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4 Modeling of a capacitive RF discharge

4.1 PIC MCC model for capacitive RF discharge

Capacitive radio frequency (RF) discharges are very popular, both in laboratory

research for the production of low-temperature plasmas, and industry, where they

are commonly used for thin film deposition and surface etching [Lieberman, 1994;

Raizer, 1995; Bogaerts, 2002]. The typical capacitive RF discharge consists of two

parallel electrodes, placed in a vacuum vessel. The electrodes are powered with

voltage from a RF power source. The working gas, fed into a system, gets ionized by

electrons, accelerated in the RF electric field, producing the weekly-ionized plasma

with an ionization degree of about 10-6–10-4. The typical distance between the

electrodes is of order 1-10 cm. The driving RF voltage is usually about 100-1000 V

with a frequency between 1 and 100 MHz. The pressure of the working gas,

depending on application of discharge, varies in the range of 1-1000 Pa.

In order to enhance the transfer of knowledge and insight gained in RF

discharge studies and make easier the comparisons between results obtained in

different experiments, the ‘Gaseous Electronics Conference Radio-Frequency

Reference Cell’ (GEC) was developed in 1988 [Hargis, 1994]. Now this standardized

set-up is used by a large number of experimental groups, working with capacitive

RF discharges. In Fig 4.1 we present the scheme of one such set-up, which is used

in the Laser and Plasma Physics Group, Bochum University for the investigation of

plasma-chemistry processes in low temperature plasmas [Bush, 1999; Bush, 2001;

Möller, 2003]. In this set-up the disk electrodes with spacing d = 4 cm are powered

with RF voltage with frequency 13.56 MHz. RF power input in the system varies in

the range of 5-100 W. Mixtures of methane and oxygen at pressures 10-1000 Pa are

normally used as the working gas. For the typical operation parameters a plasma

with density en ~ 109-1010 cm-3 and electron temperature eT ~ 3 eV is obtained in

the discharge.

4 Modeling of capacitive RF discharge

68

Although the experimental set-up for capacitive RF discharge seems to be

rather simple, the discharge itself is inherently complex. In such discharges the

physics of a non-equilibrium non-stationary plasma is combined with the

complexity of reactive plasma processes, including the surface interaction, which

makes the modeling of such systems a real challenge. Despite the numerous

experimental and theoretical studies performed on RF capacitive discharges (see, for

example, [Raizer, 1995] and references contained therein), the understanding of its

behavior is still far from complete. The appropriate model for capacitive RF

discharges should be able to resolve the dynamics of a non-Maxwellian bounded

plasma in a varying RF field coupled with the kinetics of chemical processes

between plasma species. The Particle-in-Cell model with Monte Carlo Collisions (see

Chapter 2) meets these requirements. Particle models were recently successfully

applied for modeling of RF discharges in helium [Surendra, 1990], hydrogen

[Vender, 1992] and argon [Vahedi, 1993a; Turner, 1993] and proved to be a

promising tool for simulation of such plasmas, providing insight into discharge

parameters which are difficult to measure experimentally.

Figure 4.1 Capacitively coupled plasma reactor. Figure from [Möller, 2003].

We applied the 2d3v PIC-MCC model, described in Chapter 2, to a capacitive RF

discharge in a methane-hydrogen mix, similar to one described in [Bush, 1999],


69

making special emphasis on accurate treatment of the relevant electron-neutral

collisions and their influence on the electron energy distribution. In the simulations

the initial electron density and temperature were chosen as en 0 = 1010 cm-3 and

eT 0 = 20 eV respectively. The mix of CH4 and H2 was used as a background gas. The

gas temperature, 500nT = K, and densities, = ⋅CHn4

14 -37 10 cm ,

= ⋅Hn2

14 -39.2 10 cm , were chosen close to those used in [Busch 1999], the total

pressure of the background gas was p = 11.24 Pa (0.085 Torr). A rectangular

domain with the length Dd Y λ= = =max 0128 4.25 cm and the width

max 08 0.19DX λ= = cm was used. In the Y direction at positions of the electrodes Y

= 0 and Y = Ymax the absorbing wall boundary conditions were applied. The potential

at Y = Ymax was fixed at zero, corresponding to the grounded electrode. At the

position of the powered electrode at Y = 0 the potential was assumed to oscillate

harmonically according to applied RF voltage: ( )0, sin( )RF RFt U tϕ ω= with

2 13.56RFω π = MHz. In order to reach equilibrium discharge conditions, the

amplitude of applied RF voltage RFU was automatically adjusted using the feedback

control (Chapter 2.9). At the boundaries in the X direction a periodic boundary

condition was applied. As neutral species densities are much larger than densities

of charged species, the neutral species were treated as background with fixed

density and temperature. Only the dynamics of charged particles was followed. In

order to obtain an accurate electron energy distribution, the comprehensive list of

electron-neutral reactions for CH4 and H2 was added in the model, including the

rotational, vibrational and electronic excitation as well as dissociation and

ionization collisions and the elastic scattering. Cross-sections for these collisional

processes were collected from the compilation used in [Bush, 1999]. In Appendix B

we present the energy dependent plots of cross-sections for electron-neutral

collisions used in the model.

In calculations a grid size 0 2 0.017λ∆ = =Dx cm and time step

110.2 3.55 10pet ω −∆ = = ⋅ s was used. The number of computational particles per

Debye cell was chosen as Nd = 1000, totaling about 4⋅106 computational particles

used in the simulation. The calculations were carried out on a 16-processor Linux

cluster in about 50 hours.


70

a) b)

c) d)

e) f)

Figure 4.2 Dynamics of the potential profile during one RF cycle; RFf =13.56 MHz,

p = 0.085 Torr, d = 4.25 cm, 10 -310 cmen = , = ⋅CHn4

14 -37 10 cm ,

= ⋅Hn2

14 -39.2 10 cm . The phase of RF cycle RFtϕ ω= is a) 2ϕ = ° b) 43ϕ = °c)

92ϕ = ° d) 182ϕ = ° e) 223ϕ = ° f) 272ϕ = ° .


71

Below we present the results of the simulation. In Fig 4.2 the potential

dynamics in the system during the RF cycle is presented. The potential profiles

calculated for 6 different times during the RF cycle are plotted. In Fig. 4.3 we

present the potential profile averaged over the RF period. We can see that a steep

potential drop, up to max 1100ϕ∆ ≈ V , takes place near the electrodes within

oscillating positive space-charge layers of about 032λ≈s DL thick - the RF sheaths.

The electric field in the bulk plasma is negligible in comparison with the field in the

sheaths, where it is 450≈�RFE V cm on average. This strong electric field in the RF

sheath regions is directed toward the electrodes, preventing electrons from leaving

the plasma for most of the RF cycle. The electrons are able to escape to electrodes

only during a short time, when the RF sheath collapses (Figs. 4.2c, 4.2f).

Figure 4.3 Potential, averaged through one RF cycle; RFf =13.56 MHz, p = 0.085

Torr, d = 4.25 cm, 10 -310 cmen = , 4

14 -37 10 cmCHn = ⋅ , 2

14 -39.2 10 cmHn = ⋅ .


72

a) b)

c) d)

Figure 4.4 Dynamics of the density profile of CH4+ ion during one RF cycle;

RFf =13.56 MHz, p = 0.085 Torr, d = 4.25 cm, 10 -310 cmen = , = ⋅CHn4

14 -37 10 cm ,

= ⋅Hn2

14 -39.2 10 cm . The phase of RF cycle is a) 2ϕ = ° b) 43ϕ = °c) 92ϕ = ° d)

133ϕ = ° .

We can follow the space charge dynamics in Figs. 4.4, 4.5, where the time

evolution of CH4+ ion and electron densities during the period of RF oscillation is

presented. As we can expect, the ions due to high inertia (ion plasma frequency


73

being smaller than RF frequency 2

0ω ω

ε= <i

pi RFi

n em

) are not able to react to the

fast changing RF electric field. On their timescale ions respond only to the electric

field averaged over RF cycle (Fig. 4.3), such that the ion density stays constant over

the RF cycle. Thus, the flux of energetic ions, accelerated in the sheath electric field

to energies of about average sheath potential drop, constantly flows to the

electrodes. The electrons, being much more mobile, follow the changes of electric

field during the RF cycle, oscillating between the electrodes on the static

background of the positive space charge of the ions. In the bulk plasma the electron

density during the RF cycle stays equal to the total ion density, maintaining the

quasi-neutrality. In the sheath regions the positive space charge of the ions during

most of the RF period remains uncompensated because the electrons reach the

electrode only for a short time, during the collapse of the sheath potential (Figs.

4.5d, 4.5e), to balance the ion current on the wall. The change of the net space

charge near the electrode during the RF cycle, resulting from the different response

of ions and electrons to the applied RF voltage, is responsible for the dynamics of

the RF sheath electric field.

More information about particle behavior can be extracted from their velocity

(energy) distribution dynamics. In Fig. 4.6 we plot the spectrum of the parallel CH4+

ion energy depending on the longitudinal coordinate Y, calculated at 16 different

times during the RF period. As we can see, in the bulk region the ions stay cold,

their mean energy of chaotic motion 4

0.07+ ≈� chCHE eV is close to the thermal energy

of the background gas. In the sheath regions low-energy ions from the bulk plasma

are sharply accelerated in the strong electric field toward the electrodes up to the

maximum energy of max 480≈E eV, which is close to the average sheath potential

drop 450≈�sU V (see Fig. 4.3). Thus, the ion energy distribution has the shape of a

narrow ridge aligned along the 0-energy axis in the bulk region and bent from 0 to

max±E in the sheath regions. Due to RF modulation the maximum ion energy at the

electrode position is oscillating through the RF cycle in the range of 430 – 530 eV.

In the sheath regions we can also distinguish additional branches in the ion energy

distribution corresponding to lower ion energies. These secondary branches in our

case are conditioned by electron impact ionization taking place in the sheath region,

resulting in the appearance of the low-energy ions in the sheaths. In Figs 4.6 d-e we

can see the ions with the energies close to zero appearing in the sheath ( 32 λ< DY ).


74

a) b)

c) d)

Figure 4.5 Dynamics of the electron density profile during one RF cycle; RFf =13.56

MHz, p = 0.085 Torr, d = 4.25 cm, 10 -310 cmen = , = ⋅CHn4

14 -37 10 cm ,

= ⋅Hn2

14 -39.2 10 cm . The phase of RF cycle is a) 2ϕ = ° , b) 22.5ϕ = ° , c) 43ϕ = ° ,

d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g) 133ϕ = ° , h) 161ϕ = ° (continued on p. 75).


75

e) f)

g) h)

Figure 4.5 (Continued).


76

a) b)

c) d)

Figure 4.6 Dynamics of the parallel CH4+ ion energy spectrum during one RF cycle;

RFf =13.56 MHz, p = 0.085 Torr, d = 4.25 cm, 10 -310 cmen = , = ⋅CHn4

14 -37 10 cm ,

= ⋅Hn2

14 -39.2 10 cm . The phase of RF cycle is: a) 2ϕ = ° , b) 22.5ϕ = ° , c) 43ϕ = ° ,

d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g) 133ϕ = ° , h) 161ϕ = ° , i) 182ϕ = ° , j)202.5ϕ = ° , k) 223ϕ = ° , l) 251ϕ = ° , m) 272ϕ = ° , n) 292.5ϕ = ° , o) 313ϕ = ° ,

p) 341ϕ = ° . Continued on p.p. 77-79.


77

e) f)

g) h)



78

i) j)

k) l)



79

m) n)

o) p)



80

0 100 200 300 400 500 6000

1

2

3

4

ion

flux

(a.u

.)

ion energy (eV)

Figure 4.7 Ion energy distribution averaged over one RF cycle CH4+ calculated atthe position of the grounded electrode; RFf =13.56 MHz, p = 0.085 Torr, d = 4.25

cm, 10 -310 cmen = , 4

14 -37 10 cmCHn = ⋅ , 2

14 -39.2 10 cmHn = ⋅ .

This corresponds to the time interval when the RF sheath collapses and

energetic electrons are able to penetrate deep in the sheath, ionizing neutrals and

generating low-energy ions directly in the sheath region. During one RF cycle this

group of low-energy ions is accelerated to the energy about 100 eV at the wall

position, producing the next secondary branch in the ion energy distribution (see

Fig. 4.6d). The time that the CH4+ ion takes to traverse the sheath is about

4

4

2290

2τ +

+

≈��

sCH

s CH

L

U m ns, when the RF period is

273.75

πτω

= =RFRF

ns. Thus it

takes about 4 RF periods for low-energy ions generated in the sheath to escape to

the electrode. During this time these low-energy ions are gradually accelerated up

to energies of about maxE , finally contributing to the primary branch of the ion

distribution. As a result, a fan-like structure in the ion energy distribution forms in

the sheath region, where the high-energy branch is contributed by ions being

accelerated from the sheath edge, and secondary peaks correspond to ions

generated inside the sheath. The number of the secondary branches should

correspond to the number of RF periods that an ion takes to cross the sheath. In


81

total, 4 secondary branches in the ion energy distribution in the sheath region can

be distinguished in Fig. 4.6.

Figure 4.8 Ion energy distributions measured in a capacitively coupled RFdischarge in argon. Figure from [Wild, 1991].

In Fig. 4.7 we plot the CH4+ ion energy distribution averaged over the RF cycle

calculated at the position of the grounded electrode. In the high-energy part of the

distribution we can distinguish a saddle-like structure with two peaks at energies

430 eV and 530 eV, caused by RF modulation of the ions which experience the full

sheath potential drop. The peaks at lower energies are contributed by low-energy

ions produced due to collisions inside the sheath. Similar structures in the ion

distribution were observed in experiments and obtained in models [Wild, 1990],

[Snijkers, 1993; Kawamura, 1999]. In Fig. 4.8 we present ion energy distribution


82

measured in argon discharge for various pressures from [Wild, 1990]. At lower

pressures we can see both the saddle structure in high-energy part of the spectrum

and the secondary peaks at lower energies. Going to higher pressures the collisional

effects in the sheath start to dominate, populating the low energy part of the ion

energy spectrum.

In Fig. 4.9 we present profiles of the electron parallel velocity component

distribution function along the system, calculated at 8 different times during the RF

half-period. The velocity is scaled by the electron thermal velocity 00

ete

e

kTv

m= ,

calculated for 0eT = 20 eV. We can clearly distinguish two groups of electrons on

these plots: the time independent formation of cold electrons in the middle of

system, and the tail of high-energy electrons oscillating between the electrodes. The

electrons from the low energy group (with average energy about 0.6 eV) are not

energetic enough to overcome the ambipolar potential barrier and penetrate the

sheath region, where they could be accelerated by the strong electric field. Thus

they are locked in the middle of the system. The energy of these electrons is far

below the energy threshold for the majority of inelastic collision processes, thus

they are not participating in collisions with neutrals, except for elastic scattering.

Eventually, due to elastic collisions, these electrons diffuse to the sheath regions,

where they are accelerated in the strong sheath electric field. This group of

electrons is most likely populated by the low-energy secondary electrons, produced

in electron-neutral ionization collisions.

Unlike the electrons from the low-temperature group, the electrons from the

high-energy tail can easily overcome the ambipolar potential barrier and penetrate

into the region of strong electric field in the sheath. As in our case the mean free

path for electron-neutral elastic collisions 1

0.5enn enn

λσ

≈� cm is the same order

of magnitude as the system length =d 4.25 cm, these electrons can oscillate

between the RF sheaths, getting reflected from them by the strong retarding electric

field. Although during single reflection from the sheath, an electron can both gain

and loose energy, depending on the phase of the RF field, but on average, electrons

can be accelerated due to stochastization of their motion, following the Fermi

acceleration mechanism [Lieberman, 1998].


83

a) b)

c) d)

Figure 4.9 Dynamics of the distribution of the parallel component of electronvelocity during one RF cycle; RFf =13.56 MHz, p = 0.085 Torr, d = 4.25 cm,

10 -310 cmen = , = ⋅CHn4

14 -37 10 cm , = ⋅Hn2

14 -39.2 10 cm . The phase of RF cycle

is: a) 2ϕ = ° , b) 22.5ϕ = ° , c) 43ϕ = ° , d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g)133ϕ = ° , h) 161ϕ = ° , i) 182ϕ = ° , j) 202.5ϕ = ° , k) 223ϕ = ° , l) 251ϕ = ° , m)272ϕ = ° , n) 292.5ϕ = ° , o) 313ϕ = ° , p) 341ϕ = ° . Continued on p.p. 84-86.


84

e) f)

g) h)



85

i) j)

k) l)



86

m) n)

o) p)


Originally, Fermi proposed the idea of stochastic heating to explain the origin of

cosmic rays [Fermi, 1949] – the flux of charged particles with super high energies E

~ 108 –1020 eV. In 1949 Fermi suggested, that ‘cosmic rays are originated and

accelerated primarily in the interstellar space of the galaxy by collisions against

moving magnetic fields’. A cosmic ray particle can gain energy from such collisions

if the ‘magnetic cloud’ is moving toward the particle. In the opposite case, when the


87

region of high magnetic field is moving away from particle, it will lose energy due to

the collision. ‘The net result will be average gain, primarily for the reason that head-

on collisions are more frequent than overtaking collisions because the relative

velocity is larger in the former case’. Ulam suggested [Ulam, 1961] a simple model

problem to illustrate the mechanism of stochastic acceleration: a ball bouncing

between one fixed and one regularly oscillating horizontal wall (also known as the

Ulam-Fermi problem). Although during one reflection from the oscillating wall the

ball can both loose and gain energy, depending on the phase of the oscillating wall,

due to dynamic randomization of the collision phase the motion of the ball can

become stochastic and after a series of reflections it can be accelerated as it was

shown in [Zaslavsky, 1965; Lieberman, 1998]. Godyak applied the Fermi

acceleration to electron heating in RF discharges [Godyak, 1971], proposing that

stochastic heating of electrons oscillating between sheaths becomes the dominant

heating mechanism in low-pressure capacitive RF discharges. Randomization of

electron motion in RF discharges can also arise due to electron-neutral collisions.

Collisions with neutrals can become the dominant randomization mechanism, when

condition for dynamic stochasticity is not satisfied [Lieberman, 1998]. Such

collisional randomization can be responsible for stochastic heating of electrons by

RF sheaths even in case of low collisionality: en dλ � [Kaganovich, 1996;

Lieberman, 1998]. In our system en dλ < , thus collisions with neutral gas should

play an important role in the randomization of electron motion and hence the

stochastic heating.

In Fig 4.10 we present the time-averaged electron energy probability function

(EEPF), calculated in the middle of the system. EEPF is defined as:

( ) 3 2 24 2e e e

e

EF E m f

mπ

� �= � ��

, (4.1)

with normalization:

( )0

e eF E EdE n∞

=� . (4.2)


88

Here ( )ef v is the electron velocity distribution function, E and en – electron

energy and local density respectively. Representation of the energy distribution in

this form is convenient as it shows Maxwellian distribution as a straight line. As we

can see in Fig. 4.10 the electron distribution can be quite well represented as a sum

of two Maxwellian distributions with temperatures T1 = 0.39 eV, T2 = 3 eV and

densities n1 = 1010 cm-3, 2n = 109 cm-3. The low temperature part corresponds to the

static group of cold electrons in the bulk region, whereas the high-temperature

component is contributed by the energetic electrons, oscillating between sheaths.

Similar bi-Maxwellian electron distributions were experimentally found in low-

pressure capacitive RF discharges [Godyak, 1990; Turner, 1993; Mahony, 1999]. In

Fig. 4.11 we present EEPF measured in the middle of the capacitive RF discharge in

argon at p = 0.1 Torr with electrode spacing d = 1.2 cm [Godyak, 1990]. This figure

clearly indicates two groups of electrons in the discharge: low energy bulk and high-

energy tail, resulting from stochastic heating.

The electrons from the high-energy tail bouncing between sheaths have enough

energy to participate in relevant inelastic collisions with neutrals: ionization,

dissociation and excitation. (The ionization energy for methane is 4iCHE = 12.6 eV

and dissociation energy - 4dCHE = 10 eV). As the mean free path for such collisions

is in our case longer than the system length λ ≈ > =i cm d cm6 4.25 , the inelastic

processes are distributed throughout the whole bulk region. In Fig. 4.12 we present

a profile of the calculated ionization rate along the system axis:

( ) ( )0

ei n i en n v E F E EdEσ∞

′ = � , (4.3)

where nn is neutral density, ( )i Eσ is the electron impact ionization cross-section,

and v is the absolute electron velocity. As we can see in Fig. 4.12, ionization is

spread rather uniformly in the bulk region, but decreases fast in the sheaths

because electrons penetrate the sheath only during short time when the sheath

collapses.


89

0 5 10 15 20

106

107

108

109

1010

1011

T2 = 3 eV, n

2 = 109 cm

T1 = 0.39 eV, n

1 = 1010 cm

electron energy (eV)

T2

T1

eepf

(eV

-3/2 cm

-3)

Figure 4.10 The time-spatial averaged electron energy probability functioncalculated for a capacitive RF discharge; RFf =13.56 MHz, p = 0.085 Torr, d = 4.25

cm, 10 -310 cmen = , = ⋅CHn4

14 -37 10 cm , = ⋅Hn2

14 -39.2 10 cm .

`

Figure 4.11 The electron energy probability function measured in a capacitive RFdischarge in argon: RFf =13.56 MHz, p = 0.1 Torr, d = 2 cm. Figure from [Godyak,1990].


90

0 32 64 96 1280

2x1014

4x1014

6x1014

8x1014

n'ei , cm-3

��c-1

Y, λλλλD

Figure 4.12 Ionization rate along the system; RFf =13.56 MHz, p = 0.085 Torr,

d = 4.25 cm, 10 -310 cmen = , 4

14 -37 10 cmCHn = ⋅ , 2

14 -39.2 10 cmHn = ⋅ .

In order to study the influence of the electron-neutral collisionality on discharge

behavior, we performed a simulation for background gas pressure, increased by a

factor of 10 ( p = 0.85 Torr), with neutral densities 4

15 -37 10 cmCHn = ⋅ and

2

15 -39.2 10 cmHn = ⋅ . The results of this simulation are presented in Figs. 4.13-

4.20. In Fig. 4.13 we present the dynamics of the potential during the RF cycle,

showing the potential profile calculated at 6 different phases of RF cycle. Comparing

the potential dynamics with the case of lower gas pressure (Fig. 4.2), we can see

that, in the high pressure case, the collapse of the RF sheath leads to the reversal of

electric field in the sheath region (Fig 4.13 b, e). During the short interval of the RF

cycle, the electric field in the sheath region changes direction and accelerates

electrons toward the electrode. Because electrons reach the electrode only during

the short time when the sheath collapses (see dynamics of electron density in Fig.

4.16), in the case of high working gas pressure, when electron mobility is reduced

due to electron-neutral collisions, the accelerating electric field is necessary at this

time to provide electron current sufficient to keep balance with generally constant

ion current.


91

a) b)

c) d)

e) f)

Figure 4.13 Dynamics of the potential profile during one RF cycle; RFf =13.56 MHz,

p = 0.85 Torr, d = 4.25 cm, 10 -310 cmen = , 4

15 -37 10 cmCHn = ⋅ ,

2

15 -39.2 10 cmHn = ⋅ . The phase of RF cycle is: a) 2ϕ = ° b) 43ϕ = ° c) 92ϕ = ° d)

182ϕ = ° e) 223ϕ = ° f) 272ϕ = ° .


92

Figure 4.14 Potential, averaged during one RF cycle; RFf =13.56 MHz, p = 0.85

Torr, d = 4.25 cm, 10 -310 cmen = , 4

15 -37 10 cmCHn = ⋅ , 2

15 -39.2 10 cmHn = ⋅ .

The time evolution of the CH4+ ion density profile is plotted in Fig 4.15. The ion

density shows the same static behavior as in the case of lower neutral gas pressure

(see Fig. 4.3), but now the ion density in the sheath region has grown about an

order in magnitude. Because the averaged sheath potential drop 380≈�sU V (see

averaged through the RF cycle potential profile in Fig. 4.14) did not change much in

comparison with the case of lower gas pressure, the strong increase of the ion

density in the sheath shows the presence of intensive ionization in this region.


93

a) b)

c) d)

Figure 4.15 Dynamics of the density profile of CH4+ ion during one RF cycle;

RFf =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, 10 -310 cmen = , = ⋅4

15 -37 10 cmCHn ,

= ⋅2

15 -39.2 10 cmHn . The phase of RF cycle is: a) 2ϕ = ° b) 43ϕ = °c) 92ϕ = ° d)

133ϕ = ° .


94

a) b)

c) d)

Figure 4.16 Dynamics of the electron density profile during one RF cycle;


15 -37 10 cmCHn ,

= ⋅2

15 -39.2 10 cmHn . The phase of RF cycle is: a) 2ϕ = ° , b) 22.5ϕ = ° , c) 43ϕ = ° ,

d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g) 133ϕ = ° , h) 161ϕ = ° . Continued on p.95.


95

e) f)

g) h)



96

a) b)

c) d)

Figure 4.17 Dynamics of the parallel CH4+ ion energy spectrum during one RF

cycle; RFf =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, 10 -310 cmen = ,

= ⋅4

15 -37 10 cmCHn , = ⋅2

15 -39.2 10 cmHn . The phase of RF cycle is: a) 2ϕ = ° , b)

22.5ϕ = ° , c) 43ϕ = ° , d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g) 133ϕ = ° , h)

161ϕ = ° , i) 182ϕ = ° , j) 202.5ϕ = ° , k) 223ϕ = ° , l) 251ϕ = ° , m) 272ϕ = ° , n)

292.5ϕ = ° , o) 313ϕ = ° , p) 341ϕ = ° . Continued on p.p. 97-99.


97

e) f)

g) h)



98

i) j)

k) l)



99

m) n)

o) p)



100

In Fig. 4.17 the dynamics of the parallel energy of CH4+ ions during the RF

period is presented. We can see that the ion energy distribution shows a pattern

similar to the lower pressure case (see Fig. 4.6) with cold bulk ions and fan-like

structures in the sheaths. But now the low-energy branches corresponding to ions

produced within the sheaths are considerably higher due to higher ionization rate

in the sheath regions. As the ion transit time through a sheath is τ + ≈4CH

158 ns , i.e.

about two RF periods, only two secondary branches in the ion distribution can be

seen in addition to a primary branch contributed by ions which are experiencing

acceleration from the sheath edge.

In Fig 4.18 we present the dynamics of the profile of the electron parallel

velocity component distribution as a function of longitudinal coordinate Y. As we

can see, the behavior of electrons has changed considerably in comparison with the

low-pressure case (see Fig. 4.9). Now all heating of electrons is taking place in the

sheath regions near the electrodes. The mean free path of electron-neutral elastic

collisions is now 0.05en cmλ ≈ , which is much smaller than the system length

=d 4.25 cm and considerably smaller then the sheath width 016 0.5λ≈ ≈s DL cm.

Thus, the Ohmic heating in the sheath region, when the electrons are accelerated in

the strong electric field between successive elastic collisions with neutrals, becomes

the dominating mechanism of electron heating. As we can see, electrons are heated

in both half-periods of the RF cycle: during sheath expansion (Fig 4.18 a-d at

maxY Y= ) and reversal of the sheath electric field (Fig 4.18 b-d at 0Y = ). In Fig.

4.19 we present the electron parallel velocity component distribution averaged

during the RF cycle. Here we can clearly distinguish two zones in the sheath where

the electrons are heated. The broadening of the velocity distribution at the distance

~16 Dλ from the electrode corresponds to electrons heated by the retarding field at

the edge of the expanding sheath. The electrons, accelerated in the reversed electric

field during the sheath collapse contribute to the hot group directly in front of the

electrode.

The mean free path for electron-neutral inelastic collisions 1 0.5i n inλ σ ≈� cm

is close to the sheath width, so electrons should be quickly cooled down by inelastic

collisions after they leave the sheath region. In Fig. 4.19 we can see that electrons

in the bulk region are considerably colder than in the sheaths, and the cooling

down takes place on a characteristic length of about the sheath width.


101

a) b)

c) d)

Figure 4.18 Dynamics of the distribution of parallel component of electron velocityduring one RF cycle; RFf =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, 10 -310 cmen = ,

4

15 -37 10 cmCHn = ⋅ , 2

15 -39.2 10 cmHn = ⋅ . The phase of RF cycle is: a) 2ϕ = ° , b)

22.5ϕ = ° , c) 43ϕ = ° , d) 71ϕ = ° , e) 92ϕ = ° , f) 112.5ϕ = ° , g) 133ϕ = ° , h)161ϕ = ° , i) 182ϕ = ° , j) 202.5ϕ = ° , k) 223ϕ = ° , l) 251ϕ = ° , m) 272ϕ = ° , n)292.5ϕ = ° , o) 313ϕ = ° , p) 341ϕ = ° . Continued on p.p. 102-104.


102

e) f)

g) h)



103

i) j)

k) l)



104

m) n)

o) p)



105

Figure 4.19 Distribution of parallel component of electron velocity averaged overone RF cycle; RFf =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, 10 -310 cmen = ,

= ⋅4

15 -37 10 cmCHn , = ⋅2

15 -39.2 10 cmHn .

In Fig. 4.20 we present a profile of the ionization rate averaged over the RF

cycle. We can see that in contrast to the previous case (see Fig. 4.11) the ionization

is taking place only in the sheath regions. We can also distinguish at each sheath

two maxima, contributed by electrons accelerated at different phases of the sheath

electric field. We can see this effect in more detail in Fig. 4.21, were the time-

resolved ionization rate close to the electrode during one RF cycle is plotted. Here

we can clearly see that the ionization peak near the electrode takes place at time

when the electric field near the wall is reversed, and the ionization peak at the

sheath edge appears when the repulsive RF sheath builds up.


106

0 32 64 96 1280.0

2.0x1015

4.0x1015

6.0x1015

8.0x1015

1.0x1016

1.2x1016

n'ei , cm-3

��c-1

Y, λλλλD

Figure 4.20 Ionization rate profile along the system averaged over one RF cycle;


15 -37 10 cmCHn ,

= ⋅2

15 -39.2 10 cmHn .

0 20 40 60 80 100 120 1400

2

4

6

8

10n'

ei , cm-3

��c-1

time, ns

Y, m

m

0

1.2E16

2.4E16

3.6E16

4.8E16

6E167E16

Figure 4.21 Spatiotemporal distribution of ionization rate in the vicinity of thepowered electrode of the capacitive RF discharge; RFf =13.56 MHz, p = 0.85 Torr,

d = 4.25 cm, 10 -310 cmen = , = ⋅4

15 -37 10 cmCHn , = ⋅2

15 -39.2 10 cmHn .


107

Figure 4.22 (a) Spatially and temporally resolved Balmer-alpha (653.3 nm)emission from a hydrogen plasma operating at 133 Pa (1 Torr) with a RF power of15 W. This corresponds to a RF voltage amplitude of 116 V. D is the distance fromthe driven electrode. (b) Spatially and temporally resolved 653.3 nm excitation ratederived from (a). (c) The temporal dependence of the electrode potential relative tothe plasma for four different RF powers. Figures from [Mahony, 1997].


108

0 5 10 15 20105

106

107

108

109

1010

1011

0.85

0.028

0.042

0.17

0.085

neutral g

as pressure (Torr)

electron energy (eV)

eepf

(eV

-3/2cm

-3)

0.34

Figure 4.23 The electron energy probability functions calculated for variouspressures of CH4 - H2 mix, RFf =13.56 MHz, en = 1010 cm-3, d = 3 cm.


109

Figure 4.24 The electron energy probability function evolution with pressuremeasured in a capacitively coupled RF argon plasma. Figure from [Godyak, 1990].

A similar effect of double hot electron layers was observed in experiments with

capacitive RF discharges as double emissive layers near the electrodes. In Fig. 4.22

we plot results of a time resolved measurement of Balmer-alpha (653.3 nm)

emission from a RF discharge in hydrogen at 1 Torr from [Mahony, 1997]. Fig 4.22a

shows the measured intensity of the radiation during 2 RF cycles in the vicinity of

the electrode. In Fig. 4.22b the electron-impact excitation rate, deduced from these

measurements is presented. We can again see two zones of high excitation rate,

corresponding to regions of hot electrons, which are correlated with the sheath


110

potential dynamics shown in Fig. 4.22c. The region of high excitation near the

electrode appears at moments when the plasma potential becomes lower than the

electrode potential (reversed electric field in the sheath). The second region of high

excitation rate further from the electrode is caused by electrons heated at the edge

of the growing sheath.

In order to investigate the transition of the electron heating mechanism with the

increase of background gas pressure, we performed simulations for the same H2-

CH4 1.3:1 mix, changing the gas pressure. In Fig. 4.23 we summarize these

simulations, presenting EEPF’s averaged over the RF cycle in the bulk plasma for

gas pressures 0.028 Torr, 0.042 Torr, 0.085 Torr, 0.17 Torr, 0.34 Torr and 0.85

Torr. As we can see in Fig. 4.23, at low pressures the EEPFs are essentially bi-

Maxwellian, revealing the stochastic electron heating mechanism, leading to the

formation of cold bulk and oscillating hot tail electrons. With increase of the neutral

gas pressure between 0.34 Torr and 0.85 Torr EEPF transforms to a convex,

Druyvesteyn–type distribution with a high-energy part depleted by inelastic

collisions, corresponding to a regime when the Ohmic heating in the sheath regions

is the dominant mechanism of the electron heating.

Such changes of electron energy distribution were observed experimentally in

capacitive RF discharges [Godyak, 1990; Godyak, 1992; Turner, 1993]. In Fig. 4.24

we present the EEPFs measured by [Godyak, 1990] in an argon capacitive RF

plasma for different pressures. We can see that the bi-Maxwellian distribution

measured at low neutral gas pressures changes to a Druyvesteyn–type distribution

at pressure of 0.2-0.3 Torr. This pressure is somewhat lower than values obtained

in our simulations, which can be explained by larger cross-sections for electron-

neutral elastic collisions for argon.

4.2 Principal results

A capacitively coupled radio-frequency (RF) discharge was studied in

collaboration with an experimental group at the University of Bochum. This

discharge type has special importance for plasma technology (etching, deposition).

We were able to follow the time and spatially resolved dynamics of the plasma

particles. For the low pressure of the neutral gas the electron distribution is a sum

of two Maxwellian distributions. The low temperature part corresponds to the static


111

group of cold electrons in the bulk region, whereas the high-temperature

component is contributed by the stochastically heated electrons, oscillating between

sheaths. Similar bi-Maxwellian electron distributions were experimentally found in

low-pressure capacitive RF discharges.

In the high-pressure case the Ohmic heating in the sheath region becomes the

dominating mechanism of electron heating. Due to electric field reversal in the

sheath region, electrons are heated in both half-periods of the RF cycle, which

cause the two peaks in the intensity of electron-induced inelastic processes

(ionization, excitation) near the electrodes. A similar effect of double hot electron

layers was observed in experiments with capacitive RF discharges as double

emissive layers near the electrodes.

As the neutral gas pressure increases, electron distribution transforms from bi-

Maxwellian (resulting from the stochastic heating) to a Druyvesteyn–type

distribution (corresponding to Ohmic heating). Such changes of electron energy

distribution were observed experimentally in capacitive RF discharges.

The electron energy distribution function (EEDF) determines the chemical

kinetic processes in the reactive discharges and hence is of vital importance for

reactive plasmas (film deposition). The ability to calculate the EEDFs provides

improved possibilities for predictive modeling of such systems [Möller, 2003].

The ion energy distribution (IED) at the wall position calculated within our

model indicates a multi-peak structure due to ion modulation in the RF sheath.

This is confirmed by the IED observed in experiments. The knowledge of IED is of

special interest for plasma etching processes.

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