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Why plasma processing? (1)
Accurate etching of fine features
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The Coburn-Winters experiment
Ion bombardment greatly enhances chemical etching
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Why plasma processing? (2)
• Plasma enhanced chemical vapor deposition (PECVD)
• Sputtering
• Ashing
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Low-temperature plasma physics
• Plasmas are collisional
• At least 3 species: ions, electrons, neutrals
• Always boundaries and sheaths
• Many effects not present in hot plasmas; e.g.
dissociation of molecules
electron attachment to form negative ions
charged particulates (dust)
interaction with surface layers
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The nature of sheaths
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The sheath’s E-field accelerates the ions
and makes them go straight
+
PLASMA
+
+
Cl atom
+
+
SHEATH
+
+
+
+ ION
Photoresist
Silicon
or SiO2
Substrate
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The Debye length and quasineutrality
E  V
  D     E  e(ni  ne )
2V  (e /  0 )(ne  ni )
2
(
n

n
)
e
eV
 L2 e i
KTe
 0 KTe
F
 KT I
G J
Hn e K
Poisson eqn.
Gradient scale length L
1/ 2
Define
D
0
e
2
e
Then
eV
L2  ni 
 2 1  
KTe D  ne 
The LHS cannot be very large, so ni ~ ne unless L ~ D
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Sheaths are very thin
numerically,
Let
Then
Te (eV )
 D  7.4
ne (1018 m3 )
m
ne  1011 cm3 (1017 m3 ), Te  4 eV
D  50  m
Debye sheaths are approximately 5D thick
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The Child-Langmuir Law
When the potential of an electrode is very negative, the
sheath drop is so large that ne can be neglected near the
electrode. The sheath thickness then follows a simple law.
ions
only
quasi-neutral
ions and
electrons
d  V 3/4
+
+ + 
+ 
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Presheath and Bohm criterion
For a sheath to form, the ions entering the sheath must have a
minimum velocity of cs = (KTe/M)1/2, or E = ½KTe. This means
that there must be a presheath where ions are accelerated.
Vb
V= 0
Plasma
n = n0
V = Vs
Vp
Presheath
PROBE
-V
Debye
sheath
C-L
sheath
x = xs = 0
n = ns ~ no/2
b
d
Presheaths are hard to calculate, so we assume a sheath edge at x = xs.
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Log (V) vs. x (exact calculation)
100
Exact
C-L
Vf
-eV / KTe
10
1
0.1
0.1
1
x D
10
The Child-Langmuir slope is not followed unless V is very large.
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Applying a voltage
• The sheath barrier for electrons at the wall
or large electrode must be about 5 KTe to
make ion and electron currents equal.
• If two walls are at different potentials, the
more negative one will have a larger sheath
and smaller electron current.
• The plasma follows the potential of the
most positive electrode. It must always
be more positive than the walls.
• If the voltages oscillate, the electron
current will flow alternately to one or the
other electrode.
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Neutral collisions
Ions and electrons make “billiard ball”
collisions with neutral atoms, so the i-n
and e-n cross sections are about the
same.
The e-n collision rate is
 en  nn   v 
where the average is over the electrons’
Maxwellian distribution
An atom has a radius of about 10-8 cm
(= 1Ǻ or 0.1 nm), so the cross section
is about 10-16 cm2. A hydrogen atom
has a  of 0.88 Ǻ.
Particles diffuse by random walk.
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Resonant charge exchange
HOWEVER, ions and atoms of the same
species have much larger cross sections
because of charge exchange.
Suppose a fast ion encounters a slow
neutral. An electron can simply jump
from the neutral to the ion, making a slow
ion and a fast neutral. The ion appears to
have suffered a large collision even if the
energy exchange is very small, so the
cross section is very large.
Charge exchange cross sections, e.g. Ar+ Ar, can be 100 times larger (~ 10-14 cm2).
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Momentum transfer collisions in Argon
100.00
Argon Momentum Transfer Cross Section
1.00
(10
-16
-3
cm )
10.00
<< Ramsauer minimum
0.10
0.01
0.00
0.01
0.10
1.00
Electron energy (eV)
10.00
100.00
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Collision frequencies
 en  nn   v 
 en (Ar)  12.55e(3.85/ T ) / TeV 0.036
eV
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Argon ionization
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Ionization cross sections
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Charge exchange cross sections
J.W. Sheldon, Phys. Rev. Lett. 8, 64 (1962)
Argon
Xenon
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Charged-particle (Coulomb) collisions
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Coulomb collisions with and w/o a B-field
No B-field
Together, these cause a 90 scatter a factor ln L
times more often than a single 90 scatter.
In “weak” B-field
“Spitzer” resistivity
These are for 90 deflections
in multiple collisions.
The Coulomb logarithm ln L
can be approximated by 10.
Electrons driven through a plasma by an E-field are slowed down by
collisions with ions, resulting in this resistivity:
Density cancels out and plasma resistivity is independent of n.
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When are electron-ion collisions important?
Electron-neutral collisions:  en  nn  v
nn  3.3 1013 p(mTorr)
en
 nn en ve
ve  vth,e  (2KTe / m)1/2
  1016 cm2
1/ 2
6 107 TeV
cm/ sec
Then
Electron-ion collisions:
(ln L ~ 10)
Hence,
For 3 mTorr and Te = 3 eV, ncrit  2  1011/cm3
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Mobility and Diffusion ( || to B or B = 0)
u j    j E ,  j  e / M j jn , ( j  i, e)
u is the drift velocity due to an
E-field, and  is the mobility
u j   D j n / n , where D j  KT j / M j jn
Here u is the drift velocity in a
pressure gradient, and D is the
diffusion coefficient
The fluxes to the walls are:
Quasineutrality
requires Gi = Ge
An E-field will set up to retard the electrons and accelerate the ions.
This is not in the sheath; it is in the collisional body of the plasma.
Result:
Γ   Dan
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Diffusion and mobility perpendicular to B
Cyclotron frequency
Larmor radius
Diffusion and mobility across B is slowed
down by a factor ~ wc2/2, which can be
large for electrons but is usually negligibly
small for ions.
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The Simon Short Circuit Effect (1)
B
i
e
e
e
i
e
i
i
In a magnetic field, ambipolarity does not have to be
obeyed in either the || or the  direction.
More electrons will flow to the endplates, and more
ions to the sidewalls.
Only the total fluxes have to be equal.
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The Simon Short Circuit Effect (2)
B
e
e
e
e
e
e
The sheath drop at the endplates can vary with radius,
allowing a few more electrons to leave at large r than
at the center. Electrons appear to have moved radially
outwards, although they are lost axially.
The ambipolar field is not observed. The electron
density tends to be Maxwellian even in the r direction.
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Particle balance in gas discharges
Ions diffuse out normally at the
Bohm rate, and electrons follow by
the short circuit effect. The g’s are
geometric factors (r, L, etc.)
Ion-electron pairs are replenished by
ionization. Here V is the volume,
and the <v> is the ionization rate.
Equating input and output, we see
that the plasma density n cancels
out, leaving only a relation
between pressure and electron
temperature.
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Te rises as pressure decreases
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Power balance in plasma sources
The energy leaving the plasma is the
sum of three terms.
This is the energy carried out by
each ion leaving the plasma through
the sheath.
This is the energy carried out by
each electron leaving the plasma,
including the perpendicular part.
1.61
Ec  23exp(3.68/ TeV )
Wc is the energy lost by line radiation and
used in ionization. It is the function
Ec(Te), which is the energy required to
make an ion-electron pair (next slide).
The density produced at given RF power
absorbed is Wtot times the loss rate of ions
through the wall sheaths.
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The Vahedi curve
1000
Ec (eV)
Argon
100
10
1
KT e (eV)
10
This includes all losses in inelastic collisions
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