4.4.1 Plasma Expansion in Vacuum
If the plume expansion takes place in vacuum, in the absence of atomic collision effects, the shape and velocity distribution in the plume will reach asymptotically constant values. The electron beam energy is rapidly converted into kinetic energy with plasma attaining extremely high expansion velocities (Kumar et al., 2010).
Physically, the pressure gradient within the plasma drives its expansion. The plasma, as it absorbs the pulse electron energy, can be simulated as a high temperature-high pressure gas, which is initially confined in small dimensions and is suddenly allowed to expand in vacuum. Because of the large pressure gradients initially present near the outer edge of the ablation spot, very high expansion velocities are induced at the edges of initial plume.
The model of Anisimov et al. (1996) is adopted here in order to model plasmas generated by pulsed electron beams in vacuum. This model is applicable for the expansion, the isentropic, adiabatic, and ionized plumes in vacuum. It was first applied to pulsed laser plasmas. The model is being used here to estimate the temperature and pressure of ions on impact during pulsed electron beam ablation. The model treats the adiabatic expansion of a one-component vapor cloud into vacuum using a particular solution of the gas-dynamic equations. The model applies to flows that are self- similar. It is assumed that the formation time of the vapor cloud is much less than its expansion time and that the focal spot of the pulsed electron beam has an elliptical shape with semiaxes Xo
and Yo, as shown in Fig. 4.3. The expansion is modeled as a triaxial gaseous semiellipsoid whose
semiaxes are initially equal to Xo, Yo, and Zo ≈ cs.τp, where τp is the duration of the pulse electron
and cs is the speed of sound in the vaporized material and given by cs = [γ(γ-1)ε]1/2. According to
this model, the temperature and pressure of ions on impact on the substrate under vacuum conditions can be calculated using Equations (4.14 - 4.16) (Anisimov et al., 1996).
Figure 4.3: Diagram showing the plume at the end of the pulse when the dimensions are Xo,
Yo, and Zo, and after a time t when the dimensions are X(t), Y(t), Z(t) (Doggett and Lunny,
𝑇 = (
𝜀(5𝛾−3)(𝛾−1) 2𝛾) (
𝑋𝑜𝑌𝑜𝑍𝑜 𝑋𝑌𝑍)
𝛾−1[1 − (
𝑧 𝑍)
2] ,
(4.14)𝑃 = (
𝐸𝑝 𝐼2(𝛾)𝑋𝑌𝑍) (
𝑋𝑜𝑌𝑜𝑍𝑜 𝑋𝑌𝑍)
𝛾−1[1 − (
𝑥 𝑋)
2− (
𝑦 𝑌)
2− (
𝑧 𝑍)
2]
𝛾 𝛾−1 , (4.15)𝐼
2(𝛾) = (
𝜋 3 2 2(𝛾−1)) (
Γ(α+2) Γ(𝛼+72))
, (4.16)γ is a thermodynamic property, whereby for a monatomic ideal gas γ = 5/3. For low temperature plasmas the value is lower. In this work, a value of 1.25 is used.
ε is the thermal energy per unit mass of ablated material ε = Ep/Mp (J/kg). It is calculated using
Strikovski’s model (Strikovski et al., 2010). Ep is the thermal energy of the initial plume and Mp
is its mass, in J and kg, respectively.
Xo, Yo, and Zo are taken as 0.76 mm, 0.7 mm, and 0.1 mm, respectively (Strikovski et al., 2010).
X, Y, and Z (uppercase) are the full dimensions of the expanding plasma. They are taken as 1.1 cm, 2.5 cm, and 5 cm (or 7 cm), respectively. x, y, and z (lowercase) are the coordinates of any ion or particle within the full range plasma, in cm.
I2(γ) is a function of the adiabatic index and is dimensionless. It is calculated using Equation (4.16)
as 120.
Γ(z) is the Gamma-function, and α = 1/(γ-1) is a thermodynamic property
The calculated pressures and temperatures of ions on impact in vacuum (at 5 cm substrate-target distance) are listed in Table 4.2. Apparently, for all values of accelerating voltage, in the range of 10 kV to 20 kV, ablation would produce extremely high energy bombarding ions that are energetic enough to end up lying inside the diamond region in the carbon phase diagram upon condensation, as discussed in section 4.5.
Table 4.2: Temperature and pressure of ions on impact at different accelerating voltage estimated using the shockwave model – vacuum condition and at 5 cm substrate-target distance.
Voltage (kV) T(ablation) (K) Ε (GJ) T (on impact)(K) P (on impact) (GPa)
10 92 876 0.0961 11 062 74.7 12 107 948 0.111 12 858 86.9 14 113 119 0.117 13 474 91.0 16 107 948 0.112 12 858 86.9 18 104 876 0.109 12 492 84.4 20 102 066 0.106 12 157 82.1
4.4.2 Plasma Expansion in Ambient Gas
In this work, ablation is carried out in a background gas, namely Argon at ~ 0.53 Pa. Therefore, the pressure and temperature of bombarding carbon ions on impact must be corrected using ambient gas conditions, as explained next.
With a background gas present, the plume species will undergo not only collisions within the plasma but also with the ambient gas. These collisions between the plasma and the gas particles, where a transfer of internal and kinetic energy will occur, result in a reduced expansion velocity of the plasma particles. The slowing and attenuation dynamics of pulsed electron ablation plasmas in low-pressure background gases are of significant interest for film growth by PEBA. The magnitude and kinetic energy of the species arriving at the substrate are key processing parameters in the case of diamond deposition as explained earlier.
Compared to plume expansion in vacuum, the interaction of the plume with the background gas is a far more complex dynamical process due to the presence of many physical processes such as deceleration, attenuation, thermalization of the ablated species, diffusion, recombination, formation of shock waves, reactive scattering and clustering. Apparently, the background gas acts as a regulator of ablated plume energetics and strongly determines the composition and dynamical behavior of the plume material ablated.
In this section, the expression of the plasma pressure in an ambient gas, i.e., Equation (4.17), is used to further modify the results of pressure on impact obtained in vacuum in the previous section.
𝑝
𝑜=
𝑃𝑜𝑋𝑜3(𝜂𝑜𝜉𝑜)1−𝛾𝐸 , Eqn (4.17) (Anisimov et al., 1996)
ξo and ηo are dimensionless quantities, where ξo = Zo/Xo= 0.132 and ηo = Yo/Xo= 0.921.
po (lowercase) is the ratio of pressure of bombarding ions on impact in vacuum to the pressure of
bombarding ions on impact in a background gas, i.e., Pvacuum/Pbackground. The calculated value of po
is 1.36. Pvacuum is the pressure of bombarding ions on impact in vacuum (Pa). It is calculated using
the aforementioned vacuum model, viz., Equation (4.15).
Po (uppercase) is the pressure of background gas in the deposition chamber (Pa), viz., 0.53 pa.
Equation (4.17) has been used to estimate the pressure of ions on impact on the substrate in the presence of an ambient gas in the PEBA chamber. My estimated pressure data of ions on impact (at 5 cm substrate-target distance) are listed in Table 4.3, and shown on the carbon phase diagram in Fig. 4.4. As can be appreciated, for all values of the accelerating voltage in the range 10 kV to 20 kV, ablation would result in ions energetic enough to be lying in the diamond region upon condensation.