Conductive Effect of Increased Crystallinity of Single-Walled Carbon Nanotubes as Field Emitter

2.1 Power spectrum in the inelastic tunneling model

In the inelastic tunneling model, the energy states of electrons passing through an energy barrier can be estimated by the energy transfer before and after electron tunneling. The power spectrum with the flow of FE current for the inelastic tunneling model was proposed by Kirtley et al. [24], Bardeen [25], and Watanabe et al. [26] based on Eqs. (12) and (13):

where ρ(E) is the state density of electrons in one-dimensional space, and ni^h (h = l (for left region) or r (for right region)) is the occupation number of electrons within state i (i = k or q).

We attempted to obtain the direct-current (DC) component of the power spectrum of the inelastic electron tunneling model using a rectangular potential barrier and Eq. (13). As mentioned above, the electrons passing through the CNTs without crystal defects are ballistic [5]. The CNTs with crystal defects are reported to have an energy bandgap that depends on the position of the defects, and electrons rarely pass through CNTs without energy loss. In order to analyze the electrical conductivity of SWCNTs, we attempted to obtain the tunneling matrix Mkq and the power spectrum of inelastic electron tunneling when electrons inelastically pass through the local area corresponding to crystal defects. The conductivity of electrons passing into SWCNTs with crystal defects was explained employing the electron tunneling model below.

For Mkq of an energy barrier having a rectangular waveform, the state of wave function transfers that in the right region (r) to the left region (l) of the energy barrier or vice versa, φrq is the wave function having wavenumber q localized in the right region of the barrier called as wave-B, and φlk is the wave function having wavenumber k localized in the left region of the barrier called as wave-A. The Fermi level of an SWCNT is represented by E_F. When electrons pass through the rectangular potential barrier from the left region to the right region, the wave function shifts from state k to state q, and the tunneling matrix Mkq is given by:

where V(z) is the potential height of the energy barrier having a rectangular distribution on the z axis shown in Figure 2, and m is the effective mass of an electron.

The wave functions φk and φq localized in the left region (wave-A) and in the right region (wave-B) are represented by:

where S is the area in which electrons pass through an SWCNT; and k_||, q_||, and x_|| are the components, along the direction perpendicular to the z axis, of the wavenumber k, wavenumber q, and position x, respectively. The wave functions χk and χq have coefficients An, Bn and Cn, Dn, respectively, and are defined as:

The energy transfer between waves-A and -B, ħω, is included with the wavenumber in Eq. (16), and it corresponds to the energy loss of an electron passing through the energy barrier from the left region to the right region:

Using Eqs. (16) and (17), Mkq can be rewritten as:

The power spectrum of inelastic electron tunneling, |I_in(ω)|², is obtained from Eqs. (13)–(18). The power spectrum at ω = 0 represents the magnitude of the direct current of the power spectrum |I_in(ω)|².

2.2 Power spectrum in F-N tunneling regime passing into an SWCNT with crystal defect

The thickness of the rectangular energy barrier and the energy transfer which means the energy difference between the left region and the right region are important factors in this simulation. Results of research on tunnel junction devices indicate that the thickness of the rectangular potential barrier for inelastic electron tunneling is only ∼3 nm [20]. The energy loss caused by the tunneling of electrons through the energy barrier depends on the bandgap associated with the crystal defects [27]. The theoretical value of difference in the energy between waves A and B is calculated by fitting to be 0.027 eV, which corresponds to a temperature of 313 K for the energy loss observed when the electrons pass through SWCNTs with crystal defects.

Eq. (12) should be transformed into a more suitable form for a field-emission cathode using metallic SWCNTs. Field-emission measurements were carried out in the F-N tunneling regime, and a further simplified form is expressed as follows:

where ϕB, shown in Figure 3, is the barrier-height potential that indicates the difference between the vacuum-level energy and the energy at the bottom of the Fermi level of metallic SWCNTs; and Eis the applied electric field on the SWCNTs enhanced by the tip shape. The enhancement factor β of Ecan be calculated by the surface-charge method of simulation using a three-dimensional SWCNT model that protrudes from an indium-tin oxide (ITO) film [28, 29, 30]:

where E0represents the real electric field between the cathode and anode. Moreover, the area of the field-emission site A on the SWCNTs of the cathode was estimated from a comparison between the simulated and measured bright spots [30].

4.1 Field-emission characteristics of hc-SWCNTs

Tunneling electron microscopy and scanning electron microscopy were used to obtain images of the dispersed hc-SWCNT bundles and the ITO film which was subjected to activation treatment. As one can clearly see in the TEM image in Figure 6(a), the hc-SWCNT bundles were well dispersed in the liquid medium. Meanwhile, the SWCNT bundles shown in the SEM image in Figure 6(b) were exposed in the scratched grooves on the activated ITO film and marked within the white circles; the dispersed SWCNTs in a scratched area are also illustrated with the red circles indicating the ends of the SWCNT bundles on the right. The number of dispersed and exposed SWCNTs per unit area could be controlled via the density of the SWCNTs. Table 1 shows the summary of the state of the exposed SWCNTs in terms of (1) the number of SWCNTs protruding from the grooved walls of the scratched ITO film per unit area, (2) the mean diameter of the exposed SWCNT bundles, and (3) the mean distance between neighboring SWCNT bundles for two types of SWCNTs, i.e., SWCNTs with crystal defects and hc-SWCNTs. The states of exposure of the two types of SWCNTs fabricated by our wet-coating process were similar, and they were homogeneously dispersed in the films.

Figure 7 shows the relationship between the current density and electric field for the SWCNTs, obtained using the system for FE measurements (Figure 5). The threshold field decreased from 2.01 V μm⁻¹ for the SWCNTs with crystal defects to 1.07 V μm⁻¹ for the hc-SWCNTs. As shown in Table 1, the dispersion density of SWCNTs and the mean diameter of SWCNT bundles in the films were similar for the two types of the SWCNTs. It means that the probability densities of SWCNTs that induced FE were similar for the two types of SWCNTs. Therefore, it can be discussed and assumed that the above improvement in electrical properties was caused by the increase in crystallinity of the SWCNTs.

Figure 8 shows the distribution of bright spots in the hc-SWCNTs and SWCNTs with crystal defects. As can be seen in Figure 7, the threshold and the driving field depended on the crystallinity of the SWCNTs. When the applied field was controlled so that the current density was uniform, the densities of the high- and low-brightness spots on the measured area were almost the same between the hc-SWCNTs and the SWCNTs with crystal defects, respectively. Previous reports showed that in the SWCNTs synthesized by arc discharge, SWCNTs with metallic and semiconductive properties coexisted [31]. Therefore, the high-brightness spots can be attributed to the FE from metallic SWCNTs, and the low-brightness spots can be attributed to the FE from semiconductive SWCNTs; these phenomena and their origins are not discussed in this chapter. When the crystallinity of SWCNTs increased, the chirality of the carbon sheets making up the SWCNTs did not change, and the mixed ratio of metallic to semiconductive SWCNTs remained unchanged [32, 33, 34, 35, 36, 37].

The results in Figure 6 and Table 1 indicate that the diameter, protruded length, and dispersion density of the SWCNT bundles protruding from the grooved ITO film were similar for the two types of the SWCNTs as FE electron sources. However, the difference in the plots of current density versus electric field of Figure 7 is impossible to be analyzed only by the exposed uniformity of the dispersed SWCNTs on the activated ITO film. Therefore, we surmised that the difference in the crystallinity of SWCNTs caused the difference in electric properties of SWCNTs. Fransen et al. and Nilsson et al. find that the interruption of carbon network of an SWCNT with crystal defects originates a local energy band that affects the electrical conductivity by scanning tunneling microscopy (STM) [22, 23]. On the basis of the above results, an electric-physical model in an SWCNT was developed from the inelastic electron tunneling model. Therefore, the energy difference originated by the energy bandgap yielding from the interruption of carbon network in an SWCNT with crystal defects is surmised to become an energy barrier which impedes the electrons pass through the SWCNT.

Figure 9 shows the FE properties obtained by experiment in Figure 7 and simulation. For the hc-SWCNTs, the results obtained by the experiment are represented by red circles and those obtained by simulation using Eq. (12) at ω = 0 of the F-N tunneling model without inelastic tunneling model are represented by the yellow dashed line. Moreover, the experimental results of the SWCNTs with crystal defects are represented by blue circles and those obtained by the following method are represented by the dark-red dashed line. We can find the fitting between experiment and simulation is in good agreement. The simulation results represented by the dark-red dashed line were obtained by combining the F-N tunneling model of electrons emitted from the end of the SWCNT with the power spectrum with inelastic tunneling model given by Eqs. (7)–(12) at ω = 0. Table 2 shows the summary of the parameters employed in the theoretical calculations on the basis of F-N tunneling and inelastic electron tunneling. The total electron emission site area (α) and the field-enhancement factor (β) for FE are important parameters in evaluating the FE characteristics. These two parameters were the same for the two types of the SWCNTs.

The inelastic electron tunneling through the SWCNTs with crystal defects, as shown in Figure 10, is a phenomenon based on the energy loss associated with the electron transfer in the SWCNTs. Although many researchers have reported that CNTs exhibit ballistic electrical conductivity, such property has not been demonstrated in any experiment that we know of. We can express successfully the first report of the development using an inelastic electron tunneling model for an SWCNT with crystal defects based on the evaluation of FE properties. The hc-SWCNTs are suitable materials for electronic devices to save energy because the energy loss associated with electron transfer in the SWCNT is suppressed.

4.2 Current fluctuations of hc-SWCNTs as field emitters: theoretical predictions versus experimental results

Figure 11 shows the relationship between the current density and electric field for the SWCNTs obtained using the FE-measurement system shown in Figure 5. The threshold field of the metallic SWCNTs was 4.1 V μm⁻¹ at a line-current density of 0.1 mA cm⁻², as shown in Figure 11. Moreover, the planar lighting observed at 2 mA cm⁻² was homogeneous, as shown in the inset in Figure 11.

Figure 12(a) shows the stability of the FE current of metallic SWCNTs when different values of DC voltage were supplied for 50 s; Figure 12(b) shows the power spectrum of the frequency based on the FE current fluctuations by Fourier transform calculation. The current density obtained from the sample biased using the DC power supply was measured as 1.2 mA cm⁻² at 6 kV, 2.0 mA cm⁻² at 6.6 kV, 4.6 mA cm⁻² at 7.1 kV, and 10 mA cm⁻² at 7.6 kV. The FE current fluctuations were observed to be stable owing to the desorption of gases from the SWCNT surface caused by annealing before the FE measurements. Some damped current distributions in the FE current were expected if the annealing of the sample was insufficient. However, Figure 12(a) shows no damping, and the SWCNT surfaces were expected to be clean for the FE measurements. The spectrum of each FE current in Figure 12(a) is normalized by the peak at 0 Hz, and it indicates that periodic fluctuations were recorded. A peak appears at 50 Hz, as shown in Figure 12(b), and it originated from the noise of the DC power supply, which represents the frequency of the commercial power supply in the eastern region in Japan. It can be seen that the stable DC current had alternating values, especially in the low-frequency domain of <100 Hz. Moreover, the spectrum distribution depends on the FE current, and the contents of each spectrum increased with the DC power.

When the experimental results and the theoretical predictions explained in the previous theoretical section are compared, the parameters that determined the tunneling characteristics of the metallic hc-SWCNTs used as field emitters are listed in Table 3. The enhanced field E in Eq. (20) was determined from the surface potential calculations with an SWCNT model. The area of the field-emission site, A, at the SWCNT surface was calculated from the analysis that determined the FE properties. Each SWCNT model protruded from the ITO wall, which included the minimum area for calculating the electric field using the surface-charge method. The potential parameter ϕ_B was estimated as the work function value of bulk carbon.

The power spectrum |I(ω)|² obtained from Eq. (19) simplified by the WKB approximation shows the dependence of the applied field on a cathode using hc-SWCNTs, as shown in Figure 13. Each calculated spectrum was normalized, and it can be seen that |I(ω)|² increased with the applied voltage at low frequency. For the metallic hc-SWCNTs, the experimental results are shown in Figure 12, and those obtained by simulation are shown in Figure 13; the two sets of results are in good agreement. From the above comment, the experimental results could be fitted well by the composition of the inelastic tunneling model and the F-N tunneling model employing the power spectrum of current fluctuation represented by Eq. (19). These results shown in Figures 12 and 13 indicate that the electrons passing into a barrier in the F-N model exhibited the electron flow fluctuation in the inelastic tunneling model, and the shape of the power spectrum depended on the increased component of current fluctuation obtained from FE.

Field emission using metallic SWCNTs was modeled as a phenomenon based on inelastic electron tunneling, and the calculated power spectra are shown in Figure 13. Although many researchers have reported that field emitters exhibit F-N tunneling with elastic electrical conductivity, the phenomenon has not been demonstrated in any reported experimental work. This chapter presents successfully the first report of the development of an inelastic electron-tunneling model with WKB approximation for SWCNTs from the FE properties. Moreover, the FE current from electron sources like SWCNTs contains time-depended fluctuations, and the I-V characteristics of the FE electron sources were expressed as the power spectrum at ω = 0, which gives the magnitude of the current originating from the electrons passing through the SWCNTs, in accordance with the inelastic electron-tunneling model.