Terahertz and mid-infrared plasmons in three-dimensional nanoporous graphene

Two-dimensional (2D) graphene emerged as an outstanding material for plasmonic and photonic applications due to its charge-density tunability, high electron mobility, optical transparency and mechanical flexibility. Recently, novel fabrication processes have realised a three-dimensional (3D) nanoporous configuration of high-quality monolayer graphene which provides a third dimension to this material. In this work, we investigate the optical behaviour of nanoporous graphene by means of terahertz and infrared spectroscopy. We reveal the presence of intrinsic 2D Dirac plasmons in 3D nanoporous graphene disclosing strong plasmonic absorptions tunable from terahertz to mid-infrared via controllable doping level and porosity. In the far-field the spectral width of these absorptions is large enough to cover most of the mid-Infrared fingerprint region with a single plasmon excitation. The enhanced surface area of nanoporous structures combined with their broad band plasmon absorption could pave the way for novel and competitive nanoporous-graphene based plasmonic-sensors.

The conductivity model used in this work is based on two components. The plasmonic peak is modelled with a Lorentz oscillator, its real part being [2]: where B is the oscillator strength, ν pl the plasmon frequency and Γ pl its linewidth. The second component takes into account the interband electronic transition in graphene [3,4]: where T is the temperature while C defines the intensity of the interband transitions, and depends on both the thickness and the effective number of graphene layers in the film.
A spatial inhomogeneity of the doping level and then of the Fermi Energy E F , probably related to the nanostructuring of 3D graphene, is necessary to describe experimental data (see main manuscript). The Fermi energy inhomogeneity can be taken into account through a probability distribution P(E F ). An infrared spot size of a few millimeters in the far-field limit, corresponds mathematically to averaging Eq. 2 over P(E F ) through the equation An analytic expression of the previous integral can be obtained assuming for P(E F ) a flat distribution between two limiting values, E F1 and E F2 , which reads: The final expression of the optical conductivity, as analitically obtained from the previous Equation and then fitted to experimental data is: where A takes into account an absorption background also observed in single-layer graphene [5]. An example of fit through Eq. 5 is plotted in Fig. 1  In Supplementary Table I, we report the result of fitting the optical conductivity model to the experimental data reported in Fig. 2 and 4 of the main manuscript. NP and NPN are relative to undoped and N-doped samples, respectively. The II, III, IV and Vth columns report the pore size p, the plasmon frequency ν pl and widths Γ pl , the average Fermi energy E F and the corresponding statistical width. Let us note that data reported in Supplementary  Table I correspond to an average on several samples having, nominally, the same physical properties.

Supplementary Note 3: Variation of optical properties
Several samples with nominally identical pore-size and doping were measured from the same batch obtaining consistent results (see Supplementary Figure 3). By taking into account both the interband threshold variation, and the fitting procedure (see Supplementary Equation 5), a relative uncertainty of about 10 % on the average Fermi energy is finally obtained. The relative uncertainty on the plasmon characteristic frequency is on the same order of magnitude.
Supplementary Table I While plasmons in 2D graphene exhibit an extinction peak from a few percent up to 25%, plasmonic excitation measured in 3D nanoporous graphene show an enhanced extinction, with values greater than 95% (see Fig.1 of the main manuscript). This enhancement is ascribable both to the large effective number of layers that constitute the 3D graphene configuration and on NPG microscopic polarizability. In order to quantitatively discuss the plasmon strength, we will compare the real part of the optical conductivity plasmons in both single-layer and 3D NPG systems. The strength of the plasmon peak in the optical conductivity (i.e. its spectral weight), is also a measure of the corresponding polarizabilities.
In Supplementary Figure 4 we show both the NPG plasmon band (for the NPG sample characterized by p=200 nm and E F =260 meV, see Fig 2g of the main manuscript), and for a single layer plasmon peak representative of data available in literature (corresponding to an extinction peak of 12%, a linewidth of Γ=120 cm −1 , and a central frequency of 750 cm −1 ). The real part of the optical conductivity of these bands is normalized to their interband high frequency value σ HF .
The plasmon band in the optical conductivity provides a similar spectral weight (area under the plasmon peak) in both cases. This demonstrates that oscillator strengths of plasmons in NPG are comparable to that found in singlelayer graphene although distributed over a larger spectral window due to geometrical and doping inhomogeneity. The broader footprint of NPG plasmons could be a useful feature when considering applications in Surface Enhanced Infrared Absorption.

Supplementary Note 5: Optical Isotropy
The isotropy of the optical response has been checked by measuring the extinction spectra in the mid-IR through a linear infrared polarizer at different polarizations. The data yield no appreciable variations (<1%) for two extreme angles, as reported in Supplementary Figure 5. This is in full agreement with photoemission measurements on samples of the same batch [1], which also show an isotropic response.