2.1. Model description

These kinds of devices are part of a larger sort of plasma devices (plasma accelerators, ion magnetrons, thrusters, plasma lenses, etc.) that use a discharge in crossed electric and magnetic fields with closed electron drift for the generation, formation, and manipulation of intense ion beams and ion plasma flows. In accordance with the basic idea of plasma optics [1], spatial over thermal E-fields can be introduced in the plasma medium of an intense ion beam, which makes possible high-current ion beams manipulation and focusing including beams of heavy ions. An idea to use the space charge for that purpose appeared to be very fruitful and successful [2, 3]. A number of effective plasma lenses for positive ion beams focusing were made and tested. The robust construction, low-energy consumption, and high-cost-effectiveness make these tools attractive for prac-

268 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Some new ideas for using these plasma-optical principles for creation axial symmetric highcurrent mass separation devices were described firstly in [4]. Note also that this approach is appropriate for the creation of linear or curved magneto-electrostatic plasma guiding ducts for use in vacuum-arc plasma filtering system. Following these plasma-optical principles, changing the magnetic field line configuration and the distribution of electric potential enables the formation and control of high-current ion beams while maintaining their quasi-neutrality. This makes the application of such devices attractive for the manipulation of high-current beams of

The plasma lens configuration of crossed electric and magnetic fields provides a suitable and attractive method for establishing a stable discharge at the low-pressure. Using plasma lens configuration in this way were elaborated, explored and developed some cost efficiency, low-maintenance plasma devices for ion treatment, and deposition of exotic coatings with given functional properties. These devices make using of permanent magnets and possess considerable flexibility with respect to spatial configuration. They can be operated as a stand-alone tool for ion treatment of substrates, or as part of integrated processing system together with cylindrical magnetron sputtering system, for coating deposition. The cylindrical plasma-optical magnetron sputtering device with virtual anodes and cylindrical plasma production device for the ion treatment of substrates with complicated cylindrical was proposed and created [5–7]. These devices can be applied both for fine ion cleaning and activation of substrates before deposition and

One particularly attractive result of this background work was observation of the essential positive potential at the floating substrate treated by cylindrical ion cleaning device. This suggested to us the possibility of an electrostatic plasma lens for focusing and manipulating high-current beams of negatively charged particles (electrons and negative ions) that is based on the use of the cloud of positive space charge in conditions of magnetic insulation electrons. The idea of the plasma lens based on electrostatic electron isolation for creation positive space charge was first proposed in [8]. Later, it was proposed to use magnetic electron insulation for

Here we describe computer modeling for some of these novel plasma-optical systems.

tical applications.

heavy ions.

for sputtering.

creation of a stable positive space charge cloud [9].

The plasma lens is a cylindrical plasma accelerator with an anode layer and used as a device with magnetic insulation of electrons for creation of the dynamic cloud of positive space charge. The scheme of the plasma lens with magnetic insulation used for creation of the dynamic cloud of positive space charge is shown in Figure 1a.

The lens has a system of permanent magnets that produce an axially symmetric magnetic field between the poles of a magnetic circuit serving as cathode. The magnetic field is controlled by varying the number of magnets. The magnetic field configuration is typical for the single magnetic lens configuration, because of the lens could focus the transported electron beams. When a positive potential is applied to the anode, a discharge in the axial magnetic, and radial electric crossed fields is ignited between the anode and the cathode. The electrons are magnetized in anode layer and drift along closed trajectories in the azimuthal direction, repeatedly ionize atoms of the working gas, and gradually diffuse to the anode. The ions thus formed are accelerated in the strong electric field created by the electron space charge and leave the ion source through a hole in the acceleration channel. The fast ions reach the system axis and accumulate in the region around it, as it schematically shown in Figure 1b. In this way, the axially converging ion beam creates a positive space charge. In the experiments, the energy of the argon ions converging beam could reach 2.5 kV. Maximum potential will be in the center on cylindrical axis. Ions are stored in the cylinder volume until their own space charge creates a critical electric field. This field forces ions to leave the volume, and the system comes to dynamic equilibrium after some relaxation time.

Electrons are magnetized in the anode layer, so their influence on ion dynamics can be neglected. The ion flow coming through the cylindrical surface will be equal to ions streaming down from the axis and leaving the cylindrical volume under action of the Coulomb force of their own space charge. Therefore, the set of equations describing this process in the cylindrical

Figure 1. (a) Scheme of the plasma lens with magnetic electron insulation: 1 – cathode; 2 – anode; 3 – magnetic system based on permanent magnets; (b) Scheme the positive space charge cloud creation.

coordinate system can be written in the form that includes the Poisson, particles motion, and continuity law equations:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial Ul}{\partial r}\right) + \frac{\partial^2 Ul}{\partial z^2} = -4\pi q\_i n\_i \tag{1}$$

solving it we get that potential and space charge density in layer is proportional to next

Substitute character system parameters we obtain estimations for ion density: n�10<sup>10</sup> cm�3, and for electric field strength about 1000 V/cm, that is sufficient for high-current electron beam

Eqs. (1)-(3) were solved numerically by particle in cell (PIC)-method [10]. Every time interval <sup>Δ</sup><sup>t</sup> (that corresponded to the actual time interval approximately equal to 4�10�<sup>8</sup> sec) <sup>N</sup> new particles of charge qi and mass Mi come to the volume considered. The magnitudes of N, Δt, qi

where ε<sup>0</sup> = εmax/2 and < ε > is average energy. They moved from cylinder surface to the system axis with the angular distribution according to cosine law: N(Θ)/N(0) ≈ cos (Θ), where N(Θ) are quantities of ions going out under angle Θ, N(0) is angular distribution amplitude. It should be noted that the distributions above are inherent to this kind of plasma accelerators with anode layer. As the first step the potential was specified at the anode, and the Laplace equation was solved. After that, the particles were launched and the equations of motion (Eq. (2)) for particles in the magnetic and calculated electric field were solved. The time step for motion equation solving was Δτ <<Δt (about 10�<sup>11</sup> sec). After time Δtby collecting of all particles with the use the "cloud in cell" method [10], the densities distributions of ions were calculated. The Poisson equation has been solved and electric field potential U(r, z) was calculated for this time moment. Electric field was calculated by the distribution of total space charge. After that in corrected electric field, the calculation of particles motion was resumed, and introducing the new portion of ions was performed. Equations of motion were solved both for "new" particles and for those that still left in the volume. Figure 2 shows calculated

, <sup>r</sup> <sup>≈</sup> 2 4<sup>π</sup> <sup>j</sup>

in r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi =Mi

S. Let us suppose that particles with energies from 0 to εmax are

2

2h iε

!

exp � ð Þ <sup>ε</sup> � <sup>ε</sup><sup>0</sup>

(5)

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(6)

p !<sup>2</sup>=<sup>3</sup>

φð Þ r; z ≈ j

in=<sup>r</sup> � �<sup>2</sup>=<sup>3</sup>

<sup>N</sup>ð Þ<sup>ε</sup> <sup>≈</sup> <sup>1</sup>

Figure 2. Positive space charge cloud formation and Ar+ ions trajectories for different time step.

ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> h i<sup>ε</sup>

z2

expressions:

focusing.

satisfy the relation: Nqi

distributed according to law:

<sup>Δ</sup><sup>t</sup> ¼ j i

$$M\_i \frac{dv\_i}{dt} = q\_i E + \frac{1}{c} [v\_i \times B] \tag{2}$$

$$V\_i \cdot \left(\frac{\partial n\_i}{\partial t} + \text{div}(\mathbf{j}\_{\text{out}})\right) = S \cdot j\_{\text{in}} \tag{3}$$

where Mi, qi, vi, ni are ion mass, charge, velocity, and ion density, respectively, E – electric field: Er = �∂U=∂r, Ez = �∂U=∂z, U – potential, B – magnetic field, V – cylindrical volume, jin – current density at the boundary of current-collecting surface S of radius r and height h, and jout– the ion current density leaving the cylindrical volume V. Knowing the space charge distribution, we can determine the expulsive force that acts on the particle on the boundary of space charge volume and calculate ions trajectories.

We can obtain estimations for electrical field and space charge in cloud in stationary case with simplifying assumptions. Let us suppose uniform ion distribution and that they enter to the cylinder perpendicular to its lateral surface and leave it along z-axis under action of electric field created by the ions own space charge. Thus considering of one-dimensional ions motion along the radius, we can write equation for potential in form:

$$\varphi\_r'' = \frac{4\pi j\_{\rm in}}{\sqrt{2q\_i(\varphi\_0 - \varphi(r))/M\_i}} \tag{4}$$

Solving it, we get solution similar to the low 3/2:

$$\left(\varphi\_0 - \varphi(r)\right)^{3/2} \approx \frac{9\pi \dot{j}\_{\rm in}}{\sqrt{2q\_i/M\_i}}r^2$$

and obtain estimation for electric field:

$$E\_r = -\frac{\partial \rho}{\partial r} \approx \frac{4}{3} \left( \frac{9\pi j\_{in}}{\sqrt{2q\_i/M\_i}} \right)^{2/3} r^{1/3}.$$

Let us now consider a cylindrical layer of ion space charge with radius r and assume that the ions leave it along z-axis, then neglecting radial coordinate we can write Poisson equation in layer in form: φ<sup>00</sup> <sup>z</sup> ¼ 4πr, where jout = rυ. In stationary case from (Eq.(3)), we obtain: j in ¼ r ∂j out <sup>2</sup>∂<sup>z</sup> . Integrating this expression with using expression <sup>υ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi φð Þ� z φ z<sup>0</sup> ð Þ ð Þ =Mi p for velocity of ions leaving the layer, we obtain expression for the distribution of ion space charge density in form: r ≈ <sup>2</sup><sup>j</sup> in <sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi φð Þ�z φ z<sup>0</sup> ð Þ ð Þ =Mi p z. Substituting last expression in Poisson equation for the layer and solving it we get that potential and space charge density in layer is proportional to next expressions:

coordinate system can be written in the form that includes the Poisson, particles motion, and

þ ∂2 U <sup>∂</sup>z<sup>2</sup> ¼ �4πqi

<sup>∂</sup><sup>t</sup> <sup>þ</sup> div jout � � � �

E þ 1 c

where Mi, qi, vi, ni are ion mass, charge, velocity, and ion density, respectively, E – electric field: Er = �∂U=∂r, Ez = �∂U=∂z, U – potential, B – magnetic field, V – cylindrical volume, jin – current density at the boundary of current-collecting surface S of radius r and height h, and jout– the ion current density leaving the cylindrical volume V. Knowing the space charge distribution, we can determine the expulsive force that acts on the particle on the boundary of space charge

We can obtain estimations for electrical field and space charge in cloud in stationary case with simplifying assumptions. Let us suppose uniform ion distribution and that they enter to the cylinder perpendicular to its lateral surface and leave it along z-axis under action of electric field created by the ions own space charge. Thus considering of one-dimensional ions motion

> in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>qi <sup>φ</sup><sup>0</sup> � <sup>φ</sup>ð Þ<sup>r</sup> � �=Mi

> > <sup>≈</sup> <sup>9</sup>π<sup>j</sup> in ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi =Mi <sup>p</sup> <sup>r</sup>

9πj in ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi =Mi

<sup>z</sup> ¼ 4πr, where jout = rυ. In stationary case from (Eq.(3)), we obtain: j

p z. Substituting last expression in Poisson equation for the layer and

!<sup>2</sup>=<sup>3</sup>

p

Let us now consider a cylindrical layer of ion space charge with radius r and assume that the ions leave it along z-axis, then neglecting radial coordinate we can write Poisson equation in

ions leaving the layer, we obtain expression for the distribution of ion space charge density in

2

r 1=3 :

<sup>q</sup> (4)

2qi φð Þ� z φ z<sup>0</sup> ð Þ ð Þ =Mi

p for velocity of

<sup>r</sup> <sup>¼</sup> <sup>4</sup>π<sup>j</sup>

<sup>φ</sup><sup>0</sup> � <sup>φ</sup>ð Þ<sup>r</sup> � �<sup>3</sup>=<sup>2</sup>

<sup>∂</sup><sup>r</sup> <sup>≈</sup> <sup>4</sup> 3

Integrating this expression with using expression <sup>υ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ S � j

ni (1)

in (3)

in ¼ r ∂j out <sup>2</sup>∂<sup>z</sup> .

½ � vi � B (2)

1 r ∂ ∂r r ∂U ∂r � �

270 Plasma Science and Technology - Basic Fundamentals and Modern Applications

along the radius, we can write equation for potential in form:

Solving it, we get solution similar to the low 3/2:

and obtain estimation for electric field:

in <sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qi φð Þ�z φ z<sup>0</sup> ð Þ ð Þ =Mi

layer in form: φ<sup>00</sup>

form: r ≈ <sup>2</sup><sup>j</sup>

φ<sup>00</sup>

Er ¼ � <sup>∂</sup><sup>φ</sup>

Mi dvi dt <sup>¼</sup> qi

Vi � <sup>∂</sup>ni

continuity law equations:

volume and calculate ions trajectories.

$$\log(r, z) \approx \left( j\_{in} / r \right)^{2/3} z^2, \rho \approx 2 \left( 4 \pi \frac{j\_{in}}{r \sqrt{2 \eta\_i / M\_i}} \right)^{2/3} \tag{5}$$

Substitute character system parameters we obtain estimations for ion density: n�10<sup>10</sup> cm�3, and for electric field strength about 1000 V/cm, that is sufficient for high-current electron beam focusing.

Eqs. (1)-(3) were solved numerically by particle in cell (PIC)-method [10]. Every time interval <sup>Δ</sup><sup>t</sup> (that corresponded to the actual time interval approximately equal to 4�10�<sup>8</sup> sec) <sup>N</sup> new particles of charge qi and mass Mi come to the volume considered. The magnitudes of N, Δt, qi satisfy the relation: Nqi <sup>Δ</sup><sup>t</sup> ¼ j i S. Let us suppose that particles with energies from 0 to εmax are distributed according to law:

$$N(\varepsilon) \approx \frac{1}{\sqrt{2\pi}\langle\varepsilon\rangle} \exp\left(-\frac{\left(\varepsilon - \varepsilon\_0\right)^2}{2\langle\varepsilon\rangle}\right) \tag{6}$$

where ε<sup>0</sup> = εmax/2 and < ε > is average energy. They moved from cylinder surface to the system axis with the angular distribution according to cosine law: N(Θ)/N(0) ≈ cos (Θ), where N(Θ) are quantities of ions going out under angle Θ, N(0) is angular distribution amplitude. It should be noted that the distributions above are inherent to this kind of plasma accelerators with anode layer. As the first step the potential was specified at the anode, and the Laplace equation was solved. After that, the particles were launched and the equations of motion (Eq. (2)) for particles in the magnetic and calculated electric field were solved. The time step for motion equation solving was Δτ <<Δt (about 10�<sup>11</sup> sec). After time Δtby collecting of all particles with the use the "cloud in cell" method [10], the densities distributions of ions were calculated. The Poisson equation has been solved and electric field potential U(r, z) was calculated for this time moment. Electric field was calculated by the distribution of total space charge. After that in corrected electric field, the calculation of particles motion was resumed, and introducing the new portion of ions was performed. Equations of motion were solved both for "new" particles and for those that still left in the volume. Figure 2 shows calculated

Figure 2. Positive space charge cloud formation and Ar+ ions trajectories for different time step.

Ar + ions trajectories in different time step. The calculation continued until reaching a selfconsistent solution. The calculation time comprised 10<sup>5</sup> sec. For that time the stationary state of the lens operation was achieved.

heavier Xe + ions (Figure 4 right) is at the axis. This can be explained by a smaller influence of

and electric field strength was up to 600 V/cm that is sufficiently for focusing intensive negative charged particle beams and correspond to our estimations. Thus the possibilities formation of the stable space charge cloud was demonstrated, and next task is to consider negative charge particle

We investigated transport electron beam with energy from 5 to 20 keV through the plasma lens. As first step was solved equations for ion's part and as result – obtaining stable positive charge cloud inside plasma lens. Next step was launch e-beam through the lens with cloud. For correct description, we must solve equations for ions and electrons parts together, so we must include electron motion equations in our consideration and modify Poisson equation to form:

For simulation high-current electron beam transport need also taking into account the space charge of the particle and the magnetic self-field that may affect the dynamic beam particles in addition to the external fields. The possibility of ionization residual gas by electron beam is

Equations of motion for electrons are solving by current tubes of variable width with central trajectory. A shape of trajectories in an electromagnetic field is calculated using Boris scheme [11]. A space charge beam density is calculated using equation of continuity: div(reve) = 0.A self-consistent solution can be found by repeated solving of Poisson equation, motion equations for all particles, and re-determination of the space charge distribution on every time step.

Numerical simulations results show clearly that for electron beam current less than 1A the electrostatic beam focusing occurs [12, 13]. The results of simulation for space charge plasma lens and magnetic lens (ML) with the same magnetic field are shown in Figure 5. The comparison shows that beam compression is stronger, and beam divergence is less in PL case than in ML case. The experimental results [12, 13] confirm our simulations. The lens can be used even more effectively for negative ions beam focusing. The simulation results for H-beam with passing through plasma lens and magnetic lens with the same magnetic field are shown in Figure 6. One can see that lens effectively compresses H-ions beam, whereas the magnetic lens

However, it should be noted, some part of cloud ions can be captured by a beam and carried away from the cloud. It is positive for beam transport, but as a result of this, cloud potential decreases, and its focusing properties deteriorate. It is not critical for electron beams with current up to 1 A, because a new Ar + ions come to the cloud and renew its focusing properties.

dz<sup>2</sup> <sup>¼</sup> <sup>4</sup>πe nð Þ <sup>e</sup> <sup>þ</sup> neb � ni (7)

–2.7�1010 cm�3,

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the momentum aberration on the converging Xe + ion beam dynamics.

beam transport through space charge PL.

necessary taking into account also.

does not focus it at all.

The calculated ion density that could accumulate around system axis reaches 109

2.3. Negative charge particle beams focusing on space charge plasma lens

d2 φ dr<sup>2</sup> <sup>þ</sup>

An iteration method with relaxation was used for faster convergence.

d2 φ

1 r dφ dr þ

#### 2.2. Simulation results

The model was applied for calculating the lens volume based on the local area with diameter of 80 mm and a height of 50 mm. In our simulations, we considered Ar<sup>+</sup> and Xe+ ion beams with maximal energy from 1 to 3 keV and total current of 20 mA that moved in the magnetic field. Magnetic field is similar to the experimental one and changes from 0.07 T near electrodes to 0.01 T on the system axis. The results of the calculations of the potential distribution when steady-state dynamical equilibrium is reached for Ar<sup>+</sup> ions with maximal energy 1.2–2.4 keV are shown in Figure 3. One can see that with ions energy increasing, the spatial distribution shape changes markedly. Whereas maximum of potential for ion beam's energy 0–1.5 keV (see left Figure 3) is double-humped situated in the coaxial region around the axis, the maximum for energy 0–2.4 keV (right) is single-humped onto the axis.

The same result we can see with ion mass increasing (see Figure 4). The maximum of potential for Ar + ion beam (Figure 4 left) is in the coaxial region around the axis, but the maximum for

Figure 3. Potential distribution in plasma lens midplane for Ar + ion beam with total current 20 mA and Emax = 1.2 keV (left) and 2.4 keV(right).

Figure 4. Potential distribution for Ar+ ion beam (left) and Xe+ ion beam (right) with total current 20 mA and Emax = 1.4 keV.

heavier Xe + ions (Figure 4 right) is at the axis. This can be explained by a smaller influence of the momentum aberration on the converging Xe + ion beam dynamics.

The calculated ion density that could accumulate around system axis reaches 109 –2.7�1010 cm�3, and electric field strength was up to 600 V/cm that is sufficiently for focusing intensive negative charged particle beams and correspond to our estimations. Thus the possibilities formation of the stable space charge cloud was demonstrated, and next task is to consider negative charge particle beam transport through space charge PL.
