## 4 Results and Discussion  
Figure 4 shows the relationship between the current density and electric field of the SWCNTs that were obtained using the FE measurement system (Figure 2). The threshold field of the metallic SWCNTs was 4.1 V/\(\upmu\)m at a line-current density of 0.1 mA/cm\({}^{2}\), as shown in Figure 4. Moreover, the planar lighting at 2 mA/cm\({}^{2}\) was shown in the inset of Figure 4.  
Figure 3: Primitive model of the FE using an SWCNT emitter based on Fowler–Nordheim tunneling.  
Figure 5 shows the FE current density of metallic SWCNTs with different supplied DC voltages over 40 s (a) and the distribution of the power spectrum of the frequency based on information on the FE current fluctuation, as obtained by the Fourier transform method (b). The sample was biased using a DC power supply, and the current density from the DC power supply was measured to be 1.2 mA/cm\({}^{2}\) at 5.2 V/\(\upmu\)m (6 kV), 2.0 mA/cm\({}^{2}\) at 5.5 V/\(\upmu\)m (6.6 kV), 4.6 mA/cm\({}^{2}\) at 5.9 V/\(\upmu\)m (7.1 kV), and 10 mA/cm\({}^{2}\) at 6.3 V/\(\upmu\)m (7.6 kV). We observed a high stability of the FE current, owing to the desorption of gases on the SWCNTs that was caused by the annealing before the FE measurement. We would have observed some damped current distributions in the FE current if the annealing of the sample had been insufficient. However, Figure 5a shows no damping, and the SWCNT surfaces were considered to have been sufficiently cleaned before the FE measurement. The spectrum of each FE current, as shown in Figure 5a, is normalized by the peak at 0 Hz, and indicates that periodic current fluctuation is present in the FE. A peak exists at 50 Hz, as shown in Figure 5b, originating from the noise of the DC power supply, which corresponds to the frequency of the commercial power supply in the east of Japan. We observed that the stable DC current in the FE images obtained from the power spectrum, as shown in Figure 5a, has an alternating-current component, particularly at low frequencies of under approximately 100 Hz. Moreover, the spectrum distribution depends on the magnitude of the FE current, and the alternating-current component of each spectrum increases with the DC power.  
Figure 4: Current density–electric-field characteristics of metallic SWCNTs. The inset shows the planar lighting.  
Figure 5: _Cont._  
The  
FE using metallic SWCNTs was thus successfully modeled as a phenomenon based on inelastic electron tunneling using metallic SWCNTs, as shown in Figure 6. Although many researchers have reported that field emitters realize Fowler-Nordheim tunneling with elastic electrical conductivity, no demonstrative experiments have been presented to the best of our knowledge. The current paper presents the first successful development of an inelastic-electron-tunneling model with the WKB approximation for metallic SWCNTs based on the evaluation of FE properties. The FE current from field emitters, for example, SWCNTs, fluctuates, and the \(I\)-\(V\) characteristics of FEs were examined by considering the power spectrum at \(\omega\) = 0 in Equation (13), which represents the magnitude of the current due to the electrons passing through a potential barrier from the SWCNTs, in accordance with Equation (13), constructed by the inelastic-electron-tunneling model.