Page 66 - Semiconductor Manufacturing Handbook

Geng(SMH)_CH06.qxd 04/04/2005 19:37 Page 6.5

PLASMA PROCESS CONTROL

PLASMA PROCESS CONTROL 6.5

C , L , and R represent the parasitic elements of the ground electrode. L is the self-inductance
GE GE GE W
of the cavity between the chamber wall and the outer surface.
The inherent stray capacitance C between the lower and upper electrodes and chamber side
chuck
wall is not included in the circuit model in Fig. 6.4. The electrical properties of the chamber (L and
W
C ) and the transmission line that powers the electrode do not change with respect to the input RF
chuck
power and plasma parameters. Specifically, C and the inductance of the rod L can be measured
chuck rod
with no plasma present. Obtaining a base pressure in the chamber and with an RF source set to a
power that is insufficient to ignite plasma but drives the chamber with a voltage, the electrical char-
acteristics of the chamber can be calculated from the measured voltage and current obtained from
the RF metrology.
w
jL rod +1/ j C chuck = V I m
/
w
m
V and I are the voltage and current measured by the RF metrology. V and I are the true voltage
m m e e
and current at the powered electrode. V , I , V , and I are all time-varying complex numbers. V and
m m e e e
I are related to V and I by
e m m
V = V − j L I
w
e m rod m
w
I = (1 −w 2 L C chuck ) I − j C chuck V m
m
rod
e
The electrical properties of the chamber (L and C ) and the transmission line have an
W chuck
insignificant role in the analysis of plasma for parallel plate systems and are often omitted. C is used
s
to define the equivalent capacitance of the plasma sheaths on the powered and grounded electrodes
and R to define the equivalent resistance of the two sheathes. Since the plasma is typically operated
s
at a frequency of 13.56 MHz, the dominant current-transport species in the plasma are the highly
mobile electrons. This is represented as a resistive component R that is defined by
b
md ν
R = b
b
Ane 2
where A = electrode area
d = bulk thickness
b
e, m = electron charge and mass
n = bulk plasma density
n = electron-atom collision frequency
The bulk resistance is inversely proportional to the mobility and density of electrons, and pro-
portional to the thickness of the bulk region. The sheath capacitance determines the effective sheath
thickness S as
0
C = Ae 2 S 0
/
0
s
where e is the vacuum dielectric constant and 2S + d = L the gap distance between the two elec-
0 0 b
trodes. The parallel sheath resistance represents the ion acceleration power (P ) in the sheath in terms
i
of the RF voltage drop (V ) across the sheath as
sh
R = V 2 P = V 4 I V
2
2
/
/
s sh i sh i dc
where I is the ion current to the electrode, and V is the dc voltage across each sheath.
i dc
Another way to account for ion power loss is through an equivalent series resistance in terms of
the discharge current (I)
2
2
/
/
R = 2 P I = 4 I V I = R s
i dc
s
i
1 + (w CR ) 2
s
s
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