An important physical quantity in electron field emission is the surface work function, which is well documented for elemental materials. For the emitters such as carbon

NTs, most of the electrons are emitted from the tips of the carbon NTs, and it is the local work function that matters to the properties of the NT field emission. The work function measured from the ln(J/E2) versus 1/E characteristics curve, where E is the macroscopic applied electric field that is an average over all of the aligned carbon NTs that are structurally divers in diameters, lengths and helical angles. We have developed a technique for the measurement the work function at the tip of a single carbon nanotube [39].

Our measurement is based on the electric field induced mechanical resonance ofcar-bon nanotubes. The principle for work function measurement is schematically shown in Figure 14a. We consider a simple case in which a carbon nanotube, partially soaked in a carbon fiber produced by arc-discharge, is electrically connected to a gold ball. Due to the difference in the surface work functions between the NT and the counter Au electrode, a static charge Q0 exists at the tip of the NT to balance this potential difference even at zero applied voltage [40]. The magnitude of Q0 is proportional to the difference between work functions of the Au electrode and the NT tip (NTT), Q0 = a(W^u — Wntt), where a is related to the geometry and distance between the NT and the electrode.

The measurement relies on the mechanical resonance ofthe carbon NT induced by an externally applied oscillating voltage with tunable frequency. In this case, a constant voltage Vdc and an oscillating voltage Vac cos 2nft are applied onto the NT, as shown in Figure 14b, where f is the frequency and Vac is the amplitude. The total induced charge on the NT is

The force acting on the NT is proportional to the square of the total charge on the nanotube

= a20 { [(Wau - Wntt + e Vdc )2 + e2 V^/ 2 ] + 2eVAc (Wau - Wntt + e Vdc)cos2nft + e2 V2AC/2cos4nft} (10)

where 0 is a proportional constant. In Eq. (10), the first term is constant and it causes a static deflection of the carbon NT; the second term is a linear term, and the resonance occurs if the applied frequency f approaches the intrinsic mechanical resonance frequency f of the carbon NT (Figure 15a). The last term in Eq. (10) is the second harmonics. The most important result of Eq. (10) is that, for the linear term, the resonance amplitude A of the NT is proportional to Vac (Wau -Wntt + eFoc). By fixing the Vdc and measuring the vibration amplitude as a function of Vac, a linear curve is received (Figure 15c).

Experimentally, we first set Vdc = 0 and tune the frequency/to find the mechanical resonance induced by the applied field. Secondly, under the resonance condition of keeping f=f and Vac constant, slowly change the magnitude of Vdc from zero to a value that satisfies Wau-Wntt + eVDC0 = 0 (Figure 15b); the resonance amplitude A should be zero although the oscillating voltage is still in effect. Vdc0 is the x-axis interception in the A ~ Vdc plot (Figure 15d). Thus, the tip work function of the NT is Wntt = Wau + eVDC0 [39].

Several important factors must be carefully checked to ensure the accuracy of the measurements. The true fundamental resonance frequency must be examined to avoid higher order harmonic effects. The resonance stability and frequency-drift of the carbon nanotubes must be examined prior and post each measurement to ensure that the reduction of vibration amplitude is solely the result of Vdc. The NT structure suffers no radiation damage at 100 kV, and the beam dosage shows no effect on the stability of the resonance frequency. Figure 16 gives the plot of the experimentally measured Vdc0 as a function of the outer diameter of the carbon NTs. The data show two distinct groups: -0.3 to -0.5 eV and ~ +0.5 eV. The work function shows no sensitive dependence on the diameters of the NTs at least in the range considered here. 75% of the data indicate that the tip work function of carbon NTs is 0.3 to 0.5 eV lower than the work function of gold (Wau = 5.1 eV), while 25% of the data show that the tip work function is ~0.5 eV higher than that of gold. This discrepancy is likely due to the nature of some nanotubes being conductive and some being semiconduc-tive, depending on their helical angles. In comparison to the work function of carbon ( Wc = 5.0 eV), the work function at the tip of a conductive multiwalled carbon NT is 0.2—0.4 eV lower. This is important for electron field emission.

Figure 15. (a) Mechanical resonance of a carbon nanotube induced by an oscillating electric field; (b) Halting the resonance by meeting the condition of W^Au— WNTT + eVDC0 = 0. (c) A plot of vibration amplitude of a carbon nanotube as a function of the amplitude of the applied alternating voltage Vac. (d) A plot of vibration amplitude of a carbon nanotube as a function of the applied direct current voltage Vdc, while the applied frequency is 0.493 MHz and Vac = 5 V.

Figure 15. (a) Mechanical resonance of a carbon nanotube induced by an oscillating electric field; (b) Halting the resonance by meeting the condition of W^Au— WNTT + eVDC0 = 0. (c) A plot of vibration amplitude of a carbon nanotube as a function of the amplitude of the applied alternating voltage Vac. (d) A plot of vibration amplitude of a carbon nanotube as a function of the applied direct current voltage Vdc, while the applied frequency is 0.493 MHz and Vac = 5 V.

Figure 16. The experimentally measured VdC0 as a function of the outer diameter of the carbon nanotube.

The technique demonstrated here has also been applied to measure the work function at the tips ofZnO nanobelts [41].

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