Simulation on Field Emission Display (FED)

In this section, FED is simulated using the PIC method without considering space charge-effect and with considering space-charge effect in turn. Thus a completed parallel Poisson’s equation solver with parallel adaptive mesh refinement is used to compute the electric field distribution of a CNT-based field emitter without considering space-charge effect as the first simulation case.

The generally accepted Fowler-Nordheim theory [Fowler and Nordheim, 1928]

for a clean metal surface relates the field emission’s current density, J, to the electric field at the tip surface of the emitter, E, in volts/nm and the work function of the emitter, φ, in electron volts (eV) by the equation,

( ) ( )

( )

( )

( )

( )

5.1.1. FED Simulation Without Space-Charge Effect

The Electron trajectory from the emitter surface to the anode surface is traced on the unstructured mesh based on the computed electric field distribution from the Poisson’s equation solver, by using the cell-by-cell particle tracking technique. The current density is then computed as the time average of the accumulated charges due to electron flow reaching the anode surface. In the following, two different cases are studied. The first studying case is to predict the FED emission current and investigate the spatial distribution of electron trajectory under different applied voltages. The second studying case is to add the external uniform magnetic field into FED to demonstrate the focus-ability and FED emission current are also strongly influenced by the magnetic field.

CaseⅠ

Fig. 5.1 depicts the simulation domain for a typical CNT triode-type field emitter

within a periodic cell. Only ¼ of the full emitter is used due to the intrinsic symmetry with Neumann boundary conditions applied at all symmetric planes. Important geometrical conditions (also summarized in part in table 5) include a tip radius of 10 nm, an emitter height of 600 and 400 nm, a distance of 0.5 µm between the gate and the cathode, a gate radius of 0.5µm above the emitter, a distance of 50 µm between the anode and the cathode, a thickness of the gate of 0.2 µm, and the half width of each cell measuring 25 µm. The applied voltage of the gate ranges from 110 to 190 volts, while the cathode and anode are grounded and applied with 400 volts, respectively. The refined final number of nodes used for the simulation is approximately 90,000. The typical results of the predicted potential distribution along with electric field distribution (gate voltage=150 volts, height= 600 nm) are shown in Fig. 5.2a and Fig. 5.2b, respectively. The maximal value of the electric field can reach

up to ~11.47 V/nm at the emitter tip when the gate voltage is 150 volts.

The predicted current and voltage data with an emitter height of 600 nm are presented in Fowler-Nordheim format in Fig. 5.3, with an anode voltage of 400 volts.

It is clear that the computed I-V data follow the Fowler-Nordheim law very well as the gate voltage varies from 110 to 160 volts. The fitted field enhancement factor

( V E d

β = ) is 26.1, where V is the applied cathode voltage, and d is the vacuum gap in

the field emission diode configuration. The corresponding electron trajectories are illustrated in Fig. 5.4 at two different gate voltages (110 and 160 volts) with a height of 600 nm. The results show that the spreading angle of electrons from the tip increases with the increasing gate voltage. This is attributed to the fact that the area of the tip surface which has a larger local electric field increases as the applied voltage increases, which results in the greater emission of electrons from the side of the emitter near the tip. As will be shown later, adding a focusing gate can help to effectively reduce the spreading angle.

The effects of CNT height and gate voltage to the emission current under an applied voltage of 400 volts are presented in Fig. 5.5, with the CNT measuring 400 and 600 nm, respectively. The results show that the turn-on voltage increases with the decreasing height of the CNT emitter. Also, the emission current increases dramatically with the given CNT height. This is reasonable since the larger the height of the CNT, the larger the local electric field which results at the tip surface (shorter anode-cathode distance with the same voltage difference), which in turn induces greater emission of electrons.

Fig. 5.6 shows schematically the same field emitter as shown in Fig. 5.1 with an additional focusing gate in-between the gate electrode and anode. Most geometrical

conditions (also summarized in part in table 6) are the same as those in Fig. 5.1, except for the distance between the focusing electrode and the gate electrode measuring 0.5 µm, the thickness of the focusing electrode measuring 0.2 µm, and the radius of the hole in the center of the focusing electrode which is 1.5 µm. Similar to that in the previous case without the focusing gate, only ¼ of a periodic cell is used for the simulation. Fig. 5.7b to Fig.5.7d present a comparison of the focusing effects of electron trajectories using different focusing electrode voltages (5, 0, –5 volts).

Likewise, data involving the absence of focusing electrode are presented for the purpose of comparison (Fig. 5.7a). The results show that the addition of a focusing electrode above the gate electrode can effectively reduce the spreading angle of the electron trajectories, which can possibly increase the resolution and the intensity at the anode. Among the cases simulated, focusing the electrode with 5 volts represents the best choice in focusing the electron flows at the anode.

Case Ⅱ

Another simulation case for parallel Poisson’s equation solver is the magnetic focusing structure consists of a solenoid (or a permanent magnet) outside of the FE device, as shown in Fig. 5.8, which is used to induce the tunable magnetic flux density (Bz), which is assumed uniformly in space. A 1/4 simulation domain of a single gated cathode structure is shown in Fig. 5.9, while a typical final adaptive

refined mesh (91930 nodes) is shown in Fig. 5.10. Important geometrical conditions include a tip radius of 10 nm, emitter height of 600 nm, distance of 0.5 µm between the gate and the cathode, gate radius of 0.5µm above the emitter, distance of 900 µm between the anode and the cathode, thickness of the gate of 0.2 µm, and the half width of each cell measuring 300 µm. The applied voltage of the gate ranges from 50 to 120 volts, while the cathode and anode are grounded and applied with 1,000 volts, respectively.

Without the externally applied magnetic focusing field, the simulated anode current versus gate voltage (I-V) curve is shown in Fig. 5.11, which displays a turn-on voltage of approximate 95V. Note the turn-on voltage is defined as the gate voltage at which the current to anode is 1 µA. The anode current plotted in Fowler-Nordheim coordinate (FN plot) is also shown as an inset to Fig. 5.11 .The linearity of FN plot clearly shows that the computed I-V data follow the Fowler-Nordheim model very well. The corresponding electron snapshots and trajectories with the gate voltage of 120 V are illustrated in Fig. 12 (a), which will be explained shortly.

Furthermore, we simulate the electron trajectories considering the presence of the externally applied downward magnetic field in the range of 0-1 Tesla to study influence of magnetic field to the electron focusing. In Fig. 12 (a)~(d) several 3-D

electron snapshots and trajectories are presented at the gate voltage of 120V, the anode voltage of 1kV, and the different magnetic flux density of 0T, -0.2 T, -0.5T, -1T, respectively. Based on the simulated electron trajectories, the maximum diameter of beam spot on the anode plane can be estimated. The dependence of electron beam diameter on the magnetic flux density is shown in Fig. 5.13, which demonstrates an Airy-function like structure. It is clear that the electron beam diameter rapidly decreases from 500µm down to less than 100µm as the magnetic flux density increases from zero to ~0.3T. At Bz = -0.35 T the beam spot size is estimated as 52µm, which is a minimum in the present simulation conditions. The over focusing of electron beam, as shown in Fig. 5.12(c), is observed in some high magnetic flux density region and the oscillation amplitude in electron beam diameter diminishes as the magnetic field becomes very large. At very large value of magnetic field the electron beam size eventually converges to ~70µm. The total emission current and anode current with magnetic focusing field shown in table 6 are the same as the results without magnetic field. From the simulation, we can find that this magnetic focusing design can optimally suppress the electron beam dispersion under a well-controlled magnetic field and the emission current to anode will not decrease by using this magnetic focusing method.

The above computational examples only serve to demonstrate the capability of

the current parallel Poisson’s equation solver using FEM with parallel adaptive mesh refinement in predicting field emission properties with complicated geometries.

5.1.3. FED Simulation With Space-Charge Effect

In this subsection, PIC method is used for considering the space-charge effect in simulating the silicon field emission diode. Fig. 5.14 shows the SEM image and surface mesh distribution for a typical silicon based field emitter within a periodic cell.

Only 1/4 of the full emitter is used due to the intrinsic symmetry with Neumann boundary conditions applied on all symmetric planes. A conical etched single emitter has been used for our modeling. Important geometrical conditions include an emitter height of 400 nm, a distance of 20 nm between the anode and the cathode, and the half width of each cell measuring 25 µm. The applied voltage of the anode probe ranges from 140 to 320 volts, while the cathode are grounded The refined final number of nodes used for the simulation is approximately 96,326. Fig. 5.15 shows the simulated potential and electric field profile with anode voltage 200 volts.

The simulated and experimental anode current versus gate voltage (I-V) curve is shown in Fig. 5.16. Fig.5.16 shows simulations with work function is 4.5eV agree very well with measurement after turn-on. And before turn-on, simulations using work function is 4.9eV agree very well with measurements probably due to the contamination on the tip surface. It also displays a turn-on voltage of approximate 175

volts. Note the turn-on voltage is defined as the gate voltage at which the current to anode is 1 µA. The anode current plotted in Fowler-Nordheim coordinate (FN plot) is also shown as an inset to Fig. 5.16. The linearity of FN plot clearly shows that the computed and experimental I-V data follow the Fowler-Nordheim model very well.

The above computational examples serve to demonstrate the capability of the current PIC-FEM code with parallel adaptive mesh refinement in predicting field emission properties with complicated geometries.