Child-Langmuir Sheaths

Basic Plasma Physics

3.7 Sheaths at the Boundaries of Plasmas

3.7.3 Child-Langmuir Sheaths

fallen to a value of -kTe 12e relative to the plasma potential where the density is no (far from the edge of the plasma). The electron density at the sheath edge is then

= 0.606 no.

Therefore, the plasma density at the sheath edge is about 60% of the plasma density in the center of the plasma.

The current density of ions entering the sheath at the edge of the plasma can be found from the density at the sheath edge in Eq. (3.7-27) and the ion velocity at the sheath edge in Eq. (3.7-25):

Ji = 0.6 noevi =

1

^{n e g ,}

2 (3.7-28)

where n is the plasma density at the start of the pre-sheath, which is normally considered to be the center of a collisionless plasma or one collision-mean-free path from the sheath edge for collisional plasmas. It is common to write Eq. (3.7-28) as

(3.7-29) where A is the ion collection area at the sheath boundary. This current is called the Bohm current. For example, consider a xenon ion thruster with a 1 0 ' k 3 plasma density and an electron temperature of 3 eV. The current density of ions to the boundary of the ion acceleration structure is found to be 11 8 Aim2, and the Bohm current to an area of 1 O-* m2 is 1.18 A .

(3.7-30) J~ = n i e v = n i e -

[$,-$I

112 ^.

Solving Eq. (3.7-30) for the ion density, Poisson’s equation in one dimension and with the electron density contribution neglected is

/:

The first integral can be performed by multiplying both by d$ / dx and integrating to obtain

(3.7-3 1) sides of this equation

Assuming that the electric field ( d # / & ) is negligible at x = 0, Eq. (3.7-32) becomes

(3.7-33) Integrating this equation and writing the potential across the sheath of thickness d as the voltage V gives the familiar form of the Child-Langmuir law:

J.=%(”) 112

d2

^v312

‘ 9 M ^(3.7-34)

This equation was originally derived by Child [ 1 11 in 191 1 and independently derived by Langmuir [12] in 1913. Equation (3.7-34) states that the current per unit area that can pass through a planar sheath is limited by space-charge effects and is proportional to the voltage to the 312 power divided by the sheath thickness squared. In ion thrusters, the accelerator structure can be designed to first order using the Child-Langmuir equation where d is the gap between the accelerator electrodes. The Child-Langmuir equation can be conveniently written as

v3l2

J , = 2 . 3 3 ~ 10" - electrons d2

J. = 5.45 x lo4

-

V3l2 singly charged ions (3.7-35) d2

312

= 4.75x10-9

V

xenonions, dL

where M a is the ion mass in atomic mass units. For charge-limited xenon ion current density across a planar

1000 V applied is 15 mAkm2.

example, the space- 1-mm grid gap with

3.7.4 Generalized Sheath Solution

To find the characteristics of any sheath without the simplifying assumptions used in the above sections, the complete solution to Poisson's equation at a boundary must be obtained. The ion density through a planar sheath, from Eq. (3.7-19), can be written as

-112

n i = n o ( l - i ) ,

and the electron density is given by Eq. (3.5-9), n, = no exp(

g) ^.

(3.7-36)

(3.7-3 7)

Poisson's equation (3.7-8) for singly charged ions then becomes

-112

d2q e

- _dx2 = - - _E,

(

n , i- n e = -

5

_Eo

[(

^{1 -}

i)

^{- exp}

(211 ^.

^{(3.7-3 8)}

Defining the following dimensionless variables,

{=-, X

AD Poisson's equation becomes

(3.7-39) This equation can be integrated once by multiplying both sides by the first derivative of

x

and integrating from 51 = 0 to

t1

⁼

5

^:

where is a dummy variable. The solution to Eq. (3.7-40) is

Since the electric field ( d$ I d x ) is zero away from the sheath where

5

^{= 0,}

rearrangement of Eq. (3.7-41) yields

To obtain a solution for

~ ( 6 ) '

Eq. (3.7-42) must be solved numerically.

However, as was shown earlier for Eq. (3.7-22), the right-hand side must always be positive or the potential will have an inflection at or near the sheath edge. Expanding the right-hand side in a Taylor series and neg!ecting the higher-order terms, this equation will also produce the Bohm sheath criterion and specify that the ion velocity at the sheath edge must equal or exceed the ion acoustic (or Bohm) velocity. An examination of Eq. (3.7.42) shows that the Bohm sheath criterion forces the ion density to always be larger than the

electron density through the pre-sheath and sheath, which results in the physically realistic monotonically decreasing potential behavior through the sheath.

Figure 3-4 shows a plot of the sheath thickness d normalized to the Debye length versus the potential drop in the sheath normalized to the electron temperature. The criterion for a Debye sheath derived in Section 3.7.1 was that the potential drop be much less than the electron temperature ( e4 << kTe ), which is on the far left-hand side of the graph. The criterion for a Child- Langmuir sheath derived in Section 3.7.3 is that the sheath potential be large compared to the electron temperature ( e$ >> kTe ), which occurs on the right- hand side of the graph. This graph illustrates the rule-of-thumb that the sheath thickness is several Debye lengths until the full Child-Langmuir conditions are established. Beyond this point, the sheath thickness varies as the potential to the 312 power for a given plasma density.

The reason for examining this general case is because sheaths with potential drops on the order of the electron temperature or higher are typically found at both the anode and insulating surfaces in ion and Hall thrusters. For example, it will be shown later that an insulating surface exposed to a xenon plasma will self-bias to a potential of about 6Te , which is called the floating potential. For a plasma with an electron temperature of 4 eV and a density of 10'8m'-3, the Debye length from Eq. (3.7-12) is 1.5 x 10-5m. Since the potential is actually

8 7 6 5

$ 4

0 3 2 1 0

0 2 4 6 8 10 12 14 16 18 20

x ^eglkT,

Fig. 3-4. Normalized sheath thickness as a function of the normalized sheath potential showing the transition to a Child-Langmuir sheath as the potential becomes large compared to the electron temperature.

significantly greater than the electron temperature, the sheath thickness is several times this value and the sheath transitions to a Child-Langmuir sheath.