3.4 Plasma Behaviour
3.4.1 Effects of Gas Composition on Plasma Formation
Despite the fact that many common and inert gases behave similarly under normal conditions, their behaviour while ionized as a plasma can vary dramatically, as can
3.4 Plasma Behaviour 35
their effects on the surrounding equipment. Gas properties such as ionization energy, breakdown potential (i.e. the ease of a plasma to be ignited), thermal conductivity, skin depth, microwave absorption, and chemical reactivity are discussed with relevance to the plasma’s startup behaviour, steady-state behaviour, and interaction with other species or materials that may be present. Each of these properties are examined in greater detail below.
Ionization energy: The ionization energy of a gas corresponds to the amount of energy required to liberate an electron from its host atom. If electrons can be liberated sufficiently quickly before recombination, a plasma can be sustained. Among all the elements in the periodic table, the non-metals - particularly the gases - have the highest ionization energies. The elements with the highest ionization energies are those in the top right of the table - light elements with nearly complete electron shells. Light elements possess valence electrons close to their nuclei and have fewer complete electron shells below their valence shell. As a result, the attractive force between the valence electrons and atomic nucleus is strong and largely unshielded. The highest first ionization energy is that of helium (24.6 eV), followed by neon (21.6 eV). For comparison, the element with the lowest ionization energy is caesium at only 3.9 eV. As electron excitation is a type of latent heat, more energy will be required to produce plasmas from gases with higher ionization energies. An initial conclusion may therefore be that practically speaking, helium is the most difficult gas to ionize; however, this is not the case. A gas’s ability to be ionized is in fact dependent on more than just its ionization energy as is explored in the following material properties. In fact, the amount of power required specifically for ionization is relatively small. Using helium with a flow rate of 5 SLPM and an approximate ionization fraction of 0.1%, only 8 W are required. Despite the low power consumption, ionization energy has a large influence on the plasma’s electron density. From the Saha equation in section 1.4.1, the ionization energy is contained in an exponential term and thus variations in this property can alter the electron density by orders of magnitude for a given temperature. Therefore, ionization energy on its own is not a significant source of energy consumption nor is it a practical limitation for plasma generation, though it does significantly affect the plasma’s electron density.
Breakdown potential: One method of determining the propensity of a gas to undergo macroscopic electrical breakdown is from how easily a spark can be generated in the gas across two electrodes. Paschen’s law can be used to determine this break- down voltage for a particular pressure (P) and electrode spacing (d), and curves for several candidate gases are displayed in Figure 3.3 (using coefficients from Lieberman and Lichtenberg [153]). For high Pd values it is in fact helium that possesses the lowest breakdown voltage, followed by neon. This is an interesting result considering these two gases also possess the highest ionization energies. The explanation for this
counter-intuitive phenomenon is that other metastable electron energy states exist within the neutral atoms, and in the case of helium and neon, these metastable states are near the ionization energy. As a result, a small number of atoms can be easily ionized and their liberated electrons then initiate the cascade of collisions leading to breakdown. Conversely, nitrogen exhibits a high breakdown voltage, making it difficult to start the plasma system. Also note that these trends apply for high pressure or large electrode spacing, whereas the trends are entirely reversed at low pressure or a small electrode spacing. The distinction must be made that Paschen’s law applies to direct current voltages and cannot be quantitatively be applied to this microwave- based system. Modifications exist for conditions such as alternating current [154] or small electrode spacing [155]. Still, Paschen’s law gives qualitative insight into the breakdown characteristics of various gases and can still be useful for development of the setup used in this work.
Fig. 3.3 Paschen curves for various potential plasma gases
Thermal conductivity: The thermal conductivity of a gas has a large influence on the components in close proximity to the plasma and heated gas. A high thermal conductivity can be detrimental for several reasons. First, the gas transfers heat very effectively to the quartz reactor tube (and any other components nearby such as an injection torch). This heating can be severe and may result in accumulated wear on the tube surface or complete crack failure. Secondly, since the gas conducts its heat easily, it is also difficult to maintain energy in the flow downstream of the plasma. Ideally, the plasma and its byproducts are very hot but the equipment containing the plasma is relatively cool. Highly conductive gases produce the opposite effect: excessive heating of the reactor tube, yet poor retention of heat in the gas downstream since the energy is conducted away too quickly. Hydrogen and helium have extremely high thermal con- ductivities (180.5 mW K−1 m−1 and 151.3 mW K−1 m−1 respectively) in comparison
3.4 Plasma Behaviour 37
to other gases such as nitrogen (25.8 mW K−1 m−1) or argon (17.7 mW K−1 m−1). Unfortunately, these gases (particularly hydrogen) are useful for CNT production as discussed in future chapters.
A second property influenced by thermal conductivity is the plasma size. As conductive gases transmit heat to adjacent atoms more readily, and as the temperature of the adjacent atoms increases, so too does their likelihood of ionization (recall from the Saha equation that ionization fraction scales with temperature). Therefore, conductive gases occupy larger volumes than non conductive gases. In this work, the contrast between helium and argon is exemplary, since the former exists as a diffuse plasma (up to ∼ 30 mm in diameter) that often occupies the majority of the tube cross-section whereas the latter exists as thin, unstable filaments <3 mm in diameter. For comparison, the reactor tube inner diameters used in this work range from 22 mm to 46 mm. Details about plasma size and shape are discussed in section 3.4.2.
Skin depth: As plasmas contain positive ions and their corresponding free elec- trons, they are inherently electrically conductive. Like any conductor, exposure to electromagnetic fields results in movement of these charges which in turn produces a screening effect whereby the electromagnetic field is not able to penetrate beyond a certain distance into the conductive medium. The characteristic penetration distance of an oscillating electromagnetic field is known as the skin depth (Ls) and can be
determined from: Ls= c r ε0me e2N e , (3.1)
where c is the speed of light, ε0 is the permittivity of free space, me is the electron
mass, e is the electron charge, and Neis electron density. Note that the skin depth is
independent of gas properties and is only influenced by electron density. Furthermore, this equation neglects the translational motion (thermal energy) of the electrons. Using a typical electron density value for an atmospheric pressure microwave plasma of Ne= 3 × 1020 m−3[69], the skin depth is approximately 0.31 mm. This indicates that
the microwave energy powering the plasma is only active on the outer layer of the plasma. Given that the scale of the skin depth is that of hundreds of micrometres to single millimetres, yet the plasma itself has a length scale of centimetres, energy must be able to to permeate farther into the plasma, beyond the skin depth. Specifically, the high translational energy of the electrons allows them to penetrate deeper into the plasma and sustain it, and it is for this reason that this type of plasma is known as "collisionally-dominated" since the majority of its volume is powered purely from collisions with free electrons in the relative absence of an external field. Had the electron density been lower (∼ 1017 m−3 for the plasmas considered here), the skin
depth would approach the plasma length scale and the microwave field would therefore be capable of permeating the entire plasma volume.
Microwave absorption: At a given electromagnetic input frequency, some plas- mas are more suited to power absorption than others. Since the plasmas considered here are collisionally-dominated, particularly by electron-neutral interactions, the power absorbed by the plasma normalized by volume and electric field strength can be determined from Leins et al. [69] as:
p V E2 = Ne Ncε0ω νen/ω 1 + νen/ω 2. (3.2)
Here, p is absorbed power, V is volume, E is electric field strength, Ne is electron
density, Nc is cutoff density, ε0is vacuum permittivity, νenis the collision frequency
between electrons and neutrals, and ω is the microwave angular frequency (ω = 2π f , where f is 2.45 GHz). The cutoff density is equal to Nc= ε0meω2/e2 where me is
electron mass and e is electron charge. The electron neutral collision frequency can also be expanded as νen = N0u< σen>, where N0 is neutral particle density, u is
relative electon-neutral velocity, and < σen> is the electron-neutral collision cross-
section. Also from Leins et al. [69], each of these constituent variables can be explored further to identify the behaviour of the collision frequency. The neutral particle density can be determined simply from the ideal gas law (assuming the charge fraction is small), N0= P/kb/Tgas, where P is pressure, kb is Boltzmann’s constant, and Tgas
is gas temperature. The electron-neutral velocity can be determined by examining both the velocity of the neutral particles and that of the electrons. The most probable (thermal) velocity of a population of particles is equal to:
uth= r
2kbT
m , (3.3)
where m is particle mass. As an example, given a temperature of 5000 K, the neutral particle velocity of an argon atom is 1.44 × 103 m s−1 and the electron velocity is 3.89 × 105 m s−1. Clearly, due to the electron’s smaller mass, its velocity is much higher than that of the heavy neutral atoms; therefore, the relative velocity between these two species can be simplified as the velocity of only the electrons. Combining the above results gives the complete equation describing the electron-neutral collision frequency: νen= P Tgas r 2Te kbme < σen> . (3.4)
Note that Tgasand Teare gas/neutral and electron temperature respectively. Next, it is
3.4 Plasma Behaviour 39 to νengives: ∂ p V E2 ∂ νen = Ne Ncε0 1 − νen/ω 2 1 + νen/ω22 . (3.5)
The positive root of this equation (i.e. the maximum power density) is achieved when νen/ω = 1. In other words, the highest power absorption or "best tuned" plasma will
be that which has an electron-neutral collision frequency equal to that of the microwave angular frequency.
Chemical activity: The chemical activity of the plasma gas is important largely because it can be damaging to the reactor tube and any other nearby components. Fortunately, inert gases are most commonly used, including nitrogen, argon, and he- lium. Depending on the properties discussed above (particularly thermal conductivity), inert gases can still be damaging to equipment; however, they do not pose any risk of damage via chemical mechanisms. Conversely, reactive gases such as oxygen and hydrogen can have detrimental chemical effects. Most of the materials used near the plasma are oxides (i.e. quartz, alumina, etc.) and as these are already oxidized, they do not readily react with oxygen in the plasma. A non-oxide that has been used in this work is boron nitride, which is only stable in oxidizing environments up to approximately 800°C despite its impressive stability in inert environments in excess of 2000°C. Therefore, when appreciable quantities of oxygen are used such as with an air plasma, boron nitride components should be avoided.
Conversely, hydrogen is a reducing agent and will readily react with oxides. Some such as alumina are resistant to reduction from hydrogen although others, such as quartz, easily react with hydrogen and this can result in severe damage to the tube within seconds. If hydrogen is to be used, quartz components should be avoided or the plasma should be constrained in such a way that it is not able to contact the quartz.