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5 High-frequency discharges I

The phenomenological dierence in breakdown behavior between DC discharges and HF driven plasmas is used to discuss the processes of carrier gain, predominantly Ohmic heating, and carrier loss (diusion) which are required for the understanding of HF breakdown. The conditions for equilibrium of gain and loss in the plasma bulk are extensively discussed. Next, power coupling is investigated from an electrical viewpoint on two levels. The chapter is nished with a discussion of matching networks and waveguides for the dierent methods of coupling.

5.1 Phenomenological introduction

In a DC discharge, an electrode which is covered by an electrically isolating plate will be charged to the weakly negative oating potential Φf ; the uxes of both types of carriers (electrons and ions) become equal in magnitude, irrespective of whether the potential is applied at the rear side of the insulator (cf. Sect. 3.4). At its surface, ions and electrons will be terminated or will recombine, and there is no need to draw o an electric current (which would be impossible). At plasma densities of 1010 cm3 , a voltage of 10 20 V develops across the sheath. The sheaths represent capacitancies which can pile up charges. Capacitance is dened as C = Q/V ; since Q ∝ V , and it takes a certain time to charge up the capacitancies (Q = Idt), the voltage does not alter instantaneously, i. e. without any time of retardation. After ignition, both sides of the insulator drop to the (negative) cathode potential. During bombardment with positively charged ions, the potential becomes more positive (less negative) because electrons are consumed to neutralize the ions: The potential at the surface facing the plasma gradually reaches Φf , which is sucient to generate an ion bombardement which can clear o the surface from weakly bounded contaminations. However, the energy of the ions incident on the surface is too low to make substantial sputtering feasible [154]. To solve this problem, we can use alternating current (AC). During the positive part of the cycle, the negative charging of the electrode will be removed by ion bombardement. The frequency which is required for successful neutralization can be roughly estimated as follows, provided the electrode current is assumed to remain constant (in fact, it will diminish): C = Q/V = I t/V, t = C V /I.

G. Franz, Low Pressure Plasmas and Microstructuring Technology, DOI 10.1007/978-3-540-85849-2 5, c Springer-Verlag Berlin Heidelberg 2009 103

104

1 p [mTorr]

5 High-frequency discharges I

10-1

10-2

Fig. 5.1. For rising operating frequency, the pressure which is required to ignite a plasma steeply drops up to some hundreds of kHz to settle down for even higher frequencies [155] ( c Elsevier).

100 f [kHz] 1000

5 4

14 MHz

S [W/cm2]

3 2

7.5 MHz

3.7 MHz

1 0 1

Fig. 5.2. For rising operating frequency, the eciency of power coupling increases [157].

10 p [mTorr] 100 1000

For a quartz 3 mm in thickness, its specic capacitance amounts to about 1 pF cm2 ; for V = 1 000 V and j≈ ≈ 1 mA cm2 (the current density is estimated from DC measurements and sputtering rates) we calculate a rise time of about 1 μsec. Since the electrode does not charge up instantaneously because the current does not remain constant but drops with falling potential, AC discharges with isolating cathodes can already be operated at frequencies beyond some ten kHz. Koenig and Maissel were the rst to demonstrate in 1970 that plasmas can be ignited with rising frequency [156]. Norstrm extended these investigations [157] (Fig. 5.1). It turned out that the coupling of RF energy close to the ignition voltage is almost independent of pressure but rises steeply for larger pressures. The pressure dependence of power coupling is larger for higher frequencies which indicates further ionizations or even the onset of additional ionization mechanisms (Fig. 5.2). However, the slope of the I(V ) curves is almost the same

5.2 Generation of carriers

105

from which we infer similar electron temperatures (ln je = const e0 V /kB Te ). This eect can be detected up to some MHz, from which we further conclude that the discharge is sustained by electrons which are not generated by electrode processes (for more details see Sects. 6.1 and 14.4)!

5.2 Generation of carriers

To explain this fact, we consider the movement of a free, undamped electron which is accelerated by an oscillating electric eld. According to Eqs. (5.1) and (5.2) (a property which is multiplied by “i” exhibits a phase shift of π/2 : eiπ/2 = cos π/2 + i sin π/2 = i), the velocity exhibits a shift of π/2, and the amplitude a shift of π against the exerting force: me ue = due = e0 E 0 eiωt dt (5.1)

e0 1 e0 E 0 eiωt . (5.2) E 0 eiωt ∧ xe = m iω mω 2 On the time average, the power absorption is determined by the integral over 2π iωt 0 e dt = 0: Neglecting the losses by radiation damping, the electron cannot absorb any energy! Hence, another mechanism must be responsible for power absorption. This is suggested also by the fact that the maximum amount of energy that can be absorbed by a carrier within one half of a microwave cycle (2.45 GHz), is less than 0.1 eV at the measured eld for breakdown of 100 V/cm. This is more than two orders of magnitude less than the ionization potential of argon [Eion (Ar) = 15.76 eV, Table 5.1 and Fig. 5.3].

Table 5.1. Velocity, kinetic energy, and maximum amplitude of an electron which oscillates freely in an HF eld at the operating frequencies of 13.56 MHz and 2.45 GHz. ν 13.56 MHz E [V/cm] 1 10 100 10 100 1000 v [cm/sec] 2.07 × 107 2.07 × 108 2.07 × 109 1.14 × 106 1.14 × 107 1.14 × 108 Ekin [eV] 0.12 12.2 1217 3.6 × 104 3.6 × 102 3.6 ψ0 [cm] 2.42 24.2 242 7.4 × 105 7.4 × 104 7.4 × 103

2.45 GHz

We have seen that an electric eld exerts an accelerating force on an electron as long as it collides with a second particle. During the impact, the eld drift is destroyed and kinetic energy is transferred. This amount is negligible for an

106

5 High-frequency discharges I

elastic collision with a heavy gas constituent. After this incident, the electron will be accelerated again, but on a higher energy level. By this repeatable process, the electron can accumulate a tremendous amount of energy until it is by a wall collision. Hence, the motion of the electron consists of a large random component and a small drift component. As we have extensively discussed (similarity rules, Sect. 4.10), the energy transferred to the electron scales with E/p with E the electric eld and p the discharge pressure. Although the operating frequency in an RF discharge is typically lower by a factor of 10 to 50 compared to a microwave discharge, the pressure is lower by about the same factor. Hence, the picture is similar: We observe many collisions during one oscillation period [at a pressure of 100 mTorr (13 Pa), νm = 9 × 108 sec1 for 12 eV electrons] and a so-called random phase movement. The elastic collisions between electrons and neutral molecules are mandatory for this mechanism of energy transfer to occur. The (directed) energy which is piled up by the electrons during the drift motion exerted by the electric eld is transformed into a random phase motion. Albeit in the two half-waves the electron is accelerated to and fro the electrode, the electron can gain energy in both parts of the AC cycle. In a mathematical sense, the absorbed energy scales with the eld squared, and it becomes independent of the sign: Ohmic heating, [Eqs. (5.4) and (5.9)]. For the absorption of power and the increase in conductivity, the ratio ω/νm with ω the operating angular frequency and νm the frequency of elastic collisions and σm the cross section of elastic scattering of electrons with neutrals (m for momentum) or the cross section of momentum transfer νm = σm ue ne (5.3)

is of paramount importance. This ratio expresses how rapidly a concerted electron movement will be damped by elastic collisions with neutrals. The energy gain of an electron amounts to 1 (5.4) Pabs = e0 E 0 ue 2 with ue the drift velocity of the electron, which can be derived from the second Newtonian axiom for harmonic distortion: me yielding e0 1 E 0 eiωt . (5.6) m iω + νm We dene the AC mobility as connecting factor between drift velocity and electric eld ue = due + νm ue = me (iωue + νm ue ) = e0 E 0 eiωt dt (5.5)

5.2 Generation of carriers μAC =

107

e0 1 , (5.7) me iω + νm which diers from the DC mobility by the imaginary summand iω in the denominator by which electron inertia a/(due /dt) is accounted for. For suciently low frequencies, this term can be neglected against the damping term mνm ue by which the dissipative losses by elastic collisions are taken into consideration; in this range, μ is real and constant.1 Hence, the AC mobility is a complex scalar function, and its real part describes the energy transfer. This can be readily understood by splitting the function into its real and imaginary part, respectively: iω νm 2 ; (5.8.1) 2 2 + νm ω + νm e0 ω νm e0 ; (μ) = ; (5.8.2) (μ) = 2 + ν2 2 + ν2 me ω me ω m m and inserting into Eq. (5.4). For the power absorption of N electrons in the volume V (n = N/V ), we obtain for one HF cycle μ= ω2 1 P = V T

T 0

e0 me

j e E dt

(5.9.1) (5.9.2)

1 1 P = ne0 ue E ne0 μAC E 2 , V 2 2 which yields, taking the real part of μ [Eq. (5.8)]:

2 2 P νm 2 E0 νm 2 E0 ne2 0 = = σDC 2 , (5.9.3) 2 + ω2 2 2 2 V mνm νm νm + ω with σDC the DC conductivity, and further provided that νm is independent of the electron velocity (this holds true only for hydrogen and √ helium and approximately for the high-energy tail of the EEDF), E = E0 / 2 denotes the RMS eld. From Eq. (5.9.3), we see that the ratio ω/νm , which is in fact the ratio of the imaginary part over the real part (μ)/ (μ), mainly determines the eciency of power coupling into the discharge.

For small values of ω/νm (low operating frequency ω and/or high pressure causing high νm ), the mean free path of the electrons decreases, and the power gain per one mean free path decreases with the mean free path squared. Vice versa,

1 Free electrons would exhibit a purely imaginary mobility (and because σ = e0 nμ a purely imaginary conductivity as well) due to the phase shift of 90 = 1/2 π against the exciting electric eld.—To compare with metals: Even against microwaves, the conductivity of metals remains almost completely real, i. e. eld and conduction current are in phase. For example, for copper, n = 8 × 1022 electrons/cm3 ; σ = 5 × 1017 sec1 = 5, 5 × 107 Ω1 cm1 and νm —it is denoted as damping constant g or γ—about 3 × 1013 Hz. However, appreciable deviations are observed in the IR or VIS range which occur in the tenuous plasmas treated here some orders of magnitude lower.

108

5 High-frequency discharges I

with rising operating frequency f = ω/2π, the amplitude of the electric eld must be increased to keep the power input constant (cf. Sect. 6.5). Hence, with constant power input and xed electron density ne , the energy transferred to the electrons would decrease with rising operating frequency 1 [158], since the time of electron acceleration is νm rather than ω 1 (Fig. 5.3). ω, causes the mean free path to grow, and Dropping the pressure, νm the absorbed power can be approximated by ne2 P 0 ≈ V 2me νm νm ω

2 2 E0 :

(5.10)

Higher plasma densities or higher electric elds of some kV are required to maintain the discharge, and below a certain pressure (number density) threshold, the discharge will extinguish. For vanishing collision frequency νm , the absorbed power equals zero.

p [mTorr] 30 40

p [mTorr] 2000 3000

0 0.8

10

20

50

60

0 0.8 0.6 0.4 0.2 0.0

1000

4000

5000

0.6 Eeff/E 0

Eeff /E 0

0.4

hydrogen

w = nm

w= nm

0.2

E 0/2 13.56 MHz 27.12 MHz

hydrogen

E 0/2 2.45 GHz

0.0 0

50

100

150 200 250 nm [10 6 sec-1]

300

350

0

5

10

15

20

25

30

nm [10 9 sec-1]

Fig. 5.3. In the case of RF discharges through hydrogen, driven at f = 13.56 MHz, the collision frequency νm equals the angular frequency at 85 MHz, in the case of MW discharges, νm equals the angular operating frequency at 15 GHz. In both cases, the eective eld is considerably smaller than the RMS eld. For vanishing collision frequency, the eective eld becomes zero: No power transfer takes place.

These eects are often taken into account by dening an eective eld by

2 Ee = 2 2 νm E0 ; 2 νm + ω 2 2

(5.11.1)

this eld would transfer the same amount of energy as a steady eld E0 , and the denominator eectively averages over these two frequencies. Writing

5.2 Generation of carriers

2 Ee = 2 1 E0 , ω2 2 1 + ν2

m

109 (5.11.2)

we readily see that the more ecient coupling with rising operating frequency is attained at the expense of the intensity of the oscillating electric eld which ω, i. e. for high will in all cases be smaller than the steady eld E0 . For νm discharge pressures, the eective eld will √ eventually turn into the RMS eld √ ω, we obtain Ee ≈ E0 / 2 νm /ω (Figs. 5.3 and 5.4 showing E0 / 2, for νm two dierent presentations). The evenness ω = νm marks the lower limit for ecient power transfer. In RF driven discharges, the eective eld is reduced to only a few percent at 10 mTorr in argon, which increases to some ten percent at 100 mTorr. In microwave discharges, the pressure should exceed 1 000 mTorr for an eective discharge. Striking the discharge becomes dicult below a certain pressure threshold.

100

100 mTorr

100

10 Torr

10-1 Eeff/E 0

argon

Eeff /E0

10-2

13.56 MHz 40.68 MHz

10 mTorr

10-2

1000 mTorr

100 mTorr

argon 2.45 GHz

10-3 0

2

4

6 Ekin [eV]

8

10

10-4 0

2

4

6 Ekin [eV]

8

10

Fig. 5.4. In the case of discharges through argon, driven at two RF frequencies at 13.56 and 40.68 MHz (LHS) and at the MW freqency of 2.45 GHz (RHS), the relative eective eld is plotted vs. the mean energy of the electrons. At 10 mTorr, the eective eld is reduced to only a few percent, which makes ignition more dicult and eventually prevents striking the discharge; and also MW discharges should be operated at pressures above 1 Torr.

By this derivation, it has been shown that the power transfer per HF cycle depends on the ratio (μ)/ (μ). For xed pressure (xed number density), the eciency of energy input should decline with rising frequency. However, this approach does not take into account an important fact: With rising frequency, the eciency of ionization steeply increases, and this eect more than makes up for the decrease in power transfer. The mechanism of Ohmic heating is the result of interaction between an electron, the force which is exerted by an electric eld, and neutrals which collide with the accelerated electron. This mechanism is mainly active in the plasma bulk provided there is a considerable electromagnetic eld. But this can happen

110

5 High-frequency discharges I

only at low plasma densities or at operating frequencies which exceed the plasma frequency of the electrons, which is the case in low-density microwave-driven discharges. These discharges are driven “electrodeless” (they can be operated through a so-called “microwave window”, mainly made of quartz) and thus have very low plasma potential. This, in turn, causes the formation of very thin plasma sheaths. In the theory of continua which is extensively dealt with in Sect. 14.6, we see that this absorption of power can be expressed by a dielectric constant which is larger than zero but less than unity [eq. (14.171)]. For rising plasma density, however, we enter the regime of evanescence, which is characterized by a negative dielectric constant (ωp,e > ω = 2πf , this will happen for an operating frequency of 2.45 GHz at 1.54×1010 Hz or a plasma density of 7.45×1010 cm3 ). In the high-pressure regime (νm > ω), the plasma impedance shows a complex behavior between inductive and capacitive characteristics, in the low-pressure regime (νm < ω), the impedance is inductive (positive imaginary). As main phenomenological result, the electric eld of the incoming wave is damped to 1/e within the skin depth, but this volume is not static as in the DC case but instead, it is vigorously moving. We speak of the creation of electron-decient breathing sheaths, and Ohmic heating in the plasma becomes less eective. These sheaths are responsible for three eects, displacement current heating, Ohmic heating, and stochastic heating. As they are mainly associated with capacitively coupled plasmas, they are discussed in Chap. 6.

5.3 Operating frequency and the EEDF

At the end of the last section, we pointed out that this approach does not take into account the frequency dependence of the dierent excitations. What we are interested in is high an ionization eciency but not a waste of energy. It was shown by the groups around Moisan and Wertheimer that in fact it is the eciency of ionization which increases with rising frequency, albeit the power input declines. This can happen because the electron energy distribution function EEDF will change its slope with increasing frequency (see also Sect. 14.1). In particular, the convex shaped EEDF exhibiting a typical Druyvesteynian characteristic with an extremely low portion of high-energy electrons beyond 15 eV which is characteristic for the DC case gradually reinforces this part of the EEDF which is so important for ionization (cf. Sect. 4.6). This is shown in Fig. 5.5 for same mean electron energy of 3.5 eV and rising operating frequency from DC (A) to microwaves (D) [159, 160]. This adjustment of the EEDF, however, is self-limiting, since for high plasma densities, Coulombic collisions between electrons become more probable and eventually become dominant, and every distribution function transforms to a Maxwellian distribution. The distribu√ tion functions are normalized to 0∞ f (E) E dE = 1.

5.4 Loss mechanisms

111

1 10-1 f(E) 10-2

MB

DC

DC

0 1.25 2 DC

10-3 10-4

Fig. 5.5. Keeping the total energy constant, the portion of higher-energetic electrons increases with rising operating frequency. Mean energy < E > = 3.5 eV; various νm /ω-ratios: A: → ∞ (DC); B: 2; C: 1.25; D: 0 (microwave plasma). (pa = 0.15 Torr cm (20 Pa cm), after [159, 160].

0

5 10 electron energy [eV]

15

Since the total absorbed power at constant discharge pressure (number density, which means constant νm ) decreases with rising operating frequency, especially in microwave plasmas, the plasma density is larger than the other types operated at DC or RF for the same absorbed power, Moisan and Wertheimer concluded that less energy is spent in atomic excitation. But this means: Since the generation of carriers depends on the operating frequency, also within the same band (e. g. the RF band between 1 and 100 MHz), the plasma density can be increased by raising the operating frequency. This was rst pointed out by Surendra and Graves [161] and is the basic principle of dual-frequency operation (Sect. 6.7).

5.4 Loss mechanisms

We have already identied two mechanisms of electron loss, diusion and attachment. 5.4.1 Diusion The loss mechanism of diusion is of paramount importance in the low-pressure regime (cf. Sect. 4.7). However, this mechanism is also modied in HF discharges and has to be analyzed precisely. We have noted that the diusion coecients of electrons and ions are mutually coupled after ignition of the discharge. As a rst result, the electric eld required for maintenance is signicantly reduced compared to the breakdown eld. In discharges of electropositive gases, the ambipolar diusion coecient Da happens to be smaller by orders of magnitude compared with that of the electrons (Fig. 5.6). The solutions of the Helmholtz equation [139]

112

10

7

5 High-frequency discharges I

106 nion/p [s Torr ]

-1 -1

105 104 10

3

10

2

Fig. 5.6. Field dependence of the diusion coecients in an RF discharge (upper value: breakdown, lower value: sustaining) after [58].

0 10 20 30 40 E/p [V/Torr cm] 50 60

d2 n νion + n = 0, dx2 Da

(5.12)

simplied for a one-dimensional problem, are set by the boundary condition of vanishing plasma density at the wall, which is justied by the fact that the diusion constant scales inversely with pressure. Since no odd function can be a solution, Eq. (5.12) is simply solved by the cosine function: n = n0 cos νion x Da (5.13)

with the boundary condition n = 0 at x = ±1/2 L the gap between the electrodes. This represents the solution for the dominant (lowermost) diusion mode. Higher diusion modes are solved by the higher harmonics of the cosine function. Moreover, this yields for the ionization frequency and the number density: νion = π 2 Da νion π ∧ = 2 L Da L n = n0 eνion t cos

2

=

1 , Λ2

(5.14)

x . (5.15) Λ with Λ the diusion length of the electrons. Inspecting Eqs. (5.13) (5.15), we remark that the constant in the spatial-dependent part, 1/Λ, equals the ratio of the two complicated functions D and νion . Whereas the former depends on gaseous pressure (in fact: on number density and gas temperature), the latter depends exponentially on the temperature of the electrons. To obtain a constant result, the two dependencies must cancel exactly. Equation (5.12) reads, in cylindrical coordinates taking the boundary conditions into account (electron density and its derivations vanish at the walls and at the electrodes):

5.4 Loss mechanisms

113

d2 n 1 dn νion +n + =0 2 dr r dr Da which yields n = n0 eνion t J0 νion r. Da

(5.16)

(5.17)

To have n vanish at the wall rmax = R, we demand that R νion /Da = 2.405, which opens the possibility to determine the ionization frequency νion . This represents the solution for the dominant (lowermost) diusion mode. Higher diusion modes with several maxima will be described by Bessel function of higher order [162]. Inserting Eq. (5.17) into the diusion equation yields for a sustaining discharge νion 2.405 1 = = Λ2 Da R

2

+

π L

2

(5.18)

with R the radius of the cylinder and L the distance between the electrodes. The diusion to the walls is described by the rst part of the RHS, the diusion to the electrodes is described by the second part of the RHS of Eq. (5.18). For a uniform electric eld, breakdown will occur when the losses by diusion are at least counterbalanced by ionizations within the plasma, i. e. when νion = Da /Λ2 or Λ2 = 5.4.2 Recombination Whereas diusion losses are heterogeneous reactions which happen at surfaces, other loss mechanisms take place in the gaseous phase of the plasma. In either case (recombination or attachment), a bimolecular reaction is required. With rising pressure, recombination processes become more likely to occur, and the condition for loss has to be extended by a term which takes the diminishing of carriers due to recombination into account. Since this must be a reaction of second order, Eq. (5.12) has to be extended by a term which is of second order in the ion density: d2 n νion + n kr n2 = 0 : i dx2 Da (5.20) Δn . n (5.19)

The negative sign is because of carrier loss, ni squared because we assume an electropositive plasma with electron density and ion density equal in magnitude. From this equation, we see that recombination processes are likely to occur only at high densities where they readily become the dominating loss mechanism.

114 5.4.3 Attachment in electronegative gases

5 High-frequency discharges I

In plasmas with electronegative molecules, we observe strong deviations from this conduct. A simple mass balance which takes into account generation (ionization) and loss (electron attachment), both considered as bimolecular reactions under general consideration of losses by diusion [Eqs. (5.15) and (5.18)], can be described as follows: Da ne = kion ne nn ka ne nn (5.21) Λ2 with kion and ka the rate coecients of ionization and electron detachment, respectively, ne and nn the densities of electrons and neutrals, Da the ambipolar diusion coecient and Λ the diusion length (approximate distance between the electrodes). This equation can be simplied to kion ka = Da , nn Λ2 (5.22)

provided the mobilities of positive and negative ions are considered equal and are small compared with that of the electrons. Mobility and diusion coecient are joint via the Einsteinian relation 2 D = < εk > μ 3 (5.23.1)

where < εk > denotes the averaged electron energy which is known in this connection as characteristic energy [58] and can be expressed for a Maxwellian distribution as D = kB Te μ with μ= Da can be expressed by [163]: Da = 1 + 2α De 1 + 2α + μe /μIon (5.25) σDC . ne0 (5.24) (5.23.2)

with α = nn /ne (nn : density of the negative ions). Of course, α is zero for argon, but becomes about 100 for SF6 : Hence, the nature of discharges through electronegative gases (i. e. Da ) can be determined predominantly by the density of the negative ions which can far surpass the density of the electrons appreciably, sometimes, by some orders of magnitude [56]. In discharges through electropositive or inert gases, we observe equilibrium between gain (ionization) and loss (ambipolar diusion). Therefore, the electron

5.4 Loss mechanisms

115

density can be described with cosine functions or Bessel functions (maximum in the middle of the reactor, zero at the walls). In discharges through strongly electronegative gases, equilibrium is achieved by electron detachment, and diusion can be neglected, and to rst order, the electron density becomes spatially independent. From equation kion ka = Da , nn Λ2 (5.26)

we can further extract that reducing the discharge pressure (i. e. diminishing the particle density nn ) will lead to a rise in Da since μ scales inversely with the density of the neutrals. In discharges of electropositive molecules (kion ka ), this means an increase of kion , which can be only realized for rising mean energy of the electrons (electron temperature, cf. Sect. 3.5). In discharges of electronegative gases, ka is of comparable magnitude of kion ; kion and ka will change in the same sense, and the electron temperature will change considerably less. Additionally, in this equation, neither the densities of the charged carriers neither their energies (temperatures) will directly appear (the densities are hidden in the ambipolar diusion coecient); and to rst order, these properties are independent of the coupled power. Within certain limits, Te depends only on gas pressure and on the ratio Te /Ti . 5.4.4 Decay The generation of carriers by electronic impact can be described by the ionization rate νion which depends exponentially on the electron temperature. When the power is turned o, the decay rate of the carriers is determined by the ratio of the Bohm velocity over the critical dimensions of the reactor (real volume over eective surface). The Bohm velocity scales with the square root of Te , and the time-dependent part of the continuity equation becomes ln ne (5.27) = νloss . t The electron density is expected to decay exponentially after having switched o the external power (“afterglow”, Fig. 5.7), which is the main advantage of pulsed plasmas: The mutual coupling of the diusion coecients will be lost, the electrons evade the plasma bulk which, in turn, causes an enrichment of positive ions. When the next HF cycle starts by plasma ignition, the reactions at the substrate surface can be tremendously dierent from the behavior associated during “normal” operation (Chaps. 11 + 12).

116

5 High-frequency discharges I

3 n e [10 9cm -3]

1

Fig. 5.7. The electron density in a cavity resonator exponentially decays after having switched o the HF power [164] ( c J. Wiley & Sons, Inc.).

2 4 6 8 t [msec] 10 12

5.5 Breakdown

Having discussed extensively the mechanisms for charge gain and charge loss, we want to tackle the question of breakdown in HF discharges, when the electric eld is still uniform throughout the plasma volume, for example for the operating frequency f = 13.56 MHz (λ ≈ 22 m). A discharge can be ignited when the electron losses by diusion, recombination or electron attachment can be compensated by ionization mechanisms. If the losses are solely due to diusion (cf. Sects. 4.7 and 5.4), the diusion current can be determined by ρ (5.28) ≥ ne0 νion j t with νion the frequency of ionization, and ρ the charge density. Typical for a mechanism which is controlled by diusion is the dependence of the breakdown voltages on pressure with distinct minima. We can readily understand this conduct by deriving Eq. (5.9.3) yielding

2 ne2 E 2 ν 2 + ω 2 2νm P = 0 0 m 2 νm me 2 (νm + ω 2 )2

(5.29)

which peaks at ω = νm (Fig. 5.8). We can readily identify two limiting cases.

2 ω 2 (many collisions during one oscilla1. High discharge pressure νm tion period): The energy which is transferred from the eld to the electron will be dissipated into the plasma by elastic collisions between electrons and neutral molecules, the energy loss per collision Δε is dened via Langevin’s energy loss parameter 2me /(mi + me ), approximately 2me /mi :

5.5 Breakdown

117

power coupling efficiency [a.u.]

w = nm

f = 13.56 MHz

0

50

100 nm [106 sec-1]

150

200

Fig. 5.8. The eciency of power coupling peaks when the angular operating frequency matches the collision frequency νm .

Δε ≈

2 1 e2 Ee 2e0 me 0 = < εe > | νm | m e ν m mi

(5.30)

with < εe > the mean energy of the electrons [165], which gives for the electric eld Ee = νm 2me 2 < εe >, e0 mi (5.31)

the eective eld must be linearly increased with rising pressure (νm ∝ p). Therefore, the dependence is similar to the DC case (energy which is piled up is proportional to E/p with E the electric eld, cf. Fig. 4.12).

2 ω 2 (many oscillations per collision): Since in this 2. Low gas pressure νm range, λe will increase, the probability of gaining energy from the eld will diminish, which will lead to a rise by the amount of the eective eld. Simplifying facts by the assumption that all inelastic collisions will cause an ionization,2 the absorbed power becomes Pabs = νion < εion >, which means for the frequency of ionization: 2 2 νm Pabs e2 E0 0 = , 2 εion me εIon νm νm + ω 2

νion = simplied for νm ω:

(5.32.1)

νion =

Pabs e2 E 2 νm = 0 0 2. εion me εIon ω

(5.32.2)

2 Again, the model gas is helium with mercury vapor (Heg gas); He exhibits a metastable level at 19.8 eV which has a lifetime of some msec: Almost every collision of a metastable He atom, He , with an Hg atom will lead to an ionization; the eective ionization potential amounts to about 19.8 eV; νm (Hg) then becomes 2.37 × 109 sec1 (p in Torr).

118

5 High-frequency discharges I The condition for breakdown is given by νion = D/Λ2 . We note that Λ, the diusion length, solely depends on the geometry of the reactor, and D can be evaluated from kinetic gas theory according to D = λ √ v > /3 < (with λ the mean free path). More precisely, we have to write < v 2 > instead of < v >, this yields νion = √ λ < v2 > . 3Λ2 (5.32.3)

√ √ Since < v 2 > can be written as < v 2 > = λνm , we eventually obtain for the ratio between the frequencies of collision and ionization: νion λ2 = νm 3Λ2 or νion νm = < v2 > . 3Λ2 (5.33.2) (5.33.1)

2 With < εe >= 1/2 me < ve >, the eld eventually becomes

E0 =

ω e0 Λνm

2 εIon < εe >. 3

(5.34)

The minimum of the eld can be located where the frequency for momentum transfer νm equals the operating angular frequency ω. The electric eld for breakdown scales with the inverse frequency of elastic collisions νm [and approximately with the inverted discharge pressure (cf. Sect. 14.1)], and with the operating angular frequency ω (Fig. 5.9). Furthermore, this dependence holds also true for maintenance of the discharge. This causes the time-independent EEDF to vary with the operating frequency [160], and the eciency of power input becomes dependent on pressure. For example, for microwave discharges through helium (σ ∝ 1/v) the range for largest power input can be conned between 4.5 and 9 Torr (600 and 1 200 Pa), and the maximum can be located at about 6 Torr (900 Pa). The breakdown at high frequencies is solely determined by the (primary) α-ionization. Knowing the ionization coecient, it is feasible to determine the strength of the electric eld required for breakdown. To put in another way, the ionization coecient α can be evaluated from this breakdown experiment. From the spatial dependent diusion equation (5.12), it is evident that νion /D is equivalent to the rst Townsend’s coecient α squared (dependent on the gas, its pressure, the electric eld, and, additionally, the operating

5.5 Breakdown

104

119

E [V/cm]

103

2 L = 0.159 cm

102

2 L = 0.475 cm

10 -1 10

1

10 p [Torr]

102

103

Fig. 5.9. Microwave breakdown in He/Hg (Heg gas), L: distance between the electrodes after [166]. Raising the pressure and the collision frequency weakens the eective electric eld. Hence, the amount of E which is required for breakdown must be increased.

frequency). α describes the growth of the electron current n/n0 as function of the cathodic distance: n = n0 exp(αx), i. e. the number of ionizations per cm which can also expressed by νion < ue > with νion the frequency of ionization and < ue > the drift velocity (< ue > = μE) or α= (5.35)

νion . (5.36) μE α, however, refers to an ionization which is caused by a drift in the DC eld, whereas νion /D describes a motion in an oscillating HF eld, which is signicantly smaller than the motion caused by a steady DC eld. Alternatively, this can be expressed by an ionization (creation of an electron-ion pair) by an electron that falls across a voltage of 1 V (instead across a distance of 1 cm). Both the coecients are related by η= νIon α νion = ημE 2 ; [η] = 1/V, [νion /D] = 1/cm2 . = E μE 2 (5.37)

By denition of ζ ζ= the coecients are related via ζ=η μ 1 1 = 2 2 . D Λ Ee (5.38.2) νion 1 , 2 D Ee (5.38.1)

120

5 High-frequency discharges I

From the Einsteinian relation, we see that the ratio D/μ measures the mean energy of the electrons and depends on E/p. Therefore, it is possible to determine η from measurements of ζ and vice versa (Fig. 5.10). However, the mean energy of the electrons can be calculated in a dierent way for the AC case and the DC case, which can lead to diculties when the mean energies, i. e. the electron temperatures, are compared.

10-2 h[ionizations/V]

Ayers (DC)

Hale (DC)

10-3

Varnerin and Brown (AC)

10-4 10

-1 Eeff/p [V Torr-1 cm ]

100

Fig. 5.10. Comparison of the coecients for ionization α and ζ in a discharge through hydrogen after [167].

The increase of the breakdown eld with rising pressure occurs for the same reason as in the DC case: λe drops, and the kinetic energy which can be piled up by the electron between two collisions, decreases as well. Therefore, the eld must scale inversely to λe , i. e. direct proportional to the discharge pressure. For low pressures, the mechanisms become dierent. The eciency of the energy transfer deteriorates, thereby causing an increase of the breakdown eld. In the limit of collisionless plasma, no energy can be absorbed at all. Hence, this minimum is no Paschen minimum [168]. In contrast to DC discharges, widening the distance between the electrodes will cause a reduction of the breakdown eld: The equilibrium between electron loss and electron generation is achieved more readily (Fig. 5.11). As in DC discharges, the minimum is the result of two competing mechanisms. 5.5.1 Microwave discharges: model for breakdown To shed light on this breakdown problem from another side and to discuss the limits of energy gain, Brown and Bell investigated the inuence of all the parameters which are involved in microwave breakdown [58, 169]. Since it is of high heuristic value and conventional and highly sophisticated microwave discharges have regained much interest as plasma sources for ion beam processing (SLAN), it will be presented here.

5.5 Breakdown

60 50 E/p [V Torr cm ]

-1

121

40 30 20 10 0 0 1 2 3 4 pL [Torr cm] 5 6 7

-1

Fig. 5.11. Reducing the gap between the electrodes (reduction of Λ) causes higher losses by diusion, and the eective eld which is required for breakdown must rise (H2 , after [58]).

We will use the property pd (p: gas pressure, d: distance between the electrodes) which behaves as almost invariant against alterations of the dimensions of the reactor [similarity rules (Sec. 4.10)]. As far as the inuence of the HF eld is concerned, pλ (with λ the wavelength of the operating eld) is the wellmatched parameter. The energy is denoted by ε instead of E to avoid confusion with the electric eld, and for the mean free path λ, we use l to avoid confusion with the wavelength λ. For all parameters, the inuence on the ratio Λ/λ has to be considered. 5.5.1.1 Frequency. As we have stated in the introductory remarks, a breakdown will occur at the state of equilibrium between generation by ionization and loss by diusion. At low frequencies, d λ: The eld across the dimensions of the reactor is uniform, this is the border of uniform eld. For high frequencies, however, a threshold is set by the condition d = πΛ ≥ λ/2 with Λ the diusion length, yielding pλ . (5.40) 2π We note that this restriction comes into play only for microwave discharges where λ ≈ Λ. pλ = 2π(pΛ) pΛ = 5.5.1.2 Pressure. Next, we nd that the pressure will conne the range of validity of the kinetic gas theory, that is, when the mean free path le becomes equal to the diusion length Λ. The probability for collision Pm is inversely proportional to the discharge pressure p, and le = Λe , yielding (5.39)

122 pΛ = 1 , Pm

5 High-frequency discharges I (5.41)

this property does depend on the (mean) energy of the electrons, Te . This denes the border of the mean free path. 5.5.1.3 Oscillation amplitude. If the amplitude of the HF oscillation becomes suciently large, the electrons can reach the electrodes at the peak of every HF cycle and are instantaneously annihilated. The amplitude can be inferred from the second Newtonian law: e0 dx d2 x + νm = E0 eiωt dt2 dt m yielding x= e0 e0 E0 E0 eiωt = eiωt . m iνm ω ω 2 m iω(iω + νm ) ω iνm 2 2 ω 2 + νm ω 2 + νm e0 E0 mω 1 ω2

2 + νm

(5.42)

(5.43)

The amplitude ψ0 of this oscillation is the amount of the complex number e0 E0 mω : (5.44) (5.45.1)

ψ0 =

,

which can be written with Eqs. (5.11) for the eective eld as √ 2e0 Ee ψ0 = . mωνm

(5.45.2)

The amplitude cannot exceed half the distance between the electrodes: √ L 2e0 Ee . (5.46) = 2 mωνm Substituting Pm = we obtain pλ = √ mπc pL Pm v ∨ pλ = 2 . e0 Ee /p (5.48) 1 2πc v ∧ω = ∧ νm = , ple λ le (5.47)

Inserting the constant yields (with L = πΛ): pλ = 106 pΛ , Ee /p (5.49)

5.5 Breakdown

123

which can be solved for known breakdown eld. This is the border of the oscillation amplitude. When entering this regime, all the electrons are lost within one half of a HF cycle by recombination with the wall (electrodes), and the eld must be increased tremendously to compensate for these losses (Fig. 5.12).

350 300 250 E [V/cm] 200 150

500 mTorr 400 mTorr 300 mTorr

100 50 0 0 50 100 l [cm] 150 200

Fig. 5.12. At the border of oscillation limit, the breakdown eld must be increased tremendously (after [170]).

5.5.1.4 Low gas pressure. Next, there exists the transition from high pressure to low pressure (many collisions per oscillation → many oscillations between two collisions). The borderline can evidently be drawn at √ < v >2 . (5.50) ω = νm = le For hydrogen [νm = 5.9×109 p0 (p in Torr) or 7.89×1011 p0 (p in Pa)], pλ = 4246. This is the border of the transition of the collision frequency. 5.5.1.5 Breakdown. Eventually, we nd the optimum border of the breakdown [Eq. (5.34)] with ω = 2πc/λ yielding 2πc e0 Λλνm Inserting the values for hydrogen: E= νm = 7.89 × 1011 p0 [in Pa] εion = 15.76 eV, assuming < ε >= εion , we obtain pλ = This borderline is drawn in Fig. 5.13. 0.56 . pΛ (5.52) 2 εIon < εe >. 3 (5.51)

124

104

5

5 High-frequency discharges I

10 pL [Pa cm] 10

3

2

3 1 4

10 1 10-1 10

2

Fig. 5.13. Meeting the borders of stability in a HF discharge through hydrogen, (1) border of the uniform eld, (2) border of the mean free path, (3) border of the oscillation amplitude, (4) border of low gaseous pressure, (5) optimum border of breakdown (after [171]).

107 108

102

103 104 105 pl [Pa cm]

106

5.5.1.6 Conclusion. For an HF discharge, there exists an island of stability, which is completely conned by diusion processes with respect to every (primary) parameter. These boundaries encircle this island of stability perfectly for discharges driven by microwaves (300 MHz or higher), whereas for RF discharges, some conditions become obsolete.

5.6 Maintenance

5.6.1 EEDF and the electric eld We rst note that νion = nkion = nσion < ve > (5.53)

using σion , the cross section for ionization by electron impact, and < ve >, the mean electron velocity. For athermal plasmas, the mean energy of the electrons is considerably greater than that of the ions. Balancing generation and loss, and remembering that the diusion coecients of both carriers are mutually coupled by the ambipolar diusion coecient, Da can be expressed approximately as Da ≈ μ i ε k (5.54)

with μi the ion mobility and εk the so-called “characteristic electron energy” which is obtained via the Einsteinian relation [Eqs. (5.23)] εk = De . μe (5.55)

Approximating Da with the diusion coecient dened by Eq. (5.54), we eventually obtain

5.6 Maintenance

125

nkion ≈

μi εe kion μi → ≈ : 2 Λ εk n Λ2

(5.56)

Although both quantities on the LHS (n and kion ) depend on the EEDF, its ratio can be expressed as a function of specic constants of the plasma-constituting gas (which determine μi ), its pressure (which mainly determines n) and the reactor geometry (by which Λ is xed). In particular, this ratio does not depend on the operating frequency. The independent properties which determine the shape of the EEDF adapt to those values to ensure the ratio kion /εk being kept constant. Following Moisan and Wertheimer [160], we can nd two limiting cases: For low plasma densities, the Coulombic collisions between electrons (i. e. the electron density itself) and the stepwise ionization can be neglected in shaping the EEDF, both kion and < εk > remain unaected and become sole functions of E/n and E/ω. – In particular, the averaged electron energy in ultrahigh frequency plasmas (some hundreds of MHz or microwave) should always by smaller than that in an RF plasma, even for higher intensity of the electric eld. – RF discharges follow neither the E/n-dependence (DC discharges) nor the E/ω (microwave) but show a somewhat mixed behavior. Thus Eq. (5.31) denes the magnitude of the discharge maintenance eld (which diers from the externally applied eld with operating angular frequency ω), and for suciently high plasma densities, the EEDF becomes Maxwellian, the electron temperature is well-dened and is given by kB Te = 2/3 < εe >. For a given operating frequency ω at a certain number density n, the electron temperature Te will determine the electric eld required for maintenance. 5.6.2 Collision frequency In a high-frequency discharge, the frequency of elastic collisions between electrons and neutrals, can be obtained by measuring the complex conductivity (which is connected to the mobility via σ = e0 nμ) (Fig. 5.14): νm σr = . σi ω However, two objectives make matters complicated: (5.57)

126

5 High-frequency discharges I

In all other gases except hydrogen and helium, the frequency for elastic collisions νm varies with electron energy and consequently, the concept of the eective eld becomes questionable, if we are not interested in the high-energy tail of the EEDF. The EEDF depends complexly on the operating frequency [172]. The power transfer exhibits a maximum at νm = ω which is a manifestation of the frequency dependence of the (time-dependent) EEDF.

10 -1 sr sr si si

0,05 eV 1 eV

10 -2 s [W -1 cm -1]

10 -3

10 -4

10 -5

Fig. 5.14. Comparison of the real part and the imaginary part of the electric conductivity of a plasma after [173] ( c J. Wiley & Sons, Inc.).

107 n e [cm -3] 10 9 10 11

Therefore, it is worthwhile to get an idea of the deviations from the simple theory. For chlorine, νm rises from 6.3 × 106 sec1 at about 1 eV to 135 × 106 sec1 at about 15 eV. The rst value is small even against the angular FCC frequency (13.56 MHz) of 85 MHz, and so we expect the characteristic electron energy to depend solely on the square of E0 /ω, whereas the averaged energy of the tail electrons should show some pressure dependence (Sect. 14.1).

5.7 High-frequency coupling: qualitative approach

Considering the generation of carriers, the operating frequency should be 1. lower than the plasma frequency to enter the cuto regime or regime of evanescence: ω < ωp , and should be 2. matched to the discharge pressure: ω = νm . Also to use dielectric electrodes, high operating frequencies would be advantageous. However, inductivities are then severely damped. It is for this reason that the upper operating limit is given by severe grounding problems which are dicult to overcome. Furthermore, below 100 MHz, the reactor can be designed very exibly; electrodes and coils can be shaped into almost all geometries. The

5.7 High-frequency coupling: qualitative approach

127

most serious disadvantage is the lack of condition (2). That is why the range between 10 and 20 MHz is a good compromise; and in the 1950s, 13.56 MHz was the only frequency in the RF band which was permitted by the Federal Communications Commission (FCC) to avoid interference with the networks used for communication [174]. Non-linear eects, however, cause the generation of numerous strong higher harmonics: The sixth one is located in the VHF band, the seventh and eighth are located in the band used for airspace surveillance. Moreover, there are several frequency-dependent eects which have to be discussed in the following. As a rst summary, we have found that the high cathode voltage which is required in a DC discharge to generate secondary electrons can be considerably reduced at the same power input in an HF discharge: γ-electrons are not necessary any more to sustain the discharge (so-called α-regime [175]). The discharge pressure can be signicantly lowered. As main result, we can get rid of the electrodes as constitutive elements within the discharge, and this discharge is spoken of an electrodeless discharge. The coupling occurs through the dielectric wall of a bell-jar or a window by means of condensor plates (capacitive coupling) or coils (inductive coupling) (Figs. 5.15, cf. Sect. 5.8).

Fig. 5.15. The coupling can happen capacitively, placing the electrodes within the discharge (LHS) or externally (M); alternatively, coupling can occur in an inductive way (electrode always located externally, RHS), here shown for a barrel reactor [176].

Hence, high-frequency discharges can be divided into the following three classes which are characterized by the interaction between the electromagnetic eld and the plasma: 1. E-type: Capacitive or voltage coupling between the RF electrodes which are placed within or out of the discharge reactor, respectively: High a capacitance is required between electrode and plasma, and the sheaths resemble the dielectric layer of a capacitor between the two plates, the electrode

128

5 High-frequency discharges I and the plasma. That the electrodes can be placed outside the plasma reactor is a clear manifestation of the fact that emission of γ-electrons is not required for the maintenance of the plasma. The orientation of the exciting electric eld is normal to the surfaces of the electrodes, the reactor walls and their sheaths. The degree of ionization depends on the amplitude of the RF voltage. This is the vast area of parallel-plate reactors (Chaps. 6, 10 and 11, Figs. 5.16). With increasing degree of ionization, i. e. larger plasma density, plasma voltage and plasma resistance decrease dramatically, and we observe an abrupt change to the

2. H-type: The time-varying current induces a time-varying magnetic eld in the plasma zone which, in turn, generates a second time-varying electric eld which is oriented in a parallel direction with respect to the reactor walls and which sustains the plasma. This excitation requires high conductance between the electrode and the plasma. This regime is denoted inductive or current coupling with large RF currents. The electric eld loops will close within the plasma which allows an electrodeless operation. Very high plasma densities (≈ 10%) can be achieved. This is the area of the ICP reactors which are now in common use (Chaps. 7, 10 and 11). For types (1) and (2), the wavelength exceeds the dimensions of the reactor appreciably. For example, the corresponding vacuum wavelength of an RF wave with f = 13.56 MHz (ω = 85.20 MHz) amounts to 22.12 m. Hence, we speak of “quasistationary” discharges or steady-state discharges. Albeit the plasma density achieved with the method of inductive coupling is at least higher by one order of magnitude than that obtained with capacitive coupling, it suers from spatial inhomogenities; therefore, the plasma is applied in the “downstream” mode [95].

RF excitation target + dark space shield gas shower head

dss RF excitation gas shower head dss

plasma

plasma

plasma

electrode with substrates

electrode with substrates dark space shield HF, RF

electrode with substrates

vacuum system

vacuum system

vacuum system

Fig. 5.16. In a countless number of modied parallel-plate reactors, capacitively coupled plasmas serve to modify surfaces either by diode sputtering (LHS), ion etching (middle) or by plasma etching and deposition (RHS). dss denotes dark space shield.

5.7 High-frequency coupling: qualitative approach

129

3. With further rise of the operating frequency, the wavelengths of the operating eld become comparable or even smaller than the dimensions of the plasma apparatus. For example, the corresponding vacuum wavelength for the common microwave frequency of 2.450 GHz is only 12.25 cm. Following our considerations in Sect. 5.5, the topmost eciency is reached for pressures of about 1500 mTorr (200 Pa) in discharges through argon, but microwave discharges can also be operated at elevated pressures, sometimes even at atmospheric pressure, however, at the expense of spatial homogenity. According to Eqs. (14.170), waves with frequencies exceeding the plasma frequency will propagate through the plasma almost undisturbed, the plasma acts as a low-loss dielectric, and the amount of dissipated power is very small still; for a driving frequency of 2.45 GHz, this will happen at plasma densities below 8 × 1010 cm3 . Beyond this threshold (cuto: ω = ωp ), the evanescent or cuto regime is entered, and the penetration depth of the wave is restricted to a zone which is determined by the skin depth with a thickness a several centimeters [eq. (14.166)], the heating zone. According to this equation, the skin eect is far more pronounced in microwave-driven discharges (cf. also Figs. 14.28). In this frequency regime, plasma generation appyling capacitive or inductive coupling is almost impractible, and instead, waveguides, coaxial cables and antennae are used (cf. Sect. 5.10). In principle, widely spread, simple microwave reactors consist of four parts: the power supply with the waveguiding system (for low powers: coaxial cables, for high powers: waveguides), an applicator for power matching and a circulator to annihilate the reected power, and the plasma reactor, a geometrically simple device with either cylindrical or rectangular symmetry, constructed of metallic or dielectric walls (Fig. 5.17).

magnetron

TE10

dielectric tube with plasma

rectangular waveguide

Fig. 5.17. Simplied schematic of a microwave discharge in a reactor (here: dielectric tube) which is transparent for electromagnetic waves which are generated in a magnetron and guided via a rectangular waveguide. For better coupling, the reactor can be constructed as resonator with a sliding short. The main sources for power loss are waveguide losses, impedance matching, and radiation.

The power supply itself consists of a microwave source (magnetron or clystron) and an attached waveguide or resonant cavity to allow easy striking of the discharge. Waveguide losses can be minimized

130

5 High-frequency discharges I by highly-reecting coatings (silver or gold), the impedance matching can be signicantly improved by terminating the waveguide with a horn, gradually ared over several wavelengths. This design also enhances the directivity of the radiation pattern. The applicator, in its simplest form, consists of the microwave window by which the vacuum recipient is sealed against atmosphere. Application of an antenna improves the characteristics of the radiation pattern, however, at the expense of contamination. If the window is tted with a converging lens, the refraction index of the plasma which is less than unity has to be taken into account (Sect. 14.6). Therefore, they have to be concave to focus. To prevent local heating due to the above mentioned skin eect behind the microwave window which consists in most cases of quartz, the plasma vessel has to be designed to act as a resonant cavity. Now, a standing wave is generated in the reactor, and the maximum intensity of the eld has moved away from the microwave window and can be found somewhere inside the reactor. A combination of both of them has been realized by Engemann et al. with their slotted antenna (SLAN) [177]. The microwave eld is coupled into an annular structure via a tuning element (rod antenna) which is located within a conventional rectangular waveguide with perpendicular orientation. To match the varying conditions of the capacitive plasma, not only can this antenna be varied in length but also the rectangular waveguide can be adapted by a sliding short. The microwave power is selectively radiated into the interior of the reactor through slots which are located at the nodes of the electric eld. The circulator serves to annihilate the reected power by a dummy load [178]. The problems associated with the skin depth and poor performance at low discharge pressures can also be circumvented by application of an intense static magnetic eld which guides the wave (plasmas generated by whistler waves, cf. Chap. 7 and Sects. 14.6 + 14.7).

In these high-frequency discharges, the plasma density is considerably higher than in DC discharges for the same power input. For convenience, we compile some basic properties of the most common plasmas in Table 5.2.

5.8 High-frequency coupling: quantitative approach

To ensure a comprehensive understanding of the two most important coupling modes (capacitive and inductive), we will focus on some of the electrical considerations. We will begin with the limiting cases of the two resonant circuits,

5.8 High-frequency coupling: quantitative approach

131

Table 5.2. Plasma sources and the most important parameters discharge pressure, plasma density, and electron temperature along with their most common application. type DC glow RF CCP, low p RF CCP, high p RF ICP RF helicon MW convent. MW SLAN MW ECR E beam p [mTorr] 1 102 1 102 102 103 1 103 0.1 10 10 103 100 103 0.1 5 0.01 0.1 Te [eV] 0.1 10 1 10 15 15 <13 <13 25 3 10 <1 np [cm3 ] 108 1011 108 1011 108 1010 1010 1012 1010 1013 108 1011 109 1011 1010 1013 108 1012 application sputtering, etching, chem. syntheses, radiation sputtering, ion etching deposition, plasma etching real ion etching, chemistry real ion etching chemistry, depos., etching chemistry, depos., etching real ion etching, chemistry sputtering, ion beam etch.

either in series or in parallel, which will be followed by coupled parallel circuits and the dual wiring of the approximated capacitively coupled plasma. 5.8.1 Absorption circuit According to Kirchhoff’s second law, the magnitude of the voltage which is drawn from the voltage source is the sum of the voltage drops within the circuit: V = VL + VC + VR with VL = iωLI and iI VC = ωC which yields for V 1 V = I R + i ωL ωC The amounts of current and voltage are then I= R2 V + ωL

1 ωC 2

(5.58)

(5.59.1)

(5.59.2)

= Z I.

(5.60)

∧ V = I R2 + ωL

1 ωC

2

.

(5.61)

From Eq. (5.60), it will become evident that Z is real for

132

5 High-frequency discharges I

ωL =

1 : ωC

(5.62)

Resonance: I and V are in phase, since the reactance Zs = iXs vanishes; the L and VC can exceed the source voltage V by the quality factor; and we voltages V observe a voltage resonance. According to Thomson, the resonance frequency of the oscillation circuit is given by ωr,series = √ 1 . LC (5.63)

Plotting the characteristic ω → I, we can evaluate the damping of the resonance. For R = 0, I → ∞, for R → ∞, a maximum cannot be identied because the initial sharp peak will broaden (cf. Figs. 5.18).

1.0 0.8 0.6

Z [W]

1 mH, 0.01mF 1 mH, 0.1mF

100 80 60

10 mH, 0.01mF 10 mH, 0.1mF

10 mH, 0.1mF 10 mH, 0.01mF 1 mH, 0.01mF 1 mH, 0.1mF

I [A] 0.4 0.2 0.0 0.0

40 20 0 0.0

2.5

5.0

7.5 w [MHz]

10.0

12.5

2.5

5.0

7.5 w [MHz]

10.0

12.5

Fig. 5.18. LHS: resonance curves of a series circuit for various values of L and C at constant R = 1 Ω. The current show a maximum at resonance, but is strongly damped. This is due to the variation in impedance (RHS).

5.8.2 Eliminator circuit According to Kirchhoff’s rst law, the total current which ows through the circuit equals the (geometrical) sum of the partial currents: I = I L + IC + IR with the coil current V V IL = = i , iωL ωL and the capacitance current (5.65.1) (5.64)

5.8 High-frequency coupling: quantitative approach

133

IC = V iωC :

(5.65.2)

the inductance causes the current intensity to lag behind the voltage (in an ideal case by 90 ), whereas the capacitance causes the current intensity to take the lead (in an ideal case 90 against the voltage). V yields again

10

1 mH, 1mF

1.00

1 mH, 0.01mF

0.75

I [mA]

50 mH, 0.5mF

10mH, 0.1mF

1/Z [W -1]

0.50

50 mH, 0.5mF

10 mH, 0.1mF

0.25

1 mH, 1mF

1 0.0

1 mH, 0.01mF

2.5

5.0

7.5 w [MHz]

10.0

12.5

0.00 0.0

2.5

5.0

7.5 w [MHz]

10.0

12.5

Fig. 5.19. LHS: In a parallel resonant circuit, the current skims a minimum for vanishing rst derivative of the I(V) curve; here displayed for the inverse impedance [several values for L and C at constant R (1 Ω)]. RHS: For minimum damping, the ratio C/L should be kept as low as possible.

1 V = I R + i ωL ωC and the amount of the current is given by I=V 1 R

2

= IZ

(5.66)

+ ωC

1 ωL

2

.

(5.67)

From Eq. (5.66), we see that Z becomes real for 1 . (5.68) ωC The condition for resonance in a parallel circuit is identical with that in a series circuit. Current and voltage are in phase; however, the current maximum is replaced by a minimum in current at resonance (Figs. 5.19). ωL = 5.8.3 Coupled parallel circuits After these preliminaries, we will deal with 1. inductive voltage coupling;

134

5 High-frequency discharges I

Fig. 5.20. Two oscillation circuits which are inductively coupled with primary series feed (transformer coupling).

C1 L 1 U1 R1

L2 C2 R2 U2

2. inductive current coupling; 3. capacitive voltage coupling; 4. capacitive current coupling; 5. transformer coupling (transducer). For our considerations, the odd numbers are of importance. These three cases can be depicted using the same equivalent circuit (Fig. 5.20). As model, we will focus on transformer coupling, which we shall meet again when we deal with inductive coupling; it is easily adapted to the two other cases. 5.8.3.1 Transformer coupling. Applying Kirchhoff’s second law, the oscillator circuit (index 1) and resonance circuit (index 2) follow the equations V = I1 Z1 + I2 iωM 0 = I2 Z2 + I1 iωM with M the coecient of mutual induction M = k L1 L2 , (5.70) (5.69)

with k the coupling factor. In the oscillator circuit, the source voltage V equals the voltage drop across the impedance Z1 + the voltage which is transmitted back from the resonance circuit to that circuit; in the resonance circuit, the transmitted voltage equals that voltage dropped across impedance Z2 . For the two impedances, it it generally valid that Z1,2 = R1,2 + i ωL1,2 From Eq. (5.69.2), we see that iωM I2 = I1 Z2 and for the other properties ω2M 2 V1 = I1 Z1 + 2 Z2 (5.73) (5.72) 1 ωC1,2 (5.71)

5.8 High-frequency coupling: quantitative approach

135

I1 = I2 =

V1 2M 2 Z1 + ω Z 2

(5.74)

iωM V1 . Z1 Z2 + ω 2 M 2

(5.75)

The voltage, which can be measured at the capacitance C2 , is I2 V2 = iωC2 which yields V2 = V1 M C2 (Z1 Z2 + ω 2 M 2 ) (5.77) (5.76)

together with Eq. (5.75). Regarding both circuits as equal in size, R1 = R2 = R; C1 = C2 = C; L1 = L2 = L; M = kL, we nd furthermore R 1 ω2 (5.79) ∧ x=1 2 ∨x=1 0 ωL ω LC ω2 with d the damping. This yields for the impedance Z of both the circuits, the current I2 which ows in the oscillator circuit, and the voltage V2 which is induced into the oscillator circuit: d= Z =r 1+i V1 I2 = i R 1+ x , d

k d k2 d2 x2 d2 k d k2 d2

(5.78)

(5.80)

+ 2i x d

,

(5.81)

V1 V2 = ωCR 1 +

x2 d2

+ 2i x d

;

(5.82)

from which we obtain for the amounts for V2 and I2 : V2 = kV1 V1 k Rd 1 (d2 + k2 x2 )2 + 4x2 d2 1 1+

k2 d2 2

,

(5.83.1)

I2 =

x2 d2

+ 4 x2 d

2

.

(5.83.2)

Deriving Eq. (5.83.1), we nd that the extrema are located at

136 x(k 2 d2 x2 ) = 0 with the roots and Eq. (5.79) for x

5 High-frequency discharges I (5.84)

√ x1 = 0 ∧ x23 = ± k 2 d2 , ω1 = ω0 ∧ ω23 = ω0 . √ 1 ± k 2 d2

(5.85)

(5.86)

Besides the resonance frequency at ω0 , two more maxima in current will occur which are due to the feedback of the resonator circuit to the oscillator circuit, and they are dened by the coupling factor k: Loose or undercritical coupling: k < d. Critical coupling: shallow maximum with three roots (k = d). Tight or overcritical coupling: k > d. Very tight coupling: k to d k2 d2 , by which Eq. (5.86) is simplied

ω23 = √

ω0 . 1±k

(5.87)

Using Eq. (5.83.2), the currents which ow in the resonance circuit can be written as I2,1 = and I2,23 = V1 ; 2R (5.88.2) V1 kd , R d2 + k 2 (5.88.1)

in the case of the critical coupling (k = d), eventually I2,1 = V1 2R (5.88.3)

√ as well. For the detunings k = d, k = d, k = d2 and k = d3 , the resonant characteristics are displayed in Fig. 5.21.

5.8 High-frequency coupling: quantitative approach

0.50

k = d3 k=d k=d

2

137

I2

0.25

k=d

2

k=d

1/2

0.00 -10

Fig. 5.21. Resonance circuit: resonant characterics of the owing current I2 for for different detunings of k/d.

5 10

-5

0 x

5.8.4 Capacitive and inductive coupling The transformer coupling is a special case of voltage coupling which can be derived from Eqs. (5.69) for the two remaining cases by replacing iωM for 1 capacitive coupling by iωC or iωL for inductive coupling: V = I 1 Z1 + 0 = I 2 Z2 +

I2 iωC I1 . iωC

(5.89)

Then M → C 1 2 , which yields for capacitive coupling ω k 1 B 1 → = with B = d ωCR R ωC and for inductive coupling k ωL B → = with B = ωL, d R R and we nd for the resonance circuit: U1 I2 = ± i R 1+

B R B2 R2

(5.90.1)

(5.90.2)

x2 d2 B R

+ 2i x d

x2 d2

, .

(5.91.1) (5.91.2)

U1 i U2 = ± iωCR 1 +

B2 R2

+ 2i x d

The ⊕ sign is valid for inductive coupling, the pling.

sign holds for capacitive cou-

5.8.5 Dual circuit of the capacitively coupled plasma The equivalent circuit of a glowing plasma bulk exhibiting a complex impedance which is separated from the conning reactor walls by capacitive sheaths is

138

5 High-frequency discharges I

Fig. 5.22. Equivalent circuit of a capacitively coupled discharge with a complex impedance of the plasma bulk which is isolated by two capacitive sheaths.

Cs,1

Rp

Lp

sheath 1

plasma bulk

Cp

sheath 2

Cs,2

shown in Fig. 5.22. We see that two resonances will occur, one series resonance and one parallel resonance. Neglecting the real part of the plasma impedance Zp , the parallel resonance is simply

2 ωparallel =

1 . Lp Cp

(5.92)

The resistance of the series resonance can be calculated according to i R= + ωCs,1 1

1 Rp

+ i ωCp

1 ωLp

i . ωCs,2

(5.93)

5.8.5.1 First approximation (symmetric discharge). Neglecting the plasma resistance Rp and assuming that the two sheath capacitancies are equal in size (Cs,1 = Cs,2 = C) yields at resonance: 1 2i + ωC ωCp

1 ωLp

= 0,

(5.94)

and the two resonance frequencies are connected via

2 2 ωseries = ωparallel

2Cp . 2Cp + C

(5.95.1)

Considering further C = ε0 A , this equals d

2 2 ωseries = ωparallel

2d . dp + 2d

(5.95.2)

5.8.5.2 Second approximation (asymmetric discharge). Neglecting the plasma resistance Rp and assuming that the two sheath capacitancies are different in size (Cs,2 = Cs,2 ) yields at resonance: which yields 1 i + ωCs,1 ωCp

1 ωLp

i = 0, ωCs,2

(5.96)

5.9 Matching networks

139

2 2 ωseries = ωparallel

(Cs,1 + Cs,2 )Cp . Cs,1 Cs,2 + Cp (Cs,1 + Cs,2 )

(5.97)

For the limiting case Cs,1

Cs,2 , both the resonance frequencies become equal: ωparallel = ωseries . (5.98)

5.9 Matching networks

For DC discharges, the maximum of power which can be transferred into the plasma is limited by the source resistance and by the character of the load (V : source voltage, Rs : source resistance, Rp : plasma resistance) P = V I = I 2 R; I = V Rp ;P = V 2 . Rs + Rp (Rs + Rp )2 (5.99)

Hence, the power depends on the ratio of the resistances and exhibits a maximum for Rs = Rp . To generate a given power for decreasing plasma load, the current must rise to the current limit (P = I 2 R I = P/R), for increasing plasma load, the voltage has to be increased to the voltage limit √ (P = V 2 /R V = P R). In all these cases, no matter whether we reach the current limit or the voltage limit at maximum power output, the voltage along the line is uniform. The same consideration is valid for the AC case. However, we have to take into account the complex character of the load impedance. In nearly all cases, the load impedance does not match the characteristic impedance of the generator (usually 50 Ω), and a standing wave will be set up along the line (Sect. 5.10). Even the load without plasma is complex; in this simple case, it is purely capacitive. Its value can be evaluated according to the simple formula for a parallel-plate capacitor according to C = ε A with ε the dielectric constant of d the gas within the reactor, A the surface of the electrodes, and d their distance apart. Since the plasma impedance depends on several parameters, it is customary to match the source impedance with the plasma impedance by inserting a network in the line to meet the requirement R ± ix = Rs . The source impedance Rs is usually terminated with 50 Ω which is less by orders of magnitude than the plasma impedance Rp (between 5 and 15 kΩ, inductive plasma impedance, capacitive plasma impedance, capacitive impedance of the sheaths, a typical value at 13.56 MHz: 0.1 pF is a capacitive load of 0.12 MΩ). The RF generator should not only be perfectly matched to the discharge during plasma operation but it should also provide a substantial contribution to the ignition voltage. This is best accomplished by coupling the circuit to the AC current resonance.

140

5 High-frequency discharges I

According to the discussion in Sect. 5.8.5, the capacitive discharge can be regarded as a dual circuit: plasma as a circuit connected in parallel, with parallel resonance according to Eq. (5.92), where the voltage peaks, and the second one, series resonance or geometric resonance according to Eqs. (5.95.1) and (5.97) where the current peaks (Fig. 5.22). From the theory of networks, we know that the output impedance of a well-matched network equals the conjugate-complex of the connected system, i. e shifted by π and equal in size. This is solved practically either by installation of an L-matching network, consisting of two variable capacitancies (one of them connected in series with the capacitive plasma load) and one coil connected in series, or of a π-matching network, consisting of two variable capacitancies (both of them connected in parallel with the capacitive plasma load) and again one coil connected in series between generator output and the powered electrode [89] (Fig. 5.23). The advantage of the π-network is a large tuning range. Since the L-network consists mainly of a blocking capacitor connected in series to the plasma, which easily facilitates the built-up of a so-called DC bias (see Sect. 6.1).3

Lmatch C (var.)

C (var.)

RF generator

C (var.)

capacitively coupled discharge

RF generator

Lmatch C (var.)

inductively coupled discharge

Fig. 5.23. LHS: a capacitively coupled discharge with a matched L-network which is connected between the generator output and the powered electrode. RHS: an inductively coupled discharge with a matched π-network which is connected between the generator output and the powered electrode.

With this means, the system can be tuned to resonance. A double-tuned transducer-coupled output is very useful since it can be coupled slightly overcritical which results in a resonance characteristic with a broad and at peak, and the power output remains nearly constant. A matching network will be required only for the frequency range beyond approximately 1 MHz (more exactly: beyond the angular plasma frequency of the ions, ωp,i ), for lower frequencies, a (coupling) transformer is sucient.

3 In the case of metal electrodes, a blocking capacitor has to be inserted between generator and electrode to suppress the DC current.

5.10 Transmission line

141

5.10 Transmission line

The HF source is connected to the matching network via the transmission line which exhibits the impedance ZL = with Z0 = 377 Ω, (5.101) Ev E Er = = = H Hv Hr μ0 μr μr = Z0 ε0 εr εr (5.100)

the characteristic wave impedance of free space and in air (this is the characteristic wave impedance of the far eld). Since μr in cables is close to unity, but εr signicantly exceeds unity, the impedance of the transmission line is Z0 ZL = √ < 377 Ω. εr Because c= l ω =√ k LC (5.103) (5.102)

√ and the phase velocity vph in the line with refractive index n = εR with εR the real part of the dielectric constant (see Sect. 14.6 for an extensive discussion) vph = c c =√ , εR n (5.104)

the connection between the characteristic impedance of the transmission line ZL and the capacitance per unit length C and the inductance per unit length l L , respectively, is given by l √ l εR (5.105.1) ZL = cC and Lc ZL = √ : l εR (5.105.2)

The characteristic impedance ZL is inversely proportional to the capacitance per unit length C , but scales with the inductance per unit length L . l l In a transmission line which is terminated with the characteristic impedance of the generator (usually 50 Ω) and lacks any imaginary fraction, Z0 = L0 , C0 (5.106)

142

5 High-frequency discharges I

there will be no reexions, since the load will completely absorb the delivered power. In all other cases, the load impedance will not match the source impedance, a reected wave will result and a standing wave will be set up along the line. In contrast to the normal process of transmittance, on a non-matched line, steady peaks of voltage can cause arcing, and steady peaks of current can result in overheating. Voltage and current are sampled by a directional coupler which is inserted into the transmission line. They are converted to two quantities which are denoted as forward and reected voltage Vf and Vr , respectively: V +Z I V Z I ∧ Vr = . (5.107) 2 2 The topmost and lowermost voltage, respectively, is the sum or the dierence of the maximum voltages of the forward and reected wave. At the topmost voltage, the impedance exhibits its maximum and the current its minimum; the extremes of voltage and current are separated apart by half a wavelength, the extremes of impedance by a quarter of a wavelength, respectively. From Eqs. (5.107), the two powers Vf = | Vf,r |2 (5.108) Z can be obtained. We see from Eqs. (5.107/5.108) that Pr will vanish for V = Z × I, but Pr takes large values for extremes in the load impedance (voltage limit or current limit). It is these considerations which will limit the capabilities of an RF generator to handle but a certain amount of reected power. The power which is transmitted into the complex load has to be corrected by the power which is consumed by the transmission line. The reected power, however, is not absorbed by the generator which can be easily seen from the extreme case, termination by an open circuit. In this case, all the currents will vanish, and no power can be transmitted. But there is a standing wave on the line whose components are mutually extinguished by the equal amplitudes of forward wave and reected wave, assuming no loss in the transmission line. In fact, we simply measure but the forward voltage and the reected voltage, and on the dials of the directional coupler, power is displayed which would be dissipated in a 50 Ω resistor [P = V 2 /(R = 50 Ω)].4 In fact, conductive losses and dielectric losses occur in the load which will throw the phases of voltage and current out of step. Conductive losses are Ohmic resistance and frequency dependent resistance which is described by the skin eect. Dielectric losses are caused by the directing eect on molecules by the Pf,r =

4 HiFi enthusiasts labor under a similar misapprehension when they connect their brandnew super amplier to real critical impedances, e. g. electrostatic speakers, which will become i very low-ohmic in the pitch range because of R = ωC , and due to the acoustic short-circuit, at the very rst trumpet signal, blue smoke will soar from the amplier . . .

5.10 Transmission line

143

electric eld; they are dicult to calculate. In total, they are proportional to the excitation frequency, whereas conductive losses scale with the square root of the frequency: At low frequencies, the conductive losses will dominate, at high frequencies, the dielectric losses will take over. In dielectric media, timely varying elds (E and H) are orientated perpendicular with respect to the direction of propagation. In the limiting case ω → 0, the elds are longitudinal: They can be derived from scalar potentials and are orientated in parallel fashion with respect to the direction of propagation. For non-vanishing conductivity, we expect a mixed behavior, i. e. modes with either a longitudinal component of the electric eld E or the magnetic eld H, respectively. The mode with the lowest cuto frequency will dominate this propagation. Below this frequency, propagation cannot occur because the wave will be damped to 1/3 of its initial intensity within the skin depth. The TEM mode does not exhibit a cuto, but its tangential components Et and Ht must simultaneously vanish at the equipotential surface of the conducting medium. But vanishing eld at the equipotential surface entails vanishing everywhere: Propagation of TEM waves within this waveguide is impossible and can be achieved only in coaxial cables and waveguides, respectively.

5.10.1 Coaxial cable The transmission is made feasible by applying coaxial cables. In these devices, the TEM mode is the dominant mode, E and H are mutually perpendicular, the current in the walls is normal with respect to the magnetic eld. The currents are considered to ow along the outside surface of the inner conductor and to ow back along the inside surface of the outer conductor or shield which is larger in size. They can be either rigid, using a central rod as inner conductor and a solid metallic tubing for the outer conductor, or they can be exible with one or more layers of conducting braid for the outer conductor (coaxial cable). In the latter case, the inner conductor can be supported by an elastomeric polymer, or by spaced dielectric beads. Since the dielectric loss limits the power that can be transmitted, the second construction is superior to a complete lling by a continuous dielectric between the two conducting lines. The impedance of these coaxial cables strongly depends on the operating frequency; in particular, standing waves can occur not only at the design frequency, which leads to considerable loss especially at the dielectric beads. In solid coaxial cables, the dielectric loss can be further reduced by stubs, a section of a coaxial line 1/4 λ long and short-circuited at the inner conductor. Since the input admittance of this device vanishes at the design frequency of the coaxial cable, this stub does not inuence the operation of the transmission line. In its active version, it can be used to vary the characteristic impedance of the cable. In particular, a mismatch which is caused by standing waves can be

144

5 High-frequency discharges I

eliminated, either only at the design frequency, or, equipped with a broadband stub, over a broader frequency range. 5.10.1.1 Characteristic impedance. To calculate the velocity of phase propagation of a transmission line, we should know the capacitance per unit length C , and the inductance per unit length L according to Eqs. (5.105). The magl l netic energy which is stored in a coaxial cable can be simply calculated to (ε for energy, E for electric eld strength) ε0 c2 1 B 2 d3 x. (5.109) ε = LI 2 = 2 2 For a cylindrical volume element of length l and thickness r d3 x = l2πrdr, this ` yields with Ampere’s law B = 1/2πε0 c2 I/r ε= or ε= ε0 c2 2

D d

I 2ε0 c2 πr

2

l 2πrdr

(5.110.1)

I 2l D ln 2 4πε0 c d

(5.110.2)

(with D the inner diameter of the outer conductor and d the diameter of the inner conductor), and we obtain for L l D ln 2πε0 c2 d and the inductance per unit length L0 L= L0 = (5.111)

1 D ln . (5.112) 2πε0 c2 d Calculating the charge which has been stored upon the surface of the cylinder with area 2πrl Q = 2πε0 rlE or Q = 2πε0 rl we obtain Q= 2πε0 l (V1 V2 ). ln D/d (5.113.3) dV , dr (5.113.2) (5.113.1)

For the capacitance per unit length, this yields C0 = Q 2πε0 = , Ul ln D/d (5.114)

5.10 Transmission line and eventually, for the characteristic impedance, Z0 = ln D/d . 2πε0 c

145

(5.115)

Since the geometric factor does depend on the dimensions of the cable only logarithmically, and 1/(2πε0 ) amounts to approximately 60 Ω, the characteristic impedance can be found to range between 50 and several hundreds of ohms. 5.10.2 Waveguide Removing the central conductor from a coaxial cable leads to the second type of transmission line, the waveguide. This is made use of in the coaxial to waveguide couplings. For a given power, the electric eld is less intense in a waveguide compared with a coaxial cable. Therefore, the transmission of higher powers is feasible until breakdown occurs. Compared with a coaxial cable, the losses, especially at higher frequencies, are considerably lower: Dielectric loss scales with frequency, Ohmic loss, however, only with the square root of frequency. Additionally, the current densities across the inner conductor are very large, and since the loss scales with the current density squared, the lower current densities in a waveguide are of advantage once more. To reduce these Ohmic losses further, its inner surfaces are often coated with silver or gold. Again, the characteristic impedance can be varied with broadband stubs, especially for coupling microwaves using a three-stub tuner. For practical reasons, the transmission through waveguides is feasible only in the wavelength range between 1 and 30 cm; but within this interval, the FCC frequency of 2.45 GHz is located which is radiated by the widely used microwave magnetrons (λ = 12.25 cm). 5.10.3 Mode patterns in transmission lines 5.10.3.1 Coaxial cable. The dominant mode is always the mode with the lowest cuto frequency. Below this frequency, the transmitted energy is rapidly attenuated (Figs. 14.25/26). The rst cylindrical mode of higher order is the TE11 , which can propagate if λ = (D d)/(2n) ≈ πD (Fig. 5.24). The rst subscript denotes the number of nodes of the radial component, the second subscript denotes the number of nodes of the azimuthal component. These modes degenerate to similar modes for a cylindrical waveguide for vanishing diameter of the inner conductor (Fig. 5.24). The frequency separation of the higher modes is a function of D/d. 5.10.3.2 Waveguide. For rectangular waveguides, the subscripts denote the number of maxima: the rst index for the wide or a dimension, the second one for the narrow or b dimension. The dominant mode is the TE10 with only one

146

5 High-frequency discharges I

E H

Fig. 5.24. The TE11 mode in a coaxial cable (LHS) will transform to the TE11 mode of a cyclindrical waveguide for vanishing diameter of the inner conductor.

maximum and the longest cuto wavelength λc . The modes next to follow are the TE20 or TE02 ; but in the vast range between cuto and double cuto, the TE10 dominates (Fig. 5.25).

E E

Fig. 5.25. The TE10 mode in a rectangular waveguide is the dominant mode in the vast range between λc = 2a and λc = a.

5.10.4 Electrode The transmission line is terminated by the electrode (in the case of ion etching). In the case of sputtering, this is identical with the target. The impedance should be as low as possible to avoid stray capacitancies which reduce the eciency of power input. For a target, this is most easily accomplished for a large ratio area over thickness—a tradeo between long-term stability, mechanical stability and electrical properties. Smooth, rounded edges reduce high eld intensities; but one of the most important issues is a perfect metallization of the backside of the target since the power loss per volume is dened by P = ε0 εω E 2 tan δ, (5.116) V with E the intensity of the electric eld across the thickness of the target and tan δ the energy dissipation factor of the electrode, which depends on the quality

5.11 Shielding

Faraday shield

147

interference circuit

CS

interfered circuit

interference circuit

interfered circuit

Fig. 5.26. The capacitive coupling of an interference circuit is eectively prevented by installation of a Faraday shield between interference circuit and and interfered circuit.

of the target material (grain size, purity, . . . ) [179]. The decisive parameter is E.

5.11 Shielding

The devices within a plasma are subject to at least three disturbing inuences: Capacitive coupling (straying of electric elds). Inductive coupling (straying of magnetic elds). HF elds (straying of electromagnetic waves). Capacitive coupling by the presence of disturbing electric elds which are generated by the storage of electric charges in capacitors will cause oscillations of a voltage signal when the electric ux is altered.5 By wrapping the circuit to be protected in a metallic shield, the penetration of electric elds is eectively prevented. This electrostatic shield is the well-known Faraday shield. In most cases, this shield is merely grounded for mechanical reasons, and stray capacitancies between circuit and shield are eliminated rather than acting as sources for crosstalk (Fig. 5.26). In the case of suspecting inductive coupling, the magnetic ux must be prevented from penetrating into the circuit. According to Faraday’s law, the magnitude of the induced surge depends on the penetrated area of the circuit to be protected, the strategies are either drilling of the wires (f < 1 MHz) or the use of coaxial cables (f < 1 GHz). Since the near eld of HF elds [characteristic wave impedance signicantly dierent from the value of free space (377 Ω)] amounts to about 1/6 λ, RF sources with an operating frequency of 13.56 MHZ threaten a double hazard capacitive

5 Among disturbing subjects, the experimenting scientist with rubber-soled shoes standing on a oor made of highly non-conductive, organic polymers should not be forgotten.

148

5 High-frequency discharges I

and inductive in nature within a distance of about 6 m. This eld is neutralized most elegantly by induced eddy currents in a Faraday shield, i. e. aimed inductive coupling in a sacricial circuit. On the other hand, the skin depth of an electric eld scales inversely with its frequency and the conductivity of the substrate; the remaining eld will not be absorbed but will be reected: aimed capacitive coupling in a sacricial circuit. Harmonic generation considered, the power density of the losses caused by eddy currents with their vortex diameters in the order of magnitude of the thickness of the shield d can be approximated by [180] dB 1 , (5.117) jE = σE 2 ≈ σd2 ω 2 4 dt this is proportional to the thickness and the operating frequency, both squared, but scales only linearly with the relative permeability (B = μ0 μH) (Fig. 5.27).

150

steel d = 3 mm

100

copper d = 3 mm

50

copper d = 0.06 mm

0 10 102 103 104 n [Hz] 105 106 107

Fig. 5.27. The losses by eddy currents depend on the thickness of the metallic sheets, their conductivity and the relative permeability, but above all on the frequency of the penetrating wave eld [181].

The skin depth of the electric eld in good metallic conductors is given by [182] d= √ which scales at high frequencies with jE ∝ d ω /2 .

3

absorption loss [dB]

1 , πμ0 μωσ

(5.118)

(5.119)

This makes readily comprehensible the fact that a thin, highly conducting sheet is more than sucient for eective capacitive coupling in the sacricial circuit; for eective inductive coupling, high frequencies, good conductivities and relative permeabilities are advantageous. The last two properties show opposite behavior; however, steel is often favored over copper, especially at frequencies below 100 kHz, which displays its advantage beyond 1 MHz (Fig. 5.27). The generation of eddy currents is much more ecient in bubble-free metals; therefore, the shielding metal used against magnetic elds should be chosen very carefully.

5.11 Shielding

149

In conclusion, it should be emphasized that the coupling of HF power is considerably more dicult than the generation of DC plasmas. To obtain reproducible results, ground loops should be avoided and a well-matched network should respond to slight but inevitable variations of primary process parameters.

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