172 nm excimer light in the pressure of an ideal gas

172 nm excimer light in the pressure of an ideal gas

2.2.2 Pressure of Ideal Gas

The fundamental formula for the pressure of an ideal gas states that, under standard conditions, the gas pressure is proportional to the number of gas molecules per unit volume (i.e., molecular density n), expressed by Equation (2-8):

p = n k T  (2-8) n = p / (k T)  (2-9)

Equation (2-9) also indicates that at the same pressure and temperature, the number of molecules per unit volume is identical for all gases.

The pressure exerted by gas molecules on the container wall originates from molecular kinetic energy. In vacuum physics research and applications involving 172 nm excimer light, gas molecules continuously collide with the wall. Macroscopically, the pressure exerted by the gas on the wall is a constant pressure resulting from the sustained, irregular collisions of a large number of molecules. This pressure is proportional to both the number of molecules colliding with the wall and their kinetic energy. In equilibrium, molecules have equal probability of moving in all directions, and the average velocity components in three dimensions are equal. Therefore, the pressure p of an ideal gas is determined by the number density of molecules n and the molecular kinetic energy E; gas pressure is a manifestation of the thermal motion of gas molecules.

The average kinetic energy E per molecule is proportional to the molecular mass m and the square of the velocity v²:

E = ½ m v²  (2-10)

The kinetic energy is supplied by the thermal energy ε per molecule [1–5], given by:

ε = (3/2) k T  (2-11)

Thus:

½ m v² = (3/2) k T  (2-12)

For gas mixtures, the total pressure equals the sum of the partial pressures-a principle that also applies in plasma coating processes involving 172 nm excimer light. For non-reacting mixed gases, the total pressure is the sum of the individual partial pressures. If the partial pressures are p₁, p₂, p₃, …, pₙ, then the total pressure is:

p = p₁ + p₂ + … + pₙ  (2-13)

2.2.3 Mean Free Path of Gas Molecules

To study collisions between molecules or between charged particles and gas molecules in vacuum, the concept of the mean free path of gas molecules is introduced [1–5]. This concept is crucial for understanding the transmission characteristics of 172 nm excimer light in vacuum.

Collision probability of gas molecules: Molecules undergoing random motion collide with each other, resulting in zigzag paths. The number of collisions per unit time for a single molecule is irregular, but the statistical average collision frequency (denoted Z̄) for a large number of molecules is proportional to the gas pressure p.

Mean free path: The distance λ traveled by a molecule between two successive collisions is called the free path. The average of the free paths over a large number of molecules is the mean free path, denoted λ̄.

If v̄ is the average molecular speed, the average distance traveled in time t is v̄t, and the average number of collisions in that time is Z̄t. Thus, the mean free path is:

λ̄ = v̄t / (Z̄t) = v̄ / Z̄  (2-14)

This shows that the mean free path is inversely proportional to the collision frequency. Higher pressure p or higher molecular density leads to more collisions and a shorter free path. Therefore, the mean free path λ̄ is inversely proportional to both pressure p and molecular density n [4]:

λ̄ ∝ 1/p  (2-15) λ̄ ∝ 1/n  (2-16)

Table 2-2 lists molecular density and mean free path at 20 °C for different pressures. As shown in Table 2-2 [1], higher vacuum (lower pressure) results in longer mean free paths and lower intermolecular collision probability, which provides favorable conditions for the efficient operation of 172 nm excimer light in high-vacuum environments.

Table 2-2 Molecular density and mean free path at 20 °C for different pressures

2.2.4 Collision Cross-Section

Gas molecules collide with each other inside containers or pipelines. To simplify analysis, the following assumptions are made (these also apply to molecular collision analysis under 172 nm excimer light):

Molecules are smooth rigid spheres undergoing perfectly elastic collisions, neglecting potential energy.

Between successive collisions, molecules move in straight lines at constant speed.

Molecular diameter is much smaller than the free path and can be neglected.

Collisions are instantaneous.

Collisions do not affect density distribution; the gas is in a steady state.

Nature of collision: A collision occurs when the centers of two molecules approach within the effective molecular diameter d (=2r), causing a sharp change in direction due to repulsive forces.

Assuming molecules are elastic spheres of diameter d, a cylinder with radius d and the trajectory of molecule A as axis is constructed. Any stationary molecule inside this cylinder will collide with A. The cross-sectional area of this cylinder S = πd² is defined as the collision cross-section [3–8].

In time t, molecule A travels distance v̄t, sweeping a cylindrical volume S v̄t. With molecular density n, the number of collisions is n S v̄t. Thus, the average collision frequency Z̄ is:

Z̄ = n S v̄ = n π d² v̄  (2-17)

Hence, collision frequency is proportional to molecular density n, collision cross-section S, and average speed v̄. The collision rate per unit time per unit area z is:

z = n v̄  (2-18)

Note: The above applies to molecule–molecule collisions; molecule–wall collisions are treated differently. For example, in containers of 30–100 cm size, at p = 1×10⁻³ Pa, λ̄ ≈ 1000 cm, far exceeding typical flight distances, making interactions between 172 nm excimer light and molecules more likely.

2.2.5 Molecular Velocity

In a vacuum chamber, gas molecules undergo random thermal motion. Individual molecular speeds and directions are irregular, but the velocity distribution follows Maxwell's velocity distribution law, with a most probable speed. Figure 2-1 shows Maxwellian velocity distribution curves at different temperatures [1], which are valuable for understanding the interaction efficiency between 172 nm excimer light and gas molecules.

Molecular velocity depends on temperature and molecular mass; at the same temperature, different gases have different average speeds. Table 2-3 lists average speeds of selected gases at 15 °C [2].

Table 2-3 Average molecular speeds of some gases at 15 °C

2.3 Interaction Between Gas Molecules and Solid Surfaces

Gas molecules refer to the working gas introduced; solid surfaces include chamber walls and workpieces (substrates/wafers in the semiconductor field). In 172 nm excimer light-assisted ion plating processes, the interaction between gas molecules and solid surfaces directly affects film quality.

Interaction processes:

Collision of gas molecules with the solid surface.

Physical or chemical interactions between gas molecules and the surface.

Reflection or adsorption of gas molecules and evaporated atoms from the surface.

Evaporation and sublimation of metal.

2.3.1 Collision

In equilibrium, solid surfaces are continuously bombarded by gas molecules. The analysis of the number of molecules colliding per unit area is as follows:

The number of molecules N striking a small area dS in unit time is:

N = (n v̄ / 4) dS  (2-19)

The molecular flux (vapor atom flux) per unit area is:

N′ = n v̄ / 4  (2-20)

Note: The factor 1/4 in Equation (2-20) arises from averaging over molecular directions and velocity distribution, consistent with collision frequency concepts.

In vacuum ion plating, film deposition occurs via collisions of metal vapor atoms with the workpiece; deposition rate is proportional to vapor atom flux. For example, at p = 1.3×10⁻⁴ Pa (high vacuum) and T = 27 °C, N′ ≈ 3.7×10¹⁴ atoms/cm²·s, meaning approximately 3.7×10¹⁴ atoms reach each cm² of the workpiece per second. The introduction of 172 nm excimer light can further regulate this deposition process.