# Temperature Comparison of Looped and Vertical Carbon Nanotube Fibers during Field Emission
## 3 Results and Discussion
### Temperature Distribution Measurement and Analysis  
Due to Joule heating, the temperature of the CNT fibers will be increased. The temperature distributions of both the looped and single vertical CNT fibers were measured directly with an IR camera at the emission current of 3 mA, as shown in Figure 6. It is important to note that even though a higher applied voltage is needed to achieve 3 mA emission current for the looped CNT fiber, its maximum temperature is only around 300 \({}^{\circ}\)C, which is approximately half of the maximum temperature of about 600 \({}^{\circ}\)C in the single vertical CNT fiber. The location of the maximum temperature is very close to the top for the looped fiber emitter, whereas for the vertical fiber emitter, besides the hotspot near the emitter tip, there is a second hotspot around 1 mm below the tip. It is important to note that, at a given current level, the exact location of the hottest spot depends on the following two factors. The first one is the local emission site, which is related to surface defects, roughness, and/or dangling fibrils along the CNT fiber. The other factor is the self-consistent coupled electrical-thermal conduction along the fiber, which is governed by Equation (2) below. For the looped fiber, both ends are attached to the same substrate with the same temperature and thus, by symmetry, the maximum temperature occurs at the middle point (i.e., the tip) of the looped fiber. For the vertical fiber, Joule heating produces excessive heat inside the fiber, which can only be conducted away  
Figure 5: The Fowler–Nordheim (FN) plot of the FE data in Figure 4, for both the single vertical CNT fiber and the looped CNT fiber emitters. (ineffectively and unsymmetrically as compared to the looped configuration) from the two ends of the fiber, resulting in heat built up inside the fiber, with a maximum temperature that is located apart from its ends. Emission stability was measured for both emitters, as shown in Figure 7. The emission current was measured for 120 min at the current level of 0.3 mA for the vertical CNT configuration, while the looped fiber emission was monitored for 300 min under the current level of 1.5 mA. We observed the long-term stability of both of the emission configurations. Figure 6: Optical image of the (**a**) looped CNT fiber and (**b**) single vertical CNT fiber. Temperature distribution of the (**c**) looped CNT fiber and (**d**) vertical CNT fiber, obtained from the IR camera. Figure 7: Long-term stability measurement for both the vertical and looped CNT fiber emitters. To better understand Joule heating and the thermal properties of the CNT fiber emitters, the temperature dependent electrical (\(\sigma\)) and thermal (\(\kappa\)) conductivities of the CNT fiber (sample from the same fabrication batch as those used in the emitters) were measured using the 3 Omega method, as shown in Figure 8. As temperature \(T\) increases, the electrical conductivity decreases linearly; whereas the thermal conductivity decreases strongly before \(T=500\) K and then gradually saturates afterwards. We next compared the experimental measured temperature distribution with a recent theory on the Joule heating of a one dimensional conductor, with the above measured temperature dependent electrical and thermal conductivities as inputs [27]. From the steady-state heat conduction equation and the electrical current continuity equation, we have [27]  
\[\frac{d}{ds}\bigg{(}\kappa(T,s)\frac{dT}{ds}\bigg{)}+\frac{J_{c}^{2}}{\sigma(T, s)}=0, \tag{2}\]  
where \(J_{c}=\sigma dV/ds=\) constant is the current density in the conductor, which satisfies \(\int_{0}^{L}(J_{c}/\sigma)ds=V_{0}\), where \(s\) is the position along the CNT fiber, \(L\) is the total length (half length) of the single vertical fiber (looped fiber), and \(V_{0}\) is the voltage drop along the fiber from the top of the emitter to the cathode base, and is much smaller than the gap voltage. For simplicity, we assume \(L\) to be the same as the height of the emitter, \(L=h\) (Figure 2).