^{1}

^{*}

^{2}

The field emission current from a carbon fiber is considered. As a model of emission of an elementary carbon tube, tunnel ionization of an electron from a short-range potential is taken. The exact solution for the wave function in such a model allows obtaining an asymptotic expression for electron current. A computer model of transverse distribution of emission current of a carbon fiber is built on the basis of the Monte Carlo method that allows taking into account the random character of distribution of local emitter sources and the distribution of gains of an electric field in carbon nanotubes.

Carbon nanotubes (CNT) can be grown in the form of small sharp spikes capable of withstanding considerable electric current densities. This assumes high potentialities of application of CNT as field emission cathodes in highpower vacuum devices. Such devices with field emission cathodes seem to be ideal for space applications [

The emission properties of an individual nanotube are described on the basis of the Fowler-Nordheim model [4,5] based on the phenomenon of quantum-mechanical tunneling of an electron under a barrier under the action of a constant electric field. The current density in such a model is determined by the dependence

In practice, field emitters in the form of a CNT array contain a very great number of individual nanotubes that differ from one another by their geometry, degree of alignment, electronic features, and other parameters [8,9]. Due to different dependence of emission currents of individual CNT on the electric field strength near a tip, the main contribution to emission is made by a relatively small number of nanotubes with the highest electric field gain

Investigations carried out earlier showed that emitter nonuniformities influenced the current density, but they did not give an answer to the question about the transverse nonuniformity of the current itself in its propagation from the cathode to the anode. At the same time, this nonuniformity directly affects also the nonuniformity of secondary radiation caused by the emission current in a target. To solve this problem, we will consider a model based on taking into account the contribution of electronic amplitudes to the expression for the total current in case of three-dimensional tunnel ionization of different ways arranging point sources.

To describe the propagation of an electron wave in a constant uniform electric field, we will choose a model, in which each individual nanotube is a point source of electron waves. Let us consider a point source, in which an electron is bound by a short-range potential [

at the energy

The solution of the Equation (2) is expressed in terms of the Green function

The function

The Equation (2) can be written as the equivalent integral equation

In the three-dimensional

Substituting the undisturbed wave function (4) in the right side of the Equation (5), we will obtain

and the current density at a great distance from the source is proportional to the squared absolute value of the wave function:

For convenience of calculations it is advisable to write the solution in the asymptotic form:

At great distances from the source in the paraxial region

As a result, for the current from one point source we have the transverse distribution

Shown in

The comparison of the curves shows the high accuracy of the asymptotic representation at a specified distance.

At longer distances the accuracy becomes still higher. So for practical calculations it is advisable to use just asymptotic expressions.

For several coherent centers the current density is

where summation is made over all coherent sources.

In

The statistical treatment of the values of the work function for nanotubes gives an average value of the work function of 5.3 eV. It does not differ greatly from a corresponding value for graphite. In this case a usual electron energy spread is 0.3 eV. Moreover, if the source of field emission electrons is a carbon fiber, the emitter surface is very nonuniform and consists of randomly oriented carbon nanotubes, or has a flaky fibrillar structure [

The value typical of experiment treatment is

The nonuniformity of distribution of field gain over the emitter surface is confirmed by the results of direct measurement with the use of a scanning anode tunnel emission microscope [

To calculate the transverse distribution of electron emission current over a target, the Monte Carlo method was used [

In this case it is necessary, besides coordinates, to specify the field gain

An ordinary random number generator gives a uniform distribution in the interval (0,1). For an arbitrary density of distribution

Since the density of distribution is

if

For the normal distribution (13)

which in view of the determination of the error function [

can be written as

Shown in

The results of calculations show high nonuniformity of transverse distribution of current density that decreases with growing number of point sources, field strength, and distance from the emitter to the screen. The ratio of the minimum current density to the average value

For comparison, shown in

To find out the influence of partial disordering on the current structure, simulation was carried out with random shifts at a level of 20% distortion of a lattice constant. The results of simulation are presented in

The investigation carried out has shown that it is convenient to describe propagation of electrons from a field carbon emitter on the basis of exact Green functions of the Schrödinger equation in a uniform electric field. Each protruding element of the carbon fiber end surface can be simulated by a point source of electron waves. The high nonuniformity of sources results in loss of coherence for different sources and in nonuniform density of electron

current distribution, and in calculations of field emission it is possible to sum densities of currents from individual sources.

Simulation by the Monte Carlo method allows obtaining characteristic patterns of current distribution for different densities of sources, field gains, and distances to the screen. Going from the random distribution of sources to their regular arrangement, even in case of partial loss of the order, considerably increases the current uniformity. The developed model allows choosing preferable parameters to increase the efficiency and life of radiation sources based on field carbon emitters.

The work has been done with the financial support of Russian Foundation for Basic Research (grant 13-07- 00270) and RF Government Contract No. 14.513.11.0133.

The authors declare that there is no conflict of interests regarding the publication of this article.