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

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 [2–11]. A plasmon is the collective carrier excitation behavior due to an external electric field. In the conventional metal nanoparticles, the localized plasmon is occurred depending on their shapes (aspect ratio (length/diameter)) and carrier densities [12, 13]. On the other hand, in the carbon nanomaterials, the plasmonic behaviors are more directly depending on the geometry of carrier conduction channel. In the carbon nanotubes (CNTs), this strongly confined one-dimensional material shows clear length-dependent plasmonic behaviors [14–17]. The patterned graphene also shows the geometrical (namely, length and width) dependent plasmonic behaviors [7, 9, 11, 18–21]. 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 [22–24].

Such behavior is also unique in dimensionality and carrier-density dependent [25]. In the noninteracting and long-wavelength condition, the polarizability is simply calculated by using linear energy dispersion relationship. Then, the plasmon frequency is also easily calculated as dimensionality depending forms. It has recently been reported that the two-dimensionally defined graphene plasmons show weaker dependence on carrier density as $\sqrt[4]{n},$ not $\sqrt[2]{n}$ [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 DC-conductance 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 reducing the gas-adsorption effect on the transport properties. Figure 1(c) shows atomic force microscopy (AFM) images of one of our devices. In the literatures [7, 11, 18–21], the many works done by the nano-ribbon structures connected external electrodes. In these structures, a cavity itself has the function of carrier injection/ejection to the plasmon resonance. Therefore, it is difficult to control the strength of confinement. In this time, we adopted the geometry, which has the carrier injection line connected to the center of cavity avoiding the disturbance the carrier concentration near the resonance edge. This geometry enables to control the strength of confinement by changing the width of devices. The carrier density is also continuously controlled by using the carrier injection line and applying gate voltage to back gate electrode. These devices were fabricated using electron beam lithography and conventional lift-off process. We used the same resonance length of 1.0 μm but varied the widths from 186 to 990 nm. The incident light polarization was parallel to the cavity-length directions. Therefore, the plasmon resonance should also be excited in the length direction.

Zoom In Zoom Out Reset image size

Figure 2(a) shows the electrical transport properties depending on the back-gate voltage (V_g ) at room temperature. In air, the charge neutral point (CNP) shifted to a positive higher gate voltage due to gas adsorptions, such as oxygen, onto the graphene surface. The blue curves show the transport properties under vacuum. These curves showed a clear minimum point (V_CNP ) of the source-drain current (maximum resistance) corresponding to the Dirac point in our devices. Figure 2(b) shows the normalized transmittance corrected by the spectra at V_CNP (T_CNP ). In these spectra, the Drude absorption behaviors were clearly observed in both gate-voltage directions with non-patterned regions. Each gate voltage was corrected as effective gate voltages (∆Vg) estimated from the differences between applied gate voltages and V_CNP .

Zoom In Zoom Out Reset image size

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 [26, 27]. The relationship between surface conductivity and optical transmittance is given by [26]

$$\begin{eqnarray}&&-\left(T-{T}_{CNP}\right)/{T}_{CNP}\left(\omega \right)=4\pi /c\cdot Re\left\{{\rm{\Delta }}\sigma \left(\omega \right)\times L\right\},\end{eqnarray} \tag{ 1 }$$

where $\left(T-{T}_{CNP}\right)/{T}_{CNP}$ is normalized transmittance, as mentioned above, $c$ is the speed of light, ${\rm{\Delta }}{\rm{\sigma }}$ is induced conductivity change due to the gate voltage, and $L$ is local field factors depending on the device configuration. The complex surface conductivity and local field factor are described as ${\rm{\Delta }}{\rm{\sigma }}={\rm{\Delta }}{\sigma }^{{\prime} }+i{\rm{\Delta }}{\sigma }^{{\prime} ^{\prime} }$ and $L={L}^{{\prime} }+i{L}^{{\prime} ^{\prime} },$ 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 [26]

$$\begin{eqnarray}&&L=1+{r}_{ag}+{t}_{ag}{r}_{gs}{t}_{ga}exp\left(2ik{n}_{Si{O}_{2}}d\right)/\left(1-exp\left(2ik{n}_{Si{O}_{2}}d\right){r}_{gs}{r}_{ga}\right),\end{eqnarray} \tag{ 2 }$$

where d is the thickness of the SiO₂ layer, k is the wave number of irradiated light, ${n}_{Si{O}_{2}}$ is the complex refractive index of SiO₂, and $r$ and $t$ 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, ${r}_{ag}$ is described as $\left({n}_{air}-{n}_{Si{O}_{2}}\right)/\left({n}_{air}+{n}_{Si{O}_{2}}\right).$ The relationship between normalized transmittance and surface conductivity is given by combination of equations (1) and (2) as:

$$\begin{eqnarray}&&\begin{array}{l}-\left(T-{T}_{CNP}\right)/{T}_{CNP}\left(\omega \right)=4\pi /c\cdot Re\left\{{\rm{\Delta }}\sigma \left(\omega \right)\times L\right\}\\ =4\pi /c\cdot \left({\rm{\Delta }}{\sigma }^{^{\prime} }\left(\omega \right){L}^{^{\prime} }-{\rm{\Delta }}{\sigma }^{{\prime} ^{\prime} }\left(\omega \right){L}^{{\prime} ^{\prime} }\right).\end{array}\end{eqnarray} \tag{ 3 }$$

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

$$\begin{eqnarray}&&{\rm{\Delta }}{\sigma }^{{\prime} }\left(\omega \right)=c/4\pi {L}^{{\prime} }\cdot \left\{-\left(T-{T}_{CNP}\right)/{T}_{CNP}\left(\omega \right)\right\},\end{eqnarray} \tag{ 4 }$$

where ${L}^{{\prime} }$ is 0.46 for the FIR region in our device configurations calculated from equation (2). Figure 2(c) shows the real part of surface-conductivity difference as a function of frequency (s⁻¹). The surface conductivity is clear due to Drude absorption phenomena. Consequently, we can describe surface conductivity as the following Drude absorption formula [26]

$$\begin{eqnarray}&&{\rm{\Delta }}{\sigma }^{{\prime} }\left(\omega \right)=Re\left\{\sigma -{\sigma }_{CNP}\right\}=Re\left\{iD/\pi \left(\omega +i{\rm{\Gamma }}\right)-i{D}_{CNP}/\pi \left(\omega +i{{\rm{\Gamma }}}_{{\rm{CNP}}}\right)\right\},\end{eqnarray} \tag{ 5 }$$

where $D$ is the Drude weight and ${\rm{\Gamma }}$ is the scattering rate. After unit conversion from angular frequency to frequency, the above equation is simply described as

$$\begin{eqnarray}&&{\rm{\Delta }}{\sigma }^{{\prime} }\left(f\right)=D{\rm{\Gamma }}/\pi \left(4{\pi }^{2}{f}^{2}+{{\rm{\Gamma }}}^{2}\right)-{D}_{CNP}{{\rm{\Gamma }}}_{CNP}/\pi \left(4{\pi }^{2}{f}^{2}+{{\rm{\Gamma }}}^{2}\right)+C,\end{eqnarray} \tag{ 6 }$$

where C is a constant and is added for background correction. In figure 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 ${D}_{CNP}$ = $6.13\times {10}^{14}{e}^{2}/h\cdot {s}^{-1}$ and ${{\rm{\Gamma }}}_{CNP}$ = $2{\rm{\pi }}c\times {10}^{2}\,{s}^{-1}.$ This scattering rate was commonly reported for transferred CVD-growth graphene devices [18, 20, 26]. The relationship between carrier density and Drude weight is given by

$$\begin{eqnarray}&&{\rm{D}}={V}_{F}{e}^{2}/h\sqrt{\pi \left|n\right|},\end{eqnarray} \tag{ 7 }$$

where ${V}_{F}$ is the Fermi velocity, $e$ is elementary charge, h is the Planck constant, and $n$ is the carrier density at each gate voltage. Figure 2(d) shows the estimated carrier density and scattering rate as a function of an effective gate voltage. These values are also in good agreement with a simple theoretical prediction curve based on the parallel plate capacity model $\left({\rm{n}}=CV/e\right).$

The localized plasmon resonance of the 990-nm sample is indicated in figure 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 figure 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 3(c) indicates the peak position as a function of the effective gate voltages in each sample. The 330-to 990-nm samples had almost the same slope dependence on the changing carrier density. However, only the narrowest sample had less dependence on gate voltage and carrier density.

Zoom In Zoom Out Reset image size

For a more detailed comparison, the peak position is plotted as a function of $\sqrt[4]{n}$ predicted as two-dimensional plasmons in figure 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 $\sqrt[4]{n}.$ These results strongly indicate that the localized plasmon resonance clearly originate from the mass-less Dirac Fermions in these resonance cavities. For the 186-nm-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 figure 4(b). These unique behaviors are clearly explained with the model suggested by Sarma and Hwang [25]. In this theory, the dimensionality of plasmon frequency was estimated under the condition of long-wavelength limit. In our case, the resonance peaks are existed around 150 cm⁻¹, and these wavenumber are corresponding to the 66.7 μm wavelength. The plasmon cavity length is 1.0 mm. Therefore, the cavity length is enough short for satisfaction the long-wavelength conditions. For more understanding the dimension transition, the relationship between confinement strength and Fermi wavelength is important to realize the quantized carrier motions in one dimension [28, 29]. In graphene devices, the quantized conductance in a quantum wire configuration is observed in width- and carrier-density-controllable devices [30, 31]. For the narrowest sample in one of these studies (230 nm), the quantized conductance was observed in a wide carrier density range from the CNP to $1.5\times {10}^{12}$ cm⁻² at 4.2 K [31]. In our narrowest device (186 nm), the carrier density modulated from $2.6\times {10}^{11}$ cm⁻² to $1.0\times {10}^{13}$ cm⁻². The quantized conductance might be less than ten channels with this device under the lowest carrier density. These estimations are consistent with our dimensionality transition observations. Similar behavior was recently observed in carbon nanotube (CNT) plasmon resonances [14–16]. These CNTs had strongly confined one-dimensional carriers due to naturally rolled-up structures of less than 10 nm in diameter. 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.

Zoom In Zoom Out Reset image size

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 $\sqrt[4]{n}$ 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.

The data that support the findings of this study are available upon reasonable request from the authors.