Broadband focusing of acoustic plasmons in graphene with an applied current

Letter

Broadband focusing of acoustic plasmons in graphene with an applied current

Michael Sammon, Dionisios Margetis, E. J. Mele, and Tony Low

Phys. Rev. B 104, L161409 – Published 29 October 2021

Abstract

Nonreciprocal plasmons in current-driven, isotropic, and homogenous graphene with proximal metallic gates are theoretically explored. Nearby metallic gates screen the Coulomb interactions, leading to linearly dispersive acoustic plasmons residing close to its particle-hole continuum counterpart. We show that the applied bias leads to spectral broadband focused plasmons whose resonance linewidth is dependent on the angular direction relative to the current flow due to Landau damping. We predict that forward focused nonreciprocal plasmons are possible with accessible experimental parameters and setup.

Received 11 May 2021
Revised 25 September 2021
Accepted 30 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L161409

Physics Subject Headings (PhySH)

Plasmons Surface plasmon polariton

Graphene

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Michael Sammon ^1,*, Dionisios Margetis ², E. J. Mele ³, and Tony Low^1,†

¹Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
²Institute for Physical Science and Technology and Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, Maryland 20742, USA
³Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*sammo017@umn.edu
^†tlow@umn.edu

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Issue

Vol. 104, Iss. 16 — 15 October 2021

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Images

Figure 1
(a) Illustration of the acoustic plasmon dispersion in graphene. Left: Acoustic plasmon dispersion in graphene with no current bias. Right: Acoustic plasmon dispersion in graphene with a current bias in the positive $x$ direction. The plasmon dispersion is illustrated by the red cone, while the teal cone is the onset of the particle-hole continuum. The acoustic plasmon cone is tilted in the direction of the applied bias, resulting in the shift of the plasmon dispersion. Solid circles illustrate the Fermi energy. For comparison, we have included the Fermi energy at zero bias as a dashed line in the right plot. (b) Color map of the loss function $S (q, ω) = {[1 - U (q) Π (q, ω)]}^{- 1}$ for a graphene-insulator-metal system with an applied current obtained by numerical integration. The blue shows a region where gain is achievable. Both the current and the wave number $q$ are assumed to be in the $x$ direction. Calculations were made using $E_{F} = 0.2 eV$ , $k_{s} = 0.6 k_{F}$ , and $κ = 1$ . Both the acoustic plasmon for $d = 3$ Å and the plasmon for $d \to \infty$ are shown. For comparison we have included the analytical dispersion at zero bias shown by the green lines.
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Figure 2
Dimensionless acoustic plasmon velocity $c_{s}$ as a function of the shifted Fermi surface center $k_{s}$ for several different values of $E_{F}$ . Negative values of $k_{s}$ correspond to the $q < 0$ branch. Inset: The dimensionless function $f (A) \equiv k_{s c} / k_{F}$ vs the dimensionless parameter $A$ . The range of $A$ is for $0.01 \leq E_{F} \leq 0.20$ and for the dielectric constant $κ = 3$ . The dashed line in the inset is the best linear fit.
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Figure 3
Ratio of the damping factor $γ$ of the red-shifted ( $γ_{-}$ ) and blue-shifted ( $γ_{+}$ ) plasmon branches for $E_{F} = 0.2 eV$ , $k_{s} = 0.6 k_{F}$ , and $κ = 1$ .
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Figure 4
(a) Temperature plot of the loss function $S (q, θ)$ for $E_{F} = 100 meV$ and $κ = 3$ at a fixed frequency $ℏ ω = 0.5 E_{F}$ for $k_{s} = 0.9 k_{F}$ . The dashed black line illustrates the boundary of the particle-hole continuum. The green line shows the angle $δ θ$ in which $S (q, θ)$ is depleted. (b) Left: Plot of the isofrequency contour $S (θ)$ for $E_{F} = 100 meV$ , $κ = 3$ , and $ℏ ω = 0.5 E_{F}$ at several $k_{s} = 0.3 k_{F}$ (blue), $k_{s} = 0.5 k_{F}$ (red), $k_{s} = 0.7 k_{F}$ (orange), and $k_{s} = 0.9 k_{F}$ . Each line is normalized to the maximum value of $S (θ)$ at fixed $k_{s}$ . The dashed line shows the value we use to define the depletion angle $δ θ$ . Right: Plot of the range $δ θ$ of angles around the current bias direction in which the isofrequency contour $S (θ)$ is exponentially depleted as function of the relative shift of the Fermi surface center. Plasmons emitted at this bias are focused in a range $2 π δ θ$ opposite the applied current bias.
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Figure 5
(a) The critical value $k_{s c}$ of the shift of the Fermi surface center vs the Fermi energy $E_{F} (e V)$ . $k_{s c}$ is defined as the minimum shift in which the angle of depletion $δ θ > π / 2$ . The dashed line is the best linear fit of the large $E_{F}$ dependence, given in the text. (b) The critical current $j_{s c}$ vs the Fermi energy $E_{F}$ . The values of $j_{s c}$ are obtained from $k_{s c}$ according to Eq. (16).
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