Velocity Distribution of Ions Incident on a
Radio-Frequency Biased Wafer
G. Wakayama and K. Nanbu
Institute of Fluid Science, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai, Japan 980-8577
Abstract. The ion velocity distribution (IVD) is important in plasma etching of microfeatures. IVD at a rf
biased wafer is studied, first analytically using probability theory and then numerically by using a particle
simulation method. The analytic expression shows that IVD is governed by the parameter qVrf/miU)l, where
q is the charge of ion, VTf is the rf bias amplitude, uj is the rf bias angular frequency, / is the penetration
depth of bias potential, and mi is the mass of ion. The analytical expression is applicable to the case when
the ion collisions in the penetration depth are negligibly few and the rf period of biasing is much shorter
than the time that ions take in traversing the depth I. The IVDs for general conditions are also examined
using the self-consistent particle-in-cell/Monte Carlo simulation.
I
INTRODUCTION
Weakly ionized plasmas used in materials processing generally consist of positive and negative ions, electrons,
and neutral species. In the case when electrons are more abundant than negative ions, a negative potential
is formed near the surface of the floating wafer immersed in the plasma. In processing plasmas the electron
temperature is few electron volts, while the ion temperature is roughly equal to the background gas temperature.
To keep the balance between the negative and positive currents on the floating surface, low energy electrons
should be reflected in the sheath region back to the plasma bulk. This is the physical reason why the potential
of the floating wafer is lower than the plasma potential.
Negative sheath potential (measured from the plasma bulk) accelerates the positive ions to the wafer. If
no collision is assumed in the sheath region, the positive ions impinge almost perpendicularly onto the wafer.
However, electrons are decelerated and lose the velocity component normal to the wafer.
When ions and electrons come into a microscopic trench, electrons are easily sticked near the top of the
side of the trench wall while ions can reach the bottom of the trench. This kind of charge separation inside
microfeatures causes a charge build-up. The strongly localized electric field due to the charge build-up distorts
the trajectories of incident ions, which results in an anomalous etch profile called "notching". The localized
charge build-up also causes the electrical breakdown induced by fatal tunneling currents through gate-oxides.
In order to suppress the so-called charging damages, new ideas on plasma sources are proposed, e.g., timemodulation of the source power [1] and use of ultra-high frequency (500 MHz) power [2]. These approaches are
intended to lower the electron temperature and hence generate more negative ions. To understand the role of
negative ions, let us consider the plasma including only positive and negative ions; when the wafer is biased
at low frequency (< 3MHz), the positive and negative ions impinge on the bottom of the trench alternately.
Thus no charge build-up occurs.
The ion energy distribution (IED) and ion angle distribution (IAD) of ions that impinge on a biased wafer
are important to clarify the mechanism of etching and charging damage of microfeatures. In low gas pressure
and high plasma density, the rf wafer biasing is employed for controlling incident ion energy independently of
the source power. Sobolewski et al. [3] measured the lEDs for various bias amplitudes and frequencies in a
planar inductively-coupled plasma (ICP) reactor. Hoekstra and Kushner [4] calculated the lEDs in an ICP.
They discussed the effects of the bias amplitude, gas pressure, and source power on the IED. Notching can be
explained from charging of the surfaces of microfeatures. Kinoshita et al. [5] presented the relationship between
the charging potential and the IED. Hasegawa et al. [6] reported that an appropriate rf biasing suppresses the
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
254
charging damage.
Nasser et al. [7] found that the penetration depth of the potential oscillation due to rf biasing is much larger
than the Debye length. Thus the IED is greatly affected by the potential oscillation. In this report we consider
two approaches for obtaining the IVD at a rf biased wafer. First, using the probability theory, we obtain the
IVD analytically under certain simplified situations. The obtained expression shows that the IVD is governed
by a similarity plasma parameter. Second, using self-consistent particle-in-cell/Monte Carlo simulation, we
determine the oscillatory sheath potential due to rf biasing and hence obtain the IVDs.
II
ANALYTICAL ION VELOCITY DISTRIBUTION
First we obtain the IVD analytically using the probability theory [8]. Ahn [9] measured the instantaneous
plasma potential in the ICP. The potential that is time-modulated by high frequency biasing is well approximated by the exponentially attenuating sine curve. In this case, the velocity of ion normal to the wafer, vw,
can be expressed as vw = vw+vw, where vw is the contribution from a time-averaged potential and vw is that
from an oscillatory part of the potential [10]. For high frequency biasing, we found that the time when the
ion impinges on the wafer is uniformly distributed. Assuming no collision, we can then obtain the probability
density function G(vw) for the time- modulated velocity vw
G(vw) =
_L——
(1)
where A is given by
A*
rri[(jjl
.
(2)
Here mi is the mass of ion, q is the charge of ion, Vrf is the amplitude of the rf bias, I is the penetration depth
of the biasing potential, and u is the bias angular frequency. The probability density function H(vw) for vw
can be obtained by assuming that ions in the plasma bulk are in equilibrium. Now the probability density
function F\\(vw) for the sum vw = vw + vw is given by the convolution of G(vw) and H(vw)
F\\ (vw) =
G(vw - v)H(v)dv.
(3)
To check the validity of the analytical IVD, we separately determined the IVD from numerical solutions of the
equation of motion and compared the two results. The best fitting of Ahn's [9] experimental data of potential
becomes 0(x, i) = 4.5 + 0.019x + 2.4 x 10~4x2 — 16exp(— x/2.1) + 55exp(— x/2.8) sin(o;t), where x in units of
mm is the normal distance measured from the wafer at x = 0. The rf bias frequency is 15 MHz. We calculated
the motion of ClJ ions. Figure 1 shows the comparison of the analytical and numerical IVDs. The analytical
expression shows a good agreement with the numerical solution. In Fig. 1, the distance between the two peaks
is 2A(= 2qVrf/m[ujl).
Ill
SELF-CONSISTENT PIC/MC SIMULATION
It is clear that the IVD is a functional of the sheath potential (/>(x, t), which depends on the wafer biasing and
collisions. In the case when there is no experimental data on 0(x, t), it is necessary to calculate 0(x, i) and IVD
simultaneously. In order to examine the effects of bias frequency and ion collisions on IVD, the self-consistent
particle-in-cell/Monte Carlo simulation is employed.
The background gas is argon which is assumed to be uniform and in equilibrium at temperature of 300 K.
Although e~, Ar + , and Ar are taken into consideration in the present calculation, we have only to examine
the motion of e~ and Ar+ . The velocity of Ar is necessary only when determining the postcollisional velocity
of e~ or Ar+ .
Since the number density of charged species is less than 1017 m~3 in the present problem, Coulomb collisions
(e~-e~, Ar+-Ar+ and e~-Ar + ) can be neglected and therefore we consider e~-Ar and Ar+-Ar collisions. For
e~-Ar collisions, elastic collision, ionizing collision, and also 25 exciting collisions are taken into consideration
255
Ion Velocity Distribution (km−1s)
OS
rH
I
Cross Section σk (10−20 m2)
I
NumericalSol.
Sol.
Numerical
Analytical
Sol.
Analytical Sol.
OS
+3
*0
O
6.5
0U
1010
−2 o
rS1010 ~2
2
I 06.5
o
2
i
o
I 44
s>>
10
77
7.5
7.5
Ion
Ion Velocity
Velocity(km/s)
(kni/s)
U
4
−4
1010 −2
n-2
101<T
88
elastic
ionization
excitation
0
1010°
2
10102
4
10 104
Electron
ε (eV)
ElectronEnergy
Energy
e (eV)
FIGURE
FIGURE 1.
1. Ion
Ion velocity
velocity distributions
distributions(no
(nocollision).
collision).
FIGURE
sections
forfor
e−e~-Ar
–Ar collision.
FIGURE2.2.Cross
Cross
sections
collision.
based
and
Allen’s
[12][12]
based on
on aa set
set of
ofthe
the cross
crosssection
sectiondata
dataofofKosaki
Kosakiand
andHayashi
Hayashi[11].
[11].AsAsforforionization,
ionization,Peterson
Peterson
and
Alien's
integral
number
of of
integral cross
cross section
section isis multiplied
multipliedby
by0.9
0.9totomatch
matchititwith
withKosaki
Kosakiand
andHayashi’s
Hayashi'sdata.
data.The
Thetotal
total
number
collisional
27); k k==1–25
forforexcitation,
k =
2626
forfor
collisional events
events isis 27.
27. Each
Each event
event isisexpressed
expressedby
byk k(k(k==1,1,2,2,· ·•·•, -,27);
1-25
excitation,
k
=
−
ionization,
–Ar collision
2. 2.The
k for
ionization, and
and kk =
= 27
27 for
for elastic
elastic collision.
collision. The
The cross
crosssection
sectionσa^
fore e~-Ar
collisionis isshown
shownininFig.
Fig.
The
probability
of
occurrence
of
the
kth
event
in
time
∆t
is
probability of occurrence of the kth event in time At is
2ε 1/2
1/2
PPe (k)
=
N
σ
(ε)
∆t,
(4)
B
k
Ai,
e (fc)=AT B a fe (m
£ )(^)
(4)
e
where
is the number density of background argon gas, ε is the electron energy, and
the
mass.
e is is
where N
NB
andmra
theelectron
electron
mass.
B is the number density of background argon gas, e is the electron energy,
e
+
Elastic
collision
and
resonant
charge
exchange
are
considered
for
the
Ar
–Ar
collision.
The
simple
+
Elastic collision and resonant charge exchange are considered for the Ar -Ar collision. The simpleand
and
computationally
totoreproduce
computationally efficient
efficient model
model proposed
proposedininour
ourprevious
previouswork
work[13],
[13],the
theuse
useofofwhich
whichwas
wasshown
shown
reproduce
the measured data of ion drift velocity and transverse diffusion coefficient, is employed. The model is a
the measured data of ion drift velocity and transverse diffusion coefficient, is employed. The model is a
combination of isotropic scattering and charge switching, both occurring with the same probability. The
combination of isotropic scattering and charge switching, both occurring with the same probability. The
probability Pi that an ion Ar++ collides with atom Ar in time ∆t is given by
probability PI that an ion Ar collides with atom Ar in time At is given by
Pi = NB ḡπd2AB
2 ∆t,
Pi = NBg7rd
+
ABAt,
(5)
(5)
where ḡ is the relative velocity between Ar + and Ar averaged over the Maxwellian distribution for Ar with
where g is the
relative
velocity
between
and cross
Ar averaged
over asthelarge
Maxwellian
for Ar with
temperature
TB (=
300 K)
and πd2AB
is the Ar
collision
section twice
as πd2BB , ddistribution
BB being the viscosity
temperature
T
(=
300
K)
and
^d^
is
the
collision
cross
section
twice
as
large
as
7nig
,
^BB
being
the
viscosity
B
B
diameter of Ar.B
diameter
of
Ar.
−
+
One time step of the simulation consists of solving the equation of motion for e and Ar , +obtaining the
One density,
time step
of thethe
simulation
consists of
the equation
of The
motion
for e~ and isArcalculated
, obtaining
charge
solving
Poisson equation,
andsolving
calculating
collisions.
time-evolution
untilthe
charge
density,
solving
the
Poisson
equation,
and
calculating
collisions.
The
time-evolution
is
calculated
until
a periodic steady state is obtained.
a periodic
steady
state
is
obtained.
The one-dimensional computational domain from the biased wafer (x = 0) to the plasma bulk (x = L) is
The in
one-dimensional
computational
domainproblem
from thearebiased
the plasma
= L) is
shown
Fig. 3. The unknowns
in the present
the dcwafer
bias V(xdc =at0)
thetowafer
and thebulk
flow (x
velocity
shown
in x
Fig.
3. The
The electron
unknowns
the present
problem
are to
thethe
dcion
biasflow
V^cvelocity
at the wafer
and
flowcurrent
velocity
of
ions at
= L.
flowinvelocity
at x =
L is equal
because
of the
no net
of the
ionsplasma
at x = bulk.
L. The
electron
flow
velocity at xfrom
= Lthe
is equal
to the
ion flowthe
velocity
because
of noone
netperiod
current
in
The
bias Vdc
is determined
floating
condition;
net current
during
+
−
in
the
plasma
bulk.
The
bias
V^
is
determined
from
the
floating
condition;
the
net
current
during
one
c
of rf biasing is zero in periodic steady
state. Ar + and e come from x > L and cross the presheath edgeperiod
at
of
rf
biasing
is
zero
in
periodic
steady
state.
e~the
come
fromdensity
x > L isand
cross theatpresheath
x = L. The flow velocity is determined in
such Ar
a wayand
that
plasma
continuous
x = L. edge at
x=
L. length
The flow
is determined
in such
a way
plasma
density
continuous
= direction
L.
The
L ofvelocity
the computational
domain
is 100
mm,that
as is the
shown
in Fig.
3. Letisthe
x–axis be at
in xthe
The length
of thewith
computational
is 100
is shown
in Fig.grids
3. Let
the =
x-axis
be inThat
the direction
normal
to the Lwafer
the origin atdomain
the wafer.
Wemm,
usedasequally
spaced
of ∆x
0.1 mm.
is, the
normal to
wafer into
with 1000
the origin
thethat
wafer.
usedlength
equally
of Ax = density
0.1 mm.of That
is,−3the
domain
wasthe
divided
cells. at
Note
theWe
Debye
is spaced
0.11 mmgrids
for electron
1016 m
16
domain
was
divided
into
1000
cells.
Note
that
the
Debye
length
is
O.llmm
for
electron
density
of
10
m~3
and electron temperature of 2 eV. The electron time step ∆te was determined so as to satisfy the condition
and electron
of 2 eV.for
The
electron
timethan
step0.03.
Ate This
was condition
determined
someets
as to the
satisfy
the condition
that
the total temperature
collision probability
electron
is less
also
requirement
for
that the method
total collision
probability
for electron
is less
than 0.03.events
This must
condition
also
meets
thewhere
requirement
Nanbu’s
[14] that
the maximal
probability
of collisional
be less
than
1/27,
27 is thefor
−11than 1/27, where 27 is the
Nanbu's
method
[14] that events.
the maximal
of collisional
be10less
total
number
of collisional
From probability
the requirement
we choseevents
∆te =must
9.26 ×
s at the background gas
−11
−11
total number
collisional
we chose
9.26
x×
10~
s at
thepBbackground
gas
pressure
pB =of
5 mTorr,
∆teevents.
= 5.29 From
× 10 thes requirement
at pB = 10 mTorr,
andAt∆t
2.65
1011
s at
= 20 mTorr.
e =
e =
n
+
The
ion time
∆ti was
∆tin s=at
30∆t
, for
which the
ArAt
–Ar
probability
than
pressure
pB =step
5mTorr,
Atechosen
= 5.29toxbe10~
pB e=
lOmTorr,
and
2.65 x 10"
s at pB is=less
20mTorr.
e = collision
The ion time step Ati was chosen to be Ati = 30Ate, for which the Ar+-Ar collision probability is less than
256
Electric Potential φ (V)
0
Substrata
O
LL==100
100mm
mm
x
−5
5 mTorr
mTorr
10
10 mTorr
20 mTorr
−10
-10
Bulk-Presheath
Boundary
00
50
50
Distance
Distance x
x (mm)
(mm)
FIGURE
FIGURE3.3. One-dimensional
One-dimensionalcomputational
computationaldomain.
domain.
100
4
20mTorr
mTorr-""
20
4
Ion Flux (1018 m−2 s−1 )
Elastic Coll. Rate (1023 m−3s−1 )
FIGURE
FIGURE 4.
4. Electric
Electric potential
potential φ.
(f>. Wafer is at
at x
x = 0.
'o
10mTorr
10
I2 2
O
U
3
mTorr
55mTorr
0o
00
50
50
Distance
(mm)
Distance xx(mm)
mTorr
55 mTorr
10 mTorr
2
20 mTorr
mTorr
20
x,
0
100
100
50
50
0
100
100
Distance x
x (mm)
(mm)
Distance
FIGURE5.5. e−
e~-Ar
elasticcollision
collisionrate.
rate.
FIGURE
–Ar elastic
FIGURE 6.
6. Ion
Ion flux.
flux.
FIGURE
0.05.Since
Sincethe
theplasma
plasmabulk
bulkisischarge-neutral,
charge-neutral, aa common
common value
value of
of 10
1016 m
m−3 was
was chosen
chosen as
as the
the ion
ion and
and electron
electron
0.05.
densities
in
the
plasma
bulk.
In
processing
plasmas
the
electron
temperature
T
is
much
higher
than
densities in the plasma bulk. In processing plasmas the electron temperature Tee is much higher than the
the ion
ion
temperature
7]
due
to
the
power
deposition
to
electrons.
Therefore
we
set
Tj
=
500
K
and
k&T
=
1
eV
Q
temperature Ti due to the power deposition to electrons. Therefore we set Ti = 500 K and kB Te = 2 eV in
in the
the
plasmabulk
bulk(x(x>>L),
L),where
wherek&B
is the Boltzmann constant.
plasma
B is the Boltzmann constant.
IV RESULTS
RESULTS AND
AND DISCUSSION
DISCUSSION
IV
Thevalidity
validityofofthe
themethod
methoddescribed
describedabove
above was
wasfirst
first examined
examined by
by calculating
calculating the
the structure
structure of
of the
the potential
The
potential
in
the
case
of
no
rf
biasing,
the
wafer
being
floating.
Figure
4
shows
the
electric
potential
</>.
in the case of no rf biasing, the wafer being floating. Figure 4 shows the electric potential φ. The
The region
region
L(=100
100mm)
consistsofofthe
the sheath
sheath plus
plus presheath.
presheath. In
In the
the presheath
presheath the
the charge
xx<<L(=
mm) consists
charge neutrality
neutrality is
is kept.
kept. Note
Note
thatthe
thepotential
potentialininthe
the plasma
plasma bulk
bulk isis set
set zero.
zero. Assuming
Assuming no
no collision
collision for
for ions
ions in
in the
the sheath
sheath and
that
and presheath
presheath
resultsininthe
thefloating
floatingpotential
potentialφfa on
on the
the wafer
wafer as
as [15]
[15]
results
f
z
1/2
mi v/
1/2
Tlee£ - T e l n( —-!-)
m
i
&
=
—
(6)
(6)
φf = − 2 − Te ln V 27rme / ..
2
2πme
As is shown later, the electron temperature in x < L is almost constant and close to the value in the plasma bulk
As is shown later, the electron temperature in x < L is almost constant and close to the value in the plasma bulk
(2 eV). The floating potential fa obtained from eq. (6) is —10.4 V. The floating potential in Fig. 4 is —12.29 V at
(2 eV). The floating potential φf obtained from eq. (6) is −10.4 V. The floating potential in Fig. 4 is −12.29 V at
the background gas pressure PB = 5 mTorr, —12.58V at p& = 10 mTorr, and —12.77V at p& = 20 mTorr. The
the background gas pressure pB = 5 mTorr, −12.58 V at pB = 10 mTorr, and −12.77 V at pB = 20 mTorr. The
corresponding flow velocity at x = L is 327 m/s at 5 mTorr, 112m/s at 10 mTorr, and —124 m/s at 20 mTorr.
corresponding flow velocity at x = L is 327 m/s at 5 mTorr, 112 m/s at 10 mTorr, and −124 m/s at 20 mTorr.
The negative flow velocity at 20 mTorr is caused by a considerable increase of the number of electrons and
The negative flow velocity at 20 mTorr is caused by a considerable increase of the number of electrons and
257
Number Density ni, ne (1016 m−3)
Ion Density ni (1016 m−3)
1
®
rH
O
rH
00.55
I'
Cfl
mTorr
55mTorr
10mTorr
mTorr
10
20mTorr
mTorr
20
i
Q
§
0.4
10mTorr
mTorr
10
,0.2
0.2
CD
Q
0
ni
ne
0
I
0
50
50
Distance xx (mm)
(mm)
Distance
100
100
Electron Temperature (eV)
Ion Velocity Distribution (km−1s)
55mTorr
mTorr
10
10mTorr
mTorr
20
20 mTorr
mTorr
2
1
I
SH
S 0.5
CJ
O
I0
§
44
FIGURE 8.8. Number
Number densities
densities ofof ion
ion and
and electron
electron
FIGURE
(10mTorr).
mTorr).
(10
FIGURE 7.
7. Ion
Iondensity.
density.
FIGURE
1
22
Distancexx(mm)
(mm)
Distance
0
00
44
55mTorr
mTorr
10
10mTorr
mTorr
20
20mTorr
mTorr
0
00
88
Ion
Ion Velocity
Velocity (km/s)
(km/s)
FIGURE 9.
9. Ion
Ion velocity
velocity distribution.
distribution.
FIGURE
50
50
Distance
Distancexx(mm)
(mm)
100
100
FIGURE
FIGURE10.
10. Electron
Electrontemperature.
temperature.
ions
–Ar and
ions in
in xx << LL due
due to
to ionization.
ionization. The
The difference
difference of
ofφ0ff between
betweenour
ourdata
dataand
and eq.
eq.(6)
(6)isisascribable
ascribabletotoe−
e~-Ar
and
++
−
Ar
–Ar elastic
Ar –Ar
-Ar collisions
collisions in
in xx <<L.
L. Figure
Figure 55shows
shows the
the rate
rate for
foree~-Ar
elastic collision.
collision. The
Therate
rateisisproportional
proportionalto
tothe
the
gas
gas pressure
pressure and
and the
the electron
electron density
density nnee approximately.
approximately. The
The rate
rate decreases
decreaseswith
withxx because
becausenne edecreases
decreaseswith
with
x,
x, as
as isis shown
shown later.
later. The
The ionization
ionization rate
rate isisabout
about 44orders
orderssmaller
smallerthan
thanthe
the elastic
elasticcollision
collisionrate.
rate. As
Aswe
wehave
have
seen,
seen, however,
however, ionization
ionization has
has aa large
large effect
effect near
near the
the outer
outer boundary
boundary ofofthe
the computational
computationaldomain.
domain. Also
Alsoion
ion
(electron)
–Ar
(electron) flux
flux isis not
not constant
constant due
due to
to ionization.
ionization. This
Thisisisshown
shownininFig.
Fig.6.6.Electrons
Electronsare
aredecelerated
deceleratedby
bye−
e~-Ar
collisions
collisions and
and ions
ions are
are by
by Ar
Ar++–Ar
-Ar collisions.
collisions. The
The fact
fact that
that φ0ff determined
determined from
from the
the simulation
simulation isislower
lowerthan
than
++
the
–Ar
the collisionless
collisionless φ0ff given
given by
by eq.
eq. (6)
(6)means
meansthat
that the
the collisional
collisionaldeceleration
decelerationisisstronger
strongerfor
forAr
Ar
-Arcollision
collisionthan
than
−
ee~-Ar
–Ar collision,
collision, thus
thus requiring
requiring aa lower
lower φ0ff to
to accelerate
accelerate ions
ions and
and satisfy
satisfy the
the floating
floating condition
condition on
on the
the wafer.
wafer.
The
The larger
larger collisional
collisional deceleration
deceleration for
forAr
Ar++–Ar
-Ar isisdue
due to
to charge-exchange
charge-exchangecollisions.
collisions.
Where
i . The
Where isisthe
the boundary
boundary between
between the
the sheath
sheath and
and presheath?
presheath? Figure
Figure77shows
showsthe
theion
iondensity
densitynn\.
Theelectron
electron
density
density nnee almost
almost agrees
agrees with
with nn\i except
except the
the region
region close
close to
to the
the wafer.
wafer. This
This region
region isis enlarged
enlarged inin Fig.
Fig. 88 at
at
ppBB = 10
mm. This
10mTorr.
mTorr. We
We see
see that
that the
the charge
charge neutrality
neutrality breaks
breaks near
near xx == 22mm.
This location
location isis the
the boundary
boundary
between
between the
the sheath
sheath and
and presheath.
presheath. Figure
Figure 99 shows
shows the
the ion
ion velocity
velocity distribution
distribution function.
function. The
The distribution
distributionisis
more
more localized
localized as
as pressure
pressure and
and hence,
hence, the
the ion
ion collision
collision frequency
frequency decreases.
decreases. The
Theupper
upperlimit
limitininthe
theright
rightside
side
of
of the
the peak
peak represents
represents the
the ion
ion velocity
velocity in
in the
the case
case of
of no
no collision.
collision. We
Weshow
show the
the electron
electron temperature
temperaturein
inFig.
Fig.10.
10.
The
The change
change in
in the
the temperature
temperature isisvery
very small
smallininthe
the presheath.
presheath. As
Aspressure
pressureincreases,
increases,the
theelectron
electrontemperature
temperature
−
in
–Ar collisions.
in the
the presheath
presheath slightly
slightly decreases
decreases due
due to
to the
the increase
increase of
of the
the rate
rate of
of inelastic
inelastic ee~-Ar
collisions. The
The rapid
rapid
decrease
near
the
wafer
is
due
to
the
sharp
decrease
of
the
potential
in
the
sheath.
decrease near the wafer is due to the sharp decrease of the potential in the sheath. The
The ion
iontemperature
temperatureisisalso
also
258
i
i
i
I
i
i
i
i
Electric Potential φ (V)
Electric Potential φ (V)
i
00
x" __ ————— - ———— - ——— ---- ——— - ———
-0-
X/
15
MHz
15MHz
3
1 −50
-50
-4-5
o
g−100
3-100
-4-5
/
/
•
/
/
/
−50
-50
0
π/2
π
3π/2
37T/2
/
/
. /
−100
3-100
7
5
Distance
Distance xx (mm)
(mm)
(a)
(a) 15
15MHz
MHz
1 MHz
o
0
———— π/2
°
πTT
------....
3π/2
67T/2
Q __ /O
0
0
10
10
/
y
0
5
Distance
Distance xx (mm)
(mm)
(b)
MHz
(b) 11MHz
10
10
0
Ion Density ni (1016 m−3)
Time Averaged Potential φ̄ (V)
FIGURE
mTorr).
FIGURE 11.
11. Time-modulation
Time-modulation of
ofpotential
potential φ$ (10
(lOmTorr).
−20
-20
−40
-40
15
MHz
15MHz
11MHz
MHz
H −60
-60
1
0.5
15 MHz
1 MHz
0
00
50
50
Distance
Distance xx (mm)
(mm)
100
100
0
50
50
Distance
Distance xx (mm)
(mm)
100
100
FIGURE
mTorr).
FIGURE 13.
13. Ion
Ion density
density (10
(lOmTorr).
FIGURE
mTorr).
FIGURE 12.
12. Time-averaged
Time-averaged potential
potential φ̄<£ (10
(lOmTorr).
almost
almost constant
constant in
in the
the presheath.
presheath. These
These results
results show
show that
that the
the present method is
is applicable
applicable to the numerical
analysis
analysis of
of the
the sheath
sheath and
and presheath.
presheath.
We
We then
then applied
applied our
our method
method to
to the
the case
case when the floating
floating wafer is under rf biasing. We fixed
fixed the
the gas
pressure
pressure at
at 10
10mTorr
mTorr and
and examined
examined the
the effect
effect of
of the
the bias
bias frequency
frequency on
on the
the sheath structure. Our choice is
15
MHz. The
15MHz
MHz and
and 11MHz.
The peak-to-peak
peak-to-peak voltage
voltage applied
applied to
to the wafer is fixed
fixed at 100
100 V.
V. Figures
Figures 11(a)
11 (a) and
and (b)
(b)
show
show the
the time-modulation
time-modulation of
of the
the potential
potential φ(x,
4>(x, α),
a), where α(=
a(= ωt)
ut) is the phase of the bias potential. As is
shown
mm. Figure 11
shown later,
later, the
the sheath
sheath and
and presheath boundary is
is near
near xx =
= 55mm.
11 shows
shows that
that the
the sheath
sheath is
is timetimemodulated
modulated at
at both
both low
low and
and high
high frequency.
frequency. Figure
Figure 11(a)
11 (a) shows
shows that
that in
in case
case of
of high
high frequency
frequency not
not only
only_ the
the
sheath
sheath but
but also
also the
the presheath is somewhat time-modulated. Figure 12
12 shows
shows the time-averaged potential φ̄(x).
0(x).
The
V for
V
The effect
effect of
of the
the bias
bias frequency
frequency on
on φ̄<^> is
is small.
small. We
We see
see that
that the
the dc
dc bias
bias is
is −59.27
—59.27V
for 15
15MHz
MHz and
and −57.86
—57.86V
for
for 11MHz.
MHz. Figure
Figure 13
13 shows
shows the
the ion
ion density.
density. The
The higher
higher ion
ion density
density at
at 15
15MHz
MHzisisprobably
probablydue
dueto
to aalarger
larger rate
rate
of
of ionization.
ionization. The
The ion
ion density
density isis hardly
hardly time-modulated,
time-modulated, as
as isis shown
shown in
in Figs.
Figs. 14(a)
14(a) and
and (b).
(b). Strictly
Strictly speaking,
speaking,
the
the ion
ion density
density is
is slightly
slightly time-modulated
time-modulated at
at the
the low
low frequency,
frequency, as
as is
is seen
seen from
from Fig.
Fig. 14(b).
14(b). The
The electron density
nnee agrees
agrees with
with the
the ion
ion density
density nnii in
in the
the presheath. However, ne is time-modulated in the sheath, as is shown
in
in Figs.
Figs. 15(a)
15(a) and
and (b).
(b). The
The electron
electron density
density isis non-zero
non-zero at
at the
the phase
phase of
of π/2,
?r/2, at
at which
which the
the wafer
wafer works
works as
as an
an
instantaneous
instantaneous anode.
anode. The
The time-averaged
time-averaged electron
electron and
and ion
ion densities
densities are
are shown
shown in
in Figs.
Figs. 16(a)
16(a) and
and (b).
(b). The
charge
charge neutrality
neutrality breaks
breaks at
at xx =
= 55mm,
mm, which
which is
is the
the edge
edge of
of the
the sheath.
sheath. Although
Although the
the plasma density is
is higher
higher
at
at 15
15MHz,
MHz, we
wecan
can see
see no
no effect
effect ofofthe
the bias
bias frequency
frequency on
on the
the sheath
sheath thickness.
thickness.
259
0.4
0.4
Ion Density ni (1016 m−3)
Ion Density ni (1016 m−3)
0.4
0.4
15
MHz
15MHz
0.2
0
π/2
π
3π/2
CD
Q
fl
o
0
0
5
Distance
Distancexx(mm)
(mm)
(a) 15
15MHz
MHz V
'
(a)
11MHz
MHz
0.2
0.2
0
π/2
π
3π/2
0
10
10
0
5
Distance
Distance xx(mm)
(mm)
(b) 11MHz
MHz V
'
(b)
10
10
15MHz
15
MHz
0.2
MHz
11MHz
0
0.2
I
°'2
CP
Q
Number Density ni, ne (1016 m−3)
0.4
0.4
e
I
°'2
cp
H
Electron Density ne (1016 m−3)
0.4
0.4
Q
0
π/2
π
3π/2
0
π/2
π
3π/2
g
0
10
5
10
0
5
Distancexx(mm)
(mm)
Distance xx(mm)
(mm)
Distance
Distance
(a) 15 MHz
(b) 1MHz
(a) 15 MHz
(b) 1 MHz
FIGURE 15. Time-modulation of electron density (lOmTorr).
FIGURE 15. Time-modulation of electron density (10 mTorr).
0
Number Density ni, ne (1016 m−3)
Electron Density ne (1016 m−3)
FIGURE14.
14. Time-modulation
Time-modulationofofion
iondensity
density(10
(lOmTorr).
FIGURE
mTorr).
10
10
<? 0.4
0.4
<? 0.4
0.4
s
S
15MHz
15
MHz
0.2
0.2
MHz
11MHz
0.2
0.2
CD
0)
Q
Q
ni
0
0
ne
55
Distancexx(mm)
(mm)
Distance
(a) 15
15MHz
MHz
(a)
10
10
^
ni
00
00
ne
55
Distance xx(mm)
(mm)
Distance
(b) 11MHz
(b)
MHz
FIGURE16.
16. Time-averaged
Time-averagednumber
numberdensities
densitiesofofion
ionand
andelectron
electron(10
(lOmTorr).
FIGURE
mTorr).
260
10
10
Ion Velocity Distribution (km−1s)
0.4
0.4
1
15MHz
15
MHz
1 MHz
1 MHz
0.2
Q
>>
0.2
O
I
§
0
0
10
20
10
20
Ion Velocity
(km/s)
FIGURE 17. Ion velocity
distribution
function
(lOmTorr).
Ion Velocity (km/s)
0
FIGURE 17. Ion velocity distribution function (10 mTorr).
Lastly, the ion velocity distribution function is shown in Fig. 17. The gas pressure is lOmTorr as before.
Owing to Ar+-Ar collisions, the function does not take the form of twin peaks in Fig. 1. Since the highest
Lastly,
ion velocity
function
is shown
Fig.
17.of the
Thepeak
gas can
pressure
is 10interpreted
mTorr as from
before.
velocitythe
is realized
in the distribution
case of no collision,
however,
the in
right
side
be partly
Owing
to Ar+ –Ar
collisions,
function
does
take ofthe
twin
Since
theeq.highest
the analytic
expression
of eq.the
(3).
In Fig. 1,
the not
location
theform
rightofpeak
is peaks
near toinvwFig.
+ A.1.Since
A in
(2)
velocity
is realized
in the case
collision,
however,
the right
side of thethe
peak
can
be partly interpreted from
decreases
with increasing
a;, of
thenopeak
in Fig.
17 is shifted
with increasing
bias
frequency.
the analytic expression of eq. (3). In Fig. 1, the location of the right peak is near to vw + A. Since A in eq. (2)
decreases with increasing ω, the peak in Fig. 17 is shifted with increasing the bias frequency.
v CONCLUSION
The IVD was obtained using the self-consistent
particle-in-cell/Monte Carlo simulation. In the case of no rf
V CONCLUSION
biasing the dependence of IVD on the background gas pressure was investigated. We found the IVD is more
localized toward the upper side of the distribution as pressure decreases. We then calculated the oscillatory
The IVD was obtained using the self-consistent particle-in-cell/Monte Carlo simulation. In the case of no rf
sheath potential due to rf biasing. The IVD under wafer biasing is broad due to ion collisions. The shape of
biasing the dependence of IVD on the background gas pressure was investigated. We found the IVD is more
IVD in high velocity regime is analogous to the analytic IVD for no collision; the location of the peak is shifted
localized
toward
the as
upper
side frequency
of the distribution
to a lower
velocity
the bias
increases. as pressure decreases. We then calculated the oscillatory
sheath potential due to rf biasing. The IVD under wafer biasing is broad due to ion collisions. The shape of
IVD in high velocity regime is analogous to the analytic IVD for no collision; the location of the peak is shifted
to a lower velocity as the bias frequency increases.
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