Simulation of Field Emission Microtriodes

Vacuum microtriode RF amplifier performance, based upon a unit cell with a conical field emitter tip, gate, and anode, was evaluated via computer simulation. Electron emis­ sion was calculated from the Fowler-Nordheim equation. The dependence of emitted current, transconductance, and field en­ hancement upon geometrical factors, e.g., tip sharpness, tip height, cone half-angle, and gate hole radius, is shown. The device design parameters of transconductance, cutoff fre­ quency, small signal gain, and efficiency, have been calculated. Electron streamlines and current flux are shown for time-de­ pendent RF input. Because a compact electron beam source has wide application, the normalized beam emittance, brightness, and beam quality are calculated for a typical case. Potential difficulties with anode power deposition are noted.


I. INTRODUCTION
M ICROMETER-SCALE vacuum electronics [1] con structed by solid-state fabrication techniques is at tracting increased interest [2]. Currently the most ad vanced vacuum microelectronics devices are based upon field emission processes [3]. Recent measurements of thin film field-emitting cathodes ("Spindt" cathode [4]) pro duced current densities "" 1 kA / cm2 and transconduc tance g per unit cathode area "" 100 S / cm2 (g = (IlIA/ Il Vg) VA) for sample 44-234-2Q at SRI. This exceeds the performance of gridded thermionic cathodes by 10 and 100, respectively. The application of such cathode struc tures to electron-beam sources and RF amplification is therefore potentially useful. A three-terminal amplifi er or triode is the simplest device type for this latter purpose. Therefore, a triode amplifi er geometry based upon the "Spindt" cathode is discussed in this paper.
The cathode structure being simulated is based upon the structures from SRI currently being used in experiments at NRL [5] and SRI [6]. The Spindt gated cathode array [4] consists of many conical field emitter tips each with its own self-aligned surrounding gate. The addition of an anode surface forms a triode. Azimuthal symmetry about the axis of the cone is assumed. Geometrical factors, for example, tip sharpness, tip height, cone half-angle, and gate hole radius, which influence the fi eld may be varied by control in processing steps. The geometry of the unit cell is shown in Fig. 1. The emitted current depends upon

II. SIMULATION METHOD
A space-charge-free model has previously been used by Feeney et at. [7] to provide the variation of electric field in a triode confi guration as a function of tip radius, cone angle, and tip height. A diode structure has been studied by Marcus et at. [8] using the electron optics simulation code SIMION 4.0. Emission currents, which depend upon tip radius, tip height, and cone angle, were calculated and a good discussion of tip heating effects was included.
To provide a more complete model of the device per formance based upon the unit cell approach, the simula tions used a modified version of the widely used MAGIC particle-in-cell simulation code [9] on the NRL Cray X-MP computer. The program is a two-dimensional, U.S. Government work not protected by U.S. copyright finite-difference time-domain code for the self-consistent simulation of the interaction of charged particles and elec tromagnetic fields. The full set of Maxwell equations and the complete Lorentz force equation is utilized. Space charge effects are automatically included. Boundary con ditions include conductors and electron emitters. which need not be conformal to the grid mesh. A variable spatial mesh. shown in Fig. 2, resolved the tip geometry while using a manageable number of mesh points for the simu lation of the entire unit cell without piecing solutions to gether. Spatial resolution was varied from below 5 to 50 A with checks for consistency in going to the larger meshes which were used for the RF dynamic simulations. As can be seen in Fig. 2, the emission surface was ap proximated by patching together conical sections, or fac ets, to approximate a hemispherical surface with small surface roughness, blending smoothly to the principal conical support. The superposition of additional micro structures or surface roughness was unnecessary for the comparison of the unit ccll with thc triodc charactcriza tion of interest.
A Fowler -Nordheim emission [10] module was devel oped and inserted into the MAGIC code to model electron field emission from the surface. The current J was cal culated from plied locally at each cell along the emitter. Although cur rent could be emitted from both the spherical and conical portions of the tip, the contribution from the sides of the cone was several orders of magnitude down and could be neglected. The electric field at the surface is then adjusted by Gauss's law to allow for field perturbation due to the emitted charge. The simulation proceeded, after this mod ule, by pushing the particles, updating the fields, and then self-consistently cvolving in time to determine the new emitted current required using the updated fi elds; this pro cess continuing for the specified number of loops.

III. GEOMETRICAL DEPENDENCE OF EMISSION CURRENT
Unit cell parameters were varied about the nominal di mensions of current experiments at NRL, which are ( Fig.  1): the basegate distance, dbg = 1.15 /lm; gate thickness, dg = 0.3 /lm; and a large gateanode distance of dga = lOO p.m. Tip-to-tip distance was 5.0 p.m, with the effects of adjacent tips allowed in the unit cell by the application of a mirror simulation boundary. Four other geometrical dimensions were also varied individually about a base case with the tip height, t" = 1.25 p.m; a tip radius, r = 250 A; gate hole radius, a = 0.55 /lm; and tip half-angle, ex = 15°. The work function was taken as that of clean Mo, 4.35 eV [11]. For the purposes of simulation, the tip height was defined as the distance from the base plane of the emitting tip to the center of curvature of the spherical segment. This spherical surface was blended smoothly to the conical tip sides.
The match to experimental data was good for samples studied at NRL. The data for the sample denoted 405A [5] has been previously published and is shown for a typ ical comparison. A one parameter fi t on tip radius gave the results shown in Fig. 3 for r = 185 A which is within the error limits for the electron microscope used [12]. Consequently, an additional tip structure was not imposed upon the simulation as has been done by Herrmannsfeldt et al.
[13] and Rosengreen et al. [14]. Due to the extreme sensitivity of the emitted current upon the tip radius, a variation of tip radius by itself can account for current experimental data in the parameter regime of interest to RF power generation. The effects of geometrical varia tions in the simulated unit cell are described below.
This strong dependence of the emitted current upon the tip radius is shown in Fig. 4 for variation from 100 to 400 A. A spherical diode has a 1/ r variation in electric field at the inner surface which translates into a large difference in emitted current for a Fowler-Nordheim emitter over this range of emitter radii. In the unit cell geometry cho sen, for a gate voltage of 140 V, the simulation exhibited a variation of scvcn orders of magnitude in emitted cur rent! For a 90-V gate voltage, almost eleven orders of magnitude current variation was noted. As the tip radius was decreased, the difference in current at the selected gate voltages became much smaller. For the 140-V case, the dependency on tip radius may be approximated as 10 7 A/300 A = 3. 3 X 10 4 A/ A. The electric field enhancement may be defined as the peak electric field at the tip divided by the electric field for parallel plates with separation equal to the base-gate distance. Fig. 5 shows the very large enhancement with decreased tip radius similar to the results of Feeney et al. [7] .
The variation of emitted current with tip half-angle is not so dramatic. As seen in Fig. 6, for the standard case of 140 V, changing the tip half-angle 150 to 1 0 changed the current by only one order of magnitude. At the some what extreme half-angle of 450, the emitted current is fi ve orders of magnitude smaller than the 10 case. The depen dency of current on half-angle is approximately linear be tween 10 and 300 with a slope of 2 x 10 3 AI degree. The electric field enhancement exhibits approximately linear behavior over a wide range of tip half-angle as seen in Fig. 7.     The dependence of emitted current upon gate hole ra dius is shown in Fig. 8. For the radii of interest (0.4 to 0.6 /Lm) only about one order of magnitude variation in the emitted current was observed for a 140-V gate volt age. The current sensitivity to gate radius is 5 x 10 1 A / /Lm. The electric field is enhanced only on the ordcr of 10% in going from 0.55 to 0.4 /Lm as shown in Fig. 9.
A variation of the "tip height" within the gate thick ness (Fig. 10) varied by only about two orders of mag nitude with the smallest tip height providing the most cur rent in the chosen geometry. The tip is at least as tall at the gate bottom to avoid gate interception of the current.
The current dependence on tip height 4 x 10 2 A/ /Lm is relatively weak with a somewhat different dependence than that seen by Feeney et al. (7], probably due to their much larger gate hole diameter.

IV. BEAM CHARACTERIZATION
A new approach for the creation and acceleration of a high-brightness, high-quality electron beam with a com pact form factor of intercst for applications such as free electron lasers, accelerators, and gyrotrons may be pos sible with fi eld emission arrays. The use of field emission arrays in electron-beam sources is of interest due to their small size ( "" 10 /Lm per unit cell) and high current density ( "" 1 kA / cm 2 ). Their robustness in such a hostile envi ronment would need to be addressed. Photocathodes have recently been utilized effectively: however, an external laser driver is required to address concerns of efficiency and system complexity. Roberson (15] has defined a frce electron laser beam quality as the current density divided by the total relative energy spread, BQ = J /(I1'Yz/'Y) where "I is the relativistic energy to rest energy ratio. The extent to which the beam-wave interaction sees the beam as monoenergetic may be thus quantified. This quantity is related to the normalized brightness, where the spread in energy is due to the emittance (16], Bn = J /21f(/).'Y,/'Y), = I / 1f2E�. A look at a typical unit cell can provide some of the required beam characterization. Cell-to-cell uniformity may be an· important factor to determine overall beam quality; however, it is not addressed here. Consider a typ ical unit triode cell with a base-gate distance db R = 1.15 /Lm, a gate thickness dg = 0.3 /Lm, a closer gate-anode distance of dga � 5.0 /Lm, tip height th = 1.25 /Lm, a tip radius r = 150 A, gate hole radius a = 0.55 /Lm, and tip half-angle ex = 15°, a tip current of 4.1 x 10-5 A was emitted. More than 97 % of this current was contained within a radius r = 2.44 x 10 -6 m at the plane z = 6.15 X 10 -6 m yielding a normalized emittance En = 3.97 X 10 -8 mrad. The normalized brightness for this unfocused beam at this plane is 2.63 x 10 9 A / (mrad) 2 . Approxi mating the current density by a uniform beam yields J = 2.2 X 10 6 A /m 2 . As the space charge is negligible for the chosen case, the emittance is the principal cause of energy spread from the unit emittin g cell. The bealTl qual ity may be taken as B Q = 27rBn = 1.65 X 1010 A/m2• An electrostatic focusing electrode above the gate hole may significantly improve these unit cell numbers by con trolling emittance spread. With this current confined with the same energy spread to a beam radius of 0.7 /-1m, nor malized brightness would be increased by more than 12 times to 3.2 X 1010 A/(mrad)2. The use of an array of unit cells suggests that a beam diameter per tip compa rable to the tip-to-tip spacing is a more useful approxi mation. Emittance growth due to the combining of beams from multiple cells [17] would need to be examined for any realistic system.

V. TRIODE CHARACTERIZATION
Several fi gures of merit may be calculated for RF triodes. In order to achieve high gain in an amplifi er con figuration, the transconductance, g = (t!.IA/ t!. Vx) VA' needs to be high. This permits a large amount of current to be switched with small voltages on the grid. The variation of transconductance with geometrical parameters is of inter est in practical design and is thus shown in Figs. 4, 6, and 8 for a gate voltage of Vg = 140 V. For the standard ge ometrical values used herein, the transeonductances for tips of radii 250 and 100 A are g = 0.00212 IlS and g = 7.75 IlS, respectively. Often the transconductance is de sired per unit area. For the 100 A case, g = 7.75 /-IS / (5 X 10-6 mi = 3.1 x 10 5 S/m2• The corresponding cutoff frequencies (fr = g /27rCg) for these tips are fr = 1.75 MHz andfr = 6.39 GHz, respectively, for a gate capac itance 0.193 fF.
High transconductances have been achieved to date, but with high currents. In order to avoid difficulties with ther mal dissipation and tip disruption, one may wish to op erate with a more modest current per tip of tens of mi croamperes. High transconductance then needs to be achieved by greater sensitivity on gate voltage. Geomet rical dependencies of the gate voltage and transconduc tance are therefore calculated for a fixed current of 10 IlA by application of a Fowler-Nordheim emission formula fitted to the previous 140-and 90-V curves. In Fig. 11, the transconductance per tip increased by almost a factor of two with a decrease in radius from 250 to 100 A. The transconductance is approximately linear with tip half-an gle, as seen in Fig. 12. The second derivative of the re quired gate voltage as a function of tip half-angle remains positive. The gate voltage departs only slightly from lin ear behavior as a function of gate radius (Fig. 13).
RF amplifier operation using an emitter array is de sired. The current-voltage characteristics of a 150-A triode are shown in Fig. 14. The emitted current is shown as a function of the anode voltage for various gate volt ages. The extreme flatness of the characteristics for VA just a few tens of volts larger than Vg indicates a high output impedance. For a 100-kO load resistance and 400-V bias source, a load line for a 100-tip array may be 1 013   To examine the dynamic characteristics of a high-per formance triode, a simulation using approximately the ge ometry of the 44-234-2Q structure was performed_ As properties (such as power deposition) associated with the anode are of interest, the gateanode distance was de creased to a value of dga = l.65 /Lm, approximately three times the gate hole radius. A planar anode was used. Other applicable dimensions are: a base-gate distance of dbg = 1.15 /Lm, a gate thickness of dg = 0.3 /Lm, an effective tip-to-tip distance of 3.0 /Lm, a tip height of th = l.25 /Lm, a tip radius of r = 235 A, a gate hole radius of a = 0.50 j.tm, and a tip half-angle of 0' = 150• With an anode tip voltage VA = 500 V and a sinusoidal variation of the gate-tip voltage with an amplitude of VgMax = 212 V, the beam trajectories and anode power deposition were ex amined as a function of time. Fig. 15 shows the behavior of the electron current emanating from the tip as the sin usoidal voltage is turned on. There is one streamline from each facet of the emitting surface. It is important to em phasize that each streamline denotes the flow of electrons from a particular facet and does not indicate the magni tude of the current along the streamline. A 1-THz simu lation frequency was used to accentuate any transit-time effects and to keep computational costs down. The current flux through the gate top plane and to the anode plane is shown in Fig. 16. There is no discernible distortion of the waveform. Of particular interest is the anode power de position, which is shown in Fig. 17 for different times as a function of distance along the anode from the tip axis. The peak power of 4.00 X 105 W 1m2 is significantly large. The reduction of tip radius to 100 A would in crease the current by three orders of magnitude resulting in a peak power in excess of 108 W I m 2 which is well beyond the limits of known materials and indicates the necessity of alternate geometries and materials for anode construction.

VI. DISCUSSION AND CONCLUSIONS
The dependence of emitted current on 1) tip height, 2) tip radius, 3) tip half-angle, and 4) gate hole radius has been determined and shows an extreme variation with tip radius in the 100-to 400-A range. Device characteristics of transconductance, cutoff frequency, gain, and effi ciency have been calculated from the unit cell and are in dicative of the possibility of useful triode devices in the few tens of gigahertz range if the parasitic capacitances and resistive losses can be minimized. With thousands of tips per emitter array, the total transconductance can be quite large (e.g., g = 7.75 ILS x 10 000 tips). The nor- malized beam brightness and current densities calculated for the unit cell are similar to pulse line and RF linac beam sources currently in use. At only slightly below photocathode beam brightness, without the laser require ments, similar focused structures could be considered for compact beam source requirements. The dynamic behavior in a microtriode has been ex amined including beam trajectory spreading, anode power deposition, and transit-time current distortion. The transit time distortion is seen to be negligible and the effect of space charge on the spreading of the beam is small. The anode power deposition is seen to present such a major problem that alternative geometries and materials need to be examined for this application.

A CKNOWLEDGMENT
The author wishes to gratefully acknowledge helpful discussions with H. Gray, P. M. Phillips, A . K. Ganguly, K. L. Jensen, J. P. Calame, and R . K. Parker.