Low Temperature Plasma Technology Laboratory

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1 Low Temperature Plasma Technology Laboratory The Floating Potential o Cylindrical Langmuir Probes Francis F. Chen and Donald Arnush Electrical Engineering Department LTP-15 May, 1 Electrical Engineering Department Los Angeles, Caliornia UNIVERSITY OF CALIFORNIA LOS ANGELES

2 The loating potential o cylindrical Langmuir probes Francis F. Chen* and Donald Arnush** Electrical Engineering Department, University o Caliornia, Los Angeles Los Angeles, Caliornia ABSTRACT The loating potential o a cylindrical probe is computed numerically, and the results are itted to analytic unctions. They dier signiicantly rom the plane approximation. The ormula normally used to calculate the space potential rom the measured loating potential is derived or plane probes and is erroneous when applied to cylindrical probes in the low-density plasmas (n < 1 1 cm -3 ) used in industrial plasma processing. The loating potential V is that at which the collected ion and electron luxes are equal. I A p is the probe area, n the density in the body o the plasma, and V = the potential there, the electron lux ΓA p to the probe is I = A nv exp( V / KT ), v ( KT / π m). (1) e p th e th e [Note: I total particle current or plane probes and current per unit length or cylindrical probes; the electrical current ±ei is not used here.] For a plane probe, the ion current is given by the Bohm criterion at the sheath edge, deined as the point where the ions have an inward drit velocity c s, having allen to the potential V = V sh = ½ KT e, where the density n is n s = n exp( ½). Thus, Ii = Apnscs = αapn cs, cs ( KTe/ M), () M being the ion mass and α has the value exp( ½) =.61. A spread in ion energies can bring α closer to the convenient value o.5. Setting I i = I e yields the usual ormula or the loating potential: ev 1 M = ln 5.18 KTe α πm in argon. (3) The ion collection area or a cylindrical probe, however, depends on the radius R sh o the sheath, which is not known a priori. In this case, there is no need or the artiice o a sharp sheath edge, since solutions o Poisson s equation can be extended to ininity. Two collisionless theories are available or calculating V(r): the Bernstein-Rabinowitz 1 (BR) theory, which takes into account the angular momentum o the ions which orbit the probe; and the Allen-Boyd-Reynolds (ABR) theory, which neglects orbiting, so that ions move only radially and axially. Chen 3 has recently shown that the BR theory overestimates the ions angular momentum in partially ionized plasmas because o collisions in the presheath.

3 Hence, or plasma processing purposes, we shall employ the ABR equation as modiied by Chen or cylindrical probes: where η -½ ξ = Jη ξe η ξ ξ, () ( ) ½ η ev / KT, ξ r / λ, λ ε KT / n e. (5) The normalized ion current J is deined by e D D e 1 I 1 J i π n λ c. (6) For each value o J, Eq. () can be solved to yield η(ξ) or all ξ. The constraint that the probe be loating can be expressed as ollows. The radius ξ p o a probe at loating potential can be ound rom the condition I i = I e at the probe surace, where D s Ie = πrpnvthexp( η ), Ii = πrpnpvi. (7) Here n p is the ion density at the surace o the loating probe, and v i is the ion velocity there, given rom energy conservation by ½ v = ( η ) c. (8) Setting I i = I e yields n p i s M η π m ½ ( η ) = n Substituting Eqs. (8) and (9) into I i and I i into Eq. (6) gives exp( ). (9) so that 1 M η J = ξ p e π m, (1) η ξ p M = ln J π m. (11) Solution o Eq. () yields the potential distribution η = η( J, ξ ). (1) Integration o Eq. () is non-trivial, and care must be taken to join smoothly to the quasineutral solution at large radii. For each J, Eqs. (11) and (1) give two curves whose intersection yields a pair o values (η, ξ p ), as illustrated in Fig. 1. Varying J generates the unction η (ξ p ), shown in Fig. or argon, which approaches the plane limit o I we now deine α J / ξ p, (13)

4 1 3 8 η, η 6 ξ p sheath edge ξ Fig. 1. The potential proile η(ξ) ( ) and the loating potential condition η (ξ) ( ) or the case J = 1, ξ p = 15 in argon. The Bohm criterion is met at the sheath edge where η = ½ ( ) α 3 η, α /α ξ p Fig.. Decrease o η ( ) with decreasing ξ p = R p /λ D due to the increase in sheath area as measured by α ( ) and α /α ( ). The line through the η points is an analytic it. Eq. (11) takes the same orm as Eq. (3), with α in place o α. Thus, rom Eq. (), αa p is the eective collection area o a loating probe, and the ratio α /α expresses the expansion o this area as ξ p is decreased. The unctions α and α /α are also shown in Fig.. There is no need to deine a sheath edge ; but i one is deined at the radius R sh where η = ½, as in Fig. 1, conservation o current requires I i = πr sh n s c s. However, n s is not.61n as in the plane case, since quasineutrality has not been assumed at R sh, and n i n e there. Using Eqs. (13) and (6), we can conveniently express the ion current to a loating probe in terms o the unction α(ξ p ):

5 Ii = πrpαn cs, (1) with α acting as an eective Bohm coeicient. The ollowing analytic its to the computed curves may be useul or probe analysis: 1 = 1 + 1, (15) ( η ) ( Aln ξ + B) ( Cln ξ + D) p p where A =.583, B = 3.73, C =.7, and D = 5.31; and α Rsh G 1 Eexp( Fξ p ) α R = +, (16) p where E =, F = 7.1, and G =.96. In the plane probe limit ξ p, η approaches the value o 5.18 or argon, and α and α/α approach.61 and 1, respectively. In the range ξ p = 1 1 commonly encountered in r discharges, η is o order or argon, signiicantly less than the usual value o 5.. The reason is that the sheath thickness at V causes a cylindrical probe o given area to collect more ion current than a plane probe, and thus the sheath drop has to be lowered to permit more electron low. FIGURE CAPTIONS Fig. 1. The potential proile η(ξ) ( ) and the loating potential condition η (ξ) ( ) or the case J = 1, ξ p = 15 in argon. The Bohm criterion is met at the sheath edge where η = ½ ( ). Fig.. Decrease o η ( ) with decreasing ξ p = R p /λ D due to the increase in sheath area as measured by α ( ) and α /α ( ).The line through the η points is an analytic it. REFERENCES * chen@ee.ucla.edu ** darnush@ucla.edu 1 I.B. Bernstein, and I.N. Rabinowitz, Phys. Fluids, 11 (1959). J.E. Allen, R.L.F. Boyd, and P. Reynolds, Proc. Phys. Soc. (London) B7, 97 (1957). 3 F.F. Chen, Phys. Plasmas (1). F.F. Chen, Plasma Physics 7, 7 (1965).