PHYSICS OF PLASMAS 13, 123503 共2006兲
Self-excitation of the plasma series resonance in radio-frequency
discharges: An analytical description
U. Czarnetzkia兲
Institute for Plasma and Atomic Physics, Ruhr-University Bochum, 44780 Bochum, Germany
T. Mussenbrock and R. P. Brinkmann
Institute for Theoretical Electrical Engineering, Ruhr-University Bochum, 44780 Bochum, Germany
共Received 17 July 2006; accepted 20 October 2006; published online 11 December 2006兲
Self-excited plasma series resonances 共PSR兲 are observed in capacitve discharges as high-frequency
oscillations superimposed on the normal rf current. This high-frequency contribution to the current
is generated by a series resonance between the capacitive sheath and the inductive and ohmic bulk
of the plasma. The nonlinearity of the sheath leads to a complex dynamic. The effect is applied, e.g.,
as a diagnostic technique in commercial etch reactors where analysis is performed by a numerical
model. Here a simple analytical investigation is introduced. In order to solve the nonlinear equations
analytically, a series of approximation is necessary. Nevertheless, the basic physics is conserved and
excellent agreement with numerical solutions is found. The model provides explicit and simple
formula for the current waveform and the spectral range of the oscillations. In particular, the
dependence on the discharge parameters is shown. Further, the model gives insight into an
additional dissipation channel opened by the high-frequency oscillations. With decreasing pressure,
the ohmic resistance of the bulk decreases as well, while the amplitude of the PSR oscillations
grows. This results in substantially higher power dissipation that exceeds the contribution of
classical stochastic heating. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2397043兴
I. INTRODUCTION
Bounded low-temperature discharges show a variety of
collective resonance modes that do not exist in homogeneous
plasmas.1 In this work, we are concerned with a particular
resonance in capacitively driven radio frequency 共rf兲 discharges, which arises from a periodic exchange between the
kinetic electron energy in the plasma bulk and the electric
field energy in the plasma sheath. A sketch of such a discharge is shown in Fig. 1. Because of the analogy to a series
circuit consisting of a lossy inductance 共the bulk兲 and one or
two capacitors 共the sheaths兲, it has become known as
“plasma series resonance,” “plasma-sheath resonance,” or
“electron series resonance” 共Fig. 2兲. In this work we will use
the first term and the abbreviation “PSR.” The effective inductance of the bulk is a consequence of the electron inertia
and the ohmic resistance due to electron-neutral collisions
leads to power dissipation. The space charge in the sheaths
provides a capacitive characteristic, although the chargevoltage relation differs from a normal capacitor.
Under typical conditions, the frequency ␻PSR of this
resonance lies well below the electron plasma frequency ␻ p.
The order of magnitude formula,
␻PSR ⬇ 冑s/L␻ p ,
共1兲
which estimates its value in terms of a typical sheath
thickness s Ⰶ L and a typical bulk extension L, has motivated
the synonym “geometric resonance.”2 According to Eq. 共1兲
frequencies of the order of 100 MHz can be expected for
typical low-pressure discharge conditions.
a兲
Author to whom correspondence should be addressed.
1070-664X/2006/13共12兲/123503/16/$23.00
The opportunity to drive a discharge at this resonance
frequency has motivated much of the previous work on this
topic.2–18 However, it has been observed that in capacitively
driven radio frequency 共rf兲 discharges, the PSR can be excited even when the driving frequency ␻rf is much lower
than PSR frequency.19,20 Due to the strong nonlinearity of the
charge-voltage characteristic of the sheaths, harmonics of the
basic frequency are generated up to considerable order. Some
of these harmonics may lie in approximate resonance with
the PSR frequency and excite it to sufficiently high amplitudes that it becomes experimentally detectable. The PSR
then represents itself as high-frequency oscillations superimposed on the normal rf current and sheath voltage. The amplitude of these oscillations is limited by collisions of the
oscillating bulk electrons with the neutral background.
A possible way of detection is to integrate a small sensor
into the reactor wall that picks up a fraction of the rf current
as indicated in Fig. 1. An example of an experimentally determined current is shown in Fig. 3. This method to observe
the self-excited oscillations in capacitive rf plasmas forms
the core of a robust and noninvasive process monitoring
technique that has recently gained general acceptance, especially in industrial applications: Self-excited electron resonance spectroscopy 共SEERS兲.19,20 In particular, it allows detecting small changes in the electron neutral collision
frequency. Therefore, the technique is sensitive to small deviations in the process gas composition or density; e.g., in
etching tools used in the semiconductor industry.
In this paper, however, we are not concerned with applications but focus on the physics of the PSR self-excitation
mechanism itself. We re-investigate a simple global model of
the nonlinear rf behavior of a capacitively coupled plasma
13, 123503-1
© 2006 American Institute of Physics
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123503-2
Czarnetzki, Mussenbrock, and Brinkmann
FIG. 1. Schematic of an asymmetric capacitively coupled rf discharge with
a small current sensor integrated into the wall at the right. The arrows
indicate the direction of the current flow between the electrode and the wall.
19
that was already studied elsewhere. We go beyond the numerical analysis presented there and offer a closed-form approximation, which leads to scaling laws and some new
physical insights. The effect of a resonantly sustained discharge, where the driving rf frequency is close to or identical
with the PSR frequency is not in the main focus of this work.
It is discussed briefly in a separate section.
The analytic solution presented here can be viewed as
being complementary to the analytical solution derived in
Refs. 22 and 23. There, an analytical solution is obtained
formally by expansion into several nested infinite power series. This has the advantage of mathematical exactness but
convergence of the series is slow. Therefore, it is not possible
in that approach to derive a solution in terms of basic functions or to provide scaling laws. Here, we solve the same
problem as in Ref. 22, but use an entirely different approach.
The analytic techniques applied here lead only to approximate solutions but with the advantage that explicit and
simple formulas can be derived. Further, the solutions as well
as the way of deriving them provide additional insight into
the physics of the problem.
The paper is organized as follows: The next section 共Sec.
II兲 will first introduce a spatially resolved model of the highfrequency behavior of a capacitive discharge. The conceptual
reduction to the one-dimensional case is then motivated. Investigation of the nonlinear characteristics of the sheath
shows that it can be modeled, to a good approximation, by a
quadratic form. Finally, the combination of the bulk and the
sheath equations with an assumed sinusoidal external voltage
FIG. 2. Schematic of the equivalent series resonance circuit. Note that the
capacitors are nonlinear.
Phys. Plasmas 13, 123503 共2006兲
FIG. 3. Measured rf current to the wall in an experimental reactor similar to
the schematic shown in Fig. 1. The rf frequency is 13.56 MHz and two
periods are shown 共Ref. 21兲.
leads to the global PSR equation, an ordinary differential
equation without spatial resolution. Sections III and IV represent the main part of our investigation. In Sec. III, the
nonlinear PSR equation is solved analytically by applying a
series of approximations. The various stages of the derivation are compared with numerical solutions. Explicit scaling
laws for the frequency spectrum of the current, the damping
behavior, and the parameter dependence of the total dissipated power are derived in Sec. IV. It will be shown that the
power dissipation by the PSR oscillations can contribute significantly to the total power dissipation. The case of quasiresonant excitation is discussed briefly in Sec. V. The last
section 共Sec. VI兲 summarizes the results and closes with a
discussion of the consequences and an outlook on possible
future work. Some technical aspects of the calculations are
moved to the appendix in order to provide better readability.
Throughout the paper, we concentrate on discharges
driven by a single sinusoidal voltage source Vrf at a frequency ␻rf, such as that depicted in Fig. 1. More complex
configurations, like discharges driven by multiple frequencies or nonsinusoidal voltages are excluded to make the derivations more transparent. Nevertheless, many of the following arguments can be suitably generalized.
II. DERIVATION OF THE PSR EQUATION
The setup for a capacitive discharge consists in general
of two electrodes and the wall of the confining chamber.
Here, only the case of a strongly asymmetric discharge as
sketched in Fig. 1 is considered where one electrode is on
ground potential and the grounded area is much larger than
the area of the powered electrode. This reduces not only the
complexity of the problem but keeps also the focus on the
essential effect caused by the nonlinearity of the rf sheaths.
This nonlinearity cancels out effectively in a symmetric discharge as has been pointed out by Lieberman and
Lichtenberg.24 In the discharge, the quasi-neutral bulk and
the space charge sheaths can be distinguished. The space
charge sheaths connect the electrodes and the wall to the
bulk.
We assume that the Debye length ␭D and the maximum
sheath thickness sm are both small compared to the typical
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123503-3
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
bulk dimension L. Separate models can then describe the
bulk and the sheath, with the quasi-neutral bulk filling the
discharge volume and the sheath forming a thin layer free of
electrons at the electrode and the walls. The small transition
region between the bulk and the sheaths is neglected here.
The externally applied voltage is assumed to be sinusoidal
and in the sheaths only the displacement current is considered. Electron and ion currents as well as release of additional electrons from the electrode by the ␥-effect or metastable atoms are considered as negligible for the effects
studied in this work.19
The first task is now to derive separately global chargevoltage or current-voltage relations for the bulk and the
sheath. Combining these relations will lead to the basic PSR
equation. Solutions of this equation are then derived in Secs.
III and V.
A. Bulk model
In the bulk, quasi-neutrality holds and the rf current is
carried by electron conduction. The current-voltage characteristic of the electrons in this region is determined by the
momentum balance equation for electrons of mass m, charge
−e, and velocity v. The so-called inertia term 共v · ⵜ兲v is neglected and the electron-neutral elastic collision frequency ␯
is assumed to be independent of the drift velocity v. It is
assumed that induction effects can be neglected and the electric field can be described by a potential function ⌽:
冉 冊
m ⳵
+ ␷ v = ⵜ⌽.
e ⳵t
共2兲
In agreement with the two-term approximation, inelastic
collisions are neglected. Further, in capacitive rf discharges,
the current has only a high-frequency ac component and on
the corresponding time scale, ions can be considered as
static.
The equation can now be integrated over an arbitrary
path between the edges of the bulk at the two sheath boundaries. Here, ␵1 and ␵2 denote two points on the sheath surface
at the powered and the grounded electrode, respectively. This
leads on the right side to the bulk voltage:
冉 冊冕
m ⳵
+␷
e ⳵t
␵2
␵1
v · d␵ = ⌽21 .
共3兲
The surface of the sheath at the powered electrode can
be considered in good approximation as an equipotential surface. At the grounded side, this is not necessarily the case.
Nevertheless, there are two important cases: Firstly, when the
discharge geometry has cylindrical or spherical geometry.
Secondly, when it shows a very strong asymmetry between
the grounded and powered electrode area and the sheath,
voltage over the grounded electrode might be neglected so
that the integration can be extended approximately to
ground. In these cases, the bulk voltage in Eq. 共3兲 becomes
independent of the particular starting and end points on the
two surfaces. If the particular path follows the direction of
the electron velocity, the scalar product can be replaced by
an ordinary product. One can now replace formally the velocity v by the total current I共t兲:
冉 冊 冕
m ⳵
+ ␷ I共t兲
e2 ⳵t
␵2
␵1
1
d␵ = ⌽21共t兲.
n共␵兲A共␵兲
共4兲
Here, n is the plasma density along the path and A is a
surface area that also depends on the particular coordinate of
the path. The total current can be moved out of the integral
since it is a constant with respect to the spatial coordinates.
The integral depends on the particular starting and end
points, but is independent of time. Formally, one can write
Eq. 共4兲 in the following form:
冉
冊
L m␯ 1 ⳵
+ 1 I共t兲 = ⌽21共t兲.
2
n̄Ā e ␷ ⳵t
共5兲
The first term on the left side is equivalent to the voltage
over an inductor and the second term describes a resistor.
This motivates the equivalent circuit of the bulk shown in
Fig. 2. L, n̄, and Ā denote effective values of the discharge
length, density, and area, respectively. They are defined as
L=
冕
␵2
1
d␵,
=
n̄
␵1
1
L
冕
␵2
␵1
1
d␵,
n
and
1
=
Ā
n̄
L
冕
␵2
␵1
1
d␵.
nA
共6兲
Apparently, the coefficient on the left side of Eq. 共5兲 has
the dimension of a voltage and depends on the particular
path, indicating that different oscillation modes can develop
in the discharge. This is an interesting topic in itself but is
not in the focus of this paper. A first approach to the investigation of mode structures can be found in Ref. 23. Further
on, we will discuss Eq. 共5兲 for one particular value of the
coefficient only. This is exact in case of simple geometries
such as an infinitely long cylinder or a sphere where the
discharge can be described by an effective coordinate as onedimensional and the area A scales by r or r2, respectively.
Then, only one value of the integral exists. Nevertheless,
even in these geometrically simple situations, explicit calculation of the coefficients is only possible if the spatial density
distribution is known. The calculation of the density distribution is a difficult task of its own and goes beyond the
scope of this work. Therefore, the aim is not to solve the
problem for a particular discharge or geometry but to choose
parameters that are in a reasonably range for typical cases.
B. Sheath model
The sheath region is described by a static ion-density
profile ni共z兲 and a step-function for the electron density. A
planar geometry is assumed. This is justified even above a
curved surface if the sheath thickness is much smaller than
the radius of curvature. The coordinate z is zero at the surface of the electrode and points towards the plasma. The
potential drop between the electrode and the temporary
sheath edge s共t兲 is given by Poisson’s equation:
冕冕
s
⌽S共s兲 = −
0
s
z
e
e
ni共z⬘兲dz⬘dz = −
␧0
␧0
冕
s
zni共z兲dz.
共7兲
0
The sheath voltage depends only implicitly on time by
the temporal sheath position s共t兲:
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123503-4
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
d⌽s ⳵⌽s ds
.
=
dt
⳵s dt
共8兲
Evaluation of Eq. 共7兲 leads to
⳵⌽s
e
= − ni共s兲s.
⳵s
␧0
共9兲
The temporary sheath thickness s共t兲 can be related to the
surface charge density ␴共␶兲:
␴共t兲 = e
冕
s共t兲
ni共z兲dz.
共10兲
0
Here, the initial condition is s共t = 0兲 = 0. Equations 共9兲
and 共10兲 relate the area charge density ␴共t兲 to the sheath
voltage ⌽s共t兲. However, in order to solve these equations, the
spatial ion density profile ni共z兲 has to be known. The simplest approximation is an effective constant ion density in the
sheath, the so-called matrix sheath approximation
␴共t兲 ⬇ en̄ss共t兲.
共11兲
Solving for s, inserting the result into Eq. 共9兲, and integrating with the boundary condition ⌽共s = 0兲 = 0 yields
⌽s ⬇ −
1
2e␧0n̄s
␴2 .
共12兲
Assuming a homogeneous sheath over an electrode with
area As yields the final relation
⌽s ⬇ −
1
2e␧0n̄sAs2
共13兲
Q2 .
Here, Q共t兲 is the integral over the current and represents
the charge. Its value is always negative or zero:
Q共t兲 =
冕
t
0
I共t⬘兲dt⬘ = − eAs
冕
C. Basic PSR equation
Now the sheath and the bulk voltages can be combined.
The applied voltage ⌽ is then
⌽=−
1
Q
2e␧0n̄sAs2
2
冉
冊
⳵Q
⳵ 2Q
.
2 +␷
2
⳵t
⳵
t
e n̄Ā
mL
+
共15兲
The applied voltage is assumed to be sinusoidal with
⌽0 ⬎ 0:
⌽=−
⌽0
关1 − cos共␻rft兲兴.
2
共16兲
The particular choice of the self-bias in Eq. 共16兲 is consistent with the previous assumption that the discharge is
strongly asymmetric and the sheath at the grounded electrode
can be neglected; i.e., in the limit of an infinite area ratio.
The choice of the bias amplitude is discussed in connection
with the dynamic floating potential condition in more detail
in Appendix B. Note that ⌽共t = 0兲 = 0. The assumption of a
sinusoidal voltage is in agreement with the assumptions
made in earlier work on the PSR.19,22 We would like to point
out that this is also in well agreement with the experience we
have from operation of various rf discharges in our laboratory. It can be shown easily that the combination of a standard 50 ⍀ power supply with a proper L-type matching unit
共series inductance, parallel capacitance兲 represents to a good
approximation a voltage source for the discharge 共mean
sheath impedanceⰇ bulk resistance兲. The particular choice
made in Eq. 共16兲 on the voltage form and the bias is essential
for the entire calculation. It is discussed in more detail in
Sec. III, where the consequences will become more obvious.
By a linear transformation in Q, the constant self-bias
can be removed from the equation:
Q̃ = Q − Q0
with
Q0 = 冑2e␧0n̄s⌽0 .
共17兲
Equation 共15兲 then becomes
s共t兲
nsdz 艋 0.
共14兲
0
The choice s共t = 0兲 = 0 is related to the initial voltage
⌽共t = 0兲 = 0. Equation 共13兲 is the well-known quadratic
charge-voltage relation for the sheath. The validity of the
quadratic relation is discussed in more detail in Appendix A.
It is important to note that the sheath voltage scales by
the inverse electrode area squared. Therefore, comparing the
sheath voltages at the powered and the grounded electrode
one can conclude that for identical sheath densities, the ratio
between the two voltages scales as the inverse area ratio
squared. This motivates the neglect of the sheath at the
grounded electrode in strongly asymmetric discharges where
the grounded area is much larger than the electrode area.
Only this case is considered here. Further, it might be noted
that the small Ohmic resistance of the sheath is neglected
since it is smaller than the capacitive impedance by approximately a factor 共␻ / ␻ps兲2 Ⰶ 1, where ␻ps is the electron
plasma frequency in the sheath.
1
⌽0
共2Q0Q̃ + Q̃2兲
cos共␻rft兲 = −
2
2e␧0n̄sAs2
+
冉
冊
⳵Q̃
⳵2Q̃
¯
.
2 +␷
2
⳵t
⳵
t
e n̄Ā
mL
共18兲
This form of the PSR equation is particularly useful for
the analytical analysis by an infinite series as performed in
Ref. 22, while the above form given by Eqs. 共15兲 and 共16兲 is
more useful for the approximate analytical solution carried
out in Sec. III. It should be noted that the transformation has
no effect on the current and is similar to the removal of a
constant potential in mechanics. The analysis carried out in
Ref. 22 is more general since it allows for an arbitrary bias
on the sheath voltage, e.g., a small rf modulation on a dc
sheath, by allowing for an arbitrary Q0. The case considered
in this work relates to the special but important case of the
fully collapsing sheath in a strongly asymmetric discharge.
In this case, Eq. 共15兲 becomes
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123503-5
−
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
冉
1
⌽0
mL ⳵2Q
2
关1 − cos共␻rft兲兴 = −
2 Q + 2
2
2
2e␧0As ns
e Ān̄ ⳵t
+␷
冊
⳵Q
.
⳵t
III. APPROXIMATE ANALYTICAL SOLUTION
OF THE PSR EQUATION
共19兲
By introducing dimensionless quantities, Eq. 共19兲 reduces to the final form discussed as the PSR equation in this
work:
sin2
冉 冊
⍀␶
= q2 − 2共q̈ + ␬q̇兲.
2
共20兲
The dimensionless quantities are defined by
q=
Q
,
Qm
␶ = ␻0t,
⍀=
␻rf
,
␻0
␬=
␷
.
␻0
共21兲
Therefore,
q is the charge normalized to Qm
the space charge in the sheath when the
maximum negative voltage −⌽0 is applied over the sheath. ns
is the effective density in the sheath. The time ␶, the external
excitation frequency ⍀, and the collision rate ␬ are all normalized by the frequency ␻0, where ␻0 is the plasma frequency ␻ p at n̄ reduced by a dimensionless geometry parameter ␥ ⬍ 1:
= As冑2e␧0ns⌽0,
␻0 = ␥␻ p with ␥ =
冑
smĀ
and sm =
LAs
冑
2␧0⌽0
.
ens
共22兲
Here, sm represents the maximum sheath extension.
Typical values of ␥ are of the order of 10−1 since sm Ⰶ L but
Ā ⬎ AS. The plasma frequency is defined as usual:
␻p =
冑
e2n̄
.
␧ 0m
共23兲
The normalization reduces the eight initial parameters
⌽0, L, Ā, As, n̄, n̄s, ␯, and ␻ to two determining parameters:
⍀ and ␬. Both parameters are small compared to 1 and are
typically in the range of 10−1 – 10−2 for frequencies, densities,
and pressures used in most experimental and industrial applications. Limits to the normalized parameters are discussed
in more detail in connection with the results obtained in Sec.
III and IV. It should be noted that for a given density profile
in the discharge, the normalized rf frequency ⍀ scales with
the density as n−1/4.
A normalized current j can be defined as
j=
2
I
q̇ =
⍀
I0
with
I0 =
Qm␻rf
.
2
共24兲
This means that the current is normalized to the amplitude of a current I0 corresponding to half of the maximum
charge oscillating at the rf frequency. The normalization factor will become more obvious later in this section.
Equation 共20兲 is the basic equation by which the plasma
series resonance excitation is discussed in this work. Further
on, it will be referred to as the PSR equation. The nonlinear
term q2 introduced by the sheath does not allow an explicit
analytical solution. In the following section, it is shown how
the equation can be integrated approximately.
In this section, the PSR equation is solved analytically
by applying certain approximations. The calculation is limited to the case ␬, ⍀ Ⰶ 1. A more detailed discussion on the
limits follows below. The quasi-resonant case ␬, ⍀ 艌 1 is
analyzed in Sec. V.
The functions representing the charge and the current
densities can be split into two terms:
q = q0 + q1
and
共25兲
j = j0 + j1 .
The first term with index 0 is the solution of the PSR
equation when the bulk contribution is neglected. The second
term with index 1 is the necessary correction that takes the
bulk contribution into account. The equation for q0 is
sin2
冉 冊
⍀␶
= q2o .
2
共26兲
When solving this for q0, one has to take into account
that q0 ⬍ 0 according to the definition by Eq. 共14兲:
冏 冉 冊冏
q0 = − sin
⍀␶
2
共27兲
.
As expected, the solution without the plasma bulk has an
amplitude equal to the maximum charge in the sheath at full
voltage. The current j0 is
冉 冊 冠 冉 冊冡
j0 = − cos
⍀␶
⍀␶
sign sin
2
2
.
共28兲
Here, sign共x兲 is −1 for a negative value of the argument
and +1 otherwise. By choice of the normalization in Eq.
共24兲, the current j0 has an amplitude of 1. This solution
shows the characteristic sawtooth current form of a capacitive rf discharge with a sudden change in sign at the time of
maximum current amplitude. Since the solution is periodic,
only the first period has to be considered here where the sign
function is 1. The current j0 will hereinafter be called the
base current.
Inserting Eq. 共27兲 into the PSR equation leads to a nonlinear differential equation for q1:
q̈1 + ␬q̇1 − 共q0 + q1兲q1 = − 共q̈0 + ␬q̇0兲.
共29兲
Firstly, we attempt finding an analytic solution to the
linearized equation where the term with q21 is neglected. This
simplification leads, especially, to a certain phase shift in the
calculated current. In a second step, this phase shift will be
determined and the validity range of the analytic solution
will be discussed.
In Fig. 4, Eq. 共20兲 and the linearized form of Eq. 共29兲 are
both solved numerically for ⍀ = 0.1 and ␬ = 0.2 as a typical
example. In addition to the numerical solutions, the current
j0 as given by Eq. 共28兲 is also displayed. In order to obtain a
numerical solution of Eq. 共20兲 with periodic boundary conditions, the integration is started nine rf periods before ␶ = 0
with q = q̇ = 0. For the above parameters, the periodic solution
is obtained already in the second period. Equation 共29兲 is
integrated with the initial condition q1 = 0 and q̇1 = ⍀ at
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123503-6
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
FIG. 4. Numerical solution of the PSR equation or ⍀ = 0.1 and ␬ = 0.2. Solid
line: exact solution; dashed line: neglect of the nonlinear term q21; dashdotted line: base current j0.
␶ = 0. These initial conditions are discussed in the next paragraph. All numerical calculations are carried out by the commercial software MATHEMATICA 4.1 共Wolfram Research兲.
One can easily identify the high-frequency oscillations superimposed on the smooth current j0. Both solutions are already very similar to the experimental current show in Fig. 3.
Compared to the exact solution, the linearized solution exhibits a phase shift and some smaller deviations at the beginning and the end of the period.
A close inspection of the numerically calculated trace of
q shows that due to the high-frequency oscillations, it can
exhibit for a time of a few 10−2 of the rf interval shortly after
␶ = 0 a slightly positive value of maximum a few times 10−2.
This unphysical behavior can be avoided by adding an additional bias of the order of 10−3. However, this does not affect
at all the solution for the current and the term is of the same
order or smaller than other terms we will neglect below in
the derivation of an analytical solution. Therefore, this effect
is ignored further on except for the quasi-resonant case discussed in Sec. V, where ⍀ is of the order of 1, which changes
the characteristics drastically.
Linearization by the neglect of q21 leads to a damped
nonlinear oscillator under an external periodic force represented by the derivatives of q0 on the right-hand side of Eq.
共29兲. The external force term contains a ␦-function by the
second derivative of q0 or first derivative of the current j0. In
a further approximation, all other contributions are neglected
compared to this ␦-function. Physically, this means that the
sudden change of sign in the current j0 and the related broadband power spectrum is the major source for the excitation
of PSR oscillations. With these approximations, Eq. 共29兲 reduces to
q̈1 + ␬q̇1 − q0q1 = ⍀␦共␶兲.
共30兲
It is important to note that the occurrence of a deltafunction is closely related to the earlier assumption made on
the bias voltage. For a bias voltage smaller than the rf amplitude, the delta-function turns into a smooth function with
a final width and peak height and the amplitude of the PSR
oscillations is reduced. The bias is reduced if the sheath to
ground has to be taken into account. Ultimately, in a sym-
metric discharge, the PSR oscillations vanish totally. However, it can be shown that the effect of a reduced bias is still
negligible for typical area ratios in the range of about 3 to 6,
depending on discharge conditions. The calculations are
rather involved and might be the topic of a forthcoming publication. Here we just want to motivate that although the
basic assumptions are related to special conditions, they are
still general enough to describe real experimental situations.
The slight deviation of the current j from j0 at the end of
the period is a consequence of the terms neglected here. Integration of Eq. 共30兲 around zero with lim ␧ → 0 leads to the
initial condition q̇1共+␧兲 = ⍀ + q̇1共−␧兲. The numerical solution
of Eq. 共20兲 shows that the slope just before zero is finite but
small compared to ⍀. Therefore, this slope as well as the
small but finite value of q1 at zero are neglected here. Apparently, these assumptions will break down at low damping
where oscillations are still strong at the end of an rf period.
This restricts the following calculation to the case ␬ 艌 ⍀.
The finite value of q1 will be discussed again in connection
with the power dissipated in the discharge. The problem to
be solved now is
q̈1 + ␬q̇1 − q0q1 = 0
with
q1共0兲 = 0
and
q̇1共0兲 = ⍀.
共31兲
The second initial condition shows already that the sudden jump at ␶ = 0 in the current j0 calculated without the
plasma bulk is transformed into a smooth transition of the
total current j by j1; i.e., lim ␧ → 0 : j共−␧兲 = j共+␧兲 = 1.
The equation can be further simplified by removing the
first derivative:
冉 冊
␬
q1共␶兲 = f共␶兲exp − ␶ .
2
共32兲
This gives the equations
f̈ + ⍀̃2 f = 0
冉
⍀̃2 = − q0 +
with
f共0兲 = 0
冊
␬2
⬇ − q0 .
4
and
ḟ共0兲 = ⍀,
共33兲
共34兲
The oscillation frequency of this oscillator is a harmonic
function in time by the term q0 and with q0 艋 0. The term
␬2 / 4 is a very small number, typically in the range 10−3 to
10−5. Therefore, the term is negligible already after about
10−4 or less of the rf period and typically 10−3 of the period
of the PSR oscillation and will be neglected further on. It
should be noticed that without linearization in Eq. 共29兲, q0
would have to be supplemented by q1 / 2. We will return to
this later. Equation 共33兲 can now be solved by a WKB approximation:
f 共WKB兲 = D共␶兲ei⌿共␶兲 + c.c.
共35兲
In the frame of the WKB approximation, the amplitude
D is assumed to vary only slowly so that the second derivative of D can be neglected in comparison to other terms. The
resulting differential equations for D and ⌿ can then be
separated easily and the approximate solution to Eq. 共33兲 is
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123503-7
冉冕
␶
sin
f
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
共WKB兲
0
=G
⍀̃d␶⬘ + ␪
冑⍀̃
冊
冋冕 冑 冉 冊 册
␶
sin
0
=G
⍀
␶⬘ d␶⬘ + ␪
2
sin
sin
冉 冊
⍀
␶
2
1/4
共36兲
.
Due to the nature of the WKB approximation, Eq. 共36兲
cannot be applied at early times; especially the determination
of the constants G and ␪ by the initial conditions is not
possible. Both the function and its first derivative vanish at
␶ = 0. Generally, the constants could be found by fitting Eq.
共36兲 to the numeric solution of the original Eq. 共20兲. However, this would not reveal the analytic dependence of the
constants on the parameter ⍀. Therefore, another approximation that is valid at early and intermediate times will now be
derived. This solution can be fitted to the initial conditions at
␶ = 0, and at intermediate times a comparison will allow determination of the constants G and ␪.
At early and intermediate times 共⍀␶ ⬍ ␲ / 2兲, the term q0
can be expanded in time to first order. The time-varying oscillation frequency then becomes
⍀
⍀̃ ⬇ ␶ .
2
共37兲
2
In this approximation, Eq. 共33兲 is the Airy differential
equation with the solution
f 共Airy兲 = A Ai共␰兲 + B Bi共␰兲,
␰=−
冉冊
⍀
2
1/3
共38兲
␶.
共39兲
Ai共␰兲 and Bi共␰兲 are the Airy functions and here the new
variable ␰ is negative. A and B are integration constants. By
the initial conditions, the result for f 共Airy兲 is
f 共Airy兲 = ⌫共1/3兲31/3
冉冊冉
⍀
2
2/3
Ai共␰兲 −
1
冑3 Bi共␰兲
冊
.
共40兲
⌫ is the gamma-function with ⌫共1 / 3兲 ⬇ 2.678 94.
The Airy solution compares quite well with the numerical solution of the linearized form of Eq. 共29兲 at early and
intermediate times but at later times the error caused by the
additional linearization of q0 becomes apparent by too high
oscillation frequencies 共Fig. 5兲.
The Airy functions can be represented by sine and cosine
functions already for relatively small values of the argument
at about −␰ ⬎ 0.5 or ⍀␶ ⬎ 共⍀ / 2兲2/3. For typical values of the
parameters, the approximation is valid already after about a
few 10−2 or less of the rf period. This is still well before the
maximum of the first PSR oscillation. The asymptotic form
of the Airy solution is then
FIG. 5. Comparison of the normalized current densities obtained by the
numerical solution of the linearized differential equation and the Airy solution for ⍀ = 0.1 and ␬ = 0.2. Solid line: Airy solution; dashed line: numerical
solution.
冉冑
冉 冊
f 共asy兲 = G共asy兲
冊
⍀ 3/2
␶ + ␪共asy兲
2
.
⍀ 1/4
␶
2
2
3
sin
共41兲
The constants are
G共asy兲 =
冉冊
2⌫共1/3兲 ⍀
31/6冑␲ 2
5/6
and
␪共asy兲 =
␲
.
12
共42兲
If now the sine function in the WKB solution given by
Eq. 共36兲 is expanded to first order as well, one immediately
identifies the identity between the asymptotic forms. Therefore, the constants in the WKB solution are determined as
G = G共asy兲 and ␪ = ␪共asy兲.
The current can now be found as the first derivative of
Eqs. 共32兲 and 共36兲. Here, to a very good approximation the
derivative of the slow exponential can be neglected in comparison to the fast oscillating function f. Further, in the derivative of f 共WKB兲, the derivative of the slow denominator can
be neglected as well for times after the first zero of the PSR
oscillations at about ⍀␶ = 共3␲⍀ / 4兲2/3. This is nearly identical
to the beginning of the validity region of the asymptotic
solution. Therefore, further on only this simplified form will
be used for the current:
j1 ⬇
冉 冊
冉 冊
2G
⍀
sin
␶
⍀
2
1/4
␬
⫻exp − ␶ .
2
冋冕 冑 冉 ⬘ 冊
␶
sin
cos
0
⍀
␶
2
d␶
⬘ +␪
册
共43兲
It might again be noted at this point that in contrast to
the asymptotic Airy solution, the WKB solution is also valid
at large times; i.e., when the linearization of the sinus breaks
down. Comparison of this solution to the numerical solution
of the linearized form of Eq. 共29兲 shows generally good
agreement for all values of the two parameters ⍀ and ␬. An
example for ⍀ = 0.1 and ␬ = 0.2 is shown in Fig. 6.
The nonlinear term q21 can now formally be considered
in the WKB solution by explicitly adding q1 / 2 as a known
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123503-8
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
FIG. 6. Comparison of the normalized current obtained by the numerical
solution of the linearized differential equation and the WKB solution for
⍀ = 0.1 and ␬ = 0.2. Solid line: WKB solution, dashed line: numerical
solution.
function to q0 in Eq. 共34兲 for ⍀̃. The general form of the
WKB solution 关共36兲 and 共43兲兴 does not depend on the particular form of the term ⍀̃. This modifies both, the integrand
in the nominator and the term in the denominator. The difference of the two forms of the integrant in Eqs. 共36兲 and
共43兲 is then
⌬ = 冑− 共y 0 + y 1/2兲 − 冑− y 0 .
共44兲
The difference ⌬ and the integral over ⌬ can now be
calculated by either using the Airy solution as an approximation or by inserting the exact numerical solution of Eq. 共20兲
as q1 = q − q0. A comparison is shown in Fig. 7 for the same
particular set of parameters as in the examples above. Both
curves agree rather well except for the phase shift. The deviation of ⌬ from zero becomes negligibly small already
after the first zero and the integral S共␶兲 = 兰0␶ ⌬共␶⬘兲d␶⬘ becomes
a constant. The behavior of ⌬ at the end of the rf period is of
no importance here, since at that time, q1 and j1 have become
already negligibly small by the exponential damping. In the
frame of the Airy solution, it can be shown analytically that
the integral between ␶ = 0 and any of the first few zeros of ⌬
is a number independent of any parameters if the exponential
damping can be neglected over this interval 共Appendix C兲.
The integral becomes a constant approximately after the
first zero of Ai共−␰兲, which is at about −␰ = 2.3 or ⍀␶
= 4.6共⍀ / 2兲2/3. These results hold also in good approximation
for the numerical solution. After this time, the nonlinear term
affects the solution simply by adding a constant phase to the
phase already given by the integration constant. In the denominator of Eq. 共36兲 the additional contribution becomes
insignificantly small and can be neglected.
Since the WKB solution is valid only after the first zero
anyway, one can conclude now that the solution holds also
for the nonlinear equation at times larger than ⍀␶ = 3⍀2/3.
The phase will not depend on any parameter and the amplitude will scale approximately in the same way as in the linearized case. By comparison with the numerical solution, the
phase is then identified as shifted by about −␲ / 2 compared
to the Airy solution. Further, the amplitude can be improved
by multiplying G by a factor of 1.3. For ⍀␶ ⬎ 3⍀2/3, the
FIG. 7. Comparison of ⌬ 共a兲 and its integral S共␶兲 = 兰␶0⌬共␶ 兲d␶ 共b兲 calcu⬘ ⬘
lated by the exact numerical solution and the Airy solution for ⍀ = 0.1 and
␬ = 0.2. Solid line: Airy solution, dashed line: numerical solution.
result is then very close to the exact solution over the entire
range of realistic parameters: ⍀ = 0.02 to 0.2. At the higher
value only about two full PSR oscillations occur within one
rf period and at the lower value already an oscillation frequency of about 0.5 GHz is obtained for an rf frequency of
13.56 MHz. Frequencies of this order or even higher are
very likely to be strongly damped in any practical rf circuit.
Since ⍀ scales with the plasma density as n−1/4, the one order
of magnitude variation in ⍀ corresponds to four orders of
magnitude variation in density. This should be sufficient for
all situations of practical interest.
Examples for two sets of parameters are presented in
Fig. 8. Excellent agreement is found for the higher value of
⍀ after the first oscillation extremum. For the lower value of
⍀ generally the agreement is of similar quality except for a
slight underestimation of the amplitude. In conclusion, the
current at times ⍀␶ ⬎ 3⍀2/3 is represented in very good approximation by Eq. 共43兲 with G = 1.3G共asy兲 and ␪ = −5␲ / 12
and G共asy兲 given by Eq. 共42兲. Correspondingly, the same applies for the charge by use of Eqs. 共32兲 and 共36兲.
IV. RESULTS DERIVED FROM THE ANALYTICAL
SOLUTION
A. Frequency spectrum and damping
The oscillatory characteristics of the current are analyzed by numerical calculation of the Fourier spectrum. The
amplitude of the Fourier transform of the solutions presented
in Fig. 8 is shown in Fig. 9. The low-frequency peak results
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123503-9
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
FIG. 8. Comparison of the normalized current densities obtained by the
numerical solution of the PSR equation and the WKB solution for 共a兲
⍀ = 0.1 and ␬ = 0.2 and 共b兲 ⍀ = 0.02 and ␬ = 0.03. Solid line: WKB solution,
dashed line: numerical solution.
from the fundamental rf frequency. The PSR spectrum is
broad and continuous without sharp resonances. For ⍀ = 0.1,
the WKB solution first underestimates the Fourier amplitude
at lower frequency and then overestimates the main broad
peak caused by the PSR oscillations. The reason comes from
the error in the waveform at early times. Since at ⍀ = 0.1
only a few oscillations occur, the error is relatively large.
With increasing number of oscillation periods, the agreement
becomes much better, as shown for ⍀ = 0.02. Nevertheless, in
both cases the frequency of maximum oscillation amplitude
and also the rapid decay at a maximum frequency is well
represented. Also shown in Fig. 9 is the Fourier spectrum of
the base current j0, which scales approximately with the reciprocal frequency. The contribution of the PSR oscillations
is the difference between the total spectrum and the spectrum
of j0.
The PSR oscillation frequency ⍀PSR can be calculated
analytically from the WKB solution. The corresponding
high-frequency period ⌻ can be defined by
2␲ =
冕 冑 冉 冊 冕 冑 冉 冊
冑 冉 冊
␶+⌻/2
sin
0
⬇⌻
sin
⍀
t dt −
2
⍀
␶ .
2
␶−⌻/2
sin
0
⍀
t dt
2
共45兲
The right-hand side of the equation follows from the fact
that the integral varies only slowly over one high-frequency
FIG. 9. Fourier transform spectrum of the current for ⍀ = 0.1 and ␬ = 0.2 共a兲
and ⍀ = 0.02 and ␬ = 0.03 共b兲. The frequency scale is normalized to the
fundamental rf frequency. The Fourier spectra exist only at discrete integers
of the rf frequency. For better visibility, the dots are connected by straight
lines. Solid line: WKB solution, dashed line: numerical solution, dashdotted line: spectrum of the current j0. The analytically calculated limits of
the spectrum related to the PSR oscillations are indicated by the vertical
bars.
period. Alternatively, one can derive the same result by a
close inspection of Eq. 共22兲 where the formal frequency of
the “harmonic oscillator” is ⍀̃2 = sin共⍀␶ / 2兲. The oscillation
frequency is limited by the maximum value that appears in
the middle of the rf period:
⍀PSR共␶兲 =
冑 冉 冊
sin
⍀
␶ 艋 1.
2
共46兲
Therefore, the maximum PSR frequency is given simply
by the plasma frequency times the geometrical factor ␥.
Since ␥ depends also on the plasma density by the sheath
thickness, this frequency scales with the plasma density for a
fixed density profile in the discharge as n1/4.
The minimum PSR frequency can be defined by the following condition: If the frequency is measured at a time ␶0 in
the middle of the high-frequency period, then apparently the
time ␶0 must be larger than half the high-frequency period:
⌻PSR共␶0兲 ⬍ 2␶0. From this condition, the time ␶0 and then the
corresponding frequency can be derived. The result is
⍀PSR共␶兲 艌
冉 冊
␲⍀
2
2/3
.
共47兲
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123503-10
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
This minimum frequency scales with the plasma density
as n1/6. Both limits agree very well with the Fourier spectra
of Fig. 9. Additional contributions to the spectrum arise from
the actual current waveform around ␶ = 0. Therefore, the
lower limit defined by Eq. 共47兲 cannot be identified as a
pronounced minimum in the spectrum. One consequence of
the conditions 共46兲 and 共47兲 is that the spectral range of the
oscillation frequencies is small for large ⍀ and wide for
small values of ⍀. This is also obvious in Fig. 9. The mean
value might be defined as the typical PSR frequency that has
about the highest amplitude in the spectrum:
⍀̄PSR共␶兲 =
冋 冉 冊册
1
␲⍀
1+
2
2
2/3
.
共48兲
For this quantity, the scaling with density is a mixture of
powers of 1 / 4 and 1 / 6 as both limits contribute.
The rf frequency stays always well below the spectral
range of the PSR frequency if it is below the limit defined by
Eq. 共47兲; i.e., ⍀ Ⰶ 共␲ / 2兲2. In order to keep the lower limit
below the upper limit, the even stricter condition ⍀ ⬍ 2 / ␲ is
obtained. Since only the case ⍀ Ⰶ 1 is considered in this
section, this is equivalent to excluding resonance effects.
A condition for the observability of PSR oscillations can
be derived in terms of limits for the damping constant ␬. An
upper limit of ␬ follows from Eq. 共47兲 with the demand that
over the first PSR oscillation the amplitude should not decay
by the exponential factor to less than 1 / e2. The lower limit is
given by the fact that sufficient damping of the oscillations to
the end of the rf period is a basic requirement for the analysis
of this work, although solutions might exist also for weaker
damping. Here, the demand is that the amplitude decreases at
the end of the period to a value of about 1 / e2 of the initial
value. In summary, this leads to
冉 冊
2
2
␬
⬍ ⬍
␲ ⍀
␲⍀
1/3
.
共49兲
This condition agrees also very well with the numerical results whereby the oscillations vanish for higher values of the
damping constant.
The damping of the oscillation amplitude of the current
is not exactly exponential due to the term sin共⍀␶ / 2兲1/4.
However, a comparison with the numerical solution shows
that the term affects very little the decay and that in very
good approximation, the decay is in fact exponential with a
time constant ␶d = 2 / ␬ 共Fig. 10兲. Therefore, the damping depends only on the electron-neutral collision frequency. The
reason for this slightly surprising result is that the occurrence
of the first negative maximum of the current is approximately at ␶1 = 4.6/ ⍀1/3, while the maximum of the analytic
current envelope is at ␶e = 1 / 共2␬兲. Therefore, the current
maximum appears after the envelope maximum for ␬ / ⍀
⬎ 共9␲⍀兲−2/3. On the other hand, Eq. 共49兲 requires ␬ / ⍀
⬎ 2 / ␲. This shows that except for low damping at small ⍀,
i.e., at high PSR frequencies, the decay falls well into the
exponential wing of the envelope. For times much larger
than ␶e, the decay is approximately exponential, as can be
seen in the left part of Fig. 10. In case of low damping, the
FIG. 10. Decay behavior of the current j1: 共a兲 ⍀ = 0.1 and ␬ = 0.2 and 共b兲
⍀ = 0.02 and ␬ = 0.03. The current is calculated from the exact
numerical solution by j1 = j − j0. Solid line: current waveform; dashed line:
A exp共−␬␶ / 2兲; dash-dotted line: B sin共⍀␶ / 2兲1/4 exp共−␬␶ / 2兲. Here, A and B
are constants that were fitted to the waveform in each figure.
current decays slowly over most of the rf period and the sine
term has little effect anyway, as can be seen in the right part
of Fig. 10.
B. Power dissipation
The period-averaged power dissipated in the discharge
can be calculated directly by the current:
⌳ = ␬具j2典
with
⌳=
P
P0
and
P 0 = ⌽ 0I 0 .
共50兲
Here the brackets denote the period average value. One calculates directly 具j20典 = 21 . The term 具j21典 cannot be expressed by
an analytic function. However, a rough approximation can be
derived from the WKB solution by 共a兲 averaging over the
high-frequency oscillatory contributions in the integral by
replacing that term by 21 and 共b兲 developing the sine-function
of the envelope to first order:
具j21典 ⬇
G2
␲⍀
冕 冑
2␲/⍀
0
⍀ −␬␶
G2
␶e d␶ ⬇
2
␲冑2⍀
冕冑
⬁
␶e−␬␶d␶ .
0
共51兲
The upper boundary of the integral can be extended to infinity in good approximation since the calculation is restricted
to the ␬ ⬎ ⍀. The result is then
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123503-11
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
具j21典 ⬇ C1
⍀7/6
␬3/2
with
C1 =
关1.3⌫共1/3兲兴2
.
31/327/6␲3/2
共52兲
The occurrence of the term ␬−3/2 is directly related to the
exponential damping and the factor ␶1/4 resulting from the
expansion of the sine function in Eq. 共51兲.
The mixed term can be split by partial integration into
two terms where the first one is much larger than the second
one:
2具j0 j1典 =
4
⍀
q1共0兲 +
␲
␲
−
冕
2␲/⍀
q 0q 1d ␶ ⬇
0
4
q共0兲
␲
31/3⌫共1/3兲 2
⍀ .
4␲
共53兲
Here, the integral was solved by applying the Airy solution
for q1. This choice is motivated by the fact that the integral
converges to a constant value already at early times where
the Airy solution provides a better approximation. Accordingly, q0 is expanded to first order. The result agrees quite
well with the numerical result. Nevertheless, this second part
is negligible in all relevant cases compared to the first part
given by the value of q1共0兲 = q共0兲 关since q0共0兲 = 0兴, as shown
below. In the analytic calculation, the small but finite value
was approximated by zero. However, at large values of ⍀
this term can have a noticeable influence on the power. Although, the analytic calculation of the zero value is beyond
the scope of this work, if it is possible at all, an empirical fit
to the numerical data can represent the correct trend. We find
for the interesting range of values for ⍀ as an approximate fit
共Appendix D兲:
6
q共0兲 = − ⍀7/6 + g共⍀, ␬兲.
5
共54兲
Here, g共⍀ , ␬兲 is a function that depends on the frequency
and the damping constant in a complicate way but that is
smaller than the first term. The function g, as well as the
second term in Eq. 共53兲 that is even smaller, will be neglected further on. The final result for the power scaling is
then
⌳⬇
⍀7/6
␣␬
+ C1
冑␬
2
with
␣⬇1−
48 7/6
⍀ .
5␲
共55兲
The first term results from the current j0 and the mixed term
j0 · j1, and the second term from the current j1. In all cases,
the power is dissipated by ohmic heating in the bulk. The
reason for the second term is the occurrence of the highfrequency current at low damping. Although the resistance of
the bulk decreases at lower pressures, the damping of the
PSR oscillation is also reduced and as a net effect, the dissipated power increases. The power decreases slower than expected without the PSR and can even increase at lower values of the damping constant. The mixed term reduces the
slope of the linear term in Eq. 共55兲 at higher values of the
normalized rf frequency ⍀. For the maximum value of
⍀ = 0.2, the correction is a remarkable 47%, while for the
lowest value ⍀ = 0.02 it is only a negligible 3%. This behavior compares also well with the numerical result 共Fig. 11兲.
FIG. 11. Normalized power dissipated in the discharge as a function of the
normalized collision rate for 共a兲 ⍀ = 0.1 and ␬ = 0.2, and 共b兲 ⍀ = 0.02 and
␬ = 0.03. Solid line: analytical formula 共37兲 with ␣ = 1; dash-dotted line:
analytical formula 共55兲; dashed line: numerical integration.
Despite the long chain of approximations, the analytic result
is surprisingly close to the exact solution. Differences are
related mainly to a slight overestimation of the contribution
by 具j21典 in the analytical approximation. The power has a
minimum at ␬m, which is about of the same order as ⍀:
␬m =
冉 冊
C1
␣
2/3
⍀7/9 =
0.77⍀7/9
.
共1 − 3.1⍀7/6兲2/3
共56兲
The occurrence of such a minimum and especially an increase at lower values of the collision frequency has been
demonstrated recently by an analysis based on the exact series expansion.22 A basic requirement for the analytical
analysis is sufficiently large damping so that the PSR oscillations have vanished at the end of a period. Therefore, the
analytical result does not apply for values of ␬ much smaller
than about ⍀. Equation 共55兲 can be rearranged:
2⌳
␬
⬇
+2
␣␬m ␬m
冑
␬m
.
␬
共57兲
This shows that the scaling of the dissipated power by the
collision frequency can be represented on a universal scale if
both quantities are renormalized. On this new normalized
scale, the minimum is at a collision frequency of 1 with a
value of 3 for the power. At this minimum, the contribution
by the high-frequency oscillations is twice as high as the
linear contribution. However, this normalization scale de-
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123503-12
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
pends on the plasma density and experimentally it might be
difficult to verify Eq. 共57兲 directly since it is usually not easy
to keep the applied voltage and the plasma density constant
while varying the pressure.
Finally, this result can be compared with the stochastic
heating as defined by Eq. 共11.2.34兲 in Ref. 24. The formula is
based on the assumption of a single frequency sinusoidal
current. Transformed to the normalized scale of this work,
the stochastic heating power becomes
冑
⌳Stoch = 0.63
kTe
␥⍀.
e⌽0
共58兲
Here, ␥ is the geometrical factor defined in Eq. 共22兲. This
relation can now be compared to the minimum of the dissipated power ⌳min. At ⌳min, already 2 / 3 of the power is contributed by the PSR current:
⌳min = 1.1␣ ⍀
1/3
7/9
.
共59兲
Since the numerical factors and ␣1/3 are about 1 and the
exponent of 7 / 9 is also close to 1, the major difference is
given by the factor ␥共kTe / e⌽0兲1/2. This factor will typically
be of the order of 0.1. Therefore, the stochastic heating will
always be substantially smaller than the contribution of the
PSR oscillations. As already done in Ref. 22, where this
effect related to the PSR was investigated by the infinite
power series approach we propose the name “nonlinear electron resonant heating” 共NERH兲. Recently experimental results supporting the interpretation of enhanced power dissipation by NERH are reported in Ref. 25. A self-consistent
model that takes into account all heating mechanisms active
at low pressure, as discussed in the literature,26–31i.e., Fermi
heating, pressure heating, electrostatic waves, and NERH, is
certainly strongly required.
Finally, it should be noted again that the effects discussed here in relation to power dissipation are based on the
assumption of a strongly asymmetric discharge. If the discharge becomes more symmetric, the sheath to ground can
no longer be neglected, the self-bias is reduced, the delta
function excitation is smoothed, and the amplitudes of the
PSR oscillations are becoming smaller. Therefore, enhanced
power dissipation by PSR oscillations does not occur in symmetric discharges.
冉 冊
sin
⍀␶
2
The quasi-resonant case, where ⍀ 艌 1, can be treated in
a much simpler way. The term “quasi-resonant” is used here
to point out that a sharp resonance frequency does not exist.
However, the current waveform in this regime is substantially different from the case discussed in the previous sections. Numerical investigations show that now the dc bias in
Eqs. 共16兲 and 共20兲 has to be reduced by a constant ␩ in order
to ensure that the sheath voltage briefly touches zero and
allows electrons to compensate the positive charge of the ion
flux on the electrode or the blocking capacitor:
− ␩ = q2 − 2共q̈ + ␬q̇兲.
共60兲
Further, the numerical investigations show that the solution
is practically purely sinusoidal. This motivates a simple ansatz:
qR = − AR关1 − sin共⍀␶ − ␸R兲兴.
共61兲
The index R indicates that this solution holds only in the
quasi-resonant regime and is used for all quantities discussed
in this section. Although the nonlinearity of the sheath still
leads to higher harmonics, a numerical investigation shows
that their contribution is actually very small and is further
decreasing with increasing ⍀. This is equivalent to neglecting terms with AR2 compared to AR⍀2 and AR⍀␬ on the
right-hand side of Eq. 共60兲, which is justified if AR Ⰶ 1 and
␬,⍀ 艌 1. The approximate analytical solution is then
AR ⬇
1
1
,
2
冑
4⍀ ⍀ + ␬2
冉冊
␸R ⬇ arctan
⍀
,
␬
␩⬇
1 3 2
− A .
2 2 R
共62兲
Since ⍀ 艌 1 is assumed, it follows consistently that AR
Ⰶ 1. For ⍀ ⬍ 1, the coefficient grows quickly and the higher
harmonics can no longer be neglected. Since AR Ⰶ 1, the dc
bias of the external voltage vanishes almost completely except of a very small residual value of the order AR2 . However,
the sheath still possesses a self-bias since the sheath voltage
is given by −qR2 and the bias is included in the ansatz of Eq.
共61兲. Apparently, the major part of the voltage is now distributed over the bulk and forces the self-bias of the external
voltage to nearly zero. The normalized current is
jR = 2AR cos共⍀␶ − ␸R兲.
共63兲
This approximate analytical result is compared with the
numerical result for ⍀ = 1 and 2 in Fig. 12. The figure shows
good agreement and similar agreement is found for other
values of the parameters. It is quite remarkable that in the
quasi-resonant case, the current is approximately sinusoidal
with a frequency equal to the rf frequency, although the nonlinearity of the sheath is still active. It should be noted that in
the intermediate regime 共0.2⬍ ⍀ ⬍ 1兲, the current has approximately a sawtooth waveform.
The normalized dissipated power averaged over one period is
⌳R =
V. THE QUASI-RESONANT CASE
2
1
␬
.
2 2
8⍀ ⍀ + ␬2
共64兲
The dissipated power has a maximum of ⌳R共max兲
= 1 / 共16⍀3兲 at ␬R共max兲 = ⍀. For a fixed collision frequency ␬,
the dissipated power decreases monotonically with increasing ⍀. Clearly, there is no resonance in ⍀ but simply an
optimum in ␬. In an asymmetric discharge, the nonlinearity
of the sheath does not allow for a resonant behavior as expected for the case of a simple linear capacitor.
Quasi-resonant operation of the discharge 共⍀ ⬇ 1兲 will
usually require rf frequencies in the range of several
100 MHz. Optimum power dissipation is then achieved for
typical collision frequencies in the range of several 10 to
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123503-13
Self-excitation of the plasma series resonance…
Phys. Plasmas 13, 123503 共2006兲
FIG. 13. Oscillating sheath voltage by the PSR effect for ⍀ = ␬ = 0.1 共solid
line兲. For comparison, the smooth sinusoidal voltage for the case when the
PSR effect is neglected is shown as a dashed line.
FIG. 12. Comparison of the normalized current densities obtained by
the numerical solution and the analytical solution for ⍀ = ␬ = 1.0 共a兲 and
⍀ = ␬ = 2.0 共b兲. Solid line: analytical solution; dashed line: numerical solution. In the figure 共b兲, the two curves are on top of each other.
100 Pa. However, spatial mode structures that are practically
unavoidable at high frequency operation might modify this
simplified result.
VI. SUMMARY AND OUTLOOK
A model describing the effect of self-excited plasma series oscillations in capacitive rf discharge has been derived.
The current is determined by a single nonlinear differential
equation in case of a simplified one-dimensional geometry.
The basic physics of the effect is fully contained in this reduced description. More complex geometries or complicate
spatial mode structures are not in the focus of this work. The
problem is solved and discussed for a sinusoidal external
voltage and the case of a strongly asymmetric discharge
where the sheath at ground can be neglected and the sheath
at the electrode is fully modulated.
The nonlinearity of the sheath is essential for the excitation of the plasma series resonance oscillations by an rf frequency well below resonance. The sudden change of sign of
the current at the time of sheath collapse leads to the excitation of the high frequency oscillations. Without the nonlinearity of the sheath only oscillations at the rf frequency
would be possible. The equation is solved analytically by a
series of approximations. The approximate analytic solution
compares well with the exact numerical solution.
From the solution, the frequency spectrum is obtained.
The spectrum is broad and does not show a sharp resonance.
Although the Fourier transform is calculated numerically,
analytical formula for the limits of the spectrum are derived.
With increasing rf frequency or decreasing plasma density
the spectral range becomes narrower. The oscillation amplitude is damped exponentially and the decay time is inversely
proportional to the elastic electron-neutral collision frequency.
In the frame of the present model, power is dissipated
only by the ohmic resistance of the bulk. However, the highfrequency oscillations of the current open a new channel not
anticipated initially. Typically the dissipated power decreases
linearly with the collision frequency or pressure. In case of
the PSR, the amplitude of the high-frequency oscillations
grows strongly and overcompensates the decreasing resistance of the bulk. This leads to strong deviations from the
linear behavior and a minimum of the dissipated power as a
function of collision frequency. At this minimum, the highfrequency oscillations contribute about twice as much to the
power than the normal rf current.
It is remarkable that there is a certain similarity between
the PSR oscillations and a dual-frequency discharge.32 In a
dual-frequency discharge, an additional high-frequency voltage is superimposed on a lower-frequency rf voltage. This
leads to a sheath voltage that basically follows the lowfrequency rf voltage but exhibits some smaller modulations
by the high-frequency contributions. Although the highfrequency voltage is smaller, it oscillates faster and dominates the current and the power dissipation in the
discharge.33–35 The same happens in case of PSR oscillations,
except that damping leads to significant oscillation amplitudes only over about half the rf cycle 共Fig. 13兲. In this half,
the calculated sheath voltage shows the same characteristic
waveform as a dual-frequency discharge. However, while in
a dual-frequency discharge the lower frequency is typically
of the order of 1 to 2 MHz, and the upper frequency is at 13
to 27 MHz, both frequencies are about one order of magnitude higher in typical PSR oscillations.
There is no explicit resonance of the PSR. Therefore, the
term “resonance” is slightly misguiding. Nevertheless, various regimes can be distinguished. The major part of this
work deals with the case where the rf frequency is well be-
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123503-14
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
low the range of PSR frequencies. However, if the rf frequency comes into this range, the form of the current
changes strongly. The superimposed high-frequency oscillations vanish and the current oscillates sinusoidal at the driving rf frequency. Between these two extremes there is a transition region where the current waveform is approximately
sawtooth like.
Although a number of problems could be addressed in
this work there are still open questions that require investigation. First of all it would be desirable to compare the
theory with the experiment for a simple geometry like a cylinder. The concept outlined here bears the potential to extend
it to the case of multifrequency excitation. The question of
spatial mode structures is certainly very interesting, both
from a physical and from a practical point of view. Finally,
the stability region of the model at low damping needs further investigation. Last but not least, it would be desirable to
compare this fluid-dynamic model to a particle-in-cell simulation.
␹=
ACKNOWLEDGMENTS
This work is funded by the Deutsche Forschungsgemeinschaft 共DFG兲 in the frame of the Sonderforschungsbereich
SFB 591 “Universal behaviour of plasmas far from equilibrium.” The authors are very much indebted to E. Semmler
and P. Awakowicz for providing a plot of an experimental
current waveform obtained with a SEERS sensor. Further,
we want to thank T. Gans for pointing out the similarity to
the dual-frequency discharge.
APPENDIX A: CHARGE-VOLTAGE CHARACTERISTIC
OF THE SHEATH
The approximation of a constant ion density in the
sheath leads to a quadratic relation between the sheath voltage and the charge density in the sheath. The validity of this
relation can be tested by assuming more realistic density profiles:
ni共z兲 =
n0
z
1−
a
冉 冊
k
with
FIG. 14. Charge-voltage characteristics of various sheath models compared
to the quadratic relation given by Eq. 共A4兲 and by replacing k by 共2 / 3兲k in
order to get an optimized fit. Exact solution: solid lines; quadratic approximation: dashed lines. For k = 0, both solutions are identical.
0艋z⬍a
and
0 艋 k ⬍ 1.
In the dc case, k = 2 / 3 corresponds to a collisionless
sheath and k = 1 / 3 to a charge exchange sheath. A constant
sheath density is represented by k = 0. The divergence at
z = a comes in artificially since the electron density is neglected here. Therefore, the point of divergence is excluded
and the profile becomes unrealistic close to this point. Although Eq. 共A1兲 is still not the correct profile for a rf sheath,
it is nevertheless much closer to reality than the convenient
but artificial assumption of a constant density. Derivation of
an analytic relation between the sheath voltage and the space
charge in the sheath is straightforward and the result is
The dimensionless quantities ␹, Us, and qs are
共A2兲
Us = ␧0
1−k
⌽,
ea2n0
and
qs =
1−k
Q. 共A3兲
ean0
The normalized quantity for the charge qs defined here
should not be confused with the normalized charge q used in
the main text. In lowest order in qs Eq. 共A2兲 becomes
Us ⬇ −
1 1 2
q .
21−k s
共A4兲
This is precisely the result obtained under the assumption of constant density with the effective mean density set
identical to the density at the electrode. Higher orders in qs
or Q become important only at large sheath expansions for s
close to a. However, here the divergence in density given by
Eq. 共A1兲 is not realistic anyway. In order to provide a good
fit over the entire range of possible qs values between −1 and
0, the coefficient in Eq. 共A4兲 might be slightly modified, e.g.,
by replacing k by 32 k. This factor or any other fitting factor
can always be considered in an effective density used in the
quadratic relation, e.g.,
n̄s ⬇ n0
共A1兲
1
Us = q − 关共1 + q兲␹ − 1兴.
␹
2−k
,
1−k
1 − 2k/3
艌 n0 .
1−k
共A5兲
As shown in Fig. 14, the quadratic relation represents
also more realistic current-voltage or charge-voltage relations quite well. Nevertheless, it should be noted again that
in the sheath model applied in this work only a second-order
nonlinearity is included. In general, for more realistic density
profiles, higher order terms will occur. But even then, clearly
the second-order term will be dominant. In a symmetric discharge, all terms of even order cancel out exactly, including
the nonlinearity considered in this work. Terms of odd order
remain but do not contribute to the excitation of series resonance oscillations. Their main effect is giving the current
waveform a sawtooth-like profile.
APPENDIX B: RF FLOATING POTENTIAL CONDITION
The dynamic floating potential condition for an rf sheath
allows calculating the relation between the rf amplitude and
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123503-15
Phys. Plasmas 13, 123503 共2006兲
Self-excitation of the plasma series resonance…
FIG. 15. The relative deviation of the bias voltage from the rf amplitude
共␸b / ␸ˆ − 1兲 as a function of the normalized dimensionless rf amplitude ␸ˆ for
the case of argon according to Eq. 共B3兲.
the dc bias. In order to avoid net charging of the electrode or
the blocking capacitor, the period-averaged electron conduction and ion currents must balance. The ion current is determined by the incoming ion flux at the sheath edge where the
ions reach the Bohm velocity and the electron conduction
current, for a Maxwellian distribution function, by the socalled Hertz-Langmuir current.36 The potential over the
sheath consists of a negative bias and a harmonic oscillatory
part:
␾ = − ␾b + ␾ˆ cos共␻rft兲.
共B1兲
The PSR effect leads in fact to a nonsinusoidal sheath
voltage 共see Fig. 15 in Sec. V兲. However, the superimposed
high-frequency oscillations affect the electron current to the
electrode only at the time of the sheath collapse. Here, certain corrections might become important. Nevertheless, as a
first approximation for the order of magnitude a simple sinusoidal voltage form might still be appropriate. The timeaveraged current balance for a sinusoidal sheath voltage as
given in Eq. 共B1兲 leads to a modified floating potential condition with modified Bessel function of first kind:19,37
冑
2␲m ␸ˆ
e兩␸ˆ 兩
e b = I0共␸ˆ 兲 ⬇
冑2␲兩␸ˆ 兩 .
M
共B2兲
Here, M is the ion mass and m the electron mass. On the
right-hand side, advantage is taken of the asymptotic form of
the Bessel function for large arguments, which is already a
very good approximation when the argument is larger than 2.
The dimensionless potential variables are normalized by kTe
ˆ / 共kTe兲. Rearranging the above equation allows exas ␸ˆ = e␾
pressing the bias voltage in terms of the rf amplitude:
冉 冊
兩␸ˆ 兩
1
␸b = 兩␸ˆ 兩 − ln
.
2
␸ˆ C
共B3兲
For rf amplitudes ␸ˆ larger than the critical amplitude ␸ˆ C,
the amplitude exceeds the bias and a temporal field reversal
occurs:
␸ˆ C =
M
= 46M A .
共2␲兲2m
共B4兲
Here, M A is the atomic mass number of the ion. For H+3
ions 共M A = 3兲, one calculates ␸ˆ C = 138 and, in fact, hydrogen
discharges are well known for a strong field reversal.38 However, gases used in processing plasmas and also all noble
gases with the exception of helium have substantially higher
masses than hydrogen. For instance, in case of argon
共M A = 40兲 a much larger factor of 1840 results. Therefore, for
practically all relevant gases, a field reversal can occur only
at unrealistically high amplitudes, in agreement with common experimental observations. Consequently, the rf amplitude is generally much smaller than the critical amplitude.
Since ␸ˆ is typically of the order of 100,24 the second
term in Eq. 共B3兲 is of the order of 1. Then the relative deviation of the bias voltage from the rf amplitude 共␸b / ␸ˆ − 1兲 is
typically of the order of 10−2 共Fig. 15兲. This small correction
is of the same order as other terms that are neglected in the
present analysis of the PSR equation and must consequently
be neglected as well.
APPENDIX C: CALCULATION OF THE INTEGRAL
OVER THE PHASE ⌬
The phase difference ⌬ is defined in Eq. 共44兲. Since ⌬
differs from zero only at early times, one can use the linearized form of q0. Further, the assumption is made that the
damping is low enough and the oscillations frequent enough
so that for the first one or two oscillations the exponential
damping can be neglected in q1. If the Airy solution is used
for f then ⌬ has the following form:
⌬=
=
冑 冉 冊 冋冉 冊 册 冑
冉 冊 关冑 冑 兴
⍀
⍀
␶+
2
2
⍀
2
⍀
2
2/3
h
1/3
␶ −
⍀
␶
2
1/3
x + h共x兲 − x .
共C1兲
Here, the new variable is x = 共⍀ / 2兲1/3␶ and h共x兲 is identical to f 共Airy兲共␶兲, as given by Eq. 共40兲, except for the factor
containing ⍀ which is explicitly moved out. A zero of ⌬ is
therefore given by k = 共⍀ / 2兲1/3␶␬, where k is a number independent of any parameters. The integral of ⌬ up to this zero
is
S共␶k兲 =
冕
␶k
冕 关冑
k
⌬d␶ =
0
0
x + h共x兲 − 冑x兴dx.
共C2兲
Apparently, the integral is independent of all parameters.
APPENDIX D: APPROXIMATE DETERMINATION OF
Q„0…
The value q共0兲 = q1共0兲 is calculated numerically by solving the PSR differential equation. In Fig. 16, these values are
plotted as function of ⍀ for ratios of ␬ / ⍀ = 1, 2, 3, and 4.
The data are fitted by a function
6
q共0兲 ⬇ − ⍀7/6 .
5
共D1兲
In addition to Eq. 共D1兲, there is a further but smaller
contribution by an unknown function g共⍀ , ␬兲. However,
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123503-16
Phys. Plasmas 13, 123503 共2006兲
Czarnetzki, Mussenbrock, and Brinkmann
J. V. Parker, J. C. Nickel, and R. W. Gould, Phys. Fluids 7, 1489 共1964兲.
J. Taillet, Am. J. Phys. 37, 423 共1969兲.
8
V. A. Godyak, Sov. J. Plasma Phys. 2, 78 共1976兲.
9
V. A. Godyak and O. A. Popov, Sov. J. Plasma Phys. 5, 400 共1979兲.
10
R. L. Stenzel, Phys. Fluids B 1, 2273 共1989兲.
11
B. Annaratone, V. P. T. Ku, and J. E. Allen, J. Appl. Phys. 77, 5455
共1995兲.
12
V. P. T. Ku, B. Annaratone, and J. E. Allen, J. Appl. Phys. 84, 6536
共1995兲.
13
V. P. T. Ku, B. Annaratone, and J. E. Allen, J. Appl. Phys. 84, 6546
共1995兲.
14
D. J. Cooperberg and C. K. Birdsall, Plasma Sources Sci. Technol. 7, 96
共1998兲.
15
Y. P. Bliokh, J. Felsteiner, Y. Z. Slutsker, Europhys. Lett. 46共6兲, 735
共1999兲.
16
N. Tanizuka, J. E. Allen, J. Plasma Phys. 61, 469 共1999兲.
17
K. E. Orlov and A. S. Smirnov, Plasma Sources Sci. Technol. 10, 541
共2001兲.
18
W. D. Qiu, K. J. Bowers, and C. K. Birdsall, Plasma Sources Sci. Technol.
12, 57 共2003兲.
19
M. Klick, J. Appl. Phys. 79, 3445 共1996兲.
20
M. Klick, M. Kammeyer, W. Rehak, W. Kasper, P. Awakowicz, and G.
Franz, Surf. Coat. Technol. 98, 1395 共1998兲.
21
E. Semmler and P. Awakowicz, private communication 共2006兲.
22
T. Mussenbrock and R. P. Brinkmann, Appl. Phys. Lett. 88, 151503
共2006兲.
23
T. Mussenbrock and R. P. Brinkmann, Proceedings of the 12th International Congress on Plasma Physics (ICPP), Nice, France, 2004, published
online by IUPAP: http://hal.ccsd.cnrs.fr/ICPP2004/en/.
24
M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges
and Materials Processing 共Wiley, New York, 1994兲.
25
G. Franz and M. Klick, J. Vac. Sci. Technol. A 23, 917 共2005兲.
26
V. A. Godyak, Sov. J. Plasma Phys. 2, 78 共1976兲.
27
O. A. Popov and V. A. Godyak, J. Appl. Phys. 57, 53 共1985兲.
28
M. A. Lieberman and V. A. Godyak, IEEE Trans. Plasma Sci. 26, 955
共1998兲.
29
L. D. Kaganovich, V. I. Kolobov, and L. D. Tsendin, Appl. Phys. Lett. 69,
3818 共1996兲.
30
M. M. Turner, Phys. Rev. Lett. 75, 1312 共1995兲.
31
G. Gozadinos, M. M. Turner, and D. Vender, Phys. Rev. Lett. 87, 135004
共2001兲.
32
T. Gans, private communication 共2006兲.
33
P. C. Boyle, A. R. Ellingboe, and M. M. Turner, Plasma Sources Sci.
Technol. 13, 493 共2004兲.
34
T. Mussenbrock, D. Ziegler, and R. P. Brinkmann, Phys. Plasmas 13,
083501 共2006兲.
35
J. Schulze, T. Gans, D. O’Connell, U. Czarnetzki, R. Faulkner, A. R.
Ellingboe, and M. M. Turner, Appl. Phys. Lett., accepted for publication
共2006兲.
36
K. U. Riemann, IEEE Trans. Plasma Sci. 16, 638 共1988兲.
37
Handbook of Mathematical Functions, edited by M. Abramowitz and I. A.
Stegun 共Dover, New York, 1972兲.
38
U. Czarnetzki, D. Luggenhölscher, and H. F. Döbele, Plasma Sources Sci.
Technol. 8, 230 共1999兲.
6
7
FIG. 16. q共0兲 as function of ⍀ for various values of ␬ / ⍀: 1 共stars兲, 2
共circles兲, 3 共squares兲, 4 共triangles兲. The solid line represents the fit function
given by Eq. 共D1兲.
since ⍀ is limited to a maximum value of 0.2, the latter
contribution might be neglected in a first rough approximation.
The finite value can be motivated by the fact that for
small values of ␶ before zero, the differential equation for q1
reduces after expansion of q0 to first order in ␶ to
q̈1 + ␬q̇1 +
⍀
⍀␬
.
␶q1 =
2
2
共D2兲
At larger negative times the derivatives can be neglected
and q1 rises from zero proportional to the reciprocal of ␶:
q1 ⬇
␬
.
␶
共D3兲
Very close to zero, the derivates become important and
the divergence is avoided. This shows that a finite value is
reached. However, it does not allow calculating this value
analytically unless the differential Eq. 共D2兲 can be solved.
Such a solution is beyond the scope of this work.
1
P. E. Vandenplas, Electron Waves and Resonances in Bounded Plasmas
共Wiley, New York, 1968兲.
2
L. Tonks, Phys. Rev. 37, 1458 共1931兲.
3
L. Tonks, Phys. Rev. 38, 1219 共1931兲.
4
F. Schneider, Z. Angew. Phys. 6, 459 共1954兲.
5
A. Dattner, Ericsson Technics 2, 309 共1957兲.
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