Possible dimensionality transition behavior in localized plasmon resonances of confinement-controlled graphene devices

We investigated the dimensionality transition behavior of graphene localized plasmon resonances in confinement-controlled graphene devices. We first demonstrated a possibility of dimensionality transition, based on the devices carrier-density dependence, from a two-dimensional plasmon resonance to a one-dimensional plasmon one. We fabricated optical transparent devices and electrical transport devices on the same optical transparent wafer. These devices allow detailed control and analysis between carrier density and plasmon resonance peak positions. The carrier density from square root n (two-dimensional) to constant (one-dimensional) is consistent with the theoretical predictions based on the Dirac Fermion carriers in linear-band structure materials.


Introduction
The optical properties of low-dimensional carbon nanomaterials are attractive due to their unique behaviors in various frequency regions [1]. These materials show unique plasmonic behavior from far-infrared (FIR) to terahertz (THz) regions due to strongly confined-carrier dynamics [2][3][4][5][6][7][8][9][10][11]. Because of their high sensitivity to charge variations, surface modifications, and molecular adsorptions, plasmonic behavior is also important for highly efficient and sensitive optical components and detection devices [12]. Such behavior is also unique in dimensionality and carrier-density dependent [13]. It has recently been reported that the twodimensionally defined graphene plasmons show weaker dependence on carrier density as √ 4 , not √ 2 [7,10]. This behavior originated from truly mass-less Dirac Fermion natures in graphene devices. In the theoretical model, a graphene plasmon should show the disappearance of carrier density derivations in the one-dimensional plasmon resonance.
However, there has not yet been any experimental demonstration of one-dimensional plasmon resonance in well-defined samples.
We present the dimensionality dependence originating from the confinement differences defined by the channel width of resonant cavities. These low-dimensional graphene plasmons showed clear transition from two dimensions to one dimension. The vanishing carrier-density dependence of plasmon peak positions are consistent with the theoretical predictions based on the mass-less Dirac fermion descriptions. These results are important for the fundamental understanding of low-dimensional localized plasmons. Moreover, these carrier density robustness are also important for fabricating controllable plasmon devices based on low-dimensional materials. Figure. 1(a) shows a schematic picture of our device configuration. The single-layer graphene was transferred on a SiO2 (285 nm)/Si substrate. The conductivity of the Si substrate was carefully set to 10 Ωcm for FIR transparent measurement and back-gate configuration transport measurement on all device. The carrier density at each gate voltage was estimated from two parameters, Drude absorption in the FIR region and DCconductance difference from the Dirac point at a conductance minimum in each device. Figure. 1(b) shows the hand-crafted measurement sample holder for this study. The graphene devices were mounted on the Fourier transform infrared spectroscopy (FT-IR) sample holder with three electrodes for electric transport measurements with conventional back-gate configurations. These devices were set into a vacuumed FT-IR system (Bruker: Vertex 80v)   For precise estimation of the carrier density at each gate voltage from these optical absorptions, Drude behaviors are considered as optical responses due to the electric field modified by environmental correction factors [14,15]. The relationship between surface conductivity and optical transmittance is given by where ( − )⁄ is normalized transmittance, as mentioned above, is the speed of light, ∆σ is induced conductivity change due to the gate voltage, and is local field factors depending on the device configuration. The complex surface conductivity and local field factor are described as ∆σ = ∆ ′ + ∆ ′′ and = ′ + ′′ , respectively. Our graphene devices were fabricated on the SiO2/Si substrate with a 285-nm insulating layer.
Therefore, the electric field near the devices should be modified, and this local field factor is given by where d is the thickness of the SiO2 layer, k is the wave number of irradiated light, 2 is the complex refractive index of SiO2, and and represent the Fresnel reflection coefficient and transmission coefficient, respectively. In the above equation, the subscripts correspond to air (a), glass (g), and silicon (s). For example, is described as The relationship between normalized transmittance and surface conductivity is given by In the FIR regions, the imaginary part of each material used in this study is negligibly small.
Therefore, the last term in the above equation is also negligibly small in this measurement.
Finally, the real part of surface conductivity is estimated as where ′ is 0.46 for the FIR region in our device configurations calculated from Eq. (2). we can describe surface conductivity as the following Drude absorption formula where is the Drude weight and is the scattering rate. After unit conversion from angular frequency to frequency, the above equation is simply described as where C is a constant and is added for background correction. In Fig. 2(c), the fitting curves are indicated as solid curves at each ∆Vg. These fitting curves showed good agreement with the surface conductivity curves. We used = 6.13 × 10 14 2 ℎ ⁄ • −1 and Γ = 2πc × 10 2 −1 . This scattering rate was commonly reported for transferred CVD-growth graphene devices [14,16,17]. The relationship between carrier density and Drude weight is given by where is the Fermi velocity, is elementary charge, h is the Planck constant, and is the carrier density at each gate voltage. Figure.  The localized plasmon resonance of the 990-nm sample is indicated in Fig. 3(a). The peak intensity significantly increased by increasing carrier density. The peak position also clearly shifted to a higher frequency depending on the increase in carrier density. For the narrowest sample as shown in Fig. 3(b) (186 nm), the peak position was more stable against changing carrier densities. To estimate the transition point of this behavior, the medium-wide samples were also tested with the same measurement and analysis procedures. Figure.   However, only the narrowest sample had less dependence on gate voltage and carrier density.
For a more detailed comparison, the peak position is plotted as a function of √ 4 predicted as two-dimensional plasmons in Fig. 4(a). The peak positions are normalized by smallest carrier density value and the horizontal axis is also shown as differences of carrier density from smallest position. The widest sample (990 nm) showed almost completely linear dependence on √ 4 . These results strongly indicate that the localized plasmon resonance clearly originate from the mass-less Dirac Fermions in these resonance cavities. For the 186nm-wide sample, however, the slope of the peak shift was suddenly suppressed, and the value of the slope was less than half that of the other wider samples, as shown in Fig. 4(b).  Therefore, these CNT plasmons are robust against chemical doping, which is stronger than electrical back-gate control of carrier densities. In the case of CNTs, however, it is difficult to precisely control both confinement strength and carrier density. In graphene devices, the plasmon behavior is completely controllable by changing the sample dimensions and external electric field such as the gating effect. Therefore, this result is important for more controllable and higher efficient optical devices based on localized plasmon resonance.

Conclusion
We first demonstrated the possibility of dimensionality transition behavior of graphene localized plasmon resonances from two dimensional plasmon to a one-dimensional one.
From a detailed analysis based on carrier density, which is estimated from the Drude absorption behavior on the same wafer and graphene sheet, the wide devices showed √ 4 dependence on the peak shift of plasmon resonance. For the narrowest sample (186 nm), the carrier-density dependence was strongly suppressed. These behaviors are consistent with theoretical prediction based on the mass-less Dirac Fermion in the linear dispersion relationship. These results also indicate that we can control the carrier-density dependence and robustness against environment effects. This is important to fabricate highly efficient optical devices based on low-dimensional plasmons.