Emission of plasmons by drifting Dirac electrons: A hallmark of hydrodynamic transport

Dmitry Svintsov

Phys. Rev. B 100, 195428 – Published 25 November 2019

Abstract

Direct current in clean semiconductors and metals was recently shown to obey the laws of hydrodynamics in a broad range of temperatures and sample dimensions. However, the determination of frequency window for hydrodynamic phenomena remains challenging. Here, we reveal a phenomenon being a hallmark of high-frequency hydrodynamic transport, the Cerenkov emission of plasmons by drifting Dirac electrons. The effect appears in a hydrodynamic regime only due to reduction of plasmon velocity by electron-electron collisions below the velocity of carrier drift. To characterize the Cerenkov effect quantitatively, we analytically find the high-frequency nonlocal conductivity of drifting Dirac electrons across the hydrodynamic-to-ballistic crossover. We find the growth rates of hydrodynamic plasmon instabilities in two experimentally relevant setups: parallel graphene layers and graphene covered by subwavelength grating, further showing their absence in ballistic regime. We argue that the possibility of Cerenkov emission is linked to singular structure of nonlocal conductivity of Dirac materials and is independent on specific dielectric environment.

Received 8 August 2019
Revised 14 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.195428

Physics Subject Headings (PhySH)

Ballistic transport Drift waves Electrical conductivity Electron relaxation Flow instability Fluid-particle interactions Hydrodynamic waves Interparticle interactions Optical conductivity Plasma instabilities Plasmonics Plasmons Sound waves Terahertz generation in plasmas

Condensed Matter, Materials & Applied PhysicsFluid DynamicsPlasma Physics

Authors & Affiliations

Dmitry Svintsov ^*

Laboratory of 2d Materials for Optoelectronics, Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny 141700, Russia

^*svintcov.da@mipt.ru

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Issue

Vol. 100, Iss. 19 — 15 November 2019

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Images

Figure 1
Two possible graphene-based setups where hydrodynamic plasmon instabilities can be observed: (a) parallel layers with counterstreaming electrons with velocities $\pm u_{0}$ , (b) graphene covered with subwavelength plasmonic grating and a highly-conducting substrate, and (c) schematic of changes in plasmon frequency (blue circles) with increasing drift velocity $u_{0}$ in Dirac materials at different Knudsen numbers $Kn = q v_{0} τ_{ee}$ . Black solid line shows the position of singularity in ballistic nonlocal conductivity. The singularity becomes softer (dashed black line) in the hydrodynamic regime.
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Figure 2
Plasmon spectra (top) and damping (bottom) in a single graphene layer in the presence of e-e collisions at different drift velocities. Positive wave vectors correspond to waves co-propagating with drift. Damping disappears at $q = 0$ and $ω u_{0} = q v_{0}^{2}$ (the latter set of frequencies and wave vectors is marked with dashed lines). Fermi energy $μ = 25$ meV, background dielectric constant $κ = 5$ , scattering rate $γ_{e e} = 6 \times 10^{12} s^{- 1}$ .
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Figure 3
Plasmons in graphene double layers with counterstreaming flows (a) Evolution of plasmon spectra in HD regime with increasing drift velocity $β$ leading to the shift of acoustic mode frequency down to zero and subsequent instability (b) Damping/growth rates of plasmons at different values of e-e scattering rate, demonstrating suppression of instability with reduced scattering. Blue line corresponds to damping of optical mode being (almost) insensitive to drift; orange and dark red lines show the damping/gain of a couple of acoustic modes (c) Stability diagram of the double-layer setup: critical value of velocity and inverse Knudsen number at which plasma waves become unstable. Dashed lines in (c) are critical velocities calculated using Eq. (15). In panels (a) and (b), interlayer distance $d = 1$ nm, Fermi energy $μ = 25$ meV. In panel (c), interlayer coupling $e^{- 2 q d}$ is set to 0.86.
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Figure 4
Evolution of reflectance spectra of graphene covered by plasmonic grating with increasing carrier drift velocity in the hydrodynamic (a) and ballistic (b) regimes. Bright regions correspond to the excitation of plasmon and Fabry-Perot modes. At large drift velocity, plasmon-enhanced absorption ( $R < 1$ ) turns to plasmon enhanced amplification ( $R > 1$ ). The reflection coefficient further diverges at the crossing of Fabry-Perot and plasmon resonance. Parameters: $μ = 50$ meV, $d = 3$ nm, $D = 100 μ m, κ = 12$ , grating period is $a = 2 μ m$ , width of metal gate $W = a / 2 = 1 μ m$ .
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