# Field Emission from Carbon Nanostructures
## 1 Introduction
### Field Emission  
In order to escape from a metal surface, electrons need sufficient energy to overcome the potential barrier at the metal-vacuum interface, i.e., the potential difference between the Fermi level of the metaland the vacuum. Such energy is known as the metal work-function and denoted by \(\phi\) in the following. It is a material property and has a crucial role in its electron emission capability. Figure 1 shows the effective potential on an electron at a distance \(x\) from the emitting surface when an electric field \(E\) is applied perpendicularly to the surface. The applied electric field \(E\) modifies the potential barrier that becomes thinner (triangular) and the effective barrier height is lowered. Typically, appreciable tunneling is expected for an applied electric field above \(3\times 10^{7}\) V/cm [9]. The width of the tunneling barrier ranges from 4.5 nm at 3 \(\times\) 10\({}^{7}\) V/cm to 0.5 nm at 3 \(\times\) 10\({}^{8}\) V/cm for a metal surface having a \(\phi\approx\) 4.5 eV. From a theoretical viewpoint, the FE is described by the Fowler-Nordheim (F-N) theory [10] based on a simplified one-dimensional free electron model and Wentzel-Kramers-Brillouin (WKB) approximation to calculate tunneling probability for a planar metallic emitter [11]. According to this model, the FE current density \(J\) at 0 K depends on the local electric field \(E_{local}\) and on the work function \(\phi\) as  
\[J=a\frac{E_{local}^{2}}{\phi}\cdot exp\left(-b\frac{\phi^{3/2}}{E_{local}}\right) \tag{1}\]  
where \(a=1.54\times 10^{-6}\) A V\({}^{-2}\) eV and \(b=6.83\times 10^{7}\) cm\({}^{-1}\) V eV\({}^{-3/2}\) are constants. The macroscopic electric field \(E\) and the local field are related by the field enhancement factor \(\beta\) as \(E_{local}=\beta\cdot E\), to take into account the emitter geometry, when the emitter is not flat. For flat surfaces, impractically high macroscopic fields of the order of several kV/um must be applied in order to switch on emitted density current of the order of few mA/cm\({}^{2}\). On the other hand, tip-shaping of the emitter can result in a high enhancement factor, that is a high local electric field, and trigger field emission under a very low macroscopic field. From Equation (1), the FE is favored for materials with low work function. Although the F-N model has been developed for emission under a uniform macroscopic electric field, typical of the parallel plate setup of Figure 1b, it has been demonstrated that it can be satisfactorily applied to more complex geometries, where a non-uniform electric field is generated for example by a tip-shaped anode, as shown in Figure 1c [12; 13]. In a more advanced approach, corrections are necessary to consider non-uniform field enhancement factor or work-functions, curved surfaces, non-zero temperature, series resistance, etc. [14; 15]. [PERSON] et al. [16; 17] proposed a more accurate model to deal with metallic nanowire large-area field emitters, but the corrections introduced therein do not seem very relevant in the case of carbon nanotubes [18]. More rigorous approaches have been also reported in order to consider either the atomic geometry or electronic structure of the emitters [19; 20; 21; 22], including first-principles calculations [23; 24]. However, the original F-N formulation is still widely applied by the scientific community to analyze experimental data with good approximation and gain an initial insight into the emission phenomena from new emitters. Examples are recent studies on FE  
Figure 1: (**a**) Potential energy of the electron as function of the distance \(x\) from the metal-vacuum interface. The barrier for electron field emission is thinned and lowered by the electric field and the image force, which add to \(\phi\) the position-dependent terms \(-eEx\) and \(-e^{2}/4x\), respectively. Experimental setup for field emission measurements in the (**b**) parallel plate and (**c**) tip configuration for average and local characterization, respectively.