Plasmon losses due to electron-phonon scattering: The case of graphene encapsulated in hexagonal boron nitride

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Plasmon losses due to electron-phonon scattering: The case of graphene encapsulated in hexagonal boron nitride

Alessandro Principi, Matteo Carrega, Mark B. Lundeberg, Achim Woessner, Frank H. L. Koppens, Giovanni Vignale, and Marco Polini

Phys. Rev. B 90, 165408 – Published 8 October 2014

Abstract

Graphene sheets encapsulated between hexagonal boron nitride (hBN) slabs display superb electronic properties due to very limited scattering from extrinsic disorder sources such as Coulomb impurities and corrugations. Such samples are therefore expected to be ideal platforms for highly tunable low-loss plasmonics in a wide spectral range. In this article we present a theory of collective electron density oscillations in a graphene sheet encapsulated between two hBN semi-infinite slabs (hBN/G/hBN). Graphene plasmons hybridize with hBN optical phonons forming hybrid plasmon-phonon modes. We focus on scattering of these modes against graphene's acoustic phonons and hBN optical phonons, two sources of scattering that are expected to play a key role in hBN/G/hBN stacks. We find that at room temperature the scattering against graphene's acoustic phonons is the dominant limiting factor for hBN/G/hBN stacks, yielding theoretical inverse damping ratios of hybrid plasmon-phonon modes of the order of 50–60, with a weak dependence on carrier density and a strong dependence on illumination frequency. We confirm that the plasmon lifetime is not directly correlated with the mobility: In fact, it can be anticorrelated.

Received 7 August 2014
Revised 16 September 2014

DOI:https://doi.org/10.1103/PhysRevB.90.165408

Authors & Affiliations

Alessandro Principi^1,*, Matteo Carrega^2,3, Mark B. Lundeberg⁴, Achim Woessner⁴, Frank H. L. Koppens⁴, Giovanni Vignale¹, and Marco Polini^2,5

¹Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
²NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
³SPIN-CNR, Via Dodecaneso 33, I-16146 Genova, Italy
⁴ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, E-08860 Castelldefels, Barcelona, Spain
⁵Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy

^*principia@missouri.edu

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Vol. 90, Iss. 16 — 15 October 2014

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Figure 1
The quantity $r (ω)$ as defined in Eq. (9) with $ɛ_{s} (ω)$ given in Eqs. (23) and (24). The parameters used in this plot are given in Table 1.
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Figure 2
(a) The RPA dispersion relation $ω_{p} (q)$ of the HPP mode in hBN/G/hBN, as obtained from the solution of Eq. (25). Units are clearly indicated in the axes. Note the three branches of the HPP collective mode and the two reststrahlen bands (shaded regions). (b) Illustrates the same quantity in the limit in which one neglects the uniaxial anisotropy of hBN by forcing $ɛ_{x} (ω) \to ɛ_{z} (ω)$ in Eq. (24) for all values of the illumination frequency $ω$ . In this case we see only two HPP branches.
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Figure 3
Feynman diagrams for the proper density-density response function ${\tilde{χ}}_{n n} (q, ω)$ at second order in the strength of the electron-phonon interaction. (a) Shows the vertex correction, while (b) shows one of the two time-reversal-conjugate self-energy insertions. The oriented solid line represents the bare electronic Green's functions, while the wavy line is the frequency-dependent effective electron-electron interaction $V (q, ω)$ . Finally, the filled circle represents a density vertex, while the cross stands for the electron-phonon interaction (21). Each wavy line therefore comes with two powers of $u_{q, ν}$ . Since the effective electron-electron interaction $V (q, ω)$ depends on frequency, the contribution of these diagrams does not vanish outside the particle-hole continuum, as one could naively expect from the resemblance of diagrams (a) and (b) to the diagrams at first order in the strength of the bare electron-electron interaction $v_{q}$ .
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Figure 4
(a) The inverse damping ratio $Q$ of the HPP mode in a hBN/G/hBN stack. In this plot we show the impact of acoustic phonon scattering (solid line). The quantity $Q$ is plotted as a function of carrier density $n$ and for a fixed illumination frequency $ω = 110 meV$ . As a comparison, we also plot the quantity $Q_{tr}$ , which is the inverse damping ratio $Q$ calculated from Eq. (11) by replacing $τ (q_{1}, ω)$ with the dc transport scattering time $τ_{tr} = {lim}_{ω \to 0} τ (0, ω)$ (dashed line). (b) The electron mobility in a hBN/G/hBN stack as a function of carrier density, as calculated from Eq. (33) by considering scattering of electrons against graphene's acoustic phonons. (c) The ratio $Q / Q_{tr}$ , as extracted from (a). (d) Same as in (a) but with $Q$ plotted as a function of the illumination frequency and for a fixed carrier density $n = 7.2 \times 10^{12} {cm}^{- 2}$ . Note the two “gaps” due to the hBN reststrahlen bands. All data in this figure have been calculated at $T = 300 K$ .
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Figure 5
The damping rate of HPP modes of Eq. (30), $1 / τ (q, ω)$ , due to the scattering against acoustic phonons. Different curves, plotted as a function of the frequency $ω$ (in eV), correspond to different values of the wave vector $q$ . In this plot the electron density is fixed at $n = 7.2 \times 10^{12} {cm}^{- 2}$ . Shaded regions correspond to reststrahlen bands, while solid dots indicate the frequency $ω = v_{F} q$ below which the RPA particle-hole continuum starts.
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Figure 6
(a) The inverse damping ratio of graphene due to the scattering with optical phonons of the substrate (solid line). The curve is plotted as a function of density $n$ in units of $10^{12} {cm}^{- 2}$ and for fixed illumination wavelength $λ = 10.6 μ m$ (corresponding to $ω = 110 meV$ ) and temperature $T = 0 K$ . Note that the contribution $Im [ɛ_{s} (ω)] / Re [ɛ_{s} (ω)]$ completely dominates the HPP damping rate. As a comparison we plot the inverse damping ratio obtained by removing the anisotropy of the substrate (dashed line). (b) Same as (a) but plotted as a function of the illumination frequency (measured in $eV$ ) and for fixed density $n = 7.2 \times 10^{12} {cm}^{- 2}$ and temperature $T = 0 K$ .
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