and antiparallel (blue dashed) to the drift velocity, are compared with their no-drift counterpart (black dotted curves). The agreement with the $\sqrt{q}$ behavior predicted by the local approximation (green dash-dotted) in the $q ≪ k_{F}$ limit can be seen. The impact of the disorder-induced electron scattering is taken into account by Mermin's approach using a phenomenological scattering rate of $ℏ / τ = 5 meV$ .
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Figure 4
Depiction of the $k$ -space dynamics [57] of the $π$ electron gas (the straight arrows along the cross section of the Dirac cones with the $k_{y} = 0$ plane) in an undoped graphene channel subjected to a drain-source voltage. The drifting occupants of the valence band whose group velocity has an opposite component to $E_{ds}$ migrate to the conduction band through the Dirac crossing. Ultimately, the migrants backscatter to the valence band after losing a quantum of energy $ℏ Ω$ via the emission of a phonon mode (wavy arrow).
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Figure 5
Color-mapped values of the energy-loss function, $S (q, ω)$ , of the $π$ electrons in a suspended and undoped graphene channel along which a high drain-source voltage is applied. The negative energy loss for the TM-SPP modes with $q_{min} \leq q \leq q_{max}$ indicates the possibility of the amplification of these modes through the current saturation mechanism ( $ℏ Ω \approx 149 meV$ ). The negative and positive $q_{x}$ axes, respectively, correspond to the P and AP cases.
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Figure 6
The analytic solution for $Δ Π (q, ω)$ can be specified in each of the six regions in the $(\tilde{q}, \tilde{ω})$ plane which are outlined by the straight lines of $\tilde{ω} = \tilde{q}$ (solid), $\tilde{ω} = 2 + \tilde{q}$ (dash-dotted), $\tilde{ω} = 2 - \tilde{q}$ (dotted) and $\tilde{ω} = \tilde{q} - 2$ (dashed).
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