Simulation of Self Organized Electron Beams

in Vacuum Microdiodes

Marjan Ilkov

, Andreas Pedersen

, Andrei Manolescu

and Ágúst Valfells

School of Science and Engineering, Reykjavík University, Menntavegi 1, Reykjavík, Iceland

Science Institute, University of Iceland, Dunhaga 3, Reykjavík, Iceland

Keywords: Vacuum Microelectronics, Molecular Dynamics, Electron Beam, Space-charge Limited Flow, Terahertz.

Abstract: Simulation of vacuum micro-diodes can be done quite effectively using molecular dynamics methods. This

results in much higher fidelity simulations than can be obtained using the particle-in-cell method (PIC),

which is most common for plasma and particle beam simulations. In particular, this method is superior to

the PIC method when treating systems where collisional effects are important. In this paper we present some

results concerning electron beam structure in a vacuum micro-diode obtained using a molecular dynamics

code developed at Reykjavík University.

1 INTRODUCTION

Vacuum electronics devices have been in continuous

use for over a hundred years although their

applications have varied (Barker et al, 2002);

(Eichmeier and Thumm, 2008). As the name implies

they are based on transporting electrons, or other

charged particles, through a vacuum, often guided

by electric and magnetic fields. Prior to the advent

of the solid state transistor vacuum tubes were

ubiquitous in electronics, but since then their use has

become more specialized. In particular, vacuum

electronic devices have been used for high power

and high frequency applications where their superior

performance characteristics have seen them keep the

solid-state devices at bay.

The main drawback of vacuum electronic

devices has been the complexity of their

construction and the need to maintain a good

vacuum for them to function properly. However,

power efficiency and robustness are examples of

their advantages. With the advent of modern

fabrication techniques it is now possible to

manufacture three dimensional microstructures in an

inexpensive and reliable manner. An advantage of

operating vacuum devices at the micro-scale is that

the mean free path of electrons is of the same order

as the characteristic dimension of the device itself

even at atmospheric pressure. Thus the “vacuum”

characteristics of the device are due to the small size

rather than the low pressure. As a result of these

developments the possibility arises for vacuum

electronics devices to become an attractive option in

many applications where they had been too costly

before.

When modelling vacuum devices at this small

scale it is important to take into account several

physical effects that can, to a large degree be

ignored in macro-scale devices, e.g. scattering and

surface inhomogeneity. An important advantage is

that the number of free electrons in such small

systems is typically on the order of 10

to 10

and as

a result it is practical to use a molecular dynamics

method where the repulsive Coulomb interaction

between every single free electron is taken into

account. This means that discrete effects can be

modelled with high fidelity –something that is not

possible to the same extent with the particle-in-cell

(PIC) methods that prevail in the plasma physics and

vacuum electronics community.

Our previous work on a single emitter has shown

that in the case of photoemission, when many

electrons are released at the same time from a

nanometric cathode, one may expect an oscillatory

current through the device (Pedersen, 2010). Similar

results have been obtained (de Lara et al., 2006), but

with emitters larger than few millimeters. These

oscillations are a result of the repulsive Coulomb

forces between the electrons in the micro diode. A

bunch of electrons just released from the cathode

will be accelerated by the electric field across the

224

Ilkov M., Pedersen A., Manolescu A. and Valfells Á..

Simulation of Self Organized Electron Beams in Vacuum Microdiodes.

DOI: 10.5220/0004595502240228

In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2013),

pages 224-228

ISBN: 978-989-8565-69-3

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

diode, but will also create an opposite electric field

that may block the release of further electrons from

the cathode temporarily. This novel beam bunching

mechanism may be useful as a tunable oscillator

operating in the THz domain. The frequency of the

current at the anode is determined by the vacuum

field and by the size of the emitter (Jonsson, 2013).

These unexpected results have motivated us to

expand the study to investigate the effects of having

an emitter area which is non-continuous, e.g. two

circular emitters separated by a given distance or an

arrangement of square emitters on the cathode. This

is a preliminary step in studying several diodes

working in parallel, which is required to increase the

output power of a corresponding device. In this

paper a brief outline on how the molecular dynamics

approach has been used to model vacuum micro-

diodes will be given, as well as preliminary results

regarding beam structure that emerges in the diode

due to nonlinear Coulomb interaction between the

particles in the electron beam released by two

cathode areas. We will discuss both longitudinal (or

temporal) and transverse structure in the beam due

to these types of arrangements.

2 SIMULATION APPROACH

The system under consideration consists of a parallel

plate vacuum diode. The cathode is grounded, and

the anode is at a potential, . The spacing between

the anode and cathode is D. In the absence of any

electrons in the gap between the cathode and anode

the electric field is uniform and perpendicular to the

electrodes and given by E

= /D. This is referred to

as the vacuum field. Electrons originate at the

cathode and are accelerated in the gap by the electric

field. Once electrons are present in the gap they

bring a new component of the electric field into

existence due to their own charge (the space-

charge). In the event that the number of electrons in

the gap becomes large enough the space-charge field

can become a dominant factor in the beam

dynamics, even to such an extent that it will inhibit

further electrons from being emitted. This is known

as the space-charge limited regime, and when

operated under this condition the beam dynamics is

highly nonlinear. In our model we assume that

electrons are emitted from a cold cathode via

photoemission, with negligible initial velocity.

In the simulation each time step is broken down

into three sub-steps: Particle emission, particle

advancement and absorption. A brief description

will be given here, but a more complete description

may be found in (Jonsson 2013).

2.1 Particle Emission

At the beginning of each time step a random point

on the emitter area is selected. If the electric field at

that point is favourable in terms of pushing an

electron into the diode gap an electron will be placed

slightly above that point. If the field is unfavourable

a failure of placement will be registered. While

keeping time constant, this process is repeated,

taking into account the electric field stemming from

those electrons that have been placed in the system

already. The iteration of this process is continued

until 100 consecutive placement failures have been

registered, at which time it is deemed that space-

charge limited emission has been reached.

The emission can also be source limited, in

which case the number of electrons per time step is

limited to a predefined number, but that regime will

not be discussed further in this paper.

2.2 Particle Advancement

Once the space-charge limit has been reached, the

force acting on each individual free electron in the

system is calculated, taking into account both the

external electric field and the Coulomb force from

every single other free electron in the diode gap.

Subsequently a velocity-Verlet method is used to

calculate each electrons position at the end of the

time step.

2.3 Particle Absorption

After calculating the final position of all the free

electrons at the end of the time step, we determine

which electrons have passed beyond the boundary of

the diode and will be absorbed. The number of

electrons absorbed at the anode per time step is used

as a basis for calculating the time dependent current

in the diode, which is obtained using a smoothed

average of the absorption rate. Additionally the point

of absorption is also registered.

3 RESULTS

In our simulations we consider circular (disk) or

square emitters on the cathode. The spacing between

the cathode and anode is 500nm, and the electric

potential is 2V. The time step used in the simulation

is 0.25fs.

SimulationofSelfOrganizedElectronBeamsinVacuumMicrodiodes

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3.1 Temporal Structure

We consider two circular emitters of radius 150 nm.

We want to observe whether and how the Coulomb

oscillations of the current are established in a device

with two emitters working in parallel. We assume

that in a realistic setup the two emitters may not be

synchronized. Therefore we consider in the

simulations a certain relative time delay, and the

current begins to be drawn from one emitter before

the other one. We set the time delay to one fourth of

the period of the space charge induced oscillation,

i.e. the current from the two circular emitters is

initially 90° out of phase. Figures 1 and 2 show the

temporal profile of the current at the anode for two

different values of the spacing between the emitting

disks. In Fig. 1 the distance between the centers of

the disks is 305 nm, i.e. the disks are almost

touching. In this case the onset of regular current

oscillations starts after about 10 000 time steps.

After this time the emission from the two disks on

the cathode becomes synchronized and the current is

qualitatively similar to that corresponding to a single

disk. In Fig. 2 the distance between the centers of

the disks is increased to 340 nm. In this case we see

a much longer transient phase of fairly weak and

chaotic oscillations. But, as time goes by the current

from the two disks becomes synchronized and the

amplitude of the oscillation grows to the same level

as in Fig. 1. We have observed that the

synchronization occurs even for larger spacing

between the emitters, but only after a longer

transient time.

Figure 1: Current at the absorber vs. time steps for two

identical circular emitters, each one with radius of 150nm

The distance between the centers of the disks is 305 nm,

i.e. the disks are almost touching. . The current is given

arbitrary units and one time step is 2.510

-16

Figure 2: Current at the absorber vs. time steps for two

identical circles (disks) with radius of 150 nm. The

distance between the centers of the disks is 305 nm, i.e.

the disks are almost touching. The current is given

arbitrary units and one time step is 2.510

-16

3.2 Transverse Structure

We are interested in time-integrated charge

absorption at the anode and its distribution

perpendicular to the axis of beam propagation. This

is the equivalent of what would be measured using

the well-known experimental method of inserting a

fluorescent screen into the path of an electron beam.

Thus it is a useful method of comparison.

Figure 3: Two identical circles with radius of150nm with

centers separated by 305nm. Here the delay of 90

was

introduced in the emission start between the two circular

emitters.

SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

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We begin by looking at the same configuration

as was used for investigation of the temporal

structure, namely two emitting disks of radius

150nm with their centers separated by 340nm.

Figure 3 shows the resulting pattern for absorption at

the anode as well as that for emission at the cathode.

We note that on the cathode, the current density is

highest along the outer edges. This is in agreement

with previous studies for two-dimensional space

charge limited emission (Ragan-Kelley, 2009). We

also note that at the anode, the beam takes on a

circular cross section with a prominent halo. This is

in agreement with the general characteristics of

space charge dominated electron beams (Reiser,

2008). An interesting aspect is the bright line that

dissects the circle on the upper part of Figure 3

(corresponding to the anode). This is along the line

of the system and is due to interference of the

current from the two emitters.

Figure 4: Two identical squares with side of 250nm each

touching at the corners.

Next we conduct a similar investigation with

slightly different configurations. First, we have two

square emitters of side length 250nm touching at the

corners, as depicted in Figure 4. Second a matrix of

four identical squares with a side length of 250nm,

touching at the corners as depicted in Figure 5. Once

again we see that the beam pattern evolves into a

circular form with a prominent halo. At the cathode

the current is most intense at the outer edges of the

emitter area. As before, we see that this is in

character with expected beam behaviour. Similarly

we also find that the charge density is greater along

the lines of symmetry in the system.

Figure 5: Four identical squares with side of 250nm each

touching at the corners.

4 CONCLUSIONS

We have developed a code for conducting high

fidelity molecular dynamics simulations of vacuum

microsystems with full representation of Coulomb

forces. With this code we have been able to observe

emergence of both temporal and spatial structure in

the beam due to space charge forces. This holds

some promise for the development of tuneable THz

radiators and for the design of electron beamlets

with a specified microstructure.

ACKNOWLEDGEMENTS

This research has been funded by the Icelandic

Research Fund (Rannis), project grant number

120009021.

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