## 1 Introduction  
Field emitters can be used as cathodoluminescence (CL) devices, which radiate rays with a wide range of frequencies from X-ray to far infrared rays. However, they have a serious problem in that the field emission (FE) current depends on the degree of vacuum, owing to the adhesion of contaminants such as residual gas or water on the field emitter surface [1; 2]. The emitter surface needs to be clean because the FE and carbon nanotubes (CNTs) are expected to enable field emitters to realize stable emission. However, the emission current from field emitters, including CNTs, has been reported to fluctuate even in a high-vacuum atmosphere [3; 4]. A CL device that uses a field emitter as a cathode electrode radiates rays that considerably fluctuate or flicker. The fluctuation or flicker from a CL device mainly has low frequencies of under 200 Hz [5; 6; 7; 8; 9; 10]. On the basis of the above problems, a cathode structure that can reduce the fluctuation of emission has been studied [9; 10]. However, a technique to reduce the fluctuation is not yet well established. In the case of FE, the electrons that are emitted from field emitters probabilistically and quantum-mechanically pass through a quantum barrier with a thickness of nm order before they are emitted in vacuum. This distance is estimated from the thickness of the electron-tunneling barrier when electrons cause Fowler-Nordheim tunneling [11; 12]. In the case of Fowler-Nordheim tunneling, electrons have been considered to inelastically pass through a barrier, and the electrons that are emitted from an FE cathode exhibit energy fluctuation.  
Therefore, we have carried out basic research to develop FE devices using single-walled CNTs (SWCNTs), and, for the first time, succeeded in employing SWCNT field emitters as the cathode of a planar lighting device with good emission stability at a high current density of over 10 mA/cm\({}^{2}\)[13; 14]. Since CNTs were first reported by [PERSON] [15], it has been shown that they exhibit one-dimensional confinement effects and can be used as coherent quantum wires [16; 17; 18]. Among the CNTs, the Young's modulus of SWCNTs is particularly high [19]. SWCNTs have been used in a wide range of applications, including field emitters, probes in scanning microscopes [20], gas-storage materials [21], and electrode materials for secondary batteries [22], and have been studied in a wide range of applied research fields. In recent studies of applications using CNTs, the development of electrical stability as field emitters requires the fabrication of efficient CL devices, such as flat panel displays with high brightness efficiency and low power consumption [23].  
We considered that an FE cathode comprising well-dispersed SWCNTs in a high-vacuum atmosphere is indispensable for decreasing the current fluctuation of field emitters and for improving the homogeneity of planar emission from a planar lighting device [13]. However, we consider that the FE fluctuation cannot be reduced to zero. In this study, to demonstrate the above consideration, the fluctuation effect on the properties of field emitters that use metallic SWCNTs with high purity and crystallinity was examined. We developed field emitters using these SWCNTs, and theoretically analyzed the fluctuation of their power spectrum in terms of their electrical conductivity by determining their electrical properties (FE properties).