Mode Transition and Hysteresis in Inductively Coupled Plasma Sources

In this chapter, the characteristics of low-temperature inductively coupled plasma sources, that is, non-equilibrium, weakly ionized and bounded plasma, are described. The phenomenon of mode transition and hysteresis is one of the main physics aspects that happens in this source. Via a hybrid model, the behaviors of plasma parameters, electron kinetics and neutral species during mode transition are presented. Still, the role of metastables and multistep ionization on triggering the hysteresis is investigated. Using a fluid model that couples the equivalent circuit module, the discontinuity of mode transition and hysteresis are observed by tuning the matching network impedance. The work indicates the mutual interaction between the plasma and the circuit excites hysteresis. Besides these findings, the other important aspects of this phenomenon are briefly discussed. To the author, the exploration on the precursors that trigger hysteresis is the most attractive topic. The investigations advance the improvement of analytical theory, numerical modeling and experimental diagnostics of low-temperature plasma physics.


Introduction
The inductively coupled plasma (ICP) source is one of the most important low-temperature plasma sources that find widespread applications in many fields [1], such as plasma photonic crystals, synthesis of nanomaterials and nanostructured materials, atomic layer processing, agriculture and innovative food cycles, medicines, environments, plasma-assisted combustion and chemical conversion and aerospace application (propulsion and flow control) and so on. Driven within the domains of radio frequency electromagnetic and rather low-pressure (mTorrs) ranges, the ICP sources present several advantages, such as high-plasma density, high anisotropy in the sheath, independent control of incident flux density and energy and simple low-cost reactor configuration (unwanted for the static magnetic devices) over some other plasma sources, such as capacitively coupled plasma and electron cyclotron resonance reactor [2]. As compared to the atmospheric discharges, this sort of low-pressure radio frequency plasma sources are known for their non-equilibrium properties, that is, T e ≫ T i > T n , where T e , T i and T n are temperatures of electrons, ions and neutrals, respectively [3], due to the low-temperature peculiarity of this type of plasma source. Another essential feature is its weak ionization degree that ensures the abundance of collisions and reactions between charged species and neutrals, which is quite different with the high-temperature fully ionized plasmas where only the Coulomb interactions between charged species are important [4]. Of great importance is the diversity in the mutual interactions among charged and neutral species, which are classified into elastic and inelastic collisions with respect to the principle of kinetic energy balance. Regarding species specialty and colliding outcomes, the inelastic collisions can be described as type (1) ionization, dissociation, electronic, rotational, vibrational excitation, attachment, detachment and de-excitation, which mainly occur between electrons and neutrals; type (2) recombination, associations, charge exchange, excitation transfer and penning ionizations, which mainly happen among heavy species (meant to all species except for electrons); and type (3) the spontaneous radiation from excited state species (without a trigger) [5]. The elastic scattering to some extent determines plasma transport process and hence spatial characteristics of plasma via the parameter of momentum transfer collision frequency, while the inelastic collisions that sustain the weakly ionized plasma mainly determine the energy loss mechanism and give steady-state plasma components optical emission. Finally, all low-temperature plasma sources are generated in chambers with their respective fixed configurations and more importantly with limit space dimension. This means that all the plasmas are bounded plasmas, as compared with the space plasma; therefore, the sheath, produced on all bound surfaces, forms one important constituent of low-temperature plasma physics [6]. In a word, non-equilibrium, weak ionization and plasma bounds characterize the low-pressure radio frequency plasma source as complicated and multi-disciplinary.
Even with the above complexity, rich and fruitful interesting physics phenomena and mechanisms are already revealed in these low-pressure and radio frequency plasma sources via present efforts. In particular, in the ICP sources, pulsed radio frequency power source [7], standing wave effects [8], nonlinear harmonics [9], double coil discharges [10], anomalous skin effects [11], nonlocal electron kinetics [12], mode transition and hysteresis [13] and so on are still or have been hot research frontiers that draw attention. In this chapter, the mode transitions and hysteresis topic is focused upon. This topic has been historically studied well and continually occupies people's attention due to its complexity of the multi-factor interactions and potential application in achieving stable plasma sources for the processing technique. The ICP source is famous for its capacity of operating at two different modes, that is, capacitive and inductive modes. The capacitive mode is sustained by radial and axial electromagnetic fields, analogous to conventional capacitively coupled plasma source that is excited by the electrostatic field and hence is abbreviated as the E mode. The inductive mode is sustained by the azimuthal electromagnetic field caused by coil current and is abbreviated as H mode. Remember that the power source applied to the coil is temporally varied in the range of radio frequency. At low-coil power, the ICP source is maintained at E mode, where the plasma density and optical emissions are weak, and the glow area of discharge is more localized under the coil. As we increase the coil power, the discharge transfers abruptly or smoothly toward H mode, where the plasma density and current are significantly increased and the optical emission is strengthened. Moreover, the discharge is more uniform. Interestingly, at certain circumstances, when cycling the power source, the trajectories of plasma parameters versus upward and downward powers don't coincide; hence, hysteresis is formed and the ICP source is therefore famous for its other feature, that is, the existence of two stable states at one fixed power value. In labs of academia or enterprise, the ICP sources are triggered from the E mode at the beginning and then transferred to H mode. Most of the plasma processing techniques prefer to be conducted in the H mode due to its better plasma properties. Therefore, understanding the E-H mode transition and hysteresis is very meaningful to the related industry.
This chapter is outlined as follows. In Section 2, the major achievements of the author on this topic are presented. Three subtopics and the used methodology are discussed and described, aimed at demonstrating to readers the characteristics of plasma parameters, electron kinetics and neutral species during mode transition and excitation of discontinuous mode transition and hysteresis by the external circuit. Finally, the conclusion and further remarks are given in Section 3.

Characteristics of basic plasma parameters
In this part, the characteristics of electron parameters, density, temperature and energy distribution function and plasma potential at two modes are presented via the two-dimensional hybrid model [14]. The hybrid model consists of three parts, that is, fluid module, electron Monte Carlo module and electromagnetic module. Species density and momentum, together with the electrostatic field generated by net charge density (analogous to ambipolar diffusion field), are given by the fluid module. Electron transport and collision coefficients, and the effective electron temperature, are calculated through the Monte Carlo method and then transferred to the fluid module. The electromagnetic module calculates the electromagnetic field generated via the coil current and voltage through the Maxwellian's equations, based on the electron conductivity from the fluid module. Both the electrostatic and electromagnetic fields are sent to the Monte Carlo module to push the electrons via Newton's law. The interactions of three modules are illustrated by the model flowchart in Figure 1. The three modules are iterated with each other until a final steady state is achieved. In this chapter, a cylindrical inductively coupled plasma reactor with planar coil is used, as shown in Figure 2.
In Figures 3 and 4, the calculated electron density and temperature profiles versus coil current at the pressure of 20 mTorr are presented. In Figure 3, at low-coil current, 10 A, the density magnitude is low and the profile is smooth. At high coil current, 40 A, the density magnitude is high, more or less four factors higher than the 10 A case. Meanwhile, the density is peaked under the coil, as referred to the reactor in Figure 2. The E-H mode transition happened along with increase in the coil current. In Figure 4, at E mode, that is, 10 A, the electron temperature is high around the plasma chamber bound but sinks at the discharge center region. This is because the ambipolar diffusion potential barrier suppresses the electrons from entering the sheath for heating due to the lack of elastic collisions at low pressure. As known, this is a representative feature for the capacitive discharges [15]. At the H mode, that is, 40 A, the temperature profile is substantially changed. It peaks under the coil and more or less decreases toward the center, bottom and sidewall. Besides, the sink area of the temperature profile is significantly shrunk and moves toward the coil, as compared to the E mode. The appearance of temperature sink at different coil currents and its alteration with coil current is related to the  spatial potential distribution in Figure 5, where the potential barrier is shifted from the discharge center to the coil with the coil current and meanwhile the area of potential barrier is decreased. In Figure 6, the electron energy distribution functions (EEDFs) of E and H modes, sampled at the discharge center, are compared at different pressures. At low pressure, that is, 20 mTorr, a prominent low-energy peak is found in the EEDF of the E mode due to the suppression of potential barrier, and it disappears at H mode because the barrier shifts toward the coil. At high pressures, that is, 50 and 100 mTorr, the EEDFs evolve to an opposite way, that is, lowenergy electrons' amount of the H mode is higher than the E mode. This is because at high pressures the suppression of the potential barrier is not important anymore due to the frequent elastic collisions between electrons and neutrals at high electron densities. Hence in the electron temperature profile (see Figure 7), the sink region disappears. The temperature profiles of E and H mode are representative of the capacitive and inductive discharges, and, as is well known, the temperature value in the E mode is higher than in the H mode [16]. To demonstrate the electron kinetics better, in Figure 8, the electron energy probability function (EEPF) variation with coil current at low pressure of 20 mTorr is shown. Clearly, at the E mode, that is, with the coil current less than and equal to 20 A, the obvious three-temperature Maxwellian distribution is observed. The low-energy electron peak, as mentioned before, is formed by the suppression of the potential barrier, while the depletion of high-energy electrons tail is caused by the inelastic electronneutral collisions, such as excitations and ionizations. At H mode, that is, with coil current equal to and larger than 25 A, the two-temperature Maxwellian EEDF in the elastic collision energy range, that is, less than 11.56 eV (the excitation threshold), now disappears due to high electron density and frequent collisions, and the high-energy electrons' depletion via inelastic collisions still exists because the electron density is not high enough for the e-e Coulomb collisions thermalizing these two electron swarms [17,18].
In a word, the hybrid model successfully captures the main characteristics of plasma parameters during the mode transition, including both the macroscopic plasma properties and microscopic electron kinetics, and all these predictions presented here agree well with the experimental measurements.

Behavior of metastable neutrals
The behavior of metastable neutrals during the mode transition is investigated by the above hybrid model, with the advanced reaction set that includes the metastables and all relevant reactions [19]. In Figure 9, the metastable densities, sampled at the discharge center, versus applied power at different pressures are presented. The densities at low and high pressures, that is, 30 and 300 mTorr, both first increase and then decrease with the power, and the decreasing trend at high pressure is more obvious. Hence, the metastables density increases with power at E mode while decreases with power at H mode, which is different with the electron density trend in Section 2.1. In Figure 10, the metastables density profiles at different coil currents are presented. It is shown that the peak density keeps increasing with coil current, however, the peak location basically shifts from the discharge center at E mode toward the coil at H mode, thus leading to the non-monotonic varying trends of metastables' densities at the discharge center in Figure 9. The localizing trend of metastables density to the coil with coil current is caused by the fact that multistep ionization becomes more and more important as the plasma density is increased, to an extent when the negative source, that is, multistep ionization rate larger than the excitation rate, is formed. The stationary metastables continuity equation with a negative source can be characterized as Bessel's equation with imaginary argument that shows spatial characteristics analogous to the localized profile. This localizing effect is more important at high pressure due to the prevalence of multistep ionizations; hence, the decreasing trend of metastables density with power is more obvious at high pressures. Last, the model predicted a non-monotonic variation of metastables density during mode transition which agrees well with the experiment [20].
Besides the exploration of metastables evolution along with mode transition, the role of metastables in determining the hysteresis is still investigated through the hybrid model. The behind mechanisms that generate hysteresis are difficultly identified since it is a process that is interplayed by so many elements. In the literature, many papers ascribed the hysteresis to the multistep ionizations [13,21]. To assess this argument, in Figure 11, the influence of metastables on electron density and temperature variations versus power is presented. Inclusion of metastables and multistep ionization overall elevates electron densities and meanwhile reduces electron temperatures against the power; however, it does not trigger hysteresis. Besides, the metastables change the trend of electron temperature with power at the H mode. The decrease of temperature with power when including metastables is caused by the fact that ionizations consume electron kinetic energy more effectively than excitations, as revealed by a novel electron mean energy Equation [22].

Discontinuous mode transition and hysteresis excited by matching network
The discontinuity feature of mode transition and interesting hysteresis phenomena have attracted people's attention for years. They are easily observed in the experiments [16,23] and can be analytically predicted by stationary zero-dimensional global model [24,25]. However, it is difficult for the self-consistent multidimensional models to capture the discontinuity and hysteresis unless the external circuit module is taken into account. In this chapter, the conventional fluid model that describes the pure inductive mode is extended by including the capacitive mode and advanced by introducing an equivalent circuit module [26,27]. The diagram of equivalent circuit is illustrated in Figure 12. It consists of radio frequency (RF) power source, matching network that consists of parallel and series capacitors and capacitive and inductive coupling branches. The capacitive coupling components include dielectric window capacitor, sheath transferred capacitor and bulk plasma-transferred resistance. The inductive coupling branch is actually based on a transformer model [13], where the coil itself and plasma-transferred inductor and resistor are included and the relations between the coil and plasma-transferred impedances are illustrated in the square of Figure 12. The plasma resistances in the capacitive and inductive branches are both transferred through the Ohm's heating mechanism but the capacitive resistance is based on radial and axial plasma current components [3] and the inductive resistance on the azimuthal current component [28].
Via the circuit module, the coil current and voltage, boundary conditions for the electromagnetic module to calculate fields can be given through Kirchhoff's law. More importantly, after considering the circuit module the discontinuity of mode transition and hysteresis can be captured by a fluid model since the mutual interacting details between the circuit and plasma, probably nonlinear, are contained. Of more significance is that two excitation means of mode transitions that have widely been seen in experiments, that is, by means of varying power Figure 11. Electron densities (a) and temperatures (b) versus power at two cases, i.e., (1) no metastables and (2) with metastables in the model. [16,23] and matching network [29], can be both captured by this advanced fluid model that couples an external circuit module.
In Figure 13, the discontinuous mode transition at a low pressure of 20 mTorr and hysteresis at high pressure of 100 mTorr is perfectly generated by the fluid model, via the alteration of electron density versus series capacitance of matching network. This prediction agrees well with the experimental observations that hysteresis mostly appears at relatively high pressures [21,23]. Accordingly, the electron temperature just displays mode transition at low pressure, but hysteresis at high pressure, as shown in Figure 14. The variations of electron density and temperature with E-H mode transition predicted by the fluid model are in agreement with the hybrid model in Section 2.1.  Interestingly, the plasma-transferred impedance evolves similarly to the plasma parameters, that is, discontinuously jumping at low pressure and displaying hysteresis at high pressure. In Figures 15 and 16, the plasma resistance and inductance of inductive branch and sheath width and capacitance of capacitive branch are plotted against the series capacitance, respectively, at the high pressure of 100 mTorr. In Figure 15, at the E-H mode transition, the plasma resistance and inductance both increase because of high-plasma density and strong azimuthal current density. The high-plasma inductance at the H mode weakens the system inductance according to the formula in Figure 12, as determined by the law of electromagnetic induction. In Figure 16, the sheath width significantly decreases with E-H mode transition due to the scaling law [6] and the sheath transferred capacitance, inversely proportional to mean sheath width, increases substantially. At the H-E mode transition of the hysteresis loop, the opposite cases happen.

Conclusion and further remarks
The low-pressure radio frequency ICP source is characterized as non-equilibrium, weakly ionized and bounded plasma and finds wide applications in many fields. It holds many interesting physical phenomena and mechanisms. One is the mode transition and hysteresis that happen at two operating modes, that is, inductive and capacitive modes. In this chapter, the characteristics of plasma parameters and neutrals during mode transition are presented by a hybrid model. Moreover, the discontinuity feature of mode transition and hysteresis excited by adjusting the matching network are predicted by a fluid model that couples an external equivalent circuit module. Still, the role of metastables on triggering hysteresis is discussed and the interesting hysteresis loop formed by plasma-transferred impedance is analyzed. The present chapter indicates that the mutual interaction of plasma with circuit is the reason which excites the hysteresis.
Note that the mode transitions and hysteresis of ICP sources are very complicated. Besides the above representative features, it still exhibits research values in the topics of reactive gas mixtures, such as O 2 [30], CF 4 /Ar [31], SO 2 [32], ammonia [33] and so on and double hysteresis loop [29], inverse hysteresis [34], spatial characteristics [35], optical emission [36], electrical diagnostics [37], instability of electronegative plasma source [38] and so on. To the author, the exploration of precursors that triggers hysteresis, for instance, metastables and multistep ionizations [13,21], electron energy distribution function [39], power coupling efficiency [40], sheath [24,41], external circuit [26,27] and nonlinear mechanisms [13] and so on, is the most attractive topic. The investigations greatly advance the improvements of analytical theory, numerical modeling, and experimental diagnostics of low-temperature plasma physics.

Acknowledgements
Gratitude is given due to the support by National Natural Science Foundations of China (Grant No. 11305023), and the useful discussions with Hui-Jing Xu are especially acknowledged.