# A Review Paper on \"Graphene Field Emission for Electron Microscopy\"
## 2 Development of a Single-Tip Cathode Cold Field Emission Source
### Source Electron Optics Parameters  
For successful electron microscopy applications, a high source brightness (\(B_{\rm r}\)) and a low energy spread (\(\Delta E\)) are always needed to provide a high-resolution focused electron beam. The calculation of \(B_{\rm r}\) requires knowledge of the virtual source size (\(d_{\rm v}\)), which is given by the following expression [35]:  
\[B_{\rm r}=\frac{4I^{\prime}}{\pi d_{\rm v}^{2}V_{\rm ext}}, \tag{2}\]  
where \(I^{\prime}\) is the angular intensity, obtained by dividing the emission current over the acceptance solid angle subtending by the source. [PERSON] et al. derived an analytical formula to calculate the virtual source size [35], given by the following expression:  
\[d_{\rm v}=1.67\frac{r_{\rm tip}}{m_{\rm a}}\sqrt{\frac{<E_{\rm t}>}{eV_{\rm ext }}}, \tag{3}\]  
where \(r_{\rm tip}\) is the tip radius and \(<E_{\rm t}>\) is an initial average tangential energy defined as \(<E_{\rm t}>\ =\phi hF/\sqrt{(8m\phi)}\), with \(F\) the local electric field strength, \(\phi\) the work function, \(m_{\rm a}\) the angular magnification and \(h\) the reduced Planck constant. In these formulas, \(m_{\rm a}\) and \(F\) can be estimated from direct ray tracing simulations and when used together with an experimentally measured angular intensity profile, the source brightness \(B_{\rm r}\) can be estimated. For high brightness electron beam applications, statistical Coulomb interactions lead to radial broadening of the virtual source size due to lateral electron-electron interactions, which in turn, lower the brightness [36]. The calculation of statistical Coulomb interactions within an electron source is not straightforward, since the electron potential and beam size vary in the gun region. A semi-analytical technique known as the slice method was introduced to do this, as reported in a recent paper by [PERSON] et al. [37]. First, the electric potential distribution in the gun region is solved numerically, and then the beam size in the region is determined by the simulated direct ray tracing of electron trajectories that leave the cathode tip. The gun region is subsequently divided into small segments over which the voltage and beam size is assumed to remain constant, and an analytical expression is used to estimate the trajectory displacement effect due to the lateral electron-electron interactions. It should be noted that the method just outlined avoids the need to measure the virtual source diameter \(d_{\rm v}\) experimentally, and this is because direct measurements of the virtual source diameter are not trivial. One way to achieve it is to put the source in a transmission electron microscope as the emitter and obtain a magnified source image [7]. A different method to measure \(d_{\rm v}\) is to use a de-magnifying lens to focus the electron beam from the gun unit into a spot which is source-size dominated (relatively low contributions from lens aberrations), and then measure the spot size of the beam by scanning it across a knife edge [38; 39]. Another method is to place a knife edge at a small distance (less than a few micrometers) from an emitter and to measure the angular width of the Fresnel fringes patterns [13; 40]. In this technique, the intensity distribution shape of the source is assumed to be a Gaussian function, and the values of magnification and electron wavelength are needed to extract \(d_{\rm v}\). Another important electron gun parameter, apart from brightness, is its energy spread. The energy spread, through the chromatic aberration of the objective lens, degrades the final spatial resolution of an electron optical system, particularly at high relative beam energy spreads. Ideally, the energy distribution (\(\Delta E\)) of the emitted electron beam should be as narrow as possible while maintaining sufficient brightness.