Acoustic plasmons at the crossover between the collisionless and hydrodynamic regimes in two-dimensional electron liquids

Iacopo Torre, Luan Vieira de Castro, Ben Van Duppen, David Barcons Ruiz, François M. Peeters, Frank H. L. Koppens, and Marco Polini

Phys. Rev. B 99, 144307 – Published 18 April 2019

Abstract

Hydrodynamic flow in two-dimensional electron systems has so far been probed only by dc transport and scanning gate microscopy measurements. In this work we discuss theoretically signatures of the hydrodynamic regime in near-field optical microscopy. We analyze the dispersion of acoustic plasmon modes in two-dimensional electron liquids using a nonlocal conductivity that takes into account the effects of (momentum-conserving) electron-electron collisions, (momentum-relaxing) electron-phonon and electron-impurity collisions, and many-body interactions beyond the celebrated random phase approximation. We derive the dispersion and, most importantly, the damping of acoustic plasmon modes and their coupling to a near-field probe, identifying key experimental signatures of the crossover between collisionless and hydrodynamic regimes.

Received 1 January 2019
Revised 25 March 2019
Corrected 21 June 2019

DOI:https://doi.org/10.1103/PhysRevB.99.144307

Physics Subject Headings (PhySH)

Optoelectronics Plasma oscillations Plasmonics Plasmons

Condensed Matter, Materials & Applied Physics

Corrections

21 June 2019

Correction: The ERC grant number contained an error and has been fixed.

Authors & Affiliations

Iacopo Torre^1,*, Luan Vieira de Castro^2,3, Ben Van Duppen², David Barcons Ruiz¹, François M. Peeters², Frank H. L. Koppens^1,4, and Marco Polini^5,6

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
²Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
³Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, Fortaleza, Ceará 60455-900, Brazil
⁴ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys 23, 08010 Barcelona, Spain
⁵Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
⁶School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

^*iacopo.torre@icfo.eu

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Vol. 99, Iss. 14 — 1 April 2019

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Images

Figure 1
Sketch of the $q - ω$ plane showing the relevant frequency and length scales for the problem at hand, and the plasmon dispersion (red and orange lines) for two different values of the screening parameter $Λ$ defined in Eq. (9). Red line: $Λ ≫ 1$ . Orange line: $Λ < 1$ . The blue solid line is the electron dispersion $ω = v_{F} q$ while the blue dashed line is the sound dispersion $ω = v_{F} q / \sqrt{2}$ (ignoring here many-body corrections). Different regimes of linear response are highlighted. In the hydrodynamic regime (blue shaded region) the Navier-Stokes equation (1) is applicable. In the overdamped regime (magenta shaded region), Eq. (1) is still applicable but plasmons are strongly damped. In the viscoelastic regime (green shaded region) Eq. (1) can still be applied considering a frequency-dependent complex viscosity [27, 34].
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Figure 2
(a) Screening parameter $Λ$ as a function of electronic density $n$ and spacer thickness $d$ for a single-layer graphene/hBN/metal heterostructure like the one used in Ref. [8]. Results in this figure have been obtained by setting ${\bar{ε}}_{z z} = 3.5$ and $Z = 0$ . Contour lines have been drawn for $Λ = 0.25$ (blue), $0.5$ (orange), and $0.25$ (green). (b) Same as in (a) but for bilayer graphene.
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Figure 3
AP phase velocity normalized to the Fermi velocity (a) and AP damping, normalized to the extrinsic damping $γ$ , (b) as functions of the frequency $f = ω / (2 π)$ , for different values of the screening parameter: $Λ = 0.25$ (blue), $0.5$ (orange), and 2 (green). Results in this figure have been obtained by setting $γ = 10^{12} s^{- 1}, γ_{ee} = 10^{13} s^{- 1}$ , and neglecting, for the sake of simplicity, many-body renormalizations by setting $v_{F}^{*} / v_{F} = K^{*} / K = D^{*} / D = 1$ . For each value of $Λ$ , the solid line denotes the result of the solution of $ε_{L} (q, ω) = 0$ , while the dashed (dash-dotted) line represents the asymptotic collisionless (hydrodynamic) result. The vertical black lines mark the frequency $2 π f = γ_{ee}$ around which the crossover occurs.
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Figure 4
Coupling efficiency $η_{z} (ω)$ as a function of frequency. Results in this figure refer to SLG separated from a metal gate by an hBN spacer of thickness $d = 4 nm$ , having ${\bar{ε}}_{x x} = {\bar{ε}}_{y y} = 6.68$ and ${\bar{ε}}_{z z} = 3.56$ . Squares correspond to excitation in the center of the spacer $z = - 2 nm$ , while circles correspond to $z = 10 nm$ , above SLG. Solid (dashed) lines represent the approximate result of Eq. (19) for excitation at $z = - 2 nm$ ( $z = 10 nm$ ), setting $Z = - 0.5$ . Different colors refer to different values of the screening parameter: $Λ = 0.25$ (blue), $0.5$ (orange), and 2 (green). All other parameters are as in Fig. 3.
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