1    Capacitively coupled discharges — basic characterisation
Basic literature: [9, 1]
I  < A
2   RF plasma conductivity
1 v + \uj
£0 ^pe — uj2 + wo)
-0 E2 ne2 ->    o -Š
3   RF. sheath
Literature: [1, 8]
L).
1
a
m(v + ílj)
m(v + \<jj) ne2 + ibJEQiniv + iu)
n
meo
Ohmic heating:
. .      1 1    n     . .       E2 ne2v
(p) = -jlEl cos a. = -El Re(a) = — ,2+(J2)
1
DC sheath voltage:
eno    8kTe ^-euah(r)/(kTe) 4   V TTm
en0
Let us assume Ush = Uq + U\ sin cut:
2tt 2^ J
0
2irm
We will use
2tt
^     /      sin ojt
2tt J
0
dt = I0(a),
where In (a) is the modified Bessel function of the order zero. We will obtain
kTP      2 \2irmJ
In
\kTe
(8)
4   Series plasma-sheath resonance
^6
sh
eoiüS
ieoSio
.2
Slot
-Slot
•4e ,.,2
The resonance (Z ~ 0) occurs for the frequency
ujsr  — Wpe y ——, (9)
where /(, is the length of the bulk plasma, stot is the total thickness of the both sheaths and / = h + stot is the distance of electrodes.
2
5   Discharge asymmetry
Symmetrical discharge:
**
zem
Asymmetrical discharge: i
<5
zem
First approach:
u.
1/k
We will use Ush oc sK and Ush oc ^— oc ^s, so that Ush oc   3^   and we will obtain
Ush. oc
The index (7 marks the sheath at the grounded electrode, the index v sheath at the powered electrode and the index e will mark the voltage at the powered electrode. We can write
uv	(Sg
	\sv
	k
a =	
	k —
(10)
e (i;4),
frequently a pa 2 (valid for matrix sheath, k = 2). The DC component of the voltage at the powered electrode can be calculated by
1
Ue0  =  UgO - UVQ  = —Uel
1
Sy
(11)
where the notation U = Uq + U\ sinwt was used.
3
Second approach:
E(x)
— / nl{y)Ay £0 J 0
s(t)
U.
sh
e
£0
For rij
da; J nj(y)dy 0 0
konst. we would get
eriiS2 Q2
(12)
(13)
U.
sh
2eo 2enl£()S2
where Q = eriiSs. We can write the eq. (13) by means of the total sheath charge (Q) even in a more general case. We will define £ = x/s and we will label the actual averaged ion concentration in a sheath n7:
U.
sh
Q2
2e iii Eq s2 l 5
2 fat
0 0
X
(14)
Because the total charge in both sheaths Qm = Qg(t) + Qv(t) is constant and because each sheath must almost collapse during each RF period, the maximum charge of each sheath is approximately equal to Qm- we can write
uK
u,
Qg=0
Qg(0
r
Qg=Q
r
g max
2eniq eqS2
Q
m
2e nlv e0 S2
T
and we can define the asymmetry parameter £
£
U,
g max
u„
iv -£g ig X;
QV=Q
M
Qv(t)
(15)
(16)
(17)
For the bias calculation we will mark Ue = Ueo + UeRF(t), where Ueo is the spontaneously generated DC component of the voltage at the powered electrode (bias) and UeRF is the known RF voltage supplied from the RF generator. When we neglect the bulk-plasma voltage, we can write Ue = Ug — Uv. We can take advantage of the situation in the two extremes of the supplied
4
voltage
U,
g max
-u„
and we can get
U,
eO
RFmax ;    =    UeO + UeRFmin
Ue RF max     &Ue ftp min
For symmetrical supplied voltage {UeRp1 will get
l+S
-UeRFmin, e.g. UeRF
u,
eO
-u,
e RF max
l-S
i + s
(18)
UeRFmaxS'mUjt) we
(19)
6   Electric asymmetric effect
For symmetric discharges the eq. (18) has the following form:
Ue RF max      Ue min
u,
eO
Consequently, if we use asymmetric waveform of the supplied voltage, we can generate an electric asymmetry (DC bias) in spite of geometric symmetry of the discharge [2].
u
e rf max
e rf
<Uerf>=0 Ue rf min
-<UV>
Electric asymmetrical effect for a strongly asymmetric supplied voltage.
An example of the electric asymmetric effect for UeRp = U cos (cut + <l)) + U cos(2ujt).
5
7   Nonlinear nature of sheaths
Literature: [3, 5]
For rii = konst. we can approximate
nes £0
nes2
Ush =
2e0
dE ds 7 = £0 -r- = ne — J dt dt
The two simplest cases are:
1) Monofrequency current I = Iicosut:
One sheath:
h
Sneoj
(sin cut + 1)
E/ah   =   -^—(-zr-)        + sin Lut-1-cos 2ujt) (20) eoen \bujJ    \4 4 /
Two sheaths:
C/e(o;i)   = - Uv(ut + ir) (21)
1 //ix2
3/1      1\     / 1      1\             1/1 1\ t I t^o"    TTo- I + h^ + ^r   smw(--; h^-^r   cos 2ut
A\SI    SV     \SI    SV A\SI SZ
(22)
1-1 ^
Ueo   =   -\uel (23) 2) Monofrequency voltage Ush = Uq + U± cosujt:
8   Ion energy distribution function (IEDF)
Basic parameters:
• Ratio between the mean transit time of an ion through the sheath and the RF period:
Tj     3su>   / rrii
T      2tt V 2eU0
• Number of ion collisions inside the sheath pa 1/ (i^Ti)
(26)
Literature: [7]
8.1    Collisionless low-frequency regime
Tj <C T, ion energy corresponds to the actual sheath voltage in the moment of ion impact. Let us assume Ush = Uq + U\ coswt.
|di|
e (C/o + U\ cosuot) dE
l
E-eU0
(27)
£7   G   (e(tfo-tfl); e([/0 + C/i))
8.2   Collisionless middle-frequency regime
The saddle structure narrows with the increase of the Tj/T ratio.
e(U0+Ul)
7
8.3   Collisionless high-frequency regime
Ti » T, for ^ —> oo the saddle structure fg transforms itself to a single peak at the energy E eU0.
I Ei
- - ^(1)
E   E   (eUo-AE; eU0 + AE)
8.4 Collisions
• elastic collisions - continuous decrease of ion energy
• charge transfer - generation of new peaks in the EEDF (/ei)
• broadening of the ion angle distribution
(28)
9   Matching box
For example [1]:
stray
7^ C-
10   Local/non-local plasma character
Local regime:
j(r,t)   = a{r,t)E(r,t)
fE(r,t)   = fE\E(r,t)
Nonlocal regime:
dp j d/' rr ( F- r'. I - I') /•; (/'
//'</
fE(r,t)   =   fE E(r',t'
r' £ V, t' < t
8
11   Plasma heating
• collisional (Ohmic) heating [9, 1, 6]
• stochastic heating [9, 1, 16]
— bounce resonance [12]
• field reversal [15]
• 7-heating, a and 7 regimes [14, 18, 11, 13] 11.1   7 regime
Potential emission - electron emission from electrode caused by an ion impact, ~ 0,01. Electron avalanche in the sheath:
da;
" [E(x,t)] je
fa[E(x',t)]dx' 3e{x)    = je(0)eO
Je(0)    = 7Ji
s(t)
J a[E(x,t)]dx
0'i)   =    ( 13i { e 0 + envb (29)
For negligible uvb we can use the last equation for calculation of the ignition voltage for the transition of the sheath from the a to the 7 regime. The current density for this transition can be estimated by means of
ds      £0 dU
]   =   ne — pa--— (60)
J dt       s   dt v '
LUU!
31   ~ £0-
s
a->7 transition:
• increase of electron concentration, plasma conductivity and current density
• decrease of sheath thickness, generation of a discharge structure that is analogous to the structure of the DC glow discharge
• discharge contraction to a smaller area, VA characteristics with a constant discharge voltage
• increase of supplied power
• in some cases an abrupt transition with hysteresis
• EEDF variations (shift to the Maxwell EEDF, increase of electron concentration, decrease of electron temperature)
9
12    Global models
Input parameters: pressure, electrode distance (/), angular frequency of the electric field (lu), RF current amplitude (ii), gas composition (Ki, Ei, v, Kexc, Eexc, Kei, mn) Output parameters: electron concentration (n), electron temperature (Te), mean sheath thickness (s) [1].
• Balance of the number of electrons:
nnnKl (I - s)   =   2hincuB (31) Ki   =   Kl0e-^ (32)
ub   =   \ — (33) V m
(nn is the concentration of neutrals, n is the means electron concentration in the bulk plasma, Ki is the rate constant for ionization, nc is the electron concentration in the discharge center, hi is the ratio between the electron concentration at the bulk-sheath border and in the plasma center, ub is the Bohm velocity, Ei is the ionization energy of neutrals.)
• Balance of the mean electron energy:
77 (.Rohm
+ 2Rstoch + 2RohmiSh) I( = 2hlncuBET{Te)S (34)
ET   =   El + ^Eexc + —^kTe + 2kTe + eA$ (35) K% mn A,
Rstoch   =   0.72 (mkTe)1/2 (36)
e/i
/      \ 3/2
Rohm   =   1.55W(/-2s) (^-J     (Se0skTe)1/2 (37)
lus
Rohm,sh   =   0.33mi/s— (38) eh
{Rohmi Rstoch and R0hm,sh are resistivities caused by the collisional heating in the bulk plasma, stochastic heating and collisional heating in the sheath, Ex is the mean energy supplied to one electron, Kexc is the rate constant for excitation of neutrals, Eexc is the excitation energy, Ke\ is the rate constant for elastic collisions between electrons and neutrals, mn is the mass of neutrals, A<J> is the average voltage that must be overcome by an electron that leaves plasma.)
Sheath thickness:
' h
12eh2n2c£0kTe \S<
(39)
The equations listed above are valid for low-pressure plasma with no negative ions and with no collisions in the sheaths.
10
13   Independent control of the reactive species concentration and ion energy
• DC + RF [19, 10]
( >
• Capacitive biasing of an electrode in a different discharge (ICP, MW)
• Double-frequency CCP [1, 4]
• Electric asymmetric effect [2, 4, 17]
14   CCP ignition
P
n
Breakdown voltage in the high-pressure branch:
dn
~dt dn
~dt
U
> 0
D
. d2n dx2
77-centr Sin
K\pe
-K2
i_p
In
K2Pl < Pi i
ifiTlTT2 D
12
Used marking of quantities
ojpi plasma frequency of ions
cj (angular) frequency of el. field
ujpe plasma frequency of electrons
/ electrode separation
A wavelength
a plasma conductivity
n electron concentration
e elementary charge
m electron mass
v mean collisional frequency of transition of momentum of the electron (to neutrals)
Eq vacuum permittivity
p power density
j current density
jl current density amplitude
E electric field intensity energy
Ei amplitude of electric field intensity
tiq electron concentration at the plasma-sheath border
rii ion concentration
k Boltzmann constant
Te electron temperature
Ush sheath voltage
m, ion mass
s sheath thickness
Uo DC voltage component
U± amplitude of the fundamental voltage component
Io modified Bessel function of the 1. kind, order 0
Zf, bulk plasma impedance bulk plasma length
S electrode
Zsh sheath impedance
Csh sheath(s) capacity
stot total thickness of both sheaths
Z discharge impedance
ivsr (angular) frequency of the plasma-sheath resonance
Ug sheath voltage (at the grounded electrode)
Uv sheath voltage (at the powered electrode)
Ue powered electrode voltage
k power in the dependence of the sheath voltage on the sheath thickness
a phase difference between current and voltage
power in the dependence of the electric and geometric discharge asymmetry
13
	designation of the CCP regime with negligible role of 7-electrons
	1. Towsend coefficient
Sg	grounded electrode area
Sv	powered electrode area
Q	el. charge
Q9	sheath charge at the grounded electrode
Qv	sheath charge at the powered electrode
Qm	total charge in both sheaths
$	el. potential
Ux max	maximum value of the voltage Ux
Ux min	minimum value of the voltage Ux
UeRF	RF component of the powered electrode voltage (i.e. Ue — Ueo)
1	quantity that describes profile of ion concentration inside a sheath
8	discharge asymmetry parameter
I	electric current
Ti	mean transit time of an ion through a sheath
T	discharge period
Vi	mean collisional frequency of an ion
ÍEi	IEDF
ÍE	EEDF (electron energy distribution function)
Vi	ionization frequency
D	coefficient of diffusion
References
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[10] V. A. Lisovskiy, N. D. Kharchenko, and V. D. Yegorenkov.   J. Phys. D: Appl. Phys., 41:125207, 2008.
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