Plasma Sources Sci. Technol. 1 (1 992) 179-1 86 Printed in the UK
`
`I
`
`A simple analysis of an inductive
`
`I
`
`R B Piejak, V A Godyak and B M Alexandrovich
`GTE Laboratories Inc. 40 Sylvan Road, Waltham. MA 02254, USA
`
`Received 15 May 1992, in final form 17 July 1992
`
`Abstract. The electrical properties of an inductive low pressure RF discharge
`have been analysed by considering the discharge to be a one-turn secondary of
`an air-core transformer. Expressions for spatially averaged quantities representing
`familiar discharge parameters such as the voltage, current and electric field have
`been determined as functions of measured electrical parameters of the primary
`circuit. Based on an analvtical expression relating the coupling between t h e
`electrical characteristics of the primary coil and the plasma load, scaling laws for
`plasma parameters and the RF power distribution between the inductor coil and
`the discharge have been determined. The analysis developed here was applied to a
`collisionally dominated inductive RF discharge in a mercury-rare gas mixture. It
`may also be applied as a practical design and optimization tool for a plasma
`processing source based on an inductive discharge.
`
`1. Introduction
`
`Inductively coupled (electrodeless) discharges hold much
`promise as plasma sources for electrical discharge light-
`ing and plasma processing. Since they do not depend
`large voltages to drive displacement current
`upon
`through the powered w sheaths, ion energies in inductive
`discharges are considerably lower than those found in
`capacitively coupled RF discharges (especially at high
`power density). Relatively low ion energies in inductive
`discharges result in a decrease in ion-wall interactions
`(e.g. sputtering, etching and a variety of energetic
`ion-induced chemical reactions). In many instances,
`inductively coupled discharges efficiently provide a high
`density plasma with relatively small ion power loss in the
`sheaths. Moreover, plasma generation and ion accelera-
`tion processes can he independently controlled for in-
`ductive discharges in plasma processing reactors through
`RF biasing of the remote substrate, resulting in independ-
`ent control of the ion energy.
`Inductively coupled discharges have been known for
`over a century [1,2] and many authors have analysed
`their operation. A short concise literature review dealing
`with modelling low-pressure, collisional discharges main-
`tained by an RF current applied to an induction coil is
`contained in recent papers by Lister and Cox [3] and by
`Denneman [4]. In these works the spatial distribution of
`the plasma RF field and the current density are numeri-
`cally calculated by solving a coupled set of Maxwell
`equations for an internal (referenced to the plasma) [31
`and an external [4] inductor coil. Assuming diffusion-
`controlled plasma density profiles (one with zero hound-
`ary values) for cylindrical and coaxial plasmas and
`0963-0252/92/030179+08 $07.50 0 1992 IOP Publishing
`
`collisional RF power transfer to electrons, the authors
`[3,4] were able to couple electrical parameters at the
`primary coil with given plasma parameters for discrete
`(measurement) points. Such analyses of a real experiment
`are somewhat qualitative in nature because of the
`assumptions mentioned above and also because a one-
`dimensional model was assumed. Note that generally, in
`applications, the inductor coil length is comparable to (or
`even shorter than) the plasma dimension and the coil
`diameter.
`In this work, as in previous works, a low-pressure
`inductive RF discharge is studied by considering it to be
`the secondary of an air-core transformer. However,
`numerical solutions of Maxwell equations and most of
`the underlying assumptions about the plasma density
`distribution and the mechanism of w power dissipation
`needed to solve these equations are avoided by taking
`advantage of well known electrical circuit principles
`(which follow from Maxwell's equations) governing
`air-core transformers and solenoidal inductors. In the
`present approach only integral plasma parameters are
`considered (such as discharge current, voltage and
`power) regardless of their particular spatial distributions.
`The spatial distribution
`is formally accounted for
`through the coupling coefficient between the primary coil
`and the plasma which can be determined from experi-
`ment. An integral representation of plasma parameters,
`rather than the differential representation given in [3,4],
`is quite advantageous since it considerably simplifies the
`analysis and enables one to generate analytical formulae
`for the relationships between external electrical and
`plasma characteristics. Furthermore, since the spatial
`discharge characteristics are not essential to this analysis,
`
`179
`
`Ex.1017 p.1
`
`

`
`R B Piejak et a/
`
`the relationships obtained here can he applied to different
`inductive discharge configurations such as external,
`internal and immersed inductor coils of finite length,
`without knowledge of the particular mechanisms invol-
`ved in forming the plasma density profile and in w power
`transfer.
`In the experimental part of this work the external
`electrical characteristics of inductive discharges in a
`mercury-rare gas mixture (typical for a fluorescent lamp)
`were measured over a range of discharge power, and
`internal plasma parameters were inferred from analytical
`formulae obtained here. It is assuring to note that as a
`function of discharge current the plasma parameters
`determined in this way appear to have gmiera; character-
`istics that are similar to those possessed by all low
`pressure gas discharges (DC,
`low
`frequency and
`capacitively coupled RF). For example, over a wide range
`of discharge current the plasma electric field is almost
`constant, thereby resulting in a plasma density that is
`proportional to the discharge current and RF power.
`
`2. Transformer formalism
`
`To model a low-pressure inductively coupled discharge,
`the discharge is regarded as the secondary coil of an air-
`core transformer as shown in figure 1. The primary of the
`transformer is the induction coil itself, which is composed
`of n turns with values of inductance Lo and resistance R,
`(or Q-factor, Q = wLo/Ro) that are measured hefore-
`hand. A sinusoidal RF voltage with an RMS value V, and
`radian frequency o is applied to the primary coil result-
`ing in a primary coil current I,. Since the discharge is an
`electrically conductive fluid that surrounds the coil, it is
`considered to be a multitude of filamentary discharges
`that essentially run in parallel, forming, in practice, a one
`turn secondary winding of an air-core transformer with
`an inductance E, and a plasma resistance R,.
`The discharge inductance E, consists of two compo-
`nents: the electron inertia inductance Le, which follows
`from the plasma conductivity formula U = e2nn,/m(v +
`jo), where e is electron charge, m is electron mass, n. is
`plasma density and v is the effective electron collision
`frequency, and the geometric (or magnetic) inductance
`L,, which is due to the discharge current path. L , is
`inductively coupled to the primary coil through mutual
`
`I IC
`
`secondary circuit
`primarycnil
`Figure 1. Electrical circuit representation of an inductive
`RF discharge as the secondary of an air-core transformer.
`
`180
`
`Figure 2. Equivalent circuit of an inductively coupled
`discharge where the secondary circuit has been
`transformed into its series equivalent in terms of the
`primary circuit current.
`
`inductance M while L, is considered to he the imaginary
`part of the plasma load impedance, L, = R,/v.
`In the analysis presented here it is assumed that the
`discharge is in a purely inductive mode, implying that the
`capacitive mode of operation, which appears to dominate
`upon discharge initiation and at relatively low power,
`can be ignored and the power transfer to the plasma
`electrons can be represented by the plasma conductivity
`formula.
`Inductive discharges are generally initiated in a
`capacitive mode between adjacent turns of a coil to
`which voltage is applied. When the coil current is large
`enough to induce an azimuthal RF field that can maintain
`the ionization process, a clearly visible increase in dis-
`charge light intensity occurs and the main mechanism
`driving the discharge shifts from a capacitive to an
`inductive mode. For a driving frequency where the
`wavelength is much larger than the discharge dimen-
`sions, the magnetic induction is in phase with the current
`and the EMF is in phase at all points occupied by the
`plasma.
`The coupled circuits shown in figure 1 can be trans-
`formed through a straightforward circuit analysis into
`that shown in figure 2 using the classic approach given by
`Termin [SI. In this representation the secondary circuit
`elements are written in terms of the primary circuit
`current. From the point of view of the primary circuit, the
`effect of the coupled secondary circuit is to add the
`impedance ( W M ) ~ / Z , to the primary circuit, where Z, is
`the complex (vector) impedance of the secondary circuit.
`With the coupled secondary impedance separated into
`real and imaginary parts, the equivalent resistance of the
`primary R , is simply the sum of the resistance of the
`primary coil Ro and a term representing the coupled
`secondary resistance R,, and it may be written as
`Rl = Ro f w2k2LoL,RilZ:
`(1)
`where M’ = kZLoL,, Z: = [oL, + ( w / v ) R , ] ~ + R: arid
`k is the coupling coefficient. The equivalent inductance
`L , is of the same general form as R , and is the sum of the
`inductance of the primary and a negative inductance
`
`Ex.1017 p.2
`
`

`
`term representing the total reactance of the coupled
`secondary. L , may be written as
`L , = - wk2LoL,[oLz + (o/v)R,]/Z:.
`(2)
`The negative sign associated with the secondary reac-
`tance can be simply understood by noting that current in
`the secondary causes some of the magnetic flux in the
`primary to be neutralized, thus resulting in a lower total
`magnetic flux in the primary circuit [5].
`Note that in an ideal transformer the coupling is
`perfect (k = 1) and (wM/Z)* = n’, and the secondary
`inductance is L, = Lo/nz. Thus, for an ideal transformer,
`the inductive reactance of the secondary totally offsets
`the reactance of the primary and the resulting equivalent
`primary circuit is purely resistive (RI = R,nz). In general,
`the coupling of an air-core transformer is not perfect and
`(oM/z)’ # n2.
`Equations (1) and (2) are independent equations
`containing three unknown quantities (R2, L , and k). A
`third equation (needed for an analytical solution) fol-
`lows directly from the basic premise of this work that the
`plasma is a one-turn current path that surrounds the
`primary coil and can be considered as a secondary of an
`air-core transformer. Since the primary coil is sur-
`rounded by a conductive fluid (the plasma) and makes a
`closed electrical path about the primary coil, the premise
`of a single-turn secondary appears entirely reasonable.
`To take advantage of this premise it must be recalled !hat
`the inductance L, of a solenoid, with h. > rs, is propor-
`tional to n,’r.’, where n,, h, and rs are the number of turns,
`the length and the radius of the solenoid respectively.
`Thus, Lo oc n2ri and L, 4 r’, where ro is the radius of the
`primary coil, r is the equivalent radius of the discharge
`path and n, = 1 for L,. Assuming the lengths of inductors
`Lo and L , to be equal, we can write
`L, = Lor’/(n2r;).
`For equal length coaxially oriented inductors k sz ri/rZ
`for r t ro and k x rz/ri for r < ro. In what follows,
`corresponding to the geometry of our experiment, we will
`consider the case r t ro, although this analysis is equally
`applicable to either case. Thus, the discharge inductance
`can be written as
`
`(3)
`L , = Lo/knz.
`Equations (I), (2) and (3) can now be solved analytically
`thus coupling the electrical characteristics of the primary
`coil with the electrical characteristics of the plasma load
`kZ = [ d ( L o - Ll)’ + (RI - Ro)zl/~Lo[~(Lo - L,)
`- o/v(Ri - Roll
`Rz = (oLo/kn’)(Ri - Ro)/C(@(Lo - L I )
`- w/v(Ri - Roll
`1, = llwL~,(Jk/n)[(wLo/kn2 + Rzw/V)2 + R:]-’”
`V, = 12R2
`Pd = Vi/R,
`
`(5)
`
`(6)
`
`(7)
`(8)
`
`(4)
`
`Simple analysis of an inductive RF discharge
`
`where I , is the RMS discharge current, V, is the ohmic part
`of the plasma voltage (in phase with the plasma current)
`and Pd is the RF power dissipated in the discharge.
`
`3. Plasma parameter evaluation
`
`Using the relationships determined in equations (4)-(8)
`one can evaluate the electrical characteristics of the
`plasma load and find some integral and spatially aver-
`aged discharge parameters through given experimental
`constants Lo, Q, ro, w and v and a measured set of
`electrical parameters such as L,, R I and P or l / V and the
`phase characteristics of the primary coil. These measu-
`reable parameters are coupled through the simple
`expressions:
`P = I,V,cosrp
`R I = cosrpVl/l,
`L , = sinrpV,/wI,.
`Having inferred the values of the plasma voltage V, and
`coupling coefficient k one can readily evaluate the
`efiective plasma RF field along the discharge path aver-
`aged across a zone where the current is localized (about
`radius r):
`
`E. = VJ2nr = VpJk/2nr,.
`(9)
`Note that E, = Ep(l + wz/v2)-1’z where E, is the RMS
`value of the RF electric field in the plasma at radius r.
`After evaluating E,, the total number of electrons Ne,
`the averaged plasma density (n.), the averaged plasma
`conductivity (uo) = e’ <ne > /mv and the skin depth
`(8) can be estimated from a power balance equation:
`
`Pd = (e’n,/mv)E’dA = (e2/mv) < E 2 > N,
`
`= (e’/mv)cE:N,
`
`c = (E2)/E:
`
`N, = Pdmv/ceZE:
`
`(ne) = Pdmv/ce2E:A
`
`( u o ) = Pd/cAE:
`
`( 8 ) = E,(2cA/op0Pd)‘”
`
`(10)
`
`(1 1)
`
`(12)
`
`(1 3)
`
`(14)
`
`(15)
`
`where E is the effective local RF field in the plasma, c is a
`form factor accounting for the difference between E: and
`<E2> which is averaged over the entire plasma volume, A
`is the plasma volume and po is the vacuum magnetic
`permeability. Form-factor c depends on the radial and
`axial distributions of the plasma density and the local RF
`field. Although in this form both plasma conductivity
`and skin depth appear to depend on discharge power
`density Pd/A, regardless of the particular value of v and
`consequently gas pressure p. gas pressure indirectly
`influences the inferred values <uo> and (8) through the
`dependence of E, on p.
`
`181
`
`I
`
`Ex.1017 p.3
`
`

`
`R B Piejak et al
`
`4. Power transfer efficiency and minimum
`maintenance power
`The power transfer efficiency 5, defined as the ratio of the
`RF discharge power P, to total RF power P needed to
`maintain an inductive discharge, is an especially import-
`ant parameter for applications where energy efficiency is
`desirable. The total power P consists of two parts:
`P = P , + P,, where P, is the power dissipated in the
`primary coil. Thus, 5 = P,/P = [ l + P,/Pd] The ratio
`P,/Pd (see figure 2) can be written as
`Pt/P, = ROAR, - Ro)
`= (R2k2QoL2)-1[(wL2 +(w/v)RJ2 + R f ]
`
`(16)
`A more convenient representation of PJP, is in terms of
`experimental constants Lo, Q and n and measureable
`parameters P,, V, and k, since (as will be shown in the
`experimental section) the last two are essentially constant
`over a wide range of power:
`P,/Pd = n2P,(kQwLo V:)-'[(wL0/knZ
`+ (w/v)V:/Pd)2 + V;/P:l.
`(17)
`From this representation the qualitative behaviour at the
`extremes of the discharge power can be readily under-
`stood with little knowledge of the actual discharge
`parameters. With the normalization t = kZQP,/Pd and
`Y = X , / R , = (wLo/kn2)(P,/V:), equation (17) can be
`written as
`t = Y - I [ ( Y + o/v)Z + 11.
`(18)
`In this form P,/P, can be easily visualized as shown in
`figure 3.
`It follows from equation (17) and figure 3 that,
`for decreasing discharge power when Y << o / v , t %
`(w2/v2 + l)Y-', the ratio P,/P, + 0 and 5 + 0 while
`P, % Pmim approaches some minimal value Pmi, given by
`Pmi, = n2Vi(kQoL0)-'(1 + 0 2 / v 2 ) .
`(19)
`
`10''
`
`1 02
`
`IO'
`100
`Y = X , / R , = Pd
`Figure 3. A normalized ratio PJP, shown a s a function
`of Y = X./R, for ofv = 0, 1 and 10. The discrete points
`shown in the figure are based on experimental data with
`o / v = 0.18 and indicate the range of the experimental
`data in terms of Y. The broken line represents a boundary
`between efficient (t > 0.5) and inefficient (5 < 0.5)
`power transfer at kZQ = 20.
`
`182
`
`represents the power loss in the primary coil needed
`P,,,
`to create a circular RF field equal to the plasma RF field
`but with no plasma present (P, = 0). Below this value of
`RF power an inductive steady-state discharge is impossi-
`ble to maintain. The existence of a minimal maintenance
`power is fundamental to all inductive discharges and
`directly follows from Maxwell's equations and the ion-
`ization and energy balance in a weakly ionized gas.
`Indeed, to sustain an inductive plasma, an electric field
`must be created around a closed path with a magnitude
`that satisfies the ionization and electron energy balances.
`This electric fieid is induced by the primary RF current
`which dissipates energy due to the resistance of the
`primary coil.
`Noting that the inductance of a cylindrical coil with
`length k can be written as:
`Lo = n2m&o/k(l + 0.88r0/h)
`
`(20)
`
`(21)
`
`one obtains
`Pmi. = 107kE:(l + 0.88r0/k)(l + 02/v2)/wk2Q
`with units Pmio(W), E,(V/m), k(m) and ro(m).
`In the opposite limit of large discharge power and
`large plasma conductivity, t % Y and P,/P, + 0 while
`P % P, grows quadratically with discharge power:
`P,,, = YPd/k2Q = P,2wL0/k3n2QVi
`(22)
`which corresponds to the limit of infinite conductivity in
`the secondary (plasma) circuit with inefficient power
`transfer to the plasma (5 + 0).
`Note that the limiting cases of small and large
`discharge power considered here correspond (and have a
`profound analogy) to the well known property of an
`ordinary transformer with an open and with a shorted
`secondary winding respectively. In both cases no power
`is dissipated in the load but in both cases there is power
`dissipated in the primary winding due to inherent
`resistive losses in the winding itself.
`Between the two extremes there is a point of max-
`imum power transfer efficiency which occurs when
`X 2 = wLo/kn2 = Vi(1 + w2/v2)li2/Pd = (Z,J
`(23)
`which corresponds to an equality between the secondary
`reactance X, and the plasma impedance IZ,I. At this
`point the ratio PJP, is minimal which corresponds to
`maximal possible power transfer efficiency,
`+ ~ / v ] / k ' Q . (24)
`(P,/Pd),in = 2[(1 +
`At the low frequency limit (w/v << 1 ) which is typical for a
`fluorescent lamp having a gas pressure around 1 Torr
`and at a driving frequency of 13.56 MHz
`(Pt/Pd)l,min = 2/k2Q
`while at the high-frequency limit (w2/v2 >> I ) ,
`(Pt/P&,in = 4(w/v)/k2Q >> (f'a/Pd)i.min.
`(26)
`The increase in coil loss at large o / v is due to a rise in the
`RF field in the plasma E, (and also in the primary current)
`needed to maintain the RF discharge. It should be
`remembered that E, = E,(1 + w2/v2)Li2 and E, remains
`
`(25)
`
`Ex.1017 p.4
`
`

`
`nearly constant for a given discharge geometry and gas
`pressure independent of frequency. The coil loss can be a
`significant (or even dominant) portion of the total RF
`power delivered to an inductively coupled RF plasma
`source; this may be the case in low pressure (around a
`mTorr) inductive discharges with o'jv' >> I. In any case,
`the issue of coil loss should be properly addressed when
`determining the RF power dissipated in the plasma from
`the measurement of total RF power.
`From equation (17) and figure 3 one can find the
`conditions at which energy transfer into an inductive RF
`discharge is inefficient. Assuming the criterion for effi-
`cient operation to be PJP, < 1 which corresponds to
`> 0.5, one can see in figure 3 that, for a given w/v and
`device parameters k2Q (ranging in real devices some-
`where between 5 and IOO), there is only a limited range of
`discharge power (Y-parameter) where efficient operation
`is possible. In figure 3 this is shown for kzQ = 20 and
`o / v = 1, where efficient operation is achieved for Y
`between 0.1 and 20. For o / v = 10, efficient operation is
`impossible (no crossing between the line of kzQ = 20 and
`t(Y)). For this last case, according to equation (26),
`(P,/Pd)min = 2 and maximal efficiency 5 = 1/3 only for a
`single point Y = 10.
`
`5. Experimental results and discussion
`
`This experiment was carried out in an inductively
`coupled RF discharge in a Hg-rare gas mixture typical
`for a fluorescent lamp discharge and corresponding to
`o / v % 0.18. The discharge was contained by a glass vessel
`with a spherical external shape having an outer diameter
`of 10cm. A partial cross-sectional view of the discharge
`configuration is shown in figure 4. A cylindrical re-
`entrant cavity with a 3.6cm inner diameter enclosed the
`primary coil, thus the discharge surrounded the primary
`coil but was not in direct contact with it. The primary coil
`was composed of seven turns of silver-plated wire of
`2 mm OD having a coil length of 3 cm and radius of 1.7 cm.
`The measured coil inductance Lo was 1.7 pH with a Q-
`factor of 120. To maintain constant mercury pressure the
`cold-spot temperature of the discharge was controlled
`
`Simple analysis of an inductive R F discharge
`
`and kept at 40 * 1 "C during discharge operation over
`
`the range of power.
`The voltage, current and phase shift at the primary
`coil were measured with the set-up described in [6] and
`[7] where similar electrical parameters were measured in
`capacitive RF discharges. A 1 3 . 5 6 M H z ~ ~ source was
`connected through an electrically symmetric matching
`network to the primary coil which initiated and main-
`tained the discharge. In comparison with an asymmetric
`(e.g., coaxial) configuration, symmetric driving of the
`inductor coil reduces the RF potential between the coil
`ends and the plasma by a factor of two (there is a virtual
`ground at the centre of the coil) likewise reducing the
`capacitive coupling between the coil and plasma due to
`stray capacitive current paths through the glass. The
`small physical contact area between the coil and the glass
`also limits the capacitive coupling between the coil and
`ground.
`The voltage was measured directly across the primary
`coil through a voltage divider and the current was
`measured with a current transformer. The measurement
`5%. The
`accuracy of voltage and current was within
`relative phase shift between the voltage and current was
`measured using a vector voltmeter with accuracy +0.2".
`The measured primary voltage V,, current I, and
`power factor cosq are shown in figures 5 and 6 as
`functions of total power P. The total primary resistance
`
`3 ,
`
`I 3 0 0
`
`
`
`0
`
`-
`
`0
`
`0 20 40
`60 80 100
`total power (W)
`Figure 5. The current and voltage on the primary coil
`versus total RF power.
`
`z
`2
`z
`z
`- 0.05 O
`a
`
`0
`
`.
`
`l
`
`I
`
`
`
`Figure 4. An axisymmetric view of partial cross section
`of the experimental configuration is shown here along
`with a qualitative diagram of the radial distribution of the
`RF field E, the plasma density ne and the discharge current
`density J. The coil radius is ro and the effective discharge
`radius is r.
`
`0 20 40 60 80 100
`total power (W)
`Figure 6. The phase shift between voltage and current in
`the primary coil versus total RF power.
`
`183
`
`Ex.1017 p.5
`
`

`
`R B Piejak et a/
`
`25 I
`
`5 0 .E !
`
`1250 -
`2 0 0 c_
`m
`0 c
`m
`?
`
`150
`
`100
`
`~ z ..
`
`0
`
`0
`80 100
`40
`20
`60
`total power (W)
`Figure 7. The real and imaginary components of the
`primary impedance versus total power.
`
`R I and reactance XI versus total RF power are shown in
`figure 7. At low power, (P < IOW), XI is nearly constant
`(XI % lSOi2) and slightly larger than do = 145R, while
`R I steadily grows, starting from R,= 1.2R. At low power
`the discharge is apparently in a capacitive mode. This
`was clearly evidenced by the plasma glowing locally near
`the coil ends, as is typical for a capacitive RF discharge at
`relatively high (p > 0.1 Torr) gas pressure.
`The slight shift in primary reactance (XI > wL,) at
`low power suggests capacitive coupling, mainly between
`the end windings of the coil through the glass, wall
`sheaths and plasma, resulting in an effective capacitance
`CO connected in parallel with the coil ends. Evaluation of
`to
`CO = (~-~OL,/X,)W-~LL,~ = 2.7pF can he used
`make a rough estimate of the effect of capacitive
`coupling.
`With increasing RF power the discharge shifts from a
`capacitive mode through a capacitive-inductive
`trans-
`ition regime and finally to an inductive mode. The
`transition occurs between IO and 20 W. At about 20 W
`the discharge abruptly switches to an inductively domi-
`nated mode, accompanied by a sharp increase in the
`visible discharge intensity. Consequently, the regime
`above 20W will be considered here with the assumption
`that the discharge is totally governed by the inductive
`mode.
`Values of plasma resistance R , and plasma current I,
`(inferred from the electrical measurements and evalua-
`tion of equations (4)-(6)) are shown in figure 8 as
`
`100
`Io' discharge power (W)
`Figure 8. The discharge resistance and discharge current
`versus discharge power based on o l ~ ' = 0.1 8.
`
`1 0'
`
`184
`
`functions of the discharge power P,. Figure 8 shows that
`(except at lower P, where capacitive effects may play a
`minor role) R, oc P; and I, a P,. This behaviour is in
`complete agreement with behaviour generally found in
`plasma of DC or RF capacitive discharges at low gas
`pressure. These functional relationships imply that the
`effective plasma RF field E, remains constant with respect
`to the discharge current or power, which is a well known
`consequence of ionization and energy balance of a
`weakly ionized gas discharge plasma.
`The effective electric field, inferred from experiment, is
`shown in figure 9 and indeed demonstrates that, above a
`certain discharge current, E, is essentially independent
`of current as is nui'maiiy loulrd iii DC discliargas. Ii
`is of interest to note that the absolute value of E,=
`0.95Vcm-' is not too far from the value of the elec-
`tric field En,=0.5Vcm-'
`found in the positive col-
`umn of DC discharges under similar conditions based on
`the numerical solution of the Boltzmann equation 181.
`At the conditions of the present work, collisional electron
`1, and the electron energy
`heating dominates, 0 2 / v z
`relaxation length is larger than the plasma thickness d.
`Therefore the electron energy distribution function (EEDF)
`should coincide with that in a DC positive column plasma
`at a similar pd and discharge current density. Moreover,
`at a discharge current density of the order of 1 Acm-*
`and a plasma density of the order of 10'2cm-3, as in the
`present work, the EEDF is expected to be Maxwellian as it
`is in the case of a DC discharge under similar conditions
`[SI. Thus the rate of the electron energy losses in both
`discharges should be the same and consequently result in
`(E2) z EL. The fact that E, found in the inductive
`is a
`discharge appears to be somewhat larger than E,
`consequence of non-local electron heating in the inhomo-
`geneous RF field which occurs because ofthe cylindrical
`geometry of the primary coil and perhaps also because of
`the skin effect in the KF inductive discharge. The inferred
`value of E, corresponds to an electric field in the zone
`where discharge current flows at the equivalent discharge
`radius r through a cross section smaller than the cross
`section of the discharge vessel filled with plasma. Thus a
`larger field is required (than that for the homogeneous
`
`3 ,
`
`13
`
`0-0
`
`0
`
`6
`
`2
`4
`discharge current (A)
`Figure 9. The effective radius and effective ~Ffield in the
`current zone a s a function of discharge current for
`wlv = 0.18. The full circles and triangles give E, based on
`LOIV = 0 and 0.3 respectively.
`
`Ex.1017 p.6
`
`

`
`field of a DC discharge) in the cross section corresponding
`to current flow to compensate for electron energy losses
`over the entire plasma volume. In figure 9, E, is given as a
`function of discharge current for w/v = 0, 0.18 and 0.3.
`The resulting points demonstrate the insensitivity of the
`inferred value of E, to changes in w/v for w/v i 1. The
`inferred value of E, in the inductively dominated mode
`allows us to estimate the plasma parameters using
`equations (11)-(15). Thus, having A = 400cm3, w =
`(calculated for
`this gas
`8.5 x 10' S-I, w/v = 0.18
`mixture [SI) and c = E&/E: = 0.28,
`the following
`values of plasma parameters were obtained for a
`discharge power of 5 0 W N , z 3.2 x
`(n,) % 8.3
`x l O " ~ m - ~ , (uo) z 0.5 (ilcm)-' and ( 6 ) z 2cm. The
`skin depth estimated here appears to be close to the
`effective current radius r z 2.2cm. According to [3], this
`suggests that the skin effect plays only a small role under
`the conditions found in this experiment.
`The equivalent discharge (current path) radius r is
`also shown in figure 9 as a function of discharge current.
`The slight influence that the discharge current has on the
`equivalent discharge radius implies that the skin effect is
`negligibly small in the ranges considered here of dis-
`charge current between 2 and 5 A and discharge power
`between 30 and 70 W. Nonetheless, at higher discharge
`current the skin effect should be more pronounced and
`results in r approaching ro, thereby increasing the cou-
`pling coeffcient k. Over the range of discharge conditions
`in the present experiment, k = 0.5-0.6 and this is gener-
`ally considered to be close coupling for an air-core
`transformer so configured.
`The power dissipated in the primary coil P, and the
`power transfer efficiency 5 have been inferred from the
`measured electrical characteristics and the result is
`shown in figure 10. It is essential to note that these data
`are independent of the transformer formalism used in the
`analysis ofthe plasma parameters and independent of the
`particular discharge operation mode (whether inductive
`or capacitive) since at all times P, = I:Ro. In the mainly
`inductive (P > 20 W) regime 5 varies between 0.93 and
`0.95 with a broad maximum between 20 and 70 W. This
`happens because in this power range the corresponding
`values of PJPd appear to be in the vicinity of their
`minimum as seen in figure 3. The power transfer effici-
`
`- 0-0
`
`0
`
`60 80
`40
`2 0
`total power (W)
`Figure 10. The power transfer efficiency and the primary
`coil loss as a function of total RF power.
`
`t o 0
`
`Simple analysis of an inductive RF discharge
`
`ency 5 should eventually decrease when P is great
`enough; the onset of this condition has been observed in
`our measurements with other inductive discharges at
`higher power density. Coil losses are also given in figure
`10 and are seen to increase very rapidly at low power,
`decrease to a local minimum at about 20 W and increase
`linearly afterwards. The data in figure 10 demonstrate
`an excellent power transfer efficiency (93-95%) in the
`inductive mode which corresponds to optimal matching
`of the plasma load to the induction coil for given range of
`discharge power. This can be seen in figure 3 where
`experimental data are presented in normalized form.
`
`6. Concluding remarks
`
`In this work an inductive discharge is analysed as
`secondary winding of an air-core transformer. In a simple
`and direct manner, expressions coupling the electrical
`characteristics of the primary induction coil and the
`plasma load have been obtained. From these analytical
`expressions one can infer integrated discharge para-
`meters, such as P,, r and N , and spatially averaged
`plasma parameters, such as E,, (ne>, (ao> and (6).
`The accuracy of the inferred integral and spatially
`averaged plasma parameters depends on the correctness
`of the analytical expression for the coupling coefficient k.
`For a cylindrical coil surrounded coaxially by a disch-
`arge current layer of equal axial length (which is clearly
`seen from the plasma luminosity in the present experi-
`ment when in the inductive mode) the expression
`is quite adequate [SI. For a more complicated
`k = r&'r2
`coil/plasma configuration some additional speculation
`about the coupling coefficient is needed to use the
`approach developed here. A fruitful solution could be a
`numerical calculation of the coupling coefficient for a
`given experimental set-up used with the analytical for-
`mulae obtained in the present work. This seems to be a
`non-trivial calculation since both the plasma and the RF
`field