Nonlocal plasmonic response of doped and optically pumped graphene, ${\mathrm{MoS}}_{2}$, and black phosphorus

Nonlocal plasmonic response of doped and optically pumped graphene, ${MoS}_{2}$ , and black phosphorus

René Petersen, Thomas Garm Pedersen, and F. Javier García de Abajo

Phys. Rev. B 96, 205430 – Published 22 November 2017

Abstract

Plasmons in two-dimensional (2D) materials have emerged as a new source of physical phenomena and optoelectronic applications due in part to the relatively small number of charge carriers on which they are supported. Unlike conventional plasmonic materials, they possess a large Fermi wavelength, which can be comparable with the plasmon wavelength, thus leading to unusually strong nonlocal effects. Here, we study the optical response of a selection of 2D crystal layers (graphene, ${MoS}_{2}$ , and black phosphorus) with inclusion of nonlocal and thermal effects. We extensively analyze their plasmon dispersion relations and focus on the Purcell factor for the decay of an optical emitter in close proximity to the material as a way to probe nonlocal and thermal effects, with emphasis placed on the interplay between temperature and doping. The results are based on tight-binding modeling of the electronic structure combined with the random-phase approximation response function in which the temperature enters through the Fermi-Dirac electronic occupation distribution. Our study provides a route map for the exploration and exploitation of the ultrafast optical response of 2D materials.

Received 9 August 2017
Revised 26 October 2017

DOI:https://doi.org/10.1103/PhysRevB.96.205430

Physics Subject Headings (PhySH)

Surface plasmons

2-dimensional systems Graphene Phosphorene Transition metal dichalcogenides

Transient absorption spectroscopy Ultrafast pump-probe spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

René Petersen^1,2,*, Thomas Garm Pedersen^1,2, and F. Javier García de Abajo^3,4,†

¹Department of Physics and Nanotechnology, Aalborg University, DK-9220 Aalborg East, Denmark
²Center for Nanostructured Graphene (CNG), DK-9220 Aalborg East, Denmark
³ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
⁴ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys, 23, 08010 Barcelona, Spain

^*rp@nano.aau.dk
^†javier.garciadeabajo@nanophotonics.es

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Vol. 96, Iss. 20 — 15 November 2017

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Figure 1
Schematic band structure diagrams and electron populations in three different scenarios for parabolic semiconductor (left) and massless Dirac-fermion (right) 2D materials. (a) Undoped system at low temperature and zero doping. (b) Doped system at low temperature. (c) Optically pumped system with conduction (electrons) and valence (holes) subsystems at elevated temperatures in instantaneous thermal equilibrium, with chemical potentials $μ_{e}$ and $μ_{h}$ , respectively. The systems are probed with a light pulse of energy and momentum $ℏ ω$ and $ℏ q$ .
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Figure 2
(a)–(c) Band structure (left) and electronic density of states (right) for graphene, ${MoS}_{2}$ , and BP, calculated using tight-binding models (see main text). (d)–(o) Optical dispersion diagrams calculated in the nonlocal-RPA for the selected 2D materials. For BP, we consider $q$ oriented along either $Γ X$ or $Γ Y$ directions. For all materials, we show the dispersion at room temperature without doping (d)–(g), at room temperature with high doping (h)–(k), and at high temperature (200 meV $\approx 2300$ K) under different doping conditions (l)–(o). The chemical potential in (h)–(k) is chosen to yield a similar plasmon dispersion relation as that obtained through optical pumping at the temperatures selected in (l)–(o). The dashed lines correspond to the energies chosen for the Purcell factor calculations in Fig. 6.
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Figure 3
Chemical potential $μ^{eq}$ required to obtain a plasmon dispersion relation at room temperature equivalent to that observed as a function of temperature $T$ for several fixed values of the electron chemical potential $μ_{e}$ (see labels). Results for $μ_{e} = E_{c}$ (the conduction band edge) are shown as green thick curves in all four panels. All results are obtained within the local-RPA approximation.
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Figure 4
Imaginary part of the Fresnel reflection coefficient for p-polarization $r_{p}$ as obtained in the local-RPA (dashed curves) and the nonlocal-RPA (solid curves) for a ${MoS}_{2}$ atomic monolayer. We compare results with either doping and heating (red curves, $μ_{e} = 925$ meV, $k_{B} = 200$ meV) or only doping (blue curves, $μ = 1116$ meV, room temperature). Vertical lines mark the integration limit $q_{cut}$ that accounts for 90% of the Purcell factor at various dipole-surface distances (top labels in Å, see text).
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Figure 5
Spectral dependence of the Purcell factor for a dipole emitter oriented perpendicularly to the surface and placed at a distance of 1 nm from monoatomic graphene, ${MoS}_{2}$ , and BP layers. For BP, we present an average of the Purcell factor for the two independent directions considered in Fig. 2 (see discussion in text). We compare local-RPA (dashed curves) and nonlocal-RPA (solid curves) calculations for the material either at room temperature and high doping (blue curves) or at high temperature (red curves).
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Figure 6
Emitter-surface distance dependence of nonlocal effects in the Purcell factor. We compare local-RPA and nonlocal-RPA calculations with three different photon emission energies, chosen at the valley and peak values of the Purcell factor marked by arrows in Fig. 5, as well as at an energy in which interband transitions are the dominant contribution to absorption for each of the materials under consideration.
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Physical Review B

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Nonlocal plasmonic response of doped and optically pumped graphene, ${MoS}_{2}$ , and black phosphorus

René Petersen, Thomas Garm Pedersen, and F. Javier García de Abajo

Phys. Rev. B 96, 205430 – Published 22 November 2017

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Physical Review B

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Nonlocal plasmonic response of doped and optically pumped graphene, MoS2, and black phosphorus

René Petersen, Thomas Garm Pedersen, and F. Javier García de Abajo

Phys. Rev. B 96, 205430 – Published 22 November 2017

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Nonlocal plasmonic response of doped and optically pumped graphene, ${MoS}_{2}$ , and black phosphorus