A high-brightness large-diameter graphene coated point cathode field emission electron source

There have been several long-standing problems of cold field emission sources for electron microscopy and lithography that have prevented their widespread use, such as their inherent ultrahigh vacuum condition requirement (<10–9 torr), relatively poor current stability and rapid emission decay. This paper presents a cold field emission electron source which overcomes these problems based upon using a graphene-coated nickel point cathode. Preliminary experiments demonstrate that it provides stable emission for relatively large tip diameters (micron sizes), can operate in high vacuum conditions (>10−8 torr) and has an ultralow work function value of 1.10 ± 0.07 eV. It has an estimated reduced brightness value of 1.46 × 109 A m−2 sr−1 V−1 for cathode tip-radius of 170 nm and the measured energy spread ranges from 0.246 eV to 0.420 eV for a tip radii range of 260 nm to 500 nm, which is comparable to state-of-the-art conventional cold field emission sources.

equipotential plot in equal voltage intervals is shown in Supplementary Fig. 4(a). The cathode is biased at -1200 V relative to the grounded anode 1 which is placed at 0.5 mm away. The resulting local field strength was found to be 0.68 V nm -1 on the tip as shown in Supplementary Fig. 4(b). In this diagram, the field enhancement factor is defined as the ratio of the local electric field strength at the apex over the applied field E, which is equal to the cathode voltage, U, divided by the cathode-tip to anode distance, d, i.e. The applied voltage is -1200 V and the resulting local field strength is 0.68 V nm -1 on the tip.
A plot of ln( / 2 ) against 1/ will have a slope of = −(6.44 × 10 9 1.5 / ). This slope depends on ϕ, d, and β. Since the value of d is fixed in the experimental setup, and m is obtained from the slope of the F-N plot, the work function value ϕ can be estimated if the field enhancement factor β is known. The field enhancement factor β can be extracted from performing direct ray tracing of electron trajectory paths by simulation as shown above (Supplementary Eq. 1), and the work function can be calculated by: However, it is important to first validate the accuracy of this approach by using it to experimentally measure the work function of the bare Ni tip, before it is coated with graphene. This procedure assumes that the addition of graphene does not change the tip geometry (confirmed by SEM imaging). From the ratio of the two F-N slopes, the effective work function is calculated from: Where Graphene+Ni and Ni are the work functions of graphene coated pointed cathode and bare Ni cathode, Graphene+Ni and Ni are the slopes of the F-N plot for graphene coated point cathode and bare Ni cathode, respectively. The local electric field strength F and β were obtained by numerically solving for the electric potential distribution using the Lorentz-2EM software.

Supplementary Note 2 | Determination of the source reduced brightness B r .
Trajectory ray-tracing simulation is used to compute the exact value of angular magnification = / , where is the final extraction angle and θ is the initial emission angle. The cathode emission area S p ( Supplementary Fig. 5) is given by: The source reduced brightness is defined by the following relationship 4 : where ′ is the angular current density, d v is the virtual source size, and V ext is the extraction voltage. The virtual source size can be calculated using the derived formula 4 : (8) In this formula, tip is the tip radius and < t > is defined as < t >= ħ /√ (8 ), with the local electric field strength, ϕ the work function, and ħ the reduced Planck constant. Using Supplementary Eqs. 7 and 8 gives the following formula for source reduced brightness for cold field emitters 4 :

Supplementary Note 3 | Coulomb interactions.
It is well known that a high brightness electron source may encounter adverse statistical Coulomb effects. These effects manifest as energy broadening or more commonly known as where I' is the angular current density and V ext is the extraction voltage.
Radial broadening and brightness correction. The slice method 7 is used to calculate the Coulomb interactions, in which the region to be calculated is divided into small segments over which the voltage and beam size is assumed to remain constant. The trajectory displacement is calculated by applying this slice method to the analytical approximation in the gun region. The integration is done with the Simpson's 1/3 rule applied to unequal intervals. Radial broadening has the effect of increasing the intrinsic virtual source size d v , which in turn will lower the brightness estimate by a correction factor K given by: The values of d blur and K are given in Supplementary

Supplementary Note 4 | Measurements of the emission repeatability.
Eighteen cycles of ′ -V curves were obtained from a cathode of tip radius of 700 nm ( Supplementary Fig. 6a), where the angular current density was kept below 2 µA sr -1 . Beyond the initial eight runs, a shift of the ′ − curve to the right was observed after which it remained stable with no further shift. After leaving the tip in the HV chamber for 25 days without emission, another round of twenty cycles of ′ − curves were measured, as shown in Supplementary Fig. 6b. The ′ − characteristics largely look similar in both rounds of testing.
The shift of the ′ − curve to the right after some initial cycles may be attributed to the expelling of the adsorbate molecules over time and can be considered as a part of a "preconditioning" process. These results confirm that the graphene coated point cathode has highly repeatable field emission characteristics.
where kT = 0.155 eV at room temperature, J FN is the well-known Fowler-Nordheim emission current density and d is the tunneling parameter (in eV) given by: where F and ϕ are the electric field strength (in V m -1 ) and work function (in eV) respectively.
The variable t(y) is a slowly-varying function of y = 3.79×10 -5 F 1/2 /ϕ and can be approximated E-E f (eV)

W (310) G-Ni
Values of ΔE intrinsic , ΔE Boersch , and ΔE total are given in Supplementary Table 3  The results in Supplementary Table 3 show that the energy spread caused by the Boersch effect is predicted to be larger for the Graphene-Ni cathode compared to a typical W(310) cold field emitter (by a factor of around 20% higher for the 170 nm radius tip), but the total estimated energy spread from the combined TED distribution and Boersch effect is approximately the same. These considerations indicate that for the smaller tip sizes (around 170 nm radius), the smaller energy spreads expected for the Graphene-Ni cathode compared to conventional tungsten cold field emitters (of comparable tip size) based upon the TED distribution, will be approximately off-set by the Boersch effect, and the total energy spread for the two emitters is therefore expected to be comparable.
It is interesting to note that since both the TED distribution and Boersch effect on energy spread decrease with increasing tip radius, a significantly smaller energy spread is predicted for the 800 nm radius Graphene-Ni tip (a factor of two small than that of the 170 nm radius tip). This would ordinarily not be possible for conventional large field emitters (tip-diameters over one micron), such as the Schottky emitter, since the Schottky field emitter only functions by heating the tip up to 1800 K, enlarging the energy spread by thermal effects to around 0.5 eV. These preliminary simple analytical considerations point towards new opportunities for obtaining smaller energy spreads with the Graphene-Ni cathode, which comes from its ability to produce stable field emission from relatively large cathode-tip radii.