Enhancing two-photon spontaneous emission in rare earths using graphene and graphene nanoribbons

Colin Whisler, Gregory Holdman, D. D. Yavuz, and Victor W. Brar

Phys. Rev. B 107, 195420 – Published 10 May 2023

Abstract

The enhancement of two-photon spontaneous emission (2PSE) from trivalent and divalent rare earth ions in proximity to graphene and graphene nanoribbons is calculated for achievable experimental conditions using a combination of finite difference time domain simulations and direct computation of transition rates between energy levels in rare earths. For ${Er}^{3 +}$ , we find that the 2PSE rate is initially 8 orders lower than the single-photon spontaneous emission rate but that, with enhancement, 2PSE can reach 2.5% of the overall decay. When graphene nanoribbons are used, we also show that the emission of free-space photon pairs from ${Er}^{3 +}$ at 3–3.2 $μ m$ via 2PSE can be increased by $\sim 400$ . Our calculations show significantly less relative graphene-enhanced 2PSE than previous works, and we attribute this variation to differences in emitter size and assumed graphene mobility. We also show that the internal energy structure of the ion can have an impact on the degree of 2PSE enhancement achievable and find that divalent rare earths are more favorable.

Received 2 January 2023
Revised 24 March 2023
Accepted 14 April 2023

DOI:https://doi.org/10.1103/PhysRevB.107.195420

Physics Subject Headings (PhySH)

Electronic structure of atoms & molecules Spontaneous emission Surface plasmons

Graphene Nanoribbon Rare-earth ions

Finite-difference time-domain method

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Colin Whisler, Gregory Holdman, D. D. Yavuz, and Victor W. Brar ^*

Department of Physics, University of Wisconsin–Madison, 1150 University Avenue, Madison, Wisconsin 53706, USA

^*vbrar@wisc.edu

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Vol. 107, Iss. 19 — 15 May 2023

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Images

Figure 1
Left: schematic of an emitter beneath a graphene nanoribbon, with single-transition decay shown in blue and 2PSE shown in red. Right: energy levels for the processes involved in two-photon spontaneous emission for ${Er}^{3 +}$ . An electron relaxes from the ${}^{4}I_{13 / 2}$ state to the ${}^{4}I_{15 / 2}$ state, but the matrix elements that determine the process's strength are defined by the electric dipole transitions to the higher $4 f^{10} 5 d$ and $4 f^{10} 6 s$ states.
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Figure 2
Results for a single-dipole orientation. All results consider an ${Er}^{3 +}$ emitter 5 nm away from graphene structures, with the dipole moment oriented along the short dimension of the ribbons (or parallel to the sheet). (a) Purcell enhancement for the emitter near 20 nm graphene ribbons. Horizontal lines correspond to the Fermi energies plotted in (b), while vertical lines show the doubly resonant frequencies for photon emission. Electric field profiles at $E_{F} = 0.385$ eV are shown below, calculated at a lateral distance of 7 nm along the ribbon with the field strength normalized to the value with no graphene present. (b) Differential 2PSE enhancement $γ / γ_{0, max}$ for the emitter near graphene ribbons for doubly resonant Fermi energies (pink dotted and orange solid lines) and a nonresonant Fermi energy (blue dashed line). (c) Integrated 2PSE enhancement $Γ / Γ_{0}$ for the emitter near 20 nm wide graphene ribbons. (d) Differential 2PSE enhancement $γ / γ_{0, max}$ at $E_{F} = 0.565$ eV for the emitter with no graphene present (pink dotted line), modified by graphene ribbons (blue dashed line) and modified by a graphene sheet (orange solid line). Differential 2PSE enhancement $γ / γ_{0, max}$ for the emitter near (e) graphene ribbons or (f) a graphene sheet. The results in (b), (d), (e), and (f) are normalized by the maximum value of free-space differential 2PSE.
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Figure 3
Comparison of single-transition decay and two-photon emission. All results consider an ${Er}^{3 +}$ emitter 5 nm away from graphene structures, with all dipole orientations averaged according to Eq. (4). (a) Enhancement $Γ / Γ_{0}$ of total two-photon emission relative to a dipole with no graphene present. (b) Enhancement of single-transition decay at 1538 nm relative to a dipole with no graphene present. (c) Total single-transition decay rate compared to total two-photon emission rate in the presence of 20 nm ribbons and a continuous graphene sheet. (d) Fraction of total decay rate that can be attributed to 2PSE for 20 nm graphene ribbons.
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Figure 4
Enhancement of two-photon emission. All results consider an ${Er}^{3 +}$ emitter 5 nm away from graphene structures, with all dipole orientations averaged according to Eq. (4). (a) Enhancement of purely radiative (photon-photon) emission. (b) Enhancement of purely radiative (photon-photon) emission in the 3–3.2 $μ m$ range.
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Figure 5
Comparison of total single-transition decay and 2PSE rates for (a) ${Sm}^{2 +}$ and (b) ${Tm}^{2 +}$ when modified by 20 nm graphene ribbons. All results consider an emitter 5 nm away from graphene structures, with all dipole orientations averaged according to Eq. (4).
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