Plasma breakdown

FUNDAMENTAL CONCEPTS OF PLASMA

2.7 Basic Plasma Properties

2.7.1 Plasma breakdown

In the following sub-sections we will consider the breakdown processes that precede the formation of DC and RF glow discharges.

2.7.1.1 DC breakdown

We will examine DC breakdown by considering, as an example, Ar at 30 mTorr

III a system, shown in Fig. 2.5, comprising two electrodes connected to a DC power

Ar P = 30 mtorr Vex)

o

^d

••

_{E V/d}

V^p,

Figure 2.5 Schematic of arc discharge prior to breakdown [35].

supply with voltage Vps through a ballast resistor R. Initially, the resistance of the neutral gas will be much greater than that of R, so the voltage across the discharge V

=

Vps' Let us assume that there is one free electron, formed perhaps by a cosmic ray or some UV photon, near the cathode. The electric field will accelerate the electron towards the anode. Let

a

be the probability per unit length that ionization will occur.

The quantity

a

is called Townsend's first ionization coefficient, and represents the net

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ionization probability, including losses. As a result of the acceleration by the electric field, the electron will gain energy and produce ionization [35]. This will lead to a multiplication of the number of electrons as shown in Fig. 2.6.

Cathode

uv

Anode

/.G

)-Electrons

--G '1

"'e~/ ~--....

E=V/d

v

R

Figure 2.6 Behaviour of the discharge at breakdown [35].

The current at the anode arising from an electron current 10emitted from the cathode is given by [32],

1d

=

1⁰exp(ad) 2.6)

The electric field will also accelerate ions, and when ions strike the cathode, electrons will be emitted by ion impact secondary electron emission with a probability y.The total number of ions created by the first electron multiplication is (exp(ad)-1). This will give rise to y(exp(ad)-1) secondary electrons, which will also be accelerated by the electric field and cause more ionization and consequently more ions [40]. If we add up this sequence of successive generation of secondary electrons giving rise to more ions giving rise to more secondaries etc., we find that the total current arriving at the anode is [35],

10exp(ad) 1d

=

-1--y-[e-x-p(-ad-)---1]

If exp(ad»> 1, we can write Eq. (2.7) as 10exp(ad)

1d=---

1- yexp(ad)

(2.7)

(2.8)

When 1-yexp(ad)

=

0 the current I^d tends to increase rapidly and a condition arises referred to as breakdown.

2.7.1.2 RF breakdown

Breakdown in an RF field is actually somewhat simpler than that for the DC case, if most of the electrons are able to undergo their oscillatory motion without colliding with a wall. In this case, the oscillating electric field puts directed energy into the electrons, which then heat up by undergoing collisions with neutrals [35]. In this way the electrons are heated up sufficiently to produce the required amount of ionization which must balance the losses due to diffusion to the walls, volume recombination, electron attachment, etc.

2.7.2 Quasi neutrality of plasma

In general, the characteristics of plasmas will differ greatly depending on things like the constituent atoms and molecules, densities, energies, and degree of ionization [40]. There is, however, one universal plasma characteristic- the free charges in the plasma will move in response to any electric field in such a way to decrease the effect of the field. In particular, electrons are usually lighter and more mobile in response to electric fields, and the ions are assumed stationary. That means the tendency of plasma electrons is always to decrease any presence of electric fields. There will not be any regions of plasma with excess positive or negative charge, because if there were, an electric field would arise that would move electrons to effectively eliminate any charge imbalance. This feature is called quasi-neutrality [40].

2.7.3 Debye shielding

If a test positive charge is inserted in plasma (Fig. 2.7), the charge will attract a cloud of electrons, and repel the local ions, so that it is completely shielded from the rest

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of the plasma. Outside the cloud there will be no electric field. This is the phenomenon of Debye shielding [35]. We can find a self-consistent solution for the electrostatic potential <P,which arises from the test charge Q.

Unscreened

potentia

_R-

Figure 2.7 Schematic of the unshielded and shielded electrostatic potentials from a point charge Q [35].

Using Boltzmann's equation we have [35],

ni

=

noexp(-e<P / kT) ne =no exp(e<P / kT)

(2.9) (2.10) where. no represents the value of both particles at equilibrium and k is the Boltzmann's constant. The charge density p is given by [35],

p

=

e (n; - ne)

=

-2noe sinh(e<P / kT) (2.11)

(2.12) Hhere, we have used the identity sinh( x)

=.!-(

eX - e-x ) .Using Poisson's [36] equation,

2 V2<p=_£

£0

Substituting Eq. (2.11) in Eq. (2.12), we obtain, V2<p

=

2noe sinh(e<P /kT)

£0

(2.13)

Near the edge of the cloud, and beyond, the electrostatic energy etPassociated with Q is much less than the electron thermal energy kT i.e., etP/kT«1. We can then approximate

sinh(e tP / kT)= etP / kT, so that Eq. (2.13) becomes,

VZtP =_2_tP (2.14)

ADZ Here we define the Debye length, AD as [35],

(2.15)

The solution ofEq. (2.14) is,

(2.16)

(2.17) By Eq. (2.15), the Debye length AD is a measure of the range of the effect of the test charge Q. It follows from Eq. (2.14) that this range is greater in a hot diffuse plasma than in a cool dense plasma. This is to be expected: if Tis large, more electrons in the cloud at a given distance from Q will be able to escape, so that Q is less efficiently screened; ifno is small electrons will have to be drawn from a larger volume in order to shield a given charge Q [35].