3,- ôOtJatt úl .Inewr.
-4
fi^j.        / .niujjj&s ÍL^nÉL/áuvmdu, Afe#t ^ ^ ****** •
0~ q/n B - ^-.m/rb^xZ , -/ 'C)
'gh. Ä /nu^uf^yi^/TTeLs AL
n; ->
3.12
P&fM d&tfdbd Oto Ml ýonJc^KAtLj      £22- ^ y,p ^ Q
3 -    ýřyj >*j
ô*   . cUMka (kw&u, -
3>4-3 (hn^A-^u dilate _
c-
r, - /"e
E- ^'~^
■VÍL ^>vl-
ŕd-t
dp
1*1
0
'3.2-  Rto&wj ctija&u''9i&PU4£ ^
3. 2, j     J D &K>' tMT. B^JurU
y QU r^-r
r o/ŕ - x dx1
(X-
' AS
dl
1   Z      r-rJL ^
t = - ±    «)   í - 'oJL
fx
dx2
3>ť /
JL-
-o TT
1
/n
vát ,^^í//é^6>
í—' O"2
a -Q
tóO--
IT'S
- /4? ha-jL ten ^—
p
1n
4%
■ram
ir) ^Äiýy ^   WTUÁJi-iUL iwyvilo  /^tdmo,^ oM&tIa
O
ßv&- Oviiy*£i*<7if otguiť /^^^ ...........< 0
Ts   = ĚlL ^   & Z? Ts
a,
Onto- Q
—>   .ÓO —•> CO
jUJhcÄ tyle- o&uß&ff. fiC^nJ^Uj sn^xatf A.pJh^ptismf .
on, s
2)
3
p
Al. 2
0(0)
3.3,i fn&dt£ji^^iMwjí
f)
i 1
>y$ts)Ľ knelt' V/z,      ■=>     M?.=<U>A>E ^
(Jö- od^xJWTčUt ^Idý? . ) k% ßiAwi&Lsu jr-ß dusené/ ßoeiM\> ßaAlensi*?? - ssil ü>y?, ji./nt £ ~t V/ty - 0 'fiter lernM^ /nžvm&^njL.   /wmôu' u^n/una.í-Li A>" M^^^lj^m^p^A^
4    y   mA TT M, die    = ^8
/T"
L , dm
7ľ dx.
4    n     Ír, ^'f/'
jiéUU
oč0 = í/33
6ÍZ
~ A   )     3   /n,lo) Co
* '210 ' 1     / ^4^^^^
4
1>U
dkr jt äkvvwMaUn*"' i&rvL^j kž /f f&tyé: HQPuiS ^Mň^ne m&xl- U&tj   ^^"^ A-
it
rMuki   &  a/ ^^/^^ ^ ,  (fasuji jJúsel**'^ d^O^u' ^/«^W MC!>«^\
sn<íMM> }
y/     _  j.      C        "h   rt>M   i <-<Jc
/
/TKAJ j fíi £     // X-> Y/n,
Dl-a  "  —-■---
^aô- <^_i>i' í>^ie jkkjF^ /pQ. faßt, /^ô^"^ ^ cLenüwK^u1      1)^ = Ife [/j-j- ž5~ ^) (.Udy stn^hsrü! /r^ Mi- sv*^- /Mí ^ /^uiýc^Ä ' ,ffe
77 « Tq   l-M   Dj cv ± DdJL,
DIFFUSION ACROSS A MAGNETIC FtELD 141
The density gradient points radially inward, and the ambipolar electric field, to contain the weakly magnetized ions, also points inward. When an electron gyrating around a line offeree suffers a collision, it changes its direction, which would tend to move its center of gyration, on the average, by a gyration radius r^. This process is random, and therefore diffusive, with rce replacing Ae as the diffusion mean tree path when rce < Ac.
To derive the perpendicular diffusion coefficient, we write the perpendicular component of the fluid equation for either species from (2.3.15):
0 = qn(E + ui X B0) - kTVn - mnv^as_
where we have again assumed an isothermal plasma and taken vm sufficiently large that the inertial (time-derivative) term is negligible. It is convenient to express the vector equation in terras of the rectangular components (taken to be x and y):
dn
mnvmtix = qnEx - kT— + qnuyBo (5.4.1a)
and
mnfmur = qnEv — kT-r— qnuxB§ (5.4.16)
« nf ii =inrl D ftvwr, /« \ Ai ft 1        «Jn---i—
The assumption that the diffusion takes place only across the magnetic field is almost never satisfied. Even for finite length systems in which / (along Bt)) ?> cl (alross £o), the more rapid diffusion along So is usually important. We therefore confider the regime in which / — d, as shown in Fig. 5.4. For simplicity, rectangular coordinates are used and the y direction is taken to be uniform and of infinite extent, Since the walls are conducting, it is clear that the fluxes across and along Bit are coupled, and ambipolariiy requires only that the total electron and ion fluxes integrated over the wall surfaces to be equal.
Magnetic field
Conducting box
jL
/
/ ,
;
''S.
' Plasma .
FIGURE 5.4. A plasma-filled conducting box in a dc magnetic field, illustrating the calculation ofambipoiar diffusion in a magnetized plasma.
Thus, the ambipolar diffusion coefficients are
parallel to the field, and
on
W + Me
ju.|Dlc + figDu
(M + Me
3)
4) ta-
)SS
i a
3>
to
on is,
5)
(5.4.16)
(5.4.17)
perpendicular to the field. We see that the parallel diffusion is the same as the case without an applied magnetic field. However, (5.4.17) and/5.4.11) are not the same. Since tic 2> fii and normally Du & D±ti (5.4.17) simplifies to
With this approximation the diffusion equation (5.4.15) becomes
dn „ d2n „ d2n dt dz2 dx2
(5.4.18)
(5.4.19)
DIFFUSION ACROSS A MAGNETIC FIELD
145
such that the perpendicular loss of ions is by free (not ambipolar) diffusion alone. Physically this corresponds to a situation in which the electrons, flowing along field 1 ines, almost completely remove the negative charge that produces Ex. Since electrons preferentially flow out along the field and ions flow out perpendicular to the field, T; + rc and currents must flow in the wall.
If electron flow along field lines is impeded by inertial or collisional effects or if the axial sheath voitagaJ^varies with x, then there can be a substantial ion acceleration potentia]forf awn this case the perpendicular ion diffusion term in (5.4.14) is smaller than tneTnobility term and the preceding derivation of Di-d is invalid. There is experimental evidence (see Lieberman and Gottscho, 1994, Section VIII.D.2) and also computer simulations (Porteous et a!., 1994) that indicate the existence of these radial potentials in magnetized processing discharges such as ECR's (see Section 13.1). Measurements and simulations both show that ions are lost radially from the bulk plasma with a characteristic loss velocity of order the Bohm velocity «b = (eTe/M)^2. However, radial expansion of field lines might affect the results. If an electric fieldexists across field lines with magnitudes — Te/d, then we can estimate ~ (i^nT^/d. Then defining D±R through F±, = — Dlitdn/6x ~ Dl3n/d, we obtain
Tj
in place of (5.4.18). For d ~ I, this can lead to substantial perpendicular ion losses in magnetized discharges, as observed in ECR measurements and simulations.
It is well known that plasmas not in thermal equilibrium are subject to instabilities. This is a major subject of fully ionized, near collisionless plasmas, and is treated in detail in most texts on plasma physics (see, for example, Chen, 1984). Magnetic field confinement is one source of such disequilibrium that leads to various instabilities which tend to destroy the confinement. Large-amplitude disturbances can lead to turbulent diffusion, which has the upper limit of the Bohm diffusion coefficient,
Iii 16 B
(5.4.20)
The scaling with B makes Bobm diffusion increasingly important as a source of cross-field diffusion at high magnetic fields, since from (5.4.10), we see that classical cross-field diffusion scales as D L « 1 /S2. Bohm diffusion tends to be less important at high collisionality (low temperature and high pressure) both due to the comparative scaling of DB to D± and also due to the fact that high collisionality tends to inhibit some of the instabilities. We have not considered nonclassical diffusion in this text. The reader wishing to explore the subject further can turn to Chen or other texts on high-temperature plasmas.
144
DIFFUSION AND TRANSPORT
DIFFUSION ACROSS A MAGNETIC FIELD
145
The diffusion is obtained from the continuity equations for electrons and ions:
d2n
dt dz az axA ox
It ~    Ik? ~ * Jz ^ + 0jJ5? ~ ^Tx ("£J
(5.4.13)
(5.4.14)
Exact two-dimensional solutions to these two coupled nonlinear diffusion equations have not been obtained. Letting Vs± and Vs|| be the potential drops across the perpendicular and parallel sheaths, then because the plasma is surrounded by a conducting wall, the potential in the center can be estimated as
* ~ V,„ + ^BJ - Vs± + \Exd
Two limiting cases can be considered depending on the size of Ex. For\Exd 5 Tj, Che perpendicular mobility terms in (5.4.13) and (5.4.14) are small corrrpMed^fo the perpendicular diffusion terms. Dropping the mobility terms, as done by Simon (1959), multiplying (5.4,13) by p\ and (5.4.14) by p^ and adding the two equations, we obtain
dn It
ju.iDe + ptD\ d2n    ßiDLc + jisDjA d7n
+ jte     dz1 fa + fie
Thus, the ambipolar diffusion coefficients are
dx1
Pi +
paratlet to the field, and
'Pi + f*e
(5.4.15)
(5.4.16)
(5.4.17)
perpendicular to the field. We see that the parallel diffusion is the same as the case without an applied magnetic field. However, (5.4.17) and/5.4.11) are not the same. Since fie > pi and normally On ^ Dlt, (5.4.17) simplifies to
With this approximation the diffusion equation (5.4.15) becomes
dn
<92h
(5.4.18)
(5.4.19)
such that the perpendicular loss of ions <s by free (not ambipolar) diffusion alone. Physically this corresponds to a situation in which the electrons, flowing along field lines, almost completely remove the negative charge that produces Ex. Since electrons preferentially flow out along the field and ions flow out perpendicular to the field, Y\ ^ rc and currents must flow in the wall.
If electron flow along field lines is impeded by inertial or collisional effects or if the axial sheath vohageJil^varies with x, then there can be a substantial ion acceleration potentially sT\^ln this case the perpendicular ion diffusion term in (5.4.14) is smaller thanTfleTnobility term and the preceding derivation of £>j.a is invalid. There is experimental evidence (see Lieberman andGottscho, 1994, Section VIII.D.2) and also computer simulations (Porteous et al, 1994) that indicate the existence of these radial potentials in magnetized processing discharges such as ECR's (see Section 13.1). Measurements and simulations both show that ions are lost radially from the bulk plasma with a characteristic loss velocity of order the Bohm velocity hb = {eYs/M)y/1. However, radial expansion of field lines might affect the results. Ifan electric field exists across field lines with magnitude/?., — Te/rf,then we can estimate r±i — pLinTe/d. Then defining D±s through Tr, = -Dltldn/dx — D±aft/d, we obtain
in place of (5.4,18). For^ ~ /, this can lead to substantial perpendicular ion losses in magnetized discharges, as observed in ECR measurements and simulations.
It is well known that plasmas not in thermal equilibrium are subject to instabilities. This is a major subject of fully ionized, near collisionless plasmas, and is treated in detail in most texts on plasma physics (see, for example, Chen, 1984). Magnetic field confinement is one source of such disequilibrium that leads to various instabilities which tend to destroy the confinement. Large-amplitude disturbances can lead to turbulent diffusion, which has the upper limit of the Bohm diffusion coefficient,
1 Te
(5.4.20)
The scaling with B makes Bohm diffusion increasingly important as a source of cross-field diffusion at high magnetic fields, since from (5.4.10), we see that classical cross-field diffusion scales as D L « I /B2. Bohm diffusion tends to be less important at high collisionality (low temperature and high pressure) both due to the comparative scaling of 0B to Dx and also due to the fact that high collisionality tends to inhibit some of the instabilities. We have not considered nonclassica! diffusion in this text. The reader wishing to explore the subject further can turn to Chen or other texts on high-temperature plasmas.
146       DIFFUSION AND TRANSPORT
5.5   MAGNETIC MULTIPOLE CONFINEMENT
In magnetic mmtipole confinement, a set of alternating rows of north and south pole permanent magnets is placed around the surface of a discharge chamber. A typical configuration, with the rows arranged around the circumference of a cylindrical chamber, is shown in Fig. 5.5. In some cases, one or both cylindrical endwalis are also covered with rows of magnets. Commonly, each row is composed of a set of many permanent magnets (diameter - length ~ 1 inch, fl0 - I kG). The alternating rows of magnets generate a line cusp magnetic configuration in which the magnetic field strength B is a maximum near the magnets and decays with distance into the chamber, as shown in Fig. 5.5. Hence most of the plasma volume can be virtually magnetic field free, while a strong field can exist near the discharge chamber wall, inhibiting plasma loss and leading to an increase in plasma density and uniformity.
Magnetic Fields
The structure of the magnetic field can be understood by unwrapping the circumference to obtain the alternating periodic arrangement of magnet rows in rectangular geometry shown in Fig. 5.6. Assuming that each row of magnets has a width A < d, the separation of the rows, then By at y - 0 can be approximated as
FIGURE 5,5. Magnetic muitipoie confinement in cylindrical geometry, illustrating the magnetic field lines and the |B| surfaces near the circumferential wails.
MAGNETIC MULTIPOLE CONFINEMENT        147
N S N S
FIGURE 5.6.   Schematic for determining multipole fields in rectangular geometry.
where S is the Dirac delta function. Introducing the Fourier transform,
0) = £AB1 sin
(5.5.2)
and equating (5.5.1) and (5.5.2), then if we multiply by sin{irx/d) and integrate from 0 to d, we obtain the fundamental (m = 1) Fourier mode amplitude A,, such that
Bj.,Cv,0) = _~s,n — a d
(5.5.3)
Because V • B = 0 and V X B = 0 for y > 0, By] satisfies Laplace's equation
d%{ J2By,
dx2 dy1
= 0
(5.5.4)
The solution to (5.5.4) with boundary conditions that 5^ (x, 0) is given by (5.5.3) and that Byi(x, y -»co) is not infinite is
Sv.(*,)')=^sin^e-^ a d
From the z component of V X B = 0, we have
mx] _ 3Byl
dy dx
Using (5.5.5) in (5.5.6) and integrating with respect to y, we obtain
(5.5.5)
(5.5.6)
&x\(x, v) = - —^— cos — ^ d d
(5.5.7)
m
148       DIFFUSION AND TRANSPORT
MAGNETIC MULTIPOLE CONFINEMENT 149
The field amplitude is Bx = (B2xl 4- B)t)lf2. Using (5.5.5) and (5.5.7), we obtain
(5.5,8)
showing an exponential decay that is independent of x into the discharge column with decay length d/ir. The smooth B\ surfaces, as well as the alternating By] and Bs\ components can be clearly seen in Fig. 5.5. The higher-order Fourier modes with nonzero coefficients (m = 3, 5,..,) have even shorter decay lengths (d/3ir, d/5ir, ...), and their effect is negligible a short distance from the chamber wall. Thus, we expect this picture to hold at distances significantly greater than d/tr within the plasma chamber. Midway between the magnets (at x — 0, ±d,...), the magnetic field is zero aty =0 and rises to a maximum value
Br,
7t
.2^2
BQ
at v => 0.28 d, after which it decays exponentially with y. The diffusion across this region is important in determining the confinement properties of the multipoles.
Plasma Confinement
Experimentally (Leung et al., 1975, 1976), multipole fields have been found to have three important effects on low-pressure plasma confinement:
1. Hot electrons, having energies ~' dc sheath potential, can be efficiently confined, provided there is end confinement either with magnetic minors, multipoles, or negative electrostatic potentials. These, electrons, if created and trapped at low pressures (large mean free path compared to the discharge size) can be the main ionization source for a discharge.
2. Significant (but not large) improvements can be obtained in the confinement of the bulk (low-temperature) plasma in a discharge.
3. Significant improvements in radial plasma uniformity can be obtained.
The effects can, at least partly, be understood in terms of magnetic mirroring in the cusps as governed by (4.3.15). The energetic electrons that are not lost by moving parallel to field lines are mirrored as they move into the higher field near the cusp. Their velocity vectors with respect to the magnetic field at the wall are randomized within the central plasma chamber, where (4.3.15) does not hold. The number of reflections from the cusp then depends on the size of the "loss cone" angle in velocity space compared to the possible solid angle of 4tt within which the velocity vector can be found. At lower velocities (or higher pressures), the scattering can take place collisionally on the outward flight, greatly increasing the loss rate. Ambipolar fields also play apart, but in a complicated manner. The improvement in plasma uniformity
follows because the diffusion is inhibited in the region of strong magnetic field, as described in Section 5.4, Thus, most of the density gradient occurs at the plasma edge, where the diffusion coefficient is small, leading to a relatively uniform central region.
As an example (Leung et al., 1975), a low-pressure dc argon discharge was created in a 30-cm-diameter, 33-cm-iong chamber by primary energetic electrons emitted from a hot filament placed inside the chamber and biased at -60 V. With multipoles and at p = 0.8 mTorr, the energetic electrons were confined for up to 70 bounces within the chamber, and the plasma density was increased by approximately a factor of 100. Of this increase, roughly a factor of 30 was measured to be due to the increased confinement of the energetic electrons, and an additional factor of three increase was due to the improvement in confinement for the bulk plasma. However, in most processing discharges the ionization is not produced by a class of very energetic electrons, and the second and third effects listed above are most significant.
A useful concept to discuss confinement is the effective leak width w of a line cusp. If there are M cusps of width w, then the effective circumferential loss width is Nw and the fraction f{aii of diffusing electron-ion pairs that will be lost to the wall is
loss
TttR'
Mw < 2ttR
(5.5.9)
The boundary condition at the wall (y » 0) for the ambipolar diffusion of plasma within the field-free discharge volume is then
r\va!l — JlossnsKB
(5.5.10)
We return to the example in Section 5.2 of steady-state diffusion in a plasma slab of length / with an ionization source proportional to the density. The density profile is given by (5.2.15). Equating T(l/2) in (5.2.17) to rwa,s in (5.5.10), we obtain, for a thin sheath,
_ „ = tan — D*8 2
(5.5.11)
This transcendental equation for B must in general be solved numerically. However, 'Moss is not too small, such that the left-hand side of (5.5.11) stilt remains much greater than unity, then we can approximate j3 = ti/l on the left-hand side to obtain
tan
lit
(5.5.12)
This is the usual regime for most processing discharges. Taking the ratio of ns = «(//2) to «0 = n(0), and using (5.2.15) to substitute for tan(£//2) in terms of we
(Q)
®)   Seje /S- *nk*£rw /nVíc^í ?
_____-----■----       " k£lm4L{jlq ./Ut AfTÁtt
jj'   JLVg E -ŕ v/% =0
tfieU-cí&u nkw . pT)íí^l^y   —- /mnPiž^U^    ,,, čl2q^ fy$4 m&brM^ ßi^nA^uUJ^ Á/Au, A&riMŕ
Sly   - Vi2   ^p- h.3
"í
J)   íaM- ÍLyCde^ pA^e stán , ffnfa^ ßusnoi, at č^e > 'cbd^' cLwsu i^^f
"~~        /^WeW ^i-^.nlp^ihTu^y feáhmeč&w {Jeúd&iax&n
Aj ůétstL, j^utdioßj flMJiiSL j^Jj^&Mpm /vyučte? p^Um^"^'p&äu&( ß bstt ^ny^z^        , k»Su^S^ jtT a Lordu      . xV j    .    , , - , ^   ■    , i
k) ,-tcnM, ^WW" T,'-O \ ~>>C
H
—t
U&MJÍ0iIl&ÍA ■imiZoř        4pUU7Ust4 /lusty, .4JP~>C(J
&r kid, " (%Müm, ítóÁ ^é-
/% (x ) í /rb^
^ í \
dx1 '-STL*-     " J i
(A)
t
ÍL &rvímo M^^-^J áT
z i
n   ßcUimo b™J*™*w prúty, lžl2.4 (BcÁ^.pyfiM^il ,
x
---- , - —_-- / I-" - _
, Of) .Ä  *nUl ^^TfK - ,^r.^V- X
f
f
.JoXs- A&r>UsW ^       .tmAJ-othu jmMu^OrfU. /KM^mM^^ ; ůJjíerm dcnĹ^Ať 0íx) ,
fofyß, %ád&Lt clo 2. h^eL(
(kr,-
/fl,K   /n<Ĺ      Gif Ol/L
est k ryd^^e- M ,, .
i c/J,' ' ' 2Ha^-2M«s--A',
vjL^r jj-mii-mJi rtfMi&WLt ry"^^.       " 7 '   >^'^-—-f ^"-—--^-c^ , — - - - ^
■    few Si /nul. ££,h#g>^
sheatn-presheatb edge ts at the right (after Riemann, 1991).
14
~ pshwu&jp pjLdsi/'uUiL M-lASjs&^ts Jk &&-?uYu
$ 3X
/ t (Jic^m^^ Hilton
2.
0^pv0 *r />^a p © oJk /Kate djfriCu 0 \0)=0 a* 4<0 /Lr
1
\ M
' (u  \V? r toiter- H> Stifle,
^dk r>hm^s cse&jkáj M^4^/pmieĹ í.^tt dJj^u'' ödnUiwt 4v3j)
nu
CJX '
o
0 S ' r- ^i.^,
-JEck - ^ , /
fide. äPn^". ^ledminÁ^ ^ °
L........................................................—
/ä^W   /WEfcě &yti/ A%U£ í^r M&uôAt/ n&é?. Arnto-H^